The Basel II Risk Parameters. Second edition

The Base II Risk Parameters Second edition

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Bernd Engemann Editors Robert Rauhmeier The Base II Risk Parameters Estimation, Vaidation, Stress Testing with Appications to Loan Risk Management

Editors Dr. Bernd Engemann bernd.engemann@quantsoutions.de Dr. Robert Rauhmeier robert.rauhmeier@arcor.de ISBN 978-3-642-16113-1 e-isbn 978-3-642-16114-8 DOI 10.1007/978-3-642-16114-8 Springer Heideberg Dordrecht London New York Library of Congress Contro Number: 2011924881 # Springer-Verag Berin Heideberg 2006, 2011 This work is subject to copyright. A rights are reserved, whether the whoe or part of the materia is concerned, specificay the rights of transation, reprinting, reuse of iustrations, recitation, broadcasting, reproduction on microfim or in any other way, and storage in data banks. Dupication of this pubication or parts thereof is permitted ony under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must aways be obtained from Springer. Vioations are iabe to prosecution under the German Copyright Law. The use of genera descriptive names, registered names, trademarks, etc. in this pubication does not impy, even in the absence of a specific statement, that such names are exempt from the reevant protective aws and reguations and therefore free for genera use. Cover design: WMXDesign GmbH, Heideberg, Germany Printed on acid-free paper Springer is part of Springer ScienceþBusiness Media (www.springer.com)

Preface to the Second Edition The years after the first edition of this book appeared have been very turbuent. We have seen one of the argest financia crisis in the history of the goba financia system. Banks which existed since more than one century have disappeared or had to be rescued by the state. Athough Base II has been impemented by many banks so far and sti a ot of effort is spent in improving credit risk management by buiding up rating systems and procedures for estimating the oan oss parameters PD, LGD, and EAD, there is sti a feeing that this is insufficient to prevent the financia system from further crisis. There are ongoing discussions how the financia system can be stabiized by either improving the reguatory framework or the interna risk management of banks. During the time when we worked on this second edition, the reguatory framework Base III has been discussed. The basic idea behind Base III is extending the capita basis of banks. It is not the aim of Base III to improve the methods and processes of banks interna credit risk management but simpy to improve system stabiity by increasing capita buffers. Since we did not view this book as a book on reguation (athough it was motivated by a reguatory framework) but as a book on risk management, we do not discuss the current reguatory ideas in this edition. Instead, we focus on one of the causes for the financia crisis, the ending behaviour of banks in the retai sector. By retai, we mean ending to debtors wherenomarketinformationontheircredit quaity, ike asset swap or credit defaut swap spreads, is avaiabe. This is the case for amost a oans except for oans to arge corporations, states or banks. In the recent financia crisis one of the origins was that American banks granted mortagage oans to too many debtors with ow income. By assuming that house prices coud not fa sharpy it was thought that the vaue of the oan s coatera wi be sufficient in the case of a defaut to ensure that no oss occurs. A arge number of bankruptcies among the banks which had invested in the American housing sector and expensive rescue programs for banks that were considered as too important to fai are the resut of this wrong assumption. The consequences of the financia crisis are not yet cear. The question how an optima financia system has to ook ike is difficut to answer. On the one hand the ending behaviour of banks shoud not be too restrictive because this wi obstruct the rea economy. On the other hand it must be restrictive enough to prevent the v

vi Preface to the Second Edition creation of bubbes. The same considerations are true for the spectrum of financia products. There shoud be enough vehices for banks and corporations to manage their risks but the compexity and the voume of derivative instruments shoud not ead to a ess stabe financia system. We do not attempt to give an answer to this compex question. Contrary to some opinions in the aftermath of the crisis that bamed mathematica modes as its main driver, we sti beieve that mathematics and statistics are vauabe toos to quantify risks. However, one has to be aware that this cannot be done with arbitrary precision. The roe of a mode in our view is more to increase the transparency of a bank s business and to identify key risks. We want to iustrate this view by presenting a pricing framework for retai oans that shows how the Base II risk parameters can be used in buiding a simpe and transparent framework for the pricing and the risk management of oan portfoios. In our view an increase in transparency in the oan market is a necessary prerequisite of any risk management or reguatory action. Compared to the first edition, we have extended the book by three new chapters. In Chap.6 estimation techniques for transition matrices are presented and their properties are discussed. A transition matrix is a natura extension of a 1-year defaut probabiity since it measures a transitions of a rating system not ony the transitions to defaut. It is an important buiding bock of the oan pricing framework that is presented in Chaps.17 and 18. In Chap.17 it is shown how the Base II risk parameters can be used to buid a risk-adjusted pricing framework for oans that can be appied to compute a oan s term based on RAROC (risk-adjusted return on capita) as performance measure and to cacuate genera oss provisions for a oan portfoio in an economicay sensibe way. Furthermore, this framework aows for an easy stress testing and answering of questions ike: What happens if the vaue of coatera turns out to be 10% ower than assumed? In Chap.18, the pricing framework is extended in a consistent way to oans with embedded options using option pricing theory. Often a oan contains prepayment rights, i.e. a debtor has the right to pay back parts or a of the notiona at certain dates or throughout the oan s ifetime without penaty. We demonstrate that the vaue of such an option is too arge to be negected and show further how to incude embedded options into the RAROC framework of Chap.17. Finay, we woud ike to thank Martina Bihn from Springer-Verag again for her support of this second edition and ast but not east our famiies for their support when we again spent a ot of time working on it. Questions and comments on this book are wecome. The editors can be reached under their e-mai addresses bernd.engemann@quantsoutions.de and robert.rauhmeier@arcor.de. Frankfurt am Main, Germany Munich, Germany December 2010 Bernd Engemann Robert Rauhmeier

Preface to the First Edition In the ast decade the banking industry has experienced a significant deveopment in the understanding of credit risk. Refined methods were proposed concerning the estimation of key risk parameters ike defaut probabiities. Further, a arge voume of iterature on the pricing and measurement of credit risk in a portfoio context has evoved. This deveopment was party refected by supervisors when they agreed on the new revised capita adequacy framework, Base II. Under Base II, the eve of reguatory capita depends on the risk characteristics of each credit whie a portfoio context is sti negected. The focus of this book is on the estimation and vaidation of the three key Base II risk parameters, probabiity of defaut (PD), oss given defaut (LGD), and exposure at defaut (EAD). Since the new reguatory framework wi become operative in January 2007 (at east in Europe), many banks are in the fina stages of impementation. Many questions have arisen during the impementation phase and are discussed by practitioners, supervisors, and academics. A best practice approach has to be formed and wi be refined in the future even beyond 2007. With this book we aim to contribute to this process. Athough the book is inspired by the new capita framework, we hope that it is vauabe in a broader context. The three risk parameters are centra inputs to credit portfoio modes or credit pricing agorithms and their correct estimation is therefore essentia for interna bank controing and management. This is not a book about the Base II framework. There is aready a arge voume of iterature expaining the new reguation at ength. Rather, we attend to the current state-of-the-art of quantitative and quaitative approaches. The book is a combination of coordinated stand-aone artices, arranged into 15 chapters so that each chapter can be read excusivey. The authors are a experts from science, supervisory authorities, and banking practice. The book is divided into three main parts: Estimation techniques for the parameters PD, LGD and EAD, vaidation of these parameters, and stress testing. The first part begins with an overview of the popuar and estabished methods for estimating PD. Chapter 2 focuses on methods for PD estimation for sma and medium sized corporations whie Chap.3 treats the PD estimation for the retai segment. Chapters 4 and 5 dea with those segments with ony a few or even no defaut data, as it is often the case in the arge corporate, financia institutions, vii

viii Preface to the First Edition or sovereign segment. Chapter 4 iustrates how PD can be estimated with the shadow rating approach whie Chap.5 uses techniques from probabiity theory. Chapter 6 describes how PDs and Recovery Rates coud be estimated under considerations of systematic and idiosyncratic risk factors simutaneousy. This is a perfect changeover to the chaps.7 10 deaing with LGD and EAD estimation which is quite new in practice compared to ratings and PD estimation. Chapter 7 describes how LGD coud be modeed in a point-in-time framework as a function of risk drivers, supported by an empirica study on bond data. Chapter 8 provides a genera survey of LGD estimation from a practica point of view. Chapters 9 and 10 are concerned with the modeing of EAD. Chapter 9 provides a genera overview of EAD estimation techniques whie Chap.10 focuses on the estimation of EAD for faciities with expicit imits. The second part of the book consists of four chapters about vaidation and statistica back-testing of rating systems. Chapter 11 deas with the perspective of the supervisory authorities and gives a gance as to what is expected when rating systems wi be used under the BaseII framework. Chapter 12 has a critica discussion on measuring the discriminatory power of rating systems. Chapter 13 gives an overview of statistica tests for the dimension caibration, i.e. the accuracy of PD estimations. In Chap.14 these methods are enhanced by techniques of Monte- Caro-Simuations which aows e.g. for integration of correation assumptions as is aso iustrated within a back-testing study on a rea-ife rating data sampe. The fina part consists of Chap.15, which is on stress testing. The purpose of stress testing is to detect imitations of modes for the risk parameters and to anayse effects of (extreme) worse scenarios in the future on a bank s portfoio. Concepts and impementation strategies of stress test are expained and a simuation study reveas amazing effects of stress scenarios when cacuating economic capita with a portfoio mode. A artices set great vaue on practica appicabiity and mosty incude empirica studies or work with exampes. Therefore we regard this book as a vauabe contribution towards modern risk management in every financia institution, whereas we steadiy keep track on the requirements of Base II. The book is addressed to risk managers, rating anayst and in genera quantitative anaysts who work in the credit risk area or on reguatory issues. Furthermore, we target interna auditors and supervisors who have to evauate the quaity of rating systems and risk parameter estimations. We hope that this book wi deepen their understanding and wi be usefu for their daiy work. Last but not east we hope this book wi aso be of interest to academics or students in finance or economics who want to get an overview of the state-of-the-art of a currenty important topic in the banking industry. Finay, we have to thank a the peope who made this book possibe. Our sincere acknowedgements go to a the contributors of this book for their work, their enthusiasm, their reiabiity, and their cooperation. We know that most of the writing had to be done in vauabe spare time. We are gad that a of them were wiing to make such sacrifices for the sake of this book. Specia thank goes to Water Gruber for bringing us on the idea to edit this book.

Preface to the First Edition ix We are gratefu to Martina Bihn from Springer-Verag who wecomed our idea for this book and supported our work on it. We thank Dresdner Bank AG, especiay Peter Gassmann and Dirk Thomas, and Quanteam AG for supporting our book. Moreover we are gratefu to a our coeagues and friends who agreed to work as referees or discussion partners. Finay we woud ike to thank our famiies for their continued support and understanding. Frankfurt am Main, Germany Munich, Germany June 2006 Bernd Engemann Robert Rauhmeier

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Contents 1 Statistica Methods to Deveop Rating Modes... 1 Eveyn Hayden and Danie Porath 2 Estimation of a Rating Mode for Corporate Exposures... 13 Eveyn Hayden 3 Scoring Modes for Retai Exposures... 25 Danie Porath 4 The Shadow Rating Approach: Experience from Banking Practice... 37 Urich Erenmaier 5 Estimating Probabiities of Defaut for Low Defaut Portfoios... 75 Katja Puto and Dirk Tasche 6 Transition Matrices: Properties and Estimation Methods... 103 Bernd Engemann and Konstantin Ermakov 7 A Muti-factor Approach for Systematic Defaut and Recovery Risk... 117 Danie Rösch and Harad Scheue 8 Modeing Loss Given Defaut: A Point in Time -Approach... 137 Afred Hamere, Michae Knapp, and Nicoe Widenauer 9 Estimating Loss Given Defaut: Experience from Banking Practice... 151 Christian Peter 10 Possibiities of Estimating Exposures... 185 Ronny Hahn and Stefan Reitz xi

xii Contents 11 EAD Estimates for Faciities with Expicit Limits... 201 Gregorio Mora 12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective... 247 Stefan Bochwitz and Stefan Hoh 13 Measures of a Rating s Discriminative Power: Appications and Limitations... 269 Bernd Engemann 14 Statistica Approaches to PD Vaidation... 293 Stefan Bochwitz, Marcus R.W. Martin, and Carsten S. Wehn 15 PD-Vaidation: Experience from Banking Practice... 311 Robert Rauhmeier 16 Deveopment of Stress Tests for Credit Portfoios... 349 Voker Matthias Gundach 17 Risk Management of Loans and Guarantees... 373 Bernd Engemann and Water Gruber 18 Risk Management of Loans with Embedded Options... 391 Bernd Engemann About the Authors... 415 Index... 421

Contributors Stefan Bochwitz Deutsche Bundesbank, Stefan.Bochwitz@bundesbank.de Bernd Engemann Independent Consutant, bernd.engemann@quantsoutions.de Urich Erenmaier KfW Bankengruppe, Urich.Erenmaier@gmai.com Konstantin Ermakov Independent Consutant, konstantin@ermakov.de Water Gruber 1 PLUS i GmbH, water.gruber@1pusi.de Voker Matthias Gundach THM University of Appied Sciences, Giessen- Friedberg, matthias.gundach@mni.th-mittehessen.de Ronny Hahn 1 PLUS i GmbH, ronny.hahn@1pusi.de Afred Hamere Universität Regensburg, Afred.Hamere@wiwi.uni-regensburg.de Eveyn Hayden Raiffeisen Bank Internationa, Eveyn.Hayden@univie.ac.at Stefan Hoh Bank for Internationa Settements, stefan.hoh@bis.org Michae Knapp Risk Research Prof. Hamere GmbH & Co. KG, michae.knapp@ risk-research.de Marcus R.W. Martin University of Appied Sciences, Darmstadt, marcus.martin@ h-da.de Gregorio Mora Banco de España, Gregorio.Mora@bde.es Christian Peter KfW Bankengruppe, Christian.Peter@Web.de Katja Puto HSBC Hodings pc, Katja.Puto@gmx.de Danie Porath University of Appied Sciences, Mainz, danie.roesch@finance. uni-hannover.de Danie Rösch University of Hannover, danie.roesch@finance.uni-hannover.de Robert Rauhmeier UniCredit Bank AG, robert.rauhmeier@arcor.de Stefan Reitz University of Appied Sciences, Stuttgart, stefan.reitz@hft-stuttgart.de xiii

xiv Contributors Harad Scheue University of Mebourne, hscheue@unimeb.edu.au Dirk Tasche Loyds Banking Group, dirk.tasche@gmx.net Carsten S. Wehn DekaBank, wehn@gmx.de Nicoe Widenauer Commerzbank AG, Nicoe.Widenauer@commerzbank.com

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Chapter 1 Statistica Methods to Deveop Rating Modes Eveyn Hayden and Danie Porath 1.1 Introduction The Interna Rating Based Approach (IRBA) of the New Base Capita Accord aows banks to use their own rating modes for the estimation of probabiities of defaut (PD) as ong as the systems meet specified minimum requirements. Statistica theory offers a variety of methods for buiding and estimation rating modes. This chapter gives an overview of these methods. The overview is focused on statistica methods and incudes parametric modes ike inear regression anaysis, discriminant anaysis, binary response anaysis, time-discrete pane methods, hazard modes and nonparametric modes ike neura networks and decision trees. We aso highight the benefits and the drawbacks of the various approaches. We concude by interpreting the modes in ight of the minimum requirements of the IRBA. 1.2 Statistica Methods for Risk Cassification In the foowing we define statistica modes as the cass of approach which uses econometric methods to cassify borrowers according to their risk. Statistica rating systems primariy invove a search for expanatory variabes which provide as sound and reiabe a forecast of the deterioration of a borrower s situation as possibe. In contrast, structura modes expain the threats to a borrower based on an economic mode and thus use cear causa connections instead of the mere correation of variabes. The opinions expressed in this chapter are those of the author and do not necessariy refect views of Raiffeisen Bank Internationa. E. Hayden Raiffeisen Bank Internationa e-mai: Eveyn.Hayden@univie.ac.at D. Porath (*) University of Appied Sciences, Mainz e-mai: danie.porath@wiwi.fh.mainz.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_1, # Springer-Verag Berin Heideberg 2011 1

2 E. Hayden and D. Porath The foowing sections offer an overview of parametric and nonparametric modes generay considered for statistica risk assessment. Furthermore, we discuss the advantages and disadvantages of each approach. Many of the methods are described in more detai in standard econometric textbooks, ike Greene (2003). In genera, a statistica mode may be described as foows: As a starting point, every statistica mode uses the borrower s characteristic indicators and (possiby) macroeconomic variabes which were coected historicay and are avaiabe for defauting (or troubed) and non-defauting borrowers. Let the borrower s characteristics be defined by a vector of n separate variabes (aso caed covariates) x ¼ x 1,..., x n observed at time t L. The state of defaut is indicated by a binary performance variabe y observed at time t. The variabe y is defined as y ¼ 1 for a defaut and y ¼ 0 for a non-defaut. The sampe of borrowers now incudes a number of individuas or firms that defauted in the past, whie (typicay) the majority did not defaut. Depending on the statistica appication of this data, a variety of methods can be used to predict the performance. A common feature of the methods is that they estimate the correation between the borrowers characteristics and the state of defaut in the past and use this information to buid a forecasting mode. The forecasting mode is designed to assess the creditworthiness of borrowers with unknown performance. This can be done by inputting the characteristics x into the mode. The output of the mode is the estimated performance. The time ag L between x and y determines the forecast horizon. 1.3 Regression Anaysis As a starting point we consider the cassica regression mode. The regression mode estabishes a inear reationship between the borrowers characteristics and the defaut variabe: y i ¼ b 0 x i þ u i (1.1) Again, y i indicates whether borrower i has defauted (y i ¼ 1) or not (y i ¼ 0). In period t, x i is a coumn vector of the borrowers characteristics observed in period t L and b is a coumn vector of parameters which capture the impact of a change in the characteristics on the defaut variabe. Finay, u i is the residua variabe which contains the variation not captured by the characteristics x i. The standard procedure is to estimate (1.1) with the ordinary east squares (OLS) estimators of b which in the foowing are denoted by b. The estimated resut is the borrower s score S i. This can be cacuated by S i ¼ Ey ð i jx i Þ ¼ b 0 x i : (1.2) Equation (1.2) shows that a borrower s score represents the expected vaue of the performance variabe when his or her individua characteristics are known.

1 Statistica Methods to Deveop Rating Modes 3 The score can be cacuated by inputting the vaues for the borrower s characteristics into the inear function given in (1.2). Note that S i is continuous (whie y i is a binary variabe), hence the output of the mode wi generay be different from 0 or 1. In addition, the prediction can take on vaues arger than 1 or smaer than 0. As a consequence, the outcome of the mode cannot be interpreted as a probabiity eve. However, the score S i, can be used for the purpose of comparison between different borrowers, where higher vaues of S i correate with a higher defaut risk. The benefits and drawbacks from mode (1.1) and (1.2) are the foowing: OLS estimators are we-known and easiy avaiabe. The forecasting mode is a inear mode and therefore easy to compute and to understand. The random variabe u i is heteroscedastic (i.e. the variance of u i is not constant for a i) since Varðu i Þ ¼ Varðy i Þ ¼ Ey ð i jx i Þ½1 Ey ð i jx i ÞŠ ¼ b 0 x i ð1 b 0 x i Þ: (1.3) As a consequence, the estimation of b is inefficient and additionay, the standard errors of the estimated coefficients b are biased. An efficient way to estimate b is to appy the Weighted Least Squares (WLS) estimator. WLS estimation of b is efficient, but the estimation of the standard errors of b sti remains biased. This happens due to the fact that the residuas are not normay distributed as they can ony take on the vaues b 0 x i (if the borrower does not defaut and y therefore equas 0) or (1 b 0 x i ) (if the borrower does defaut and y therefore equas 1). This impies that there is no reiabe way to assess the significance of the coefficients b and it remains unknown whether the estimated vaues represent precise estimations of significant reationships or whether they are just caused by spurious correations. Inputting characteristics which are not significant into the mode can seriousy harm the mode s stabiity when used to predict borrowers risk for new data. A way to cope with this probem is to spit the sampe into two parts, where one part (the training sampe) is used to estimate the mode and the other part (the hod-out sampe) is used to vaidate the resuts. The consistency of the resuts of both sampes is then taken as an indicator for the stabiity of the mode. The absoute vaue of S i cannot be interpreted. 1.4 Discriminant Anaysis Discriminant anaysis is a cassification technique appied to corporate bankruptcies by Atman as eary as 1968 (see Atman 1968). Linear discriminant anaysis is based on the estimation of a inear discriminant function with the task of separating individua groups (in this case of defauting and non-defauting borrowers) according to specific characteristics. The discriminant function is

4 E. Hayden and D. Porath S i ¼ b 0 x i : (1.4) The Score S i is aso caed the discriminant variabe. The estimation of the discriminant function adheres to the foowing principe: Maximization of the spread between the groups (good and bad borrowers) and minimization of the spread within individua groups Maximization ony determines the optima proportions among the coefficients of the vector b. Usuay (but arbitrariy), coefficients are normaized by choosing the pooed within-group variance to take the vaue 1. As a consequence, the absoute eve of S i is arbitrary as we and cannot be interpreted on a stand-aone basis. As in inear regression anaysis, S i can ony be used to compare the prediction for different borrowers ( higher score, higher risk ). Discriminant anaysis is simiar to the inear regression mode given in (1.1) and (1.2). In fact, the proportions among the coefficients of the regression mode are equa to the optima proportion according to the discriminant anaysis. The difference between the two methods is a theoretica one: Whereas in the regression mode the characteristics are deterministic and the defaut state is the reaization of a random variabe, for discriminant anaysis the opposite is true. Here the groups (defaut or non-defaut) are deterministic and the characteristics of the discriminant function are reaizations from a random variabe. For practica use this difference is virtuay irreevant. Therefore, the benefits and drawbacks of discriminant anaysis are simiar to those of the regression mode: Discriminant anaysis is a widey known method with estimation agorithms that are easiy avaiabe. Once the coefficients are estimated, the scores can be cacuated in a straightforward way with a inear function. Since the characteristics x i are assumed to be reaizations of random variabes, the statistica tests for the significance of the mode and the coefficients rey on the assumption of mutivariate normaity. This is, however, unreaistic for the variabes typicay used in rating modes as for exampe financia ratios from the baance-sheet. Hence, the methods for anayzing the stabiity of the mode and the pausibiity of the coefficients are imited to a comparison between training and hod-out sampe. The absoute vaue of the discriminant function cannot be interpreted in eves. 1.5 Logit and Probit Modes Logit and probit modes are econometric techniques designed for anayzing binary dependent variabes. There are two aternative theoretica foundations. The atent-variabe approach assumes an unobservabe (atent) variabe y* which is reated to the borrower s characteristics in the foowing way:

1 Statistica Methods to Deveop Rating Modes 5 y i ¼ b 0 x i þ u i (1.5) Here b, x i and u i are defined as above. The variabe y i * is metricay scaed and triggers the vaue of the binary defaut variabe y i : y i ¼ 1 if y i >0 (1.6) 0 otherwise This means that the defaut event sets in when the atent variabe exceeds the threshod zero. Therefore, the probabiity for the occurrence of the defaut event equas: Py ð i ¼ 1 Þ ¼ Pu ð i > b 0 x i Þ ¼ 1 Fð b 0 x i Þ ¼ Fðb 0 x i Þ: (1.7) Here F(.) denotes the (unknown) distribution function. The ast step in (1.7) assumes that the distribution function has a symmetric density around zero. The choice of the distribution function F(.) depends on the distributiona assumptions about the residuas (u i ). If a norma distribution is assumed, we are faced with the probit mode: F(b 0 x i Þ¼ pffiffiffiffiffiffiffi 1 2 p bð 0 x i 1 e t2 2 dt (1.8) If instead the residuas are assumed to foow a ogistic distribution, the resut is the ogit mode: eb0 x i F(b 0 x i Þ¼ (1.9) 1 þ e b0 x i The second way to motivate ogit and probit modes starts from the aim of estimating defaut probabiities. For singe borrowers, defaut probabiities cannot be observed as reaizations of defaut probabiities. However, for groups of borrowers the observed defaut frequencies can be interpreted as defaut probabiities. As a starting point consider the OLS estimation of the foowing regression: p i ¼ b 0 x i þ u i (1.10) In (1.10) the index i denotes the group formed by a number of individuas, p i is the defaut frequency observed in group i and x i are the characteristics observed for group i. The mode, however, is inadequate. To see this consider that the outcome (which is E(y i x i ) ¼ b 0 x i ) is not bounded to vaues between zero and one and therefore cannot be interpreted as a probabiity. As it is generay impausibe to assume that a probabiity can be cacuated by a inear function, in a second step the inear expression b 0 x i is transformed by a noninear function (ink function) F: p i ¼ Fðb 0 x i Þ: (1.11)

6 E. Hayden and D. Porath An appropriate ink function transforms the vaues of b 0 x i to a scae within the interva [0,1]. This can be achieved by any distribution function. The choice of the ink function determines the type of mode: with a ogistic ink function (1.11) becomes a ogit mode, whie with the norma distribution (1.11) resuts in the probit mode. However, when estimating (1.10) with OLS, the coefficients wi be heteroscedastic, because Var(u i ) ¼ Var(p i ) ¼ p(x i )(1 p(x i )). A possibe way to achieve homoscedasticity woud be to compute the WLS estimators of b in (1.10). However, abeit possibe, this is not common practice. The reason is that in order to observe defaut frequencies, the data has to be grouped before estimation. Grouping invoves considerabe practica probems ike defining the size and number of the groups and the treatment of different covariates within the singe groups. A better way to estimate ogit and probit modes, which does not require grouping, is the Maximum-Likeihood (ML) method. For a binary dependent variabe the ikeihood function ooks ike: LðbÞ ¼ Y i Pðb 0 x i Þ y i 1 Pðb 0 x i Þ 1 y i : (1.12) For the probit mode P(.) is the norma density function and for the ogit mode P(.) is the ogistic density function. With (1.12) the estimation of the mode is theoreticay convincing and aso easy to hande. Furthermore, the ML-approach ends itsef for a broad set of tests to evauate the mode and its singe variabes (see Hosmer and Lemeshow (2000) for a comprehensive introduction). Usuay, the choice of the ink function is not theoreticay driven. Users famiiar with the norma distribution wi opt for the probit mode. Indeed, the differences in the resuts of both casses of modes are often negigibe. This is due to the fact that both distribution functions have a simiar form except for the tais, which are heavier for the ogit mode. The ogit mode is easier to hande, though. First of a, the computation of the estimators is easier. However, today computationa compexity is often irreevant as most users appy statistica software where the estimation agorithms are integrated. What is more important is the fact that the coefficients of the ogit mode can be more easiy interpreted. To see this we transform the ogit mode given in (1.9) in the foowing way: P i 1 P i ¼ e b0 x i (1.13) The eft-hand side of (1.13) is the odds, i.e. the reation between the defaut probabiity and the probabiity of surviva. Now it can be easiy seen that a variation of a singe variabe x k of one unit has an impact of e b k on the odds, when bk denotes the coefficient of the variabe x k. Hence, the transformed coefficients e b are caed odds-ratios. They represent the mutipicative impact of a borrower s characteristic on the odds. Therefore, for the ogit mode, the coefficients can be interpreted in a pausibe way, which is not possibe for the probit mode. Indeed, the most important weakness of binary modes is the fact that the interpretation of the coefficients is not straightforward.

1 Statistica Methods to Deveop Rating Modes 7 The strengths of ogit and probit modes can be summarized as: The methods are theoreticay sound. The resuts generated can be interpreted directy as defaut probabiities. The significance of the mode and the individua coefficients can be tested. Therefore, the stabiity of the mode can be assessed more effectivey than in the previous cases. 1.6 Pane Modes The methods discussed so far are a cross-sectiona methods because a covariates are reated to the same period. However, typicay banks dispose of a set of covariates for more than one period for each borrower. In this case it is possibe to expand the cross-sectiona input data to a pane dataset. The main motivation is to enarge the number of avaiabe observations for the estimation and therefore enhance the stabiity and the precision of the rating mode. Additionay, pane modes can integrate macroeconomic variabes into the mode. Macroeconomic variabes can improve the mode for severa reasons. First, many macroeconomic data sources are more up-to-date than the borrowers characteristics. For exampe, financia ratios cacuated from baance sheet information are usuay updated ony once a year and are often up to 2 years od when used for risk assessment. The oi price, instead, is avaiabe on a daiy frequency. Secondy, by stressing the macroeconomic input factors, the mode can be used for a form of stress-testing credit risk. However, as macroeconomic variabes primariy affect the absoute vaue of the defaut probabiity, it is ony reasonabe to incorporate macroeconomic input factors into those casses of modes that estimate defaut probabiities. In principe, the structure of, for exampe, a pane ogit or probit mode remains the same as given in the equations of the previous section. The ony difference is that now the covariates are taken from a pane of data and have to be indexed by an additiona time series indicator, i.e. we observe x it instead of x i. At first gance pane modes seem simiar to cross-sectiona modes. In fact, many deveopers ignore the dynamic pattern of the covariates and simpy fit ogit or probit modes. However, ogit and probit modes rey on the assumption of independent observations. Generay, cross-sectiona data meets this requirement, but pane data does not. The reason is that observations from the same period and observations from the same borrower shoud be correated. Introducing this correation in the estimation procedure is cumbersome. For exampe, the fixed-effects estimator known from pane anaysis for continuous dependent variabes is not avaiabe for the probit mode. Besides, the modified fixed-effects estimator for ogit modes proposed by Chamberain (1980) excudes a non-defauting borrowers from the anaysis and therefore seems inappropriate. Finay, the random-effects estimators proposed in the iterature are computationay extensive and can ony be computed with speciaized software. For an econometric discussion of binary pane anaysis, refer to Hosmer and Lemeshow (2000).

8 E. Hayden and D. Porath 1.7 Hazard Modes A methods discussed so far try to assess the riskiness of borrowers by estimating a certain type of score that indicates whether or not a borrower is ikey to defaut within the specified forecast horizon. However, no prediction about the exact defaut point in time is made. Besides, these approaches do not aow the evauation of the borrowers risk for future time periods given they shoud not defaut within the reference time horizon. These disadvantages can be remedied by means of hazard modes, which expicity take the surviva function and thus the time at which a borrower s defaut occurs into account. Within this cass of modes, the Cox proportiona hazard mode (cf. Cox 1972) is the most genera regression mode, as it is not based on any assumptions concerning the nature or shape of the underying surviva distribution. The mode assumes that the underying hazard rate (rather than surviva time) is a function of the independent variabes; no assumptions are made about the nature or shape of the hazard function. Thus, the Cox s regression mode is a semiparametric mode. The mode can be written as: h i ðtjx i Þ ¼ h 0 ðtþe b0 x i ; (1.14) where h i (t x i ) denotes the resutant hazard, given the covariates for the respective borrower and the respective surviva time t. The term h 0 (t) is caed the baseine hazard; it is the hazard when a independent variabe vaues are equa to zero. If the covariates are measured as deviations from their respective means, h 0 (t) can be interpreted as the hazard rate of the average borrower. Whie no assumptions are made about the underying hazard function, the mode equation shown above impies important assumptions. First, it specifies a mutipicative reationship between the hazard function and the og-inear function of the expanatory variabes, which impies that the ratio of the hazards of two borrowers does not depend on time, i.e. the reative riskiness of the borrowers is constant, hence the name Cox proportiona hazard mode. Besides, the mode assumes that the defaut point in time is a continuous random variabe. However, often the borrowers financia conditions are not observed continuousy but rather at discrete points in time. What s more, the covariates are treated as if they were constant over time, whie typica expanatory variabes ike financia ratios change with time. Athough there are some advanced modes to incorporate the above mentioned features, the estimation of these modes becomes compex. The strengths and weaknesses of hazard modes can be summarized as foows: Hazard modes aow for the estimation of a surviva function for a borrowers from the time structure of historica defauts, which impies that defaut probabiities can be cacuated for different time horizons. Estimating these modes under reaistic assumptions is not straightforward.

1 Statistica Methods to Deveop Rating Modes 9 1.8 Neura Networks In recent years, neura networks have been discussed extensivey as an aternative to the (parametric) modes discussed above. They offer a more fexibe design to represent the connections between independent and dependent variabes. Neura networks beong to the cass of non-parametrica methods. Unike the methods discussed so far they do not estimate parameters of a we-specified mode. Instead, they are inspired by the way bioogica nervous systems, such as the brain, process information. They typicay consist of many nodes that send a certain output if they receive a specific input from the other nodes to which they are connected. Like parametric modes, neura networks are trained by a training sampe to cassify borrowers correcty. The fina network is found by adjusting the connections between the input, output and any potentia intermediary nodes. The strengths and weaknesses of neura networks can be summarized as: Neura networks easiy mode highy compex, noninear reationships between the input and the output variabes. They are free from any distributiona assumptions. These modes can be quicky adapted to new information (depending on the training agorithm). There is no forma procedure to determine the optimum network topoogy for a specific probem, i.e. the number of the ayers of nodes connecting the input with the output variabes. Neura networks are back boxes, hence they are difficut to interpret. Cacuating defaut probabiities is possibe ony to a imited extent and with considerabe extra effort. In summary, neura networks are particuary suitabe when there are no expectations (based on experience or theoretica arguments) on the reationship between the input factors and the defaut event and the economic interpretation of the resuting modes is of inferior importance. 1.9 Decision Trees A further category of non-parametric methods comprises decision trees, aso caed cassification trees. Trees are modes which consist of a set of if-then spit conditions for cassifying cases into two (or more) different groups. Under these methods, the base sampe is subdivided into groups according to the covariates. In the case of binary cassification trees, for exampe, each tree node is assigned by (usuay univariate) decision rues, which describe the sampe accordingy and subdivide it into two subgroups each. New observations are processed down the tree in accordance with the decision rues vaues unti the end node is reached, which then represents the cassification of this observation. An exampe is given in Fig. 1.1.

10 E. Hayden and D. Porath Sector Construction Other Years in business EBIT Less than 2.. Equity ratio Less than 15% More than 15% Risk cass 2 Risk cass 3 Fig. 1.1 Decision tree One of the most striking differences of the parametric modes is that a covariates are grouped and treated as categorica variabes. Furthermore, whether a specific variabe or category becomes reevant depends on the categories of the variabes in the upper eve. For exampe, in Fig. 1.1 the variabe years in business is ony reevant for companies which operate in the construction sector. This kind of dependence between variabes is caed interaction. The most important agorithms for buiding decision trees are the Cassification and Regression Trees agorithms (C&RT) popuarized by Breiman et a. (1984) and the CHAID agorithm (Chi-square Automatic Interaction Detector, see Kass 1978). Both agorithms use different criteria to identify the best spits in the data and to coapse the categories which are not significanty different in outcome. The genera strengths and weaknesses of trees are: Through categorization, noninear reationships between the variabes and the score can be easiy modeed. Interactions present in the data can be identified. Parametric methods can mode interactions ony to a imited extent (by introducing dummy variabes). As with neura networks, decision trees are free from distributiona assumptions. The output is easy to understand. Probabiities of defaut have to be cacuated in a separate step. The output is (a few) risk categories and not a continuous score variabe. Consequenty, decision trees ony cacuate defaut probabiities for the fina node in a tree, but not for individua borrowers. Compared to other modes, trees contain fewer variabes and categories. The reason is that in each node the sampe is successivey partitioned and therefore continuousy diminishes. The stabiity of the mode cannot be assessed with statistica procedures. The strategy is to work with a training sampe and a hod-out sampe.

1 Statistica Methods to Deveop Rating Modes 11 In summary, trees are particuary suited when the data is characterized by a imited number of predictive variabes which are known to be interactive. 1.10 Statistica Modes and Base II Finay, we ask the question whether the modes discussed in this chapter are in ine with the IRB Approach of Base II. Prior to the discussion, it shoud be mentioned that in the Base documents, rating systems are defined in a broader sense than in this chapter. Foowing } 394 of the Revised Framework from June 2004 (cf. BIS 2004) a rating system comprises a the methods, processes, contros, and data coection and IT systems that support the assessment of credit risk, the assignment of interna ratings, and the quantification of defaut and oss estimates. Compared to this definition, these methods provide one component, namey the assignment of interna ratings. The minimum requirements for interna rating systems are treated in Part II, Section III, H of the Revised Framework. A few passages of the text concern the assignment of interna ratings, and the requirements are genera. They mainy concern the rating structure and the input data, exampes being: A minimum of seven rating casses of non-defauted borrowers (} 404) No undue or excessive concentrations in singe rating casses (}} 403, 406) A meaningfu differentiation of risk between the casses (} 410) Pausibe, intuitive and current input data (}} 410, 411) A reevant information must be taken into account (} 411) The requirements do not revea any preference for a certain method. It is indeed one of the centra ideas of the IRBA that the banks are free in the choice of the method. Therefore the modes discussed here are a possibe candidates for the IRB Approach. The strengths and weaknesses of the singe methods concern some of the minimum requirements. For exampe, hazard rate or ogit pane modes are especiay suited for stress testing (as required by }} 434, 345) since they contain a timeseries dimension. Methods which aow for the statistica testing of the individua input factors (e.g. the ogit mode) provide a straightforward way to demonstrate the pausibiity of the input factors (as required by } 410). When the outcome of the mode is a continuous variabe, the rating casses can be defined in a more fexibe way (}} 403, 404, 406). On the other hand, none of the drawbacks of the modes considered here excudes a specific method. For exampe, a bank may have a preference for inear regression anaysis. In this case the pausibiity of the input factors cannot be verified by statistica tests and as a consequence the bank wi have to search for aternative ways to meet the requirements of } 410. In summary, the minimum requirements are not intended as a guideine for the choice of a specific mode. Banks shoud rather base their choice on their interna

12 E. Hayden and D. Porath aims and restrictions. If necessary, those components that are ony needed for the purpose to satisfy the criteria of the IRBA shoud be added in a second step. A modes discussed in this chapter aow for this. References Atman EI (1968), Financia Indicators, Discriminant Anaysis, and the Prediction of Corporate Bankruptcy, Journa of Finance 23 (4), pp 589 609. BIS (2004), Internationa Convergence of Capita Measurement and Capita Standards, Base Committee on Banking Supervision, June 2004. Breiman L, Friedman JH, Oshen RA, Stone SJ (1984), Cassification and Regression Trees, Wadsworth, Bemont. Chamberain G (1980), Anaysis of Covariance with Quaitative Data, Review of Economic Studies 47, 225 238. Cox DR (1972), Regression Modes and Life Tabes (with Discussion), Journa of Roya Statistica Society, Series B 34, pp 187 220. Greene W (2003), Econometric Anaysis, 5th ed., Prentice-Ha, New Jersey. Hosmer W, Lemeshow S (2000), Appied Logistic Regression, New York, Wiey. Kass GV (1978), An Exporatory Technique for Investigating Large Quantities of Categorica Data, Appied Statistics 29 (2), pp. 119 127.

Chapter 2 Estimation of a Rating Mode for Corporate Exposures Eveyn Hayden 2.1 Introduction This chapter focuses on the particuar difficuties encountered when deveoping interna rating modes for corporate exposures. The main characteristic of these interna rating modes is that they mainy rey on financia ratios. Hence, the aim is to demonstrate how financia ratios can be used for statistica risk assessment. The chapter is organised as foows: Sect. 2.2 describes some of the issues concerning mode seection, whie Sect. 2.3 presents data from Austrian companies that wi iustrate the theoretica concepts. Section 2.4 discusses data processing, which incudes the cacuation of financia ratios, their transformation to estabish inearity, the identification of outiers and the handing of missing vaues. Section 2.5 describes the actua estimation of the rating mode, i.e. univariate and mutivariate anayses, muticoinearity issues and performance measurement. Finay, Sect. 2.6 concudes. 2.2 Mode Seection Chapter 1 presents severa statistica methods for buiding and estimating rating modes. The most popuar of these mode types in the academic iterature as we as in practice is the ogit mode, mainy for two reasons. Firsty, the output from the ogit mode can be directy interpreted as defaut probabiity, and secondy, the mode aows an easy check as to whether the empirica dependence between the potentia expanatory variabes and defaut risk is economicay meaningfu (see Sect. 2.4). Hence, a ogit mode is chosen to demonstrate the estimation of interna rating modes for corporate exposures. The opinions expressed in this chapter are those of the author and do not necessariy refect views of Raiffeisen Bank Internationa. E. Hayden Raiffeisen Bank Internationa e-mai: Eveyn.Hayden@univie.ac.at B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_2, # Springer-Verag Berin Heideberg 2011 13

14 E. Hayden Next, the defaut event must be defined. Historicay, rating modes were deveoped using mosty the defaut criterion bankruptcy, as this information was reativey easiy observabe. However, banks aso incur osses before the event of bankruptcy, for exampe, when they aow debtors to defer payments without compensation in hopes that ater on, the troubed borrowers wi be abe to repay their debt. Therefore, the Base Committee on Banking Supervision (2001) defined a reference definition of defaut that incudes a those situations where a bank ooses money and decared that banks woud have to use this reguatory reference definition of defaut for estimating interna rating-based modes. However, as demonstrated in Hayden (2003), rating modes deveoped by excusivey reying on bankruptcy as the defaut criterion can be equay powerfu in predicting the comprising credit oss events provided in the new Base capita accord as modes estimated on these defaut criteria. In any case, when deveoping rating modes one has to guarantee that the defaut event used to estimate the mode is comparabe to the event the mode sha be capabe to predict. Finay, a forecast horizon must be chosen. As iustrated by the Base Committee on Banking Supervision (1999), even before Base II for most banks it was common habit to use a modeing horizon of one year, as this time horizon is on the one hand ong enough to aow banks to take some action to avert predicted defauts, and on the other hand the time ag is short enough to guarantee the timeiness of the data input into the rating mode. 2.3 The Data Set The theoretica concepts discussed in this chapter wi be iustrated by appication to a data set of Austrian companies, which represents a sma sampe of the credit portfoio of an Austrian bank. The origina data, which was suppied by a major commercia Austrian bank for the research project described in Hayden (2002), consisted of about 5,000 firm-year observations of baance sheets and gain and oss accounts from 1,500 individua companies spanning 1994 to 1999. However, due to obvious mistakes in the data, such as assets being different from iabiities or negative saes, the data set had to be reduced to about 4,500 observations. Besides, certain firm types were excuded, i.e. a pubic firms incuding arge internationa corporations that do not represent the typica Austrian company and rather sma singe owner firms with a turnover of ess than 5 m ATS (about 0.36 m EUR), whose credit quaity often depends as much on the finances of a key individua as on the firm itsef. After eiminating financia statements covering a period of ess than tweve months and checking for observations that were incuded twice or more in the data set, amost 3,900 firm-years were eft. Finay, observations were dropped where the defaut information (bankruptcy) was missing or dubious. Tabe 2.1 shows the tota number of observed companies per year and spits the sampe into defauting and non-defauting firms. However, the data for 1994 is not depicted, as we are going to cacuate dynamic financia ratios (which compare

2 Estimation of a Rating Mode for Corporate Exposures 15 Tabe 2.1 Number of observations and defauts per year Year Non-defauting firms Defauting firms Tota 1995 1,185 54 1,239 1996 616 68 684 1997 261 46 307 1998 27 2 29 1999 23 1 24 Tota 2,112 171 2,283 current to past eves of certain baance sheet items) ater on, and these ratios cannot be cacuated for 1994 as the first period in the sampe. 2.4 Data Processing Section 2.4 discusses the major preparatory operations necessary before the mode estimation can be conducted. They incude the ceaning of the data, the cacuation of financia ratios, and their transformation to estabish inearity. 2.4.1 Data Ceaning Some of the important issues with respect to data ceaning were mentioned in Sect. 2.3 when the Austrian data set was presented. As described, it was guaranteed that: The sampe data was free of (obvious) mistakes The data set comprised ony homogeneous observations, where the reationship between the financia ratios and the defaut event coud be expected to be comparabe The defaut information was avaiabe (and reiabe) for a borrowers In addition, missing information with respect to the financia input data must be propery managed. Typicay, at east for some borrowers, part of the financia information is missing. If the number of the observations concerned is rather ow, the easiest way to hande the probem is to eiminate the respective observations competey from the data set (as impemented for the Austrian data). If, however, this woud resut in too many observations being ost, it is preferabe to excude a variabes with high numbers of missing vaues from the anaysis. Once the mode has been deveoped and is in use, the missing information needed to cacuate the mode output can be handed by substituting the missing financia ratios with the corresponding mean or median vaues over a observations for the respective time period (i.e. practicay neutra vaues) in order to create as undistorted an assessment as possibe using the remaining input factors.

16 E. Hayden 2.4.2 Cacuation of Financia Ratios Once the quaity of the basic financia data is guaranteed, potentia expanatory variabes have to be seected. Typicay, ratios are formed to standardise the avaiabe information. For exampe, the ratio Earnings per Tota Assets enabes a comparison of the profitabiity of firms of different size. In addition to considering ratios that refect different financia aspects of the borrowers, dynamic ratios that compare current to past eves of certain baance sheet items can be very usefu for predicting defaut events. Overa, the seected input ratios shoud represent the most important credit risk factors, i.e. everage, iquidity, productivity, turnover, activity, profitabiity, firm size, growth rates and everage deveopment. After the cacuation of the financia input ratios, it is necessary to identify and eiminate potentia outiers, because they can and do severey distort the estimated mode parameters. Outiers in the ratios might exist even if the underying financia data is absoutey cean, for exampe, when the denominator of a ratio is aowed to take on vaues cose to zero. To avoid the need to eiminate the affected observations a typica procedure is to repace the extreme data points by the 1% respectivey the 99% percentie of the according ratio. Tabe 2.2 portrays the expanatory variabes seected for use for the Austrian data and presents some descriptive statistics. The indicators chosen comprise a sma set of typica business ratios. A broader overview over potentia input ratios as we as a detaied discussion can be found in Hayden (2002). The ast coumn in Tabe 2.2 depicts the expected dependence between the accounting ratio and the defaut probabiity, where + symboises that an increase in the ratio eads to an increase in the defaut probabiity and symboises a decrease in the defaut probabiity given an increase in the expanatory variabe. Tabe 2.2 Seected input ratios Financia ratio Risk factor Mean Stand. Dev. Min. Max. Hypo. 1 Tota Liabiities/Tota Assets Leverage 0.89 0.18 0.02 1.00 + 2 Equity/Tota Assets Leverage 0.04 0.34 0.92 0.98 3 Bank Debt/T. Assets Leverage 0.39 0.26 0.00 0.97 + 4 Short Term Debt/Tota Assets Liquidity 0.73 0.25 0.02 1.00 + 5 Current Assets/Current Liquidity 0.08 0.15 0.00 0.72 Liabiities 6 Accounts Receivabe/Net Saes Activity 0.13 0.12 0.00 0.41 + 7 Accounts Payabe/Net Saes Activity 0.12 0.12 0.00 0.44 + 8 (Net Saes Materia Costs)/ Productivity 2.56 1.85 1.03 8.55 Person. Costs 9 Net Saes/Tota Assets Turnover 1.71 1.08 0.01 4.43 10 EBIT/Tota Assets Profitabiity 0.06 0.13 0.18 0.39 11 Ordinary Business Income/ Profitabiity 0.02 0.13 0.19 0.33 Tota Assets 12 Tota Assets (in 1 Mio. EUR) Size 35.30 72.98 0.22 453.80 13 Net Saes/Net Saes ast year Growth 1.06 0.34 0.02 2.03 /+ 14 Tota Liabiities/Liabiities ast year Leverage Growth 1.00 1.03 0.07 1.23 +

2 Estimation of a Rating Mode for Corporate Exposures 17 Whenever a certain ratio is seected as a potentia input variabe for a rating mode, it shoud be assured that a cear hypothesis can be formuated about this dependence to guarantee that the resuting mode is economicay pausibe. Note, however, that the hypothesis chosen can aso be rather compex; for exampe, for the indicator saes growth, the hypothesis formuated is /þ. This takes into account that the reationship between the rate at which companies grow and the rate at which they defaut is not as simpe as that between other ratios and defaut. Whie it is generay better for a firm to grow than to shrink, companies that grow very quicky often find themseves unabe to meet the management chaenges presented by such growth especiay within smaer firms. Furthermore, this quick growth is unikey to be financed out of profits, resuting in a possibe buid up of debt and the associated risks. Therefore, one shoud expect that the reationship between saes growth and defaut is non-monotone, what wi be examined in detai in the next section. 2.4.3 Test of Linearity Assumption After having seected the candidate input ratios, the next step is to check whether the underying assumptions of the ogit mode appy to the data. As expained in Chap. 1, the ogit mode can be written as eb0 x i P i ¼ P(y i ¼ 1Þ ¼F(b 0 x i Þ¼ ; (2.1) 1 þ e b0 x i which impies a inear reationship between the og odd and the input variabes: P i Log odd ¼ n ¼ b 0 x i (2.2) 1 P i This inearity assumption can be easiy tested by dividing the indicators into groups that a contain the same number of observations, cacuating the historica defaut rate respectivey the empirica og odd within each group, and estimating a inear regression of the og odds on the mean vaues of the ratio intervas. When appied to the Austrian data (by forming 50 groups), this procedure permits the concusion that for most accounting ratios, the inearity assumption is indeed vaid. As an exampe the reationship between the variabe EBIT/Tota Assets and the empirica og odd as we as the estimated inear regression is depicted in Fig. 2.1. The regression fit is as high as 78.02%. However, one expanatory variabe, namey saes growth, shows a non-inear and even non-monotone behaviour, just as was expected. Hence, as portrayed in Fig. 2.2, due to the inearity assumption inherent in the ogit mode, the reationship between the origina ratio saes growth and the defaut event cannot be correcty captured by such a mode.

18 E. Hayden R2:.7802 4 Empirica Log Odd 4.5 5 5.5 6 6.5.2 0 EBIT / Tota Assets.2.4 Log Odds Fitted vaues Fig. 2.1 Reationship between EBIT/Tota Assets and og odd 4.5 Empirica Log Odd 5 5.5 6 6.5.6.8 1 1.2 1.4 Net Saes / Net Saes Last Year Empirica Log Odd Linear Prediction Smoothed Vaues Fig. 2.2 Reationship between Saes Growth and og odd

2 Estimation of a Rating Mode for Corporate Exposures 19 Therefore, to enabe the incusion of the indicator saes growth into the rating mode, the ratio has to be inearized before ogit regressions can be estimated. This can be done in the foowing way: the points obtained from dividing the variabe saes growth into groups and potting them against the respective empirica og odds are smoothed by a fiter, for exampe the one proposed in Hodrick and Prescott (1997), to reduce noise. Then the origina vaues of saes growth are transformed to og odds according to this smoothed reationship, and in any further anaysis the transformed og odd vaues repace the origina ratio as input variabe. This test for the appropriateness of the inearity assumption aso aows for a first check as to whether the univariate dependence between the considered expanatory variabes and the defaut probabiity is as expected. For the Austrian data the univariate reationships between the investigated indicators and the defaut event coincide with the hypotheses postuated in Tabe 2.2, i.e. a ratios behave in an economicay meaningfu way. 2.5 Mode Buiding 2.5.1 Pre-seection of Input Ratios After verifying that the underying assumptions of a ogistic regression are vaid, the mode buiding process can be started. However, athough typicay a huge number of potentia input ratios are avaiabe when deveoping a rating mode, from a statistica point of view it is not advisabe to enter a these variabes into the ogit regression. If, for exampe, some highy correated indicators are incuded in the mode, the estimated coefficients wi be significanty and systematicay biased. Hence, it is preferabe to pre-seect the most promising expanatory variabes by means of the univariate power of and the correation between the individua input ratios. To do so, given the data set at hand is arge enough to aow for it, the avaiabe data shoud be divided into one deveopment and one vaidation sampe by randomy spitting the whoe data into two sub-sampes. The first one, which typicay contains the buk of a observations, is used to estimate rating modes, whie the remaining data is eft for an out-of-sampe evauation. When spitting the data, it shoud be ensured that a observations of one firm beong excusivey to one of the two sub-sampes and that the ratio of defauting to non-defauting firms is simiar in both data sets. For the Austrian data, about 70% of a observations are chosen for the training sampe as depicted in Tabe 2.3. The concrete pre-seection process now ooks as foows: At first, univariate ogit modes are estimated in-sampe for a potentia input ratios, whose power to identify defauts in the deveopment sampe is evauated via the criterion of the accuracy ratio (AR), a concept discussed in detai in Chap. 13. Afterwards, the pairwise correation between a expanatory variabes is computed to identify subgroups of highy correated indicators, where by rue of thumb ratios with absoute

20 E. Hayden Tabe 2.3 Division of the data into inand out-of-sampe subsets Year Training sampe Vaidation sampe Non-defauting Defauting Non-defauting Defauting 1995 828 43 357 11 1996 429 44 187 24 1997 187 25 74 21 1998 20 2 7 0 1999 17 1 6 0 correation vaues of above 50% are pooed into one group. Finay, from each correation sub-group (that usuay contains ony ratios from one specific credit risk category) that expanatory variabe is seected for the mutivariate mode buiding process that has got the highest and hence best accuracy ratio in the univariate anaysis. Tabe 2.4 dispays the accuracy ratios of and the correation between the financia ratios cacuated for the Austrian data set. As can be seen, expanatory variabe 1 is highy correated with indicator 2 (both measuring everage) and ratio 10 with variabe 11 (both refecting profitabiity). Besides, the input ratios 2 and 11 have got better (higher) accuracy ratios than the indicators 1 respectivey 10, hence, the atter ones are dropped from the ist of expanatory variabes for the mutivariate anaysis. 2.5.2 Derivation of the Fina Defaut Prediction Mode Those ratios pre-seected in the previous step are now used to derive the fina mutivariate ogit mode. Usuay, however, the number of potentia expanatory variabes is sti too high to specify a ogit mode that contains a of them, because the optima mode shoud contain ony a few, highy significant input ratios to avoid overfitting. Thus, even in our sma exampe with ony 12 indicators being eft, we woud have to construct and compare 2 12 ¼ 4,096 modes in order to determine the best econometric mode and to entirey resove mode uncertainty. This is, of course, a tough task, which becomes infeasibe for typica short ists of about 30 to 60 pre-seected input ratios. Therefore, the standard procedure is to use forward/ backward seection to identify the fina mode (see Hosmer and Lemeshow 2000). For the Austrian data set backward eimination, one possibe method of these statistica stepwise variabe seection procedures that is impemented in most statistica software packages, was appied to derive the fina ogit mode. This method starts by estimating the fu mode (with a potentia input ratios) and continues by eiminating the worst covariates one by one unti the significance eve of a remaining expanatory variabes is beow the chosen critica eve, usuay set at 90% or 95%. Tabe 2.5 describes two ogit modes derived by backward eimination for the Austrian data. It depicts the constants of the ogit modes and the estimated coefficients

2 Estimation of a Rating Mode for Corporate Exposures 21 Tabe 2.4 Pairwise correation of a potentia input ratios Ratio AR in % 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 32.0 1 0.81 +0.49 +0.50 0.48 +0.05 +0.25 0.05 0.05 0.25 0.36 0.17 +0.08 +0.38 2 34.6 1 0.42 0.40 +0.39 +0.10 0.21 +0.10 +0.13 +0.28 +0.38 +0.22 0.12 0.25 3 20.7 1 0.03 0.33 +0.02 +0.01 +0.06 0.30 0.10 0.24 0.07 +0.06 +0.14 4 26.5 1 0.32 +0.13 +0.20 0.09 +0.20 0.16 0.18 0.15 +0.07 +0.24 5 17.2 1 0.12 0.17 +0.09 +0.14 +0.14 +0.20 +0.04 0.01 0.14 6 16.0 1 +0.29 +0.02 0.21 0.03 0.02 0.01 +0.10 +0.03 7 25.4 1 +0.11 0.32 0.24 0.24 +0.02 +0.18 +0.10 8 25.5 1 0.05 +0.28 +0.25 0.01 +0.02 0.11 9 2.1 1 +0.25 +0.25 0.19 0.12 0.05 10 19.7 1 +0.96 0.08 0.18 0.25 11 24.1 1 0.02 0.18 0.28 12 6.3 1 0.06 +0.00 13 14.2 1 0.01 14 1.4 1

22 E. Hayden Tabe 2.5 Estimates of mutivariate ogit modes Financia ratio Risk factor Mode 1 Mode 2 Hypo. (fina M.) 2 Equity/Tota Assets Leverage 0.98** 0.85** 3 Bank Debt/Tota Assets Leverage 1.55*** 1.21*** + 4 Short Term Debt/Tota Assets Liquidity 1.30** 1.56*** + 6 Accounts Receivabe/Net Saes Activity 1.71* + 7 Accounts Payabe/Net Saes Activity 2.31** 1.53* + 8 (Net Saes Materia Costs)/Personne Productivity 0.23*** 0.23*** Costs 9 Net Saes/Tota Assets Turnover 0.26** Constant 1.18 0.95 for a those financia ratios that enter into the respective mode. The stars represent the significance eve of the estimated coefficients and indicate that the true parameters are different from zero with a probabiity of 90% (*), 95% (**) or 99% (***). Mode 1 arises if a 12 pre-seected variabes are entered into the backward eimination process. Detaied anaysis of this mode shows that most signs of the estimated coefficients correspond to the postuated hypotheses, however, the mode specifies a positive reationship between the ratio number 9 Net Saes/Tota Assets, whie most empirica studies find that arger firms defaut ess frequenty. What s more, even for our data sampe a negative coefficient was estimated in the univariate anaysis. For this reason, a coser inspection of input ratio 9 seems appropriate. Athough the variabe Net Saes/Tota Assets does not exhibit a pairwise correation of more than 50%, it shows absoute correation eves of about 30% with severa other covariates. This indicates that this particuar ratio is too highy correated (on a mutivariate basis) with the other expanatory variabes and has to be removed from the ist of variabes entering the backward eimination process. Mode 2 in Tabe 2.5 depicts the resuting ogit mode. Here a coefficients are of comparabe magnitude to those of mode 1, except that the ratio Accounts Receivabe/Net Saes becomes highy insignificant and is therefore excuded from the mode. As a consequence, a estimated coefficients are now economicay pausibe, and we accept mode 2 as our (preiminary) fina mode version. 2.5.3 Mode Vaidation Finay, the derived ogit mode has to be vaidated. In a first step, some statistica tests shoud be conducted in order to verify the mode s robustness and goodness of fit in-sampe, and in a second step the estimated mode shoud be appied to the vaidation sampe to produce out-of-sampe forecasts, whose quaity can be evauated with the concept of the accuracy ratio and other methods depicted in Chap. 13.

2 Estimation of a Rating Mode for Corporate Exposures 23 The goodness-of-fit of a ogit mode can be assessed in two ways: first, on the basis of some test statistics that use various approaches to measure the distance between the estimated probabiities and the actua defauts, and second, by anaysing individua observations which can each have a certain strong impact on the estimated coefficients (for detais see Hosmer and Lemeshow 2000). One very popuar goodness-of-fit test statistic is the Hosmer-Lemeshow test statistic that measures how we a ogit mode represents the actua probabiity of defaut for groups of firms of differenty perceived riskiness. Here, the observations are grouped based on percenties of the estimated defaut probabiities. For the Austrian data 10% intervas were used i.e. ten groups were formed. Now for every group the average estimated defaut probabiity is cacuated and used to derive the expected number of defauts per group. Next, this number is compared with the amount of reaised defauts in the respective group. The Hosmer-Lemeshow test statistic then summarises this information for a groups. In our case of ten groups the test statistic for the estimation sampe is chi-square distributed with 8 degrees of freedom, and the corresponding p-vaue for the rating mode can then be cacuated as 79.91%, which indicates that the mode fits quite we. However, the Hosmer-Lemeshow goodness-of-fit test can aso be regarded from another point of view for the appication at hand. Unti now we ony deat with the deveopment of a mode that assigns each corporation a certain defaut probabiity or credit score, which eads towards a ranking between the contempated firms. However, in practice banks usuay want to use this ranking to map the companies to an interna rating scheme that typicay is divided into about ten to twenty rating grades. The easiest way to do so woud be to use the percenties of the predicted defaut probabiities to buid groups. If for exampe ten rating casses sha be formed, then from a observations the 10% with the smaest defaut probabiities woud be assigned the best rating grade, the next 10% the second and so on ti the ast 10% with the highest estimated defaut probabiities woud enter into the worst rating cass. The Hosmer-Lemeshow test now tes us that, given one woud appy the concept described above to form rating categories, overa the average expected defaut probabiity per rating grade woud fit with the observed defaut experience per rating cass. What s more, as depicted in Tabe 2.6, the in-sampe accuracy ratio is about 44%, which is a reasonabe number. Usuay the rating modes for corporate exposures presented in the iterature have an accuracy ratio between 40% and 70%. As discussed in Chap. 13 in detai, AR can ony be compared reiaby for modes that are appied to the same data set, because differences in the data set such as varying reative amounts of defauters or non-equa data reiabiity drives this measure heaviy, hence, an AR of about 44% seems satisfactory. Tabe 2.6 Vaidation resuts of the fina ogit mode Fina mode (mode 2) Accuracy ratio s ÂR 95% conf. interva Hosmer-Lemeshow test statistic p-vaue In-sampe 0.4418 0.0444 [0.3574, 0.5288] 79.91% Out-of-sampe 0.4089 0.0688 [0.2741, 0.5438] 68.59%

24 E. Hayden Finay, the out-of-sampe accuracy ratio amounts to about 41%, which is amost as high as the in-sampe AR. This impies that the derived rating mode is stabe and powerfu aso in the sense that it produces accurate defaut predictions for new data that was not used to deveop the mode. Therefore, we can now eventuay accept the derived ogit mode as our fina rating too. 2.6 Concusions This chapter focused on the specia difficuties that are encountered when deveoping interna rating modes for corporate exposures. Athough the whoe process with data coection and processing, mode buiding and vaidation usuay takes quite some time and effort, the job is not yet competed with the impementation of the derived rating mode. The predictive power of a statistica modes depends heaviy on the assumption that the historica reationship between the mode s covariates and the defaut event wi remain unchanged in the future. Given the wide range of possibe events such as changes in firms accounting poicies or structura disruptions in certain industries, this assumption is not guaranteed over onger periods of time. Hence, it is necessary to revaidate and eventuay recaibrate the mode reguary in order to ensure that its predictive power does not diminish. References Base Committee on Banking Supervision (1999), Credit Risk Modeing: Current Practices and Appications, Bank for Internationa Settements. Base Committee on Banking Supervision (2001), The Interna Ratings-Based Approach, Bank for Internationa Settements. Hayden E (2002), Modeing an Accounting-Based Rating Mode for Austrian Firms, unpubished PhD dissertation, University of Vienna. Hayden E (2003), Are Credit Scoring Modes Sensitive to Aternative Defaut Definitions? Evidence from the Austrian Market, Working Paper, University of Vienna. Hodrick R, Prescott C (1997), Post-War U.S. Business Cyces: An Empirica Investigation, Journa of Money, Credit and Banking 29, pp. 1 16. Hosmer W, Lemeshow S (2000), Appied Logistic Regression, Wiey, New York.

Chapter 3 Scoring Modes for Retai Exposures Danie Porath 3.1 Introduction Rating modes for retai portfoios deserve a more detaied examination because they differ from other bank portfoios. The differences can mainy be attributed to the specific data structure encountered when anayzing retai exposures. One impication is that different statistica toos have to be used when creating the mode. Most of these statistica toos do not beong to the banker s standard toobox. At the same time and stricty speaking for the same reason the banks risk management standards for retai exposures are not comparabe to those of other portfoios. Banks often use scoring modes for managing the risk of their retai portfoios. Scoring modes are statistica risk assessment toos especiay designed for retai exposures. They were initiay introduced to standardize the decision and monitoring process. With respect to scoring, the industry had estabished rating standards for retai exposures ong before the discussion about the IRBA emerged. The Base Committee acknowedged these standards and has modified the minimum requirements for the interna rating modes of retai exposures. The aim of this chapter is to discuss scoring modes in the ight of the minimum requirements and to introduce the non-standard statistica modeing techniques which are usuay used for buiding scoring tabes. The discussion starts with an introduction to scoring modes comprising a genera description of scoring, a distinction of different kinds of scoring modes and an exposure of the theoretica differences compared to other parametric rating modes. In Sect. 3.3, we extract the most important minimum requirements for retai portfoios from the New Base Capita Framework and consider their reevance for scoring modes. Section 3.4 is dedicated to modeing techniques. Here, specia focus is paced on the preiminary univariate anaysis because it is competey different from other portfoios. We concude with a short summary. D. Porath University of Appied Sciences, Mainz e-mai: danie.porath@wiwi.fh.mainz.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_3, # Springer-Verag Berin Heideberg 2011 25

26 D. Porath 3.2 The Concept of Scoring 3.2.1 What is Scoring? Like any rating too, a scoring mode assesses a borrower s creditworthiness. The outcome of the mode is expressed in terms of a number caed score. Increasing scores usuay indicate decining risk, so that a borrower with a score of 210 is more risky than a borrower with a score of 350. A comprehensive overview about scoring can be found in Thomas et a. (2002). The mode which cacuates the score is often referred to as a scoring tabe, because it can be easiy dispayed in a tabe. Tabe 3.1 shows an extract of two variabes from a scoring mode (usuay scoring modes consist of about 7 up to 15 variabes): The tota customer score can be cacuated by adding the scores of the borrower s severa characteristics. Each variabe contains the category neutra. The score of this category represents the portfoio mean of the scores for a variabe and therewith constitutes a benchmark when evauating the risk of a specific category. Categories with higher scores than neutra are beow the average portfoio risk and categories with ower scores are more risky than the average. For exampe, divorced borrowers dispay increased risk compared to the whoe portfoio, because for the variabe marita status the score of a divorced borrower (16) is ower than the score for the category neutra (19). Scoring modes usuay are estimated with historica data and statistica methods. The historica data invoves information about the performance of a oan ( good or bad ) and about the characteristics of the oan some time before. The time span between the measurement of the characteristic on the one hand and the performance on the other hand determines the forecast horizon of the mode. Estimation procedures for scoring modes are ogistic regression, discriminant anaysis or simiar methods. The estimation resuts are the scores of the singe Tabe 3.1 Extract from a scoring tabe Variabe Score of the variabes attributes Marita status of borrower Unmarried 20 Married or widowed 24 Divorced or separated 16 No answer 16 Neutra 19 Age of borrower 18 24 14 24 32 16 32 38 25 38 50 28 50 65 30 65 or oder 32 Neutra 24

3 Scoring Modes for Retai Exposures 27 characteristics. Usuay the scores are rescaed after estimation in order to obtain round numbers as in the exampe shown in Tabe 3.1. More detais regarding estimation of the scores are shown in Sect. 3.4. 3.2.2 Cassing and Recoding Scoring is a parametric rating mode. This means that modeing invoves the estimation of the parameters b 0,...,b N in a genera mode S i ¼ b 0 þ b 1 x i1 þ b 2 x i2 þ...þ b N x in : (3.1) Here S i denotes the Score of the oan i ¼ 1,...,I and x 1,...,x N are the input parameters or variabes for the oan i. The parameters b n (n ¼ 0,...,N) refect the impact of a variation of the input factors on the score. Scoring differs from other parametric rating modes in the treatment of the input variabes. As can be seen from Tabe 3.1, the variabe marita status is a quaitative variabe, therefore it enters the mode categoricay. Some vaues of the variabe have been grouped into the same category, ike for exampe married and widowed in order to increase the number of borrowers within each cass. The grouping of the vaues of a variabe is a separate preiminary step before estimation and is caed cassing. The genera approach in (3.1) cannot manage categorica variabes and therefore has to be modified. To this end, the (categorica) variabe x n has to be recoded. An adequate recoding procedure for scoring is to add the category neutra to the existing number of C categories and repace x n by a set of dummy variabes d xn(c), c ¼ 1,...,C which are defined in the foowing way: 8 < 1 for x n ¼ c d xnðcþ ¼ 1 for x n ¼ neutra : 0 ese: (3.2) The recoding given in (3.2) is caed effect coding and differs from the standard dummy variabe approach where the dummies ony take the vaues 0 and 1. The benefit from using (3.2) is that it aows for the estimation of a variabe-specific mean which is the score of the category neutra. As can be seen from (3.2), the vaue of the category neutra is impicity given by the vector of dummy vaues ( 1,..., 1). The coefficients of the other categories then represent the deviation from the variabe-specific mean. This can be iustrated by recoding and repacing the first variabe x i1 in (3.1). Mode (3.1) then becomes S i ¼ b 0 þ b 10 þ b 11 d x11;i þ b 12 d x12;i þ...þ b 1C d x1c;i þ b2 x i2 þ...þ b N x in : (3.3)

28 D. Porath Here (b 10 b 11 b 12 b 1C ) is the variabe-specific average ( neutra ) and the coefficients b 11,...,b 1C represent the deviation of the individua categories from the average. The scores of the singe categories (see Tabe 3.1) are given by the sums b 10 þ b 11, b 10 þ b 12,..., b 10 þ b 1C. Apart from the specia recoding function (3.2), the procedure discussed so far is the standard procedure for handing categorica variabes. The major characteristic of scoring is that the same procedure is conducted for the quantitative variabes. This means that a variabes are cassed and recoded prior to estimation and therefore are treated as categorica variabes. As a consequence, the overa mean b 0 in (3.3) disappears and the mode can be rewritten as: S i ¼ b 10 þ b 11 d x11;i þ...þ b 1C d x1c;i þ...þ bn0 þ b N1 d xn1;i þ...þ b NC d xnc;i : (3.4) With an increasing number of variabes and categories, equation (3.4) soon becomes unmanageabe. This is why scoring modes are usuay dispayed in tabes. The effect of cassing and recoding is twofod: On the one hand, the information about the intercass variation of the quantitative variabe disappears. As can be seen from Tabe 3.1, an increasing age reduces risk. The mode, however, does not indicate any difference between the age of 39 and 49, because the same score is attributed to both ages. If the variabe age entered the mode as a quantitative variabe with the estimated coefficient b age, any difference in age (Dage) woud be captured by the mode (its effect on risk, i.e. the score, ceteris paribus, being b age Dage). On the other hand, categorization aows for fexibe risk patterns. Referring again to the exampe of age, the impact on risk may be strong for the ower age categories whie diminishing for increasing ages. Such a noninear impact on the score S i can be modeed by seecting narrow casses for ower ages and broad casses for higher ages. The quantitative mode, on the contrary, attributes the same impact of b age to a one-year change in age starting from any eve. Thus, cassing and recoding is an easy way to introduce noninearities in the mode. The theoretica merits from cassing and recoding, however, were not pivota for the wide use of scoring modes. The more important reason for cassing and recoding is that most of the risk-reevant input variabes for retai customers are quaitative. These are demographic characteristics of the borrower (ike marita status, gender, or home ownership), the type of profession, information about the oan (type of oan, intended use) and information about the payment behaviour in the past (due payment or not). The reason for transforming the remaining quantitative variabes (ike age or income) into categorica variabes is to obtain a uniform mode. 3.2.3 Different Scoring Modes Banks use different scoring modes according to the type of oan. The reason is that the data which is avaiabe for risk assessment is oan-specific. For exampe, the

3 Scoring Modes for Retai Exposures 29 scoring of a mortgage oan can make use of a the information about the rea estate whereas there is no comparabe information for the scoring mode of a current account. On the other hand, modes for current accounts invove much information about the past payments observed on the account (income, drawings, baance) which are not avaiabe for mortgage oans. For mortgage oans, payment information generay is restricted to whether the monthy instament has been paid or not. As a consequence, there are different modes for different products and when the same person has two different oans at the same bank, he or she generay wi have two different scores. This is a crucia difference to the genera rating principes of Base II. Scoring modes which are primariy based on payment information are caed behavioura scoring. The prerequisite for using a behavioura score is that the bank observes information about the payment behaviour on a monthy basis, so that the score changes monthy. Furthermore, in order to obtain meaningfu resuts, at east severa monthy payment transactions shoud be observed for each customer. Since the behavioura score is dynamic, it can be used for risk monitoring. Additionay, banks use the score for risk segmentation when defining strategies for retai customers, ike for exampe cross-seing strategies or the organization of the dunning process ( different risk, different treatment ). When payment information is sporadic, it is usuay not impemented in the scoring mode. The score then invoves static information which has been queried in the appication form. This score is caed an appication score. In contrast to the behavioura score, the appication score is static, i.e. once cacuated it remains constant over time. It is normay cacuated when a borrower appies for a oan and heps the bank to decide whether it shoud accept or refuse the appication. Additionay, by combining the score with dynamic information it can be used as a part of a monitoring process. 3.3 Scoring and the IRBA Minimum Requirements Interna Rating systems for retai customers were in use ong before Base II. The reason is that statistica modes for risk assessment are especiay advantageous for the retai sector: on the one hand, the high granuarity of a retai portfoio aows banks to reaize economies of scae by standardization of the decision and monitoring processes. On the other hand, the database generay consists of a broad number of homogenous data. Homogeneity is owed to standardized forms for appication and monitoring. As a consequence, the database is particuary suited for modeing. In fact, statistica procedures for risk forecasting of retai oans have a history of severa decades (cf. Hand 2001), starting with the first attempts in the 1960s and coming into wide use in the 1980s. Today, scoring is the industria standard for the rating of retai customers. Since these standards have deveoped independenty from the New Base Capita Approach, there are some differences to the IRBA minimum requirements. The Capita Accord has acknowedged these differences

30 D. Porath and consequenty modified the rues for retai portfoios. Hence most banks wi meet the minimum requirements, possiby after some sight modifications of their existing scoring systems. In the foowing subsections we discuss the meaning of some seected minimum requirements for scoring and therewith give some suggestions about possibe modifications. The discussion is restricted to the minimum requirements, which according to our view, are the most reevant for scoring. We refer to the Revised Framework of the Base Committee on Banking Supervision from June 2004 (cf. BIS 2004) which for convenience in the foowing is caed Capita Framework. 3.3.1 Rating System Design Foowing } 394 of the Capita Framework, a rating system comprises the assignment of a rating to credit risk and the quantification of defaut and oss estimates. However, scoring modes ony provide the first component, which is the score S i. The defaut and oss estimates (which in the Capita Framework are PD, LGD, and EAD) usuay are not determined by the scoring mode. When a bank intends to use a scoring mode for the IRBA, these components have to be assessed separatey. 3.3.2 Rating Dimensions Generay, the IRBA requires a rating system to be separated by a borrower-specific component and a transaction-specific component (see } 396 of the Capita Framework). However, in the previous section we have seen that scoring modes typicay mix variabes about the borrower and the type of oan. In order to render scoring modes eigibe to the IRBA, the Base Committee has modified the genera approach on the rating dimensions for retai portfoios. According to } 401 of the Capita Framework both components shoud be present in the scoring mode, but need not be separated. Consequenty, when referring to the risk cassification of retai portfoios, the Capita Framework uses the term poo instead of rating grade. With } 401, banks have greater fexibiity when defining poos, as ong as the pooing is based on a risk-reevant information. Poos can be customer-specific or oan-specific (ike in a scoring mode) or a mixture of both. A further consequence of } 401 is that one the same borrower is aowed to have two different scores. 3.3.3 Risk Drivers Paragraph 402 of the Capita Framework specifies the risk drivers banks shoud use in a scoring mode. These cover borrower characteristics, transaction characteristics

3 Scoring Modes for Retai Exposures 31 and deinquency. As seen in the previous section, borrower and transaction characteristics are integra parts of a scoring tabe. Deinquency, on the other hand, is not usuay integrated in a scoring mode. The rationae is that scoring aims at predicting deinquency and that therefore no forecast is needed for a deinquent account. However, a correct impementation of a scoring mode impies that deinquent accounts are separated (and therefore identified), so that the cacuation of the score can be suppressed. Hence, when using a scoring mode, normay a risk drivers mentioned in } 402 of the Capita Framework are integrated. 3.3.4 Risk Quantification Risk quantification in terms of Base II is the assessment of expected oss as the product from PD, LGD and EAD. Since the expected oss of a oan determines the risk weight for the capita requirement, the reguatory capita framework contains precise definitions for the quantification of these components. This means that the underying time horizon is fixed to 1 year and that the underying defaut event is expicity defined. Scoring modes generay do not foow these definitions since their primary aim is not to fufi the supervisory requirements but to provide interna decision support. The appication score, for exampe, tes whether an appication for a oan shoud be accepted or refused and for this decision it woud not suffice to know whether the oan wi defaut in the foowing year ony. Instead, the bank is interested to know whether the oan wi defaut in the ong run, and therefore scoring modes generay provide ong-run predictions. Additionay, the defaut event sets as soon as the oan becomes no onger profitabe for the bank and this is usuay not the case when the oan defauts according the Base definition. It depends, instead, on the bank s interna cacuation. To sum up, scoring modes used for interna decision support generay wi not compy with the requirements about risk quantification. A strategy to conserve the power of an interna decision too and at the same time achieve compiance with the minimum requirements is: Deveop the scoring mode with the interna time-horizons and definitions of defaut. Define the poos according to } 401 of the Capita Framework. Estimate the poo-specific PD, LGD and EAD foowing the Base definitions in a separate step. Finay, it shoud be noted that the time horizon for assigning scores is not specified in the Base Accord. In paragraph 414 of the Capita Framework it is stated that the horizon shoud be generay onger than 1 year. The ong-term horizon normay used by scoring systems therefore is conforming to the minimum requirements.

32 D. Porath 3.3.5 Specia Requirements for Scoring Modes In } 417 the Capita Framework expicity refers to scoring modes (and other statistica modes) and specifies some additiona requirements. The rationae is that the impementation of a scoring mode eads to highy standardized decision and monitoring processes where faiures may be overooked or detected too ate. Therefore, the requirements given in } 417 refer to specia quaitative features of the mode and specia contro mechanisms. These requirements wi generay be met when banks foow the industria standards for the deveopment and impementation of scoring modes. The most important standards which have to be mentioned in this context are: The use of a representative database for the deveopment of the mode Documentation about the deveopment incuding univariate anaysis Preparation of a user s guide Impementation of a monitoring process 3.4 Methods for Estimating Scoring Modes The statistica methods which are suitabe for estimating scoring modes comprise the techniques introduced in Chap. 1, e.g. ogit anaysis, or discriminant anaysis, with the specia feature that a input variabes enter the mode as categorica variabes. This requires an extensive preiminary data anaysis which is referred to as univariate anaysis. Univariate anaysis generay is interesting for rating anaysis because it serves to detect probems concerning the data and heps to identify the most important risk-drivers. However, for retai portfoios, univariate anaysis is more compex and more important than in the genera case. There are severa reasons for this: Univariate anaysis determines the casses on which the recoding is based (see Sect. 3.2) and hereby becomes an integra part of the mode-buiding process. In retai portfoios, quaitative information is predominant (e.g. a person s profession, marita status). In retai portfoios, many quaitative variabes are hard factors and do not invove human judgement. Exampes incude a person s profession, marita status and gender. Note that quaitative information encountered in rating systems for corporate oans, often require persona judgement on part of the anayst (e.g. a company s management, the position in the market or the future deveopment of the sector where the company operates). For retai portfoios, a priori, it is often unknown whether a variabe is reevant for the risk assessment. For exampe, there is no theory which tes whether a borrower s profession, gender or domicie heps in predicting defaut. This is different for the corporate sector where the main information consists of financia ratios taken from the baance sheet. For exampe, EBIT ratios measure the

3 Scoring Modes for Retai Exposures 33 profitabiity of a firm and since profitabiity is inked to the firm s financia heath, it can be cassified as a potentia risk factor prior to the anaysis. For retai portfoios, univariate anaysis repaces a priori knowedge and therefore heps to identify variabes with a high discriminatory power. Often, the risk distribution of a variabe is unknown a priori. This means that before anayzing a variabe, it is not cear which outcomes correate with high risks and which outcomes correate with ow risks. This is competey different from the corporate sector, where for many financia ratios, the risk patterns are we-known. For exampe, it is a priori known that ceteris paribus, high profitabiity eads to ow risk and vice versa. For retai portfoios, the risk distribution has to be determined with the hep of univariate anaysis. The consequences are twofod: On one hand, univariate anaysis is particuary important for repacing a priori knowedge. On the other hand, the statistica methods appied in the univariate anaysis shoud be designed to hande quaitative hard factors. The basic technique for creating a scoring mode is crosstabuation. Crosstabs dispay the data in a two-dimensiona frequency tabe, where the rows c ¼ 1,...,C are the categories of the variabe and the coumns are the performance of the oan. The ces contain the absoute number of oans incuded in the anaysis. Crosstabuation is fexibe because it works with quaitative data as we as quantitative data quantitative information simpy has to be grouped beforehand. A simpe exampe for the variabe marita status is dispayed in Tabe 3.2. The crosstab is used to assess the discriminative power. The discriminative power of a variabe or characteristic can be described as its power to discriminate between good and bad oans. However, it is difficut to compare the absoute figures in the tabe. In Tabe 3.2, the bank has drawn a sampe of the good oans. This is a common procedure, because often it is difficut to retrieve historica data. As a consequence, in the crosstab, the number of good oans cannot be compared to the number of bad oans of the same category. It is therefore reasonabe to repace the absoute vaues by the coumn percentages for the good oans P(c Good) and for the bad oans P(c Bad), see Tabe 3.3. Tabe 3.2 Crosstab for the variabe Marita status Marita status of borrower No. of good oans No. of bad oans Unmarried 700 500 Married or widowed 850 350 Divorced or separated 450 650 Tabe 3.3 Coumn percentages, WoE and IV Marita status of borrower P(c Good) P(c Bad) WoE c Unmarried 0.3500 0.3333 0.0488 Married or widowed 0.4250 0.2333 0.5996 Divorced or separated 0.2250 0.4333 0.6554 IV 0.2523

34 D. Porath The discriminative power can be assessed by regarding the risk distribution of the variabe which is shown by the Weight of Evidence WoE c (see Good 1950). The Weight of Evidence can be cacuated from the coumn percentages with the foowing formua: WoE c ¼ nðpcjgood ð ÞÞ nðpcjbad ð ÞÞ: (3.5) The interpretation of WoE c is straightforward: Increasing vaues of the Weight of Evidence indicate decreasing risk. A vaue of WoE c > 0(WoE c < 0) means that in category c good (bad) oans are over-represented. In the above exampe, the Weight of Evidence shows that oans granted to married or widowed customers have defauted with a ower frequency than those granted to divorced or separated customers. The vaue of WoE c cose to 0 for unmarried customers dispays that the risk of this group is simiar to the average portfoio risk. The Weight of Evidence can aso be interpreted in terms of the Bayes theorem. The Bayes theorem expressed in og odds is n PðGoodjcÞ PðBadjcÞ PðcjGoodÞ PðGoodÞ ¼ n þ n PðcjBadÞ PðBadÞ : (3.6) Since the first term on the right hand of (3.6) is the Weight of Evidence, it represents the difference between the a posteriori og odds and the a priori og odds. The vaue of WoE c therefore measures the improvement of the forecast through the information of category c. Hence it is a performance measure for category c. A comprehensive performance measure for a categories of an individua variabe can be cacuated as a weighted average of the Weights of Evidence for a categories c ¼ 1,...,C. The resut is caed Information Vaue, IV (cf. Kuback 1959) and can be cacuated by: IV ¼ XC c¼1 Woe c PðcjGoodÞ PðcjBadÞ : (3.7) A high vaue of IV indicates a high discriminatory power of a specific variabe. The Information Vaue has a ower bound of zero but no upper bound. In the exampe of Tabe 3.3, the Information Vaue is 0.2523. Since there is no upper bound, from the absoute vaue we cannot te whether the discriminatory power is satisfactory or not. In fact, the Information Vaue is primariy cacuated for the purpose of comparison to other variabes or aternative cassings of the same variabe and the same portfoio. The Information Vaue has the great advantage of being independent from the order of the categories of the variabe. This is an extremey important feature when anayzing data with unknown risk distribution. It shoud be noted that most of the better-known performance measures ike the Gini coefficient or the power curve do not share this feature and therefore are of imited reevance ony for the univariate anaysis of retai portfoios.

3 Scoring Modes for Retai Exposures 35 Crosstabuation is a means to generate cassings which are needed for the recoding and estimation procedures. There are three requirements for a good cassing. First, each cass shoud contain a minimum number of good and bad oans, otherwise the estimation of the coefficients b in (3.4) tend to be imprecise. Foowing a rue of thumb there shoud be at east 50 good oans and 50 bad oans in each cass. Probaby this is why in the above exampe there is no separate category widowed. Second, the categories grouped in each cass shoud dispay a simiar risk profie. Therefore, it is feasibe to combine the categories separated and divorced to one singe cass. Third, the resuting cassing shoud revea a pausibe risk pattern (as indicated by the Weight of Evidence) and a high performance (as indicated by a high Information Vaue). Fixing a cassing is compex, because there is a trade-off between the requirements. On one hand, the Information Vaue tends to increase with an increasing number of casses, on the other hand, estimation of the coefficients b tends to improve when the number of casses decreases. In order to fix the fina cassing anaysts produce a series of different crosstabs and cacuate the corresponding Weights of Evidence and Information Vaues. Finay, the best cassing is seected according to the criteria above. The fina cassing therefore is the resut of a heuristic process which is strongy determined by the anayst s know-how and experience. 3.5 Summary In this section, we briefy summarise the ideas discussed here. We have started from the observation that for retai portfoios, the methods for deveoping rating modes are different from those appied to other portfoios. This is mainy due to the different type of data typicay encountered when deaing with retai oans: First, there is a predominance of hard quaitative information which aows the integration of a high portion of quaitative data in the mode. Second, there is itte theoretica knowedge about the risk reevance and risk distribution of the input variabes. Therefore, anayzing the data requires specia toos. Finay, there is a high amount of comparaby homogenous data. As a consequence, statistica risk assessment toos were deveoped ong before rating modes for other banks portfoios have boosted and the standards have been setted independenty from Base II. The standard modes for the rating of retai portfoios are scoring modes. Generay, scoring modes compy with the IRBA minimum requirements as ong as they fufi the industria standards. However, usuay they ony constitute risk cassification systems in terms of the IRBA and it wi be necessary to add a component which estimates PD, EAD and LGD. The estimation of a scoring mode requires the cassing of a individua variabes. This is done in a preiminary step caed univariate anaysis. The cassings can be defined by comparing the performance of different aternatives. Since risk distribution of the variabes is often competey unknown, the univariate anaysis

36 D. Porath shoud rey on performance measures which are independent from the ordering of the singe casses, ike for exampe the Weight of Evidence and the Information Vaue. Once the cassing is setted the variabes have to be recoded in order to buid the mode. Finay, the mode can be estimated with standard techniques ike ogit anaysis or discriminant anaysis. References BIS (2004), Internationa Convergence of Capita Measurement and Capita Standards, Base Committee on Banking Supervision, June 2004. Good IJ (1950), Probabiity and the Weighing of Evidences, Chares Griffin, London. Hand DJ (2001), Modeing consumer credit risk, IMA Journa of Management Mathematics 12, pp. 139 155. Kuback S (1959), Information Theory and Statistics, Wiey, New York. Thomas LC, Crook J, Edeman D (2002), Credit Scoring and its appications, Siam Monographs on Mathematica Modeing and Computation.

Chapter 4 The Shadow Rating Approach: Experience from Banking Practice Urich Erenmaier 4.1 Introduction In this artice we wi report on some aspects of the deveopment of shadow rating systems found to be important when re-devising the rating system for arge corporations of KfW Bankengruppe (KfW banking group). The artice focuses on genera methodoogica issues and does not necessariy describe how these issues are deat with by KfW Bankengruppe. Moreover, due to confidentiaity we do not report estimation resuts that have been derived. In this introductory section we want to describe briefy the basic idea of the shadow rating approach (SRA), then summarise the typica steps of SRA rating deveopment and finay set out the scope of this artice. The shadow rating approach is typicay empoyed when defaut data are rare and externa ratings from the three major rating agencies (Standard & Poor s, Moody s or Fitch) are avaiabe for a significant and representative part of the portfoio. As with other approaches to the deveopment of rating systems, the first modeing step is to identify risk factors such as baance sheet ratios or quaitative information about a company that are supposed to be good predictors of future defauts. The SRA s objective is to choose and weight the risk factors in such a way as to mimic externa ratings as cosey as possibe when there is insufficient data to buid an expicit defaut prediction mode (the atter type of mode is e.g. described in Chap. 1. To make the resuting rating function usabe for the bank s interna risk management as we as for reguatory capita cacuation, the externa rating grades (AAA, AA, etc.) have to be caibrated, i.e., a probabiity of defaut (PD) has to be attached to them. With these PDs, the externa grades can then be mapped to the bank s interna rating scae. The opinions expressed in this artice are those of the author and do not refect views of KfW Bankengruppe (or modes appied by the bank). U. Erenmaier KfW Bankengruppe e-mai: Urich.Erenmaier@gmai.com B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_4, # Springer-Verag Berin Heideberg 2011 37

38 U. Erenmaier The foowing moduar architecture is typica for SRA but aso for other types of rating systems: 1. Statistica mode 2. Expert-guided adjustments 3. Corporate group infuences/sovereign support 4. Override The statistica mode constitutes the basis of the rating system and wi most ikey incude baance sheet ratios, macroeconomic variabes (such as country ratings or business cyce indicators) and quaitative information about the company (such as quaity of management or the company s competitive position). The statistica mode wi be estimated from empirica data that bring together companies risk factors on the one hand and their externa ratings on the other hand. The mode is set up to predict externa ratings more precisey, externa PDs as efficienty as possibe from the seected risk factors. The second modeing ayer of the rating system, that we have termed Expertguided adjustments wi typicay incude risk factors for which either no historica information is avaiabe or for which the infuence on externa ratings is difficut to estimate empiricay. 1 Consequenty, these risk factors wi enter the mode in the form of adjustments that are not estimated empiricay but that are determined by credit experts. The third modeing ayer wi take into account the corporate group to which the company beongs or probaby some kind of government support. 2 This is typicay done by rating both the obigor on a standaone basis and the entity that is supposed to infuence the obigor s rating. Both ratings are then aggregated into the obigor s overa rating where the aggregation mechanism wi depend on the degree of infuence that the corporate group/sovereign support are assessed to have. Finay, the rating anayst wi have the abiity to override the resuts as derived by steps 1 3 if she thinks that due to very specific circumstances the rating system does not produce appropriate resuts for a particuar obigor. This artice wi focus on the deveopment of the rating system s first modue, the statistica mode. 3 The major steps in the deveopment of the statistica mode are: 1 This occurs e.g. when a new risk factor has been introduced or when a risk factor is reevant ony for a sma sub-sampe of obigors. 2 There aso might be other types of corporate reationships that can induce the support of one company for another one. For exampe, a company might try to bai out an important suppier which is in financia distress. However, since this issue is ony a minor aspect of this artice we wi concentrate on the most common supporter-reationship in rating practice, i.e. corporate groups and sovereign support. 3 We wi, however, aso incude a short proposa for the empirica estimation of corporate group infuences/sovereign support (step 3).

4 The Shadow Rating Approach: Experience from Banking Practice 39 1. Depoyment of software toos for a stages of the rating deveopment process 2. Preparation and vaidation of the data needed for rating deveopment (typicay externa as we as interna data sets) 4 3. Caibration of externa ratings 4. Sampe construction for the interna rating mode 5. Singe (univariate) factor anaysis 6. Muti factor anaysis and vaidation 7. Impact anaysis 8. Documentation This artice deas with steps 3 6, each of which wi be presented in one separate section. Nevertheess, we want to provide comments on the other steps and emphasise their reative importance both in quaitative as in quantitative terms for the success of a rating deveopment project: Initia project costs (i.e. interna resources and time spent for the initia deveopment project) wi be very high and mainy driven by steps 1 3 (but aso 8) with step 1 being the singe biggest contributor. In contrast, foow-up costs (future refinement projects reated to the same rating system) can be expected to be much ower and more equay distributed across a steps with step 2 most ikey being the singe biggest contributor. The importance of step 2 for the statistica anayses that buid on it must be stressed. Moreover, this step wi be even more important when externa data sets are empoyed. In this case, it wi aso be necessary to estabish compatibiity with the interna data set. Step 7: Once a new rating system has been deveoped and vaidated, it wi be important to assess the impact of a change to the new system on key interna and reguatory portfoio risk measures, incuding for exampe, expected oss or reguatory and economic capita. Regarding step 8 we found it very hepfu and time saving to transfer a number of the resuts from statistica anayses to appendices that are automaticay generated by software toos. Finay, we want to concude the introduction with some comments on step 1, the depoyment of software toos. The objective shoud be to automate the compex rating deveopment process as competey as possibe through a the necessary steps, in order to reduce the manpower and a-priori know how required to conduct 4 In this artice, the term externa data sets or externa data wi aways refer to a situation where additiona to internay rated companies a typicay much arger sampe of not internayrated companies is empoyed for rating deveopment. This externa data set wi often come from an externa data provider such as e.g. Bureau van Dijk but can aso be the master sampe of a datapooing initiative. In such a situation, usuay ony quantitative risk factors wi be avaiabe for both, the interna and the externa data set whie quaitative risk factors tend to be confined to the interna data set. In this situation, a number of specific probems arise that have to be taken into account. The probems we found most reevant wi be deat with in this artice.

40 U. Erenmaier a deveopment project. Therefore, different, inter-connected toos are needed, incuding: Datamarts: Standardised reports from the bank s operating systems or data warehouse covering a information reevant for rating deveopment/vaidation on a historica basis Data set management: to make externa data compatibe with interna data, for sampe construction, etc. Statistica anaysis toos: taior made for rating deveopment and vaidation purposes. These toos produce documents that can be used for the rating system s documentation (step 8). These documents comprise a major anayses as we as a reevant parameters for the new rating agorithm. Generic rating agorithm too: Aows the appication of new rating agorithms to the reevant sampes. It shoud be possibe to customise the too with the resuts from the statistica anayses and to buid competey new types of rating agorithms. 4.2 Caibration of Externa Ratings 4.2.1 Introduction The first step in buiding an SRA mode is to caibrate the externa agencies rating grades, i.e. to attach a PD to them. The foowing ist summarises the issues we found important in this context: Externa rating types: which types of ratings shoud be empoyed? Probabiity of defaut (PD)/Expected oss (EL) ratings, Long-/Short-term ratings, Foreign/Loca currency ratings Externa rating agencies: pros and cons of the different agencies ratings with respect to the shadow rating approach Defaut definition/defaut rates: differences between externa and interna definitions of the defaut event and of defaut rates wi be discussed Sampes for externa PD estimation: which time period shoud be incuded, are there certain obigor types that shoud be excuded? PD estimation technique: discussion of the pros and cons of the two major approaches, the cohort and the duration-based approach Adjustments of PD estimates: if PD estimates do not have the desired properties (e.g. monotonicity in rating grades), some adjustments are required Point-in-time adjustment: externa rating agencies tend to foow a through-thecyce-rating phiosophy. If a bank s interna rating phiosophy is point-in-time then either The externa through-the-cyce ratings must be adjusted to make them sensitive to changes in macroeconomic conditions or,

4 The Shadow Rating Approach: Experience from Banking Practice 41 The effects of deveoping on a through-the-cyce benchmark must be taken into account The above mentioned issues wi be addressed in the foowing sections. 4.2.2 Externa Rating Agencies and Rating Types For SRA ratings systems, typicay the ratings of the three major ratings agencies Standard & Poors (S&P), Moody s and Fitch are empoyed. Two questions arise: 1. For each rating agency, which type of rating most cosey matches the bank s interna rating definition? 2. Which rating agencies are particuary we suited for the purpose of SRA deveopment? Regarding question 1 issuer credit ratings for S&P and Fitch and issuer ratings for Moody s were found to be most suitabe since these ratings assess the obigor and not an obigor s individua security. Moreover, it wi usuay make sense to choose the ong-term, oca currency versions for a rating agencies and rating types. 5 Regarding question 2 the major pro and cons were found to be the foowing: Length of rating history and track record: S&P and Moody s dominate Fitch. See e.g. Standard and Poor s (2005), Moody s (2005), and Fitch (2005). Rating scope: whie both S&P and Fitch rate an obigor with respect to its probabiity of defaut (PD), which is consistent with banks interna ratings as required by Base II, Moody s assesses its expected oss (EL).This concusion draws on the rating agencies rating definitions (cf. Standard and Poor s (2002), Moody s (2004), and Fitch 2006), discussions with rating agency representatives and the academic iterature (cf. G utter 2004). Are differences between oca and foreign currency ratings (LC and FC) aways identifiabe? Whie S&P attaches a oca and foreign currency rating to amost every issuer rating, this is not aways the case for Moody s and Fitch. Based on an assessment of these pros and cons it has to be decided whether one agency wi be preferred when more than one externa rating is avaiabe for one obigor. The foowing sections wi dea with PD estimations for externa rating grades. In this context we wi for the sake of simpicity focus on the agencies S&P and Moody s. 5 Long-term ratings because of the Base II requirements that banks are expected to use a time horizon onger than one year in assigning ratings (BCBS (2004), } 414) and because amost a anayses of externa ratings are conducted with ong-term ratings. Loca currency ratings are needed when a bank measures transfer risk separatey from an obigor s credit rating.

42 U. Erenmaier 4.2.3 Definitions of the Defaut Event and Defaut Rates For the PD estimates from externa rating data to be consistent with interna PD estimates, (a) the definition of the defaut event and (b) the resuting definition of defaut rates (defaut counts in reation to obigor counts) must be simiar. Whie there might be some minor differences regarding the cacuation of defaut rates, 6 the most important differences in our opinion stem from different definitions of the defaut event. Here are the most important deviations 7 : Different types of defauts (bank defauts vs. bond market defauts): a company that has probems meeting its obigations might e.g., first try to negotiate with its bank before exposing it to a potentia defaut in the bond market. Differences in quaitative defaut criteria: according to Base II, a company is to be cassified as defaut when a bank considers that the obigor is unikey to pay its credit obigations in fu. This coud easiy appy to companies that are in the owest externa non-defaut rating grades. 8 Number of days of deayed payment that wi ead to defaut Base II: 90 days S&P: defaut when payments are not made within grace period which typicay ranges from 10 to 30 days Moody s: 1 day Materiaity: Whie externa agencies wi measure defauts without respect to the size of the amount due, under Base II, payment deays that are sma with respect to the company s overa exposure wi not be counted as defauts. In order to assess the effects of these and other differences in defaut definition on estimated PDs, the defaut measurement of S&P and Moody s has to be compared with the bank s interna defaut measurement. In a first step S&P and Moody s coud be compared with each other (a) If the differences between the two externa agencies are not significant, interna defauts can be compared with the pooed externa defauts of S&P and Moody s (b) The foowing technique might be usefu for steps (a) and (b): 6 Exampes: (a) Whie the externa agencies count the number of obigors ony at the beginning of the year and then the resuting defauts from these obigors over the year, a bank might count on a finer basis (e.g., monthy) in order to track as many obigors as possibe; (b) defauts that occur because of foreign currency contros and not because the individua obigor is not abe to meet its obigations shoud not be counted as defaut for the purpose of PD-estimation if a bank quantifies transfer risk separatey. 7 The Base II defaut definition is given in (BCBS (2004), } 452). The rating agencies defaut definitions are described in their respective defaut reports (cf. Standard and Poor s (2005), Moody s (2005), and Fitch 2005). 8 This assessment draws on externa agencies verba definitions of those rating grades (cf. Standard and Poor s (2002), Moody s (2004), and Fitch 2006).

4 The Shadow Rating Approach: Experience from Banking Practice 43 1. Estimation of the ratio of Moody s defauts for each S&P defaut and the ratio of externa defauts for each interna defaut respectivey. 2. This ratio can be interpreted as an adjustment factor with which (a) PDs derived for Moody s have to be scaed in order to arrive at PDs compatibe with S&P and (b) with which externa PDs have to be adjusted in order to be comparabe with internay derived PDs. 3. Cacuation of confidence intervas for the resuting estimators using a mutinomia mode and a Chi-square-type test statistic 9 Depending on the estimation resuts it has to be decided whether an adjustment factor shoud be appied. If estimators prove to be very voatie, additiona defaut data (e.g. form data pooing initiatives) might be needed to arrive at more confident estimates. 4.2.4 Sampe for PD Estimation For the estimation of externa PDs the obigor sampes of S&P and Moody s as used by these agencies to derive defaut rates in their annua defaut reports can be empoyed. 10 The foowing two dimensions of sampe construction shoud in our opinion be cosey anaysed: 1. Obigor sector and country: shoud a obigor types be incuded irrespective of industry sector and country? 2. Length of time series With respect to (4.1) one can start with the hypotheses that as ratings agencies caim externa ratings are comparabe across industry sectors and countries. 11 Consequenty, for those rating types (S&P and Fitch) that aim to measure an obigor s PD, PD estimates woud ony have to be conditiona on an obigor s rating grade, not its industry sector or country. Where ratings measure an obigor s EL for senior unsecured obigations (Moody s), however, PD estimates woud aso have to be conditiona on a obigor characteristics that affect the LGD on these obigations, as coud for exampe be the case for a company s industry sector or home country. But if LGD differences across obigors are sma compared to PD 9 For exampe, for the comparison of externa and interna defauts, the mutinomia random variabe woud for each defauted company indicate one of three potentia outcomes: (1) Externa and interna defaut, (2) Externa defaut but no interna defaut, (3) Interna defaut but no externa defaut. Moreover, due to the typicay sma amount of data, no arge-sampe approximation but the exact Chi-square distribution shoud be empoyed. Confidence imits can be estimated by appying the test statistic on a sufficienty fine grid for the parameters of the mutinomia distribution. 10 See Standard and Poor s (2005) and Moody s (2005). 11 See agencies rating definitions: Standard and Poor s (2002) and Moody s (2004) respectivey.

44 U. Erenmaier differences between rating grades, estimates based ony on the rating grade might be toerabe for pragmatic reasons. To address the first issues (comparabiity of ratings across countries/sectors), the iterature on differences between externa defaut rates across industry sectors and countries shoud be reviewed. We found ony three papers on the defaut rate issue. 12 None identified country specific differences whie they were inconcusive with respect to sector specific differences. 13 Regarding the second issue (reative size of the LGD effect), the bank s interna LGD methodoogy shoud be anaysed with respect to differences between senior unsecured LGDs across industries and countries. 14 Based on the assessment of both issues it shoud be decided as to whether country or industry sector specific estimates are needed. We now turn to the second dimension of sampe construction, i.e. the ength of the time series. On the one hand, a ong time series wi reduce statistica uncertainty and incude different states of the business cyce. On the other hand, there is the probem that because of structura changes, data coected earier, might not refect current and future business conditions. A sensibe starting point wi be the time horizon that is most often used by both the rating agencies and the academic iterature (starting with the years 1981 and 1983 respectivey). One can then anayse changes in rating grade defaut rates over time and assess whether structura changes in the defaut rate behaviour can be identified or whether most of the variabiity can be expained by business cyce fuctuations. 4.2.5 PD Estimation Techniques Once the sampe for PD estimation has been derived, the estimation technique must be specified. Typicay, the so caed cohort method (CM) is appied where the number of obigors at the beginning of each year in each rating grade and the number of obigors that have defauted in this year are counted respectivey. Both figures are then summed over a years within the time horizon. The resuting PD estimate is arrived at by dividing the overa number of defauts by the overa number of obigors. 15 The duration-based (DB) approach aims to improve on the cohort-method by incuding information on rating migration in the estimation process. The underying 12 See Ammer and Packer (2000), Cantor and Fakenstein (2001), and Cantor (2004). 13 Ammer and Packer (2000) found defaut-rate differences between banks and non-banks. However, they pointed out that these differences are most ikey attributabe to a specific historic event, the US Savings and Loans crisis, and shoud therefore not be extrapoated to future defaut rates. Cantor and Fakenstein (2001), in contrast, found no differences in the defaut rates of banks and non-banks once one contros for macroeconomic effects. 14 For a discussion of LGD-estimation methods we refer to Chapter VIII of this book. 15 This method can be improved on by counting on a monthy or even finer base.

4 The Shadow Rating Approach: Experience from Banking Practice 45 idea is to interpret defaut events as the resut of a migration process. In the simpest setting where the migration process can be assumed to foow a stationary Markov process, a T-year migration matrix M T can be derived by appying the one year migration matrix M T T times: M T ¼ M 1 T (4.1) The continuous time anaogue of (4.1) is M t ¼ Expðm tþ; (4.2) where m is the margina migration matrix, t the time index and Exp(.) the matrix exponentia. 16 Hence, M 1 (incuding in particuar 1-year defaut probabiities) can be derived by first estimating m from transition counts and then appying the matrix exponentia to the estimated margina transition matrix. A detaied description of the duration-based approach (DB) and the cohort method (CM) can be found in Schuermann and Hanson (2004). They aso state the major differences between CM and DB estimates, in particuar, that the atter produce PDs that spread more widey across the rating scae, i.e. PDs for good rating grades wi be much ower and PDs for bad ratings wi be much higher under DB than under CM. Both estimation techniques have their pros and cons: DB makes more use of the avaiabe information by aso taking into account rating migrations. For this reason, the DB method can aso produce positive PD estimates for the best rating grades where no defaut observations are avaiabe. CM is more transparent and does not rey on as many modeing assumptions as the DB method. As ong as there is no cear-cut empirica evidence on the reative performance of both methods, it seems therefore sensibe to appy both techniques and compare the resuting estimates. However, it is ikey that in the future such comparisons wi become avaiabe and therefore it wi be hepfu to keep an eye on the corresponding reguatory and academic discussion. 4.2.6 Adjustments Because the PD estimates resuting from the appication of the estimation methods as described in the previous section wi not aways be monotonic (i.e. not aways wi PD estimates for better rating grades be ower than for worse rating grades), the estimates have to be adapted in these non-monotonic areas. One option is to regress 16 The matrix exponentia appies the exponentia series to matrices: exp (m) ¼ I+m 1 / 1! +m 2 /2! +...,wherei is the identity matrix.

46 U. Erenmaier the ogarithm of the PD estimates on the rating grades and to check whether the interpoations that resut for the non-monotonic areas are within confidence imits. Here are some comments on the underying techniques: Regression In order to perform the regression, a metric interpretation has to be given to the ordina rating grades. Pots of PD estimates against rating grades on a ogarithmic scae suggest that this approach is sensibe from a pragmatic point of view (cf. Atman and Rijken 2004). It may make sense to weight the regression by the number of observations avaiabe for each rating grade since the precision of PD estimates is dependent on it. Confidence intervas (CI) For the cohort approach, confidence intervas can be derived from the binomia distribution by assuming independent observations. 17 It is usuay assumed that defaut observations are correated because of macroeconomic defaut drivers that affect the defaut behaviour of different obigors. Hence, binomia confidence intervas wi be a conservative estimate (they are tighter then they woud be under correated defauts). CIs derived from a Merton stye simuation mode (cf. Chap. 15 of this book) coud be the ogica next step. In the context of the duration-based method, CIs are typicay derived via Bootstrap methods (cf. Schuermann and Hanson 2004). These tend to be even tighter. The topic of correated defauts/migrations has to our knowedge not yet been addressed in this context. 4.2.7 Point-in-Time Adaptation In the context of Base II, a bank s rating system is supposed to measure an obigor s probabiity of defaut (PD) over a specific time horizon (the next T years). In practice, the objective of rating systems differs, particuary with respect to: 1. The time horizon chosen by a bank 2. Whether PDs are conditiona on the state of the business cyce (through-thecyce phiosophy, TTC) or not (point-in-time phiosophy, PIT) Whie the first point can be taken into account by correspondingy adjusting the time horizon for defaut rate estimation, a bank that foows a PIT approach wi have 17 For an efficient derivation and impementation of exact confidence imits for the binomia distribution see Day (1992).

4 The Shadow Rating Approach: Experience from Banking Practice 47 Tabe 4.1 Comparison of point-in-time and through-the-cyce rating systems Issue Point-in-time (PIT) Through-the-cyce (TTC) What does the rating system measure? Stabiity of an obigor s rating grade over the cyce Stabiity of a rating grade s unconditiona PD Unconditiona PD Pro-cycica: Rating improves during expansions and deteriorates in recessions Stabe: Unconditiona PDs of ratings grades do not change. Obigor s higher unconditiona PDs during recession are accounted for by migrations to ower rating grades and vice versa PD conditiona on the state of the business cyce. The PD estimate might be either conditiona on a worst case ( bottom of the cyce scenario ) a or on an average business cyce scenario Stabe: Rating grades tend to be unaffected by changes in the business cyce Pro-cycica: PDs improve during expansions and deteriorate during recessions a This has for exampe been suggested by a survey of bank rating practices by the Base Committee s Mode Task Force (cf. BCBS 2000). to appy PIT-adjustments to the PD estimates derived for externa rating grades since externa rating agencies tend to foow a TTC-approach. 18 In the remainder of this section we wi (a) anayse the effects resuting from the deveopment of ratings systems on TTC-PDs and (b) outine a technique for PIT adjustments of externa rating grades. To address both points, we first summarise the most important properties of PIT and TTC rating systems in Tabe 4.1. These properties foow straightforwardy from the above definitions. A detaied discussion can be found in Heitfied (2004). Turning to the first point of investigation, we now ist the most important consequences when deveoping a rating system on a TTC-PD benchmark: Pure macroeconomic risk factors that focus on business cyce information wi expain ony the (typicay quite sma) PIT-part of externa ratings and wi therefore tend to receive very ow weights in statistica modes. This effect shoud be ess pronounced for mixed factors that contain both business cyce information and non-business cyce eements, for exampe baance sheet ratios or country ratings. A bank that foows a PIT rating approach but has not yet finaised a fuy-fedged PIT-adaptation of externa ratings might therefore manuay adjust regression resuts 18 The TTC-property of externa ratings has been observed in the academic iterature (cf. L offer 2004) and has aso been proved to be significant by our own empirica investigations. It must, however, be stressed that in practice rating systems wi neither be competey TTC or PIT but somewhere in between.

48 U. Erenmaier in order to attach higher weights to pure business-cyce risk factors. For banks that aready want to impement a statisticay founded PIT-adaptation of externa ratings, the foowing approach coud be considered: Estimation of a cassic defaut prediction mode, for exampe via ogistic regression (see Chap. 1), with externa PDs and business cyce factors (on a regiona, country or industry eve) as risk factors The dependent variabe is the company s defaut indicator as measured by the externa ratings agencies defaut definition (or, where avaiabe, the bank s own defaut definition). Accordingy, data from externa rating agencies wi be needed on a singe obigor eve whie for TTC-PD estimation, aggregate obigor and defaut counts are sufficient. When estimating such a mode, the foowing chaenges are pertinent: Different countries have different macroeconomic indicators that might not be comparabe. Because estimating separate modes for separate countries wi not be feasibe due to data restrictions, it wi be important to use indicators that are approximatey comparabe across countries. To get a picture of the reated effects, it might be sensibe to start by buiding a mode for the US (where data avaiabiity is high) and see how parameter estimates change when other countries are added. Probaby separate regiona modes can hep. An aternative approach woud be to use externa point-in-time rating systems for the PIT-adaptation of through-the-cyce agency ratings. An exampe of a pointin-time externa rating is Moody s KMV s EDF credit risk measure that buids on a Merton stye causa defaut prediction mode. 19 Anaysis is then required as to whether it woud not be better to skip the through-the-cyce agency ratings atogether and repace them with the externa point-in-time ratings. In deciding on which approach to take, a bank must trade off the associated costs with the avaiabiity of the respective benchmarks. 20 4.3 Sampe Construction for the SRA Mode 4.3.1 Introduction Once externa PDs have been caibrated, and a interna and externa data required for the deveopment of the SRA mode have been compied, it is necessary to 19 See http://www.moodyskmv.com/. 20 For exampe, market-based measures such as Moody s KMV s EDF are ony avaiabe for pubic companies.

4 The Shadow Rating Approach: Experience from Banking Practice 49 construct sampes from this data. As we wi see, different sampes wi be needed for different types of statistica anaysis. In this section we mention these anaysis techniques in order to map them to the corresponding sampes. The techniques wi be described in Sects. 4.4 and 4.5. In this section, the foowing issues wi be deat with: Which types of sampes are needed? How can these sampes be constructed? Weighted observations: If the information content of different observations differs significanty, it might be necessary to aow for this by attaching different weights to each observation. Correated observations: We discuss the correation structure that may resut from the described sampe construction technique and discuss the consequences. It shoud be noted that some parts of the sampe construction approach described in this section might be too time consuming for an initia deveopment project. Nevertheess, it can serve as a benchmark for simper methods of sampe construction and coud be graduay impemented during future refinements of the initia mode. 4.3.2 Sampe Types The sampes reevant for the deveopment of SRA rating systems can be cassified by the foowing dimensions: Sampes for singe (univariate) factor anaysis (e.g. univariate discriminatory power, transformation of risk factors) vs. muti factor anaysis sampes (e.g. regression anaysis, vaidation) Sampes that incude ony externay rated obigor vs. sampes that incude externay and ony internay rated obigors Externa data vs. interna data 21 Deveopment vs. vaidation sampe We wi start with the first dimension. Univariate anaysis investigates the properties of each singe risk factor separatey. Therefore, for this type of anaysis each change of the one anaysed factor wi generate a new observation in the data set; for the muti factor anaysis, each change of any risk factor wi produce a new observation. This can be taken into account by the foowing approach to sampe construction: 21 Externa data are often empoyed for the deveopment of SRA rating systems in order to increase the number of obigors and the number of points in time avaiabe for each obigor. See Sect. 4.1 for more detais.

50 U. Erenmaier 1. Risk factors are divided in different categories. A factors for which changes are triggered by the same event are summarised into the same risk factor category. 22 2. The sampes for the univariate risk factor anaysis are constructed separatey for each category. A compete series of time intervas is buid that indicates which risk factor combination is vaid for the category in each time interva or whether no observation was avaiabe in the interva. The time intervas are determined by the points in time where the risk factors of the category under consideration change. This is done separatey for each obigor. 3. A singe category sampes from step 2 are merged into a new series of time intervas. Each interva in the series is defined as the argest interva for which the risk factors in each category remain constant. This is done separatey for each obigor. In the foowing tabe we give an exampe comprising two risk factor categories (baance sheet data and quaitative factors) and hence two different sampes for univariate factor anaysis. Tabe 4.2 dispays the observations for one singe obigor. For each of the sampe types described above, two sub-types wi be needed, one that incudes ony externay rated obigors and one that contains a obigors. The first sub-type wi be needed e.g. for discriminatory power anaysis, the second e.g., for risk factor transformation or vaidation. A third dimension is added when externa as we as interna data are empoyed. Typicay, for SRA modes, externa data wi be used to estimate the quantitative mode (comprising baance sheet factors as we as macroeconomic indicators) whie the compete mode, consisting of both, quantitative and quaitative risk factors wi be cacuated on the interna data set because quaitative risk factors are not avaiabe for the externa data set. A fourth dimension comes with the need to distinguish between deveopment and vaidation sampes. Moreover, vaidation shoud not ony rey on the externa PD but shoud aso incude defaut indicator information, i.e. the information whether Tabe 4.2 Styised exampe for different sampes and observations invoved in rating deveopment Sampe Trigger ID Vaid from Vaid unti Baance sheet data Accounts 1 Jan 03 Dec 03 2 Jan 04 Dec 04 Quaitative factors Interna rating 1 May 03 March 04 2 Apri 04 Dec 04 Muti factor (merged) Accounts 1 Jan 03 Apri 03 Interna rating 2 May 03 Dec 03 Accounts 3 Jan 04 March 04 Interna rating 4 Apri 04 Dec 04 22 One category might for exampe incude a baance sheet factors (triggered by the reease of a company s accounts). Another category wi be quaitative factors as assessed by the bank s oan manger. They are triggered by the interna rating event. A third category might be macroeconomic indicators.

4 The Shadow Rating Approach: Experience from Banking Practice 51 a company has or has not defauted within a specific period of time after its rating has been compied. When vaidating with respect to the defaut indicator, the need for the separation of deveopment and vaidation sampes is not so pressing since the benchmarks empoyed for deveopment and vaidation are different. Due to the typica scarcity of interna defaut data (the rationae for the SRA approach), it is sensibe to perform this type of vaidation on the compete interna data set. However, when vaidating with respect to externa PDs, a separation between deveopment and vaidation sampe is desirabe. If the quantitative mode has been deveoped on externa data, the interna data set shoud typicay be an appropriate vaidation sampe. 23 For the vaidation of the compete mode, depending on the number of observations avaiabe reative to the number of risk factors, the foowing options can be considered: Constructing two competey different sampes (preferaby out-of-time 24 ) Deveoping on the compete interna sampe and vaidating on a subset of this sampe, e.g. the most recent observations for each obigor or some randomy drawn sub-sampe Appication of bootstrap methods 25 Summarising the issues raised in this section, Tabe 4.3 gives a simpe exampe of the different sampes invoved in SRA rating deveopment and the types of statistica anaysis performed on these sampes. For simpicity, our exampe comprises ony two input categories of which ony one (baance sheet data) is avaiabe for the externa and the interna data set and the other (quaitative factors) is ony avaiabe for the interna data set. 4.3.3 Externa PDs and Defaut Indicator For those sampes consisting ony of externay rated obigors (EX) and for those sampes that are empoyed for vaidation on the defaut indicator (VAL-D), an externa PD or the defaut indicator have to be attached to each ine of input variabes respectivey. At east two different approaches to achieve this can be considered: 23 Note that the externa sampe wi typicay aso incude some or amost a interna obigors. To construct two competey different sets, interna obigors woud have to be excuded from the externa data. However, if the externa data set is much arger than the interna data set, such excusion might not be judged necessary. 24 Out-of-time means that deveopment and vaidation are based on disjoint time intervas. 25 For an appication of bootstrap methods in the context of rating vaidation see Appasamy et a. (2004). A good introduction to and overview over bootstrap methods can be found in Davison and Hinkey (1997).

52 U. Erenmaier Tabe 4.3 Styised exampe for the different sampes and corresponding types of anaysis that are needed for the deveopment of SRA type rating systems ID a Input categories Sampe type b Type of anaysis E1 Baance sheet data SC EX DEV Representativeness, Fiers for missing vaues, Univariate discriminatory power, Estimation of the quantitative muti factor mode c I1a Baance sheet data SC ALL DEV Representativeness, Truncation and standardisation of risk factors, Fiers for missing vaues I1b EX VAL-E/DEV Univariate discriminatory power, Vaidation of the quantitative muti factor mode deveoped on sampe E1 I2a Quaitative factors SC ALL DEV Standardisation of risk factors, Fiers for missing vaues I2b EX Score cacuation, Univariate discriminatory power I3a Baance sheet data and quaitative factors M EX Risk factor correations/muticoinearity, Estimation of the compete muti factor mode (quantitative and quaitative) I3b ALL DEV/VAL-D Risk factor correations/muticoinearity, defaut indicator vaidation of the compete muti factor mode deveoped on sampe I3a I4 EX VAL-E Separate vaidation sampe, for exampe most recent observations for a obigors from sampe I3a or a randomy drawn sub-sampe a E denotes externa and I denotes interna data. b We write SC for singe-category sampes and M for merged sampes. ALL and EX are standing for a obigors and ony externay rated obigors respectivey. DEV denotes deveopment sampe, VAL-E and VAL-D denote vaidation sampes where vaidation is performed on externa PDs and on the defaut indicator respectivey. c Note that in this case it is not necessary to merge different singe-factor sampes in order to perform the muti-factor anaysis, because ony one input-category exists. Moreover, a separate vaidation sampe for the externa data is not necessary since vaidation is performed on the interna data set. 1. Externa PDs/the defaut indicator are treated as yet another risk factor category, i.e. a series of time intervas is constructed for each externa rating agency/for the defaut indicator indicating the time spans for which a specific externa rating/defaut indicator reaisation had been vaid. These intervas are then merged with the reevant singe factor or merged factor sampes in the same way as singe factor sampes are merged with each other. 26 If there are competing 26 Note that the time intervas of input factors and defaut indicator are shifted against each other by the ength of the time horizon for which the rating system is deveoped. For exampe, if the horizon is 1 year and the defaut indicator is equa to zero from Jan 2003 to Dec 2004 then this vaue wi be mapped to the risk-factor interva from Jan 2002 to Dec 2003.

4 The Shadow Rating Approach: Experience from Banking Practice 53 PDs from different externa agencies at the same time, an aggregation rue wi be appied. We wi discuss this rue in the second part of this section. 2. For each risk factor time interva, a weighted average is determined for each externa agency PD and for the defaut indicator respectivey. The weights are chosen proportionay to the ength of the time interva for which the externa rating/the defaut indicator has been vaid. As under 1), an aggregation rue is appied to transate the PDs of different externa agencies into one singe externa PD. For the defaut indicator the first approach seems to be more adequate, since with the second approach the 0/1 indicator variabe woud be transformed into a continuous variabe on the interva [0,1] and many important anaytica toos (e.g. the ROC curve) woud not be directy appicabe. This argument, obviousy does not appy to the externa PDs since they are aready measured on the interva [0,1]. Moreover, externa PDs tend to change more frequenty than the defaut indicator and hence the number of observations woud increase markedy compared to the corresponding risk factor sampes. Additionay, the PDs of not ony one but three different rating agencies woud have to be merged, further increasing the number of observations. Since the information content of different observations beonging to the same risk factor combination wi tend to differ ony sighty, such a procedure wi produce many highy correated observations which is not desirabe (see Sect. 4.3.5). Consequenty the second approach appears to be more adequate for externa PDs. As mentioned above, an aggregation rue has to be devised for cases where more than one externa rating is vaid at some point in time. The most straightforward choice wi be weighted averages of the different externa PDs with a preferentia treatment of those rating agencies that are assessed to be most suitabe for SRA deveopment (see Sect. 4.2.2). 4.3.4 Weighting Observations The information content of a singe observation in the different sampes depends on the ength of the time interva it is associated with. If, for exampe, a particuar baance sheet B is vaid from Jan 04 to Dec 04 and we observe two corresponding sets of quaitative factors, Q1 (vaid unti Feb 04) foowed by Q2 (vaid from Feb 04 unti Dec 04) we woud obviousy ike to put a much higher weight on the observation (B, Q2) than on (B, Q1). The most straightforward way is to choose weights that are proportiona to the ength of the time interva associated with a specific observation. In this context, the foowing issues are of particuar interest: Stochastic interpretation of weighted observations: The weight attached is a measure for the size of the error term associated with each observation, i.e. its standard deviation: the ower the weight, the higher the standard deviation.

54 U. Erenmaier Practica impementation: Most statistics software packages incude options to perform statistica computations with weighted observations. This usuay appies for a techniques mentioned in this artice. 4.3.5 Correated Observations Correated observations (or, more precisey, correated error terms) are a genera probem in singe and muti factor anaysis. Basic techniques assume independence. Using these techniques with correated observations wi affect the vaidity of statistica tests and confidence intervas, probaby aso reducing the efficiency of estimators. To resove this probem, information about the structure of the correations is necessary. In this artice, the correation issue wi be deat with in two steps: 1. In this section we wi address the specific correations structure that may arise from the method of sampe construction described above 2. In Sect. 4.5.3 we wi anayse the statistica techniques that can be used to address this or other correation structures in the context of muti factor anaysis. When constructing sampes according to the method described above, the degree of correation in the data wi rise when the time intervas associated with each observation become smaer. It wi aso depend on the frequency and intensity of changes in the risk factor and the externa rating information empoyed. It is worth noting that the resuting type of correation structure can be best described within a pane data setting where the correations within the time series observations for each singe obigor wi be different to the cross-sectiona correation between two obigors. Cross-sectiona correations in SRA deveopment may resut from country or industry sector dependencies. Time series correations wi typicay be due to the fact that there are structura simiarities in the reationship between a singe company s risk factors and its externa rating over time. Since modes for crosssectiona correations are widey appied in credit portfoio modes, 27 we wi focus on time series correations in this artice. In what foows we propose some options for deaing with correations in the time series parts. The options are isted in order of rising compexity: For simpicity, basic statistica techniques are empoyed that do not account for correated error terms. With this option, as much correation as possibe can be eiminated by dropping observations with sma weights. If a observations have approximatey the same weight, a sub-sampe can be drawn. Here, the appropriate baance has to be found between osing too much information in the sampe and retaining a degree of correation that sti appears to be compatibe with not modeing these correations expicity. In any case, the remaining correation in 27 See Erenmaier (2001).

4 The Shadow Rating Approach: Experience from Banking Practice 55 the data shoud be measured and the modeer shoud be aware of the resuting consequences, in particuar with respect to confidence intervas (they wi tend to be too narrow) and with respect to statistica tests (they wi tend to be too conservative, rejecting the nu too often). Simpe modes of autocorreation in the time series data are empoyed, the most obvious being a first order autoregressive process (AR1) for the time series error terms. Of course, higher order AR processes or more compex correation modes might aso be considered appropriate. 28 A continuous time mode for the reation between risk factors and externa ratings is buit (e.g. Brownian motion or Poison process type modes) and the resuting correation structure of the discrete observations error terms is derived from this mode. This of course is the most compex option and wi most probaby be seen as too time consuming to be appied by most practitioners. It might, however, be a road for academic researchers that in turn coud make the method avaiabe for practitioners in the future. 4.4 Univariate Risk Factor Anaysis 4.4.1 Introduction Before buiding a muti factor mode, each risk factor has to be anaysed separatey in order to determine whether and in which form it shoud enter the muti factor mode. This type of anaysis is referred to as univariate risk factor anaysis. The foowing issues shoud be deat with in this context: Measurement of a risk factor s univariate discriminatory power Transformation of risk factors to (a) improve their inear correation as assumed by the muti factor regression mode with the og externa PD 29 or (b) to make different risk factors comparabe with each other Checking whether the sampes on which the rating system is deveoped are representative for the sampes to which the rating system wi be appied (deveopment vs. target sampe) Treatment of missing vaues Each of these issues wi be deat with separatey in the foowing sections. 28 For an introduction to such modes and further references see Greene (2003). 29 See Sect. 4.5.2. Throughout this artice we wi use the term og externa PD to denote the natura ogarithm of the PD of an obigor s externa rating grade. How PDs are derived for each externa rating grade has been described in Sect. 4.2.

56 U. Erenmaier 4.4.2 Discriminatory Power A rating system is defined as having a high discriminatory power if good rating grades have a comparativey ow share of obigors that wi defaut ater on and vice versa. Accordingy, its discriminatory power wi deteriorate with an increase in the reative share of ater on defauted obigors in good rating grades. There are severa statistica measures for this important attribute of a rating system, the Gini coefficient being the most popuar. 30 Due to the ack of a sufficient number of defaut observations in SRA modes, these types of discriminatory power measurement wi usuay ony be appied as an additiona vaidation measure. In the deveopment stage, discriminatory power wi be defined in terms of the usefuness of the rating system or in the context of univariate factor anaysis a singe risk factor in predicting an obigor s externa PD: The better a rating system or a risk factor can be used to predict an obigor s externa PD, the higher its discriminator power for the SRA approach. 31 The foowing techniques can be hepfu to measure a risk factor s discriminatory power for the SRA approach: Linear and rank-order correations of the risk factors with the og externa PD 32 Bucket pots Whie the correation measures are straightforward, the bucket pots require further comment. The underying rationae for appying bucket pots is to visuaise the compete functiona form of the reationship between the risk factor and the externa PD in contrast to the correation measures that aggregate this information into a singe number. This is done to make sure that the risk factors indeed dispay an approximatey inear reationship with externa PDs as is required by the muti factor mode. Bucket pots for continuous risk factors can for exampe be constructed in the foowing way: Each risk factor range was divided into n separate buckets, where we chose the 0, 1/n, 2/n,...,(n-1)/n, 1 quanties of each risk factor s distribution as interva boarders. 30 For an overview on measures of discriminatory power see Deutsche Bundesbank (2003) or Chap. 13. 31 A good discriminatory power of the interna rating system in terms of predicting externa ratings and a good discriminatory power of the externa ratings in terms of predicting future defauts wi then estabish a good discriminatory power of the interna rating system in terms of predicting future defauts. 32 Linear correations are typicay termed Pearson correations whie rank-order correations are associated with Spearman. Linear correations are important since they measure the degree of inear reationship which corresponds with the inear mode empoyed for the muti-factor anaysis. Rank-order correations can be compared with inear correations in order to identify potentia scope for risk factor transformation.

4 The Shadow Rating Approach: Experience from Banking Practice 57 For each bucket we cacuated the average associated externa PD. By constructing the bucket boarders using quanties it can be made sure that each interva contains the same number of observations. The number n of intervas has to be chosen with regard to the overa number of PD observations avaiabe for each risk factor: with increasing n it wi be possibe to observe the functiona form of the reationship on an ever finer scae. However, the precision of the associated PD estimates for each bucket wi decrease and their voatiity wi increase. In order to quantify the degree of uncertainty, confidence intervas for the PD estimates of each bucket can be cacuated. The resuting PD estimates and confidence intervas are then potted against the mean risk factor vaue of each bucket. If a ogarithmic scae is used for the PD axis, an approximatey inear reationship shoud resut when the risk factor has been appropriatey transformed. Figure 4.1 shows an exampe of a bucket pot for a continuous risk factor. Bucket pots for discrete risk factors can be devised according to the same method as described above with ony one difference: for discrete factors, each reaisation shoud represent one bucket irrespective of the number of observations avaiabe. PD 12.10 % 7.34 % 4.45 % 2.70 % 1.64 % 0.99 % 0.60 % 0.37 % PLOT 0.22 % 80 70 60 50 40 30 20 10 0 10 20 30 40 Risk factor Average externa PD Lower confidence imit Upper confidence imit Fig. 4.1 Exampe of a bucket pot. It iustrates the functiona reationship between a risk factor and corresponding externa PDs where the atter are measured on a ogarithmic scae. The reationship on this scae shoud be approximatey inear

58 U. Erenmaier 4.4.3 Transformation The foowing types of transformation typica for the deveopment of rating modes wi be considered in this section: Truncation Other non-inear transformations of continuous risk factors (e.g. taking a risk factor s ogarithm) Attaching a score to discrete risk factors Standardisation, i.e. a inear transformation in order to achieve the same mean and standard deviation for each risk factor We wi discuss each of these types of transformations in turn. Truncation means that continuous risk factors wi be cut off at some point on the eft and right, more precisey, x trunc ¼ minfx u ; maxfx ; xgg where x u is the upper and x the ower border at which the risk factor x is truncated. Note that the truncation function described above can be smoothed by appying a ogit-type transformation instead. Truncation is done mainy for the foowing reasons: To reduce the impact of outiers and to concentrate the anaysis on a risk factor s typica range 33 To reduce a risk factor to the range on which it has discriminatory power Other types of non-inear transformations are typicay appied to continuous risk factors to achieve an approximatey inear reationship with the og externa PD. An overview of methods to achieve inearity can be found in Chap. 2. These methods wi therefore not be discussed here. In contrast to continuous risk factors, discrete factors (such as quaitative information about the obigor, e.g. its quaity of management or competitive position) do not have an a priori metric interpretation. Therefore, a score has to be attached to each of the discrete risk factor s potentia reaisations (e.g., exceent, good, medium or poor quaity management). As with the non-inear transformation for the continuous risk factors, the scores have to be chosen in such a way as to achieve the inear reationship of risk factors with og PDs. This can typicay be achieved by cacuating the mean externa PD for each risk factor reaisation and then appying the ogarithm to arrive at the fina score. However, the resuting scores wi not aways be monotonic in the underying risk factor (i.e., the average PD may not aways decrease when the assessment with respect to this risk factor improves). In such cases it has to be decided whether the effect is within statistica confidence eves or indeed indicates a probem with the 33 This is often necessary for sensibe visuaization of the risk factor s distribution.

4 The Shadow Rating Approach: Experience from Banking Practice 59 underying risk factor. If the first hods true (typicay for risk factor reaisations where ony very few observations are avaiabe), interpoation techniques can be appied augmented by expert judgements. In the second case, depending on the severity of the effects identified, it may be necessary (a) to anayse the reasons for this effect, or (b) to merge different reaisations of the risk factor to a singe score, or (c) to eiminate the risk factor from subsequent anaysis. A transformations that have been described up to now have been performed in order to improve the risk factor s inear correation with og externa PDs. The remaining transformation (standardisation) has a inear functiona form and wi therefore not ater inear correations. It is performed in order to unify the different risk factor s scaes and, accordingy, improve their comparabiity, primariy in the foowing two respects: How good or bad is a risk factor reaisation compared with the portfoio average? Interpretabiity of the coefficients resuting from the inear regression as weights for the infuence of one particuar risk factor on the rating resut Typicay, the risk factors are standardised to the same mean and standard deviation. This transformation ony makes sure that the risk factors are comparabe with respect to the first and second moment of the distribution. Perfect comparabiity wi ony be achieved when a moments of the standardised risk factor s distribution wi be roughy the same, i.e. if they foow a simiar probabiity distribution. This wi typicay not be the case, in particuar since there are risk factors with continuous and discrete distributions respectivey. However, some degree of overa distributiona simiarity shoud be achieved by the need to estabish an approximatey inear reationship between each risk factor and the og externa PD. Moreover, we wi comment on the rationae of and the potentia probems with the interpretation of regression estimates as weights of infuence in Sect. 4.5.4 where we dea with muti factor anaysis. 4.4.4 Representativeness Representativeness, whie important for other types of rating systems, shoud be treated with particuar care when deveoping SRA rating systems. 34 The foowing two types of comparisons are of specific interest: Comparison of the interna sampes types IE (incuding ony externay rated obigors) and IA (comprising a interna obigors) with each other. 34 An SRA-rating system wi aways face the probem that due to the reative rareness of defaut data it is difficut to vaidate it for obigors that are not externay rated. Whie some vaidation techniques are avaiabe (see Sect. 4.5.8), showing that the data for externay rated obigors is comparabe with that of non-externay rated obigors wi be one of the major steps to make sure that the derived rating system wi not ony perform we for the former but aso for the atter.

60 U. Erenmaier This comparison is necessary since SRA rating systems are deveoped on sampes that incude ony externay rated obigors but are aso appied to obigors without externa ratings. Comparison of the externa data set (E) with the interna data set IA. This comparison arises from the need to increase the avaiabe number of observations for rating deveopment by incuding externa data. Representativeness can be anaysed by comparing the distribution of the risk factors and some other key factors (such as countries/regions, industry sectors, company type, obigor size, etc.) on each sampe. In this context frequency pots (for continuous factors, see Fig. 4.2) and tabes ordered by the frequency of each reaisation (for discrete factors) can be particuary usefu. These toos can be suppemented with basic descriptive statistics (e.g. difference of the medians of both sampes reative to their standard deviation or the ratio of the standard deviations on both sampes). Forma statistica tests on the identity of distributions across sampes were not found to be usefu since the question is not whether distributions are identica (typicay they are not) but whether they are sufficienty simiar for the extrapoation of resuts and estimates derived on one sampe to the other sampe. 20.0 Externa Data Interna Data 17.5 15.0 Percent 12.5 10.0 7.5 5.0 2.5 0 0.3 0.2 0.1 0 0.1 0.2 0.3 0.2 0.1 0 0.1 0.2 Risk Factor Fig. 4.2 Exampe for a frequency pot that compares a risk factor s distribution on the externa data set with its distribution on the interna data set

4 The Shadow Rating Approach: Experience from Banking Practice 61 What can be done when data is found to be unrepresentative? First, it has to be ascertained whether the probem occurs ony for a few risk factors/key figures or for the majority. In the first case, the reasons for the differences have to be anaysed and the deveopment sampes adjusted accordingy. One reason, for exampe, might be that the distribution of obigors across regions or industry sectors is extremey different. The deveopment sampe can then be adjusted by reducing the amount of obigors in those regions/industry sectors that are over-represented in the deveopment sampe. In the second case, a variety of approaches can be considered, depending on the specific situation. Exampes incude: The range of the risk factors can be reduced so that it ony incudes areas that are observabe on both the deveopment and the target sampe. The weight of a risk factor found to be insufficienty representative can be reduced manuay or it can be excuded from the anaysis. 4.4.5 Missing Vaues A missing vaue anaysis typicay incudes the foowing steps: Decision as to whether a risk factor wi be cassified as missing for a particuar observation Cacuation of fiers for missing vaues/excusion of observations with missing vaues Whie for some risk factors such as quaitative assessments (e.g., management quaity), the first issue can be decided immediatey, it is not aways that cear-cut for quantitative risk factors such as baance sheet ratios that may be cacuated from a number of different singe positions. Typica exampes are baance sheet ratios that incude a company s cash fow that in turn is the sum of various singe baance sheet items. The probem typicay arising on the externa data set is that for a arge proportion of observations at east one of these items wi be missing. Hence, in a first step the reative sizes of the baance sheet items have to be compared with each other and based on this comparison, rues must be devised as to which combination of missing vaues wi trigger the overa position to be cassified as missing: if components with a arge absoute size are missing, the risk factors shoud be set to missing; if not, the aggregate position can be cacuated by either omitting the missing items or using fiers which, however, shoud be chosen conditiona on the size of the argest components. We now come back to the second issue raised at the beginning of this section, i.e., the cacuation of fiers for missing vaues on the risk factor eve. It is, of course, reated to the issue of cacuating fiers on the component eve. However, the need to

62 U. Erenmaier empoy conditiona estimates is not so severe. Typicay, there wi be quite a ot of risk factors that are correated with each other. Hence, making estimates for missing vaues of one risk factor conditiona on other risk factors shoud produce more accurate fiers. However, it wi aso be time consuming. Therefore, in practice, ony some very simpe bits of information wi typicay be used for conditioning, e.g., the portfoio to which an obigor beongs (externa or interna data set). Moreover, different quanties of the distribution might be empoyed for the cacuation of fiers on the externa and interna data set respectivey. For the externa sampe, a missing vaue may not constitute a significant negative signa in itsef. For the interna sampe, on the other hand, missing vaues usuay are negative signas, since a company coud be expected to provide to the bank the information it needs to compete its interna rating assessment. Therefore, missing vaues on the interna sampe wi typicay be substituted by more conservative quanties than missing vaues on the externa data set. Finay, depending on the reative frequency of missing vaues in the sampe, it might be necessary to excude some observations with missing vaues to avoid biases in statistica estimates. 4.4.6 Summary Concuding this section we want to summarise the techniques that we have presented for univariate risk factor anaysis and map them to the sampes on which they shoud be performed. Since we have aready deat with the sampe issue in Sect. 4.3, here we wi focus on those two sampe dimensions that we think are most important for univariate factor anaysis, i.e. externay rated obigors versus a obigors and externa versus interna data set. As in Sect. 4.4.4 we use the foowing shortcuts for these sampe types: E: Externa data set, ony externay rated obigors, IE: Interna data set, ony externay rated obigors. IA: Interna data set, a obigors, The univariate anaysis techniques and corresponding sampe types are summarised in Tabe 4.4. 4.5 Muti-factor Mode and Vaidation 4.5.1 Introduction Once the univariate anaysis described in Sect. 4.4 has been competed, the mutifactor mode has to be estimated and the estimation resuts communicated, adjusted (if necessary), and vaidated. These issues wi be deat with in Sect. 4.5 in this order:

4 The Shadow Rating Approach: Experience from Banking Practice 63 Tabe 4.4 Univariate anaysis techniques and corresponding sampe types Type of univariate Sampe Description anaysis type Factor transformation IE,IA a (a) Truncation (b) other non-inear transformations of continuous risk factors (e.g., taking a risk factor s ogarithm) (c) cacuating scores for discrete risk factors (d) standardisation: inear transformation in order to achieve the same median (mean) and standard deviation for a risk factors Discriminatory power E,IE (a) Correation (rank order and inear) with externa PD (b) Bucket pots Representativeness IE,IA (a) Comparison of interna sampes with each other (IE and IA) E,IA (b) Comparison of externa sampe (E) with interna sampe (IA) Missing vaues E,IA Fiers for missing vaues in the externa and interna sampes respectivey a IE is ony needed to derive the scores for the quaitative risk factors. A other types of anaysis are performed on IA. Mode seection: which type of mode is chosen and which risk factors wi enter the mode? Mode assumptions: Statistica modes typicay come with quite a few modeing assumptions that guarantee that estimation resuts are efficient and vaid. Therefore, it has to be anaysed whether the most important assumptions of the seected mode are vaid for the data and if not, how any vioations of modeing assumptions can be deat with. Measuring the infuence of risk factors: We wi discuss how the reative infuence of singe risk factors on the rating resut can be expressed in terms of weights to faciitate the interpretation of the estimated mode. In a second step, we comment on the probems associated with the cacuation and interpretation of these weights. Manua adjustments and caibration: We discuss the rationae and the most important issues that must be deat with when mode estimates are adjusted manuay and describe how the resuting mode can be caibrated. Two-step regression: It is briefy noted that with externa data the regression mode wi typicay have to be estimated in two steps. Corporate groups and government support: We propose a simpe method to produce an empirica estimate for the optima absoute infuence of supporters on an obigor s rating. Vaidation: We briefy itemise the vaidation measures that we found most usefu for a short-cut vaidation in the context of rating deveopment. 4.5.2 Mode Seection The issue of mode seection primariy has two dimensions. First, the mode type has to be chosen and then it has to be decided which risk factors wi be incuded in

64 U. Erenmaier the mode. Regarding the first question the most simpe and most frequenty used mode in muti factor anaysis is inear regression. A typica inear regression modes for SRA type rating systems wi have the foowing form 35 : LogðPD i Þ¼b 0 þ b 1 x i1 þþb m x im þ e i ði ¼ 1;...; nþ; (4.3) where PD i denotes the externa PD, x ij the vaue of risk factor j, e i the regression mode s error term for observation i, and b 0,...,b m are the regression coefficients that must be estimated from the data. Note that each observation i describes a specific firm over a specific time span. Risk factors are regressed on og PDs because on the one hand, this scae is typicay most compatibe with the inear reationship assumed by the regression mode and on the other hand, because interna master scaes that transate PDs into rating grades, are often ogarithmic in PDs. We now turn to the second issue in this section, the seection of those risk factors that wi constitute the fina regression mode empoyed for the rating system. The foowing types of anaysis are usefu for risk factor seection: Univariate discriminatory power (on interna and externa data set) Representativeness Correations/muticoinearity between risk factors Forma mode seection toos We have aready deat with the issues of discriminatory power and representativeness in Sect. 4.4. For correations between risk factors and muticoinearity we refer the reader to Chap. 2. In this section we wi add some comments on typica forma mode seection toos in the context of inear regression: Forma mode seection toos are no substitute for a carefu singe factor and correation anaysis. There are quite a variety of forma mode seection methods. 36 We found the R 2 maximisation method that finds the mode with the best R 2 for each given number of risk factors particuary usefu for the foowing reasons: It aows to trade off the reduction in muticoinearity against the associated oss in the mode s R 2 on the deveopment sampe. The R 2 measure is consistent with the inear correation measure empoyed in the singe factor anaysis. 37 35 Throughout this artice Log denotes the natura ogarithm with base e. 36 For reviews on forma mode-seection methods see Hocking (1976) or Judge et a. (1980). 37 R 2 is the square of the inear correation between the dependent variabe (the og externa PD) and the mode prediction for this variabe.

4 The Shadow Rating Approach: Experience from Banking Practice 65 4.5.3 Mode Assumptions Three crucia stochastic assumptions about the error terms e constitute the basis of inear regression modes 38 : Norma distribution (of error terms) Independence (of a error terms from each other) Homoscedasticity (a error terms have the same standard deviation) For a three issues there are a variety of statistica tests (e.g., Greene 2003). If these tests reject the above hypotheses, it is up to the modeer to decide on the severity of these effects, i.e., whether they can be accepted from a practica point of view or not. As for normaity, ooking at distribution pots of the residuas 39 we found that they often came very cose to a norma distribution even in cases where statistica tests reject this hypothesis. Moreover, even under the vioation of the normaity assumption, estimators are sti efficient (or, more precisey, BLUE). 40 Ony the reated statistica tests and confidence intervas are no onger vaid. But even here convergence is achieved for arge sampe size. Vioations of the two other assumptions (independence and homoscedasticity) tend to be more severe. They can be summarised as deviations from the regression mode s error term covariance matrix which is assumed to have identica vaues for each entry of the diagona (homoscedasticity) and zeros for each entry that is not on the diagona (independence). If statistica tests reject the hypotheses of independence/homoscedasticity, this probem can be deat with when a) pausibe assumptions about the structure of the covariance matrix can be made and b) when this structure can be described with a sufficienty sma set of parameters. If this is the case these parameters and hence the covariance matrix can be estimated from the data (or, more precisey, from the residuas). The east square method empoyed for parameter estimation in the regression mode can then be adjusted in such a way that the origina desirabe properties of the ordinary east square estimators (OLS) can be restored. In the iterature (e.g., Greene 2003) this method is referred to as generaised east square (GLS). In order to proceed, hypotheses on the structure of the covariance matrix have to be derived. In Sect. 4.3 deaing with sampe construction, we have aready described one possibe source of heteroscedasticity 41 and correation in the data respectivey. 38 For a comprehensive overview on appied inear regression see Greene (2003). 39 Residuas (e) are the typica estimators for the (unobservabe) theoretica error terms (e). They are defined as the difference between the dependent variabe and the mode predictions of this variabe. 40 BLUE stands for best inear unbiased estimator. 41 The term heteroscedasticity refers to cases where standard deviations of error terms are different as opposed to the assumption of identica standard deviations (homoscedasticity).

66 U. Erenmaier We argued that the size (i.e., the standard deviation) of the error term might sensiby be assumed to be proportiona to the ength of the time interva to which the observation is attached. Hence, we proposed to weight each observation with the ength of the corresponding time interva. In the context of regression anaysis, weighting observations exacty means to assume a specific type of heteroscedastic covariance matrix and appication of the corresponding GLS estimation. We aso concuded that autocorreation in the time series part of the data might we increase when time intervas become smaer and smaer. One of the simpest and most commony empoyed structures for correated error terms assumes an AR (1) correation structure between subsequent error terms: e t ¼ re t 1 þ u t ðt ¼ 1;...; TÞ ; (4.4) where the variabes u t are independent of each other. Hence, the issue coud be deat with by estimating the parameter r from the data, deriving the correation matrix and appying GLS. 42 There is, however, one crucia probem with this procedure: it is not ogica to assume this correation structure for the compete data set as woud be done in a standard time series regression setting. Rather, the rating deveopment data set at hand wi typicay have a pane data structure where the correation structure of the cross section s error terms (different obigors) wi most ikey be different from the correation structure of the time series part (different points in time for the same obigor). Appying a pane data mode with an AR(1) structure in the time series part coud be a sensibe first approximation. Corresponding error term modes offered by statistics software packages are often of the type e it ¼ r i e i;t 1 þ u it ðt ¼ 1;...; T; i ¼ 1;...; nþ : (4.5) Note that the AR parameter r is estimated separatey for each cross section (i.e. firm): r ¼ r i. Therefore, quite a few time series observations are required for each singe obigor to make confident estimates, which often wi not be feasibe for rating deveopment data. A more practicabe mode woud estimate an average AR parameter r for a obigors: e it ¼ re i;t 1 þ u it ðt ¼ 1;...; T; i ¼ 1;...; nþ : (4.6) There might be other sources of correation or heteroscedasticity in the data requiring a different structure for the covariance matrix than the one described above. If no specific reasons can be thought of from a theoretica point of view, one wi usuay ook at residua pots to identify some patterns. Typicay, residuas wi be potted (a) against the independent variabe (og PD in our case), (b) against those dependent variabes (risk factors) with the highest weights or (c) against some other structura variabe, such as the ength of the time interva associated with each 42 Indeed, a standard procedure for deaing with autocorreated error terms in the way described above is impemented in most statistica software packages.

4 The Shadow Rating Approach: Experience from Banking Practice 67 observation. If effects can be identified, first a parametrica mode has to be devised and then the associated parameters can be estimated from the residuas. That wi give a rough picture of the severity of the effects and can hence provide the basis for the decision as to whether to assess the deviations from the mode assumptions as acceptabe or whether to incorporate these effects into the mode either by weighting observation (in the case of heteroscedasticity) or by devising a specific correation mode (in the case of deviations from independence). 4.5.4 Measuring Infuence Once a specific regression mode has been chosen and estimated, one of the most important aspects of the mode for practitioners wi be each risk factor s infuence on an obigor s rating. Hence, a measure of infuence has to be chosen that can aso be used for potentia manua adjustments of the derived mode. To our knowedge, the most widey appied method is to adjust for the typicay different scaes on which the risk factors are measured by mutipying the estimator for the risk factor s coefficient in the regression mode by the risk factor s standard deviation and then deriving weights by mapping these adjusted coefficients to the interva [0,1] so that the absoute vaues of a coefficients add up to 1. 43 What is the interpretation of this approach to the cacuation of weights? It defines the weight of a risk factor x j by the degree to which the og PD predicted by the regression mode wi fuctuate when a other risk factors (x k ) k6¼j are kept constant: the more og PD fuctuates, the higher the risk factor s infuence. As a measure for the degree of fuctuation, the predictor s standard deviation is used. Hence, the weight w j of a risk factor x j with coefficient b j can be cacuated as w j ¼ w 1 w j þþ w ; (4.7) m where n o w j ¼ STD LogðPDÞ ðx k Þ k6¼j ¼ STDðb j x j Þ¼b j STDðx j Þ; (4.8) and STD denotes the standard deviation operator. 43 Note that this method is aso suggested by standard regression outputs. The associated estimates are typicay termed standardized coefficients. Moreover, if the risk factors have aready been standardized to a common standard deviation as described in Sect. 4.4 they aready have the same scae and coefficients ony have to be mapped to [0,1] in order to add up to 1.

68 U. Erenmaier However, when using this type of infuence measure, the foowing aspects have to be taken into account: The standard deviation shoud be cacuated on the interna data set containing a obigors, not ony the externay rated obigors. The master rating scae wi typicay be ogarithmic in PDs. Therefore, measuring the risk factor s infuence on predicted og PDs is approximatey equivaent to measuring its infuence on the obigor s rating. This shoud usuay be what practitioners are interested in. However, if the infuence on an obigor s predicted PD is to be measured, the above ogic wi not appy anymore since predicted PDs are an exponentia function of the risk factor and hence their standard deviation cannot be factored in the same fashion as described above. Moreover, the standard deviation of the externa PD wi depend on the reaisations of the other risk factors (x k ) k6¼j that are kept constant. The probems described in the previous point aso arise for the og-pd infuence when risk factors are transformed in a non-inear fashion, e.g. when a risk factor s ogarithm is taken. In this case, the above interpretation of infuence can ony be appied to the transformed risk factors which usuay have no sensibe economic interpretation. Aso, the above mentioned interpretation does not take into account the risk factor s correation structure. The correation between risk factors is usuay not negigibe. In this case the conditiona distribution (in particuar, the conditiona standard deviation) of the og-pd predictor, given that the other risk factors are constant, wi depend on the particuar vaues at which the other risk factors are kept constant. Making the risk factor s distributions comparabe ony by adjusting for their standard deviation might be a crude measure if their distributiona forms differ a ot (e.g., continuous versus discrete risk factors). 44 The weights described above measure a risk factor s average infuence over the sampe. Whie this may be suitabe in the mode deveopment stage when deciding, e.g., about whether the resuting weights are appropriate, it may not be appropriate for practitioners interested in the infuence that the risk factors have for a specific obigor. Other toos can be appied here, e.g., potting how a change in one risk factor over a specified range wi affect an obigor s rating. Despite the above cited theoretica probems standard deviation based measures of infuence have proved to work quite we in practice. However, there appears to be some scope for further research on aternative measures of infuence. Moreover, it shoud be noted that, when correations between risk factors are non-negigibe, a risk factor s correation with predicted og PDs can be quite high, even if the weight as defined above is not. We therefore found it important for the interpretation of the 44 Additionay, the standard deviation tends to be a very unstabe statistica measure that can be very sensitive to changes in the risk factor s distribution. However, this probem shoud be reduced significanty by the truncation of the risk factors which reduces the infuence of outiers.

4 The Shadow Rating Approach: Experience from Banking Practice 69 derived regression mode, to evauate these correations for a risk factors and report them together with the weights. 4.5.5 Manua Adjustments and Caibration There may be quite a variety of rationaes for manuay adjusting the estimation resuts derived from the statistica mode, for instance, expert judgements that deviate significanty from those estimations, insufficient empirica basis for specific portfoio segments, insufficient representativeness of the deveopment sampe, or excessivey high weights of quaitative as opposed to quantitative risk factors. 45 When manua adjustments are made, the foowing subsequent anayses are important: 1. Ensuring that the ratings system s discriminatory power is not reduced too much 2. Re-estabishing the caibration that statistica modes provide automaticay in the SRA context Regarding the first issue, the standard vaidation measures as briefy described in Sect. 4.5.8 wi be appied. The second issue can be addressed by regressing the score resuting from the manuay adjusted weights o 1,...,o n against og PDs: LogðPD i Þ¼c 0 þ c 1 ½o 1 x i1 þþo m x im Šþe i ði ¼ 1;...nÞ: (4.9) Note that c 0 and c 1 are the coefficients that must be estimated in this second regression. The parameter c 0 is reated to the average PD in the portfoio whie c 1 contros the rating system s impicit discriminatory power, i.e., the degree to which predicted PDs vary across the obigors in the portfoio. 46 The estimates for c 0 and c 1 wi give additiona evidence for the degree to which the manua adjustments have changed the rating system s overa properties: If changes are not too big, then c 0 shoud not differ much from b 0 and c 1 shoud be cose to b S ¼ ½jb 1 jþþjb m jš if a risk factors have been standardised to the same standard deviation. 47 Finay, for each observation i, a PD estimate can be derived from the above regression resuts by the foowing formuas: 45 With the SRA approach to rating deveopment, there is the probem that the oan manager may use quaitative risk factors in order to make interna and externa ratings match. If that is the case, the reative weight of quaitative factors as estimated by the statistica mode wi typicay be too high compared to the weights of quantitative risk factors. The vaidation measures that are not inked to externa ratings (see Sect. 4.5.8) and aso expert judgement may then hep to readjust those weights appropriatey. 46 More formay, the impicit discriminatory power is defined as the expected vaue of the (expicit) discriminatory power as measured by the Gini coefficient (cf. Chap. 13). 47 This can be derived from (4.7) and (4.8).

70 U. Erenmaier E½PD i jx i Š¼expðm i þ s i 2 =2Þ (i ¼ 1;:::,n), where m i ¼ E½ogðPD i ÞjX i Š¼c 0 þ c 1 ½o 1 x i1 þþo m x im Š and s i 2 ¼ Varðe i Þ: (4.10a) (4.10b) (4.10c) Note that X i denotes the vector of a risk factor reaisations for observation i and E[.] is the expectation operator. The resut is derived from the formua for the mean of og-normay distributed random variabes. 48 For the formua to be vaid, the error terms e i have to be approximatey normay distributed which we found typicay to be the case (see Sect. 4.5.3). Moreover, the most straightforward way to estimate s i from the residuas woud be to assume homoscedasticity, i.e. s i ¼ s (i ¼ 1,...,n). If homoscedasticity cannot be achieved, the estimates for s i wi have to be conditiona on the structura variabes that describe the sources of heteroscedasticity. 4.5.6 Two-step Regression In this section we note that when externa data are empoyed it wi typicay be necessary to estimate two modes and, therefore, go through the process described in the previous sections twice. If, for exampe, ony baance sheet ratios and macroeconomic risk factors are avaiabe for the externa data set, then a first quantitative mode wi have to be estimated on the externa data set. As a resut, a quantitative score and corresponding PD can be cacuated from this mode that in turn can be used as an input factor for the fina mode. The fina mode wi then incude the quantitative score as one aggregated independent variabe and the quaitative risk factors (not avaiabe for the externa data set) as the other independent variabes. 4.5.7 Corporate Groups and Sovereign Support When rating a company, it is very important to take into account the corporate group to which the company beongs or probaby some kind of government support (be it on the federa, state or oca government eve). This is typicay done by rating both the obigor on a standaone basis (¼standaone rating) and the entity that 48 If X is normay distributed with mean m and standard deviation s, then E [exp (X)] ¼ exp (m + s 2 /2), where E is the expectation operator (Limpert et a. 2001).

4 The Shadow Rating Approach: Experience from Banking Practice 71 is supposed to infuence the obigor s rating (¼supporter rating). 49 The obigor s rating is then usuay derived by some type of weighted average of the associated PDs. The weight wi depend on the degree of infuence as assessed by the oan manager according to the rating system s guideines. Due to the huge variety and often idiosyncratic nature of corporate group or sovereign support cases, it wi be very difficut to statisticay derive the correct individua weight of each supporter, the average weight, however, coud we be vaidated by estimates from the data. More precisey, consider that for the deveopment sampe we have i ¼ 1,...,n obigors with PDs PD i, corresponding supporters with PDs PD S i and associated supporter weights w i > 0 as derived by the rating anayst s assessment. 50 Then, a regression mode with [(1 w i ) PD i ] and [w i PD S i ] as independent variabes and PDex i (the obigor s externa PD) as dependent variabe can be estimated to determine as to whether the average size of the supporter weights w i is appropriate or whether it shoud be increased or decreased. 4.5.8 Vaidation The vaidation of rating systems is discussed at ength in Chaps. 12 15. Specific vaidation techniques that are vauabe in a ow defaut context (of which SRA portfoios are a typica exampe) are discussed in BCBS (2005) and in Chap. 5. During rating deveopment it wi typicay not be possibe to run through a fuyfedged vaidation process. Rather, it wi be necessary to concentrate on the most important measures. We wi therefore briefy itemise those issues that we found important for a short-cut vaidation of SRA rating systems in the context of rating deveopment: Vaidation on externa ratings/externa PDs Correations of interna and externa PDs (for a modues of the rating system 51 ) Case-wise anaysis of those companies with the argest differences between interna and externa ratings Comparison of average externa and interna PDs across the entire portfoio and across sub-portfoios (such as regions, rating grades, etc.) Vaidation on defaut indicators Gini coefficient (for a modues of the rating system) 49 Note that for the sake of simpicity, the expression supporter is used for a entities that infuence an obigor s rating, be it in a positive or negative way. 50 The standaone and supporter PDs have of course been derived from the regression mode of the previous sections, probaby, after manua adjustments. 51 The typica modues of a SRA-rating system (statistica mode, expert-guided adjustments, corporate-group infuence/government support, override) have been discussed in Sect. 4.1.

72 U. Erenmaier Comparison of defaut rates and corresponding confidence intervas with average interna PDs. This is done separatey for a rating grades and aso across a rating grades Forma statistica tests of the rating system s caibration (such as e.g. Spiegehater, see Chap. 15) Comparison of the new rating system with its predecessor (if avaiabe) Comparison of both rating system s vaidation resuts on externa ratings and the defaut indicator Case-wise anaysis of those companies with the argest differences between od and new rating system There are aso some other vaidation techniques not yet discussed but that coud enter a short-cut vaidation process in the rating deveopment context, in particuar addressing the reative rareness of defaut data in SRA portfoios (see BCBS 2005): Using the owest non-defaut rating grades as defaut proxies Comparison of SRA obigors with the obigors from other rating segments that have the same rating Estimation of interna PDs with the duration-based approach, i.e. incuding information on rating migration into the interna PD estimation process Data pooing 4.6 Concusions In this artice we have reported on some aspects of the deveopment of shadow rating (SRA) systems found to be important for practitioners. The artice focused on the statistica mode that typicay forms the basis of such rating systems. In this section we want to summarise the major issues that we have deat with: We have stressed the importance both, in terms of the quaity of the resuting rating system and in terms of initia deveopment costs of The depoyment of sophisticated software toos that automate the deveopment process as much as possibe and The carefu preparation and vaidation of the data that are empoyed. Externa PDs form the basis of SRA type modes. We have outined some major issues that we found to be important in this context: Which externa rating types/agencies shoud be used? Comparison between bank interna and externa defaut definitions and consequences for resuting PD estimates Sampe construction for the estimation of externa PDs (which time period, which obigor types?) PD estimation techniques (cohort method vs. duration-based approach) Point-in-time adjustment of externa through-the-cyce ratings In Sect. 4.3 we pointed out that different sampes wi be needed for different types of anaysis and made a proposa for the construction of such sampes.

4 The Shadow Rating Approach: Experience from Banking Practice 73 In this context we aso deat with the issues of weighted and correated observations. Univariate risk factor anaysis is the next deveopment step. In Sect. 4.4 we have described the typica types of anaysis required measurement of a risk factor s discriminatory power, transformation of risk factors, representativeness, fiers for missing vaues and have mapped them to the sampes on which they shoud be performed. In Sect. 4.5 we deat with muti factor modeing, in particuar with Mode seection The vioation of mode assumptions (non-normaity, heteroscedasticity, error term correations) The measurement of risk factor infuence (weights) Manua adjustments of empirica estimates and caibration A method to empiricay vaidate the average infuence of corporate groups or sovereign supporters on an obigor s rating Finay, in the same section, we gave a brief overview over the vaidation measures that we found most usefu for a short-cut vaidation in the context of SRA rating deveopment. Whie for most modeing steps one can observe the emergence of best practice toos, we think that in particuar in the foowing areas further research is desirabe to sharpen the instruments avaiabe for SRA rating deveopment: Data pooing in order to arrive at more confident estimates for adjustment factors of externa PDs that account for the differences between bank interna and externa defaut measurement Empirica comparisons of the reative performance of cohort-based versus duration-based PD estimates and reated confidence intervas Point-in-time adjustments of externa through-the-cyce ratings Pane type correation modes for SRA sampes and software impementations of these modes Measurement of risk factor infuence (weights) References Atman E, Rijken H (2004), How Rating Agencies achieve Rating Stabiity, Journa of Banking and Finance 28 (11), pp. 2679 2714. Ammer J, Packer F (2000), How Consistent Are Credit Ratings? A Geographic and Sectora Anaysis of Defaut Risk, FRB Internationa Finance Discussion Paper No. 668. Appasamy B, Hengstmann S, Stapper G, Schark E (2004), Vaidation of Rating Modes, Wimott Magazine, May, pp. 70 74. Base Committee on Banking Supervision (BCBS) (2005), Vaidation of Low-defaut Portfoios in the Base II Framework, Base Committee Newsetter No. 6. Base Committee on Banking Supervision (BCBS) (2004), Internationa Convergence of Capita Measurement and Capita Standards, Bank for Internationa Settements, Base.

74 U. Erenmaier Base Committee on Banking Supervision (BCBS) (2000), Range of Practice in Banks Interna Ratings Systems, Bank for Internationa Settements, Base. Cantor R, Fakenstein E (2001), Testing for Rating Consistency in Annua Defaut Rates, Moody s Investors Service, New York. Cantor R (2004), Measuring the Quaity and Consistency of Corporate Ratings across Regions, Moody s Investors Service, New York. Day L (1992), Simpe SAS Macros for the Cacuation of Exact Binomia and Poisson Confidence Limits, Computers in Bioogy and Medicine 22 (5), pp. 351 361. Davison A, Hinkey D (1997), Bootstrap Methods and their Appication, Cambridge University Press, Cambridge. Deutsche Bundesbank (2003), Vaidierungsans atze f ur interne Ratingsysteme, Monatsbericht September, pp. 61 74. Erenmaier U (2001), Modes of Joint Defauts in Credit Risk Management: An Assessment, University of Heideberg Working Paper No. 358. Fitch (2005), Fitch Ratings Goba Corporate Finance 2004 Transition and Defaut Study. Fitch Ratings Credit Market Research. Fitch (2006), Fitch Ratings Definitions. http://www.fitchratings.com/corporate/fitchresources. cfm?detai¼1 [as at 18/02/06] Greene W (2003), Econometric Anaysis, Pearson Education, Inc., New Jersey. G utter A (2004), Using a Bootstrap Approach to Rate the Raters, Financia Markets and Portfoio Management 19, pp. 277 295. Heitfied E (2004), Rating System Dynamics and Bank-Reported Defaut Probabiities under the New Base Capita Accord, Working Paper, Board of Governors of the Federa Reserve System, Washington. Hocking R (1976), The Anaysis and Seection of Variabes in Linear Regression, Biometrics 32, pp. 1 50. Judge G, Griffiths W, Hi R, Lee T (1980), The Theory and Practice of Econometrics, John Wiey & Sons, Inc., New York. Limpert E, Stah W, Abbt M (2001), Lognorma Distributions Across the Sciences: Keys and Cues, BioScience 51 (5), pp. 341 352. L offer G (2004), An Anatomy of Rating Through the Cyce, Journa of Banking and Finance 28 (3), pp. 695 720. Moody s (2004), Moody s Rating Symbos & Definitions. Moody s Investors Service, New York. Moody s (2005), Defaut and Recovery Rates of Corporate Bond Issuers, 1920 2004, Specia Comment, New York. Schuermann T, Hanson S (2004), Estimating Probabiities of Defaut, FRB of New York Staff Report No. 190. Standard & Poor s (2002), S&P Long-Term Issuer Credit Ratings Definitions. http://www2. standardandpoors.com/servet/sateite?pagename¼sp/page/fixedincomeratingscriteriapg&r¼ 1&¼EN&b¼2&s¼21&ig¼1&ft¼26 [as at 18/02/06] Standard & Poor s (2005), Annua Goba Corporate Defaut Study: Corporate Defauts Poised to Rise in 2005, Goba Fixed Income Research, New York.

Chapter 5 Estimating Probabiities of Defaut for Low Defaut Portfoios Katja Puto and Dirk Tasche 5.1 Introduction A core input to modern credit risk modeing and managing techniques is probabiities of defaut (PD) per borrower. As such, the accuracy of the PD estimations wi determine the quaity of the resuts of credit risk modes. One of the obstaces connected with PD estimations can be the ow number of defauts, especiay in the higher rating grades. These good rating grades might enjoy many years without any defauts. Even if some defauts occur in a given year, the observed defaut rates might exhibit a high degree of voatiity due to the reativey ow number of borrowers in that grade. Even entire portfoios with ow or zero defauts are not uncommon. Exampes incude portfoios with an overa good quaity of borrowers (e.g. sovereign or bank portfoios) as we as highexposure ow-number portfoios (e.g. speciaized ending). Usua banking practices for deriving PD vaues in such exposures often focus on quaitative mapping mechanisms to bank-wide master scaes or externa ratings. These practices, whie widespread in the industry, do not entirey satisfy the desire for a statistica foundation of the assumed PD vaues. One might beieve that the PDs per rating grade appear correct, as we as thinking that the ordina ranking and the reative spread between the PDs of two grades is right, but find that there is insufficient information about the absoute PD figures. Lasty, it coud be questioned whether these rather quaitative methods of PD caibration fufi the minimum requirements set out in BCBS (2004a). The opinions expressed in this chapter are those of the authors and do not necessariy refect views of their respective empoyers. K. Puto HSBC Hodings pc e-mai: Katja.Puto@gmx.de D. Tasche (*) Loyds Banking Group e-mai: dirk.tasche@gmx.net B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_5, # Springer-Verag Berin Heideberg 2011 75

76 K. Puto and D. Tasche This issue, amongst others, has recenty been raised in BBA (2004). In that paper, appications of causa defaut modes and of exogenous distribution assumptions on the PDs across the grades have been proposed as soutions. Schuermann and Hanson (2004) present the duration method of estimating PDs by means of migration matrices (see aso Jafry and Schuermann 2004). This way, nonzero PDs for high-quaity rating grades can be estimated more precisey by both counting the borrower migrations through the ower grades to eventua defaut and using Markov chain properties. We present a methodoogy to estimate PDs for portfoios without any defauts, or a very ow number of defauts in the overa portfoio. The proposa by Schuermann and Hanson (2004) does not provide a soution for such cases, because the duration method requires a certain number of defauts in at east some (usuay the owquaity) rating grades. For estimating PDs, we use a avaiabe quantitative information of the rating system and its grades. Moreover, we assume that the ordina borrower ranking is correct. We do not use any additiona assumptions or information. 1 Our methodoogy deivers confidence intervas for the PDs of each rating grade. The PD range can be adjusted by the choice of an appropriate confidence eve. Moreover, by the most prudent estimation principe our methodoogy yieds monotonic PD estimates. We ook both at the cases of uncorreated and correated defaut events, in the atter case under assumptions consistent with the Base risk weight mode. Moreover, we extend the most prudent estimation by two appication variants: First we scae our resuts to overa portfoio centra tendencies. Second, we appy our methodoogy to muti-period data and extend our mode by time dependencies of the Base systematic factor. Both variants shoud hep to aign our principe to reaistic data sets and to a range of assumptions that can be set according to the specific issues in question when appying our methodoogy. The paper is structured as foows: The two main concepts underying the methodoogy estimating PDs as upper confidence bounds and guaranteeing their monotony by the most prudent estimation principe are introduced by two exampes that assume independence of the defaut events. The first exampe deas with a portfoio without any observed defauts. For the second exampe, we modify the first exampe by assuming that a few defauts have been observed. In a further section, we show how the methodoogy can be modified in order to take into account non-zero correation of defaut events. This is foowed by two sections discussing extensions of our methodoogy, in particuar the scaing to the overa portfoio centra tendency and an extension of our mode to the muti-period case. The ast two sections are devoted to discussions of the scope of appication and of 1 An important exampe of additiona assumptions is provided by a-priori distributions of the PD parameters which ead to a Bayesian approach as described by Kiefer (2009). Interestingy enough, Dwyer (2006) shows that the confidence bound approach as described in this paper can be interpreted in a Bayesian manner. Another exampe of an additiona assumption is presented in Tasche (2009). In that paper the monotonicity assumption on the PDs is repaced by a stronger assumption on the shape of the PD curve.

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 77 open questions. We concude with a summary of our proposa. In Appendix A, we provide information on the numerics that is needed to impement the estimation approach we suggest. Appendix B provides additiona numerica resuts to Sect. 5.5. We perceive that our most prudent estimation principe has been appied in a wide range of banks since the first edition of this book. However, appication has not been imited to PD estimation, as intended by us. Rather, risk modeers seem to have made generous use of the methodoogy to vaidate their rating systems. We have therefore added another short section at the end of this paper that expains the sense and non-sense of using our principe for vaidation purposes, and carify what the methodoogy can and cannot do. 5.2 Exampe: No Defauts, Assumption of Independence The obigors are distributed to rating grades A, B, and C, with frequencies n A, n B, and n C. The grade with the highest credit-worthiness is denoted by A, the grade with the owest credit-worthiness is denoted by C. No defauts occurred in A, B or C during the ast observation period. We assume that the sti to be estimated PDs p A of grade A, p B of grade B, and p C of grade C refect the decreasing credit-worthiness of the grades, in the sense of the foowing inequaity: p A p B p C (5.1) The inequaity impies that we assume the ordina borrower ranking to be correct. According to (5.1), the PD p A of grade A cannot be greater than the PD p C of grade C. As a consequence, the most prudent estimate of the vaue p A is obtained under the assumption that the probabiities p A and p C are equa. Then, from (5.1) even foows p A ¼ p B ¼ p C. Assuming this reation, we now proceed in determining a confidence region for p A at confidence eve g. This confidence region 2 can be described as the set of a admissibe vaues of p A with the property that the probabiity of not observing any defaut during the observation period is not ess than 1 g (for instance for g ¼ 90%). If we have got p A ¼ p B ¼ p C, then the three rating grades A, B, and C do not differ in their respective riskiness. Hence we have to dea with a homogeneous sampe of size n A þ n B þ n C without any defaut during the observation period. Assuming unconditiona independence of the defaut events, the probabiity of 2 For any vaue of p A not beonging to this region, the hypothesis that the true PD takes on this vaue woud have to be rejected at a type I error eve of 1-g (see Casea and Berger 2002, Theorem 9.2.2 on the duaity of hypothesis testing and confidence intervas).

78 K. Puto and D. Tasche observing no defauts turns out to be ð1 p A Þ n Aþn B þn C. Consequenty, we have to sove the inequaity 1 g ð1 p A Þ n Aþn B þn C (5.2) for p A in order to obtain the confidence region at eve g for p A as the set of a the vaues of p A such that p A 1 ð1 g Þ 1= ð n Aþn B þn C Þ (5.3) If we choose for the sake of iustration n A ¼ 100; n B ¼ 400; n C ¼ 300; (5.4) Tabe 5.1 exhibits some vaues of confidence eves g with the corresponding maximum vaues (upper confidence bounds) ^p A of p A such that (5.2) is sti satisfied. According to Tabe 5.1, there is a strong dependence of the upper confidence bound ^p A on the confidence eve g. Intuitivey, vaues of g smaer than 95% seem more appropriate for estimating the PD by ^p A. By inequaity (5.1), the PD p B of grade B cannot be greater than the PD p C of grade C either. Consequenty, the most prudent estimate of p B is obtained by assuming p B ¼ p C. Assuming additiona equaity with the PD p A of the best grade A woud vioate the most prudent estimation principe, because p A is a ower bound of p B. If we have got p B ¼ p C, then B and C do not differ in their respective riskiness and may be considered a homogeneous sampe of size n B þ n C. Therefore, the confidence region at eve g for p B is obtained from the inequaity 1 g ð1 p C Þ n Bþn C (5.5) (5.5) impies that the confidence region for p B consists of a the vaues of p B that satisfy p B 1 ð1 g Þ 1= ð n Bþn C Þ (5.6) If we again take up the exampe described by (5.4), Tabe 5.2 exhibits some vaues of confidence eves g with the corresponding maximum vaues (upper confidence bounds) ^p B of p B such that (5.6) is sti fufied. Tabe 5.1 Upper confidence bound ^p A of p A as a function of the confidence eve g. No defauts observed, frequencies of obigors in grades given in (5.4) g 50% 75% 90% 95% 99% 99.9% ^p A 0.09% 0.17% 0.29% 0.37% 0.57% 0.86%

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 79 Tabe 5.2 Upper confidence bound ^p B of p B as a function of the confidence eve g. No defauts observed, frequencies of obigors in grades given in (5.4) g 50% 75% 90% 95% 99% 99.9% ^p B 0.10% 0.20% 0.33% 0.43% 0.66% 0.98% Tabe 5.3 Upper confidence bound ^p C of p C as a function of the confidence eve g. No defauts observed, frequencies of obigors in grades given in (5.4) g 50% 75% 90% 95% 99% 99.9% ^p C 0.23% 0.46% 0.76% 0.99% 1.52% 2.28% For determining the confidence region at eve g for p C we ony make use of the observations in grade C because by (5.1) there is no obvious upper bound for p C. Hence the confidence region at eve g for p C consists of those vaues of p C that satisfy the inequaity 1 g ð1 p C Þ n C (5.7) Equivaenty, the confidence region for p C can be described by p C 1 ð1 gþ 1=n C (5.8) Coming back to our exampe (5.4), Tabe 5.3 ists some vaues of confidence eves g with the corresponding maximum vaues (upper confidence bounds) ^p C of p C such that (5.8) is sti fufied. Comparison of Tabes 5.1 5.3 shows that besides the confidence eve g theappicabesampesizeisamaindriver of the upper confidence bound. The smaer the sampe size, the greater wi be the upper confidence bound. This is not an undesirabe effect, because intuitivey the credit-worthiness ought to be the better, the greater the number of obigors in a portfoio without any defaut observation. As the resuts presented so far seem pausibe, we suggest using upper confidence bounds as described by (5.3), (5.6) and (5.8) as estimates for the PDs in portfoios without observed defauts. The case of three rating grades we have considered in this section can readiy be generaized to an arbitrary number of grades. We do not present the detais here. However, the arger the number of obigors in the entire portfoio, the more often some defauts wi occur in some grades at east, even if the genera quaity of the portfoio is very high. This case is not covered by (5.3), (5.6) and (5.8). In the foowing section, we wi show sti keeping the assumption of independence of the defaut events how the most prudent estimation methodoogy can be adapted to the case of a non-zero but sti ow number of defauts.

80 K. Puto and D. Tasche 5.3 Exampe: Few Defauts, Assumption of Independence We consider again the portfoio from Sect. 5.2 with the frequencies n A, n B, and n C. In contrast to Sect. 5.2, this time we assume that during the ast period no defaut was observed in grade A, two defauts were observed in grade B, and one defaut was observed in grade C. As in Sect. 5.2, we determine a most prudent confidence region for the PD p A of A. Again, we do so by assuming that the PDs of the three grades are equa. This aows us to treat the entire portfoio as a homogeneous sampe of size n A þ n B þ n C. Then the probabiity of observing not more than three defauts is given by the expression X 3 i¼0 n A þ n B þ n C p i i A ð1 p AÞ n Aþn B þn C i (5.9) Expression (5.9) foows from the fact that the number of defauts in the portfoio is binomiay distributed as ong as the defaut events are independent. As a consequence of (5.9), the confidence region 3 at eve g for p A is given as the set of a the vaues of p A that satisfy the inequaity 1 g X3 i¼0 n A þ n B þ n C p i i A ð1 p AÞ n Aþn B þn C i (5.10) The tai distribution of a binomia distribution can be expressed in terms of an appropriate beta distribution function. Thus, inequaity (5.10) can be soved anayticay 4 for p A. For detais, see Appendix A. If we assume again that the obigors numbers per grade are as in (5.4), Tabe 5.4 shows maximum soutions ^p A of (5.10) for different confidence eves g. Athough in grade A no defauts were observed, the three defauts that occurred during the observation period enter the cacuation. They affect the upper confidence bounds, which are much higher than those in Tabe 5.1. This is a consequence of the precautionary assumption p A ¼ p B ¼ p C. However, if we aternativey considered grade A aone (by re-evauating (5.8) with n A ¼ 100 instead of n C ), we woud obtain an upper confidence bound of 1.38% at eve g ¼ 75%. This vaue is sti much higher than the one that has been cacuated under the precautionary assumption p A ¼ p B ¼ p C a consequence of the ow frequency of obigors in grade A in this exampe. Nevertheess, we see that the methodoogy described by (5.10) yieds fairy reasonabe resuts. 3 We cacuate the simpe and intuitive exact Copper-Pearson interva. For an overview of this approach, as we as potentia aternatives, see Brown et a. (2001). 4 Aternativey, soving directy (5.10) for p A by means of numerica toos is not too difficut either (see Appendix A, Proposition A.1, for additiona information).

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 81 Tabe 5.4 Upper confidence bound ^p A of p A as a function of the confidence eve g. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4) g 50% 75% 90% 95% 99% 99.9% ^p A 0.46% 0.65% 0.83% 0.97% 1.25% 1.62% Tabe 5.5 Upper confidence bound ^p B of p B as a function of the confidence eve g. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4) g 50% 75% 90% 95% 99% 99.9% ^p B 0.52% 0.73% 0.95% 1.10% 1.43% 1.85% In order to determine the confidence region at eve g for p B, as in Sect. 5.2, we assume that p B takes its greatest possibe vaue according to (5.1), i.e. that we have p B ¼ p C. In this situation, we have a homogeneous portfoio with n B þ n C obigors, PD p B, and three observed defauts. Anaogous to (5.9), the probabiity of observing no more than three defauts in one period then can be written as: X 3 i¼0 n B þ n C p i i B ð1 p BÞ n Bþn C i (5.11) Hence, the confidence region at eve g for p B turns out to be the set of a the admissibe vaues of p B which satisfy the inequaity 1 g X3 i¼0 n B þ n C p i i B ð1 p BÞ n Bþn C i (5.12) By anayticay or numericay soving (5.12) for p B with frequencies of obigors in the grades as in (5.4) we obtain Tabe 5.5 with some maximum soutions ^p B of (5.12) for different confidence eves g. From the given numbers of defauts in the different grades, it becomes cear that a stand-aone treatment of grade B woud yied sti much higher vaues 5 for the upper confidence bounds. The upper confidence bound 0.52% of the confidence region at eve 50% is amost identica with the naïve frequency based PD estimate 2/400 ¼ 0.5% that coud aternativey have been cacuated for grade B in this exampe. For determining the confidence region at eve g for the PD p C, by the same rationae as in Sect. 5.2, the grade C must be considered a stand-aone portfoio. According to the assumption made in the beginning of this section, one defaut 5 At eve 99.9%, e.g., 2.78% woud be the vaue of the upper confidence bound.

82 K. Puto and D. Tasche Tabe 5.6 Upper confidence bound ^p C of p C as a function of the confidence eve g. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4) g 50% 75% 90% 95% 99% 99.9% ^p C 0.56% 0.90% 1.29% 1.57% 2.19% 3.04% occurred among the n C obigors in C. Hence we see that the confidence region for p C is the set of a admissibe vaues of p C that satisfy the inequaity 1 g X1 i¼0 n C i p i C ð1 p CÞ nc i ¼ð1 p C Þ n C þ n C p C ð1 p C Þ n C 1 (5.13) For obigor frequencies as assumed in exampe (5.4), Tabe 5.6 exhibits some maximum soutions 6 ^p C of (5.13) for different confidence eves g. So far, we have described how to generaize the methodoogy from Sect. 5.2 to the case where non-zero defaut frequencies have been recorded. In the foowing section we investigate the impact of non-zero defaut correation on the PD estimates that are effected by appying the most prudent estimation methodoogy. 5.4 Exampe: Correated Defaut Events In this section, we describe the dependence of the defaut events with the one-factor probit mode 7 that was the starting point for deveoping the risk weight functions given in BCBS (2004a) 8. First, we use the exampe from Sect. 5.2 and assume that no defaut at a was observed in the whoe portfoio during the ast period. In order to iustrate the effects of correation, we appy the minimum vaue of the asset correation that appears in the Base II corporate risk weight function. This minimum vaue is 12% (see BCBS 2004a, } 272). Our mode, however, works with any other correation assumption as we. Likewise, the most prudent estimation principe coud potentiay be appied to other modes than the Base II type credit risk mode as ong as the inequaities can be soved for p A, p B and p C, respectivey. 6 If we had assumed that two defauts occurred in grade B but no defaut was observed in grade C, then we woud have obtained smaer upper bounds for p C than for p B. As this is not a desirabe effect, a possibe conservative work-around coud be to increment the number of defauts in grade C up to the point where p C woud take on a greater vaue than p B. Nevertheess, in this case one woud have to make sure that the appied rating system yieds indeed a correct ranking of the obigors. 7 According to De Finetti s theorem (see, e.g., Durrett (1996), Theorem 6.8), assuming one systematic factor ony is not very restrictive. 8 See Gordy (2003) and BCBS (2004b) for the background of the risk weight functions. In the case of non-zero reaized defaut rates Bathazar (2004) uses the one-factor mode for deriving confidence intervas of the PDs.

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 83 Tabe 5.7 Upper confidence bounds ^p A of p A, ^p B of p B and ^p C of p C as a function of the confidence eve g. No defauts observed, frequencies of obigors in grades given in (5.4). Correated defaut events g 50% 75% 90% 95% 99% 99.9% ^p A 0.15% 0.40% 0.86% 1.31% 2.65% 5.29% ^p B 0.17% 0.45% 0.96% 1.45% 2.92% 5.77% ^p C 0.37% 0.92% 1.89% 2.78% 5.30% 9.84% Under the assumptions of this section, the confidence region at eve g for p A is represented as the set of a admissibe vaues of p A that satisfy the inequaity (cf. Buhm et a. 2003, Sects. 2.1.2 and 2.5.1 for the derivation) ð 1 p 1 g ðyþ 1 F F 1 ð A Þ ffiffiffi n r y A þn B þn C pffiffiffiffiffiffiffiffiffiffiffi dy; (5.14) 1 r 1 where and F stand for the standard norma density and standard norma distribution function, respectivey. F 1 denotes the inverse function of F and r is the asset correation (here r is chosen as r ¼ 12%). Simiary to (5.2), the right-hand side of inequaity (5.14) tes us the one-period probabiity of not observing any defaut among n A þ n A þ n A obigors with average PD p A. Soving 9 (5.14) numericay 10 for the frequencies as given in (5.4) eads to Tabe 5.7 with maximum soutions ^p A of (5.14) for different confidence eves g. Comparing the vaues from the first ine of Tabe 5.7 with Tabe 5.1 shows that the impact of taking care of correations is moderate for the ow confidence eves 50% and 75%. The impact is much higher for the eves higher than 90% (for the confidence eve 99.9% the bound is even six times arger). This observation refects the genera fact that introducing unidirectiona stochastic dependence in a sum of random variabes entais a redistribution of probabiity mass from the centre of the distribution towards its ower and upper imits. The formuae for the estimations of upper confidence bounds for p B and p C can be derived anaogousy to (5.14) [in combination with (5.5) and (5.7)]. This yieds the inequaities ð 1 p 1 g ðyþ 1 F F 1 ð B Þ ffiffiffi nb þn r y C pffiffiffiffiffiffiffiffiffiffiffi dy (5.15) 1 r 1 9 See Appendix A, Proposition A.2, for additiona information. Taking into account correations entais an increase in numerica compexity. Therefore, it might seem to be more efficient to dea with the correation probem by choosing an appropriatey enarged confidence eve in the independent defaut events approach as described in Sects. 5.2 and 5.3. However, it remains open how a confidence eve for the uncorreated case, that appropriatey adjusts for the correations, can be derived. 10 The more intricate cacuations for this paper were conducted by means of the software package R (cf. R Deveopment Core Team 2003).

84 K. Puto and D. Tasche Tabe 5.8 Upper confidence bounds ^p A of p A, ^p B of p B and ^p C of p C as a function of the confidence eve g. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Correated defaut events g 50% 75% 90% 95% 99% 99.9% ^p A 0.72% 1.42% 2.50% 3.42% 5.88% 10.08% ^p B 0.81% 1.59% 2.77% 3.77% 6.43% 10.92% ^p C 0.84% 1.76% 3.19% 4.41% 7.68% 13.14% and 1 g ð 1 1 p ðyþ 1 F F 1 ð C Þ ffiffiffi n r y C pffiffiffiffiffiffiffiffiffiffiffi dy; (5.16) 1 r to be soved for p B and p C respectivey. The numerica cacuations with (5.15) and (5.16) do not deiver additiona quaitative insights. For the sake of competeness, however, the maximum soutions ^p B of (5.15) and ^p C of (5.16) for different confidence eves g are isted in rows 3 and 4 of Tabe 5.7, respectivey. Secondy, we appy our correated mode to the exampe from Sect. 5.3 and assume that three defauts were observed during the ast period. Anaogous to (5.9), (5.10) and (5.14), the confidence region at eve g for p A is represented as the set of a vaues of p A that satisfy the inequaity 1 g ð 1 zðyþ ¼ X3 1 i¼0 ðyþzðyþdy; n A þ n B þ n C i where the function G is defined by Gðp A ; r; yþ i ð1 Gðp A ; r; yþþ n Aþn B þn C i ; (5.17) p Gðp; r; yþ ¼F F 1 ðpþ ffiffiffi r y pffiffiffiffiffiffiffiffiffiffiffi : (5.18) 1 r Soving (5.17) for ^p A with obigor frequencies as given in (5.4), and the respective modified equations for ^p B and ^p C yieds the resuts presented in Tabe 5.8. Not surprisingy, as shown in Tabe 5.8 the maximum soutions for ^p A, ^p B and ^p C increase if we introduce defauts in our exampe. Other than that, the resuts do not deiver essentia additiona insights. 5.5 Extension: Caibration by Scaing Factors One of the drawbacks of the most prudent estimation principe is that in the case of few defauts, the upper confidence bound PD estimates for a grades are higher than the average defaut rate of the overa portfoio. This phenomenon is not surprising,

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 85 Tabe 5.9 Upper confidence bound ^p A;scaed of p A, ^p B;scaed of p B and ^p C;scaed of p C as a function of the confidence eve g after scaing to the centra tendency. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Uncorreated defaut events g 50% 75% 90% 95% 99% 99.9% Centra Tendency 0.375% 0.375% 0.375% 0.375% 0.375% 0.375% K 0.71 0.48 0.35 0.30 0.22 0.17 ^p A 0.33% 0.31% 0.29% 0.29% 0.28% 0.27% ^p B 0.37% 0.35% 0.34% 0.33% 0.32% 0.31% ^p C 0.40% 0.43% 0.46% 0.47% 0.49% 0.50% Tabe 5.10 Upper confidence bound ^p A;scaed of p A, ^p B;scaed of p B and ^p C;scaed of p C as a function of the confidence eve g after scaing to the centra tendency. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Correated defaut events g 50% 75% 90% 95% 99% 99.9% Centra Tendency 0.375% 0.375% 0.375% 0.375% 0.375% 0.375% K 0.46 0.23 0.13 0.09 0.05 0.03 ^p A 0.33% 0.33% 0.32% 0.32% 0.32% 0.32% ^p B 0.38% 0.37% 0.36% 0.36% 0.35% 0.35% ^p C 0.39% 0.40% 0.41% 0.42% 0.42% 0.42% given that we incude a defauts of the overa portfoio in the upper confidence bound estimation even for the highest rating grade. However, these estimates might be regarded as too conservative by some practitioners. A remedy woud be a scaing 11 of a of our estimates towards the centra tendency (the average portfoio defaut rate). We introduce a scaing factor K to our estimates such that the overa portfoio defaut rate is exacty met, i.e. ^p A n A þ ^p B n B þ ^p C n C n A þ n B þ n C K ¼ PD Portfoio : (5.19) The new, scaed PD estimates wi then be ^p X;scaed ¼ K^p X ; X ¼ A; B; C: (5.20) The resuts of the appication of such a scaing factor to our few defauts exampes of Sects. 5.3 and 5.4 are shown in Tabes 5.9 and 5.10, respectivey. The average estimated portfoio PD wi now fit exacty the overa portfoio centra tendency. Thus, we remove a conservatism from our estimations. Given the poor defaut data base in typica appications of our methodoogy, this might be seen as a disadvantage rather than an advantage. By using the most prudent estimation 11 A simiar scaing procedure was suggested by Benjamin et a. (2006). However, the straightforward inear approach as described in (5.19) and (5.20) has the drawback that, in principe, the resuting PDs can exceed 100%. See Tasche (2009, Appendix A) for a non-inear scaing approach based on Bayes formua that avoids this issue.

86 K. Puto and D. Tasche principe to derive reative PDs before scaing them down to the fina resuts, we preserve the soe dependence of the PD estimates upon the borrower frequencies in the respective rating grades, as we as the monotony of the PDs. The question of the appropriate confidence eve for the above cacuations remains. Athough the average estimated portfoio PD now aways fits the overa portfoio defaut rate, the confidence eve determines the distribution of that rate over the rating grades. In the above exampe, though, the differences in distribution appear sma, especiay in the correated case, such that we woud not expore this issue further. The confidence eve coud, in practice, be used to contro the spread of PD estimates over the rating grades the higher the confidence eve, the higher the spread. However, the above scaing works ony if there is a nonzero number of defauts in the overa portfoio. Zero defaut portfoios woud indeed be treated more severey if we continued to appy our origina proposa to them, compared to using scaed PDs for ow defaut portfoios. A variant of the above scaing proposa that takes care of both issues is the use of an upper confidence bound for the overa portfoio PD in ieu of the actua defaut rate. This upper confidence bound for the overa portfoio PD, incidentay, equas the most prudent estimate for the highest rating grade. Then, the same scaing methodoogy as described above can be appied. The resuts of its appication to the few defauts exampes as in Tabes 5.9 and 5.10 are presented in Tabes 5.11 and 5.12. Tabe 5.11 Upper confidence bound ^p A;scaed of p A, ^p B;scaed of p B and ^p C;scaed of p C as a function of the confidence eve g after scaing to the upper confidence bound of the overa portfoio PD. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Uncorreated defaut events g 50% 75% 90% 95% 99% 99.9% Upper bound for 0.46% 0.65% 0.83% 0.97% 1.25% 1.62% portfoio PD K 0.87 0.83 0.78 0.77 0.74 0.71 ^p A 0.40% 0.54% 0.65% 0.74% 0.92% 1.16% ^p B 0.45% 0.61% 0.74% 0.84% 1.06% 1.32% ^p C 0.49% 0.75% 1.01% 1.22% 1.62% 2.17% Tabe 5.12 Upper confidence bound ^p A;scaed of p A, ^p B;scaed of p B and ^p C;scaed of p C as a function of the confidence eve g after scaing to the upper confidence bound of the overa portfoio PD. No defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Correated defaut events g 50% 75% 90% 95% 99% 99.9% Upper bound for 0.71% 1.42% 2.50% 3.42% 5.88% 10.08% portfoio PD K 0.89 0.87 0.86 0.86 0.86 0.87 ^p A 0.64% 1.24% 2.16% 2.95% 5.06% 8.72% ^p B 0.72% 1.38% 2.39% 3.25% 5.54% 9.54% ^p C 0.75% 1.53% 2.76% 3.80% 6.61% 11.37%

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 87 In contrast to the situation of Tabes 5.9 and 5.10, in Tabes 5.11 and 5.12 the overa defaut rate in the portfoio depends on the confidence eve, and we observe scaed PD estimates for the grades that increase with growing eves. Nevertheess, the scaed PD estimates for the better grades are sti consideraby ower than the corresponding unscaed estimates from Sects. 5.3and 5.4, respectivey. For the sake of comparison, we provide in Appendix B the anaogous numerica resuts for the no defaut case. The advantage of this atter variant of the scaing approach is that the degree of conservatism is activey manageabe by the appropriate choice of the confidence eve for the estimation of the upper confidence bound of the portfoio PD. Moreover, it works in the case of zero defauts and few defauts, and thus does not produce a structura break between both scenarios. Lasty, the resuts are ess conservative than those of our origina methodoogy. 5.6 Extension: The Muti-period Case So far, we have ony considered the situation where estimations are carried out on a 1 year (or one observation period) data sampe. In case of a time series with data from severa years, the PDs (per rating grade) for the singe years coud be estimated and coud then be used for cacuating weighted averages of the PDs in order to make more efficient use of the data. By doing so, however, the interpretation of the estimates as upper confidence bounds at some pre-defined eve woud be ost. Aternativey, the data of a years coud be pooed and tacked as in the 1-year case. When assuming cross-sectiona and inter-tempora independence of the defaut events, the methodoogy as presented in Sects. 5.2 and 5.3 can be appied to the data poo by repacing the 1-year frequency of a grade with the sum of the frequencies of this grade over the years (anaogous for the numbers of defauted obigors). This way, the interpretation of the resuts as upper confidence bounds as we as the frequency-dependent degree of conservatism of the estimates wi be preserved. However, when turning to the case of defaut events which are cross-sectionay and inter-temporay correated, pooing does not aow for an adequate modeing. An exampe woud be a portfoio of ong-term oans, where in the inter-tempora poo every obigor woud appear severa times. As a consequence, the dependence structure of the poo woud have to be specified very carefuy, as the structure of correation over time and of cross-sectiona correation are ikey to differ. In this section, we present two muti-period extensions of the cross-sectiona one-factor correation mode that has been introduced in Sect. 5.4. In the first part of the section, we take the perspective of an observer of a cohort of obigors over a fixed interva of time. The advantage of such a view arises from the conceptua separation of time and cross-section effects. Again, we do not present the methodoogy in fu generaity but rather introduce it by way of an exampe.

88 K. Puto and D. Tasche As in Sect. 5.4, we assume that, at the beginning of the observation period, we have got n A obigors in grade A, n B obigors in grade B, and n C obigors in grade C. In contrast to Sect. 5.4, the ength of the observation period this time is T >1. We consider ony the obigors that were present at the beginning of the observation period. Any obigors entering the portfoio afterwards are negected for the purpose of our estimation exercise. Nevertheess, the number of observed obigors may vary from year to year as soon as any defauts occur. As in the previous sections, we first consider the estimation of the PD p A for grade A. PD in this section denotes a ong-term average 1-year probabiity of defaut. Working again with the most prudent estimation principe, we assume that the PDs p A, p B, and p C are equa, i.e. p A ¼ p B ¼ p C ¼ p. We assume, simiar to Gordy (2003), that a defaut of obigor i ¼ 1,..., N ¼ n A þ n B þ n C in year t ¼ 1,..., T is triggered if the change in vaue of their assets resuts in a vaue ower than some defaut threshod c as described beow by (5.22). Specificay, if V i,t denotes the change in vaue of obigor i s assets, V i,t is given by p V i;t ¼ ffiffiffi p r St þ ffiffiffiffiffiffiffiffiffiffiffi 1 r x i;t ; (5.21) where r stands for the asset correation as introduced in Sect. 5.4, S t is the reaisation of the systematic factor in year t, and x i,t denotes the idiosyncratic component of the change in vaue. The cross-sectiona dependence of the defaut events stems from the presence of the systematic factor S t in a the obigors change in vaue variabes. Obigor i s defaut occurs in year t if The probabiity V i;1 >c;...; V i;t 1 >c; V i;t c: (5.22) P[V i;t cš ¼p i;t ¼ p (5.23) is the parameter we are interested to estimate: It describes the ong-term average 1-year probabiity of defaut among the obigors that have not defauted before. The indices i and t at p i,t can be dropped because by the assumptions we are going to specify beow p i,t wi neither depend on i nor on t. To some extent, therefore, p may be considered a through-the-cyce PD. For the sake of computationa feasibiity, and in order to keep as cose as possibe to the Base II risk weight mode, we specify the factor variabes S t, t ¼ 1,...,T, and x i,t, i ¼ 1,...,N, t ¼ 1,...,T as standard normay distributed (cf. Buhm et a. 2003). Moreover, we assume that the random vector (S 1,...,S T ) and the random variabes x i,t, i ¼ 1,...,N, t¼ 1,...,T are independent. As a consequence, from (5.21) it foows that the change in vaue variabes V i,t are a standard-normay distributed. Therefore, (5.23) impies that the defaut threshod 12 c is determined by 12 At first sight, the fact that in our mode the defaut threshod is constant over time seems to impy that the mode does not refect the possibiity of rating migrations. However, by construction of the

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 89 c ¼ F 1 ðpþ; (5.24) with F denoting the standard norma distribution function. Whie the singe components S t of the vector of systematic factors, generate the cross-sectiona correation of the defaut events at time t, their inter-tempora correation is affected by the dependence structure of the factors S 1,...,S T.We further assume that not ony the components but aso the vector as a whoe is normay distributed. Since the components of the vector are standardized, its joint distribution is competey determined by the correation matrix 0 1 1 r 1;2 r 1;3 r 1;T r 2;1 1 r 2;3 r 2;T..... : (5.25) B C @ A r T;1 r T;T 1 1 Whereas the cross-sectiona correation within 1 year is constant for any pair of obigors, empirica observation indicates that the effect of inter-tempora correation becomes weaker with increasing distance in time. We express this distancedependent behaviour 13 of correations by setting in (5.25) r s;t ¼ # js tj ; s; t ¼ 1; ; T; s 6¼ t; (5.26) for some appropriate 0 <#<1 to be specified beow. Let us assume that within the T years observation period k A defauts were observed among the obigors that were initiay graded A, k B defauts among the initiay graded B obigors and k C defauts among the initiay graded C obigors. For the estimation of p A according to the most prudent estimation principe, therefore we have to take into account k ¼ k A þ k B þ k C defauts among N obigors over T years. For any given confidence eve g, we have to determine the maximum vaue ^p of a the parameters p such that the inequaity 1 g P½No more than k defauts observedš (5.27) is satisfied note that the right-hand side of (5.27) depends on the one-period probabiity of defaut p. In order to derive a formuation that is accessibe to numerica cacuation, we have to rewrite the right-hand side of (5.27). mode, the conditiona defaut threshod at time t given the vaue V i,t-1 wi in genera differ from c. As we make use of the joint distribution of the V i,t, therefore rating migrations are impicity taken into account. 13 Bochwitz et a. (2004) proposed the specification of the inter-tempora dependence structure according to (5.26) for the purpose of defaut probabiity estimation.

90 K. Puto and D. Tasche The first step is to deveop an expression for obigor i s conditiona probabiity to defaut during the observation period, given a reaization of the systematic factors S 1,...,S T. From (5.21), (5.22), (5.24) and by using the conditiona independence of the V i,1,..., V i,t given the systematic factors, we obtain P½Obigor i defauts js 1 ; ;S T Š ¼ P min V i;t F 1 ðpþjs 1 ;...; S T t¼1;...;t p ¼ 1 P x i;1 > F 1 ðpþ ffiffiffi r S1 p ffiffiffiffiffiffiffiffiffi ;...; x i;t > F 1 ðpþ ffiffiffi r pffiffiffiffiffiffiffiffiffi 1 r 1 r ¼ 1 YT ð1 Gðp; r; S t ÞÞ; t¼1 p ST js 1 ;...; S T (5.28) where the function G is defined as in (5.18). By construction, in the mode a the probabiities P[Obigor i defauts S 1,..., S T ] are equa, so that, for any of the i, we can define pðs 1 ; :::; S T Þ¼P[Obigor i defauts j S 1 ; :::; S T Š ¼ 1 YT ð1 Gðp; r; S t ÞÞ t¼1 (5.29) Using this abbreviation, we can write the right-hand side of (5.27) as P[No more than k defauts observed] ¼ Xk ¼0 ¼ Xk ¼0 E[P[Exacty obigors defaut j S 1 ; :::; S T ŠŠ N E[pðS 1 ; :::; S T Þ ð1 pðs 1 ; :::; S T ÞÞ N Š: (5.30) The expectations in (5.30) are expectations with respect to the random vector (S 1,...,S T ) and have to be cacuated as T-dimensiona integras invoving the density of the T-variate standard norma distribution with correation matrix given by (5.25) and (5.26). When soving (5.27) for ^p, we cacuated the vaues of these T-dimensiona integras by means of Monte-Caro simuation, taking advantage of the fact that the term X k ¼0 N E[pðS 1 ; :::; S T Þ ð1 pðs 1 ; :::; S T ÞÞ N Š (5.31) can be efficienty evauated by making use of (5.35) of Appendix A.

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 91 In order to present some numerica resuts for an iustration of how the mode works, we have to fix a time horizon T and vaues for the cross-sectiona correation r and the inter-tempora correation parameter #. We choose T ¼ 5 as BCBS (2004a) requires the credit institutions to base their PD estimates on a time series with minimum ength 5 years. For r, we chose r ¼ 0.12 as in Sect. 5.4, i.e. again a vaue suggested by BCBS (2004a). Our feeing is that defaut events with a 5 years time distance can be regarded as being neary independent. Statisticay, this statement might be interpreted as something ike the correation of S 1 and S 5 is ess than 1%. Setting # ¼ 0.3, we obtain corr[s 1,...,S T ] ¼ # 4 ¼ 0.81%. Thus, the choice # ¼ 0.3 seems reasonabe. Note that our choices of the parameters are purey exempary, as to some extent choosing the vaues of the parameters is rather a matter of taste or judgement or of decisions depending on the avaiabe data or the purpose of the estimations. 14 Tabe 5.13 shows the resuts of the cacuations for the case where no defauts were observed during 5 years in the whoe portfoio. The resuts for a the three grades are summarized in one tabe. To arrive at these resuts, (5.27) was first evauated with N ¼ n A þ n B þ n C,thenwithN ¼ n B þ n C, and finay with N ¼ n C. In a three cases we set k ¼ 0in(5.30) in order to express that no defauts were observed. Not surprisingy, the cacuated confidence bounds are much ower than those presented as in Tabe 5.7, thus demonstrating the potentiay dramatic effect of expoiting onger observation periods. For Tabe 5.14 we conducted essentiay the same computations as for Tabe 5.13, the difference being that we assumed that over 5 years k A ¼ 0, defauts were observed in grade A, k B ¼2 defauts were observed in grade B, and k C ¼ 1 Tabe 5.13 Upper confidence bounds ^p A of p A, ^p B of p B and ^p C of p C as a function of the confidence eve g. No defauts during 5 years observed, frequencies of obigors in grades given in (5.4). Cross-sectionay and inter-temporay correated defaut events g 50% 75% 90% 95% 99% 99.9% ^p A 0.03% 0.06% 0.11% 0.16% 0.30% 0.55% ^p B 0.03% 0.07% 0.13% 0.18% 0.33% 0.62% ^p C 0.07% 0.14% 0.26% 0.37% 0.67% 1.23% Tabe 5.14 Upper confidence bounds ^p A of p A, ^p B of p B and ^p C of p C as a function of the confidence eve g. During 5 years, no defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Crosssectionay and inter-temporay correated defaut events g 50% 75% 90% 95% 99% 99.9% ^p A 0.12% 0.21% 0.33% 0.43% 0.70% 1.17% ^p B 0.14% 0.24% 0.38% 0.49% 0.77% 1.29% ^p C 0.15% 0.27% 0.46% 0.61% 1.01% 1.70% 14 Benjamin et a. (2006) propose a simiar methodoogy that poos muti-year data into one arge poo of customers. Effectivey, they impicity assume identica cross-borrower and intra-tempora correations and disregard borrower dupication within the observation period.

92 K. Puto and D. Tasche defauts were observed in grade C (as in Sects. 5.3 and 5.4 during 1 year). Consequenty, we set k ¼ 3 in(5.30) for cacuating the upper confidence bounds for p A and p B, as we as k ¼ 1 for the upper confidence bounds of p C. Compared with the resuts presented in Tabe 5.8, we observe again the very strong effect of taking into account a onger time series. The methodoogy described above coud be christened cohort approach as cohorts of borrowers are observed over mutipe years. It does not take into account any changes in portfoio size due to new ending or repayment of oans. Moreover, the approach ignores the information provided by time custers of defauts (if there are any). Intuitivey, time-custering of defauts shoud be the kind of information needed to estimate the cross-sectiona and time-reated correation parameters r and # respectivey 15. A sighty different muti-period approach (caed mutipe binomia in the foowing) aows for variation of portfoio size by new ending and amortization and makes it possibe, in principe, to estimate the correation parameters. In particuar this approach ignores the fact that most of the time the portfoio composition this year and next year is amost identica. However, it wi turn out that as a consequence of the conditiona independence assumptions we have adopted the impact of ignoring the amost constant portfoio composition is reasonaby weak. Assume that the portfoio size in year t was N t for t ¼ 1,...,T, and that d t defauts were observed in year t. Given reaisations S 1,...,S T of the systematic factors, we then assume that the distribution of the number of defauts in year t conditiona on S 1,...,S T is binomia as in (5.17) and (5.18), i.e. P½d t defauts in year tjs 1 ;...; S T Š ¼ N t Gðp; r; S t Þ d t ð1 Gðp; r; S t ÞÞ N t d t (5.32) d t Under the additiona assumption of conditiona independence of defaut events at different moments in time conditiona on a reaisation of the systematic factors, (5.32) impies that the unconditiona probabiity to observe d 1 defauts in year 1,..., d T defauts in year T is given by P½d t defauts in year t; t ¼ 1;...; TŠ ¼ EPd ½ " ½ t defauts in year t; t ¼ 1;...; TjS 1 ;...; S T ŠŠ ¼ E YT N # t Gðp; r; S t Þ d t ð1 Gðp; r; S t ÞÞ N t d t t¼1 d t (5.33) 15 Indeed, it is possibe to modify the cohort approach in such a way as to take account of portfoio size varying due to other causes than defaut and of time-custers of defaut. This modification, however, comes at a high price because it requires a much more compicated input data structure that causes much onger cacuation time.

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 93 As (5.33) invoves a binomia distribution for each point in time t we ca the approach the mutipe binomia approach. If we assume that the atent systematic factors foow a T-dimensiona norma distribution with standard norma marginas as specified by (5.25) and (5.26), then cacuation of the right-hand side of (5.33) invoves the evauation of a T-dimensiona integra. This can be done by Monte- Caro simuation as in the case of (5.31). By means of an appropriate optimisation method 16, the right-hand side of (5.33) can be used as the ikeihood function for the determination of joint maximum ikeihood estimates of the correation parameters r and # and of the ong-run PD parameter p. It however requires at east one of the annua defaut number observations d t to be positive. Otherwise the ikeihood (5.33) is constant equa to 100% for p ¼ 0 and it is not possibe to identify unique parameters r and # that maximise the ikeihood. In the context of Tabe 5.14, if we assume that the three defauts occurred in the first year and consider the entire portfoio, the maximum ikeihood estimates of r, # and p are 34.3%, 0%, and 7.5 bps respectivey. In the case where vaues of the correation parameters are known or assumed to be known, it is aso possibe to use the mutipe binomia approach to compute confidence bound type estimates of the ong-run grade-wise PD estimates as was done for Tabe 5.14. To be abe to do this cacuation with the mutipe binomia approach, we need to cacuate the unconditiona probabiity that the tota number of defauts in years 1 to T does not exceed d ¼ d 1 þþd T. As the sum of binomiay distributed random variabes with different success probabiities in genera is not binomiay distributed, we cacuate an approximate vaue of the required unconditiona probabiity based on Poisson approximation: P½No more than d defauts in years 1 to TŠ " # X d I r;p ðs 1 ;...; S T Þ k E exp I r;p ðs 1 ;...; S T Þ ; k! k¼0 I r;p ðs 1 ;...; s T Þ ¼ XT N t Gðp; r; S t Þ: t¼1 (5.34) The expected vaue in (5.34) again has to be cacuated by Monte-Caro simuation. Tabe 5.15 shows the resuts of such a cacuation in the context of Tabe 5.13 [i.e. Tabe 5.13 is recacuated based on (5.34) instead of (5.31)]. Simiary, Tabe 5.16 dispays the recacuated Tabe 5.14 [i.e. Tabe 5.14 is recacuated based on (5.34) instead of (5.31)]. Both in Tabe 5.15 and Tabe 5.16 resuts seem hardy different to the resuts in Tabe 5.13 and Tabe 5.14 respectivey. Hence the use of (5.34) instead of (5.31) in order to aow for different portfoio sizes due to new ending and amortisation appears to be justified. 16 For the numerica exampes in this paper, the authors made use of the R-procedure nminb.

94 K. Puto and D. Tasche Tabe 5.15 Upper confidence bounds ^p A of p A, ^p B of p B and ^p C of p C as a function of the confidence eve g. No defauts during 5 years observed, frequencies of obigors in grades given in (5.4). Cross-sectionay and inter-temporay correated defaut events. Cacuation based on (5.34) G 50% 75% 90% 95% 99% 99.9% ^p A 0.02% 0.05% 0.10% 0.15% 0.29% 0.53% ^p B 0.03% 0.06% 0.12% 0.17% 0.32% 0.60% ^p C 0.06% 0.13% 0.26% 0.36% 0.66% 1.19% Tabe 5.16 Upper confidence bounds ^p A of p A, ^p B of p B and ^p C of p C as a function of the confidence eve g. During 5 years, no defaut observed in grade A, two defauts observed in grade B, one defaut observed in grade C, frequencies of obigors in grades given in (5.4). Crosssectionay and inter-temporay correated defaut events. Cacuation based on (5.34) g 50% 75% 90% 95% 99% 99.9% ^p A 0.12% 0.21% 0.33% 0.42% 0.68% 1.12% ^p B 0.13% 0.23% 0.37% 0.47% 0.76% 1.24% ^p C 0.14% 0.26% 0.44% 0.59% 0.99% 1.66% 5.7 Appications The most prudent estimation methodoogy described in the previous sections can be used for a range of appications, both within a bank and in a Base II context. In the atter case, it might be specificay usefu for portfoios where neither interna nor externa defaut data are sufficient to meet the Base requirements. A good exampe might be Speciaized Lending. In these high-voume, ow-number and ow-defaut portfoios, interna data is often insufficient for PD estimations per rating category, and might indeed even be insufficient for centra tendency estimations for the entire portfoio (across a rating grades). Moreover, mapping to externa ratings athough expicity aowed in the Base context and widey used in bank interna appications might be impossibe due to the ow number of externay rated exposures. The (conservative) principe of the most prudent estimation coud serve as an aternative to the Base sotting approach, subject to supervisory approva. In this context, the proposed methodoogy might be interpreted as a specific form of the Base requirement of conservative estimations if data is scarce. In a wider context, within the bank, the methodoogy might be used for a sorts of ow defaut portfoios. In particuar, it coud compement other estimation methods, whether this be mapping to externa ratings, the proposas by Schuermann and Hanson (2004) or others. As such, we see our proposed methodoogy as an additiona source for PD caibrations. This shoud neither invaidate nor prejudge a bank s interna choice of caibration methodoogies. However, we tend to beieve that our proposed methodoogy shoud ony be appied to whoe rating systems and portfoios. One might think of caibrating PDs of individua ow defaut rating grades within an otherwise rich data structure.

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 95 Doing so amost unavoidaby eads to a structura break between average PDs (data rich rating grades) and upper PD bounds (ow defaut rating grades) which makes the procedure appear infeasibe. Simiary, we beieve that the appication of the methodoogy for backtesting or simiar vaidation toos woud not add much additiona information. For instance, purey expert-based average PDs per rating grade woud normay be we beow our proposed quantitative upper bounds. 5.8 Open Issues For appications, a number of important issues need to be addressed: Which confidence eves are appropriate? The proposed most prudent estimate coud serve as a conservative proxy for average PDs. In determining the confidence eve, the impact of a potentia underestimation of these average PDs shoud be taken into account. One might think that the transformation of average PDs into some kind of stress PDs, as done in the Base II and many other credit risk modes, coud justify rather ow confidence eves for the PD estimation in the first pace (i.e. using the modes as providers of additiona buffers against uncertainty). However, this concusion woud be miseading, as it mixes two different types of stresses : the Base II mode stress of the singe systematic factor over time, and the estimation uncertainty stress of the PD estimations. Indeed, we woud argue for moderate confidence eves when appying the most prudent estimation principe, but for other reasons. The most common aternative to our methodoogy, namey deriving PDs from averages of historica defaut rates per rating grade, yieds a comparabe probabiity that the true PD wi be underestimated. Therefore, high confidence eves in our methodoogy woud be hard to justify. At which number of defauts shoud users deviate from our methodoogy and use norma average PD estimation methods, at east for the overa portfoio centra tendency? Can this critica number be anayticay determined? If the reative number of defauts in one of the better ratings grades is significanty higher than those in ower rating grades (and within ow defaut portfoios, this might happen with ony one or two additiona defauts), then our PD estimates may turn out to be non-monotonic. In which cases shoud this be taken as an indication of an incorrect ordina ranking? Certainy, monotony or non-monotony of our upper PD bounds does not immediatey impy that the averagepdsaremonotonicornon-monotonic. Under which conditions woud there be statistica evidence of a vioation of the monotony requirement for the PDs? Currenty, we do not have definite soutions to above issues. We beieve, though, that some of them wi invove a certain amount of expert judgment rather than anaytica soutions. In particuar, that might be the case with the first item. If our

96 K. Puto and D. Tasche proposed approach were used in a supervisory say Base II context, supervisors might want to think about suitabe confidence eves that shoud be consistenty appied. 5.9 Estimation Versus Vaidation We have been somewhat surprised to see the methodoogy described in this chapter being often appied for PD vaidation rather than PD estimation. This new section for the second edition of the book sets out principes as to when and when not appy the methodoogy for PD estimation, as we as exampes where appication might be usefu in practice. First, the ow defaut estimation methodoogy based on upper confidence bounds has a high degree of inbuit conservatism. Comparing defaut rates or PDs estimated by other methodoogies against confidence-bound-based PDs must take this estimation bias into account having observed defaut rates not breaching our upper confidence bounds shoud not be regarded as a particuar achievement, and observing defaut rates above the confidence bounds may indicate a serious PD underestimation indeed. Second, spreading the centra tendency of a portfoio across rating grades via the most prudent estimation principe has the grade PDs, in effect, soey driven by grade popuation and the confidence eve. There are imits as to how wide the centra tendency can be statisticay spread, impying that the sope of the most prudent PDs over rating grades tends to be much fatter than PDs curves derived by aternative methods (e.g. benchmarking to externa ratings). So which benefits can be derived from vaidation via benchmarking against ow defaut estimates based on upper confidence bounds? As the ow defaut methodoogy deivers conservative PD estimates, it can offer some insight into the degree of conservatism for PDs caibrated by another method. For a given PD estimate (derived, for exampe, by benchmarking to externa ratings) and an observed number of defauts, an intermediate step of the cacuation of upper confidence bounds gives an impied confidence eve that woud have deivered the same PD from the defaut rate via the confidence bound cacuation. Indeed, using (5.27) with the given PD estimate to determine g generates an impied confidence eve as desired. Whie there is no test as to which confidence eve is too conservative in this context, the approach offers an opportunity for the quantification of conservatism that might be hepfu in bank interna and reguatory discussions. The approach is most usefu for centra tendency comparisons appication at grade eve may resut in very different confidence eves across the rating scae due to the ow number of defauts. The interpretation of such fuctuating eves then becomes somewhat of a chaenge. The approach might yied usefu resuts over time, however, as the impicit confidence eve changes. The voatiity can give some quaitative indication as to how much point in time or through the cyce a rating

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 97 system is the atter shoud resut in higher voatiity as observed defaut rates are aways point in time. 5.10 Concusions In this artice, we have introduced a methodoogy for estimating probabiities of defaut in ow or no defaut portfoios. The methodoogy is based on upper confidence intervas by use of the most prudent estimation. Our methodoogy uses a avaiabe quantitative information. In the extreme case of no defauts in the entire portfoio, this information consists soey of the absoute numbers of counter-parties per rating grade. The ack of defauts in the entire portfoio prevents reiabe quantitative statements on both the absoute eve of average PDs per rating grade as we as on the reative risk increase from rating grade to rating grade. Within the most prudent estimation methodoogy, we do not use such information. The ony additiona assumption used is the ordina ranking of the borrowers, which is assumed to be correct. Our PD estimates might seem rather high at first sight. However, given the amount of information that is actuay avaiabe, the resuts do not appear out of range. We beieve that the choice of moderate confidence eves is appropriate within most appications. The resuts can be scaed to any appropriate centra tendency. Additionay, the muti-year context as described in Sect. 5.6 might provide further insight. Appendix A This appendix provides additiona information on the anaytica and numerica soutions of (5.10) and (5.14). Anaytica soution of (5.10). If X is a binomiay distributed random variabe with size parameter n and success probabiity p, then for any integer 0 k n, we have X k i¼0 Ð 1 n p i ð1 pþ n i p ¼ P[X kš¼1 P[Y pš¼ tk ð1 tþ n k 1 dt Ð i 1 0 tk ð1 tþ n k 1 dt (5.35) with Y denoting a beta distributed random variabe with parameters a ¼ k þ 1 and b ¼ n k (see, e.g., Hinderer (1980), Lemma 11.2). The beta distribution function and its inverse function are avaiabe in standard numerica toos, e.g. in Exce. Direct numerica soution of Equation (5.10). The foowing proposition shows the existence and uniqueness of the soution of (5.10), and, at the same time, provides initia vaues for the numerica root-finding [see (5.38)].

98 K. Puto and D. Tasche Proposition A.1. Let 0 k < n be integers, and define the function f n,k : (0, 1)! R by f n;k ðpþ ¼ Xk i¼0 Fix some 0 < v < 1. Then the equation n p i ð1 pþ n i ; p 2ð0; 1Þ (5.36) i f n;k ðpþ ¼v (5.37) has exacty one soution 0 < p ¼ p(v) < 1. Moreover, this soution p(v) satisfies the inequaities 1 np ffiffi v pðvþ Proof. A straight-forward cacuation yieds df n;k ðpþ dp np ffiffiffiffiffiffiffiffiffiffiffi 1 v (5.38) n ¼ ðn kþ p k ð1 pþ n k 1 : (5.39) k Hence f n,k is stricty decreasing. This impies uniqueness of the soution of (5.37). The inequaities f n;0 ðpþ f n;k ðpþ f n;n 1 ðpþ (5.40) impy the existence of a soution of (5.37) and the inequaities (5.38). Numerica soution of (5.14). For (5.14) we can derive a resut simiar to Proposition A.1. However, there is no obvious upper bound to the soution p(v) of (5.42) asin(5.38). Proposition A.2. For any probabiity 0 < p < 1, any correation 0 < r < 1 and any rea number y define p F r ðp; yþ ¼F F 1 ðpþþ ffiffiffi r y pffiffiffiffiffiffiffiffiffiffiffi ; (5.41) 1 r where we make use of the same notations as for (5.14). Fix a vaue 0 < v < 1 and a positive integer n. Then the equation v ¼ ð 1 1 ðyþð1 F r ðp; yþþ n dy; (5.42) with denoting the standard norma density, has exacty one soution 0 < p ¼ p (v) < 1. This soution p(v) satisfies the inequaity pðvþ 1 np ffiffi v : (5.43)

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 99 Proof of Proposition (A.2) Note that for fixed r and y the function F r (p, y) is stricty increasing and continuous in p. Moreover, we have 0 ¼ im p!0 F r ðp; yþ and 1 ¼ im p!1 F r ðp; yþ (5.44) Equation (5.44) impies existence and uniqueness of the soution of (5.42). Define the random variabe Z by Z ¼ F r ðp; YÞ; (5.45) where Y denotes a standard normay distributed random variabe. Then Z has the we-known Vasicek distribution (cf. Vasicek 1997), and in particuar we have Using (5.45), (5.42) can be rewritten as E[ZŠ ¼p: (5.46) v ¼ E½ð1 ZÞ n Š: (5.47) Since y! (1 y) n is convex for 0 < y < 1, by (5.46) Jensen s inequaity impies v ¼ E½ð1 ZÞ n Šð1 pþ n : (5.48) As the right-hand side of (5.42) is decreasing in p, (5.43) now foows from (5.48). Appendix B This appendix provides additiona numerica resuts for the scaing extension of the most prudent estimation principe according to Sect. 5.5 in the case of no defaut portfoios. In the exampes presented in Tabes 5.17 and 5.18, the confidence eve for deriving the upper confidence bound for the overa portfoio PD, and the confidence eves for the most prudent estimates of PDs per rating grade have Tabe 5.17 Upper confidence bound ^p A;scaed of p A, ^p B;scaed of p B and ^p C;scaed of p C as a function of the confidence eve g after scaing to the upper confidence bound of the overa portfoio PD. No defaut observed, frequencies of obigors in grades given in (5.4). Uncorreated defaut events g 50% 75% 90% 95% 99% 99.9% Centra tendency 0.09% 0.17% 0.29% 0.37% 0.57% 0.86% K 0.61 0.66 0.60 0.58 0.59 0.59 ^p A 0.05% 0.11% 0.17% 0.22% 0.33% 0.51% ^p B 0.06% 0.13% 0.20% 0.25% 0.39% 0.58% ^p C 0.14% 0.24% 0.45% 0.58% 0.89% 1.35%

100 K. Puto and D. Tasche Tabe 5.18 Upper confidence bound ^p A;scaed of p A, ^p B;scaed of p B and ^p C;scaed of p C as a function of the confidence eve g after scaing to the upper confidence bound of the overa portfoio PD. No defaut observed, frequencies of obigors in grades given in (5.4). Correated defaut events g 50% 75% 90% 95% 99% 99.9% Centra tendency 0.15% 0.40% 0.86% 1.31% 2.65% 5.29% K 0.62 0.65 0.66 0.68 0.70 0.73 ^p A 0.09% 0.26% 0.57% 0.89% 1.86% 3.87% ^p B 0.11% 0.29% 0.64% 0.98% 2.05% 4.22% ^p C 0.23% 0.59% 1.25% 1.89% 3.72% 7.19% aways been set equa. Moreover, our methodoogy aways provides equaity between the upper bound of the overa portfoio PD and the most prudent estimate for p A according to the respective exampes of Sects. 5.2 and 5.4. References Bathazar L (2004), PD Estimates for Base II, Risk, Apri, pp. 84 85. Base Committee on Banking Supervision (BCBS) (2004a), Base II: Internationa Convergence of Capita Measurement and Capita Standards: a Revised Framework. http://www.bis.org/pub/ bcbs107.htm. Base Committee on Banking Supervision (BCBS) (2004b), An Expanatory Note on the Base II IRB Risk Weight Functions. http://www.bis.org. Benjamin N, Cathcart A, Ryan K (2006), Low Defaut Portfoios: A Proposa for Conservative Estimation of Probabiities of Defaut. Discussion Paper, Financia Services Authority. British Bankers Association (BBA), London Investment Banking Association (LIBA) and Internationa Swaps and Derivatives Association (ISDA) (2004), The IRB Approach for Low Defaut Portfoios (LDPs) Recommendations of the Joint BBA, LIBA, ISDA Industry Working Group, Discussion Paper. http://www.isda.org/speeches/ pdf/isda-liba-bba- LowDefauPortfoioPaper080904-paper.pdf. Bochwitz S, Hoh S, Tasche D, Wehn C (2004), Vaidating Defaut Probabiities on Short Time Series. Capita & Market Risk Insights, Federa Reserve Bank of Chicago, December 2004. http://www.chicagofed.org/banking_information/capita_and_market_risk_insights.cfm. Buhm C, Overbeck L, Wagner C (2003), An Introduction to Credit Risk Modeing, Chapman & Ha/CRC, Boca Raton. Brown L, Cai T, Dasgupta A (2001), Interva Estimation for a Binomia Proportion, Statistica Science 16 (2), pp. 101 133. Casea G, Berger RL (2002), Statistica Inference, second edition, Duxbury Press, Caifornia. Durrett R (1996), Probabiity: Theory and Exampes, Second Edition, Wadsworth, Bemont. Dwyer D (2006), The Distribution of Defauts and Bayesian Mode Vaidation, Working Paper, Moody s KMV. Gordy M (2003), A Risk-Factor Mode Foundation for Ratings-Based Bank Capita Rues. Journa of Financia Intermediation 12 (3), pp. 199 232. Hinderer K (1980), Grundbegriffe der Wahrscheinichkeitstheorie. Zweiter korrigierter Nachdruck der ersten Aufage, Springer, Berin. Jafry Y, Schuermann T (2004), Measurement, Estimation, and Comparison of Credit Migration Matrices, Journa of Banking & Finance 28, pp. 2603 2639. Kiefer NM (2009) Defaut Estimation for Low-Defaut Portfoios, Journa of Empirica Finance 16, pp. 164 173.

5 Estimating Probabiities of Defaut for Low Defaut Portfoios 101 R Deveopment Core Team (2003), R: A Language and Environment for Statistica Computing, R Foundation for Statistica Computing, Vienna. http://www.r-project.org Schuermann T, Hanson S (2004), Estimating Probabiities of Defaut, Staff Report no.190, Federa Reserve Bank of New York. Tasche D (2009), Estimating Discriminatory Power and PD Curves when the Number of Defauts is Sma, Working Paper. Vasicek O (1997), The Loan Loss Distribution. Working Paper, KMV Corporation.

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Chapter 6 Transition Matrices: Properties and Estimation Methods Bernd Engemann and Konstantin Ermakov 6.1 Introduction In Chaps. 1 3 estimation methods for 1-year defaut probabiities have been presented. In many risk management appications a 1-year defaut probabiity is not sufficient because muti-year defaut probabiities or defaut probabiities corresponding to year fractions are needed. Practica exampes in the context of retai oan pricing and risk management are presented in Chaps. 17 and 18. In other appications, ike credit risk modeing, rating transitions, i.e. the probabiity that a debtor in rating grade i moves to rating grade j within a period of time, are of importance. In a cases, a 1-year transition matrix serves as the starting point. In this chapter, we wi assume a rating system with n rating grades where the n-th grade is the defaut grade. A 1-year transition matrix is a nxnmatrix with the probabiities that a debtor in rating grade i migrates to rating grade j within 1 year. We start with exporing the properties of transition matrices. Under the assumption that rating transitions are Markovian, i.e. that rating transitions have no memory, and that transition probabiities are time-homogeneous it is possibe to compute transition matrices for arbitrary time periods. We wi show the formuas for this cacuation in detai. These concepts wi be iustrated with a numerica exampe where a 6-month transition matrix is computed. We wi see from this exampe that a straightforward appication of the formuas for computing transition matrices for arbitrary time frames can ead to impausibe resuts. We wi aso see that this is the case for most practica exampes. To make the cacuation of arbitrary transition matrices work in practice, a reguarization agorithm has to be appied to the origina 1-year transition B. Engemann (*) Independent Consutant e-mai: bernd.engemann@quantsoutions.de K. Ermakov Independent Consutant e-mai: konstantin@ermakov.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_6, # Springer-Verag Berin Heideberg 2011 103

104 B. Engemann and K. Ermakov matrix. A number of reguarization agorithms exist in the iterature. We wi present one of them that is easy to impement and deivers reasonabe resuts. After that, two different estimation methods for transition matrices are presented, the cohort method and the duration method. Whie the cohort method directy estimates a 1-year transition matrix, the duration matrix estimates the generator of the transition matrix, i.e. its matrix ogarithm. Whie in the iterature it is occasionay caimed that the duration method offers an advantage over the cohort method (Jafry and Schuermann 2004), we wi show using a simpe simuation study that this is not the case. 6.2 Properties of Transition Matrices A 1-year transition matrix P is a matrix of the form 0 1 p 1;1 p 1;2... p 1;n. P ¼....... B C @ p n 1;1 p n 1;2 p n 1;n A 0 0 0 1 (6.1) where p i,j is the probabiity that a debtor migrates from rating grade i to grade j within 1 year. The fina grade n is the defaut state which is absorbing, i.e. once a debtor has defauted he cannot migrate back to an aive state but wi stay in the defaut state forever. A transition matrix P is characterized by the four properties: A entries are probabiities, i.e. 0 p i,j 1, i, j ¼ 1,..., n. The sum of the entries of each row is one P n j¼1 p i;j ¼ 1. The most right entry of each row p i,n is the defaut probabiity of rating grade i. The defaut grade is absorbing, p n,j ¼ 0, j < n, p n,n ¼ 1. The second property can aso be interpreted intuitivey. If a debtor is in rating grade i at the beginning of a period he must be either sti in rating grade i, orin some other rating grade, or in defaut at the end of the period. Therefore, a row probabiities have to sum to one. In practice it can happen that a debtor disappears from the data sampe because it is no onger rated. This is not considered in a modeing approach. Typicay these cases are excuded from the data sampe or an additiona rating grade Non-rated is introduced to measure the proportion of annua transitions into this cass. However, when the transition matrix is used in a practica appication the Nonrated grade has to be removed and the transition probabiities have to be rescaed to sum to one. Typicay transition matrices refer to a time period of 1 year. In severa risk management appications muti-year defaut probabiities are needed. If we assume

6 Transition Matrices: Properties and Estimation Methods 105 that the process of rating transitions is stationary and Markovian, it is possibe to compute muti-year transition matrices. The first property means that the probabiity for a migration from i to j depends on the ength of the observation period ony, not on its starting point in time. A transition matrix describing rating migrations from 1/1/2010 to 1/1/2011 and a matrix corresponding to the time period from 1/1/2012 to 1/1/2013 are identica. Rating transitions are caed Markovian if the migration probabiities depend ony on the current rating of a debtor, but not on the rating path a debtor has passed through during the past years. Both properties of rating processes are questionabe from an empirica point of view but ead to very convenient mathematica structures. This wi be iustrated with a simpe exampe. Consider a rating system with two rating grades and a 1-year transition matrix Pð1Þ ¼ 0:95 0:05 0:00 1:00 We compute the 2-year transition matrix. If a debtor survives year one the transition matrix for year two is again equa to P because of the stationarity property. The possibe rating paths are iustrated in Fig. 6.1. Fig. 6.1 Possibe rating paths of a debtor in the simpe rating system Year 1 Year 2 0.95 0.05 0.95 0.05 A debtor in grade 1 can defaut after year 1, he can survive year one and defaut in year two, and he can survive both years. The sum of the first two paths eads to the 2-year defaut probabiity 0.05 + 0.05 0.95 ¼ 0.0975, whie the ast path eads to the 2-year surviva probabiity 0.95 0.95 ¼ 0.9025. This eads to the 2 year transition matrix Pð2Þ ¼ 0:9025 0:0975 0:00 1:00 A coser ook to the cacuations we have carried out reveas that the 2-year transition matrix is the resut of the mutipication of the 1-year transition matrix with itsef. 0:95 0:05 0:95 0:05 Pð1ÞPð1Þ ¼ 0:00 1:00 0:00 1:00 0:95 0:95 0:95 0:05 þ 1:00 0:05 0:9025 0:0975 ¼ ¼ 0:00 1:00 0:00 1:00 Therefore, arbitrary muti-year transition matrices can be computed by iterative mutipication of the 1-year transition matrix with itsef. Using this, defaut

106 B. Engemann and K. Ermakov probabiities for m years can be computed from the 1-year transition matrix. They can be read directy from the ast coumn of the m-year transition matrix. In some appications more genera transition matrices are needed, e.g. a transition matrix corresponding to a time period of 3 months. An exampe is the pricing of oans with embedded options which is described in Chap. 18. It is not possibe to compute transition matrices for arbitrary year fractions with the methods presented so far. Suppose we woud ike to compute a 6-months transition matrix. This is equivaent to computing a square root of the 1-year transition matrix because we know that a mutipication of the 6-months transition matrix with itsef must resut in the 1-year transition matrix. Therefore, we can write 1 Pð1Þ ¼Pð0:5ÞPð0:5Þ ¼ðPð0:5ÞÞ 2 p Pð0:5Þ ¼ ffiffiffiffiffiffiffiffiffi Pð1Þ ¼ Pð1Þ 0:5 ¼ exp og Pð1Þ 0:5 ¼ expð0:5 ogðpð1þþþ (6.2) In principe, (6.2) can be generaized to arbitrary year fractions t. If the ogarithm of the 1-year transition matrix woud be known arbitrary transition matrices coud be computed from the exponentia. It remains to expain how to compute a ogarithm and an exponentia of a matrix. Both functions are defined for an arbitrary matrix X by their Tayor series expansions expðxþ ¼I þ X þ 1 2 X2 þ 1 3! X3 þ... (6.3) ogðxþ ¼X I 1 2 ðx IÞ2 þ 1 3 ðx IÞ3... (6.4) where I is the identity matrix. Both series have to be evauated unti a reasonabe accuracy for the ogarithm and the exponentia is achieved. As an exampe, we compute the 6-months transition matrix of the 1-year matrix M given in Fig. 6.2. The matrix is based on Moody s average 1-year etter rating from 1920 to 2007 (Moody s 2008). In the origina document Moody s (2008), the fraction of companies that migrated into the without rating state is reported. To get the matrix M in Fig. 6.2, this category has to be removed and a probabiities have to be rescaed that each row sums to one. The matrix has nine rating grades where the ninth grade is the defaut grade. To compute the 6-months transition matrix, the ogarithm of M has to be computed using (6.4) as a first step. The resut is given in Fig. 6.3. For this cacuation we have used 50 terms in the Tayor expansion (6.4). Finay, this matrix has to be mutipied with 0.5 and the exponentia of the resuting matrix has to be computed using (6.3). This eads to the 6-months 1 Note that by og(x) we mean the inverse of exp(x), not the ogarithm to the base ten.

6 Transition Matrices: Properties and Estimation Methods 107 1 2 3 4 5 6 7 8 D 1 2 3 4 5 6 7 8 D 0.911200 0.078020 0.008779 0.001743 0.000251 0.000010 0.000000 0.000000 0.000000 0.013430 0.907400 0.068850 0.007316 0.001864 0.000394 0.000021 0.000043 0.000671 0.000859 0.031110 0.902300 0.056180 0.007349 0.001145 0.000202 0.000085 0.000806 0.000454 0.003170 0.049960 0.877800 0.055250 0.008395 0.001623 0.000173 0.003170 0.000079 0.000921 0.005346 0.066460 0.827100 0.078360 0.006256 0.000573 0.014870 0.000080 0.000613 0.001965 0.007155 0.071460 0.811600 0.056910 0.005702 0.044490 0.000000 0.000317 0.000418 0.002442 0.010240 0.100800 0.709900 0.040120 0.135700 0.000000 0.000000 0.001338 0.000000 0.005466 0.037360 0.088770 0.637900 0.229100 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Fig. 6.2 Moody s average 1-year etter rating migrations from 1920 to 2007 1 2 3 4 5 6 7 8 D 1 2 3 4 5 6 7 8 D 0.093630 0.085740 0.006387 0.001426 0.000132 0.000022 0.000002 0.000003 0.000033 0.014750 0.099110 0.075980 0.005729 0.001653 0.000312 0.000008 0.000050 0.000642 0.000680 0.034320 0.105900 0.062890 0.006411 0.000739 0.000135 0.000095 0.000657 0.000463 0.002540 0.056010 0.134600 0.064520 0.006779 0.001563 0.000137 0.002605 0.000059 0.000838 0.003879 0.077820 0.196500 0.095460 0.004454 0.000295 0.013730 0.000083 0.000618 0.001888 0.004980 0.087120 0.217800 0.074710 0.005705 0.042700 0.000009 0.000340 0.000233 0.002322 0.007298 0.130600 0.351500 0.059400 0.150200 0.000002 0.000071 0.001691 0.000595 0.004852 0.043130 0.130600 0.453700 0.274200 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Fig. 6.3 Logarithm of the matrix M of Fig. 6.2 1 2 3 4 5 6 7 8 D 1 2 3 4 5 6 7 8 D 0.954400 0.040890 0.003824 0.000789 0.000096 0.000003 0.000001 0.000001 0.000008 0.007036 0.952100 0.036150 0.003286 0.000878 0.000176 0.000004 0.000023 0.000328 0.000387 0.016330 0.949100 0.029710 0.003462 0.000472 0.000084 0.000045 0.000365 0.000229 0.001438 0.026440 0.935900 0.029830 0.003840 0.000793 0.000078 0.001444 0.000035 0.000439 0.002336 0.035930 0.907900 0.043200 0.002750 0.000222 0.007154 0.000040 0.000307 0.000959 0.003091 0.039410 0.898900 0.032560 0.002887 0.021860 0.000002 0.000163 0.000166 0.001182 0.004503 0.057570 0.840700 0.024390 0.071340 0.000001 0.000016 0.000749 0.000139 0.002598 0.020130 0.053770 0.797800 0.125100 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Fig. 6.4 Six-months transition matrix corresponding to the matrix M of Fig. 6.2 transition matrix given in Fig. 6.4. Again 50 terms of the Tayor expansion (6.3) are used. We find that the 6-month transition matrix does not fufi a the necessary properties of transition matrices because it contains negative probabiities. The transition probabiities between ow grades and high grades are very sma but

108 B. Engemann and K. Ermakov negative numbers. At a first gance, one might suppose that the sma negative numbers are the resut of a numerica instabiity or inaccuracy in the evauation of (6.3) and (6.4). However, this is not the case. There is an economic reason why it is impossibe to compute a meaningfu 6-month transition matrix from the matrix M. In Fig. 6.2 we see that the matrix M contains severa transition probabiities equa to zero. In the data sampe no transitions from rating grade 1 to a grade worse than 6 have been observed. Simiary, no rating improvement from grade 7 directy to grade 1 has been reported. However, it is possibe to migrate within 1 year, for instance, from grade 1 to grade 4. For reasons of consistency, a migration probabiities that are zero in the 1-year matrix have to be zero in the 6-months matrix. The same is true for the positive probabiities. Under these restrictions, however, it is impossibe to compute a vaid 6-months matrix. In a 6-months matrix a transition from grade 1 to grade 4 must have a positive probabiity and a transition from grade 4 to grade 7 must have a positive probabiity. This impies that the 1-year transition probabiity from grade 1 to grade 7 must be positive because a debtor can migrate in 6 months from grade 1 to grade 4 and in the foowing 6 months from grade 4 to grade 7. In the matrix M a migration from grade 1 to grade 7 has a probabiity of zero which is a contradiction. From this exampe, we see that whenever a 1-year transition matrix contains zero entries there is no vaid transition matrix for time periods beow 1 year. 2 From the theory of Markov chains it is known that transition matrices for arbitrary time periods can be computed if the ogarithm of the 1-year transition matrix resuts in a generator matrix. A matrix G ¼ (g i,j ) i,j¼1,..., n is caed a generator matrix if it has the three properties: A diagona entries are not positive, g i,i 0, i ¼ 1,..., n. A other entries are not negative, g i,j 0, i, j ¼ 1,..., n and i 6¼ j. A row sums are zero P n j¼1 g i;j ¼ 0, i ¼ 1,..., n. From the generator matrix, an arbitrary transition matrix P(t) corresponding to a time period t is computed as PðtÞ ¼expðt GÞ From Fig. 6.3 we see that the ogarithm of the matrix M is no generator matrix. Some off-diagona entries corresponding to rating transitions from very high to very ow grades and from very ow to very high grades are negative. However, the absoute vaue of these negative numbers is sma compared to the remaining entries of the matrix. 2 This is not true in genera because there are cases where it is sti possibe to compute transition matrices for arbitrary time periods if the 1-year matrix contains zeros. The simpest exampe is the identity matrix. However, basicay for a practicay reevant cases it is true that no consistent t-year transition matrix can be computed from the one-year matrix where t is an arbitrary year fraction.

6 Transition Matrices: Properties and Estimation Methods 109 One idea to sove the probems associated with the non-existence of a generator matrix is repacing the ogarithm of the 1-year transition matrix by a generator matrix that is cose to the ogarithm matrix. Repacing the ogarithm of the 1-year matrix by a simiar matrix which has the properties of a generator matrix, or equivaenty, repacing the origina 1-year transition matrix by a simiar transition matrix that aows the cacuation of a generator matrix, is caed reguarization. Severa suggestions for reguarization agorithms have been made in the iterature. Exampes are Israe et a. (2001) and Kreinin and Sidenikova (2001). A very simpe reguarization agorithm is proposed by Kreinin and Sidenikova (2001). It can be summarized by three steps: 1. Compute the ogarithm of M, G ¼ og(m). 2. Repace a negative non-diagona entries of G by zero. 3. Adjust a non-zero eements of G by: g i;j P n i¼1 g i;j jg i;j j g i;j P n i¼1 jg i;jj It is easy to check that the resuting matrix of the above reguarization agorithm indeed fufis a properties of a generator matrix. In our exampe, we have seen that the cacuation of the ogarithm of the matrix M of Fig. 6.2 does not ead to a generator matrix. Appying the Steps 2 and 3 of the above reguarization agorithm to the matrix of Fig. 6.3 eads to the generator matrix in Fig. 6.5 beow. From this generator matrix, we can compute the 6-months transition matrix again. The resut is presented in Fig. 6.6. We see that now the resuting matrix is indeed a transition matrix. A entries are rea probabiities taking vaues inside the interva [0, 1]. Finay, we recomputed the 1-year transition matrix from the generator matrix of Fig. 6.5 by appying the exponentia function to get an impression how far the origina data have been changed by the reguarization agorithm. The resut is shown in Fig. 6.7. Comparing Figs. 6.2 and 6.7 we see that the reguarization 1 2 3 4 5 6 7 8 D 1 2 3 4 5 6 7 8 D 0.093660 0.085720 0.006385 0.001426 0.000132 0.000000 0.000000 0.000000 0.000000 0.014750 0.099110 0.075980 0.005728 0.001653 0.000312 0.000000 0.000050 0.000642 0.000680 0.034320 0.105900 0.062890 0.006411 0.000739 0.000135 0.000095 0.000657 0.000463 0.002540 0.056010 0.134600 0.064520 0.006779 0.001563 0.000137 0.002605 0.000059 0.000838 0.003879 0.077820 0.196500 0.095460 0.004454 0.000295 0.013730 0.000083 0.000618 0.001888 0.004980 0.087120 0.217800 0.074710 0.005705 0.042700 0.000000 0.000340 0.000233 0.002322 0.007297 0.131800 0.351500 0.059400 0.150200 0.000000 0.000000 0.001690 0.000000 0.004849 0.043100 0.130500 0.454100 0.274000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Fig. 6.5 Reguarization of the matrix og(m) of Fig. 6.3

110 B. Engemann and K. Ermakov 1 2 3 4 5 6 7 8 D 1 2 3 4 5 6 7 8 D 0.954400 0.040870 0.003823 0.000789 0.000096 0.000007 0.000001 0.000001 0.007036 0.952100 0.036150 0.003286 0.000878 0.000176 0.000007 0.000023 0.000387 0.016330 0.949100 0.029710 0.003462 0.000472 0.000084 0.000045 0.000229 0.001438 0.026440 0.935900 0.029830 0.003840 0.000793 0.000078 0.000035 0.000439 0.002336 0.035930 0.907900 0.043200 0.002750 0.000222 0.000041 0.000308 0.000959 0.003092 0.039410 0.898900 0.032560 0.002887 0.000002 0.000164 0.000166 0.001186 0.004503 0.057570 0.840700 0.024380 0.000001 0.000015 0.000752 0.000118 0.002600 0.020110 0.053730 0.797700 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000008 0.000328 0.000365 0.001444 0.007154 0.021860 0.071340 0.125000 1.000000 Fig. 6.6 Six-months transition matrix computed from the generator matrix of Fig. 6.5 1 2 3 4 5 6 7 8 D 1 2 3 4 5 6 7 8 D 0.911200 0.077990 0.008776 0.001743 0.000251 0.000029 0.000003 0.000002 0.013430 0.907400 0.068850 0.007316 0.001864 0.000395 0.000027 0.000043 0.000859 0.031110 0.902300 0.056180 0.007349 0.001145 0.000202 0.000085 0.000454 0.003170 0.049960 0.877800 0.055250 0.008395 0.001623 0.000173 0.000079 0.000921 0.005346 0.066460 0.827100 0.078360 0.006256 0.000573 0.000080 0.000614 0.001965 0.007157 0.071460 0.811600 0.056910 0.005701 0.000008 0.000319 0.000419 0.002455 0.010240 0.100800 0.709900 0.040120 0.000003 0.000055 0.001352 0.000446 0.005476 0.037330 0.088690 0.637700 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000033 0.000672 0.000806 0.003170 0.014870 0.044490 0.135700 0.228900 1.000000 Fig. 6.7 One-year transition matrix computed from the generator matrix of Fig. 6.5 agorithm had a very mid infuence on the input data ony. The changes in the origina data are we beow typica statistica errors when transition matrices are estimated. It basicay repaces the zero transition probabiities by very sma positive probabiities and adjusts the remaining entries to make sure that a row entries sum to one. We remark that there might be situations where the reguarization agorithm s infuence on the origina data is much arger. Especiay it might change the 1-year defaut probabiities what is unwanted because they are typicay tied to a master scae that is used in many appications of a bank. Therefore, when computing a generator matrix by a reguarization agorithm, an additiona requirement might be to keep the defaut probabiities unchanged. This can be obtained by adding a fourth step to the above reguarization agorithm. It is a property of generator matrices that if a generator matrix G is mutipied with a diagona matrix D from the eft, then the matrix product DG is sti a generator matrix. Therefore, the defaut probabiities can be eft unchanged by finding an appropriate matrix D using some optimization agorithm. A good reference on transition matrices and their generators for further reading is Buhm et a. (2003).

6 Transition Matrices: Properties and Estimation Methods 111 6.3 Estimation of Transition Matrices Having discussed the mathematica properties of transition matrices in the ast section, the focus in this section is on the estimation of 1-year transition matrices. A good reference on the estimation of 1-year transition matrices is Jafry and Schuermann (2004). There are two simpe methods of estimating a 1-year transition matrix, the cohort method and the duration method. The cohort method directy estimates a 1-year transition matrix. In practice this might ead to transition matrices containing zero probabiities what makes the direct cacuation of a generator matrix infeasibe and the appication of a reguarization agorithm necessary. To avoid the need for a reguarization agorithm for cacuating the generator matrix, it is aso possibe to estimate the generator matrix directy. This is done by the duration method. To expain both estimation techniques, we assume that we have a data sampe avaiabe that contains a portfoio of firms and the rating history of each firm, i.e. the dates where upgrades or downgrades have occurred are stored in a data base. An excerpt of the data sampe is iustrated in Fig. 6.8. In the data history certain reference dates have to be defined that are used to define transition periods. For the estimation of a 1-year transition matrix, the ength of these periods is equa to 1 year. For each firm in the sampe and each time period, the rating at the period s start is observed and the rating at the period s end. This defines one empirica rating transition. We iustrate the concept with some exampes in Fig. 6.8. Firm 2 is at Y1 in rating grade 2 and at Y2 in rating grade 3. Therefore, this is an observation of a rating transition from grade 2 to grade 3. In the remaining time, Firm 2 stays in rating grade 3, i.e. from Y2 to Y3 and from Y3 to Y4 Firm 2 contributes to observations of firms staying in rating grade 3. The treatment of Firm 1 4 2 2 Firm 2 Firm 3 2 6 3 3 6 Firm 4 4 6 Firm 5 Firm 6 Firm 7 3 1 7 2 3 2 4 4 2 2 0 Y1 Y2 Y3 Y4 Fig. 6.8 Rating transitions in a hypothetica data sampe

112 B. Engemann and K. Ermakov rating transitions during the year as in the case of firm 5 between Y2 and Y3 depends on the specific estimation method. Both estimation techniques, the cohort method and the duration method, are the resut of the maximum ikeihood estimation principe. The transition matrix (or the generator matrix in case of the duration method) is estimated to maximize the probabiity associated with the empirica data sampe. In the case of the cohort method, the transition probabiity from grade i to grade j is estimated as ^p i;j ¼ N i;j N i ; (6.5) where N i is the number of debtors in rating grade i at the beginning of each time period and N i,j is the number of rating transitions from rating grade i to rating grade j that are observed during the time period. To carify this concept, we consider again Fig. 6.8. In this data sampe Firm 5 is downgraded from grade 2 to grade 3 between Y2 and Y3 and shorty after the downgrade the company is upgraded again to grade 2. In the estimation (6.5) the period from Y2 to Y3 for Firm 5 is counted as an observation of a firm that stays in rating grade 2 during this time interva, i.e. intermediate observations during the year are ignored by the cohort method. The duration method is different in this respect. In this estimation method a rating transitions are used in the estimator. The estimator for the generator matrix G is given by ^g i;j ¼ K i;jðtþ R T 0 K iðsþds ; (6.6) where K i,j is the number of a transitions from rating grade i to rating grade j in the data sampe, T is the ength of data set s time horizon, and K i (s) is the number of firms in rating grade i at time s. In contrast to the cohort method, for the duration method the spitting of the time frame into 1-year periods in Fig. 6.8 is not necessary. One simpy has to count a transitions in the data sampe and approximate the integra in (6.6) by counting a firms in each rating grade at a given time grid and use the resut for cacuating the integra. In the iterature (e.g. Jafry and Schuermann, 2004), it is often considered as an advantage of the duration method over the cohort method that a transitions in the data sampe, aso the transitions during the year, can be used in the estimator. In a simpe simuation study we woud ike to measure this advantage. We use the transition matrix of Fig. 6.7 as a starting point. The basic idea is to simuate rating paths from this matrix, estimate the 1-year transition matrix from the simuated rating paths, and measure the estimation error using some matrix norm. Since the estimation resut is known (the transition matrix of Fig. 6.7) we can measure the estimation error exacty and can compare the accuracy of the cohort method with the accuracy of the duration method. Note, since the duration method estimates the generator matrix, we have to compute the 1-year transition matrix from the generator matrix before computing the estimation error.

6 Transition Matrices: Properties and Estimation Methods 113 We expain how rating paths are simuated. First, we have to define a time grid 0 ¼ t 0, t 1,..., t s where ratings shoud be observed. We wi use a homogeneous time grid t v ¼ v Dt, v ¼ 0,..., s, Dt ¼ t s /s. Since each time interva is identica, it is sufficient to compute the transition matrix P(Dt) ¼ (p i,j (Dt)) corresponding to the time ength Dt. To simuate a rating path, the foowing steps have to be carried out: 1. Definition of the initia rating k 2. Simuation of an uniformy distributed random number u on [0, 1] 3. Finding the index with P 1 j¼1 p k;jðdtþ u P j¼1 p k;jðdtþ 4. Setting the rating of the next time point to k ¼ 5. Repeating the steps 2 4 unti a rating is assigned to a time points We simuate the rating paths for a portfoio of firms and end up with a data sampe simiar to the iustration in Fig. 6.8. Finay, we have to define the matrix norm that we use to measure the estimation error. If P ¼ (p i,j ) and Q ¼ (q i,j ) are two matrices, we define the difference of these two matrices by the matrix norm kp Qk ¼ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n X n n p 2 2 i;j q i;j : (6.7) i¼1 To compare the two estimation methods, cohort method and duration method, we carry out the foowing steps: 1. Definition of a portfoio, i.e. definition of the tota number of firms N and the rating decomposition N i 2. Definition of a time grid t v where ratings shoud be observed 3. Simuation of a rating path for each firm 4. Estimation of the 1-year transition matrix using the cohort method and estimation of the generator matrix using the duration method together with the cacuation of the 1-year transition matrix from the resut 5. Cacuation of the estimation error for both methods using (6.7) 6. Carrying out the simuation for severa times and cacuating average estimation errors By varying the portfoio size N we can check the dependence of the estimation quaity of transition matrices on portfoio size. Further, by refining the time grid we can measure the advantage of the duration method over the cohort method if there is any. We expect that the duration method is the more accurate the more frequenty the firm ratings are observed. We have used portfoios with 1,000, 5,000, 10,000, 25,000, 50,000, and 100,000 debtors in the first eight rating grades and no debtors in the defaut grade. We have simuated rating paths over a time interva of 3 years and we have used six different observation frequencies for the rating, annuay, semi-annuay, quartery, monthy, weeky, and every 2 days. Our expectation is that the duration method wi be the more efficient the more ratings are observed during the year. To measure the estimation error we have carried out 50 simuations for each combination of portfoio j¼1

114 B. Engemann and K. Ermakov size and observation frequency, computed the estimation error in each simuation scenario from (6.7) and averaged over the 50 scenarios. The resuts are reported in Tabes 6.1 6.6. Tabe 6.1 Average estimation errors for annuay observation frequency Tabe 6.2 Average estimation errors for semiannuay observation frequency #Debtors Error cohort method Error duration method 1,000 0.000394 0.001414 5,000 0.000165 0.001393 10,000 0.000119 0.001390 25,000 0.000076 0.001387 50,000 0.000050 0.001387 100,000 0.000037 0.001388 #Debtors Error cohort method Error duration method 1,000 0.000377 0.000784 5,000 0.000181 0.000750 10,000 0.000123 0.000735 25,000 0.000076 0.000729 50,000 0.000053 0.000734 100,000 0.000037 0.000733 Tabe 6.3 Average estimation errors for quartery observation frequency #Debtors Error cohort method Error duration method 1,000 0.000357 0.000484 5,000 0.000171 0.000396 10,000 0.000119 0.000386 25,000 0.000076 0.000386 50,000 0.000053 0.000375 100,000 0.000039 0.000376 Tabe 6.4 Average estimation errors for monthy observation frequency #Debtors Error cohort method Error duration method 1,000 0.000367 0.000348 5,000 0.000166 0.000186 10,000 0.000113 0.000163 25,000 0.000075 0.000141 50,000 0.000053 0.000133 100,000 0.000037 0.000130 Tabe 6.5 Average estimation errors for weeky observation frequency #Debtors Error cohort method Error duration method 1,000 0.000366 0.000335 5,000 0.000163 0.000149 10,000 0.000126 0.000115 25,000 0.000079 0.000076 50,000 0.000051 0.000054 100,000 0.000038 0.000045

6 Transition Matrices: Properties and Estimation Methods 115 Tabe 6.6 Average estimation errors for bi-daiy observation frequency #Debtors Error cohort method Error duration method 1,000 0.000383 0.000354 5,000 0.000176 0.000159 10,000 0.000121 0.000111 25,000 0.000072 0.000064 50,000 0.000054 0.000051 100,000 0.000037 0.000035 We see that the average estimation error of the cohort method converges to zero with increasing portfoio size. We aso observe that this is not the case for the duration method uness the observation frequency is arge. If ratings are observed annuay the duration method contains a substantia estimation bias that cannot be reduced by increasing the portfoio size. To reduce the bias in the duration method at east weeky observations of ratings are required, a condition hardy met in practice. The reason for the poor performance of the duration method is that the theory behind this estimator reies on continuous rating paths. Our simuations have shown that vioating this continuity conditions introduces a simuation bias that can be substantia. Therefore, we recommend using the cohort method in practice because we do not trust a method that does not converge under practicay reevant observation frequencies. 3 We remark that in this artice we have presented the theory and the estimation of transition matrices assuming Markovian rating transitions and time-homogeneous transition probabiities. There has been research recenty on reaxing one of these assumptions or both in modeing rating transitions. An exampe is Buhm and Overbeck (2007). In some appications muti-year defaut probabiities are known in addition to a 1-year transition matrix. They show how removing the timehomogeneity assumption can ead to a satisfactory modeing of rating transitions in this situation. References Buhm C, Overbeck L (2007), Caibration of PD Term Structures: To be Markov or not to be, Risk 20 (11), pp. 98 103. Buhm C, Overbeck L, Wagner C (2003), An Introduction to Credit Risk Modeing, Chapman & Ha/CRC, Boca Raton. 3 Note that our concusion is contrary to Jafry and Schuermann (2004). In their anaysis they end up recommending the duration method. However, it is not so cear in their artice why the duration method shoud be superior. They basicay show that in practica appications ike portfoio credit risk modeing it makes a considerabe difference if one uses a transition matrix based on the cohort method or based on the duration method. They do not anayze, however, if this difference comes from an estimation bias in the resuts for the duration method which is our conjecture.

116 B. Engemann and K. Ermakov Israe RB, Rosentha JS, Wei JZ (2001), Finding Generators for Markov Chains via Empirica Transition Matrices, with Appications to Credit Ratings, Mathematica Finance 11, pp. 245 265. Jafry Y, Schuermann T (2004), Measurement and Estimation of Credit Migration Matrices, Journa of Banking and Finance 28 (11), pp. 2603 2639. Kreinin A, Sidenikova M (2001), Reguarization Agorithms for Transition Matrices, Ago Research Quartery 4, pp. 23 40. Moody s (2008), Corporate Defauts and Recovery Rates 1920 2007.

Chapter 7 A Muti-factor Approach for Systematic Defaut and Recovery Risk Danie R osch and Harad Scheue 7.1 Modeing Defaut and Recovery Risk Banks face the chaenge of forecasting osses and oss distributions in reation to their credit risk exposures. Most banks choose a moduar approach in ine with the current proposas of the Base Committee on Banking Supervision (2004), where seected risk parameters such as defaut probabiities, exposures at defaut and recoveries given defaut are modeed independenty. However, the assumption of independence is questionabe. Previous studies have shown that defaut probabiities and recovery rates given defaut are negativey correated [Carey (1998), Hu and Perraudin (2002), Frye (2003), Atman et a. (2005), or Cantor and Varma (2005)]. A faiure to take these dependencies into account wi ead to incorrect forecasts of the oss distribution and the derived capita aocation. This paper extends a mode introduced by Frye (2000). Modifications of the approach can be found in Pykhtin (2003) and D umann and Trapp (2004). Our contribution is origina with regard to the foowing three aspects. First, we deveop a theoretica mode for the defaut probabiities and recovery rates and show how to combine observabe information with random risk factors. In comparison to the above mentioned modes, our approach expains the defaut and the recovery rate by risk factors which can be observed at the time of the risk assessment. According to the current Base proposa, banks can opt to provide their own recovery rate forecasts for the reguatory capita cacuation. Thus, there is an immediate industry need for modeing. This artice originay appeared in the September 2005 issue of The Journa of Fixed Income and is reprinted with permission from Institutiona Investor, Inc. For more information pease visit http:// www.iijournas.com. D. R osch (*) University of Hannover e-mai: danie.roesch@finance.uni-hannover.de H. Scheue University of Mebourne e-mai: hscheue@unimeb.edu.au B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_7, # Springer-Verag Berin Heideberg 2011 117

118 D. R osch and H. Scheue Second, we show a framework for estimating the joint processes of a variabes in the mode. Particuary, the simutaneous mode aows the measurement of the correation between the defauts and recoveries given the information. In this mode, statistica tests for the variabes and correations can easiy be conducted. An empirica study reveas additiona evidence on the correations between risk drivers of defaut and recovery. Cantor and Varma (2005) anayze the same dataset and identify seniority and security as the main risk factors expaining recovery rates. This paper extends their approach by deveoping a framework for modeing correations between factor-based modes for defaut and recovery rates. Third, the impications of our resuts on economic and reguatory capita are shown. Note that according to the current proposas of the Base Committee, ony the forecast defaut probabiities and recovery rates but no correation estimates, enter the cacuation of the atter. We demonstrate the effects of spuriousy negecting correations in practica appications. The rest of the paper is organized as foows. The theoretica framework is introduced in the second section ( Mode and Estimation ) for a mode using historic averages as forecasts and a mode taking time-varying risk factors into account. The third section ( Data and Resuts ) incudes an empirica anaysis based on defaut and recovery rates pubished by Moody s rating agency and macroeconomic indices from the Conference Board. Section four ( Impications for Economic and Reguatory Capita ) shows the impications of the different modes on the economic capita derived from the oss distribution and the reguatory capita proposed by the Base Committee. Section five ( Discussion ) concudes with a summary and discussion of the findings. 7.2 Mode and Estimation 7.2.1 The Mode for the Defaut Process Our basic framework foows the approach taken by Frye (2000) and Gordy (2003). We assume that n t firms of one risk segment are observed during the time periods t (t ¼ 1,..., T). For simpicity, these firms are assumed to be homogenous with regard to the reevant parameters and a atent variabe describes each obigor i s (i ¼ 1,..., n t ) credit quaity p S it ¼ w F t þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w 2 U it (7.1) (w 2 [0,1]). F t ~ N(0,1) and U t ~ N(0,1) are independent systematic and idiosyncratic standard normay distributed risk factors. The Gaussian random variabe S it may be interpreted as the return on a firm s assets and therefore w 2 is often caed asset correation. A defaut event occurs if the atent variabe crosses a threshod c S it < c (7.2)

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 119 which happens with probabiity p ¼ F(c) where F(.) is the standard norma cumuative density function. If an obigor is in defaut, the indicator variabe D it equas one and zero otherwise: 8 < 1 obigor i defauts in period t D it ¼ : (7.3) : 0 ese Conditiona on the reaization f t of the systematic risk factor, defaut events are assumed to be independent between obigors, i.e., each firm defauts with the conditiona defaut probabiity pðf t Þ ¼ PD ð it ¼ 1jF t ¼ f t Þ ¼ F c w f t pffiffiffiffiffiffiffiffiffiffiffiffiffi : (7.4) 1 w 2 7.2.2 The Mode for the Recovery In modeing the recovery rate R it of a defauted obigor, we foow Sch onbucher (2003) and D umann and Trapp (2004) and use a ogistic norma process: R it ¼ exp ~Y it (7.5) 1 þ exp ~Y it with the transformed recovery rate ~Y it ¼ m þ b X t þ Z it (7.6) where X t ~N(0,1), Z it ~N(0,d 2 ) are independent systematic and idiosyncratic factors and m and b are parameters. These idiosyncratic factors are independent from the idiosyncratic factors which drive the atent defaut variabe. Compared to the norma distribution assumption for recovery rates Frye (2000), the chosen transformation has the advantage that recovery rates are bounded between 0 and 100%. Note that any other cumuative density function coud be used. As a matter of fact, we estimated modes using a standard norma transformation and received simiar resuts. If we observe a homogenous segment of borrowers, the transformed recovery rate is given by ~Y t ¼ 1 n t X n t i¼1 ~Y it ¼ m þ b X t þ 1 n t X n t i¼1 Z it (7.7) P n t with Z t ¼ n 1 t Z it which is normay distributed with mean zero and variance i¼1 d 2. The variance converges for arge nt to zero: n 2 t

120 D. R osch and H. Scheue! im Var 1 X nt Z it ¼ 0 (7.8) n t!1 n t i¼1 Therefore, we approximate the average transformed recovery rate by ~Y t Y t ¼ m þ b X t ; (7.9) which is driven ony by a systematic risk factor and normay distributed Y t ~N(m,b 2 ). The ink between the recovery and defaut process is introduced by modeing the dependence of the two systematic risk factors. Since both F t and X t are marginay norma distributed, we mode their dependence by assuming that they have a bivariate norma distribution with correation parameter r. Aternative, a copua which is different from the Gaussian coud have been assumed. It then foows that the average transformed recovery rate and atent defaut triggering variabe have a correation CorrðS it ; Y t Þ ¼ w r (7.10) The correation equas one in the specia case that a singe systematic factor drives both the defaut events as we as the recoveries given these events. 7.2.3 A Muti-factor Mode Extension So far, we presented a mode for systematic risk in defauts and recoveries where systematic risk is driven by common factors which are not directy observabe. These unobservabe factors induce uncertainties into the forecasts of oss distributions. The higher their impact, ceteris paribus, the more skewed the resuting distributions are and the higher key risk measures such as the Vaue-at-Risk or the Conditiona Vaue-at-Risk wi be. Since the true parameters of the modes are unknown, the severity of the impact must be estimated from observabe data. As an aternative to the modes above, we anayze a mode, which has aready been used in the context of defaut modeing. Exampes are R osch and Scheue (2004) and Hamere et a. (2003). These modes show that part of the cycica fuctuations in defaut rates can be attributed to observabe systematic risk factors. Once these factors are identified and incorporated into the mode, a arge part of uncertainty from unobservabe factors can be expained. These types of modes are aso exhibited in Heitfied (2005) and are reated to a concept broady known as a point-in-time approach because osses are forecast based on information on the prevaiing point of the business cyce. In our extension, it is assumed that the defaut threshod for the factor mode of the defaut process fuctuates over time. Aternativey, we coud introduce a factor mode with time-varying mean. This variation with time is introduced by K

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 121 observabe macroeconomic risk factors, such as GDP growth or interest rates. We assume that these state variabes are observed in prior time periods and denote them by z D t 1 ¼ zd t 1;1 ;...; zd t 1; K. As a resut, the conditiona defaut probabiity for each borrower within the risk segment is modified (compare R osch (2003) and Heitfied (2005) who additionay condition defaut probabiities on firm-specific factors): p z D t 1 ; f t ¼ PDit ¼ 1jz D t 1 ; f t g þ g0 zd ¼ F 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 1 w f p t 1 w 2 (7.11) where g ¼ (g 1,..., g K ) denotes a vector of exposures to the common observabe factors and g 0 is a constant. The mean of this conditiona defaut probabiity with respect to the unobservabe standard normay distributed factor f t is given by 1 ð p z D t 1 ¼ 1 p z D t 1 ; f t df ft ð Þ ¼ F c þ g 0 z D t 1 (7.12) In a simiar way, we assume that the mean of the og-transformed systematic recovery rate depends on common macroeconomic factors z R t 1 ¼ z R t 1;1 ;...; z R t 1;L. This vector may or may not contain factors which aso describe the defaut process: Y t ¼ b 0 þ b 0 z R t 1 þ b X t (7.13) where b ¼ ðb 1 ;...; b L Þ denotes a vector of exposures and b 0 the constant. If modes (12) and (13) hod, i.e., defauts and recoveries are driven by observabe agged systematic risk factors, it can be shown that their means are fuctuating with the change of the economy. Moreover, if these modes hod, then mode (4) and (9) with constant mean are misspecifications. Consequenty, fitting mode (4) and (9) to observabe data wi have the effect that a time variation is captured in the estimates of the exposures to the unobservabe random factors F t and X t.on the other hand, attributing time variation to observabe factors wi ead to ower parameter estimates for the infuences of the unobservabe factors, thereby reducing uncertainty with regard to the forecasts of the oss distributions. We wi demonstrate these effects on the economic and reguatory capita beow. 7.2.4 Mode Estimation Once the modes are specified, an agorithm for estimating the parameters from observabe data is needed. Foowing work by Frye (2000) we choose the Maximum- Likeihood method. Extending these studies, we suggest an ML-procedure which

122 D. R osch and H. Scheue aows the joint estimation of a coefficients, incuding those of modes (11) and (13) with observabe factors. Let us consider a reaization f t of the unobservabe random factor F t. Given this reaization the defaut events are independent and the number of defauts D t ¼ Pn t D it is conditionay binomia distributed with probabiity distribution i¼1 PD ð t ¼ d t j f t 8 < d t pðf Þ ¼ n t Þ d t ½1 pðf t ÞŠ n t d t d t ¼ 0; 1; :::; n t t : ese 0 (7.14) with p(f t )asin(7.4). Note that the transformed recovery rate can aso be modeed given a reaization f t. It hods that the random vector (F t,y t ) is normay distributed with F t N Y t 0 m ; 1 br br b 2 From the aw of conditiona expectation it foows that Y t has conditiona mean mðf t Þ ¼ EY ð t j f t Þ ¼ m þ b r f t (7.15) : and conditiona standard deviation p sðf t Þ ¼ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi VarðY t j f t Þ ¼ b 1 r 2 (7.16) Hence, the joint density g(.) of d t defauts and a transformed recovery rate g t given f t is simpy the product of the density of g t and the probabiity of d t, i.e., gd ð t ;y t j ¼ sðf t f t Þ ( ) 1 ½ pffiffiffiffiffiffiffiffi exp y t mðf t ÞŠ 2 Þ 2 p 2 ½sðf t ÞŠ 2 d t n t pðf t Þ d t ½1 pðf t ÞŠ n t d t (7.17) Note, g(.) depends on the unknown parameters of the defaut and the recovery process. Since the common factor is not observabe we estabish the unconditiona density gd ð t ;y t Þ ¼ ð1 1 sðf t ( ) 1 ½ pffiffiffiffiffiffiffiffi exp y t mðf t ÞŠ 2 Þ 2 p 2 ½sðf t ÞŠ 2 d t n t pðf t Þ d t ½1 pðf t ÞŠ n t d t dfðf t Þ (7.18)

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 123 Observing a time series with T periods eads to the fina unconditiona ogikeihood function ðm; b; c; w; rþ ¼ XT t¼1 nðgd ð t ; y t ÞÞ (7.19) This function is optimized with respect to the unknown parameters. In the appendix we demonstrate the performance of the approach by Monte-Caro simuations. For the second type of modes which incude macroeconomic risk factors, we repace pðf t Þ from (7.4) by p z D t 1 ; f t from (7.11) and m ð ft Þ from (7.15) by b 0 þ b 0 z R t 1 þ b r f t and obtain the og-ikeihood ðb 0 ; b; b; g 0 ; g; w; rþ. 7.3 Data and Resuts 7.3.1 The Data The empirica anaysis is based on the goba corporate issuer defaut rates and issue recovery rates (cf. Moody s 2005). In this data set, defaut rates are cacuated as the ratio of defauted and tota number of rated issuers for a given period. According to Moody s (2005), a defaut is recorded if Interest and/or principa payments are missed or deayed Chapter 11 or Chap. 7 bankruptcy is fied or Distressed exchange such as a reduction of the financia obigation occurs Most defauts are reated to pubicy traded debt issues. Therefore, Moody s defines a recovery rate as the ratio of the price of defauted debt obigations after 30 days of the occurrence of a defaut event and the par vaue. The recovery rates are pubished for different eves of seniority such as tota (Tota), senior secured (S_Sec), senior unsecured (S_Un), senior subordinated (S_Sub), subordinated (Sub) and junior subordinated debt. We excuded the debt category junior subordinated from the anaysis due to a high number of missing vaues. In addition, the composite indices pubished by The Conference Board (http:// www.tcb-indicators.org) were chosen as macroeconomic systematic risk drivers, i.e., the Index of four coincident indicators (COINC) which measures the heath of the U.S. economy. The index incudes the number of empoyees on non-agricutura payros, persona income ess transfer payments, index of industria production and manufacturing as we as trade saes.

124 D. R osch and H. Scheue Index of ten eading indicators (LEAD) which measures the future heath of the U.S. economy. The index incudes average weeky hours in manufacturing, average weeky initia caims for unempoyment insurance, manufacturers new orders of consumer goods and materias, vendor performance, manufacturers new orders of non-defence capita goods, buiding permits for new private housing units, stock price index, money suppy, interest rate spread of 10-year treasury bonds ess federa funds and consumer expectations. The indices are recognized as indicators for the U.S. business cyce. Note that for the anaysis, growth rates of the indices were cacuated and agged by 3 months. Due to a imited number of defauts in previous years, the compied data set was restricted to the period 1985 2004 and spit into an estimation sampe (1985 2003) and a forecast sampe (2004). Tabes 7.1 and 7.2 incude descriptive statistics and Bravais-Pearson correations for defaut rates, recovery rates and time agged macroeconomic indicators of the data set. Note that defaut rates are negativey correated with the recovery rates of different seniority casses and macroeconomic variabes. Figure 7.1 shows that both, the defaut and recovery rate fuctuate over time in opposite directions. This signas that defaut and recovery rates show a considerabe share of systematic risk which can be expained by time varying variabes. Figure 7.2 contains simiar graphs for the recovery rates of the different seniority casses. Note that the recovery rates increase with the seniority of a debt issue and show simiar patterns over time. This indicates that they may be driven by the same or simiar systematic risk factors. Tabe 7.1 Descriptive statistics of the variabes Variabe Mean Median Max. Min. Std. dev. Skew. Kurt. Defaut rate 0.0176 0.0144 0.0382 0.0052 0.0103 0.6849 2.2971 Recovery rate (Tota) 0.4208 0.4300 0.6170 0.2570 0.0902 0.2883 3.0464 Recovery rate (S_Sec) 0.5794 0.5725 0.8360 0.3570 0.1379 0.2631 2.0440 Recovery rate (S_Un) 0.4481 0.4450 0.6280 0.2310 0.1158 0.1816 2.2725 Recovery rate (S_Sub) 0.3703 0.3695 0.5190 0.2030 0.0984 0.1868 1.7668 Recovery rate (Sub) 0.2987 0.3245 0.4620 0.1230 0.1117 0.2227 1.7387 COINC 0.0215 0.0245 0.0409 0.0165 0.0160 0.9365 3.0335 LEAD 0.0130 0.0154 0.0336 0.0126 0.0151 0.4568 1.9154 Tabe 7.2 Bravais-Pearson correations of variabes Variabe Defaut rate Tota S_Sec S_Un S_Sub Sub COINC LEAD Defaut rate 1.00 0.67 0.72 0.72 0.53 0.34 0.75 0.47 Recovery rate (Tota) 1.00 0.78 0.68 0.72 0.29 0.32 0.54 Recovery rate (S_Sec) 1.00 0.66 0.48 0.37 0.33 0.55 Recovery rate (S_Un) 1.00 0.56 0.42 0.49 0.48 Recovery rate (S_Sub) 1.00 0.24 0.20 0.40 Recovery rate (Sub) 1.00 0.41 0.17 COINC 1.00 0.28 LEAD 1.00

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 125 0.05 0.70 0.04 0.60 Frequency defaut rate 0.04 0.03 0.03 0.02 0.02 0.01 0.50 0.40 0.30 0.20 Frequency recovery rate 0.01 0.10 0.00 1985 1990 1995 Year 2000 0.00 Defaut Rate Recovery Rate (Tota) Fig. 7.1 Moody s defaut rate vs. recovery rate 0.90 0.80 0.70 0.60 Frequency 0.50 0.40 0.30 0.20 0.10 0.00 1985 1990 1995 Year Recovery Rate (S_Sec) Recovery Rate (S_Sub) 2000 Recovery Rate (S_Un) Recovery Rate (Sub) Fig. 7.2 Moody s recovery rates by seniority cass Next to the business cyce and the seniority, it is pausibe to presume that recovery rates depend on the industry, the coatera type, the ega environment, defaut criteria as we as the credit quaity associated with an obigor. Tabes 7.3 and 7.4 show the recovery rates for different industries and issuer credit ratings

126 D. R osch and H. Scheue Tabe 7.3 Recovery rates for seected industries (Moody s 2004) Tabe 7.4 Recovery rates for seected issuer credit rating categories (Moody s 2005) Industry Recovery rate (1982 2003) Utiity-Gas 0.515 Oi 0.445 Hospitaity 0.425 Utiity-Eectric 0.414 Transport-Ocean 0.388 Media, broadcasting and cabe 0.382 Transport-surface 0.366 Finance and banking 0.363 Industria 0.354 Retai 0.344 Transport-Air 0.343 Automotive 0.334 Heathcare 0.327 Consumer goods 0.325 Construction 0.319 Technoogy 0.295 Rea estate 0.288 Stee 0.274 Teecommunications 0.232 Misceaneous 0.395 Issuer credit rating Recovery rate (1982 2004) Aa 0.954 A 0.498 Baa 0.433 Ba 0.407 B 0.384 Caa-Ca 0.364 (cf. Moody s 2004, 2005). Refer to these documents for a more detaied anaysis of the properties of recovery rates. 7.3.2 Estimation Resuts Based on the described data set, two modes were estimated: Mode without macroeconomic risk factors [(7.4) and (7.9)]: we refer to this mode as a through-the-cyce mode because the forecast defaut and recovery rate equa the historic average from 1985 to 2003 Mode with macroeconomic risk factors [(7.11) and (7.13)]: we refer to this mode as a point-in-time mode because the forecast defaut and recovery rates fuctuate over time Within the credit risk community, a discussion on the correct definition of a through-the-cyce and point-in-time mode exists, in which the present artice

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 127 does not intend to participate. We use these expressions as styized denominations, being aware that other interpretations of these rating phiosophies may exist (cf. Heitfied 2005). Due to the imitations of pubicy avaiabe data, we use Moody s goba defaut rates, tota recoveries, and recoveries by seniority cass. Tabe 7.5 shows the estimation resuts for the through-the-cyce mode (4) and (9) and Tabe 7.6 for the point-in-time mode (11) and (13) using the variabes COINC and LEAD as Tabe 7.5 Parameter estimation resuts for the through-the-cyce mode Parameter Tota S_Sec S_Un S_Sub Sub c 2.0942 *** 2.0951 *** 2.0966 *** 2.0942 *** 2.0940 *** (0.0545) (0.0550) (0.0546) (0.0544) (0.0549) w 0.2194 *** 0.2212 *** 0.2197 *** 0.2191 *** 0.2210 *** (0.0366) (0.0369) (0.0367) (0.0366) (0.0369) m 0.3650 *** 0.2976 ** 0.2347 * 0.5739 *** 0.8679 *** (0.0794) (0.1284) (0.1123) (0.0998) (0.1235) b 0.3462 *** 0.5598 *** 0.4898 *** 0.4351 *** 0.5384 *** (0.0562) (0.0908) (0.0795) (0.0706) (0.0873) r 0.6539 *** 0.7049 *** 0.7520 *** 0.5081 ** 0.3979 * (0.1413) (0.1286) (0.1091) (0.1799) (0.2013) Annua defaut and recovery data from 1985 to 2003 is used for estimation Standard errors are in parentheses *** Significant at 1% eve ** Significant at 5% eve * Significant at 10% eve Tabe 7.6 Parameter estimation resuts for the point-in-time mode Parameter Tota S_Sec S_Un S_Sub Sub g 0 1.9403 *** 1.9484 *** 1.9089 *** 1.9232 *** 1.9040 *** (0.0524) (0.05210) (0.0603) (0.05660) (0.0609) g 1 8.5211 *** 8.1786 *** 10.078 *** 9.2828 *** 10.134 *** (1.8571) (1.7964) (2.2618) (2.0736) (2.2884) COINC COINC COINC COINC COINC w 0.1473 *** 0.1522 *** 0.1485 *** 0.1483 *** 0.1508 *** (0.0278) (0.0286) (0.0276) (0.0277) (0.0279) b 0 0.4557 *** 0.1607 0.5576 *** 0.6621 *** 1.1883 *** (0.0867) (0.1382) (0.1635) (0.1194) (0.1845) b 1 7.4191 * 11.1867 * 15.0807 ** 7.2136 14.9625 ** (4.1423) (6.4208) (6.1142) (6.0595) (6.8940) LEAD LEAD COINC LEAD COINC b 0.3063 *** 0.4960 *** 0.4260 *** 0.4071 *** 0.4820 *** (0.0513) (0.0838) (0.0691) (0.0673) (0.0279) r 0.6642 *** 0.7346 *** 0.6675 *** 0.4903 ** 0.1033 (0.1715) (0.1520) (0.1481) (0.2088) (0.2454) Annua defaut and recovery data from 1985 to 2003 is used for estimation Standard errors are in parentheses *** Significant at 1% eve ** Significant at 5% eve * Significant at 10% eve

128 D. R osch and H. Scheue expanatory variabes. In the atter mode we choose both variabes due to their statistica significance. First, consider the through-the-cyce mode. Since we use the same defaut rates in each mode, the estimates for the defaut process are simiar across modes, and consistent to the ones found in other studies (compare Gordy (2000) or R osch 2005). The parameter estimates for the (transformed) recovery process refect estimates for the mean (transformed) recoveries and their fuctuations over time. Most important are the estimates for the correation of the two processes which are positive and simiar in size to the correations between defaut rates and recovery rates found in previous studies. Note that this is the correation between the systematic factor driving the atent defaut triggering variabe asset return S it and the systematic factor driving the recovery process. Therefore, higher asset returns (ower conditiona defaut probabiities) tend to come aong with higher recovery. A positive vaue of the correation indicates negative association between defauts and recoveries. The defaut rate decreases whie the recovery rate increases in boom years and vice versa in depression years. Next, consider the point-in-time mode. The defaut and the recovery process are driven by one macroeconomic variabe in each mode. The parameters of a macroeconomic variabes show a pausibe sign. The negative sign of the COINC index in the defaut process signas that a positive change of the index comes aong with subsequent ower number of defauts. The positive signs of the variabes in the recovery process indicate that higher recoveries foow a positive change in the variabe. In addition, most variabes are significant at the 10% eve. The ony exception is the parameter of the macroeconomic index LEAD for the senior subordinated recovery rate, which indicates ony a imited exposure to systematic risk drivers. Note that the infuence of the systematic random factor is reduced in each process by the incusion of the macroeconomic variabe. Whie we do not mean to interpret these indices as risk drivers themseves, but rather as proxies for the future state of the economy, these variabes are abe to expain part of the previousy unobservabe systematic risk. The remaining systematic risk is refected by the size of w and b and is sti correated but cannot be expained by our proxies. Once the point estimates for the parameters are given, we forecast separatey the defauts and recoveries for year 2004. Tabe 7.7 shows that the point-in-time mode eads to forecasts for the defaut and recovery rates that are coser to the reaized vaues than the ones derived from the through-the-cyce mode. Tabe 7.7 Forecasts and reaizations for year 2004 (through-the-cyce versus point-in-time) Parameter Tota S_Sec S_Un S_Sub Sub Defaut rate Forecast TTC 0.0181 0.0181 0.0180 0.0181 0.0181 Forecast PIT 0.0162 0.0162 0.0160 0.0162 0.0162 Reaization 0.0072 0.0072 0.0072 0.0072 0.0072 Recovery rate Forecast TTC 0.4097 0.5739 0.4416 0.3603 0.2957 Forecast PIT 0.4381 0.6159 0.4484 0.3867 0.3014 Reaization 0.5850 0.8080 0.5010 0.4440 0.1230

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 129 7.4 Impications for Economic and Reguatory Capita Since the main contribution of our approach ies in the joint modeing of defauts and recoveries, we now appy the forecast defaut rates, recovery rates for the year 2004 as we as their estimated correation to a portfoio of 1,000 obigors. To simpify the process, we take the senior secured cass as an exampe and assume a credit exposure of one monetary unit for each obigor. Figure 7.3 and Tabe 7.8 compare two forecast oss distributions of the throughthe-cyce mode. To demonstrate the infuence of correation between the processes we compare the distribution which assumes independence to the distribution which is based on the estimated correation between the defaut and recovery rate transformations of 0.7049. Economic capita or the credit portfoio risk is usuay measured by higher percenties of the simuated oss variabe such as the 95-, 99-, 99.5- or 99.9- percentie (95%-, 99%-, 99.5%- or 99.9%-Vaue-at-Risk). It can be seen that these percenties are consideraby higher if correations between defaut and recovery rates are taken into account. If we take the 99.9%-Vaue-at-Risk as an pct 100 Loss Distribution 80 60 40 20 0 0 10 Independence Dependence 20 30 40 50 Loss Fig. 7.3 Loss distributions for the through-the-cyce mode (S_Sec) Tabe 7.8 Descriptive statistics of oss distributions for the through-the-cyce mode Mean Std. dev. Med 95 99 99.5 99.9 Base II capita (standardized) Base II capita (foundation IRB) Base II capita (advanced IRB) Ind. factors 7.82 5.59 6.53 18.55 27.35 31.92 39.02 80.00 74.01 70.08 Corr. factors 8.73 7.59 6.62 23.81 36.04 42.43 58.75 80.00 74.01 70.08 Portfoios contain 1,000 obigors with an exposure of one monetary unit each, 10,000 random sampes were drawn for each distribution with and without correation between systematic factors

130 D. R osch and H. Scheue exampe, the percentie under dependence exceeds the percentie under independence by approximatey 50%. In other words, if dependencies are not taken into account, which is a common feature in many of today s credit risk modes, the credit portfoio risk is ikey to be seriousy underestimated. Forecast defaut and recovery rates can be used to cacuate the reguatory capita for the hypothetica portfoio. For corporate credit exposures, the Base Committee on Banking Supervision (2004) aows banks to choose one of the foowing options: Standardized approach: reguatory capita is cacuated based on the corporate issuer credit rating and resuts in a reguatory capita between 1.6 and 12% of the credit exposure. The reguatory capita equas 8% of the credit exposure if firms are unrated Foundation Interna Ratings Based (IRB) approach: reguatory capita is cacuated based on the forecast defaut probabiities and a proposed oss given defaut for senior secured caims of 45% (i.e., a recovery rate of 55%) and for subordinated caims of 75% (i.e., a recovery rate of 25%) Advanced IRB approach: reguatory capita is cacuated based on the forecast defaut probabiities and forecast recovery rates For the through-the-cyce mode, the Standardized approach and the Foundation IRB approach resut in a reativey cose reguatory capita requirement (80.00 vs. 74.01). The reason for this is that the forecast defaut rate (0.0181) is cose to the historic average which was used by the Base Committee when caibrating reguatory capita to the current eve of 8%. The Advanced IRB approach eads to a ower reguatory capita (70.08 vs. 74.01) due to a forecast recovery rate which is higher than the assumption in the Foundation IRB approach (57.39% vs. 55%). Note that Foundation IRB s recovery rate of 55% is comparabe to the average recovery rate of the senior secured seniority cass but is proposed to be appied to both the senior secured (uness admitted coatera is avaiabe) as we as the senior unsecured caims. This coud indicate an incentive for banks to favour the Foundation approach over the Advanced IRB approach especiay for senior unsecured credit exposures. Simiar concusions can be drawn for the Foundation IRB s recovery rate of 25% which wi be appied for both senior subordinated as we as subordinated caims. Figure 7.4 and Tabe 7.9 compare the respective oss distributions with and without correations using the point-in-time mode. It can be observed that the economic capita, expressed as Vaue-at-Risk, is consideraby ower for the point-in-time mode than for the through-the-cyce mode. The reasons are twofod. First, the incusion of macroeconomic variabes eads to a ower forecast of the defaut rate (1.62%), a higher forecast of the recovery rate (61.59%) for 2004 and therefore to ower expected osses. Second, the exposure to unknown random systematic risk sources is reduced by the incusion of the observabe factors. This eads to ess uncertainty in the oss forecasts and therefore to ower variabiity (measured, e.g., by the standard deviation) of the forecast distribution. Moreover, the reguatory capita is the owest for the

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 131 pct 100 Loss Distribution 80 60 40 20 0 0 10 Independence Dependence 20 30 40 50 Loss Fig. 7.4 Loss distributions for the point-in-time mode (S_Sec) Tabe 7.9 Descriptive statistics of oss distributions for the point-in-time mode Mean Std. dev. Med 95 99 99.5 99.9 Base II capita (standardized) Base II capita (foundation IRB) Base II capita (advanced IRB) Ind. factors 6.33 3.61 5.64 13.10 18.01 20.43 25.77 80.00 71.16 60.74 Corr. factors 6.78 4.71 5.64 16.03 22.78 25.60 31.77 80.00 71.16 60.74 Portfoios contain 1,000 obigors with an exposure of one monetary unit each, 10,000 random sampes were drawn for each distribution with and without correation between systematic factors Advanced IRB approach which takes both the forecast defaut and recovery rate into account. We aso notice another important effect. The economic capita, measured by the higher percenties of the credit portfoio oss, increases if the estimated correation between the defaut and recovery rates is taken into account. This increase is not as dramatic as in the through-the-cyce mode, athough the correation between risk factors of defauts and recoveries has sighty increased. The incusion of macroeconomic factors renders the systematic unobservabe factors ess important and diminishes the impact of correations between both factors. To the extent that recoveries and defauts are not exposed at a to unobservabe random factors, the correations between these factors are negigibe for oss distribution modeing. Figure 7.5 shows this effect. We assumed constant exposure of b ¼ 0.5 to the recovery factor and varied the exposure to the systematic factor for the defauts (asset correation) for given correation between the systematic factors. The benchmark case is a correation of zero between the factors. Here, we notice a reduction of economic capita from 44 (i.e., 4.4% of tota exposure) for an asset correation of 0.1 to 13 (1.3%) when the asset correation is zero. In the case of a correation

132 D. R osch and H. Scheue 70 60 50 99.9 % - VaR 40 30 20 10 0 0 0.02 0.04 0.06 0.08 0.1 asset correation rho = 0 rho = 0.4 rho = 0.8 Fig. 7.5 Economic capita gains from decrease in impied asset correation for correated risk factors; Figure shows 99.9 percenties of oss distributions for the senior secured seniority cass depending on asset correation and correation of systematic risk factors. Portfoio contains 1,000 obigors each with defaut probabiity of 1%, exposure of one monetary unit, and expected recovery of 50% between the factors of 0.8, the Vaue-at-Risk is reduced from 61 (6.1%) to 13 (1.3%). Thus, the higher the correation of the risk factors, the higher the economic capita gains are from owering the impied asset correation by the expanation with observabe factors. 7.5 Discussion The empirica anaysis resuted in the foowing insights: 1. Defaut events and recovery rates are correated. Based on an empirica data set, we found a positive correation between the defaut events and a negative correation between the defaut events and recovery rates. 2. The incorporation of the correation between the defaut events and recovery rates increases the economic capita. As a resut, most banks underestimate their economic capita when they fai to account for this correation. 3. Correations between defauts decrease when systematic risk drivers, such as macroeconomic indices are taken into account. In addition, the impact of correation between defauts and recoveries decreases. 4. As a resut, the uncertainty of forecast osses and the economic capita measured by the percenties decreases when systematic risk drivers are taken into account.

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 133 Most empirica studies on recovery rates (incuding this artice) are based on pubicy avaiabe data provided by the rating agencies Moody s or Standard and Poor s and naturay ead to simiar resuts. The data sets of the rating agencies are biased in the sense that ony certain exposures are taken into account. Typicay, arge U.S. corporate obigors in capita intensive industries with one or more pubic debt issues and high credit quaity are incuded. Thus, the findings can not automaticay be transferred to other exposure casses (e.g., residentia mortgage or credit card exposures), countries, industries or products. Moreover, the data is imited with regard to the number of exposures and periods observed. Note that our assumption in (7.8) of a arge number of firms is crucia since it eads to the focus on the mean recovery. If idiosyncratic risk can not be fuy diversified the impact of systematic risk in our estimation may be overstated. Due to the data imitations, we cannot draw any concusions about the cross-sectiona distribution of recoveries which is often stated to be U-shaped (see, e.g., Schuermann 2003). In this sense, our resuts ca for more detaied anayses, particuary with borrower-specific data which possiby incudes financia ratios or other obigor characteristics and to extend our methodoogy to a pane of individua data. As a resut, we woud ike to ca upon the industry, i.e., companies, banks and reguators for feedback and a sharing of their experience. In spite of these imitations, this paper provides a robust framework, which aows creditors to mode defaut probabiities and recovery rates based on certain risk drivers and simutaneousy estimates interdependences between defauts and recoveries. It can be appied to different exposure types and associated information eves. Contrary to competing modes, the presence of market prices such as bond or stock prices is not required. Appendix: Resuts of Monte-Caro Simuations In order to prove the reiabiity of our estimation method, a Monte-Caro simuation was set up which comprises four steps: Step 1: Specify mode (1) and mode (9) with a given set of popuation parameters w, c, b, m, and r. Step 2: Draw a random time series of ength T for the defauts and the recoveries of a portfoio with size N from the true mode. Step 3: Estimate the mode parameters given the drawn data by the Maximum- Likeihood method. Step 4: Repeat Steps 2 and 3 for severa iterations. We used 1,000 iterations for different parameter consteations and obtained 1,000 parameter estimates which are compared to the true parameters. The portfoio consists of 10,000 obigors. The ength of the time series T is set to T ¼ 20 years. We fix the parameters at w ¼ 0.2, m ¼ 0.5, and b ¼ 0.5 and set the correations between the systematic factors to 0.8, 0.1, and 0.5. In addition, we anayze three

134 D. R osch and H. Scheue rating grades A, B, and C where the defaut probabiities and threshods c in the grades are: A: p ¼ 0:005, i.e., c ¼ 2:5758 B: p ¼ 0:01, i.e., c ¼ 2:3263 C: p ¼ 0:02, i.e., c ¼ 2:0537 Tabe 7.10 contains the resuts from the simuations. The numbers without brackets contain the average of the parameter estimates from 1,000 simuations. The numbers in round (.)-brackets represent the sampe standard deviation of the estimates (which serve as an approximation for the unknown standard deviation). The numbers in square [.]-brackets give the average of the estimated standard deviations for each estimate derived by Maximum-Likeihood theory. It can be seen in each consteation that our ML approach for the joint estimation of the defaut and recovery process works consideraby we: the averages of the estimates are cose to the originay specified parameters. Moreover, the estimated standard deviations refect the imited deviation for individua iterations. The sma downward bias resuts from the asymptotic nature of the ML-estimates and shoud be toerabe for practica appications. Tabe 7.10 Resuts from Monte-Caro simuations c w m b r Grade r A 0.8 2.5778 0.1909 0.4991 0.4784 0.7896 (0.0495) (0.0338) (0.1112) (0.0776) (0.1085) [0.0468] [0.0317] [0.1070] [0.0756] [0.0912] 0.1 2.5789 0.1936 0.4970 0.4824 0.1139 (0.0484) (0.0336) (0.1154) (0.0788) (0.2269) [0.0475] [0.0322] [0.1079] [0.0763] [0.2185] 0.5 2.5764 0.1927 0.5048 0.4826 0.4956 (0.0492) (0.0318) (0.1116) (0.0798) (0.1923) [0.0472] [0.0320] [0.1078] [0.0763] [0.1697] B 0.8 2.3287 0.1927 0.4999 0.4852 0.7951 (0.0480) (0.0327) (0.1104) (0.0774) (0.0920) [0.0460] [0.0306] [0.1084] [0.0765] [0.0856] 0.1 2.3291 0.1906 0.4927 0.4831 0.0861 (0.0472) (0.0306) (0.1105) (0.0778) (0.2330) [0.0456] [0.0305] [0.1080] [0.0764] [0.2152] 0.5 2.3305 0.1900 0.4988 0.4805 0.4764 (0.0479) (0.0324) (0.1115) (0.0806) (0.1891) [0.0453] [0.0303] [0.1074] [0.0759] [0.1703] C 0.8 2.0536 0.1935 0.4972 0.4855 0.7915 (0.0489) (0.0315) (0.1104) (0.0804) (0.0956) [0.0448] [0.0297] [0.1080] [0.0763] [0.0843] 0.1 2.0542 0.1943 0.5030 0.4851 0.1067 (0.0580) (0.0382) (0.1168) (0.0782) (0.2374) [0.0448] [0.0298] [0.1085] [0.0770] [0.2128] 0.5 2.0554 0.1923 0.4998 0.4833 0.4898 (0.0510) (0.0359) (0.1085) (0.0852) (0.1815) [0.0443] [0.0295] [0.1076] [0.0766] [0.1656]

7 A Muti-factor Approach for Systematic Defaut and Recovery Risk 135 References Atman E, Brady B, Resti A, Sironi A (2005), The Link between Defaut and Recovery Rates: Theory, Empirica Evidence and Impications, Journa of Business 78 (6), pp. 2203 2228. Base Committee on Banking Supervision (2004), Internationa Convergence of Capita Measurement and Capita Standards A Revised Framework, Consutative Document, Bank for Internationa Settements. Cantor R, Varma P (2005), Determinants of Recovery Rates on Defauted Bonds and Loans for North American Corporate Issuers: 1983 2003, Journa of Fixed Income 14 (4), pp. 29 44. Carey M (1998), Credit Risk in Private Debt Portfoios, Journa of Finance 53, pp. 1363 1387. D umann K, Trapp M (2004), Systematic Risk in Recovery Rates- An Empirica Anaysis of U.S. Corporate Credit Exposures, Discussion Paper, Series 2: Banking and Financia Supervision, No 02/2004. Frye J (2000), Depressing Recoveries, Risk 13 (11), pp. 106 111. Frye J (2003), A Fase Sense of Security, Risk 16 (8), pp. 63 67. Gordy M (2003), A Risk-Factor Mode Foundation for Ratings-Based Bank Capita Rues, Journa of Financia Intermediation 12, pp. 199 232. Gordy M (2000), A Comparative Anatomy of Credit Risk Modes, Journa of Banking and Finance 24, pp. 119 149. Hamere A, Liebig T, R osch D (2003), Benchmarking Asset Correations, Risk 16, pp. 77 81. Heitfied A (2005), Dynamics of Rating Systems, in: Base Committee on Banking Supervision: Studies on the Vaidation of Interna Rating Systems, Working Paper No. 14, February, pp. 10 27. Hu T, Perraudin W (2002), The Dependence of Recovery Rates and Defauts, Working Paper, Birkbeck Coege. Moody s (2004), Defaut and Recovery Rates of Corporate Bond Issuers 1920 2003. Moody s (2005), Defaut and Recovery Rates of Corporate Bond Issuers 1920 2004. Pykhtin M (2003), Unexpected Recovery Risk, Risk, 16 (8), pp. 74 78. R osch D (2005), An Empirica Comparison of Defaut Risk Forecasts from Aternative Credit Rating Phiosophies, Internationa Journa of Forecasting 21, pp. 37 51. R osch D (2003), Correations and Business Cyces of Credit Risk: Evidence from Bankruptcies in Germany, Financia Markets and Portfoio Management 17, pp. 309 331. R osch D, Scheue H (2004), Forecasting Retai Portfoio Credit Risk, Journa of Risk Finance 5 (2), pp. 16 32. Sch onbucher J (2003), Credit Derivatives Pricing Modes: Modes, Pricing and Impementation, John Wiey and Sons, New York. Schuermann T (2003), What Do We Know About Loss-Given-Defaut?, Working Paper, Federa Reserve Bank of New York.

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Chapter 8 Modeing Loss Given Defaut: A Point in Time -Approach Afred Hamere, Michae Knapp, and Nicoe Widenauer 8.1 Introduction In recent years the quantification of credit risk has become an important topic in research and in finance and banking. This has been acceerated by the reorganisation of the Capita Adequacy Framework (Base II). 1 Previousy, researchers and practitioners mainy focused on the individua creditworthiness and thus the determination of the probabiity of defaut (PD) and defaut correations. The risk parameter LGD (oss rate given defaut) received ess attention. Historica averages of LGD are often used for practica impementation in portfoio modes. This approach negects the empirica observation that in times of a recession, not ony the creditworthiness of borrowers deteriorates and probabiities of defaut increase, but LGD aso increases. Simiar resuts are confirmed in the empirica studies by Atman et a. (2003), Frye (2000a), and Hu and Perraudin (2002). If LGD is ony integrated in portfoio modes with its historica average, the risk tends to be underestimated. Hence, adequate modeing and quantification of LGD wi become an important research area. This has aso been advocated by Atman and Kishore (1996), Hamiton and Carty (1999), Gupton et a. (2000), Frye (2000b), and Schuermann (2004). The definitions of the recovery rate and the LGD have to be considered when comparing different studies of the LGD, since different definitions aso cause different resuts and concusions. Severa studies distinguish between market LGD, impied 1 Base Committee on Banking Supervision (2004). A. Hamere University of Regensburg e-mai: Afred.Hamere@wiwi.uni-regensburg.de M. Knapp Risk Research Prof. Hamere GmbH & Co. KG e-mai: michae.knapp@risk-research.de N. Widenauer (*) Commerzbank AG e-mai: Nicoe.Widenauer@commerzbank.com B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_8, # Springer-Verag Berin Heideberg 2011 137

138 A. Hamere et a. market LGD and workout LGD. 2 This paper uses recovery rates from Moody s defined as market recovery rates. In addition to studies which focus ony on data of the bond market or data of bonds and oans, 3 there are studies which focus on oans ony. 4 Loans generay have higher recovery rates and therefore ower vaues of LGD than bonds. 5 This resut reies especiay on the fact that oans are more senior and in many cases aso have more coectibe coateras than bonds. Studies show different resuts concerning the factors potentiay determining the LGD which are presented briefy beow. The iterature gives inconsistent answers to the question if the borrower s sector has an impact on LGD. Surveys such as Atman and Kishore (1996) confirm the impact of the sector. Gupton et a. (2000) concude that the sector does not have an infuence on LGD. They trace this finding back to the fact that their study ony examines oans and not bonds. The impact of the business cyce is approved by many authors, e.g. Atman et a. (2003), Varma and Cantor (2005), Acharya et a. (2007), Grunert and Weber (2009), and Bruche and Gonzaez-Aguado (2010). In contrast, Asarnow and Edwards (1995) concude that there is no cycica variation in LGD. Comparing these studies one has to consider that different data sources have been used, and the atter ony focused on oans. Severa studies support the infuence of the borrower s creditworthiness or the seniority on LGD. 6 Neary a studies anaysing LGD using empirica data cacuate the mean of the LGD per seniority, per sector, per rating cass or per year. Sometimes the means of the LGD per rating cass and per seniority are cacuated. We refer to the atter prices as matrix prices sometimes enabing a more accurate determination of LGD than the use of simpe historica averages. 7 The authors agree that the variance within the casses is high and there is a need for more sophisticated modes. Atman et a. (2003) suggest a first extension of the mode by using a regression mode with severa variabes as the average defaut rate per year or the GDP growth to estimate the average recovery rate. The present paper makes severa contributions. A dynamic approach for LGD is deveoped which aows for individua and time dependent LGDs. The mode provides point-in-time predictions for the next period. The unobservabe part of systematic risk is modeed by a time specific random effect which is responsibe for dependencies between the LGDs within a risk segment in a fixed time period. 2 For a definition of these vaues of LGD see Schuermann (2004) and Base Committee on Banking Supervision (2005). 3 Schuermann (2004). 4 Asarnow and Edwards (1995), Carty and Lieberman (1996), Carty et a. (1998). 5 Gupton et a. (2000). 6 Carty and Lieberman (1996), Carty et a. (1998), Gupton et a. (2000), Atman (2006), Roesch and Scheue (2008). 7 Araten et a. (2004), Gupton et a. (2000), Schuermann (2004).

8 Modeing Loss Given Defaut: A Point in Time -Approach 139 Furthermore, the reationship between issuer specific rating deveopments and LGD can be modeed adequatey over time. The rest of this chapter is organised as foows: Sect. 8.2 states the statistica modeing of the LGD. Section 8.3 describes the dataset and the mode estimations. Section 8.4 concudes and discusses possibe fieds for further research. 8.2 Statistica Modeing The dataset used in this chapter mainy uses bond data. Recovery rates wi be cacuated as market vaue of the bonds 1 month after defaut. The connection between LGD and recovery rate can be shown as: LGD tðiþ ¼ 1 R tðiþ : Here, LGD t(i) and R t(i) denote the LGD and recovery rate of bond i that defauts in year t, i¼1,...,n t. The number of defauted bonds in year t, t¼1,...,t is denoted with n t. The resuting recovery rates and oss rates normay range between 0 and 1, athough there are exceptions. 8 Firsty, the LGDs wi be transformed. The transformation used in this chapter is LGD tðiþ y tðiþ ¼ og : 1 LGD tðiþ Written in terms of the recovery rate, the foowing reation is obtained: y tðiþ ¼ og 1 R tðiþ R tðiþ R tðiþ ¼ og : 1 R tðiþ This ogit transformation of the recovery rate is aso proposed by Sch onbucher (2003) and D umann and Trapp (2004). 9 The LGD can be written as: LGD tðiþ ¼ expðy tðiþþ 1 þ expðy tðiþ Þ : 8 Recovery rates greater than one are unusua. In these cases the bond is traded above par after the issuer defauts. These vaues are excuded from the dataset in the empirica research, see Sect. 8.3.1. 9 This transformation ensures a range between 0 and 1 of the estimated and predicted LGD.

140 A. Hamere et a. Anaogous to the mode used in Base II, the foowing approach for the transformed vaues y t(i) is specified (D umann and Trapp 2004): p y tðiþ ¼ m þ s ffiffiffiffi p o ft þ s ffiffiffiffiffiffiffiffiffiffiffiffi 1 o e tðiþ (8.1) The random variabes f t and e t(i) are standard normay distributed. A random variabes are assumed to be independent. The parameter s is non-negative and vaues of o are restricted to the interva [0, 1]. Other specifications are aso discussed. Frye (2000a) suggests an approach according to (8.1) for the recovery rate itsef. Pykthin (2003) assumes og-normay distributed recovery rates and chooses a specification ike (8.1) for og(r t(i) ). In the next step, mode (1) is extended incuding firm and time specific observabe risk factors. The dependence upon the observabe risk factors is specified by the foowing inear approach: m tðiþ ¼ b 0 þ b 0 x t 1ðiÞ þ g 0 z t 1 ; (8.2) where i¼1,...,n t, t¼1,...,t. Here x t-1(i) characterises a vector of issuer and bond specific factors observed in previous periods. Exampes for these issuer and bond specific variabes are the issuer rating of the previous year or the seniority. By z t 1 we denote a vector of macroeconomic variabes representing potentia systematic sources of risk. The macroeconomic variabes are incuded in the mode with a time ag. Generay it can be assumed that regression equation (8.2) hods for a predefined risk segment, e.g. a sector. Regarding (8.1) and (8.2) it can be seen that the ogit transformed vaues of LGD are normay distributed with mean m t(i) and variance s 2. The random time effects f t cause a correation of the transformed vaues of LGD y t(i) of different bonds defauting in year t. This correation shows the infuence of systematic sources of risk which are not expicity incuded in the mode or which affect LGD contemporariy. If fundamenta factors are having an impact on the LGD of a defauted bonds at east in one sector a correation of LGD is obtained as a resut (as ong as these systematic risk factors are not incuded in the mode). It can be seen that the factors have different effects in different segments, e.g. different time ags or sensitivities in different sectors. If in contrast, the reevant systematic risk factors are incuded in the vector z t-1 and if no other risk factors infuence LGD contemporariy, the impact of time effects shoud be reduced significanty. The unknown parameters in (8.1) and (8.2) are estimated by maximum ikeihood considering (8.1) extended by (8.2) as a pane regression mode with random effects, (Batagi 1995, Chap. 3). Note that a bond specific random effect does not enter the mode, since defauted bonds in different periods t and s (t6¼s) are different. Parameter estimates are obtained using PROC MIXED in SAS. 10 10 Wofinger et a. (1994).

8 Modeing Loss Given Defaut: A Point in Time -Approach 141 For the covariance and correation of the transformed vaues of LGD in year t, the foowing reationships hod: Covðy tðiþ ; y tðjþ Þ¼s 2 o Corrðy tðiþ ; y tðjþ Þ¼o; i 6¼ j: 8.3 Empirica Anaysis 8.3.1 The Data A dataset from Moody s Defaut Risk Service is used for empirica anayses. It contains data from about 2,000 defauted debt obigations, i.e. bonds, oans and preferred stock from 1983 to 2003. More than 1,700 debt obigations are from American companies. The dataset incudes information about the recovery rates of defauted bonds. The LGD and the transformed LGD used in this anaysis can be cacuated from the recovery rate as described in Sect. 8.2. When a borrower defauted for the first time, this event was recorded and a defaut events after the first one are not considered in this study. 11 About 90% of these debt obigations are bonds. To ensure a homogenous dataset, ony bonds are used in this study. For the same reason, ony data from companies in the sector industry 12 are used in the fina anaysis. In this sector there are 84% of the bonds. In the sectors financia service providers and sovereign/pubic utiity there are fewer defauted borrowers and therefore fewer defauted bonds. After restricting the data to American bonds in the (aggregated) sector industry, there are 1,286 bonds in the dataset. Additionay, the dataset is imited to bonds with a debt rating of Ba3 or worse. The reason for this constraint was that the rating categories A3 to Ba2 have sparse observations in severa years of the period 1983 2003. In addition, severa defauted issuers hod five or more bonds. Some of 11 This constraint naturay ony affects borrowers who defauted severa times. Furthermore, observations with LGD equa to zero and negative LGD are excuded from the anaysis, because the transformed LGD y t(i) cannot be cacuated. If the recovery rate is greater than 1, i.e. if the market vaue of a bond one month after defaut is greater than the nomina vaue of the bond, the LGD becomes negative. In the dataset this was the case in 0.5% of a observations. 12 The (aggregated) sector industry contains the sectors industria, transportation and other non-bank of Moody s sectora cassification (with 12 sectors) in Moody s Defaut Risk Service (DRS) database. For reason of competeness one has to know that there are two other aggregated sectors. On the one hand there is the (aggregated) sector financia service providers containing the sectors banking, finance, insurance, rea estate finance, securities, structured finance and thrifts and on the other hand the (aggregated) sector sovereign/pubic utiity containing the sectors pubic utiity und sovereign. This aggregation was made as severa sectors did not have enough observations.

142 A. Hamere et a. these bonds have the same LGD at the time of defaut athough they have distinct debt ratings or distinct seniorities. Other bonds have a different LGD athough they dispose of the same issuer and debt rating and the same seniority. These differences cannot be expained with the data at hand. Probaby they can be traced back to issuer s attributes not avaiabe in the dataset. For this reason, ony issuers with four or fewer bonds remain in the dataset. 13 Additionay, bonds of companies with obvious cases of fraud ike Enron or Wordcom were eiminated from the dataset to ensure a homogenous poo. Subsequenty, the dataset is adjusted marginay. On the one hand, there is ony one bond with a rating B2 defauting in 1996. This bond has a very sma LGD and is removed from the dataset because it coud cause a biased estimation of random effects. On the other hand, four bonds having a bond rating of Ca and C in the years 1991, 1992 and 1995 are eiminated from the dataset because they aso have ony one or two observations per year. Consequenty, there are 952 bonds from 660 issuers remaining in the dataset. The random effect f t and the error term e t(i) are assumed to be independent, with a standard norma distribution as described in Sect. 8.2. The transformed LGD y t(i) is tested for an approximatey normay distribution. As a resut, a norma distribution of the data can be assumed. This distribution can aso be confirmed when the distribution of y t(i) by year is tested. In the anaysis, the infuence of issuer- and bond-specific variabes x t 1(i) is examined as mentioned in Sect. 8.2. In the anayses the foowing variabes are tested: Issuer rating: Moody s estimated senior rating has 21 grades between Aaa (highest creditworthiness) and C (ow creditworthiness). 14 An aggregation of the rating categories is tested as we. A possibe cassification woud be the distinction between investment grade ratings (rating Aaa to Baa3 ) and specuative grade ratings (rating Ba1 to C ). Besides this reativey rough cassification the ratings are cassified into the categories Aaa to A3, Baa1 to Baa3, Ba1 to Ba3, B1 to B3, Caa, 15 Ca and C. The issuer rating has a time ag of 1 year in the anayses. Debt rating: Its cassification is anaogous to the issuer rating and has a time ag of 1 year. In addition to the cassifications mentioned above, the ratings are cassified into the categories Ba3 to B3 and Caa to C. 13 In principe, ony issuers with one bond coud be eft in the dataset if the effect of severa bonds per issuer shoud be eiminated. As this restriction woud ead to reativey few observations, ony issuers with five or more bonds are excuded. Hence the dataset is ony diminished by 4%. 14 For withdrawn ratings, Moody s uses a cass WR. Because of the agged consideration of rating there are no bonds in the dataset with rating WR one year before defaut. 15 Moody s used to name this rating cass with Caa unti 1997. Since 1998, this cass has been separated into the three rating casses Caa1, Caa2 and Caa3. To use the data after 1998, the atter three ratings have been aggregated in one rating cass which is named Caa in the foowing.

8 Modeing Loss Given Defaut: A Point in Time -Approach 143 Difference between issuer and debt rating: the fact that the issuer rating is one, two, three or more than three notches better than the debt rating is tested on its impact on the transformed LGD. Additionay, the impact of the fact that the issuer rating is better or worse than the debt rating is tested. The rating cassification of an issuer and a bond can differ if the bond finances a certain project which has a different risk and sovency appraisa compared to the issuer. Seniority: Starting with Moody s cassification, the casses senior secured, senior unsecured, senior subordinated, subordinated and junior subordinated are distinguished. 16 To distinct these seniority casses from the reative seniority, they are sometimes referred to as absoute seniority. Reative seniority: According to Gupton and Stein (2005) the reative importance of the seniority is surveyed. This variabe can be best expained by an exampe: If issuer 1 has two bonds one is secured subordinated and the other junior subordinated and issuer 2 has three bonds one with seniority senior secured, another with senior subordinated and the third bond with seniority subordinated then the subordinated bond from issuer 1 is going to be served first and possiby has a ower LGD than the bond with seniority subordinated from issuer 2 which is served after the two other bonds from issuer 2. Additiona backing by a third party: If the bond is secured additionay by a third party beside the protection by the issuer emitting the bond, then this information is aso used in the anayses. Maturity (in years): The maturity of the bond is cacuated as the difference of the maturity date and the defaut date. It indicates the remaining time to maturity if the bond woud not have defauted. Voume of defauted bond (in miion doars): The number of outstanding defauted bonds times the nomina of this defauted bond denotes the voume of the defauted bond. It quantifies the infuence of the voume of one defauted bond, not the infuence of the voume of defauted bonds in the market atogether. Certain companies ike insurances are not aowed to hod defauted bonds. On the other hand, there are specuative investors who are interested in buying defauted bonds. The higher the voume of the defauted bond, the higher the suppy of defauted bonds on the market. Therefore it can be more difficut for the defauted issuers to find enough buyers or to caim high prices for the defauted bond. Issuer domicie: The country of the issuer is impicity considered by the imitation on American data. This imitation can be important because different countries may be in different stages of the economic cyce in the same year. If the data is not imited to a certain country, the macroeconomic condition of a countries incuded in the dataset shoud be considered. Additionay, different ega insovency procedures exist in different countries, so that a country s ega procedure can infuence the eve of recovery rates and LGD. 16 For a consideration of the hierarchy of seniority casses see Schuermann (2004, p. 10).

144 A. Hamere et a. 80 % 70 % 60 % LGD (in percent) 50 % 40 % 30 % 20 % 10 % 0% 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Year Fig. 8.1 Average LGD per year for bonds in the (aggregated) sector industry In Fig. 8.1 the average (reaised) LGD for bonds in the (aggregated) sector industry per year in the period 1983 2003 are depicted: As can be seen from Fig. 8.1, the LGD is obviousy underying cycica variabiity. This is why the cycica variations of LGD are expained with the hep of macroeconomic variabes in the vector z t-1. Therefore, a database with more than 60 potentia macroeconomic variabes is estabished. It contains interest rates, abour market data, business indicators ike gross domestic product, consumer price index or consumer sentiment index, infation data, stock indices, the Leading Index etc. 17 In addition, the average defaut rate per year of the bond market is taken into account. A variabes are incuded contemporariy and with a time ag of at east 1 year. The consideration of these variabes shoud enabe a point-in-time mode. 8.3.2 Resuts Two different mode specifications for the (aggregated) sector industry are examined. 18 In contrast to mode (8.1), another (but equivaent) parameterisation is used. The modes can be instantaneousy estimated with the procedure MIXED in the statistica program SAS. In the next step, the parameter estimates for s and o can be determined from the estimates for b 1 and b 2. Tabe 8.1 summarises the resuts. 17 A ist of potentia macroeconomic factors can be found in the appendix. 18 Additionay, modes for a sectors are estimated containing dummy variabes for the different sectors in addition to the variabes mentioned beow. The use of a singe sector eads to more homogenous data.

8 Modeing Loss Given Defaut: A Point in Time -Approach 145 Tabe 8.1 Parameter estimates and p-vaues (in parentheses) for modes I and II (ony bonds of the (aggregated) sector industry ) Mode I Mode II AIC 3,224.3 3,222.8 b 2 2 1.7336 (<0.0001) 1.7327 (<0.0001) b 2 1 0.3421 (0.0052) 0.2859 (0.0064) Constant 0.3868 (0.1146) 0.8697 (0.0164) Debt rating Ba3 to B3 (t 1) 0.1938 (0.0463) 0.1783 (0.0672) Seniority senior unsecured 0.6194 (0.0004) 0.6064 (0.0005) Seniority senior subordinated 0.7061 (0.0002) 0.6909 (0.0002) Seniority subordinated and junior subordinated 1.0487 (<0.0001) 1.0443 (<0.0001) Reative Seniority 2 and 3 0.5041 (0.0001) 0.5084 (<0.0001) Additiona backing by a third party 0.2717 (0.0325) 0.2697 (0.0338) Bond maturity (in years) 0.03407 (0.0020) 0.03546 (0.0013) Voume of defauted bonds (in miion doars) 0.001118 (0.0001) 0.001087 (0.0002) Average defaut rate (in percent) (t 1) 0.2186 (0.0358) Mode I: y tðiþ ¼ b 0 þ b 0 x t 1ðiÞ þ b 1 f t þ b 2 e tðiþ. Mode II: y tðiþ ¼ b 0 þ b 0 x t 1ðiÞ þ g 0 z t 1 þ b 1 f t þ b 2 e tðiþ. The resuts of modes I and II can be interpreted as foows 19 : If a bond is rated Ba3, B1, B2 or B3 1 year before defaut, it has a significanty smaer LGD than a bond with rating Caa, Ca or C. In addition to the debt rating, the seniority aso affects LGD. Bonds with seniority senior unsecured as we as bonds secured senior subordinated, subordinated or junior subordinated have a significanty higher LGD than senior secured bonds. When the seniority casses are compared, it can be stated that senior unsecured bonds have a smaer LGD than senior subordinated bonds. Bonds secured subordinated or junior subordinated have the highest LGD. Using we secured bonds a creditor can expoit better securities than a creditor secured with ower ranked bonds resuting in ower osses. Generay, this resut sustains the resuts pubished by Moody s. 20 However, not ony the (absoute) seniority, but aso the reative seniority affects LGD. If a bond is ranked second or third in terms of coateraisation, the LGD of this bond is significanty higher than the LGD of a bond secured at first rank. If the company is going to be commerciaised, the atter are served before the bonds ranking second or third and therefore have to bear fewer osses. Regarding the coherence between absoute and reative seniority and LGD, it must be recognised that besides the creditworthiness of the bond, the seniority aso pays a roe for the determination of LGD. The fact that in addition to the absoute seniority, reative seniority aso infuences LGD is an interesting resut. This coherence is aso detected in the modes of Gupton and Stein (2002, 2005). 19 In genera, a interpretations according to the quoted mode refer to the transformed LGD y t(i). As y t(i) is the resut of a stricty monotonic transformation of LGD a interpretations hod as we for LGD. 20 Hamiton and Carty (1999).

146 A. Hamere et a. If in addition to the coateraisation by the direct issuer, the bond is protected by a third party, these bonds have a significanty ower LGD than bonds without this additiona backing. These additiona providers of coatera coud fi in for the defauted company if the atter does not have a substantia vaue. Therefore, it can reduce the oss of these bond creditors. Another impact on LGD is given by the maturity of the bond. A onger maturity eads to higher LGDs. This resut can possiby be expained by the fact that future payments are insecure. The recovery rate and LGD are cacuated as the market price 1 month after defaut. If maturity is onger, higher cash fows are achieved in the future which are generay more insecure. This is refected in ower recovery rates and higher LGDs. Gupton and Stein (2005) negate the infuence of maturity on LGD in their recent paper. In their opinion the maturity does not pay a roe for defauted bonds. Ony the risk horizon matters, which is 1 year in their anaysis. However, Gupton and Stein (2005) negect the uncertainty of future cash fows. Additionay, the voume of the defauted bonds infuences LGD as a factor of the suppy side. As mentioned above, a higher voume of defauted bonds eads to a higher suppy and to ower prices for these bonds, i.e. to ower recovery rates and higher LGDs. 21 The incorporation of macroeconomic factors in mode II tries to expain the cycica variations of LGD. These factors can be interpreted as foows: The average defaut rate of the bond market (in percent) with a time ag of 1 year is taken into account in the mode as a possibe proxy for the cycica infuence. An increasing agged average defaut rate eads to significanty higher LGDs. This resut is supported by Atman et a. (2003) who detected a positive reationship between the defaut rate and the (average) LGD as we. The cycica variation in LGD (see Fig. 8.1), can be expained by the fact that more borrowers and therefore more bonds are defauting during a recession. More companies and coateras have to be commerciaised eading on the one hand to a greater suppy of coatera and therefore ower coatera prices. On the other hand, the demand for these commerciaised coateras decines because the non-defauted companies are not abe to invest the same amount of money during a recession as during an expansion. Macroeconomic variabes ike the agged defaut rate try to expain these cycica variations. Apart from the modes described above, severa other modes were tested: A potentia variabe is the difference between issuer and debt rating in the year before defaut. 22 If the issuer rating is better than the debt rating, the LGD of this bond is expected to be smaer than the LGD of bonds with an issuer rating equa to or worse than the debt rating. Because issuers with an issuer rating better than the debt rating dispose of a higher borrower s creditworthiness, we can expect that 21 Atman et a. (2003) aso detected a reationship between the average LGD per year and the voume of defauted bonds. 22 For exampe the issuer rating coud be Aaa and the debt rating A.

8 Modeing Loss Given Defaut: A Point in Time -Approach 147 there is an additiona protection of the bond by the issuer. However, this variabe did not infuence LGD significanty. Aongside, the interactions between absoute and reative seniority were tested. As they are ony partiay significant they are not incuded in the mode. The interactions between issuer rating and absoute and reative seniority were incuded as we but do not show a significant infuence on LGD. Additionay, a finer sectora cassification is tested to distinguish the impact of severa sectors. This finer cassification does not have sufficient observations for a sectors so a mode with this fine cassification cannot be estimated. Moreover, other macroeconomic factors are integrated in the mode. They comprise the GDP (gross domestic product) growth and the index of eading indicators which are incuded in the modes contemporariy and with a time ag of 1 or 2 years. Furthermore, severa macroeconomic variabes such as the unempoyment rate, the consumer sentiment index, the yied of the consumer sentiment index and different interest rates are tested with severa ags. The average LGD per year is incuded with a time ag of 1 year in the mode. These variabes do not affect LGD significanty when the defaut rate 1 year before defaut is aso incuded in the modes. Atman et a. (2001, 2003) receive simiar resuts. They concude that fundamenta macroeconomic variabes do not have a significant infuence on the average LGD in a mutivariate context if the mode contains the defaut rate. The variance of the error term b 2 2 is 1.9266 if a mode without expanatory variabes is used. Ony the constant term refecting the average eve of the transformed vaue of LGD is taken into account in this mode. In modes I and II the variance of the error term decines sighty to about 1.7336 and 1.7327, respectivey. This can be attributed to the improved estimation of LGD incuding issuer specific and macroeconomic variabes and thus to a decreasing prediction risk. In mode II, the variance of the random time effect b 1 2 decreases because appropriate macroeconomic factors have been integrated compared to mode I. This resut indicates that the integration of the defaut rate eads to a decrease in the variance of the random effect. Taking (8.1) into account, the variance of the transformed LGD s 2 and the correation o for two different borrowers in the same year are examined. A standard deviation ^s of 1.4883 for a mode without expanatory variabes, 1.4407 for mode I and 1.4208 for mode II is obtained. 23 The correation ^o between the predicted LGDs for next year is 16.48% in mode I. It decines to 14.16% in mode II because of the effect of systematic economic risk factors. 24 Finay, it shoud be mentioned that the variance estimates ^s 2 for modes I and II are sti high. This resut indicates that there may be further important issuer specific variabes which expain the variation of LGD. Exampes are baance sheet variabes not avaiabe in Moody s dataset. 23 s 2 ¼ b 2 1 þ b2 2. 24 o ¼ b 2 1 =s2.

148 A. Hamere et a. 8.4 Concusions In most empirica anayses concerning LGD, the distribution of LGD is impied to be constant and LGD is generay estimated using historica averages. Therefore, the individua vaues of LGD of issuers within a certain time period as we as the vaues of LGD over time shoud deviate ony randomy from a certain mean. Such an assumption seems to be unreaistic given the fact that in times of a recession, not ony the creditworthiness of the borrowers decines and PDs rise, but that aso LGD is systematicay higher. In this chapter a dynamic approach which generaises other approaches is presented. LGD is modeed depending on issuer and bond specific as we as macroecomic factors. As the variabes are agged, the LGDs for the next year can be predicted on the basis of vaues that are known at the time the prediction is made. Reduced uncertainty in the prediction of LGD is important for the determination of LGD, not ony for Base II but aso for interna risk management using credit portfoio modes. At a given state of the economy, more precise predictions about the economic capita can be made than using historica averages. Furthermore, in a credit portfoio mode, the prediction uncertainty can be taken into account at the simuation of the predicted oss distribution, e.g. resuting from the estimation of the parameters ^b and ^g. In a next step, further bond specific performance figures that coud not be reproduced in the dataset at hand wi be anaysed. This coud ead to a further reduction of prediction uncertainty, which is reativey high in comparison to PD predictions. If banks have a database which is arge enough to estimate individua LGDs, the mode presented in this chapter can be used. Athough there may be other factors infuencing LGD in a bank, e.g. type of coatera (financia coateras, rea estate etc.), the LGD can be estimated individuay using an econometric approach. The point-in-time predictions of LGD can aso be used to predict downturn LGDs demanded by Base II using downturn states of the macroeconomic variabes. At present there are reativey few studies for the determination of recovery rates and LGD on the basis of individua data. Moreover, the avaiabiity of data is restricted. Therefore, further research is necessary in this area. Appendix: Macroeconomic Variabes Interest Rate Fed Fund monthy Interest Rate Treasuries, constant maturity 6 months, nomina, monthy Interest Rate Treasuries, constant maturity 1 year, nomina, monthy Interest Rate Treasuries, constant maturity 5 years, nomina, monthy Interest Rate Treasuries, constant maturity 7 years, nomina, monthy Interest Rate Treasuries, constant maturity 10 years, nomina, monthy Interest Rate Conventiona mortgages, fixed rate monthy (continued)

8 Modeing Loss Given Defaut: A Point in Time -Approach 149 Commercia bank interest rates, 48-month new car, quartery Commercia bank interest rates, 24 months persona, quartery Commercia bank interest rates, a credit card accounts, quartery Commercia bank interest rates, Credit card accounts, assessed interest Interest Rate, new car oans at auto finance companies, monthy Interest Rate, bank prime oan, monthy Civiian Labour Force Leve Empoyment Leve Unempoyment Leve Unempoyment rate Initia Caims for Unempoyment Insurance Chaenger Report, Announced Layoffs Mass Layoffs Manufacturing Data: Shipments Tota Manufacturing New Orders Tota Manufacturing Unfied Orders Tota Manufacturing Inventory Tota Manufacturing Inventory to shipments Tota Manufacturing Capacity Utiization tota Business Bankruptcy Fiings Non-business Bankruptcy Fiings Tota Bankruptcy Fiings Dow Jones Industria Index S&P500 NASDAQ100 Price Indices: GDP Impicit Price Defator (2000 ¼ 100) Consumer Price Index, A Urban Consumers; U.S. city average, a items Producer Price Index; U.S. city average, Finished Goods Gross Domestic Product Gross Private Domestic Investment Percent Change From Preceding Period in Rea Gross Domestic Product Pubic Debt Tax Revenues Uni Michigan Consumer Sentiment Index PMI (Purchase Manager Index, Institute for Suppy Management) Retai Saes tota (exc. Food Services) Revised Estimated Monthy Saes of Merchant Whoesaers Business Cyce Indicator: Index of Leading Indicators (The Conference Board) Average crude oi import costs (US$/barre) Average defaut rate of issuers at the bond market References Acharya V, Bharath S, Srinivasan A (2007), Does Industry-Wide Distress Affect Defauted Firms? Evidence from Creditor Recoveries, Journa of Financia Economics 85, pp. 787 821. Atman E (2006), Defaut Recovery Rates and LGD in Credit Risk Modeing and Practice: An Updated Review of the Literature and Empirica Evidence, Working Paper, New York University.

150 A. Hamere et a. Atman E, Kishore V (1996), Amost Everything You Wanted to Know About Recoveries on Defauted Bonds, Financia Anaysts Journa 52, pp. 57 64. Atman E, Resti A, Sironi A (2001), Anayzing and Expaining Defaut Recovery Rates. A Report Submitted to The Internationa Swaps & Derivatives Association, December. Atman E, Brady B, Resti A, Sironi A (2003), The Link between Defaut and Recovery Rates: Theory, Empirica Evidence and Impications.http://pages.stern.nyu.edu/~eatman/Link_between_Defaut_and_Recovery_Rates.pdf. Araten M, Jacobs M, Varshney P (2004), Measuring LGD on Commercia Loans: An 18-Year Interna Study, The RMA Journa 86, pp. 28 35. Asarnow E, Edwards D (1995), Measuring oss on defauted bank oans: A 24-year study, The Journa of Commercia Lending 77, pp. 11 23. Batagi B (1995), Econometric Anaysis of Pane Data, Chichester, Wiey. Base Committee on Banking Supervision (2004), Internationa Convergence of Capita Measurement and Capita Standards: A Revised Framework, June. Base Committee on Banking Supervision (2005), Studies on the Vaidation of Interna Rating Systems Revised version, Working Paper No. 14, May. Bruche M, Gonzaez-Aguado C (2010), Recovery Rates, Defaut Probabiities, and the Credit Cyce, Journa of Banking and Finance 34, pp. 754 764. Carty L, Lieberman D (1996), Defauted Bank Loan Recoveries, Moody s Specia Report, November. Carty L, Hamiton D, Keenan S, Moss A, Muvaney M, Marshea T, Subhas M (1998), Bankrupt Bank Loan Recoveries, Moody s Specia Comment, June. D umann K, Trapp M (2004), Systematic Risk in Recovery Rates An Empirica Anaysis of US Corporate Credit Exposures, Deutsche Bundesbank Discussion Paper, 02/2004. Frye J (2000a), Depressing Recoveries, Risk 13(11), pp. 108 111. Frye J (2000b), Depressing Recoveries, Poicy Studies, Federa Reserve Bank of Chicago. Grunert J, Weber M (2009), Recovery Rates of Commercia Lending: Empirica Evidence for German Companies, Journa of Banking and Finance 33, pp. 505 513. Gupton G, Stein R (2002), LossCacTM: Mode for Predicting Loss Given Defaut (LGD), Moody s Rating Methodoogy, February. Gupton G, Stein R (2005), LossCac V2: Dynamic Prediction of LGD, Moody s Rating Methodoogy, January. Gupton G, Gates D, Carty L (2000), Bank Loan Loss Given Defaut, Moody s Specia Comment, November. Hamiton D, Carty L (1999), Debt Recoveries for Corporate Bankruptcies, Moody s Specia Comment, June. Hu Y, Perraudin W (2002), The Dependence of Recovery Rates and Defauts, Working Paper, Birkbeck Coege, February. Pykthin M (2003), Unexpected Recovery Risk, Risk 16, pp. 74 78. Roesch D, Scheue H (2008), The Empirica Reation between Credit Quaity, Recoveries, and Correation in a Simpe Credit Risk Mode, Working Paper, University of Hannover and University of Mebourne. Sch onbucher P (2003), Credit Derivatives Pricing Modes: Modes, Pricing and Impementation, Chichester, Wiey. Schuermann T (2004), What Do We Know About Loss-Given-Defaut? London, Risk Books. Varma P, Cantor R (2005), Determinants of Recovery Rates on Defauted Bonds and Loans for North American Corporate Issuers: 1983 2003, Journa of Fixed Income 14, pp. 29 44. Wofinger R, Tobias R, Sa J (1994), Computing Gaussian Likeihoods and their Derivatives for Genera Linear Mixed Modes, SIAM Journa of Scientific and Statistic Computing 15(6), pp. 1294 1310.

Chapter 9 Estimating Loss Given Defaut: Experience from Banking Practice Christian Peter 9.1 Introduction Modern credit risk measurement and management systems depend to a great extend on three key risk parameters: probabiity of defaut (PD), exposure at defaut (EAD), and oss given defaut (LGD). PD describes the probabiity that the ending institution wi face the defaut of some obigor or transaction. EAD gives an estimate of the exposure outstanding at the time of the defaut, aso indicating the maximum oss on the respective credit products. 1 Finay, LGD measures the credit oss a bank is ikey to incur due to an obigor defaut. In its advanced interna rating based approach (IRBA), the New Base Accord (Base II) underpins the importance of these key parameters by aowing financia institutions to appy their own estimates for PD, EAD, and LGD in the computation of reguatory capita. Since the risk weight of a credit faciity is inear in LGD, the bank s abiity to appropriatey estimate LGDs for its portfoios wi directy affect the amount of reguatory capita required under Base II. LGD numbers may, however, not ony pay a significant roe in interna credit risk management and future reguatory reporting, but may aso be used in accounting. For exampe, a bank may want to appy modified LGDs in its fair vaue as we as impairment computations required for IAS/IFRS. 2 Despite a these fieds of appication, LGD estimation has gained reativey itte attention in the iterature. 3 1 This artice wi use the expressions credit product and (credit) faciity interchangeaby as generic terms for a credit risk bearing instruments of a bank. 2 Internationa Accounting Standard/Internationa Financia Reporting Standard. 3 See for exampe Atman et a. (2005) or the artices avaiabe at http://www.defautrisk.com The materia and opinions presented and expressed in this artice are those of the author and do not necessariy refect views of KfW Bankengruppe or modes appied by the bank. I woud ike to thank a coeagues working with me on the LGD topic for the many fruitfu discussions during that time. C. Peter KfW Bankengruppe e-mai: Christian.Peter@Web.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_9, # Springer-Verag Berin Heideberg 2011 151

152 C. Peter This artice approaches LGD estimation from a perspective gained in banking practice, intending to address not ony the estimation probem itsef but aso to touch on some aspects of the deveopment process as we as the ater appication of these numbers. By doing so the artice rather concentrates on practica aspects of the topic than on statistica detais. The artice is organized as foows: The first section discusses the requirements arising from different domains of appication for LGD estimates. Economic oss and LGD are introduced next. The foowing section presents a short survey of different approaches for LGD estimation. A mode for workout LGD as we as the design of an LGD mode for performing and defauted exposures is discussed in the next three sections. Finay, the artice coses with some concuding remarks. 9.2 LGD Estimates in Risk Management A bank may appy LGD estimates for different domains of appication, which often impose different requirements on the definition of the performance number and its estimation procedures. Reguatory requirements as defined in BCSB (2004) are surveyed in Sect. 9.2.1. Afterwards, Sect. 9.2.2 outines further requirements which may be raised from risk management and accounting perspective. 9.2.1 Base II Requirements on LGD Estimates: A Short Survey BCBS (2004) defines severa requirements on LGD estimates eigibe for determining reguatory capita. The foowing provides a short survey: 4 Scope. Appication of foundation IRB approach requires LGD estimates for retai exposures ony (} 331). The advanced IRB approach aso aows banks to use their own estimates for corporate, sovereign, and bank exposures (}} 297 and 298). 5 Defaut definition (}} 452 457). The reference definition of defaut given in BCBS (2004) provides the basis for LGD estimation. When using interna or externa oss data inconsistent with this definition, appropriate adjustments have to be made. 4 See BCBS (2004) for the fu text as we as additiona rues not mentioned here (for exampe, concerning documentation, stress tests, overrides, etc.). The reader shoud aso take the respective reguations of nationa supervisors into account. 5 For purchased receivabes, see }} 364 and 367.

9 Estimating Loss Given Defaut: Experience from Banking Practice 153 Loss definition (} 460). LGD is based on economic oss; see Sect. 9.3 for detais. LGD estimates (}} 468 471). A bank must estimate an LGD for each faciity that aims to refect economic downturn conditions where necessary to capture the reevant risks (downturnlgd).the ong-time, defaut-weighted average of oss rate given defaut cacuated based on the average economic oss of a observed defauts [...] for that type of faciity provides a ower imit for LGD estimates. If existent, cycica variation has to be taken into account. Any significant dependence between the risk of the borrower and the coatera or its provider as we as the effect of currency mismatches must be considered in a conservative manner. LGD estimates must be grounded in historica recoveries and, where appicabe, must not soey be based on the coatera s estimated market vaue. An institute must fufi certain requirements on its coatera management processes for a coatera that is recognized in the bank s LGD estimates. For defauted exposures, banks have to determine a best estimate LGD, which is based [...] on the current economic circumstances and faciity status, as we as a conservative estimate refecting [...] the possibiity that the bank woud have to recognize additiona, unexpected osses during the recovery period. Data requirements (}} 472 473). The data basis shoud ideay cover at east one economic cyce, but must be no shorter than 7 years for sovereign, bank, and corporate exposures or 5 years for retai exposures, respectivey. Assessing the effect of guarantees and credit derivatives (}} 480 489). Banks are aowed to refect the effect of guarantees through adjustment of either PD or LGD estimates. The respective adjustment criteria must be ceary specified, pausibe, and appropriate. The bank must adopt the chosen technique in a consistent way (both over time and across different types of guarantees). Furthermore, it must assign a rating to each guarantor, fufiing a minimum requirements defined for borrower ratings. Except for certain types of obigors, guarantors, and instruments, the adjustment of PD or LGD is restricted in a way such that the risk weight of the guaranteed exposure need not be ower than the risk weight of a comparabe direct exposure to the guarantor (no recognition of doube-defaut effects). There are no restrictions on eigibe guarantors. Guarantees must fufi certain standards (for exampe, evidenced in writing, noncanceabe on the part of the guarantor, etc.) to be eigibe. Vaidation (}} 500 505). Banks must have a robust system in pace to vaidate the accuracy and consistency of rating systems, processes and a reevant risk components. Comparisons between reaized and estimated LGDs must be performed reguary (at east annuay) to demonstrate that reaized LGDs are within the expected range. Banks must aso use other quantitative vaidation toos and comparisons with reevant externa data sources. They must demonstrate that methods do not vary systematicay with the economic cyce. Furthermore, the bank must define reaction standards for the case that deviations between reaized and estimated LGDs turn to be significant enough to question the vaidity of the estimates.

154 C. Peter Interna Risk Management Interna Reporting Limit Management Credit Portfoio Mode Credit Risk Reports Credit Approva Authority Pricing Performance Reports Reguatory Reporting Reguatory Capita (Tier 1) Discosure (Tier 3 ) EAD / LGD Engine Accounting Information Coatera and Guarantee Information Master Data Credit Loss Information... NPL Process Accounting IAS Fair Vaue IAS Impairment Fig. 9.1 LGD estimates data sources and domains of appication 6 9.2.2 LGD Estimates in Risk Management and Other Appications Whie Base II provides the focus of this book, banks may use LGD numbers in many appications apart from reguatory reporting. Figure 9.1 depicts some of these appications as we as the various connections between them. A bank s interna credit risk reporting and management processes require LGD estimates for different purposes: Interna reporting (risk bearing abiity, performance measurement, etc.), pricing, the bank s credit approva authority reguations, and imit management may be some of these appications. Accounting can become another fied of appication for LGD estimates or derivatives of them. When considering IAS/IFRS, LGD figures may enter fair vaue computations and impairment tests. IAS asks banks to discose fair vaues for financia assets and iabiities at east in the notes of the annua statement. 7 These numbers can, for exampe, be computed appying a discounted cash fow mode, with LGD numbers used to adjust cash fows for credit risk. 8 Impairment tests provide further possibiities for connecting accounting and credit risk management processes. Genera provisions can be computed using a modified 9 LGD number based on the finding that the concepts of incurred oss as 6 NPL is used as an abbreviation for non-performing oan. 7 See IAS 39.8, IASB (2005), for a definition of fair vaue. 8 As an aternative to cash fow adjustment, one may appy a discount rate adjustment approach. In this case, one may refer under certain circumstances to simiar risk-adjusted discount rates as used for LGD estimation; see Sect. 9.6.2.4. 9 Some of the necessary modifications are addressed beow.

9 Estimating Loss Given Defaut: Experience from Banking Practice 155 defined by IAS/IFRS and expected oss as used for credit risk measurement are quite simiar. 10 Furthermore, best estimate LGDs as required for reguatory purpose and specific provisions computed foowing the rues of IAS/IFRS are both based on expectations about future cash fows from a defauted faciity, its coatera, and guarantees. Therefore, one may derive both specific provisions and best estimate LGDs from the same information base. This wi be discussed in more detai in Sect. 9.7. A great part of the functionaity required for these three domains of appication, i.e. reguatory reporting, interna risk reporting and management, as we as accounting, is identica. However, there are differences due to diverging intentions stabiity of the bank in case of Base II and objective reporting of the bank s assets in case of IAS/IFRS. This may concern the definition of EAD as we as the definition of LGD. For exampe, impairment considers book vaue as EAD. Fair vaue computations may not take future drawings into account, whie these are part of Base II compiant exposure at defaut. Risk management, on the other hand, may recognize future redemption to a arger extent than reguatory requirements aow. In addition to the impact of different EAD definitions, the oss definition underying LGD may sighty vary with the domain of appication. The eve of conservatism underying the estimates wi be different due to diverging intentions. Definition of oss components can differ; for exampe, interna costs may not be part of IAS numbers, whie Base II and interna appications wi recognize them. Furthermore, one may decide to consider separate LGDs for different credit events, for exampe, poitica risks in interna modes. 11 In addition to the 1-year horizon considered in Base II, a bank may be interested (at east for some appications, possiby incuding reguatory capita) in a dynamic, muti-period projection of risk numbers. Another potentia fied of deviations is the assessment of risk mitigation effects. Deaing with different definitions of EAD and LGD can cause some confusion in interna communication despite their different domains of appication and therefore requires bridging one EAD or LGD number into the other in order to expain the differences. Furthermore, the compexity of an LGD engine, which takes a these different requirements into account, can be high, aso resuting in increased costs of deveopment and maintenance. Before stating bank specific additiona requirements, one shoud therefore carefuy check whether the expected gain in expanatory power rectifies the corresponding effort and costs. 9.3 Definition of Economic Loss and LGD Base II requires measuring economic oss as a basis for LGD estimation. [...] When measuring economic oss, a reevant factors shoud be taken into account. This must incude materia discount effects and materia direct and indirect costs 10 Due to restricted data avaiabiity, differences might be greater in theory than in banking practice. 11 This wi be necessary if a bank defines its PD ratings as oca currency ratings.

156 C. Peter associated with coecting on the exposure. [...] (see BCBS (2004), } 460). The directive ony mentions basic components of economic oss whie eaving the exact definition to the banks. One may think of economic oss as the change in a faciity s vaue due to defaut, 12 i.e. EcoLoss j ðt DF Þ ¼ V j ðt DF ; pþ V j ðt DF ; npþ (9.1) with V(t DF,(n)p) describing the vaue of a (non)performing faciity j in t DF, the time of defaut. Foowing the current discussion, the vaue of the performing faciity, V j (t DF,p), is generay approximated by the amount outstanding at defaut pus eventua further drawings after defaut, i.e. by EAD. 13, 14 The residua vaue of the defauted faciity, V j (t DF,np), can be expressed as the net present vaue of a recoveries from the exposure diminished by a direct and indirect costs arising from defaut. The LGD of a faciity j then foows as the ratio of economic oss to exposure at defaut, i.e. LGD j ðt DF Þ¼ EAD jðt DF Þ NPV Rec j ðtþ;t t DF EAD j ðt DF Þ þ NPV Costsj ðtþ;t t DF (9.2) with NPV(.) the net present vaue, Rec j (t) and Costs j (t) a recoveries and costs observed at t, respectivey. Negative economic oss or LGD indicate a gain. Whie negative LGDs are sometimes observed in practice, LGD estimates are generay required to be greater than or equa to zero. This artice wi refer to reaisations of LGD as ex-post LGDs, whie estimates of oss quotas wi aso be named ex-ante LGDs. Recoveries after defaut resut from faciity or coatera sae, guarantees, bankrupt s assets, as we as restructured or cured exposures. Further unexpected sources of recoveries may sometimes aso be observed. Whie ex-post LGDs may incude a types of recoveries received for a defauted exposure, the reference dataset (RDS) for mode deveopment shoud generay not refect extraordinary recoveries, for exampe stemming from non-eigibe coatera or guarantees, in order to avoid distortion. 15 Materia direct and indirect costs arising from the handing of a defauted exposure are, for exampe, externa and interna abour costs, ega costs, costs for forced administration, insurance fees, costs for storage, maintenance, repairs of 12 Note that differences in defaut definition wi therefore affect economic oss. 13 As an aternative, one might define V(t, p) as the net present vaue of a future recoveries and costs of the faciity in case of no defaut in t. Whie theoreticay appeaing, such a definition can be difficut to impement in practice. Furthermore, it woud aso require a respective definition of EAD as might be done in interna modes ony. 14 See, for instance, Chaps. 10 and 11 for more detais on EAD estimation. 15 Exemptions may be possibe if such extraordinary recoveries are observed on a reguar basis.

9 Estimating Loss Given Defaut: Experience from Banking Practice 157 assets, etc. Furthermore, one shoud incude ongoing costs, for exampe, corporate overhead. Refinancing costs resuting from incongruence of cash fows due to defaut may aso be considered if materia. 16 On the other hand, osses of future earnings (e.g., interest income) are generay not considered as part of economic oss. With respect to (9.1), one may recognize ony additiona costs, i.e. the difference between costs arising from the performing and the defauted exposure, respectivey. As mentioned above, economic oss and LGD used for IAS purpose shoud not incude interna costs. In order to recognize discount effects, a recoveries and costs have to be discounted. Since workout processes can be time demanding, the chosen discount rate may significanty affect the resuting economic oss and LGD; see Sect. 9.6.2.4. 9.4 A Short Survey of Different LGD Estimation Methods The foowing provides a short survey of main approaches for LGD estimation currenty discussed among academia and practitioners. When cassifying different LGD approaches, a first distinction can be made between subjective and objective methods. A bank may have insufficient data to rey soey on quantitative methods. This can occur for ow defaut portfoios, new products, or during the introduction of LGD methodoogy. In these situations, the bank may think of subjective methods primariy based on expert judgment as a vauabe source of information. Whie there seems to be no specia iterature on subjective methods in LGD estimation, techniques known from other fieds of appication can easiy be adopted. Interviews with experts from different units of the financia institute, comparisons with simiar portfoios, or scenario techniques may hep to deveop an idea of the oss quotas one shoud expect to observe. As far as possibe, the bank shoud incorporate a kinds of avaiabe oss (reated) information into subjective methods. Subjective methods may aso prove vauabe for a vaidation of the resuts obtained from appying one of the objective methods described next. Objective methods can be further cassified as being either expicit or impicit, depending on the characteristics of the data sources on which they are based. Datasets anaysed in expicit methods aow for a direct computation of LGDs. The so-caed market LGD approach, a first expicit method, is appied by comparing market prices of bonds or marketabe oans shorty after defaut with their par vaues. To compute workout LGDs, it is necessary to discount a recoveries and costs observed after defaut to determine the vaue of the defauted faciity, which is then compared with the defauted exposure. 16 Incongruence can ead to osses or gains depending on the eve of interest rates at the time of credit granting and defaut. It is therefore sometimes argued that gains wi offset osses due to the mean reversion property of the interest rate.

158 C. Peter Different from expicit approaches, impicit methods rey on data sources which do not aow for a direct LGD computation but impicity contain LGD reevant information. This information has to be extracted appying appropriate procedures. Two approaches which have been discussed in banking practice and in the iterature are impied market LGD and impied historica LGD method, respectivey. The idea of the impied market LGD approach is to derive LGD estimates from market prices of non-defauted bonds. 17 The spreads observed for these instruments at the market express among other things the oss expectation of the market, which may be broken down into PD and LGD. Whie theoreticay appeaing, it may be difficut to separate adequatey the credit risk component of the spread and break it down into PD and LGD. The computation of impied historica LGDs is described in the Base II framework as one approach to determine LGDs for retai portfoios (see BCBS (2004), } 465). This approach invoves deriving LGDs from reaized osses and an estimate of defaut probabiities. Except for impied market LGDs, which may deiver at east theoreticay directy (or with minor modifications) estimates for non-performing faciities, a other concepts considered before at first hand deiver ex-post LGDs. The rest of this section wi consider different approaches for estimating ex-ante LGDs. The main interest of a bank is generay to derive estimates for workout LGDs, since these best refect its osses. Ex-post observations of market LGDs may aso be used in mode deveopment; however, doing so may require appropriate adjustments since market LGDs incude components as risk premiums for unexpected osses, which may not be considered in workout LGDs. Furthermore, required components ike the institute s specific workout costs are not part of these oss quotas. As a first, simpe approach, one may consider an ex-ante LGD estimation procedure where LGDs are assigned top-down to exposures based on faciity grades or poo characteristics. Such a procedure requires a segmentation of the portfoio under consideration into a sma number of, in terms of their oss quota, reativey homogeneous groups of faciities. Statistica anaysis as we as expert judgment provides the basis to identify these segments and to deveop the necessary assignment rues. Since individua characteristics of faciities can ony be recognized to a imited extent in such a two-stage approach, 18 one wi expect reasonabe performance especiay for highy standardized oan programs or retai portfoios. For portfoios of ess standardized faciities, one may presuppose better performance from direct or bottom-up estimation approaches. Higher individua credit voumes and smaer portfoio sizes wi often be other arguments rectifying the deveopment and appication of more sophisticated estimation procedures. The basic idea of direct estimation techniques is to estimate LGDs based on a mode, 17 If avaiabe, one may aso consider market vaues of oans or credit derivate instruments. 18 See CEBS (2005), } 234.

9 Estimating Loss Given Defaut: Experience from Banking Practice 159 which takes individua characteristics of each faciity, its coateraization, as we as other important risk factors expicity into account. As for PD prediction, empirica statistica or simuation-based modes may be appied. Simuation approaches are often used for speciaized ending transactions where the abiity of the borrower to fufi his obigations primariy depends on the cash fows generated by the financed object. An individua mode of the transaction that describes the free cash fows generated by the financed object and therefore the abiity to pay interest and principa as a function of important risk factors provides the basis for the simuation. By simuating different scenarios of the transaction s progress, an institute wi be abe to derive estimates for PD, EAD, and LGD. Whie such approaches provide great fexibiity, costs for modeing a specific transaction and performing the simuation can be high, depending on the structure of the simuation too. LGD estimates based on empirica statistica modes can be generated by appying a singe equation or a component-based approach. Whie the first approach intends to describe LGD by a singe (for exampe regression) mode, the atter one consists of a set of submodes each describing a certain component of LGD, e.g. the recovery rate for a certain coatera type or costs of certain workout activities. LGD estimates are then generated by appropriatey aggregating the resuts of the estimates for these components. Statistica modes for LGD or singe LGD components can aso be used in simuations. Banks may appy different techniques depending on the characteristics of the respective portfoio segment, its importance with respect to the whoe portfoio, and the avaiabiity of oss data. This aows on the one hand measuring LGDs for different products with customized estimation procedures. On the other hand, however, it can make a consistent measurement of credit risk over the whoe portfoio more difficut. 9.5 A Mode for Workout LGD Consider the situation that a bank faces after a borrower s defaut. Whie defaut itsef marks a unique reference point for oss measurement, the workout of a defauted credit faciity as we as the resuting oss can vary substantiay. However, one wi probaby observe a certain pattern of typica deveopments, caed after-defaut scenarios in this paper. Tabe 9.1 provides a reasonabe set of such scenarios. Depending on the banks portfoio as we as its workout strategy, the number and definition of after-defaut scenarios may sighty differ. Whie the oss observed within a certain scenario may be simiar for different (comparabe) faciities, it wi generay be impossibe to know the after-defaut scenario in advance. One may therefore consider the oss quota of a faciity j, LGD j, as a random variabe foowing a mixture distribution. With SC j a discretevaued random variabe describing the occurrence of after-defaut scenarios and LGD j (sc i ) a second, continuous-vaued random variabe describing the oss of

160 C. Peter Tabe 9.1 A set of possibe after-defaut scenarios Scenario sc i Definition and expanation a Cure The defauted entity cures after a short time and continues to fufi its contractua obigations. No significant osses; no changes in the structure or conditions of the credit faciities. Restructuring The defauted entity recovers after a restructuring of its faciities. Repossession and sae of coatera may sometimes be part of the restructuring. Loss amount may vary; customer reationship maintained. Liquidation A credit products of the defauted entity are iquidated, i.e. sae of oans, coatera (if avaiabe), etc. Loss amount generay higher than observed for restructuring; end of customer reationship. a Scenarios wi generay be defined with respect to the defauted entity (i.e. for borrower or guarantors) and may therefore not aways correspond with what is observed for a singe credit product a faciity depending on the scenario sc i and d(.) the indicator function, 19 LGD j can be defined as 20 LGD j ¼ X i d sci SC j LGDj ðsc i Þ (9.3) Coatera and guarantees wi generay have a strong impact on the oss quota reaized for a defauted faciity. Consider a faciity, which is secured by n 1 risk mitigation instruments. 21 Each of these instruments k coateraizes sq k percent of the exposure. One can now break down the exposure into m n subexposures, each coateraized by at east one instrument and an additiona part, sq 0, which remains unsecured. The percentage of oss reaized on each subexposure sq,0 m, may depend on the respective risk mitigation instrument as we as the afterdefaut scenario currenty under consideration. The tota oss quota in scenario sc i is therefore given by LGD j ðsc i Þ ¼ X sq j; LGD j; ðsc i Þ (9.4) 0m where LGD j, (sc i ) describes the percentage of oss observed on an (un)secured subexposure of size sq j,. Since the breakdown in (9.4) is equivaenty performed for each of the after-defaut scenarios, one may aternativey write 19 I.e. d sc (SC) ¼ 1 for SC ¼ sc and d sc (SC) ¼ 0 otherwise. 20 To simpify the presentation, time references are eft out in (9.3) as we as in most of the formuas foowing. It is generay assumed in this artice that one intends to predict the oss quota for a defaut occurring within a time interva T ¼ [t a,t e ) given the information up to t 0 (the time where the computation takes pace), i.e. LGD j ¼ LGD j (T t 0 ). 21 This artice uses the expression risk mitigation instrument (rmi) as a genera notion for a kind of coatera and guarantees.

9 Estimating Loss Given Defaut: Experience from Banking Practice 161 LGD j ¼ X sq j; LGD j; (9.5) 0m with LGD j; ¼ X i d sci SC j LGDj; ðsc i Þ (9.6) With respect to (9.2), LGD j, (sc i ) can be expressed as LGD j; ðsc i Þ ¼ max 0 ; 1 RR j; ðsc i ÞþCosts j; ðsc i Þ ; (9.7) with RR j, (sc i ) and Costs j, (sc i ) the percentage of recovery and costs on exposure sq of faciity j in scenario sc i. Equation (9.5) foows the structure of the formua provided in BCBS (2004) for risk mitigation. The extension of considering after-defaut scenarios may prove hepfu as a theoretica mode as we as for anaysing the characteristics of observed economic oss or mode deveopment. The reativey simpe structure of the mode, which demonstrates the main idea whie hiding most of the compexity of the underying statistica modes, wi aso be easy to communicate within the bank. This may increase acceptance of the estimation procedures, which may appear as a back box for credit anaysts. Ex-ante estimates, however, are often generated based on a reduced form of the mode presented here. 9.6 Direct Estimation Approaches for LGD The foowing considers direct estimation approaches for LGD. Setting up such a procedure requires a description of the components of economic oss, i.e. recoveries on secured and unsecured exposures as we as costs, in terms of appropriate expanatory variabes with respect to the requirements imposed by different domains of appication, i.e. Base II, IAS, or interna risk management, respectivey. The deveopment process for an LGD estimation procedure can generay be structured aong the foowing steps: 1. Data coection, pre-processing and anaysis 2. Mode design and estimation 3. Mode vaidation Some steps of the deveopment process may have to be repeated severa times before a satisfactory soution is found. Figure 9.2 depicts a typica series of projects a bank may set up in order to deveop an LGD engine. The impementation of a credit oss database is often the first step. It creates the basis for a systematic coection of oss data required for mode deveopment. The respective project generay incorporates (or is foowed by) activities to transfer (a part of) the bank s oss history

162 C. Peter Time Project Vaidation Processes Deveopment Impementation Test Vaidation (annua) Additiona requirements new products new reguation rues etc. According to demand Project LGD Methodoogy Deveopment Impementation Test Project Credit Loss Database Deveopment Impementation Test LGD Engine Enhancement & Maintenance of LGD & Vaidation Methodoogy Deveopment Impementation Test Further processes oss data coection etc. Initia project phase Continuous process Fig. 9.2 Typica structure of an initia phase to set up an LGD engine and the foowing annua vaidation process from paper fies to the database. The LGD estimation mode as we as the required vaidation procedures and processes can be deveoped afterwards. Foowing the initia project phase, the LGD engine wi be subject to reguar enhancement and maintenance activities. Such activities may be triggered, for exampe, through the introduction of new products or reguatory changes as we as the resuts of the annua vaidation. The foowing concentrates on the first two steps of the deveopment process for an LGD engine. The presentation starts with a short discussion of some aspects of data coection, mainy through a description of typica eements of a credit oss database. Afterwards, different aspects of mode deveopment are discussed. 9.6.1 Coecting Loss Data: The Credit Loss Database One wi generay consider the bank s own past oss experience as the most vauabe information avaiabe for the deveopment of an LGD estimation procedure, since it directy refects the characteristics of the institute s credit products and processes (e.g., origination, monitoring, and workout processes). Banks therefore often set up a credit oss database in order to coect a reevant information concerning defauted entities and their credit exposures. The aggregate of a information concerning a defauted entity and its exposures is often caed a oss fie. A oss fie wi generay incude not ony information about the time after the occurrence of a defaut but aso information about the time before. Information about the time after defaut occurrence consists of

9 Estimating Loss Given Defaut: Experience from Banking Practice 163 Possibe further drawings after defaut A recoveries reated to the defauted entity, its credit faciities, and risk mitigation instruments A costs arising from the workout process Additiona information about the workout process (for exampe, events and remarks as we as identifiers of restructured faciities and repossessed assets, which ater aows to identify these objects within the bank s IT-systems) Further information coected within the credit oss database incudes cash fows before defaut (or exposure at the time of defaut), master data, rating history, coatera vaues, etc. The ater mode deveopment and estimation process generay requires additiona information, for exampe, time series of macroeconomic variabes or version numbers of the appied risk measurement toos (ratings toos, coatera vauation toos, etc.), which may aso be incorporated in the database. It wi often take some time to reaize a cash fows from cured or restructured credit faciities as we as from repossessed assets. Since workout usuay ends much earier and credit products or assets are then transferred from the workout unit to another unit within the bank or an externa service provider, oss fies wi often be cosed by the end of the respective workout activities. Cured or restructured credit faciities as we as repossessed assets are vaued by that time and the resut stored as non-cash recovery in the oss fie. 22 Since the number of oss observations is often sma and oss data coming from the atest defauts aso contains the most up-to-date information about current oss quotas, it appears attractive to incude incompete oss fies as eary as possibe in the reference dataset for mode deveopment. The decision as to whether an incompete oss fie shoud be incorporated in the reference dataset wi generay be made on a case-by-case basis and can aso depend on the appication. A reasonabe decision criterion may often be defined based on the uncertainty sti inherent in the vaue of economic oss due to the incompeteness of the oss case. Often, the end of the workout process is a reasonabe time to incude a oss fie into the reference dataset. A component-based estimation approach may provide possibiities for even earier usage of incompete oss data; for exampe, by considering incompete oss fies in the reference dataset of some LGD components ony. 23 Whie the use of incompete oss fies wi make oss data avaiabe more quicky, this data, sti incorporating estimates, can ony be used to a imited extent, which may imit the benefit. 22 However, as mentioned above one shoud incude references into the oss fie in order to aow for a ater repacement of non-cash recoveries by the corresponding cash recoveries reaized from the respective cured or restructured faciities. Note that non-cash recoveries are generay estimates of future, uncertain cash fows. 23 For exampe, repossession and sae of coatera might aready be finished for a defauted credit product. The respective information can then be used to update the estimate of the recovery rate for the respective coatera type(s) whie at the same time the information required to re-estimate the recovery rate for unsecured exposure might sti be incompete.

164 C. Peter A number of further aspects shoud be considered during data coection and preprocessing; the foowing outines a few of them: Most of the requisite data can generay be found in existing IT-systems, aowing for the automatic coection of oss data. However, manua inputs are probaby necessary during the workout process. These wi incude most information about the workout process, i.e. events, remarks, etc. Whie remarks aow entering information in an unstructured way, events provide the possibiity of marking specified states and decisions, miestones, or turning points in order to structure the workout process for ater anaysis. 24 The extent to which such data must be added shoud be specified carefuy in order to get informative oss fies without causing too much extra work and costs. Since estimation procedures wi improve over time, it wi often be beneficia to coect a superset of the oss data currenty required for mode estimation. The degree of detai may be different depending on the business ine or credit product. This may, for exampe, resut in more detaied oss data coection for arge corporate than for retai exposures. Assuring the quaity of oss data can be more time-consuming than expected at first gance. Simpe automatic consistency checks might hep to detect irreguarities in the data; however, a arger part of the checks requires a deeper understanding of the workout processes as we as the oss cases themseves and therefore has to be done in coaboration with experts from restructuring and workout units. 9.6.2 Mode Design and Estimation The genera structure of an LGD estimation procedure often consists of the foowing three steps: 1. Data coection. Identification and coection of a data required to estimate LGD. 2. Pre-processing. Transformation of raw data into a form suitabe for the estimation of LGD or LGD-reated numbers. This may aready incude estimates for singe LGD components. 3. Generating estimates. Generation of LGD estimates by appropriatey assembing the resuts of pre-processing. In particuar, this incudes recognizing the risk mitigation effect of guarantees and coatera. As a by-product, the procedure may aso provide other usefu information, as for exampe statistics on the concentration in risk mitigation instruments, etc. Figure 9.3 shows the basic structure of an LGD engine as it might be impemented within a bank s IT-systems. Depending on the IT-infrastructure, institutes 24 An exampe of how this information may be used in LGD estimation is given in Sect. 9.6.2.3.

9 Estimating Loss Given Defaut: Experience from Banking Practice 165 Onine computation (vauation of singe transactions) Controer Data Coection Pre-processing Computation of LGD Data Bases Coatera Accounting data Ratings Master data... Fig. 9.3 Structure of an LGD engine Batch computation (vauation of portfoios) Data Warehouse may run various engines for different appications or portfoio segments or refer to a centra engine as depicted here. In the ater case, a controer may organize the computation of LGD estimates for different appications. Regression-type modes are generay preferred as a fexibe approach for modeing LGD or its components. Such approaches have been considered in severa pubications on LGD estimation; see for exampe Atman et a. (2003) or Chap. 8. Banks wi often suffer, at east during the initia years after introducing LGD estimation procedures, from an insufficient number of oss observations at east for certain parts of their portfoio. The need to rey on information from various sources, sometimes foowing different definitions of defaut and oss, and aso having different quaity characteristics, can make other, more simpe 25 approaches attractive. Capacity as we as time restrictions or priority settings among different portfoio segments are additiona reasons why banks may start with these approaches for some portfoios. Lookup-tabe based approaches wi often provide the basis for LGD estimation procedures in such situations. The idea here is to tabuate possibe vaues of some variabe of the mode, for exampe, a recovery or cost rate, or the resuting LGD numbers themseves, together with the respective seection criteria. For instance, a bank may tabuate recovery rates for unsecured exposures depending on customer type, faciity type, seniority, and region (see aso Tabe 9.2). Given such a tabe, the 25 Whie being simpe from a pure statistica point of view, setting up a procedure that generates reasonabe LGD predictions based on different types of information wi nevertheess often remain a demanding task.

166 C. Peter bank can easiy generate an estimate for the recovery rate of some exposure by reading the recovery vaue corresponding to these four characteristics. The deveopment of such a tabe requires first the identification and description of segments of simiar vaues for the considered variabe in terms of appropriate expanatory variabes. Afterwards, a representative vaue for the variabe under consideration has to be estimated for each segment. Both steps can be supported by expert judgment or other externa information sources if the bank s reference dataset is insufficient. One shoud expect such modes to capture ony a part of the (expicabe) variabiity of LGD numbers observed in practice. For exampe, it wi be difficut to describe the dynamics with respect to changes in macroeconomic variabes. This can resut in higher margins of conservatism and therefore rather conservative LGD estimates. On the other hand, ookup-tabe based approaches are more intuitivey understandabe, thus supporting interna communication and acceptance within the bank, which can be advantageous especiay during the introduction phase. They may therefore serve as a starting point for some portfoio segments when introducing an LGD estimation procedure. It is then a matter of further deveopments to successivey repace ookup-tabes with more sophisticated statistica modes wherever sufficient oss data can be made avaiabe and one expects significant improvements in the quaity of LGD estimates. However, designing an LGD engine in a way that easiy supports the migration from a simpe to a more sophisticated estimation procedure at a ater point in time can be compicated and may ead to increased foow-up costs. The foowing sections consider some aspects of the design of an LGD estimation procedure. The first section considers basic expanatory variabes for LGD. Afterwards, approaches to estimate the two main components of LGD, recoveries and costs, are described. The choice of appropriate discount rates is considered next. A ast section concudes this part with a short discussion on how the Base II requirements concerning the conservatism of LGD estimates can be recognized. It is beyond the scope of this artice to describe the whoe deveopment procedure in detai; the foowing wi therefore skip many technica detais which may be found in most statistica textbook. 9.6.2.1 Possibe Expanatory Variabes for LGD Estimation To identify appropriate expanatory variabes, aso named risk factors or risk drivers, one may start with a ist of possibe risk factors resuting from expert judgment, which are then tested during mode deveopment for their individua and joint expanatory power. In practice, the imited number of oss observations wi sometimes make a statistica anaysis difficut or even impossibe, and may therefore restrict the set of risk drivers that can be considered in an LGD mode. Tabe 9.2 summarizes some possibe expanatory variabes generay considered as possibe risk drivers when deveoping LGD estimation procedures. Most of them

9 Estimating Loss Given Defaut: Experience from Banking Practice 167 Tabe 9.2 Exampes of possibe expanatory variabes grouped by categories Category Expanatory variabes Borrower Customer type (sovereign, private entity, SME, corporate,...), country or region, industry, ega structure and capita structure of the entity, rating, etc. Credit faciity Seniority (senior, junior,...), debt type (oan, bond,...), transaction type (syndicated oan,...) and number of financing entities, exposure, financing purpose, degree of standardization, coateraization (LTV,...), etc. Coatera Type, current book or market vaue, vaue depreciation, age, mobiity (immobie, nationa or internationa mobie), producer, technica characteristics (for exampe, engine type of an airpane or gauge of a ocomotive), etc. Guarantee Guarantor (see ist of expanatory variabes required for borrowers as provided above), coverage, warranty causes, etc. Macroeconomic and other GDP growth rate, unempoyment rates, interest rates, FX rates, externa factors Bank interna factors price indices, ega system and institutions, etc. Versions of vauation procedures and toos, workout strategy, coateraization strategy, etc. can easiy be justified by intuition. 26 Furthermore, one wi expect some of these variabes to have expanatory power not ony for (singe components of) LGD but aso for PD and EAD, indicating dependences between these key parameters of credit risk. Borrower and credit product characteristics as for exampe industry, capita structure, and seniority may expain recovery rates on unsecured exposures in iquidation scenarios (i.e. from bankrupt s assets). They may aso indicate workout intensity as a proxy for workout costs. Depending on the regiona distribution of the portfoio, it coud be necessary to consider region or country as expanatory variabes. 27 Recoveries from coatera wi depend on the possibiity of repossessing and seing the respective assets. Depending on the market size and structure observed for a certain asset cass, the bank may have to accept discounts for distressed sae. Technica characteristics of the respective assets coud serve as an indicator for the eve of such discounts and may aso expain in part the costs of sae. Anaogousy, the vaue of a guarantee depends on the credit standing of the respective guarantor as we as on specific warranty causes. In case the guarantor defauts, recoveries can be expected to depend to a arge degree on the same expanatory variabes as mentioned above for unsecured exposures (i.e. borrower characteristics). 26 A comprehensive survey of empirica anayses can be found in Bennett et a. (2005); the foowing mentions ony a few of them. 27 Atman and Kishore (1996) and Acharya et a. (2004) found significant differences in recoveries of defauted bonds beonging to different seniority casses. The same authors report significant differences for ony some industry sectors, whie Araten et a. (2004) coud not find significant impact of industry (or region) on LGDs observed for oans.

168 C. Peter The macroeconomic situation at defaut wi generay infuence LGD, as was demonstrated by severa authors. 28 Base II expicity asks to take economic cyces into consideration. Depending on the regiona distribution of the institution s portfoio and the considered recovery source (e.g., a certain asset type), one may consider different economic variabes. Since defaut and recoveries from bankrupt s assets and coatera may both depend on the same macroeconomic variabes, an appropriate recognition of these dependences wi be important to avoid overestimating recoveries. 29 Other externa factors as jurisdiction and ega system can aso pay a roe when expaining engths and costs of workout activities as we as amount of recoveries. 30 As a ast group of expanatory variabes for LGDs, one shoud consider bank interna characteristics. Loss experience as we as LGD estimates wi refect to a certain degree characteristics of the bank s interna processes (e.g., origination, monitoring, and workout processes). For instance, a bank s workout strategy has a strong impact on the magnitude of recoveries and costs. Therefore, any change in the strategy may require modifications in the LGD estimation procedures in order to recaibrate them. For exampe, the a modification of a coatera vauation procedure may require a transformation of historica vauations and adjustments in estimated recovery rates for the respective asset type as we as modifications of the LGD estimation procedure. 31 9.6.2.2 Estimating Recoveries Recoveries are generay the main driver of LGD. With respect to (9.3) (9.7), one may define recovery rates as RR j; ðsc i Þ ¼ NPV CF j;ðsc i Þ (9.8) sq j; EAD j for (un)secured exposures of size sq j, EAD j observed for a oss case 32 j in the respective after-defaut scenario sc i. sq j, may be defined in different ways as wi be 28 Araten et a. (2004) report correation of unsecured exposures (but not of secured exposures) with economic cyce. Severa authors report dependences found in bond data, see for exampe Hamiton et a. (2006) or Atman et a. (2003). 29 Severa authors have anaysed the ink between defaut and LGD; see for exampe Frye (2000a, b), Atman et a. (2003), and D umann and Trapp (2004). 30 See for exampe Franks et a. (2004) for an anaysis of recovery processes and rates in the U.K., France, and Germany. Usefu information about doing business in different countries may aso be found at http://www.doingbusiness.org. 31 See Sect. 9.6.2.2 for more detais. The exampe aso demonstrates why the version number of a coatera vauation too may be important information within the credit oss database; see Sect. 9.6.1. 32 A oss case wi generay comprise a credit products of a defauted entity.

9 Estimating Loss Given Defaut: Experience from Banking Practice 169 Tabe 9.3 Recovery sources with respect to different after-defaut scenarios Scenario sc i Unsecured exposure Secured exposure Cure Recovery ¼ cured faciity Restructuring (I) no usage of rmi (II) usage of rmi Liquidation Recovery ¼ restructured faciity Recovery from bankrupt s assets Recovery ¼ restructured faciity C E Recovery from eigibe coatera or guarantee Recovery from eigibe coatera or guarantee A B D F discussed beow. NPV(CF) again denotes the net present vaue of a cash fows which are observed on the respective exposure. Assume for this section that recovery rates are determined without taking costs into consideration. Equation (9.8) can be used to generate ookup-tabes based on historica oss information or may aso be computed as part of the estimation procedure. The ater case may be attractive if the bank pans to consider the discount rate as an input parameter of the estimation procedure. 33 Tabe 9.3 summarizes recovery sources for the after-defaut scenarios shown in Tabe 9.1. Base II requires assigning LGDs to each faciity (see Sect. 9.2.1); however, in practice, recoveries can be observed on different, often more aggregated eves. These are generay credit entities (i.e. borrower and guarantors), faciities, and risk mitigation instruments. 34 For exampe, two oans of a defauted obigor may be coateraized by the same asset. In this case, the distribution of the asset s saes proceeds onto the oans is often ambiguous. Ex-post LGD computation as we as ex-ante estimation on a oan eve therefore require appropriate procedures to aocate recoveries to faciities. Since guarantees require, at east under Base II, a sighty different treatment, the foowing considers first exposures, which are either unsecured or secured by coatera. Afterwards, the risk mitigation effect of guarantees is considered in a separate subsection. A concuding third section outines additiona aspects of recovery rate estimation. Unsecured Exposures and Exposures Secured by Coatera Consider an exposure coateraized by an asset A i having a reference vaue V t (A i )at time t. For ex-ante estimation, the reference vaue wi ater generay be the resut of the most recent vauation of the asset. Ex-post, one may use either the ast vauation 33 Detais of this approach are considered in the next section for coatera recoveries. 34 The same hods true for other components of LGD, see for exampe Sect. 9.6.2.3.

170 C. Peter Tabe 9.4 Two approaches for estimating recovery rates of (un)secured exposures Approach 1 Approach 2 Secured exposure Unsecured exposure Estimate RR k of asset type k based on recoveries of A, B, D, and F a Estimate RR 0 based on recoveries of A, B, Estimate RR k of asset type k based on recoveries of D and F a Estimate RR 0 based on recoveries of C, and E C, and E a Tota exposure Estimate RR 0 based on recoveries of A and B a a A references with respect to Tabe 9.3 Estimate P A, B and P C, E, the probabiities of after-defaut scenarios without or with usage of risk mitigation instruments a before defaut or if avaiabe a vauation performed after defaut. 35 Given these information, the recovery rate RR k for a certain coatera type k can be estimated ex-post as the ratio of the net present vaues NPV(CF) of a sod assets of type k to the respective coatera vauations V(A) before defaut. Now assume that asset A i is of coatera type k(i) and that oss experience indicates a recovery rate RR k(i) for this coatera type or for an exposure coateraized by it, respectivey. The bank woud then expect to reaise a recovery of V t (A i ) RR k(i) for the respective reference object, i.e. for a secured exposure or the asset itsef. The reference size in (9.8) is given by sq j,(i) ¼ min{1; V t (A i )/EAD j }. Aternativey, one may define the respective reference size in (9.8) with respect to the recovery of asset A i, i.e. sq j,(i) ¼ min{1; V t (A i ) RR k(i) /EAD j }. The recovery on subexposure sq j,(i) EAD j wi then be 100%. In both cases, one may proceed simiary for unsecured exposures, considering the respective exposure size sq 0 ¼ max{0; 1 1 sq j, } as the asset vaue. The genera mode described in (9.3) (9.7) defines LGD as the weighted sum of LGDs observed in different after-defaut scenarios on a set of subexposures. Estimation of ex-ante LGDs may foow this ine, i.e. first estimate LGD j, (sc i ) for each subexposure in each scenario and afterwards aggregate these numbers to determine the LGD estimate for the exposure under consideration. However, one may want to simpify the procedure by aggregating as many of these LGD j, (sc i ) estimations as possibe in order to ower computationa compexity. Tabe 9.4 demonstrates two possibe approaches to do so. The idea of the first approach is to estimate LGDs for a subexposure without taking expicity into account after-defaut scenarios. However, recoveries on secured exposures may not ony depend on coatera but aso on faciity and borrower characteristics. In principe, this probem can be overcome by partitioning the poo into homogeneous groups of obigors and estimating parameters for each partition 35 In order to estimate PD, EAD, and LGD in a consistent way, one wi often appy a cohort approach for a three variabes. Therefore the ast vauation before defaut is the more appropriate reference vaue.

9 Estimating Loss Given Defaut: Experience from Banking Practice 171 separatey. A imited number of oss observations often hinder a partitioning in practice. The approach therefore appears especiay appeaing for arge, homogeneous portfoios. The second approach disaggregates recoveries which depend on asset characteristics and those which do not. In fact, it can be considered as a generaisation of the first approach. Mainy coatera-independent recoveries of the after-defaut scenarios cure and restructuring without rmi usage are estimated for the whoe exposure whie recoveries in scenarios with rmi usage are estimated separatey for secured and unsecured subexposures (as was the case in variant 1). The compexity of the approach is therefore ony sighty higher; however, one has to estimate more parameters. Instead of modeing different components for (un)secured exposures and/or different after-defaut scenarios, one may aso try to describe tota recoveries on an exposure by a singe recovery component. This might be done, for exampe, by considering the sum of expected asset recoveries as an expanatory variabe. If exposures are secured by ony one asset, as wi often be the case, one may aso try to incorporate asset vaues directy as expanatory variabes into a recovery mode. Since recovery rates generay depend on the respective asset type, such modes wi probaby require considering asset type as an additiona expanatory variabe. Furthermore, one may face the same probems as aready discussed above. Expicit consideration of after-defaut scenarios foowing the approach outined in Sect. 9.5 and discussed in more detai above may be appied in oss data anayses as we as LGD estimation for defauted exposures (see Sect. 9.7). Furthermore, expicit consideration of scenarios can sometimes be usefu when combining different interna and externa data sources or when oss data is missing for some parts of the bank s portfoio. Incorporating externa data into the mode may require different techniques depending on type and data source. For exampe, probabiity of cure depends on the bank s defaut definition. A separate description of the cure scenario may therefore be of interest for LGD caibration if externa data (for exampe, from a data pooing) is used for estimation purposes or if the bank itsef has changed its defaut criteria over time. 36 As a second exampe, assume that the bank has a ow number of observations for some portfoio segments. It may then try to derive estimates (for exampe, considering after-defaut scenarios) for these segments by comparing key characteristics of this portfoio segment with those of other segments where oss observations are avaiabe. Thus, the institute may obtain an idea of the recoveries it can expect on the respective portfoios. However, subjective methods as previousy outined can generay ony suppement the anaysis of externa data. 36 This may sometimes be the case during the introduction of Base II compiant processes.

172 C. Peter As a third exampe, consider the estimation of recovery rates for assets where the bank does not have own workout experience. 37 A possibe approach for deriving recovery rates for coatera (in part) from externa data can be stated as foows: 1. Estimate the time series of vaue depreciation for the specified asset type. Sources of information on vaue depreciation can be market data as we as data from brokers or appraisers. 2. Estimate the time Dt required for repossession or sae. In practice, one may observe time series of cash fows, for exampe rents or easing rates foowed by one or severa cash fows from the observed asset sae. Whie such cash fow patterns may theoreticay aso be recognized in a mode, it wi often be sufficient to assume that the tota cash fow arises at one point in time. An exposureweighted average time often provides a reasonabe reference time. If no recovery observations are avaiabe, one may refer to experience from simiar asset types or rey on expert judgement. 3. Estimate haircuts D for vaue voatiity, distress sae, etc. Again, market data can often be a main source of information. Experience from repossession or sae of simiar assets may aso provide usefu information for estimating haircuts. In addition, one has to determine an appropriate discount factor; see Sect. 9.6.2.4. Having determined these parameters, recovery estimates can be generated as NPV (V(t DF þ Dt) (1 D)). To obtain a better idea of the magnitude of recoveries, one may aso perform scenario anayses or simuations where the input parameters determined in the three steps above are varied in order to refect certain economic scenarios. Any substantia dependence between the vaue of an asset and the defaut event of the borrower shoud carefuy be taken into account, since they may substantiay decrease the effect of risk mitigation (see aso BCBS (2004), } 469). It is often hepfu to distinguish between genera and specific dependences. The first named recognizes norma dependences which shoud be refected in the recovery rates discussed so far. The second type addresses an individua characteristic of a faciity-coatera reation, which is generay difficut to detect automaticay. It is therefore often reasonabe to give credit anaysts the possibiity to grade such dependences manuay. These grades can then be used to adjust haircuts on recovery rates in an appropriate manner. Exposures Secured by Guarantees or Credit Derivatives 38 Since the risk mitigation effect of a guarantee essentiay consists of a (partia) transfer of credit risk to a different entity, one may expicity mode the guarantor s 37 For unsecured exposures, recovery estimates may be derived from market LGDs; see Sect. 9.4. 38 The foowing considers guarantees to simpify the presentation. Credit derivatives can often be treated in a simiar way.

9 Estimating Loss Given Defaut: Experience from Banking Practice 173 defaut probabiity as a major driver of the guarantee s vaue, i.e. recoveries from a guarantee can be described as 39 RR j; ¼ PDðGjBÞRR DD j; þ ð1 PDðGjBÞÞRR SD j; (9.9) with PD(G B) the conditiona probabiity of defaut of the guarantor given the defaut of the borrower. The parameters RR SD and RR DD are the recovery rates a bank may observe in case of an isoated defaut (SD) of the borrower or a doube defaut (DD) of both the borrower and guarantor. One may extend (9.9) anaogousy for cases where an exposure is secured by more than one guarantee (for exampe, in case of a counter-guarantee). The size of a secured exposure, sq j,, can be determined in a simiar way as described for coatera above, taking into account that the reference vaue of a guarantee is generay defined as a maximum amount, V max (Gar), and/or a certain percentage sq Gar of the exposure. 40 When pubished first in June 2004, the Base II Framework restricted risk mitigation effects of guarantees by requiring that the risk weight resuting from an exposure secured by a guarantee shoud not be ess than that of a comparabe exposure with the guarantor in pace of the borrower. This approach is known as the substitution approach, indicating the basic idea of repacing the borrower by the guarantor. It has often been criticized for being too conservative. To understand why, consider for a moment the borrower as a first guarantor of the contractua cash fows. The guarantor then in fact provides a counter-guarantee for these cash fows. Therefore, the bank faces substantia osses ony if the guarantor is unabe to pay at the time of the borrower s defaut, i.e. in case of a doube defaut. Ony if one assumes perfect dependence between the two defauts, which wi generay not be the case, a substitution mechanism wi describe the credit risk appropriatey. With its update in 2005, Base II now aows for a imited recognition of doube defaut effects in both IRB approaches. Restrictions are defined on the set of eigibe instruments, obigors, and guarantors as we as on the method and the correation parameters. 41 A Merton-stye defaut mode [see Merton (1974)] is considered to determine joint defaut probabiities of guarantor and obigor. Let Y i be the appropriatey normaized asset vaue of a borrower or guarantor i at a 1-year horizon, respectivey. With X a systematic risk factor, Z BG a risk factor shared by borrower 39 Again, j indicates the faciity and the exposure part secured by the guarantee. 40 In practice, the vaue of a guarantee may depend on further warranty causes. To mention a few, guarantees may cover ony a subset of the borrower s obigations, for exampe ony interest rate payments or redemption. They may aso be restricted to protect certain risk casses ony (for exampe, no poitica risks). Furthermore, they may (party) protect residua oss after recovery of other coatera and the bankrupt s assets ony. This artice does not consider the modifications necessary to adequatey vaue such guarantees. Note that some characteristics mentioned above may aso be incompatibe with Base II requirements for eigibe guarantees and can therefore ony be considered in interna modes. 41 See BCBS (2004), }} 284 (i) (iii) and 307 (i), (ii).

174 C. Peter and guarantor, and E i a counterparty-specific risk factor, the asset vaues of both entities can be described as pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y i ¼ X r i þ ZBG 1 r i c BG þ Ei 1 r i 1 c BG (9.10) X, Z BG, and E i are considered as independent random variabes foowing a standard norma distribution. Furthermore, one assumes that counterparty i defauts if its asset vaue, Y i, fas beow a threshod k i. Given the defaut probabiities of both entities, the joint probabiity can therefore be computed as JPDðB; GÞ ¼ FF 1 ðpdðbþþ; F 1 ðpdðgþþ; r BG (9.11) With F 1 (PD(i)) ¼ k i and r BG ¼ (r B r G ) 0.5 þ c BG ((1 r B ) (1 r G )) 0.5 the correation between borrower and guarantor. Stressed defaut probabiities are determined by conditioning on the systematic risk factor X. For technica detais see BCBS (2005) and Heitfied and Barger (2003). Both the substitution and the doube defaut approach of the Base II Framework are defined in a way that is most easiy impemented in a two-step procedure. Firsty, it is necessary to estimate the LGD of borrower and guarantor considering the risk mitigation effect of coatera (if avaiabe) ony. Afterwards, risk mitigation effects of guarantees are recognized in a second step by appropriatey modifying the risk-weight of the secured exposure foowing the substitution rue or doube-defaut formua. For interna purposes, banks may want to reax the restrictions of Base II or appy their own approach for recognizing doube defaut effects. This can be done, for exampe, by computing recovery rates based on (9.9) (9.11) or, whenever components of LGD are used as input parameters of some simuation mode, by directy simuating the risk mitigation effect of guarantees within the simuation. 42 The required information about the dependence structure (i.e. correations) may often be avaiabe through the bank s credit portfoio mode. Depending on the eve of conservatism underying these correation estimates, one may want to impose additiona margins of conservatism in order to avoid overestimating the effect of risk mitigation by guarantees. As for coatera, a bank may aow credit anaysts to grade any specific correation between guarantor and borrower, which may then, for exampe, resut in a modified vaue of c BG in (9.10). 43 Estimates of the recovery 42 In fact, a bank may use both techniques simutaneousy for different purposes. For exampe, expicit simuation of guarantees may sometimes be too time-consuming so that LGD numbers aready incuding the risk mitigation effect have to be appied instead. 43 It may sometimes be possibe to detect certain types of dependences automaticay. For exampe, knowedge on economic interdependence of different addresses, which might be avaiabe in the institute s IT-systems (for exampe, in form of borrower units), can be used to decide whether (or to what extent) a guarantee is eigibe for a faciity of a certain borrower.

9 Estimating Loss Given Defaut: Experience from Banking Practice 175 rates RR SD and RR DD can be obtained with ony sight changes on the procedures described above for assets. Further Aspects of Estimating Recovery Rates Concuding Sect. 9.6.2.2, the foowing outines additiona aspects of recovery rate estimation not yet considered. Participation effects. Having compensated the ender under a guarantee for any obigation due by the borrower, the guarantor generay acquires the right to ask for repayment paid by the borrower (which is generay not possibe due to its defaut) or from the recoveries of the borrower s coatera and bankrupt s assets. Furthermore, guarantees may sometimes ony protect residua oss after recoveries from other risk mitigation instruments, etc. Taking these aspects into account compicates recovery rate estimation since simpy adding the recoveries of different instruments may ead to distortion. Furthermore, recovery times of singe instruments can change significanty, depending on whether other risk mitigation instruments aso protect the same exposure. A tree representation of the transaction and its risk mitigation instruments can be hepfu in describing these effects and deriving the respective recovery rates. Optima aocation of risk mitigation instruments. Whenever risk mitigation instruments are not ceary assigned to singe faciities, the bank may want to optimize the aocation. 44 This can be done foowing simpe heuristics or by soving a (non) inear optimization probem for minimizing risk-weighted assets; see Beckmann and Papazogou (2004) and G urter and Heithecker (2005). Muti-period estimation. An institute may want (at east for some appications) to generate a muti-period projection of its credit risk numbers. Different techniques ike simuation or scenario computation may be appied for this purpose. As a first step, one may aso decide to rey on the conservative assumptions of Base II (i.e. appying downturn LGDs as time-independent estimate of future LGDs). To derive future recoveries from coatera, one can proceed simiary as aready discussed for estimating recovery rates of assets based on externa data. The depreciation profie of the respective asset type provides the basis for estimating a time series of the asset s vaue. Depending on whether recovery rates are defined as the net present or nomina vaue of recoveries, estimates can be performed directy by mutipying recovery rate with the predicted future asset vaue or firsty estimating the time of recovery cash fows. For guarantees, it is necessary to estimate rating migration and cumuative defaut probabiity of the guarantor up to the (assumed) defaut time of the borrower. Furthermore, the 44 The potentia for optimization stems from the joint effect of different risk mitigation instruments, possibe currency mismatches, changes in exposure cass due to risk mitigation, etc.

176 C. Peter vaue of a guarantee in terms of V max (Gar) and sq Gar may sometimes change over time. Maturity and currency mismatches. Maturity or currency mismatches between faciity and risk mitigation instruments have to be considered in ex-ante estimates. Maturity mismatches may be recognized by computing a time-weighted average of LGD estimates with and without recoveries of the respective instrument. Currency mismatches are generay recognized by haircuts, which can be derived from an anaysis of the voatiity of FX rates. This may aso require taking individua conversion agreements into account. 9.6.2.3 Estimating Costs Simiary, as described for recoveries, costs can generay be assigned to entities (i.e. borrower and guarantors), credit products, and risk mitigation instruments (coatera and guarantees). It therefore often makes sense to break down the workout costs of a faciity j arising in an after-defaut scenario sc j into two basic components: (1) genera costs Costs g j(sc i ), which refect a costs of the workout process not reated to risk mitigation instruments, and (2) specific costs Costs s j,k(sc i ), which refect a costs (on a secured exposure part sq j, ) reated to the handing of risk mitigation instruments; for exampe, costs arising during from the repossession of an asset. With respect to (9.7) one then has Costs j; ðsc i Þ ¼ Costs g j ðsc i Þþ X Costs s j;kð sc iþ (9.12) k secures sq j; Aternativey, one may decide to offset any costs attributed to coatera or guarantees directy from the respective recoveries on secured exposures eading to RR ¼ max{0; RR Costs s }. If a bank pans to use its LGD estimates for IAS/ IFRS purposes as we, the equation shoud be impemented in a way that aows separating interna and externa costs since ony the second are generay aowed entering the respective IAS cacuations. In practice, measuring direct and especiay indirect costs can be difficut. The required steps wi depend on the institute s cost accounting system, which may not necessariy suits the requirements of LGD estimation. Interna costs may at east in part be known ony on a eve, which is more aggregated than required (for exampe for workout or restructuring units but not for defauted entities), causing extra compications for mode deveopment. 45 Anaysis of the institute s workout processes may often serve as a starting point for modeing the cost component of LGD. This comprises firsty identifying key activities or processes causing workout reated costs, their respective cost units as 45 Information on externa costs wi generay be coected in the CLDB. This assures its avaiabiity.

9 Estimating Loss Given Defaut: Experience from Banking Practice 177 we as possibe expanatory variabes, for a after-defaut scenarios. A rough cassification according to expected cost amounts might be hepfu in guiding further deveopment. Expert judgment can pay an important roe at this stage. Whie externa costs may be assigned directy to processes, further assumptions are usuay required to determine interna costs. Estimates of time required for a certain activity, the number of persons invoved, as we as work intensity (i.e. percentage of daiy or weeky working hours spend on the task) together with the institute s cost rate per working hour can provide a basis to derive cost estimates for workout activities. Ideay, key activities are recorded by appropriate events within the oss fie so that the institute is abe to estimate the engths of its workout activities from past experience. Cost accounting and expert judgment can deiver at east a first estimate of the other parameters, whereas a fina mode may require a more detaied anaysis of a sampe of oss cases. Once key costs have been modeed, residua costs can often be distributed proportionay. If costs are modeed on a borrower, coatera, or guarantee eve, which may often be appropriate, LGD computation requires breaking them down to the faciity eve. This can either be done within the estimation procedure itsef, i.e. individuay for each entity and credit faciity, or a priori during mode deveopment. Reasonabe distribution keys are in both cases the tota exposure of an address faciities as we as coatera and guarantee vaues or, aternativey, the number of the respective objects. More reaistic estimates can generay be expected from an individua cost distribution during the estimation process. However, the computationa effort may be too high with respect to the expected improvement of the quaity of LGD estimates. 9.6.2.4 Determining Discount Rates Both recovery and cost estimates require net present vaue computations to take materia discount effects into account. The choice of discount rate(s) wi affect the resuting LGD numbers especiay when recovery periods are ong. Different approaches have been appied and discussed in the iterature. Basic characteristics for a categorization are: historica vs. present rates, singe rates vs. interest rate curves as we as the procedure appied to determine the rates or curves, respectivey. Simpe approaches, for exampe, discounting with the contractua oan rate, the effective origina oan rate, 46 or ender s cost of capita, have been appied in many artices. From a theoretic point of view, it appears most appropriate to discount each cash fow using a discount rate that refects the respective eve of risk as we as the time required for reaizing it. Determining an appropriate discount rate curve for each risk cass, however, can be difficut. Macachan (2004) suggested a procedure based on the CAPM that may be usefu in this context. 46 Since IAS requires the appication of the effective origina oan rate, a bank may think about appying this rate in its estimates if LGD numbers are used for IAS purposes as we.

178 C. Peter Discount rates appied in ex-post and ex-ante estimates may differ. Ex-post LGD numbers are generay computed using historica interest rates observed at the time of defaut. Discount rates chosen for ex-ante estimates wi depend on the appied discounted cash fow method. If cash fows are adjusted by margins of conservatism, the risk-adjusted rate shoud refect the ower risk profie, i.e. it can sometimes be (amost) the risk free rate. Discount rates appied in downturn LGD estimates shoud aso refect downturn conditions. For a point-in-time LGD estimate, the current interest rate curves can be reied on to use the most up-to-date information. Combining current interest rates with past oss experience, however, may ead to distorted estimates if dependences between interest rate and nomina recoveries are not considered adequatey. 9.6.2.5 Determining the Leve of Conservatism for LGD Estimates The Base II Framework asks for conservative LGD estimates: LGD has to be estimated so as to refect economic downturn conditions where necessary to capture the reevant risk (BCBS (2004), } 468). LGD cannot be ess than the ong-time defaut-weighted average (BCBS (2004), } 468). Banks must add a margin of conservatism to their LGD estimates that is reated to the ikey range of unpredictabe errors (BCBS (2004), } 451). Institutes must consider dependences between the risk of the borrower and that of the coatera as we as the coatera provider. Furthermore, currency mismatches have to be considered conservativey (BCBS (2004), } 469). 47 The kind of (conditiona) LGD expectation defined by Base II wi not aways correspond to the concepts that banks may have defined for their interna risk measurement. Specificay, the required downturn characteristic can be questioned for interna appication where one generay wants to recognize the economic cyce in an expicit manner (point-in-time estimate). Depending on the compexity that a bank is wiing to accept in its methods, diverging requirements may ead to different modes or parameterizations of LGD components appied for reguatory and interna purposes, respectivey. One possibiity is to appy the concept, proposed in BCBS (2004) for non-performing exposures, to performing positions as we, i.e. to refer to a best estimate LGD for interna credit risk management 48 whie appying a conservative LGD for reguatory purposes. This artice wi not discuss this rather institution-specific question in more detai. If LGD estimates are composed from estimates of their components as discussed in this artice (see Sect. 9.4), each of the modes for these components has to fufi 47 Means of fufiing this requirement were discussed in Sect. 9.6.2.2. 48 Voatiity of LGD then has to be recognized separatey in unexpected oss estimates.

9 Estimating Loss Given Defaut: Experience from Banking Practice 179 the requirements mentioned above. When determining the eve of conservatism for components, the impact on the resuting eve of conservatism for the fina LGD estimate shoud carefuy be considered to avoid too conservative estimates. Downturn conditions can be recognized foowing different approaches. A first approach is to identify the subset of oss observations refecting economic downturn and to deveop estimation procedures based ony on this reference (sub) dataset. Time series of macroeconomic variabes may be used to identify the respective time periods refecting economic downturn. However, with imited oss observations avaiabe, this approach wi often be a rather theoretica option. Aternativey, one may restrict considerations to the margina distribution of oss observations for the considered component, i.e. impicity recognize economic downturn by choosing an appropriatey conservative quantie. If the bank intends to deveop an LGD mode, which expicity recognizes the impact of economic cyces, a more eegant soution might be to estimate downturn LGDs by appying this mode with downturn parameters instead of input parameters refecting the current economic situation. Margins of conservatism can be derived as percenties from empirica distributions, based on appropriate parametric distribution assumptions or, for exampe, from appying resamping techniques as bootstrapping. In practice, observed voatiities can be arge, eading to arge margins even for ower confidence eves. Practica probems aso arise where oss history may not refect the characteristics of future osses. If, for exampe, a bank redesigns its workout processes or changes its workout strategy, future osses may differ from what has been observed in the past. Depending on the portfoio it can take severa years unti the effect of a structura break may become visibe in oss observations. During that time the bank has the difficut task of recognizing the unknown effect of the modification in its oss estimates in a conservative manner. Simiar probems of data aging may arise due to changes in aws, etc. 9.7 LGD Estimation for Defauted Exposures When estimating LGDs for defauted faciities, an institute faces a sighty different situation than for performing exposures. Besides differences in reguatory requirements (e.g., the need to generate a best estimate and conservative estimate of LGD; see Sect. 9.2.1) and possibe synergies from coaboration with provisioning processes (see Sect. 9.2.2), the bank wi often aso be abe to estimate LGDs for defauted entities based on better information. Defauted exposures are generay monitored more intensivey than performing faciities, resuting in more up-to-date and often aso more precise information about its current status. The bank wi aso receive additiona information, not avaiabe before defaut. This can be expicit information, for exampe decisions taken during the workout process or updates of market data, or impicit information, as for exampe time passed after defaut. Expicit information generay repaces the estimate for some components of LGD. One may therefore think of LGD estimates for defauted exposures as a

180 C. Peter transition from ex-ante LGD estimation to ex-post LGD observation. Updateprocedures can differ depending on whether the bank keeps the time of defaut as reference time or considers the current date instead. More interesting for practica appication is generay the second variant, which considers ony residua oss, i.e. EADðtÞ ¼EADðt DF Þ X LGDðtÞ ¼1 t2½t DF ;tþ NPV CFRec t cf Rec t ; t t NPV CF Costs t ; t t EADðtÞ (9.13) with cf and CF the reaized or expected recovery cash fows and costs. Whie the update-scheme itsef has a simpe structure, its impementation can become compicated. In particuar, the update of EAD and LGD requires that the sources of a cash fows can be automaticay identified. Impicit information, for exampe, time after defaut or certain events observed after defaut, may be used in estimates of NPL-LGDs by considering (abstract) states of information as additiona expanatory variabes or, more generay, state space modes. As an exampe, consider cure probabiity as a decreasing function of time after defaut in a mode foowing (9.3) (9.7). Whie theoreticay appeaing, estimating such modes requires arge reference datasets and reativey homogenous portfoios if not (party) parameterised by expert judgement. Portfoios of standardized retai exposures may therefore be the main fied of appication. Purey statistica approaches wi often not be abe to capture a information avaiabe for individua defauted faciities. For exampe, recoveries from bankrupt s assets or coatera as we as costs or payment dates can often be estimated more precisey based on the specific information avaiabe for a defauted entity. An LGD estimate may therefore be improved by aowing overrides for some of the mode s input parameters or the purey statistica LGD estimate. This can aso affect the mode design. Since provisioning requires simiar information to oss estimation, it may be reasonabe to ink the two processes in order to use a consistent set of information, avoid process redundancies, and et provisions and LGD estimates converge as far as possibe, which may aso simpify interna communication of these numbers. Links may be estabished in both directions, as depicted in Fig. 9.4: A statistica LGD mode may deiver information concerning the oss distribution of a defauted entity or credit product as we as other usefu information, for exampe, expected coatera recoveries etc. These may serve as a basis or reference for determining provisions in a provisioning too. During the provisioning process, the responsibe anayst may then modify or suppement estimation parameters based on her information or expectations about the respective oss case. 49 These inputs can afterwards be used to improve LGD estimates. 49 For exampe, she may eect the respective after-defaut scenario or modify the time structure of future recovery cash fows.

9 Estimating Loss Given Defaut: Experience from Banking Practice 181 Controer Data Bases Coatera Accounting data Ratings Master data... Data Coection Pre-processing Computation of PL-LGD NPL-LGD best estimate conservative Data Warehouse Estimated recoveries based on past oss experience (scenario dependent) Recovery information for coatera and guarantees etc. Provisioning too Scenario information Adjusted / additiona cash fows etc Fig. 9.4 Connection of LGD estimation for nonperforming exposures and provisioning 50 Before any input from the provisioning process may enter NPL-LGD estimates, it is necessary to anayse any differences in the respective vauation approaches appied by the bank. It shoud be kept in mind that couping the two processes can further compicate the impementation of the estimation process. The decision of whether and how the two processes shoud cooperate often depends on the respective portfoio. For exampe, oss experience for standardized or retai portfoios wi generay provide a sufficient basis to deveop and appy more advanced machinedriven estimation procedures. In this case, the bank may want to derive its provisions from LGD estimates (but not vice versa). The opposite wi probaby become true for customized credit products where expert judgment may prove more vauabe than imited empirica oss experience. Due to reguatory requirements, institutes have to determine a best estimate of oss (LGD BE ) as we as a conservative estimate (LGD CE ) for Base II purpose. As discussed in Sect. 9.6.2.5, some banks may appy a simiar scheme for performing faciities as we. A procedures described so far in this artice can be considered to generate best estimate LGDs (in the sense of the best possibe estimate of expected oss quotas). Conservative estimates may be generated by appropriatey stressing the best estimate. This may be done be stressing input parameters of the estimation procedure (for exampe, recovery rates for coatera, workout periods, etc.) or the resuting estimate. Empirica distributions of historicay observed parameter vaues (e.g. recovery rates of certain coatera types, etc.) or oss quotas may hep to define stress factors. Sometimes, the same or simiar stress factors as aready used for performing oans may be appied for non-performing oans as we. Sometimes, one may expect stress factors to be smaer after defaut due to more precise information about the economic situation of a defauted entity. However, appropriatey stressing 50 (N)PL-LGD is used as an abbreviation for an LGD of a (non)performing exposure.

182 C. Peter human judgment (which may enter when appying procedures as outined above) in LGD estimates can be difficut. Depending on its impact on the estimate, one may simpy ignore them in conservative LGD estimates. 51 9.8 Concuding Remarks This artice provides a genera survey of LGD modeing from a practica point of view. Due to the scope of the artice, various aspects incuding most technica detais coud not be covered. Severa aspects of LGD estimation are sti topics of discussion and current research. Two important exampes are Lack of oss history. Estimating LGD for exposures of portfoios with itte or no defauts is a difficut but common probem. But even for portfoios where oss data is in principe avaiabe, it may not aways be representative for the future due to interna or externa changes, for exampe, modifications in workout strategy or reevant aws. Some simpe approaches to dea with this situation have been outined in this artice; however, additiona research is recommended. Vaidation. Whie not considered in detai within this artice, mode vaidation forms an important part of LGD methodoogy. BCBS (2004) requires a banks appying the advanced IRB approach to vaidate their rating systems and processes on an annua basis. Litte has been pubished on the vaidation of LGD modes; see for exampe Bennett et a. (2005). Some methods may be taken from PD vaidation, which aready provides more advanced concepts 52 ; however, specific characteristics of LGD estimation approaches wi probaby require adjustments or the deveopment of new vaidation approaches. The ack of oss data wi again compicate the appication of quantitative toos for some portfoio segments. A unification of vaidation techniques, processes, and reports for the risk parameters PD, EAD, and LGD appears reasonabe to reduce costs and promote an understanding of the resuts within the institute; however, itte can be found in the iterature on this topic. Many further, ess prominent topics arise from daiy work within the conficting fieds of statistica significance, degree of detai desired for different appications, and cost-benefit aspects. One may therefore expect and ook forward to see further interesting deveopments within the fied of LGD estimation. 51 One may aso think about aowing anaysts to judge the uncertainty of recoveries as we, giving them the possibiity to infuence stress factors, etc. Any degree of freedom in the appied procedure, however, may not ony improve the quaity of estimates but aso bears the danger of deterioration and generay aso compicates the whoe procedure from impementation and workfow aspects up to a ater vaidation. 52 Cf. Chaps. 14 and 15.

9 Estimating Loss Given Defaut: Experience from Banking Practice 183 References Acharya VV, Bharath ST, Srinivasan A (2004), Understanding the Recovery Rates of Defauted Securities, Working Paper, Moodys KMV, http://www.moodyskmv.com/conf04/pdf/papers/ understdg_rec_rates_def_sec.pdf Atman E, Brady B, Resti A, Sironi A (2003), A Link between Defaut and Recovery Rates: Theory, Empirica Evidence and Impications, http://www.journas.uchicago.edu/doi/pdf/ 10.1086/497044. Atman E, Kishore V (1996), Amost Everything You Wanted to Know about Recoveries on Defauted Bonds, Financia Anaysts Journa, November/December, pp. 57 64. Atman E, Resti A, Sironi A (eds.) (2005), Recovery Risk The Next Chaenge in Credit Risk Management, Risk Books, London. Araten M, Jacobs M, Varshney P (2004), Measuring LGD on Commercia Loans: An 18-Year Interna Study, The RMA Journa, May, pp. 28 35. Base Committee on Banking Supervision (BCBS) (2004, updated 2005), Internationa Convergence of Capita Measurement and Capita Standards: A Revised Framework, Bank for Internationa Settements, http://www.bis.org. Base Committee on Banking Supervision (BCBS) (2005), The Appication of Base II to Trading Activities and the Treatment of Doube Defaut, Bank For Internationa Settements, http:// www.bis.org. Committee of European Banking Supervisors (CEBS) (2005, revised 2006), Guideines on the Impementation, Vaidation and Assessment of Advanced Measurement (AMA) and Interna Rating Based (IRB) Approaches, CP10, http://www.c-ebs.org. Beckmann C, Papazogou P (2004), Sicherheitenoptimierung nach Base II, Kreditwesen 3, pp. 146 150. Bennett RL, Catarineu E, Mora G (2005), Loss Given Defaut Vaidation, in: Base Committee on Banking Supervision, Studies on the Vaidation of Interna Rating Systems, Working Paper No. 14, Bank For Internationa Settements, pp. 60 93. D umann K, Trapp M (2004), Systematic Risk in Recovery Rates An Empirica Anaysis of US Corporate Credit Exposures, Deutsche Bundesbank Discussion Paper, 02/2004. Franks J, de Servigny A, Davydenko S (2004), A Comparative Anaysis of Recovery Process and Recovery Rates for Private Companies in the U.K., France, and Germany, Working Paper, Standard & Poor s Risk Soutions. Frye J (2000a), Coatera Damage, Risk, Apri, pp. 91 94. Frye J (2000b) Depressing Recoveries, Risk, November, pp. 106 111. G urter M, Heithecker D (2005), Sicherheitenoptimierung: ad aquate Anrechnung von B urgschaften, Kreditwesen 17, pp. 926 930. Hamiton D, Varma P, Ou S, Cantor R (2006), Defaut and Recovery Rates of Corporate Bond Issuers, 1920 2005, Moody s Investor Service, New York. Heitfied E, Barger N (June 2003), Treatment of Doube-Defaut and Doube-Recovery Effects for Hedged Exposures under Piar I of the Proposed New Base Capita Accord, A White Paper by the Staff of the Board of Governors of the Federa Reserve System in Support of the Forthcoming Advance Notice of Proposed Ruemaking, http://www.federareserve.gov/generainfo/base2/ docs2003/doubedefaut.pdf. Internationa Accounting Standard Board (IASB) (2005), Internationa Financia Reporting Standard (IFRS), IASCF Pubications Department, London, UK. Macachan I (2004), Choosing the Discount Factor for Estimating Economic LGD, http://members. dodo.net.au/~macach/lgddiscount.pdf. Merton RC (1974), On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journa of Finance, 2, pp. 49 71.

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Chapter 10 Possibiities of Estimating Exposures Ronny Hahn and Stefan Reitz 10.1 EAD Estimation in Line with the Loss-Parameter- Estimation of Base II 10.1.1 Definition of Terms The exposure at defaut (EAD) is defined as the expected amount of a receivabe at the time a defaut happens. In order to describe the borrower-reated-risk the EAD has to be set economicay before provisions are considered. 1 Provisions that are utiizabe bank-internay ony serve to cover the equity in the baance sheet in case of osses as possibe osses aready have reduced the risk bearing capacity of the bank at the moment of risk identification by the reaization of provisions. This definition shows that in a first step the EAD is determined by the exact time of defaut. If observed economicay the EAD to be expected depends on the horizon of defaut, i.e., it makes a difference of this horizon consists of 1 or of 2 years. According to reguatory prescriptions the EAD must not be ower than the book vaue of a baance sheet receivabe. 2 Therefore a reguatory necessity to estimate future EADs for such positions is not given. Credit conversion factors (CCF) have to be estimated for non-baance-sheet transactions and credit approvas. They describe the percentage rate of credit ines 1 Cf. BCBS (2006), }308. 2 Cf. BMF (2006), }100 seqq. R. Hahn 1 PLUS i GmbH e-mai: ronny.hahn@1pusi.de S. Reitz (*) University of Appied Sciences, Stuttgart e-mai: stefan.reitz@hft-stuttgart.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_10, # Springer-Verag Berin Heideberg 2011 185

186 R. Hahn and S. Reitz amount CL EAD CCF t 0 t1 = D time Fig. 10.1 Difference between exposure and credit approva (CL) that have not been paid out yet, but that wi be utiized by the borrower unti the defaut happens. Therefore the EAD is defined as: EAD ¼ CL CCF: For credit ines that aready have been paid out (baance sheet receivabes) the CCF is defined as 100%. The estimation procedure for credit conversion factors is further iustrated in Fig. 10.1. 10.1.2 Reguatory Prescriptions Concerning the EAD Estimation Reguatory prescriptions concerning estimations of oss parameters and therefore aso the EAD are mainy given in the reguatory rues reated to Base II. Within the capita provisioning requirements that are defined here, three separate approaches for the fixing of risk assets are distinguished. In the Standardized Approach (SA) and the Foundation Interna Ratings-Based Approach (FIRB) there is no freedom as far as the estimation of the EAD/CCF is concerned. This is due to the fact that the CCF reated to casses of receivabes is prescribed by reguatory entities. Specific minimum requirements on eigibe EAD estimates are defined in the Advanced Interna Ratings-Based Approach (AIRB) 3 : EAD for an on-baance sheet or off-baance sheet item is defined as the expected gross exposure of the faciity upon defaut of the obigor. For on-baance 3 Cf. BCBS (2006), }474 seqq.

10 Possibiities of Estimating Exposures 187 sheet items, banks must estimate EAD at no ess than the current drawn amount, subject to recognizing the effects of on-baance sheet netting as specified in the foundation approach... Advanced approach banks must have estabished procedures in pace for the estimation of EAD for off-baance sheet items. These must specify the estimates of EAD to be used for each faciity type. Banks estimates of EAD shoud refect the possibiity of additiona drawings by the borrower up to and after the time a defaut event is triggered. Where estimates of EAD differ by faciity type, the deineation of these faciities must be cear and unambiguous. Advanced approach banks must assign an estimate of EAD for each faciity. It must be an estimate of the ong-run defaut-weighted average EAD for simiar faciities and borrowers over a sufficienty ong period of time, but with a margin of conservatism appropriate to the ikey range of errors in the estimate. If a positive correation can reasonaby be expected between the defaut frequency and the magnitude of EAD, the EAD estimate must incorporate a arger margin of conservatism. Moreover, for exposures for which EAD estimates are voatie over the economic cyce, the bank must use EAD estimates that are appropriate for an economic downturn, if these are more conservative than the ong run average. For banks that have been abe to deveop their own EAD modes, this coud be achieved by considering the cycica nature, if any, of the drivers of such modes. Other banks may have sufficient interna data to examine the impact of previous recession(s). However, some banks may ony have the option of making conservative use of externa data. The criteria by which estimates of EAD are derived must be pausibe and intuitive, and represent what the bank beieves to be the materia drivers of EAD. The choices must be supported by credibe interna anaysis by the bank. The bank must be abe to provide a breakdown of its EAD experience by the factors it sees as the drivers of EAD. A bank must use a reevant and materia information in its derivation of EAD estimates. Across faciity types, a bank must review its estimates of EAD when materia new information comes to ight and at east on an annua basis. Due consideration must be paid by the bank to its specific poicies and strategies adopted in respect of account monitoring and payment processing. The bank must aso consider its abiity and wiingness to prevent further drawings in circumstances short of payment defaut, such as covenant vioations or other technica defaut events. Banks must aso have adequate systems and procedures in pace to monitor faciity amounts, current outstandings against committed ines and changes in outstandings per borrower and per grade. The bank must be abe to monitor outstanding baances on a daiy basis. Apart from that specific minima estimation periods are defined concerning casses of receivabes. Reguatory prescriptions ceary show the high quaitative and quantitative demands banks have to meet when using the AIRB. In practice the utiization of a bank s interna mode to bring the estimation method for the EAD

188 R. Hahn and S. Reitz into unison with the other oss parameters, probabiity of defaut (PD) and oss given defaut (LGD), is independent from the question whether the mode is utiized for interna or for reguatory requirements or for both. 10.1.3 Deimitation to Other Loss-Parameters Next to a cear definition of the specific parameters and the best possibe data quaity a uniform definition of defaut is most important for a methodoogicay correct interna estimation of oss parameters for credit risk. The first primary oss parameter is the PD which describes the probabiity of defaut of a borrower within a predefined period usuay 1 year. The statistica methods for estimating the PD are described esewhere in this book in Chaps. 1 3 and 5. The estimation period of the PD is identica with the estimation period of the EAD 1 year. The difference between those two parameters ies in the data basis that is needed. Concerning the PD estimation the genera question is how many of the origina customers (at time t 0 ) wi defaut. Therefore the overa portfoio has to be considered. Concerning the CCF estimation the data basis is reduced and ony those credit ines where a defaut took pace within the period of observation (1 year) are incuded ex post. Within an economic focus and taking amortization effects into consideration a receivabes have to be considered for the EAD estimation. LGD, that describes the fraction of the defauted amount of receivabes (EAD) that finay eads to a oss for the creditor, is the third major component of credit risk and expected oss EL (EL ¼ EAD PD LGD). The LGD estimation, simiar to the CCF estimation, depends on defauts that aready have taken pace. Ony on the basis of the defauted receivabes it can be measured empiricay which part of the defaut-voume wi ead to an economic oss for the bank. The biggest difficuty concerning the empirica LGD estimation is the reativey imited data amount and the ong duration of the estimation period. Whie PD and EAD/CCF estimation with an estimation horizon of 1 year aready requires a very ong period compared to the estimation of market price risk parameters, LGD estimation periods even reach on average 3 5 years. Experience within banks shows that the iquidation of defauted engagements incuding the reaization of coatera takes such ong periods. For this reason, a backtest of LGD estimations becomes increasingy difficut or neary impossibe. The foowing Fig. 10.2 shows the reations between EAD-CCF-LGD in a scheme. The figure shows that LGD aways refers to the EAD. Concerning estimations a simpe motto might be derived from this fact. Everything that takes pace unti the defaut happens has to be considered in the EAD estimation. A payments after this event ony infuence the estimation of LGD.

10 Possibiities of Estimating Exposures 189 amount CCF LGD CL EAD CCF LGD t 0 t 1 = D t n time Fig. 10.2 Reation between EAD-CCF-LGD 10.1.4 Reguatory EAD Estimation for Derivative Products If we ook at derivative products ike interest rate swaps, caps, foors, swaptions, cross currency swaps, equity swaps, or commodity swaps, two kinds of counterparty risks have to be considered: settement and pre-settement risks. Settement risks occur if the payments are not synchronous: This is for exampe the case if Bank A has paid a EUR cash fow in a cross currency swap to Bank B before it has received the USD cash fow. So the risk consists in the missing USD cash fow. If Bank B defauts a oss in the amount of this cash fow woud occur. Settement risks obviousy mosty have a short-term character. Much more important are the pre-settement risks. Characteristic for pre-settement risks is the foowing situation: Bank A expects in the future a rising interest rate curve. For hedging its oan portfoio Bank A makes a payer swap with Bank B. This transaction eiminates the interest rate risk but creates a counterparty risk. If Bank B defauts during the ifetime of the interest rate swap, Bank A has to ook for a new counterparty to make the same payer swap with this new counterparty. If in the meantime the interest rate curve has moved up, the repacement with an identica swap wi ony be possibe by paying an upfront payment to the new counterparty. Pre-settement risks have a ong term nature because they may occur during the whoe ifetime of the derivative product. For cacuating the EAD for derivative products it has to be noted that the EAD consists of two parts: The current exposure (CE): This is the repacement cost of a derivative transaction if the counterparty defauts immediatey, and is given by the actua market vaue of the instrument if this market vaue is positive. If the market vaue is negative it is zero: CE ¼ max{market vaue; 0}.

190 R. Hahn and S. Reitz EAD Potentia Future Exposure + "Current Exposure" = max{market vaue; 0} time to maturity Interest rate foreign exchange and god equity precious metas (not god) commodities up to 1 year 0% 1% 6% 7% 10 % 1 to 5 years 0,5 % 5 % 8 % 7% 12% over 5 years 1,5 % 7,5 % 10 % 8% 15% Fig. 10.3 EAD cacuation for derivative products in the reguatory context The potentia future exposure (PFE): This is an estimate for the increase in market vaue to a pre-specified time horizon (e.g., 1 year). It shoud be cacuated using probabiity anaysis based upon a specific confidence interva. So the EAD is given by: max{market vaue; 0} þ PFE. For cacuating the PFE for reguatory purposes fixed add-on factors, which have to be appied to the nomina amount of the contract, are used. Figure 10.3 demonstrates the cacuation of the EAD for derivative products in the reguatory context. If there is a netting agreement with the counterparty, the negative and positive market vaues of a derivative contracts, which are incuded in the netting agreement, can be offset against each other and the current exposure of a these contracts is therefore given by: ( ) CE ¼ max market vaue i ; 0 (10.1) X i For the potentia future exposure a tota offsetting of the various PFE s of the various contracts is not aowed. A so caed PFE-foor which is given by 40% of the sum of the PFE s of the various derivative contracts must be provided. The remaining 60% depend on the market vaue structure of the biatera derivative portfoio. Overa, the PFE under a netting agreement is given by: PFE ¼ 0;4 X ( ) X max P marketvaue i ; 0 i PFE i þ 0;6 PFE i P maxfmarketvaue i i i ; 0g : (10.2) If there are further coatera agreements then the EAD can be reduced by the amount of the coatera. i

10 Possibiities of Estimating Exposures 191 Overa, the foowing shortcomings of the way how the EAD for derivative portfoios is cacuated can be stated: The add-on-factors are static. The actua voatiities and correations of the economicay reevant risk factors are not taken into account. The specific product structure is negected in the add-on-factors, e.g., for an interest rate swap and an interest rate cap the add-on-factor is the same as ong as they fa in the same maturity time band. There is no offsetting between negative market vaues and the PFE aowed. So an interest rate swap with market vaue 0 and an interest rate swap with a negative market vaue wi ead to the same EAD. There is ony a very rough differentiation between the add-on-factors for products with different maturities. For instance, an interest rate swap with a maturity of 6 years has the same add-on-factor as an interest rate swap with a maturity of 30 years, athough the two products react competey different to changes in the interest rate curve. No amortization effects are recognized. In the described reguatory proceeding the cash fows of a product, which are paid before the proposed defaut point (e.g., 1 year) shoud not be considered in the CE. Banks who use the AIRB may use more eaborated techniques for cacuating the EAD of derivative portfoios. These techniques wi be expained in Sect. 10.2.2. 10.2 Banks Own Methods of EAD Estimation 10.2.1 Introduction The method for EAD estimation depends on the product category. In the case of credit ines banks wi use empirica methods (see Sect. 10.2.2) and for derivative products their own interna approaches (see Sects. 10.2.3 and 10.2.4). Under the Interna Mode Method (IMM) of Base II banks are aowed to derive estimations of EAD for derivatives using their own interna approaches. The key eements of such approaches are statistica methods for simuating future distributions of credit exposures resuting from (netted) portfoios of financia instruments. The benefit from using the IMM instead of other methods such as the Current Exposure Method or the Standardised Method is the fact that the IMM is more risksensitive since it aows banks to appy very sophisticated and portfoio specific techniques which improve the measurement of counterparty credit exposure. 10.2.2 Empirica Modes for Credit Products The deimitation between the oss parameters fufis the main requirement concerning the creation of an interna empirica data coection mode for the EAD.

192 R. Hahn and S. Reitz 1 1 1 1 1 1 1 1 1 1 1 1 Credit A 1 year Capita Interest Stop Repayments 12 Defaut Credit B Fig. 10.4 Reaized cash fows in the exampe 1 year It is advisabe to contain baance sheet exposures in addition to open ines in an interna empirica mode even if the respective information cannot be integrated into the reguatory exposure estimation. In genera, variations in the EAD shoud not be underestimated even in the case of baance sheet exposures. This wi be shown in a short exampe. Two credits A and B that are utiized at a eve of 100 and bear an interest rate of 6% sha be given. For Credit A an annua repayment of 12% and for Credit B a monthy repayment of 1% are agreed upon. If we assume that both borrowers stop paying their annua repayments after 11 months, meaning that ninety days after this a defaut in accordance with the defaut definitions of Base II occurs (Fig. 10.4). This means for Credit B that at the time of defaut a tota amount of receivabes of 100 is given. Apart from the 6% interest ¼ 6 for 1 year and the interest for the 90-days-period of about 1.6 have not been paid. This sums up to a tota receivabes amount (EAD) of 107.6. For Credit A this means that 11 repayments and interest payments have been duy effected and that therefore the remaining amount of receivabes remains at 89. Interest payments for the 12th month and for the excess period of 1.75 have to be added. The tota amount of receivabes that is to be demanded from the customer sums up to about 90.75. This deimitation is of prior importance aso concerning the LGD estimation. If we assume that the debtor of Credit A pays back the tota receivabe amount of 90.75 and additionay a costs reated to administration, no oss occurs for the bank neither economicay nor concerning the baance sheet. But if we assume the reguatory EAD definition for baance sheet receivabes, the EAD sums up to 100 and a oss (LGD) of about 10% occurs. The reguatory approach that the parameter effects (EAD vs. LGD) of cases A and B cance out each other can be accepted as far as capita provisioning is concerned but shoud not be accepted as far as an economic observation is given.

10 Possibiities of Estimating Exposures 193 EAD CCF t 0 t 1 =D time Fig. 10.5 Deveopment of the exposure reated to receivabes on the baance sheet This is due to the fact that seen from a risk perspective the parameter PD or defaut correations have to be added. Figure 10.1 therefore has to be adapted concerning baance sheet receivabes as foows (Fig. 10.5). When constructing empirica modes it makes sense to define the EAD in market vaues. In Case B the customers pays interests of 6%. If we assume that 1 year before defaut an interest rate of 5% is in ine with the market for this customer, a cash-vaue of more than 100% resuts. Within an economic observation this caim to profits is not reaizabe for the bank. A potentia refinancing oss starting from the date of defaut has to be considered in the frame of the LGD. After fixing the mentioned methodoogica framework for an interna empirica mode its creation is reativey simpe. In genera, the foowing requirements have to be met: Storing a EAD and CL reated information at east for 1 year and concerning a accounts, if necessary incuding market interest rates, conditions, cash fow structures, etc. Segmentation of casses of receivabes to create poos for the EAD/CCF estimation, i.e., oans for home construction, current account overdrafts, or guarantees. The practica experience shows that it is important to differentiate by products where the credit commitment wi be canceed in the defaut event (i.e., investment finance) and products which aow further drawings after defaut (i.e., guarantees). Cassification according to casses of receivabes, ratings, etc. Segmentation by the drawn eve 1 year before defaut (10% of ine or 90% of ine). The ast two points refer to the necessity of a cear definition of the aggregation eve of the survey. The schemes of those are depicted in the foowing Fig. 10.6.

194 R. Hahn and S. Reitz banks firms Rating Rating Rating A B C D A B C D A B C D Kind of Credit 1 Kind of Credit 2 Kind of Credit 3 Fig. 10.6 Leves of aggregation of the EAD survey 10.2.3 Exposure Measurement for Derivative Products In the event that a counterparty in a derivative transaction wi defaut, the position wi be cosed out and there wi be no future contractua payments. Depending on the mark-to-market (MtM) vaue of the transaction at the time of defaut, two cases are possibe: If MtM is positive, a oss is reaised. The size of this oss is the MtM vaue at defaut time minus any recovery vaue. If MtM is negative, no oss is made. The EAD as seen from today is thus a random variabe defined by EAD ¼ maxðmtm; 0Þ: As ony positive MtM vaues are reevant, it is natura to define the expected exposure (EE), which is the expected vaue of max(mtm,0) of a singe transaction or a portfoio of transactions (incuding netting effects and coateras). Within credit risk management an important question is what the worst exposure coud be at a certain time in the future. This question is answered by an exposure measure caed potentia future exposure (PFE) which was aready mentioned in Part 1. In terms of statistics, the PFE is an exposure at a certain time in the future that wi be exceeded with a probabiity of no more than a%. We reaize that PFE is a quantie (the 1 a% quantie) of the distribution of future MtM vaues (Fig. 10.7): In the simpest case, when MtM is a normay N(m, s 2 ) distributed random variabe, we have EE ¼ ð1 1 maxðm þ s x; 0Þ dx ¼ ¼ m Fðm=sÞþs ðm=sþ: ð1 m=s ðm þ s xþ ðxþ dx

10 Possibiities of Estimating Exposures 195 Expected MtM EE PFE MtM 0 Fig. 10.7 EE and PFE In reaity, MtM distributions of compex derivative portfoios have to be simuated by a Monte Caro simuation. This means that a market variabes (incuding their correations) that infuence future portfoio vaues have to be simuated incuding a portfoio characteristics such as path dependencies, netting agreements and coateraisation. In principe the foowing steps have to be carried out (cf. Gregory 2010 and Cesari 2009): Choice of Risk Factors: The set of risk factors typicay incudes (depending on the type of transactions in the portfoio) a underyings and their voatiities: FX rates, interest rates, credit spreads, equity and commodity prices, or impied voatiities. These factors may be modeed in a simpe one-factor mode or a more compex muti-factor approach, e.g., a muti-factor interest rate mode. Of course there wi aways be a trade-off between mode sophistication and tractabiity of the simuation. Whatever mode is chosen, the key issue wi be that future mutivariate distributions of market parameters are predicted in a reasonabe and efficient way and that the mode is we caibrated to current market data. Generation of Scenarios: In order to generate scenarios of the risk factors, a time grid has to be defined, which incudes a future points in time t i,forwhich risk factors reaisations are needed. The number and spacing of simuation points depends on the structure of derivatives within the portfoio. In practice, exposure profies can be highy discontinuous over time due to maturity dates, option exercise, cash fow payments and amortisation. The risk of missing jumps in exposure is caed the ro-off risk. The fina simuation date t n hat to be greater than the maturity of the instrument with the ongest maturity within the portfoio. Typica vaues n for the number of simuation points are within the range 50 200. The foowing Fig. 10.8 iustrates a simuated set of MtM scenariosasweascacuatedeeandpfe vaues (shown in the bottom chart).

196 R. Hahn and S. Reitz 1.05 1 0.95 MtM 0.9 0.85 0.8 0.75 0.00 1.00 2.00 3.00 4.00 5.00 Time (years) 1.02 0.98 PFE Exposure 0.94 0.9 0.86 EE 0.82 0.00 1.00 2.00 3.00 4.00 5.00 Fig. 10.8 MtM paths, EE and PFE paths Portfoio Vauation: A positions within the portfoio under consideration have to be revaued in every scenario and at each point in time t i. It is important to avoid extremey compex vauation modes here as the number of instrument revauations is enormous. If the number of counterparties is denoted by x and the (average) number of trades per counterparty by y we have (for n ¼ 100 and 10,000 scenarios per time step) to perform x y 100 10,000 revauations severa biion of revauations for arge portfoios! The need of (crude) anaytica approximations for pricing formuas is obvious. Aggregation: As a resut of the scenario generation and portfoio revauation we wi have a (huge) matrix of MtM-vaues for each singe transaction of our portfoio. For each point in time t i and scenario k a transactions beonging to a specific netting set (a set of transactions with a counterparty under certain netting conditions) the exposure E i,k is defined as ( ) E i;k ¼ max Xp PV ;i;k ; 0 : ¼1 Here, PV,i,k is the PV of trade in t i and scenario k, where a the trades with indices 2f1;...; pg beong to one netting set.

10 Possibiities of Estimating Exposures 197 Consideration of Coatera Effects: For each exposure path, we have to appy effects from coatera agreements which can reduce the exposure dramaticay. If the credit exposure against a counterparty is uncoateraised, it is necessary to mode the future distribution of risk factors over the fu time horizon of a transactions. In this case, typica ong-term assumptions such as mean reversion and drift have to be carefuy considered. In case we have a partia (or fu) coateraisation we wi have to mode counterparty exposure over much shorter periods (the remargin frequency). This can be done by VaR-methodoogies know form market risk. Cacuation of Risk Measures: Using the simuated paths of exposure figures E ik, a number of different statistica measures can be determined, for exampe the expected exposure EE i and the PFE i of a netting set for time t i : EE i ¼ 1 K XK k¼1 E i;k ; PFE i ¼ q 1 a% ðe i;k : k 2f1;...KgÞ; Figure 10.9 shows PFE i, EE i and the maximum exposure for (1) a singe interest rate swap and (2) a portfoio of two interest rate swaps. There are a number of additiona exposure measures which pay an important roe in EAD estimation. For our purposes we wi need the foowing parameters: The expected positive exposure (EPE) is defined as the average expected exposure through time and can be interpreted as a singe number representation of exposure; its forma definition is EPE :¼ X t i 1 EE i Dt i : It is the time weighted sum for a time points ess than or equa to 1 year. In the simuation of future scenarios one typicay observes that the number of remaining trades within a netting set and the number of remaining cash fows wi decrease. As a consequence, the amortisation effect starts dominating the diffusion effect at some point t i and so EE j decreases for increasing vaues of t j. As a certain percentage of expiring trades are ikey to be repaced by new trades (esp. short-term trades), within Base II the parameter EPE has been repaced by the so caed effective expected positive exposure (EEPE) that is cacuated as foows: In the first step the effective expected exposure (EEE) is cacuated by the foowing recursive definition (vaid for t i 1): EEE i ¼ maxfeee i 1 ; EE i g; EEE 0 ¼ EE 0 : The idea behind this definition is to avoid the amortisation effect mentioned above expiring trades wi not reduce the vaue of the effective EE. For t j > 2we define EEE j :¼ EE j.

198 R. Hahn and S. Reitz 140.000,00 120.000,00 Maximum Exposure 100.000,00 Exposure 80.000,00 60.000,00 40.000,00 20.000,00 PFE EE 0,00 0 1 2 3 4 5 6 Years 200.000,00 Receiver Swap 6 Years 150.000,00 100.000,00 Exposure 50.000,00 Tota PFE 0,00 0 1 2 3 4 5 6 50.000,00 Payer Swap 3.5 Years 100.000,00 Years Fig. 10.9 Exposures from swap positions After having defined EEE we can cacuate the effective EPE (EEPE) in the same way we derived EPE from EE: EEPE ¼ X Dt i 1 EEE i Dt i : This definition minimizes the ro-over risk coming from short term OTC derivatives or repo stye transactions which ead to an underestimation of EPE. 10.2.4 Estimation of EAD for Derivative Products The IMM approach in Base II aows banks to estimate EAD using cross-product netting. This means that within a predefined netting set of transactions, an EE profie

10 Possibiities of Estimating Exposures 199 (incuding portfoio effects) has to be created and from this the EEPE profie is cacuated as described above. After that, the EAD is defined as EAD ¼ a EEPE; where a is a mutipier, which refects the granuarity and concentration of the portfoio under consideration. In the (hypothetica) case of a portfoio with infinite diversification, a wi be 1. In reaity, portfoios consist of a finite number of counterparties. Therefore, there are non-zero correations between exposures and there might be wrong-way risk (i.e., a non-zero correation between exposures and defaut events) which eads to an a factor arger than 1. Why do we need the factor a? As we repace the set of a possibe future exposure-paths for each counterparty by a singe number (the EPE), we are cacuating the economic capita by repacing random exposure distributions through non-random EPE figures per counterparty. It can be shown (cf. Wide 2001) that in the case of a portfoio with an infinite number of counterparties with sma exposures (infinite diversification), where zero correation among exposures and between exposures and defaut events can be assumed, the economic capita of the actua portfoio equas the economic capita of a hypothetica portfoio which consists of non-random exposures of the size EPE for each counterparty: economic capita (actua portfoio) ¼ economic capita (portfoio with EPE exposures): In this context, EPE is an accurate oan-equivaent measure for cacuating economic capita (the term oan equivaent is used for a fixed amount that repaces a random exposure in the process of capita cacuation). Now, for rea portfoios the above mentioned conditions are not satisfied. This means that the foowing ratio is arger than 1: a ¼ economic capita ðrea portfoioþ economic capita (portfoio with EPE exposures) : The IMM approach in Base II aows banks to define the factor a by an own estimation instead of using the fixed vaue of a ¼ 1.4. There is a foor of a ¼ 1.2 for bank interna estimations of a in order to imit mode risk. A procedure for an estimation of a for a given portfoio coud be as foows: Consider a portfoio with a given number y of counterparties with an average probabiity of defaut PD and a given asset correation r. Specify a MtM distribution for a given time horizon for each counterparty in the portfoio. Cacuate the EPE based on the MtM distribution for each counterparty. The EPE for the whoe portfoio is the sum of the individua EPE vaues.

200 R. Hahn and S. Reitz Cacuate the distribution of osses in two cases: 1. Random exposures at the defaut time point; independent exposures for different counterparties 2. Fixed individua exposures (EPE-vaue) for each counterparty at the defaut time point Compare any economic capita measure (e.g., the 99% quantie of the oss distribution) in both cases; the ratio of both numbers defines the factor a. References Base Committee on Banking Supervision (BCBS) (2006), Internationa Convergence of Capita Measurement and Capita Standards. A Revised Framework Comprehensive Version. BMF (2006), Verordnung uber die angemessene Eigenmitteausstattung von Instituten, Institutsgruppen und Finanzhoding-Gruppen (Sovabiit atsverordnung SovV). Cesari G, Aquiina J, Charpion N, Fiipovic Z, Lee, Manda I (2009), Modeing, Pricing, and Hedging Counterparty Credit Exposure: A Technica Guide, Springer, Berin. Gregory J (2010), Counterparty Credit Risk. The New Chaenge for Goba Financia Markets, Wiey, London. Wide T (2001), ISDA s Response to the Base Committee on Banking Supervision s Consutation on the New Capita Accord, Annex 1.

Chapter 11 EAD Estimates for Faciities with Expicit Limits Gregorio Mora 11.1 Introduction The estimation of exposure at defaut, EAD, for a faciity with credit risk, has received a ot of attention, principay in the area of counterparty risk and has focused on situations where the variabiity of the exposure is due to: the existence of variabiity in the underying variabes of a derivative; the use of a fixed nomina amount not expressed in the presentation currency; or the existence of coatera whose vaue (variabe over time), reduces the exposure. Less attention has been given to the case of oan commitments with expicit credit imits. In this case, the source of variabiity of the exposure is the possibiity of additiona withdrawas when the imit aows this. The impementation of Base II is forcing credit institutions to address this probem in a rigorous, transparent and objective manner. Moreover, Base II imposes a set of minimum conditions on the interna EAD estimates in order to aow the use of these as inputs in the cacuation of the minimum capita requirement. Currenty, credit institutions have probems meeting the requirements of both the data and the methodoogies. This chapter anayses various methods for estimating EAD for faciities with expicit imits and tries to assess their optimaity from both an interna and a reguatory point of view. It focuses on objective methods, based on a reference data set (RDS) extracted from observed defauted faciities, which are frequenty used in practice by banks. Section 11.2 presents the definition of reaised conversion factors (reaised CFs) that are the basic input in most of the estimation procedures. Section 11.3 describes severa approaches for computing reaised CFs: Fixed Time Horizon, Cohort Approach, and Variabe Time Horizon and summarises their pros and cons. Section 11.4 expores issues that have to be 1 The views expressed in this paper are the responsibiity of the author and do not necessariy refect those of Banco de España. G. Mora Banco de España 1 e-mai: Gregorio.Mora@bde.es B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_11, # Springer-Verag Berin Heideberg 2011 201

202 G. Mora addressed before estimating EADs such as: structure and scope of the reference data set (RDS); data ceaning; treatment of observations with negative or greater than one CFs; and risk drivers. Section 11.5 focuses on EAD estimates. First, it estabishes the equivaence between EAD estimators and CF estimators under certain conditions. Second, the most common methods used by banks in practice are presented as specia cases of optimisation probems. It concudes that these methods are soutions for regression probems with quadratic and symmetric oss functions. Section 11.6 discusses issues reated to the optimaity of the estimates and introduces a different kind of oss function, one that is inear and asymmetric. These oss functions are naturay inked to Base II capita requirements and they are used to derive optima estimators that, consequenty, coud be more appropriate when the estimates are used for computing capita requirements under Advanced Interna Ratings-Based approaches (AIRB). Section 11.7 iustrates issues discussed in the previous sections and the consequences of using different estimation methods with a styised but reaistic exampe. Finay, Sect. 11.8 summarises the current practice on CF and EAD estimation, highights probematic aspects, suggests possibe improvements and concudes that traditiona methods, based on averages, are ess conservative than those based on quanties. 11.2 Definition of Reaised Conversion Factors In practice, when estimating the EAD for a non-defauted faciity, f, with an expicit credit imit, 2 there are two main casses of methods in terms of the basic equation used to ink the estimated EAD with the imit: In the first cass, estimates of the EAD are based on a suitabe conversion factor for the tota imit of the faciity, EAD( f ) ¼ CCF( f ) Limit( f ). In the second cass, estimates of the EAD are based on another factor 3 appied to the undrawn part of the imit, EAD( f ) ¼ Current Exposure( f ) þ LEQ( f ) Undrawn Limit( f ). 4 2 For exampe, credit ines which are committed, i.e. the borrower can draw additiona amounts unti a imit L(t) is reached. 3 In the Revised Framework and the Capita Directive such factors are caed Credit Conversion Factors (CCFs) and Conversion Factors (CFs) respectivey. In the drafts of Rues for Impementation of Base II in the US the factor used is caed LEQ factor and the Guideines by CEBS uses the term Conversion Factors (CFs). In this chapter, for carity, conversion factors that are appied to the undrawn amount are caed Loan Equivaent (LEQ) factors and the term Credit Conversion Factor, CCF, is reserved for the factor reated to the tota imit. 4 This is the approach required for these types of faciities in the Revised Framework, the Capita Directive, the drafts of Rues for Impementation of Base II in the US, and in the CEBS Guideines.

11 EAD Estimates for Faciities with Expicit Limits 203 L(t) EAD = E(td) E(tr) E(t) tr td Fig. 11.1 Definition of reaised LEQ factor As it is shown in Sect. 11.5, both approaches are equivaent and the probem of EAD( f ) estimation can be reduced to the estimation of suitabe conversion factors CF( f )(CCF( f )orleq( f )). In order to obtain the CF estimates, banks use as basic data, a set of observations at specific dates prior to the defaut time, of defauted faciities. Most of the estimation methods used are based on certain statistics, reated to the increase in the usage of the faciity 5 between a reference date and the defaut date, computed from the former observations. One of these statistics is caed the reaised LEQ factor and is defined beow. Consider a defauted faciity g with an exposure variabe over time, given by E(t) and a credit imit given by L(t). Figure 11.1 presents the evoution of the exposure. If the faciity has a defaut date td, given a reference date tr < td, the pair i ¼ {g, tr}is caed index of the observation. If EAD i stands for the observed exposure at defaut, 6 E(td), this can be expressed in terms of the exposure and the imit of the faciity observed at the reference date, assuming that L(tr) 6¼ E(tr), as: Where LEQ i is given by: EAD i ¼ EðtrÞþLEQ i ðlðtrþ EðtrÞÞ (11.1) LEQ i ¼ EðtdÞ EðtrÞ LðtrÞ EðtrÞ (11.2) 5 Throughout this chapter the term usage refers to the usage of the faciity in euros (sometimes the terms exposure, drawn amount or utiization are used with the same meaning). 6 In this chapter, it is assumed that a precise definition of observed EAD for defauted faciities, EAD i, has been estabished previousy and that it is appied consistenty across faciities and over time for different interna purposes. To understand why an expicit definition of observed EAD is necessary see Araten and Jacobs (2001, p. 37), where two situations are cited when the simpe definition of EAD i ( fina amounts shown at the time of defaut ) is not adequate: charge-offs or seizures of coatera occurred just prior to the defaut date.

204 G. Mora or: LEQ i ¼ EðtdÞ LðtrÞ EðtrÞ LðtrÞ 1 EðtrÞ ¼ ead i eðtrþ 1 eðtrþ LðtrÞ (11.3) Therefore, given an observation, O i, characterised by a pair i ¼ {g, tr}, with L (tr) 6¼ E(tr), the former formuae can be used to compute a reaised LEQ factor. We denote the reaised LEQ factor associated with the observation O i by LEQ i, and by LEQ(tr) when the focus is on the reference date tr. There are three imitations when using this statistic as the basic input for estimation procedures: It is not defined when L(tr) ¼ E(tr). This impies that it is not possibe to estimate directy EAD( f ) based on the vaue of this statistic for faciities that at the current date exhibit percent usage, e(tr), equa to one. 7 It is not stabe when L(tr) ffi E(tr). This means that reaised LEQ factors are not very informative when percent usage is cose to one. As shown in Sect. 11.4.2.2, the different behaviour of reaised LEQ factors, depending on the eve of credit percent usage at the reference date, has important practica consequences. It does not take into account changes in the imit over time. In formuae (11.2) and(11.3) reaised LEQ factors have been defined without taking into account possibe changes in the imit of the faciity between the reference date and the defaut date. 8 As it is shown in detai in Sect. 11.4.2.3, thisis ony one of the causes that justifies the existence of reaised LEQ i factors greater than one. For these reasons, banks sometimes use other statistics as their basis for estimating EADs. For exampe, an obvious possibiity is to define reaised CCFs simiary to reaised LEQ factors. By using an equation anaogous to (11.1) the expression for this statistic is given by the percent exposure at defaut: CCF i ¼ EAD i LðtrÞ ¼ ead i (11.4) Athough this statistic is ess used in practice than LEQ i for these types of faciities, it has two advantages: 7 This imitation appies when the estimates are used for interna purposes because, in principe, interna uses do not need to assume that LEQ( f ) 0, or equivaenty, that the EAD( f ) estimate has to be greater or equa than the current exposure of this faciity, E( f ). 8 Some banks define reaised LEQ factors by using E(td)/L(td), percent usage at defaut, instead of ead i ¼ E(td)/L(tr), percent exposure at defaut, in (11.3). The aim of this definition is to take into account changes in the credit imit after the reference date and to avoid computing reaised LEQ factors greater than 1. It is straightforward to show that the former definition is consistent with (1) if EAD i is mutipied by the factor L(tr)/L(td).

11 EAD Estimates for Faciities with Expicit Limits 205 The reaised CCF is we defined even when L(tr) ¼ E(tr). It is stabe even when L(tr) ffi E(tr). Sometimes it is said that with this statistic, if the faciity g had a constant imit L(g), it is not necessary to specify a reference date. However, as it is shown in Sect. 11.4, data sets for estimating procedures need to incude the vaues of certain risk drivers that vary over time and therefore it is necessary to consider an expicit reference date. Additiona usefu statistics are introduced in Sect. 11.5; unti then it is assumed that reaised LEQs are used as the basis for the estimation process. 11.3 How to Obtain a Set of Reaised Conversion Factors Given a set of defauted faciities, there are severa approaches frequenty empoyed by banks to obtain reaised conversion factors or other statistics 9 that can be used, in addition with other information, to obtain estimates for the EAD of non defauted faciities. A these approaches are based on observations of defauted faciities at specific reference dates previous to the defaut date. Depending on the rue used for seecting these reference dates we refer to these approaches as: Fixed Time Horizon, Cohort Approach or Variabe Time Horizon. 11.3.1 Fixed Time Horizon In this approach, first a time horizon, T, is seected and second, for each defauted faciity with L(td T) 6¼ E(td T), a reaised LEQ factor is computed by using td T as the reference date: LEQðtd TÞ ¼ EðtdÞ Eðtd TÞ Lðtd TÞ Eðtd TÞ (11.5) In practice, T is frequenty set to 1 year (Fig. 11.2). Drawbacks: The fixed time horizon, T, is conventiona. It is not possibe to incude directy defauted faciities when the age of the faciity at the date of defaut is ess than T. 9 As is shown in Sect. 11.5, in addition to the reaised CFs, the percent increase in usage between the reference date and the defaut date or the increase in exposure between those dates are statistics that can be used to estimate CFs or EADs.

206 G. Mora Reaised LEQs L(td1-T) E(td1-T) E(td) LEQ1 LEQ2 td1-t td2-t E(td2-T) L(td2-T) td1 td2 E(td2) DF1 DF2 Defauted faciities Observation period Fig. 11.2 Reaised LEQ with the fixed time horizon approach It does not take into account a the reevant information because for each faciity g defauted during the observation period, ony the observation {g, td T}isused. It does not take into account the possibiity that current exposures can defaut at any moment during the foowing year. Impicity, estimates based on this approach assume that the defaut date for each faciity that wi defaut over the foowing 12 months, wi be the end of this period. This assumption coud introduce bias into the estimates. Advantages: Dispersion of reference dates. The use of a common horizon, T ¼ td tr, contributes to the homogeneity of the reaised LEQs. 11.3.2 Cohort Method First, the observation period 10 is divided into intervas of a fixed ength (cohorts), for exampe 1-year intervas. Second, the faciities are grouped into cohorts according to the interva that incudes their defaut dates. Third, in order to compute a reaised LEQ factor associated with each faciity, the starting point of the time interva that contains its defaut date is used as the reference date, {t1,t2,..., ti,..., tn}: LEQðtiÞ ¼ EðtdÞ EðtiÞ LðtiÞ EðtiÞ (11.6) This is iustrated in Fig. 11.3. 10 The period of time covering the data is the observation period.

11 EAD Estimates for Faciities with Expicit Limits 207 Reaised LEQs t1 t2 t3 LEQ1 td1 DF1 LEQ2 LEQ3 LEQ4 td4 td2 td3 DF2 DF3 DF4 Defauted faciities Observation period Fig. 11.3 Reaised LEQ with the cohort approach Drawbacks: The ength of cohorts is conventiona The reference dates are conventiona It does not use a the reevant avaiabe information because for each faciity g defauted during the observation period (and incuded in a cohort with initia date t j ) ony the observation {g, t j } is used The reference dates are concentrated The reaised LEQs are ess homogenous than those computed by using a fixed time horizon. The reason is that this approach computes LEQ i factors with very different vaues for the horizon (td tr) Advantages: It does take into account the possibiity that current exposures can defaut at any moment during the foowing year. 11.3.3 Variabe Time Horizon First, a range for horizon vaues (e.g., 1 year) for which we are going to compute LEQ i factors is fixed. Second, for each defauted faciity we compute the reaised LEQ factors associated with a set of reference dates, (for exampe, 11 1 month, 2 months,..., 12 months before defaut). 11 Athough with this approach, in theory, it is not necessary to use monthy observations, from now on it is assumed that the reference dates are the end of each month from the first month before the defaut date (td tr ¼ 1) to 12 months before (td tr ¼ 12). This choice may be adequate for most of the product types and, in many cases, compatibe with the information currenty avaiabe in banks.

208 G. Mora The rationae for this method is to take into account a broader set of possibe defaut dates than in the other approaches when estimating a suitabe LEQ factor for a non-defauted faciity conditiona on the defaut during the foowing year. LEQðtd jþ ¼ EðtdÞ Eðtd jþ ; j ¼ 1;...;12 months (11.7) Lðtd jþ Eðtd jþ In principe, 12 tweve reaised LEQs coud be associated with each defauted faciity (Fig. 11.4). However, these LEQ factors are ceary not homogenous in the sense that some of these vaues are computed by using observations very cose to the defaut date (i ¼ {g, td 1}) and others are based on observations 1 year before defaut (j ¼ {g, td 12}). This means that it is necessary to recognise these differences via risk drivers. As shown in Sect. 11.4.3, the key point is to take into account when the bank identified the faciities as non-norma and, consequenty for the purpose of obtaining estimates for faciities in a norma status, to use ony observations meeting this requirement. The main reason is that near to defaut, borrowers are in genera, cassified in a non-norma interna cass (in the foowing the variabe that identifies these different interna casses is caed status ). This means that a faciity is subject to cose monitoring and, in genera, the borrower can not make additiona drawdowns under the same conditions as before. For exampe, in retai portfoios during the ast 3 months before defaut, since the first impairment, it is very difficut for the borrower to make further drawdowns and, in genera, ony interest and other interna charges are aowed. Therefore, it is necessary to identify when a defauted faciity was abeed as non-norma and ony use the reaised LEQs associated with previous dates when estimating LEQ factors to norma faciities. In practice, for retai portfoios, at east six dates can frequenty be used, and as a maximum, nine dates. On the other hand, for corporate portfoios, the status of the faciities is cosey inked to the interna rating of the borrower, and therefore there coud be cases in which the norma status appies unti it is known that the borrower has defauted. In genera, it is necessary to take into account the tweve separate LEQ i factors associated with the same faciity because the vaues of the risk drivers can be different for each reference date. Advantages: It takes into account more observations than the previous methods. Those faciities with L(tr) ¼ E(tr), that in the previous methods were not taken into account, can now be used for those reference dates when L(td i) 6¼ E(td i) for some i ¼ 1,...,12. Each faciity coud produce up to tweve LEQ i associated with tweve different observations. 12 For exampe, if a faciity is ony 4 months od when it defauts, then we wi have at most four associated LEQ factors.

11 EAD Estimates for Faciities with Expicit Limits 209 Reaised LEQs LEQ1 LEQ2 LEQ3 LEQ4 LEQ12 td-12 td-4 td-3 td-2 td-1 td DF1 DF2 Defauted faciities Observation period Fig. 11.4 Reaised LEQs with the variabe time horizon method In principe, estimate procedures based on these data shoud produce more stabe (it uses more observations) and accurate (it uses more information) estimates. Drawbacks: Banks have to store more data for each defauted faciity (up to tweve observations). It is necessary to use a variabe (status) that contributes to identifying homogenous LEQ i factors. 11.4 Data Sets (RDS) for Estimation Procedures This section discusses the idea requirements for the reference data set (RDS) which incudes the avaiabe information that can be used for estimation procedures. It focuses on those RDS based on historica information from faciities that defauted over an observation period. First, it presents a genera structure for this RDS that faciitates the impementation of estimation procedures and then it enumerates some fieds that shoud be incuded in the RDS. Second, it ists certain scope requirements. Finay, it comments on severa adjustments and decisions that have to be made before the estimation phase. 11.4.1 Structure and Scope of the Reference Data Set 11.4.1.1 Structure Given the focus on estimation procedures based on observations of defauted faciities at certain reference dates, it is usefu to have a structure for the reference

210 G. Mora data set adapted to this approach. Consequenty, the data structure shoud contain the reevant information on the basis of observations O i which have associated a unique pair formed by a defauted faciity, g, and a vaid reference date, tr < td, (more specificay, the mentioned pair i ¼ (g, tr) shoud be the primary key of the reference data set). Each of these observations, O i, incudes: The vaues of certain static characteristics of g, I(g) The vaues of a set of observabe variabes reated to g at the reference date tr, that are going to be used as expanatory variabes or Risk Drivers, RD(tr) The observed EAD i and defaut date td In summary, a very genera structure for the RDS is given by: RDS ¼ O i¼ðg;trþ ; Oi¼ðg;trÞ ¼ fðg; trþ; IðgÞ; RDðtrÞ; EAD i ¼ EðtdÞg (11.8) with regard to the fieds that contain the information associated with each observation, in practica impementations, as a minimum, the foowing data are required: Static characteristics, I(g): identifier of faciity, NF; type of faciity, TF; identifier of portfoio, TP; and identifier of borrower, NB Risk drivers, RD: reference date, tr; defaut date, td; reference exposure, E(tr); reference imit, L(tr); faciity status, S(tr); and rating cass or poo, R(tr) If other potentia risk drivers for the EAD were identified, the RDS shoud contain fieds for the vaues of these potentia RD at the reference date tr. For exampe, it is worth considering the incusion of macroeconomic indicators, MI that can be used to increase the forward ooking character of the estimates and the predictive abiity of the estimators. In symbos: RDðtrÞ ¼ fe; L; S; R; td; MI; Otherg IðgÞ ¼ðNF; NB; TF; TP; OtherÞ (11.9) Risk drivers are discussed in more detai in Sect. 11.4.3. 11.4.1.2 Scope and Other Requirements on the RDS In addition to a structure for the RDS suitabe for the estimation procedures, the RDS has to meet certain interna and externa requirements reated to the scope of the RDS. The scope of the RDS has to be defined without ambiguity. As a minimum, it is necessary: To define the type of faciities, type of borrowers and type of portfoios To make expicit the definition of defaut used and the observation period covered To identify and describe the source (or sources) of the data

11 EAD Estimates for Faciities with Expicit Limits 211 The RDS shoud incude observations for a the faciities that have defauted during the observation period and meet the other scope requirements (type of faciities, portfoios, etc). A the excusions shoud be identified and justified The definition of defaut used shoud be consistent with the ones used for PD and LGD estimation purposes The observation period shoud be ong enough to incude observations of faciities defauted under very different genera economic circumstances, ideay covering an entire economic cyce Additionay, to use the estimates in capita requirements under AIRB approaches: The definition of defaut shoud be consistent with the IRB defaut definition The observation period shoud cover at east 7 years for corporate portfoios and five for retai portfoios When necessary, the observation period shoud contain a period with downturn conditions 11.4.2 Data Ceaning As we as other more genera issues reated to data ceaning (identification and treatment of outiers, eimination of poor quaity data, etc.), before to the estimation phase it is necessary to make certain decisions that coud affect the observations incuded in the RDS. Some of these issues are anaysed in the next sections. 11.4.2.1 Treatment of Mutipe Credit Faciities with a Singe Obigor Athough it is cear that reaised CFs and the other reevant information incuded in the RDS are computed or observed at faciity eve, under certain circumstances, to produce sensibe estimates, it coud be necessary or appropriate to group together, within the same observation, information from different faciities associated with the same borrower. There are at east two situations to be considered: If there are two or more observations of simiar credit faciities with the same borrower and the same risk drivers vaues, excuding current usages and other vaues that are a function of L(t) and E(t), then it coud be appropriate to group these observations in a new observation as 13 : fðh; trþ; Eh; ð trþ; Lh; ð trþ; BðhÞ; RDðh; trþg ) fðh þ g; trþg fðg; trþ; Eg; ð trþ; Lg; ð trþ; BðgÞ; RDðg; trþg fðh þ gþg ¼ fðh þ g; trþ; Eh; ð trþþeg; ð trþ; Lh; ð trþþlg; ð trþ; B; RDðtrÞg (11.10) 13 This procedure is mentioned in Araten and Jacobs (2001, p. 36).

212 G. Mora For certain portfoios and faciities, it is common for the maturity to be 1 year. However, in most cases, the bank approves a new faciity (maybe with a different imit), when the od faciity expires. In these circumstances, faciities defaut with age ess than 12 months and therefore it is not possibe to obtain tweve observations for the RDS. However, if this faciity was approved at the time of the expiration of a non-defauted faciity of the same type with the same borrower, it coud be usefu to chain these faciities together. Using this procedure, more observations can be incuded in the RDS. Depending on the characteristics of the portfoio, these decisions coud be made on a case by case basis or foowing a mechanica rue. 11.4.2.2 Treatment of Observations with Negative Reaised LEQ Factors 14 As Fig. 11.5 shows, it is possibe to obtain negative reaised LEQ factors associated with defauted faciities. Arithmeticay, negative reaised LEQs arise when EAD i ¼ E(td) < E(tr). This situation is especiay frequent when td tr is arge and the credit percent usage at the reference date, e(tr), is cose to one, moreover some of these vaues are very arge in absoute vaue. It is very important to note that: The empirica distributions of reaised LEQ factors conditiona on the percent usage at the reference date, e(tr), are very different These empirica distributions are highy asymmetrica, especiay for percent usage vaues cose to one. L(t) E(tr) > EAD + EAD = E(td) E(t) Fig. 11.5 Negative reaised LEQ factors tr td 14 From a forma point of view, this discussion is simiar to that reated to reaised LGDs. However, there are substantia differences in the reasons that justify the existence of negative reaised vaues between both cases.

11 EAD Estimates for Faciities with Expicit Limits 213 To iustrate these points, from definition (11.3) it can be seen that a sma increment of e(tr) affects the reaised LEQ factor foowing: @LEQ i @eðtrþ 1 eðtdþ ¼ ð1 eðtrþþ 2 (11.11) Therefore, the sensitivity of reaised LEQ factors to sma changes in the percent usage at the reference date depends criticay on the eve of e(tr). The smaer is (1 e(tr)) 2, the arger tends to be the variabiity of LEQ conditiona on e(tr). Moreover, if LEQ i is expressed in terms of a percent reaised exposure at defaut ead i proportiona to the percent usage at the reference date, from definition (11.3) the foowing is obtained: eðtrþð1 þ DÞ eðtrþ LEQ i ðdþ ¼ ¼ D eðtrþ ð1 eðtrþþ 1 eðtrþ (11.12) and for arge vaues of e(tr) there is no possibiity of arge vaues of D, but it is possibe to find negative arge vaues for D. The former asymmetries among LEQ i for ow and arge percent usage vaues and the existence of more observations with arge negative LEQ i than with arge positive vaues have practica importance. The main reason is that, as is shown in Sect. 11.5, banks frequenty use averages of LEQ i as estimators for LEQ( f ) and these sampe means are severey affected by both circumstances. The former points suggest that, as a minimum, these averages shoud be restricted to those observations with simiar percent usage eves or, in other words, percent usage eve shoud be a risk driver for LEQ( f ). As a consequence, it is important to carify the treatment of those observations with negative reaised LEQ factors. In practice, there are severa possibiities: Censoring 15 the data (the LEQ i factors) to impose certain restrictions: Some banks change the definition of reaised LEQ to force the non-negativity: LEQ þ i ¼ max[0, LEQ i ] In other cases, banks change the definition of the reaised EAD used in LEQ i computations directy (observed EAD): EAD þ i ¼ max[ead i, E(tr)]. As discussed previousy, negative LEQ i can be associated with vaid observations of defauted faciities. To justify this practice, banks argue that, ceteris paribus, this adjustment introduces a conservative bias into the estimates. 15 It is necessary to use of this terminoogy (censoring and truncation) carefuy because these words are not used consistenty in the iterature. For exampe, Araten and Jacobs (2001, p. 36), uses the term truncation for describing what in this paper is referred to as censoring. The terminoogy empoyed in the text foows that used in Working Paper No. 14 BCBS (2005, p.66).

214 G. Mora Truncation: this practice consists of the remova of the observations associated with negative LEQ i factors. It is difficut to find a rationae for the truncation of observations with negative or zero reaised LEQs. In principe, this truncation coud be a practica method to generate a stressed distribution of LEQ i factors. However, this procedure presents at east two important drawbacks: The eimination of observations with LEQ i 0 coud introduce inconsistencies with the RDS used for obtaining LGD estimates because some of those observations coud be associated with faciities with high reaised osses. When the estimation method uses sampe averages, the LEQ estimates based on a truncated RDS coud be very unstabe with changes in the RDS depending on the number of observations with LEQ i factors cose to zero. Do nothing with the reaised LEQ factors (but set a foor to the estimates, 16 LEQ( f ) 0).This is the most natura decision. As proved in Sect. 11.6.3.1, if the constraint on the estimators given by LEQ( f ) 0 is imposed and a specific mode for the estimated LEQ based on minimising the estimation errors (measured in terms of a specia oss function) is adjusted then the same estimates are produced by using the origina or the censored data. 11.4.2.3 Treatment of Observations with Reaised LEQ Factors Greater than One In principe, given the definition of LEQ i factors (11.2) it woud be natura to expect LEQ i factors to be ess or equa to one in a bank with an adequate contro environment. However, the existence of LEQ i factors greater than one is not in a cases an indicator of a faiure in the contros estabished by the bank to ensure that credit imits are effective. There are situations in which LEQ i factors greater than one naturay arise. For exampe: In some cases, banks use unadvised imits 17 instead of the nomina imits of the faciities to manage the risk internay. The possibiity of additiona drawdowns for the borrower ony stops when the exposure is greater than the unadvised imit In some products, for exampe credit cards or current account overdrafts, such probems are difficut to avoid because there is typicay a time ag between the current exposure and the figure used by the bank to estabish contros Sometimes the exposure at defaut incudes the ast iquidation of interest (and fees) and this amount is charged to the account even when the imit had been previousy reached. 16 As a minimum, this foor is a requirement when the estimates are used for reguatory purposes. 17 Frequenty, these unadvised imits are computed as a percentage or a fixed amount above the expicit advised imits.

11 EAD Estimates for Faciities with Expicit Limits 215 The former excesses over the nomina imits are typicay sma. In these cases, it woud be appropriate to treat these observations as any other cases. However, in other circumstances there are observations with arge reaised LEQ factors that are the resut of severa causes competey different, such as: Changes in the imit after the reference date and previous to the knowedge of difficuties in the faciities Expicit or impicit change of imit at the date of defaut or when difficuties with the faciity have aready arisen Inadequate contro environment and existence of human errors or frauds that coud be treated as operationa risk events. In spite of the diversity of the former circumstances, some banks cap a the reaised EADs at one. In genera, this rue is neither adequate for interna use nor for reguatory use and, on the contrary, a detaied anaysis of the causes behind these observations is necessary before making acceptabe decisions for each situation. In any case, coherence with the procedures used when cacuating reaised LGDs is a prerequisite. 11.4.3 EAD Risk Drivers In practice, risk drivers (RD) affect the estimates in two different ways. First, certain quaitative and quantitative characteristics are used to segment the portfoio under anaysis into homogenous casses. Among these risk drivers, different studies state as a minimum: Faciity type: the importance of this characteristic is because there is a spectrum of faciities with expicit imits and different conditions for drawdowns, ranging from faciities with unconditiona imits, to faciities in which each drawdown requires approva. Covenants: frequenty the bank can deny additiona drawdowns when specific circumstances occur. The causes which detai these circumstances are caed covenants. 18 Typicay, these covenants are reated to objective situations that are indicators of credit deterioration of the borrower such as: downgrades, drops in profitabiity or changes in certain key financia ratios beow expicit threshods. 19 Second, once we have identified a cass incuding faciities that, in principe, are homogenous enough for the proposa of designing a common expanatory EAD mode, it is necessary to seect an appropriate set of expanatory (quantitative) 18 Sometimes these causes are caed Materia Adverse Changes (MAC) causes. See Lev and Rayan (2004, p. 14). 19 For more detais on covenants, see Sufi (2005, p.5).

216 G. Mora variabes (risk drivers). Among these quantitative risk drivers, different studies, based on private data bases, suggest it is convenient to consider as a minimum: Commitment size L(tr) The drawn and undrawn amounts, E(tr) and L(tr) E(tr) The credit percent usage at the reference date e(tr). As discussed in 11.4.2.2, this percent usage vaue has discriminative power with regard to reaised LEQ factors The time to defaut td tr: ex-post anaysis shows that this variabe has significant expanatory power, at east cose to defaut The rating cass at the reference time R(tr): this variabe is in genera reevant, but different studies have found a significant positive correation between credit quaity and CF in some cases and a significant negative correation in others. It seems that the roe of the rating as a reevant risk driver is inked to the type of portfoio, the dynamic of each rating system and the uses of the rating for interna purposes Status of the faciity at the reference date S(tr): most banks, in addition to rating or scoring systems, have warning systems that focus on eary identification of iquidity probems and other short term borrower difficuties. The basic difference with the rating is that these warning systems are more dynamic and identify probems before the rating 20 does. As a resut of these systems, certain faciities are cassified into certain broad casses, typicay: norma status and a few grades under specia vigiance. This means that once a faciity has been identified as inked to a probematic borrower the eve of monitoring and, in some cases; the practica conditions for additiona drawdowns are changed. 21 Therefore, the status is a critica risk driver when estimating EAD Macro indicators. For the observations in the RDS, the vaues of the above isted risk drivers are in genera, known. For a non-defauted faciity, the vaues of these variabes are computed using the current date t, as the reference date tr. With regard to the time to defaut, there is a probem because, for a non defauted faciity, the time to defaut is unknown. In the Base II context, the interest is in EAD estimates subject to the condition that the faciity defauts during a period of 1 year. Therefore, in this context, the interest is in the infuence of this variabe when the vaue ranges from 1 to 12 months. 20 The most common reationship between these eary warning systems and the ratings is that certain changes of status trigger the processes for a new evauation of the borrower rating. 21 } 477. Due consideration must be paid by the bank to its specific poicies and strategies adopted in respect of account monitoring and payment processing. The bank must aso consider its abiity and wiingness to prevent further drawings in circumstances short of payment defaut, such as covenant vioations or other technica defaut events. Banks must aso have adequate systems and procedures in pace to monitor faciity amounts, current outstandings against committed ines and changes in outstandings per borrower and per grade. The bank must be abe to monitor outstanding baances on a daiy basis., BCBS (2004).

11 EAD Estimates for Faciities with Expicit Limits 217 11.5 EAD Estimates 11.5.1 Reationship Between Observations in the RDS and the Current Portfoio This section presents different methods of assigning a 1-year EAD estimate to a non-defauted faciity f at the date t, incuded in the current portfoio, based on a subset of a RDS which comprises observations (of defauted faciities) simiar to f at t. We denote this subset by RDS( f ). The process of assigning a subset of the RDS to each faciity in the portfoio is caed mapping and this aows the current portfoio to be cassified by grouping faciities with the same or simiar RDS( f ). Conversey, some banks segment the portfoio of current exposures into casses comprising simiar faciities. This approach coud be reduced to the previous one because after this cassification of exposures, each cass C has to be mapped into a RDS(C) which is used to estimate EAD( f ) for a f incuded in C. 11.5.2 Equivaence Between EAD Estimates and CF Estimates Given a non-defauted faciity f and an estimator EAD( f ), if L( f ) 6¼ E( f ), the estimate can be expressed in terms of a LEQ( f ) factor foowing the equation: if LEQ( f ) is given by: EADð f Þ¼Eð f ÞþLEQð f ÞðLð f Þ Eð f ÞÞ (11.13) LEQð f Þ¼ EADð f Þ Eð f Þ Lð f Þ Eð f Þ ¼ eadð f Þ eð f Þ 1 eð f Þ (11.14) Additionay, if we are interested in EAD( f ) estimates that satisfy EAD( f ) E( f ), then from (11.13): If L( f ) > E( f ) then EAD( f ) E( f ) if and ony if LEQ( f ) 0 If L( f ) < E( f ) then EAD( f ) E( f ) if and ony if LEQ( f ) 0 Therefore, without any additiona hypothesis, for faciities that verify L( f ) 6¼ E( f ), it has been shown that to estimate EAD( f ), it is sufficient to focus on methods that estimate suitabe conversion factors LEQ( f ) based on the observations incuded in the reference data set, RDS( f ) and afterwards to empoy (11.13) to assign individua EAD estimates. Finay, the simpest procedure to estimate a cass EAD is to add the individua EAD estimates for a the faciities incuded in the cass.

218 G. Mora For exampe, for certain faciity types, some banks assign EADs by using a previousy estimated CCF( f ), and then appying the formua: EADðf Þ¼CCFðf ÞLðf Þ (11.15) This method is sometimes caed Usage at Defaut Method. 22 If e( f ) 6¼ 1, this case can be reduced to the genera method, given in (11.13), by assigning a LEQ( f ) factor given by: LEQðf Þ¼ CCFðf ÞLðf Þ Eðf Þ Lðf Þ Eðf Þ ¼ CCFðf Þ eðf Þ 1 eðf Þ (11.16) Conversey, if a LEQ( f ) is avaiabe, from (11.16), an expression for an equivaent CCF( f ) can be found, given by: CCFðf Þ¼LEQðf Þð1 eðf ÞÞþeðf Þ (11.17) Therefore, the EAD estimation method based on LEQ( f ) and the one based on CCF( f ) are equivaent, with the exception of those faciities with e( f ) ¼ 1. In the foowing sections, severa methods that are normay used in practice by banks to estimate LEQ factors are presented from a unified perspective. This is used ater to anayse the optimaity of the different approaches. Additionay, the formuae most used in practice are derived as specia cases of the previous methods when a specific functiona form has been assumed for LEQ( f ). 11.5.3 Modeing Conversion Factors from the Reference Data Set This section presents severa methods for estimating conversion factors based on regression probems starting with the foowing basic equation: EADðf Þ Eðf Þ¼LEQðf ÞðLðf Þ Eðf ÞÞ (11.18) These methods try to expain the observed increases in the exposure between the reference date and the defaut date and they can be grouped into three approaches depending on how these increases are measured: as a percentage of the avaiabe amount (focus on reaised LEQ factors); as a percentage of the observed imit (focus on percent increase in usage); or finay in absoute vaue (focus on increase in exposure). 22 This method is caed Momentum Method in CEBS Guideines (2006, }} 253 and 254).

11 EAD Estimates for Faciities with Expicit Limits 219 Mode I. Focus on reaised LEQ factors Dividing (11.18) byl( f ) E( f ), it is obtained: eadðf Þ eðf Þ 1 eðf Þ ¼ EADðf Þ Eðf Þ ðlðf Þ Eðf ÞÞ ¼ LEQðf Þ (11.19) In this approach, the rationae is to determine a function of the risk drivers LEQ (RD) which expains the LEQ i factors associated with RDS( f ), LEQ i ¼ (EAD i E i )/(L i E i ), in terms of LEQ(RD i ). This can be made starting with an expression for the error associated with LEQ i LEQ(RD i ) and soving a minimisation probem. In practice, a quadratic and symmetric error function is amost universay used. As a consequence of this choice, the minimisation probem to sove is given by (Probem P.I): Min LEQ ( ) X ðleq i LEQðRD i ÞÞ 2 i ¼ Min LEQ ( X EAD i E ) 2 i LEQðRD i Þ L i E i i (11.20) Or: LEQðf Þ¼Min LEQ ( ) X 1 ðl i E i Þ 2 ð EAD i E i LEQðRD i ÞðL i E i ÞÞ 2 i (11.21) Mode II. Focus on the increase of the exposure as a percentage of the observed imit (focus on percent increase in usage). Dividing the basic (11.18) byl( f ), it is obtained: EADðf Þ Eðf Þ Lðf Þ Lðf Þ Eðf Þ ¼ LEQðf Þ Lðf Þ (11.22) Therefore, using this approach, the observabe amounts to be expained are (EAD i E i )/L i and the expanatory vaues are LEQ(RD i ) (L i E i )/L i. Foowing the same reasoning as in the previous approach, the minimisation probem to sove is given by (Probem P.II): Min LEQ ( X EAD i E i LEQðRD i Þð L ) i E 2 i Þ L i L i i (11.23) Or: LEQðf Þ¼Min LEQ ( ) X 1 2 L ð EAD i E i LEQðRD i ÞðL i E i ÞÞ 2 i i (11.24)

220 G. Mora Mode III. Focus on increases in the exposure Directy from the basic equation, it is obtained: EADðf Þ Eðf Þ¼LEQðf ÞðLðf Þ Eðf ÞÞ (11.25) In this case, the amounts to expain are EAD i E i and the expanatory variabe is LEQ(RD i ) (L i E i ). As in the other cases, the associated minimization probem is given by (Probem P.III): LEQðf Þ¼Min LEQ ( ) X ðead i E i LEQðRD i ÞðL i E i ÞÞ 2 i (11.26) From (11.21), (11.24) and (11.26), these probems can be reduced to a more genera (Probem P.IV): Min LEQ ( X ) EAD i E i ðl i E i Þ 2 LEQðRD i Þ o i o i i (11.27) where o i stands for L i E i in Mode I, L i in Mode II, and 1 in Mode III. If F* denotes the empirica distribution of (EAD E)/o associated with the observations incuded in RDS( f ), the Probem P.IV can be expressed as: LEQðf Þ¼Min LEQ ( * + ) 2 EAD E ðl EÞ E LEQðRDÞ F o o (11.28) In the most genera case, assuming that (L E)/o is constant for observations in RDS( f ), the soution to (11.28) is given by 23 : LEQðf Þ¼E F * + EAD E o RDðf Þ oðf Þ Lðf Þ Eðf Þ (11.29) As a consequence, the practica probem is to find out methods to approximate these conditiona expectations. If a parametric form for LEQ is assumed, the probem becomes: LEQðf Þ¼LEQð^a; ^b; ( :::Þ; * + ) EAD E ðl EÞ 2 (11.30) f^a; ^b; :::g ¼ Min E LEQða; b; :::Þ fa;b;:::g F o o 23 See Appendix B.

11 EAD Estimates for Faciities with Expicit Limits 221 If the parametric functiona form is inear in the parameters, the probem becomes a inear regression probem. In summary, traditiona methods can be cassified as regression modes that focus on the minimization of quadratic errors in the forecasts of: LEQ i factors; EAD i in percentage of the imit; or EAD i. These methods produce different EAD( f ) estimates based on LEQ( f ) estimates proportiona to conditiona expectations. At first gance, the approach that focuses directy on LEQ factors (Mode I) seems the most natura, the method that focuses on percent increases in usage (Mode II) seems more stabe than the previous one and, as is shown in detai in Sect. 11.6, the approach based on EAD increases (Mode III), coud present advantages when the estimates are used in reguatory capita computations because of the ink between capita requirements and EAD. 11.5.4 LEQ ¼ Constant 11.5.4.1 Probem P.I: The Sampe Mean In practice, 24 banks frequenty use, as an estimator for LEQ( f )att, the sampe mean of reaised LEQ i, restricted to those observations i ¼ {g, t} simiar to {f, t, RD}. Assuming that the conversion factor is a constant for observations simiar to {f, t}, LEQ(f) ¼ LEQ, and soving the Probem P.I the foowing is obtained: ( X ) 2 EAD i E i LEQ ¼ Min LEQ2R ðl i E i Þ LEQ ¼ 1 n i X EADi E i ðl i E i Þ ¼ 1 X LEQi n (11.31) In other cases, banks use a sampe weighted mean that tries to account for a possibe reationship between size of the exposures (or imits) and LEQ. If in Probem P.I a weight w i is introduced, and it is assumed that LEQ is constant for observations simiar to {f, t}, then: LEQ ¼ Min LEQ2R ( X ) EAD i E 2 i w i ðl i E i Þ LEQ ¼ i P wi LEQ i P wi (11.32) When the reason for incorporating the weighting is to take into account a LEQ risk driver, this approach is inconsistent. The reason for this is that the weighted average is the optimum soution ony after assuming that LEQ ¼ constant, i.e. no risk drivers are considered. 24 At east this is the case in modes appied by some Spanish banks at present (2006).

222 G. Mora 11.5.4.2 Probem P.II: The Regression Without Constant Another method widey used by banks is to use the regression estimator for the sope of the regression ine based on Mode II, assuming that LEQ is a constant. Under these conditions the expression for the regression estimator is given by: ( X ) EAD i E i L i E 2 i LEQ ¼ Min LEQ LEQ2R L i L i i P ðead i E i ÞðL i E i Þ P L 2 ¼ i ð eadi e i Þð1 e i Þ P L i E 2 ¼ P i ð 1 ei Þ 2 L i (11.33) 11.5.4.3 Probem P.III: Sampe Weighted Mean If in P.III it is assumed that LEQ ¼ constant it can be expressed as: ( X ) LEQðf Þ¼ Min ðl i E i Þ 2 EAD i E 2 i LEQ2R ðl i E i Þ LEQ i (11.34) And the optimum is given by: LEQ ¼ P wi LEQ P i ; with w i ¼ ðl i E i Þ 2 (11.35) wi Therefore, using this approach, a weighted mean naturay arises. However, it is worth noting that these weights (L i E i ) 2 are different from those currenty proposed by some banks (based on L i or E i ). 11.5.5 Usage at Defaut Method with CCF ¼ Constant (Simpified Momentum Method) This method is sometimes used by banks that try to avoid the expicit use of reaised negative LEQ factors, or for faciities for which the current usage has no predictive power on EADs. It estimates the EAD for a non-defauted faciity, EAD(f), by using (11.15) directy and a rough CCF estimate, for exampe, the sampe mean of the reaised CCFs computed from a set of defauted faciities C. EADðf Þ¼CCFðCÞLðf Þ (11.36)

11 EAD Estimates for Faciities with Expicit Limits 223 From (11.16) and assuming that CCF ¼ constant, a specific functiona form for LEQ(e( f )) is founded, given by: LEQðf Þ¼ CCF Lðf Þ Eðf Þ Lðf Þ Eðf Þ ¼ CCF eðf Þ 1 eðf Þ (11.37) In genera, two faciities with the same estimated CCF and with different vaues for current percent usage, e(t), wi have different LEQ estimates foowing the former (11.37). Themaindrawbackwiththeprocedurebasedon(11.36) is that experience shows that, in genera, drawn and undrawn imits have strong expanatory power for the EAD. For this reason, this method (with CCF ¼ constant) does not seem to meet the requirement of using a the reevant information 25 (because it does not take into account the drawn and undrawn amounts as expanatory variabes in the EAD estimating procedure) for most of the types of faciities that arise in practice. 11.6 How to Assess the Optimaity of the Estimates To assess the optimaity of the different CF estimates associated with a reference data set and a portfoio, it is necessary to be more precise about some eements in the basic probem. The first eement requiring carification is the type of estimates according to the roe of macroeconomic risk drivers in the estimation method. The second eement is how to measure the errors associated with the estimates and to motivate that particuar choice. This can be done by introducing a oss function that specifies how the differences between the estimated vaues for the EAD and the actua vaues are penaised. 11.6.1 Type of Estimates Focusing on the use of the macroeconomic risk drivers, the foowing types of estimates can be distinguished: Point in Time estimates (PIT): these estimates are conditiona on certain vaues of the macroeconomic risk drivers, for exampe, vaues cose to the current ones. This aows the estimates to be affected by current economic 25 } 476. The criteria by which estimates of EAD are derived must be pausibe and intuitive, and represent what the bank beieves to be the materia drivers of EAD. The choices must be supported by credibe interna anaysis by the bank. [...] A bank must use a reevant and materia information in its derivation of EAD estimates. [...], BCBS (2004).

224 G. Mora conditions and to vary over the economic cyce. In theory, this is a good property for the interna estimates banks need for pricing and other management purposes. The main probem with PIT estimates is that theyarebased on ess data than ong-run estimates (LR estimates, defined beow) and therefore, in practice, they are ess stabe than LR estimates and harder to estimate. Long-run estimates (LR): These are unconditiona macroeconomic estimates, i.e. the macroeconomic risk drivers are ignored. The main advantage is that they are more robust and stabe than PIT estimates. These LR estimates are required in AIRB approaches, 26 except for those portfoios in which there is evidence of negative dependence between defaut rates and LEQ factors. Currenty, these LR estimates are aso used by banks for interna purposes. Downturn estimates (DT): these are specific PIT estimates based on macroeconomic scenarios (downturn conditions) in which the defaut rates for the portfoio are deemed to be especiay high. When there is evidence of the existence of adverse dependencies between defaut rates and conversion factors, this coud be the type of estimates that, in theory, shoud be used in IRB approaches. 27 In practice, the use of DT estimates is difficut because, in addition to the difficuties associated with PIT estimates, it is necessary to identify downturn conditions and to have sufficient observations in the RDS restricted to these scenarios. In the foowing, it is assumed that the focus is on ong run estimates. 11.6.2 A Suitabe Cass of Loss Functions The objective of this section is to determine a type of oss function that meets the basic requirements for the EAD estimation probem when it is necessary to obtain EAD estimates adequate for IRB approaches. Therefore, it makes sense to specify the oss associated with the difference between the estimated vaue and the rea one in terms of the error in the minimum reguatory capita (computed as the difference between the capita requirements under both vaues). By using the reguatory formua, at the eve of the faciity, the oss associated with the difference between 26 } 475. Advanced approach banks must assign an estimate of EAD for each faciity. It must be an estimate of the ong-run defaut-weighted average EAD for simiar faciities and borrowers over a sufficienty ong period of time, [...] If a positive correation can reasonaby be expected between the defaut frequency and the magnitude of EAD, the EAD estimate must incorporate a arger margin of conservatism. Moreover, for exposures for which EAD estimates are voatie over the economic cyce, the bank must use EAD estimates that are appropriate for an economic downturn, if these are more conservative than the ong-run average., BCBS (2004). 27 This can be interpreted in the ight of the carification of the requirements on LGD estimates in Paragraph 468 of the Revised Framework, BCBS (2005a, b).

11 EAD Estimates for Faciities with Expicit Limits 225 the capita requirement under the estimated vaue of the exposure K(EAD( f )) and the rea one K(EAD), coud be expressed as foows 28 : LðDKðf Þ Þ ¼ LðKðEADÞ KðEADðf ÞÞÞ ¼ LðfðPDÞLGD ðead EADðf ÞÞÞ ¼ LðfðPDÞLGD DðEADðf ÞÞÞ (11.38) Additionay, at east from a reguatory point of view, underestimating the capita requirement creates more probems than overestimating such a figure. For this reason, it is appropriate to use asymmetric oss functions that penaises more an underestimation of the capita requirement than an overestimation of the same amount. The simpest famiy of such functions is given by (11.39), where b > a: a DK iff DK 0 LðDKÞ ¼ b DK iff DK<0 (11.39) These oss functions quantify the eve of conservatism. The arger b/a (reative oss associated with an underestimation of K), the arger is the eve of conservatism imposed. For exampe, if a ¼ 1andb ¼ 2, the oss associated with an underestimation of the capita requirement (DK < 0) is twice the oss for an overestimation of the same amount. 29 The graphic of the oss function is presented in Fig. 11.6. L=2 ΔK L=ΔK Fig. 11.6 Linear asymmetric oss function ΔK 0 ΔK 28 In the foowing it is assumed that a PD ¼ PD( f ) and an LGD ¼ LGD( f ) have been estimated previousy. 29 To the best of my knowedge, the first appication of such a oss function in the credit risk context was proposed in Mora (1996). In that paper the oss function is used to determine the optima eve of provisioning as a quantie of the portfoio oss distribution.

226 G. Mora By using this specific type of oss function (11.39), and assuming that LGD 0, a simper expression for the error in K in terms of the error in EAD is obtained: LðDKðf ÞÞ ¼ fðpdþlgd LðDðEADðf ÞÞÞ (11.40) The oss associated with an error in the capita requirement is proportiona to the oss associated with the error in terms of exposure and the units of the oss are the same as those of the exposure ( ). 11.6.3 The Objective Function Once the oss function has been determined, it is necessary to find the most natura objective function for the estimation probem. 11.6.3.1 Minimization at Faciity Leve of the Expectation in the Capita Requirement Error If the expected error in the minimum capita requirement at the eve of exposure is used as an objective function, by using (11.40) the foowing is obtained: Min LEQ feldkðf ½ ð ÞÞŠg ¼ fðpd ÞLGD Min f ELD ½ ð ð EADðf Þ ÞÞŠg (11.41) LEQ This means that Probem P.III in Sect. 11.5.3 arises with a different oss function: E h L ð EAD E LEQðRDÞ ð L E ÞÞi (11.42) Min LEQ F or in terms of the sampe Min LEQ ( ) X LðEAD i E i LEQðRD i ÞðL i E i ÞÞ i (11.43) and a soution is given 30 by: * + b LEQðf Þ¼Q EAD E; 1 F a þ b Lðf Þ Eðf Þ ; (11.44) RDðf Þ 30 See Appendix B.

11 EAD Estimates for Faciities with Expicit Limits 227 where Q(x, b/(a þ b)) stands for a quantie of the distribution F(x) such that 31 F(Q) ¼ b/(a + b). When a ¼ b, the oss function (11.39) is symmetric and the former quantie is the median and for vaues of b/a > 1 the associated quantie is paced to the right of the median and, therefore, a more conservative estimate of LEQ( f ) is obtained. It is interesting to note that (11.44), with b > a, penaises uncertainty. 32 An important consequence of using the former oss function L is that the probems M.I and M.II described in (11.45) and (11.46) are equivaent. 33 Probem M.I: Min LEQ ( ) X LðEAD i E i LEQðRD i ÞðL i E i ÞÞ i Subject to: 0 LEQðRDÞ 1 (11.45) Probem M.II: Min LEQ ( ) X LðMin½Max½EAD i ; E i Š; L i Š E i LEQðRD i ÞðL i E i ÞÞ i Subject to: 0 LEQðRDÞ 1 (11.46) This means that an estimator meeting the constraint 0 LEQ( f ) 1 that is optima when using the origina data is aso optima when using data censored to show reaised LEQ factors between zero and one. 11.6.3.2 Minimization of the Error in the Capita Requirement at Faciity Leve for Reguatory Casses Sometimes, in spite of the existence of interna estimates for LEQ factors at faciity eve, it coud be necessary to associate a common LEQ with a the faciities incuded in a cass comprising faciities with different vaues for the interna risk drivers. This coud occur due to difficuties in demonstrating with 31 In practice, it is necessary to be more precise when defining a q-quantie because the distribution F(x) is discrete. A common definition is: a q-quantie of F(x) is a rea number, Q(x,q), that satisfies P[X Q(x,q)] q and P[X Q(x,q)] 1 q. In genera, with this definition there is more than a q-quantie. 32 } 475. Advanced approach banks must assign an estimate of EAD for each faciity. It must be an estimate [...] with a margin of conservatism appropriate to the ikey range of errors in the estimate., BCBS (2004). 33 The proof foows from the proposition in Appendix A.

228 G. Mora the avaiabe data, that discrimination at an interna eve of granuarity is justified. In this case, for reguatory use, it isnecessarytoadoptaessgranuar structure for the risk drivers than the existing interna one. Therefore, the probem of finding an optima estimator for reguatory use can be soved by using the reguatory structure for the risk drivers. In other words, the procedure is to compute new estimates using the same method and a ess granuar risk driver structure. In genera, the new estimator is not a simpe or weighted average of the former more granuar estimates. 11.7 Exampe 1 This exampe 34 iustrates the pros and cons of using the methods expained in the former sections for estimating LEQ factors and EADs. The focus is on ong run estimates for the EAD of a faciity f in norma status by using as basic risk drivers the current imit L( f ) and exposure E( f ). 11.7.1 RDS 11.7.1.1 Characteristics The main characteristics of the reference data set, used in this exampe, are described beow: Source of the RDS: the observations were obtained from a set of defauted faciities from a portfoio of SMEs Observation period: 5 years Product types: credit ines with a committed imit of credit, that is known for the borrower, given by L(t) Excusions: It does not incude a the interna defauts which took pace during the observation period because severa fiters had been appied previousy. As a minimum, the foowing faciities were excuded from the data set: defauted faciities with L(td 12) < E(td 12) and those with ess than 12 monthy observations before the defaut date Number of observations, O i : #RDS ¼ 417 12 ¼ 5,004 observations, which are associated with 417 defauted faciities and dates 1, 2,...,12 months before the defaut date 34 Athough this exampe coud be representative for certain SME portfoios comprising credit ines, it is not a portfoio taken from a bank.

11 EAD Estimates for Faciities with Expicit Limits 229 Structure of the reference data set: the structure proposed in (11.8) but, for simpicity, ony a basic set of risk drivers is considered: O i ¼ fi; ðf ; trþ; RD i ¼ flðtrþ; EðtrÞ; SðtrÞg; EAD; td; trg (11.47) Status of a faciity at the reference date, S(tr): there is no information about the status of the faciities. The bank has impemented a warning system that cassifies the exposures on four broad casses: N ¼ norma monitoring and contros; V ¼ under cose monitoring for drawdowns; I ¼ current exposure greater than the imit and impies tight contros making additiona drawdowns impossibe without a previous approva; D ¼ defauted, no additiona drawdowns are possibe, but sometimes there are increases in the exposures due to the payment of interest and fees. However, in this exampe, in order to take into account the status, S(tr), as a risk driver, observations with S(tr) ¼ N are approximated using the foowing procedure: First, a the observations with L(tr) < E(tr) are marked as in a non-norma status Second, after anaysing the empirica distributions of reaised LEQ factors (and other information) it was decided to consider a the observations with td tr ess than 5 months as if they were in a non-norma status and to eiminate a the observations with td tr ¼ 7 months (see next section). In practice, the use of the vaues of the variabe status is necessary, because the eary identification of probematic borrowers and the subsequent changes in the avaiabiity of access to the nomina imit have important consequences in the observed EADs. For this reason, observations up to 5 months before defaut for which E(tr) L(tr) are considered in norma status. In this case, the number of observations with S(tr) ¼ N is: #RDS(N) ¼ 2,919. 11.7.1.2 Empirica Distributions of Certain Statistics Distribution of Reaised LEQ Factors Figure 11.7 summarises the empirica variabiity of the reaised LEQ factors associated with 2,442 observations for which it is possibe to compute this statistic. 35 It shows that the distribution is asymmetric with a high number of observations outside of [0,1] which is the natura range for LEQ factors. The sampe mean is about 525 due to the existence of many observations with arge negative vaues 35 Observations associated with, the horizon vaue, td tr ¼ 7 were removed from the RDS as it is expained ater on.

230 G. Mora #O i 120 100 80 Q 50 % 60 40 Q 25 % Q 75 % 20 0-4 -2 0 2 4 LEQ i Fig. 11.7 Histogram of reaised LEQ factors and it highights one of the main issues when a sampe mean is used as the estimator. The median is 0.97 and this vaue, in contrast with the former sampe mean vaue, highights the advantages of using statistics ess dependent on the extreme vaues of the distribution for estimation purposes. Joint Distribution of Reaised LEQ Factors and Percent Usage at the Reference Date To reduce the variabiity in the observed reaised LEQ factors, it is necessary to consider a variabe that exhibits expanatory power, at east, for the range of vaues of reaised LEQ factors. For exampe, the joint empirica distribution presented in Fig. 11.8 shows that the variabe percent usage at the reference date is important for imiting the variabiity of reaised LEQ factors. Back points at the top of Fig. 11.8 represent the observations in the space {1 e(tr), LEQ i }. Infuence of td tr in the Basic Statistics Figure 11.9 presents the empirica distributions of reaised LEQs associated with a fixed distance in months between the defaut and reference dates for td tr ¼ 1,...,12.

11 EAD Estimates for Faciities with Expicit Limits 231 10 5-5 LEQ i 0-10 40 30 #O i 20 10 0 0.2 0.4 1-e(tr) 0.6 0.8 1 Fig. 11.8 Joint distribution of LEQ i and percent usage at the reference date tr 1 0.8 7 F*(LEQ i ) 0.6 0.4 0.2 0 4 3 2 10 8 6 4 2 0 2 4 LEQ i Fig. 11.9 Empirica distributions of LEQ i conditiona on different td tr vaues

232 G. Mora 1 F*(ead i -e(tr)) 0.8 0.6 0.4 7 2 0.2 0 4 3 1 0.5 0 0.5 1 ead i -e(tr) Fig. 11.10 Empirica distributions of percent increase in usage since the reference date 1 0.8 7 F*(EAD i -E(tr)) 0.6 0.4 0.2 0 3 2 1-40000 -20000 0 20000 40000 EAD i -E(tr) Fig. 11.11 Empirica distributions of the increase in exposure from tr to td The distributions associated with td tr ¼ 1, 2, 3, 4 are very different from the others. The distribution conditiona on td tr ¼ 7 months is totay anomaous and the reason for that is an error in the processes that generated these data. Figure 11.10 presents the empirica distributions of the percent increase in usage between the reference and the defaut dates, ead i e(tr), associated with a fixed distance in months between the defaut and reference dates for td tr ¼ 1,...,12. Again, the differences among the distributions conditiona on reference dates near to defaut and far from defaut are obvious and the existence of anomaous vaues for the case td tr ¼ 7 is evident. Finay, Fig. 11.11 shows the empirica distributions of the increase in exposure, EAD i E(tr), between the reference and the defaut dates.

11 EAD Estimates for Faciities with Expicit Limits 233 11.7.2 Estimation Procedures 11.7.2.1 Mode II Origina Data and Fixed Time Horizon Some banks use Mode II assuming a constant LEQ, and a fixed time horizon approach, T ¼ 12 months. This means that they adjust a inear regression mode without an independent term, given by: EAD i Ltd ð 12Þ Etd 12 ð Þ Etd ¼ k 1 ð 12 Þ Ltd ð 12Þ Ltd ð 12Þ (11.48) Therefore, in these cases, the bank s approach focuses on the minimisation of the quadratic error in the increase of the exposure expressed in percentage terms of the imit. The resuts with this method are summarised beow: By using the origina data, the estimated LEQ factor is LEQ ¼ 0.637 and the adjusted R 2 is 0.13. Therefore, the fina estimate for the EAD of a faciity, f, in norma status is given by the formua: EADðf Þ¼EðtÞþ0:637 ðlðtþ EðtÞÞ (11.49) Figure 11.12 presents, for each observation in the RDS(td 12), the vaues of the pairs {1 e(td 12), ead i e(td 12)}. The upper shadow zone in Figs. 11.12 11.14 are associated with points with LEQ i > 1. From anaysis of the distribution of these points and the resuts of the regression it is cear that, at east: 4 3 ead i -e(tr) 2 1 0-1 0 0.2 0.4 0.6 0.8 1 1-e(tr) Fig. 11.12 Percent increase in usage from tr, to td and percent usage at the reference date

234 G. Mora 1 0.8 ead i -e(tr) 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1-e (tr) Fig. 11.13 Linear regression in Mode II and censored data 4 3 ead i -e(tr) 2 1 0-1 0 0.2 0.4 0.6 0.8 1 1-e(tr) Fig. 11.14 Linear regression in Mode II and variabe time approach 1. It is necessary to carry out an expicit RDS ceaning process before the estimation phase. For exampe, it is necessary to anayse the observations associated with the points above the ine y ¼ x and afterwards to make decisions about which observations have to be removed from the RDS. 2. The degree of adjustment is very ow. Most of the points (those with 1 e(tr) coser to zero) have itte infuence on the resut of the regression mode because of the constraint that there is no independent term. 3. In order to assess the reiabiity of the estimated LEQ it is necessary to identify outiers and infuentia observations and to perform stabiity tests. In this case, given the functiona form of the mode, y ¼ k x, and the ow number of points associated with arge vaues of 1 e(tr), these observations are infuentia

11 EAD Estimates for Faciities with Expicit Limits 235 points. 36 It is easy to understand that changes in these points affect the resut of the regression and therefore the LEQ estimate. 4. In order to get more stabe resuts, it is necessary to get more observations (for exampe by using a variabe time horizon approach). Censored Data and Fixed Time Horizon Sometimes banks use censored data to force reaised LEQ factors to satisfy the constraint 0 LEQ i 1. Using censored data, the estimated LEQ factor is 0.7 and the R 2 increase to 0.75. In this case, a the points are in the white trianguar region of Fig. 11.13 and it is cear that the existence of very infuentia points (those with arge vaues of 1 e(r)) introduces instabiity. Figure 11.13 presents the censored observations and the regression ine. The EAD estimator is in this case: EADðf Þ¼EðtÞþ0:7 ðlðtþ EðtÞÞ (11.50) Origina Data and Variabe Time Approach By using a variabe time approach, based on observations with tr ¼ td {12, 11, 10, 9, 8}, the estimated LEQ factor is LEQ ¼ 0.49 and the R 2 is 0.06. Figure 11.14 presents, for each observation in the RDS, the pairs {1 e(tr), ead i e(tr)} and the regression ine associated with this extended data set and Mode II. In Mode II, the use of a time variabe approach does not increase the degree of adjustment (which is very ow due to the functiona form assumed in the mode), but increases the stabiity of the resuts. The EAD estimator in this case is: EADðf Þ¼EðtÞþ0:49 ðlðtþ EðtÞÞ (11.51) 11.7.2.2 The Sampe Mean and the Conditiona Sampe Mean If Mode I is used and a constant LEQ for faciities simiar to f is assumed, an estimate for EAD( f ) is obtained by computing the sampe mean of the reaised LEQ conditiona on observations in RDS( f ) as the LEQ( f ) estimate and then appying (11.13). With regard to RDS( f ), in this exampe, two possibiities are anaysed: 36 Infuentia points have a significant impact on the sope of the regression ine which, in Mode II, is precisey the LEQ estimate.

236 G. Mora RDS( f ) ¼ RDS or equivaenty to use a goba sampe mean as estimator. RDS( f ) ¼ {O i such as percent usage e i is simiar to e( f )} or equivaenty to use as estimator a function based on different oca means depending on e( f ). Case RDS( f) ¼ RDS, Use of a Goba Sampe Mean If the sampe mean of a the reaised LEQ factors associated with the observations in the RDS is computed, the resut is a nonsensica figure: LEQðf Þ¼LEQ ¼ 1 X LEQ i ¼ 578 (11.52) n The probems that arise when using this goba average are due to: 1. Instabiity of certain reaised LEQ factors: when 1 E( f )/L( f ) is sma the reaised LEQs are not informative. 2. Very high vaues for certain observations, in some cases severa times L(tr) E(tr). The treatment of these observations needs a case by case anaysis. 3. Asymmetries in the behaviour of positive and negative reaised LEQ factors. 4. Evidence of a non-constant LEQ i sampe mean depending on the vaues of 1 E( f )/L( f ). Figure 11.15 represents the distribution of the reaised LEQ factors and undrawn amounts as a percentage of the imit, 1 E( f )/L( f ) and it can hep to increase understanding of the main probems associated with this method: Figure 11.16 focuses on observations associated with vaues of reaised LEQ factors ess than 2. It is cear that there are observation reaised LEQ factors greater than one, (upper shadow zones in Figs. 11.16 and 11.17) across the range of percent i 40 20 LEQ i 0-20 -40-60 0 0.1 0.2 0.3 1-e(tr) 0.4 Fig. 11.15 Reaised LEQ factors and percent usage at the reference date

11 EAD Estimates for Faciities with Expicit Limits 237 2 1 0 LEQ i -1-2 -3-4 0 0.2 0.4 0.6 0.8 1 1-e(tr) Fig. 11.16 Reaised LEQ factors smaer than two 1.5 1 LEQ i 0.5 0-0.5 0.5 0 0.2 0.4 0.6 0.8 1-e(tr) 1 Fig. 11.17 Approximation for E[LEQ 1 e(tr)] and the adjusted regression curve usage vaues, athough such observations are much more common when the percent usage vaues are arge (sma vaues of 1 e(tr)). For these reasons, before using this procedure, it is necessary to make some decisions after anaysing the observations in the RDS, for exampe: To eiminate from the RDS those anomaous observations with arge LEQ i factors To censor other observations associated with LEQ i factors greater than one To remove observations with very ow vaues of E( f ) L( f ) from the RDS, because their LEQ i vaues are not informative. In this exampe, observations with 1 E(tr)/L(tr) 0.1 and those with LEQ i 2 were removed from the reference data set. After these modifications of the RDS, the new LEQ i sampe mean is:

238 G. Mora LEQðf Þ¼LEQ ¼ 1 X LEQ i ¼ 0:08 (11.53) m It is cear that this goba estimate of 8% is very ow for most of the faciities in the portfoio because of the weight in the goba average of the negative reaised LEQ factors associated with observations with ow vaues of 1 e( f ). An improvement to the former estimate is to eiminate outiers, i.e. observations associated with very arge (in absoute terms) reaised LEQ factors. If observations with LEQ factors beow the tenth percentie and above the ninetieth are considered outiers, the average restricted to the RDS without outiers is about 33% and this vaue is stabe when the former percenties are changed. i LEQðf Þ¼LEQ ¼ 1 r X LEQ i ¼ 0:33 (11.54) i However, it is cear that oca averages are very different and therefore this goba estimate of 33% for the LEQ is not adequate. For this reason, it is necessary to consider different estimates for the LEQ factor for different vaues of 1 E( f )/L( f ). Case RDS( f) ¼ {O i Such as Percent Usage e i is Simiar to e(f)} In this case, the RDS(f) comprises a the observations O i with 1 e(tr) 2 [1 e( f ) 0.2, 1 e(f) + 0.2] and the average of the reaised LEQ factors restricted to observations in the RDS( f ) is used as the estimate of LEQ( f ). To seect a functiona form for LEQ( f ), first the estimated vaues for different 1 e(tr) vaues are computed and second, a regression mode is adjusted using 1 e(tr) as the expanatory variabe, and the oca sampe mean as the dependent variabe. After rejecting different modes and using intervas of width p0.4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an expression for the oca 37 sampe mean of LEQ factors based on a þ b ð1 etr ð ÞÞ is obtained as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LEQðf Þ¼ 0:82 þ 1:49 1 Eðf Þ=Lðf Þ (11.55) with an adjusted R 2 equa to 0.94. Figure 11.17 represents the reaised LEQ factors, the oca averages and the adjusted function (with the constraint LEQ ( f ) 0). Therefore an estimator for EAD( f ) of a faciity f in norma status is given by: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EADðf Þ¼Eðf ÞþMax 0; 0:82 þ 1:49 1 Eðf Þ=Lðf Þ ðlðf Þ Eðf Þ i Þ (11.56) 37 The oca condition is to consider ony those observations in an interva centred on 1 E( f )/L ( f ) and with ength 0.4.

11 EAD Estimates for Faciities with Expicit Limits 239 11.7.2.3 The Median and the Conditiona Quanties The rationae under Mode III is to expain directy the increase in exposure from the reference date to the defaut date. Therefore, it is necessary to expain EAD i E(tr) in terms of LEQ(RD i ) (L(tr) E(tr)). For simpicity, it is assumed that RD i ¼ {S(tr),L(tr) E(tr)} and the focus is on observations with status S(tr) ¼ norma and the ony quantitative variabe that is taken into account is the current undrawn amount L( f ) E( f ). Moreover, the oss function proposed in (11.39) is used to determine the optima estimates and therefore as shown in Sect. 11.6.3.1, the soution is to approximate the quantie Q[b/(a þ b)] of the distribution of EAD i E(tr) conditiona on those observations which satisfy L(tr) E(tr) ¼ L( f ) E( f ). To approximate that quantie for each vaue of EAD( f ) E( f ), the process is simiar to the one expained in the previous section. First, RDS( f ) is defined as a the observations such as (L(tr) E(tr)) 2 [(L( f ) E( f )) 0.8, (L( f ) E( f )) 1.2]. Second, for each vaue of L(tr) E(tr) the optima quantie is computed. Third, a inear regression mode that uses L(tr) E(tr) as the expanatory variabe and the optima quantie as the dependent variabe is adjusted and, finay, the estimator for LEQ( f ) is obtained by using formua (11.44). Figure 11.18 represents, for each observation in the RDS with tr ¼ td {12, 11, 10, 9, 8}, the pairs {L(tr) E(tr), EAD i E(tr)} in the range of vaues of L(tr) E(tr) given by [0, 17000], for which it is considered there exists sufficient number of observations. The shadow zones in Figs. 11.18 and 11.19 are defined as EAD i L(tr). The resuts of the regression mode for the oca medians (case a ¼ b) and for the 66.6th percentie (case 2 a ¼ b) produces the foowing resuts: 20000 10000 EAD i -E(tr) 0 10000 20000 30000 0 2500 5000 7500 10000 12500 15000 L(tr)-E(tr) Fig. 11.18 Observations in Mode III

240 G. Mora 17500 15000 EAD-E(tr) 12500 10000 7500 5000 Q 66 % Q 50 % 2500 0 0 2500 5000 7500 10000 12500 15000 L(tr)-E(tr) Fig. 11.19 Quanties conditiona on the undrawn amount and adjusted EAD E(tr) vaues Median½EADð f Þ Eð f ÞŠ ¼ 86:8 þ 0:76 ðlð f Þ Eð f ÞÞ Quantie½EADð f Þ Eð f Þ; 0:666Š ¼ 337:8 þ 0:92 ðlð f Þ Eð f ÞÞ (11.57) With adjusted R 2 equa to 0.95 and 0.99 respectivey. Therefore, the associated LEQ estimates, obtained dividing (11.57)byL( f ) E( f ), are amost constant (cose to 0.76 and 0.92 respectivey) and have vaues arger than the previous estimates. Figure 11.19 represents the oca medians (Q 50% ine) and oca 66.6 percenties (Q 66% ine) obtained from the origina points, the regression ines associated with (11.57) (dotted ine for the adjusted 66.6 percenties, thick ine for the adjusted oca medians). 11.8 Summary and Concusions The foowing points summarise the current practice on CF and EAD estimates and highight some probematic aspects: The CF and EAD estimators appied by banks can be derived from specia cases of regression probems, and therefore these estimators are based on conditiona expectations Impicity, the use of these estimators assumes the minimisation of prediction errors by using a quadratic and symmetric oss function that is neither directy correated with the errors in terms of minimum capita requirements nor

11 EAD Estimates for Faciities with Expicit Limits 241 penaises uncertainty. The way in which these errors are measured is crucia because they are very arge In most of the cases, the EAD estimates are based on the unreaistic assumption of a constant LEQ factor mean Frequenty, the basic statistics for the estimation process are censored to obtain reaised LEQ factors between zero and one Banks frequenty use Cohort Approaches or Fixed Time Horizon Approaches to seect the observations incuded in the estimation process. These approaches do not take into account a the reevant information because they ony focus on a conventiona reference date for each defauted faciity With regard to risk drivers, the focus is on the rating at the reference date. Other approaches and some comments on different aspects: For reguatory use, it seems appropriate for the estimators to be soutions to optimisation probems that use a oss function directy reated with errors in terms of capita requirements For exampe, a ogica choice is to use a simpe inear asymmetric oss function appied at the eve of faciity. This oss function enabes banks or supervisors to quantify the eve of conservatism impicit in the estimates Using this oss function, the derived estimators are based on conditiona quanties (for exampe, the median for interna purposes and a more conservative quantie for reguatory use) If the estimates are based on sampe means LEQ factors, as a minimum, shoud depend on the eve of the existing avaiabiity of additiona drawdowns: LEQ(1 e(tr)) The common practice of censoring the reaised LEQ factors to [0, 1], is not justified and, in genera, it is not possibe to concude ex ante if the associated LEQ estimates are biased in a conservative manner However, under certain hypotheses, the use of censored data does not change the optima estimator for LEQ The estimates shoud be based on observations at a the reevant reference dates for defauted faciities, Variabe Time Approach With regard to risk drivers, if there is a warning system for the portfoio, it is important to focus on the status of the faciity at the reference date rather than on the rating The exampe presented here suggests that: Estimates based on sampe means are ess conservative than those based on conditiona quanties above the median The CF estimates obtained by using these conditiona quanties, are so arge that the use of downturn estimates in this case might not be a priority.

242 G. Mora Appendix A. Equivaence Between Two Minimisation Probems Proposition: Consider a set of observations O ¼ fðx i ; y i Þg i¼1;...;n and the probem G.I given by: hx n i Minimise g2g Ly ð i¼1 i gðx i ÞÞ (11.58) Subject to f ðxþ gðxþ hðxþ where the error is measured in terms of the function L that satisfies: Lðx þ yþ ¼LðxÞþLðyÞ if x y 0 (11.59) then, g is a soution of Probem G.I if and ony if it is a soution of Probem G.II given by: hx n i Minimise g2g L ð Min ½ Max ½ y i¼1 i; hðx i ÞŠ; f ðx i ÞŠ gðx i ÞÞ (11.60) Subject to f ðxþ gðxþ hðxþ Proof: The set O can be partitioned into three casses O ¼ O þ O O ¼, where: For observations in O + : O þ ¼fðx i ; y i Þjy i >f ðx i Þg; O ¼fðx i ; y i Þjy i <hðx i Þg (11.61) ðy i fðx i ÞÞðfðx i Þ gðx i ÞÞ 0 (11.62) Therefore, from (11.59) and (11.62), the error in Probem G.I associated with an observation in O + can be expressed in terms of the error in Probem G.II pus an amount independent of g: err½gi; ðx i ; y i ÞŠ ¼ Ly ð i gx ð i ÞÞ ¼ Ly ð i fðx i Þþfðx i Þ gx ð i ÞÞ ¼ Ly ð i fðx i ÞÞþLðfðx i Þ gx ð i ÞÞ ¼ Ly ð i fðx i ÞÞþLðMin½Max½y i ; hx ð i ÞŠ; fðx i ÞŠ gx ð i ÞÞ ¼ Ly ð i fðx i ÞÞþerr½GII; ðx i ; y i ÞŠ (11.63) But the O + set does not depend on the functiong, therefore for these observations, and for a g, the error in Probem G.I can be decomposed in a fixed amount, independent of the g function, given by P Ly ð i fðx i ÞÞ, where the index i appies at the observations in O + and the error in Probem G.II. Simiary, for observations in O, the error in Probem G.I is equa to the error in Probem G.II pus the fixed amount P Lhx ð ð i Þ y i Þ. Finay, for the observations in O ¼ the errors in Probem G.I and in Probem G.II are the same.

11 EAD Estimates for Faciities with Expicit Limits 243 Appendix B. Optima Soutions of Certain Regression and Optimization Probems Let X and Y be random variabes with joint distribution given by F(x,y), then we get in the case of a quadratic oss function d ðxþ ¼EYX h j i ¼ Min dðxþ E F D E ðy dðxþþ 2 : (11.64) In the case of the inear asymmetric oss function, with a > 0 and b > 0: LðxÞ ¼ a x iff x 0 (11.65) b x iff x<0 The foowing is found d b ðxþ ¼Q Y X; a þ b ¼ Min dðxþ E h LY ð dðxþ Þi F (11.66) See, for exampe, Pratt et a. (1995, pp. 261 263). Therefore, a soution for (11.28) can be obtained from (11.64), and taking into account: Y ¼ EAD E o ; dx¼ ð RDÞ ¼ LEQðRDÞhðRDÞ; where hðrdþ ¼ L E o (11.67) Then, d* is given by (11.64) and assuming that h(rd) ¼ h( f ) for observations in RDS( f ): d ðx ¼ RDð f Þ Þ ¼ E EAD E jrd o Lð f Þ Eð f Þ ¼ LEQðRDð f ÞÞ oð f Þ The resut showed in (11.29) is obtained from the former equation. (11.68)

244 G. Mora Appendix C. Diagnostics of Regressions Modes Mode II (Sect. 11.7.2.1) By using origina data: EAD i Ltd ð 12Þ Etd 12 ð Þ Etd ¼ 0:64 1 ð 12 Þ Ltd ð 12Þ Ltd ð 12Þ (11.69) By using censored data: EAD i Ltd ð 12Þ Etd 12 ð Þ Etd ¼ 0:7 1 ð 12 Þ Ltd ð 12Þ Ltd ð 12Þ (11.70) By using a variabe time approach: EAD i Ltr ð Þ Etr ð Þ Ltr ð Þ ¼ 0:49 1 Etr ð Þ Ltr ð Þ (11.71) Mode I (Sect. 11.7.2.2) By using Mode I, variabe time approach: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LEQð f Þ¼ 0:82 þ 1:49 1 Eð f Þ=Lð f Þ (11.72) The diagnostics for this regression mode are:

11 EAD Estimates for Faciities with Expicit Limits 245 Mode III (Sect. 11.7.2.3) By using a variabe time approach: Median½EADð f Þ Eð f ÞŠ ¼ 86:8 þ 0:76 ðlð f Þ Eð f ÞÞ Quantie½EADð f Þ Eðf Þ; 0:666Š ¼ 337:8 þ 0:92 ðlð f Þ Eð f ÞÞ (11.73) With the diagnostics given by: and for the quantie: Appendix D. Abbreviations AIRB CCF CF EAD EAD i ¼ E(td) EAD( f ) ead i E(t) e(t) e i ¼ e(tr) f g i ¼ {g, tr} IRB LEQ LEQ( f ) LEQ i LGD L(t) O i PD Q a ¼ Q(x, a) RDS RDS( f ) RD S(tr) Advanced interna ratings-based approach Credit conversion factor Conversion factor Exposure at defaut Reaised exposure at defaut associated with O i EAD estimate for f Reaised percent exposure at defaut, associated with O i Usage or exposure of a faciity at the date t Percent usage of a faciity at the date t Percent usage associated with the observation O i¼{g, tr} Non-defauted faciity Defauted faciity Index associated with the observation of g at tr Interna ratings-based approach Loan equivaent exposure LEQ estimate for f Reaised LEQ factor associated with the observation O i Loss given defaut Limit of the credit faciity at the date t Observation associated with the pair i ¼ {g, tr} Probabiity of defaut Quantie associated with the a% of the distribution F(x) Reference data set RDS associated with f Risk drivers Status of a faciity at the reference date tr

246 G. Mora t td tr td tr Current date Defaut date Reference date Horizon References Araten M, Jacobs M (2001), Loan Equivaents for Revoving Credits and Advised Lines, The RMA Journa, 83 pp. 34 39. Base Committee on Banking Supervision (2005), Guidance on Paragraph 468 of the Framework Document. Base Committee on Banking Supervision (2004), Internationa Convergence of Capita Measurement and Capita Standards, a Revised Framework. Base Committee on Banking Supervision (2005), Studies on the Vaidation of Interna Rating Systems, Working Paper No. 14 Revised version. CEBS (2006), Guideines on the impementation, vaidation and assessment of Advanced Measurement (AMA) and Interna Ratings Based (IRB) Approaches, CP 10 revised. Lev B, Rayan S (2004), Accounting for commercia oan commitments. Mora G (1996), Pérdida atente, incertidumbre y provisión óptima, Banco de España, Boetín de a Inspección de ECA. Pratt J, Raiffa H, Schaifer R (1995), Introduction to Statistica Decision Theory. The MIT Press. Sufi A (2005), Bank Lines of Credit in Corporate Finance: An Empirica Anaysis, University of Chicago Graduate Schoo of Business.

Chapter 12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective Stefan Bochwitz and Stefan Hoh 12.1 Base II and Vaidating IRB Systems 12.1.1 Base s New Framework (Base II and Further Work) Base II and further work is associated with the work undertaken by the Base Committee on Banking Supervision (BCBS). 1 This aimed to secure internationa convergence on revisions to supervisory reguations on capita adequacy standards of internationay active banks. The main objective of the 1988 Accord 2 and its revision is to deveop a risk-based capita framework that strengthens and stabiises the banking system. At the same time, it shoud provide for sufficient consistency on capita adequacy reguation across countries in order to minimize competitive inequaity among internationa banks. In June 2004, the BCBS issued Base II, tited Internationa Convergence of Capita Measurement and Capita Standards: 1 The Base Committee on Banking Supervision is a committee of banking supervisory authorities that was estabished by the centra bank governors of the Group of Ten countries in 1975. Up to 2009 it consisted of senior representatives of bank supervisory authorities and centra banks from Begium, Canada, France, Germany, Itay, Japan, Luxembourg, the Netherands, Spain, Sweden, Switzerand, the United Kingdom, and the United States. The membership of the Base Committee on Banking Supervision was broadened in June 2009. The new members are representatives from the G20 countries that were not in the Base Committee before. These are Argentina, Indonesia, Saudi Arabia, South Africa and Turkey. In addition, Hong Kong SAR and Singapore had aso been invited to become BCBS members. 2 see Base Committee on Banking Supervision (1988). The views expressed are those of the authors and do not necessariy refect those of the Bank for Internationa Settements (BIS), the Base Committee of Banking Supervision, or the Deutsche Bundesbank. S. Bochwitz (*) Deutsche Bundesbank e-mai: Stefan.Bochwitz@bundesbank.de S. Hoh Bank for Internationa Settements e-mai: stefan.hoh@bis.org B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_12, # Springer-Verag Berin Heideberg 2011 247

248 S. Bochwitz and S. Hoh A Revised Framework, carefuy crafting the baance between convergence and differences in capita requirements. In December 2009 the enarged BCBS issued a consutative document tited Strengthening the Resiience of the Banking Sector as part of its reform package to address essons from the financia crisis starting in 2007. The proposas aim at strengthening goba capita and iquidity reguations to promote a more resiient banking sector in the future. Accordingy, enhancing risk coverage as we as reducing procycica ampification of financia shocks throughout the financia system are among its key objectives. Both may have an impact on issues reated to vaidation of bank s interna risk management systems. For exampe, one of the proposed future requirements is the use of stressed inputs for determining bank s capita requirement for counterparty credit risk. The cycicaity in minimum capita requirements over time has aways been a key consideration for the BCBS during the design of the Base II framework. As such, the BCBS had introduced a number of safeguards to address this issue incuding the requirement to use ong term data horizons to estimate probabiities of defaut (PD) and the introduction of downturn oss-given-defaut (LGD) estimates. This paper presents pragmatic views on vaidating IRB systems. It discusses issues reated to the chaenges facing supervisors and banks of vaidating the systems that generate inputs into the interna ratings-based approach (IRBA) used to cacuate the minimum reguatory capita for credit risk, based on interna bank information. The key roe of Banks as financia intermediaries highights their core competences as ending, investing and risk management. In particuar, anaysing and quantifying risks is a vita part of efficient bank management. An appropriate corporate structure is vita to successfu risk management. Active credit risk management is indispensabe for efficienty steering a bank through the economic and financia cyces, despite the difficuties stemming from a ack of credit risk data. A we-functioning credit risk measurement system is the key eement in every bank s interna risk management process. It is interesting to note that the debate about the new version of the Base Capita Accord (Base II and further work), which estabishes the internationa minimum requirements for capita to be hed by banks, has moved this topic back to the centre of the discussion about sound banking. The proper impementation of the IRBA is one key aspect of a ivey debate among bankers, academics and reguators. At the same time a paradigm shift in credit risk management has taken pace. Previousy, credit risk assessment used ony the experience, intuition and powers of discernment of a few seect speciaists. The new process is more formaised, standardised and much more objective by bank s interna rating systems. The human eement has not been entirey discounted, however; now both human judgement and rating systems each pay an equay important roe in deciding the credit risk of a oan. Since the IRBA approach has been impemented in most of the G10-countries in the past and wi be impemented in amost a G20-countries in the near future, the debate on the IRBA has shifted its accent. More emphasis is now given to the

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 249 probem of vaidating a rating system, rather than how to design a rating system. Both the private sector and banking supervisors need we-functioning rating systems. This overap of interests and objectives is refected in the approach towards vaidation of rating systems; even if different objectives impy different priorities in quaifying and monitoring the pausibiity of such systems. We wi discuss some of the chaenges faced by banks and supervisors, aware that we have ony scratched the surface. This is foowed by a discussion of some of the responses given by the BCBS. We then wi discuss a pragmatic approach towards vaidating IRB systems whie touching on some issues previousy raised. However, we woud ike to stress that impementation of Base II, and especiay the vaidation of IRB systems (and simiary AMA modes for operationa risk) requires ongoing diaogue between supervisors and banks. This artice, incuding its imitations, offers a conceptua starting point to dea with the issues reated to the vaidation of IRB systems. 12.1.2 Some Chaenges The discussion on vaidation has to start with a discussion of the structure and usage of interna rating systems within banks. The two-dimensiona risk assessment for credit risk as required in Base II, aims to quantify borrower risk, via the probabiity of defaut (PD) for a rating grade, and the faciity risk by means of the Loss Given defaut (LGD). The third dimension is the faciity s exposure at defaut (EAD). The broad structure of a bank-interna rating system is shown in Fig. 12.1. First, the information, i.e. the raw data on the borrower to be rated have to be coected in accordance with estabished banking standards. Accordingy, the data is used to determine the potentia borrower s credit risk. In most cases, a quantitative rating method which draws on the bank s previous experience with credit defauts is initiay used to determine a credit score. Borrowers with broady simiar credit scores, refecting simiar risk characteristics, are typicay aocated to a preiminary risk category, i.e. rating grade. Usuay, a oan officer then decides the borrower s fina rating and risk category, i.e. this stage invoves the appication of quaitative information. A we-working rating system shoud demonstrate that the risk categories differ in terms of their risk content. The quantification of risk parameters is based on the bank s own historica experience, backed by other pubic information and to certain extent, private information. For exampe, the share of borrowers in a given rating category who have experienced an occurrence defined as a credit defaut 3 within a 3 What constitutes credit defaut is a matter of definition. For banks, this tends to be the occurrence of an individua vaue adjustment, whereas at rating agencies, it is insovency or evident payment difficuties. The IRBA incuded in the new Base Capita Accord is based on an estabished definition of defaut. Compared with individua vaue adjustments, the Base definition of defaut provides for a forward-ooking and therefore reativey eary warning of defaut together with a retrospective fagging of payments that are 90 days overdue.

250 S. Bochwitz and S. Hoh Bank Controing and Management Bank Input-Data Pricing Quantitative Method Rating Proposa Rating System Extraordinary Effects Portfoio Modes Fina Rating Overview Interna Reporting Fig. 12.1 Schematic evoution of a rating process and its integration in the bank as a whoe given time-frame, usuay 1 year, wi be used for the estimation process. The described standardisation of ratings aows the use of quantitative modes where sufficient borrower data is avaiabe and highights the need for high-quaity, informative data. For consumer oans, the BCBS aso aows risk assessment on the eve of homogenous retai portfoios that are managed accordingy by the bank on a poo basis. The chaenge for banks is to identify such homogenous portfoios exhibiting simiar risk characteristics. This eads to the importance of using bank-interna data, which pays a crucia roe in both the segmentation process used to find homogenous portfoios, and the quantification process used for the risk parameters. One of the techniques used for segmentation and quantification is the utiisation of so-caed ro rates, 4 where different deinquency stages are defined (for exampe 30 days, 60 days etc.). Counting the ro rate from one deinquency stage to another and fiing the migration matrix woud serve as a basis for estimating the PDs for those exposures. There are a coupe of issues reated to this procedure. Firsty, there is the issue of segmentation, i.e. do ro rates take into account a reevant risk drivers as required in the Base II framework? Secondy, for quantification purposes, how wi ro rates be transated into PDs, more specificay, which deinquency cass shoud be used (to compy with the Base II framework), and to what extent can these PDs be vaidated? Lasty, in many instances a quicker reaction of current conditions, sometimes couped with a onger time horizon, might be needed for purposes of risk management and pricing, especiay for retai exposures. How woud such 4 Fritz et a. (2002).

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 251 quantification processes for PDs (and LGDs) be rectified with the appication of the use-test as required in Base II? Another issue reates to the modification performed by a credit officer of the automated rating proposa, i.e. a quaitative adjustment. This may question the rigidity needed for vaidation, especiay in cases where documentation may be insufficient, and the information used is more quaitativey based, the atter being a genera probem in credit assessments. A simpe, but important, question is who has the responsibiity for vaidating a rating system in the context of Base II, given that the cacuation of minimum reguatory capita is egay binding and set by the supervisors. In addition, a vaid point in this regard is that some requirements, for exampe, the quantification process focussing on ong-term averages to reduce voatiity of minimum reguatory capita requirements, are not fuy in ine with bank practice. This may ead to a different quantification process, i.e. a second process for the soe purpose of meeting supervisory standards, or even to a different risk management process as suggested above in the retai portfoios. In sum, the use-test requirement, the extent to which an interna rating system is used in daiy banking business, wi pay a crucia roe in assessing compiance with Base II impementation incuding the vaidation of IRB systems. Since a bank s interna rating systems are individua, and in the best case, fuy taiored to the bank s necessities; vaidation techniques must be as individua as the rating system they are used for. As an exampe, we highight the so-caed Low- Defaut-Portfoios. As the IRB framework in Base II is intended to appy to a asset casses, there are naturay portfoios which exhibit reativey ow or even no defaut at a. 5 This makes the quantification, required to be grounded in historica experience, of PDs and LGDs, extremey chaenging. Thus, a straightforward assessment based on historic osses woud not be sufficienty reiabe for the quantification process of the risk parameters, but conservative estimates serving as an upper benchmark may be derived (cf. Chap. 5). Some of the issues raised in this section have been discussed by the BCBS. 12.1.3 Provisions by the BCBS The Subgroup on Vaidation (AIGV) 6 of the BCBS Accord Impementation Group (AIG) was estabished in 2004. The objective of the AIGV is to share and exchange 5 See BCBS newsetter No 6, for exampe, some portfoios historicay have experienced ow numbers of defauts and are generay but not aways considered to be ow-risk (e.g. portfoios of exposures to sovereigns, banks, insurance companies or highy rated corporate borrowers). 6 The Vaidation Subgroup is focusing primariy on the IRB approach, athough the principes shoud aso appy to vaidation of advanced measurement approaches for operationa risk. A separate Subgroup has been estabished to expore issues reated to operationa risk (see BCBS newsetter No 4.).

252 S. Bochwitz and S. Hoh views reated to the vaidation of IRB systems. To the extent possibe, the groups shoud aso narrow gaps between different assessments of the New Framework by the different supervising agencies represented in the AIGV. The objective of the vaidation of a rating system is to assess whether a rating system can and utimatey does fufi its task of accuratey distinguishing and measuring credit risk. The common view describes the term vaidation as a means to combine quantitative and quaitative methods. If appied together, it shoud indicate whether a rating system measures credit risks appropriatey and is propery impemented in the bank in question. The BCBS newsetter No. 4, January 2004, informs about the work of the AIGV in the area of vaidation in Base II. The most important information provided was the reativey simpe answer to the question, what aspects of vaidation wi be ooked at? Despite the importance of vaidation as a requirement for the IRB approach, the New Framework does not expicity specify what constitutes vaidation. Consequenty, the Subgroup reached agreement on that question. In the context of rating systems, the term vaidation encompasses a range of processes and activities that contribute to assessing whether ratings adequatey differentiate risk, and importanty, whether estimates of risk components (such as PD, LGD, or EAD) appropriatey characterise and quantify the reevant risk dimension. Starting with this definition, the AIGV deveoped six important principes (see Fig. 12.2), on vaidation that resut in a broad framework for vaidation. The vaidation framework covers a aspects of vaidation, incuding the goa of vaidation (principe 1), the responsibiity for vaidation (principe 2), expectations on vaidation techniques (principes 3, 4, and 5), and the contro environment for vaidation (principe 6). Pubishing these principes was a major step in carifying the ongoing discussions between banks and their supervisors about vaidation for at east three reasons: 1. The principes estabish a broad view on vaidation. Quite often, vaidation was seen as being restricted to ony deaing with aspects reated to backtesting. The estabished broad view on vaidation reinforces the importance of the minimum requirements of the IRBA, as we as highighting the importance of riskmanagement. The debate around the IRBA was too often restricted to soey risk quantification or measurement aspects. We think that this baanced perspective, incuding the more quaitative aspects of the IRBA, refects the shortcomings in estabishing and vaidating rating systems, especiay given the data imitations. This carification aso formed the basis for the deveopment of vaidation principes for the so-caed Low Defaut Portfoios (LDPs) as proposed in the BCBS newsetter No. 6 from August 2005. 2. The responsibiity for vaidation and the deegation of duties has aso been carified. The main responsibiity ies rightfuy with the bank, given the importance of rating systems in the bank s overa risk management and capita aocation procedures. Since vaidation is seen as the utimate sanity-check for a rating system and a its components, this task ceary must be performed by the bank itsef, incuding the fina sign-off by senior management. It shoud be

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 253 Principe 1: Vaidation is fundamentay about assessing the predictive abiity of a bank s risk estimates and the use of ratings in credit processes The two step process for ratings systems requires banks to firsty discriminate adequatey between risky borrowers (i.e. being abe o discriminate between risks and its associated risk of oss) and secondy caibrate risk (i.e. being abe to accuratey quantify the eve of risk). The IRB parameters must, as aways with statistica estimates, be based on historica experience which shoud form the basis for the forward-ooking quaity of the IRB parameters. IRB vaidation shoud encompass the processes for assigning those estimates incuding the governance and contro procedures in a bank. Principe 2: The bank has primary responsibiity for vaidation The primary responsibiity for vaidating IRB systems ies with the banks itsef and does not remain with the supervisor. This certainy shoud refect the sef-interest and the need for banks having a rating system in pace refecting its business. Supervisors obviousy must review the bank s vaidation processes and shoud aso rey upon additiona processes in order to get the adequate eve of supervisory comfort. Principe 3: Vaidation is an iterative process Setting up and running an IRB system in rea ife is by design an iterative process. Vaidation, as an important part of this circe, shoud therefore be an ongoing, iterative process foowing an iterative diaogue between banks and their supervisors. This may resut in a refinement of the vaidation toos used. Principe 4: There is no singe vaidation method Many we-known vaidation toos ike backtesting, benchmarking, repication, etc are a usefu suppement to the overa goa of achieving a sound IRB system. However, there is unanimous agreement that there is no universa too avaiabe, which coud be used across portfoios and across markets. Principe 5: Vaidation shoud encompass both quantitative and quaitative eements Vaidation is not a technica or soey mathematica exercise. Vaidation must be considered and appied a broad sense, its individua components ike data, documentation, interna use and the underying rating modes and a processes which the rating system uses are equay important. Principe 6: Vaidation processes and outcomes shoud be subject to independent review For IRB systems, there must be an independent review within the bank. This specifies neither the organigram in the bank nor its reationship across departments, but the review team must be independent of designers of the IRB system and those who impement the vaidation process. Fig. 12.2 The six principes of vaidation noted that ony banks can provide the resources necessary to vaidate rating systems. 3. Principes 3 5 estabish a comprehensive approach for vaidating rating systems. This approach proposes the key eements of a broad vaidation process, on which we wi eaborate more in the next section.

254 S. Bochwitz and S. Hoh 12.2 Vaidation of Interna Rating Systems in Detai According to the BCBS eaboration on the term vaidation, we consider three mutuay supporting ways to vaidate bank interna rating systems. This encompasses a range of processes and activities that contribute to the overa assessment and fina judgement. More specificay, this can be directy reated to the appication of principe 4 and principe 5 of the BCBS newsetter as discussed above. 1. Component-based vaidation: anayses each of the three eements data coection and compiation, quantitative procedure and human infuence for appropriateness and workabiity. 2. Resut-based vaidation (aso known as backtesting): anayses the rating system s quantification of credit risk ex post. 3. Process-based vaidation: anayses the rating system s interfaces with other processes in the bank and how the rating system is integrated into the bank s overa management structure. 12.2.1 Component-Based Vaidation 12.2.1.1 Avaiabiity of High-Quaity Data Ensuring adequate data quaity is the key task which, for at east two reasons, must be addressed with the greatest urgency. First, as the rating is based primariy on individua borrowers current data, it can ony be as good as the underying data. Second, the quantitative component of the rating process requires a sufficienty reiabe set of data, incuding a cross-sectiona basis, which is crucia for caibration of the risk parameters. Accordingy, both historica data and high-quaity recent data are essentia to ensure that a rating system can be set up adequatey, and wi aso be successfu in the future. Ceary, the avaiabiity of data, i.e. financia versus account specific information, and its use for different borrower characteristics, whoesae versus consumer is dissimiar. Activities in consumer finance may produce more bank-specific behavioura data whereas financia information for arge whoesae borrowers shoud be pubicy avaiabe. However, the avaiabiity of reiabe and informative data, especiay for the mid-size privatey owned borrowers, may frequenty not be met for at east severa reasons: Data compiation and quaity assurance incur high costs because they require both quaified staff and a high-performance IT infrastructure. In addition, these tasks seem to have itte to do with origina banking business in its strict sense, and their usefuness may ony become apparent years ater. Ceary, proper investment is needed, adding pressure to costs and competition. Simiary, it is a costy exercise in staffing and resource aocation in credit departments. However, the Base II efforts may have heped to aocate more resources to capturing adequate and reiabe credit data.

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 255 In reaity, borrowers aso are often reuctant to suppy the requested data. This may be because, especiay at smaer enterprises, this data is not readiy avaiabe. Admittedy, because of the predominant cassic house bank system in Germany, this information historicay had not been requested. Aso, potentia misuse of data and reuctance on the part of firms to provide information on their own economic situation seems to be a widespread concern. Sometimes, data is passed on to a very imited number of third parties ony. 7 Further concentration in the banking industry is aso contributing to the probem. Owing to the ack of uniform standards for banks, in the event of a merger, different sets of data have to be synchronized this adds a new dimension to the probem and is, again, no quick and easy task to do. A thorough knowedge of the IT systems underying the rating approach is necessary for the proper assessment of data quaity; in addition the foowing may hep to provide a reaistic evauation: Ensuring data quaity: The sheer existence and quaity of bank interna guideines, incuding tests around them, is an indication of the importance banks pace on good data quaity. Whether a bank takes its own guideines seriousy can be gauged from day-to-day appications. For instance, data quaity assurance can reasonaby be expected to be as automated as possibe to ensure that a uniform standard is appied throughout the bank. Aso, comparison with externa sources of data seems to be necessary to ensure data pausibiity. Bank-wide use of the data: The extent to which data are used aows assessing the confidence that the bank has in its data. This eads to two consequences. On the one hand, frequent and intensive use of specific data within a bank exposes inconsistencies which might exist. On the other hand, where arger numbers of peope are abe to manuay adjust data, the more ikey is its potentia contamination, uness suitabe countermeasures are taken. 12.2.1.2 The Quantitative Rating Modes The second facet of the rating process, in the broadest sense, is the mathematica approach which can be used to standardise the use of data. The aim is to compress data coected in the first stage to prepare and faciitate the oan officer s decision on the credit risk assessment of a borrower. In recent years, the anaysis and deveopment of possibe methods has been a focus of research at banks and in microeconomics. The second stage methods attribute to each borrower, via a rating function f Rat, a continuous or discrete risk measure Z, a rating score, which is dependent on both the 7 An indication of this attitude, which is widespread in Germany, is, for exampe, the approach that is adopted to the obigation aid down in Section 325 of the German Commercia Code for corporations to pubish their annua accounts. No more than 10% of the enterprises concerned fufi this statutory obigation.

256 S. Bochwitz and S. Hoh individua features of each borrower X 1, X 2,..., X N aso denoted as risk factors and free, initiay unknown mode parameters a 1,a 2,..., a M : Z ¼ f Rat ða 1 ; ; a M ; X 1 ; ; X N Þ: The vaue of Z permits the suggested rating to be derived from the quantitative anaysis of the borrower concerned, in that each vaue of Z is aocated precisey to one of Y various rating categories. The methods suitabe for this kind of quantitative component can be cassified as: Statistica methods: This is probaby the best known and the most widespread group of methods. They are used by amost a banks in both corporate and private sector business. The best known of such methods are discriminatory anayses (primariy in corporate business) and ogit regressions (used mainy as scorecards in private sector business). Generaised regression and cassification methods (such as neura networks) aso beong in this category, even if they are rarey used in practice. Rue-based systems: Such systems mode the way in which human credit experts reach a decision and are used in corporate business. They comprise a set of predetermined if... then rues (i.e. expert knowedge). Each enterprise is first graded according to these rues. The next stage is for the rues matched by the firm to be aggregated in order to give a risk rating. Benchmarking methods: In these methods, observabe criteria, such as bond spreads, are used to compare borrowers with unknown risk content with rated borrowers with known risk content the so-caed benchmarks. Appied economic modes: Option price theory modes are the most widey known. They enabe, for exampe, an enterprise s equity capita to be modeed as a ca option on its asset vaue and thus the concepts used in option price theory to be appied to credit risk measurement. The starting point for the deveopment of these modes was the Merton mode; KMV has been successfu in its further deveopment offering its Pubic Firm Mode for isted enterprises and a Private Firm Mode for unisted enterprises (Crosbie and Bohn 2001), now marketed under the Moody s KMV -abe. Another cassification distinguishes between empirica modes, where the parameters are determined from data of known borrowers by using mathematica or numerica optimisation methods, and expert methods, where the parameters are specified by credit experts based on their experience. Basicay, the difference ies in the specification of the mode parameters a 1,a 2,..., a M. 12.2.1.3 The Mode Itsef Transparency, inteigibiity and pausibiity are crucia for vaidating the appropriateness of the rating process. Ceary, either with the set of rues for expert systems or with the underying mode in the case of benchmarking methods and appied economic modes, these requirements seem to be easiy fufied. The situation

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 257 regarding statistica modes is somewhat more compex as there is no economic theory underying these modes. However, certain basic economic requirements shoud aso be incorporated in using statistica modes. For exampe, experience has shown that many risk factors are invariaby more marked among good borrowers than bad borrowers. Likewise, if a requirement of risk measure Z is invariaby arger among better borrowers than among worse borrowers, the direct consequence is that the monotony of the risk factor must aso be evident in the monotony of the risk measure. Therefore, for the i-th risk factor X i, the foowing appies: @Z ¼ @f Ratða 1 ; ; a M ; X 1 ; ; X N Þ >0: @X i @X i Economic pausibiity eads to the excusion of non-monotonous risk factors in inear modes. Non-monotonous risk factors are, for exampe, growth variabes, such as changes in the baance sheet tota, changes in turnover etc. Experience shows that both a decine and excessivey high growth of these variabes impy a high risk. Such variabes cannot be processed in inear modes, i.e. in modes ike Z ¼ a 0 þa 1 X 1 þþa N X N, because, owing to @Z @X i ¼ a i ¼ const:; the pausibiity criterion in these modes cannot be fufied for non-monotonous features. 8 Further economic pausibiity requirements and sensitivity anaysis shoud be considered in a causa reationship with economic risk, for exampe the creditworthiness of an enterprise cannot be derived from the managing director s shoe size! The commony appied statistica standards must be observed for a empirica modes (statistica modes, specific expert systems and appied economic modes). Non-compiance with these standards is aways an indication of design defects, which generay exhibit an adverse effect when appied. Without caiming competeness, we consider the foowing aspects to be vita when deveoping a mode: Appropriateness of the random sampe for the empirica mode: The appropriateness of the random sampe is the decisive factor for a empirica and statistica modes. This is aso reevant to appied economic modes, as is iustrated by the KMV modes. These modes have been based on data on US firms, meaning that they draw on deveopments in the US markets ony and soey refect US accounting standards. Not a data which is important in this system is avaiabe when other accounting standards are used, with the resut that when the modes are transferred to other countries, one has to work with possiby questionabe approximations. This has a bearing on certain characteristics of the modes such as ack of ambiguity and the stabiity of the resuts. 8 Moody s RiskCac (Fakenstein et a. 2000) provides one way of processing non-monotonous risk factors by appropriate transformation in inear modes. Another one can be found in Chap. 2.

258 S. Bochwitz and S. Hoh Over-parameterising the mode: A mistake, frequenty observed, is to incude too many risk factors in the design of a rating system. The reasons for this incude an overy cautious approach when deveoping the system, i.e. each conceivabe risk factor, or those which credit experts seem to consider obvious, are to be fed into the system. On the other hand, rating systems are often deveoped by committees and these woud naturay ike to see their particuar babies (mosty a favourite variabe or a specia risk factor) refected in the rating design. Neither approach is optima from the statistica perspective as there is an upper imit to the number of parameters to be cacuated, depending on the size of the sampe and the mode used. If this rue is breached, errors are made which are caed overfitting. Statistica performance of the estimated mode: The performance of the mode in a statistica sense is generay provided as a type-1 or a type-2 error, appying measures of inequaity such as Gini coefficients or entropy measures (Fakenstein et a. 2000), or other statistica measures which can be determined either for the sampe or the entire popuation. These variabes quantify the rating system s abiity to distinguish between good and bad borrowers and thus provide important information about the capabiity of the rating mode with regard to discriminating between risks. These variabes are especiay important during the deveopment of a rating system as they aow comparison of the performance of various modes within the same data set. However, we think that these toos are ony of minor importance for ongoing prudentia monitoring. First, owing to the concave form of the risk weighting function in the new Base Accord, which provides ogica incentives so that systems which discriminate more finey, are ess burdened by reguatory capita than coarser systems. Second, the absoute size of the probabiity of defaut is the variabe reevant for banking supervision as it is inked to the size of the reguatory capita. Modeing errors, precision and stabiity: Certain modeing errors are inevitaby part of every mode because each mode can depict ony a part of economic reaity in a simpified form. In order to be abe to use a mode correcty, one has to be aware of these imitations. However, in addition to these imitations, which are to a certain extent a natura feature of each mode, the modeing errors caused by using an optimisation or estimation procedure aso need to be considered. These estimation errors can be quantified for the mode parameters from the confidence eves of the mode parameters. Given certain distribution assumptions, or with the aid of cycica or rotation methods, these confidence eves can be determined anayticay from the samedatawhichisusedtoestimatetheparameters(fahrmeireta.1996). If error cacuation methods frequenty used in the natura sciences are appied, it is possibe to estimate the extent to which measurement bias of the individua mode parameters affects the credit score Z. The stabiity of a mode can be derived from the confidence eves of mode parameters. Determining the stabiity of a mode seems to be particuary important, i.e. the responsiveness to portfoio changes. A more critica issue is mode precision. In some methods, mode parameters are determined though iogicay with a precision that is severa orders of magnitude higher than for the risk parameters.

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 259 12.2.1.4 Roe of the Loan Officer-or Quaitative Assessment Loan officers pay an important roe in both setting up a rating system as we as using it in practice. We think that quaitative assessments shoud be incuded in the fina rating assignment, by aowing the oan officer to modify the suggested credit rating provided by the quantitative mode. 9 This is certainy necessary for exposures above a certain size; retai oans may be dependent on the business activities and risk management structures in the bank. The sheer size of mass financing of consumer oans certainy resuts in ess infuence for the oan officer, rather, they rey on correct procedures to check the automated rating proposa and the input provided by the saes officer. We discuss three important aspects accordingy: The oan officer s powers: Any manua modification of the automated rating proposa shoud be contained within a controed and we-documented framework. The oan officer s discretion shoud be set within ceary defined imits which specify at east the conditions permitting a deviation from the automated rating proposa and the information that the oan officer used additionay. One way to ook at discretion is the use of a deviation matrix of fina and suggested ratings, showing for each rating category, how many suggested ratings (generated by the quantitative rating too) are changed by manua override: more specificay, the share M ij of borrowers assigned by the quantitative system to the i-th category which oan officers finay pace in category j. In a we-defined, practicabe rating system, a high match between suggested ratings and fina ratings shoud be expected in most cases, so in each ine the vaues of M ii shoud be the argest and M ij shoud decrease the more the fina ratings diverge from the suggestions. Ceary, greater deviations shoud ead to carefu anaysis of the shortcomings of the rating mode, either indicating data issues or probems with the mode itsef. Monitoring the ratings over time: Any rating system must ideay be monitored continuousy and be abe to process incoming information swifty; however, ratings must be updated at east annuay. This does aso appy for consumer oans. However, the focus is on ensuring that oans and borrowers are sti assigned to the correct poo, i.e. sti exhibiting the oss characteristics and the deinquency status of the previousy assigned poo. As such, different methodoogies may be used, for exampe by using an account-specific behavioura score. For whoesae oans, it may be hepfu to anayse the frequency distribution of the time-span between two successive ratings of a borrowers in a specific portfoio. The expected pattern is shown in Fig. 12.3: most borrowers are reevauated at reguar intervas, roughy once every 12 months, but in between, ad hoc ratings are based on information deemed to be important and their frequency increases with the amount of time that has eapsed since the first rating. Between the two reguar re-ratings, a whoe group of the same type of borrowers 9 The norma transformation of quaitative information ike famiy status, gender, etc into numerica variabes for the assessment of consumer oans woud not repace such a quaitative oversight.

260 S. Bochwitz and S. Hoh 40 % 35 % Ratio of rerated cients 30 % 25 % 20 % 15 % 10 % 5% Ad-hoc reratings caused by materia new information Reguar rating review Review of ratings of a group of simiar borrowers, eg. of a certain sector due to a specific event 0% 1 2 3 4 5 6 7 8 9 10 11 12 Time between two subsequent ratings in months more than 12 Fig. 12.3 Frequency distribution of the time-span between two successive ratings for a borrowers in one portfoio (e.g. enterprises in one sector) may occasionay be re-rated because information reevant to the rating of this group has been received. It shoud be possibe to expain any divergent behaviour which, in any case, provides insights into the quaity of the rating process. Feedback mechanisms of the rating process: A rating system must take account of both the justified interests of the user i.e. the oan officer whose interest is driven by having a rating process which is ean, easy to use, comprehensibe and efficient. On the other hand, the mode deveoper is interested in a rating mode which is theoreticay demanding and as comprehensive as possibe. Where interests confict, these wi need to be reconcied. It is a the more important that a rating system is checked whist in operationa mode, to ascertain whether the mode which the process is based on is appropriate and sufficienty understood by the users. In any case, procedures must be impemented according to which a new version or at east a new parameterisation of the rating mode is carried out. 12.2.2 Resut-Based Vaidation In 1996, the pubication of capita requirements for market risk for a bank s trading book positions as an amendment to the 1988 Base Accord, was the first time that a bank s interna methodoogy coud be used for purposes of reguatory capita. The output of bank interna modes, the so-caed Vaue-at-Risk (VaR) which is the most popuar risk measure in market risk, is transated into a minimum capita requirement, i.e. three times VaR. The supervisory chaenge for most countries, certainy Germany, was to estabish an appropriate supervisory strategy to finay

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 261 permit these bank interna modes for cacuating reguatory capita. In addition to the supervisory assessment of the quaitative market risk environment in a bank, another crucia eement of the strategy was the impementation of an efficient top-down monitoring approach for banks and banking supervisors. The reativey simpe comparison between ex-ante estimation of VaR and ex-post reaisation of the cean P&L 10 of a trading book position, excuding extraneous factors such as interest payments, was the foundation for the quantitative appraisa. The concept for backtesting in the IRBA as introduced in paragraph 501 of the New Framework is reativey simiar. In the IRB approach, according to market risk, the probabiity of defaut (PD) per rating category or, in specia cases, the expected oss (EL) in the case of consumer oans, must be compared with the reaised defaut rate or osses that have occurred. Despite the basic features common to market risk and credit risk, there are aso important differences, most importanty the foowing two. First, the conceptua nature is different; in market risk the forecasted VaR is a percentie of the cean P&L distribution. This distribution can be generated from the directy observabe profit and osses, and thus the VaR can be directy observed. By contrast, in credit risk ony reaised defauts (and osses) according to a specific definition can be observed directy instead of the forecasted PD (and EL). A common and widespread approach for credit risk is the appication of the aw of arge numbers and to infer from the observed defaut rate, the probabiity of defaut. 11 To our knowedge, amost a backtesting techniques for PD (or EL) rey on this statistica concept. However, a proper appication requires that borrowers are grouped into grades exhibiting simiar defaut risk characteristics. 12 This is necessary even in the case of direct estimates of PD, when each borrower is assigned an individua PD. The second main difference reates to the avaiabe data history on which the comparison is based. In market risk, the frequency is at east 250 times a year in the case of daiy data. By contrast, in credit risk there is ony one data point per annum to be assumed. To make it more compex, there is an additiona probem arising from measuring credit defaut, which is the key variabe for the quantification and therefore the vaidation. The definition of credit defaut is argey subjective. The New Framework suggests retaining this subjective eement as the basis of the IRB 10 There are different interpretations among different supervisors on this issue. 11 Beside the fact, that an appication of the aw of arge numbers woud require that defauts are uncorreated, there is another subte vioation in the prerequisites for appying the aw of arge numbers. It is required that the defauts stem from the same distribution. This requirement cannot be seen to be fufied for different borrowers. To give a picture: The difference for the task of determining the probabiity of throwing a six is ike approximating this probabiity either by throwing the same dice 1,000 times and cacuating the ratio of sixes to the tota number of throws or throwing 1,000 dices once and cacuating the ratio of sixes to the number of dices thrown. 12 We beieve that vaidation of rating systems, i.e. the caibration of PDs is amost impossibe without the grouping of borrowers to grades with the same risk profie; which is aso one of the key requirements of Base II.

262 S. Bochwitz and S. Hoh approach, abeit with a forward-ooking focus and a back-stop ratio of 90 days past due. This may be justified, not east by the fact that a significant number of defauted borrowers seem to have a considerabe infuence on the timing of the credit defaut. Correspondingy, the criteria and more importanty the appied methodoogy are aso different. In market risk, the chaenge is to provide a cean P&L and to store the corresponding data. This differs significanty from the necessary compiation of the rating history and credit defauts over time. Depending on the required reproducibiity of the resuts, considerabe time and effort may be needed and it is difficut to estimate what requirement is most important for what area, thus entaiing higher costs for the bank. Owing to the voume of avaiabe data points in market risk, the simpicity and mutipicity of the appicabe methods are impressive. This naturay poses an apparenty insuperabe chaenge for credit risk. A further probem is the impact of a rating phiosophy on backtesting. The rating phiosophy is what is commony referred to as either Point-in-Time (PIT) or Through-the-Cyce (TTC) ratings. PIT-ratings measure credit risk given the current state of a borrower in its current economic environment, whereas TTC-ratings measure credit risk taking into account the (assumed) state of the borrower over a fu economic cyce. This means assuming an underying two-step-rating process as described above a TTC rating system requires (1) constant PDs per grade over time and (2) a structure that et migrate borrowers on idiosyncratic effects ony, i.e. no migration of borrowers between grades reated to cyce effects. Consequenty, a TTC-rating system must ensure that there is virtuay no correation between grade migration and the cyce. Simiary, PD estimates for each rating grade must not change in a way which is correated with the cyce An aternative might be to make some adjustments to the basic outputs in order to achieve an acycica effect. An exampe for that is given by the UK-FSA s scaing approach. 13 PIT and TTC mark the ends of the spectrum of possibe rating systems. In practice, neither pure TTC nor pure PIT systems wi be found, but hybrid systems, which are rather PIT or rather TTC. Agency ratings are assumed to be TTC, whereas current bank interna systems at east in most cases in Germany and many other countries are ooked at as PIT. This is pausibe because for the purpose of managing credit risk a PIT-system, that detects borrowers deterioration eary, seems to be more reasonabe than a TTC-system. The increased focus on reducing excess cycicaity in minimum capita requirements by supervisors may ead banks to possiby promote the use of TTC ratings versus PIT ratings. A bank then may decide to expicity use two different PD caibrations, one for interna purposes (PIT for exampe for pricing, margining and remuneration) and one for reguatory purposes (TTC for exampe for reguatory capita). In this case a very important question to be asked is whether this may be appropriate in the ight of the requirement of the use-test. To this end, as ong as the interna processes, i.e. the credit granting process as we as the rating assignment 13 Cf. Financia Services Agency (2006, 2009).

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 263 process, stay the same for both caibrations we woud suggest that the use-test criteria may be acceptabe. In addition, importance shoud aso be given to the fact that broader system-wide economic changes on the PD estimate shoud not be refected in the TTC estimates in order to reduce its idiosyncratic risk induced voatiity. The rating phiosophy has an important impact on backtesting. In theory, for TTC systems borrower ratings, i.e. its rating grade, are stabe over time, refecting the ong-term fu cyce assessment. However, the observed defaut rates for the individua grades are expected to vary over time in accordance with the change in the economic environment. The contrary is the case for PIT systems. By more quicky reacting to changing economic conditions, borrower ratings tend to migrate through the rating grades over the cyce, whereas the PD for each grade is expected to be more stabe over time, i.e. the PD is more independent from the current economic environment. The Base Committee did not favour a specia rating phiosophy. Both PIT systems as we as TTC systems are fit for the IRBA. However, it seems to be reasonabe to ook at risk parameters as a forecast for their reaisations which can be observed within a 1 year time horizon. This reasoning is refected in the first vaidation principe of the AIGV, where a forward ooking eement is required to be incuded in the estimation of Base s risk parameters. However, vaidation of TTC-ratings is extremey chaenging if it is ooked at from the perspective of backtesting since for TTC ratings the target for PD caibration refects an average for the cyce. If statistica testing techniques are to be appied, then the requirement for the ength of a time series wi be increased by the factor of a cyce ength in years. Additionay, backtesting requires the integration of fu cyces ony. Therefore the accuracy of the risk quantified in TTC ratings is difficut to evauate and straightforward backtesting techniques, as sketched out in many artices of this book, are expected to be of imited vaue. In the specia case of consumer oans, the estimation and vaidation of key parameters is extremey dependent on the approach taken by a bank. A simiar rating system as used for whoesae borrowers, eads to an anaogous assessment for purposes of vaidation. In contrast, instead of rating each borrower separatey, the BCBS custers oans in homogenous portfoios during the segmentation process (see above). This segmentation process shoud incude assessing borrower and transaction risk characteristics ike product type etc., as we as identifying the different deinquency stages (30 days, 60 days, 90 days etc.). Subsequenty, the risk assessment on a (sub-) portfoio eve coud be based on its ro rates, transaction moving from one deinquency stage to another. The impications of these rather genera considerations and possibe soutions for the probems raised here are discussed in detai in Chap. 9. 12.2.3 Process-Based Vaidation Vaidating rating processes incudes anaysing the extent to which an interna rating system is used in daiy banking business. The use test and associated risk estimates

264 S. Bochwitz and S. Hoh is one of the key requirements in the BCBS fina framework. There are two different eves of vaidation. Firsty, the pausibiity of the actua rating in itsef, and secondy, the integration of ratings output in the operationa procedure and interaction with other processes: Understanding the rating system: It is fundamenta to both types of anaysis that empoyees understand whichever rating methodoogy is used. The earning process shoud not be restricted to oan officers. As mentioned above, it shoud aso incude those empoyees who are invoved in the rating process. In-house training courses and other training measures are required to ensure that the process operates propery. Importance for management: Adequate corporate governance is crucia for banks. In the case of a rating system, this requires the responsibiity of executive management and to a certain extent the supervisory board, for authorising the rating methods and their impementation in the bank s day-to-day business. We woud expect different rating methods to be used depending on the size of the borrower, 14 and taking account of the borrowers different risk content and the reevance of the incoming information foowing the decision by senior management. Interna monitoring processes: The monitoring process must cover at east the extent and the type of rating system used. In particuar, it shoud be possibe to rate a borrowers in the system, with the fina rating aocated before credit is granted. If the rating is given after credit has been granted, this raises doubts about the usefuness of interna rating. The same appies to a rating which is not subject to a reguar check. There shoud be a check at east annuay and whenever new information about the debtor is received which casts doubt on their abiity to cear their debts. The stabiity of the rating method over time, baanced with the need to update the method as appropriate, is a key part of the vaidation. To do this, it is necessary to show that objective criteria are incorporated so as to ay down the conditions for a re-estimation of the quantitative rating mode or to determine whether a new rating mode shoud be estabished. Integration in the bank s financia management structure: Uness rationa credit risk is recorded for each borrower, it is impossibe to perform the proper margin cacuation taking into account standard risk costs. If this is to be part of bank management by its decision-making and supervisory bodies, a reationship must be determined between the individua rating categories and the standard risk costs. However, it must be borne in mind that the probabiity of defaut is simpy a component of the cacuation of the standard risk costs and, simiary to the credit risk modes, other risk parameters, such as the rate of debt coection and the size of the exposure in the event of a credit defaut, the maturity of the oan, transfer risk and concentration risks shoud aso be recorded. Utimatey the gross margin, which approximates to the difference between ending rates and 14 In the Base Committee s new proposas in respect of the IRB approach, sma enterprises may, for reguatory purposes, be treated as retai customers and, unike arge corporate customers, sma and medium-sized enterprises are given a reduced risk weighting in ine with their turnover.

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 265 refinancing costs, can act as a yardstick for incuding the standard risk costs. In order to determine the concentration risks at portfoio eve more appropriatey, it seems essentia to use credit risk modes and thus to be in a position to aocate venture capita costs appropriatey. Therefore, if net commission income is added to the gross margin, the operationa costs netted out, and aso the venture capita costs taken into account, it is possibe to cacuate the resut of ending business. It is naturay advisabe to incude as part of the management of the bank, a other conventiona instruments of credit risk measurement, such as empoyee bonus systems, portfoio optimisation. In principe, the Base Committee requires these mainy portfoio-based methods in the second piar of the new Accord as part of the sef-assessment of capita adequacy required of the banks in the Capita Adequacy Assessment Process (CAAP). This frequenty eads to probems when integrating banks own rating systems into credit risk modes purchased from speciaist providers. In our view, this may utimatey increase the compexity for banks and banking supervisors and at the same time entai considerabe competitive distortions if the rating is ess objective. 12.3 Concuding Remarks To set up and to vaidate bank-interna rating systems is a chaenging task and requires a considerabe degree of sensitivity (Neae 2001). Our anaysis started with the comparativey more difficut data situation and the avaiabiity of pubic and private information in order to quantify credit risk of banks borrowers in a structured way incuding its subsequent vaidation. The advantage of the structured credit risk assessment, when appying an automated rating process, is its objectivity. This is true for the rating method and for the seection of the risk factors in the rating mode, incuding their effectiveness in generating a rating proposa. The fina integration of the quaitative credit assessment, based on a subjective decision by the oan officer, is more difficut in the structured assessment. The fina rating outcome comprises an array of individua observations, which may provide very different resuts. Utimatey, our suggested approach to vaidation takes this compexity into account by highighting the importance of the rating process. This interdependence is refected in the ongoing cyce of setting up and monitoring the rating system. Individua observations during the monitoring process are frequenty integrated quicky into a revision of the methodoogica process. The vaidation method is anaogous to a jigsaw puzze. Ony if the many individua pieces are being assembed propery, wi the desired resut be achieved. The individua pieces of the puzze seem unimpressive and often unattractive at first, but they eventuay contribute to the utimate picture. This may, for exampe, be an appropriate description when setting up the system and conducting ongoing checks on the quaity of the data management or the ongoing adjustment of banks interna credit standards. Each piece of the puzze is crucia, to both component-based and process-based vaidation. One cruciay important piece is the process-based

266 S. Bochwitz and S. Hoh component. A conventiona methods of quantitative vaidation shoud encompass the assessment of the rating too s economic meaningfuness as we as its compiance with statistica standards. Transparency and comprehensibiity of the chosen methods at each stage of deveopment, as we as its pausibiity, are fundamenta requirements of a sound rating system. The advantage of using empirica statistica approaches is that these modes are comprehensibe and that defects or statistica shortcomings can be detected by simpe statistica tests. By contrast, rue-based systems and appied economic modes are more heaviy mode-dependent and therefore point to mode risk. In the case of benchmarking methods; however, the choice of the peer group with known risk content is decisive, athough the instabiity of such modes, in particuar, can be a probem. Despite the differences, most appied methods can fufi a requirements initiay, abeit to a differing degree. The broad use and the interpay of different quantitative pausibiity and vaidation methods is the basis of a quantitative anaysis of the methods used. Backtesting using a simpe retrospective comparison of estimated defaut probabiities with actua defaut rates is crucia, and therefore a decisive eement in the vaidation of the resuts. 15 Compementary methods are aso needed, particuary in the deveopment stage of rating modes, in order to ensure the pausibiity of the seected methods. These incude techniques which underscore the stabiity and accuracy of the methods, athough caution is required with regard to quantification and especiay with regard to methods used to measure accuracy. The vaidation of interna rating systems underscores the importance of using a formaised process when devising them and in their daiy appication. This covers both the formaised keying in of data and the criteria for subjectivey overruing the rating proposa. Uness interna ratings are used on a reguar basis and in a structured manner over time, banks and banking supervisors by referring to the use-test wi find difficuties in accepting such a rating system. References Base Committee on Banking Supervision (1988), Internationa Convergence of Capita Measurement and Capita Standards. http://www.bis.org/pub/bcbs04a.htm Base Committee on Banking Supervision (2005), Internationa Convergence of Capita Measurement and Capita Standards: A Revised Framework, BIS, Updated November 2005. http:// www.bis.org/pub/bcbs107.htm Base Committee on Banking Supervision (2005a), Vaidation, Newsetter No. 4. http://www.bis. org/pub/bcbs_n4.htm Base Committee on Banking Supervision (2005b), Vaidation of Low Defaut Portfoios, Newsetter No. 6. http://www.bis.org/pub/bcbs_n4.htm Base Committee on Banking Supervision (2005c), The Appication of Base II to Trading Activities and the Treatment of Doube Defaut Effects. http://www.bis.org/pub/ bcbs116.htm 15 We thus concur with the Base Committee on Banking Supervision.

12 Vaidation of Banks Interna Rating Systems: A Supervisory Perspective 267 Base Committee on Banking Supervision (2009), Strengthening the Resiience of the Banking Sector. http://www.bis.org/pub/bcbs164.pdf Crosbie PJ, Bohn JR (2001), Modeing Defaut Risk, KMV LLC. http://www.kmv.com/ insight/ index.htm Fahrmeir L, Hamere A, Tutz G (1996), Mutivariate Statistische Verfahren, de Gruyter, Berin - New York Fakenstein E, Bora A, Carty LV (2000), RiskCac TM for Private Companies: Moody s Defaut Mode, Moody s Investor Service May 2000. http://www.moodys.com/cust/research/venus/ Pubication/Rating%20Methodoogy/noncategorized_number/56402.pdf Financia Services Authority (2006), Use of ong run Probabiities of Defaut, counter-cycica scaing factors, and the interaction of these with economic cyce stress testing, http://www.fsa. gov.uk/pubs/internationa/crsg_ong_run_pds.pdf Financia Services Authority (2009), Variabe Scaars Approaches to Estimating Through the cyce PDs, http://www.fsa.gov.uk/pubs/internationa/variabe_scaars.pdf Fritz S, Popken L, Wagner C (2002), Scoring and vaidating Techniques for Credit Risk Rating Systems, in Credit Ratings, Risk Books, London Neae C (2001), The Truth and the Proof, Risk, 13 (3), pp. 18 19

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Chapter 13 Measures of a Rating s Discriminative Power: Appications and Limitations Bernd Engemann 13.1 Introduction A key attribute of a rating system is its discriminative power, i.e., its abiity to separate good credit quaity from bad credit quaity. Simiar probems arise in other scientific discipines. In medicine, the quaity of a diagnostic test is mainy determined by its abiity to distinguish between i and heathy persons. Anaogous appications exist in bioogy, information technoogy, and engineering sciences. The deveopment of measures of discriminative power dates back to the eary 1950s. An interesting overview is given in Swets (1988). Many of the concepts deveoped in other scientific discipines in different contexts can be transferred to the probem of measuring the discriminative power of a rating system. Most of the concepts presented in this artice were deveoped in medica statistics. We wi show how to appy them in a ratings context. Throughout this chapter, we wi demonstrate the appication of a concepts on two prototype rating systems which are deveoped from the same data base. We consider ony rating systems which distribute debtors in separate rating categories, i.e., the rating system assigns one out of a finite number of rating scores to a debtor. For both rating systems, we assume that the tota portfoio consists of 1,000 debtors, where 50 debtors defauted and 950 debtors survived. Both rating systems assign five rating scores 1, 2, 3, 4, and 5 to debtors where 1 stands for the worst credit quaity and 5 for the best. Tabe 13.1 summarizes the rating scores that were assigned to the surviving debtors by both rating systems. Tabe 13.1 tes us precisey the distribution of the non-defauting debtors on the two rating systems. For exampe, we can read from Tabe 13.1 that there are 40 nondefauting debtors who were assigned into rating category 4 by Rating 1 whie they were assigned into rating category 5 by Rating 2. The other numbers are interpreted anaogousy. The distribution of the defauting debtors in the two rating systems B. Engemann Independent Consutant, e-mai: bernd.engemann@quantsoutions.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_13, # Springer-Verag Berin Heideberg 2011 269

270 B. Engemann Tabe 13.1 Distribution of the non-defauting debtors in Rating 1 and Rating 2 Rating 1 1 2 3 4 5 Tota Rating 2 1 90 60 15 10 5 180 2 45 90 30 20 15 200 3 10 35 100 45 20 210 4 5 10 30 100 70 215 5 0 5 10 40 90 145 Tota 150 200 185 215 200 Tabe 13.2 Distribution of the defauting debtors in Rating 1 and Rating 2 Rating 1 1 2 3 4 5 Tota Rating 2 1 20 5 0 3 0 28 2 4 7 0 0 0 11 3 3 0 2 0 0 5 4 0 0 0 2 2 4 5 0 2 0 0 0 2 Tota 27 14 2 5 2 is given in Tabe 13.2. Both tabes provide a information needed to appy the concepts that wi be introduced in the subsequent sections of this chapter. We introduce the notation that we wi use throughout this chapter. We assume a rating system which consists of discrete rating categories. The rating categories 1 are denoted with R 1,...,R k where we assume that the rating categories are sorted in increasing credit quaity, i.e., the debtors with worst credit quaity are assigned to R 1 whie the debtors with the best credit quaity are assigned to R k. In our exampe in Tabes 13.1 and 13.2 we have k ¼ 5 and R 1 ¼ 1,...,R 5 ¼ 5. We denote the set of defauting debtors with D, the set of non-defauting debtors with ND, and the set of a debtors with T. The number of debtors in the rating category R i is denoted with N(i) where the subscript refers to the group of debtors. If we discuss a specific rating we make this cear by an additiona argument, e.g., for Rating 1 the number of defauters in rating category 4 is N D (4;1) ¼ 5, or the tota number of debtors in rating category 2 is N T (2;1) ¼ 214. Since the event Defaut or Non-defaut of a debtor is random, we have to introduce some random variabes. With S we denote random distribution of rating scores whie the subscript wi indicate the group of debtors the distribution function corresponds to, e.g., S D denotes the distribution of the rating scores of the defauting debtors. The empirica distribution of the rating scores, i.e., the distribution of the rating scores that is reaised by the observed defauts and non-defauts is denoted by ^S, where the subscript again refers to the group of debtors. For exampe, for Rating 1 1 The terminoogy rating category or rating score is used interchangeaby throughout this chapter.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 271 ^S D ð3;1þ ¼2=50 ¼ 0:04; ^S ND ð2;1þ ¼200=950 ¼ 0:21; ^S T ð5;1þ ¼202=1000 ¼ 0:20: The cumuative distribution of S is denoted with C, i.e., C(R i ) is the probabiity that a debtor has a rating score ower than or equa to R i. The specific group of debtors the distribution function is referring to is given by the corresponding subscript. The empirica cumuative distribution function is denoted by ^C, e.g., the empirica probabiity that a non-defauting debtor s rating score under Rating 2 is ower than or equa to 4 is given by ^C ND ð4;2þ ¼ð180 þ 200 þ 210 þ 215Þ=950 ¼ 0:847: Finay, we define the common score distribution of two rating systems Rating 1 and Rating 2 by S 12. The expression S 12 (R i,r j ) gives the probabiity that a debtor has rating score R i under Rating 1 and a rating score R j under Rating 2. Again the index D, ND, T refers to the set of debtors to which the score distribution corresponds. The cumuative distribution is denoted with C 12, i.e., C 12 (R i,r j ) gives the probabiity that a debtor has a rating score ess than or equa to R i under Rating 1 and ess than or equa to R j under Rating 2. Again, exampes are given for the corresponding empirica distributions using the data of Tabes 13.1 and 13.2: ^S 12 D ð2;2þ ¼7=50 ¼ 0:14; ^S 12 NDð2;4Þ ¼10=950 ¼ 0:0105; ^C 12 D ð2;3þ ¼ð20 þ 5 þ 4 þ 7 þ 3 þ 0Þ=50 ¼ 0:78: Having defined the notation, we give a short outine of this chapter. In Sect. 13.2 we wi define the measures, Cumuative Accuracy Profie (CAP) and Receiver Operating Characteristic (ROC), which are the most popuar in practice and show how they are interreated. In Sect. 13.3 we wi focus on the statistica properties of the summary measures of the CAP and the ROC. The fina section discusses the appicabiity and the correct interpretation of these measures. 13.2 Measures of a Rating System s Discriminative Power We wi define the measures of discriminative power that are of interest to us in this section. We wi focus on the Cumuative Accuracy Profie (CAP) and the Receiver Operating Characteristic (ROC). These are not the ony measures described in the iterature but the most important and the most widey appied in practice. Exampes of measures that are not treated in this artice are entropy measures. We refer the

272 B. Engemann reader to Sobehart et a. (2000) for an introduction to these measures. Besides the basic definitions of the CAP and the ROC and their summary measures, we wi show how both concepts are connected and expore some extensions in this section. 13.2.1 Cumuative Accuracy Profie The definition of the Cumuative Accuracy Profie (CAP) can be found in Sobehart et a. (2000). It pots the empirica cumuative distribution of the defauting debtors ^C D against the empirica cumuative distribution of a debtors ^C T. This is iustrated in Fig. 13.1. For a given rating category R i, the percentage of a debtors with a rating of R i or worse is determined, i.e., ^C T ðr i Þ. Next, the percentage of defauted debtors with a rating score worse than or equa to R i, i.e., ^C D ðr i Þ, is computed. This determines the point A in Fig. 13.1. Competing this exercise for a rating categories of a rating system determines the CAP curve. Therefore, every CAP curve must start in the point (0, 0) and end in the point (1, 1). There are two specia situations which serve as imiting cases. The first is a rating system which does not contain any discriminative power. In this case, the CAP curve is a straight ine which haves the quadrant because if the rating system contains no information about a debtor s credit quaity it wi assign x% of the defauters among the x% of the debtors with the worst rating scores ( Random Mode in Fig. 13.1). The other extreme is a rating system which contains perfect information concerning the credit quaity of the debtors. In this case, a defauting debtors wi get a worse rating than the surviving debtors and the resuting CAP curve raises straight to one and stays there ( Perfect Forecaster in Fig. 13.1). Ĉ D 1 Perfect Forecaster a P Ĉ D (R i ) A a R Rating Mode Random Mode 0 0 Ĉ 1 T (R i ) Ĉ T Fig. 13.1 Iustration of cumuative accuracy profies

13 Measures of a Rating s Discriminative Power: Appications and Limitations 273 The information contained in a CAP curve can be summarised into a singe number, the Accuracy Ratio (AR) (this number is aso known as Gini Coefficient or Power Statistics). It is given by AR ¼ a R a P ; (13.1) where a R is the area between the CAP curve of the rating mode and CAP curve of the random mode (grey/back area in Fig. 13.1) and a P is the area between the CAP curve of the perfect forecaster and the CAP curve of the random mode (grey area in Fig. 13.1). The ratio AR can take vaues between zero and one. 2 The coser AR is to one, i.e., the more the CAP curve is to the upper eft, the higher is the discriminative power of a rating mode. We finish this subsection by cacuating the CAP curves of Rating 1 and Rating 2. Since both rating systems have five rating categories, we can compute four points of the CAP curve in addition to the points (0,0) and (1,1). To get a rea curve, the six points of each CAP curve have to be connected by straight ines. We iustrate this procedure for Rating 1. Starting at the eft, we have to compute ^C T ð1;1þ and ^C D ð1;1þ, which we get from Tabes 13.1 and 13.2 as ^C T ð1;1þ ¼177=1000 ¼ 0:177; ^C D ð1;1þ ¼27=50 ¼ 0:540: In the next step, we compute ^C T ð2;1þ and ^C D ð2;1þ which resuts in ^C T ð2;1þ ¼ð177 þ 214Þ=1000 ¼ 0:391; ^C D ð2;1þ ¼ð27 þ 14Þ=50 ¼ 0:820: The remaining points are computed anaogousy. The procedure for Rating 2 is simiar. The resuting CAP curves are iustrated in Fig. 13.2. We see that the CAP curve of Rating 1 is aways higher than the CAP curve of Rating 2, i.e., the discriminative power of Rating 1 is higher. This is aso refected in the AR vaues of both rating modes. For Rating 1, we find an AR of 0.523 whie for Rating 2, the AR is cacuated as 0.471. 13.2.2 Receiver Operating Characteristic The concept of the Receiver Operating Characteristic (ROC) was deveoped in signa detection theory, therefore the name. It was introduced to rating systems in Sobehart and Keenan (2001). The concept is iustrated in Fig. 13.3. This figure 2 In principe, AR coud be negative. This woud be the case when the ranking of the debtors by the rating system is wrong, i.e., the good debtors are assigned to the rating categories of the poor debtors.

274 B. Engemann Percentage of defauting debtors CAP Curves 1.00 0.80 0.60 0.40 0.20 0.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Percentage of tota debtors Rating 1 Rating 2 Random Rating Perfect Forecaster Fig. 13.2 CAP curves for Rating 1 and Rating 2 Frequency V Defauters Non-defauters Fig. 13.3 Rating score distributions for defauting and non-defauting debtors Rating Score shows the distributions of the rating scores for defauting and non-defauting debtors. It can be seen that the rating system has discriminative power since the rating scores are higher for surviving debtors. A cut-off vaue V provides a simpe decision rue to cassify debtors into potentia defauters and non-defauters. A debtors with a rating score ower than V are considered as defauters whie a debtors with a rating score higher than V are treated as non-defauters. Under this decision rue four scenarios can occur which are summarised in Tabe 13.3.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 275 If a debtor with a rating score beow V defauts, the rating system s prediction was correct. We ca the fraction of correcty forecasted defauters the hit rate. The same is true for non-defauters with a rating score above V. In this case, a nondefauter was predicted correcty. If a non-defauter has a rating score beow V, the decision was wrong. The rating system raised a fase aarm. The fourth and fina case is a defauter with a rating score above V. In this case the rating system missed a defauter and made a wrong prediction. For a given cut-off vaue V, a rating system shoud have a high hit rate and a ow fase aarm rate. The Receiver Operating Characteristic curve is given by a pairs (fase aarm rate, hit rate), which are computed for every reasonabe cut-off vaue. It is cear that the ROC curve starts in the point (0, 0) and ends in the point (1, 1). If the cut-off vaue ies beow a feasibe rating scores both the hit rate and the fase aarm rate is zero. Simiary, if the cut-off vaue is above a feasibe rating scores, the hit rate and the fase aarm rate are equa to one. The concept of the ROC curve is iustrated in Fig. 13.4 beow. In our setting, the cut-off points V are defined by the rating categories. Therefore, we get in tota k-1 cut-off points. Consider the point B in Fig. 13.4. To compute this point we define the decision rue: A debtor is cassified as a defauter if he has a rating of R i or worse, otherwise he is cassified as a non-defauter. Under this Tabe 13.3 Outcomes of the simpe cassification rue using the cut-off vaue V Defaut Non-defaut Rating score Beow cut-off vaue Correct prediction (hit) Wrong prediction (fase aarm) Above cut-off vaue Wrong prediction (error) Correct prediction (correct rejection) Ĉ D Rating Mode Perfect Forecaster 1 Random Mode Ĉ D (R i ) B AUROC 0 0 Ĉ 1 ND (R i ) Ĉ ND Fig. 13.4 Iustration of receiver operating characteristic curves

276 B. Engemann decision rue, the hit rate is given by ^C D ðr i Þ, which is the fraction of a defauters with a rating of R i or worse. Simiary, the fase aarm rate is given by ^C ND ðr i Þ, which is the fraction of a non-defauters with a rating of R i or worse. The ROC curve is obtained by computing these numbers for a rating categories. Again, we have the two imiting cases of a random mode and the perfect forecaster. In the case of a random mode where the rating system contains no discriminative power, the hit rate and the fase aarm rate are equa regardess of the cut-off point. In the case of the perfect forecaster, the rating scores distributions of the defauters and the non-defauters of Fig. 13.3 are separated perfecty. Therefore, for every vaue of the hit rate ess than one the fase aarm rate is zero and for every vaue of the fase aarm rate greater than zero, the hit rate is one. The corresponding ROC curve connects the three points (0, 0), (0, 1), and (1, 1) by straight ines. Simiar to the CAP curve, where the information of the curve was summarized in the Accuracy Ratio, there is aso a summary statistic for the ROC curve. It is the area beow the ROC curve (AUROC). This statistic can take vaues between zero and one, where the AUROC of the random mode is 0.5 and the AUROC of the perfect forecaster is 1.0. The coser the vaue of AUROC is to one, i.e., the more the ROC curve is to the upper eft, the higher is the discriminative power of a rating system. 3 We appy the concept of the ROC curve to the exampe in Tabes 13.1 and 13.2. We proceed in the same way as in the previous subsection, when we computed the CAP curve. Since we have five rating categories, we can define four decision rues in tota which gives us four points in addition to the points (0, 0) and (1, 1) on the ROC curve. To get a curve, the points have to be connected by straight ines. We compute the second point of the ROC curve for Rating 2 to iustrate the procedure. The remaining points are computed in an anaogous way. Consider the decision rue that a debtor is cassified as a defauter if he has a rating of 2 or worse and is cassified as a non-defauter if he has a rating higher than 2. The corresponding hit rate is computed as ^C D ð2;2þ ¼ð28 þ 11Þ=50 ¼ 0:78; whie the corresponding fase aarm rate is given by ^C ND ð2;2þ ¼ð180 þ 200Þ=950 ¼ 0:40: The remaining points on the ROC curve of Rating 2 and Rating 1 are computed in a simiar fashion. The ROC curves of Rating 1 and Rating 2 are iustrated in Fig. 13.5. Computing the area beow the ROC curve, we get a vaue of 0.762 for Rating 1 and 0.735 for Rating 2. 3 A rating system with an AUROC cose to zero aso has a high discriminative power. In this case, the order of good and bad debtors is reversed. The good debtors have ow rating scores whie the poor debtors have high ratings.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 277 1.00 ROC Curves 0.80 Hit Rate 0.60 0.40 0.20 0.00 0.00 0.20 0.40 0.60 0.80 1.00 Fase Aarm Rate Rating 1 Rating 2 Random Rating Perfect Forecaster Fig. 13.5 ROC curves for Rating 1 and Rating 2 We finish this subsection by exporing the connection between AR and AUROC. We have seen that the CAP curve and the ROC curve are computed in a simiar way. In fact, it can be shown that both concepts are just different ways to represent the same information. In Appendix A, we proof the simpe reation between AR and AUROC AR ¼ 2 AUROC 1: (13.2) From a practica perspective, both concepts are equivaent and it is a question of preference as to which one is used to evauate the discriminative power of a rating system. In Sect. 13.3, we wi see that AUROC aows for an intuitive probabiistic interpretation which can be used to derive various statistica properties of AUROC. By (13.2) this interpretation carries over to AR. However, it is ess intuitive in this case. 13.2.3 Extensions CAP curves and ROC curves ony aow a meaningfu evauation of some rating function s abiity to discriminate between good and bad if there is a inear reationship between the function s vaue and the attributes good and bad. This is iustrated in Fig. 13.6. The figure shows a situation where the rating is abe to discriminate perfecty between defauters and survivors. However, the score distribution of the defauters is bimoda. Defauters have either very high or very ow score vaues. In practice, when designing corporate ratings, some baance sheet variabes ike growth in saes have this feature.

278 B. Engemann Frequency Defauters Non-defauters Rating Score Fig. 13.6 Score distribution of a non-inear rating function 1 ROC Curves Hit Rate 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fase Aarm Rate Rating System Perfect Forecaster Random Rating Fig. 13.7 ROC curve corresponding to the score distribution of Fig. 13.6 A straight forward appication of the ROC concept to this situation resuts in a miseading vaue for AUROC. The ROC curve which corresponds to the rating distribution of Fig. 13.6 is shown in Fig. 13.7. It can be seen that the AUROC corresponding to the score distribution in Fig. 13.6 is equa to 0.5. In spite of the rating system s abiity to discriminate perfecty between defauters and non-defauters, its AUROC is the same as the AUROC of a rating system without any discriminative power. This is due to the non-inearity in the reationship between the rating score and credit quaity of the debtors. To obtain meaningfu measures of discriminatory power aso in this situation, Lee and Hsiao (1996) and Lee (1999) provide severa extensions to the AUROC measure we have introduced in Sect. 13.2.2. We discuss ony one of these extensions, the one which coud be most usefu in a rating context.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 279 Lee (1999) proposes a simpe modification to the ROC concept which deivers meaningfu resuts for score distributions as iustrated in Fig. 13.6. For each rating category the ikeihood ratio L is computed as LðR i Þ¼ S DðR i Þ S ND ðr i Þ : (13.3) The ikeihood ratio is the ratio of the probabiity that a defauter is assigned to rating category R i to the corresponding probabiity for a non-defauter. To iustrate this concept, we compute the empirica ikeihood ratio ^L which is defined as ^LðR i Þ¼ ^S D ðr i Þ ^S ND ðr i Þ ; (13.4) for the rating systems Rating 1 and Rating 2. The resuts are given in Tabe 13.4. In the next step, the ikeihood ratios are sorted from the highest to the east. Finay, the ikeihood ratios are inverted to define a new rating score. 4 In doing so, we have defined a new rating score that assigns ow score vaues to ow credit quaity. The crucia point in this transformation is that we can be sure that after the transformation, ow credit quaity corresponds to ow score vaues even if the origina data ooks ike the data in Fig. 13.6. We compute the ROC curves for the new rating score. They are given in Fig. 13.8. Note that there is no difference to the previous ROC curve for Rating 2 because the sorting of the ikeihood ratios did not change the order of the rating scores. However, there is a difference for Rating 1. The AUROC of Rating 1 has increased sighty from 0.7616 to 0.7721. Furthermore, the ROC curve of Rating 1 Tabe 13.4 Empirica ikeihood ratios for Rating 1 and Rating 2 Rating category 1 2 3 4 5 Rating 1 ^S D ðr i ;1Þ 0.54 0.28 0.04 0.10 0.04 ^S ND ðr i ;1Þ 0.16 0.21 0.19 0.23 0.21 ^LðR i ;1Þ 3.42 1.33 0.21 0.44 0.19 Rating 2 ^S D ðr i ;2Þ 0.56 0.22 0.10 0.08 0.04 ^S ND ðr i ;2Þ 0.19 0.21 0.22 0.23 0.15 ^LðR i ;2Þ 2.96 1.05 0.45 0.35 0.26 4 The inversion of the ikeihood ratios is not necessary. We are doing this here just for didactica reasons to ensure that ow credit quaity corresponds to ow rating scores throughout this chapter.

280 B. Engemann 1.00 ROC Curves (after Transformation of Scores) 0.80 Hit Rate 0.60 0.40 0.20 0.00 0.00 0.20 0.40 0.60 0.80 1.00 Fase Aarm Rate Rating 1 (LR) Random Rating Rating 2 (LR) Perfect Forecaster Fig. 13.8 ROC curves for the transformed rating scores of Rating 1 and Rating 2 is concave everywhere after the transformation. As pointed out by Tasche (2002), the non-concavity of a ROC curve is a cear sign that the rating mode does not refect the information contained in the data in an optima way. With this simpe transformation, the quaity of the rating mode can be improved. A practica probem in the construction of rating modes is the incusion of variabes that are non-inear in the credit quaity of debtors (e.g., Fig. 13.6). As pointed out in Chap. 2, these variabes can offer a vauabe contribution to a rating mode provided that they are transformed prior to the estimation of the rating mode. There are severa ways to conduct this transformation. Computing ikeihood ratios and sorting them as was done here is a feasibe way of producing inear variabes from non-inear ones. For further detais and an exampe with rea data, refer to Engemann et a. (2003b). 13.3 Statistica Properties of AUROC In this section we wi discuss the statistica properties of AUROC. We focus on AUROC because it can be interpreted intuitivey in terms of a probabiity. Starting from this interpretation we can derive severa usefu expressions which aow the computation of confidence intervas for AUOC, a rigorous test if a rating mode has any discriminative power at a, and a test for the difference of two rating systems AUROC. A resuts that are derived in this section carry over to AR by appying the simpe reation (13.2) between AR and AUROC.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 281 13.3.1 Probabiistic Interpretation of AUROC The cumuative distribution function of a random variabe evauated at some vaue x, gives the probabiity that this random variabe takes a vaue which is ess than or equa to x. In our notation, this reads as C D ðr i Þ¼PðS D R i Þ; C ND ðr i Þ¼PðS ND R i Þ; (13.5) or in terms of the empirica distribution function ^C D ðr i Þ¼Pð^S D R i Þ; ^C ND ðr i Þ¼Pð^S ND R i Þ; (13.6) where P(.) denotes the probabiity of the event in brackets (.). In Appendix B, we show that AUROC can be expressed in terms of empirica probabiities as AUROC ¼ Pð ^S D < ^S ND Þþ 1 2 Pð^S D ¼ ^S ND Þ: (13.7) To get further insight, we introduce the Mann-Whitney statistic U as 1 X U ¼ u D;ND ; N D N ND ðd;ndþ 8 1; if ^s D < ^s ND >< 1 u D;ND ¼ 2 ; if ^s D ¼ ^s ND >: 0; if ^s D > ^s ND (13.8) where ^s D is a reaisation of the empirica score distribution ^S D and ^s ND is a reaisation of ^S ND. The sum in (13.8) is over a possibe pairs of a defauter and a non-defauter. It is easy to see that U ¼ Pð^S D < ^S ND Þþ 1 2 Pð ^S D ¼ ^S ND Þ; (13.9) what means the area beow the ROC curve and the Mann-Whitney statistic are measuring the same quantity. This gives us a very intuitive interpretation of AUROC. Suppose we draw randomy one defauter out of the sampe of defauters and one survivor out of the sampe of survivors. Suppose further we shoud decide from the rating scores

282 B. Engemann of both debtors which one is the defauter. If the rating scores are different, we woud guess that the debtor with the ower rating score is the defauter. If both scores are equa we woud toss a coin. The probabiity that we make a correct decision is given by the right-hand-side of (13.9), i.e., by the area beow the ROC curve. Throughout this artice, we have introduced a concepts and quantities with the data set given in Tabes 13.1 and 13.2. However, the data set of Tabes 13.1 and 13.2 is ony one particuar reaisation of defauts and survivas from the underying score distributions which are unknown. It is not the ony possibe reaisation. In principe, other reaisations of defauts coud occur which ead to different vaues for AUROC and U. These different possibe vaues are dispersed about the expected vaues of AUROC and U that are given by E½AUROCŠ ¼E½UŠ ¼PðS D < S ND Þþ 1 2 PðS D ¼ S ND Þ: (13.10) To get a feeing of how far the reaised vaue of AUROC deviates from its expected vaue, confidence intervas have to be computed. This is done in the next subsection. Finay, we remark that the AR measure can aso be expressed in terms of probabiities. Appying (13.2) we find E½ARŠ ¼PðS D < S ND Þ PðS D > S ND Þ: (13.11) The expected vaue of AR is the difference between the probabiity that a defauter has a ower rating score than a survivor and the probabiity that a defauter has a higher rating score than a survivor. It is not so cear how to give an intuitive interpretation of this expression. 13.3.2 Computing Confidence Intervas for AUROC To get a feeing for the accuracy of a measure obtained from a data sampe, it is customary to state confidence intervas to a confidence eve a, e.g., a ¼ 95%. In the first papers on the measurement of the discriminative power of rating systems, confidence intervas were aways computed by bootstrapping. 5 These papers mainy used the measure AR. Bootstrapping requires the drawing of ots of portfoios with repacement from the origina portfoio. For each portfoio, the AR has to be computed. From the resuting distribution of the AR vaues, confidence intervas can be computed. The main drawback of this method is its computationa inefficiency. 5 Efron and Tibshirani (1998) is a standard reference for this technique.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 283 A more efficient method is based on the appication of we-known properties of the Mann-Whitney statistic introduced in (13.8). The connection between AR and a sighty modified Mann-Whitney statistic is ess obvious 6 than for AUROC which might be the reason for the inefficient techniques that were used in those eary papers. From mathematica statistics it is known that an unbiased estimator of the variance s 2 U of the Mann-Whitney statistic U in (13.8) is given by ^s 2 U ¼ 1 4 ðn D 1ÞðN ND 1Þ : ^P D6¼ND þðn D 1Þ ^P D; D; ND þðn ND 1Þ ^P ND; ND; D 4 ðn D þ N ND 1ÞðU 0:5Þ 2 ; (13.12) where ^P D6¼ND, ^P D;D;ND, and ^P ND;ND;D are estimators for the probabiities P(S D 6¼S ND ), P D,D,ND, and P ND,ND,D where the atter two are defined as P D; D; ND ¼ PS D;1 ; S D;2 < S ND PSD;1 < S ND < S D;2 PS D;2 < S ND < S D;1 þ PSND < S D;1 ; S D;2 ; (13.13) P ND; ND; D ¼ PS ND;1 ; S ND;2 < S D PSND;1 < S D < S ND;2 PS ND;2 < S D < S ND;1 þ PSD < S ND;1 ; S ND;2 ; where S D,1 and S D,2 are two independent draws of the defauter s score distribution and S ND,1 and S ND,2 are two independent draws of the non-defauter s score distribution. Using (13.12) confidence intervas can be easiy computed using the asymptotic reationship AUROC E½AUROCŠ ^s U N D ; N ND!1! Nð0; 1Þ: (13.14) The corresponding confidence intervas to the eve a are given by AUROC ^s U F 1 1 þ a 2 ; AUROC þ ^s U F 1 1 þ a 2 ; (13.15) where F denotes the cumuative distribution function of the standard norma distribution. The asymptotic reation (13.14) is vaid for arge numbers N D and N ND. The question arises how many defauts a portfoio must contain to make the asymptotic vaid. In Engemann et a. (2003a, b) a comparison between (13.14) and 6 In (13.8) the ½ has to be repaced by 0, and the 0 has to be repaced by 1 to get the corresponding Mann-Whitney statistic for the AR.

284 B. Engemann bootstrapping is carried out. It is shown that for 50 defauts a very good agreement between (13.14) and bootstrapping is achieved. But even for sma numbers ike 10 or 20 reasonabe approximations for the bootstrapping resuts are obtained. We finish this subsection by computing the 95% confidence interva for the AUROC of our exampes Rating 1 and Rating 2. We start with Rating 1. First we compute ^P D6¼ND. It is given by the fraction of a pairs of a defauter and a survivor with different rating scores. It is computed expicity as! ^P D6¼ND ¼ 1 ^P D¼ND ¼ 1 X5 ^S D ði;1þ ^S ND ði;1þ i¼1 ¼ 1 ð2 200 þ 5 215 þ 2 185 þ 14 200 þ 27 150Þ=ð50 950Þ¼0:817 The estimators for P D,D,ND and P ND,ND,D are more difficut to compute than for P D6¼ND. To estimate P D,D,ND it is necessary to estimate three probabiities, P(S D,1, S D,2 < S ND ), P(S D,1 < S ND <,S D,2 ) (which is equa to P(S D,2 < S ND <,S D,1 )), and P(S ND < S D,1,S D,2 ). We iustrate the procedure for P(S D,1,S D,2 < S ND ). The other probabiities are computed anaogousy. A naïve way to compute P(S D,1,S D,2 < S ND ) is to impement a tripe oop, two oops over a defauters and one oop over a survivors. For each tripe, one has to check if the scores of both defauters are ess than the score of the survivor. The probabiity P(S D,1,S D,2 < S ND ) is then estimated as the number of tripes where this condition is fufied by the tota number of a tripes. However, this procedure is very time consuming when the number of survivors is arge. It is much more efficient to expoit the sorting of the debtors in their score vaues. We get the resuts X P ^S D;1 ; ^S D;2 < ^S 5 ND ¼ i¼2 X P ^S D;1 < ^S ND < ^S 5 D;2 ¼ i¼1 X P ^S ND < ^S D;1 ; ^S 4 D;2 ¼ i¼1 ^C 2 D ði 1Þ ^S ND ðiþ; ^C D ði 1Þ ^S ND ðiþ 1 ^C D ðiþ ; ^S ND ðiþ 1 ^C 2: D ðiþ Simiar estimation formuas can be derived for P(S ND,1,S ND,2 < S D ), P(S ND,1 < S D < S ND,2 ), and P(S D < S ND,1,S ND,2 ). Appying these formuas to the rating system Rating 1 we get Pð^S D;1 ; ^S D;2 < ^S ND ;1Þ ¼0:554; Pð ^S D;1 < ^S ND < ^S D;2 ;1Þ ¼0:051; Pð^S ND < ^S D;1 ; ^S D;2 ;1Þ ¼0:044; ^P D; D; ND ¼ 0:497; Pð ^S ND;1 ; ^S ND;2 < ^S D ;1Þ ¼0:069; Pð ^S ND;1 < ^S D < ^S ND;2 ;1Þ ¼0:046; Pð^S D <^S ND;1 ; ^S ND;2 ;1Þ ¼0:507; ^P ND; ND; D ¼ 0:483:

13 Measures of a Rating s Discriminative Power: Appications and Limitations 285 Finay, we have a ingredients for (13.12) and compute the variance of U as ^s 2 U ¼ 0:001131. Finay we compute the confidence interva to the eve 95% which resuts in [0.69573, 0.82754]. A simiar cacuation for Rating 2 eads to a 95% confidence interva of [0.66643, 0.80431]. We see that both confidence intervas are rather broad. This is due to the reativey ow number of debtors in our exampe rating systems. 13.3.3 Testing for Discriminative Power The 95% confidence intervas of the AUROC of Rating 1 and Rating 2 are far away from the vaue 0.5. This suggests that the discriminative power of both rating systems is statisticay significant. To confirm this we appy a rigorous statistica test. The nu hypothesis of our test is that a rating system does not contain any discriminative power. Under this nu hypothesis, (13.12) can be simpified consideraby. If a rating system has no discriminative power, the score distributions of the defauters and the survivors are identica. We get the identity PðS D 6¼ S ND Þ=3 ¼ PðS D;1 ; S D;2 < S ND Þ¼PðS D;1 < S ND < S D;2 Þ ¼ PðS ND < S D;1 ; S D;2 Þ¼PðS ND;1 ; S ND;2 < S D Þ ¼ PðS ND;1 < S D < S ND;2 Þ¼PðS D < S ND;1 ; S ND;2 Þ (13.16) This eads to the simpified formua for the variance of the Mann-Whitney statistic s 2 U ¼ PðD 6¼ NDÞð1 þ N D þ N ND Þ 12 ðn D 1ÞðN ND 1Þ (13.17) If we make a two-sided test the p-vaue of this test given by soving (13.15) for one minus the confidence eve a. This cacuation resuts in p-vaue ¼ 2 2 F U 0:5 : (13.18) ^s The appication of (13.18) with the variance of (13.17) eads to a p-vaue of 8.23 10 12 for Rating 1. The corresponding vaue for Rating 2 is 5.36 10 10. This means both rating systems have a highy significant discriminatory power. This confirms our conjecture at the beginning of this subsection. 13.3.4 Testing for the Difference of Two AUROCs Throughout this artice we aways considered two rating modes, Rating 1 and Rating 2. We have seen so far that Rating 1 has a sighty higher AUROC than

286 B. Engemann Rating 2. The question arises whether this difference is significant from a statistica point of view. To answer this question, we discuss a test on the difference of two AUROCs that was deveoped by DeLong et a. (1988). Comparing the confidence intervas of the AUROC of Rating 1 and Rating 2, we find that they overap widey. Therefore, we woud suppose that there is no significant difference between both AUROCs. However, when comparing confidence eves ony, we are negecting correations between both AUROCs. To carry out a rigorous statistica test, we need in addition to the variances of both AUROCs, the covariance between them. The estimator for the covariance is more compex than the estimator for the variance. It is given by ^s U1 ;U 2 ¼ 1 h ^P 12 D;D;ND;ND 4 ðn D 1ÞðN ND 1Þ þðn D 1Þ ^P 12 D;D;ND þðn ND 1Þ ^P 12 ND;ND;D 4 ðn D þ N ND 1ÞðU 1 0:5ÞðU 2 0:5Þ ; (13.19) where ^P 12 D;D;ND;ND, ^P 12 D;D;ND, and ^P 12 ND;ND;D P 12 D;D;ND;ND, P12 D;D;ND P 12, and P12 ND;ND;D are estimators for the probabiities which are defined as D;D;ND;ND ¼ PS1 D > S1 ND ; S2 D > S2 ND ÞþPðS1 D < S1 ND ; S2 D < S2 ND PS 1 D > S1 ND ; S2 D < S2 ND PðS 1 D < S 1 ND ; S2 D > S2 ND Þ; P 12 D;D;ND ¼ P S1 D;1 > S1 ND ; S2 D;2 > S2 ND þ P S 1 D;1 < S1 ND ; S2 D;2 < S2 ND P S 1 D;1 > S1 ND ; S2 D;2 < S2 ND P S 1 D;1 < S1 ND ; S2 D;2 > S2 ND ; P 12 ND;ND;D ¼ P S1 D > S1 ND;1 ; S2 D > S2 ND;2 ÞþPðS1 D < S1 ND;1 ; S2 D < S2 ND;2 P S 1 D > S1 ND;1 ; S2 D < S2 ND;2 PðS 1 D < S1 ND;1 ; S2 D > S2 ND;2 Þ; i (13.20) where the quantities S i D, Si D;1, Si D;2 are independent draws from the score distribution of the defauters. The index i indicates whether the score of Rating 1 or of Rating 2 has to be taken for this defauter. The meaning of S i ND, Si ND;1, Si ND;2 is anaogous for the score distributions of the of non-defauters under Rating 1 and Rating 2. Under the nu hypothesis that both AUROCs are equa it is shown in DeLong et a. (1988) that the test statistic T which is defined as T ¼ ðu 1 U 2 Þ 2 ^s 2 U 1 þ ^s 2 U 2 2^s U1 ;U 2 : (13.21) is asymptoticay w 2 distributed with one degree of freedom. This asymptotic reationship aows us the computation of critica vaues given a confidence eve a.

13 Measures of a Rating s Discriminative Power: Appications and Limitations 287 We finish this section by computing the p-vaue of the test for difference of the two AUROCs of Rating 1 and Rating 2. The variances for both AUROC vaues have aready been computed in Sect. 13.3.2. It remains to compute the covariance between both AUROCs. We show expicity how to compute estimators for P 12 D;D;ND;ND, and P12 D;D;ND. The estimator for P12 ND;ND;D is computed in a simiar way as for P 12 D;D;ND. We start with the computation of ^P 12 D;D;ND;ND. To compute this estimator, four probabiities have to be cacuated from the sampe. Consider the probabiity PS 1 D > S1 ND ; S2 D > S2 ND. A naïve way to cacuate this probabiity woud be to impement a oop over a defauters and a oop over a survivors. This probabiity is then given by the fraction of a pairs where both Rating 1 and Rating 2 assign a higher rating score to the defauter. This can be done in a more efficient way by using the sorting of debtors in score vaues. The four probabiities needed for the computation of ^P 12 D;D;ND;ND can be cacuated by P ^S 1 D > ^S 1 ND ; ^S 2 D > X ^S 5 X 5 2 ND ¼ i¼1 j¼1 P ^S 1 D < ^S 1 ND ; ^S 2 D < X ^S 5 X 5 2 ND ¼ i¼1 j¼1 P ^S 1 D > ^S 1 ND ; ^S 2 D < X ^S 5 X 5 2 ND ¼ i¼1 j¼1 P ^S 1 D < ^S 1 ND ; ^S 2 D > X ^S 5 X 5 2 ND ¼ i¼1 j¼1 ^S 12 X5 NDði; jþ X 5 k¼iþ1 ¼jþ1 X ^S 12 NDði; jþxi 1 j 1 k¼1 ¼1 ^S 12 X5 X j 1 NDði; jþ k¼iþ1 ¼1 ^S 12 NDði; jþxi 1 k¼1 X 5 ¼jþ1 ^S 12 D ðk; Þ; ^S 12 D ðk; Þ ^S 12 D ðk; Þ ^S 12 D ðk; Þ Evauating these formuas with the data of Tabe 13.1 eads to P ^S 1 D > ^S 1 ND ; ^S 2 D > ^S 2 ND ¼ 0:0747; P ^S 1 D < ^S 1 ND ; ^S 2 D < ^S 2 ND ¼ 0:5314; P ^S 1 D > ^S 1 ND ; ^S 2 D < ^S 2 ND ¼ 0:0357; P ^S 1 D < ^S 1 ND ; ^S 2 D > ^S 2 ND ¼ 0:0506; ^P 12 D;D;ND;ND ¼ 0:5197 In the next step we consider the estimation of P 12 D;D;ND. Again, four probabiities have to be estimated. A naïve way to estimate for instance, the probabiity P S 1 D;1 > S1 ND ; S2 D;2 > S2 ND is the impementation of a tripe oop, two oops over the defauters and one oop over the survivors. This probabiity is then estimated as the fraction of a tripes where the first defauter has a higher rating score than the survivor under Rating 1 and the second defauter has a higher score than the survivor under Rating 2. A more efficient procedure is given by the formuas

288 B. Engemann P ^S 1 D;1 > ^S 1 ND ; ^S 2 D;2 > ^S 2 ND ¼ X5 X 5 i¼1 i¼1 P ^S 1 D;1 < ^S 1 ND ; ^S 2 D;2 < ^S 2 ND ¼ X5 i¼1 X 5 i¼1 P ^S 1 D;1 > ^S 1 ND ; ^S 2 D;2 < ^S 2 ND ¼ X5 X 5 i¼1 i¼1 P ^S 1 D;1 < ^S 1 ND ; ^S 2 D;2 > ^S 2 ND ¼ X5 X 5 i¼1 i¼1 ^S 12 ND ði; jþ 1 ^C D ði;1þ 1 ^C D ðj;2þ ; ^S 12 ND ði; jþ ^C D ði 1;1Þ^C D ðj 1;2Þ; ^S 12 ND ði; jþ 1 ^C D ði;1þ ^C D ðj 1;2Þ; ^S 12 ND ði; jþ ^C D ði 1;1Þ 1 ^C D ðj;2þ : An appication of these formuas to the data of Tabe 13.1 eads to P ^S 1 D;1 > ^S 1 ND ; ^S 2 D;2 > ^S 2 ND ¼ 0:0389; P ^S 1 D;1 > ^S 1 ND ; ^S 2 D;2 < ^S 2 ND ¼ 0:0620; ^P 12 D;D;ND ¼ 0:3930: P ^S 1 D;1 < ^S 1 ND ; ^S 2 D;2 < ^S 2 ND ¼ 0:4949; P ^S 1 D;1 < ^S 1 ND ; ^S 2 D;2 > ^S 2 ND ¼ 0:0790; A simiar cacuation for P 12 ND;ND;D eads to P12 ND;ND;D ¼ 0:3534. Taking everything together and evauating (13.21) eads to T ¼ 0.57704. This corresponds to a p-vaue of 0.4475. This means that the difference in the AUROC vaues of Rating 1 and Rating 2 is not statisticay significant. This resut is not surprising given the ow number of debtors. 13.4 Correct Interpretation of AUROC In this section we want to give some guideines on how to interpret AUROC vaues. 7 When discussing rating systems, one is often confronted with the opinion that a good rating system shoud have some minimum vaue for the AUROC. Sometimes peope are happy that their rating system has a higher AUROC than the rating mode of others or a company wants to achieve an AUROC of x% during the next 5 years for its rating systems. In this section we expain why a these opinions and goas are unreasonabe. Consider a hypothetica portfoio with identica debtors ony, e.g., a portfoio of companies with identica baance sheets. No rating mode has a chance to discriminate anything in this situation because there is nothing to discriminate. This means that the AUROC does not depend on the rating mode ony, but aso the portfoio. 7 See aso Bochwitz et a. (2005).

13 Measures of a Rating s Discriminative Power: Appications and Limitations 289 This can be proven formay. Hamere et a. (2003) show that for a portfoio of N debtors the expected AUROC is given by EðAUROCÞ¼ 0:5 2 1 PD P NT 2 PD ð1 PD 1 þ 2 PD 2 þþn T PD NT Þ PD P 1 P N T (13.22) where debtor i has a defaut probabiity PD i and the average defaut probabiity of the portfoio is denoted with PD P. Furthermore, it is assumed that the debtors are sorted from the worst credit quaity to the best. A further exampe is provided. Consider two rating systems with two rating categories for different portfoios. They are given in Tabe 13.5. Tabe 13.5 Two rating systems on different portfoios Rating category 1 2 Rating A Number of debtors 500 500 Defaut probabiity 1% 5% Rating B Number of debtors 500 500 Defaut probabiity 1% 20% We assume that both rating modes are perfect, i.e., they assign the correct defaut probabiity to each debtor. Then we find for the expected AUROC vaues E½AUROC A Š ¼ 0:6718; E½AUROC B Š ¼ 0:7527: We see that there is a huge difference in the AUROC vaues in spite of the fact that both ratings are perfect. This demonstrates that a comparison of AUROC vaues for different portfoios is meaningess. The same appies to a comparison of the AUROC on the same portfoio in different time points. Because of changes in the portfoio structure over time, i.e., changes in the defaut probabiities of the debtors, the rating mode is being compared on different portfoios. However, this anaysis coud be hepfu in spite of this. If the AUROC of a rating mode worsens over time, one shoud find out if this is due to changes in the portfoio structure or if the quaity of the rating mode has indeed deteriorated and a new estimation is necessary. We concude that a comparison of the AUROC of two rating modes is meaningfu ony if it is carried out on the same portfoio at the same time. It does not make sense to compare AUROCs over different portfoios or to try to achieve a target AUROC. As demonstrated in the exampe in Tabe 13.5, achieving a higher AUROC coud require the incusion of more poor debtors into the portfoio, a business strategy not every credit institution might want to foow.

290 B. Engemann Appendix A. Proof of (13.2) We introduce the shortcut notation ^C i D ¼ ^C D ðr i Þ, ^C i ND and ^C i T have a simiar meaning. Furthermore, we denote the sampe defaut probabiity by ^p. Note that ^C i T can be written in terms of ^C i ND and ^C i D as ^C i T ¼ ^p ^C i D þ ð1 ^p Þ ^C i ND : (13.23) By computing simpe integras, we find for AUROC, a R þ 0.5, and a P the expressions AUROC ¼ Xk i¼1 a R þ 0:5 ¼ Xk i¼1 0:5 ^C i D þ ^C D i 1 ^C i ND ^C ND i 1 ; 0:5 ^Ci D þ ^Ci T ; a P ¼ 0:5 ð1 ^p Þ: ^C i 1 D ^C i 1 T (13.24) Pugging (13.23) into the expression for a R þ 0:5 and simpifying eads to a R þ 0:5 ¼ Xk i¼1 ¼ Xk i¼1 0:5 ^C i D þ ^C D i 1 ^C i T ^C T i 1 0:5 ^Ci D þ ^p ^Ci D ¼ ð1 ^p Þ Xk i¼1 þ ^p Xk i¼1 ^C i 1 D 0:5 ^C i D þ ^C i 1 D ^C i 1 D ^C i ND ^C i 1 0:5 ^C i D þ ^C D i 1 ^C i D ^C D i 1 þ ð1 ^p Þ ^Ci ND ¼ ð1 ^p ÞAUROC þ 0:5 ^p Xk ^C i 2 2 D ^Ci 1 D i¼1 ¼ ð1 ^p ÞAUROC þ 0:5 ^p: ND ^C i 1 ND (13.25) Taking (13.24) and (13.25) together eads to the desired resut AR ¼ a R ð ¼ 1 ^p Þ ð AUROC 0:5 Þ ¼ 2 AUROC 1: a P 0:5 ð1 ^p Þ

13 Measures of a Rating s Discriminative Power: Appications and Limitations 291 Appendix B. Proof of (13.7) Using the same shortcut notation as in Appendix A, we get AUROC ¼ Xk i¼1 ¼ Xk i¼1 ¼ Xk i¼1 ¼ Xk i¼1 0:5 ^C i D þ ^C D i 1 ^C i ND ^C ND i 1 0:5 P ^S D R i þ P ^S D R i 1 P ^S ND ¼ R i P ^S D R i 1 þ 0:5 P ^S D ¼ R i P ^S ND ¼ R i P ^S D R i 1 X P ^S k ND ¼ R i þ 0:5 ¼ P ^S D < ^S ND þ 0:5 P ^S D ¼ ^S ND which proves (13.7). i¼1 P ^S D ¼ R i P ^S ND ¼ R i References Bochwitz S, Hamere A, Hoh S, Rauhmeier R, R osch D (2005), Myth and Reaity of Discriminatory Power for Rating Systems, Wimott Magazine, January, pp. 2 6. DeLong E, DeLong D, Carke-Pearson D (1988), Comparing the Areas under Two or More Correated Receiver Operating Characteristic Curves: A Nonparametric Approach, Biometrics 44, pp. 837 845. Efron B, Tibshirani RJ (1998), An Introduction to the Bootstrap, Chapman & Ha, Boca Raton, FL. Engemann B, Hayden E, Tasche D (2003a), Testing Rating Accuracy, Risk 16 (1), pp. 82 86. Engemann B, Hayden E, Tasche D (2003b), Measuring the Discriminative Power of Rating Systems, Working Paper. http://www.bundesbank.de/downoad/bankenaufsicht/dkp/200301dkp_b.pdf. Hamere A, Rauhmeier R, R osch D (2003), Uses and Misuses of Measures for Credit Rating Accuracy, Working Paper. Lee WC, Hsiao CK (1996), Aternative Summary Indices for the Receiver Operating Characteristic Curve, Epidemioogy 7, pp. 605 611. Lee WC (1999), Probabiistic Anaysis of Goba Performances of Diagnostic Tests: Interpreting the Lorenz Curve-Based Summary Measures, Statistics in Medicine 18, pp. 455 471. Sobehart JR, Keenan SC (2001), Measuring Defaut Accuratey, Risk 14, pp. S31 S33. Sobehart JR, Keenan SC, Stein RM (2000), Benchmarking Quantitative Defaut Risk Modes: A Vaidation Methodoogy, Moody s Investors Service. Swets JA (1988), Measuring the Accuracy of Diagnostic Systems, Science 240, pp. 1285 1293. Tasche D (2002), Remarks on the Monotonicity of Defaut Probabiities, Working Paper. http:// www-m4.ma.tum.de/pers/tasche/monoton.pdf.

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Chapter 14 Statistica Approaches to PD Vaidation Stefan Bochwitz, Marcus R.W. Martin, and Carsten S. Wehn 14.1 Introduction When deveoping an interna rating system, besides its caibration, the vaidation of the respective rating categories and associated probabiities of defaut pays an important roe. To have a vaid risk estimate and aocate economic capita efficienty, a credit institution has to be sure of the adequacy of its risk measurement methods and of the estimates for the defaut probabiities. Additionay, the vaidation of rating grades is a reguatory requirement to become an interna ratings based approach bank (IRBA bank). We discuss different methods of vaidating estimates for probabiities of defauts (PDs). We start by outining various concepts used to estimate PDs and the assumptions in rating phiosophies incuding point-in-time and through-the-cyce approaches, a distinction necessary for a proper vaidation. Having discussed this, severa common statistica tests used for the vaidation of PD estimates are introduced. These tests incude the binomia test, the norma test and goodness-of-fit-type tests ike the w 2 -test. Aso, the incorporation of descriptive measures inked to density forecast methods is discussed. For every test, the question of respective quaity is raised. An aternative approach starts with the one factor mode and gives an intuitive vaidation too, the so-caed extended traffic ight approach. We concude with This chapter represents the persona opinions of the authors and cannot be considered as representing the views of the Deutsche Bundesbank, the University of Appied Sciences at Darmstadt or DekaBank. S. Bochwitz Deutsche Bundesbank e-mai: Stefan.Bochwitz@bundesbank.de M.R.W. Martin University of Appied Sciences, Darmstadt e-mai: marcus.martin@h-da.de C.S. Wehn (*) DekaBank e-mai: wehn@gmx.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_14, # Springer-Verag Berin Heideberg 2011 293

294 S. Bochwitz et a. a discussion of the approaches introduced, especiay with respect to possibe imitations for the use in practice and to their respective usefuness. 14.2 PDs, Defaut Rates, and Rating Phiosophy The meaning of Vaidation of PDs or backtesting in credit risk can be described quite simpy. In the words of the Base Committee on Banking Supervision, it is to compare reaized defaut rates with estimated PDs for each grade of a rating system and to assess the deviation between the observed defaut rates and the estimated PD, (cf. Base Committee on Banking Supervision (2004), } 501). Here, backtesting is defined as a statistica task which hopefuy can be soved with the existing means and toos. However, performing such backtesting in practice raises some issues. Before we discuss the statistica means we want to draw readers attention to some more genera aspects: Recognition of defauts: Vaidation of PDs is fundamenta in the recognition of defauts. A correct count of defauts is a necessary prerequisite for a correcty determined defaut rate, and the measurement of defaut events is the underying concept of risk for determining PDs. A defaut of a borrower, however, is not objective event. On the one hand, there is the fact that a reasonabe number of defauted borrowers seem to have a considerabe infuence on the timing of the credit defaut. On the other hand, there is the observation that decaring a borrower as defauted eaves room for judgement. Therefore, the definition of credit defaut is to a considerabe degree, subjective, and even the new Base framework retains this subjective eement as the basis of the IRBA. However, a forward-ooking focus and a imit of 90 days past due which is objective, is impemented into the definition of defaut, (cf. Base Committee on Banking Supervision (2004), }} 452 and 453). The requirement is that the definition of defaut with a its subjective eements has to be appied consistenty to guarantee that the concusions drawn from the vaidation of PDs are correct. Inferring from defaut rates to PDs: A common and widespread approach for credit risk is the appication of the aw of arge numbers, and to infer from the observed defaut rate the probabiity of defaut. An appication of the aw of arge numbers woud require that the defauts are independent and occur in the same distribution. This requirement cannot be seen to be fufied for different borrowers. To te it in a picture: The difference for the task of determining the probabiity of throwing a six is ike approximating this probabiity either by throwing the same dice 1,000 times and cacuating the ratio of sixes to the tota number of throws or throwing 1,000 dices once and cacuating the ratio of sixes to the number of dices thrown. In any case, a proper appication requires that borrowers are grouped into grades exhibiting simiar defaut risk characteristics. Thus, the vaidation of PDs in most cases is preceded by grouping the borrowers to grades with the same risk profie (for an exempary exception, the

14 Statistica Approaches to PD Vaidation 295 Spiegehater statistics, cf. Chap. 15) This is necessary even in the case of direct estimates of PD, when each borrower is assigned an individua PD. PDs in their context: An immediate consequence of the issues raised is that PDs have a meaning just in a certain context, namey in the portfoio. In our opinion, there is no such thing as an objective PD which can be measured with a rating system ike temperature can be measured with a thermometer. Let us assume we rate the same borrower with two different rating systems: One with good discriminatory power resuting in different grades, which are assumed to be caibrated perfecty, and the other very simpe system assigning a borrowers to the same grade, caibrated with the portfoio PD. Appying these two rating systems to the same borrower woud resut in different PDs; either in the PD of the respective grade or in the portfoio PD. However, both systems can caim to be right and there is no method of deciding what the true PD of that borrower might be. The exampe works exacty the same for two rating systems with simiar discriminatory power and the same numbers of grades, providing both systems are caibrated with two different portfoios. Let us assume there is a subset of borrowers, which appears in both portfoios. If the remainder of the respective portfoios is different in terms of risk, then the same borrower in genera wi be assigned to grades with different PDs, and again, both systems can caim to be right. Rating phiosophy: Rating phiosophy is what is commony referred to as either point-in-time (PIT) or through-the-cyce (TTC) ratings. PIT-ratings measure credit risk given the current state of a borrower in its current economic environment, whereas TTC-ratings measure credit risk taking into account the (assumed) state of the borrower over a whoe economic cyce. PIT and TTC mark the ends of the spectrum of possibe rating systems. In practice, neither pure TTC nor pure PIT systems wi be found, but hybrid systems, which are rather PIT or rather TTC. Agency ratings are assumed to be TTC, whereas bank interna systems at east in most cases in Germany are ooked at as PIT. The underying rating phiosophy definitey has to be assessed before vaidation resuts can be judged, because the rating phiosophy is an important driver for the expected range for the deviation between PDs and defaut rates. Jafry and Schuermann (2004) have introduced the equivaent average migration as a too for assessing rating phiosophy. According to Jafry and Schuermann (2004), the rescaed Eucidean distance mobiity metric is equa to the average migration, which describes the average number of borrowers migrating from one rating grade to another grade. This average migration gives an impression at which end of the spectrum a rating system can be found, if it is 0, then the rating system has no migration at a a PIT system in its purest form if it is 1, then on average, no borrower stays in a rating grade. To eve off credit risk measurement for PIT systems as we as for TTC systems, the Base Committee has carified that estimation of PDs for reguatory purposes needs to incude a forward ooking eement (cf. Principe 1 of Newsetter No. 4, Base Committee on Banking Supervision 2005a). In practice, this woud mean that for reguatory purposes in respect of risk quantification of their grades, PIT and TTC systems are a bit coser.

296 S. Bochwitz et a. 14.3 Toos for Vaidating PDs This section is devoted to a brief overview on statistica tests that can be performed to vaidate the so-caed caibration of a rating system, i.e. the assignment of a probabiity of defaut (PD) to a certain rating grade or score vaue. In order to draw the right concusions, in most cases due to insufficient obigors or defauts to obtain reiabe statistica impications a purey statistica vaidation of a rating system is not sufficient to ensure the vaidity of the rating system. It has to be compemented by aternative quaitative approaches such as, e.g., shadow rating systems or pausibiity checks by credits experts (cf. OeNB/FMA (2004, pp. 94), or Base Committee on Banking Supervision 2004 and 2005b). Furthermore, we impicity assume that the vaidation of the rating system s discriminatory power and stabiity is to be aso checked by a vaidation procedure which shoud be part of an integrated process covering caibration, discriminatory power and stabiity of the rating system (cf. Bochwitz and Hoh (2007), Tasche (2005, pp. 32), or OeNB/FMA (2004) which aso incudes some numerica exampes). For various techniques for caibrating rating systems we refer to D oher (2010) and van der Burgt (2007). We describe the rating system to be vaidated as foows: Let N denote the tota number of borrowers cassified within a portfoio by appication of the rating system. Moreover, N k denotes the number of obigors in this portfoio which were associated to the rating grade k 2 {1,..., K}. Hence, we have N ¼ X K k¼1 N k: Finay, et each rating grade be assigned a probabiity of defaut forecast PD k. The statistica tests presented in this section can be cassified rather approximatey either by vaidation period (singe- versus muti-period tests) or by the number of rating grades undergoing the test (singe- versus muti-grade tests). By construction, TTC rating systems are based on much onger time horizons than PIT rating systems. Therefore, the vaidation methodoogies set out in this section wi, in practice, be more appicabe to PIT rather than to TTC rating systems. 14.3.1 Statistica Tests for a Singe Time Period We start by considering tests that are usuay appied to a singe time period case, i.e. starting about one and a haf years after the first introduction of a rating system and in the annua vaidation process that foows. The most prominent exampe for this kind of test is the binomia test (as we as its norma approximation) which is the most often appied singe-grade singeperiod test in practice. On the other hand, the Hosmer-Lemeshow- or w 2 -test

14 Statistica Approaches to PD Vaidation 297 provides an exampe of a singe-period muti-grade test that can be used to check the adequacy of PD forecasts for severa rating grades simutaneousy. 14.3.1.1 Binomia Test To appy the binomia test, we consider one singe rating grade over a singe time period, usuay 1 year. Therefore, we fix a certain rating grade by (B.1) choosing a fixed rating grade k 2 {1,..., K} throughout this subsection, and, additionay, (B.2) assume independence of defaut events between a credits within the chosen rating grade k. The ast assumption readiy impies that the number of defauts in rating grade k 2 {1,..., K} can be modeed as a binomiay distributed random variabe X with size parameter n:¼n k and success probabiity p:¼pd k. Thus, we can assess the correctness of the PD forecast for one time period by testing the nu hypothesis H0: The estimated PD of the rating category is conservative enough, i.e. the actua defaut rate is ess than or equa to the forecasted defaut rate given by the PD against the aternative hypotheses H1: The estimated PD of the rating category is ess than the actua defaut rate. Thereby, the nu hypothesis H0 is rejected at a confidence eve a whenever the number of observed defauts d in this rating grade is greater than or equa to the critica vaue ( X N k ) N d a ¼ min d : k PD j j k 1 PD j Nk j k 1 a : j¼d According to Tasche (2005), the binomia test is the most powerfu test among a tests at a fixed eve and the true type I error (i.e. the probabiity to reject erroneousy the hypothesis of an adequate PD forecast) can be much arger than the nomina eve of the test if defaut events are correated. In fact, assumption (B.2) is not reaistic at a and turns out to competey disagree with a empirica experiences: In practice, defaut correations in a range between 0 and 3% do occur. The Base II framework assumes asset correation between 12 and 24%. Despite this, we shoud particuary mention two recent resuts e.g.: For we diversified German retai portfoios, indications exist that asset correations are in a range between 0 and 5% which in turn woud impy that defaut correations are even smaer fractions of these (cf. Hamere et a. (2004) and Huschens and Stah 2005). Therefore, one gets a reaistic eary warning too using the binomia test and its rather compicated expression for the critica number of defauts. Another aspect worth considering is that one shoud rey on consistency between the modeing of correation for risk measurement within the internay appied credit portfoio mode on the one hand and the vaidation on the other to derive consistent confidence intervas.

298 S. Bochwitz et a. 14.3.1.2 Norma Approximation to the Binomia Test One possibiity of obtaining an easier (but ony approximate) expression for the number of critica defauts within a fixed rating grade k 2 {1,..., K}, is to appy the centra imit theorem: In short, we take advantage of the imiting properties of the binomia distribution and assume it approaches a norma distribution in the imit as the number of obigors N k becomes arge (enough). Hence, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~d a ¼ N k PD k þ F 1 ðaþ N k PD k ð1 PD k Þ as a critica vaue where F 1 () denotes the inverse of the cumuative standard norma distribution function. To appy this asymptotic approximation by the norma distribution, we necessariy have to ensure that the condition (sometimes aso caed Lapace s rue of thumb) N k PD k ð1 PD k Þ> 9 hods. In most cases of practica importance, the approximation seems to be vaid aready for not too arge numbers of N k (whie some numerica exampes indicate that even for figures of N k as ow as 50, the approximation works reasonaby we). Note that for ow defaut probabiities and a ow numbers of credits in the individua rating casses, these prerequisites for using the norma approximation impy impausibe high numbers of obigors. The same approach as the one used to derive the norma approximation to the binomia test was appied by Stein (2003) to get a ower estimate for the number of defauts necessary for vaidating the accuracy of the PD forecasts. Stein (2003) aso discusses the question of sampe size [cosey reated to the finite popuation correction by Cochran (1977)] as we as the infuence of correated defauts which we address in the foowing subsection, too. 14.3.1.3 A Modified Binomia Test Accounting for Correated Defauts The assumption of uncorreated defauts (B.2) for the binomia test generay yieds an overestimate of the significance of deviations in the reaized defaut rate from the forecast rate. In particuar, this is true for risk underestimates, i.e. cases in which the reaized defaut rate is higher than the forecasted rate. Therefore, from a purey conservative risk assessment point of view, overestimating significance is not critica in the case of risk underestimates. This means that it is entirey possibe

14 Statistica Approaches to PD Vaidation 299 to operate under the assumption of uncorreated defauts. Ceary, persistent overestimates of significance wi ead to more frequent recaibration of the rating mode. In addition, this can have negative effects on the mode s stabiity over time. It is therefore necessary to determine at east the approximate extent to which defaut correations infuence PD estimates. Simiar to the one-factor approach underying the risk-weight functions of the IRB approach of Base II, defaut correations can be modeed on the basis of the dependence of defaut events on common (systematic) and individua (specific or idiosyncratic) random factors (cf. Tasche 2003 and 2005). For correated defauts, this mode aso enabes us to derive imits for assessing deviations in the reaized defaut rate from its forecast as significant at certain confidence eves. On a confidence eve a, the nu hypothesis H0 is rejected under the assumptions (B.1) and(b.2) whenever the number of observed defauts d in rating grade k 2 {1,..., K} is greater than or equa to the critica vaue d a : ¼ q þ 2q 1 qð1 qþ 2N p k ffiffi ð1 p 2rÞF 1 ð1 aþ ffiffiffi r F 1 ðpd k Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r F 1 ð1 aþþf 1 ðpd k Þ pffiffiffiffiffiffiffiffiffiffiffi 2N k rð1 rþ rð1 rþ where pffiffiffi q :¼ F 1 r F 1 ðaþþf 1 ðpd k Þ pffiffiffiffiffiffiffiffiffiffiffi 1 r and r denotes the defaut correation. This adjustment takes into account that due to unsystematic risk correation with the systematic risk factor, the respective quantie ies a itte further in the tai than without this further uncertainty and thus needs to be corrected. Tasche (2005) shows that assumption (B.2) is not robust for higher percenties, i.e.: Sma deviations from a zero correation aready ead to dramatic changes in the critica vaue of the test which is of course not a desirabe feature of a test. Furthermore, Tasche (2005) concudes that taking into account dependence by incorporating a one factor dependence structure generated by a Vasicek dynamic and Gordy s granuarity adjustment, yied tests of rather moderate power. This is the case even for such ow correation eves as typica for the probem of correated defauts. Ceary, the norma approximation is aso appicabe in this context and yieds an easier expression for the critica number of defauts. Up to now, ony singe rating grades k were vaidated separatey. The next test by Hosmer and Lemeshow wi cose this gap by an approach to vaidating more than a singe rating grade simutaneousy.

300 S. Bochwitz et a. 14.3.1.4 Goodness-of-Fit Type Tests: x 2 - or Hosmer-Lemeshow-test The two-sided Hosmer-Lemeshow-test provides an aternative approach in a singeperiod vaidation environment to check the adequacy of PD forecasts for severa rating grades simutaneousy. Reca that PD k denotes the PD forecast for rating grade k 2 {1,..., K}. For this purpose, et us pose the foowing assumptions: (LH.1) The forecasted defaut probabiities PD k and the defaut rates p k :¼ d k /N k are identicay distributed. (LH.2) A the defaut events within each of the different rating grades as we as between a rating grades are independent. Let us define the statistic S w2 K : ¼ XK k¼1 ðn k PD k d k Þ 2 N k PD k ð1 PD k Þ with d k ¼ p k N k denoting the number of defauted obigors with rating k 2 {1,..., K}. By the centra imit theorem, when N k!1simutaneousy for a k 2 {1,..., K}, the distribution of S K wi converge in distribution towards a w 2 -distribution with K degrees of freedom because of assumptions (LH.1) and (LH.2). Again, a imiting distribution is used to assess the adequacy of the PD forecasts of the rating system by considering the p-vaue of a w 2 K-test: The coser the p-vaue is to zero, the worse the estimation is. A further probem arises when the PD k are very sma: In this case the rate of convergence to the w 2 K-distribution may be very ow as we. Furthermore, reying on the p-vaue enabes under certain circumstances (e.g. comparabiity of underying portfoios) a direct comparison of forecasts with different numbers of rating categories. The construction of the test is based on the assumption of independence and a norma approximation again. Therefore, the Hosmer-Lemeshow-test is aso ikey to underestimate the true type I error (as the binomia test). 14.3.1.5 Brier Score Another method to vaidate a rating system across a rating grades is to cacuate the average quadratic deviation of the forecasted PD and the reaized defaut rates. Here, in contrast to the preceding statistica tests, it is about an exporatory method. The resuting score between zero and one is caed Brier score (cf. Brier 1950) and is defined in the context of N debtors associated to the K rating grades by B ¼ 1 N ¼ 1 N X K k¼1 X K N k k¼1 h i d k ð1 PD k Þ 2 þðn k d k ÞPD 2 k h i p k ð1 PD k Þ 2 þð1 p k ÞPD 2 k

14 Statistica Approaches to PD Vaidation 301 where PD k denotes the probabiity of defaut assigned to each obigor in rating grade k and p k ¼ d k /N k is the observed defaut rate within the same rating grade (cf. OeNB/FMA 2004). The coser the Brier score is to zero, the better is the forecast of defaut probabiities. Note that, by definition, the Brier score does not measure directy the difference of the defaut probabiity forecast and the true conditiona probabiity of defaut. Hence, the Brier score is in fact not a measure of caibration accuracy aone. Since the Brier score can be decomposed as B ¼ pð1 pþþ 1 N X K k¼1 N k ðpd k p k Þ 2 1 N X K k¼1 N k ðp p k Þ 2 (cf. Murphy and Winker 1992) whereby p ¼ d/n, a separate anaysis is in principe possibe: The first term describes the variance of the defaut rate observed over the entire sampe. Here, PD denotes the defaut frequency of the overa sampe. This vaue is independent of the rating procedure s caibration and depends ony on the observed sampe itsef. It represents the minimum Brier score attainabe for this sampe with a perfecty caibrated but aso trivia rating mode, which forecasts the observed defaut rate precisey for each obigor, but ony comprises one rating cass for the whoe sampe, i.e. PD ¼ PD k ¼ p k ¼ d k /N k for a k 2 {1,..., K}. In this case the expected Brier score is equa to the variance of the defaut indicator, i.e. the first of the three terms in the representation above, B ¼ B :¼ p ð1 pþ. The second term represents the average quadratic deviation of forecast and reaized defaut rates in the K rating casses. A we-caibrated rating mode wi show ower vaues for this term than a poory caibrated rating mode. The vaue itsef is thus aso referred to as the caibration. The third term describes the average quadratic deviation of observed defaut rates in individua rating casses, from the defaut rate observed in the overa sampe. This vaue is referred to as resoution. Whie the resoution of the trivia rating mode is zero, it is not equa to zero in discriminating rating systems. In genera, the resoution of a rating mode rises as rating casses with ceary differentiated observed defaut probabiities are added. Resoution is thus inked to the discriminatory power of a rating mode. An additiona caveat is the different signs preceding the caibration and resoution terms. These make it more difficut to interpret the Brier score as an individua vaue for the purpose of assessing the cassification accuracy of a rating mode s caibration. Moreover, the numerica vaues of the caibration and resoution terms are generay far ower than the tota variance. One of the main drawbacks of the Brier score is its performance for sma defaut probabiities. In this case the trivia rating mode yieds a rather sma Brier score. By trivia rating mode we mean that a debtors are assigned the reaized defaut rates p, of the overa sampe. In this case, the expected Brier score is equa to the

302 S. Bochwitz et a. variance of the defaut indicator, i.e. the first of the three terms in the representation above, B ¼ p ð1 pþ: Evidenty, for p! 0 the Brier score aso converges to zero. The ony possibiity of appying this score in a meaningfu way is to compute the Brier score reative to the trivia score B since the absoute vaues are very cose together for cases with few defauts. 14.3.2 Statistica Muti-period Tests Whie the binomia test and the w 2 -test are usuay restricted to a singe-period vaidation framework, the norma test and the extended traffic ights approach are devoted to overcoming the assumption of independence inherent to most singe-period tests by assuming a dependence structure throughout a time horizon of severa years. 14.3.2.1 Norma Test The norma test for a given rating grade k, is a muti-period test of correctness of a defaut probabiity forecast for a singe rating grade. It can be appied under the assumption that the mean defaut rate does not vary too much over time and that defaut events in different years are independent. Mathematicay speaking, the fundamenta assumptions for the norma test are given by (N) The random variabes PD k,t ¼ D t,k /N t,k that describe the forecasted probabiities of defaut for a singe rating grade k 2 {1,..., K} over the years t 2 {1,..., T} are independent with means m t,k and common variance s 2 k > 0. In this case, the centra imit theorem can be appied to prove that the standardized sum S N k with P T PD k;t m t;k S N k ¼ t¼1 pffiffiffi s k T wi converge to the standard norma distribution as T!1. Since the rate of convergence is extremey high, even sma vaues of T yied acceptabe resuts. Consequenty, to appy the norma test to the PD forecasts PD k,t and corresponding observed percentage defaut rates m t,k, one has to estimate the variance s 2 k. The cassica estimator ^s 2 0;k ¼ 1 T 1 XT t¼1 2 m t;k PD k;t

14 Statistica Approaches to PD Vaidation 303 is unbiased ony if the forecasted PDs exacty match the defaut rates m t,k. Otherwise, the cassica estimator wi be reasonaby upwardy biased, hence one shoud choose 2 ^s 2 k ¼ 1 T 1 4 X T t¼1 2 1 m t;k PD k;t T X T t¼1 ðm t;k PD k;t Þ instead. This aternative estimator ^s 2 k is unbiased under the hypothesis of exact forecasts, too, but ess upwardy biased than the cassica estimator otherwise. Now, we can test the nu hypothesis HN: None of the reaized defaut rates in the years t 2 {1,..., T} is greater than its corresponding forecast PD k,t. Therefore, the nu hypothesis HN is rejected at a confidence eve a whenever! 2 3 5 S N k > z a where z a denotes the standard-norma a-quantie. Note that cross-sectiona dependence is admissibe in the norma test. Tasche (2003, 2005) points out that the quaity of the norma approximation is moderate but exhibits a conservative bias. Consequenty, the true type I error tends to be ower than the nomina eve of the test. This means that the proportion of erroneous rejections of PD forecasts wi be smaer than might be expected from the forma confidence eve of the test. Furthermore, the norma test seems even to be, to a certain degree, robust against a vioation of the assumption that defauts are independent over time. However, the power of the test is moderate, in particuar for short time series (for exampe 5 years). 14.3.2.2 Extended Traffic Light Approach Dating back to the approva of market risk modes for reguatory purposes, the idea of using a traffic ight approach for mode vaidation seems to be a considerabe exporatory extension to statistica tests. In the Base Committee on Banking Supervision (1996), for vaue at risk outiers produced by market risk modes, a binomia test with green, yeow and red zones is impemented that eads eventuay to higher capita charges against potentia risks. Tasche (2003) picks up the idea of a traffic ight approach for the vaidation of defaut probabiities. The basic idea is to introduce probabiity eves a ow ¼ 0.95 and a high ¼ 0.999 (note, that the exempary eves are simiar to Base Committee on Banking Supervision 1996) with respective critica vaues c ow and c high, that assure with the mode used, that the ex post observed number of defauts exceeds the eve c ow by ony a probabiity of 1 a ow (and for c high by probabiity 1 a high respectivey). First, the modified binomia test is introduced as above.

304 S. Bochwitz et a. Furthermore, the Vasicek mode with asset correation r, independent standard norma random variabes X, and x 1,...,x n and a threshod c is given by (see aso Martin et a. 2006). d k ¼ XN k i¼1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1 ð 1;cŠ ð r X þ 1 rx i Þ: Now, to determine critica vaues, the choice of asset correation is of crucia importance as the critica vaues are given for a eve a by c crit ¼ minfi : Pðd k iþ 1 ag Two approaches are introduced, one based on a granuarity adjustment and one based on moment matching, see above. It can be concuded that, for high vaues of asset correation, the respective critica vaues change ceary. Bochwitz et a. (2005) propose an aternative approach for impementing a traffic ight based judgment that does not need an expicit specification of asset correations emphasizing the accessibiity for practitioners. They use a heuristic approach to the vaidation of rating estimates and to identify suspicious credit portfoios or rating grades. Starting again with assumptions (B.1) and (B.2), the number of defauts can be determined to be binomiay distributed. Using the resuts given in Sect. 14.3.1.2, they obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p max ¼ PD k þ F 1 PD k ð1 PD k Þ ða bin Þ N k for some given eve of confidence a bin. A simiar consideration for the one-factor mode (cf. Vasicek (1987) among others) with asset correation r yieds pffiffiffi r F 1 ða asset ÞþF 1 ðpd k Þ p max ¼ F pffiffiffiffiffiffiffiffiffiffiffi : 1 r The next step is to compare the second order error for the statistics of these two approaches. Using b :¼ 1 a with b bin and b asset as the vaues for the respective modes, they derive: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi PD k þ F 1 PD k ð1 PD k Þ r F 1 ð1 b ð1 b bin Þ ¼ F asset ÞþF 1 ðpd k Þ pffiffiffiffiffiffiffiffiffiffiffi N k 1 r A comparison shows that for ow eves of asset correation covering many reevant situations, there is no significant difference in the second order errors.

14 Statistica Approaches to PD Vaidation 305 Therefore, for good reason, the subsequent considerations can be based on the norma approximation. To compare the adequacy of eventuay time changing forecasts for probabiities of defaut, the appication is based on a reative distance between observed defaut rates and forecasted probabiities of defaut. Motivated by the considerations above and by taking into account the expression p sðpd k ; N k Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD k ð1 PD k Þ=N k ; Bochwitz et a. (2005) estabish four cooured zones to anayse the deviation of forecasts and reaisations by setting Green if p k < PD k Yeow if PD k p k < PD k þ K y sðpd k ; N k Þ Orange if PD k þ K y sðpd k ; N k Þp k < PD k þ K o sðpd k ; N k Þ Red if PD k þ K o sðpd k ; N k Þp k. The parameters K y and K o have to be chosen carefuy as they strongy infuence the resuts of the ater appication to a given data set. Practica considerations ead to the concusion that the respective probabiity for the coours green, yeow, orange and red to appear shoud decine. But in contrast, K o shoud not be chosen too arge as in the tai of the distribution, asset correation infuences resuts much more than in the centre of it. Hence, a proper choice coud be K y ¼ 0.84 and K o ¼ 1.64, which corresponds to a probabiity of observing green of 0.5, observing yeow with 0.3, orange with 0.15 and red with 0.05. Being in the comfortabe situation to incude more than one period into the evauation framework, a potentia enhancement is the appication to a muti period. Now, a abeing function is given by L½L g ; L y ; L o ; L r Š¼1000L g þ 100L y þ 10L o þ L r A possibe weighting function is O½L g ; L y ; L o ; L r Š¼P g L g þ P y L y þ P o L o þ P r L r with L g denoting the number of observed green periods, L y the respective yeow number and so on and P g, P y, P o, and P r the associated probabiities (i.e. 0.5, 0.3, 0.15, and 0.05 respectivey). With the hep of the weighting function, it is possibe to assign a mixed coour for more then one observed period. By numerica anaysis and by appication to rating agencies data, it is concuded that for many reevant cases, the deducted extended traffic ight approach gives cear indications for a review of the forecasts for probabiity of defauts. According to Bochwitz et a. (2004), it is aso possibe to appy a muti-period nu hypothesis which is in fact a continuation of the nu hypothesis as in the

306 S. Bochwitz et a. norma test (HN): Reject the hypothesis at a eve b if L[L g, L y, L o, L r ] c b, where c b ¼ maxfcjpðl½l g ; L y ; L o ; L r ŠcÞ < 1 bg: Numerica studies to check the robustness with respect to the adequacy of negecting correations show that the extended traffic ight approach is a usefu too in the jigsaw of vaidation. 14.3.2.3 Some Further Readings and Remarks In Chap. 5 a PD estimation method appicabe even for ow defaut portfoios is suggested. The main idea is to use the most prudent estimation principe, i.e.to estimate the PD by upper confidence bounds whie guaranteeing at the same time, a PD ordering that respects the differences in credit quaity indicated by the rating grades. Unfortunatey, the appication of the proposed methodoogy for backtesting or simiar vaidation toos woud not add much additiona information, as the (e.g. purey expert based) average PDs per rating grade woud normay be we beow the quantitative upper bounds proposed using the most prudent estimation principe. Other approaches to estimating non-zero PDs for high-quaity rating grades are based upon Markov chain properties of rating migrations matrices [cf. Schuermann and Hanson (2004) or Jafry and Schuermann (2004)]. Therefore, a quaitative study of the evoution of these transition matrices across severa years can shed ight on possibe probems in a rating system. After a, we sti ack reiabe statistica vaidation methods for ow defaut portfoios or highquaity rating grades. For further discussions concerning backtesting issues, refer to Frerichs and L offer (2003) or B uher et a. (2002) and the references therein. 14.3.3 Discussion and Concusion A the above mentioned tests focus on comparisons between the forecasted probabiities of defaut and the afterwards observed defaut rates. For a statistica tests, the eventua correation (i.e. asset or defaut correation) between different obigors pays a crucia roe and thus infuences the possibiities for the use of the test in practice. Some tests negect correation, for others, it is necessary to specify it. It is common understanding, that to test correation itsef, the database is insufficienty comprehensive. Hence, it is highy important to keep in mind the respective assumptions used by the different tests. Further work can be done on integrating different rating categories into one test and with respect to the ranking of statistica tests for their use in practice. In the

14 Statistica Approaches to PD Vaidation 307 vaidation process to be estabished in a bank, the use of the statistica tests and exporatory means introduced herein can thus ony be one piece of the puzze among others. 14.4 Practica Limitations to PD Vaidation For severa reasons, backtesting techniques of PDs, as described here, have their imitations: Precision of measurement: Caibrating a rating system is comparabe to measuring a physica property. If as a rue of thumb in measurement theory a standard deviation is taken as a reasonabe size of the measurement error, the figures are rather disappointing. A ower bound for the measurement error of the k-th rating grade is given by the standard deviation of the uncorreated binomia distribution. As a numerica exampe: Setting N k ¼ 500 and PD k ¼ 1% yieds s(pd k, N k ) ¼ 0.45%, resuting in a reative error of measurement of 45%, which is an extraordinary high error compared to physica properties measured. This argument can be turned as we: If it is assumed, that the PD had been estimated precisey, then there woud have been no surprise in defaut rates fuctuating with a standard deviation around the PD. 1 Limited data: Backtesting reies on data. A statistica methods discussed here need a certain number of defauts to be observed before they can be appied. This chaenge can be iustrated with a numerica exampe. For investment grade portfoios with PDs of ess than 10 bps, a size of more than 1,000 borrowers is necessary to observe an average one defaut per year. These portfoios often are much smaer in size, and empirica evidence shows in most years no defaut at a. In these cases, backtesting woud not provide any information, because neither evidence for a right caibration nor for an insufficient caibration can be found, because for PDs arger than zero, defaut rates of zero are observed. The impication of imited defaut data on the vaidation of rating systems and specificay on backtesting issues, are discussed in the Base Committee on Banking Supervision (2005b). Impact of stress: Rating systems are designed to work in norma times. In genera they are caibrated to a more or ess conservative estimated expected vaue of the PD for a onger time horizon. However, from time to time, unforeseeabe events often caed stress resut in a sudden increase of defaut rates, which may be interpreted as a resut of a sudden and ikewise unforeseeabe increase of PDs caused by that event. Usuay, banks utiize credit risk modes and the correations modeed therein, yieding measures ike Credit Vaue at Risk (CVar). In the Base framework, this is impemented in the risk 1 Under the settings of the norma approximation of the binomia test in Sect. 14.3.1.2 there is a more than 15%-chance, that the defaut rate exceeds the PD by more than a standard deviation.

308 S. Bochwitz et a. weight function, which can be ooked at as a kind of stressed PD: The expected vaue of the PD is transated by this function into a stressed PD, which is expected to appear once in 1,000 years, see Base Committee on Banking Supervision (2005c). If PDs are estimated as expected vaues, then in periods of stress, any vaidation technique of PDs that compares a caibrated ong run average PD to observed defaut rates wi fai, because as a resut of the stress to which the rated borrowers are exposed, the defaut rates wi exceed that type of PD heaviy. Further, when rating systems are backtested, two aspects need to be baanced: (1) One period tests make a statement about the current performance of a rating system s caibration. However, this statement must be judged carefuy, because it may be miseading for reasons aready mentioned. (2) Muti period tests as suggested in this artice provide a more robust statement about a rating system s performance, but these tests have another drawback: They need a time series of 4 years at minimum. In 4 years time, however, a rating system has undergone some revisions, triggered by the experience a bank has coected by using the rating system. That s why mutiperiod tests may infer using outdated information, and in the extreme, make a statement on a rating system which has ceased to exist. Our concusion is that backtesting techniques as described here have to be carefuy embedded into a comprehensive vaidation approach of rating systems. Vaidation of PDs shoud be the first eement in a top down vaidation approach, since a successfu keeping in mind its imits backtesting is just a necessary prerequisite for a we functioning rating system. Backtesting may revea deficiencies in a rating system, but the fina concusion as to whether the rating system works as designed or not can be drawn ony if the entire rating system is ooked at. References Base Committee of Banking Supervision (2005a), Update on Work of the Accord Impementation Group Reated to Vaidation Under the Base II Framework, Newsetter No. 4. http://www.bis. org/pub/bcbs_n4.htm. Base Committee of Banking Supervision (2005b), Vaidation of Low Defaut Portfoios, Newsetter No. 6. http://www.bis.org/pub/bcbs_n6.htm. Base Committee on Banking Supervision (1996), Supervisory Framework for the Use of Backtesting in Conjunction with the Interna Modes Approach to Market Risk Capita Requirements. http://www.bis.org. Base Committee on Banking Supervision (2004), Internationa Convergence of Capita Measurement and Capita Standards, a Revised Framework. http://www.bis.org. Base Committee on Banking Supervision (2005c), An Expanatory Note on the Base II IRB Risk Weight Functions. http://www.bis.org. Bochwitz S, Hoh S (2007), Vaidation of Bank s Interna Rating Systems: A Chaenging Task? Journa of Risk Mode Vaidation 1 (4), pp. 3 16. Bochwitz S, Hoh S, Tasche D, Wehn CS (2004), Vaidating Defaut Probabiities on Short Time Series, Capita & Market Risk Insights (Federa Reserve Bank of Chicago), December Issue.

14 Statistica Approaches to PD Vaidation 309 Bochwitz S, Hoh S, Wehn CS (2005), Reconsidering Ratings, Wimott Magazine, May, pp. 60 69. Brier G (1950), Verification of Forecasts Expressed in Terms of Probabiity, Monthy Weather Review 78 (1), pp. 1 3. B uher W, Enge C, Korn O, Stah G (2002), Backtesting von Kreditrisikomodeen, in: Oeher A (ed.): Kreditrisikomanagement Kernbereiche, Aufsicht und Entwickungstendenzen, Sch affer- Poesche, 2nd edition. Cochran WG (1977), Samping Techniques, Wiey, New York. D oher S (2010), Vaidation of Credit Defaut Probabiities via Mutipe Testing Procedures, Working Paper. http://www.defautrisk.com/pp_test_17.htm. Frerichs H, L offer G (2003), Evauating Credit Risk Modes Using Loss Density Forecasts, Journa of Risk 5 (4), pp. 1 23. Hamere A, Liebig T, Scheue H (2004), Forecasting Portfoio Risk, Deutsche Bundesbank, Discussion Paper. Huschens S, Stah G (2005), A Genera Framework for IRBA Backtesting, Bankarchiv, Zeitschrift f ur das gesamte Bank- und B orsenwesen 53, pp. 241 248. Jafry Y, Schuermann T (2004), Measurement, Estimation and Comparison of Credit Migration Matrices, Journa of Banking and Finance 28, pp. 2603 2693. Martin RW, Reitz S, Wehn CS (2006), Kreditderivate und Kreditrisikomodee: Eine mathematische Einf uhrung, Vieweg Verag. Murphy A, Winker R (1992), Diagnostic Verification of Probabiity Forecasts, Internationa Journa of Forecasting 7, pp. 435 455. OeNB/FMA (2004), Guideines on Credit Risk Management Rating Modes and Vaidation. http://www.oenb.at or http://www.fma.gv.at. Schuermann T, Hanson S (2004), Estimating Probabiities of Defaut, Staff report no. 190, Federa Reserve Bank of New York. Stein RM (2003), Are the Probabiities Right? A First Approximation on the Number of Observations Required to Test for Defaut Rate Accuracy, KMV Moody s Technica Report #030124. Tasche D (2003), A Traffic Lights Approach to PD Vaidation, Deutsche Bundesbank, Working Paper. Tasche D (2005), Rating and Probabiity of Defaut Vaidation, In: Studies on the Vaidation of Interna Rating Systems, Working Paper no. 14, Base Committee on Banking Supervision. Van der Burgt MJ (2007), Caibrating Low-defaut Portfoios Using the Cumuative Accuracy Profie, Journa of Risk Mode Vaidation 1 (4), pp. 17 35. Vasicek O (1987), Probabiity of Loss on Loan Portfoios, KMV Corporation, Working Paper.

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Chapter 15 PD-Vaidation: Experience from Banking Practice Robert Rauhmeier 15.1 Introduction This chapter deas with statistica hypothesis tests for the quaity of estimates of probabiities of defauts (PDs). The focus is on the practica appication of these tests in order to meet two main targets. Firsty, bank interna requirements have to be met, assuming that PDs from bank interna rating systems are an essentia eement of modern credit risk management. Secondy, under the future regime of the Base II framework, reguar recurrent vaidations of bank interna rating systems have to be conducted in order to get (and retain!) the approva of banking supervisors for the purpose of cacuating the reguatory capita charge. The theoretica findings are iustrated by an empirica vaidation study with rea word rating data from bank interna modes. We want to iustrate how vaidation or more accuratey, statistica backtesting coud be conducted with rea word rating data in banking practice. We organised this artice as foows. In the second section we describe briefy how rating systems are commony used in the banking industry. Some basic notation is introduced in Sect. 15.3. In the fourth section, common statistica tests ike the exact and the approximated binomia test, the Hosmer-Lemeshow test and the Spiegehater test, are discussed. These tests are suitabe for testing the absoute quaity of a rating system presuming that the fina outcome of the anayzed rating system is a forecast of defaut probabiities. For comparing two rating systems a further centra issue in rating praxis additiona tests are required. In vaidation practice, these tests can be used to anayze whether using expert human opinion, which is usuay appied subsequent to the pure machine rating, significanty improves the quaity of the rating. The appication of the tests discussed in this artice is imited by assumptions, e.g., independence of the defaut events or high The views expressed in this artice are those of the author and do not necessariy refect those of UniCredit Bank AG R. Rauhmeier UniCredit Bank AG e-mai: robert.rauhmeier@arcor.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_15, # Springer-Verag Berin Heideberg 2011 311

312 R. Rauhmeier numbers of obigors in order to fufi the centra imit theorem. Section 15.5 presents some practica guidance to tacke these chaenges by simuation techniques. Additiona research on the issue, incuding which of the suggested tests performs best under certain portfoio compositions is presented. Furthermore, resuts on the anaysis regarding the test power (b error) under practica, near to reaity conditions are shown. In Sect. 15.6, we introduce the concept of creating backtesting sampes from databases found in banking practice. Section 15.7 iustrates the theoretica considerations deveoped in previous sections by rea word rating data and Sect. 15.8 concudes. 15.2 Rating Systems in Banking Practice 15.2.1 Definition of Rating Systems Firsty, we define the outcome of a rating system. In this artice, a rating system forecasts a 1-year defaut probabiity of a (potentia) borrower. It is not just a rank order of creditworthiness, nor an estimate of overa (expected) osses, nor the prediction of specific defaut events. 1 The atter means that we suppose that defauts are the reaisation of random variabes and a rating system consequenty can at best forecast accurate probabiities for an event but not the event itsef. 2 Secondy, it needs to be specified what is meant by a defaut. In this artice and especiay in the empirica exampe we refer to the Base II defaut definition. 3 15.2.2 Moduar Design of Rating Systems Often, bank interna rating systems are designed in a moduar way, which is sketched in Fig. 15.1. The first modue is often caed machine rating, because a mechanica agorithm generates a first proposa for the borrower s PD. Typicay, this agorithm is based on statistica modes as described in the initia chapters of this book. Usuay this modue is composed of a quantitative eement, which consists of hard risk drivers (e.g., baance sheet ratios, ega form, gender, profession, age) and a quaitative eement consisting of soft risk drivers, which have to be assessed by the oan manager or rating anayst (e.g., management quaity, competitiveness of the borrower). 1 We use the phrase forecast instead of estimation in order to emphasis that at the time the rating for a certain borrower is done, the regarding event, namey the defaut, is in the future. 2 We wi come to this ater in Sect. 15.3. 3 See BCBS (2005a), }452 seqq.

15 PD-Vaidation: Experience from Banking Practice 313 Begin of rating procedure Modue 1 Machine Rating Modue 2 Expert-guided Adjustments Modue 3 Supporter Logic Modue 4 Manua Override Fig. 15.1 Moduar Design of Rating Systems End of rating procedure Resut: Forecast of the Defaut Probabiity for Borrower i expressed in rating grade The second modue, expert-guided adjustments, aows for the adjustments of the rating by the anayst subject to obigor specific detais not or not sufficienty refected in the machine rating. Usuay this is done in a standardised form, for exampe, possiby by seecting predefined menu items and evauating their severity. This is in contrast to the quaitative part of modue 1, where the weights of the respective risk drivers are fixed by the agorithms and ony the vaue has to be assessed (for exampe good, average or bad ). In modue 2, even the weight of the risk driver can be determined by upgrading or downgrading in fu rating grades (Sect. 15.2.4). As an interim resut, we obtain the stand-aone-rating of the borrower. Modue 3 supporter ogic captures effects arising from a potentia backing of a borrower cose to defaut. This modue is especiay important for rating systems designed for corporates and banks. 4 Here, often expert guided weightings of borrower ratings and potentia supporter ratings are used, fanked with some reasonabe guideines. Like the first two modues, modue 3 is aso tighty standardised. These three modues have to be subsequenty passed through and wi resut in a rue-based proposa for the PD. Since it is impossibe to foresee every eventuaity affecting the creditworthiness of a borrower in the mode buiding process and the utimate goa of the bank is to forecast the PD as accuratey as possibe for each individua borrower, the rating system might aow an override of the rue based rating. In our moduar rating approach this refers to modue 4 manua override. Overrides shoud be of exceptiona character and must be we documented, founded and approved by a senior management board. Additionay, 4 Contrary to the extensive opinion, the term supporter has not to be taken iteray because the supporter coud even have negative infuence on the PD of the borrower. Further on a parties with strong direct infuence on the PD of the borrower shoud be considered here. Popuar is the infuence of the corporate group where the regarding borrower is embedded, but aso essentia other (one-sided) dependencies coud be taken into account. For exampe an automobie manufacturer might support his most important suppier in case of an imminent defaut in order to ensure his own medium-term interests.

314 R. Rauhmeier Base II requires separate monitoring of overrides. 5 Therefore, we suggest incorporating monitoring of overrides into the annua vaidation process. Frequent reasons for overrides coud ead to a refinement of the rue-based modues of the rating system. It has to be stressed that the detaied design of the sketched moduar set-up of a rating system may strongy vary in practice and even one or more modues wi be omitted if they are irreevant, impractica or even too cost-intensive in reation to the expected benefits. A good exampe here is retai business with credit cards, where often the machine modue is used excusivey. 15.2.3 Scope of Rating Systems A rating mode is a mode of the rea word process that generates defaut events. This process is caed defaut generating process (DGP) and can be thought of as a function of various risk drivers The rating mode takes into account ony a imited number of seected key risk drivers of the DGP. Since borrowers of different portfoio segments foow different DGPs it is a consequence that there have to be as many different rating systems as portfoio segments to cover the whoe portfoio. 6 But a rating systems have the same intrinsic aim, namey to forecast the 1-year-PD of a borrower as good as possibe. With this in mind, the introduced backtesting methods are appicabe in genera for a rating systems as ong as they are forecasting 1-year-PDs and reaisations of defauts or non-defauts coud be observed. Certainy, there are constraints regarding the number of borrowers (and the number of associated defauts). 7 These constraints affect the significance of the resuts of the statistica backtesting, but not the methodoogy itsef. 15.2.4 Rating Scaes and Master Scaes It is common banking practice to use rating scaes. This means that there is ony a imited number of possibe PD forecasts (associated with the corresponding rating grades) rather than a continuum of PD forecasts. Usuay, there is a bank wide rating scae caed a master scae which a rating systems are mapped into. An exampe of a master scae is iustrated in Tabe 15.1. The tabe is to be interpreted as foows. If the machine rating modue, assuming a ogistic regression mode is used, produces a forecast PD of 0.95%, then it fits into 5 See BCBS (2005a), } 428. 6 Stricty speaking every borrower foows its own specific DGP but in practice borrowers foowing simiar DGPs can be pooed into portfoio segments. 7 See Chap. 5 where ow defaut portfoios are treated.

15 PD-Vaidation: Experience from Banking Practice 315 Tabe 15.1 Iustration of a master scae Rating grade PD range PD of grade 1............... 4... 0.11%......... 8 0.80 1.40% 1.05%......... 14...... 35 28 Forecast PD in % 21 14 7 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Rating Garde Fig. 15.2 Typica Master Scae exponentia run of the curve the PD range of rating grade 8 and for the sake of keeping things simpe, we round this forecast to 1.05% as it is the (geometrica) mean of the boundaries. We coud interpret this kind of oss of measurement accuracy simpy as rounding-off difference. Using a master scae has certain advantages. For exampe, it is easier to generate reports and figures and for bank interna communication in genera. Moreover, for some peope it is easier to think in a few discrete vaues instead of a continuous measurement scae. This is especiay reevant when adjustments of ratings coming from the pure machine rating within modue 2 are competed by upgrading or downgrading rating grades. But there are obvious pitfas accompanying the use of a master scae which arises from soey thinking in rating grades and negecting the fact that these grades are just proxies or aiases of forecast PDs. For instance, downgrading a borrower from grade 4 to grade 8 does not mean a doubing of the PD. Because of the exponentia reationship of grades and corresponding PDs, this means neary a tenfod increase in the forecast PD. As seen in Fig. 15.2, master scaes often have the attribute that the PD according to the rating grades increases roughy exponentiay. 8 Two reasons may expain 8 Thinking in a ogarithmic word, n(pd) of the master scae grows amost ineary in the grades.

316 R. Rauhmeier this. First, master scaes are sometimes inspired by the scae of the rating agencies and the derived defaut rates for these grades. Second, banks want (and supervisors caim, see BCBS (2005a), } 403) to have a meaningfu distribution of their borrowers across their rating grades. As noted above, a master scae is used group wide. Rating grades of the master scae mean the same across different portfoios. For exampe a rating grade 8 means the same namey a forecast PD of 1.05% no matter whether it is assigned to a arge corporate or a retai customer. Additionay we assume that rating grades of the master scae mean the same across time. A rating grade 8 means the same no matter if it is assigned in 1998 or 2005. This definition is often referred to as Pointin-Time (PIT) rating approach. 15.2.5 Parties Concerned by the Quaity of Rating Systems In genera we can distinguish three groups of stakehoders of a bank s interna rating system as iustrated in Fig. 15.3. First of a, there is the supervisory authority with the main objective of ensuring the stabiity of credit markets and financia markets in genera. Therefore, the sovency of the bank itsef has to be assured. Transferring this intention to the fied of testing the quaity of rating systems supervisors wi accept forecast PDs that are too high compared to the true PDs. But they wi intervene, if the defaut risk is significanty underestimated. But supervisory authority tends to foow a rather conservative approach which is understandabe from its position. The opposite hods for the (possibe) borrower, who is interested in ow interest rates and favourabe credit conditions. Assuming the price for the credit or at east the credit accommodation itsef depends on the PD (beside the other risk parameters LGD and EAD), the borrower cas for a ow PD assessment. So an underestimation of his PD is a right for the borrower, but an overestimation of his PD is not acceptabe from his point of view. Borrower Financia Institution / Bank Supervisory Authority Favourabe conditions, ow interest rates Libera approach: Assess my Risk as ow as possibe Optimisation of capita aocation, pricing, maximisation of profits Accurate Risk Assessment (systematic) under- / overestimation of risk: bad borrowers are attracted, good borrowers are ost Assurance of financia stabiity in financia markets Conservative approach: In case of doubt a bank has to estimate risk parameters in a conservative manner In the ong run, banks with the most accurate rating system wi prevai at the credit market! Fig. 15.3 Parties concerned by the quaity of rating systems

15 PD-Vaidation: Experience from Banking Practice 317 The third party is the bank itsef. Each kind of misjudgement of the creditworthiness harms an optima capita aocation, a good pricing system, and, in consequence, the maximisation of profits. Therefore, we concude that neither an underestimation nor an overestimation of risk is satisfactory. In terms of statistica test theory, supervisors and borrowers woud perform one-sided statistica hypothesis tests whereas the bank prefers two-sided tests. We introduce some notation in the next section and describe the theoretica framework. 15.3 Statistica Framework We have obigors i ¼ 1,..., N each with a true, but unfortunatey unknown probabiity of defaut p i 2 [0;1]. The main intention of the rating system is to forecast each p i as accuratey as possibe. We denote a forecast by ^p i. We want to start with the description of the theoretica framework of the defaut generating process (DGP). Therefore, we mainy refer to the we known mode of categorica regression in its variations, ogistic regression or probit regression. These topics are expained in detai in Chap. 1. The standard method used to describe a binary outcome variabe y i depending on one or more variabes x i is the categorica regression mode. The mode equation is p i ðx i Þ ¼ Py ð i ¼ 1jx i Þ ¼ Fðx 0 ibþ (15.1) The outcome variabe y i takes the vaue y i ¼ 1 if a defaut is observed and y i ¼ 0 if a non-defaut is observed. In the vector x i a kinds of risk drivers are incuded. These may be financia ratios, obigor specific indicators ike age or status of marriage, macroeconomic risk factors ike GDP-growth-rate or interest rates, and even variabes describing trends in industria sectors. These variabes mainy depend on the specific segment of obigors that is considered and on the data that is in genera avaiabe for this segment. 9 Note that in (15.1), the probabiity of defaut for obigor i, p i, is the outcome of the mode and not the forecast of the outcome event itsef. Therefore, it fits perfecty into our basic understanding what a rating system shoud do as described in Sect. 15.1. The probabiity that obigor i gets in the status non-defaut is simpy 1 p i ðx i Þ ¼ Py ð i ¼ 0jx i Þ ¼ 1 Fðx 0 ibþ (15.2) 9 In more sophisticated modes ike pane modes or hazard rate modes (see Chap. 1) the time index t has to be incorporated beside index i in order to account for the time dependency of the risk drivers. In rating practice it is often assumed that the risk drivers in x are time-agged (e.g. x t-1 ) expaining the defaut of borrower i in t. For the reason of keeping things simpe we negect this time-series component in this chapter.

318 R. Rauhmeier The specification of the cumuative distribution function F(.) denotes whether we assume a ogistic mode or a probit mode. ib p i ðx i Þ ¼ Py ð i ¼ 1jx i Þ ¼ ex0 (15.3) 1 þ e x0 ib p i ðx i Þ ¼ Py ð i ¼ 1jx i Þ ¼ Fðx 0 ibþ (15.4) where F(.) denotes the cumuative standard norma distribution function. Other specifications for F(.) exist. Often x 0 i b is caed inear predictor or simpy score. The vector b consists of the weights for the risk drivers in x used to obtain the score. Because F(.) represents a cumuative distribution function, a monotonic reationship between x 0 ib and p i is assured. Some conceptiona background shoud expain (15.3) and (15.4), the modes of the categorica regression: Suppose that behind the observabe dichotomy of the depending variabe y i, there is a non observabe, meaning atent, continuous variabe ~y i. The vaue of ~y i depends on the vaue of the risk drivers x i. If the atent variabe ~y i fas beow the aso atent threshod y i the status y i ¼ 1 is observabe, otherwise the status y i ¼ 0 is reaised: y i ¼ 1, ~y i ¼ x 0 ib þ ~e i y i y i ¼ 0, ~y i ¼ x 0 ib þ ~e i > y i (15.5) The error term ~e i aows for randomness and is needed to account for idiosyncratic risk factors not covered in x i. The random error term ~e i foows a cumuative distribution function F(.) and it is found p i ðx i Þ ¼ Py ð i ¼ 1jx i Þ ¼ P ð~y i y i Þ ¼ P ð~e i y i x 0 ibþ ¼ Fðy i x 0 ibþ ¼ F ~ (15.6) y i The atent threshod y i can be combined with the constant b 0 in b and we obtain our starting point equation (15.1). Depending on the cumuative distribution function that is assumed for ~e i, a ogit (15.3) or probit (15.4) mode is obtained. Further on, we wi restrict ourseves to the standard norma distribution function. For exampe for a borrower i with a rating grade k ¼ 8 accompanied with a probabiity of defaut p i;k¼8 ¼ 0:0105 we wi acquire ~y i;k¼8 ¼ F 1 ð0:0105þ ¼ 2:3080: So ~ y i;k is determined by the PDs of the master scae grades.

15 PD-Vaidation: Experience from Banking Practice 319 As a next step, we want to extend the mode in order to integrate the possibiity of modeing dependencies in the DGP. A widey used approach is the one-factor mode 10 which is aso the basis of the Base II formua for the risk weighted assets. We spit up the error term ~e i in equation (15.5) in the components e i and f and get y i ¼ 1, ~y i ¼ x 0 p ib þ ffiffiffi pffiffiffiffiffiffiffiffiffiffiffi r f þ 1 r e i y i y i ¼ 0, ~y i ¼ x 0 p ib þ ffiffiffi pffiffiffiffiffiffiffiffiffiffiffi (15.7) r f þ 1 r e i > y i where f ~ N(0,1) and e i ~ N(0,1) are normay distributed random variabes with mean zero and standard deviation one. The random variabe e i represents the idiosyncratic risk and f represent the so caed systematic risk. It is assumed that idiosyncratic risk and systematic risk are independent and idiosyncratic risk is independent for two different borrowers. Therefore, the integration of the systematic factor f, modes dependencies in the DGP of two borrowers and r is caed the asset correation 11 : s 2 pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffi 2 i ¼ Var ð~y i Þ ¼ r þ 1 r ¼ 1 pffiffiffi 2 s ij ¼ Cov ~y i ; ~y j ¼ r ¼ r Cov ~y i ; ~y j r ij ¼ Corr ~y i ; ~y j ¼ ¼ r Var ð~y i ÞVar ~y j (15.8) Conditiona on the reaisation f of the common random factor, the (conditiona) defaut probabiity becomes p i ðx i ; f Þ ¼ Py ð i ¼ 1jx i ; f Þ ¼ P ð~y i y i Þ ¼ P x 0 p ib þ ffiffiffi pffiffiffiffiffiffiffiffiffiffiffi r f þ 1 r e i y i ¼ F e i y i x 0 p ib ffiffiffi r f pffiffiffiffiffiffiffiffiffiffiffi 1 r p ~y i ffiffiffi! r f ¼ F pffiffiffiffiffiffiffiffiffiffiffi 1 r (15.9) Up to now, this detaied annotation may seem to be purey academic, but we wi see its practica benefits in Sect. 15.5 where we extend the (standard) statistica hypothesis test being introduced in the foowing section by using this simpe but very usefu mode variant in order to account for dependent defaut events. 10 See for exampe Finger (2001). 11 The asset correation can be transformed in defaut correations as shown in severa papers, see e.g. BCBS (2005b, Chap. III).

320 R. Rauhmeier 15.4 Centra Statistica Hypothesis Tests Regarding Caibration As shoud become apparent, the reaisation y i ¼ 1ory i ¼ 0, respectivey, is the resut of a random process (the DGP), which is expressed by incuding the random variabe e i in our approach. This means that even if the parameters of the mode b are specified perfecty correct, some unpredictabe randomness sti remains. Hence it is cear, that a certain reaization of the defaut event coud not be forecast, because this woud impy that the rating system coud exacty predict the reaization of the random variabe e i. This situation coud easiy be compared to the we known random experiment of throwing a dice. Even if you know that a six-sided dice is not bogus, you cannot predict the resut. The best you can specify is the probabiity of throwing a certain number, in this exampe this is 1/6. By anaogy, the best a rating system can do is to forecast the probabiity of defaut most exacty for each obigor i. In the foowing, sometimes the term caibration is used. In our context caibration means a property of a rating system and not an action. The ater interpretation as action to caibrate a mode means to estimate the parameter of the (statistica) mode, e.g., to estimate by means of OLS or a maximum ikeihood estimator the coefficients in the equation of the ogistic regression. But in this artice caibration is more in the sense of to be caibrated. The phrase refers to the outcomes of the rating systems and is a property of the rating system. This means that each forecast probabiity of defaut is right: ^p i ¼ p i 8 i. Therefore, we introduce severa approaches how to perform tests on caibration next. 15.4.1 Binomia Test 15.4.1.1 Exact Binomia Test Someone whose task is to vaidate the hypothesis whether the PDs predicted by a rating system are consistent with observed defaut events, wi most ikey perform the we known binomia test, as presented in standard statistica textbooks, as a first step. Suppose we have N g obigors in rating grade g, and a of them have the same (true but unknown) probabiity of defaut p g. If we assume that the reaisations are independent from each other, (we wi drop this constraint at a ater stage), then the number of defauts in grade g, N g,y ¼ 1, foows a binomia distribution with N PN g;y¼1 jn g ; p g ¼ g N g;y¼1 p N g;y¼1 Ng N g 1 p g;y¼1 g (15.10) Based on this, we coud perform a statistica hypothesis test with the nu hypothesis H 0 : p g ¼ ^p g (15.11)

15 PD-Vaidation: Experience from Banking Practice 321 and the aternative H 1 : p g 6¼ ^p g (15.12) where ^p g denotes the forecast derived from the rating system. The test statistic is the observed number of defauts N g,y ¼ 1 and we reject the nu hypothesis if the incidence of observing N g,y ¼ 1 under H 0 is too unikey. What is meant by too unikey is defined by the confidence eve a. Knowing the distribution of N g,y ¼ 1 under H 0 we can cacuate these critica region as N g;y¼1 bða= 2Þ or N g;y¼1 bð1 a= 2Þ (15.13) where b(.) 12 is the quantie of the cumuative distribution function of the binomia distribution B(N g,p g ). Figure 15.4 iustrates an exampe with N g ¼ 350 in rating grade 8. If we wi observe at east 9 defauts or no defaut at a this is too unikey under the nu hypothesis. In this case we woud reject the correctness of the nu hypothesis knowing that we made a wrong decision with probabiity of a ¼ 0.05. 0.25 0.2105 Probabiity 0.20 0.15 0.10 0.1710 0.0923 0.1937 0.1423 0.0868 N g = 8 = 350 a = 0.05 H 0 : p g = 8 = 0.0105 H 1 : p g = 8 0.0105 0.05 0.00 0.0453 0.0249 0.0206 0.0083 0.0030 0.0010 0 1 2 3 4 5 6 7 8 9 10 11 Number of Defauts Fig. 15.4 Iustrative binomia test with marked rejection areas 12 It has to hod for bða= 2Þ: Bba ð = 2ÞjN g ; p g a= 2 < Bba ð = 2Þþ1jNg ; p g and for bð1 a= 2Þ: 1 Bb1 ð a= 2Þ 1jN g ; p g a= 2 < 1 Bba ð = 2Þ 2jNg ; p g.

322 R. Rauhmeier 15.4.1.2 Norma Approximation of the Binomia Test The norma approximation of the exact binomia test is often appied in practice, using the fact that the exact discrete binomia distribution converges to the norma distribution for increasing sampe sizes. As a rue of thumb, this approximation may be sound if N g p g 10 and at the same time N g p g (1 p g ) 10 hods. 13 The number of defauts is normay distributed N g ~ N(N g p g ; N g p g (1 p g )) and the test statistic has to be constructed as Z bin ¼ q N g y g N g p g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ð 0; 1 Þ (15.14) N g p g 1 p g and foows a standard norma distribution, where y g ¼ N g;y¼1 N g denotes the observed defaut rate in rating grade g. Performing the two-sided hypothesis test, the critica vaues can easiy be derived as the a/2 and 1 a/2-quantie of the standard norma distribution. 15.4.2 Spiegehater Test (SPGH) Up to now, we have presented very standard approaches. But these approaches have a shortfa, namey they are primariy suited for testing a singe rating grade but not severa or a rating grades simutaneousy. Spiegehater (1986) introduced a further generaisation we ca Spiegehater test (SPGH). Originay it was used in the context of cinica statistics and the vaidation of weather forecasts. The starting point is the Mean Square Error (MSE), aso known as Brier Score 14 MSE ¼ 1 N X N i¼1 ðy i ^p i Þ 2 (15.15) representing the squared difference of the defaut (y i ¼ 1) and non-defaut (y i ¼ 0) indicators, respectivey, and the corresponding defaut probabiity forecast ^p i 15 averaged across a obigors. Obviousy the MSE gets sma, if the forecast PD assigned to defauts is high and the forecast PD assigned to non-defauts is ow. Generay speaking, a sma vaue of MSE indicates a good rating system. The higher the MSE the worse is the performance of the rating system (keeping other things equa). 13 This rue of thumb may vary depending on what statistica text book is consuted. 14 See Brier (1950). 15^p i ¼ ^p g if obigor i is rated in rating grade g.

15 PD-Vaidation: Experience from Banking Practice 323 The MSE can be interpreted as a weighted average of independent Bernoui distributed random variabes. Spiegehater derived an approach which aows us to test whether an observed MSE is significanty different from its expected vaue or not. Again the hypotheses are H 0 : p i ¼ ^p i 8 i and H 1 : not H 0 (15.16) Then under H 0 the MSE has an expected vaue of and variance EðMSE pi ¼^p i Þ¼ 1 N VarðMSE pi ¼^p i Þ¼ 1 N 2 X N i¼1 X N i¼1 p i ð1 p i Þ (15.17) ð1 2p i Þ 2 p i ð1 p i Þ (15.18) It is obvious from (15.17) that the expected vaue of the MSE under the nu hypothesis is greater than zero, 16 and a function of the true (but unknown) probabiities of defauts. Therefore the absoute vaue of the MSE is not a meaningfu performance index of the rating system because its vaue is constrained by the quaity of the rating system and the portfoio structure i.e., the true but unknown defaut probabiities. Using the centra imit theorem, it can be shown that under the nu hypothesis the test statistic Z S ¼ MSE EðMSE p i ¼^p i Þ VarðMSE pi ¼^p i Þ 0;5 1 X N ðy i p i Þ 2 1 X N p i ð1 p i Þ N N i¼1 i¼1 ¼ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (15.19) 1 X N N 2 i¼1 ð1 2p i Þ 2 p i ð1 p i Þ foows a standard norma distribution and the famiiar steps coming to a test decision have to be conducted. It can be shown that a forecaster (in our case the rating system) minimizes its expected MSE when he or she forecasts the probabiity of defaut for each obigor equa to its true defaut probabiity. 17 There is no way of improving the MSE by modifying the forecast probabiities away from the true probabiities. Thus it can be 16 As ong as we do not consider the specia case of a deterministic DGP, where a true PDs are zero or one. 17 See De Groot and Fienberg (1983).

324 R. Rauhmeier stated that the MSE rewards honest forecasting. This is known as a proper scoring rue. 18 As a specia case of the SPGH statistic, namey if there is just one singe probabiity of defaut in the entire portfoio, then the SPGH statistic Z S is exacty equa to the Z bin of the approximated binomia test. 19 The major advantage of the SPGH test over the binomia test is that with the former a rating grades can be tested simutaneousy on the property of caibration within one step. 20 15.4.3 Hosmer-Lemeshow-x 2 Test (HSLS) The same can be done with an approach introduced by Hosmer and Lemeshow (1980, 2000). Their test statistic has its origin in the fied of categorica regression and is often used in the process of mode finding as a performance measure for goodness-of-fit. The SPGH test penaizes squared differences between reaised event indicators (defaut or non-defaut) and PD forecasts on an individua eve. 21 In contrast, the basic idea of the Hosmer-Lemeshow test (HSLS) is to penaize squared differences of forecast defaut rates from reaised defaut rates on a group eve as coud be seen from numerator terms in (15.20). w 2 HL ¼ XG ðn g y g N g ^p g Þ 2 ¼ XG ðy g ^p g Þ 2 N g (15.20) N g ^p g 1 ^p g ^p g 1 ^p g g¼1 Originay the groups come from arranging individua forecasts into e.g., ten centies or by using the number of covariate patterns in the ogistic regression mode. In this context, the groups are defined by the rating grades. 22 When using the HSLS test statistic as a means of backtesting, w 2 HL is approximatey w2 -distributed with G degrees of freedom. 23, 24 This can easiy be seen because w 2 HL consists in fact of G independent squared standard norma distributed random variabes if g¼1 18 See e.g. Murphy and Dann (1985). 19 See Appendix A. 20 Rauhmeier and Scheue (2005) show that by factorising the MSE more rating properties coud be derived and how they infuence Base II capita. 21 See (19). 22 Hosmer et a. (1988) aude to some approximation conditions, e.g. that in about 4/5 of a groups the expected number of defauts shoud exceed the number of five and in no group the number of defauts shoud be smaer than one. 23 G denotes the number of rating grades with N g > 1, i.e. with at east one obigors being rated in this cass. 24 When using the HSLS statistic as a measure of fit in the process of mode finding, then we say in-sampe, because the mode estimation sampe and the sampe on which the measure of fit is

15 PD-Vaidation: Experience from Banking Practice 325 H 0 : p g ¼ ^p g 8 g (15.21) hods. It can be shown that in an extreme case, when there is just one rating grade at a, the HSLS test statistic and the (squared) SPGH test statistic and the (squared) approximated binomia test statistic are identica. 15.4.4 A Test for Comparing Two Rating Systems: The Redemeier Test Up to now we have introduced approaches adequate for testing whether the fina outcomes of the rating system forecasts of PDs for each obigor are statisticay in ine with their reaisations. This is unquestionaby the main objective of statistica backtesting. But, more questions arise when deaing with rating systems in practice. One might be interested in knowing whether the quaity of the rating system is significanty enhanced when e.g., using so caed human expertise in a modue subsequent to the machine rating modue. This interest might arise from a purey statistica perspective, but in banking practice, the rating systems which are to be impemented and maintained, are cost intensive. These costs may incude saaries for the rating anaysts as we as IT-reated costs for operating systems and data storage. First of a we want to stress that ony a comparison of two or more rating systems by means of the same rating data is meaningfu as mentioned in Sect. 15.4.2 and in Chap. 13. This means the same obigors (by name) in the same time period and with the same defaut indicator definition have to be used. 25 Therefore, whie it is in genera not feasibe to compare ratings across banks one shoud think of business confidentiaity and protection of data privacy this may be done in the context of pooing, 26 or especiay when comparing two rating modues of the same rating system of a bank. We may primariy attend to the atter. The basic idea of the approach introduced by Redemeier et a. (1991) is to compare two MSEs cacuated on the same data basis. A test statistic is derived which aows us to test whether the deviation of a reaised MSE from its expected vaue is significanty different of the deviation of another reaised MSE of its expected vaue derived by an other modue on the same data basis. As described in Sect. 3.2 the modue with the ower MSE is the better one. computed are identicay. In this case the distribution is w 2 with G 2 degrees of freedom. When using the HSLS statistic for backtesting, we say out-of-sampe, because there is no observation coexistent in the estimation sampe and the vaidation sampe. 25 See Chap. 13 and Hamere et a. (2005). 26 Here we mean cooperation of autonomous banks organized in a project structure with the object of gathering data in order to enarge the common data basis by merging banks individua data bases.

326 R. Rauhmeier The test statistic 27 is P N h ^p 2 i; m1 i ^p2 i; m2 2 ^p i; m1 ^p i; m2 yi i¼1 Z R ¼ P N h 2 i (15.22) 0;5 ^p i; m1 ^p i; m2 ^pi; m1 þ ^p i;m2 2 ^pi; m1 ^p i; m2 i¼1 and foows a standard norma distribution under the hypotheses: H 0 : E ½ðEðMSE m1 Þ MSE m1 Þ ðeðmse m2 Þ MSE m2 ÞŠ ¼ 0 and H 1 : E ½ðEðMSE m1 Þ MSE m1 Þ ðeðmse m2 Þ MSE m2 ÞŠ 6¼ 0 (15.23) Note that it ony makes sense to compare two MSE derived from two modues when each modue passes a test of caibration ike the SPGH test for exampe. Otherwise, comparing two MSE with respect to the property caibration is useess knowing that at east one of the two modues is not fufiing the premise to be in ine with the forecasts. We stress that we do not pay attention to theoretica considerations on statistica tests regarding discriminatory power as presented in Chap. 13, but we use them in our empirica anaysis in Sect. 15.7. 15.5 The Use of Monte-Caro Simuation Technique As mentioned previousy, the statistica tests introduced to date are based on crucia assumptions ike independent reaisations of defauts and/or a arge number of observations in order to ensure that the centra imit theorem hods. Using a simuation technique, which is sometimes referred to as Monte-Caro-Simuation, aows us to drop these imiting assumptions. Fortunatey, the basic ideas of the approaches discussed in Sect. 15.4 coud be taken up and be combined with the defaut generation process of Sect. 15.3. Furthermore, these techniques coud be used to derive some resuts on the anaysis regarding the test power (b error) under practica, near to reaity conditions. This is a fundamenta concept in order to highight the chance of a nondetection of a ow quaity rating system. 27 See Appendix B for detais.

15 PD-Vaidation: Experience from Banking Practice 327 15.5.1 Monte-Caro-Simuation and Test Statistic: Correction of Finite Sampe Size and Integration of Asset Correation The fundamenta idea is to derive the distribution of the test statistic (e.g., SPGH Z S, HSLS w 2 HL ) under the nu hypothesis by simuation, that means by repication of a random experiment severa times. If basic assumptions ike independent defaut events or infinite sampe size are not fufied, we can impement those circumstances in our simuation process and substitute the theoretica test statistic (e.g., norma distribution in the case of the SPGH test), by the one obtained by the simuation. A test decisions are then based on the new simuated distribution of the test statistic. The more simuation runs are used, the more accuratey the new simuated distribution can be determined. Our approach is very simiar to the one in Bathazar (2004) and coud be interpreted as an extension, as his focus was on tests for a singe rating grade whereas we want to use tests for a grades simutaneousy. Firsty, we consider the simuation under H 0 : ^p k ¼ p k : The simuation approach coud be best iustrated in eight steps starting with (15.6): 1. Cacuate the threshod y ~ i;k ¼ F 1 ðp k Þ depending on which rating grade the obigor i is rated into ( y ~ i:k ¼ y i x 0 ib). Constitute the asset correation r before the start of the simuation. 28 2. Generate a reaisation of the random variabe f ~ N(0,1). This represents the common factor of the DGP, the systematic risk. 3. For each obigor i ¼ 1,..., N in the examined portfoio: generate a reaisation of the random variabe e i ~ N(0,1). This represents the idiosyncratic, unsystematic risk. 4. Cacuate the vaue of ~y i under consideration of r. 5. Cacuate whether obigor i defauts in this simuation run according to (15.7). 6. Cacuate a the test statistics of interest. 7. Repeat steps two to six, say about 1 Mio times (i.e., 1 Mio simuation runs) and generate a simuated distribution of the test statistic (based on the simuated defauts). 8. Having a simuated distribution of the test statistic, the rejection areas of the H 0 can be cacuated and by comparison with the observed test statistic vaue, a test decision coud be derived for each test considered. This approach permits a very fexibe appication because according to requirements, severa vaues for the asset correation coud be anaysed with respect to their impact on the distribution of the test statistic. Secondy, the impact of the portfoio size may be studied but this is not our focus as in norma backtesting situations the portfoio is given. Nevertheess, someone might get a feeing for the 28 We wi discuss this point in detai ater.

328 R. Rauhmeier variance caused by ow numbers of obigors and/or the impact of the supposed asset correation r. 15.5.1.1 The Simutaneous Binomia Test (Sim Bin) The above described eight steps are sufficient to generate the simuated w 2 -HSLS test statistic and the simuated SPGH-Z 29 S in order to backtest a whoe rating system a grades simutaneousy under more practica situations. Considering the exact binomia test a further chaenge arises. Whereas the binomia test by means of the simuation has been extended for integration of correation, (the number of defauts under the simuation scenario divided by the number of obigors in the rating grade generates the simuated test distribution), there sti is the probem of using the resuts of the grade-wise conducted binomia tests for the backtesting of a grades simutaneousy. Our aim is to draw a concusion for the whoe rating system and not just for a singe grade. The starting point of our consideration is the fact that for a rating system of 14 grades and a binomia test done with a ¼ 0.10, we have to expect that for 1.4 grades, the correct nu hypothesis wi be rejected. Someone who assumes that a rating system is good ony if the statistica test fais for no grade, is off the track. Therefore, we suggest a two-staged approach within our simuation framework when the binomia test is used. The two steps are: 1. Generate the rejection areas for each grade individuay (maybe regarding some correation with hep of the Monte-Caro-simuation) on a certain a eve and conduct the test decision. 2. Count the number of grade-wise rejections per simuation run (introduce a step 7b in our 8 step approach) and use them to generate the distribution of the sum of grade-wise rejections. When the 1 a sb -percentie of this distribution is exceeded (i.e., the critica vaue) by the observed sum of rejections of the individua grade-wise test, the rating system as a whoe woud fai the quaity check. 30 Note that we perform a one-sided test in this second eve. The reason is that, assuming very ow numbers of grade-wise rejections indicates a high quaity of a rating system and too many grade-wise rejections are a signa of a ow quaity rating system. 29 We have to emphasis that the simuated HSLS test statistic is generay not w 2 distributed as we as the simuated Spiegehater test statistic is not standard norma distributed but for convenience we maintain the termini w 2 and Z S. 30 We use a sb to abe the simutaneous binomia test. We point out that the a - eve of the individua tests and the a sb - eve of the distribution of the sum of the grade-wise rejections (simutaneous binomia test) need not to be the same vaue.

15 PD-Vaidation: Experience from Banking Practice 329 15.5.1.2 Remarks on the Adherence of the a-leve with Using the Exact Binomia Test We woud ike to point out that because of the discreteness of the binomia distribution, the a eve that is in fact being hed is ower than the ascertained a woud suggest. We ca this phenomenon effect of diution. Therefore, a binomia test is in genera too ess conservative as coud be seen for exampe in Fig. 15.4 where the probabiity of being in the non-rejection area (1 8 defauts) is 96.24% and therefore the rea a eve is 3.76% which is evidenty ower as the composed eve of 5%. The (correct) nu hypothesis is rejected in much fewer cases than expected. This is especiay true for sampes with a ow number of borrowers. The effect disappears when the exact binomia distribution converges to the norma distribution with a growing number of borrowers or to any other continuous distribution generated by simuation as described above. The effect of diution intensifies when using the simutaneous binomia test in stage two as a discrete distribution is aso used here (see e.g., Tabe 15.2). 15.5.1.3 Simuation Study A: Impact of Portfoio Size and Correation To demonstrate our theoretica findings above, we perform a sma simuation study. We have three portfoios, each with the same reative distribution over the grades as shown in Fig. 15.5, but with different absoute size. We start with a sma portfoio, with N ¼ 200 obigors representing for exampe a portfoio of arge corporates or financia institutions, next we have a portfoio of N ¼ 1,000 acting Tabe 15.2 Resuts from the simuation study A, non-rejection areas, 1 Mio Runs, a ¼ 0.05 Portfoio size SPGH HSLS Identica Sim Bin a Exact Bin, g ¼ 8 decisions r N N g¼8 Lower bound Upper bound Upper bound in % b Upper bound c Lower bound Upper bound 0.00 200 22 1.7551 2.1437 34.40 95.74 1 0.0000 0.0455 0.01 1.8448 2.4306 35.64 95.93 1 0.0000 0.0455 0.10 2.0505 4.7912 54.98 99.45 1 0.0000 0.0909 0.00 1,000 110 1.8697 2.0403 33.05 95.48 2 0.0000 0.0364 0.01 2.5659 3.1637 36.21 96.09 2 0.0000 0.0364 0.10 4.6298 9.5255 93.89 97.51 2 0.0000 0.0455 0.00 10,000 1,100 1.9569 1.9665 28.91 95.39 2 0.0046 0.0164 0.01 6.1455 7.7193 65.90 98.05 2 0.0036 0.0200 0.10 14.0278 29.2670 527.55 97.50 4 0.0000 0.0391 a In the first and the second step we used a a ¼ 0.05 regarding the simutaneous binomia test b In percent of the 1 miion simuation runs c Marks the upper bound of the non-rejection area. For exampe in the first row (r ¼ 0.00 and N ¼ 200), simutaneous binomia test: If 2 or more grade-wise rejections are observed, the rating system as a whoe woud be rejected Exact binomia test for rating grade 8: If a defaut rate of more than 0.0455 is observed (more than 22 0.0455 ¼ 1 defaut) the nu hypothesis can be rejected

330 R. Rauhmeier 12 % 10 % Reative Frequency 8% 6% 4% 2% 0% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Rating Grades Fig. 15.5 Distribution of the borrowers over the rating grades in simuation study A as an exampe for a portfoio of midde sized corporates and finay we anayse a portfoio consisting of N ¼ 10,000 obigors which coud be seen as a portfoio of sma business cients. The distribution we appied is be-shaped 31 as coud be seen from Fig. 15.5 with an average probabiity of defaut p ¼ 0:0308 and p g s according to the master scae of Sect. 15.2.4 (e.g., p g¼8 ¼ 0.0105). A tests are done with a ¼ 0.05. In Tabe 15.2 the resuts of our simuation study are presented. We show the ower and upper bound of the SPGH for the three portfoio sizes and furthermore, for three assumed asset correations r ¼ 0.00, r ¼ 0.01 and r ¼ 0.10. For the HSLS it is sufficient that we show ony the upper bound because the ower bound is fixed at zero. We aso report in the coumn tited Identica decisions how often the SPGH and HSLS came to the same test decision as we want to anayse whether someone has to await different (and therefore confusing) test decisions when appying both tests. As we can see from our study, in 95 to >99%, the HSLS and SPGH reach the same test decision. In genera, we can state, that when r increases, the distribution gets broader and therefore the bounds of the non-rejection areas move outwards. Especiay for the exact binomia test and the simutaneous binomia test, this effect is somewhat diuted because of the discrete character of these distributions. When we ook at the SPGH under r ¼ 0.00, we ceary see how the approximation to the standard norma distribution is improved when the number of observations is increased. For N ¼ 10,000 we get very cose to the Z S ¼ F 1 (0.025) 1.96 31 Often in banking practice the master scae is constituted in the way that many obigors are rated in the grades in the midde of the master scae and fewer in the very good or very bad grades.

15 PD-Vaidation: Experience from Banking Practice 331 Tabe 15.3 Simutaneous binomia test, portfoio size N ¼ 10,000, a ¼ 0.05 Number of r ¼ 0.10 r ¼ 0.00 grade-wise rejections 0 86.1791 60.7638 1 5.9460 30.8927 2 1.6888 7.2325 3 0.9566 1.0092 4 0.6389 0.0952 5 0.5221 0.0067 6 0.4445 7 0.4462 8 0.4679 9 0.5172 10 0.5810 11 0.6455 12 0.5681 13 0.3122 (ower bound) and Z S ¼ F 1 (0.975) þ1.96 (upper bound) we expect. The same is true in principe for the HSLS but the convergence is much sower, as it hods w(0.95,14) 23.68. What is interesting is that in the presence of asset correation (r > 0.00), an increased in N eads seemingy not to a convergence of the boundaries to any vaue. Instead, when we extend from N ¼ 1,000 to N ¼ 10,000, the non-rejection area increases dramaticay from [ 4.6298; þ 9.5255] to [ 14.0278; þ 29.2670] by r ¼ 0.10. The same hods for HSLS and Sim Bin but not for the exact binomia test. Now, we turn to the Sim Bin as we reported the simuation detais in Tabe 15.3. As stated aready above, we expect using a ¼ 0.05, a number of 0.0514 ¼ 0.7 grade-wise rejections on average (expected vaue). Because of the effect of diution, this vaue was not achieved as coud be cacuated from Tabe 15.3: Forr ¼ 0.01 and N ¼ 10,000, we get 0.57 whereas the effect of diution is quite higher for r ¼ 0.00, as we get just 0.49. Therefore, the effect of diution on step one and step two is weakened when correation is taken into account. We concude this subject with the proposition that a of the three tests conducted within our simuation framework are appropriate for means of backtesting. It is somewhat a question of favour which test is preferred for banks backtesting. We tend to suggest SPGH because of its most continuous distribution generated by the simuation. 15.5.1.4 Remarks on the Asset Correation As can be seen from Tabe 15.2, the extent of the asset correation r has a very high impact on the distributions of the test statistics and therefore finay on the test decisions itsef. We fee it is worthwhie to think twice which asset correation to use. Though we do not want to describe how asset correations can be estimated in

332 R. Rauhmeier detai, we discuss some basic considerations regarding the right choice of asset correations and its impact on PD vaidation. First of a, the asset correations used in the backtesting of the bank s interna rating mode shoud be in ine with the asset correations used in other fieds of the bank wide (credit) risk management systems as in the credit portfoio mode. This guarantees a consistent bank wide risk assessment. In practice, asset correations are often not estimated on bank interna data, but based on empirica studies on externa data which serve as a guideine. For exampe, Hamere et a. (2003) report that asset correations in a point-in-time rating framework are in a range of roughy 0.01 0.02. This is sighty higher than assuming no asset correation at a the most conservative approach regarding statistica backtesting but much ower than the asset correations used in the Base II framework. In the atter, the asset correation depends on the corresponding exposure cass and varies from r ¼ 0.04 (exposure cass: Quaifying Revoving Retai) over r ¼ 0.15 (Residentia Mortgage) up to r ¼ 0.16 (Other Retai), r ¼ 0.24 (Corporates, Sovereigns and Banks), and even r ¼ 0.30 for High Voatie Commercia Rea Estate. These Base II asset correations might not be taken as best estimators of asset correations by nature, but rather are assessed by poitica reguatory concerns in the ight of being conservative. 15.5.2 Assessing the Test Power by Means of Monte-Caro-Simuation 15.5.2.1 Theoretica Background As mentioned above, a further appication of the Monte-Caro-Simuation is the assessment of the type II error or the pendant, caed test power. Our aim is to derive an approach for getting an idea of how we our tests work with respect to the test power. In genera, the power of a statistica hypothesis test measures the test s abiity to reject the nu hypothesis when it is actuay fase that is, to make a correct decision. Tabe 15.4 gives an overview of the possibiities of correct and incorrect decisions one can make with statistica hypothesis tests. The type II error (b-error) is defined as the probabiity of not rejecting H 0 when in fact H 1 is right. The power of a statistica hypothesis test is defined as the probabiity of not committing a type II error. It is cacuated by subtracting the probabiity of a type II error from one: power ¼ (1 b). Tabe 15.4 Types of test decisions and its consequences Test decision H 0 H 1 Reaity H 0 is true Correct decision Type I Error (a-error) H 1 is true Type II Error (b-error) Correct decision

15 PD-Vaidation: Experience from Banking Practice 333 0.25 H 0 : p g = 8 = 0.0105 Probabiity 0.20 0.15 0.10 H 1 : p g = 8 = 0.0210 N g = 8 = 350 a = 0.05 Power = 1 b = 0.3166 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Number of Defauts Fig. 15.6 Iustration of a-error and b-error with the exact binomia test Whereas we can contro the a-error by setting a to a specific vaue (usuay 0.01, 0.05, or 0.10), we have no contro of the b-error simutaneousy. The reason is that the b-error depends on the hypothesis H 1. We wi not go into theoretica detais, but demonstrate it with an exampe. We refresh the exampe for the exact binomia test of Sect. 15.4.1.1 with H 0 : p g¼8 ¼ 0.0105 and H 1 : p g¼8 6¼ 0.0105. With this pair of hypotheses, there are an infinite number of possiby aternative hypotheses. Therefore, we have to pick out one of these. For exampe, we can specify H 1 : p g¼8 ¼ 20.0105 ¼ 0.0210. Thus, we can cacuate the possibiity of detecting a fase H 0 when the true PD of the grade is twice as high as predicted. The grey bars in Fig. 15.6 mark the distribution under H 1. The area outside of the non-rejection area of H 0 (no defaut and at east 9 defauts) and under the H 1 - distribution determines the test power. In our exampe, we get a power of 0.3166. In genera ceteris paribus the power of a test rises if The number of borrowers rises, The distance of vaues under H 0 and H 1 (here the PDs) rises, The a eve is raised. 15.5.2.2 Simuation Study B: What is the Best Test? The concept of assessing the test power is obviousy not restricted to the exact binomia test but appicabe to other statistica tests and in particuar, the SPGH test and the HSLS test and even the simutaneous binomia test. Furthermore, the concept works we in our framework which aows correations.

334 R. Rauhmeier In the foowing, we take the simuation framework of Sect. 15.5.1 and add more steps in order to anayze the test power. Now, steps one to eight have to be done under H 0 and again under H 1. Finay the area outside the non-rejection area of H 0 has to be cacuated under the H 1 distribution. 32 The focus is twofod: First, we want to anayse how the power reacts under certain conditions such as varying numbers of borrowers and/or asset correations. Second, we want to anayse which of our tests SPGH, HSLS or simutaneous binomia performs best. We ca test A better than another test B if it has more power (a ower type II error), with respect to an aternative hypothesis H 1, but at the same time hods the 33, 34 assumed a-eve. We emphasise that we do not want to carry out a stringent mathematica proof, but merey provide an initia gance within our simuation framework. This chapter is strongy orientated towards rea banking practice and we continue this approach in this subsection: We distinguish three modes which may serve as point aternative hypothesis H 1 : Mode 1: a fraction 1 q of a borrowers is assumed to be cassified in the correct grade where the fraction q is randomy distributed over a rating grades. Mode 2: a borrowers are graded up by s grades Mode 3: a borrowers are graded down by s grades 35 Whereas Mode 2 and Mode 3 describe a systematic, monotonic error in the rating system, 36 Mode 1 represents a mixture of incorrect ratings and might be the most reaistic probem in backtesting rating systems. Tabe 15.5 shows the resut of our simuation study. As expected, an increase in portfoio size eads, ceteris paribus, generay to an increase in power. This is true for the three tests and for the three modes regarded. Further on, an increase in asset correation eaving the portfoio size constant decreases the power. 32 We assume hereby again that the reative frequency resuting from the 1 miion runs is a good enough approximation for the probabiity. 33 This is simiar but - not identica - to the concept of uniformy most powerfu test. A test is caed a uniformy most powerfu test to a eve a if under a given initia situation it maximizes the probabiity of rejecting the H 0 on a distributions or parameter vaues beonging to the aternative hypothesis H 1. 34 The atter is fufied automaticay as we derived the boundaries if the non-rejection area within the simuation. 35 Rating grade 1 (14) has an upper (ower) absorbing boundary which means that a borrower in the first (ast) rating grade remains in it and cannot become better (worse). 36 Within the master scae we use (see Sect. 15.2.4) the PD from one rating grade to the next worse grade increases by a factor between 1.75 and 2 depending on the specific grade.

15 PD-Vaidation: Experience from Banking Practice 335 Tabe 15.5 Resuts from the simuation study B, power, 1 Mio runs, a ¼ 0.05 r N SPGH HSLS Sim Bin Mode 1: q ¼ 0.5 0.00 200 0.1746 0.2408 0.1071 0.01 0.1482 0.2345 0.1066 0.10 0.0686 0.1595 0.1071 0.00 1,000 0.7644 0.9987 0.9763 0.01 0.4345 0.9954 0.9763 0.10 0.1001 0.8239 0.9759 0.00 10,000 >0.9999 >0.9999 >0.9999 0.01 0.6839 >0.9999 >0.9999 0.10 0.1111 0.9606 >0.9999 Mode 2: a borrowers graded up by s ¼ 1 0.00 200 0.1927 0.0203 0.0015 0.01 0.1863 0.0200 0.0016 0.10 0.0036 0.0204 0.0016 0.00 1,000 0.7605 0.0291 0.0139 0.01 0.4697 0.0228 0.0138 0.10 0.1369 0.0130 0.0138 0.00 10,000 >0.9999 >0.9999 0.9996 0.01 0.7510 0.6141 0.9996 0.10 0.1543 0.0078 0.9996 Mode 3: a borrowers graded down by s ¼ 1 0.00 200 0.3428 0.1699 0.1568 0.01 0.2836 0.1719 0.1563 0.10 0.1217 0.1385 0.1560 0.00 1,000 0.9119 0.4875 0.4277 0.01 0.5854 0.4275 0.4282 0.10 0.1362 0.1905 0.4295 0.00 10,000 >0.9999 >0.9999 >0.9999 0.01 0.7771 0.8669 >0.9999 0.10 0.1388 0.2212 >0.9999 It is remarkabe that when ooking at the SPGH aready at N ¼ 1,000 and by r ¼ 0.01 or ower for a three modes, a power near to or over 0.5 is achieved. But the picture is quite mixed when regarding the HSLS or Sim Bin. These two tests perform worse in comparison to SPGH especiay for Mode 2 and a sma portfoio size. Anaysing the reative competitiveness of the SPGH, HSLS and Sim Bin the picture is not unambiguous. Regarding Mode 1, which stands for an interchange of obigors assessed rating, HSLS seems to be the best choice. SPGH outperforms when the systematic up-grade by one grade is anaysed as an aternative hypothesis. Even the Sim Bin in some situations has the highest power. What can we earn from this simuation study about power and what are the consequences for practica backtesting? We concude that unfortunatey none of the statistica test we anaysed ceary outperforms the others in a circumstances. For practica issues, a tests shoud be performed when an assessment of the probabiity of non-detecting a ow quaity rating system is required.

336 R. Rauhmeier What is most important at a is that especiay the higher management shoud be aware that there is 37 a (perhaps significant) probabiity that in fact H 0 is wrong, but the statistica toos did not revea this. Our simuation approach can be interpreted as an instrument to fufi this purpose. 15.6 Creating Backtesting Data Sets: The Concept of the Roing 12-Month-Windows Up to now we have shown some concepts for statistica backtesting, but when deaing with rea data, the first step is aways to create a specific sampe on which a meaningfu anaysis can be carried out. In banking practice ratings are performed continuay over the year, for instance, when a new customer must be evauated, a credit ine requires extension, new information (e.g., financia figures) concerning a borrower aready in the portfoio comes up, or questions of any fieds regarding the creditworthiness are recognised. We propose an approach for creating backtesting sampes ceary in ine with The definition of what a rating is, namey a forecast for the 1-Year-PD. What coud be found in the IT-database at any point of time we may ook into it. The genera concept a bank manages its credit risks incuding the cacuation of Base II risk capita. From these guideines, it foows that whenever we ook into the rating database we find the bank s best assessment of the borrower s probabiity of defaut for the next year. This is irrespective of how od the rating is at the time we ook into the database. This is because when the bank has an inducement that when there is a noteworthy change in the creditworthiness of the borrower (its PD), the bank has to ater the rating immediatey. 38 This means that a re-rating just once a year, for exampe whenever new annua accounts are avaiabe, might be not adequate in the case when other, reevant information regarding the PD in any form is made avaiabe. When there is no change in the rating, it remains vaid and predicates each day the same, namey the forecast of the 1-year-PD from the day we found it in the database. In the same way, the second essentia variabe, the defauts and non-defauts, have to be coected. The termination of the backtesting sampe is done according to the principe of reporting date. We ca this approach cutting sices or roing 12-months-window (compare to Fig. 15.7). 37 This is true even if the hypothesis H 0 The rating system forecasts the PD we. coud not be rejected at a certain eve a. 38 See BCBS (2005a), } 411 and } 449.

15 PD-Vaidation: Experience from Banking Practice 337 Time Q3 01/01/2004 01/01/2005 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q1 Q2 Q3 Q4 G8 G10 n.a. G8 G14 G14 G12 G12 proportiona n.a. G4 G14 G14 G12 n.a. A G4 B n.a. C n.a. G12 D E n.a. G8 G10 G5 G4 G14 G14 G12 G6 Fig. 15.7 Concept of the roing 12-months-windows the backtesting sices We start with the first sice caed Q1/2004, which begins at January 2004. We ook in the database and find borrower A with rating grade 8. He was rated with grade 8 a few months before (and gets other ratings after First January 2004), but has grade 8 at the beginning of January 2004. Within the next 12 months (up to the end of December 2004) he did not get into defaut, this was indicated with a.he enters the sice Q1/2004, as non-defaut and rating grade 8 (y A ¼ 0; ^p g¼8 ¼ 0:0105). The second borrower B enters with grade 10 but as defaut, because he defauted somewhere in the third quarter of 2004 indicated with N (y A ¼ 1; ^p g¼10 ). Borrower C was not found in the rating database at January 1, 2004 as he was rated for the first time just before the beginning of the second quarter 2004. Therefore he is not contained in sice Q1/2004. Borrower D enters with grade 12 as non-defaut, because the defaut we observe is past the end of the 12 month period which ends by December 31, 2004. Borrower E is found in the database with a rating grade 5 but he ended the business connection with the bank (indicated by L). Therefore it is impossibe to observe if he has defauted or survived within the 12 month period. This observation for borrower E shoud be incuded in the sice Q1/ 2004 as a weighted non-defaut, where the proportion is cacuated as the quota (number of months it has been observed)/12. A non-consideration or fu consideration may cause biases. In the same way, the foowing sices have to be constructed. We show the compositions of the sices as a summary in the eft side of Fig. 15.7.

338 R. Rauhmeier For practica issues, utimo data fies can be used best. So for the sice Q1/ 2004, we use the utimo data fies from December 2003. In Fig. 15.7 we present the sice on a quartery basis but sampe creation can aso be done on a monthy basis. This has the advantage that some eements of monitoring are fufied and neary no rating and defaut is ost. The ony exception is when a rating changes within a month. Therefore, the initia rating was not seen in the utimo data fie. The same is true when a rating is competed and the rated borrower gets into defaut before he has passed his first end of month. We recommend anaysing these specia cases separatey, for exampe regarding detection of fraud. When using the introduced method of roing 12-month-windows, it is of concern that the sices greaty overap. For a tuned (entries and exits are baanced, dates of rating compiations are eveny distributed a over the year) portfoio of borrowers with ong term business reationship, two subsequent sices may overap by about 11/12. As a consequence, we expect that we get often the same test resuts for two or more subsequent sices. We wi see this in the next section, where we demonstrate our theoretica considerations by appying them to rea word rating data. 15.7 Empirica Resuts 15.7.1 Data Description In this section, we demonstrate the appication of our concepts to rea rating data. The data used is part of a rating system introduced in the beginning of 2004 for sma business cients in Germany. 39 We anaysed sices beginning in February 2004 up to January 2005. 40 So for backtesting sice Jan2005, we considered the defauts and non-defauts up to the end of December 2005. Here we can see that for backtesting a compete vintage of ratings, in fact a period of two years, is needed. The rating system foows mainy the architecture sketched in Sect. 15.2.2, and is composed of various parae sub-modes for the machine rating modue. These submodes differ according to whether there is a tradesman, freeancer/professiona 41 or a micro corporate to be rated. Micro corporates dominate with about 45% of a ratings, foowed by tradesman (about 30%) and remaining freeancer and professionas with about 25%. The basic structure of a sub-modes contains approximatey a dozen quantitative and quaitative risk drivers as it is usua for this kind of portfoio in banking 39 In order to avoid discosure of sensitive business information, the data base was restricted to a (representative) sub-sampe. 40 For the construction of e.g. the sice Feb2004 we used the utimo data store of 31st January 2004. 41 Like architects, doctors, or awyers.

15 PD-Vaidation: Experience from Banking Practice 339 practice. Within the second modue, expert guided adjustment, up or down grading of the machine rating can be done. For micro corporates a supporter ogic modue is avaiabe. In our empirica anaysis, we want to examine the sices Feb2004 to Jan2005 and in detai the comprehensive sice Jan2005. Atogether, more than 26,000 different ratings can be anaysed in the sices Feb2004 to Jan2005. Whereas sice Feb2004, consists of itte more than a hundred ratings because of the recent aunch of the rating system, the numbers in the sices increase steadiy up to more than 24,000 in Jan2005. Note that with our concept of roing 12-months-windows, the sices overap by a high degree. For exampe Jan2005 and Dec2004 have 88% observations in common, sices Jun2004 and Ju2004 about 75%. 15.7.2 The First Gance: Forecast Versus Reaised Defaut Rates When taking about the quaity of a rating system, we get a first impression by ooking at forecast defaut rates and reaised defaut rates. Figure 15.8 shows that reaised defaut rates vary between 2 and 2.5%, whereas the forecast PD underestimates the reaised defaut rate sighty for amost a sices. Furthermore, it can be seen that on average, the fina rating is more conservative than the machine rating. This means that the expert guided adjustments and supporter ogic on average ead to a downgrade of borrowers. This might be an 0.040 0.030 0.020 0.010 0.000 Feb2004 Mar2004 Apr2004 May2004 Jun2004 Ju2004 Aug2004 Sep2004 Oct2004 Nov2004 Dec2004 Jan2005 reaised Defaut Rate forecast Defaut Rate (fina rating) forecast Defaut Rate (machine rating) Fig. 15.8 Reaised defaut rate versus forecast defaut rate by sice

340 R. Rauhmeier interesting resut, because in banking practice the opposite is often assumed. The ine of thought is, rating anaysts or oan managers are primariy interested in seing oans which is easier because of bank interna competence guideines or simpy by questions regarding the credit terms if the machine rating is upgraded by the expert. The accurate rating is often assumed to be of subordinate importance for the oan manager. Here we have an exampe, which disproves this hypothesis. We wi see whether this difference of machine rating and fina rating regarding the quaity of forecasts is significant or not in Sect. 15.7.4. 15.7.3 Resuts of the Hypothesis Tests for a Sices As we are interested in whether the deviation of the fina rating from the defaut rates is significant, we focus on the SPGH and the HSLS test. Tabe 15.6 shows the resuts. For r ¼ 0.01, the SPGH rejects in no sice the nu hypothesis of being caibrated, the HSLS rejects in two sices tighty. For the very conservative approach with r ¼ 0.00, in some sices the nu hypothesis has to be rejected for Tabe 15.6 Test decisions by sice, fina rating, 1 Mio runs, a ¼ 0.05 Sice r SPGH HSLS Lower bound Upper bound Test statistic Decision Upper bound Test statistic Decision Feb2004 0.00 1.5063 2.2255 0.2075 No rej. 25.3473 5.7982 No rej. Mar2004 1.8380 2.0736 0.3214 No rej. 27.7315 6.1607 No rej. Apr2004 1.8948 2.0137 0.2490 No rej. 21.5598 6.8883 No rej. May2004 1.9512 1.9780 0.9859 No rej. 21.3653 10.8339 No rej. Jun2004 1.9549 1.9697 2.0617 Rej. 20.8402 17.1008 No rej. Ju2004 1.9544 1.9697 1.3236 No rej. 20.6058 33.3231 Rej Aug2004 1.9549 1.9673 2.0724 Rej 20.3097 67.6734 Rej Sep2004 1.9626 1.9675 2.4033 Rej 20.3765 78.3339 Rej Oct2004 1.9570 1.9691 2.1408 Rej 20.5659 68.2907 Rej Nov2004 1.9575 1.9604 1.6973 No rej. 20.6235 70.2873 Rej Dec2004 1.9592 1.9629 1.0893 No rej. 20.6672 78.3400 Rej Jan2005 1.9569 1.9620 0.9927 No rej. 20.9511 96.3306 Rej Feb2004 0.01 1.5063 2.3911 0.2075 No rej. 26.5294 5.7982 No rej. Mar2004 2.2839 2.9406 0.3214 No rej. 30.5962 6.1607 No rej. Apr2004 3.1715 4.0670 0.2490 No rej. 29.4874 6.8883 No rej. May2004 3.9862 5.1376 0.9859 No rej. 35.4975 10.8339 No rej. Jun2004 4.7208 6.1255 2.0617 No rej. 43.0297 17.1008 No rej. Ju2004 5.5315 7.2272 1.3236 No rej. 53.6896 33.3231 No rej. Aug2004 6.2755 8.2214 2.0724 No rej. 65.1878 67.6734 Rej Sep2004 6.9194 9.0275 2.4033 No rej. 76.8287 78.3339 Rej Oct2004 7.5017 9.7802 2.1408 No rej. 90.2356 68.2907 No rej. Nov2004 8.0797 10.5260 1.6973 No rej. 103.8628 70.2873 No rej. Dec2004 8.6682 11.2619 1.0893 No rej. 119.0537 78.3400 No rej. Jan2005 9.1811 11.9508 0.9927 No rej. 130.8062 96.3306 No rej.

15 PD-Vaidation: Experience from Banking Practice 341 SPGH and HSLS. The simutaneous binomia tests (resuts are not shown here expicity), shows that even for r ¼ 0.00, the nu hypothesis in no sice coud be rejected, indicating a good quaity of the rating system, too. Note the different test decisions of the consuted tests SPGH and HSLS for some sices. From Tabe 15.6, we can aso see how we the approximation of the SPGH to the standard norma under H 0, works as the number of ratings in the sices increases for r ¼ 0.00. The same is true for the HSLS, when we take into account that ony 10 of 14 grades have a arge number of observations 42 w 2 (0.95,10) ¼ 20.48. Secondy, we might find it impressive how broad the non-rejection area is when taking correation into account, even when used for a very ow asset correation of r ¼ 0.01. Notice that the non-rejection areas for r ¼ 0.01 of SPGH, HSLS and Sim Bin, get even broader when the number of ratings increases, athough the reative distribution of the borrowers over the grades ony changes negigiby. The same phenomenon was observed in the simuation study A, Tabe 15.2. 15.7.4 Detaied Anaysis of Sice Jan2005 Now we turn to a more detaied anaysis of sice Jan2005, as we can observe up to now that the rating system passes our quaity checks we. The distribution, not shown here expicity, of the observations over the rating grades, is roughy beshaped, for exampe about 900 observations in grade 4, up to 4,500 in grade 8 and 1,000 in grade 12. We can see in Fig. 15.9 that for three rating grades, the reaised defaut rate is in the rejection area for the binomia test. Hereby we assumed r ¼ 0.01. The reaised defaut rate increases in the rating grades as is assumed and therefore confirms our previous impression of the rating system we obtained from the SPGH and HSLS. Next we anaysed the power of our tests. As coud be seen from Tabe 15.7, the high number of ratings eads to a high power of a tests in a anaysed circumstances. When assuming no correation, the power is >0.9999 for each of the three tests. When assuming r ¼ 0.01 we get, e.g., for the SPGH in Mode 3, a power of 0.7548. This means that the SPGH when in fact a borrowers shoud have got a rating one grade worse woud have detected this with a probabiity of about 76%. 42 Ratings Grades 1 to 3 of the master scae are intended mainy for sovereigns, internationa arge corporates and financia institutions with exceent creditworthiness and coud ony in exceptiona cases be achieved by sma business cients. The worst rating grade is assigned to a very ow number of borrowers in the data base, what is comprehensibe because the rated portfoio mainy consists of initia ratings, so potentia borrowers with a ow creditworthiness are not accepted by the bank at a and therefore do not get into the rating database.

342 R. Rauhmeier Rating Grade 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Rating Grade reaised defaut rate Fig. 15.9 Reaised defaut rates and exact binomia test by grades, sice Jan2005, 1 Mio Runs, r ¼ 0.01, a ¼ 0.05 Tabe 15.7 Anaysis of power, fina rating, sice Jan2005, 1 Mio runs, a ¼ 0.05 r SPGH HSLS Sim Bin Mode 1: q ¼ 0.5 0.00 >0.9999 >0.9999 >0.9999 0.01 0.7894 >0.9999 >0.9999 Mode 2: a borrowers graded up by s ¼ 1 0.00 >0.9999 >0.9999 >0.9999 0.01 0.6798 0.4888 0.5549 Mode 3: a borrowers graded down by s ¼ 1 0.00 >0.9999 >0.9999 >0.9999 0.01 0.7548 0.8201 0.8227 To get a compete picture of the quaity of the rating system and the regarded portfoio, we ook at its discriminatory power. 43 Figure 15.10 dispays the ROC- Curves for the machine rating and the fina rating. For both rating modues, no discrepancies coud be observed from the ROCs. We see that the ROC-Curve of fina rating is aways atop of the ROC-Curve of the machine rating, indicating an increase in discriminatory power when human expert assessment is brought into 43 For a definition of the measures ROC-Curve and AUROC and their statistica properties, we refer to Chap. 13.

15 PD-Vaidation: Experience from Banking Practice 343 1.00 0.75 Sensitivity 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 1-Specificity Fina_Rating ROC area: 0.745 Machine_Rating ROC area: 0.7258 Reference Fig. 15.10 ROC-curve for fina rating and machine rating, sice Jan2005 Tabe 15.8 Machine rating versus fina rating MSE AUROC H 0 p-vaue Machine rating 0.0230 0.7258 MSE mach.rating ¼ MSE fin.rating <0.0001 Fina rating 0.0226 0.7450 AUROC mach.rating ¼ AUROC fin.rating <0.0001 account. The AUROC of the fina rating is therefore a bit higher (0.7450), than those of the machine rating (0.7258). As coud be seen from Tabe 15.8, the AUROC and MSE of the machine rating and fina rating differ significanty. For comparing the MSE, we used the Redemeier test described in detai in Sect. 15.4.4. 44 To draw an overa resut, the rating system passes our quaity checks very we. With the high number of ratings in the anaysed portfoio, we woud have been abe to detect potentia shortcomings, but we did not find any. As the system was introduced 2 years ago, this was the first backtest that was performed, and the more highy this good resut is to be regarded. 44 As it was a prerequisite that the machine rating shoud pass a test on caibration we conducted the SPGH and the HSLS. We find that we coud not reject the nu hypothesis of being caibrated with r ¼ 0.01, but we have to reject the nu hypothesis with r ¼ 0.00.

344 R. Rauhmeier 15.8 Concusion In this chapter we deat with vaidation of rating systems, constructed to forecast a 1-year probabiity of defaut. Hereby, we focused on statistica tests and their appication for bank interna purposes, especiay in the Base II periphery. We buit up a simuation based framework to take account of dependencies in defauts (asset correation), which additionay has the potentia to appraise the type II error, i.e., the non-detection of a bad rating system, for optiona scenarios. Hereby, the we known exact and approximated binomia test and the Hosmer-Lemeshoww 2 test are used, but we aso introduced the ess popuar Spiegehater test and an approach caed simutaneous binomia test, which aow the testing of a compete rating system and not just each grade separatey. As it is important for banks to compare the quaity of modues of their rating system, we aso refer to the Redemeier test. As for any appied statistica method, buiding test sampes is an important issue. We designed the concept of the roing 12-months-window to fufi the Base II and bank s interna risk management requirements as we as using the bank s IT-environment (rating database) effectivey and is in harmony with our definition of what a rating shoud refect, namey the bank s most accurate assessment of the 1-year-PD of a borrower. A concepts are demonstrated with a very up-to-date, rea-ife bank interna rating data set in detai. We focus mainy on statistica concepts for rating vaidation (backtesting) but it has to be emphasised that for a comprehensive and adequate vaidation in the spirit of Base II, much more is required. To name a few, these incude adherence of defined bank interna rating processes, accurate and meaningfu use of ratings in the bank s management systems and correct impementation in the IT-environment. Appendix A We show that the SPGH test statistic Z S is equa to the Z bin test statistic of the approximated binomia test in case where there is ony one singe PD. This is when a obigors are rated in the same rating grade g. We start with (15.19) and substitute ^p i ¼ ^p g respectivey p i ¼ p g because we argue under H 0 : 1 X Ng 2 1 X Ng y i p g p g 1 p g N g N i¼1 g i¼1 Z S ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 X N g t 2 1 2p g pg 1 p g ¼ P N g i¼1 N 2 g i¼1 y 2 P N g 2 p g y þ N g p 2 g PN g p g 1 p g i¼1 i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N g 1 2p g pg 1 p g

15 PD-Vaidation: Experience from Banking Practice 345 ¼ ¼ P N g i¼1 P N g i¼1 and get (15.14). P N g y 2 p g y þ N g p 2 g PN g p g 1 p g i¼1 i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N g 1 2p g pg 1 p g y 2 p g PN g y þ 2 N g p 2 g N g p g i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N g 1 2p g pg 1 p g P N g P N g 1 2p g y 1 2p g Ng p g y N p g i¼1 i¼1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N g 1 2p g pg 1 p g N g p g 1 p g ¼ N g y N p g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N g p g 1 p g Appendix B We want to derive the test statistic Z R of the Redemeier test as it is shown in (15.22) according to Redemeier et a (1991). We start with the MSE from modue 1 as MSE m1 ¼ 1 N X N i¼1 2 1 X N y i ^p i;m1 ¼ y i 2 y i ^p i;m1 þ ^p 2 i;m1 N i¼1 (15.24) Because of the randomness of the defauts the MSE wi differ from its expected vaue EðMSE m1 Þ ¼ 1 X N N i¼1 2 1 X N p i ^p i;m1 ¼ N i¼1 p i 2 p i ^p i;m1 þ ^p 2 i;m1 (15.25) The difference of the reaized and the expected MSE for modue 1 is d m1 ¼ EðMSE m1 Þ MSE m1 ¼ 1 N X N i¼1 y i 2 y i ^p i;m1 ^p i þ 2 p i ^p i;m1 (15.26)

346 R. Rauhmeier The same consideration has to be done for modue 2: d m2 ¼ EðMSE m2 Þ MSE m2 ¼ 1 N X N i¼1 y i 2 y i ^p i;m2 ^p i þ 2 p i ^p i;m2 (15.27) To determine whether two sets of judgments are equay reaistic we compare the difference between d m1 and d m2 : d m1 d m2 ¼ 2 N X N i¼1 ^p i;m1 ^p i;m2 ð pi y i Þ (15.28) As it can be seen from (15.28) the true but unknown PD p i is sti required and has therefore be assessed. A choice might be to set a p i equa to the average of the corresponding judgments (^p i;m1,^p i;m2 ) (consensus forecast). 45 This seems to be a reasonabe choice since we presumed that each modue itsef has satisfied the nu hypothesis of being compatibe with the data. Using the consensus forecast p i ¼ 0:5 ^p i;m1 þ ^p i;m2 (15.29) we get d m1 d m2 ¼ 2 N ¼ 1 N ¼ 1 N ¼ 1 N X N i¼1 X N i¼1 X N i¼1 X N i¼1 ^p i;m1 ^p i;m2 0:5 ^pi;m1 0:5 ^p i;m2 y i ^p 2 i;m1 ^p2 i;m2 2 y i ^p i;m1 ^p i;m2 h i y i 2 y i^p i;m1 þ ^p 2 i;m1 y i 2 y i ^p i;m2 þ ^p 2 i;m2 2 1 X N y i ^p i;m1 N i¼1 ¼ MSE m1 MSE m2 2 y i ^p i;m2 (15.30) It is interesting that in the case we use the consensus forecast for substituting p i the term d m1 d m2 is simpy the difference of the two reaized MSEs. In the next step we cacuate the variance using the fact that the expected vaue of d m1 d m2 is zero under the nu hypothesis, see (15.23). 45 Other approaches are possibe, e.g. one may get the true p i s from an externa source.

15 PD-Vaidation: Experience from Banking Practice 347 Varðd m1 d m2 Þ ¼ Var ¼ 4 N 2 X N 2 X N ^p i;m1 ^p i;m2 N i¼1 i¼1 Finay we get the test statistic ^p i;m1 ^p i;m2 ð pi y i Þ 2 pi ð1 p i Þ! (15.31) d m1 d m2 Z R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðd m1 d m2 Þ P N h ^p 2 i; m1 i ^p2 i; m2 2 ^p i; m1 ^p i; m2 yi (15.32) i¼1 ¼ P N h 2 i 0;5 ^p i; m1 ^p i; m2 ^pi; m1 þ ^p i;m2 2 ^pi; m1 ^p i; m2 i¼1 References Bathazar L (2004), PD Estimates for Base II, Risk, Apri, pp. 84 85. Base Committee on Banking Supervision (BCBS) (2005a), Base II: Internationa Convergence of Capita Measurement and Capita Standards: a Revised Framework Updated November 2005. http://www.bis.org/pub/bcbs118.pdf Base Committee on Banking Supervision (BCBS) (2005b), Studies on the Vaidation of Interna Rating Systems Revised Version May, Working Paper No. 14. http://www.bis.org/pub/ bcbs_wp14.pdf Brier G (1950), Verification of Forecasts Expressed in Terms of Probabiity, Monthy Weather Review 78 (1), pp. 1 3. DeGroot M, Fienberg S (1983), The Comparison and Evauation of Forecasters, The Statistician 32, pp. 12 22. Finger C (2001), The One-Factor CreditMetrics Mode in The New Base Capita Accord, RiskMetrics Journa 2 (1), pp. 9 18. Hamere A, Liebig T, R osch D (2003), Benchmarking Asset Correation, Risk 16, pp. 77 81. Hosmer D, Lemeshow S, Kar J (1988), Goodness-of-Fit Testing for Mutipe Logistic Regression Anaysis When the Estimated Probabiities are Sma, Biometrica Journa 30, pp. 911 924. Murphy A, Daan H (1985), Forecast Evauation, in: Murphy A, Katz R (eds.), Probabiity, Statistics, and Decision Making in the Atmospheric Sciences, Westview Press, Bouder, pp. 379 438. Redemeier DA, Boch DA, Hickman DH (1991), Assessing Predictive Accuracy: How to Compare Brier Score, Journa of Cinica Epidemioogy 44 (11), pp. 1141 1146. Scheue H, Rauhmeier R (2005), Rating Properties and their Impication on Base II-Capita, Risk 18, pp. 78 81. Spiegehater D (1986), Probabiistic Prediction in Patient Management and Cinica Trais, Statistics in Medicine 5, pp. 421 433.

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Chapter 16 Deveopment of Stress Tests for Credit Portfoios Voker Matthias Gundach 16.1 Introduction Advanced portfoio modes combined with naive reiance on statistics in credit risk estimations run the danger of underestimating atent risks and negecting the peri arising from very rare, but not unreaistic risk consteations. The atter might be caused by abnorma economic conditions or dramatic events for the portfoio of a singe credit institute or a compete market. This incudes events of a poitica or economic nature. To imit the impact of such sudden incidents, the study of fictiona perturbations and shock testing the robustness/vunerabiity of risk characteristics is required. This procedure is known as stress testing. It aows the review and actuaisation of risk strategies, risk capacities and capita aocation. Thus it can pay an important roe in risk controing and management in a credit institute. This view is shared by the banking supervision, in particuar by the Base Committee on Banking Supervision of the Bank for Internationa Settements (BIS). Consequenty, stress testing for credit risk pays a roe in the reguatory requirements of the Revised Framework on the Internationa Convergence of Capita Measurements and Capita Standards (Base II). Nevertheess, it has not reached the standards of stress testing for market risk estimations, which has been common practice for severa years (see Breuer and Krenn 1999). In the foowing, we describe the purpose and signification of stress testing for credit risk evauations. Then we reca the reguatory requirements, in particuar of the Base II framework. We describe how stress tests work and present some weestabished forms of stress tests, a cassification for them and suggestions how to dea with them. We aso incude exampes for iustration. To concude, we offer a concept for an evoutionary way towards a stress testing procedure. This is done in view of the appicabiity of the procedure in banks. V.M. Gundach THM University of Appied Sciences, Giessen-Friedberg e-mai: matthias.gundach@mni.th-mittehessen.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_16, # Springer-Verag Berin Heideberg 2011 349

350 V.M. Gundach 16.2 The Purpose of Stress Testing Stress testing means (reguar) expeditions into an unknown, but important territory: the and of unexpected events and osses. It requires anticipating risks which coud, but need not arise in the future and resuts in the determination of possibe unexpected osses. As the atter are of immense reevance for financia institutions, there is growing interest in this topic. Whie it is aready an intrinsic task to gain enthusiasm amongst the senior risk management for the rather theoretica vaues of unexpected osses, it is even more difficut to achieve acceptance for the quantitative output of stress tests. It makes sense to reduce such evauations to (reative) comparisons of the unexpected osses in stress and norma situations. Moreover, there are various reasons for conducting stress testing due to the expicit or impicit reation between unexpected oss and economic capita or reguatory capita, respectivey. Crucia for the understanding of and the approach towards stress testing, is the definition of unexpected oss. Though it is cear that this quantity shoud be covered by economic capita, there is no genera agreement as to how to define unexpected oss. It is quite common to regard the difference between expected oss and the vaueat-risk (VaR) of a given confidence eve, or the expected shortfa exceeding the VaR, as unexpected oss. One of the probems with this approach is that such an unexpected oss might not ony be unexpected, but aso quite unreaistic, as its definition is purey of a statistica nature. Therefore, it is sensibe to use stress tests to underscore which osses amongst the unexpected are pausibe or to use the outcome of stress tests, instead of unexpected oss to determine economic capita. Though the idea of using stress tests for estimating economic capita seems quite straight forward, it is ony rarey reaized, as it requires reiabe occurrence probabiities for the stress events. With these, one coud use the expected oss under stress as an economic capita requirement. Nevertheess, stress tests are mainy used to chaenge the reguatory and economic capita requirements determined by unexpected oss cacuations. This can be done as a simpe test for the adequacy, but aso to derive a capita buffer for extreme osses exceeding the unexpected osses, and to define the risk appetite of a bank. For new credit products ike credit derivatives used for hedging against extreme osses it might be of particuar importance to conduct stress tests on the evauation and capita requirements. Using stress tests to evauate capita requirements has the additiona advantage of aowing the combination of different kind of risks; in particuar market risk, credit and iquidity risk, but aso operationa risk and other risks such as reputationa risk. Because time horizons for market and credit risk transactions are different, and it is common for banks to use different confidence eves for the cacuation of VaRs for credit and market risk (mainy due to the different time horizons), joint considerations of market and credit risk are difficut and sedom used. Reaistic stress scenarios infuencing various kinds of risk therefore coud ead to extreme osses, which coud be of enormous importance for controing risk and shoud be refected in the capita requirements.

16 Deveopment of Stress Tests for Credit Portfoios 351 In any case, there can be strong correations between the deveopments of market, iquidity and credit risk which coud resut in extreme osses and shoud not be negected. Consequenty, investigations into events causing simutaneous increases in market and credit risk are more than reasonabe. An overview over severa types of risk reevant for stress testing can be found in Baschke et a. (2001). The quantitative outcome of stress testing can be used in severa paces for portfoio and risk management: Risk buffers can be determined and/or tested against extreme osses The risk capacity of a financia institution can be determined and/or tested against extreme osses Limits for sub-portfoios can be fixed to avoid given amounts of extreme osses Risk poicy, risk toerance and risk appetite can be tested by visuaising the risk/ return under abnorma market conditions Such approaches focusing on quantitative resuts might be of particuar interest for sub-portfoios (ike some country-portfoios), where the historic voatiity of the respective oans is ow, but drastic changes in risk reevant parameters cannot be excuded. Stress tests shoud not ony be reduced to their purey quantitative features. They can and shoud aso pay a major roe in the portfoio management of a bank, as they offer the possibiity of testing the structure and robustness of a portfoio against perturbations and shocks. In particuar they can represent a worthwhie too to Identify potentia risks and ocate the weak spots of a portfoio Study effects of new intricate credit products Guide discussion on unfavourabe deveopments ike crises and abnorma market conditions, which cannot be excuded Hep monitor important sub-portfoios exhibiting arge exposures or extreme vunerabiity to changes in the market Derive some need for action to reduce the risk of extreme osses and hence economic capita, and mitigate the vunerabiity to important risk reevant effects Test the portfoio diversification by introducing additiona (impicit) correations Question the bank s attitude towards risk 16.3 Reguatory Requirements As we have seen in the previous section, the benefits of using stress tests are manifod for the controing and portfoio management. Tribute to this fact is aso paid by the Base II Revised Framework, see Base Committee on Banking Supervision (2004). Here stress testing appears in Piar 1 (about the minimum capita requirements) and Piar 2 (about the supervisory review process) for banks using the IRB approach. The target of the requirements is improved risk management.

352 V.M. Gundach The requirements in the Base II Revised Framework are not precise. They can be summarized as 1 : Task: Every IRB bank has to conduct sound, significant and meaningfu stress testing to assess the capita adequacy in a reasonaby conservative way. In particuar, major credit risk concentrations have to undergo periodic stress tests. Furthermore, stress tests shoud be integrated in the interna capita adequacy process, in particuar, risk management strategies to respond to the outcome of stress testing. Intention: Banks sha ensure that they dispose of enough capita to meet the reguatory capita requirements even in the case of stress. Requirements: Banks shoud identify possibe events and future changes in economic conditions, which coud have disadvantageous effects on their credit exposure. Moreover, the abiity of the bank to withstand these unfavourabe impairments has to be assessed. Design: A quantification of the impact on the parameters probabiity of defaut (PD), oss given defaut (LGD) and exposure at defaut (EAD) is required. Rating migrations shoud aso be taken into account. Specia notes on how to impement these requirements incude: The use of scenarios ike: Economic or industry downturn Market-risk events Liquidity shortage is recommended. Recession scenarios shoud be considered, worst-case scenarios are not required. Banks shoud use their own data for estimating rating migrations and integrate the insight of rating migrations in externa ratings. Banks shoud buid their stress testing aso on the study of the impact of smaer deterioration in the credit environment. Though the requirements for stress testing are mainy contained in Piar 1 of Base II, the method is a fundamenta part of Piar 2, since it is an important way of assessing capita adequacy. This expains the ack of extensive reguations for stress testing in that document as Piar 2 acknowedges the abiity to judge risk and use the right means for this procedure. As another consequence, not ony reguatory capita shoud be the focus of stress tests, but aso economic capita as the counterpart of the portfoio risk as seen by the bank. Not ony the BIS (see CGFS 2000, 2001 and 2005) promotes stress testing, but aso some centra banks and reguators 2 have taken care of this topic (e.g., Deutsche Bundesbank 2003 and 2004; Fender et a. 2001), in particuar regarding the stabiity 1 The exact formuations can be found in }434-}437, }765, }775 and }777 of BIS (2004). 2 Reguators are aso interested in contagion, i.e. the transmission of shocks in the financia system. This topic is not part of this contribution.

16 Deveopment of Stress Tests for Credit Portfoios 353 of financia systems. They have pubished statements which can be regarded as suppements to the Base II Revised Framework. These pubications give a better impression of the reguatory goas and basic conditions for stress testing, which can be summarized as: Stress tests shoud consider extreme deviations from norma deveopments and hence shoud invoke unreaistic, but sti pausibe situations, i.e. situations with ow probabiity of occurrence. Stress tests shoud aso consider consteations which might occur in future and which have not yet been observed. Financia institutions shoud aso use stress testing to become aware of their risk profie and to chaenge their business pans, target portfoios, risk poitics, etc. Stress testing shoud not ony be addressed to check the capita adequacy, but aso used to determine and question imits for awarding credit. Stress testing shoud not be treated ony as an amendment to the VaR-evauations for credit portfoios, but as a compimentary method, which contrasts the purey statistica approach of VaR-methods by incuding causay determined considerations for unexpected osses. In particuar, it can be used to specify extreme osses in a quaitative and quantitative way. 16.4 Risk Parameters for Stress Testing The centra point of the procedure of stress testing aso seen in Base II is the change in risk parameters. For reguatory capita, these parameters are given by the probabiity of defaut (PD), oss given defaut (LGD) and exposure at defaut (EAD). In this connection, a superior roe is in most cases payed by the variations of PD, as LGD and EAD are asting quantities which due to their definition shoud aready be conditioned to disadvantageous situations, namey the defaut of the obigor. The possibiities of stress effects are hence restricted, especiay for EAD. The atter might be worsened by a few exogenous factors such as the exchange rate, but they shoud aso be party considered in the usua EAD. The exogenous factors affecting the EAD might ony be of interest if they aso have an impact on the other risk parameters and hence coud ead to an accumuation of risky infuence. The possibe variances for the LGD depend heaviy on the procedure used to determine this quantity. Thus, deviations which might arise from the estimation methods, shoud be determined, as we as parts of the process that might depend on economic conditions. As the determination of the LGD is conditioned by definition to the unfavourabe situation of a defaut, it shoud take into account asting vaues for coateras, and ead to vaues that can be seen as conservative. Thus, there shoud not too many factors be eft, that coud ead to extreme changes for the LGD. Mainy the evauation of coatera coud have some infuence which cannot be negected when stressing the LGD. In particuar, it might be possibe that factors affecting the vaue of the coateras aso have an impact on other risk parameters and hence shoud be taken into account.

354 V.M. Gundach For stressing derivative products ike credit defaut swaps (CDS), credit defaut obigations (CDOs) and CDO 2 it might make sense to investigate the effects on the LGD. Very often these products contain everage effects or are opposed to systematic risk. These phenomena coud be observed for exampe in the subprime crisis, when the burst of the rea estate bubbe in the US had enormous effects on the vaue of houses and hence on the LGDs of corresponding credits, eading to even higher downgrades of LGDs for respective CDSs and CDOs. This might indicate how compex the evauation of LGDs can be. The PD is by far the most popuar risk parameter which is varied in stress tests. There are two main reasons why variations in the PD of an obigor can occur. On the one hand, the assignment of an obigor to a rating cass might change due to atered inputs for the rating process. On the other hand, the reaised defaut rates of the rating casses itsef might change, e.g., because of modified economic conditions and their impact on the performance of the oans. This aows two options for the design of the integration of PDs into stress testing: modifications either of the assignment to rating casses or of the PDs of the rating casses for stress tests. Atered assignments of rating casses for obigors in stress tests have the advantage that they aso aow the incusion of transitions to non-performing oans. The change of PDs corresponds to a change of rating cass. The possibe deviation in the assignment of rating casses can be promoted by the rating procedure. Thus, the possibiities of variances and the sensitivity of the input for the rating process shoud be investigated in order to get a first estimate for possibe deviations. Consequenty, as we as the anaysis of historic data for rating transitions, expert opinions on the rating methodoogy shoud be a part of the design process for the stress test. The modification of PDs for the rating casses, coud have its origin in systematic risk, i.e. in the dependence on risk drivers, one of the main topics in designing stress tests, as wi be discussed beow. Whie it is sensibe to estimate the voatiity of PDs in a first step and use the outcome of this procedure for tests on reguatory capita, the differentiation of the effects of systematic and idiosyncratic risk on PD deviations shoud be considered in a second step. This wi ead to more advanced and reaistic stress tests, in particuar on economic capita. An anaysis of the transition structure for rating casses might aso be used to determine PDs under stress conditions. The advantage of modifying PDs against modifying the assignment of rating casses is a greater variety for the choices of changes; the disadvantage is the absence of a modified assignment to the performing and non-performing portfoio. This has to take pace on top of the modification of PDs. Estimating economic capita PD, LGD and EAD might not be sufficient to design stress tests. In addition, parameters used for dispaying portfoio effects, incuding correations between the oans or the common dependence on risk drivers are needed. 3 Investigations on historic crises for credit risk show that correations 3 The basis for widey used portfoio modes ike CreditRiskþ or CreditMetrics, which are used by banks for estimating the VaR, are provided by factor modes. The (abstract) factors are used to

16 Deveopment of Stress Tests for Credit Portfoios 355 and risk concentration exhibit huge deviations in these circumstances. In any case, their variations shoud be considered in stress tests with portfoio modes if possibe. Some advanced modes for estimating economic capita might even require more information, in particuar economic conditions. Portfoio modes such as CreditMetrics not ony consider the defaut of oans, but aso the change of vaue by using migration probabiities. In this case, the migration probabiities shoud be stressed in the same way as PDs. Stressing of risk parameters in tests need not take pace for the whoe portfoio, but ony for parts of it. Aso, the strength of the parameter modification might depend on sub-portfoios and credit products. Such approaches are used to pay tribute to different sensitivities of parts of the portfoio to risk reevant infuences or to study the vunerabiity of certain (important) sub-portfoios. They can be particuary interesting for investigations on economic capita with the hep of portfoio modes. In these cases, parameter changes for parts of the portfoio need not have a smaer impact than anaogous variations for the whoe portfoio due to effects of concentration risk or diversification, respectivey. 16.5 Evauating Stress Tests As stress testing shoud be a part of the interna capita adequacy process, there shoud be an understanding of how to use the outcome of stress tests for controing and managing portfoio risk. The starting point for this shoud be the reguatory and economic capita as output of the underying stress tests. The first task consists of checking whether the financia institution hods sufficient capita to aso cover the requirements in the stress situation. As there shoud be imits, buffers and poicies to guarantee this, the evauation of stress testing shoud be aso used to review these toos. Since the atter might be appicabe to different portfoio eves (e.g. imits for sub-portfoios, countries, or obigors) they shoud be checked in detai. The concept of stress testing woud be incompete without knowing when action has to be considered as a resut of the outcome of tests. It makes sense to introduce indicators and threshods for suggesting when To inform management about potentia critica deveopments To deveop guideines for new business in order to avoid the extension of existing risky consteations To reduce risk for the portfoio or sub-portfoios with the hep of securitisation and syndication To readjust an existing imit management system and the capita buffer for credit risk To re-think the risk poicy and risk toerance present systematic risk affecting the oans. In these modes it makes sense to stress the strength of the dependence on the factors and the factors themseves.

356 V.M. Gundach Indicators for the ca on action coud be The increase of risk indicators as expected oss, unexpected oss, expected shortfa over a threshod or by a specified factor The increase of capita requirements (reguatory or economic) over a threshod or by a specified factor The sovency ratio of capita and capita requirements under a threshod A ow sovency eve for meeting the economic capita requirements under stress A specified quantie of the oss distribution for the portfoio under stress conditions does not ie within a specified quantie of the oss distribution for the origina portfoio Expected oss for the portfoio under stress conditions overaps the standard risk costs (cacuated on the basis of expected oss for the duration of the oans) by a specified factor or gets too cose to the unexpected oss for the unstressed portfoio The risk/return ies above a specified threshod, where risk is measured in terms of unexpected oss The interpretation of the outcome of stress tests on economic capita can easiy ead to misapprehensions, in particuar if the capita requirement is estimated on the basis of a VaR for a rather arge confidence eve. The motivation for the atter approach is the avoidance of insovency by hoding enough capita, except for some very rare events. Stress tests might simuate situations coming quite cose to these rare events. Adhering to the arge confidence eves for estimating economic capita, offers the possibiity of comparing the capita requirements under different conditions, but the resuting VaR or economic capita shoud not be used to question the sovency. In fact, it shoud be considered whether to use adapted confidence eves for stress testing or to rethink the appropriateness of high confidence eves. One can see the probabiity of occurrence or the pausibiity of a stress test as a reated probem. We refer to a detaied discussion on this topic and an approach to resoution to Breuer and Krenn (2001). 16.6 Cassifying Stress Tests According to reguatory requirements, a bank shoud perform stress tests on its reguatory as we as its economic capita. This differentiation of stress tests is not essentia and mainy technica, as the input for determining these two forms of capita might be quite different as described in the previous section. Another technica reason for differentiating stress tests is the division into performing and non-performing oans, as their respective capita requirements foow different rues. For non-performing oans, oss provisions have to be made. Thus one has to consider the foowing cases for stress tests:

16 Deveopment of Stress Tests for Credit Portfoios 357 A performing oan gets downgraded but remains a performing oan the estimation of economic capita invoves updated risk parameters. A performing oan gets downgraded and becomes a non-performing oan provisions have to be estimated invoving the net exposures cacuated with the LGD. A non-performing oan deteriorates the provisions have to be increased on the basis of a decined LGD. As aready discussed in the previous section, defauts can be incuded in stress tests via a worsened assignment to rating casses. If stress tests focus on PDs rather than rating casses, then stress rates for the transition of performing to nonperforming oans are required for the same purpose. Ideay, they depend on ratings, branches, economic states, etc. and are appied to the portfoio after stressing the PDs. Moreover, the methodoogy of a bank to determine the voume of the provision for a defauted credit shoud be considered. A typica approach is to equate the oss amount given the defaut (i.e. the product of LGD with the exposure) with the provision. Typica ways to categorize stress tests can be taken over from market risk. They are we documented in the iterature (CGFS 2005 and Deutsche Bundesbank 2003 and 2004). The most important way to cassify stress tests is via the methodoogy. One can distinguish stress tests with respect to techniques in statisticay and mode based methods, and with respect to conceptua design in sensitivity anaysis and scenario anaysis. Whie the atter is based on modeing economic variances, sensitivity anaysis is statisticay founded. The common basis for a these specifications is the eementary requirement for stress tests to perturb the risk parameters. These can be the basic risk parameters (EAD, LGD, PD), of the oans as aready mentioned for the tests on the reguatory capita. However, these can aso be parameters used in a portfoio mode ike asset correations or dependencies on systematic risk drivers. The easiest way to perform stress tests is a direct modification of the risk parameters and beongs to the cass of sensitivity anaysis. The goa is to study the impact of major changes in the parameters on the portfoio vaues. For this method, one or more risk parameters are increased (simutaneousy) and the evauations are made for this new consteation. The increase of parameters shoud depend on statistica anaysis or/and expert opinion. As these stress tests are not inked to any event or context and are executed for a oans of a (sub-) portfoio, without respect to individua properties, we refer to them as fat or uniform stress tests. Most popuar are the fat stress tests for PDs, where the increase of the defaut rates can be derived from transition rates between the rating casses. An advantage of these tests is the possibiity of performing them simutaneousy at different financia institutions and aggregating these resuts to check the financia stabiity of a system. This is done by severa centra banks. Such tests are suited to checking the space and buffer for capita requirements, but it does not mean any hep for portfoio and risk management.

358 V.M. Gundach Mode based methods for stress testing incorporate observabe risk drivers, in particuar, macroeconomic variabes for representing the changes of risk parameters. In the foowing, we wi refer to these risk drivers as risk factors. The respective methods rey on the existence of a mode mainy based on econometrica methods that expains the variations of the risk parameters by changes of such risk factors. One can distinguish univariate stress tests, which are defined by the use of a singe, isoated risk factor, and mutivariate stress tests, where severa factors are changed simutaneousy. These tests can be seen as a refinement of those previousy described: stressing the risk factors eads to modified risk parameters which are finay used for the evauation of the capita requirements. Note that risk factors can have quite different effects on risk parameters throughout a portfoio. Changes in the risk factors can ead to upgrades as we as downgrades of risk parameters. For exampe, an increase in price of resources such as oi or energy can have a negative impact on PDs in the automobie or any other industry consuming ots of energy, but it coud have a positive impact on the PDs in the country trading these resources. By using univariate stress tests, banks can study specific and especiay reevant impacts on their portfoios. This has the benefit of isoating the infuence of an important observabe quantity. Consequenty, it can be used to identify weak spots in the portfoio structure. Thus, univariate stress tests represent another kind of sensitivity anaysis, now in terms of risk factors instead of risk parameters. They have the disadvantage of possiby eading to an underestimation of risk by negecting potentia effects resuting from possibe correations of risk factors. This shortcoming is aboished by using mutivariate stress tests. The price is the reiance on additiona statistica anaysis, assumptions or the estabishment of another mode describing the correation of the risk factors invoved. This is done in a framework known as scenario anaysis, where hypothetica, historica and statisticay determined scenarios are distinguished. It resuts in the determination of stress vaues for the risk factors which are used to evauate stress vaues for the risk parameters. With respect to the design of scenarios, we can discriminate approaches driven by the portfoio (bottom-up approaches) and driven by events (top-down approaches). Bottom-up approaches tend to use the resuts of sensitivity anaysis to identify sensitive dependence on risk factors as starting points. As a consequence, those scenarios are chosen which invove risk factors having the argest impact. For exampe, for a bank focusing on rea estate, GDP, empoyment rate, infation rate, spending capacity in the countries, it is acting in, wi be of more reevance than the oi price, exchange rates, etc. Thus, it wi ook for scenarios invoving the reevant risk factors. Top-down approaches start with a chosen scenario, e.g., the terror attack in New York on September 11, 2001, and require the anaysis of the impact of this scenario on the portfoio. The task in this situation is to identify those tests which cause the most dramatic and reevant changes. Historica scenarios are typica exampes of top-down approaches. They refer to extreme consteations of the risk factors which were observed in the past and in the majority of the cases can be reated to historica events and crises. They are transferred to the current situation and portfoio. This can be seen as a disadvantage

16 Deveopment of Stress Tests for Credit Portfoios 359 of this approach, as the transferred vaues may no onger be reaistic. Another drawback is that generay, it is not possibe to specify the probabiity of the scenario occurring. Aso, statisticay determined scenarios might depend on historica data. They are based on the (joint) statistica distribution of risk factors. In this approach, scenarios might be specified by quanties of such distributions. Whist it might be very difficut to produce suitabe distributions in particuar, joint distributions, the advantage is that it is possibe to evauate the probabiity of the scenario occurring as this is given by the compement of the confidence eve used for the quantie. The existence of such probabiities of occurrence aows the cacuation of expected extreme osses which can be used for the estimation of economic capita. The crucia point of this approach is the generation of a suitabe risk factor distribution. Ony if the atter is chosen compatibe with the state of economy, (hence does not rey too heaviy on historic data), can usefu concusions for the management of the portfoio be derived. Finay, there are hypothetica scenarios which focus on possibe rare events that might have an important impact on the portfoio, but have not been observed yet in the form they are considered. The crucia point is the presentation of the consequences of the event on the risk factors. For the estimation of this expert opinion, it is necessary to accompany the macro-economic modeing of the dependence of the risk parameters on risk factors. If macroeconomic parameters are not part of the input for determining the risk parameters which are stressed, there are three steps required for macro stress tests. Firsty, it is necessary to mode the dependence of the risk parameters on the risk factors. Secondy, it is necessary to choose vaues for the risk factors which are representative for stress events. Since it is intended to reproduce correations and causa interreations between risk factors and stress events, intricate (macro-economic), methods of estimation and vaidation are needed. A disadvantage of hypothetica scenarios might be having to specify the probabiity of occurrence of such hypothetica scenarios. On the other hand, there is the major advantage of having forward-ooking scenarios which do not necessariy refect historica events. Thus, hypothetica scenarios present interesting adjuncts to VaR-based anaysis of portfoio risk and are a worthwhie too for portfoio management. The use of risk factors as in the mutivariate scenario anaysis has the additiona advantage of aowing common stress tests for credit, market and iquidity risk. Here, it is necessary to consider factors that infuence severa forms of risk or scenarios that invove risk factors for them. Hypothetica scenarios can aso be produced on the basis of expert opinions. Though this approach might have the disadvantage of being mathematicay/statisticay not as precise as the one based on macro-economic modeing, it in fact can have the advantage of understanding the risk profies of a portfoio. For this it is important to discuss with experts step by step a the possibe effects a scenario might have. If this is done with a detais, a perfect risk profie and a good insight in portfoio risk can be gained. It is aso possibe to go an even onger way: one can start with a so-caed risk map describing a potentia genera risks (e.g. cassified

360 V.M. Gundach with respect to their nature ike catastrophes, war and terror, oss of financia stabiity, etc.) and their main effects. Having identified the main genera risks for the portfoio it is possibe to use a so-caed risk monitor to zoom into these risks and identify the effects on the portfoio in more detai. Further anaysis with experts can then resut into the determination of hypothetica scenarios. 16.7 Conducting Stress Tests In the foowing section we wi discuss how the stress tests we have just introduced in the previous section, can be and are, appied in financia institutions. We try to provide detais how to determine and conduct stress tests, focussing mainy on the performing part of credit portfoios. 16.7.1 Uniform Stress Tests The most popuar stress tests in banks are uniform stress tests, in particuar for the PDs. The intention is to use increased PDs for the cacuation of economic or reguatory capita. In the easiest case, there is a fat increase rate for a PDs 4 of obigors or/and countries, but in genera, the change might depend on rating casses, branches, countries, regions, etc. We suggest severa ways to derive the stress PDs: 1. Anayse the defaut data with respect to the dependence on rating casses, branches, countries, regions, etc. This data coud originate from the bank s own portfoio or from rating agencies. Determine the deviations of the defaut rates from the PD. Another way to derive such variations might arise from the anaysis of spreads for respective credit derivatives. The stress PD then can be determined from the PD by adding the standard deviation, a quantie or other reevant characteristic of the deviation distribution. It might seem to be a good idea to use the quantie to determine aso a probabiity of the stress occurring, but one shoud question the quaity and the reevance of the distribution before using this approach. 2. Use migration rates (referring to the bank s own portfoio or coming from rating agencies), to determine transitions between rating casses. These transitions might depend on branches, countries, etc. In an intermediate step, stressed 4 Such stress tests are often used by centra banks to test the stabiity of the respective financia systems. In the studies in Deutsche Bundesbank (2003) PDs are increased by 30% and 60%, respectivey. These changes approximatey correspond to downgrades of Standard and Poor s ratings by one or two casses, respectivey. The atter is seen as conservative in that paper. Banks shoud anayse their defaut data to come up with their own rates of increase, which we expect to be in the worst case arger than 60%.

16 Deveopment of Stress Tests for Credit Portfoios 361 migration matrices can be generated by omitting rating upgrades, by conditioning on economic downturns (Bangia et a. (2002)), by uniformy increasing the downgrade rates at the expense of uniformy decreasing the upgrade rates or on the basis of a time series anaysis. Next, one can derive for every origina rating cass, a stressed rating cass by evauating quanties or any other characteristics for the transition probabiities. Consequenty, it is possibe to buid the stress test on the rating casses. Now, the stress test consists of repacing the origina rating cass by the stressed rating cass. Aternativey, one can repace the origina PD by the PD of the stressed rating cass. A different approach uses the stressed migration rates. Depending on their derivation, they possiby have to be caibrated to become transition probabiities. Then they can be used to cacuate an expected PD for every rating cass, which can pay the roe of a stressed PD. The decision as to which option shoud be chosen for determining the stress PD shoud depend on the data, which is avaiabe for statistica anaysis. Aso, expert opinions coud be a part of the process to generate the stress PDs. In particuar, it makes sense to study the deviations that can be caused by the rating process due to sensitive dependence on input parameters. This coud ead to an additiona add-on when generating the stress PDs. The preference for stressed PDs or stressed rating casses shoud depend on the possibiities of reaising the stress tests. Regarding the portfoio mode, the dependence of a PD on a branch or country in a rating cass coud for exampe represent a probem. A criterion in favour of stressed rating casses is the incusion of defauts. Such a stressing might ead to assignments of oans to casses beonging to the non-performing portfoio. These can be treated respectivey, i.e. instead of the capita requirements, provisions can be cacuated. In the case that PDs are stressed, instead of rating casses, one shoud first consider the stressing of the PDs in the portfoio and then the stressing of transition rates to the non-performing part of the portfoio. In this context, Monte Caro simuations can be used to estimate capita requirements for the performing, and provisions for the non-performing part of the portfoio. Transition rates to the non-performing portfoio, usuay corresponding to defaut rates, can be stressed in the same form and with the same methods as the PDs. The same hods for migration rates between rating casses which are used in some portfoio modes. Fat stress tests for LGDs coud aso be based on statistica anaysis, in this case for oss data. The approach to determine and study deviations in oss rates is anaogous to the one for defaut rates. Expert opinion coud pay a bigger roe. An exampe of an interesting stress test coud be provided by a significant fa in rea estate prices in some markets. Uniform stressing of EAD is often not reevant. Deviations of this quantity mainy depend on individua properties of the oans. Variations of exchange rates can be seen as the most important infuence on the deviations of EAD from the expected vaues. It is commendabe to investigate this effect separatey.

362 V.M. Gundach For uniform stressing of parameters used in portfoio modes, it seems to be the best to rey on expert opinions, as it is very difficut to detect and statisticay verify, the effect of these parameters on the deviations from expected or predicted vaues of defauts and osses. Whie it is aready rather intrinsic to determine suitabe parameter vaues for the uniform tests invoving singe parameters, it even becomes more difficut to do this for severa parameters at the same time. Experience derived from historic observations and expert opinion seems to be indispensabe in this situation. 16.7.2 Sensitivity Anaysis for Risk Factors This kind of stress testing is very popuar for market risk, where risk factors can easiy be identified, but it can aso be seen as basic for scenario anaysis. This is due to the crucia task of recognising suitabe risk factors and introducing a vaid macroeconomic mode for the dependence of risk parameters on the risk factors representing the state of the business cyce. Of course, there are obvious candidates for risk factors ike interest rates, infation rates, stock market indices, credit spreads, exchange rates, annua growth in GDP, oi price, etc. (Kairai and Scheicher (2002)). Others might depend on the portfoio of the financia institute and shoud be evident for good risk managers. Using time series for the risk factors on reevant markets, as we as for the deviations of risk parameters and standard methods of statistica anaysis ike discriminant anaysis, one shoud try to deveop a macroeconomic mode and determine those factors suitabe to describe the evoution of risk parameters. Typicay, the impact of stress on the risk parameters or directy on credit oss characteristics is modeed using inear regression. One of the probems invoves determining the extent to which the risk factors must be restricted, whist aowing a feasibe mode. Discovering which risk factors have the biggest impact on the portfoio risk in terms of the VaR or whatever is used for the evauation of unexpected osses, is the target and the benefit of sensitivity anaysis. Stressing is anaogous to the uniform stress test on risk parameters. Stress vaues for a singe risk factor are fixed on the basis of statistica anaysis or expert opinion. The consequences for the risk parameters are cacuated with the hep of the macroeconomic mode and the modified vaues for the risk parameters are finay used for evauating capita requirements. Risk factors which have an impact on severa risk parameters and which aso pay a roe for stress testing market risk, might be of particuar interest. Sensitivity anaysis coud aso be used to verify the uniform stress testing by checking whether the range of parameter changes due to sensitivity anaysis is aso covered by the fat stress tests. Moreover, it can be seen as a way to pre-seect scenarios: ony those historica or hypothetica scenarios which invove risk factors showing some essentia effects in the sensitivity anaysis are worth considering.

16 Deveopment of Stress Tests for Credit Portfoios 363 16.7.3 Scenario Anaysis Having specified the reevant risk factors, one can aunch historic scenarios, statistica seection of scenarios and hypothetica scenarios. These different methods shoud party be seen as compementing each other. They can aso be used for specifying, supporting and accentuating the other. 16.7.3.1 Historica Scenarios Historica scenarios are easy to impement, as one ony has to transfer the vaues of risk factors corresponding to a historic event to the current situation. In most cases, it does not make sense to copy the vaue of the risk factors, but to determine the change of vaue (either in absoute or in reative form) which is accompanied by the insertion of the event and assume it aso appies to the actua evauation. The foowing events are most popuar for historica scenarios: Oi crisis 1973/1974 Stock market crash (Back Monday 1987, goba bond price crash 1994, Asia 1998) Terrorist attacks (New York 9/11 2001, Madrid 2004) or wars (Guf war 1990/ 1991, Iraq war 2003) Currency crisis (Asian 1997, European Exchange Rate Mechanism crisis 1992, Mexican Peso crisis 1994) Emerging market crisis Faiure of LTCM 5 and/or Russian defaut (1998) Though the impications of historica scenario anaysis for risk management might be restricted due its backward ooking approach, there are good reasons to use it. First of a, there are interesting historic scenarios which certainy woud not have been considered, as they happened by accident, i.e. the probabiity of occurrence woud have been seen too ow to ook at them. Exampes of this case are provided by the coincidence of the faiure of LTCM and the Russian defaut or the 1994 goba bond price crash. It can be assumed that both events woud rarey have contributed to the VaR at the time of their occurrence, due to the extremey ow probabiity of joint occurrence for the singe incidents. 6 5 The hedge fund Long-Term Capita Management (LTCM) with huge, but we diversified risk positions was affected in 1998 by a market-wide uprising of risk boosted by the Russia crisis. This ed to arge osses of equity vaue. Ony a joint cooperation of severa US-investment banks under the guidance of the Federa Reserve coud avoid the compete defaut of the fund and a systemic crisis in the word s financia system. 6 The movements of government bond yieds in the US, Europe and Japan are usuay seen as uncorreated. Hence their joint upward movement in 1994 can be seen as an extremey unikey event.

364 V.M. Gundach There is aso much to earn about stress testing and scenario anaysis from working with historic scenarios. On the one hand, the atter can be used to check the vaidity of the uniform stress tests and sensitivity anaysis, on the other hand, they can be very hepfu in designing hypothetica scenarios. Thus, the anaysis of historica scenarios offers the unique possibiity of earning about the joint occurrence of major changes to different risk factors and the interaction of severa types of risks, e.g., the impact of credit risk events on iquidity risk. For these reasons, we regard historica scenario anaysis as a worthwhie part of estabishing a stress testing framework, but not necessariy as an essentia part of managing and controing risk. 16.7.3.2 Statisticay Determined Scenarios A specia roe is payed by the anaysis of scenarios which are chosen on the basis of risk factor distributions. These are not directy reated to the other types of scenario anaysis. Centra to this approach is the use of (joint) risk factor distributions. Whie it shoud not be too difficut for isoated common risk to generate such distributions on the basis of historic data, a situation invoving severa factors can be far more intricate. Nevertheess, distributions generated from historic data might not be sufficient. It woud be much better to use distributions conditioned to the situation appying at the time of stress testing. This coud represent a rea probem. We woud ike to point out that ony in the case of a reiabe factor distribution, shoud this approach be used. If expected osses conditioned to a quantie are evauated in order to interpret them as unexpected osses and treat them as economica capita requirement, then the risk factor distribution shoud aso be conditioned to the given (economic) situation. 16.7.3.3 Hypothetica Scenarios Hypothetica scenario anaysis is the most advanced means of stress testing in risk management. It shoud combine experience in anaysing risk reevant events with expert opinion on the portfoio, as we as the economic conditions and statistica competency. The impementation of hypothetica scenario anaysis is anaogous to the one for historic scenarios. The ony difference is provided by the choice of vaues for the risk factors. This can be based on or derived from historica data, but expert opinion might aso be used to fix reevant vaues. The choice of scenarios shoud refect the focus of the portfoio for which the stress test is conducted and shoud have the most vunerabe parts of it as the target. Common scenarios (together with risk factors invoved) are provided by the foowing:

16 Deveopment of Stress Tests for Credit Portfoios 365 Significant rise in oi price (increased oi price, reduced annua growth in GDP to describe weakened economic growth, indices describing increased consumer prices, etc.) Major increase of interest rates (indices describing the voatiity of financia markets, increased spreads, reduced annua growth in GDP to describe weakened economic growth, voatiity of exchange rates, consumer indices, etc.) Drop in goba demand (reduced annua growth in GDP, stock market indices, consumer indices, etc.) Emerging market crisis (reduced annua growth in GDP to describe weakened economic growth, widened sovereign credit spreads, decine in stock prices, etc.) Burst of economic bubbes ike the ones on the rea estate markets in the US, UK or Spain in 2007/2008 (reduced annua growth in GDP, drop in exchange rate, widened sovereign credit spreads, reduced consumer rates, etc.) Hypothetica scenarios have the additiona advantage that they can take into account recent deveopments, events, news and prospects. Note that scenarios invoving market parameters ike interest rates are we suited for combinations with stress tests on market and iquidity risk. 16.8 Exampes In the foowing we wi present the outcome of some stress tests on a virtua portfoio to iustrate the possibe phenomena, the range of appications and advantages corresponding to the tests. The portfoio consists of 10,000 oans and exhibits a voume of 159 biion EUR. The oans are normay distributed over 18 rating casses (PDs between 0.03% and 20% and a mean of 0.6%) and LGDs (ranging from 5 to 50% with a mean of 24%). Moreover, they are gamma-distributed with respect to exposure size (ranging from 2.000 EUR to 100 miion EUR with mean 1 miion EUR). To determine economic capita, we empoy the we known portfoio mode CreditRisk þ (Gundach and Lehrbass 2004). We use it here as a six-factor-mode, this means that we incorporate six (abstract) factors corresponding to so-caed sectors (rea estate, transport, energy, resources, airpanes, manufacturing) which represent systematic risk drivers. For our version of CreditRisk +, each obigor j is assigned exacty to one sector k ¼ k(j). This is done according to a weight w j, 0 w j 1. For each sector k there is a corresponding random risk factor S k, which is used to modify the PD p j to r j via r j ¼ p j w j S kðjþ : (16.1) The random factors S k have mean 1 and are gamma-distributed with one parameter s k corresponding to the variance of the distribution. Correations in CreditRisk +

366 V.M. Gundach are thus introduced via the S k, i.e. in our CreditRisk + -version, ony correations between obigors from the same sector are sustained. The strength of the correations depends on the weights w j and the variation s k. These parameters can both be modified in stress tests, though it seems more natura to increase the s k s. The oans in the portfoio are randomy distributed over the six sectors, representing systematic risk, and 13 countries, which pay a roe in some of the scenarios. The dependence of the oans on respective systematic risk factors varies between 25 and 75% and is randomy distributed in each sector. The sectoria variation parameters s k s are cacuated from the voatiities of the PDs according to some suggestion from the origina version of CreditRisk + and range between 1.8 and 2.6. In the stress tests we ony take account of the dependence of the risk parameter PD, on risk factors b i. When modeing this interreation, we used a simpe inear regression to predict the changes of rating agencies defaut rates for the sector and country division of the portfoio and transferred this dependence to the PDs p j used in our mode p i ¼ X i x jib i þ u j : (16.2) Here the u j s represent residua variabes and the indices refer to a cassification of PDs according to sectors and countries. Due to the sma amount of data and the crude portfoio division, we ended with a rather simpe mode for the PDs with respect to their assignment to sectors and countries invoving ony an oi price index, the S&P 500-Index, the EURIBOR interest rate, the EUR/USD exchange rate and the GDP of the USA and EU. We performed severa stress tests on the virtua portfoio. The evauation of these tests takes pace in terms of expected oss, reguatory and economic capita. For the atter, we cacuate the unexpected oss as the difference between VaR for a confidence eve of 99.99% and expected oss. We focus on the outcome for the whoe portfoio, but aso report on interesting phenomena for sub-portfoios. The cacuations of reguatory capita are based on the Base II IRBA approach for corporates, whie the estimations of VaR are done with CreditRisk +. Loss provisions are aso considered in some tests. In the case that the assignment of obigors to rating casses is stressed, non-performing oans and hence candidates for oan provisions are impicity given. In other cases, they are determined for each rating cass according to a stressed PD. The voume of the respective portfoio is reduced respectivey. We have considered the foowing stress tests, incuding uniform stress tests, sensitivity anaysis, historica and hypothetica scenario anaysis: 1. Fat increase of a PDs by a rate of 50%, (a) with and (b) without oan oss provisions 2. Fat increase of a PDs by a rate of 100% (a) with and (b) without oss provisions 3. Uniform upgrade of a rating casses by one 4. Fat increase of a LGDs by 5%

16 Deveopment of Stress Tests for Credit Portfoios 367 5. Fat increase of a PDs by a rate of 50% and a LGDs by 5% 6. Fat increase of a sectoria variances s k by a rate of 50% 7. Fat increase of a LGDs by 10% for rea estates in UK and USA (burst of rea estate bubbe) 8. Drop of stock market index (S&P500-Index) by 25% 9. Rise of oi price by 40% 10. September 11 (drop of oi price by 25%, of S&P-Index by 5.5%, EURIBOR by 25%) 11. Recession USA (drop of S&P-Index by 10%, GDP of USA by 5%, GDP of EU by 2%, increase of EUR/USD-exchange rate by 20%) The outcome is summarised in the foowing tabe where a isted vaues are in miion EUR (Tabe 16.1): The incusion of oss provisions does not seem to pay a major roe in the overa outcome of stress testing, as the sum of the provisions and the economic capita is rather sma. Nevertheess, the discrimination of economic capita and provisions (in particuar with the comparison of the atter with expected oss), is quite interesting. Aso, the distinction between stressing PDs and stressing the Tabe 16.1 Outcome of stress testing on a virtua portfoio No. Stress test Reguatory capita Economic capita Expected oss 0 None (Basis 3,041 1,650 235 0 portfoio) Loss provision 1a) PD * 150% 3,715 2,458 353 0 1b) PD * 150% with 3,631 2,255 320 332 provisions 2a) PD * 200% 4,238 3,267 470 0 2b) PD * 200% with 4,151 2,996 427 332 provisions 3 Rating cass þ 1 3,451 1,911 273 376 4 LGD þ 5% 3,676 1,985 283 0 5 LGD þ 5%, 4,490 3,935 567 0 PD * 150% 6 Systematic 3,041 3,041 235 0 factor * 150% 7 Rea estate bubbe Sectoria increase of economic capita 3,106 1,686 240 0 32% for rea estates, 45% for UK and USA 8 Stock price decine 3,591 2,368 329 0 58% for USA, Western Europe, Japan 9 Rise of oi price 3,430 2,057 300 0 65% for transport and airpanes 10 Terror attack New York September 11 3,897 2,622 399 0 77% for USA, Western Europe, Japan 11 Recession USA 3,688 2,307 351 0 68% for USA and South America, 57% for airpanes

368 V.M. Gundach assignment to rating casses has a rather imited impact on the resut of the stress testing. Furthermore, it is not a surprise that stress testing has a arger impact on economic capita than on reguatory capita. The significant diversity of impact on the sectors and countries by the scenario anaysis underscores the importance of this kind of stress testing for detecting weak spots in the portfoio and for portfoio management. As the portfoio used here is rather we diversified, the effects woud be arger in a rea portfoio. Aso, the simutaneous stressing of severa risk parameters has major effects. This is underined by the joint increase of PDs and LGDs. Aso, the roe of parameters describing systematic risk cannot be overestimated, as is indicated by the test given by the increase of systematic risk factors. Some of the scenarios ack the exhibition effects one woud expect (ike a major deterioration of airpane industry in the historic scenario concerning the terrorist attacks of September 11), which coud not be indicated by the inear regression, but which coud be produced in the design of the stress test using expert opinion. 16.9 Concusion Stress testing is a management too for estimating the impact on a portfoio of a specific event, an economic consteation or a drastic change in risk reevant input parameters, which are exceptiona, even abnorma, but pausibe and can cause arge osses. It can be seen as an amendment as we as a compement to VaR-based evauations of risk. It aows the combinations of statistica anaysis and expert opinions for generating reevant and usefu predictions on the imits for unexpected osses. Stress testing shoud not ony be seen as a risk management method though it can be used in various ways, but aso as an means towards anaysing risk and risk reevant consteations. In particuar, it shoud ead to a higher awareness and sensitivity towards risk. This requires a better knowedge of risk drivers, portfoio structure and the deveopment of risk concentrations. It cannot be achieved in a standard way. Instead experience, research and sustained investigations are required. In particuar it makes sense to use an evoutionary approach to overcome the compexity of requirements for stress testing. We woud ike to make the foowing suggestion as an evoutionary way towards a reasonabe and feasibe framework of stress testing. The basis of stress tests is provided by rich data for defauts, rating transitions and osses. The starting point for the deveopment of stress tests shoud be an anaysis of the voatiities of these rates and estimations for bounds on deviations for them. The statistica anaysis shoud be accompanied by investigations of the reasons for the deviations. It shoud be studied which fraction of the deviations arise from the methodoogy of the rating processes and which from changes in the economic, poitica, etc. environment. Expert opinion shoud be used to estimate bounds for the deviations arising from the methodoogy. Statistica anaysis shoud ead to an

16 Deveopment of Stress Tests for Credit Portfoios 369 identification and quantification of the exogenous risk factors having the main impact on the risk parameters needed to determine capita requirements. The combination of these two procedures shoud enabe the estabishment of uniform stress testing. The anaysis of defaut and oss data with respect to estimating deviations from the risk parameters shoud be foowed by statistica anaysis of the dependence of these deviations on risk factors and an identification of the most reevant factors. For the atter, first considerations of historic events which are known to have a arge impact on portfoio risk shoud aso be taken into account. These investigations shoud cuminate in a macroeconomic mode for the dependence of risk parameters on risk factors. With this mode sensitivity, anaysis for risk factors can be performed. The outcome of these stress tests can be used to check whether the uniform stress tests invove sufficient variations of risk parameters to cover the resuts of univariate stress tests. As a consequence, the uniform stress tests might have to be readjusted. Moreover, the sensitivity anaysis shoud aso be used to check whether the chosen risk factors are contributing to drastic changes in the portfoio. If this not the case, they shoud be negected for further stress tests. The invovement of reevant risk factors shoud aso be a good criterion for picking historica and hypothetica scenarios. It makes sense to consider historica scenarios first in order to benefit from the experience with historica data. This experience shoud aso incude the consideration of the interpay of different kinds of risks ike market, credit, operationa, iquidity risk, etc. The design of hypothetica scenario anaysis shoud be seen as the highight and cumination point of the stress testing framework. Scenario anaysis based on statistica anaysis is a method which is not connected too cosey with the others. Nevertheess, a ot of preiminary work has to be done to generate reiabe tests of this kind. The main probem is the generation of probabiity distributions for the risk factors, in particuar joint distributions and distributions conditioned on actua (economic) situations. The evoutionary approach towards a feasibe framework for stress testing can be summarized by the chart in Fig. 16.1. Having estabished a stress testing framework, we recommend Reguar uniform stress tests for reguatory and economic capita in order to provide a possibiity for evauating the changes made to the portfoio in terms of possibe extreme osses, and Hypothetica scenario anaysis suitabe to the actua portfoio structure and the conditions provided by the economy, poitics, nature, etc. The atter shoud party be combined with stress tests on market and iquidity risk. Aso, effects on reputationa and other risks shoud not be negected. Furthermore, one shoud have in mind that a crisis might have a onger horizon than 1 year, the typica period for evauations of risk, even in the common stress scenarios.

370 V.M. Gundach Stress tests Statistica anaysis and research Anayse methods for risk Parameter determination w.r.t. possibiities for deviations Anayse defaut and oss data w.r.t. possibiities for deviations of risk parameters Design and reaisation of uniform stress tests Anayse dependence of risk parameter deviations on risk factors Anayse historic events with high impact on portfoio risk vaidation Determine risk factors and their possibe deviations reevant for the portfoio and stress testing Reaisation of sensitivity anaysis for risk factors seection Reaisation of historic scenario anaysis Reaisation of statisticay based scenario anaysis vaidation Deveop a macroeconomica mode for the dependence of risk parameters on risk factors Determine a distribution for the (historic) deviations of the reevant risk factors Investigate the interpay of credit risk, market risk and other sorts of risk Design and reaisation of hypothetica scenario anaysis Fig. 16.1 Deveopment of a stress testing framework References Bangia A, Diebod F, Schuermann T (2002), Ratings Migration and the Business Cyce, With Appication to Credit Portfoio Stress Testing, Journa of Banking and Finance 26 (2 3), March, pp. 445 474. Base Committee on Banking Supervision (2004), Internationa Convergence of Capita Measurement and Capita Standards, Bank for Internationa Settements. Baschke W, Jones M, Majnoni G (2001), Stress Testing of Financia Systems: An Overview of Issues, Methodoogies, and FSAP Experiences, IMF Working Paper. Breuer T, Krenn G (1999), Stress Testing Guideines on Market Risk, Oesterreichische Nationabank, Vo. 5. Breuer T, Krenn G (2001), What is a Pausibe Stress Scenario, in Kuncheva L et a. (eds.): Computationa Inteigence: Methods and Appications, ICSC Academic Press, Canada/ Switzerand, pp. 215 221. CGFS Committee on the Goba financia System (2000), Stress Testing by Large Financia Institutions: Current Practice and Aggregation Issues, Bank for Internationa Settements. CGFS Committee on the Goba Financia System (2001), A Survey of Stress Tests and Current Practice at Major Financia Institutions, Bank for Internationa Settements. CGFS Committee on the Goba Financia System (2005), Stress Testing at Major Financia Institutions: Survey Resuts and Practice, Bank for Internationa Settements. Deutsche Bundesbank (2003), Das deutsche Bankensystem im Stresstest. Monatsbericht Dezember.

16 Deveopment of Stress Tests for Credit Portfoios 371 Deutsche Bundesbank (2004), Stresstests bei deutschen Banken Methoden und Ergebnisse. Monatsbericht Oktober. Fender I, Gibson G, Mosser P (2001), An Internationa Survey of Stress Tests, Federa Reserve Bank of New York, Current Issues in Economics and Finance 7 (10), pp. 1 6. Gundach M, Lehrbass F (eds.) (2004), CreditRisk + in the Banking Industry, Springer, Berin. Kairai H, Scheicher M (2002), Macroeconomic Stress Testing: Preiminary Evidence for Austria. Oesterreichische Nationabank, Financia Stabiity Report 3, pp. 58 74.

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Chapter 17 Risk Management of Loans and Guarantees Bernd Engemann and Water Gruber 17.1 Introduction In previous chapters, the estimation of the key oan risk parameters was presented. In Chaps. 1 3, and 5 estimation methods for 1-year defaut probabiities were discussed. In typica credit risk appications, however, a 1-year horizon is insufficient. In Chap. 6 it was shown how to compute muti-year defaut probabiities with the hep of transition matrices. In Chap. 7 techniques to estimate defaut probabiities and oss given defaut rates simutaneousy were discussed whie Chaps. 8 and 9 presented methods for oss given defaut estimation. In recent years, banks have invested considerabe effort on buiding up data bases, constructing rating systems and estimating the credit risk parameters PD, LGD, and EAD from the coected data. This work was mainy driven from reguatory considerations. Under the Base II capita accord banks are aowed to cacuate their reguatory capita from these risk parameters if the estimation procedures fufi the quaity requirements of supervisors. This might ead to capita reductions compared to the od framework if the credit quaity of a bank s debtors and the quaity of coatera that is used to back a bank s oan portfoio is sufficienty high. In our view, it woud be a waste of effort if the estimated risk parameters woud be used mainy for reguatory purposes. In this chapter, we show how they can be used to price oans and guarantees. We show how the basic pricing formuas can be used to compute the terms of a oan, how the premium of a guarantee can be determined, and how the mode can be used to cacuate genera oss provisions in a consistent and economicay meaningfu way, i.e. how the mode can be used in managing the risk of credit osses. B. Engemann (*) Independent Consutant e-mai: bernd.engemann@quantsoutions.de W. Gruber 1 PLUS i GmbH e-mai: water.gruber@1pusi.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_17, # Springer-Verag Berin Heideberg 2011 373

374 B. Engemann and W. Gruber Often oan portfoios and bond portfoios are treated very differenty by a bank. From an economic perspective this does not make sense because the characteristics of both products are very simiar. Both products consist of a stream of deterministic cash fows that are subject to defaut risk. The main difference is that bonds are tradabe in contrast to most oans. For this reason we suggest an approach for the pricing and risk management of oans that is structuray very simiar to bonds. Since it is not possibe to observe spreads, e.g. asset swap spreads or CDS spreads, for most debtors in the oan market, we use defaut probabiities for the measurement of a debtor s credit quaity instead. This chapter consists of three sections. In the first section, we expain the pricing formuas for oans and guarantees and iustrate their use for the most popuar oan structures, buet oan, instament oan and annuity oan. In the second part we expain how to compute the terms of a oan. Our scheme is based on the RAROC (risk-adjusted return on capita) concept. Further, we show how to compute genera oss provisions for a oan portfoio dynamicay. We concude this artice with a short discussion of our oan pricing framework in the ight of the recent financia crisis. 17.2 Pricing Framework In this section we expain the pricing formuas and the input data of these formuas. In the first part we discuss the pricing of oans and in the second part we state the formuas for guarantees. 17.2.1 Pricing of Loans We expain the vauation of a oan that is characterized by interest rate payments that might be either fixed or foating and a deterministic amortization schedue. A deterministic amortization schedue impies that a oan does not contain any embedded options ike prepayment rights. This case is treated in detai in Chap. 18. Under this assumption the vaue of a oan is the discounted expected vaue of a future cash fows. The future cash fows consist on the one hand of the interest rate and amortization payments, on the other on the recovery in the case of a defaut of the debtor. We find for the vaue V of a oan the expression V ¼ Xn i¼1 þ Xn ðz i t i N i þ A i Þdf ðt i ÞqðT i Þ i¼1 R i N i ðt i df ðtþðqðtþ qðt þ dtþþ (17.1) T i 1

17 Risk Management of Loans and Guarantees 375 With T 1,...,T n we denote the future interest rate payment dates of the oan, where T n is the repayment date of the oan s outstanding notiona. With z i we denote the annuaized interest rate corresponding to period i which might be either fixed or foating, 1 with A i the amortization payment in period i, with t i the year fraction of the i-th interest rate period, with df(t) the discount factor corresponding to time T, with q(t) the surviva probabiity of the debtor up to time T, with N i the oan s notiona that is outstanding in period i, and with R i the recovery rate corresponding to period i, i.e. the percentage of the notiona that can be recovered by the creditor if the debtor defauts. This recovery rate might be period dependent. Suppose a oan is backed by coatera and the vaue of this coatera is assumed constant throughout the ifetime of the oan. If the oan is amortizing then the percentage of the oan that is coateraized is increasing in time. Therefore the recovery rate is increasing in time. The discount factors are computed from an interest rate curve that can be considered as approximatey risk free. Often the swap curve is taken as a reference curve. This curve is certainy not risk free because it refects the credit risk in the interbank market. It is nevertheess a reasonabe choice for a reference curve because it refects the funding conditions of banks. An additiona spread refecting the debtor s credit quaity is not incuded into the discounting in (17.1). The debtor s credit quaity is incuded by his time-dependent surviva probabiity ony. The two terms in (17.1) can be interpreted intuitivey. The first term is the discounted expected vaue of a interest rate and amortization payments. Each interest rate and amortization payment is discounted according to its payment date and weighted with the probabiity of its occurrence, i.e. the surviva probabiity of the debtor. The second term is a bit more compicated. It modes the recovery if the debtor defauts. In contrast to the interest rate payments the defaut time is not known in advance. Therefore, we have to compute the defaut probabiity of each sma time interva in the future, weight it with the discounted vaue of the recovery, and sum over a future sma time intervas. In its exact form this eads to an integra. It is possibe to approximate this integra by an easy to evauate formua. We assume that on average a debtor defauts in the midde of each interest rate period, i.e. at the time t ¼ 0.5 (T i + T i 1 ). We discount the recovery from the period mid and weight the resut with the probabiity that a debtor defauts in the period. This eads to R i N i ðt i df ðtþðqðtþ qðt þ dtþ T i 1 ÞR i N i df ð0:5 ðt i þ T i 1 Þ ÞðqðT i 1 Þ qðt i ÞÞ (17.2) 1 A foating interest rate is often directy inked to a Libor rate that is fixed at the beginning of each interest rate period. In this case z i can be computed as a forward rate from the discount curve that is extracted from the swap market. In other cases, the bank has some freedom to decide about when to increase or decrease a foating interest rate. Here some assumption has to be made how the bank s decision is inked to the forward rates impied from the swap curve.

376 B. Engemann and W. Gruber In a concrete impementation we have to define T 0 ¼ 0. The approximation (17.2) is easy to impement and very accurate. We remark that we do not mode the compicated process of iquidating a oan s coatera expicity. We assume that at the time of defaut the creditor wi receive the discounted vaue of a future payments from iquidating coatera minus the associated costs with the iquidation process. The compexity of the iquidation process is therefore refected in the estimation of the recovery rate R i, not in the formuas for oan vauation. Formuas (17.1) and (17.2) ook pretty simpe because of the probabiistic assumptions we have impicity used without stating them yet. First, we can write expected discounted vaues of future cash fows by weighting the cash fows with the product of discount factor and surviva probabiity of the debtor because of the assumption that defauts and interest rate dynamics are independent. Second, we can write the expected recovery in the case of a defaut as the product of recovery rate and defaut probabiity because we have impicity assumed that defaut probabiities and recovery rates are independent. The first assumption might be probematic for banks seing mosty foating rate oans because rising interest rates shoud ead to an increase in defaut rates in this context. The second assumption is aso in contradiction to empirica iterature (Frye 2000 and 2003) and to the observation of faing house prices in connection with high defaut rates during the recent financia crisis. This shoud be kept in mind when parameterizing this simpe formua. We wi come back to this point in the fina section. For the practica appication of (17.1) and (17.2) we have to estimate discount factors, recovery rates, and surviva probabiities. The easiest part is the determination of discount factors. They are computed from quotes of interbank market instruments ike deposits, interest rate futures, or swaps. These quotes are avaiabe every day and can be accessed by market data providers ike Boomberg or Reuters. As aready outined before, this interest rate curve is suitabe for oan vauation because the interbank curve serves as the reference for determining the funding conditions of a bank. More difficut is the estimation of the surviva probabiity (or equivaenty, the defaut probabiity) of a debtor. There are basicay three possibiities of estimating surviva probabiities Direct estimation of surviva probabiity term structures Extrapoation from transition matrices Extraction from bond or credit defaut swap spreads First, term structures of surviva probabiities can be estimated directy. If the surviva probabiities for different rating grades are known for a number of years and a reasonabe parameterization for the shape of this term structure is given, it might be caibrated for each rating grade separatey. Second, term structures of surviva probabiities can be easiy extrapoated from 1-year transition matrices under the assumption of Markovian rating transitions and time-homogeneity. This is expained in detai in Chap. 6. Finay, in rare cases of debtors with bond issues outstanding, the term structure of surviva probabiities can be inferred

17 Risk Management of Loans and Guarantees 377 from bond or credit defaut swap spreads. Detaied expanations on the cacuation of surviva probabiities from credit defaut swap spreads can be found in Brigo and Mercurio (2006). The fina parameter needed is the recovery rate. This parameter typicay is determined from the coatera of a oan. For each type of coatera a separate recovery rate is estimated from the data of defauted debtors. From the recovery rates of each type of coatera and the recovery rate for the unsecured part of the oan a net recovery rate for the oan wi be computed. The basic principes of estimating recovery rates are expained in more detai in Chap. 11. We concude this section with the specification of (17.1) for the three most common oan types, buet oan, instament oan, and annuity oan. The simpest oan type is the buet oan with initia notiona N. Here no amortization prior to the repayment of the notiona at the oan s expiry takes pace. We have N i ¼ N A i ¼ 0; i < n N; i ¼ n z; fixed interest rate z i ¼ f i þ m; foating interest rate where f i is the forward rate corresponding to the i-th interest rate period and m is the margin (or spread) over Libor the debtor has to pay for the oan if the interest rate is foating. Compared to the buet oan, the instament oan has a fixed amortization payment in addition to the interest rate payments in every period i. This amortization payment is specified by a constant annuaized amortization rate a. We find for the instament oan A ¼ 1 k a N N i ¼ maxð0; N ði 1ÞAÞ A i ¼ A; i < n N n ; i ¼ n z; fixed interest rate z i ¼ f i þ m; foating interest rate where k is the number of interest rate (and amortization) payments per year. Of course one has to make sure that N i is aways non-negative. For an instament oan the amortization payment is constant over its ifetime whie the interest rate payments are reduced due to the reduction in the oan s outstanding notiona. Therefore, the sum of amortization and interest rate payments is not constant over the ifetime of an instament oan. This is the case for an annuity oan. To ensure that the sum of interest and amortization payments is constant over the oan s ifetime the interest rate has to be fixed to a vaue z. We get for the annuity oan

378 B. Engemann and W. Gruber K ¼ a þ z N k A i ¼ minðk t i z N i ; N i Þ N i ¼ N; i ¼ 0 N i 1 A i 1 ; i > 0 where K is the constant sum of interest and amortization payments in each period. Note that it is not possibe to compute A i and N i independent of each other for an annuity oan. Both quantities have to be computed recursivey starting from i ¼ 0. 17.2.2 Pricing of Guarantees In this section, we expain the pricing formuas for guarantees. In a guarantee a bank or some other financia institution provides insurance against the defaut of a debtor of a oan. In the case of a defaut the guarantor wi either pay for the oss of the oan or take over the oan at par, i.e. buy the oan and pay the outstanding notiona for it. For insuring the oan against defauts the guarantor gets a premium g which is paid periodicay and is proportiona to the oan s outstanding notiona. In this respect a guarantee is very simiar to a credit defaut swap. The ony difference is that the underying of a credit defaut swap is one or more bonds of a company or a state whie the underying of a guarantee is a oan. The pricing of a guarantee is very simiar to the pricing of a oan. Its vaue is the expected discounted vaue of a future cash fows. We take the perspective of a guarantor who receives premium payments and has to buy the oan at par in the case of a defaut. We assume that premiums are paid at the end of each period and that in case of a defaut no premium has to be paid for the period where the defaut occurred. 2 The equivaent to (17.1) for the vaue G of a guarantee is G ¼ Xn i¼1 Xn g t i N i df ðt i i¼1 ð1 R i ÞN i ÞqT ð i Þ ð T i df ðtþðqðtþ qtþ ð dtþþ (17.3) T i 1 2 There might be other conventions in the market concerning premium payments, i.e. the premium might be paid at the beginning of each period or in the case of a defaut the premium for the period where the defaut occurred has to be paid up to the defaut time. In this case, the formuas we derive for the expected present vaue of premium payments have to be sighty modified to propery refect the different convention that is used.

17 Risk Management of Loans and Guarantees 379 Here, g is the annuaized premium that has to be paid for the guarantee, n is the number of future periods of the guarantee s ifetime, T i are the payment dates of the premium, N i is the oan s outstanding notiona in period i, df(t) is the discount factor and q(t) the surviva probabiity of the oan s debtor corresponding to time T. The first term in (17.3) is the expected present vaue of the premium payments that are paid if the debtor survives and the second term is the oss of the guarantor if the debtor defauts. Simiar to (17.2) the integra in (17.3) can be simpified by assuming that on average a debtor defauts in the midde of each period. This eads to ð1 R i ÞN i ð1 R i ð T i df ðtþðqðtþ qtþ ð dtþþ T i 1 ÞN i df ð0:5 ðt i þ T i 1 ÞÞðqT ð i 1 Þ qt ð i ÞÞ (17.4) Again we end up with an easy to impement formua. 17.3 Appications In this section we outine how to appy the pricing formuas that were derived in Sects. 2.1 and 2.2 in banking practice. In the next section, we expain a cacuation scheme for a oan s terms based in RAROC (risk-adjusted return on capita). After that, we show how to compute genera oss provisions for a oan portfoio in an economicay meaningfu way. Both appications are iustrated with concrete numerica exampes. 17.3.1 Cacuation of a Loan s Terms As aready outined above, we expain a scheme for cacuating a oan s terms based on the performance measure RAROC. RAROC measures the revenues of a oan in reation to its risk. In our context, risk is defined as the economic capita that is needed as a buffer against unexpected osses of the oan. Economic capita is typicay measured by the risk contribution of a oan to the tota credit risk of a bank that is typicay computed as the vaue-at-risk or the expected shortfa of a oss distribution that is generated by a credit risk mode. A good introduction to credit risk modeing is Buhm et a. (2003). Risk measures ike vaue-at-risk and expected shortfa and their properties are anayzed in Artzner et a. (1999) and Tasche (2002), whie good references for capita aocation techniques are Kakbrenner (2005), Kurth and Tasche (2003), and Kakbrenner et a. (2004).

380 B. Engemann and W. Gruber Using a simpe one-factor credit risk mode, Gordy (2003) has derived an anaytica formua for the economic capita reated to a oan exposure which is aso used in the Base II capita accord (BCBS 2004) for the cacuation of reguatory capita under the interna ratings based approach. Using the exact parameterization of Base II, the economic capita E per oan exposure is computed as rffiffiffiffiffiffiffiffiffiffiffi 1 E ¼ ð1 RÞ F pffiffiffiffiffiffiffiffiffiffiffi F 1 r (PD) þ F 1 ðaþ PD bðmþ 1 r 1 r 1 þ max 1; min M; 5 bðmþ ¼ ð ½ ½ ŠŠ 2; 5 Þ ð 0; 11852 0; 05478 ogðpdþ Þ2 1 1; 5 ð0; 11852 0; 05478 og(pdþþ 2 (17.5) where PD is the 1-year defaut probabiity of the debtor, R the recovery rate corresponding to a defaut within 1 year, r is the mean asset correation among a debtors of a bank, a the confidence eve where the vaue-at-risk of the oss distribution is computed, and F the cumuative distribution function of the standard norma distribution. For r and a it is possibe to use own estimations or the vaues given in the Base II accord. The term b is the maturity adjustment that refects the increased risk of a oan with a higher maturity. One assumption behind the derivation of this formua was the absence of concentration risk in the oan portfoio. If the oan portfoio contains significant concentrations, the formua can be adjusted by adding an additiona factor for granuarity. A possibe way to compute this add-on for voume concentration risk is described in Gordy and L utkebohmert-hotz (2007). A oan s revenue is computed as the difference of the interest earned and the costs associated with the oan. If the oan woud be riskess and a interna costs and the funding spread woud be zero, the interest rates z i woud have to be equa to the swap rate s corresponding to the oan s maturity to bear the oan s funding costs. This swap rate is computed as s ¼ 1 df ð T nþ P n i¼1 t i df ðt i Þ (17.6) Using s as a base rate, the tota margin of a oan is defined as m ¼ z eff s. Here, z eff is defined as the period-independent fixed ( effective ) interest rate that defines a fixed-rate oan that has the same vaue as the oan with interest rates z i. To be more specific, z eff is defined from the condition X n i¼1 z eff t i N i þ A i df ð Ti which can be soved expicity for z eff : ÞqT ð i Þ ¼ Xn i¼1 ðz i t i N i þ A i Þdf ðt i ÞqT ð i Þ

17 Risk Management of Loans and Guarantees 381 z eff ¼ P n i¼1 P n i¼1 z i t i N i df ðt i t i N i df ðt i ÞqT ð i Þ ÞqT ð i Þ The costs associated with a oan are on the one hand the risk costs to cover the expected oss of the oan, on the other hand a other costs of the ender (funding spread, operating costs, interna fees, etc.). The risk costs can be computed from (17.1) appying the condition V ¼ N. If the discounted expected vaue of a future cash fows equas the outstanding notiona, it is ensured that expected osses are covered. In Sect. 2.1 we considered a buet oan, an instament oan, and an annuity oan as the most popuar oan types expicity. Cacuating risk costs for a buet oan with a period-independent fixed interest rate eads to r ¼ 1 df ðt n ÞqT ð n Þ Pn i¼1 P n R i df ð0:5 ðt i þ T i 1 ÞÞðqT ð i 1 Þ qt ð i ÞÞ s t i df ðt i ÞqT ð i Þ i¼1 For an instament oan we report the formua for the case of a foating interest rate N Pn r ¼ P n i¼1 i¼1 P n ða i þ f i t i N i i¼1 t i N i df ðt i Þdf ðt i ÞqT ð i Þ ÞqT ð i Þ R i N i df ð0:5 ðt i þ T i 1 ÞÞðqT ð i 1 Þ qt ð i ÞÞ P n i¼1 t i N i df ðt i ÞqT ð i Þ In this case the risk cost r is the spread over Libor that must be charged by a bank to cover expected osses of the oan. For an annuity oan it is impossibe to cacuate the risk costs anayticay because the amortization schedue depends on the eve of the fixed interest rate. Here a zero-search agorithm has to be appied for the cacuation of risk costs. Finay a other cost components must be transformed into an annuaized cost margin per notiona. This cost margin is denoted by c. Since interna cost structures differ from bank to bank there is no genera rue how to aggregate cost components to a cost margin. It might be possibe that the argest part of the interna costs has to be paid as an upfront payment by the debtor, or that no upfront payment is required and a costs are incuded into the oan s interest margin, or that some combination

382 B. Engemann and W. Gruber of these two extremes is appied. In any case a bank has to ensure that the expected present vaue of future cost payments covers a interna and refinancing costs. To make this cear we take the funding spread s fu a bank has to pay as an exampe. We assume that a bank has to pay the funding spread for a oan unti the oan s maturity regardess if the debtor defauts. Under this assumption the cost margin c fu corresponding to funding costs can be computed from the equation X n i¼1 N i s fu t i df ðt i Þ ¼ Xn i¼1 N i c fu t i df ðt i ÞqT ð i Þ The eft hand side is the funding spread that is paid by the bank over the oan s ifetime. The right hand side equas the expected present vaue of the cost margin payment by the debtor. The cost margin wi be paid uness the debtor is in defaut. For the cost margin corresponding to funding costs we get the expicit formua c fu ¼ P n i¼1 s fu Pn i¼1 N i t i df ðt i Þ N i t i df ðt i ÞqT ð i Þ Simiar formuas have to be derived for other cost components. After that a cost margins corresponding to a cost components have to be aggregated to the tota cost margin c. Using the above margin components, we compute the RAROC of a oan as RAROC ¼ m r c E (17.7) RAROC measures the return on economic capita that is reaized by seing a oan for a tota margin of m. Typicay, a bank defines a minimum eve of the return on economic capita that must be reached to consider a oan as profitabe. This minimum eve is caed the hurde rate h. From the hurde rate, the minimum margin that must be gained by a oan investment can be computed as m min ¼ r þ h E þ c (17.8) In (17.8) the definition of the minimum margin ensures that a oan is profitabe for a bank. If for some reasons a oan is sod beow the minimum margin, the effect on the reaized return on economic capita can be measured by (17.7). The three components in (17.8) can be interpreted intuitivey. The first component is a compensation for expected osses, the second component is a compensation for unexpected osses, and the fina component is a compensation for a other costs that are reated to a oan. The RAROC scheme for a oan can aso be appied to a guarantee. The ony modification is that the risk costs of a guarantee are computed from the condition

17 Risk Management of Loans and Guarantees 383 G ¼ 0whereG is vaue of the guarantee that was defined in (17.3). A other parts of the RAROC pricing formua for a oan are identica for a guarantee. We concude this section with an iustrative exampe. We consider an instament oan with a maturity of 15 years. The oan s interest rate is 3M Libor pus a margin, i.e. the oan pays interest quartery. Further, the annua amortization rate of the oan is 4%. We assume that a bank computes economic capita according to the Base II formuas in (17.5) with the parameterization of Base II for residentia mortgages. Further, we assume that the bank uses a hurde rate h of 10% and its interna and funding costs are propery refected by a cost margin of 1%. The bank s rating system is described by the transition matrix in Fig. 6.7 of Chap. 6, i.e. the bank uses nine rating grades where the fina grade is the defaut grade. We further assume that the vaue of the coatera that is posted by a debtor equas 40% of the origina oan amount. This means that the recovery rate is increasing in time because of the amortization effects. Finay, we assume a fat zero rate of 5% with annua compounding to compute discount factors. To generate the surviva probabiities in (17.1) and(17.2) it is necessary to mutipy the 1-year transition matrix with itsef to compute muti-year defaut probabiities as described in Chap. 6. We report the resuting muti-year defaut probabiities over 15 years for each rating grade in Tabe 17.1. From these defaut probabiities we can compute the term structures of surviva probabiities that are needed in (17.1) and (17.2). Since our oan pays interest quartery we aso need surviva probabiities at intermediate points in time. They can be generated by computing transition matrices corresponding to year fractions as expained in Chap. 6. An aternative coud be a simpe interpoation scheme ike inear interpoation of the ogarithms of the surviva probabiities which approximatey eads to the same resut. Tabe 17.1 Muti-year cumuative defaut probabiities for each rating grade computed from the 1-year transition matrix in Fig. 6.7 of Chap. 6 Time (years) Rating grade 1 2 3 4 5 6 7 8 1 0.0000 0.0008 0.0009 0.0036 0.0167 0.0496 0.1490 0.2496 2 0.0002 0.0016 0.0022 0.0085 0.0359 0.1010 0.2673 0.4173 3 0.0004 0.0026 0.0040 0.0148 0.0572 0.1516 0.3612 0.5325 4 0.0007 0.0037 0.0063 0.0223 0.0799 0.2001 0.4361 0.6134 5 0.0011 0.0051 0.0091 0.0309 0.1037 0.2455 0.4964 0.6718 6 0.0017 0.0066 0.0125 0.0405 0.1279 0.2875 0.5454 0.7149 7 0.0024 0.0085 0.0164 0.0509 0.1522 0.3261 0.5856 0.7476 8 0.0033 0.0106 0.0209 0.0621 0.1762 0.3614 0.6191 0.7729 9 0.0043 0.0130 0.0259 0.0738 0.1999 0.3936 0.6472 0.7931 10 0.0055 0.0157 0.0314 0.0861 0.2229 0.4230 0.6711 0.8094 11 0.0069 0.0187 0.0374 0.0987 0.2451 0.4498 0.6917 0.8230 12 0.0086 0.0221 0.0439 0.1117 0.2665 0.4743 0.7095 0.8344 13 0.0104 0.0258 0.0508 0.1248 0.2871 0.4966 0.7252 0.8442 14 0.0125 0.0299 0.0581 0.1380 0.3068 0.5172 0.7390 0.8526 15 0.0149 0.0342 0.0658 0.1513 0.3256 0.5360 0.7514 0.8600

384 B. Engemann and W. Gruber Tabe 17.2 Cost components of the instament oan for the cacuation of the oan s terms in the exampe for each rating grade Rating grade Risk costs (%) Opp. costs (%) Minimum margin (%) 1 0.0193 0.0469 1.0662 2 0.0594 0.0888 1.1482 3 0.1137 0.1023 1.2161 4 0.3330 0.2864 1.6194 5 1.0146 0.8258 2.8404 6 2.4618 1.5762 5.0380 7 6.1548 2.5831 8.7380 8 10.8426 2.9259 14.7684 In Tabe 17.2, we report the minimum margins that must be charged by a bank to cover a its costs according to (17.8). Since the oans have a foating interest rate, the minimum margins are the spreads over Libor that must be paid by a debtor. To see the main drivers of the minimum margin we report a cost components. The cost margin (corresponding to c in (17.8)) which is not reported in Tabe 17.2 is independent of the rating grade and by assumption equa to 1%. In the tabe, we report the risk costs (r in (17.8)) and the opportunity costs of capita (h E in (17.8)). We see that the risk costs are ower than the capita costs for the good rating grades whie it is vice versa for the poor rating grades. For poor rating grades the expected oss is aready rather high because of the high defaut probabiities. Therefore, the surprise component of unexpected osses is reativey ow for these rating grades. 17.3.2 Cacuation of Genera Loss Provisions We start this section by describing a simpe framework for managing the risks of credit osses in our mode framework. We expain the genera principe without going deepy into accounting detais which depend on the specific accounting framework that is appied by a bank. The aim is to manage oan portfoios in a way that ensures that a bank does not suffer osses even if defauts in the oan portfoio occur. For this, the component of the interest margin that refects the expected oss risk of a oan is coected and stored in an expected oss account. When defauts happen, the vaues of the defauted oans are corrected to their expected recovery vaues. If the estimated defaut probabiities and recovery rates are on average cose to their reaizations, the sum of the proceeds from iquidating the coatera of defauted oans pus the money on the expected oss account are on average cose to the present vaue of the bonds that were issued to fund the oan. In this context, on average means that during a recession reaized defaut rates are higher than defaut probabiities, whie during a boom they are ower. Therefore, booms shoud be used to fi the expected oss account for the bad times when it is needed. Furthermore, on average means that this approach wi ony work

17 Risk Management of Loans and Guarantees 385 we for arge oan portfoios, e.g. in the retai sector. A bank with a highy speciaized business and a reativey ow number of oans wi have difficuties in impementing this approach because if the number of oans is sma, it is expected that reaized defaut rates are in genera ess in aignment with defaut probabiities which makes this approach of managing oan oss risks ess accurate. Usuay credit osses do not occur by surprise but downgrades of debtors signa an increase in defaut probabiities eary. If the vauation of a oan is done by (17.1) changes in the rating of a debtor and changes in interest rates are directy refected in an increase or decrease of a oan s vaue. This aows the buiding of oss provisions for a oan portfoio to make the process of reaizing unexpected osses smooth. For this, the oan portfoio has to be vaued in reguar intervas and provisions for osses have to be buid. For the cacuation of the oss provision per oan the condition V ¼ N that was used to compute the risk margin in the ast section wi be appied. Suppose a oan was sod for an interest rate z if it is a fixed rate oan or for a margin m over Libor if it is a foating rate oan. We define the interest rate z r (or the margin m r ) that contains the risk costs ony by z r ¼ z h E c m r ¼ m h E c (17.9) If the oan is sod for an interest rate computed by (17.8) then using the interest rate (17.9) and vauing the oan using (17.1) eads to a oan vaue V equa to N, the initia notiona. If the oan is vaued at a ater stage, the changes in interest rates and in the debtor s credit quaity might ead to risk costs that are not refected in (17.9). Therefore, vauing the oan using the interest rate (17.9) and subtracting the outstanding notiona of the oan from the resut gives the gain or oss in the oan s vaue. This is a reasonabe quantity for buiding a provision. We wi iustrate this concept with a simpe exampe. We consider a portfoio of ten instament oans that were a sod on March 31, 2009. A oans have the structure of the exampe oan in Sect. 3.1., i.e. instament oans with a maturity of 15 years and an amortization rate of 4%. The rating of each debtor and the initia notiona of each oan are reported in Tabe 17.3. Tabe 17.3 Exampe portfoio of 15Y instament oans Number of oan Debtor rating Initia notiona 1 1 1,000,000 2 2 500,000 3 3 750,000 4 3 750,000 5 3 1,000,000 6 4 750,000 7 4 600,000 8 5 400,000 9 5 750,000 10 5 500,000

386 B. Engemann and W. Gruber The oan portfoio has a tota notiona of seven miion. We further assume that each oan was sod for the minimum margin that was computed in Tabe 17.2. We compute the genera oss provision for this portfoio at Apri 01, 2012. We assume that interest rates at this date are given by a fat zero curve of 4% with annua compounding. Since the oans have a foating interest rate their interest rate sensitivity is moderate. Over the 3 years the ratings of some debtors have improved whie the credit quaity of other debtors has deteriorated. As in Sect. 3.1 we assume that the economic capita for each oan is computed by the Base formua (17.5). This means that the economic capita changes with the rating. When we compute the interest rate (17.9) we use the economic capita corresponding to the current rating. Finay, we assume that the bank s cost structure did not change and 1% is sti the appropriate cost margin. We report the vaue of each oan when it is priced using the interest rate (17.9) together with the outstanding notiona in Tabe 17.4. If the rating grade remains unchanged as in the case of oan 2 the vaue of the oan increases. The reason is that after 3 years the expected oss margin corresponding to a then 12-year oan is ess than the origina margin corresponding to a 15-year oan. This eads to an increase in the oan s vaue. In this exampe, however, the increase is mid because the expected oss for a debtor in rating grade 2 is rather sma. Overa we see that the credit quaity of the portfoio has deteriorated and some oans are worth consideraby ess than the outstanding notiona when the pricing is done with the interest rate (17.9). The reason is that both the opportunity cost of capita and the expected oss margin cacuated under the new rating have increased. In tota we find a oss in portfoio vaue of 147,032. This shoud be reported as the genera oss provision for this portfoio in the baance sheet of the bank. This number compares to a tota of expected oss margins that have been earned by the bank over the 3 years after the oan portfoio has been sod of 69,653. 3 Tabe 17.4 Vauation of the oan portfoio of Tabe 17.3 after interest rates and ratings have changed three years after the oans in the portfoio were sod Number of oan Debtor rating Loan vaue Outstanding notiona P & L 1 4 852,043 880,000 27,957 2 2 440,752 440,000 752 3 7 446,046 660,000 213,954 4 1 667,668 660,000 7,668 5 2 885,774 880,000 5,774 6 3 680,791 660,000 20,791 7 3 544,633 528,000 16,633 8 7 262,234 352,000 89,766 9 2 740,948 660,000 80,948 10 3 492,079 440,000 52,079 3 We have negected interest rate effects when computing this number. It is just the sum over the expected oss margins that were charged by the bank.

17 Risk Management of Loans and Guarantees 387 Overa, the expected oss margins earned so far do not cover the deterioration in the oan portfoio s vaue at this point in time. Finay, we remark that computing the vaue of a oan by (17.1) using the interest rate (17.9) is not ony usefu for the cacuation of oss provisions. This is aso the price that shoud be paid for a oan when it is sod. The reason is that the interest paid by a oan must cover the desired return on economic capita and the interna costs of a bank. Therefore, these cost components have to be subtracted from the tota margin of a oan when computing the expected discounted vaue of its future cash fows. 17.4 Discussion In this artice we have presented a framework for the risk-adjusted pricing of oans and guarantees. To concude this artice we want to discuss its usefuness for practica appications. The recent financia crisis has ed to a debate on the quaity of modes that are appied by banks. The opinions range from baming modes and their creators to be a main driver of the crisis to the other extreme that sti not enough modes are used propery by banks to measure and monitor the risk s of its business. We take one aspect of the origin of the financia crisis, the ending behaviour of American banks. Basicay oans were granted to home buyers of poor credit quaity because it was beieved that house prices cannot fa and in the case of a defaut the sae of the house wi make up for the oss. We wi anayze in the seque how the use of a mode wi affect the business of a bank under these assumptions. We use again the instament oan of Sect. 3.1 as an exampe. This time we assume that the vaue of the coatera is equa to the oan s initia notiona and we assume that if the coatera is iquidated that it cannot be worth more than the oan s outstanding notiona, i.e. that the recovery rate in (17.1) can never exceed 100%. We compute Tabe 17.2 again under this assumption. The resuts are presented in Tabe 17.5. Not surprisingy we find that the minimum margin is basicay independent of a debtor s credit risk. The ony difference comes from the risk of the creditor of osing an interest rate payment. Tabe 17.5 Recacuation of Tabe 17.2 under the assumption of a maximum recovery of 100% Rating grade Risk costs (%) Opp. costs (%) Minimum margin (%) 1 0.0011 0.0000 1.0011 2 0.0017 0.0000 1.0017 3 0.0025 0.0000 1.0025 4 0.0056 0.0000 1.0056 5 0.0142 0.0000 1.0142 6 0.0311 0.0000 1.0311 7 0.0710 0.0000 1.0710 8 0.1202 0.0000 1.1202

388 B. Engemann and W. Gruber Tabe 17.6 Recacuation of Tabe 17.3 under the assumption of recovery rates >100% Rating grade Risk costs (%) Opp. costs (%) Minimum margin (%) 1 0.0338 0.0000 0.9662 2 0.0774 0.0000 0.9226 3 0.1548 0.0000 0.8452 4 0.3602 0.0000 0.6398 5 0.7839 0.0000 0.2161 6 1.3322 0.0000 0.3322 7 1.9483 0.0000 0.9483 8 2.3547 0.0000 1.3547 If a debtor defauts the bank has in genera unimited access to the coatera. This coud in principe ead to recovery rates greater than 100% if a defaut happens after some amortization payments are made and the assumption that a house price cannot fa beow the initia notiona is true. Under this assumption we get the margins of Tabe 17.6. The resuts in Tabe 17.6 are counterintuitive. Now the mode tes us to favour debtors with ow credit quaity, and they even shoud be paid interest for the oan. The reason is that under this assumption a defaut is more profitabe than earning interest over the fu ifetime of a oan. The profit comes from the amortization payments. The bank makes the highest profit if a debtor defauts after he has made some amortization payments. This profit can be achieved more ikey with debtors of ow credit quaity. Therefore, they even shoud get a fee for entering the oan instead of paying interest. From an economic point of view Tabes 17.5 and 17.6 deiver reasonabe resuts. However, especiay the assumptions underying Tabe 17.6 resut in a business mode that no one woud undertake. In this context a mode is just a too to transate assumptions about markets into a business mode in a transparent way. It is sti the task of a risk manager to judge if the resuting business mode is reasonabe or if it is not. The atter case shoud ead to a questioning of the assumptions underying the mode. The assumptions underying the pricing mode (17.1) can be questioned in two ways. It is known from empirica studies by Fry (2000, 2003) that high defaut rates are historicay accompanied by ow recovery rates, i.e. that the assumption of independence between defaut and recoveries is wrong from an empirica point of view. This is aso anayzed in Chap. 7. To improve the pricing mode (17.1) one coud either mode the correation between defaut and recovery expicity which woud resut in a more compicated mode or use a more conservative parameterization, i.e. instead of using an average LGD one shoud use a conservative estimation of a LGD ( downturn LGD ) to acknowedge this effect. If we modify the exampe of Tabe 17.6 by restricting recovery rates to 100% and assuming that the coatera is worth 80% of the initia notiona in the case of a defaut ony, which sti resuts in a rather high coateraization, we get the minimum margins for each rating grade that are reported in Tabe 17.7. In addition we compute Tabe 17.7 for a buet oan, i.e. we set the amortization rate to zero.

17 Risk Management of Loans and Guarantees 389 Tabe 17.7 Recacuation of Tabe 17.3 under the assumption of a coatera vaue of 80% of the initia notiona Rating grade Risk costs (%) Opp. costs (%) Minimum margin (%) 1 0.0018 0.0156 1.0174 2 0.0065 0.0296 1.0361 3 0.0102 0.0341 1.0443 4 0.0341 0.0955 1.1295 5 0.1326 0.2753 1.4079 6 0.3952 0.5254 1.9206 7 1.2725 0.8610 3.1336 8 2.5336 0.9753 4.5089 Tabe 17.8 Recacuation of Tabe 17.7 under the additiona assumption of no amortization payments Rating grade Risk costs (%) Opp. costs (%) Minimum margin (%) 1 0.0139 0.0156 1.0295 2 0.0352 0.0296 1.0648 3 0.0690 0.0341 1.1031 4 0.1785 0.0955 1.2740 5 0.4690 0.2753 1.7443 6 1.0115 0.5254 2.5369 7 2.2373 0.8610 4.0983 8 3.7580 0.9753 5.7333 We see that even under mid assumptions on osses in the case of defauts the minimum margins for debtors of poor credit quaity increase consideraby. It is questionabe that a debtor of poor credit quaity, which in practice means ow income, coud afford a oan under these conditions even if Libor rates are very ow. In the case of Tabe 17.7 this means that a debtor with a rating of 7 woud have to pay Libor þ 7.13% every year (Libor þ 3.13% interest þ 4% amortization) and in the case of a buet oan in Tabe 17.8 woud have to pay Libor + 4.10% which shoud be too much for a debtor with ow income to buy a home worth a mutipe of his annua saary. These exampes iustrate how even a very simpe mode can increase the transparency of a business mode undertaken by a bank. It forces the bank to decare its assumptions on defauts and recovery rates and transates them into minimum margins that have to be charged for a debtor of a certain credit quaity. These assumptions can be verified using empirica data. Further, the consequences of sma deviations from these assumptions can be anayzed. This can hep to increase market discipine and prevent banks from charging margins that do not refect the risks of a oan propery. Beyond this increase in transparency, the use of a mode ike (17.1) aso brings the treatment of oan portfoios more in ine with the treatment of other asset casses ike bonds. Since both asset casses (oans and bonds) are vaued using expected discounted cash fows, the changes in portfoio vaue of oans and instruments that are issued to fund the oans can be compared directy and mismatches either in vaue or in maturity can be detected easiy together with a quantification of the

390 B. Engemann and W. Gruber corresponding interest rate risks. Finay, we note that the RAROC vauation approach we presented is fexibe enough to be generaized to iiquid equity investments, which again aows a bank to get a consistent view on different asset casses. This generaization is done in Engemann and Kamga-Wafo (2010). References Artzner P, Dabaen F, Eber JM, Heath D (1999), Coherent Measures of Risk, Mathematica Finance 9 (3), pp. 203 228. Base Committee on Banking Supervision (BCBS) (2004), Base II: Internationa Convergence of Capita Measurement and Capita Standards: a Revised Framework. http://www.bis.org/pub/ bcbs107.htm Buhm C, Overbeck L, Wagner C (2003), An Introduction to Credit Risk Modeing, Chapman & Ha/CRC, Boca Raton. Brigo D, Mercurio F (2006), Interest Rate Modes: Theory and Practice with Credit and Infation, 2nd Edition, Springer, Berin Heideberg New York. Engemann B, Kamga-Wafo GL (2010), Measuring the Performance of Iiquid Equity Investments: What Can We Learn from the Loan Market? Journa of Private Equity 13 (3), pp. 39 46. Frye J (2000), Depressing Recoveries, Risk 13 (11), pp. 106 111. Frye J (2003), A Fase Sense of Security, Risk 16 (8), pp. 63 67. Gordy M (2003), A Risk-Factor Mode Foundation for Ratings-Based Bank Capita Rues, Journa of Financia Intermediation 12, pp. 199 232. Gordy M, L utkebohmert-hotz E (2007), Granuarity Adjustment for Base II, Working Paper. Kakbrener M (2005), An Axiomatic Approach to Capita Aocation, Mathematica Finance 15 (3), pp. 425 437. Kakbrener M, Lotter H, Overbeck L (2004), Sensibe and Efficient Capita Aocation for Credit Portfoios, Risk 17 (1), pp. 19 24. Kurth A, Tasche D (2003), Contributions to Credit Risk, Risk 16 (3), pp. 84 88. Tasche D (2002), Expected Shortfa and Beyond, Journa of Banking and Finance 26 (7), pp. 1519 1533.

Chapter 18 Risk Management of Loans with Embedded Options Bernd Engemann 18.1 Motivation In Chap. 17 it was outined how the Base II risk parameters can be used for the risk management of oans. It was shown in detai how to appy a risk-adjusted pricing formua for the cacuation of a oan s terms and of genera oss provisions. In the framework of Chap. 17 a oan was characterized by a pre-defined structure of future interest rate and amortization payments ony. In reaity, oans are in genera much more compex products. Often oans contain embedded options. The most popuar exampe of an embedded option is a prepayment right. Here, a debtor has the right but not the obigation to bay back certain amounts of a oan in addition to the agreed amortization schedue. In Germany, often banks aow debtors to bay back 5 or 10% of the initia notiona each year. Furthermore, by aw every debtor has the right to pay back the outstanding notiona of a oan after 10 years even if the agreed maturity of the oan is onger. In the anguage of option theory these amortization rights are of European or Bermudan stye because it is ony possibe for a debtor to amortize at a discrete set of dates. In other countries, prepayment rights are even of American stye, i.e. a debtor can pay back the outstanding notiona at any time. Typicay no penaty payment by a debtor is required when he pays back a part or a of the outstanding notiona. Therefore, this right can be of considerabe vaue for a debtor. In a foating rate oan, it is possibe to define upper and ower bounds for the interest rate that has to be paid by a debtor. These bounds are caed cap and foor. This oan is therefore a mixture of a fixed rate and a foating rate oan. Part of the risk of fuctuating interest rates has sti to be taken by the debtor but this risk is capped. Whie the embedded interest rate cap is vauabe for a debtor because it protects him from rising interest rates, the foor is a protection for a bank to ensure that the interest income cannot become arbitrariy ow. Introducing an interest rate B. Engemann Independent Consutant e-mai: bernd.engemann@quantsoutions.de B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8_18, # Springer-Verag Berin Heideberg 2011 391

392 B. Engemann Tabe 18.1 Minimum interest rates of the instament oan for each rating grade Rating grade Risk costs (%) Opp. costs (%) Minimum rate (%) 1 0.0193 0.0469 5.9751 2 0.0594 0.0888 6.0571 3 0.1137 0.1023 6.1250 4 0.3330 0.2864 6.5283 5 1.0146 0.8258 7.7493 6 2.4618 1.5762 9.9469 7 6.1548 2.5831 14.6469 8 10.8426 2.9259 19.6773 foor in addition to a cap, therefore, makes the oan cheaper for a debtor. A cap can be very vauabe for a debtor if future interest rate voatiity is high. In addition to prepayment rights or caps and foors on foating interest rates, oan commitments are often part of a oan. A oan commitment is an option to draw additiona amounts during a oan s ifetime. It was aready treated in Chaps. 10 and 11 in the context of EAD modeing. In practice, a debtor pays interest and amortization payments for the part of a oan s tota notiona that is drawn aready and a commitment fee for the part that is not yet drawn. In times where banks face iquidity probems or are very risk-averse a oan commitment can be of considerabe vaue. In this chapter we treat prepayment rights in detai because they are the most common embedded options in oans. The mathematica framework presented beow can be easiy modified for caps and foors on variabe interest rates. Loan commitments, however, are a different story. Here, in addition to assumptions on interest rates and a debtor s credit quaity, assumptions have to be made on the funding conditions of a bank to derive a pricing mode for the oan commitment that resuts in a commitment fee refecting a key risks a bank is facing. This is not part of this chapter. To derive the key drivers of oan prepayment, we start with a simpe exampe. We use a simiar oan that we have aready used for iustration in Chap. 17. It is a 15-year instament oan with a fixed interest rate and an amortization rate of 4%. The oan s coatera is worth 40% of the initia notiona. The bank s rating system is described by the transition matrix of Fig. 6.7 in Chap. 6 and discount factors are computed from a fat zero rate of 5% with annua compounding. The minimum interest rates that have to be charged by a bank for each rating grade using the framework of Chap. 17 are reported in Tabe 18.1. In the cacuation of the minimum interest rates of Tabe 18.1 it was assumed, as in the exampes of Chap. 17, that economic capita is computed according to the Base II formuas for residentia mortgage oans under the advanced IRB approach (BCBS 2004). The cost margin is 1% and the hurde rate 10%. 1 Note that risk costs and opportunity costs of capita are exacty equa to the resuts in Tabe 18.2 of 1 See Chap. 17 for a precise definition of these quantities.

18 Risk Management of Loans with Embedded Options 393 Tabe 18.2 Minimum interest rate (in %) for the outstanding instament oan after 10 years Rate shift (in %) Rating grade 1 2 3 4 5 6 7 8 3.00 2.91 (P) 2.95 (P) 2.97 (P) 3.14 (P) 3.72 (P) 4.84 (P) 7.48 (P) 10.28 (C) 2.00 3.92 (P) 3.96 (P) 3.97 (P) 4.15 (P) 4.73 (P) 5.86 (P) 8.53 (C) 11.36 (C) 1.00 4.93 (P) 4.97 (P) 4.98 (P) 5.16 (P) 5.75 (P) 6.88 (P) 9.58 (C) 12.44 (C) 0.00 5.94 (P) 5.98 (P) 5.99 (P) 6.17 (P) 6.76 (P) 7.91 (C) 10.63 (C) 13.52 (C) þ1.00 6.95 (P) 6.99 (P) 7.01 (P) 7.18 (P) 7.78 (C) 8.94 (C) 11.68 (C) 14.61 (C) þ2.00 7.97 (C) 8.01 (C) 8.02 (C) 8.20 (C) 8.80 (C) 9.97 (C) 12.74 (C) 15.70 (C) þ3.00 8.99 (C) 9.03 (C) 9.04 (C) 9.22 (C) 9.83 (C) 11.00 (C) 13.80 (C) 16.79 (C) Chap. 17. This is no surprise for the opportunity cost of capita because they are computed in exacty the same way in both cases. Concerning the risk costs one might have expected a sma difference because the oan in Chap. 17 was a foating interest rate oan whie in this chapter we are using a fixed interest rate oan for iustration. However, since we have used an interest rate curve with a fat zero rate in both exampes, a forward rates are equa to the base swap rate for the oan. Therefore, the risk costs have to be exacty equa in both cases. Under a reaistic interest rate curve which contains some steepness and curvature sma differences in the risk costs of a foating rate versus a fixed rate oan wi be observed because the risk of osing an interest payment is vaued differenty depending on the variabiity in the forward rates. We assume that this 15-year oan contains a prepayment option after 10 years. To get an impression of the risk factors driving prepayment, we compute the minimum interest rate for the oan again after 10 years assuming that the oan was sod to a debtor in rating grade 5 initiay. During the 10 years interest rates and the rating of the debtor can change. Concerning interest rate changes, we assume that ony parae shifts are possibe, i.e. that the discount curve after 10 years is sti represented by a fat forward curve. The resuts under different combinations of scenarios are summarized in Tabe 18.2. In the above tabe the minimum interest rates using the framework of Chap. 17 for the outstanding 5-year instament oan under the different scenarios for rating and interest rates changes are computed. If the minimum interest rate is beow the initia minimum interest rate of the oan (7.7493%), the debtor wi prepay his oan and refinance at the cheaper rate. The cases where the debtor wi prepay are indicated by a (P), the cases where he continues his oan are marked with a (C). We see from the resuts that both the eve of interest rates and the rating of a debtor at the prepayment date have an infuence on the prepayment decision. If interest rates fa sharpy but at the same time the debtor s rating deteriorates it might be sti reasonabe to continue the oan besides the reduced interest rates. This is the case, for instance, for an interest rate reduction of 300 basis points but a simutaneous downgrade of the debtor to rating grade 8. On the other hand if interest rates rise prepayment might sti be reasonabe if at the same time the debtor s rating has improved. If interest rates rise by 100 basis points and the debtor s rating improves by at east one grade, prepayment is advantageous.

394 B. Engemann If nothing happens, i.e. interest rates and the debtor s rating stay constant, the debtor wi prepay. The reason is that for the outstanding 5-year oan the average defaut probabiity appied in the pricing formua is much ower than for the initia 15-year oan. This resuts in a ower minimum margin. Overa, we concude from the simpe exampe that both interest rate and rating changes affect the prepayment decision. Therefore, we have to extend the pricing framework of Chap. 17 by stochastic interest rate and rating changes to incude prepayment rights into the framework. Furthermore, it is known from practice that debtor do not act as rationa as, for instance, interest rate derivatives or bond traders. They might not prepay even it is advantageous for them. This behaviour of debtors shoud aso be incuded into a mode. In the next section, we wi expain a pricing framework for oans with prepayment rights. We wi expain the necessary mathematica toos on an intuitive eve without going too much into detais. Some comments on the theory behind the pricing framework in the ight of derivatives pricing and credit risk modeing are made and appications of the framework for the risk management of oans with embedded options are outined. In Sect. 18.3 the pricing agorithm wi be iustrated with an exampe. In the fina section a short concusion with possibe extensions of the framework for the risk management of oan portfoios is given. 18.2 Pricing Mode We wi derive a pricing mode for oans with embedded options in three steps. In the first step, the modeing of rating transitions wi be expained which resuts in a rating tree. In the second step, a term structure mode for the evoution of interest rates wi be introduced and we wi try to expain its basic features in an intuitive way that is aso understandabe for readers who are not famiiar with interest rate derivatives pricing theory. This wi resut in a tree mode that is used for pricing interest rate dependent products. In the fina step, we wi combine the rating tree and the interest rate tree to a three-dimensiona tree that can be used for pricing oans with prepayment rights. 18.2.1 Modeing Rating Transitions Most of the mathematics behind the modeing of rating transitions was aready deveoped in Chap. 6. Here, we wi appy the resuts deveoped in Chap. 6 ony. Suppose, we have a financia product that depends on the rating of a debtor at the times 0 ¼ T 0, T 1,..., T m. These coud be the payment times of a oan or the dates where prepayment of a or parts of the outstanding notiona of a oan is possibe. At time zero the rating of a debtor is known. At times T k, k 6¼ 0, we can compute the probabiities that a debtor wi be in rating grade i under the assumption that he was in rating grade j at time T k 1. These probabiities can be computed from the

18 Risk Management of Loans with Embedded Options 395 Rating Grade p 31 (T 1 T 0 ) p 32 (T 1 T 0 ) p 33 (T 1 T 0 ) p 34 (T 1 T 0 ) p 35 (T 1 T 0 ) T 0 T 1 T 2 T 3 Time Fig. 18.1 Rating tree for a rating system with five rating grades transition matrix corresponding to the time period T k T k 1. This is iustrated with the rating tree in Fig. 18.1. In Fig. 18.1, the notation p ij (T 1 T 0 ) denotes the probabiity that a debtor migrates from rating grade i in T 0 to rating grade j in T 1. In the rating tree it is assumed that the debtor has an initia rating of 3. Buiding the tree from T 1 to T 2 is more compicated because the rating of the debtor in T 1 is not known in T 0. Here we have to specify the transition probabiities for a rating grades (except for the defaut grade) separatey. A these probabiities can be read from the transition matrix P(T 2 T 1 ) corresponding to the time period T 2 T 1. The transition probabiities for a migration from grade i in T 1 to grade j in T 2 can be read from the i-th row of the matrix P(T 2 T 1 ). For the cacuation of this matrix we again refer to Chap. 6. In this way it is possibe to describe every possibe rating path a debtor can take in the tree and associate a probabiity to each path, the product of the transition probabiities in each time step. From a practica perspective, a rating tree is not an important too if it is considered stand-aone. The reason is that there are hardy financia products in the market that depend soey on the rating of a debtor. Therefore, a rating tree wi be amost aways appied in combination with some other vauation framework. In the context of oan vauation this wi be a short rate tree that wi be introduced in the next section. 18.2.2 Modeing Interest Rate Dynamics In this section we wi give an introduction to short rate modes, the simpest cass of term structure modes which is appied in banks. We start with a short overview of

396 B. Engemann interest rate products that are iquidy traded and that are needed for caibrating a term structure mode. After that the Gaussian one-factor mode is introduced and its mathematica properties are expained. We show how the parameters of the mode are determined from market data and how tree agorithms for pricing interest rate dependent products are constructed in this mode. 18.2.2.1 Interest Rate Markets The most important interest rates in the interbank market are Libor rates and swap rates. A Libor rate determines the interest rate at which banks end money to each other. These interest rates are avaiabe for maturities up to 12 months. A swap rate is the fixed interest rate in a swap, i.e. a contract where counterparties exchange a fixed rate of interest for a Libor rate. Swap contracts have maturities from 1 year up to 50 years. These interest rates are quoted on a reguar basis in market data systems ike Reuters or Boomberg. As we have aready expained in Chap. 17 these interest rate are needed to compute the interbank discount curve that can be used as a reference curve for oan pricing. On these two interest rates ca and put options are traded. In the case of Libor rates these options are caed caps and foors. Options on a swap rate are caed swaptions. The market conventions for these products are a bit different from ca and put options in equity markets. An exampe of a cap is iustrated in Fig. 18.2. A cap is not a singe option but a series of ca options. In the exampe of a 7Y cap on a 12M Libor rate of Fig. 18.2 the cap consists of six ca options with a strike price of 4%. Each of these options is caed a capet. Each capet has an exercise date and a payment date. At the exercise date the option hoder can decide if he wishes to exercise the option what he wi do if the option payoff is positive, i.e. if 2nd payment if exercised 4th payment if exercised 6th payment if exercised 1st payment if exercised 3rd payment if exercised 5th payment if exercised 0 1 2 3 4 5 6 7 Time 1st exercise if 12M Libor > 4 % 3rd exercise if 12M Libor > 4 % 5th exercise if 12 M Libor > 4 % 2nd exercise if 12 M Libor > 4 % 4th exercise if 12 M Libor > 4 % 6th exercise if 12 M Libor > 4 % Fig. 18.2 A 7Y cap on a 12M Libor rate with a strike price of 4%

18 Risk Management of Loans with Embedded Options 397 receiving of 4 % annuay if the swaption was exercised 0 5 15 exercise if swap rate < 4 % payments of the foating eg if the swaption was exercised Fig. 18.3 A 5Y receiver swaption on a 10Y swap with a strike price of 4% the Libor rate of this day is greater than 4%. The date where the payment is done if the option was exercised is 12 months ater. The time gap between exercise and payment corresponds exacty to the tenor of the interest rate. This is different from equity options where payment is done immediatey after exercising an option. In genera notation, the payoff of a capet with maturity T is maxð0; f KÞt (18.1) where f is a Libor rate that is fixed in T, K is the strike price, and t is the tenor of the Libor rate. The payment time of the payoff, if it is positive, is in T þ t. An exampe of a swaption is given in Fig. 18.3. Here, a 5Y receiver swaption on a 10Y swap with a strike price of 4% is iustrated where we stick to the European convention of a swap paying the fixed rate annuay and the foating rate semiannuay. In this contract the hoder has the right to enter in 5 years into a receiver swap with a maturity of 10 years. The terminus receiver swap means that the option hoder wi receive fixed payments in this swap contract. 2 Therefore, he wi exercise the option if the market swap rate at the exercise date is beow the strike price of the option contract. The payoff of a receiver swaption cannot be expressed by a simpe formua ike the payoff of a capet because the profit of exercising a receiver swaption is reaized at every payment date of the fixed rate in the swap. Therefore, this payoff at the exercise date must be written as the present vaue of a these profits X n i¼1 t i df (T i Þmax (0; K sþ (18.2) where n is the number of fixed rate payments in the swap, T 1,..., T n the payment times, df(t) the discount factor corresponding to time T, t i the year fraction of the 2 If the option hoder pays the fixed rate in the swap the contract is caed a payer swaption.

398 B. Engemann i-th fixed rate period in the underying swap, K the strike price, and s the swap rate that is fixed at the exercise date of the swaption. For both option contracts, the option premium is determined by the Back 76 formua, a market standard formua for ca and put options in many markets (see Back 1976). It is assumed that at the exercise date the underying interest rate, the Libor rate in the case of a cap and the swap rate in the case of a swaption, is distributed og-normay, i.e. that the ogarithm of the interest rate is distributed normay. This distribution is given by p n ðy T ÞNðn ðy 0 Þ 0:5 s 2 T; s ffiffiffi T Þ where y T is the vaue of the interest rate at time T, y 0 is its current forward vaue, and s its voatiity. Note that the standard deviation is proportiona to the square root of T, i.e. the uncertainty of the future vaue of the interest rate y is increasing with time. Under this assumption a simpe formua can be derived for the option premium of both contracts by cacuating the discounted expected vaue of each contract s payoff. In the case of the capet, we get for the premium V capet the expression V capet ¼ df ðt þ tþt ðf Nðd 1 Þ K Nðd 2 ÞÞ d 1 ¼ ogð f =KÞþ0:5 s2 T pffiffiffi s T pffiffiffi d 2 ¼ d 1 s T (18.3) where f is the forward of the underying Libor rate, K the capet s strike price, t the Libor s tenor, T the capet s expiry, df(t þ t) the discount factor corresponding to the capet s payment time, and N(.) is the cumuative distribution function of the norma distribution. A simiar formua exists for the fooret. For the premium of a receiver swaption V receiver we get V receiver ¼ M ðk Nð d 2 Þ s Nð d 1 ÞÞ M ¼ Xn i¼1 t i df ðt i Þ d 1 ¼ ogðs=kþþ0:5 s2 T pffiffiffi s T pffiffiffi d 2 ¼ d 1 s T (18.4) The notation used in this formua was aready expained above. For payer swaptions an anaogous formua exists. In practice, these options are traded iquidy and prices are determined by suppy and demand. The formuas (18.3) and (18.4) are used as quotation toos. For each option the voatiities are quoted at which traders are wiing to buy or se an option

18 Risk Management of Loans with Embedded Options 399 and the formuas are needed to convert voatiities into option prices. The reason is that for traders it is easier to compare option prices for different maturities and different strike prices in terms of impied voatiities. The voatiities that refect current market prices are quoted in market data systems ike Reuters or Boomberg. 18.2.2.2 The G1 Mode The simpest mode cass for modeing the term structure of interest rates are short rate modes. The short rate is an artificia mathematica quantity that cannot be observed in the market. It describes the interest rate that is vaid over the next infinitesima time period. Iustrativey, one can think of the short rate as an approximation of the overnight rate. Therefore, short rate modes describe the dynamics and the future deveopment of the overnight rate. If the distribution of the short rate at future times is norma, the corresponding short rate mode is caed Gaussian. In its simpest version the short rate is driven by one stochastic factor and is caed a Gaussian one-factor short rate mode, the G1 mode. Mathematicay, the mode is described by the dynamics rðtþ ¼xðtÞþyðtÞ; dx ¼ k xðtþdt þ sðtþdw; xð0þ ¼0; (18.5) where r is the short rate, x the stochastic factor, y a deterministic time-dependent function, k a positive constant, s the (possiby time-dependent) voatiity of x, and W a Wiener process. 3 For readers who are not famiiar with continuous-time stochastic cacuus we iustrate the short rate dynamics (18.5) by its discretized version. Starting from x ¼ 0 at time t ¼ 0 a path of the short rate can be simuated using uniform time steps Dt by rðidtþ ¼xðiDtÞþyðiDtÞ; pffiffiffiffiffi (18.6) xðidtþ ¼xðði 1ÞDtÞ k xðði 1ÞDtÞDt þ sðði 1ÞDtÞ Dt Z; where Z a is random number that is normay distributed. The short rate r at each time point is written as the sum of the stochastic factor x and the function y. The stochastic factor is driven by two components. The first component is deterministic. The second component is stochastic and modes the randomness of x. 3 The G1 mode is a mathematica transformation of the Hu and White (1990) mode. The transformed mode is more convenient from a mathematica perspective but contains exacty the same economic content.

400 B. Engemann To simuate a path of the short rate one has to simuate normay distributed random numbers and appy (18.6) iterativey. The parameter s is a measure for the uncertainty of future vaues of the short rate. The arger s the more the third term in (18.6) can fuctuate around zero. Economicay one woud expect from overnight rates that they are imited to a certain range of numbers. This is in contrast to a stock where an unimited growth over time is in principa possibe. If the overnight rate has a rather high vaue one woud expect it to fa with high probabiity in the future. The opposite is true if the overnight rate is on a historica ow vaue. This property of interest rates is caed mean reversion. In (18.6) the mean reversion property is ensured by the second term invoving k. Whenever x is positive (negative) the second term becomes negative (positive) and generates a downward (upward) drift. To price interest rate products, one has to be abe to simuate future Libor or swap rates. Both interest rates are computed from the future discount curve. Therefore, it suffices to simuate future discount curves. This can be done in a short rate mode by simuating overnight rates and mutipying the resuting overnight discount factors to get a discount factor corresponding to a arger time interva for a specific scenario. Taking expectations over a arge number of scenarios resuts in a future discount curve. To be mathematicay more precise, a future discount curve at time t can be computed conditiona on the short rate r(t) at time t from df ðtjrðtþ ¼rÞ ¼E exp ð T t rðtþdt jrðtþ ¼r (18.7) To iustrate the basic principe of product pricing using a short rate mode we take the exampe of a capet. To price a capet in the short rate mode (18.5) the foowing steps have to be carried out: 1. Simuate a short rate path from time t ¼ 0 to time T using (18.6). This path ends at time T in the vaue r(t). 2. Simuate many short rate paths from time T to time T þ t using (18.6). A paths start in r(t). 3. From a the short rate paths in step 2 compute the discount curve using (18.7) where the expectation is repaced by an arithmetic average. 4. From the discount curve in step 3 compute the reaized Libor rate f f ¼ ð1=df ðt þ tjrðtþ ¼rÞ 1Þ=t 5. Compute the discounted payoff for this scenario by df ðt þ tjrðtþ ¼rÞt maxð0; f KÞ (18.8) 6. Compute the discount factor corresponding to the path in step 1 and mutipy it with the resut in (18.8) to discount the payoff back to time t ¼ 0.

18 Risk Management of Loans with Embedded Options 401 7. Repeat steps 1 6 many times and compute the option price as the average over a simuated vaues. This procedure ooks very awkward because nested Monte-Caro simuations have to be appied to compute the price of a rather simpe instrument. The above procedure woud never be impemented in practice. It shoud just serve as an iustration how a short rate mode in principe can be used to determine the price of a financia instrument. One of the reasons for the popuarity of the Gaussian one-factor mode is its anaytica tractabiity. For instance, it is not necessary to compute (18.7) by Monte- Caro simuation because an anaytica expression exists for this expectation. In the next two subsections we wi expain the missing parts for using this mode in practice, how to caibrate the mode parameters and how to impement an efficient pricing agorithm after the mode is caibrated. We have expained the short rate mode on a rather intuitive eve. For instance, we have computed product prices as expectations over simuated scenarios. Athough this procedure seems pausibe it is not cear that this eads to reasonabe prices. In derivatives pricing theory it is shown that the absence of arbitrage in financia markets, i.e. the absence of trading possibiities that deiver risk free profits without capita investments, impy the existence of a probabiity measure, the risk-neutra measure, under which meaningfu prices can indeed be computed as expectations. For detais the reader is referred to the books of Hu (2008), Joshi (2003) and Shreve (2004). Finay, we comment on the economic interpretation of the mode. On a first gance it may seem strange to mode the dynamics of the fu term structure by the dynamics of the overnight rate. From empirica anayses of term structure movements over time it is known that the term structure dynamics can be described with very good accuracy by three components, parae shifts, changes in the steepness, and changes in the curvature of the term structure (Litterman and Scheinkman 1991). The most important component is the parae movement. Basicay, a onefactor mode describes the changes in the eve of interest rates. If these eve changes are modeed by a short-term, medium-term, or ong-term interest rate does not pay a roe. In this sense the modeing of the term structure by a very short term rate can be justified. Of course, to mode more genera movements of the term structure more factors are needed. A good reference for modern interest rate modeing approaches is Brigo and Mercurio (2006). 18.2.2.3 Caibration of the G1 Mode In the ast subsection we have expained the G1 mode and outined its economic interpretation. To use the mode its parameters y(t), k, and s(t) have to be specified from market data. If we appy (18.7) witht ¼ 0andr(0) approximatey equa to the current overnight rate, the eft hand side of (18.7) is equa to the current discount curve. This resuts in a condition for the parameter y(t). In fact, it is possibe to derive

402 B. Engemann a formua for y(t) that expresses y(t) in terms of the current discount curve and the (sti undetermined) parameters k and s(t) yðtþ ¼f ðtþþe 2kt ð t f ðtþ ¼ 0 @df ðtþ =df ðtþ: @t e 2ks s 2 ðsþds; (18.9) The function f(t) is the instantaneous forward rate, i.e. the forward rate that is vaid from time t over an infinitesima time period. It can be computed from the current discount curve as shown in (18.9). This choice of y(t) ensures that zero bond prices are correct in the G1 mode and equa to current discount factors. It remains to determine the mode parameters k and s. These parameters are caibrated from the prices of iquid options, i.e. caps or swaptions. The basic idea is to use the mode parameters that resut in mode prices matching given market prices as cose as possibe. This eads to an optimization probem min k;s X K i¼1 2 V i;mode V i;market (18.10) where K is the number of market instruments that are used for caibration, V i,mode is the mode price of the i-th instrument whie V i,market is its market price. To carry out the caibration efficienty we need pricing formuas for capets and swaptions in the G1 mode. To derive the necessary pricing formuas we wi show as a first step that in the G1 mode both the pricing of caps and the pricing of swaptions can be reduced to the pricing of options on zero bonds. We start with a capet with maturity T and payment time T þ t. The underying of the capet is a Libor rate f(t, t) that is fixed in time T and has a tenor t. By P(t,T) we denote the price of a zero bond with maturity T at time t. Note that at time t ¼ 0 it hods df(t) ¼ P(0,T). For the capet we write its price as the expected vaue of its discounted payoff. The expectation is taken over possibe paths of the short rate which are determined by its dynamics. 2 0 capet ¼ E4exp@ 2 0 ¼ E4exp@ 2 0 ¼ E4exp@ Tþt ð 0 ð T 0 ð T 0 1 rðsþdsa t ð 1 fðt; t Þ K Þ þ 3 5 rðsþdsa PT; ð T þ tþt ðfðt; tþ K Þ þ 3 5 1 rðsþdsa 1 PT; ð T þ tþ PT; ð T þ tþ 1 K t þ 3 5

18 Risk Management of Loans with Embedded Options 403 2 0 ¼ E4exp@ ð T 0 1 ¼ ð1 þ K tþe4exp@ rðsþdsa ð1 ð1 þ K tþpt; ð T þ tþ 2 ¼ put on zero bond 0 ð T 0 Þ þ 3 5 1 rðsþdsa 1 ð1 þ K tþ PT; ð T þ tþ The notation (.) þ is an abbreviation for max(0,.). By expressing the Libor rate in time T þ t by the corresponding zero bond price we end up with a formua for a put option on a zero bond. If we know a formua for the price of a put option on a zero bond in the G1 mode we can use the above reation to compute the price of a capet. To derive a simiar reationship for swaptions is more compicated. We start with the payoff of a swaption with maturity T 0. We repace the discount factors in the payoff formua (18.2) by zero bond prices P(T 0,T i ) because we have to consider the vaue of the payoff at maturity T 0 to compute the swaption s price. þ 3 5 payoff ¼ maxð0; K sþ Xn i¼1 ¼ max ¼ max ¼ max 0; K Xn 0; Xn i¼1 0; Xn i¼1 i¼1 t i PT ð 0 ; T i Þ t i PT ð 0 ; T i Þ s Xn i¼1 t i K PT ð 0 ; T i! c i PT ð 0 ; T i Þ 1 : t i PT ð 0 ; T i Þ Þ ð1 PT ð 0 ; T n ÞÞ!! (18.11) In the derivation we have used the formua for the forward swap rate s ¼ 1 PT ð 0; T n Þ P n i¼1 t i PT ð 0 ; T i Þ and have introduced the coupons c i K t; if i < n 1 c i ¼ K t n 1 þ 1; if i ¼ n 1 We have shown that a swaption is equivaent to an option on a coupon bond with a strike price of 1. This option wi be exercised if the condition

404 B. Engemann X n i¼1 c i PT ð 0 ; T i Þ 1 is fufied. To proceed with deriving a pricing formua for the swaption we need a pricing formua for zero bonds. They can be computed from expression (18.7) Pt; ð TjrðtÞ ¼r Þ ¼ At; ð TÞexpð r Bt; ð TÞÞ (18.12) The functions A(t,T) and B(t,T) can be expressed in terms of the mode parameters df ðtþ At; ð TÞ ¼ df ðtþ exp Bt; ð TÞ ¼ 1 k 0 @ Bt; ð T T t 1 e k ð Þ : ð t Þf ðtþ 1 2 Bt; ð TÞ2 e 2kt 0 1 e 2ks s 2 ðsþdsa; We see that the zero bond price in (18.12) is monotonous is the short rate r. Therefore it is possibe to find a unique vaue r * that fufis the condition X n i¼1 c i PT ð 0 ; T i jrt ð 0 Þ ¼ r Þ ¼ 1: (18.13) The vaue r * of the short rate has to be determined by a zero-search agorithm ike Newton s method or a bisection agorithm (Press et a. 1992). Combining reation (18.13) with (18.11) eads to payoff ¼ max ¼ max ¼ Xn i¼1 0; Xn i¼1 0; Xn i¼1 c i PT ð 0 ; T i Þ 1! c i PT ð 0 ; T i Þ c i PT ð 0 ; T i jrt ð 0 Þ ¼ r Þ c i maxð0; PT ð 0 ; T i Þ PT ð 0 ; T i jrt ð 0 Þ ¼ r ÞÞ: We see that simiar to the price of a capet aso the price of a swaption can be written in terms of prices of options on zero bonds because we have rewritten the payoff of a swaption as the payoff of a portfoio of ca options on a zero bond with strike prices P(T 0,T i r(t 0 ) ¼ r * ). If we know a pricing formua for options on zero bonds in the G1 mode, the prices of capets and swaptions can be cacuated easiy. The price of an option on a zero bond can be computed as an expectation over the payoff using the formua for a zero bond price (18.12) in connection with the!

18 Risk Management of Loans with Embedded Options 405 distribution of the short rate r at the option s maturity. This eads to an easy to evauate formua for the price V Ca of a ca option with maturity T on a zero bond with maturity T þ t V Ca ¼ df ðt þ tþnðhþ df ðtþnh ð s P Þ; h ¼ 1 og df ðt þ tþ þ s P s P df ðtþk 2 ; ð T s P ¼ ðbð0; T þ tþ Bð0; TÞÞ 2 e 2kT e 2ks s 2 ðsþds: 0 (18.14) The price of a put option on a zero bond can be computed from the put-ca parity V Put þ df ðt þ tþ ¼ V Ca þ K df ðtþ This gives us the fina ingredient to carry out the caibration of the G1 mode. It remains to decide which instruments (caps or swaptions) shoud be used for caibration. It depends on the product for which the mode is needed. As a genera principe, one shoud use instruments for caibration that are simiar to the product that shoud be priced. If a oan that contains prepayment rights shoud be priced, swaptions are more appropriate for caibration because a prepayment right is basicay an embedded swaption. For a oan with embedded caps and foors on a foating interest rate, interest rate caps and foors are more suitabe for the caibration of the G1 mode. 18.2.2.4 Tree Impementation of the G1 Mode In this subsection we present an efficient impementation of the G1 mode by a trinomia tree. A trinomia tree is a discrete method to price products in the G1 mode. In the Monte-Caro simuation we have used to iustrate the G1 mode in Sect. 18.2.2.1 we aready have done a time discretization but the short rate coud sti attain any rea vaue in each point of time. In a trinomia tree the set of admissibe vaues for the short rate is restricted to a finite grid of points in each time step. This is done in a structured way to construct an efficient agorithm for the cacuation of product prices. An exampe of a trinomia tree is shown in Fig. 18.4. We denote each node with r i,j which is the j-th short rate grid point at the i-th time grid point. Every node r i,j in the tree has exacty three succeeding nodes r i+1,j, r i+1,j+1, and r i+1,j+2. These nodes are buit in a way that the tree is recombining. This ensures that the number of nodes does not grow exponentiay with the number of time steps. Furthermore, associated with every node are three probabiities q d,i,j, q m,i,j, and q u,i,j that are needed to compute product prices as discounted expectations.

406 B. Engemann Short Rate q u,0,0 q m,0,0 q d,0,0 q u,1,2 q m,1,2 q d,1,2 q u,1,1 q m,1,1 q d,1,1 qu,1,0 q m,1,0 q d,1,0 t 0 t 1 t 2 t 3 t 4 Time Fig. 18.4 Iustration of a trinomia tree for the G1 mode Both the vaues r i,j of the nodes and the probabiities q d,i,j, q m,i,j, and q u,i,j are determined by a construction process. The three probabiities are computed in a way that ensures that the oca expected vaues and variances of the stochastic factor x are identica to the corresponding vaues of the continuous-time process (18.5). The vaues of the short rate are chosen to ensure that zero bonds with maturities identica to the time grid points of the trinomia tree are priced correcty. To be more specific, to construct the tree we have to start with defining a time grid 0 ¼ t 0, t 1,..., t. It has to be ensured that a dates that are reevant for product pricing ike coupon payments and exercise dates are incuded in this grid. Further, we denote with x i,j the j-th point of the factor grid at time t i. From the dynamics of the stochastic factor x in (18.5) we get m i;j ¼ Ext ð iþ1 Þjxt ð i Þ ¼ x i;j ¼ xi;j expð k ðt iþ1 t i ÞÞ v 2 i;j ¼ Var x ð t s 2 ðt i Þ iþ1þjxt ð i Þ ¼ x i;j 2 k 1 exp 2 k t (18.15) ð ð ð iþ1 t i ÞÞÞ where for the cacuation of the variance it was assumed that the voatiity s is ocay constant. The standard deviation v i,j of (18.15) is used to construct the grid for x. Intuitivey, it is cear that the step size Dx of the grid shoud be proportiona to the standard deviation. A standard choice is p ffiffi Dxðt iþ1 Þ ¼ max v i;j 3 : (18.16) j With this choice for the step size the x-grid is constructed as k Dx. The vaues for k that are needed to construct the grid at time t i+1 are defined from the mean vaues in (18.15) to ensure that the grid covers the vaue range that is attained with high probabiity

18 Risk Management of Loans with Embedded Options 407 m i;j k ¼ round : Dxðt iþ1 Þ (18.17) With this definition of the midde node of the three succeeding nodes at time t i+1 of each node at time t i, we have a ingredients to compute the tree probabiities. This is done by matching the moments of the continuous and the discrete distributions of the short rate by soving the set of equations m i;j ¼ q u;i;j x iþ1;kþ1 þ q m;i;j x iþ1;k þ q d;i;j x iþ1;k 1 ; v 2 i;j ¼ q u;i;j x 2 iþ1;kþ1 þ q m;i;j x 2 iþ1;k þ q d;i;j x 2 iþ1;k 1 2; q u;i;j x iþ1;kþ1 þ q m;i;j x iþ1;k þ q d;i;j x iþ1;k 1 1 ¼ q u;i;j þ q m;i;j þ q d;i;j : It can be shown that the choice of the step size in the x-grid (18.16) eads indeed to probabiities, i.e. that the quantities q d,i,j, q m,i,j, and q u,i,j are positive (Brigo and Mercurio 2006). The vaues of the short rate tree r i,j can be computed by adding y(t i ) to the x-tree which is computed by (18.9). Since y in (18.9) is derived from a continuous-time process there wi be a sma discretization bias when zero bonds are priced with the tree, i.e. the discount curve wi not be matched exacty by the tree. To fit discount factors exacty one coud aternativey compute the correction term y(t i ) instead of using (18.9) by an additiona numerica caibration in the tree. The detais of this cacuation can be found in Brigo and Mercurio (2006). After the construction of the tree is finished it can be used for product pricing. Product prices are computed by iterative expectations. The discretized product vaue V i,j is initiaized in time t. Depending on the specific product this can be done by initiaizing V,j by the product s payoff or by the vaue of a coupon payment. The preceding vaues of V are then computed iterativey as discounted expectations V i;j ¼ exp r i;j ðt iþ1 t i Þ qu;i;j V iþ1;kþ1 þ q m;i;j V iþ1;k þ q d;i;j V iþ1;k 1 where k was defined by (18.17). At every time point t i where either a coupon is paid or a counterparty of the product has an exercise right, the vaue of V i,j has to be modified appropriatey. We wi see this in detai in the next section when a pricing agorithm for a oan with prepayment rights is deveoped. Finay, we remark that the trinomia tree is a popuar and intuitive but not the most efficient way of pricing interest rate products. It can be shown that pricing a product in the G1 mode is equivaent to soving a partia differentia equation that is determined from the short rate dynamics (18.5). For partia differentia equations soution agorithms exist that deiver a higher accuracy for ess computationa effort than the trinomia tree. Detais can be found in Randa and Tavea (2000).

408 B. Engemann 18.2.3 A Genera Loan Pricing Framework In this section we combine the rating tree of Sect. 18.2.1 and the interest rate tree of Sect. 18.2.2 to a pricing agorithm for oans with embedded options. By assuming that interest rate changes are independent from rating changes both modes can be easiy combined to a three-dimensiona tree. This mode was aready suggested by Sch onbucher (2003) in a different context. The resuting three-dimensiona tree is iustrated in Fig. 18.5. In this exampe the tree is buit for a rating system with six rating grades where the sixth grade is the defaut grade. From every node it is possibe to reach eighteen succeeding nodes, six possibe rating changes times three possibe changes in the short rate. Because of the independence assumption of rating changes and interest rate changes the tree probabiities can be easiy computed by mutipying the probabiities of the interest rate tree with the probabiities of the rating tree. The pricing of a oan in the tree is carried out anaogousy to the pricing in the two-dimensiona trees by computing discounted expectations iterativey starting from the most-right nodes. We expain the pricing of oans with prepayment rights in detai. To mode prepayment some assumptions on the behaviour of debtors and the conditions of refinancing a oan have to be made: We assume that a debtor needs the money that was ent by a bank unti the oan s maturity. If he is abe to get a cheaper oan over the remaining maturity on a prepayment date he wi prepay with a probabiity p ex. If a debtor prepays and enters a new oan the opportunity costs of capita and the interna costs for the new oan are the same as for the od oan (cf. Chap. 17 for an expanation of these cost components). Rating Grade Short Rate Time Fig. 18.5 Iustration of the three-dimensiona tree that is used for pricing oans with embedded options

18 Risk Management of Loans with Embedded Options 409 If a debtor prepays and enters a new oan, this oan wi not have any prepayment rights. A banks have the same opinions on defaut probabiities and recovery rates. The exercise probabiity p ex is introduced to mode the irrationa behaviour of retai costumers. They do not act perfecty rationa ike interest rate derivatives or bond traders and might not prepay even if it is advantageous for them. The probabiity p ex gives the probabiity that a debtor wi prepay when the conditions are in his favour. We expain the steps that are necessary to impement a pricing agorithm for a fixed-rate buet oan with prepayment rights in this mode in detai. We assume that a time grid 0 ¼ t 0, t 1,..., t is constructed that contains a important time points, the payment times of coupons and the times where prepayment is possibe. Further, we assume that the tree of Fig. 18.5 is constructed using the steps that were expained in Sects 18.2.1 and 18.2.2. We use the notation N for the oan s notiona, z is the oan s fixed interest rate, T 1,..., T m are the interest rate payment times, c tot is the sum of the opportunity cost of capita and the interna cost margin, and t i is the year fraction of the i-th interest rate period. We compute the price V(u,r i,j,t i ) of a fixedrate buet oan with prepayment rights depending on the rating u, the short rate r i,j and time t i using the agorithm: 1. At t : Initiaize V(u,r,j,t ) and V ex (u,r,j,t ) with N. 2. At t : Add z t m N to V(u,r,j,t ) and (z c tot ) t m N to V ex (u,r,j,t ). 3. At t -1 : Compute V(u,r 1,j,t 1 ) from the vaues of V at the succeeding nodes: X n 1 V u; r 1;j ; t 1 ¼ g¼1 p ug ðt jt 1 Þ ^V g; r 1;j ; t 1 þ p un ðt jt 1 ÞRt ð 1 ÞN ^V g; r 1;j ; t 1 ¼ e r 1;j ðt t 1 Þ q u; 1;j V g; r ;kþ1 ; t þ q m; 1;j V g; r ;k ; t þ qd; 1;j V g; r ;k 1 ; t 4. Repeat step 3 for V ex. 5. Repeat steps 3 and 4 unti time t z ¼ T m 1 is reached. 6. At T m 1 : Add z t m 1 N to V(u,r z,j,t z ) and (z c tot ) t m 1 N to V ex (u,r z,j,t z ). 7. At T m 1 :IfT m 1 is a prepayment time repace V(u,r z,j,t z )by p ex N þð1 p ex ÞVðu; r z;j ; t z Þ if the condition V ex (u,r z,j,t z ) > N is fufied. 8. Repeat steps 3 7 unti t ¼ 0 is reached. To incude prepayment rights into oan structures with amortization schedues ike instament oans or annuity oans the amortization payments have to be added to the interest rate payments in the above agorithm. It is aso possibe (but a bit more compicated) to extend the pricing agorithm to oans with a foating interest rate.

410 B. Engemann The auxiiary variabe V ex is used ony to determine if prepayment is advantageous. It is computed from the interest margin excuding a cost components except for the risk costs. If the oan vaue at the prepayment date is fair, i.e. if the oan s margin is exacty equa to the then prevaiing minimum margin under the assumptions on costs used in the pricing agorithm, then the condition V ex ¼ N woud be fufied. Therefore, prepayment is advantageous if the oan under its current terms is too expensive under the actua market rates which is refected in the condition V ex > N. We wi iustrate this pricing agorithm with a numerica exampe in the next section. This section wi be concuded with comments on the theoretica properties of this pricing mode. What we have done in the tree agorithm is mixing riskneutra probabiities of the interest rate tree that are impied from the market with rea-word probabiities of the rating tree that are based on statistica information. The underying theory of derivatives pricing modes impies a trading strategy that aows the perfect hedging of interest rate risk with basic instruments in the interest rate market. The combination with statistica probabiities resuts in a mode of an incompete market, i.e. a mode that contains risks that are not tradabe and cannot be hedged. This has consequences for the risk management of prepayment rights. In principe, prepayment rights can be hedged by receiver swaptions. The number of receiver swaptions that are needed for the hedge (the hedge ratio) is determined by the pricing mode. However, if reaized defaut rates are different from defaut probabiities, these hedge ratios turn out to be wrong. In this case the hedge might ead to unexpected osses. Anayzing this mode in some detai shows that the risk of unexpected osses on the hedge of prepayment rights does not ead to an increase of economic capita for a oan portfoio because for typica oan portfoios the eve of economic capita is dominated by defaut risks. That means that unexpected osses in hedges of payment rights are aready covered by the economic capita that is needed as a buffer for unexpected osses due to defauts. The detais of these anayses are worked out in Engemann (2010). 18.3 Numerica Exampe In this section, we wi present a numerica exampe using rea market data. We use the discount curve that is presented in Tabe 18.3 and the swaption voatiity matrix of Tabe 18.4. As an exampe we consider a buet oan with a maturity of 15 years. The oan has a fixed interest rate and a prepayment right after 10 years. The debtor has the right to fuy pay back the oan after 10 years without penaty. The notiona of the oan is 1 miion. The oan is secured with coatera worth 400,000. As in the exampes of Chap. 17 the defaut probabiities are computed from the transition matrix of Fig. 6.7 in Chap. 6.

18 Risk Management of Loans with Embedded Options 411 Tabe 18.3 Discount curve used in the exampe for pricing a oan with prepayment rights Maturity (years) Discount factor Maturity (years) Discount factor 0.0027 0.999978 2.0000 0.955710 0.0833 0.999214 3.0000 0.932863 0.1667 0.997850 4.0000 0.900632 0.2500 0.996469 5.0000 0.866350 0.3333 0.995137 6.0000 0.830990 0.4167 0.993367 7.0000 0.796393 0.5000 0.991578 8.0000 0.762382 0.5833 0.989688 9.0000 0.727801 0.6667 0.987618 10.0000 0.694570 0.7500 0.985465 12.0000 0.631269 0.8333 0.983284 15.0000 0.542595 0.9167 0.981037 20.0000 0.434336 1.0000 0.978903 30.0000 0.318877 Tabe 18.4 Swaption voatiities used in the exampe for pricing a oan with prepayment rights (First coumn: swaption expiry, First row: tenor of the underying swap) 1 2 3 4 5 6 7 8 9 10 0.08 0.475 0.376 0.338 0.319 0.311 0.305 0.297 0.291 0.286 0.281 0.17 0.486 0.387 0.350 0.332 0.316 0.305 0.296 0.291 0.286 0.281 0.25 0.484 0.402 0.360 0.332 0.314 0.301 0.293 0.287 0.283 0.279 0.50 0.453 0.367 0.326 0.301 0.284 0.275 0.271 0.268 0.268 0.265 0.75 0.422 0.340 0.302 0.279 0.265 0.257 0.253 0.251 0.250 0.250 1 0.392 0.316 0.283 0.263 0.249 0.241 0.237 0.236 0.235 0.234 1.5 0.312 0.269 0.248 0.233 0.224 0.218 0.216 0.215 0.215 0.215 2 0.261 0.238 0.224 0.214 0.206 0.203 0.201 0.201 0.201 0.202 3 0.210 0.198 0.190 0.186 0.182 0.182 0.182 0.182 0.182 0.182 4 0.180 0.173 0.170 0.168 0.167 0.167 0.167 0.167 0.167 0.167 5 0.162 0.158 0.156 0.156 0.157 0.156 0.155 0.155 0.155 0.156 7 0.144 0.142 0.141 0.140 0.140 0.139 0.139 0.140 0.141 0.142 10 0.131 0.130 0.129 0.129 0.129 0.130 0.131 0.132 0.134 0.136 15 0.130 0.131 0.134 0.137 0.141 0.144 0.148 0.152 0.156 0.159 20 0.165 0.169 0.174 0.179 0.183 0.186 0.189 0.191 0.193 0.194 To measure the effect of rating migration on the pricing, we carry out the agorithm of Sect. 18.2.3 both with the fu transition matrix and with the termstructures of defaut probabiities that were computed in Tabe 17.1 of Chap. 17. In the atter case the agorithm of Sect. 18.2.3 is appied with two rating grades ony, the non-defaut grade and the defaut grade. The exercise probabiity p ex in this agorithm is set to 100%. There are two ways to extend the RAROC pricing framework of Chap. 17 to oans with amortization rights. One possibiity is increasing the risk costs by incuding prepayment risk. This is done by computing the risk costs from the condition V ¼ N. This condition was aso appied in the case without amortization rights but eads to an increased vaue of the risk costs when amortization rights are incuded. Aternativey, instead of increasing the interest margin a bank coud charge the option premium by an upfront payment. In this case the risk costs are

412 B. Engemann Tabe 18.5 Risk costs (in %) for the 15 years oan with and without prepayment right after 10 years Rating grade Risk costs (no prepay. right) Risk costs (PD term structure) 1 0.038 0.193 0.194 2 0.098 0.247 0.251 3 0.192 0.331 0.339 4 0.493 0.616 0.637 5 1.286 1.397 1.440 6 2.759 2.890 2.964 7 6.084 6.432 6.543 8 10.244 10.986 11.057 Risk costs (migration matrix) computed in the same way as for the otherwise identica oan without amortization rights. The option premium is determined by pricing the oan using the margin of formua (17.9) of Chap. 17 and computing the difference to the initia notiona. We start with caibrating the G1 mode. Since the oan has one prepayment right ony, a reasonabe caibration strategy is to caibrate the mode to the 10Y swaption into a 5Y swap which can be viewed as the underying option of the oan. Since two parameters cannot be caibrated from one instrument, we have caibrated k from the fu swaption matrix, i.e. we have soved the minimization probem (18.10) with time-independent s to determine k. 4 After that, we modify s to match the price of the caibration instrument. We find k ¼ 0.0182 and s ¼ 0.0090. In the first exampe we assume that the premium for the prepayment option resuts in an increased margin. We compute risk costs for the oan without prepayment right to get the reference rate refecting the margin for expected oss ony. After that we compute the risk costs incuding the prepayment right for the two cases expained above, using a term structure of defaut probabiities ony versus using the fu transition matrix. The resuts are presented in Tabe 18.5. In the second exampe we assume that the oan is sod with the risk margin corresponding to an otherwise identica oan without prepayment right and that the option premium is paid upfront by the debtor. The resuting option premia are reported in Tabe 18.6. From Tabe 18.5 we see that for the good rating grades the argest proportion of the risk costs corresponds to the prepayment right. For the poor rating grades it is vice versa. Here the risk costs are mainy driven by defaut risk. Further, we see that migration does not have an effect for the good rating grades. These debtors ony face the risk of downgrades which woud make their prepayment option ess vauabe. For this reason it does not make a difference if the prepayment right is priced with a term structure of defaut probabiities ony or with the fu transition matrix. The situation is different for debtors with poor rating grades. They have the change of upgrades which woud increase the vaue of their prepayment option consideraby. This chance of upgrades eads to a higher risk margin if the pricing is done with the migration matrix compared to the term structure of defaut probabiities. 4 The caibration of k from market data might be rather unstabe, i.e. the vaue of k is fuctuating strongy with changing market data. For this reason, this parameter is aternativey often estimated empiricay from historica data.

18 Risk Management of Loans with Embedded Options 413 Tabe 18.6 Prepayment option premia when the option premium is charged upfront Rating grade Option premium (PD term structure) 1 16,917 17,093 2 16,349 16,776 3 15,278 16,169 4 13,309 15,335 5 11,112 14,938 6 10,973 16,529 7 18,998 24,662 8 28,592 30,852 Option premium (migration matrix) The picture is simiar in Tabe 18.6 where the option premium is charged upfront instead of by an increased margin. We see that the effect of rating migration is sma for good rating grades and considerabe for the poor rating grades. 5 We see that option premia are not monotonous in the rating grade. Furthermore, we find that the premium increase under the incusion of rating migration is aso not monotonous in the rating grade. The option premium is the resut of severa economic effects. First, of course, there is a chance for faing interest rates. This effect is the same for a debtors. Second, for debtors with very ow defaut probabiities the option premium is basicay the premium for interest rate risk. Defaut risks do not pay a roe in this situation. Third, if defaut probabiities are increased midy this eads to a greater chance that a debtor wi defaut before the prepayment date and the prepayment right wi expire worthess. This eads to a decrease in the premium. Fourth, for debtors with high defaut probabiities the risk costs wi decrease consideraby if they survive unti the prepayment date. This has the effect that a debtor wi prepay for sure amost regardess of the interest rates at the prepayment date in this case. A these effects are incuded in the option premium. 18.4 Concusion In this chapter, we have discussed an agorithm for pricing oans with embedded options. We have focussed on prepayment rights because these are the most popuar embedded options in oan markets. However, it is aso possibe to extent the pricing agorithm to foating rate oans with embedded caps and foors. In a numerica exampe we have computed the necessary margin increase or the upfront premium depending on the way the prepayment right is charged by a bank. We have seen that option premia can be considerabe and that these options shoud not be negected when a oan is sod. 5 In fact the option premia for rating grade 1 shoud be identica because there is no possibiity for an upgrade of the debtor. The difference resuts from a numerica effect because the interpoation of defaut probabiities in the term structure eads to sighty different numbers than the exact cacuation by transition matrices corresponding to year fractions.

414 B. Engemann The presented agorithm offers further possibiities of extensions. One key assumption in the agorithm was that the cost structure of a bank remains constant in time. The recent financia crisis has shown that this is not true. In times of financia distress the funding conditions of a bank can worsen consideraby which eads to an increase of the margin of a oan. If the oan contains a prepayment right, however, the debtor might be abe to refinance his oan at a ower rate just because of the reduction in banks funding conditions when markets went back to norma. By modifying the cost assumptions in the agorithm, this effect can be incuded in the option premium. Finay, the agorithm can be used for the risk management of oan portfoios. It can be used for the cacuation of genera oss provision that was aready outined in Chap. 17. Furthermore, it can be used to hedge the embedded options by market options ike interest rate caps and interest rate swaptions. The pricing mode wi te the amount of hedging instruments needed by cacuating the so-caed greeks (deta, gamma, vega) based on information about the current market prices of interest rate options (incuded in the mode parameters of the G1 mode), the defaut probabiities of the debtors, the migration probabiities, and the product structures. In addition, it offers the possibiity to mode irrationa behaviour of debtors. Therefore, it incudes a information that is needed from an economic perspective and sti resuts in a tractabe mode that can be impemented efficienty. References Base Committee on Banking Supervision (BCBS) (2004), Base II: Internationa Convergence of Capita Measurement and Capita Standards: A Revised Framework. http://www.bis.org/pub/ bcbs107.htm Back F (1976), The Pricing of Commodity Contracts, Journa of Financia Economics 3, pp. 167 179. Brigo D, Mercurio F (2006), Interest Rate Modes: Theory and Practice with Credit and Infation, 2nd Edition, Springer, Berin Heideberg New York. Engemann B (2010), A Framework for Pricing and Risk Management of Loans with Embedded Options, Working Paper. Hu J (2008), Options, Futures, and other Derivatives, 7th edition, Prentice Ha, New Jersey. Hu J, White A (1990), Pricing Interest Rate Derivative Securities, Review of Financia Studies 3, pp. 573 592. Joshi M (2003), The Concepts and Practice of Mathematica Finance, Cambridge University Press, Cambridge. Litterman R, Scheinkman J (1991), Common Factors Affecting Bond Returns, Journa of Fixed Income, pp. 54 61. Press W, Teukosky S, Vettering W, Fannery B (1992), Numerica Recipes in C, Cambridge University Press, Cambridge. Randa C, Tavea D (2000), Pricing Financia Instruments: The Finite Difference Method, Wiey, New York. Sch onbucher P (2003), Credit Derivatives Pricing Modes, Wiey, New York. Shreve S (2004), Stochastic Cacuus for Finance II: Continuous-time Modes, Springer, Berin Heideberg New York.

About the Authors Stefan Bochwitz is Head of Section in the Department of Banking Supervision of the Bundesbank s Centra Office in Frankfurt. He is in charge for impementing and supervising Base s IRB approach in Germany and member of the AIG-subgroup on vaidation issues. His responsibiities incude setting up the supervision of credit risk modes as we as research activities in credit risk measurement and management. Prior to this position he was responsibe for the Bundesbank s assessment system for German corporates. He hods a degree in physics. Bernd Engemann works as an independent consutant for the financia industry. His areas of expertise are mathematica modes for pricing and risk management of financia instruments of various asset casses. Before that he was a founder and a managing director of Quanteam AG, a derivatives technoogy and consuting boutique in Frankfurt that focused on the deveopment of taior-made front-office and risk management soutions. Prior to that Bernd worked in the research department of the Deutsche Bundesbank where his focus was on the construction and vaidation of statistica rating modes. He has pubished severa artices in this fied. Bernd hods a dipoma in mathematics and a Ph.D. in finance from the University of Vienna. Urich Erenmaier works for KfW Bankengruppe s risk contro and management unit in Frankfurt am Main. He is responsibe for the deveopment and vaidation of rating systems. Prior to that he worked for Aarea Bank in Wiesbaden (Germany), where he supervised a project for the deveopment and impementation of new Base II compatibe rating systems. Urich hods a dipoma in mathematics and a Ph.D. in economics from the University of Heideberg. Konstantin Ermakov currenty works as an independent consutant for the financia industry. He studied mathematics and mechanics in the St. Petersburg State University (Russia), where he achieved a Master degree. After mowing towards the banking industry, he competed the Master of Quantitative Finance program at the Frankfurt Schoo of Finance and Management. His main areas of expertise are mathematica modes in finance, Monte-Caro methods, and parae and distributed computing. He has pubished severa artices on Monte-Caro methods and its appications. B. Engemann and R. Rauhmeier (eds.), The Base II Risk Parameters, DOI 10.1007/978-3-642-16114-8, # Springer-Verag Berin Heideberg 2011 415

416 About the Authors Water Gruber hods a Ph.D. in business mathematics and is managing partner of 1 PLUS i GmbH. He started his professiona career at an investment bank in the Treasury division and ALCO Management. Foowing that, Water worked as team eader, banking supervision, on the board of management of Deutsche Bundesbank in the areas of research and the principa issues of interna risk modes and standard procedures. He aso represented the Bundesbank on various internationa boards (Base, IOSCO). Foowing this he was managing director of a consuting firm where he worked as consutant and trainer in banking supervision, risk management and product assessment procedures. Water has pubished severa papers, in banking supervision, market and credit risk modes and derivative finance products. He has aso had severa standard works pubished in these fieds. Voker Matthias Gundach currenty works as a professor for mathematics at the THM University of Appied Sciences in Gießen. From 2008 to 2010 he was professor for mathematics at the business schoo of the Ostfaia University of Appied Sciences in Wofsburg. Before that he was senior project manager for KfW Bankengruppe s risk and portfoio management unit in Frankfurt am Main, Germany. There he was responsibe for the deveopment and reaisation of anticipative risk management incuding stress tests. He started his career at Aarea Bank in Wiesbaden (Germany) where he deveoped a credit portfoio mode, set up a credit risk reporting and an evauation for MBS. Matthias hods a dipoma in mathematics from the University of W urzburg (Germany), an MSc and a PhD in mathematics from the University of Warwick (UK) and a Habiitation in mathematics from the University Bremen (Germany). He edited two books for Springer edition on stochastic dynamics and the credit portfoio mode CreditRisk+. His other research activities incude work on ergodic theory, stochastic dynamica systems and mathematica bioogy. Ronny Hahn hods a degree in business administration (BA) and is managing partner of 1 PLUS i GmbH. He is speciaised in Base II and credit risk management. At the beginning of his career he worked for a savings bank as credit risk anayst in the oan department and as risk manager for the ALM-positions. Foowing that, Ronny worked as a consutant and trainer in the areas banking supervision, risk management, and risk measurement. He has pubished severa papers, particuary on banking supervision and market and credit risk management. Afred Hamere is Professor of Statistics at the Facuty of Business, Economics and Information Systems, University of Regensburg. Prior to his present position, he was Professor of Statistics at the University of Konstanz and Professor of Statistics and Econometrics at the University of T ubingen. His primary areas of research incude statistica and econometric methods in finance, credit risk modeing and Base II, and mutivariate statistics. Afred has pubished eight books and more than 80 artices in scientific journas. Eveyn Hayden works for Raiffeisen Bank Internationa s Rating Mode Vaidation and Methods unit in Vienna, where she is responsibe for the deveopment and vaidation of rating modes. Previousy she worked at the Austrian Nationa Bank

About the Authors 417 and at the University of Vienna and aso participated in banking-industry projects as freeance consutant. She has pubished severa artices in the area of risk measurement and management. Eveyn hods dipomas in internationa business administration and in statistics and a Ph.D. in finance from the University of Vienna. Stefan Hoh is a Senior Financia Sector Speciaist at the Financia Stabiity Institute (FSI) of the Bank for Internationa Settements (BIS). He is primariy responsibe for the dissemination of information on sound practices for effective banking supervision, covering a broad range of topics. Before joining the FSI, Stefan was a Senior Economist (Supervision) in the BIS Representative Office for Asia and the Pacific in Hong Kong. This foowed his work for the Deutsche Bundesbank, Frankfurt, in the department for Banking Supervision, being responsibe for the Deutsche Bundesbank s market risk modes examination and vaidation team. He is an adjunct professor at City University of Hong Kong and a quaified mathematician. Michae Knapp is member of the board of directors of Risk Research Prof. Hamere GmbH & Co. KG. Michae wrote his doctorate on Point-in-Time Credit Portfoio Modes. He is consutant for internationa financia institutions as we as medium-sized credit institutions with over 10 years experience. Prior to his current position he was academic adviser at the Chair of Statistics, Facuty of Business, at the University of Regensburg. The main focus of his research is the deveopment of credit portfoio modes, the PD / LGD/ EAD modeing and concepts for credit portfoio controing. Marcus R. W. Martin headed the professiona group for risk modes and rating systems at the banking examination department I of the Deutsche Bundesbank s regiona office in Frankfurt since 2004. He was examiner in charge and senior examiner for conducting audits for interna market risk modes as we as for interna ratings based approaches (IRBA), interna mode methods (IMM) and advanced measurement approaches (AMA). Since 2008 he is professor of financia mathematics and stochastic at the University of Appied Sciences at Darmstadt. His research is focused on (integrated) market and credit risk modeing, in particuar counterparty credit risk modeing, and pricing derivatives. Gregorio Mora is head of the Anaysis and Methodoogica Advice Unit within Credit and Operationa Risk Management Modes Division (CRMD) of the Banco de España (BE). He has contributed to the deveopment of the vaidation scheme adopted by the BE for Base II. He has deveoped review procedures, pubishes on vaidation issues and contributes to seminars on vaidation approaches. He has been working in the CRMD, focusing on credit risk modes since 2001. In his previous supervisory experience, he worked as an examiner reviewing a range of banking institutions. His research interests incude appied topics in credit risk, especiay the estimation and vaidation of risk parameters, benchmarking of capita requirements, provisioning and procycicaity issues.

418 About the Authors Christian Peter works for the securitization unit of KfW Bankengruppe in Frankfurt am Main where he is responsibe for quantitative structuring, modeing and anaysis as we as reguatory issues. Prior to that, he has worked for severa years in KfW Bankengruppe s risk controing and management unit. During that time, he was invoved among other things in the deveopment of a rating and pricing too for speciaized ending transactions and ed projects on coatera vauation as we as EAD and LGD estimation and vaidation. Christian hods a dipoma in business engineering and a Ph.D. from the University of Karsruhe (TH). Katja Puto heads the Quantitative Risk Assessment team at HSBC Hodings pc. Within this roe, she deveops the methodoogica framework for the group whoesae and retai rating modes and economic capita modes, and oversees the mode governance and vaidation framework. Moreover, she assists with the risk framework for wider poicy issues ike procycicaity or risk based oan oss provisioning. Prior to joining HSBC, Katja worked with Dresdner Bank, deveoping and impementing ratings systems and credit risk modes, and with the Banking Supervision department of the Deutsche Bundesbank. In the atter capacity, she approved bank interna market risk modes and represented the Bundesbank in various working groups of the Base Committee of Banking Supervision and the Committee of European Supervisors during the Base II negotiations. Danie Porath has deveoped and vaidated scoring modes for various banks when he was a senior anayst at the INFORMA consuting company. Afterwards, he entered the bank supervision department of the Deutsche Bundesbank where he deveoped a hazard mode for the risk-monitoring of German credit cooperatives and savings banks. He was aso invoved in the supervisory assessment of the banks risk management methods and in the on-site inspections of banks. Since 2005 he is Professor for Quantitative Methods at the University of Appied Sciences at Mainz. His research is focused on empirica studies about rating methods and the German banking market. Robert Rauhmeier works for UniCredit Bank AG Member of UniCredit Group in the Risk Instruments and Methods department in Munich since 2007. He is responsibe for the deveopment and vaidation of the retai rating systems. Previousy he was a member of Risk Architecture at Dresdner Bank where he worked on the deveopment, enhancement and vaidation of the group wide rating modes. Prior to that, he worked for KfW-Bankengruppe. In that roe he supervised the project Conception and Impementation of a Backtesting Environment. Robert studied economics and hods a Ph.D. from the University of Regensburg. His thesis invoved an anaysis of the Vaidation and Performance measuring of Bank Interna Rating Systems. Stefan Reitz hods a Ph.D. degree in mathematics and is professor for financia mathematics at the University of Appied Sciences in Stuttgart. He aso works as a consutant in the financia industry in various projects (risk controing, risk management, pricing of derivatives). Prior to his current position he was an auditor and audit supervisor within the banking examination department at the Deutsche Bundesbank

About the Authors 419 at its regiona office in Frankfurt. He conducted internationa audits at major and regiona banks in the areas portfoio risk modes, pricing of derivatives, risk management, minimum requirements for trading activities, and Base II impementation. Danie R osch is professor of finance and head of the Institute of Banking and Finance at the Leibniz University of Hannover. He received a Ph.D. from the University of Regensburg. Danie s work covers a broad range in asset pricing and empirica finance. He has pubished numerous artices on risk management, credit risk, banking and quantitative finance in eading internationa journas. Danie has aso hed numerous executive training courses and is consuting financia institutions on credit risk issues. Harad Scheue is a Senior Lecturer of Finance at the University of Mebourne. He is an expert in the areas of banking, financia risk measurement and management, fixed income securities, insurance, prudentia reguation, management of financia institutions and structured finance. He competed his Ph.D. at the University of Regensburg, Germany, on Forecasting Credit Portfoio Risk in coaboration with the German centra bank. Harad has consuted banks, insurance and reinsurance companies in Austraia, Europe and North America. Dirk Tasche is a senior risk advisor in the Risk Methodoogies & Anaytics department of Loyds Banking Group s Whoesae division. Prior to joining Loyds Banking Group, Dirk hed positions at Bayerische HypoVereinsbank, in the banking supervision division of the Deutsche Bundesbank, and at Fitch Ratings. In addition he had worked as researcher at universities in Germany and Switzerand. Dirk has pubished a number of papers on measurement of financia risk and capita aocation. Carsten S. Wehn is head of market risk contro at DekaBank, Frankfurt, where he is responsibe for the measurement of market and iquidity risk of the bank and the deveopment of risk methods and modes as we as the vaidation of the adequacy of the respective risk modes. Before joining DekaBank, he worked at Deutsche Bundesbank as an auditor and audit supervisor for reguatory examinations of banks quantitative modes for risk measurement and management. Carsten studied mathematics in Siegen, Germany, and Nantes, France, and he hods a Ph.D. in mathematics from the University of Siegen. He acts aso a ecturer for master students in mathematics at a University and reguary pubishes in we-known industria magazines as we as in books, mainy about quantitative aspects of risk modeing. He has pubished five books. Nicoe Widenauer hods a Ph.D. in economics from the University of Regensburg, where she worked as a research assistant at the Chair of Statistics. Her Ph.D. thesis investigated Loss Given Defaut Modeing in Credit Risk Management. Currenty Nicoe works at Commerzbank AG in Frankfurt in the rating methods department of the Risk Controing & Capita Management Group. She is responsibe for the LGD modes of various portfoios and aso worked on the estimation and impementation of an LGD mode in BRE Bank S.A., the Poish subsidiary of Commerzbank, in Warsaw and Lodz.