# Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Save this PDF as:

Size: px
Start display at page:

Download "Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio"

## Transcription

1 Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of loan losses: wdely used n bankng used n Basel II regulatons to set bank captal requrements Motvaton lnked to dstance-to-default analyss But, model of dependence s Gaussan Copula agan Key assumptons (apart from Gaussan dependence) homogeneous portfolo (equal nvestment n each credt) very large number of credts Merton-model Approach to Dstrbuton of Portfolo Losses 2

2 Motvaton: Merton s Model In Merton model value of rsky debt depends on frm value and default rsk s correlated because frm values are correlated (e.g., va common dependence on market factor). Value of frm at tme T: V = V exp( ( µ (1/ 2) 2 ) ) where ~ (0,1), σ T + σ T % ε %, V, ε N T V surprse n R C expected value of R C We wll assume that correlaton between frm values arses because of correlaton between surprse n ndvdual frm value (ε ι ) and market factor (m) Merton-model Approach to Dstrbuton of Portfolo Losses 3 Correlaton structure: Gaussan Copula Suppose correlaton between each frm s value and the market factor s the same and equal to sqrt() ). Ths means that we may model correlaton between the ε s as ε = m + 1 v, = 1, KN and corr( ε, ε ) = j Where m and v are ndependent N(0,1) random varables and s common to all frms Notce that f v ~ N(0,1) and m ~ N(0,1) then ε ~ N(0,1) Merton-model Approach to Dstrbuton of Portfolo Losses 4

3 Structural Approach, contd. From our analyss of dstance-to-default, we know that under the Merton Model a frm defaults when: 1 ε, µ σ, σ,, = 2 V 2 R ( ) / where ln( / ) D T V T RD B V The uncondtonal (natural) probablty of default, p, s therefore: RD, ( µ 2 σ V, ) T RD, ( µ 2 σv, ) T p Prob ε < N = σv, T σ T In ths model we assume that the default probablty, p, s constant across frms Merton-model Approach to Dstrbuton of Portfolo Losses 5 Idea: Sngle Common Factor and Large Homogeneous Portfolo Workng out the dstrbuton of portfolo losses drectly when the ε s are correlated s not easy But, f we work out the dstrbuton condtonal on the market shock, m, then we can explot the fact that the remanng shocks are ndependent and work out the portfolo loss dstrbuton Merton-model Approach to Dstrbuton of Portfolo Losses 6

4 Structural Approach, contd. The shock to the return, ε, s related to the common and dosyncratc shocks by: ε = m + 1 v Default occurs when: 1 2 RD, ( µ 2 σv, ) ε = m + 1 v < = N ( p) σ T or v < ( ) N p m 1 V, Merton-model Approach to Dstrbuton of Portfolo Losses 7 Vascek and the Intensty Model We ll see later that the Vascek model s essentally the same as the ntensty model when: the ntensty s the same for all the names; and the number of names becomes large equal nvestment n each name we use the Gaussan copula Merton-model Approach to Dstrbuton of Portfolo Losses 8

5 The Default Condton n Vascek v < N 1 ( p) m 1 A large value of m means a good shock to the market (hgh asset values) The larger the value of m the market shock the more negatve the dosyncratc shock, v, has to be to trgger default The hgher the correlaton,, between the frm shocks, the larger the mpact of m on the crtcal value of v. Merton-model Approach to Dstrbuton of Portfolo Losses 9 Condtonal Default Probablty Condtonal on the realsaton of the common shock, m, the probablty of default s therefore: Prob(default m)= Prob v < N ( p) m 1 N ( p) m = N = θ ( m), say 1 N ( p) m N m and therfore = = ( θ ( )) 1 Merton-model Approach to Dstrbuton of Portfolo Losses 10

6 The relaton between θ(m) and m For a gven market shock, m, θ(m) gves the condtonal probablty of default on an ndvdual loan H corr Lo corr Merton-model Approach to Dstrbuton of Portfolo Losses 11 Implcatons of Condtonal Independence For a gven value of m, as the number of loans n the portfolo, the proporton of loans n the portfolo that actually default converges to the probablty θ (m) <<<< KEY IDEA In the chart, f the market shock s m* then the ACTUAL proporton of defaults n the portfolo converges to 15% as # loans Merton-model Approach to Dstrbuton of Portfolo Losses 12

7 The crtcal value of m For a gven actual frequency of loss, θ, we can calculate the correspondng value of the market shock, m(θ) that wll produce exactly that level of loss: N N 1 ( p) m( θ ) θ = N 1 ( θ ) = m( θ ) = ( ) ( θ ) N p m 1 N ( p) 1 N ( θ ) Merton-model Approach to Dstrbuton of Portfolo Losses 13 The dstrbuton of portfolo loss Snce the proporton of portfolo losses decreases wth m, the probablty that the proporton of loans that default (L) s less than θ s: Prob ( L < θ ) = prob ( m > m( θ )) = prob m > Prob ( L θ ) N ( p ) 1 N = N 1 1 ( θ ) ( θ ) 1 N N ( p) < = N N ( p) 1 N ( θ ) Merton-model Approach to Dstrbuton of Portfolo Losses 14

8 Prob Loan Loss Dstrbuton ( L θ ) ( θ ) 1 N N ( p) < = N Ths result gves the cumulatve dstrbuton of the fracton of loans that default n a well dversfed homogeneous portfolo where the correlaton n default comes from dependence on a common factor Homogenety means that each loan has: the same default probablty, p (mplctly) the same loss-gven-default the same correlaton,, across dfferent loans The dstrbuton has two parameters default probablty, p correlaton, Merton-model Approach to Dstrbuton of Portfolo Losses 15 Loan Loss Dstrbuton wth p = 1% and = 12% and 0.6% p = 1.5% rho = 12.0% p = 1.5% rho = 0.6% % 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% Portfolo Loan Loss (%( Merton-model Approach to Dstrbuton of Portfolo Losses 16

9 Example of Vascek formula Appled to Bank Portfolo Source: Vascek Merton-model Approach to Dstrbuton of Portfolo Losses 17 Relatonshp between the Vascek model and the ntensty model wth the Gaussan Copula Merton-model Approach to Dstrbuton of Portfolo Losses 18

10 Fundamentally, Vascek model gves same results Intensty model and Gaussan copula (!) Default condton n Vascek model: 1 2 RD, ( µ 2 σv, ) ε = m + 1 v < = N ( p) σ T In other words, whether a normally dstrbuted N(0,1) varable s larger or smaller than a gven fxed number, N 1 ( p) V, Merton-model Approach to Dstrbuton of Portfolo Losses 19 and.. n ntensty model (wth the Gaussan copula).. the same (!) In the ntensty model default occurs when 1 τ = ln(1 U ) τ * where U = N ( ε ) λ.e., when the default tme τ s smaller than the maturty τ* Defne 1 τ* = ln(1 U*) and U* = N( ε*) λ Then default occurs when ε ε * Merton-model Approach to Dstrbuton of Portfolo Losses 20

11 Or.. n pctures.. If the value of e that we draw s smaller than the crtcal value ε ε * Then τ s less than τ* and we have a default 1 τ = ln(1 U ) τ * where U = N ( ε ) λ Merton-model Approach to Dstrbuton of Portfolo Losses 21 Intensty model wth 1000 names and equal ntensty and Vascek model wth equal default probablty and correlaton Example Merton-model Approach to Dstrbuton of Portfolo Losses 22

12 The bottom lne.. The Vascek model s the same as the ntensty model wth a Gaussan copula, dentcal default probabltes and a large number of names. Merton-model Approach to Dstrbuton of Portfolo Losses 23 Applcatons Vascek s obtans a formula for the dstrbuton of losses wth: sngle common factor homogeneous portfolo large number of credts But the approach can be generalsed to a much more realstc (mult-factor) correlaton structure and granularty n the portfolo holdngs qute wdely used n bankng for management of rsk of loan portfolo Merton-model Approach to Dstrbuton of Portfolo Losses 24

13 Takeaways Vascek s formula gves useful quck method for generatng dstrbuton of losses n large portfolo n one-factor verson fundamentally the same as Gaussan copula explotaton of condtonal ndependence s useful dea Applcatons tend to be n rsk management of actual loan losses (natural dstrbuton) rather than prcng (rsk-neutral dstrbuton) less evdence of poor performance n natural dstrbuton (same story about structural model agan) Merton-model Approach to Dstrbuton of Portfolo Losses 25

### THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

### benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

### Portfolio Loss Distribution

Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

### PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

### Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

### Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

### Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis

Copulas Modelng dependences n Fnancal Rsk Management BMI Master Thess Modelng dependences n fnancal rsk management Modelng dependences n fnancal rsk management 3 Preface Ths paper has been wrtten as part

### THE TITANIC SHIPWRECK: WHO WAS

THE TITANIC SHIPWRECK: WHO WAS MOST LIKELY TO SURVIVE? A STATISTICAL ANALYSIS Ths paper examnes the probablty of survvng the Ttanc shpwreck usng lmted dependent varable regresson analyss. Ths appled analyss

### Capital asset pricing model, arbitrage pricing theory and portfolio management

Captal asset prcng model, arbtrage prcng theory and portfolo management Vnod Kothar The captal asset prcng model (CAPM) s great n terms of ts understandng of rsk decomposton of rsk nto securty-specfc rsk

### Transition Matrix Models of Consumer Credit Ratings

Transton Matrx Models of Consumer Credt Ratngs Abstract Although the corporate credt rsk lterature has many studes modellng the change n the credt rsk of corporate bonds over tme, there s far less analyss

### Risk Management and Financial Institutions

Rsk Management and Fnancal Insttutons By John C. Hull Chapter 3 How Traders manage Ther Exposures... Chapter 4 Interest Rate Rsk...3 Chapter 5 Volatlty...5 Chapter 6 Correlatons and Copulas...7 Chapter

### An Alternative Way to Measure Private Equity Performance

An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

### Lecture 14: Implementing CAPM

Lecture 14: Implementng CAPM Queston: So, how do I apply the CAPM? Current readng: Brealey and Myers, Chapter 9 Reader, Chapter 15 M. Spegel and R. Stanton, 2000 1 Key Results So Far All nvestors should

### Chapter 4 Financial Markets

Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

### x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

### SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

### ENTERPRISE RISK MANAGEMENT IN INSURANCE GROUPS: MEASURING RISK CONCENTRATION AND DEFAULT RISK

ETERPRISE RISK MAAGEMET I ISURACE GROUPS: MEASURIG RISK COCETRATIO AD DEFAULT RISK ADIE GATZERT HATO SCHMEISER STEFA SCHUCKMA WORKIG PAPERS O RISK MAAGEMET AD ISURACE O. 35 EDITED BY HATO SCHMEISER CHAIR

### A Model of Private Equity Fund Compensation

A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

### Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

### Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

### Solutions to the exam in SF2862, June 2009

Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

### Hedging Interest-Rate Risk with Duration

FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

### The impact of bank capital requirements on bank risk: an econometric puzzle and a proposed solution

Banks and Bank Systems, Volume 4, Issue 1, 009 Robert L. Porter (USA) The mpact of bank captal requrements on bank rsk: an econometrc puzzle and a proposed soluton Abstract The relatonshp between bank

### Efficient Project Portfolio as a tool for Enterprise Risk Management

Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

### Simon Acomb NAG Financial Mathematics Day

1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve

### Can Auto Liability Insurance Purchases Signal Risk Attitude?

Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

### b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei

Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally:

### Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120

Kel Insttute for World Economcs Duesternbrooker Weg 45 Kel (Germany) Kel Workng Paper No. Path Dependences n enture Captal Markets by Andrea Schertler July The responsblty for the contents of the workng

### Stress test for measuring insurance risks in non-life insurance

PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

### An Analysis of Pricing Methods for Baskets Options

An Analyss of Prcng Methods for Baskets Optons Martn Krekel, Johan de Kock, Ralf Korn, Tn-Kwa Man Fraunhofer ITWM, Department of Fnancal Mathematcs, 67653 Kaserslautern, Germany, emal: krekel@twm.fhg.de

### Bank Credit Conditions and their Influence on Productivity Growth: Company-level Evidence

Bank Credt Condtons and ther Influence on Productvty Growth: Company-level Evdence Rebecca Rley*, Chara Rosazza Bondbene* and Garry Young** *Natonal Insttute of Economc and Socal Research & Centre For

### Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

### A Multistage Model of Loans and the Role of Relationships

A Multstage Model of Loans and the Role of Relatonshps Sugato Chakravarty, Purdue Unversty, and Tansel Ylmazer, Purdue Unversty Abstract The goal of ths paper s to further our understandng of how relatonshps

### THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

### A random variable is a variable whose value depends on the outcome of a random event/experiment.

Random varables and Probablty dstrbutons A random varable s a varable whose value depends on the outcome of a random event/experment. For example, the score on the roll of a de, the heght of a randomly

### Formula of Total Probability, Bayes Rule, and Applications

1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

### Financial Mathemetics

Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

### 9.1 The Cumulative Sum Control Chart

Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

### The OC Curve of Attribute Acceptance Plans

The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

### The Analysis of Outliers in Statistical Data

THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

### Implied (risk neutral) probabilities, betting odds and prediction markets

Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

### 9 Arithmetic and Geometric Sequence

AAU - Busness Mathematcs I Lecture #5, Aprl 4, 010 9 Arthmetc and Geometrc Sequence Fnte sequence: 1, 5, 9, 13, 17 Fnte seres: 1 + 5 + 9 + 13 +17 Infnte sequence: 1,, 4, 8, 16,... Infnte seres: 1 + + 4

### 1. Math 210 Finite Mathematics

1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

### Recurrence. 1 Definitions and main statements

Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

### Valuing Customer Portfolios under Risk-Return-Aspects: A Model-based Approach and its Application in the Financial Services Industry

Buhl and Henrch / Valung Customer Portfolos Valung Customer Portfolos under Rsk-Return-Aspects: A Model-based Approach and ts Applcaton n the Fnancal Servces Industry Hans Ulrch Buhl Unversty of Augsburg,

### The Short-term and Long-term Market

A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

### STATISTICAL DATA ANALYSIS IN EXCEL

Mcroarray Center STATISTICAL DATA ANALYSIS IN EXCEL Lecture 6 Some Advanced Topcs Dr. Petr Nazarov 14-01-013 petr.nazarov@crp-sante.lu Statstcal data analyss n Ecel. 6. Some advanced topcs Correcton for

### R&I Tranche Pad. Version 1.0 Technical Document. October 1, 2013 Structured Finance Division

R&I Tranche Pad Verson 1.0 Techncal Document October 1, 2013 tructured Fnance Dvson 1 Table of Contents 1. Credt Rsk of Rated Products...4 1.1. Product Outlne...4 1.2. tructure...4 1.3. Credt Rsk Hdden

### Exchange rate volatility and its impact on risk management with internal models in commercial banks

Banks and Bank Systems, Volume, Issue 4, 007 Devjak Sreko (Slovena), Andraž Grum (Slovena) Exchange rate volatlty and ts mpact on rsk management wth nternal models n commercal banks Abstract Fnancal markets

### DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice

DI Fund Suffcency Evaluaton Methodologcal Recommendatons and DIA Russa Practce Andre G. Melnkov Deputy General Drector DIA Russa THE DEPOSIT INSURANCE CONFERENCE IN THE MENA REGION AMMAN-JORDAN, 18 20

### 1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

### Do Banks Use Private Information from Consumer Accounts? Evidence of Relationship Lending in Credit Card Interest Rate Heterogeneity

Do Banks Use Prvate Informaton from Consumer Accounts? Evdence of Relatonshp Lendng n Credt Card Interest Rate Heterogenety Sougata Kerr, Stephen Cosslett, Luca Dunn December, 2004 Author nformaton: Kerr,

### Calculation of Sampling Weights

Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

### A Simplified Method for Calculating the Credit Risk of Lending Portfolios

A Smplfed Method MOETARY for Calculatng AD ECOOMIC the Credt STUDIES/DECEMBER Rsk of Lendng Portfolos 000 A Smplfed Method for Calculatng the Credt Rsk of Lendng Portfolos Akra Ieda, Kohe Marumo, and Toshnao

### Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

### Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

### Risk Measurement and Management of Operational Risk in Insurance Companies from an Enterprise Perspective

FRIEDRICH-ALEXANDER UNIVERSITÄT ERLANGEN-NÜRNBERG RECHTS- UND WIRTSCHAFTS- WISSENSCHAFTLICHE FAKULTÄT Rsk Measurement and Management of Operatonal Rsk n Insurance Companes from an Enterprse Perspectve

### Débats économiques et financiers N 1

Débats économques et fnancers N 1 How dfferent s the regulatory captal from the economc captal: the case of busness loans portfolos held by the major bankng groups n France Mchel Detsch * et Henr Frasse

### Stock Profit Patterns

Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

### CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

### Measuring portfolio loss using approximation methods

Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 Publshed onlne Aprl 0, 014 (http://www.scencepublshnggroup.com/j/sjams) do: 10.11648/j.sjams.01400.11 Measurng portfolo loss usng approxmaton

### Depreciation of Business R&D Capital

Deprecaton of Busness R&D Captal U.S. Bureau of Economc Analyss Abstract R&D deprecaton rates are crtcal to calculatng the rates of return to R&D nvestments and captal servce costs, whch are mportant for

### Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

### Traffic-light extended with stress test for insurance and expense risks in life insurance

PROMEMORIA Datum 0 July 007 FI Dnr 07-1171-30 Fnansnspetonen Författare Bengt von Bahr, Göran Ronge Traffc-lght extended wth stress test for nsurance and expense rss n lfe nsurance Summary Ths memorandum

### Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

### Awareness and Stock Market Participation

WORKING PAPER NO. 110 Awareness and Stock Market Partcpaton Lug Guso and Tullo Jappell November 2003 Ths verson June 2004 Unversty of Naples Federco II Unversty of Salerno Boccon Unversty, Mlan CSEF -

### INVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMA-HDR NETWORKS

21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech-2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS

### An Analysis of Factors Influencing the Self-Rated Health of Elderly Chinese People

Open Journal of Socal Scences, 205, 3, 5-20 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/ss http://dx.do.org/0.4236/ss.205.35003 An Analyss of Factors Influencng the Self-Rated Health of

### Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

### PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

### Benefits and Risks of Alternative Investment Strategies*

Benefts and Rsks of Alternatve Investment Strateges* Noël Amenc Professor of Fnance at Edhec Drector of Research and Development, Msys Asset Management Systems Lonel Martelln Assstant Professor of Fnance

### The impact of hard discount control mechanism on the discount volatility of UK closed-end funds

Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closed-end funds Abstract The mpact

### Solutions to First Midterm

rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e.

### The Application of Fractional Brownian Motion in Option Pricing

Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

### Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

### Optimal Consumption and Investment with Transaction Costs and Multiple Risky Assets

THE JOURNAL OF FINANCE VOL. LIX, NO. 1 FEBRUARY 2004 Optmal Consumpton and Investment wth Transacton Costs and Multple Rsky Assets HONG LIU ABSTRACT We consder the optmal ntertemporal consumpton and nvestment

### Chapter 11 Practice Problems Answers

Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make

### Optimal Bidding Strategies for Generation Companies in a Day-Ahead Electricity Market with Risk Management Taken into Account

Amercan J. of Engneerng and Appled Scences (): 8-6, 009 ISSN 94-700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a Day-Ahead Electrcty Market wth Rsk Management Taken nto Account

### Multiple discount and forward curves

Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

### Structural Estimation of Variety Gains from Trade Integration in a Heterogeneous Firms Framework

Journal of Economcs and Econometrcs Vol. 55, No.2, 202 pp. 78-93 SSN 2032-9652 E-SSN 2032-9660 Structural Estmaton of Varety Gans from Trade ntegraton n a Heterogeneous Frms Framework VCTOR RVAS ABSTRACT

### Multivariate EWMA Control Chart

Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

### Simple Interest Loans (Section 5.1) :

Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

### How Much to Bet on Video Poker

How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks

### Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscrete-tme rsk-mnmzng hedgng strateges An Chen Netspar, he Netherlands Department

### Naïve Bayes classifier & Evaluation framework

Lecture aïve Bayes classfer & Evaluaton framework Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Generatve approach to classfcaton Idea:. Represent and learn the dstrbuton p x, y. Use t to defne probablstc

### NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6

PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has

### Modeling Loss Given Default in SAS/STAT

Paper 1593-014 Modelng Loss Gven Default n SAS/SA Xao Yao, he Unversty of Ednburgh Busness School, UK Jonathan Crook, he Unversty of Ednburgh Busness School, UK Galna Andreeva, he Unversty of Ednburgh

### Forecasting and Stress Testing Credit Card Default using Dynamic Models

Forecastng and Stress Testng Credt Card Default usng Dynamc Models Tony Bellott and Jonathan Crook Credt Research Centre Unversty of Ednburgh Busness School Verson 4.5 Abstract Typcally models of credt

### Lecture 10: Linear Regression Approach, Assumptions and Diagnostics

Approach to Modelng I Lecture 1: Lnear Regresson Approach, Assumptons and Dagnostcs Sandy Eckel seckel@jhsph.edu 8 May 8 General approach for most statstcal modelng: Defne the populaton of nterest State

5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two

### A Probabilistic Theory of Coherence

A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

### ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA

ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA Duc Vo Beauden Gellard Stefan Mero Economc Regulaton Authorty 469 Wellngton Street, Perth, WA 6000, Australa Phone: (08)

### Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration

CED0105200808 Amerprse Fnancal Servces, Inc. 70400 Amerprse Fnancal Center Mnneapols, MN 55474 Incomng Account Transfer/Exchange/ Drect Rollover (Qualfed Plans Only) for Amerprse certfcates, Columba mutual

### Chapter 3 Research Method

Chapter 3 Research Method 3.1 Framework 3.1.1 Research Desgn A two-phase study was desgned to explore the feasblty of RM n agng socetes from both supply and demand aspect. In the aspect of the borrowers,

### Survival analysis methods in Insurance Applications in car insurance contracts

Survval analyss methods n Insurance Applcatons n car nsurance contracts Abder OULIDI 1 Jean-Mare MARION 2 Hervé GANACHAUD 3 Abstract In ths wor, we are nterested n survval models and ther applcatons on