MULTI-OBJECTIVE APPROACHES TO PUBLIC DEBT MANAGEMENT

Transcription

1 MULTI-OBJECTIVE APPROACHES TO PUBLIC DEBT MANAGEMENT A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SC IENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY EMRE BALIBEK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN OPERATIONAL RESEARCH JANUARY 2008

2 Approval of the thei: MULTI-OBJECTIVE APPROACHES TO PUBLIC DEBT MANAGEMENT ubmitted by EMRE BALIBEK in partial fulfillment of the requirement for the degree of Doctor of Philoophy in Operational Reearch Department, Middle Eat Technical Univerity by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Science Prof. Dr. Çağlar Güven Head of Department, Operational Reearch Prof. Dr. Murat Kökalan Supervior, Indutrial Engineering Dept., METU Examining Committee Member: Prof. Dr. Çağlar Güven Indutrial Engineering Dept., METU Prof. Dr. Murat Kökalan Indutrial Engineering Dept., METU Prof. Dr. Neim Erkip Indutrial Engineering Dept., Bilkent Univerity Aoc. Prof. Dr. Yaemin Serin Indutrial Engineering Dept., METU Ait. Prof. Dr. Ema Gaygıız Economic Dept., METU Date: ii

3 I hereby declare that all information in thi document ha been obtained and preented in accordance with academic rule and ethical conduct. I alo declare that, a required by thee rule and conduct, I have fully cited and referenced all material and reult that are not original to thi work. Name, Lat name : Emre BALIBEK Signature : iii

4 ABSTRACT MULTI-OBJECTIVE APPROACHES TO PUBLIC DEBT MANAGEMENT Balıbek, Emre Ph.D., Department of Operational Reearch Supervior: Prof. Dr. Murat Kökalan January 2008, 124 Page Public debt manager have a certain range of borrowing intrument varying in their interet rate type, currency, maturity etc. at their dipoal and have to find an appropriate combination of thoe while raiing debt on behalf of the government. In electing the combination of intrument to be iued, i.e. the borrowing trategy to be purued for a certain period of time, debt manager need to conider everal objective that are conflicting by their nature, and the uncertainty aociated with the outcome of the deciion made. The objective of thi thei i to propoe an approach to upport the deciion making proce regarding overeign debt iuance. We incorporate Multi-Criteria Deciion Making (MCDM) tool uing a multi-period tochatic programming model that take into account equential deciion concerned with debt iuance policie. The model i then applied for public debt management in Turkey. Keyword: Public Debt Management, Rik Management, MCDM, Stochatic Programming iv

5 ÖZ KAMU BORÇ YÖNETİMİNE ÇOK-AMAÇLI YAKLAŞIM Balıbek, Emre Doktora, Yöneylem Araştırmaı Tez Yöneticii: Prof.Dr.Murat Kökalan Ocak 2008, 124 Sayfa Kamu borç yönetimleri, kamu ektörünün borçlanma ihtiyacını karşılarken farklı faiz, döviz, vade yapıına ahip çeşitli finanal araçlar kullanmaktadır. Finanman ihtiyacının karşılanmaına yönelik olarak öz konuu finanal araçların hangi oranlarda kullanılacağına ve kamu borç portföyünün yapıına ilişkin tratejiler oluşturulurken; borç yöneticileri, kamunun ödünleşim içeren çeşitli borç yönetimi amaçlarını ve verilen kararların onuçlarına ilişkin belirizlikleri göz önünde bulundurmaktadır. Bu çalışmada, kamu borçlanmaında izlenecek tratejilere ilişkin karar verme ürecine yönelik olarak nicelikel bir yaklaşım önerilmektedir. Bu kapamda, kamu borçlanmaına ilişkin ardışık kararları dikkate alan tokatik bir program geliştirilmiş, öz konuu model kullanılarak çok-amaçlı karar verme yöntemleri içeren bir karar detek üreci oluşturulmuştur. Geliştirilen öneriler, Türkiye uygulamaını baz alan bir örnek üzerinde omutlaştırılmaktadır. Anahtar Kelimeler: Kamu Borç Yönetimi, Rik Yönetimi, Çok-Amaçlı Karar Verme, Stokatik Programlama v

6 To My Family vi

7 ACKNOWLEDGEMENTS I would like to expre my incere gratitude to my upervior Prof.Dr. Murat Kökalan for hi continuou guidance, advice and inight throughout thi tudy. Without hi contribution and encouragement, thi work could never be completed. I am alo grateful to Prof.Dr. Neim Erkip and Aoc.Prof. Yaemin Serin for their valuable uggetion and contructive criticim. I am indebted to my manager at the Turkih Treaury for providing me the opportunity to purue thi tudy and my colleague for compenating my occaional abence at work during the coure of thi tudy at the cot of extra work hour. I alo wih to thank my parent and iter for their endle upport over the year. Finally, I would alo like to thank my wife for her upport and patience during the coure of thi work and to my on for arriving jut in time to ee the end of thi diertation. The idea expreed in thi tudy are only thoe of mine and do not necearily reflect the view and policie of the Turkih Treaury. vii

8 TABLE OF CONTENTS ABSTRACT... iv ÖZ... v ACKNOWLEDGEMENTS... vii TABLE OF CONTENTS... viii LIST OF TABLES... x LIST OF FIGURES... xi LIST OF SYMBOLS... xiii CHAPTER 1. INTRODUCTION THE PUBLIC DEBT MANAGEMENT PROBLEM Public Debt Management Objective: Cot and Rik Cot of Public Debt Market Rik Liquidity Rik Macroeconomic Objective Borrowing Intrument Contraint in Public Debt Management The Effect of Uncertainty in PDM Deciion Literature Review on PDM Strategy Formulation A MULTI-OBJECTIVE STOCHASTIC PROGRAMMING APPROACH Baic of Stochatic Programming Baic Stochatic Programming Model Anticipative Model Adaptive Model Recoure Model Determinitic Equivalent Formulation Scenario Generation in Stochatic Programming Model Overview of Scenario Generation Method Evaluation of Scenario Tree Generation Method The Multi-objective Public Debt Management Model Objective of the Model Cot Market Rik Liquidity Rik Notation Parameter and Index Set Stochatic Variable: Deciion Variable: viii

9 Auxiliary Variable: Contraint Objective Function Expected Cot of Debt Market Rik Liquidity Rik A Simple Illutrative Model An Integrated Simulation/Optimization Approach MULTI-CRITERIA DECISION MAKING ANALYSIS FOR THE PUBLIC DEBT MANAGEMENT PROBLEM Obtaining Efficient Solution Definition of an Efficient Solution Identifying Efficient Solution Uing the PDM Model Viual Interactive Approach of Korhonen and Laako Accounting for Stochaticity in the SP Model Multivariate Statitical Analyi Contructing Confidence Region around Efficient Solution A Stochatic Interactive Approach The Deciion Aid Framework APPLICATION OF THE MCDM APPROACHES: AN EXAMPLE FROM TURKISH CASE Background for Public Debt Management in Turkey The Stochatic Programming Model for Turkey Scenario Tree Generation The Statitical Model Scenario Clutering Scenario Tree Generation Algorithm Aement of the Scenario Generation Algorithm Experiment on the PDM Model uing MCDM Tool Experiment on a Single Scenario Tree Application of the Stochatic Interactive Method Experimentation with the Simulation/Optimization Framework CONCLUSIONS REFERENCES APPENDIX-A: CONDITIONAL COST AT RISK APPENDIX-B: SAMPLE PROGRAM CODE FOR THE PDM-SP MODEL CURRICULUM VITAE ix

10 LIST OF TABLES TABLES Table 1 Type of Treaury Securitie Table 2 Stability Reult without Clutering Table 3 Stability Reult after Clutering over 100 data point Table 4 Stability Reult after Clutering over 1000 data point Table 5 Optimal Borrowing Strategie (Initial Stage Deciion) Table 6 Optimal Borrowing Strategie (Initial Stage Deciion) Table 7 Reult of Dynamic Game. Experiment Table 8 Reult of Dynamic Game. Experiment Table 9 Reult of Dynamic Game. Experiment Table 10 Reult of Dynamic Game. Experiment x

11 LIST OF FIGURES FIGURES Figure 1 Total Central Government Debt, in Selected OECD Countrie... 7 Figure 2 A Sample Scenario Tree Figure 3 Scenario Tree for the One-Year Interet Rate Figure 4 The Simulation/Optimization Framework Figure 5 Illutration of the Tchebycheff Program Figure 6 Illutration of the Viual Interactive Approach of Korhonen and Laako (1986) Figure 7 Variability of Efficient Solution-Billion TRY Figure 8 Projection onto the Efficient Surface Figure 9 Illutration of the Viual Interactive Approach: Obtaining Scenario-Baed Solution a θ i incremented Figure 10 Illutration of the Viual Interactive Approach: Objective Function Value and Confidence Band 64 Figure 11 The Deciion Making Proce Uing the Stochatic Interactive Algorithm Figure 12 Budget Deficit in Turkey (in percent of GNP) Figure 13 Central Government (Treaury) Debt Stock (in percent of GNP). 71 Figure 14 The Structure of the Treaury Debt Stock Figure 15 Generating a Scenario Tree Figure 16 Generating a Scenario Tree - An Example with 5 Cluter Figure 17 A Repreentation of the Efficient Frontier (billion TRY) Figure 18 A Set of Efficient Solution (billion TRY) Figure 19 Criterion Value Trajectorie (billion TRY) Figure 20 Criterion Value Trajectorie (billion TRY) Figure 21 Movement on the Efficient Surface (billion TRY) Figure 22 The Reult of Randomne in the Scenario Generation Mechanim Figure 23 Confidence Ellipoid around a Set of Efficient Solution Figure 24 An Application of the Viual Interactive Approach: Iteration xi

12 Figure 25 An application of the Viual Interactive Approach: Iteration Figure 26 Performance of Peudo-Centroid Solution i other Scenario Tree Figure 27 Empirical cdf Plot Figure 28 Conditional Value at Rik xii

13 LIST OF SYMBOLS α p z i Confidence level parameter Number of criteria The i th objective function, i=1,,p x X T S Deciion variable vector Deciion pace Deciion horizon Scenario et Scenario Index, S p Probability aociated with cenario. xiii

14 CHAPTER 1 1INTRODUCTION Public debt management (PDM) i the proce of raiing fund for the financing need of the government and managing the government financial liabilitie. Public debt manager have a certain range of borrowing intrument varying in their interet rate type, currency, maturity etc. at their dipoal and have to find an appropriate combination of thoe while raiing debt on behalf of the government. In electing the combination of intrument to be iued, i.e. the borrowing trategy to be purued for a certain period of time, debt manager need to conider everal objective that are conflicting by their nature, and the uncertainty aociated with the outcome of the deciion made. Given the budgetary and other financial requirement of the government, one of the main objective of PDM i to meet the funding need with the lowet poible cot. On the other hand, the rik aociated with the debt portfolio hould be contained to avoid any advere affect on the macroeconomic environment. Among the major rik that concern public debt manager, there are the market rik, which i defined a the rik of an increae in the cot of debt ervice due to fluctuation in market condition and the liquidity (re-funding) rik that indicate the poibility to fail in finding the required fund in order to make debt re-payment. Thank to the nature of financial market, in general, there exit a trade-off between return and rik. For a portfolio manager, achieving a higher return require inveting in high rik aet. From the government point of view, conidering the fact that the government i the iuer of financial aet (ecuritie) the dilemma i between attaining a low cot portfolio and retricting the rik aociated. Thu, the 1

15 public debt management problem, i.e. formation of the financing trategie, i a multi-objective deciion making problem. On the other hand, the level of development of a country financial market, the government credit rating, it ability to acce international market and other macroeconomic environmental condition impoe everal limitation on the activitie of debt manager. Different level of budget deficit induce different ize of borrowing requirement and not every country can iue the ame type of intrument. Some countrie have developed penion fund ytem that demand long-term government bond, while other are truggling for finding cutomer for their medium-term ecuritie. Government debt manager hould all conider thee contraint in developing their funding trategie. An important characteritic of the multi-objective PDM problem i that deciion are made under uncertainty. Debt manager are not faced with choice that have determinitic outcome. There i a degree of uncertainty aociated with the evolution of economic factor uch a interet and exchange rate that drive the cot of borrowing. Debt management deciion are concerned with future action of the government and the outcome of the deciion made depend on the realization of relevant macroeconomic variable. The tochaticity of thee factor need to be taken into account while formulating the cot-rik tructure of the debt portfolio. Given it characteritic and ignificance on a country economic life, the public debt management problem ha drawn attention of both academician and practitioner, including taff of International Financial Intitution uch a the International Monetary Fund or the World Bank and debt management office. Rik management practice are gaining prime importance in public liability management operation and everal approache adapted from technique applied by private financial intitution have been propoed for the cae of the government. Thee are generally imulation or cenario analyi baed method that aim at quantifying the cot and rik of alternative trategie. However, to the bet of our knowledge, there i little work on providing guidance to deciion maker in comparing thee quantitie, the computed cot and rik metric, and aiting them in finding efficient 2

16 olution to explore the conequence of different trategie in term of cot and rik. Thi thei ha two main objective. Firt, we aim to develop an optimization approach for the debt trategy formulation problem. In thi context, we how the applicability of a mathematical modelling paradigm, tochatic programming, in the field of public debt management. In developing the tochatic programming model, we adopt a multi-objective approach taking into account the multiple objective aociated. We incorporate relevant criteria and develop a quantitative approach that take into account equential deciion concerned with debt iuance policie, taking uncertainty into account making ue of a cenario tree. We formulate the debt management problem a a determinitic equivalent linear programming model, in which the deciion variable are the amount of different type of bond to be iued, accounting for the cah flow contraint for the government. In that etting, the government iue a certain et of treaury ecuritie to meet it overall financing requirement that arie from it debt and non-debt obligation (net of tax receipt). The exact amount of each type of bond to be iued i determined by the model baed on the deciion maker preference with regard to the debt management objective conidering the cenario et available. The econd and ultimate objective i to develop an integrated deciion upport framework to guide debt manager in developing bond iuance trategie. We how how Multi-Criteria Deciion Making (MCDM) approache can be incorporated on the SP model and how the model can be ued to ait deciion maker in analyzing the trade-off between alternative coure of action. In thi context, we identify efficient olution baed on different preference tructure and develop an interactive MCDM approach to guide the deciion maker (DM) in developing the debt trategy. We demontrate how overeign deciion maker can experiment with uch a tool in a practical etting, drawing on the cae of Turkey. While developing the MDCM framework for public debt management deciion, we bring forward the idea of contructing confidence region around efficient olution. We believe the concept can generically be applied in analye 3

17 regarding deciion under uncertainty. The tochatic interactive approach we develop in the context of public debt management i alo original, to the bet of our knowledge, in MCDM literature, and can well be adapted for deciion making problem in other area that involve multiple objective and tochaticity. The thei i organized in four main chapter. Since we bring together idea from different dicipline in the field of public debt management, literature review on everal concept and tool to which we make reference are pread over the chapter depending on the content. Chapter 2 define the public debt management problem and dicue it main feature. In thi ection, we elaborate on the general objective and contraint of PDM and introduce the main financial intrument available to debt manager. A literature review on variou approache to the PDM problem, both from practical and theoretical perpective i alo provided. In Chapter 3, we preent our generic multi-tage SP model, developed to guide iuance deciion. The chapter begin with a dicuion of the tochatic programming paradigm, touching on baic concept including different type of model and cenario generation method. Thi part alo provide ome literature review on the application of SP model to financial deciion making problem. We then continue with the dicuion of the mathematical formulation of the relevant objective and contraint in the PDM context. After preenting the notation and formulation of the model, we include a imple illutrative model to concretize the dicuion. The ection end with the Simulation/Optimization approach which i developed a a deciion aid framework to ait trategy deciion in a dynamic environment. Chapter 4 tart with a dicuion of the relevance of Multi-Criteria Deciion Making tool for the PDM problem and preent how we employ the SP model to develop a deciion making framework. The ection include the methodology for obtaining efficient olution. We then preent an interactive algorithm by which the deciion maker can experiment to explore alternative olution. The algorithm make ue of multivariate tatitical analyi tool to cope with the inherent uncertainty in the problem. 4

18 Chapter 5 i about the application of the developed method in an example problem, i.e. the cae of the Turkih Treaury, the intitution in charge of PDM in Turkey. In thi ection, we alo preent a pecific cenario generation mechanim that employ idea from cenario clutering and reduction technique developed to model the Turkih macroeconomic environment. In Chapter 6, we conclude and indicate propect for future work. 5

19 CHAPTER 2 2THE PUBLIC DEBT MANAGEMENT PROBLEM Public debt management (PDM) i concerned with meeting the funding requirement of a country that arie from budgetary and other financial liabilitie of the government. More pecifically, it can be defined a the proce of etablihing and executing a trategy for managing the government debt to raie the required amount of funding, purue it cot/rik objective, and meet any other public debt management goal the government may have et, uch a developing and maintaining an efficient and liquid market for government ecuritie (International Monetary Fund - World Bank, 2003, p.5). Almot all countrie, developed or under-developed, have a certain level of debt. State build up debt to fund extra expenditure at time of national trouble (war, natural diater etc.) or to undertake development project (to contruct road, bridge, to finance ocial project etc). Government ometime reort to borrowing even to finance current expenditure, when raiing debt i technically and/or politically eaier than to impoe additional tax. Some countrie alo borrow in foreign currencie to build-up foreign exchange reerve or to finance their international payment. Figure 1 depict the ize of Central Government Debt in ome elected OECD (Organization for Economic Co-Operation and Development) countrie in comparion to their Gro Dometic Product (GDP). In order to meet the financing requirement of the government, public debt management authoritie (organized under the National Treaury, the Minitry of Finance, Central Bank or the Debt Management Office in different countrie) iue hort-term bill or longer-term bond (ecuritie) in the financial market or ue loan from bank or multi-national/governmental intitution. Once a certain level of debt 6

20 tock i acquired, countrie generally lack the fund or do not prefer to retire entire debt all at once, ince thi would require increaing the tax level or decreaing government expenditure ubtantially. Then, they have to roll-over exiting debt to ome extent by iuing new debt to finance re-payment. Thu, debt management i a continuou proce and the government debt portfolio ha a dynamic tructure ince there are bond and bill entering and leaving the debt portfolio throughout time % Autral ia Autria Belgium Canada Czech Republic Denmark Finland France Germany Greece Hungary Iceland Ireland Italy Korea (Republic of) Luxembourg Country Figure 1 Total Central Government Debt, in Percent of GDP, in Selected OECD Countrie, a of (ource:oecd). Mexico Net herland New Zealand Norway Poland Portugal Slovak Republic Spain Sweden Switzerland Turkey United Kingdom United State Epecially in low and middle income countrie, government debt i the larget financial liability portfolio in the country. The overall tructure of public debt portfolio i key to a country macroeconomic tability, given the expoure of public ector balance and the country financial tability to public debt. Thi ha been proved by a number of recent macroeconomic crie in everal emerging countrie uch a Mexico (1994) and Turkey (2001), where the financial turmoil ha been amplified by the highly vulnerable compoition of the countrie debt liabilitie. 7

21 Once a government i in a financial problem, i.e. facing difficultie in fulfilling fical liabilitie or having to pay exceive cot when iuing debt, thi ha pillover effect on the entire economy. The banking ector, other financial intitution and individual who have extended loan to the government are all affected by the government financial trouble. The problem of intermediarie in the financial market then hamper the functioning of the production ector that relie on fund provided by thoe intitution. Generally, the government cot of borrowing, i.e. the interet rate on government ecuritie et the bai for the level of interet rate in a country ince the government i accepted a the leat riky borrower in it own economy. The private intitution that are competing with the public ector to acce fund have to pay a premium over the government borrowing cot. If a government adopt a riky debt tructure, thi i reflected in it funding cot a lender will price in a rik premium while extending credit to the public ector. Thi then affect the interet rate in the entire economy and hinder economic growth. Thu, the financial liability portfolio of the government mut be effectively managed. Public debt manager have to decide on a certain debt management trategy taking into account everal policy target. They have a range of financial intrument (ecuritie) at their dipoal and have to form a pecific portfolio combination, in term of maturity, currency and interet type, that would uit the government debt management objective. 2.1 Public Debt Management Objective: Cot and Rik Traditionally, the mot important concern of public debt management had been the cot of borrowing, or even to be able to raie the neceary fund. Tobin (1963) tate that If anyone i in the poition to be hi own inurer, it i the Secretary of the Treaury and thu argue the government hould focu on cot minimization. For ome countrie, thi cot minimization objective materialized into unbalanced debt tructure, relying too much on hort-term and/or foreign currency denominated debt, which turned out to be a ource of rik in later year. 8

22 In 1994, Mexico did not heitate to replace bulk of her public debt from peo denominated bond to hort-term dollar-linked teobono in puruit of lower interet rate and attracting foreign invetor, which led to an increaed vulnerability of the country economy to financial crii. Mexican Crii at the end of 1994 i partly attributable to the 29 billion United State Dollar (USD) teobono maturing in 1995, with 10 billion USD to be paid in the firt 3 month, while the country foreign reerve tood at a level of 6.3 billion USD (Caard et al, 1997). Gill and Pinto (2005) found out that Argentina debt increaed by 41.7% of her Gro National Product (GDP) between 2001 and 2003, while that of Ruia by 40.4% in after the financial crie in thoe countrie due to the high hare of foreigndenominated debt in thoe countrie debt tock. The increaed volatility of international fund flow, the complexity of intrument ued and the recent crie highlighted the importance of rik-related criteria, in addition to cot, while raiing public debt. Mot public debt manager are now concerned with the rik and macroeconomic iue aociated a well a cot. The public debt management objective in the United Kingdom, for example, i to minimie, over the long term, the cot of meeting the Government financing need, taking into account rik, whilt enuring that debt management policy i conitent with the aim of monetary policy (HM Treaury, 2007). In it Guideline for Public Debt Management, Italian Treaury (2007) declare that during 2007 Government bond iue will be calibrated o a to meet the financing need of the Central Government, with a medium-term view to further reducing the expoure to interet rik (nominal and real) and refinancing rik, while at the ame time containing the dynamic of interet burden a a percentage of GDP. The major rik public debt manager face are the market rik, which i defined a the rik of an increae in the cot of debt ervice a a reult of unfavorable movement in market condition and the liquidity (re-funding) rik that indicate the poibility to fail in finding the required fund in order to make debt re-payment. 9

23 2.1.1 Cot of Public Debt It i poible to meaure the cot aociated with public debt in everal way. Debt management office may adopt different cot meaure depending on their perpective in debt management and/or country characteritic uch a the accounting tandard. The mot common meaure of cot in borrowing fund i the interet rate requeted by lender. Interet rate i the time value of money and lender who extend loan to any borrower ak to be compenated for the duration of the loan ince they will not be able to ue their own fund in that period. When a government iue debt, the cot of borrowing i reflected in the government budget in term of interet expenditure. Government that employ cah accounting tandard record interet expenditure when payment are actually made, while countrie that follow an accrual accounting tandard, depict interet expene a they accrue. For countrie that iue debt in foreign currencie, the interet expenditure calculated in imple term via multiplying the principal amount of bond by the interet rate i not the ole ource of cot. The change in the value of the debt, meaured in the local currency, due to fluctuation in the exchange rate alo add to the cot of debt. Debt management office that engage in frequent econdary market activitie uch a debt buy-back or bond exchange may alo follow the marked-to-market value of their debt portfolio. Marked-to-market value of a bond how it value when meaured with repect to current market indicator, i.e. the prevailing interet and exchange rate; and the difference between it iue price and market value i the buy-back cot for the government. For a country that redeem bond at maturity at the interet rate or price et at the time of iuance, the marked-to-market value i of no relevance. When cot are ditributed over a number of year, they can be meaured in a preent value bai. They can alo be normalized with repect to a macroeconomic 10

24 magnitude uch a the GDP or the ize of debt portfolio to allow period wie comparion. Even though the relevant cot definition may differ from country to country, the common aim in PDM i to minimize the cot of debt. It will be tax payer who will be paying back the debt and one of the main objective of debt management office i to find the neceary fund at the lowet poible cot in line with the citizen expectation Market Rik A well-known characteritic of financial market i that there i a trade-off between return and rik. Generally, the higher the return from an invetment, the higher are the aociated rik. Conidering the fact that an invetor return on a financial intrument i a cot for the iuer, the rik/return trade-off concept ha it mirror image for the government a the cot-rik dilemma. The cot and market rik objective are generally conflicting by their nature, a hort-term interet rate are uually lower than longer-term rate. Thi i alo true in an economy where interet rate tend to decline. A good example to uch a ituation i the cae of countrie that went through a proce of economic convergence a they were candidate for the European Union (EU). In the acceion proce, thee countrie aw the convergence of level of the variable in their economie to EU tandard. In uch a context, it would be le cotly for the government to iue hortterm debt to make ue of lower or declining interet rate. The aim in iuing hort term bill or longer term variable rate bond indexed to hort-term interet rate i to horten the interet rate fixing period of the debt tock o that each time the interet rate are fixed there will be le cot on government debt. Thi policy will expectedly erve for cot minimization purpoe. However, rolling debt too frequently or renewing the interet rate in hort interval will then increae expoure to changing market condition. In cae of a udden climb in interet rate in financial market due 11

25 to ome external or internal reaon, the cot on a major portion of the government debt will have to increae. Thi i the market rik in public debt management. For countrie that have liabilitie denominated in foreign currencie, the volatility of exchange rate alo contitute a major portion of the market rik. Conidering the fact that, the main revenue of government are taxe which are collected in the local currency, an increae in exchange rate, i.e. a depreciation in the value of the local currency may caue a ignificant rie in the debt ervice cot while the level of revenue remain contant Liquidity Rik Liquidity rik or re-funding/re-financing rik a it i ometime called i alo a major concern for public debt manager, epecially for thoe in developing countrie. Thi type of rik often arie from concentration of debt re-payment at a certain point in time. If a country ha to pay back or refinance the bulk of it debt within a hort period, there i alway a rik concerned with acceing the required amount of fund to fulfill liabilitie. The rik may arie from the level of cah reerve of the government, for example due to a decline in tax revenue or from the lender reluctance in renewing their loan. The latter cae i imilar to liquidity rik faced by financial invetor. For an invetor who hold a certain financial intrument, liquidity rik i in failing to find potential buyer for thi intrument when he decide to ell it. For a country debt office, re-funding rik i the poibility of falling hort in finding lender who would purchae government ecuritie at a time the government i in a cah hortage. In under-developed or developing countrie that lack a well-functioning, liquid financial market with many actor, liquidity rik i more ignificant. At a time of financial turmoil, there will be limited amount of fund and a mall number of lender in the market, which in turn will amplify the level of refinancing rik. Controlling liquidity rik i crucial for a government reputation. If a government i een truggling for finance, thi ha ignificant conequence. The realization of uch a ituation will ignite ome turbulence or even panic in the entire 12

26 economy. Lender who have extended credit, other government intitution that depend on the central government to fulfil their own obligation, public employee who rely on alarie paid by the government, and in the end all the citizen will be affected from the government liquidity crie. Thu, controlling liquidity rik i an important objective of public debt management along with containing the level of cot and market rik. Unfortunately, there can alo be a trade-off between the cot and liquidity rik objective, ince reducing the liquidity rik may require long-term borrowing at high cot and/or keeping a certain level of exce cah reerve which alo induce a cot for the government. Aiming to minimize the market rik may dictate to borrow fixed rate long-term bond whoe repayment accumulate at a certain point in time which in turn induce a certain level of liquidity rik Macroeconomic Objective Debt management trategie need to be formed in harmony with the general macroeconomic policie of a country uch a the monetary and fical policie. Monetary policy i generally concerned with a country money upply and i aimed at objective of maintaining price tability, caring for the health of money market, providing ufficient liquidity to the financial ytem etc. Fical policy i about the government action and plan in etting the level and compoition of it revenue and expenditure. Central bank, the main intitution reponible for developing and implementing the monetary policy, may ometime reort to conducting open market operation in financial market in order to control the level of money upply and to reach their objective with regard to growth and inflation rate. Thee operation are often in the form of auction which aim at injecting (withdrawing) liquidity to (from) financial market. Auction in which bond are tendered are alo the main tool for public debt management office to raie the liquidity need of the government. Therefore, liquidity management require a decent co-ordination between thee organization. PDM office and central bank hould abtain from engaging in 13

27 contradictory action in financial market in order to avoid harming each other in reaching their objective. Thi require haring of information on the cah flow of the government and the level of liquidity in the financial market, and keeping away from conducting auction at the ame time. Debt management hould alo take into account the general intrument of monetary policy pecific to a country. In an economy where a fixed or pegged foreign exchange (FX) rate regime i implemented by the Central Bank, the government too much reliance on foreign currency borrowing may impact the credibility of the regime even though thi i le cotly. Harmonization of debt management trategie and fical policy i alo of vital importance. Interet expenditure that arie from the debt obligation contitute an important part of a government budget. On the other hand, tax revenue are the main ource to fulfil debt obligation. Therefore, there i a mutual relationhip between fical and debt management policie. The projection regarding revenue and non-debt outflow of the general government i a crucial input for debt management, while the tructure of debt repayment mut be known to adapt the relevant fical policie. Debt manager hould alo take into account current and planned taxing regime regarding financial intrument while deciding on what intrument to iue. 2.2 Borrowing Intrument Public debt manager have everal intrument, different type of bond and bill, at their dipoal to raie fund for the government. Thee vary in interet rate, denomination currencie and maturitie. Formally, a bond i a debt intrument requiring the iuer or the borrower to repay to the lender the amount borrowed plu interet over a pecified period of time (Fabozzi, 2000, p.1). A typical bond pecifie a certain date, the maturity date, when the amount borrowed i due and the level and timing of interet which will be paid over the borrowed amount. The amount due at the maturity i known a the 14

28 principal. The principal i alo referred to a the face value, par value, maturity value, redemption or nominal value. Creditor lend fund in return for a certain interet rate, either fixed at the tart of the loan or allowed to vary throughout the life of debt. The interet on uch variable or floating rate debt can be indexed to an external indicator uch a the price index in cae of inflation linked bond. Interet payment can alo be linked to ome commonly accepted interet rate indicator uch a the London Interbank Offered Rate (Libor) or to the interet on ome other ecurity. Turkih Treaury, for example, ue the rate on it three and ix month Treaury Bill to et the rate for longer term floating rate note (FRN). Every three or ix month, the interet rate on FRN change depending on the realization in the mot recent bill auction. Debt management office have to decide on the type of interet rate on their bond taking into account their expectation and the government preference. The timing of interet payment i alo a deciion variable for the iuer. Zero-coupon bond pay the entire interet at maturity while coupon bond have interet payment interval in which a certain portion of interet on the bond i redeemed. Typically, the coupon are paid quarterly, emi-annually or yearly. The interet the bond iuer pay in each coupon period i known a the coupon rate. The amount paid in each coupon period i calculated by multiplying the face value of the bond by the coupon rate, adjuted for the coupon period. The term to maturity of a bond i the number of month or year over which the bond iuer promie to meet her obligation. At the maturity date, the iuer redeem the bond by paying the amount borrowed and debt ceae to exit. Securitie can be iued in variou maturitie ranging from three month up to fifty year. Interet rate charged for a loan generally differ with repect to the duration of the loan. Therefore, the interet or yield on a bond depend on it maturity. The graphical repreentation of thi relationhip between the yield on debt intrument of the ame iuer and maturity i known a the yield curve. The yield curve exhibit different hape due to the tructure of financial market and the preference of invetor. Generally, invetor perceive higher rik for credit they have extended 15

29 for longer maturitie and thi caue the interet rate charged for longer term to be higher than thoe for the near-term. However, the expectation of a reduction in interet rate may caue an inverion in the hape of the yield curve. If a majority of invetor believe that the yield will decline in the future, they will tend to demand longer term bond to able to fix their invetment at the currently high yield level and ell hort term bond. Then, by force of demand and upply, the price of longer term bond will increae and yield for the long-term will be lower than thoe for the hort-term. In countrie that have a wide intitutional invetor bae, uch a penion fund who demand long-term ecuritie to able to meet their long term liabilitie, again the longer-term bond may have lower yield. Public debt manager hould conider the term tructure of interet rate and the iue that affect it when making bond iuance deciion. Bond can be iued in dometic or foreign currencie. In general, developed countrie that have developed dometic financial market prefer to iue ecuritie in their local currencie, while other countrie reort to holding ome foreign currency debt to increae maturitie and/or to obtain cot aving. Some countrie alo chooe to borrow in foreign currencie to diverify their liability portfolio and to achieve an improvement in the rik profile of public debt. Table 1 include the iue of conideration and the correponding type of ecuritie. A pecific ecurity include a dimenion from all thee four deciion iue (uch a a zero-coupon, fixed rate, local currency bill with a maturity of 3 month). Public debt management office have everal method to iue their bond. The mot common technique i to conduct frequent auction where invetor quote price or interet rate bid along with the amount of bond they would like to purchae. Thee bid are then evaluated and the amount of fund needed i covered by iuing the appropriate quantity of ecuritie. The price of a bond iued depend on the expected cah flow from the bond and the yield required by the invetor for lending fund for the maturity of the bond. Like other financial intrument, the price of a bond i the preent value of the expected cah flow from the invetment. 16

30 Therefore, if invetor require a higher yield from a bond, they reduce the price they bid in the auction. Table 1 Type of Treaury Securitie. Deciion Iue Timing of Interet Payment Type of Interet Currency Denomination Maturity Type of Borrowing Intrument Zero-coupon bill/bond Coupon Bond Fixed Rate Bill/Bond Variable Rate Bond Local Currency Bill/Bond Foreign Currency Bill/Bond Bill (3-12 month) Bond (1 year and over) Explanation Interet and principal paid at maturity Interet paid in regular coupon period, principal paid at maturity Interet fixed at iuance, remain contant until maturity Interet baed on an index uch a Libor, inflation etc. Government often announce auction chedule or financing program to publicize the amount and timing with regard to planned bond iuance chedule. Thee iuance program decribe the type of bond the government i planning to iue to meet the projected financing requirement in a certain period. The announcement frequency change from country to country. Some countrie ue monthly program while ome announce the auction calendar for a whole year. Early announcement of iuance trategie leave time for market participant, i.e. potential 17

31 invetor to aborb the information revealed and to adjut their cah-flow cheme if they would like to participate in the auction. Debt manager alo ue public offering or direct ale technique to convey their bond to a pecific group of invetor without inviting them to auction. Thee method help them diverify the invetor bae. Uing the available et of intrument, public debt manager have to find a pecific portfolio combination or develop a pecific iuance program that would embody deciion on maturity, currency and interet type tructure and timing of iuance, in line with government debt management objective. The PDM problem i about reflecting the government cot and rik preference to the public debt portfolio and electing the appropriate combination of financial intrument, i.e. etablihing the debt management trategy. With all the conflicting objective to be conidered, the public debt management problem, i.e. formation of the financing trategie, i a multi-objective deciion making problem with everal contraint. The olution to thi problem would not be trivial even without the uncertainty aociated. 2.3 Contraint in Public Debt Management The ize and efficiency of a country financial market, the government ability to acce international market and other macroeconomic environmental condition impoe everal limitation on the type of ecuritie the debt manager can iue. That i, the et of intrument available to PDM office may differ from country to country. For under-developed or developing countrie, where the level of dometic aving and efficiency of internal financial market are limited, the main option i to opt for fund from international market, often in the form of loan from international financial intitution uch a the World Bank or the International Monetary Fund (IMF). Some developing countrie have a functioning dometic financial market, but they alo have acce to international market and iue debt in foreign currency to lengthen maturity, ince dometic lender generally prefer horter 18

32 maturitie. The advanced economie, whoe market have largely integrated with international market, have more option in electing currency and maturity of loan. Given a certain intrument et, public debt manager hould alo conider market contraint with regard to the availability of fund, demand for different type of ecuritie etc. In a volatile environment, creditor may not be willing to extend long-term loan, and the government initence on lengthening maturitie may reult in a funding-crii. Intitutional invetor, uch a penion fund, may prefer longer-term bond while individual may be aking for liquid hort-term bond. Preference of bank may be different than thoe of inurance companie. Thu, the characteritic of different egment in the market may impoe different contraint on the ize of bond to be offered. The amount of bond to be iued i alo contrained by the financing requirement of the government, i.e. the amount of fund raied hould not be le than thoe required by the budget. Government generally hold a cah account which erve a buffer to cover unexpected cah need and thi allow borrowing more or le than needed for a certain period of time. However, there are alo limitation to the level of thi account, i.e. government can not over or under borrow continuouly. Thu, PDM office hould conider the inter-temporal budgetary and cah account contraint while iuing ecuritie. 2.4 The Effect of Uncertainty in PDM Deciion An important characteritic of the multi-objective PDM problem i that deciion are made under uncertainty. Debt manager are not faced with choice that have determinitic outcome. Debt management deciion are concerned with future action of the government and while making trategy deciion, debt manager are not certain about the future tate of nature for the relevant macro-economic variable. There i a degree of uncertainty aociated with the evolution of economic factor uch a interet and exchange rate that drive the cot of borrowing. For example, a debt management office may iue floating rate bond auming that the interet rate will fall in the future in order to achieve a cot 19

33 reduction in the bond iuance program. However, the actual cot of borrowing through floating rate ecuritie will be dependent on the interet rate realization during the maturity of bond. On the other hand, iuing fixed rate ecuritie carrie the rik of locking at high rate in cae of a decline in interet rate, i.e. the rik of miing the chance to make ue of more favourable market condition. Therefore, the actual outcome of the deciion made while formulating the iuance trategy are contingent on realization of macro-economic variable that exhibit different type of tochaticity. In fact, it i thi uncertainty that raie the need to conider market and liquidity rik objective. Strategie developed need to enure that the government debt management objective hould be covered under different cenario realization. Debt manager have to take into account the underlying tochaticity of macroecomic factor while formulating cot-rik tructure of the debt portfolio. On the other hand, the debt trategy i not a one-off deciion. The PDM problem embodie a equence of deciion that would allow the government debt portfolio adjut to changing environmental condition. That i, a deciion made now for the portfolio tructure i ubject to reviion in the future depending on changing outlook for the macro-economy. Therefore, debt manager hould incorporate thi need for elaticity in their deciion making procee. Deciion made a of now baed on current tate of nature and current projection about the future mut be flexible enough to be changed when needed. Debt manager need to conider the effect of the potential for adjuting deciion in the future, ince the future deciion will be contingent on the previou action and prevailing market condition. 2.5 Literature Review on PDM Strategy Formulation Given it importance, the problem of deigning the public debt management trategy, in term of etting the maturity and the type of intrument to be ued, draw attention of both practitioner and academician from variou perpective. Aleina et al (1990) elaborate on the choice of maturity of public debt and argue that iuing debt at long maturitie and evenly concentrated in time will boot 20

34 public confidence and reduce perceived likelihood of confidence crii about debt default. Miale and Blanchard (1994) claim that government can ue the maturity of debt to how her commitment to anti-inflationary policie and thu hould prefer hort-maturity or indexed debt. The tax moothing approach aume that the main reaon for the government to change taxe i to meet the long-term financing contraint, and the objective i to mooth taxe by chooing the optimal compoition of debt with repect to maturity and contingencie. The aumption i that welfare lo from taxation i higher if taxe change from one period to other than the cae they are contant. Thu taxe are ditorting and government debt hould be tructured in a way that would minimize the need for changing taxe. There i uncertainty about macroeconomic variable uch a public expenditure, tax bae etc. and therefore, the compoition of debt matter (Barro, 1995). The argument i that if the government can iue debt with cot that are lower when net tax receipt are alo lower and vice vera, then debt can erve a a buffer. In that cae, the government can keep the tax rate contant by adjuting the debt pay-off. In Luca and Stokey (1983), the government can iue debt contingent on the outcome of public revenue and pending. Barro (2003) propoe iuance of indexed ecuritie (tied to interet rate, inflation rate etc) when uch tate-contingent debt i not available. Debt management office, Treaurie or other public intitution in charge of managing overeign debt take a practical point of view and apply concept and tool derived from thoe employed by private financial intitution. Danih Central Bank (Danmark Nationalbank 2005) and Swedih National Debt Office (Bergtröm et al. 2002) are two organization that make ue of Cot-at-Rik imulation model by which cot and rik performance of alternative debt management trategie are teted under variou macroeconomic imulation cenario. Hahm and Kim (2003) apply the ame approach to Korea; Turkih Treaury (2004) alo ue a imilar model. Bolder (2003) explain the imulation model for debt trategy analyi in Canada. More recently, Bolder and Rubin (2007) try to combine imulation and optimization approache in debt trategy analyi. Their aim i to approximate the 21

35 debt management objective function through imulation, uing function approximation algorithm and to optimize on thi approximation. The imulation model of PDM office are generally derived from the Value at Rik (VaR) concept widely ued by bank and other financial firm. VaR model are developed to obtain an etimate of the maximum probable lo that the aet may uffer within a given period a certain confidence interval. For overeign, thi approach i modified into a Cot-at-Rik or Cah-Flow-at-Rik model. Countrie alo apply other method like tre teting or cenario analyi to compare different PDM trategie (ee IMF-WorldBank, 2003 and OECD, 2005 for dicuion on debt management practice of elected countrie). In thee practical method macroeconomic variable are not imulated, but everal plauible cenario are created by expert judgment. The general aim i to quantify cot and rik aociated with policy choice in conideration. Giavazzi and Miale (2004) ue the deviation between the urvey of expectation and realization a well a ordinary forecating method to judge unexpected movement in macro variable and their effect on government debt. Debt management objective are alo defined in everal different way. George (2003) and Barro (2003) concentrate on the minimization of the fluctuation of the government budget, rather than the interet burden on the debt tock and try to mooth the budget balance by uing bond that will erve a hedge to the movement of public revenue or expenditure. Goldfajn (1998) conider the objective of minimizing inflation in addition to that of moothing of the budget. A good review of theoretical and practical concept regarding public debt management can be found in Dornbuh and Draghi (1990) and Leong (1999). 22

36 CHAPTER 3 3 A MULTI-OBJECTIVE STOCHASTIC PROGRAMMING APPROACH Simulation model in ue at PDM office, generally compare time-invariant trategie, thu aume that the borrowing trategy will be kept contant until the end of the choen time horizon whatever the macroeconomic condition turn out to be in time. In real life, once a trategy i adopted, it may be ubject to reviion given the change in environmental condition. Therefore, it i ueful to develop a mechanim that would allow trategie adjut to varying macroeconomic circumtance dynamically. Since, in theory, there i an infinite number of way to contruct a borrowing compoition, the olution pace of the problem i continuou. To implify the olution, practitioner identify a certain number of plauible and applicable alternative and chooe to compare only thee, thu convert the problem into a dicrete cae. Thi eem a a reaonable approach for the imulation framework. In thi thei, we try to adapt a continuou olution pace approach by developing a tochatic programming model for PDM taking into account the aociated objective. Stochatic programming (SP) ha been widely ued for modelling multiperiod aet management problem in order to deal with the multi-tage deciion and the uncertainty involved in the parameter regarding economic factor uch a interet rate, price of ecuritie etc. A eminal contribution wa made by Bradley and Crane (1972) who propoed a multi-tage model for bond portfolio management. More recently, Carino et al (1994) applied SP to the aet-liability management problem of the inurance indutry, and Zenio et al (1998) and Topaloglou et al (2004) formulated model for a portfolio of fixed income ecuritie. Nielen and 23

37 Poulen (2004) propoed a multi-tage SP model for managing mortgage backed loan. Voloov et al (2004) developed a two-tage deciion model for foreign exchange expoure management. Grill and Ötberg (2003) have applied an optimization approach for debt management. Yu et al (2003) provide a bibliography of SP model in financial optimization. Extenive collection of tochatic programming model for financial optimization problem can alo be found in Ziemba and Mulvey (1998) and Dupacova et al (2002). While the claical Markowitz (1952) model conider the portfolio management problem a a ingle period cae in which the deciion on which intrument to include i made at the tart of the period, taking into account expected return and variance over time; multi-tage SP model allow for change in the tructure of the portfolio a time evolve 1. In the SP framework, the deciion maker tart with a certain portfolio of aet, ha knowledge on the current value of the economic/financial parameter and aee the poible movement and comovement of thoe parameter in the future. He ha a longer horizon and ha to conider the effect of the potential for adjuting hi deciion in the future on hi current deciion, ince the future deciion will be contingent on the previou action and prevailing market condition. The incorporation of adaptive deciion under changing condition provide a more realitic approach for actual problem. Fleten et al (2002) compare the performance of a multi-tage SP model againt a tatic approach and conclude that due it adaptive nature, the SP model dominate the fixed mix tatic model. Multi-tage model can alo integrate important practical iue uch a tranaction cot (in elling and purchaing ecuritie), pread between ak and bid price, trading limit, taxe etc. and allow for modelling of derivative or hedging intrument (option, future contract, interet rate cap or floor). The general approach of multi-tage SP model in repreenting uncertainty i forming a cenario tree that reflect the evolution of random variable in each tage of the deciion horizon, by dicretizing their joint probability ditribution. 1 There are alo multi-period extenion of the mean-variance framework of Markowitz. Two example are Steinbach (2001) and Draviam and Chellathurai (2002). 24

38 SP framework i a ueful approach for tackling the multi-objective debt management problem which i a real life multi-tage deciion problem under uncertainty. Although much implification may be required regarding the number of deciion tage, and the ize and cope of the cenario tree due to the complexity of the olution, the reult from the SP model olution may well erve a a benchmark. Moreover, the cenario tree formulation embodied in SP may provide a more clear repreentation of uncertainty for the deciion maker in term of explaining the dependence between the tate of tochatic variable and the deciion made at intermediate tage and thu can erve a part of the deciion upport proce. 3.1 Baic of Stochatic Programming Stochatic Programming model were formulated and propoed in mid 1950 independently by Dantzig (1955) and Beale (1955) and have been widely tudied ince then. Along with the development of conceptual modelling iue, the progre in computer technology and computational method enabled handling of large cale real life problem with a high degree of reliability and SP technique have become applicable to real life problem. We will now introduce ome pecial cae of tochatic program and elaborate on cenario generation method in the context SP model formulation Baic Stochatic Programming Model Stochatic programming provide a general purpoe framework to model deciion making under uncertainty and i regarded a a powerful modelling paradigm for different field of application. Anticipative and adaptive model are baic type of tochatic program and their combination lead to the recoure model which i widely applied in financial deciion making problem. The dicuion in thi ection i baed on Birge and Louveaux (1997), Kouwenbeg and Zenio (2001) and Yu et al (2003). 25

39 Anticipative Model Anticipative model, alo known a tatic model, are developed for cae in which the deciion doe not depend on pecific future obervation of tochatic variable, but ha to conider all poible realization for prudent planning. Once a deciion i made, there i no opportunity to adapt deciion. In uch model, feaibility i articulated in the form of probabilitic (or chance) contraint. For example, let u conider a cae where a deciion x mut be made in an uncertain environment which i decribed by a random vector w with upport Ω. If a reliability level, α ( 0 < α 1) i pecified, the contraint can be expreed in the following form: { w f x, w) = 0, j = 1 n} α P j (,..., (3.1) where x i the m-dimenional vector of deciion variable and m f j : R Ω R, j = 1,..., n. The objective function can alo be imilar: { w f 0 ( x, w) γ } m : R Ω RU P (3.2) where f { + } 0 and γ i a contant. An anticipative model identifie a deciion that meet deirable characteritic of the contraint and the objective function. In the example given, it i required that the probability of contraint violation i le than the pecified threhold level Adaptive Model In adaptive model, information related to the uncertainty become partly available before the deciion i made. The main difference to anticipative model i that deciion making take place in a learning environment. Let A be the et of all relevant information available by obervation. A i a ubfield of all poible event and the deciion x depend on the event that can be oberved. x i termed A adapted or A meaurable. Uing conditional expectation with repect to A, an adaptive SP can be formulated a follow: 26

40 Min E [ f ( x( w), w) A] 0 (3.3).t. E[ f j ( x( w), w) A] 0, j = 1,..., n = (3.4) x( w) X almot urely (3.5) The mapping x : Ω X i uch that x(w) i A meaurable. The two extreme cae occur when there i complete and no information. In the abence of any information, the model reduce to an anticipative form. When there i full information about uncertainty, the model turn into what i known a a ditribution model which characterize the ditribution of the objective function Recoure Model The recoure model combine the anticipative and adaptive model. Thi framework trie to identify a trategy that not only anticipate future realization but alo take into account temporarily available information about the tate of tochatic variable. Thu, the model can adapt by taking recoure deciion. For example, in an aet management problem, to formulate the mot profitable portfolio management trategy, a financial manager hould conider the future movement of aet return (anticipation) together with the requirement to rebalance the portfolio compoition a price change and cah flow from the aet are realized (adaptation). A two-tage SP model with recoure can be formulated a follow: Min f ( x) E[ Ψ( x, w) ].t + (3.6) Ax = b (3.7) R m 0 x + (3.8) where x i the m 0 dimenional vector of firt tage deciion made before the random variable are oberved (anticipative) and Ψ ( x, w) i the optimal value for the following program: Min g ( y, w) (3.9).t W ( w) y = h( w) T ( w) x (3.10) R m 1 y + (3.11) 27

41 In thi program, y i the m 1 dimenional vector of econd tage deciion made after the random variable are oberved, thu thee deciion are adaptive. g ( y, w) denote the cot function in the econd tage. Parameter T (w), W (w) and h(w) are function of the random vector w. T tand for the technology matrix and contain the coefficient that convert the firt tage deciion x into reource for the econd tage. W i the recoure matrix while h denote the reource vector for the econd tage. In thi formulation, the econd tage problem trie to identify a deciion, y that minimize the cot in the econd tage for a given value of x, the firt tage deciion. Once a firt-tage deciion i made, ome realization of the random variable can alo be oberved. Then, the two-tage program with recoure i about optimizing the cot of the firt-tage deciion and the expected cot of the econdtage deciion. Thi can generally be formulated a follow: f ( x) + E min m y R 1 + g( y, w) T ( w) x + W ( w) y = h( w) Min { }.t (3.12) Ax = b (3.13) R m 0 x + (3.14) The recoure problem i not retricted to two-tage formulation. It i poible that obervation about tochatic variable are made at different point in time and deciion are revied accordingly. Thi lead to the formulation of multi-tage problem where tage correpond to time intance when ome information i revealed and a deciion can be made Determinitic Equivalent Formulation Determinitic equivalent formulation conider the cae where the random 1 2 N variable w ha a dicrete ditribution with finite upport Ω = { w, w,..., w }, which i called a the cenario et. If p denote the probability of realization of the th 28

42 cenario w ( > 0 problem can be written a: vector, E expreed a: N l = 1 p and p = 1), then the expected value of the econd tage N [ Ψ x, w) ] = ( p Ψ( x, w ) (3.15) l= 1 A different econd tage deciion i made for each realization of the random w Ω, If thi i denoted by y, the reulting econd tage problem can be Min g ( y, w ) (3.16).t. W ( w ) y = h( w ) T ( w ) x (3.17) y + m R 1 (3.18) Combining the above, the determinitic equivalent formulation of the twotage model turn out to be a follow: Min.t N f ( x) + p g( y, w ) (3.19) l= 1 Ax = b (3.20) W ( w ) y = h( w ) T ( w ) x for all w Ω (3.21) R m 0 x + (3.22) y + m R 1 (3.23) Scenario Generation in Stochatic Programming Model A determinitic equivalent tochatic programming model i baed on a cenario tree (or event tree) repreentation of the movement of tochatic variable in time. Each branch of the tree denote a different path of evolution for the relevant random variable. The cenario tree ha ome deciion node that repreent the tage where the deciion maker() decide on the coure of action to be purued. The branche of the cenario tree dieminate from thee deciion node and correpond to alternative tate of nature for the tochatic variable after each deciion tage. The model i olved on thi dicretization and the olution determine an optimal 29

43 deciion for each node baed on the information et available at that point. Therefore, contructing a good cenario tree that approximate the real tochatic proce i a key iue for the ucce of the SP model Overview of Scenario Generation Method Scenario generation ha been an active field of reearch within the SP context and everal alternative method have been propoed for creating good cenario tree. Yu et al (2003) and Kaut and Wallace (2003) provide brief overview of ome common method available for cenario tree generation. The implet approach for generating cenario i to ue hitorical data regarding random variable without any modeling and claim that future will replicate the pat (e.g. ampling from pat yield from different point in time for generating cenario for bond return). Thi method allow for cenario generation without auming any pecific ditributional form for the random variable. Boottrapping hitorical data i a common method employed in Value-at-Rik analyi known a Hitorical Simulation. A drawback i that the approach i backward looking and doe not repreent expectation for the future. Thu, the reult may be dominated by a ingle, recent, pecific crie and it i very difficult to tet other aumption (Marrion, 2002, pp. 118) Another approach that doe not rely on ditributional aumption i to ue the empirical characteritic of random variable and try to create cenario that replicate thoe uch a the moment matching method of Hoyland and Wallace (2001). In thi approach, a cenario tree that matche the pecified target value for the random variable, including correlation in-between, i generated. The uer are allowed to pecify the tatitical propertie (moment) that are relevant and the idea i to minimize ome ditance meaure between thee pecified propertie and the propertie of the generated outcome on the cenario tree. Hoyland et al (2003) propoe an algorithm to peed up thi cenario generation method. A more ophiticated approach require tatitical or econometrical modelling that would capture the characteritic of the movement (and co-movement) of 30

44 random variable in time. Boender (1997) ue a vector autoregreive (VAR) time erie model to generate aet return and wage increae cenario for Dutch penion fund. Villaverde (2003) preent two VAR model including US, European and Japanee aet and exchange rate. In Pflug (2001), the method to generate the dicrete cenario tree i baed on the objective of minimizing the approximation error. Thi optimal dicretization method trie to generate the dicrete approximation in uch a way that the approximation error, i.e. the difference between the optimal value of the underlying problem and the value found by inerting the olution of approximate problem i mallet. Reearch effort in the field of cenario generation ha alo concentrated on reducing the number of cenario in a given cenario tree to control model complexity while preerving the degree of approximation. The approach of Heitch and Römich (2005) i to bundle and delete ome cenario repeatedly from a preupplied multivariate cenario tree generated from hitorical or imulated data erie. They employ a certain ditance metric and proceed by uniting or deleting cenario that are cloe to each other to obtain a tree, maller than the given cenario fan, which maintain to be a good approximation. Since a cenario tree repreentation contain a limited number of branche, the problem olved i only an approximation of the real problem and thu the quality of the cenario tree i extremely important for the quality of the olution. The model olution can be hardly relied if the cenario tree we ue i far from repreenting the true tochatic proce. Naturally, the higher the number of cenario on the cenario tree, the better i the degree of repreentation. However, that come along with an amplification in the complexity of the model, i.e. an increae in olution time, and thu, we need to retrict the ize of the tree in order to preerve the ability to olve the model. Here lie a trade-off between having a good approximation of the real tochatic proce and controlling the dimenion of the SP model. 31

45 Evaluation of Scenario Tree Generation Method Depite the importance of cenario tree generation in the SP framework, to the bet of our knowledge, there ha been little reearch on the aement of the repreentative capacity of cenario tree. Kaut and Wallace (2003) focu on thi iue and dicu the evaluation of the quality of cenario generation method, defining ome minimal requirement. Specifically, they propoe two meaure to tet the uitability of a certain generation method for a given SP model: one related with the robutne of the tree generator (tability) and the other regarding the bia it contain. If the cenario tree generation method i tochatic, it can generate different intance in different run. In that cae, we need to enure that olving the SP model on different tree, generated by the ame method, yield imilar optimal value. Thu, by what they define a in-ample tability, Kaut and Wallace (2003) propoe that the optimal objective value obtained in the SP model baed on different cenario tree intance hould be approximately identical. While in-ample tability i concerned with the variability of the optimal objective function value, out-of-ample tability i related with the performance of the optimal olution in the deciion pace. In thi regard, the author propoe the evaluation of the olution of the SP model in the true problem and tet whether olution obtained on different cenario tree yield imilar reult when plugged in the real problem. However, thi i not alway poible ince we may not have full information about the actual ditribution that drive our tochatic variable. To enure that the cenario generation method contain no bia, we need to compare the optimal value in the cenario baed problem to that of the true problem and ee whether or not they are cloe to each other. Thi i again impoible in mot cae, ince thi require olving the true problem optimally. A a proxy, Kaut and Wallace (2003) recommend the employment of a larger reference tree which i believed to have a better repreentation of the true tochatic proce and ue the reult from thi a a benchmark to tet for a poible bia. 32

46 3.2 The Multi-objective Public Debt Management Model We now formulate the Public Debt Management Problem uing the Stochatic Programming approach. We incorporate relevant criteria and develop a determinitic equivalent model baed on a cenario tree repreentation of macroeconomic factor that affect the cot of public debt. The model ha a multitage tructure that take into account equential deciion concerned with debt iuance policie. The government ha to decide on the type of borrowing intrument (bond) to be iued to meet the financing requirement in a given planning period. Our model aim to ait the formulation of the iuance calendar which include the timing and amount of bond to be iued. The objective i to pecify a equence of bond iuance deciion at dicrete point in time. We formulate the debt management problem a a linear programming model. In imulation baed approache, the general methodology i to aume that the government i to meet it funding requirement by applying a fixed trategy which dictate the proportion of intrument to be iued in each period. Thu, the PDM office elect a certain et of weight for the bond to be ued and iue the ame proportion of ecuritie in each time tep whatever the financing need i. Uing thee weight a deciion variable in an optimization framework reult in a non-linear and unfortunately non-convex problem tructure. A bond iued in the firt period i to be paid back in one of the following period which add up to the financing requirement in that phae which, in turn, i to be financed with the ame (or another) et of weight. That induce a multiplicative form for the deciion variable contained in the problem and to implify the cae and to enure the optimality of the problem, we try to develop a linear program. We preent a general n-tage model in which each period i divided into everal ub-period, t. (If the period correpond to year, ub-period can be month or quarter). Scenario unfold in each ub-period. Deciion are made at tart of each period for the ub-period contained in that period, i.e. iuance deciion are not revied in each ub-period, but only at deciion tage, the cenario between 33

47 deciion tage combine to form a equence of joint realization for a certain period. Thee equence of cenario are linked at the deciion node and we have cenario path covering the entire planning horizon. We aume that at the beginning of each year the government et a borrowing trategy, which embodie the timing and amount of bond to be offered in each month (or quarter) of the following year and revie thi trategy annually. The debt manager tart with a given liability cah flow cheme (ariing from the current debt portfolio) and a et of anticipated cenario about future tate of relevant macroeconomic variable uch a the interet and exchange rate. Baed on the given cenario et, he decide on the iuance policy for each ub-period (month or quarter) within the next year and a a reult, at the tart of next year he ha a new liability portfolio. He now ha to make a new et of deciion incorporating thi new portfolio tructure, thu the updated cah flow cheme contingent on the cenario realization in the interim (firt year) and the current cenario tree about the evolution of tochatic variable. Thu deciion, other than the firt tage deciion, are pathdependent and we have a tochatic programming problem with recoure. Figure 2 illutrate the tructure of a problem with 3 period each divided into 4 quarter. One main aumption we make for our model i that the macroeconomic environment i independent of the government policy action with regard to public borrowing. That i, the amount and the type of the bond the government decide to iue do not effect the level of prevailing interet rate in the market and the government can iue any amount of bond without changing the interet rate. Thi i not an unrealitic aumption for countrie that have deep and liquid bond market with many iuer and lender. 34

48 Revie Strategy Scenario Path Set Strategy for Year 1 quarter deciion tage Figure 2 A Sample Scenario Tree Objective of the Model We formulate the PDM problem a a tri-criteria model, accounting for the objective of minimizing the cot of raiing debt, the market rik and the liquidity rik. One can alo think of other objective of debt management uch a increaing the invetor bae for government ecuritie, improving efficiency in the local market, aligning borrowing trategie with other macroeconomic policie of the government uch a the monetary and fical policie etc. Here, we adopt a financial point of view and ee the problem a portfolio management exercie. In the following ection, we elaborate on alternative formulation of the financial objective of government debt management. 35

49 Cot It i poible to define the cot and thu the variation of cot aociated with public debt in everal way: The cot of debt can be meaured by the market value of debt tock, preent or nominal value of future interet cah flow, accrual baed interet payment etc. Each debt office track everal cot meaure depending on the prevailing accounting principle and it own market activitie. For countrie that reort to debt buy-back and exchange operation frequently, the market-value of the bond can be a relevant cot indicator, while for debt office that prefer to redeem bond at maturity, the interet expenditure i the appropriate meaure. We aume that the debt manager aim to minimize the expected value of their relevant cot meaure over the deciion horizon. In our model, we only account for the cot of bond iued during the deciion horizon a the cot of bond already in the tock will be ame for all alternative borrowing trategie. If the model i extended to include buyback and debt exchange that would allow for deciion on changing the tructure of the tarting debt tock, then the cot definition can be widened to include all liabilitie including thoe fixed before time t= Market Rik Market rik i generally defined a the rik of an increae in cot, which again can be meaured in everal way. Approximating thi rik with the tandard deviation a in the claical Markowitz model would reult in a quadratic optimization problem. Thu, we conider other meaure to preerve LP olvability (ee Manini et al for linear rik meaure ued in portfolio optimization model). The Value-at-Rik (VaR) meaure i a very popular concept which i widely ued by private bank and other financial firm. VaR model, ued to obtain a meaure of the maximum potential change in value of a portfolio of financial intrument with a given probability over a pre-et horizon (RikMetric, 1996, p.6) ha even become part of the regulatory meaure in the banking ector. Depite it 36

50 popularity, it ha been hown that VaR ha undeirable mathematical characteritic uch a non-convexity and non-ubadditivity (VaR of a portfolio can be larger than the total of that of individual aet) and it i difficult to optimize when it i calculated from cenario (ee Pflug 2000). VaR doe not either provide any information about the level of rik if the confidence level i exceeded. The Conditional Value-at-Rik (CVaR) alo referred a the mean exce lo or the expected hortfall emerged a an alternative rik meaure a a repone to the limitation of VaR. While the VaR of a portfolio i the maximum amount of lo expected over a certain horizon in a given confidence level, the portfolio CVaR i the expected lo given that the lo i greater than (or equal to) it VaR. In other word, it i the expected value of 100α % wort cot over the entire cenario et at a given level of α. Pflug (2000) ha hown that CVaR poee the required propertie of coherent rik meaure in the ene identified by Artzner et al. (1999). Rockafellar and Uryaev (2000) how that CVaR, which quantifie the conditional expectation of loe when VaR i exceeded, can be efficiently minimized uing linear programming in a cenario baed framework. More dicuion on the Conditional Value-at-Rik concept i provided in Appendix-A. For a government who i concerned with the level of interet cot rather than the value of the debt portfolio, the CVaR meaure can be turned into a Conditional Cot at Rik (CCaR) metric. The wort-cae cot can alo be ued a a meaure of the market rik. The government might alo have a target level for the debt ervice expenditure, and any deviation above thi level due to market condition can be a meaure of market rik. If the high deviation are more important than lower one, one can aign different weight to different level of exce cot and invent a piece-wie linear objective function Liquidity Rik While the cot and the market rik can be meaured in accounting term, liquidity or re-financing rik, a it i alo called, i aociated with the actual debt 37

51 ervice (or total) cah flow of the government. It can preciely be defined a the threat that at the time of debt repayment, the government will lack the neceary fund, have difficulty in raiing new debt and fail in fulfilling it objective. To avoid uch adveritie, it i common practice for public debt manager to mooth out debt repayment and try to avoid concentration of payback in certain period to control liquidity rik. For example, Sweden plan it borrowing in uch a way that no more than a certain proportion of debt mature over a 12 month period (IMF-World Bank 2003). Thi precaution can be matched with a meaure that would quantify the variability of cah flow through time. In our modeling framework, we can compute the deviation of the cah flow received and paid by the government in each time tep for every cenario and produce an expected variability magnitude over all cenario in the problem with an approach imilar to that of Bergtröm et al. (2002) Minimizing thi Mean In-Scenario Variation meaure, computed over debt amortization and interet payment may erve a a mean for moothing out debt repayment, while computing the ame for net cah-flow of the government will help match the inflow and outflow. The government may alo be intereted in containing the maximum poible in-cenario variation of cah flow rather than it expected value and thu can alo adopt a minimax type objective function. Similar to market rik, the in-cenario (net) cah flow variation can be meaured by the tandard deviation of cah flow in each time tep over the entire time horizon for a ingle cenario. To avoid non-linearity, we can again adopt an expected exce cah flow type of function for the refinancing rik. The mean abolute deviation, which average the abolute value of digreion from the mean or a target level can a well be ued a a meaure even though it attache equal importance to all degree of variation. A weighted abolute variation meaure can be a remedy if different degree of deviation from a target level bring different concern for the government. The highet poible debt ervice level in ingle time tep over all covered cenario (not in a ingle one) can alo be treated a a ignal of liquidity rik and a minimax type objective function accounting for thi highet level may erve for 38

52 preventing high concentration of debt in one period. Given that exceive amount of debt re-payed in a mall interval implie a critical level of refinancing rik for the government, the minimization of the highet poible cah outflow can be a notable intention for PDM unit. We hould note thi formulation would reult in a concentration of debt repayment in period beyond the planning horizon if we are operating in a hort deciion horizon Notation The model i baed on cah-flow equation that guarantee that the total outflow of the government match the inflow. Since our model aim to determine the amount of different type of bond to be iued conidering the given cenario et, the borrowing trategy hould cover all poible cenario. We include a cah account in our model that would aborb any exce or hort borrowing that might occur when certain cenario are realized ince the cah outflow are baed on ome parameter that are cenario pecific. Thu, the debt manager et the amount of each bond to be iued in all ub-period of the following period conidering the poibilitie for the level of the financing requirement. If, in ome cae, the total debt raied i more (le) than needed, the exce (hort) amount i injected into (withdrawn from) the cah account of the government. We firt define the parameter of the model: Parameter and Index Set T I T i t S N : deciion horizon : number of period : length of period i, i=1,,i : time index (denoting ubperiod), t=1,...,t : the cenario et : cenario index, S : deciion tage (the beginning of each period) 39

53 p : probability aociated with cenario. J 1 J 2 J 3 : et of zero-coupon bond/bill (bond that pay interet at maturity) : et of variable coupon bond/bill (with interet fixing at the tart of each coupon period) : et of fixed coupon bond/bill J : et of all bond ( J = J 1 J 2 J 3 ) m j : maturity of intrument j, j J, c j : coupon period of intrument j, j J 2 J 3 (We aume all coupon are t j emi-annual) u, : upper bound for the iuance of bond j at time t. PS t : Primary urplu(net non-debt cah-flow) at time t, t=1,...,12 τ : Coupon payment indicator for intrument j (j J 2 J 3 ) iued at timeτ for Y, t, j ( t) time t. ( Y τ, t, j = 1 if intrument j iued at timeτ pay coupon at time t.) n, : deciion node for cenario, for period t The deciion are made at the node of the cenario tree, thu node are where cenario path dieminate. The parameter n(,t) denote the node in which the iuance deciion i made for time t under cenario. For all the cenario in period 1, the deciion i made in node 0. ( t) n, = 0, for S and t T 1 The cenario dependent variable are given below: Stochatic Variable: r, : interet rate prevailing at time t for intrument j under cenario. t j e t, j : exchange rate prevailing at time t for intrument j under cenario. ( e t, j local currency intrument) =1 for 40

54 L t : Liability payment fixed before the deciion horizon (which may be cenario pecific) for time t under cenario. The deciion variable are defined for each node of the cenario tree: Deciion Variable: X : amount of intrument j to be iued in period t under cenario, decided at n(, t) t, j deciion point n(,t) Auxiliary Variable: I t : total interet paid at time t under cenario. D t : total principal (debt) paid at time t under cenario. B t : borrowing requirement at time t under cenario. TC : total cot for cenario. C t : withdrawal from cah account at time t, under cenario. CB t : level of cah account (cah balance) at time t, under cenario. VR : variable ued in the definition of CCaR equal to VaR at the optimal olution. cv : exce cot beyond VaR for cenario Contraint Some of the contraint of the model are for definitional purpoe while ome provide for the cah flow balance regarding the government payment, including amortization and interet payment, and cah receipt. There i alo an intertemporal balance equation for the cah account. We alo include contraint regarding the marketability of the bond, a there might be bond pecific limitation for the amount 41

55 42 of iuance due to the tructure of market demand. Below are the contraint of our model: Total principal paid back at time t, cenario : t D = J j m t n j m t j j X ), (, m t t j >, 0 : (3.24) Thi equation um the principal value of all the bond that mature at time t for a pecific cenario. Total interet paid at time t, cenario : t I = J j m t n j m t j j X ), (, ( j m t j t j e e,, -1) + 1 ), (, J j m t n j m t j j X j m t j r,. j m t j t j e e,, + j t J j n j t m t Y X j,, ), (, 1. 2 τ τ τ τ = j c t j r,. j j t e e,, τ + j t J j n j t m t Y X j,, ), (, 1. 3 τ τ τ τ = j r, τ. j j t e e,, τ m t t j >, 0 : (3.25) The interet cah-flow equation for cenario, conit of the interet paid on maturing zero coupon bond and the coupon paid for live fixed and floating rate bond at time t, all adjuted for change in the underlying exchange rate. The interet paid i computed by multiplying the principal value of a bond by the applicable interet rate, which i fixed at time of iuance for zero and fixed coupon bond and at the tart of coupon period for variable rate note. The change in the market value of debt due to exchange rate fluctuation i alo included in the interet definition. Thi i relevant for countrie that have foreign currency denominated debt. The cah-flow balance: = J j t n j X t 1 ), (, + t C = t D + t I + t L - t PS t, (3.26) The cah-flow balance equation indicate that the total amount of bond iued at time t (for cenario ) and the amount ued from the government cah account hould equal the um of debt repayment, including principal and interet, and the non-debt liabilitie of the government, accounting for the primary urplu available for time t. Cah account balance:

56 CB t = CB 1 t - C t t, (3.27) In thi equation, CB 0 i the tarting cah account balance. The cah account balance hould be adjuted after each time tep taking into account in and out-flow. Non-negativity: CB 0 t, (3.28) t n(, t ) X 0 t,, j (3.29) t, j We aume that the government doe not allow it cah account to deplete. The amount of bond iued can not a well be negative (no-buyback are allowed). Marketability:, t,, j (3.30) n(, t) X t j ut, j The marketability equation account for limitation on demand for different type of government bond Objective Function Our tri-criteria model i baed on minimizing the cot, and the market and refinancing rik aociated with public debt. We include ome alternative formulation for the objective function baed on the dicuion in the previou ection. Thee objective are all ubject to the contraint decribed in the previou ection Expected Cot of Debt The expected cot can be calculated by multiplying the cot aociated in each cenario with the repective probability S Min = 1 p TC (3.31) Here, the cot definition ( TC ) i to be determined taking into account the relevancy of poible alternative meaure. For example, if a country debt office i 43

57 operating on cah-accounting principle, then the relevant cot indicator may be the total interet payment made in the planning horizon: T TC = I t t= 1 (3.32) Market Rik We provide formulation for two alternative market rik meaure. The CCaR value can be computed in line with Rockafellar and Uryaev (2000). We define an auxiliary variable, cv, which take poitive value when a certain level, VR, i exceeded. In the optimal olution, the VR value equal the aociated VaR level for a given α. Min VR + 1 α S.t. cv ( p. cv ) (3.33) TC VR (3.34) cv 0 (3.35) The wort-cae cot (wcc), which can be employed a another meaure of market rik, i the highet level of cot that emerge acro the entire cenario et. Min wcc,.t. wcc TC (3.36) Liquidity Rik The liquidity rik can be meaured by the maximum liability payment made in a ingle time tep and the above objective trie to minimize thi acro the entire cenario et. Min max ( D + I +,t t t L t ) (3.37) 44

58 The below objective account for the primary urplu available for debt ervice and approximate the liquidity rik by taking into account the net cah outflow. Min max ( D + I + L - PS ) (3.38),t t t t t Minimizing the expected abolute in cenario variation, a defined above, will enure that liability payment will be moothed out over the deciion horizon a dicued before. Min T ( p. D + I + L µ ) (3.39) S t= 1 t t t where T Dt + I t + Lt µ = (3.40) T t=1 3.3 A Simple Illutrative Model In thi ection, we preent a two tage model baed on a cenario tree with two branche at each tage to concretize our modeling approach. The model horizon i two year and the year are not divided into ub-period (month or quarter). For ake of implicity, we aume that there are two financial intrument at the dipoal of debt manager: two zero-coupon bond with maturitie of one and two year repectively. The government doe not have a tarting debt tock, i.e. there are no pre-determined liabilitie to be fulfilled during the period covered by the model, other than a borrowing requirement of 10 million TRY during year one that arie from non-debt obligation (e.g. alary payment for government employee). Then in thi example, S = 1,2,3,4}, J = {1,2}, T = {1,2}, m = 1, m 2. { 1 2 = Let the expected evolution of the one-year interet rate be a depicted on the following cenario tree: 45

59 5% 10% 2.5% 10% 20% 30% Year1 Year2 Figure 3 Scenario Tree for the One-Year Interet Rate. Auming a flat yield curve, the annual rate for the two-year intrument will be ame a the rate for the one-year bond, i.e. the period rate for two year i twice a much a the one-year rate. Then the cenario pecific value for our tochatic variable are a follow: L r 1 1,1 r 3 1,1 r 1 2,1 r 2 2,1 r 3 2,1 r 4 2,1 = r = r 2 1,1 4 1,1 = 5% = 2.5% = 10% = 30% = 10% = 20% rt,2 = 2rt,1, t Auming all four cenario are equally likely, we have: p = 0.25, Since there i a pre-fixed outflow of 10 TRY in year 1, = 10, L2 = 0,. 1 The deciion node correponding to the cenario and time period are a follow: 46

60 n(,1)=0, n(,2)=1, for =1,2 and n(,2)=2, for =3,4. We have a total of ix deciion variable, which denote the amount of the two bond to be decided at each deciion node: X : amount of bond 1 to be iued in year 1, decided at node ,1 X : amount of bond 2 to be iued in year 1, decided at node ,2 X : amount of bond 1 to be iued in year 2, decided at node 1 (cenario 1 2,1 pecific deciion) X : amount of bond 2 to be iued in year 2, decided at node 1 (cenario 1 2,2 pecific deciion) X : amount of bond 1 to be iued in year 2, decided at node 2 (cenario 2 2,1 pecific deciion) 2 X 2,2 : amount of bond 2 to be iued in year 2, decided at node 2 (cenario pecific deciion) The total principal to be re-paid in period 1 and 2 are calculated a follow: D = 0, 1 D 2 0 = X, 1,1 That i, the only maturing bond in our two year horizon i the one-year bond to be iued in the firt year. The interet payment equation are a follow: I = 0, 1 0 I 2 = 0. 10X 1,1 for =1,2. 0 I = 0.20, for =3,4. 2 X 1,1 Then, the cah-flow balance equation will turn out to be a given: 0 X 1, , 2 X + C 1 = 10 (ince L 1 i fixed for all cenario, C 1 i alo fixed and thi will reduce into one equation.) 1 X 2, , 2 X + C 1 2 = D I 2, for cenario 1 47

61 1 X 2, , 2 2 X 2, , 2 2 X 2, , 2 Here, X + C 2 2 = D X + C 3 2 = D X + C 4 2 = D I 2, for cenario 2 3 I 2, for cenario 3 4 I 2, for cenario 4 C 2 will erve a a buffer that mop up over or under financing that may arie due to cenario pecific value for the interet expenditure, while the iuance amount for the bond i made at the deciion node without eeing the realization of interet rate. In our model, CB 0 =0, C t ha to atify the cah-account balance equation: CB 1 = CB0 - C 1 (Thi will again be a ingle equation ince C 1 i fixed for all cenario). CB 2 = CB 1 - C 2 Aume that we would like to minimize the um of expected interet paid in the two year period and the interet cot accrue for the bond that do no mature at the end of the deciion horizon. The bond that are till alive at the end of year two are the two-year bond iued in year 1 and all the bond that are iued in year two. Since the interet on thee bond i not paid within our planning period, we adjut our cot definition to include interet accrued on thoe intrument. Let A denote the interet to accrue in cenario. Then, auming that interet cot accrue linearly and bond are iued in the middle of each period, the equation for A are: A = 1, X X X ,1 2, A = 1, X X X ,1 2, A = 1, X X X ,1 2, A = 1, X X X Then the total expected cot will be: 48 2,1 2,2

62 z 4 1 = p.( A + I1 + I 2 ) = 1 Solving the model by minimizing z 1, we find out the following firt tage olution (note that the econd tage olution are cenario pecific): 0 X 1,1 =10, 0 X 1,2 =0, z 1=2,2 That i, the model chooe to iue a hort-term bond in the firt year and roll thi over in the econd year. Thi i expected ince in three of the four cenario for year two, the interet rate i declining. Thu, to reduce the expected interet cot, the model chooe not to lock in a fixed cot for two year by not chooing the two-year bond. We now olve the model with a rik management objective. Aume that the iuer aim to minimize the wort cae cot ( z 2 ) taking all cenario into conideration. { A + I I } z2 = max S Solving the model minimizing z 2 yield the below olution: 0 X 1,1 =0, 0 X 1,2 =10, z 1=2.25, z 2 =3.0. Thi time, the model propoe to iue a two-year bond in order not be affected by the high interet rate in cenario 4 (Note that the z 2 value in the firt olution, minimization of cot, wa Thi i due to renewed borrowing in period 2 due to hort-term borrowing in period 1). 3.4 An Integrated Simulation/Optimization Approach Debt management i a continuou proce and the government debt portfolio ha a dynamic tructure ince there are bond and bill entering and leaving the debt portfolio throughout time. Public debt manager may renew their debt trategy deciion while maturing bond and bill are rolled over by new debt iuance in line with the development in the financial market and in the macroeconomic environment. 49

63 Our SP-baed framework take thi fact into conideration. The firt tage deciion i made by conidering repercuion on the later tage deciion which are deemed to be cenario-pecific. That i, in later tage, the deciion maker can revie her trategy a cenario are realized. However, the initial tage deciion hould be robut enough to meet poible outcome contained in firt-period of the cenario et and then to allow for policy change in the upcoming period. In real-life, the PDM model i to be olved repeatedly in time. Firt the DM will olve the model baed on the exiting cenario tree that contain alternative future path for tochatic variable, emanating from their current tate. The initial tage deciion of the model will be implemented during the firt upcoming period. Meanwhile, depending on the cenario realization, the tochatic variable will move to new tate of nature. The DM will then contruct a new cenario tree that originate from thoe tate and olve the model once again, thi time tarting with an updated liability portfolio baed on the deciion from the previou tage. Again the current initial tage deciion will be implemented and thi proce will be repeated in time. In thi context, we propoe a method to tet the quality of our initial tage deciion. To thi aim, we adopt an integrated imulation/optimization approach imilar to the method in Zenio et al (1998). In thi etting, which they call a dynamic game, alternative model are compared in a rolling-horizon environment. In a ingle game, a cenario tree i generated at the beginning of the deciion horizon (t 0 ) and the firt tage deciion i applied. Then the clock i moved on (to t 1 ) and a random cenario i aumed to be realized (for the period [t 0 t 1 ]). Then baed on thi aumed realization a new cenario tree i generated and the SP model i reolved and the firt tage deciion are implemented. The clock i moved on again and the proce i repeated until the end of the deciion horizon. Thi end one run of the game and the game i repeated everal time. The imulation reult provide for an aement of the performance of the model in a dynamic etting imilar to real life. The game can be played for alternative model for a comparion of performance in a imulation etting. Figure 50

64 4 ummarize an adoption of thi approach to our PDM model on a two-tage example. The approach i illutrated in an application given in Chapter 5. Implement initial tage deciion on the realized cenario path Generate new cenario tree Realized cenario path in the imulation Scenario tree at t=t 0 Scenario tree at t=t 1 Figure 4 The Simulation/Optimization Framework. 51

65 CHAPTER 4 4 MULTI-CRITERIA DECISION MAKING ANALYSIS FOR THE PUBLIC DEBT MANAGEMENT PROBLEM Several approache derived from technique applied by private financial intitution have been adopted for the cae of the government in order to quantify the cot and rik aociated in public debt management. However, debt management objective are generally conflicting by their nature and thu public debt management i multi-objective deciion making problem. Therefore there i a need for providing upport to deciion maker to find efficient olution and to explore the conequence of different financing trategie. Exiting tudie on public debt management formulate the cae a a ingle or a bi-criteria problem taking into account a cot and/or a rik meaure. For example, the widely ued Cot-at-Rik imulation model of debt management office come up with a et of alternative debt management trategie and preent the degree of trade-off between the adopted cot and rik criteria, with rik defined a the variability of cot, omitting other deciion criteria. However, meauring rik with a ingle meaure embodie everal problem in the context of an aet/liability management problem. Hallerbach and Spronk (2002) argue that there are many factor that affect potential future variability in return (cot for our cae), extending from the tate of interet and exchange rate to pychological factor uch a the market entiment. Even the change in interet rate i derived by change in the level, lope and curvature of the yield curve. All thee factor may inflict deviation in different direction. Moreover, dependent on the preference of deciion maker, other 52

66 attribute uch a the liquidity and taxability of intrument may be deirable in a portfolio. Thu, the two parameter rik-return perpective may not be ufficient and multi-criteria approache may be of practical ue. Zopounidi (1999) encourage the ue of a multiobjective vantage point in financial deciion making problem arguing that peaking of optimality i illuory and narrow the view, and ince deciion maker are human it i neceary to take into conideration their preference, experience and knowledge in order to olve thee problem. Multicriteria deciion making (MCDM) literature contain many example which combine MCDM tool with the financial deciion making proce. Zopounidi (1999) and Steuer and Na (2003) preent extenive bibliographie on the ubject howing that method like multiobjective/goal programming, outranking relation approache, Analytical Hierarchical Proce (AHP) etc. have been applied to the field of portfolio analyi, financial planning, budgeting, rik analyi, corporate management etc. In the previou ection, we developed a multi-objective tochatic programming (SP) model that incorporate conecutive iuance deciion, taking uncertainty into account making ue of a cenario tree. We now incorporate Multi- Criteria Deciion Making tool on thi determinitic equivalent SP model to ait deciion maker (DM) in analyzing the trade-off between conflicting objective. We identify ome efficient olution baed on different preference tructure and employ an interactive MCDM approach to guide DM in making debt trategy olution. We will illutrate the ue of thi tool in an application for PDM in Turkey. 4.1 Obtaining Efficient Solution We now would like to employ our model, which i of the following form, in a deciion aid framework to guide policy analyi in the multi-objective PDM problem. Min z= { x), z ( x), z ( )}.t 1( 2 3 x z (4.1) x X (4.2) where z i, i=1, p are the objective function and x X are the deciion variable. 53

67 We ue quotation mark ince the minimization of a vector i not a well defined operation. When multiple criteria are conidered, it i unuual to have a ingle olution that i bet for all criteria. Typically, one need to acrifice in ome criteria in order to improve in other criteria. Thi i alo the cae for our PDM problem. Since, in general, hort term rate are lower than long term rate, public debt office find it le cotly to borrow in hort term maturitie. However, a we have dicued, thi then lead to an increaed expoure to changing market condition, i.e. a higher level of market rik. In order to contain market rik, the deciion maker need to make ome acrifice from their cot reduction objective. Thu, we need to identify the degree of trade-off between our objective function in order to ae alternative financing olution. On the other hand, thee trade-off do not exit for all poible olution. There might be ome olution which are wore off than other in all criteria. Some olution may be a good a other in mot criteria, while being urpaed in one or more. Thi dicuion lead u to the definition of efficient olution Definition of an Efficient Solution In general, if we have p criteria, a olution there doe not exit for at leat one i. If { x), z ( x),..., z ( )} 1( 2 x x X i aid to be efficient if x' X uch that zi ( x') zi ( x) for all i=1, p, and zi ( x') < zi ( x) x X i efficient then it image in the criterion pace z p i aid to be non-dominated. x X i weakly efficient if there doe not exit ome x * X uch that z ( * i x ) < zi ( x) for all i=1,,p. For an inefficient olution, there exit ome olution which i equally a good in all criteria while being better in at leat one. A a firt tep in our MCDM approach, we like to preent the DM a et of efficient olution to be able to communicate the exitent trade-off between the objective function 54

68 4.1.2 Identifying Efficient Solution Uing the PDM Model In our framework, the SP model form the bai on which the deciion maker can experiment making ue of MCDM approache. We experiment with our model exploring poible achievement of the objective and dicu the reult from the model olution to demontrate how overeign deciion maker can employ thee model a tool in making their deciion. We firt would like to preent the deciion maker the degree of trade-off between alternative objective to enable them to explore the outcome of alternative iuance trategie. To thi end, we try to obtain a et of non-dominated olution i.e. a portion of the efficient frontier (E) by utilizing an achievement calarizing program (See Steuer, 1986 pp for a dicuion on achievement calarizing * function). We firt identify an ideal point ( z ) in the criterion pace where each objective attain it repective minimum and then project thi reference point onto the non-dominated urface. We employ a weighted Tchebycheff metric to dicover the projected point on the urface, which i defined by the criterion vector that ha the lowet valued weighted Tchebycheff ditance to the ideal point. Thi projection i obtained by olving the following achievement calarizing program: p Min β + ε z i ( x (4.3).t. i= 1. ) [ z ( x) z ( )] * x β λ i. i=1,,p (4.4) i i x X (4.5) where ε i a very mall poitive contant. The approach i illutrated in Figure 5. The incluion of ε in the objective guarantee that the olution obtained i non-dominated and the CCaR objective i properly computed, i.e. in line with Rockafellar and Uryaev (2000). Iteratively, by changing the value of λ i, i.e. the weight aigned to the Tchebycheff ditance with repect to the three criteria and olving the above program, we end up with a et of different point on the efficient urface. 55

69 z 2 An Efficient Solution z * Tchebycheff contour z 1 Figure 5 Illutration of the Tchebycheff Program Viual Interactive Approach of Korhonen and Laako It i practically not poible to identify all alternative efficient olution in our continuou objective pace. Therefore, getting deciion maker involvement through the deciion upport proce will help ae their preference and explore ditinct alternative olution. For example, in the viual interactive approach of Korhonen and Laako (1986), the DM can interact with the model olution proce by pecifying reference direction, d=( d 1,...,d p ) that indicate the objective they would like to improve baed on a given olution, h=( h 1,...,hp ). The DM then elect a preferred olution from a et of efficient olution obtained along direction d. Thi provide an opportunity to explore part of the non-dominated olution et according to deciion maker choice and contitute a learning environment for the DM. The method i baed on the olution of the following achievement calarizing program: 56

70 p Min β + ε z i ( x) (4.6).t. i i= 1 [ z ( x) h. d ] β λ. θ i=1,...,p (4.7) i i i x X (4.8) where ε i a very mall poitive contant and θ i the tep ize along direction d. The DM are aited with a graphical diplay where the change in the objective function value are depicted baed on different d and θ value a illutrated in Figure 6. Korhonen and Laako olve the achievement calarizing program for θ going from 0 to. The kink of the objective function value trajectorie occur at θ value which correpond to bae change in the olution of the linear program. The employment of thee MCDM approache on the SP model i illutrated in a real life application for the cae of the Turkih Treaury. z i θ Figure 6 Illutration of the Viual Interactive Approach of Korhonen and Laako (1986). 57

71 4.2 Accounting for Stochaticity in the SP Model Our tochatic programming model relie on a cenario tree generated to reflect the uncertainty contained in real life uch a the evolution of interet and exchange rate. In mot real life cae, we do not know the underlying ditribution of the relevant tochatic variable, and we need to approximate thoe by a cenario generator that mimic uncertainty in real life. Even if the actual procee are exactly known, the tochaticity contained in real life induce ome randomne in the cenario generation mechanim. Thi then lead to ome variation in the cenario tree intance and in turn, in the optimal objective function and deciion variable value. Thi variation can be meaured by the tability metric of Kaut and Wallace (2003) a dicued in Chapter 3. Method like conditional or elective ampling, cenario reduction or cenario bundling help obtain more repreentative cenario tree with a limited number of branche o that the model tability i maintained. However, in a cenario tree generation mechanim that involve ome degree of randomne, it i not poible to remove the variation in the model output entirely without increaing the number of branche to infinity. In practical application, the cenario generation mechanim i deemed to be of ufficient quality if the variation in the model output i contained within a certain range (ee Di Domenica et al, 2003 for an example). The dark-colored point in Figure 7 depict a et of efficient olution in a bicriteria example we obtain from a ingle cenario tree for the cae of Turkey (detail of thi example are given in Chapter 5). The light-colored point repreent how an efficient olution may change when the ame problem i olved uing different cenario intance from the ame generator. 58

72 120 market rik expected cot Figure 7 Variability of Efficient Solution-Billion YTL (A Bi-criteria Example). We incorporate the effect of tochaticity in the cenario generation mechanim into the deciion making proce. Our aim to make ue of information obtained from olution baed on independent and identically ditributed cenario tree and tool from multivariate tatitical analyi to provide more guidance to deciion maker. We preent model reult within a certain confidence interval, i.e. contruct confidence region around identified efficient olution Multivariate Statitical Analyi We employ multivariate tatitical analyi technique to make analye about non-dominated olution. The dicuion in thi ection i not pecific to our problem, but generally applicable. Since we do not know the actual multivariate ditribution we are concerned with, we have to work with a large ample ize, and apply large ample method to make inference. It i known that large ample inference about the mean are baed on the 2 χ ditribution. Let Y j (j=1,...,q) denote p-dimenional vector independently ' 1 ampled from the ame ditribution. Then q ( Y µ ) S ( Y µ ) ha an approximate 59

73 2 χ ditribution with p degree of freedom, where vector µ and Y are the population and ample mean repectively, and S i the ample variance/covariance matrix (for neceary theory, epecially on contruction of confidence region, ee Johnon and Wichern, 2002, pp ). Y = 1 q q Y j j= 1 (4.9) S q 1 = ( Y q 1 j= 1 j Y )( Y j Y )' (4.10) Thu, provided that n i large, ' 1 2 P[ q( Y ) S ( Y µ ) χ ( α) ] = 1-α (4.11) µ p Then, a 100(1- α ) % confidence region for the mean of p-dimenional Y j i given by the ellipoid defined in the above equality. The relative weight and direction of the axe of the confidence ellipoid are determined by the eigenvalue and eigenvector of the variance-covariance matrix, S of Y i. With Y at the center, the axe of the ellipoid are given by: where 2 i χ p α ± e Λ ( ), i=1,...,p, (4.12) Se i i = Λ iei,i.e. i e and Λ i are the eigenvector and eigenvalue of S. We can alo build imultaneou confidence interval for the individual component mean, µ, i.e. the mean of our objective function value. One method to i thi end i to project the ellipoid on the axe where we depict the objective function value. We then end up with the following 100(1- α ) % imultaneou confidence tatement, where y ii ± χ p contain µ i, i=1,...,p (4.13) q i 2 ii i the element in the i th row and i th column of S, i.e. the variance for component i. We will demontrate thee further in conjunction with the PDM problem of Turkey. 60

74 4.2.2 Contructing Confidence Region around Efficient Solution The overall aim of the tochatic PDM model i to guide deciion maker in making bond iuance deciion and to help them undertand the trade-off inherent in their deciion with regard to their objective criteria, z i, i=1,...,p. With thi aim, one of the tool we employ i to preent the DM with a et of non-dominated olution that we obtain through a pecial cae of an achievement calarizing program, namely a Tchebycheff program, of the form given in ection In our tochatic etting, we exploit thi approach a follow: We create a number of (q) independent cenario tree and employ our achievement calarizing program on each of them to project their ideal point, in term of the optimal value of each objective function, onto their repective efficient urface. Figure 8 depict thi proce in a bi-criteria cae where j* z and Y j, j=1,,q denote ideal point and their projection onto the efficient frontier repectively. We firt chooe a direction, in term of λ i and project all the ideal point in thi direction. We repeat thi everal time changing the direction, thu ending up on different region of the et of efficient frontier. z 2 Y 1 Y 2 Y 3 z 1* z 2* z 3* z 1 Figure 8 Projection onto the Efficient Surface. 61

75 A a reult, we obtain et of independent and identically ditributed (a we are uing independent intance from the ame tree generator) multivariate obervation (array of objective function value) in different part of the efficient frontier to be. We will decribe thee et of point at different region of the efficient urface a efficient cluter. The light-coloured group of point in Figure 7 i an example for an efficient cluter. We can now employ multivariate tatitical tool to build our confidence region for our multi-dimenional objective function to repreent joint ditribution of it component baed on varying DM preference. Thi framework i alo illutrated in Chapter A Stochatic Interactive Approach For our problem, we apply a modified verion of the viual interactive approach of Korhonen and Laako (1986) to account for uncertainty contained in the cenario generation proce. In thi approach we do not bae our reult on one cenario tree intance, but intead ue the information from olution on different cenario tree. The procedure flow a follow: Uing the achievement calarizing program given in ection 4.1.2, auming a certain et of λ i, and working on q cenario tree intance, we identify a tarting olution for each cenario, i.e. t h 0,i, t=1,...,q, i=1, p.. Here, q hould be ufficiently high to be able to make large ample inference. We then ak the DM to pecify a deired direction. Given thee, we olve the above program on every cenario tree intance tarting from aociated h t i for everal pecific θ value and preent the trajectorie to the DM. At thi point, the DM decide whether to top, to continue in the ame direction or to change direction. We repeat thi proce by replacing h t i with olution of the current iteration when the DM change direction. The method i illutrated in Figure 9. The dot correpond to independent obervation for the objective function optimal value while the dahed arrow repreent the direction we would like to move. 62

76 At each tep, for each of the q cenario, we have an array of obervation for the value of the objective function. That i, we have q independent obervation for our p-dimenional array. Now, by mean of multivariate tatitical analyi, we can make imultaneou confidence tatement for the mean of the array component, i.e the objective function value. To thi end, we ue the variance/covariance information obtained from different cenario tree olution. We then preent the average and confidence limit of objective function value in a graphical diplay to be examined by DM. Thi approach i illutrated in Figure 10 where confidence band are depicted with dahed line. Figure 9 Illutration of the Viual Interactive Approach: Obtaining Scenario-Baed Solution a θ i incremented. Thi viual repreentation will not only enable the DM to evaluate the inherent trade-off among deciion criteria, but alo help them have an idea about the degree of uncertainty aociated with the objective function value. The entire deciion making proce, a modified verion of the Korhonen and Laako (1986) method i ummarized in Figure 11. The approach i applied for a real life example in the following Chapter. 63

77 z i θ Figure 10 Illutration of the Viual Interactive Approach: Objective Function Value and Confidence Band. 4.3 The Deciion Aid Framework In thi ection, we have dicued the applicability of ome MCDM concept and tool to the public debt management trategy formulation problem. In thi context, the tochatic programming model we have developed contitute the main intrument on which other method are incorporated. We propoe two main option for the employment of thi determinitic equivalent model to contruct a deciion aid framework. Firt alternative i to ue the SP model on a ingle cenario tree intance. Thi ingle cenario tree, which depict alternative tate of nature for relevant tochatic variable, can be generated by the method available in literature, uch a momentmatching or tatitical/econometrical modelling. Thee would both require quantitative analyi regarding hitorical time erie of model input at different level of detail. 64

78 Select a tarting olution ( t h,i 0 DM for a reference direction ( d ). Set j=0. 0 ) t=1,...,q, i=1,...,p and ak the z,θ, in For everal dicrete θ value, obtain a olution t j i j direction ( d ) for each cenario, preent the value for each objective function within confidence band on a graphical diplay. I the DM atified with any of the olution? Ye No Current direction No STOP Doe the DM like to explore a new direction? Doe the DM like to explore a new direction or continue with different θ value in the current direction? Ye New direction t Set j=j+1, Set h j, i = z t Ak for a new direction j j,θ, i d Figure 11 The Deciion Making Proce Uing the Stochatic Interactive Algorithm. 65

79 Survey done among market participant, i.e. banker or repreentative of other financial intitution to deduct market expectation can alo be ued in contructing a cenario tree. Financial data ditribution companie or governmental organization frequently conduct uch urvey and announce detailed reult about the general mean or median level and the range of expectation for future rate. Analyt can ue thi information to develop a cenario tree baed on market expectation. In thi regard, the moment-matching method of Hoyland and Wallace (2001) can alo be employed to create a cenario tree with branche that match the value pecified in thee urvey. Once a cenario tree i contructed, we can employ the achievement calarizing program to identify portion of the efficient frontier and ak the deciion maker to convey their view on alternative olution. If the deciion maker need more guidance to compare alternative olution or to move around olution on the efficient frontier, the viual interactive approach can be implemented. In thi method, the analyt will have the chance to get deciion maker involvement and to ae their preference while exploring ditinct alternative olution. When the deciion maker are atified with a ingle olution, i.e. they are content with the trade-off tructure they have obtained, the only thing that remain to be done i to identify the correponding iuance trategy. Thi will then be implemented a part of the government financing program. A econd alternative for the employment of the SP model i to ue it in a tochatic etting. Thi would epecially be ueful if the cenario generation i tochatic. In that cae, we propoe to employ multivariate tatitical analyi technique to contruct confidence region around the efficient olution identified to incorporate the inherent tochaticity. The tochatic interactive algorithm alo relie on imulation and tatitical analyi technique to guide deciion maker viually by preenting confidence band around everal objective function value. The deciion maker will then be directed to a ingle preferred efficient cluter. One main problem in uing the tochatic method i about the reflection of the elected objective function value, i.e. the preferred efficient cluter in the 66

80 deciion pace. In mot cae, different point within the elected efficient cluter will be reached by uing varying iuance trategie, depending on the tructure of the cenario tree intance that lead to thi olution. We then need to identify a ingle bond iuance program which would reflect the preference tructure of the deciion maker in the objective pace under different cenario tree intance. To thi end, we propoe to analyze the iuance deciion correponding to the elected efficient cluter. In thi regard, identifying common pattern in the olution within the ame cluter will be helpful to decide on a ingle trategy. Thi olution ha to be robut enough to yield imilar objective function value under different cenario tree intance. Thi argument i illutrated in the following Chapter. We believe the developed imulation framework can alo erve a a deciion aid tool. Each replication of the imulation mimic the actual deciion making proce, and by carrying out everal replication we can provide the expected cot and rik level aociated with a certain deciion making tyle in rolling-horizon etting. The rolling horizon etting we adapted replicate the real life deciion making proce. In Turkey, for example, debt trategy deciion made by the Debt Management Committee 2 are revied every year depending on the realization of the previou year and on the current future outlook. The committee analyze the exiting debt poition, the current tate of the economic parameter and future expectation when reviing the debt trategy. Thi trategy i implemented for a certain period, generally a year, and then i ubject to reviion if needed. Thi proce can be generalized for many countrie even though the reviion period or other dynamic of the deciion making procedure may differ.. In the experiment in Chapter 5, the deciion making tyle i baed on an optimization approach which aume that at each deciion tage the DM aim to minimize a certain utility function in an SP model looking ahead over a et of year. The imulation reult depict the poible performance of the optimization approach in a dynamic etting. One can alo imulate the performance of other deciion 2 The final deciion making authority within the Treaury on trategic deciion regarding debt management. 67

81 making tyle that can include baing trategy deciion on certain deciion rule (or rule of thumb), uing tatic deciion or even making random choice at each tage. Simulation reult can then provide ome inight for the DM while adopting a certain deciion making approach that i to be employed over and over in the dynamic environment of real life. 68

82 CHAPTER 5 5 APPLICATION OF THE MCDM APPROACHES: AN EXAMPLE FROM TURKISH CASE We now illutrate the idea dicued in the previou ection for the cae of overeign debt management in Turkey. In thi implementation, we employ the generic tochatic programming model for PDM preented in Chapter 4, adopting the relevant cot and rik definition for Turkey. The application alo include a cenario generation framework to contruct cenario tree for the evolution of macroeconomic variable that affect government financing. The tochatic programming model baed on the determinitic equivalent cenario tree mechanim i ued a a platform for Multi-Criteria Deciion Making analyi. 5.1 Background for Public Debt Management in Turkey In Turkey, cah and debt management on behalf of the government i carried out by the Underecretariat of Treaury. The Treaury iue debt ecuritie to meet the financing requirement of the o-called Central Government Budget which cover the budget of central government organization uch a the minitrie and other government intitution like the State Highway Directorate, State Waterwork Adminitration etc. The financing requirement that arie from the budget deficit and re-payment of exiting debt i mainly met by iuing Treaury Bill and Government Bond in dometic financial market and Eurobond to international invetor. There are alo other form of financing available from International Intitution uch a the IMF and from foreign governmental intitution in addition to project-baed credit that are pecifically allocated to a public invetment project 69

83 rather than the current financing need of the government cah poition. Thi application will only focu on deciion making for finance raied from financial market in term of ecurity iuance. While analyzing the accumulation of public debt in Turkey, we can identify two main reaon. One i the chronic budget deficit that became a regular phenomenon in the lat decade (Figure 12) and paved the way for regular dometic bond auction tarting from A the Treaury had to finance maturing debt a well a the deficit of the current period, thi led to increaing borrowing requirement, which in turn caued higher interet rate and higher amount of debt maturing in the upcoming period, creating ome ort of a viciou circle or namely, a debt overhang. The typical phenomenon of the late 1990 wa the iuance of hortterm dometic debt at high real interet rate a a reult of eroding public confidence. 20 Budget Deficit in % of GNP year Figure 12 Budget Deficit in Turkey (in percent of GNP) Source: SPO 3. Another reaon that added to the government financing deficit wa the iuance of debt ecuritie for bailing-out the loe incurred by public and private bank epecially after the 2001 financial crii which led to a harp leap in the debt tock (Figure 13). 3 State Planning Organization. 70

84 A of Augut 2007, the total debt tock (in term of principal value) the Treaury i reponible for managing tand at a level of 336 billion TRY (New Turkih Lira) and about 90% of thi amount conit of debt raied through bond/bill iuance, with the remaining in the form of loan. Figure 14 depict compoition of the Treaury debt tock in term of the currency and interet type tructure. 120% Treaury Debt Stock in %of GNP 100% 80% 60% 40% 20% 0% year Dometic Debt Foreign Debt Figure 13 Central Government (Treaury) Debt Stock (in percent of GNP) Source: Turkih Treaury 4. Depite the improvement in recent year, the level and tructure of the debt tock till induce a izeable vulnerability againt advere market movement Thu, when formulating the debt management trategy, Turkey hould take into account the rik objective in addition to cot. Thi fact i alo reflected in the legal framework for debt management and the Regulation on the Principle and Procedure of Coordination and Execution of Debt and Rik Management define the core objective of public debt management a follow (Article 4). 4 Source: 71

85 The execution of public debt and rik management hall be baed on the following principle: a) To follow a utainable, tranparent and accountable debt management policy that conform to monetary and fical policie, conidering macroeconomic balance, b) To meet financing requirement at the lowet poible cot in the medium and long term, taking into account the rik, in addition to dometic and international market condition (%) Aug FX Denom./Ind. TRY Floating Rate TRY Fixed Rate Figure 14 The Structure of the Treaury Debt Stock. Source: Turkih Treaury. 5.2 The Stochatic Programming Model for Turkey In the MCDM analyi for the debt trategy formulation problem of the Turkih Treaury, we employ the generic PDM model preented in Chapter 3, only modifying objective function formulation. To adopt the relevant cot definition, we evaluate interet cot in accrual term o that the interet payment can be attributed to the period they are generated. However, for Turkey, the interet charge i not the ole ource of cot for the government. There are bond iued in currencie other than the numeraire currency, which i the legal tender for the country and any increae in debt 72

86 repayment, including principal and interet component, due to change in the exchange rate, add up to the cot of debt. Thu, our cot definition cover not only the interet charge to accrue during the planning period but alo the change in the market value of foreign currency denominated debt (ee Turkih Treaury, 2004, p.59). Foreign currency linked debt that mature beyond the deciion horizon are marked to market value at the end of the model period. A a reult, we include the following accrual cot meaure in our model: Total accrued cot at T in cenario : T 1 A = j J τ = T m + 1 j X e n(, τ ) T, j τ, j ( eτ, j T 1-1)+ j J1τ = T m j + 1 X n(, τ ) τ, j e T, j e τ, j T.( τ m j r, j τ )+ T 1 j J 2τ = T m j + 1 X n(, τ ) τ, j ( 1 Yτ, T, j ) e T, j r T c j +1, j. eτ, j. 1 + c j T 1 j J 3τ = T m j + 1 X n(, τ ) τ, j ( 1 Yτ, T, j ) e T, j r τ, j. eτ, j. 1 c j t : t m j > 0, (5.1) (where c j i 2, ince the regular coupon period for Treaury bond i 6 month) The accrued cot i calculated for bond and bill that have not yet matured at time T. It conit of the change in the value of the bond due to movement in the exchange rate and the interet that ha accumulated on a bond ince it iuance (for zero-coupon bond) or it lat coupon payment (for coupon bond) up to time T. We compute the accrued interet rate by multiplying the effective interet rate by the fraction of day that have paed ince the tart of the coupon payment to the total number of day in the entire coupon period. We then adopt the following cot definition, which i the um of actual interet payment made in cah and interet cot accrued. T TC = I t t= 1 + A (5.2) A the meaure of market rik, we apply the Conditional Cot-at-Rik (CCaR) meaure, by which we quantify the expected value of the cot of public borrowing 73

87 given a certain threhold level i exceeded. In thi application we calculate the CCaR value on the 10% tail ditribution. For the liquidity rik objective, we again adopt a Conditional at-rik meaure. We firt identify the maximum cah outflow of the government for each cenario ( cof cof = ) where max ( D + I + t t t L t ) (5.3) We calculate the average of the highet 10% of the cof over the entire cenario et to obtain our Conditional Payment-at-Rik (CPaR) meaure where α =10%. CPaR Min PR + 1 α S.t. cp ( p. cp ) (5.4) cof PR (5.5) cp 0 (5.6) Here, PR i an auxiliary variable ued in the definition of CPaR that equal to 100(1-α )% percentile value of cof at the optimal olution, and cp i the exce payment beyond PR for cenario. Thu our problem turn to be: (P) Min z= {, z z } z where 1 2, 3 S z 1= = 1 p TC (5.7) z 2 = VR + 1 α S ( p cv ) (5.8) z 3 = PR + 1 α S ( p cp ) (5.9) ubject to (5.1), (5.2), (5.3), (5.5), (5.6), (3.24), (3.25), (3.26), (3.27), (3.28), (3.29), (3.30), (3.34), (3.35). The model cover a period of 3 year, in line with the central government medium term fical plan. We aume that the government prepare an annual borrowing program, thu we have a 3-period model. Each year i then divided into 4 74

88 quarter and at the beginning of each year the government decide on the iuance trategy for the following 4 quarter. Thi deciion i then to be revied at the beginning of the next year. The application i baed on data available a of December A a implification, we aume that the exiting forward liabilitie forecat a of that date are cenario-independent ( L = L t t ) a well a the non-debt cah flow of the government ( PS = PS t t ). The parameter L t are baed on debt tock data available at the Turkih Treaury web-ite ( and PSt value are in line with the government fical aumption in the Medium Term Fical Plan ( ) detail of which can be found at the Official Gazette (no ) dated (in Turkih). The numeraire currency of the model i New Turkih Lira (TRY), i.e. all cot and rik are meaured in that currency. Our model preent a election of even different kind of bond: four of which are TRY denominated zero-coupon bond ranging from 3 month to 18 month in maturity. We alo include in the model a 3 year TRY fixed rate coupon bond, and a 3 year TRY variable rate coupon bond indexed to the 6 month Treaury Bill yield, a well a a 3 year USD denominated fixed rate coupon bond (ee Table 5). All coupon are emi-annually redeemed. The variable rate bond i aumed to be iued with a fixed pread over the prevailing 6- month interet rate. We include two marketability contraint: We aume that in one quarter the amount of 3 month bill the Treaury can iue i capped at 10 billion TRY and the market availability for the USD denominated bond i 3 billion USD per quarter, conidering the ize and preference of the lender in thoe egment of the debt market. 5.3 Scenario Tree Generation Our initial approach to cenario tree generation in the context of our debt management problem wa to ue a tatitical model that decribe the relationhip among the random variable of concern, calibrated by hitorical data and then to ample randomly from the error term ditribution of our model. To be able to 75

89 maintain computational complexity, we need to work with parely branched multiperiod cenario tree and the pure random ampling method we have adopted, led to untable cenario tree when we kept the number of branche at a limited level. We tried to overcome thi tability problem by increaing the dimenion, i.e. the number of branche of the tree a much a we can, acrificing from olution time. We then improved our approach by incorporating idea from reearch on cenario reduction/bundling. We again ue our tatitical model to create random cenario, however we increae our ample ize ubtantially, ince we then try to identify imilar outcome and bundle them into dicrete cenario cluter. Thi then allow u to reduce the number of branche in our cenario tree, and adjut the probabilitie of remaining branche accordingly. Thi idea hould be credited to the work of Grill and Ötberg (2003) The Statitical Model The cenario for the SP model were generated by a modified verion of one of the macroeconomic imulation model of the Turkih Treaury, which i a vector autoregreive time erie model containing the hort and medium term local interet rate, the USD/TRY parity, the inflation rate (Conumer Price Index) and the Treaury funding rate in USD denominated iue 5. The vector autoregreive (VAR) approach model the co-movement of elected variable a function of their own lagged value a well a the lagged value of the other variable. The following equation illutrate a VAR model for a vector Yt of n variable, including l lagged value: Y t = C + l i= 1 AiY t i + e (5.10) t 5 The parameter of the model can not be dicloed due to the confidentiality reaon. However, etimation of the model parameter i a traightforward proce which can be carried out by uing commercially available econometric package. 76

90 where C i an nx1 vector of contant, A i (i=1,,l) are nxn matrice of coefficient and ε t i the vector of error term with the following propertie: E( e t ) = 0 for all t (5.11) Ω for = t ' E( e. e )= t (5.12) 0 otherwie where Ω i the variance/covariance matrix aumed to be poitive definite. The parameter of the time erie model are etimated baed on a monthly data et from 2001 to Since Turkey had been implementing a pegged currency regime in year 2000 and moved to a floating regime afterward, we tart our dataet from The three month and twelve month interet rate reflect the rate that emerged in Treaury auction for ecuritie in thoe maturitie. The inflation rate i the monthly rate of change in the 1994 baed Conumer Price Index announced by the Turkih Statitical Intitute. The USD/TRY exchange rate i the monthly average calculated over daily figure announced by the Central Bank of Turkey. We take the mid point of the official purchae and ell rate of the bank. At the end of 2005, the annualized 3 month and 12 month interet rate tood at a level of 14.2% and 14.1% repectively, while the average annual interet rate for one-year USD denominated bond wa about 4.8%. The monthly average value of the USD in December 2005 wa1.35 TRY. The annual inflation rate for 2005 wa recorded a 7.7% We create random cenario for our tochatic variable by making ue of the VAR model via impoing correlated random hock through the error term making ue of the Choleky decompoition of the variance/covariance matrix. The random hock are achieved by drawing five random variable from the tandard normal ditribution. To be able to create cenario conitent with the empirical co-movement of our macroeconomic variable, one need to obtain correlated random hock. To thi end, we make ue of the Choleky decompoition of the covariance vector Ω. Thu, we firt find a matrix F uch that F'.F = Ω (5.13) We then tranform the 5x1 vector that contain the tandard normal random variable by multiplying it by F and impoe the reulting vector to the VAR model a 77

91 a random hock. The monthly path created by the model are then converted to quarterly figure by taking average over three-month period. Once we obtain imulated value for the hort and one year interet rate, we compute the yield for maturitie in-between by linear interpolation. For maturitie longer than a year, a flat yield curve i aumed. That i, interet rate are not allowed to vary with maturity, but taken contant for maturitie over a year Scenario Clutering In order to increae the approximation capacity or our cenario tree, we need to increae to number of ample cenario obtained through the VAR model. However, there i alo a need to reduce the number of branche on our cenario tree due to computational reaon. To thi end we firt create a high number of cenario, then bundle imilar one together uing the clutering approach, which ha gained wide popularity in the field of data mining, a reearch area that trie to extract information from huge level of data and to identify any meaning or pattern that are not evident at firt ight. In it implet definition, clutering i to group imilar item together. A more formal, mathematically elegant definition Graepel (1998) make i a follow: Let X mxn R be a et of data item repreenting a et of m point x i in The goal i to partition X into K group n R. Ck uch that data that belong to the ame group are more alike than data in different group. Each of the K group i called a cluter. The reult of the algorithm i an injective mapping X to cluter C k. C of data item x i The meaure of alikene or imilarity i baed on a certain ditance metric depending on the characteritic of data and/or modeler objective. The clutering problem can be formulated in everal way. Data item can be clutered into dijoint or overlapping clae. The cluter can be determinitic o that each item i attached to a certain cluter or probabilitic, where each intance ha an aigned probability to be a member of a certain group. We can alo form 78

92 hierarchical cluter uch that low level cluter merge to form larger cluter at higher level of the hierarchy. In our cae, i.e. macroeconomic cenario clutering for the tochatic debt management model, we are concerned with obtaining a dijoint et of cenario. To thi end, we would like to cluter a large number of cenario that are obtained through Monte Carlo imulation into clae that will form our cenario et for the SP model. We are not intereted in finding a hierarchy with regard to our obtained cluter, ince the branche at the cenario tree are all at the ame level. Since at the end of VAR imulation we only have the data point and do not have any a priori knowledge about the actual cluter thee data point belong to, we are faced with an unupervied learning/clutering problem (In upervied learning, there i a et of training data whoe actual cluter are already known, and the objective i to undertand the relation between the characteritic of data and being a member of a certain cluter, and then to predict a cluter when a new data item arrive. ) There are everal clutering algorithm available. Among thoe, we adopt the K-mean algorithm (MacQueen, 1967), which i one of the earliet, probably the implet and the mot widely ued method in practical application. The aim in K- mean i to cluter data point in K mutually excluive clae, where the number of clae K i pre-upplied. Each cluter i identified by it centroid. That i the point to which the um of ditance from all data point in the cluter i minimum. Given a et of initial cluter, the clutering algorithm move item between cluter until the um of ditance within cluter can not be decreaed below a certain level. The algorithm for K-mean clutering i a follow: K-mean Algorithm: (a) Find a et of initial cluter centroid, j C 1,..,. C at iteration j=0. j k (b) j=j+1. For each x i, i=1,...,m, find the nearet cluter with repect to the choen ditance metric and aign x i to that cluter (c) For all k=1,...,k, re-compute cluter centroid, j C 1,..., C a the mean of all j k point aigned to that cluter. 79

93 (d) If the number of point re-aigned to a different cluter i le than a certain level or the um of ditance could not ignificantly be decreaed further, top, ele go to tep (b) The K-mean algorithm ha everal drawback: There i no clear guidance to find out the real number of cluter, i.e. K, and the reult can be enitive to the initial et of cluter. To chooe the right K, we have tried the algorithm with different K value and tried to ee the effect on obtained cluter and on our SP model reult. We did not code the algorithm from cratch, but ued the built-in K-mean function in MATLAB 6.5 Statitical Toolbox. Thi function offer ome choice on the election of initial cluter which can be completely random, uniformly elected over the data range or upplied by the uer. We have elected the fourth option which perform a preliminary clutering analyi over a 10% ub-ample of the data et to find out the cluter to begin with. Our econometric model contain five macroeconomic variable uch a the interet and exchange rate which take value on different cale. Thu, to cluter our five-dimenional data point, we re-caled the point on a 0-1 cale. For example, we aigned the data point (cenario) which ha the highet exchange rate a value of 1 on that dimenion and caled the other point accordingly. We did thi in every dimenion o that every cenario turned into a vector of item on a 0-1 cale. We have then adopted the Euclidian ditance metric a a meaure of ditance (imilarity) The following ection explain the entire cenario generation-clutering proce that we have adopted to create cenario tree for our multi-tage SP model Scenario Tree Generation Algorithm Our debt management model i baed on a multitage cenario tree of the form given in Figure 2. There are everal deciion tage (which may correpond to year) each divided into everal ub-period (which may correpond to quarter) in which the tochatic variable are et to evolve. That i, cenario unfold in each ubperiod, but deciion are only made or revied at certain deciion tage. Then, the 80

94 cenario between deciion tage combine to form a equence of joint realization for a certain period. Thee equence of cenario are linked at the deciion node and we have cenario path covering the entire planning horizon. In thi etting, our cenario generation algorithm proceed a follow: We firt create a populated et of cenario path emanating from the initial (or current) deciion node extending up to the beginning of next tage (or year) uing the vector autoregreive model by mean of random ampling through Monte Carlo imulation. We then cluter thee into K ditinct clae to form K deciion node at the next tage uing the method dicued in the previou ection. Here, we only take into account the lat element of cenario path (e.g. outcome in ub-period 4 for cenario in the firt tage, ub-period 8 for cenario of the econd tage etc.) and run the clutering algorithm on thee end point. We do not ue data point regarding the interior ub-period (ub-period that are in-between deciion node, e.g. ubperiod 1,2,3 for the firt tage) covered by the cenario et and accept the lo of ome information, a thi reduce the dimenion of our clutering problem. Once we have obtained the cluter, we unite the cenario path in the ame cluter by finding the centroid in each dimenion for all ub-period covered. We keep track of the ize of each cluter, i.e. the number of element in each cluter and aign each cluter a probability baed on it relative ize. For example, if we have two cluter: one with 4 and the other with 6 element, then the cenario path formed from cluter one i aigned with a probability of 40% while the path from cluter two i given a probability of 60%. We repeat thi procedure for all conequent tage. The following algorithm ummarize thi proce in an orderly manner: Scenario Tree Generation: Let i=1,...,n denote the period in our model, n=0,...,n-1 the deciion tage, T the total time horizon and T i the et of ubperiod contained in period i. Then, (a) n=0, i=1 n (b) For j=1 to K (For each node at the current deciion tage), repeat i. Obtain a et of M multivariate cenario path, covering all ubperiod in T i, by mean of random ampling through the VAR model 81

95 ii. iii. Take the lat obervation in each cenario path, convert thi M multivariate data point into a 0-1 cale over each dimenion, and cluter them into K clae, cenario path are allocated to cluter by their end (detination) point. Count the element in the cluter and join the path in the ame cluter by finding the centroid for each ub-period int i iv. Aign a probability to each newly formed k=1,...,k cenario path, by taking the ratio of the number of element in cluter k to M. (c) n=n+1, i=i+1. If n=n, (if all tage are covered), top. Ele go to tep (b) Figure 15 illutrate thi algorithm on a imple two-tage, two-cluter cae. C C Figure 15 Generating a Scenario Tree. The branch probabilitie are accumulated to get the overall path probabilitie from time 0 to T. Thee cenario probabilitie are then fed into our SP model a well a other cenario characteritic and form the bai for our expected cot and rik calculation. Figure 16 illutrate the development of a ample cenario tree for interet rate in a ingle tage problem. The probabilitie aociated with cenario path, obtained through the number of element in each cluter, are given to the right of the right pane. Note that the clutering algorithm i run on five dimenional data, while the figure only depict one (the interet rate) dimenion. 82

96 A drawback of the clutering approach i the lo of information regarding the wort-cae cenario. Thi i a eriou problem if we adopt the wort-cae cot or payment a an objective criterion to be minimized. There are problem with wortcae cenario baed meaure, even without adopting a clutering approach, ince the obtained reult i baed on a ingle obervation obtained through a tochatic etting. The following ection comment on the reult from our SP model when run on cenario tree of different ize, with variou choice for the clutering parameter K. Original Scenario Path Scenario Tree with Aociated Probabilitie after Clutering Probabilitie 4.4% Interet Rate (%) % 27.5% 35.3% 17.2% Sub-period Figure 16 Generating a Scenario Tree - An Example with 5 Cluter Aement of the Scenario Generation Algorithm We ae the quality of our cenario tree generation approach in the framework of the dicuion in To tet the in-ample tability of our method, we olved our SP model baed on different intance obtained from the ame cenario tree generator, trying to optimize our objective function eparately. We have generated 50 independent and identically ditributed cenario tree and olved our model 50 time uing thee a input to ee how the optimal value of each objective function varie due to the tochaticity included in our modelling framework. 83

97 In thi etting, we have only accounted for the expected cot and market rik meaure. We did not include our third objective. Since the SP model include bond with maturitie a long a the deciion horizon, the optimal olution for the liquidity rik minimizing model i independent of the cenario tree intance a the model chooe long maturitie regardle of the tree. The model i implemented on GAMS 2.0 uing CPLEX a the linear programming olver. A ample code developed for the experiment i included in Appendix-B. Model parameter depend on debt tock realization a of end-2005, and financing projection are baed on the medium term fical plan for year We firt preent the reult obtained without employing any clutering algorithm, i.e. by pure random ampling from the error term of the VAR model. Table 2 depict the average and tandard deviation of optimal objective function value when model are olved on tree of different ize. The reult are all baed on a three-tage model, only the number of branche differ. The notation of 10x10x10 correpond to a three-tage tree with 10 branche dieminating from each node in each tage. Thu, in the final tage there are 1,000 branche. The reult in Table 2 provide evidence about two iue regarding pure random generation: Firt, a low-branch tree tend to underetimate the objective, and econd, we need to increae the dimenionality ubtantially to achieve a ignificant reduction in the variation of the optimal objective function value. Table 2 Stability Reult without Clutering (Baed on 50 Independent Replication). Scenario Tree Objective: Min Cot Objective: Min CCaR Billion TRY Average St. Dev. Average St. Dev. 10x10x x10x x10x

98 We can now compare thee reult to thoe from our cenario generation algorithm baed on clutering of randomly generated data. We firt run our model by generating 100 random path from each node and clutering thee into 5, 8 and 10 cluter conecutively (i.e. M=100 in the algorithm preented in Section 5.3.3). The reult in Table 3 how that average and tandard deviation of optimal function value obtained from tree of different ize are quite cloe to each other, while the variance are le than thoe of higher dimenional tree in Table 2. Table 4 depict the effect of increaing the number of cenario path (M), i.e. the data point to be clutered. Working with a higher number of path before clutering provide ignificant improvement in in-ample tability. A regard to meauring the bia in our model, the olution in Table 4 are comparable to that of the larget (8,000 branch) tree given in Table 2. Table 3 Stability Reult after Clutering over 100 data point (Baed on 50 Independent Replication). Scenario Tree Objective: Min Cot Objective: Min CCaR Billion TRY Average St. Dev. Average St. Dev. 5x5x x8x x10x Table 4 Stability Reult after Clutering over 1000 data point (Baed on 50 Independent Replication). Scenario Tree Objective: Min Cot Objective: Min CCaR Billion TRY Average St. Dev. Average St. Dev. 5x5x x8x x10x

99 A dicued previouly, the K-mean algorithm doe not provide any guidance to the true number of cluter. To thi end, we have carried out viual and quantitative analyi of the cenario generated, however ince we are working with randomly generated data, it i hard to find a ingle K value that would be correct for all intance generated. Thu, we ugget uing the reult from our tability tet to ae the effect of uing different K value. A depicted in Table 4, the average for the optimal value of the two objective function are quite imilar, when we create our cenario with different K value, while there i a light improvement in variance a we increae the number of cluter. Even though it may be more appropriate to ue 8 or more cluter to be able to obtain more table reult, there i n a cot aociated a the total number of cenario i K. Conidering the fact that our MCDM approache require equential olution of the model (epecially when approximating a portion of the efficient frontier), we preent our illutrative reult in the following ection baed on a 125 branch (5 3 ) tree. Our model are olved on a Pentium 4, 728MB RAM PC. We hould note that it take around 20 minute to olve the 8,000 branch tree given in Table 2. That alo include the time pent on the cenario generation proce, baed on pure random ampling. On the other hand, the total time ued by our clutering-baed algorithm to create a ample tree of 125 branche out of 1,000 path at each node and by GAMS to olve a model of thi ize i around 100 econd. Thee time quoted are for model in which a ingle objective i minimized. The time required for etting up and olving achievement calarizing program (of the form given in the following ection) i around 120 econd for a tree with 125 branche, Pentium 4, 728MB RAM PC capacity i not ufficient for getting olution of calarizing program for 8,000 branch tree of the form given in Table 2. The model ize for a problem baed on a 125-branch tree i about 20,000 row and column before any reduction i done by the CPLEX olver. When the tree ize i increaed to 8,000, the model ize reache to a level around 1.4 million row and column. 86

100 5.4 Experiment on the PDM Model uing MCDM Tool We now preent finding from experiment on our model uing the MCDM tool dicued in the previou ection. Firt et of experiment were done baed on a ingle cenario intance, while the econd part preent application of the tochatic interactive approach on a multi-cenario-tree framework. All experiment were conducted on a tree of 125 branche. That i, there are five branche dieminating from each node. Thee et of 5 branche were created by clutering 1,000 path down to five for each node Experiment on a Single Scenario Tree We firt provide the optimal borrowing trategie generated by our model when each deciion criterion i optimized in iolation. Table 5 include the iuance policy generated for 4 quarter of year 1 with repect to each objective. Since deciion for year 2 and 3 are cenario dependent, we only include the bond to be iued in the firt year. A expected, the model chooe hort-term ecuritie for the firt year when expected cot i to be minimized ince our vector autoregreive model generally generated cenario with declining interet rate in line with the macroeconomic environment in Turkey in the recent year. Short-term rate are alo lower on average. However, a far a the market rik i concerned, all funding i raied through fixed rate bond. The model aim to extend maturitie to minimize liquidity rik, however floating rate bond are alo iued. Figure 17 depict everal point identified on the efficient frontier and a fitted urface to thoe point which portray the degree of trade off between our three criteria. Preenting the efficient frontier may ait the deciion maker in analyzing the trade-off between alternative olution and the ection of the frontier in which they are intereted can be analyzed in more detail. 87

101 Table 5 Optimal Borrowing Strategie (Initial Stage Deciion). Billion TRY Minimize Cot Minimize CCaR Minimize Liquidity Rik Bond Quarter 3 month bill 6 month bill 12 month bill 18 month bill 3 year variabl e rate bond 3 year fixed rate bond 3 year USD bond Figure 18 diplay the contour of the efficient frontier at different value of liquidity rik. The graph reflect the teepening nature of our efficient urface in z3 dimenion. In our illutrative problem, a the contraint on liquidity rik i releaed, we obtain a diminihing improvement in the level of the other two objective. Thu, it may be more preferable for the PDM deciion maker to operate on relatively lower level of re-financing rik ince thi would not require much acrifice in the cot and market rik objective. 88

102 Figure 17 A Repreentation of the Efficient Frontier (billion TRY). z2: ccar z3<=115 z3<=95 z3<=75 z3<= z1: cot Figure 18 A Set of Efficient Solution (billion TRY). We now preent an application of the viual interactive approach of Korhonen and Laako (1986). Figure 19 diplay the effect of altering the tep ize (θ ) in moving from the point where liquidity rik i at it minimum in a direction to reduce cot, i.e. d=(-3.6, 8.8, 94.0), with λ =(1/3,1/3,1/3). 89

103 Cot LiqRik CCaR Cot CCaR, LiqRik theta Figure 19 Criterion Value Trajectorie (billion TRY). Let u aume that the DM like the olution at θ =0.3 ( z 1 =56.5, z 2 =85.1, z 3 =70.6) among the olution in Figure 19. Table 6 contain the correponding iuance trategy. A a compromie olution, we have a mixture of hort term and long term bond to be iued in the upcoming year. Table 6 Optimal Borrowing Strategie (Initial Stage Deciion). Bond Quarter 3 month bill 6 month bill 12 month bill 18 month bill 3 year variable rate bond 3 year fixed rate bond 3 year USD bond Changing the direction (d) and (θ ) interactively with the deciion maker will produce different trajectorie on which the deciion maker can analyze and experiment with their deciion. Let u know uppoe the deciion maker would like to explore olution on the direction of reducing the market rik. Figure 20 plot the trajectorie auming that direction change occur at θ =0.3 in Figure 19 in a way to 90

104 reduce the market rik ( z 2 ), for example with d=(1.0, -5.4, 73.8) (θ again tart from 0, ince we are moving in a new direction). Cot LiqRik Cot CCaR CCaR, LiqRik theta Figure 20 Criterion Value Trajectorie (billion TRY). Figure 21 diplay the above trajectorie on the efficient urface. The olid line i aociated with Figure 19 while the dahed line correpond to Figure 20 (reduction of in the CCaR meaure). The experiment on the model can be extended even further in interaction with the deciion maker. Figure 21 Movement on the Efficient Surface (billion TRY). 91

105 5.4.2 Application of the Stochatic Interactive Method Like other cenario generation method, our cenario generation mechanim for the Turkih model contain a degree of randomne which reflect the underlying tochaticity of the model parameter. Thu, guidance provided one cenario tree intance may be mileading due to the variability of the optimal olution on different tree. We now demonrate an application of the modified interactive method given in ection taking into account thi tochaticity. Figure 22 depict how tochaticity in the cenario generation mechanim effect the model olution. The cluter of point repreent efficient et obtained by the method in ection 4.2.2, baed on elected a et of λ i. One can oberve different degree of variation within each et. That i the variance/covariance tructure between the objective function value varie depending on the region of the efficient urface we are operating at. We can ue thee covariance tructure to build confidence ellipoid for each efficient cluter uing idea from ection where we dicued Multivariate Statitical Analyi tool. Figure 23 contain an example for a bi-criteria cae. Thee confidence ellipoid can be projected on the axe to obtain imultaneou confidence interval for our objective function value. Figure 24 repreent an application of the viual interactive approach depicting imultaneou confidence interval for each objective function. In thi example, the DM tart from a given efficient cluter with centroid ([cot,ccar,liqrik]=[63.0, 91.58, 51.72]) and move in a direction to reduce CCaR iteratively. The figure depict the change in objective function a theta change (In thi example λ i =1/3, i=1,,3). 92

106 Figure 22 The Reult of Randomne in the Scenario Generation Mechanim CCaR cot Figure 23 Confidence Ellipoid around a Set of Efficient Solution (Billion TRY). 93

107 Cot LiqRik Cot, LiqRik CCaR CCaR theta Figure 24 An application of the Viual Interactive Approach: Iteration 1 (Billion TRY). Let u aume that the DM like the olution at θ =0.5 and wihe to explore olution that have a lower cot expectation. We then elect a new direction accordingly and re-run our method tarting from the current olution experimenting with different θ value. The reult are depicted in Figure 25. Note that θ =0.6 correpond to θ =0.1 after we change the direction. Aume that at ome point during thi interactive proce, the DM chooe a certain olution (an efficient cluter) evaluating the preented confidence interval (uch a thoe in Figure 24 and 25). One hould alo conider the reflection of thi election in the deciion pace. When we analyze deciion variable in our cenariobaed olution, we ee that there i a variation in term of elected bond and their iuance amount depending on the cenario intance. Since the DM ha to make a certain iuance deciion at the tart of the deciion horizon, thi olution ha to be robut enough to yield imilar objective function value under different cenario tree. 94

108 Cot CCaR Cot, LiqRik C CCaR LiqRik ,1 0,2 0,3 0,4 0,5 0,1 0,2 0,3 0,4 0,5 theta 86 Figure 25 An Application of the Viual Interactive Approach: Iteration 2 (Billion TRY). In thi cae, we would ugget the DM to implement the olution that ha the hortet ditance to the centroid of the elected efficient cluter, in term of objective function value. Figure 26 depict the performance of the peudo-centroid olution under other random cenario tree intance in a bi-criteria example. Here, circle repreent the actual objective function value in each cenario intance, while tar tand for the reult of peudo-centroid olution when plugged in other cenario tree intance. Teting with everal efficient cluter at different place of the frontier we ee that on average the error caued by the peudo-centroid i negligible. In Figure 26, for example, the ratio of root-mean quare error in the cot dimenion to average cot i le than 0.2%. Thu, in thi example, our iuance trategy i ufficiently robut to yield the deired objective function value. In thi context, once the DM chooe a certain efficient cluter, we recommend to carry out the robutne analyi to reflect about the actual deciion variable. 95

109 CCaR Cot Figure 26 Performance of Peudo-Centroid Solution i other Scenario Tree. In ummary, in our deciion upport procedure, we firt apply the interactive algorithm to preent the DM everal efficient olution within confidence limit and experiment in different region of the efficient urface baed on DM preference. Once they are content with a certain olution, we carry out an analyi on the deciion variable correponding to that olution to demontrate how they perform under different cenario tree intance Experimentation with the Simulation/Optimization Framework We created the integrated teting framework through a MATLAB-GAMS interface. The cenario tree input for the SP model i generated in MATLAB 6.5 a decribed in the previou ection. The model i then olved in GAMS 2.0 and the reulting iuance deciion are exported to MATLAB environment to be included in our debt portfolio. The model in MATLAB then elect a random cenario, create a cenario tree baed on thi aumed realization, can the current debt tock matrix to 96

110 evaluate the cah flow ariing from previouly made deciion and generate a cahflow cenario tree. Thi information i then exported to GAMS for further deciion. We carried out a imple tet and compared the reult from the three-tage model for Turkey to thoe of a myopic (ingle-tage) model in the imulation/optimization etting. The myopic model ha a one year horizon, i.e. make deciion conidering only the firt year cenario path. The cenario tree for the myopic model doe not re-branch after the firt year, thu there are no cenariopecific olution like thoe of the three tage model. We have 125 cenario in the myopic model o that the number cenario in the two model are equalized. In order to be able to compare the olution of the two model, we need to ue the ame et of DM preference. For thi purpoe, we aume an additive utility function that i to be optimized in both model. A a tarting point, we chooe a utility function that combine our three objective with equal weight. In the experiment, the horizon of our imulation etting i three year for both model. At the tart, the model are fed with their appropriate cenario tree generated on the ame et of tarting condition. Once the deciion for the firt year i made, we generate a random cenario realization, implement thi on the aumed cenario path and hift the model twice to obtain deciion for the remaining two year, re-generating the cenario tree baed on the aumed realization. One replication of the imulation end at the end of the third year. The proce i then repeated. We calculate the expected objective function value over imulation realization. The expected cot value i the average of all replication. The Conditional Cot-at-Rik i computed over the maximum 10% of the cot realization in all replication. The liquidity rik meaure i calculated imilarly. Table 7 depict the reult baed on 1,000 imulation replication. The reult from thi imple experiment how that the three-tage model yield better reult with repect to the aumed utility function when compared to the myopic model. In a ene, thi reult i expected ince the three-tage model cover the entire deciion horizon (in thi cae the imulation horizon) when making deciion. However, we hould note that ince the market rik value i calculated 97

111 over the cot realization in imulation replication, it i correlated with the expected cot computation. Then, it i more appropriate to conider the expected cot and liquidity rik objective when comparing alternative imulation etting. Table 7 Reult of Dynamic Game. Experiment 1: An Additive Utility Function. 3-Stage Model 1-Stage Model Expected Cot Market Rik (CCaR) Liquidity Rik (CPaR) We can ue the reult of imulation analyi to examine and compare the empirical ditribution of objective function value. Figure 27 depict the empirical cumulative ditribution function (cdf) of the cot value for our model baed on the reult of imulation analyi. The three-tage model firt order tochatically dominate the myopic model in term of their cot meaure except for the mall region at the lower end. That i, for any given cot value a, the three-tage model ha a higher probability of being le than a. We alo compare the two model uing a utility function baed on Tchebycheff ditance. In thi etting, the objective i to minimize the maximum weighted Tchebycheff ditance from a given point a explained in Chapter 4. Table 8 contain the reult from 1000 imulation replication uing an equally weighted Tchebycheff metric baed on the ideal point [cot, ccar, liqrik]=[20, 50, 40]. Simulation horizon i again three year. 98

112 Figure 27 Empirical cdf Plot. Table 8 Reult of Dynamic Game. Experiment 2: A Tchebycheff-Type Utility Function. 3-Stage Model 1-Stage Model Expected Cot Market Rik (CCaR) Liquidity Rik (CPaR) Table 9 depict the reult when the weight aociated with the Tchebycheff ditance are changed to [0.5, 0.1, 0.4] for cot, market rik and liquidity rik repectively. The three-tage model produce a better reult in the cot dimenion which i given more weight in the aumed utility function. 99

113 Table 9 Reult of Dynamic Game. Experiment 3: A Tchebycheff-Type Utility Function. 3-Stage Model 1-Stage Model Expected Cot Market Rik (CCaR) Liquidity Rik (CPaR) We now compare the two model uing the additive equally weighted utility function of Table 7 in five year etting. In thi experiment the three-year and oneyear model are ued five time at the beginning of each year to obtain the iuance trategy throughout thee five year. The reult are given in Table 10. While the cot value are comparable, the three-tage model yield lower rik value in thi five year etting under the aumed utility function. Table 10 Reult of Dynamic Game. Experiment 4: An Additive Utility Function, A Deciion Period of 5 year. 3-Stage Model 1-Stage Model Expected Cot Market Rik (CCaR) Liquidity Rik (CPaR) With thee outcome, the experiment highlight the importance of electing the right deciion horizon when making debt trategy deciion or public policy deciion in general. For our experiment, we choe to employ a three-year model in line with the regulatory legal framework in Turkey 6. However, there eem to be 6 The Comminique on Principle and Procedure of Coordination and Execution of Debt and Rik Management (Publihed in the Official Gazette dated September 1, 2002 and no ) impoe 100

114 more value in experimenting with longer deciion horizon. Baing deciion over a longer term analyi may yield better reult over the medium and long term even though the cot meaured in the hort term may eem lightly higher. The number of experiment in thi imulation/optimization etting can eaily be increaed uing alternative deciion maker preference tructure or variou modelling framework. that the debt trategy deciion are made for a three-year period. The budgetary framework i alo baed on a three-year horizon. 101

115 CHAPTER 6 6CONCLUSIONS In electing the combination of intrument to be iued, i.e. the borrowing trategy, public debt manager have to take into account variou objective and the uncertainty aociated with the outcome of the deciion made. In debt management theory and practice, everal approache have been derived and propoed for the cae of overeign debt manager; however thee are generally contented with the quantification of relevant cot and rik meaure. Therefore, there i an additional need to ait deciion maker in comparing alternative coure of action ince targeted objective are conflicting by their nature. The objective of thi thei i to propoe a framework to upport the deciion making proce regarding overeign debt iuance, drawing on the cae of Turkih Treaury, the intitution in charge of overeign debt management in Turkey. We tried to incorporate relevant criteria and develop quantitative approache that take into account equential deciion concerned with debt iuance policie. We firt preented a multi-objective multi-period tochatic programming model that aim to upport bond iuance deciion of public debt management unit. It help quantify the cot and rik aociated with alternative coure of action under uncertainty, making ue of a cenario tree. The determinitic-equivalent liability management model i concretized on an illutrative example. Thi generic model can erve for different country characteritic with mall modification. There can be a many period a relevant, and the number of ubperiod in each period can vary. For example, the model can conit of two diimilar period, the firt period correponding to year one and the econd covering all remaining year in the model horizon. It i alo poible to incorporate everal 102

116 different cot and rik meaure to the model. SP model can well be extended to include debt management tool uch a bond buy-back and debt exchange that have tarted to gain popularity among public debt management office. The addition of uch tool i traightforward and can be directly implemented if needed. Having developed a generic deciion model, we then incorporated ome MCDM approache to contruct a generic deciion aid tool. The aim wa to ait deciion maker in analyzing the trade-off between conflicting objective. In thi context, we propoed to identify ome efficient olution baed on different preference tructure. Since, it wa practically not poible to identify all alternative efficient olution in our continuou objective pace, we employed an interactive MCDM approach aimed at getting deciion maker involvement through the deciion upport proce. We believe thi will help ae preference, explore ditinct alternative olution and guide DM in making debt trategy olution. Since cenario generation mechanim try to reflect the underlying tochaticity of the model parameter, uch a interet and exchange rate, they contain a degree of randomne. Conequently, there i uually a variation in the optimal objective value when the model i olved on different cenario tree intance. In thi work, we modified the available MCDM approache to account for the poible randomne in the cenario tree generation proce and make tatitical inference. Specifically, we contructed confidence region around our efficient olution and modified the interactive procedure to cope with the uncertainty in the cenario generation method which repreent the tochaticity in real life. To thi end, we made ue of tool from multi-variate tatitical analyi. Thi framework i then applied for the cae of overeign debt management in Turkey. In thi illutrative application, we adapted the generic SP model in line with the tructure of public debt management in the country. The three-objective threetage model developed by taking into account the relevant objective and contraint wa ued for further experimentation. In thi context, the high degree of variability of the cenario tree for macro economic variable, contructed by a vector autoregreive model, impoed tability problem. A a remedy to thi problem, we adapted idea from cenario reduction 103

117 and clutering technique and employed a pecific cenario tree generation algorithm for the cae of Turkih economy baed on the K-mean clutering algorithm. In thi regard, it may alo be worthwhile to try generate the tree with other clutering method, to change to number of branche in each tage or to allow the number of cluter vary dynamically. That remain a a future work to be accomplihed. The cenario-tree baed model wa then ued a bai for the illutration of MCDM approache for the cae of the government. Our experiment with the cae of Turkey how that thi framework can be of practical ue in a real etting. The model ugget iuing hort-term bond to minimize expected cot and longer-term fixed rate ecuritie to decreae the level of market rik a expected. The deciion maker can olve the model attaching different weight to the objective and gain inight about the reulting debt trategy compoition. With the help of uch a quantitative tool, the overeign debt iuer will have the mean to ee the effect of different rik and cot preference on the debt iuance policy. Experimentation on the model can help the aement of the deciion maker preference with regard to aociated criteria, which are not only crucial in debt management policie, but alo in other financial deciion of the government. Dicloing mainline reult from the modelling work to general public opinion can alo help the debt management office convey their trategie to market participant and inform takeholder, uch a tax payer, about the objective of the public debt management policy. Similar model can alo be employed by independent organization that conduct reearch on the macroeconomic policie of the government. Thi will provide them the mean to comment on the action and plan of the government in raiing debt on behalf of the citizen. The model i teted in imulation etting, where the model i olved iteratively on a rolling horizon etting. In thi context, performance of alternative model were teted againt each other in a imulation framework that mimic the dynamic macro-economic environment and the actual deciion making proce. We believe that the developed imulation framework can well erve a a deciion aid tool by itelf and the performance of alternative deciion making tyle can be analyzed in the propoed rolling-horizon etting. 104

118 Future reearch direction may include developing the tochatic programming model in a non-linear tructure to cope with non-linear objective function, uch a non-linear rik meaure. In thi regard, the tandard deviation of funding cot can be conidered a a rik-related objective that the government would like to minimize. The model can alo be modified to allow deciion on the debt portfolio compoition rather than targeting the iuance trategie. The main aumption of the modelling approach i that the evolution of tochatic variable i independent of the deciion of the government. Epecially, in countrie where the government i the larget financial actor and dominate the financial market, the deciion of the government may have an impact on the tate of nature for financial variable. In uch cae, the volume of bond iued in a pecific maturity may affect the level of interet rate for that egment of the yield curve depending on the demand condition. Future work may alo concentrate on attaining mean to cope with thi iue. A imple olution to thi iue would be to employ penalty function that would penalize the cot of exce borrowing given the level of demand in the financial market for government bond. Providing more deciion maker involvement in the cenario tree generation proce can alo be of interet a a future reearch propect. The algorithm provided for MCDM analyi may alo be enhanced to allow further analyi. In general, thi diertation aimed to bring together idea, concept and method from different dicipline, uch a mathematical and financial modelling, rik management, imulation, clutering, tatitical analyi and multi-criteria deciion making in order to develop a quantitative framework for aiting debt trategy deciion of government. A general objective wa to demontrate the relevance and applicability of thee concept in the realm of public debt management. The exiting method employed for public debt trategy analyi rely on enumeration of cot and rik aociated with given financing trategie under variou different macro-economic cenario. Since, thee method are limited with the uer-upplied alternative to be evaluated in a cenario-baed analyi; they do 105

119 not guarantee efficient olution. In thi tudy, we innovate an optimization approach for the PDM trategy problem uing a multi-objective tochatic programming model. The developed framework help identify efficient olution and guide the deciion maker in undertanding the degree of trade-off between different debt management objective. Our experiment for the cae of Turkey how that thi tool can have important ue in a practical etting. Even though the method and tool are dicued in the context of overeign debt management, we believe that the developed deciion tool can be of practical ue not only in debt trategy deciion analyi, but alo in other deciion involving multiple objective and uncertainty. The concept of developing confidence region around efficient olution can provide an important input for deciion analyi in general. Additionally, the tochatic interactive approach we developed a part of thi work i novel in the MCDM literature and can be adapted for different multi-criteria deciion making problem that involve tochaticity. 106

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127 7APPENDIX-A 8CONDITIONAL COST AT RISK In financial invetment deciion, there i a trade-off between cot and rik, inherent in the tructure of financial market, i.e. the higher the rik aociated with return i, the greater i the expected return. Rik, in thi ene, i the threat that the elected invetment portfolio will generate a lo or a lower return than expected due to unfavorable market condition and i attributed to the uncertainty aociated with variable that drive the portfolio return. Given the dilemma in financial deciion making problem, there ha been ubtantial reearch and modelling effort to meaure the degree of rik and to upport invetment deciion. Markowitz (1952, 1959) in hi eminal work, propoed to define thi notion of rik a the variability (tandard deviation) of return of the element (intrument) in a financial portfolio and developed the rik-return efficient portfolio concept. Even though thi paradigm ha prevailed for quite a long time and formed the bai for numerou application, it i now widely accepted that it main aumption that return of aet follow normal ditribution doe not generally hold in reality. In today multifariou market condition, return are affected by many different factor and thu exhibit ditribution that are more complex than the normal ditribution. To capture the aymmetrie in the behaviour of portfolio return, everal metric have been developed and propoed. Among thoe, the Value-at-Rik (VaR) meaure ha gained wide popularity and even become part of the regulatory meaure in the banking ector. VaR i generally defined a the maximum probable lo (or minimum return) for a given portfolio for a pecified time horizon within a certain confidence interval. If we let vector r denote the tochatic variable that drive the lo(return) of a portfolio and f(x,r) tand for the lo for a portfolio 114

128 coniting of aet x, then the VaR value for thi portfolio at a certain confidence level 100(1-α )% i given by the following equation: VaR(x,1-α ) = min{u: P[f(x,r) u] 1-α } (A.1) Depite it popularity, it ha been hown that VaR ha undeirable mathematical characteritic uch a non-convexity and non-ubadditivity (VaR of a portfolio can be larger than the total of that of individual aet) and it i difficult to optimize when it i calculated from cenario (ee Pflug, 2000). VaR doe not either provide any information about the level of rik if the confidence level i exceeded, i.e about the nature of the tail of lo(return) ditribution The Conditional Value-at-Rik (CVaR) alo referred a the mean exce lo, the expected hortfall or the the tail-var emerged a an alternative rik meaure a a repone to the limitation of VaR. CVaR, which quantifie the conditional expectation of loe when VaR level i exceeded for a portfolio x, can be obtained by the given formula for pecified level ofα : CVaR(x,α ) = E[ f(x,r) f(x,r) VaR(x,1-α ) ] (A.2) That i, CVaR take into account the tail of the lo ditribution and i the conditional expectation of wort α *100% loe (Figure 28). pdf CVaR VaR Lo (-return) Figure 28 Conditional Value at Rik. Pflug (2000) ha hown that CVaR poee the required propertie of coherent rik meaure in the ene identified by Artzner et al (1999). Rockafellar 115

129 and Uryaev (2000) illutrate that CVaR can be efficiently minimized uing linear programming in a cenario baed framework. Their approach i pertinent in tochatic programming application in which uncertainty i expreed in the form of a cenario tree coniting of a finite number of dicrete cenario. The main theme i to average the loe on cenario that yield loe greater than the pre-pecified VaR level. To illutrate thi method, let S={ : =1,..., S} be the et of dicrete cenario that are built to expre the uncertainty regarding the tochatic variable r and r denote value of r under cenario. For each cenario, we have an aociated probability p and we can compute the lo of our portfolio of x, i.e. f(x, r ) for each r, =1,...,S. Then, CVaR(x,α ) = φ + α 1 { S: f ( x, r ) z } f ( x, r ). p (A.3) where φ = VaR(x,1-α ). Here, α i the probability that the VaR level i exceeded, that i f(x, r ) φ, and thu, CVaR(x,α ) turn out to be the conditional expectation of loe regarding portfolio x, given that the lo i greater than or equal to z. If we define an auxiliary variablecv for each cenario uch that cv = max { 0, f(x, r ) φ } then, CVaR can be expreed a follow: CVaR(x,α ) = φ + α 1 S (A.4) cv. p (A.5) We apply thi approach to our cenario baed tochatic programming application. Recently, Topaloglou et al (2004) have applied thi meaure for a portfolio management problem. 116

130 APPENDIX-B 9SAMPLE PROGRAM CODE FOR THE PDM-SP MODEL The developed determinitic equivalent PDM-SP model i implemented on GAMS 2.0 uing the CPLEX olver for the cae of Turkey. The program code given below i developed for the experiment regarding the application of the viual interactive approach of Korhonen and Laako (1986): $title Stochatic Programming Model for Public Debt Management $ontext The government ha to decide on the type of borrowing intrument (bond) to be iued to meet the financing requirement in a given planning period. The debt manager have a certain range of intrument at their dipoal, and they have to et a certain borrowing trategy which embodie the proportion of each intrument to be iued for the coure of the deciion horizon. We aume a three-year period at the beginning of which the government determine a borrowing mix to be implemented in the following year. The trategy i revied at the tart of year 2 and 3 depending on the macroeconomic circumtance. Each year i divided into quarter, thu the trategy et at the beginning of the year i purued for 4 quarter $offtext * The et of the model are defined below: et q dum /1/ zz criteria /1*3/ zzz /1/ zzzz /1*3/ t time /1*12/ t1(t) ub-period in the firt period /1*4/ j bond /1*7/ jtl(j) TRY bond /1*6/ j1(jtl) zero-coupon /1*4/ j2(jtl) floating coupon /5/ j3(jtl) fixed coupon /6/ jf(j) fx-linked bond /7/ 117

131 cenario /1*125/ n node /n0*n30/ alia(z,t); et tn(n,t,) time-node-cenario mapping livezero(j1,t) live zero-coupon bond at end of horizon /2.11,3.(9*11),4.(7*11)/ validfrn(j2,z,t) frn coupon payment livefrn(j2,t) live coupon bond at end of horizon validfixed(j3,z,t) fixed coupon payment livefixed(j3,z) live coupon bond at end of horizon validfx(jf,z,t) fixed coupon payment livefx(jf,z) live coupon bond at end of horizon; validfrn(j2,z,t)=ye$((ord(z)<ord(t))and mod(ord(t)-ord(z),2)=0); livefrn(j2,t)=ye$((mod((card(t)-ord(t)),2)=1)); validfixed(j3,z,t)=ye$((ord(z)<ord(t))and mod(ord(t)-ord(z),2)=0); livefixed(j3,z)=ye$((ord(z)<card(t))and mod(card(t)-ord(z),2)=1); validfx(jf,z,t)=ye$((ord(z)<ord(t))and mod(ord(t)-ord(z),2)=0); livefx(jf,z)=ye$((ord(z)<card(t))and mod(card(t)-ord(z),2)=1); tn(n,t,)$(ord(t)<5)=ye$(ceil(ord()/125)=ord(n)); tn(n,t,)$(ord(t)<9 and ord(t)>4)=ye$(ceil(ord()/25)= ord(n)-1); tn(n,t,)$(ord(t)<13 and ord(t)>8)=ye$(ceil(ord()/5)= ord(n)-6); * The main parameter of the model are defined in the following ection Parameter m(j) maturity /1 1,2 2,3 4, 4 6, 5 12, 6 12, 7 12/ cp(j) coupon period /5 2,6 2,7 2/ p(t) primary balance /1 8.8, ,3 10.1,4 5.7,5 7.2,6 10.4,7 9.9,8 3.6,9 7.8, , ,12 3.9/ r3(,t) hort interet rate r12(,t) long interet rate fx(,t) exchange rate fxr(,t) fx interet rate alpha confidence level /0.9/ l(t) current liabilitie /1 40.9,2 43.4,3 41.9,4 27.7,5 32.3,6 31.6,7 9.9,8 20.5,9 12.7, ,11 9.6, / opt(zzzz,zzz) current reference olution; 118

132 * The following ection read the cenario tree tored a text file table r3(,t) $batinclude 'data3.txt'; table r12(,t) $batinclude 'data12.txt'; table fx(,t) $batinclude 'datafx.txt'; table fxr(,t) $batinclude 'datafxr.txt'; * The following ection read the current reference olution table opt(zzzz,zzz) $batinclude 'datakorh.txt'; * The interet rate for maturitie other than 3 and 12 month are calculated in the following ection parameter r(,t,jtl) interet rate for j; r(,t,j1)$(ord(j1)<4)=(1+(r3(,t)+((r12(,t)-r3(,t))/3)* (m(j1)-1)))**(m(j1)/4)-1; r(,t,'4')=(1+r12(,t))**(3/2)-1; r(,t,j2)=r(,t,'2')+0.01; r(,t,j3)=(1+r(,t,'3'))**(1/2)-1; * The following ection read probabilitie calculated for each cenario branch parameter p(,zzz) probability; table p(,zzz) $include 'datapr.txt' * The following ection read current theta value from a text file table teta(zzz,q) $include 'tetat.txt'; * The variable of the model: variable dit ditance meaure x(n,t,j) amount of bond iued in type j decided at node n for time t y(,t,j) node-cenario mapping for TRY bond yf(,t,j) node-cenario mapping for foreign currency bond i(,t) interet paid d(,t) principal paid b(,t) borrowing requirement cb(,t) cah account balance c(,t) withdrawal from cah account cot expected interet cot ct() cot aociated with one cenario branch ac() accrued coupon in cenario 119

133 az() accrued zero-coupon at end of horizon in cenario af() accrued fx bond in cenario aff(,t) accrued fx difference in cenario afc() accrued fx coupon in cenario a() total accrued cot in cenario py() total interet paid in cenario cvar conditional value at rik for meauring market rik cv the cot over var (dummy variable in cvar calculation) var value at rik for meauring market rik liqcv(,t) conditional value at rik for meauring liquidity rik liqcvar the cot over liqvar while meauring liquidity rik liqvar value at rik for meauring market rik myobj; poitive variable x,o,i,d,b,cb,y,var,cv,liqcv; * the demand contraint yf.up(,t,'7')=3; x.up(n,t,'1')=10; * The equation that define the objective function and contraint of the model equation obj objective fucntion definition objcot cot definition cenariocot() total interet paid in cenario repay(t,) principal payment intpay(t,) interet payment iue(n,t,,j) TRY bond iued in node n iuefx(,t,j) fx bond iued cahbal(n,t,) cah flow balance pay() total interet paid cahaccountbal(t,) cah account balance accruedcoupon() accrued interet calculation accruedzero() accrued interet calculation accrued() accrued interet calculation accruedfx() accrued interet calculation accruedfxcoup() accrued interet calculation accruedfxdiff(,t) accrued interet calculation ConditionalVar conditional value at rik calculation CondVar() conditional value at rik calculation ConditionalVar2 conditional value at rik calculation CondVar2(,t) conditional value at rik calculation crit1 Thchebycheff ditance calculation crit2 Thchebycheff ditance calculation crit3 Thchebycheff ditance calculation; cahbal(tn(n,t,)).. um(j,x(n,t,j)) + c(,t) =e= d(,t)+i(,t)+l(t)-p(t); iue(tn(n,t,),j)..y(,t,j)=e=x(n,t,j); iuefx(,t,jf)..yf(,t,jf)=e=y(,t,jf)/fx(,t); repay(t,).. d(,t) =e= um(j,y(,t-m(j),j)); 120

134 intpay(t,).. i(,t) =e= um(j1,y(,t-m(j1),j1)* r(,t-m(j1),j1)) + um(validfrn(j2,z,t),y(,z,j2)* r(,t-cp(j2),j2)) + um(validfixed(j3,z,t),y(,z,j3)* r(,z,j3))+ um(validfx(jf,z,t),yf(,z,jf)* fxr(,z)/2*fx(,t)); pay()..py()=e=um(t,i(,t)); cahaccountbal(t,)..cb(,t)=e=cb(,t-1)- c(,t); accruedcoupon()..ac()=e=um(livefrn(j2,t),y(,t,j2)* r(,'11',j2)*1/2)+um(livefixed(j3,z),y(,z,j3)* r(,z,j3)*1/2); accruedfxcoup()..afc()=e=um(livefx(jf,z),yf(,z,jf)* fxr(,z)/2*1/2*fx(,'12')); accruedfxdiff(,t)..aff(,t)=e=um(jf,(yf(,t,jf)* fx(,'12')-y(,t,jf))); accruedfx()..af()=e=afc()+um(t,aff(,t)); accruedzero()..az()=e=um(livezero(j1,t),y(,t,j1)* r(,t,j1)*((card(t)-ord(t))/m(j1))); accrued()..a()=e=ac()+az()+af(); cenariocot()..ct()=e=py()+a(); objcot..cot=e=um(,p(,'1')*ct()); ConditionalVar..cvar=e=var+1/(1-alpha)*um(,p(,'1')*cv()); CondVar()..cv()=g=(py()+a()-var); ConditionalVar2..liqcvar=e=liqvar+ 1/(1-alpha)*um((,t),p(,'1')*liqcv(,t)); CondVar2(,t)..liqcv(,t)=g=(d(,t)+i(,t)+l(t)-liqvar); crit1..dit=g=1/3*(cot-opt('1','1')-teta('1','1')*(7.234)); crit2..dit=g=1/3*(cvar-opt('2','1')-teta('1','1')*(-5.521)); crit3..dit=g=1/3*(liqcvar-opt('3','1')-teta('1','1')*(12.954)); obj.. myobj =e= dit+0.001*(cot+cvar+liqcvar); model debt /all /; file output /out.dat/; file output2 /out2.dat/; option profile=0; option olprint=off; option yout=off; option limcol=0; option limrow=0; option iterlim= ; option relim= ; option lp=cplex; 121

135 olve debt uing lp minimizing myobj; put output; put put put /; put output2; loop((t1,j), put x.l('n0',t1,j):7:4 /); 122

136 10 CURRICULUM VITAE Peronal Information Surname, Name: Balıbek, Emre Nationality: Turkih (T.C.) Date and Place of Birth: 24 February 1974, Balıkeir Marital Statu: Married Phone: Fax: Education Degree Intitution Year of Graduation MS in Management Science and Operation Reearch Warwick Buine School, Univerity of Warwick, Coventry, England 1998 BS in Indutrial Engineering High School Diploma Boğaziçi Univerity, İtanbul 1996 Sırrı Yırcalı Anadolu Liei, Balıkeir 1992 Work Experience Poition Intitution Year Head of Department Turkih Treaury 2007-preent Specialit (Hazine Müteşarlığı) Directorate of Public Finance Aitant Specialit Teaching Experience Coure Intitution Year IE 496 Topic in Financial Engineering Indutrial Engineering Department, METU

137 Publication Balıbek E., Kökalan M A Stochatic Multi-criteria Approach for Deciion Making in Public Debt Management. Working Paper. Indutrial Engineering Dept. Middle Eat Technical Univerity Balıbek E., Kökalan M A Multi-Objective Multi-Period Stochatic Programming Model for Public Debt Management, ubmitted to the European Journal of Operation Reearch Balıbek E İç Borçlanmada Rik Yönetimi Türkiye İrdelemei (Rik Management in Dometic Borrowing An Analyi of Turkih Cae). Unpublihed Proficiency Thei. Turkih Treaury. Ankara, Turkey Yaman R., Balıbek E Deciion Making for Facility Layout Problem Solution. Computer and Indutrial Engineering 37. pp Conference Balıbek E Etablihing a Rik Management Function - Note from Turkih Experience. 16th OECD Global Forum on Public Debt Management. 6-7 Dec. 2006, Amterdam, The Netherland, available at 0.html Balıbek E., Kökalan M Selection of the Public Debt Management Strategy: A Multiobjective Approach. 18th International Conference on Multi Criteria Deciion Making Jun Chania, Greece Duman E., Balıbek E., Firat, A., Barla, Y A Dynamic Feedback Model for Strategic Management of an Inurance Company. Proceeding of International Sytem Dynamic Conference Aug Itanbul, Turkey 124