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Structured Credit Portfolio Analysis, Baskets and CDOs (Chapman & Hall Crc Financial Mathematics Series)
Structured Credit Portfolio Analysis, Baskets and CDOs (Chapman & Hall Crc Financial Mathematics Series)
Christian Bluhm, Ludger Overbeck,
The financial industry is swamped by credit products whose economic performance is linked to the performance of some underlying portfolio of creditrisky instruments, like loans, bonds, swaps, or assetbacked securities. Financial institutions continuously use these products for tailormade long and short positions in credit risks. Based on a steadily growing market, there is a high demand for concepts and techniques applicable to the evaluation of structured credit products. Written from the perspective of practitioners who apply mathematical concepts to structured credit products, Structured Credit Portfolio Analysis, Baskets & CDOs starts with a brief wrapup on basic concepts of credit risk modeling and then quickly moves on to more advanced topics such as the modeling and evaluation of basket products, creditlinked notes referenced to credit portfolios, collateralized debt obligations, and index tranches. The text is written in a selfcontained style so readers with a basic understanding of probability will have no difficulties following it. In addition, many examples and calculations have been included to keep the discussion close to business applications. Practitioners as well as academics will find ideas and tools in the book that they can use for their daily work.
Categories:
Economy
Year:
2006
Edition:
1
Language:
english
Pages:
376
ISBN 10:
1584886471
ISBN 13:
9781420011470
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PDF, 5.85 MB
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CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES Structured Credit Portfolio Analysis, Baskets & CDOs CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an everexpanding slice of the financial sector. This series aims to capture new developments and summarize what is known over the whole spectrum of this field. It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners. The inclusion of numerical code and concrete realworld examples is highly encouraged. Series Editors M.A.H. Dempster Centre for Financial Research Judge Business School University of Cambridge Dilip B. Madan Robert H. Smith School of Business University of Maryland Rama Cont CMAP Ecole Polytechnique Palaiseau, France Published Titles An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and Christoph Wagner Financial Modelling with Jump Processes, Rama Cont and Peter Tankov Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers AmericanStyle Derivatives; Valuation and Computation, Jerome Detemple Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck Proposals for the series should be submitted to one of the series editors above or directly to: CRC Press, Taylor and Francis Group 2425 Blades Court Deodar Road London SW15 2NU UK CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES Structured Credit Portfolio Analysis, Baskets & CDOs Christian Bluhm Ludger Overbeck Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑10: 1‑58488‑647‑1 (Hardcover) International Standard Book Number‑13: 978‑1‑58488‑647‑1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Bluhm, Christian. Structured portfolio analysis, baskets and CDOs / Christian Bluhm, Ludger Overbeck. p. cm. ‑‑ (Chapman & Hall/CRC financial mathematics series ; 5) Includes bibliographical references and index. ISBN 1‑58488‑647‑1 (alk. paper) 1. Portfolio management. 2. Investment analysis. I. Overbeck, Ludger. II. Title. HG4529.5.B577 2006 332.64’5‑‑dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 2006049133 Preface The financial industry is swamped by structured credit products whose economic performance is linked to the performance of some underlying portfolio of creditrisky instruments like loans, bonds, credit default swaps, assetbacked securities, mortgagebacked assets, etc. The market of such collateralized debt or synthetic obligations, respectively, is steadily growing and financial institutions continuously use these products for tailormade risk selling and buying. In this book, we discuss mathematical approaches for modeling structured products creditlinked to an underlying portfolio of creditrisky instruments. We keep our presentation mathematically precise but do not insist in always reaching the deepest possible level of mathematical sophistication. Also, we do not claim to present the full range of possible modeling approaches. Instead, we focus on ideas and concepts we found useful for our own work and projects in our daily business. Therefore, the book is written from the perspective of practitioners who apply mathematical concepts to structured credit products. As preknowledge, we assume some facts from probability theory, stochastic processes, and credit risk modeling, but altogether we tried to keep the presentation as selfcontained as possible. In the bibliography, the reader finds a collection of papers and books for further reading, either for catching up with facts required for an even deeper understanding of the model or for following up on further investigations in alternative approaches and concepts found useful by other colleagues and researchers working in the field of portfolio credit risk. A helpful prereading for the material contained in this book is the textbook [25], which introduces the reader to the world of credit risk models. However, there are many other suitable textbooks in the market providing introductory guidance to portfolio credit risk and we are sure that for every reader’s taste there is at least one ‘best matching’ textbook out there in the book market. It is our hope that after reading this book the reader will find the challenge to model structured credit portfolios as fascinating as we find it. We worked in this area for quite some years and still consider it as a great and intriguing field where application of mathematical ideas, concepts, and models lead to valueadding business concepts. Zurich and Munich, August 2006 Christian Bluhm and Ludger Overbeck v vii Acknowledgments Christian Bluhm thanks his wife, Tabea, and his children, Sarah Maria and Noa Rebeccah, for their continuous support and generous tolerance during the writing of the manuscript. Their love and encouragement is his most important source of energy. Ludger Overbeck is most grateful to his wife, Bettina, and his children, Leonard, Daniel, Clara, and Benjamin for their ongoing support. We highly appreciate feedback and comments on the manuscript by various colleagues and friends. We owe special thanks to Christoff Goessl (HypoVereinsbank, London), Christopher Thorpe (FitchRatings, London), Walter Mussil (Mercer Oliver Wyman, Frankfurt), and Stefan Benvegnu (Credit Suisse, Zurich) with whom we had the honor and pleasure to work for quite some years. Disclaimer This book reflects the personal view of the authors and does not provide information about the opinion of Credit Suisse and HypoVereinsbank. The contents of this book have been written solely for educational purposes. The authors are not liable for any damage arising from any application of the contents of this book. About the Authors Christian Bluhm is a Managing Director at Credit Suisse in Zurich. He heads the Credit Portfolio Management unit within the Credit Risk Management Department of Credit Suisse. Before that, he headed the team Structured Finance Analytics in HypoVereinsbank’s Group Credit Portfolio Management in Munich, where his team was responsible for the quantitative evaluation of assetbacked securities with a focus on collateralized debt and loan obligations, collateralized synthetic obligations, residential and commercial mortgagebacked securities, default baskets, and other structured credit instruments, from origination as well as from an investment perspective. His first professional position in risk management was with Deutsche Bank in Frankfurt. In 1996, he earned a Ph.D. in mathematics from the University of ErlangenNuremberg and, in 1997, he was a postdoctoral fellow at the mathematics department of Cornell University, Ithaca, New York. He has coauthored a book on credit risk modeling, together with Ludger Overbeck and Christoph Wagner, and coauthored a book on statistical foundations of credit risk modeling, together with Andreas Henking and Ludwig Fahrmeir. He regularly publishes papers and research articles on credit risk modeling and management in journals and books, including RISK magazine and RISK books. During his academic time, he published several research articles on harmonic and fractal analysis of random measures and stochastic processes in mathematical journals. He frequently speaks at conferences on risk management. Ludger Overbeck holds a professorship in mathematics and its applications at the University of Giessen in Germany. His main interests are quantitative methods in finance, risk management, and stochastic analysis. Currently, he also is a Director at HypoVereinsbank in Munich, heading the Portfolio Analytics and Pricing Department within the Active Portfolio Management unit in HypoVereinsbank’s Corporates & Markets Division. His main tasks are the pricing of structured credit products and the risk management of the investment credit portfolio. Until June 2003, he was Head of Risk Research & Development in Deutsche Bank’s credit risk function located in Frankfurt. His main responsibilities included development and implementation of the internal groupwide credit portfolio model, the operational risk model, specific (market) risk modeling, the EC/RAROCmethodology, integration of risk types, backtesting of ratings and correlations, quantitative support for portfolio management and all other risk types, and risk assessment of credit derivatives and portfolio transactions like collateralized debt and loan obligations. Prior to that, he worked for the Banking Supervision Department in the main office of Deutsche Bundesbank in Duesseldorf, Germany, mainly concerned with internal market risk models and inspections of the banks in line with the socalled ‘minimum requirements for trading businesses’. He frequently publishes articles in various academic and applied journals, including RISK, and is coauthor of a book on credit risk modeling, together with Christian Bluhm and Christoph Wagner. He is a regular speaker at academic and financial industry conferences. Ludger holds a Ph.D. in probability theory and habilitations in applied mathematics from the University of Bonn and in economics from the University of Frankfurt. ix A Brief Guide to the Book Before we send the reader on a journey through the different chapters of this book, it makes sense to briefly summarize the contents of the different sections so that readers can define their own individual path through the collected material. Chapter 1 is intended to be an introduction for readers who want to have a brief overview on credit risk modeling before they move on to the core topics of the book. For experienced readers, Chapter 1 can serve as a ‘warmingup’ and an introduction to the notation and nomenclature used in this book. The following keywords are outlined: • Singlename credit risk measures like ratings and scorings, default probabilities, exposures, and loss given default • Modeling of default risk via latent variable and threshold models • Decomposition of credit risks into systematic and idiosyncratic components by factor models • Credit portfolio loss distributions and their summary statistics like expected loss, unexpected loss, and quantilebased and expected shortfallbased economic capital • Comments and remarks regarding the tradeoff between accuracy and practicability of credit risk methodologies It is not necessary to read Chapter 1 in order to understand subsequent chapters. But experienced readers will need not more than an hour to flip through the pages, whereas nonexperienced readers should take the time until all of the keywords are properly understood. For this purpose, it is recommended to additionally consult an introductory textbook on credit risk measurement where the mentioned keywords are not only indicated and outlined but fully described with all the technical details. There are many textbooks one can use for crossreferences; see Chapter 5 for recommendations. xi xii Chapter 2 treats the modeling of basket products. The difference between basket products and collateralized debt obligations (CDOs) covered in Chapter 3 is the enhanced cash flow richness in CDO products, whereas baskets are essentially credit derivatives referenced to a portfolio of creditrisky names. The content of Chapter 2 can be divided into two broad categories: • Modeling techniques: Essential tools for credit risk modeling are introduced and discussed in detail, including term structures of default probabilities, joint default probabilities, dependent default times and hazard rates, copula functions with special emphasis on the Gaussian, Studentt, and Clayton copulas, and dependence measures like correlations, rank correlations, and tail dependence. • Examples and illustrations: Chapter 2 always tries out the justdeveloped mathematical concepts in the context of fictitious but realistic examples like duo baskets, default baskets (firsttodefault, secondtodefault), and creditlinked notes. One section in the context of examples is dedicated to an illustration on scenario analysis showing how modeling results can be challenged w.r.t. their plausibility. In general, we present techniques rather than readytogo solutions, because, based on our own experience, we found that general principles and techniques are more useful due to a broader applicability. In many cases, it is straightforward to adapt modeling principles and techniques to the particular problem the modeler has to solve. For instance, we do not spend much time with pricing concepts but provide many techniques in the context of the modeling of dependent default times, which can be applied in a straightforward manner to evaluate cash flows of creditrisky instruments. Such evaluations can then be used to derive a model price of creditrisky instruments like default baskets or tranches of collateralized debt obligations. Chapter 3 is dedicated to collateralized debt obligations (CDOs). In a first part of our exposition, we focus on a nontechnical description of reasons and motivations for CDOs as well as different types and applications of CDOs in the structured credit market. For the modeling of CDOs we can apply techniques already elaborated in Chapter 2. This is exercised by means of some CDO examples, which are sufficiently xiii realistic to demonstrate the application of models in a way ready to be adopted by readers for their own work. In addition to the dependent default times concept, Chapter 3 also includes a discussion on alternative modeling approaches like multistep models and diffusionbased first passage times. Techniques applicable for the reduction of modeling efforts like analytic and semianalytic approximations as well as a very efficient modeling technique based on the comonotonic copula function conclude our discussion on modeling approaches. At the end of Chapter 3, a comprehensive discussion on singletranche CDOs (STCDOs) and index tranches as important examples for STCDOs is included. In this section, we also discuss pricing and hedging issues in such transactions. The two last topics we briefly consider in Chapter 3 are portfolios of CDOs and the application of securitizationbased tranche spreads as building blocks in a costtosecuritize pricing component. Chapter 4 is a collection of literature remarks. Readers will find certain guidence regarding the access to a rich universe of research articles and books. The collection we present does by no means claim to be complete or exhaustive but will nevertheless provide suggestions for further reading. The Appendix at the end of this book contains certain results from probability theory as well as certain credit risk modeling facts, which are intended to make the book a little more selfcontained. Included are also certain side notes, not central enough for being placed in the main part of the book, but nevertheless interesting. One example for such side notes is a brief discussion on entropymaximizing distributions like the Gaussian distribution and their role as standard choices under certain circumstances. As mentioned in the preface of this book, the collection of material presented in this book is based on modeling techniques and modeling aspects we found useful for our daily work. We very much hope that every reader finds at least some pieces of the presented material to be of some value for her or his own work. Contents Preface v About the Authors ix A Brief Guide to the Book xi 1 From Single Credit Risks to Credit Portfolios 1.1 Modeling SingleName Credit Risk . . . . . . . . 1.1.1 Ratings and Default Probabilities . . . . . 1.1.2 Credit Exposure . . . . . . . . . . . . . . 1.1.3 Loss Given Default . . . . . . . . . . . . . 1.2 Modeling Portfolio Credit Risk . . . . . . . . . . 1.2.1 Systematic and Idiosyncratic Credit Risk 1.2.2 Loss Distribution of Credit Portfolios . . 1.2.3 Practicability Versus Accuracy . . . . . . 1 2 2 10 14 17 17 20 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Default Baskets 2.1 Introductory Example: Duo Baskets . . . . . . . . . . . 2.2 First and SecondtoDefault Modeling . . . . . . . . . 2.3 Derivation of PD Term Structures . . . . . . . . . . . . 2.3.1 A TimeHomogeneous Markov Chain Approach . 2.3.2 A NonHomogeneous Markov Chain Approach . 2.3.3 Extrapolation Problems for PD Term Structures 2.4 Duo Basket Evaluation for MultiYear Horizons . . . . 2.5 Dependent Default Times . . . . . . . . . . . . . . . . . 2.5.1 Default Times and PD Term Structures . . . . . 2.5.2 Survival Function and Hazard Rate . . . . . . . 2.5.3 Calculation of Default Time Densities and Hazard Rate Functions . . . . . . . . . . . . . . . . . 2.5.4 From Latent Variables to Default Times . . . . . 2.5.5 Dependence Modeling via Copula Functions . . . 2.5.6 Copulas in Practice . . . . . . . . . . . . . . . . 27 27 34 39 40 52 57 59 67 67 68 69 78 85 93 xv xvi 2.5.7 2.6 2.7 Visualization of Copula Differences and Mathematical Description by Dependence Measures . . 2.5.8 Impact of Copula Differences to the Duo Basket 2.5.9 A Word of Caution . . . . . . . . . . . . . . . . . NthtoDefault Modeling . . . . . . . . . . . . . . . . . 2.6.1 NthtoDefault Basket with the Gaussian Copula 2.6.2 NthtoDefault Basket with the Studentt Copula 2.6.3 NthtoDefault Basket with the Clayton Copula . 2.6.4 NthtoDefault Simulation Study . . . . . . . . . 2.6.5 Evaluation of Cash Flows in Default Baskets . . 2.6.6 Scenario Analysis . . . . . . . . . . . . . . . . . . Example of a Basket CreditLinked Note (CLN) . . . . 3 Collateralized Debt and Synthetic Obligations 3.1 A General Perspective on CDO Modeling . . . . . . . . 3.1.1 A Primer on CDOs . . . . . . . . . . . . . . . . . 3.1.2 Risk Transfer . . . . . . . . . . . . . . . . . . . . 3.1.3 Spread and Rating Arbitrage . . . . . . . . . . . 3.1.4 Funding Benefits . . . . . . . . . . . . . . . . . . 3.1.5 Regulatory Capital Relief . . . . . . . . . . . . . 3.2 CDO Modeling Principles . . . . . . . . . . . . . . . . . 3.3 CDO Modeling Approaches . . . . . . . . . . . . . . . . 3.3.1 Introduction of a Sample CSO . . . . . . . . . . 3.3.2 A FirstOrder Look at CSO Performance . . . . 3.3.3 Monte Carlo Simulation of the CSO . . . . . . . 3.3.4 Implementing an Excess Cash Trap . . . . . . . . 3.3.5 MultiStep and First Passage Time Models . . . 3.3.6 Analytic, SemiAnalytic, and Comonotonic CDO Evaluation Approaches . . . . . . . . . . . . . . . 3.4 SingleTranche CDOs (STCDOs) . . . . . . . . . . . . . 3.4.1 Basics of SingleTranche CDOs . . . . . . . . . . 3.4.2 CDS Indices as Reference Pool for STCDOs . . . 3.4.3 ITraxx Europe Untranched . . . . . . . . . . . . 3.4.4 ITraxx Europe Index Tranches: Pricing, Delta Hedging, and Implied Correlations . . . . . . . . 3.5 Tranche Risk Measures . . . . . . . . . . . . . . . . . . 3.5.1 Expected Shortfall Contributions . . . . . . . . . 3.5.2 Tranche Hit Contributions of Single Names . . . 3.5.3 Applications: Asset Selection, CosttoSecuritize 3.5.4 Remarks on Portfolios of CDOs . . . . . . . . . . 99 113 118 120 121 127 127 129 136 140 147 165 166 167 172 178 184 186 190 194 194 199 202 210 213 220 250 250 253 259 271 287 288 292 294 299 xvii 4 Some Practical Remarks 303 5 Suggestions for Further Reading 307 6 Appendix 6.1 The Gamma Distribution . . . . . . . . . . . . . . . . . 6.2 The ChiSquare Distribution . . . . . . . . . . . . . . . 6.3 The Studentt Distribution . . . . . . . . . . . . . . . . 6.4 A Natural ClaytonLike Copula Example . . . . . . . . 6.5 EntropyBased Rationale for Gaussian and Exponential Distributions as Natural Standard Choices . . . . . . . 6.6 Tail Orientation in Typical Latent Variable Credit Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 The Vasicek Limit Distribution . . . . . . . . . . . . . . 6.8 OneFactor Versus MultiFactor Models . . . . . . . . . 6.9 Description of the Sample Portfolio . . . . . . . . . . . 6.10 CDS Names in CDX.NA.IG and iTraxx Europe . . . . 311 311 312 312 314 References 339 Index 349 315 318 320 322 329 332 Chapter 1 From Single Credit Risks to Credit Portfolios We begin our exposition by a brief noninformal tour through some credit risk modeling concepts recalling some basic facts we will need later in the book as well as making sure that we are all at the same page before Chapters 2 and 3 lead us to the ‘heart’ of our topic. A good starting point is a saying by Albert Einstein who seemingly made the observation As far as the laws of mathematics refer to reality they are not certain; and as far as they are certain they do not refer to reality.1 We feel free to interpret his statement in the following way. Most of the time we cannot work with deterministic models in order to describe reality. Instead, our models need to reflect the uncertainty inherent in the evolution of economic cycles, customer behavior, market developments, and other random components driving the economic fortune of the banking business. Thanks to the efforts of probabilists over many centuries, we have a full range of tools from probability theory at our disposal today in order to model the uncertainties arising in the financial industry. By means of probability theory randomness can never be eliminated completely in order to overcome the uncertainty addressed in Einstein’s statement, but randomness can be ‘tamed’ and conclusions and forecasts with respect to a certain level of confidence can be made on which business and investments decisions can be based. It is the aim of the following sections to perform a quick but thorough walk through concepts exploiting probability theory to finally arrive at meaningful conclusions in structured finance. 1 Albert Einstein in his address Geometry and Experience to the Prussian Academy of Sciences in Berlin on January 27, 1921. 1 2 Structured Credit Portfolio Analysis, Baskets & CDOs As a reference for the sections in this chapter we refer to the book [25] where most of the introductory remarks made in this chapter are elaborated in much greater detail. In order to keep the exposition fluent, we do not interrupt the text too often to make bibliographic remarks. Instead, suggestions for further reading are provided in Chapter 5. The main purpose of Chapter 1 is to build up the basis of nomenclature, risk terms, and basic notions required for understanding the subsequent chapters of this book. 1.1 Modeling SingleName Credit Risk Based on Einstein’s observation regarding uncertainty in real world problems, we have to rely on probabilistic concepts already at the level of single borrowers. In the following brief exposition, we focus on ratings and default probabilities, loss quotes, and exposures. 1.1.1 Ratings and Default Probabilities As a standard tool for the lending business, all banks have rating systems in place today ranking the creditworthiness of their clients by means of a ranking, e.g., AAA, AA, A, BBB, BB, B, CCC, or Aaa, Aa, A, Baa, Ba, B, Caa, respectively, for best, 2ndbest, 3rdbest, ..., and worst credit quality where the first row of letter combinations is used by the rating agencies Standard & Poor’s2 (S&P) and Fitch3 and the second row of letter combinations is used by Moody’s.4 Banks and rating agencies assign default probabilities (short: PD for ‘probability of default’) with respect to certain time horizons to ratings in order to quantify the likelihood that rated obligors default on any of their payment obligations within the considered time period. In doing so, the ranking (with respect to creditworthiness) of obligors in terms of ratings (letter combinations) is mapped onto a metric scale of numbers (likelihoods) in the unit interval. Throughout the book we use oneyear default probabilities assigned to ratings (in S&P notation) according to 2 See www.standardandpoors.com See www.fitchratings.com 4 See www.moodys.com 3 From Single Credit Risks to Credit Portfolios 3 TABLE 1.1: Oneyear PDs for S&P ratings; see [108], Table 9 AAA AA A BBB BB B CCC D 0.00% 0.01% 0.04% 0.29% 1.28% 6.24% 32.35% Table 1.1. Table 1.1 can be read as follows in an intuitive way. Given a portfolio of 10,000 Brated obligors it can be expected that 624 of these obligors default on at least one of their payment obligations within a time period of one year. Kind of ‘nonacceptable’ is the zero likelihood of default for AAArated borrowers. This would suggest that AAArated customers never default and, therefore, can be considered as riskfree asset for every investor. It is certainly very unlikely that a firm with a AAArating fails on a payment obligation but there always remains a small likelihood that even the most creditworthy client defaults for unexpected reasons. Moreover, as addressed5 in research on PD calibration for low default portfolios, the zero default frequency for AAA is just due to a lack of observations of AAAdefaults. In order to assign a nonzero oneyear PD to AAArated customers, we make a linear regression of PDs on a logarithmic scale in order to find a meaningful estimated PD for AAArated borrowers. This yields a oneyear PD for AAArated obligors of 0.2 bps, which we will assume to be the ‘best guess PD’ for AAArated clients from now on. Default probabilities are the ‘backbone’ of credit risk management. In the new capital accord6 (Basel II), the socalled Internal Ratings Based (IRB) approach (see [15]) allows banks to rely on their internal 5 See Pluto and Tasche [100] and Wilde and Jackson [114]. Note that the Bank for International Settlements (BIS) is located in the Swiss city Basel such that the capital initiative, which led to the new regulatory framework is often addressed under the label ‘Basel II’; see www.bis.org 6 4 Structured Credit Portfolio Analysis, Baskets & CDOs estimates for default probabilities for calculating regulatory capital. Therefore, the better the quality of a bank’s rating systems, the more appropriate are the bank’s regulatory capital figures. But not only for regulatory purposes but, even more important, for economic reasons like competitive advantage in the lending market it is important that banks pay a lot of attention to their internal rating systems. It is beyond the scope of this book and beyond the purpose of this introductory chapter to go into great details here but the following remarks will at least provide a link to the practical side of ratings. Internal ratings can be obtained by various methodologies. In this book, where the focus is on structured credit products like default baskets and collateralized debt obligations (CDOs; see Chapter 3), ratings and PDs are typically modeled via socalled causal rating methods. In such models, underlying drivers of the default event of a client or asset are explicitly modeled as well as the functional link between risk drivers and the default event. We will come back to this principle over and over again in subsequent chapters. For a first simple example, we refer to Figure 1.3. Causal rating models are a conceptually preferred type of ratings because nothing is more desirable for a bank than understanding the ‘explicit mechanism’ of default in order to define appropriate measures of default prevention. Unfortunately, it is not always possible to work with causal rating models. For example, private companies (Small and Medium Enterprises; SME) are difficult to capture with causal models because underlying default drivers are not given in uptodate explicit form as it is in the case for listed corporate clients where public information, for instance, by means of stock prices, is available. In cases where causal models are not implementable, we need to work with more indirect approaches like socalled balance sheet scorings where balance sheet and income statement informations (as ‘explaining variables’) are used to explain default events. In such rating systems, the bank optimizes the discriminatory power (i.e., the ability of the rating system to separate defaulting from nondefaulting clients w.r.t. a certain time horizon) by means of a scheme as illustrated in Figure 1.1. Starting with a long list of financial ratios, the rating modelers attempt to find the optimal selection of ratios and optimal weights for the combination of ratios to finally arrive at a rating score best possible identifying potential defaulters in the bank’s credit approval process. An important working step in the development of scoring models From Single Credit Risks to Credit Portfolios FIGURE 1.1: 5 Rating/scoring system optimization (illustrative) is the calibration of the scores to actual default probabilities. This procedure is briefly called the PD calibration of the model. A major challenge in PD calibration and rating/scoring model development is to find a healthy balance between the socalled error of first kind (αerror) and the error of second kind (βerror), where the first mentioned refers to a defaulting client approved by the scoring system and the lattermentioned refers to a nondefaulting client erroneously rejected by the scoring system. The error of first kind typically contributes to the bank’s P&L with the realized loss (lending amount multiplied by the realized loss quote) of the engagement, whereas the second error contributes to the bank’s P&L in the form of a missed opportunity, which in monetary terms typically sums up to the lending amount multiplied by some margin. It is obvious that the first error typically can be expected to be more heavy in terms of P&L impact. However, banks with too conservative PDs will have a competitive disadvantage in the lending market. If too many clients are rejected based on ratings or scorings, the earnings situation of a bank can severely suffer, in terms of a negative P&L impact as well as in terms of lost market share. There are other rating system categories, besides causal rating and scoring systems, which are common in credit risk management. For instance, in some cases ratings are based on a hybrid methodology 6 Structured Credit Portfolio Analysis, Baskets & CDOs combining a scoring component and a causal model component into a common framework. We will not comment further on ratings in general but conclude our discussion with an example illustrating the competitive advantage banks have from better rating systems. This remark also concerns our structured credit applications since banks with more sophisticated models for, e.g., the rating or evaluation of tranches in collateralized debt obligations (CDOs), have a chance of being more competitive in the structured credit market. FIGURE 1.2: Illustration of the competitive advantage arising from discriminatory power in rating systems For our example we consider a portfolio with 100 assets or clients. We assume that the average default quote in the portfolio is 1%. If the rating/scoring model of the bank has no discriminatory or prediction power at all, then the rating outcome for a client is purely based on chance. In Figure 1.2, we indicate such a worst case rating model by a socalled receiver operating characteristic (ROC)7 on the diagonal of 7 See [43] for an introduction to receiver operating characteristic (ROC) curves. In our chart, the xaxis shows the false alarm rate (FAR) and the yaxis shows the hit rate (HR) of the scoring system. Let us briefly explain the meaning of FAR and HR. Given that credit approval is based on a score S satisfying S > c, where c denotes From Single Credit Risks to Credit Portfolios 7 the unit square. The AUROC is a common measure for discriminatory power; see, e.g., [43]. The AUROC for the worst case rating system (see Figure 1.2) equals 0.5, which corresponds to the area under the diagonal in the unit square. This means that the credit score in the credit approval process of the bank does a job comparable to a coinflipping machine with a fair coin: ‘head’ could mean approval of the loan and ‘tail’ could mean rejection of the loan. If the bank has such a rating/scoring system, then clients cannot be distinguished regarding their default remoteness. Therefore, given a portfolio default quote of 1%, every client will get assigned a PD of 1%. This is extreme case I in Figure 1.2. Extreme case II refers to a ‘crystal ball’ rating/scoring system where the model with certainty identifies the one defaulting client out of 100 clients in the portfolio who is responsible for the portfolio default quote of 1%. In such an unrealistically lucky case, it is natural to assign a zero PD to all nondefaulting clients and a PD of 100% to the one defaulting client. In extreme case I as well as in extreme case II, the average of assigned PDs in the portfolio equals 1%. If the bank, starting from the worst case rating system in extreme case I, manages to improve their ratings toward the ‘crystal ball’ rating system from extreme case II, then 99 PDs experience a significant reduction (from 1% down to 0%) and one PD is significantly increased (from 1% to 100%). Our example is purely illustrative because typical rating/scoring systems have an AUROC between, e.g., 70% and 90%, but rating revisions, which improve the discriminatory power of the system exhibit the same behavior as our extreme case evolution in the example: The benefit of improved predictive power in rating/scoring systems typically leads to lower PDs for assets/clients with higher credit quality and to higher PDs for assets/clients with worse credit quality. In this way, improved ratings allow the bank to offer a more competitive or aggressive pricing for high quality assets and, at the same time, lead to an improved rejection mechanism for low quality assets. It can be observed in practice that better ratings have a potential to substantially contribute to an a prespecified critical cutoff, then a false alarm occurs if S < c for a nondefaulting client. In contrast, the scoring system achieves a hit, if the score S shows S < c for a defaulting client, because the comparison of score and cutoff values helped us indentifying a dangerous credit applicant. Now, the ROC curve is a plot of the curve (FAR(c), HR(c))c∈{cutoffs} . A ‘crystal ball’ scoring system yields an area under the ROC curve (AUROC) of 100% because FAR=0 and HR=1 are best case situations. 8 Structured Credit Portfolio Analysis, Baskets & CDOs improved P&L distribution. After our brief excursion to the ‘practical side’ of ratings, we now come back to the development of the framework used in this book for modeling structured credit products. As already indicated, we are working in a causal modeling world when it comes to default baskets and CDOs; see Figure 3.8. But even at singleasset level, we can think of default events in a causal model context. For this purpose, we introduce for obligor i in a portfolio of m obligors a Bernoulli variable for indicating default within the time interval [0, t], (t) Li (t) (t) = 1{CWI(t) < c(t) } ∼ B(1; P[CWIi < ci ]) i (1.1) i following a socalled threshold model in the spirit of Merton [86], Black & Scholes [23], Moodys KMV’s PortfolioManager,8 and CreditMetrics9 from the Risk Metrics Group (RMG) for modeling (t) default of credit customers. Here, (CWIi )t≥0 denotes a creditworthiness index (CWI) of obligor i indexed by time t triggering a payment default of obligor i in case the obligor’s CWI falls below a certain crit(t) (t) ical threshold ci within the time period [0, t]. The threshold (ci )t≥0 is called the (time dependent) default point of obligor i. Equation (1.1) expresses the causality between the CWI of obligor i and its default (t) event. Note that  despite its notation, we do not consider (CWIi )t≥0 as a stochastic process but consider it as a timeindexed set of latent random variables w.r.t. time horizons t where t ≥ 0 denotes continuous time. It is important to keep this in mind in order to avoid misunderstandings. It needs some work10 (see Sections 3.3.5.1 and 3.3.5.2) to get from a sequence of latent variables to a stochastic process reflecting a certain time dynamics. For most of the applications discussed in this book, the ‘fixed time horizon view’ is completely sufficient. However, for the sake of completeness, we discuss in Section 3.3.5.2 a particular model based on the first passage time w.r.t. a critical barrier of a 8 See www.kmv.com See www.riskmetrics.com 10 The step from fixed horizon CWIs to a stochastic process can be compared with the construction of Brownian motion: just considering a sequence of normal random variables (Xt )t≥0 with Xt ∼ N (0, t) does not mean that (Xt )t≥0 already is a Brownian motion. It needs much more structure regarding the time dynamics and quite some work to choose (Xt )t≥0 in a way making it a Brownian motion. 9 From Single Credit Risks to Credit Portfolios 9 stochastic process instead of a sequence of (fixed horizon) latent variables w.r.t. a sequence of default points; see Figure 1.3 illustrating the concept of stochastic processes as default triggers. The process introduced in Section 3.3.5.2 will be called an ability to pay process (APP) because it reflects a stochastic time dynamics, whereas the sequence of CWIs can be seen as a perfect credit score indicating default with certainty according to Equation (1.1). FIGURE 1.3: Ability to pay process (APP) as default trigger CWIs are difficult or may be even impossible to observe in practice. In the classical Merton model, the ability to pay or distance to default of a client is described as a function of assets and liabilities.. However, the concept of an underlying latent CWI triggering default or survival of borrowers is universally applicable: not only listed corporate clients but any obligor has its own individual situation of wealth and financial liabilities w.r.t. any given time horizon. If liabilities exceed the financial power of a client, no matter if the client represents a firm or a private individual,bankruptcy and payment failure will follow. In this way, the 10 Structured Credit Portfolio Analysis, Baskets & CDOs chosen approach of modeling default events by means of latent variables is a concept meaningful for corporate as well as for private clients and can be applied to the total credit portfolio of banks. For the ‘fixed time horizon’ CWI approach, the timedependent de(t) fault points (ci )t≥0 are determined by the obligor’s PD term structure (see Section 2.3) (t) (t) (t) (pi )t≥0 = (P[CWIi < ci ])t≥0 (1.2) in a way such that (t) ci = F−1 (t) (t) CWIi (pi ) (1.3) where FZ denotes the distribution function of any random variable Z and F−1 Z denotes the respective (generalized) quantile function F−1 Z (z) = inf{q ≥ 0  FZ (q) ≥ z}. The derivation of PD term structures is elaborated in Section 2.3. Note that the beforementioned term structures and CWIs so far reflect only the onedimensional flow of the marginal distributions of singlename credit risks, not the joint distributions of CWIs in the sense of a multivariate distribution. Later, we will catchup in this point and spend a lot of time with multivariate CWI vectors reflecting dependencies between singlename credit risks. 1.1.2 Credit Exposure Despite default probabilities we need to know at least two additional informations for modeling singlename client risk, namely the exposure at default (EAD) outstanding with the considered obligor and the loss given default (LGD) reflecting the overall default quote taking place if obligors default. One could easily dedicate a separate chapter to each of both risk drivers, but again we have to restrict ourselves to a few remarks roughly indicating the underlying concepts. Exposure measurement has several aspects. For drawn credit lines without further commitment, exposure equals the outstanding11 notional exposure of the loan, whereas already for committed undrawn 11 Taking amortizations into account. From Single Credit Risks to Credit Portfolios 11 lines exposure starts to involve a random component, in this case depending on the drawing behavior of the client. Another exposure notion appears in the context of counterparty credit risk in derivative products where exposure equals potential exposure (PE; typically applied in the context of limit setting, etc.) or expected positive exposure (EPE; exposure often used for economic capital calculation, etc.); see, e.g., [79] as well as [13] for an introductory paper12 on exposure measurement. In this book we will not need PE or EPE techniques and can restrict our exposure notion to EAD; see also the new capital accord [15]. The addendum ‘at default’ in EAD refers to the fact that for the determination of realized loss we have to take into account all monetary amounts contribution to the realized loss. Hereby, it is important that we think of EAD as a conglomerate of principal and interest streams; see Figure 1.4 where the interest stream consists of coupon payments and the principal stream is made up by the repayment of capital at the maturity of the bond. In assetbacked transactions as discussed later in this book, we have to deal with EAD at two different levels, namely at the level of the underlying assets (e.g., some reference credit portfolio) as well as at the level of the considered structured security, e.g., some CDO tranche. Actually, the same twolevel thinking has to take place for LGDs where the LGD at structured security level typically depends on the LGDs of the underlying assets, loans, or bonds, respectively. We will come back to this point later in the book. Another aspect of EAD, mentioned for reasons of practical relevance in the same way as our excursion on ratings, is the potential uncertainty regarding the actual outstanding exposure at the default time of an asset. In CDOs, the amount of outstandings typically is well defined and controlled by the offering circular or the term sheet, respectively, describing the structure of the considered transaction. Nevertheless, one could easily think of situations where uncertainties in the outstanding exposure can occur, e.g., in case of residential mortgage backed securities (RMBS) where loans can be prepayed13 or in cases where the 12 Note that the nomenclature in exposure measurement exhibits some variability in the literature due to a certain lack of standardization of notions although the concepts discussed in different papers are most often identical or at least close to each other. 13 In RMBS transactions, prepayed loans are typically replaced by other mortgagebacked loans matching the eligibility criteria of the transaction; in such cases, the offering circular defines clear rules how and in which cases replenishments can be 12 Structured Credit Portfolio Analysis, Baskets & CDOs FIGURE 1.4: Exposure as a conglomerate of principal and interest streams underlying assets are PIKable where ‘PIK’ stands for payment in kind addressing more or less the option to exchange current interest payments against principal or capital (e.g., in a bond where the issuer has the option to capitalize interest at any payment date by issuing additional par amount of the underlying security instead of making scheduled interest payments). The most common uncertainty in lending exposures at single loan level is due to unforeseen customer behavior in times of financial distress; see Figure 1.5 where the wellknown fact that obligors tend to draw on their committed but so far undrawn lines in times of trouble. The most sophisticated approach to deal with exposure uncertainties is by means of causal modeling in a comparable way to causal rating models but focussing on exposure instead of default events, hereby taking the underlying drivers for changes in the outstanding exposure into account and modeling the relationship between these underlying exposure drivers and EAD, e.g., by means of Monte Carlo simulation techniques. As an example, prepayment behavior of obligors in RMBS portfolios is strongly coupled with interest rates such that the interest rate term structure as underlying driver is a reasonable starting point for a causal modeling of prepayments in an RMBS done by the collateral or asset manager in charge of the underlying asset pool. From Single Credit Risks to Credit Portfolios 13 model. In a comparable way, prepayments in corporate lending can be tackled, taking into account that in corporate lending prepayments are a function of interest rates, borrower’s credit quality, and prepayment penalties (depending on the domestic lending market); see, e.g., [68]. FIGURE 1.5: Exposure uncertainty due to unforeseen customer be havior To mention another example, the new capital accord deals with exposure uncertainties for offbalance sheet positions by means of socalled credit conversion factors in the formula EAD = OUTSTANDINGS + CCF × COMMITMENT where CCF denotes the credit conversion14 factor determined w.r.t. the considered credit product; see [15], §8289 and §310316. For instance, in the socalled Foundation Approach a CCF of 75% has to be applied to credit commitments regardless of the maturity of the underlying facility, whereas a CCF of 0% applies to facilities, which, e.g., are uncommitted or unconditionally cancellable. In practical applications and in the sequel of this book, we will always consider EAD w.r.t. some time horizon t, written as EAD(t) (where necessary) addressing the outstanding exposure at time t. 14 In general, credit conversion can address the conversion of noncash exposures into cash equivalent amounts or the conversion of nonmaterialized exposures into expected exposures at default (for example, quantifying the potential draw down of committed unutilized credit lines), etc. 14 1.1.3 Structured Credit Portfolio Analysis, Baskets & CDOs Loss Given Default We now turn our attention to LGD. In general, LGD as a concept is simply explained but far from being trivial regarding modeling and calibration. In the early times of quantitative credit management, many banks defined ‘standard LGDs’ for broad loan categories, e.g., claiming that corporate loans on average admit, e.g., a 40% recovery quote such that the LGD of such products equals 60%. Today we see an increasing trend toward more sophisticated LGD approaches, to some extent due to the new capital accord [15] where banks approved for the alreadymentioned IRB approach in its advanced form can base their capital figures not only on internally estimated PDs but also on their internal LGD estimates. In the sequel, we briefly summarize some basic facts about LGDs. The most advanced LGD approach is, in the same way as mentioned in the context of PDs and EADs  the causal modeling approach again. We will later see that, e.g., the LGD of CDO tranches is a good example for an LGD derived by a causal modeling approach; see Section 3.1. Here, the loss of a CDO tranche is linked to the performance of the underlying asset pool such that in a Monte Carlo simulation one can model the economics of the underlying pool and the causality between the pool and the creditlinked securities at the liability side of the CDO structure. By transforming each asset pool scenario into a scenario at the CDO tranche level one obtains a loss distribution for the CDO tranche from which one can derive the LGD as the mean loss quote of the full distribution of possible loss severities. At single loan level, a good compromise between sophistication and practicability is the following approach. Let us assume that the bank built up a collateral database containing comprehensive historic experience regarding the achieved value in selling collateral for loss mitigation in the context of defaulted creditrisky instruments. For example, such a database will contain information about the average market value decline of residential mortgages, different types of commercial mortgages, single stocks, bonds, structured products, cars for leasings, aircraft vessels for aircraft finance, ships for ship financing, etc. The database will also contain information about recovery quotes on unsecured exposures w.r.t. different seniorities, and so on. Based on such a database, value quotes (VQ) w.r.t. different types of collateral can be derived where the VQ incorporates the expected market value decline of the consid From Single Credit Risks to Credit Portfolios 15 ered collateral category. In addition, the collateral value volatility can be derived, useful for the calibration of stochastic approaches to LGD modeling; see Section 3.3. Given such a database exists, AIRB15 banks need to have such databases for regulatory approval of their internally calibrated LGDs, we can proceed as in the following example. FIGURE 1.6: Illustration of LGD determination Let us assume that we want to determine the LGD in a situation where some client has two loans secured by three types of collateral; see Figure 1.6. Out of the bank’s collateral database we obtain value quotes VQ1, VQ2, VQ3 (in percentage) for the collateral values. Given the market value of the collaterals is given by MV1, MV2, and MV3, the overall, on average achievable, estimated collateral value (ECV) for loss mitigation is given by ECV = VQ1 × MV1 + VQ2 × MV2 + VQ3 × MV3. Note that in most of the cases, collateral selling to the market takes place quite some time after the actual default time of the obligor such 15 Banks in the socalled advanced internal ratings based (AIRB) approach; see [15]. 16 Structured Credit Portfolio Analysis, Baskets & CDOs that an appropriate discounting of collateral values reflecting the time value of money has to be incorporated into the value quotes. Denoting the exposure of the two loans approved for the client by EAD1 and EAD2, the total exposure allocated to the client equals EAD = EAD1 + EAD2. Then, the realized loss, here denoted by LOSS, is given by LOSS = max (1 − VQunsecured) × (EAD − ECV), 0 where VQunsecured equals the percentage average recovery quote on unsecured exposures, also to be calibrated based on the collateral/loss database of the bank. Note that in Figure 1.6 we assume (for reasons of simplicity) that we lose 100% of the unsecured exposure in case of default, i.e., VQunsecured = 0. Because the value quotes already incorporate the potential market value decline of the considered collateral,16 LOSS expresses an expectation regarding the realized loss amount after taking into account all available collateral securities for loss mitigation as well as the typical recovery rate on unsecured exposures. Then, the severity of loss in case the obligor defaults can be expressed as a percentage loss quote, namely the LGD, by LGD = LOSS . EAD (1.4) Because LOSS represents the realized loss amount in units of money, Equation (1.4) is ‘compatible’ with the wellknown formula for the expected loss (EL) on singlenames EL[$] = PD × EAD × LGD. (1.5) Note that the EL can be written in the simple form of Equation (1.5) only if the Bernoulli variable indicating default of the obligor and the two quantities EAD and LGD, e.g., considered as realizations of corresponding random variables due to certain uncertainties inherent in these quantities, are stochastically independent. If EAD and LGD are used as fixed values, they can be considered as expectations of corresponding random variables. Also note that, in general, in the same 16 Equivalently to the chosen approach one could have defined LGDs for collateral securities and an LGD for the overall exposure net of recoveries and then aggregated both figures into an estimate for the ‘realized loss’ accordingly. From Single Credit Risks to Credit Portfolios 17 way as in the case of EAD, we have an additional degree of complexity in the time dimension. Because Definition 1.4 involves exposures, and exposures (EADs) are time dependent, we will write LGD(t) in the sequel where necessary and meaningful, in line with our timedependent notion of exposures, EAD(t) . Under all circumstances it is essential that LGD modeling clearly reflects the recovery and workout process of the bank. Deviations from the bankinternal workout practice will lead to distortions in the LGD calibration and will cause unwanted longterm deviations of provisions or realized losses, respectively, from forecasts based on the expected loss of the bank’s portfolio. Advanced IRB banks have to make sure that PDs and LGDs can be backtested and validated in order to achieve regulatory approval of their internal estimates. 1.2 Modeling Portfolio Credit Risk Turning from singlename to portfolio credit risk is a challenging step in credit risk modeling. For the topics covered in this book, portfolio credit risk is not only one of several issues but rather the fundamental basis for modeling structured credit products. This is confirmed by the notion correlation products sometimes used as a headline for default baskets and CDOs, addressing the fact that these products trade correlations and interdependencies between singlename credit risks in a tailormade way leading to interesting (portfolioreferenced) risk selling and buying opportunities. 1.2.1 Systematic and Idiosyncratic Credit Risk The need for a sound modeling of correlations and ‘tail dependencies’ in the context of portfolio loss distributions is the main reason why for structured credit products we prefer latent variable models indicated in Equation (1.1) enabling an explicit modeling of correlations in contrast to other wellknown models where dependencies between singlename credit risks are modeled implicitly, for example, by means of default rate volatilities in systematic sectors; see CreditRisk+ [35] and Chapter 4 in [25]. According to Equation (1.1), every obligor i is represented by 18 Structured Credit Portfolio Analysis, Baskets & CDOs a default indicator (t) Li = 1{CWI(t) < c(t) } i i such that the obligor defaults if and only if its CWI falls below its default point, always measured w.r.t. to some time interval [0, t]. Now, the starting point of dependence modeling is a decomposition of the client’s risk into a systematic and an idiosyncratic risk component; see Figure 1.7. FIGURE 1.7: Decomposition of two firm’s credit risk into systematic and idiosyncratic parts (separation approach; illustrative) In Figure 1.7, the credit risk of two German automotive manufacturers is decomposed into systematic and idiosyncratic components, here with a 100% country weight in a factor representing Germany and a 100% industry weight in a factor representing the automobile industry. The basic assumption in such models is that dependence takes place exclusively via the systematic components of counterparties. After normalizing the overall firm risk (say, the volatility of the CWI) to 1, the socalled R2 (Rsquared) of the firm captures the systematic risk, whereas the quantity 1−R2 quantifies the residual or idiosyncratic risk of the firm, which cannot be explained by means of systematic risk From Single Credit Risks to Credit Portfolios 19 drivers like countries and industries; see [25], Chapter 1 for a detailed discussion about systematic and idiosyncratic credit risk. In more mathematical terms, Figure 1.7 suggests to decompose the CWI of obligor i in Equation (1.1) into systematic and idiosyncratic components, (t) (t) (t) CWIi = βi Φi + εi (t ≥ 0) (1.6) (t) (t) where we assume that the residuals (ε1 )t≥0 , ..., (εm )t≥0 are independent and for every fixed t identically distributed as well as independent (t) of the systematic variables Φi = (Φi )t≥0 . Here, m denotes the number of obligors in the portfolio and the index Φi is called the composite factor of obligor i because it typically can be represented by a weighted sum of indices, (t) Φi = N X wi,n Ψ(t) n n=1 (i = 1, ..., m; t ≥ 0), (1.7) with positive weights wi,n . Regarding the indices (t) (t) Ψ1 = (Ψ1 )t≥0 , ... , ΨN = (ΨN )t≥0 we find two major approaches in the market and ‘best practice’ industry models, respectively. • The separation approach: Here, Ψ1 , ..., ΨNC denote country indices and ΨNC +1 , ..., ΨN are industry indices. The number of industry factors is then given by NI = N − NC , whereas NC denotes the number of country factors. An industry example for such a model is the socalled Global Correlation Model by Moodys KMV.17 • The combined approach: Here, every index Ψn refers to an industry within a country. CreditMetrics by the RiskMetrics Group (RMG) follows such an approach18 via MSCI indices.19 There are pros and cons for both approaches and it seems to us that both factor models are well accepted in the market. There are other differences in the way Moodys KMV and RMG model correlations, e.g., 17 See www.kmv.com See www.riskmetrics.com 19 Morgan Stanley Capital International Inc. 18 20 Structured Credit Portfolio Analysis, Baskets & CDOs KMV derives asset value correlations from a nondisclosed optiontheoretic model transforming equity processes and related information into modelbased asset value processes, whereas RMG works with equity correlations as a proxy for asset value correlations. For more information on factor modeling we refer to Chapter 1 in [25]. In this context, it is worthwhile to mention that correlations are subject to certain controversial discussions. One can find research papers in the market where people suddenly seem to discover that correlations are negligible. However, we are convinced from our practical experience that correlations are not only inherent/omnipresent in the credit market but play a fundamental role in the structuring and design of default baskets and CDOs. If correlations would be negligible, these kind of correlation products would not be as successful and growing as they are in the credit market. Moreover, in Section 3.4.4 we discuss implied correlations in index tranches. Here, investment banks actually quote correlations in the same way as they quote spreads and deltas. This provides strong support for the importance of correlations and dependencies in the structured credit market. For further study, we recommend to interested readers two very valuable discussion papers on the correlation topic. The first paper is from Moody’s KMV [116] and the second paper is from Fitch Ratings [6]. In these papers, the reader finds strong evidence that correlations are far from being negligible. Moreover, correlations can be measured within systematic sectors like industries (intraindustry) but also between different systematic sectors (e.g., interindustry). The type of correlation considered in the two papers is the asset correlation referring to the correlation between underlying latent variables, in our case the CWIs of clients. We come back to this issue in Chapter 2 where we introduce correlations in a more formal way, distinguishing between asset or CWI correlations and the default correlation, which refers to the correlation of Bernoulli variables as in (1.1) indicating default. As we will see later, default correlations typically live at a smaller order of magnitude than CWI or asset correlations; see (2.5) and thereafter. 1.2.2 Loss Distribution of Credit Portfolios In the early times of credit risk modeling, the ‘end product’ of credit risk models typically was the portfolio loss distribution w.r.t. some fixed time horizon. Today, and especially in structured finance, people are From Single Credit Risks to Credit Portfolios 21 often more interested in looking at dependent default times. We will come back to this issue in Section 2.5. Obviously, the fixed time horizon point of view is a ‘side product’ of default time models just by restricting the default time τ of some obligor to a fixed time interval [0, T ] and considering the Bernoulli variable 1{τ <T } . For reasons of completeness, we illustrate in Figure 1.8 the loss distribution of a credit portfolio. FIGURE 1.8: The loss distribution of a credit portfolio Based on the loss distribution of the credit portfolio, all relevant risk quantities can be identified as ‘summary statistics’ of this distribution. For instance, the expected loss (EL) of the portfolio20 (in percentage) 20 Recall Equation (1.5) for the EL of singlename credit risk in case of independence of the default indicator, exposure, and severity of loss. 22 Structured Credit Portfolio Analysis, Baskets & CDOs equals the sum of singlename ELs weighted by the total portfolio EAD, EL(t) [%] = P m 1 m X (t) j=1 EADj i=1 m X = P m 1 (t) j=1 EADj i=1 (t) (t) ELi (t) (1.8) (t) (t) E[Li × LGDi ] × EADi where Li is defined in (1.1). Note that Equation (1.8) allows for stochastic (nondeterministic) LGDs but assumes EAD to be a deterministic (nonrandom) fixed quantity. The economic capital (EC) of a credit portfolio typically is defined as the quantile of the loss distribution w.r.t. to some given target level of confidence, e.g., 99.9%, minus the EL of the portfolio, which is supposed to be fully pricedin by the front office of the bank, hP i (t) (t) (t) m qα L × EAD × LGD i=1 i i i − EL(t) [%] (1.9) EC(t) Pm α [%] = (t) j=1 EADj where qα [X] denotes the αquantile of some random variable X. Note that the quantile function qα [X] in Equation (1.9) is a much more ‘tricky’ object than the expectation E[·] arising in Equation (1.8). For instance, expectations are linear functions in their arguments whereas quantile functions are highly nonlinear in general. For capital allocation purposes better suitable than quantilebased risk measures are socalled shortfall measures; see Chapter 5 in [25] and the literature mentioned in the last chapter of this book for an explanation21 why shortfall measures are superior to quantilebased measures. Figure 1.8 illustrates a typical shortfall measure. Let us say the senior management considers a certain loss threshold q as ‘critical’ for the economic future of the bank. Such a threshold typically will be much lower than the αquantile of the loss distribution. Then, it makes sense to calculate the expectation of all losses exceeding q, weighted via the loss likelihoods illustrated by the loss distribution. Such a conditional expectation is called the expected shortfall (ESF) w.r.t. q of the credit portfolio. It is the mean loss or expected loss conditional 21 Shortfall measures are the most prominent examples of socalled coherent risk measures. 24 Structured Credit Portfolio Analysis, Baskets & CDOs So far, we kept the presentation fairly generic without making explicit assumptions regarding the distribution of involved random variables. In the subsequent sections, we will be more explicit in tail risk modeling mainly involving four different socalled copula functions, namely the Gaussian, Studentt, Clayton, and comonotonic copulas. We will see that copula functions as dependence modeling tools have a major impact on portfoliobased risk measures like EC and ESF. We will also see how copula functions influence the distribution of joint default times in credit portfolios. The consequences for the modeling of structured credit products will be discussed in great detail later on. 1.2.3 Practicability Versus Accuracy We close our brief walk through credit risk modeling aspects with Figure 1.9 illustrating the different tradeoffs credit modelers are exposed to when balancing between practicability and accuracy of their models, parameterizations and estimates. FIGURE 1.9: Finding a balance between practicability and accuracy in credit risk evaluation In vertical direction, Figure 1.9 shows at the top corner the wish for steadily improved risk parameter estimates. For instance, in the previous section we discussed how positive an increase of the discriminatory From Single Credit Risks to Credit Portfolios 25 power of a rating system can impact the bank’s P&L. The more accurate the rating systems and PD estimates of the bank the more reliable are credit decisions. In opposite vertical direction we find drivers of efficiency in the bank’s valuation and credit approval processes. For instance, reducing the number of qualitative factors in a rating system can significantly accelerate credit processes. Banks have to balance between efficiency and accuracy/sophistication in credit risk estimates, at singlename as well as at portfolio level. In horizontal direction, Figure 1.9 shows some other diametral forces in banking. At the lefthand side we find the clear demand for consequent risk/return steering. As soon as a bank has recognized the potential of sound credit risk models and steering measures these intiatives should be applied to the daytoday business of the bank. The classical ‘relationship banking paradigm’ points in the opposite direction. In times of overbanked loan markets, banks will think about it twice before they shock a longterm wellknown client by an increased risk premium as credit price component due to an increased exposure concentration with this client. The way out of the dilemma of welljustified diametral demands is active credit portfolio management (ACPM). Creating tailormade credit products for important clients and offering attractive conditions due to an active management of credit risks is a key factor of success in today’s credit business. Chapter 2 Default Baskets A default basket is a portfolio of not too many obligors, e.g., not more than 10, although there is no ‘officially agreed hard limit’ regarding the number of obligors allowed in order to speak of a basket. At least it is clear that a basket contains more than one obligor or asset, respectively, such that in case of moderate or low correlation between the intruments in the basket diversification effects will reduce the overall portfolio or basket risk. In this chapter, we model default baskets and certain related products. This chapter also serves as a preparation for the slightly more complicated Chapter 3 on CDOs. As we will see later, default baskets and CDOs are ‘close relatives’. 2.1 Introductory Example: Duo Baskets We start our discussion by the most simple basket one can think of, namely, a duo basket consisting of two creditrisky instruments only. The purpose of the following introductory discussion is twofold. First, we want to review the idea of diversification and its interplay with correlation. Second, we want to elaborate a first example illustrating the necessary balance between risk and return, a fundamental way of thinking in portfolio management and especially in the field of structured credit products. Let us assume that we consider the following two loans: • Loan A has a oneyear default probability of pA = 100 bps, loan B has a oneyear default probability of pB = 50 bps. • Both loans have a bullettype exposure profile, where ‘bullet’ refers to a nonamortizing loan with 100% of the exposure outstanding from the first to the last day of the term of the loan. 27 28 Structured Credit Portfolio Analysis, Baskets & CDOs • Both loans have an LGD of 100% such that in case of default the full outstanding exposure amount will be lost for the lending bank. Let us restrict ourselves to the oneyear horizon for the time being. According to our threshold model (see Equation (1.1)) explained in Chapter 1 we assume the existence of CWIs for obligors A and B and corresponding Bernoulli variables (1) LA = 1{CWI(1) < c(1) } A and A (1) LB = 1{CWI(1) < c(1) } B (2.1) B indicating default or survival within one year. In line with Equation (1.3), the default points of the obligors are determined by (1) cA = F−1 (1) (1) CWIA (pA ) and (1) cB = F−1 (1) (1) CWIB (pB ). (2.2) In order to come up with explicit numbers, let us assume that the CWIs of obligors A and B are standard normally distributed with a CWI correlation of 10%. If we want to embed the CWI correlation of two assets into a factor model as outlined in Section 1.2 then we can introduce a standard normally distributed random variable Y ∼ N (0, 1) and two independent variables εA , εB ∼ N (0, 1), independent of Y , and write (dropping the time index for a moment for reasons of a simplified notation) p √ (2.3) CWIA = ̺ Y + 1 − ̺ εA p √ CWIB = ̺ Y + 1 − ̺ εB such that CWIA , CWIB ∈ N (0, 1), and Corr[CWIA , CWIB ] = ̺A,B = ̺ where Corr[· , ·] denotes correlation. A CWI correlation of 10% determines ̺ = 0.1 for our working example. The default points of A and B can then easily be calculated by application of the standard normal quantile function N −1 [·], cA = N −1 [pA ] = −2.33 and cB = N −1 [pB ] = −2.58 confirming the natural expectation that the obligor with lower PD should be more bankruptcy remote; see also Appendix 6.6. Default Baskets 29 Now we are ready for studying the interplay of the two loans in our duo basket regarding default behavior. Let’s say the bank wants to allocate a certain amount of money to the duo basket, namely w percent of the available amount to obligor A and (1 − w) percent to obligor B. The question comes up which breakdown of capital by means of exposure weights wA = w and wB = (1 − w) will lead to a maximum benefit for the lending institute. This question is the classical starting point for balancing between risk and return of an investment. Regarding returns we have to make a reasonable working assumption1 . For the moment let us assume that the gross margin the bank earns with each loan is given by the formulas MarginA = w × (pA + 0.2 × pA (1 − pA ) + 0.005) MarginB = (1 − w) × (pB + 0.2 × pB (1 − pB ) + 0.005). At the oneyear horizon, pA and pB are the percentage ELs and pA (1 − pA ) and pB (1 − pB ) are the percentage default variances of loans A and B, assuming an LGD of 100%. Therefore, we assume that loans earn a gross margin incorporating a 100% charge on the expected loss plus a 20% charge on the variance of losses2 plus a uniform margin of 50 bps. For w = 1 we obtain the gross margin of loan A MarginA = pA + 0.2 × pA (1 − pA ) + 0.005 = 0.017 and for w = 0 we obtain the gross margin for loan B MarginB = pB + 0.2 × pB (1 − pB ) + 0.005 = 0.011. Due to the low default probabilities of the assets, the gross margin formula approximately yields the EL scaled by 1.2 plus an absolute offset of 50 bps. We can now calculate the portfolio profit µ as a weighted sum of margins, µ = µ(w) = MarginA + MarginB = w × (pA + 0.2 × pA (1 − pA ) + 0.005) + (1 − w) × (pB + 0.2 × pB (1 − pB ) + 0.005). 1 In the credit market, margins or spreads are not only determined by the magnitude of credit risk inherent in the considered credit instrument but also by its liquidity complemented, in case of structured investments, by some ‘complexity premium’. 2 In a portfolio context, one would rather pricein the economic capital cost based on the ECcontribution of the loan. Comparable approaches are used in RAROC (riskadjusted return over capital) and transfer pricing concepts. 30 Structured Credit Portfolio Analysis, Baskets & CDOs For measuring diversification effects in our duo basket we have to choose a risk measure sensitive to portfolio effects. The three standard measures matching this condition are EC, ESF, and UL where UL stands for unexpected loss as a synonym for the standard deviation of the portfolio’s loss distribution. For a twoname portfolio, EC and ESF are not the most useful measures so that in this particular case we decide in favor of the portfolio UL (here denotes by σ) given by the square root of the portfolio variance σ 2 w.r.t. the chosen weight w, σ 2 = σ 2 (w) = w2 pA (1 − pA ) + (1 − w)2 pB (1 − pB ) p + 2 r w(1 − w) pA (1 − pA )pB (1 − pB ) (2.4) where r = rA,B denotes the default correlation of the two obligors, (2.5) r = rA,B = Corr 1{CWI(1) < c(1) } , 1{CWI(1) < c(1) } A A B B N2 [N −1 [pA ], N −1 [pB ]; ̺A,B ] − pA pB p = pA (1 − pA )pB (1 − pB ) based on the normal distribution assumption we made for the obligor’s CWIs; see also Equation (2.3). In Equation (2.5), N2 [ · , · ; ̺] denotes the standard bivariate normal distribution function with a correlation of ̺. For ̺ = 10%, pA = 0.01 and pB = 0.005 we obtain a default correlation of 74 bps. The order of magnitude of the default correlation compared to the CWI correlation is not unusual. Note that in typical credit model approaches, default correlations are much lower than the correlation between underlying latent variables. Figure 2.1 illustrates the dependence of the default correlation r = rA,B on the PDs pA and pB and on the CWI correlation ̺ = ̺A,B . Table 2.1 shows the corresponding numbers. For every w ∈ [0, 1], we obtain a duo basket with a mix of a wportion of risk/return due to loan A and a (1−w)portion of risk/return arising from loan B. The portfolio’s risk/return can graphically be illustrated by plotting the curve of points σ(w), µ(w) 0≤w≤1 in 2dimensional risk/return space; see Figure 2.2. The case w = 0 leads to a duo basket consisting of loan B only, the case w = 1 is a basket with loan A only. Obviously, these cases are Default Baskets 31 FIGURE 2.1: Default correlation for different levels of CWI correlation and pairs of PDs not interesting because one can hardly call a single asset a basket or portfolio. For 0 < w < 1, both assets are contributing to the duo basket’s performance. With w increasing from 0 to 1 we move from the Bonly portfolio to the Aonly portfolio along the curve plotted in Figure 2.2. An interesting point on this curve is the duo basket where the portfolio UL attains its minimum. In line with classical portfolio theory we call this portfolio the minimum variance portfolio. Calculating the derivative of the portfolio variance (i.e., UL2 ) w.r.t. the weight w based on Equation (2.4), ∂ 2 σ (w) = 2 w pA (1 − pA ) − 2 (1 − w) pB (1 − pB ) + ∂w 32 Structured Credit Portfolio Analysis, Baskets & CDOs TABLE 2.1: Default correlation as a function of CWI correlation and default probabilities p + 2 r (1 − 2w) pA (1 − pA )pB (1 − pB ), we obtain (∂/∂w)σ 2 (w) = 0 for the minimum variance weight wmin = pB (1 − pB ) − r p (2.6) pA (1 − pA )pB (1 − pB ) p pA (1 − pA ) + pB (1 − pB ) − 2 r pA (1 − pA )pB (1 − pB ) yielding wmin = 0.33 if we insert pA , pB , and r into Equation (2.6). In Figure 2.2, one can see that the minimum variance portfolio corresponding to w = 0.33 offers more return at less risk (if risk is identified with volatility) than the single asset B. The benefit of w = 0.33 (basket) compared to w = 0 (single asset B) is due to the diversification effect arising from investing in two instead of in one single asset only. We also see that the diversification potential depends on the correlation of the assets in the basket. In Figure 2.2, we draw a return level (horizontal line) at µ = 150 bps. The lower the correlation of the two obligors the lower the risk (in terms of UL) of the 2asset portfolio and the higher the diversification benefit. If we consider the basket exclusively from a portfolioUL perspective, then the perfect correlation case (̺ = 1) is the worst situation one can obtain because the risk corresponding to a return of 150 bps attains its maximum. Increasing Default Baskets 33 FIGURE 2.2: Different duo baskets consisting of wmixed loans A and B in risk/return space the weight w from 0 to 1 yields the dashed (̺ = 1) or solid (̺ = 0.1) line or curve, respectively, in Figure 2.2 starting in asset B (w = 0) and ending in asset A (w = 1). In the perfect correlation case, we have no diversification benefit at all in the basket. Due to the same argument, the best possible diversification benefit can be achieved if we combine independent3 assets in a basket. Figure 2.2 is wellknown in portfolio theory and everyone involved in risk management has seen comparable pictures in the literature. Nevertheless, the duo basket case study will help us in the sequel to explore techniques for basket analysis in an easy ‘test environment’ before we turn our attention to more complicated portfolios. 3 Note that based on our model setup we always assume nonnegative CWI correlations. We do not allow for negative correlations leading to offsets in terms of hedging. 34 2.2 Structured Credit Portfolio Analysis, Baskets & CDOs First and SecondtoDefault Modeling Based on our duo basket example from Section 2.1, we can make some first steps in typical basket products. In Chapter 3, we will see that socalled equity investors in a CDO take the first loss of some reference portfolio consisting of creditrisky instruments. In the same way, the firsttodefault buyer/taker in a basket structure bears the loss arising from the first default in the basket. Analogously, the secondtodefault refers to the loss caused by the second default in the basket. Later, we will investigate first and secondtodefaults regarding their dependence on the chosen time horizon, hereby relying on dependent default times. For the moment, we restrict ourselves to the oneyear horizon in order to keep the exposition simple and reduced to certain aspects of these products. As a consequence, we suppress the time index for CWIs. For the duo basket, the question arises what first and secondtodefault means in terms of event probabilities and corresponding losses in the context of our example. In the sequel, we systematically investigate this question. First of all, we define some notation by writing (1) p1st = P 1{CWIA < cA } + 1{CWIB < cB } > 0 (2.7) for the oneyear firsttodefault probability and (1) p2nd = P 1{CWIA < cA } × 1{CWIB < cB } > 0 (2.8) (1) for the oneyear secondtodefault probability. Note that p1st is the (1) hitting probability of the basket, whereas p2nd is the wipeout probability of the basket. Equations (2.7) and (2.8) can also be written in terms of sets (i.e., ‘events’ is the language of probability theory) as (1) p1st = P {CWIA < cA } ∪ {CWIB < cB } (2.9) (1) p2nd = P {CWIA < cA } ∩ {CWIB < cB } (2.10) (1) (1) Let us now calculate p1st and p2nd for our duo basket example. 2.2.1 Proposition The oneyear firsttodefault probability equals (1) p1st = Z∞ −∞ gpA ,̺ (y) + gpB ,̺ (y)[1 − gpA ,̺ (y)] dN (y) (2.11) Default Baskets 35 where the conditional oneyear PD of A (analogously for B) is given by h c − √̺ y i A gpA ,̺ (y) = N √ . (2.12) 1−̺ Here, N [·] denotes the standard normal distribution function and cA = N −1 [pA ] represents the default point or default threshold of asset A. Proof. First, let us recall the derivation of the conditional PD gpA ,̺ (y), gpA ,̺ (y) = P[1{CWIA < cA } = 1  Y = y] p √ = P[ ̺ Y + 1 − ̺ εA < cA  Y = y] √ h i cA − ̺ Y √ = P εA < Y =y 1−̺ h c − √̺ y i A = N √ 1−̺ (see also Appendices 6.6 and 6.7). Then, Equation (2.11) can be explained as follows. The first summand is the likelihood that A defaults, the second summand is the likelihood that B defaults and A survives. (1) In settheoretic notation,4 the formula for p1st reflects the equation {CWIA < cA } ∪ {CWIB < cB } = = {CWIA < cA } ∪ {CWIB < cB }\{CWIA < cA } where M \N addresses all elements in M which are not in N . 2 Equation (2.12) can be more generally described in the context of Bernoulli mixture models; see [58], [51], Chapter 2 in [25] and Appendices 6.6 and 6.7. We will come back to this and related formulas again in the following sections. 2.2.2 Proposition The oneyear secondtodefault probability equals (1) p2nd Z∞ = gpA ,̺ (y)gpB ,̺ (y) dN (y). −∞ 4 See also Figure 2.41 for a comparable situation with three events. (2.13) 36 Structured Credit Portfolio Analysis, Baskets & CDOs Proof. Conditional on Y = y the default probabilities multiply for the joint default probability due to conditional independence in the mixture model. As usual in mixture models, we have to integrate the product of conditional PDs w.r.t. the mixing variable Y . Again, we could have worked with a settheoretic (eventbased) approach as in the proof of Proposition 2.2.1. 2 2.2.3 Remark The oneyear secondtodefault probability coincides with the joint default probability (JDP) of the two assets defined by (1) JDPA,B = JDPA,B = P[1{CWIA < cA } = 1, 1{CWIB < cB } = 1] (2.14) = 2π p 1 1− ̺2 −1 N −1 Z (pA ) N Z (pB ) 1 2 2 2 e− 2 (xA −2̺ xA xB +xB )/(1−̺ ) dxA dxB −∞ −∞ based on the joint normal distribution of the obligor’s CWIs in line with Equation (2.3). In explicit numbers, we obtain in our example (1) • p1st = 0.0149 for the oneyear firsttodefault probability and (1) • p2nd = 0.0001 for the oneyear secondtodefault probability. As a crossreference, we can check that (1) (1) p1st + p2nd = pA + pB . (1) (2.15) (1) Given p1st and p2nd we can now easily calculate the oneyear EL and other risk quantities, incorporating LGDs of the underlying assets and the exposure weight w in our duo basket. Before we close this section we want to briefly comment on the influ(1) (1) ence of PDs and the correlation impact on p1st and p2nd . Figures 2.3 and (1) (1) 2.4 illustrate the dependence of p1st and p2nd on the CWI correlation ̺. Here, we make an important structural observation based on Figures 2.3 and 2.4. 2.2.4 Remark The firsttodefault probability attains its maximum in case of zero CWI correlation, whereas the secondtodefault probability attains its maximum in case of perfect correlation. Default Baskets FIGURE 2.3: 37 (1) Influence of the CWI correlation on p1st Later in this book we will rediscover this fact and interpret it by saying that an investor taking the first default or loss in a basket or CDO is in a worst case scenario if assets default in a completely independent way because that makes defaults in a basket most unpredictable. In contrast, a secondtodefault event will only take place if both assets default. In addition, the higher the CWI correlation the higher the probability of a joint default event; see Figure 2.5. In the extreme case of perfect correlation, both default events are linearly related leading (1) to the highest possible value of p2nd . Again we have parallels to CDOs: investors in senior tranches suffer from high correlations because they make tail events (joint defaults) more likely. For reasons of completeness we briefly discuss the extreme cases ̺ = 0 and ̺ = 1 in terms of formulas. • ̺ = 0: According to Equation (2.12), conditioning on Y has no effect in case of zero correlation. Equations (2.11) and (2.13) then yield (1) p1st = pA + pB (1 − pA ) = 0.01495 (1) p2nd = pA × pB = 0.00005. 38 Structured Credit Portfolio Analysis, Baskets & CDOs FIGURE 2.4: (1) Influence of the CWI correlation on p2nd In other words, the zero correlation case corresponds to a situation where the systematic risk component is switchedoff completely. • ̺ = 1: In this case, idiosyncratic deviations from the systematic factor Y in Equation (2.3) are no longer possible. Then, we can replace CWIA and CWIB in Equation (2.14) by Y such that JDPA,B = P[1{Y < cA } = 1, 1{Y < cB } = 1] = P[Y < min{cA , cB }] = N min{N −1 [pA ], N −1 [pB ]} = min[pA , pB ]. (1) Therefore, we have p2nd = min[pA , pB ]. Since Equation (2.15) (1) must be fulfilled, we can conclude that p1st = max[pA , pB ]. We will come back to these and related aspects later in the text when we consider baskets and CDOs over multiyear periods rather than with respect to the oneyear horizon. Typically, baskets and CDOs have multiyear terms. A standard maturity for ‘real life’ baskets is 3 or 5 Default Baskets FIGURE 2.5: 39 Joint default probability (JDP) as a function of CWI correlation years so that we are motivated to leave the oneyear horizon perspective behind and turn our attention to time horizons longer than a year. In order to incorporate the time dimension, we have to introduce a concept for modeling the term structure of default probabilities. This is exercised in the following section, which then directly will lead us to dependent default times in Section 2.5. 2.3 Derivation of PD Term Structures In this section, we explain the calibration of PD term structures by means of an example based on published data from Standard and Poor’s (S&P) [108]. Note that PD term structures sometimes are also called credit curves, just to mention another keyword for the same object in order to make any search in the literature easier for interested readers. Note that, as long as nothing else is said, time in this section is 40 Structured Credit Portfolio Analysis, Baskets & CDOs measured in years. Typically, every bank has its own way to calibrate credit curves. We proceed in our exposition in three steps. First, we calibrate a socalled generator or Qmatrix Q w.r.t. an average oneyear migration matrix from S&P, describing migrations of a continuoustime Markov chain at ‘infinitesimal small’ time intervals. Next, we calculate the Markov PD term structure generated by Q and compare the result with empirical PD term structures from S&P. We will find what other people before us already discovered: although timehomogeneous Markov chains are very popular in credit risk modeling, their ability to fit empirical term structures is limited to some extent. Therefore, we leave the timehomogeneous Markov approach behind and turn our attention to nonhomogeneous Markov chains in continuous time. We will see that this approach is more fruitful than our first attempt. The empirical term structures published by S&P can be approximated by a nonhomogeneous Markov chain with surprising quality; see [29]. 2.3.1 A TimeHomogeneous Markov Chain Approach Timehomogeneous Markov chain approaches for the calibration of PD term structures are wellknown in the literature and have been applied over and over again by financial practitioners; see Israel, Rosenthal, and Wei [61], Jarrow, Lando, and Turnbull [62], and Kreinin and Sidelnikova [70], just to mention a few examples in the literature. In these papers, one finds several alternatives for the calibration of continuoustime Markov chains to a given oneyear migration matrix. For our examples in the sequel, we work with the socalled diagonal adjustment approach discussed in [70] and apply it to S&P data; see also [26]. Alternative approaches can be worked out analogously. In Section 1.1 we introduced the notion for PD term structures in (1.2) for some obligor i by the curve (t) (t) (t) (pi )t≥0 = (P[CWIi < ci ])t≥0 . (t) In other words, pi is the probability that obligor i defaults within the time interval [0, t], and the curve of these cumulative default probabilities, naturally increasing with increasing time, constitutes the PD term structure of the considered obligor. Going back to Table 1.1 in Section Default Baskets 41 (t) (t) 1.1, we find that oneyear default probabilities pi and pj for differently rated obligors i and j are naturally different so that we expect to see the same dependence of PD term structures on ratings of the corresponding obligors not only at the oneyear horizon but over the whole time axis. In fact, we will define term structures in a ratingmonotonic way such that the credit curve of obligor i is above the credit curve of obligor j at all times if obligor i has a higher oneyear PD (corresponding to a worse rating) than obligor j; see also Figure 2.13 and the discussion in the corresponding section. As mentioned at the beginning of this section, we now provide a stepbystep calibration of credit curves based on the socalled diagonal adjustment approach to timecontinuous homogeneous Markov chains. In practical applications, one can, in principal, follow a comparable approach but has to make sure that calibrated credit curves comply with the multiyear default history of the bank’s portfolio. We will come back to this point later in this section when we compare the calibrated credit curves with empirically observed cumulative default frequencies. Another difficulty we see in practice in this context are certain client segments, e.g., private clients, for which migration history cannot as easily build up as it is the case for corporate clients where balance sheet data flows into the bank at least once a year. For such segments, PD term structures are more difficult to obtain and have to be derived from historical multiyear defaults rather than from estimates based on credit migration data. A last argument worthwhile to be mentioned is the discussion about how realistic the Markov assumption for PD evolution over time possibly can be. Here, we clearly look at the problem through the ‘glasses of practitioners’. During recent years our jobs forced us to look at various approaches in the context of PD term structure calibration. Without going into details, we found, although the Markov assumption is somewhat questionable, that Markov approaches can yield fairly well approximations to historically observed cumulative default frequencies. In Section 2.3.2, we will see an example of a Markov approach that fits empirical data with outstanding quality. Rating agencies publish discrete credit curves based on cohorts of historically observed default frequencies annually, [108] is an example for such an agency report. The problem with historically observed cohorts is that the resulting multiyear PDs have a tendency to look kind of ‘saturated’ at longer horizons due to lackofdata problems. Comparing corporate bond default reports from 5 years ago with reports as of 42 Structured Credit Portfolio Analysis, Baskets & CDOs TABLE 2.2: Modified S&P average oneyear migration matrix today, one certainly recognizes major improvements on the data side. For ratings containing many clients (like BBB for instance), curves are smoother and do not imply zero forward PDs after a few years. Nevertheless, data quality still needs improvements in order to rely on historical data purely without ‘smoothing’ by some suitable model. The following Markov chain approach is an elegant way to overcome this problem and to generate continuoustime credit curves. Let us now start with the adjusted average oneyear migration matrix M = (mij )i,j=1,...,8 shown in Table 2.2 from S&P (see [108], Table 9). As said, we overcome the zero default observation problem for AAAratings by assuming an 0.2 bps default probability for AAArated customers. In addition, the matrix in Table 2.2 has been rownormalized in order to force row sums to be equal to 1 such that M is a stochastic matrix. Next, we need the following theorem based on a condition, which will be fulfilled in most of the cases, although we saw a few examples of migration matrices (typically calibrated ‘pointintime’ instead of ‘throughthecycle’) for which the condition was hurt for certain rating classes. We denote by I the identity matrix and by N the number of credit states. 2.3.1 Theorem If a migration matrix M = (mij )i,j=1,...,N is strictly diagonal dominant, i.e., mii > 12 for every i, then the logexpansion Q̃n n X (M − I)k = (−1)k+1 k k=1 (n ∈ N) converges to a matrix Q̃ = (q̃ij )i,j=1,...,N satisfying 1. PN j=1 q̃ij = 0 for every i = 1, ..., N ; Default Baskets 43 2. exp(Q̃) = M . The convergence Q̃n → Q̃ is geometrically fast. Proof. See Israel et al. [61]. 2 Theorem 2.3.1 leads us to Condition 1 of the following remark, which serves, complemented by two other conditions, as definition for a special type of square matrices known as generators or Qmatrices in Markov chain theory; see Chapter 6 in [25]. 2.3.2 Remark The generator of a timecontinuous Markov chain is given by a socalled Qmatrix Q = (qij )1≤i,j≤N satisfying the following properties: PN 1. j=1 qij = 0 for every i = 1, ..., N ; 2. 0 ≤ −qii < ∞ for every i = 1, ..., N ; 3. qij ≥ 0 for all i, j = 1, ..., N with i 6= j. For background on Markov chains we refer to the book by Noris [94]. Before we continue, let us briefly recall Theorem 2.3.1. If we calculate the logexpansion of M only for k = 1, then we obtain M −I as a firstorder approximation to the logexpansion of M . Because M is a stochastic matrix, it is immediately clear that the row sums of M − I are zero such that Condition 1 in Remark 2.3.2 naturally is fulfilled. Conditions 2 and 3 are also obvious. Therefore, M − I is the most simple generator we can obtain from a migration matrix M . Now, how are generators related to migration matrices? The next theorem, well known in Markov theory, gives an answer to this question. 2.3.3 Theorem The following conditions are equivalent (here we have the special situation Q ∈ R8×8 ): • Q satisfies Properties 1 to 3 in Remark 2.3.2. • exp(tQ) is a stochastic matrix for every t ≥ 0. 44 Structured Credit Portfolio Analysis, Baskets & CDOs TABLE 2.3: Calculation of 1storder generator approximation M − I TABLE 2.4: Exponential exp(M − I) of 1storder generator approximation Proof. See Noris [94], Theorem 2.1.2. 2 Above we mentioned that M − I is the most simple generator, which can be derived from M . Let us see what we get when we calculate the matrix exponential exp(M − I) corresponding to the situation t = 1 in Theorem 2.3.1. For this purpose, we first calculate M − I. Table 2.3 shows the result of this exercise. Next, we calculate the matrix exponential of M − I. For this, recall that the matrix exponential is given by the expansion exp(A) = ∞ X Ak k=0 k! for any square matrix A. Table 2.4 shows the result. We find that exp(M − I) already shows some similarities to the oneyear migration matrix M we started with, although the approximation is very rough. Based on this little exercise, one can imagine that with increasing k in Theorem 2.3.1 the logarithmic expansion of M applied as argument in the matrixvalued exponential function approaches the original matrix M with better and better approximation quality. However, as we will see in a moment, this does not necessarily mean that, in contrast to M − I, the logexpansion yields a generator at all. Default Baskets TABLE 2.5: Logexpansion of oneyear migration matrix M TABLE 2.6: Approximative generator (Qmatrix) for M 45 Let us now see what we get from the logexpansion (n → ∞ in Theorem 2.3.1) rather than looking at M −I (case n = 1) only. We calculate the logexpansion Q̃ = (q̃ij )i,j=1,...,8 of the oneyear migration matrix M = (mij )i,j=1,...,8 according to Theorem 2.3.1. This can be done with a calculation program like Mathematica or Matlab, but can as easily also be implemented in Excel/VBA. Table 2.5 shows the resulting matrix Q̃. Theorem 2.3.1 guarantees that Q̃ fulfills Condition 1 of generators listed in Remark 2.3.2. Condition 2 is also not hurt by Q̃, but Condition 3 is not fulfilled. So we see confirmed what we already indicated, namely, that (in contrast to M − I) the logexpansion of M not necessarily yields a Qmatrix. However, there are only three migration rates in Table 2.5 not in line with generator conditions: • q̃AAA,B = −1 bps, • q̃B,AAA = −1 bps, • q̃CCC,AA = −2 bps. Only these three entries disable Q̃ from being a generator matrix. Because these three values are very small numbers, we can follow the diagonal adjustment approach announced already several times and set the three negative migration rates equal to zero. In order for still being in line with the zero row sum condition, we then need to decrease the diagonal elements of rows AAA, B and CCC by an amount compensating 46 Structured Credit Portfolio Analysis, Baskets & CDOs for the increased row sums by setting negative entries to a zero value. The evident name diagonal adjustment for this procedure is mentioned in [70]. As a result we obtain a generator matrix Q = (qij )i,j=1,...,8 as shown in Table 2.6. From Theorem 2.3.1 we know that we get back the original migration matrix M from Q̃ by exp(Q̃). But what about getting M back from Q? Because we manipulated Q̃ in order to arrive at a generator Q, exp(Q) will not exactly equal M . What is the error distance between M and exp(Q)? Because the manipulation we did is kind of negligible, we already expect the result described in the following proposition. 2.3.4 Proposition M has an approximate Qmatrix representation by Q. The meansquare error can be calculated as v u 8 uX (mij − (exp(Q))ij )2 ≈ 0.00023. kM − exp(Q)k2 = t i,j=1 Proof. Just calculate the distance. 2 We conclude from Proposition 2.3.4 that we can work with Q instead of Q̃, hereby accepting the minor approximation error. Finding a Qmatrix representation Q for a timediscrete Markov chain represented by a transition matrix M is called an embedding of the timediscrete Markov chain represented by M into a timecontinuous Markov chain represented by its generator or Qmatrix Q in Markov theory. Of course, our embedding only holds in an approximate manner (see Proposition 2.3.4), but the error is negligibly small. Probabilists know that the existence of such embeddings is far from being obvious, and to some extent we have been very lucky that it worked so well with the S&Pbased migration matrix M . In [62], [61], and [70] one finds other ways to manage such embeddings, but as already said we found that the diagonal adjustment approach is well working in many situations. Readers interested in actually applying such embeddings in the context of their credit risk engines should take the time to experiment with different approaches in order to find their own ‘best way’ to deal with the challenge of PD term structure calibrations. For the timehomogeneous case we achieved our target. Our efforts have been rewarded by a generator Q such that exp(Q) ≈ M . The Default Baskets 47 credit curves can be read off from the collection of matrices (exp(tQ))t≥0 by looking at the default columns. More precisely, for any asset i in the portfolio with rating R = R(i) we obtain (t) pi = (etQ )row(R),8 (2.16) where row(R) denotes the transition matrix row corresponding to the given rating R. Figure 2.6 shows a chart of credit curves from t = 0 to t = 50 years on a quarterly base, calibrated as just described. FIGURE 2.6: Calibrated credit curves based on a timecontinuous and timehomogeneous Markov chain approach The curves in Figure 2.6 are kind of typical. Credit curves assigned to subinvestment grade ratings have a tendency to slow down their growth, because conditional on having survived for some time the chances for further survival improve. For good ratings, we see the opposite effect. For instance, assets with excellent credit quality have no upside potential because they are already outstanding but have a large downside risk of potential deterioration of their credit quality over time. 48 Structured Credit Portfolio Analysis, Baskets & CDOs If we calculate quarterly forward PDs via (t) p̃i (t) = (t−0.25) pi − pi (t−0.25) 1 − pi (t ≥ 0.25) (2.17) we find a mean reversion effect in PD term structures. Moody’s KMV writes in the documentation of their rating system RiskCalc for private companies the following statement: We noticed that obligors appear to exhibit mean reversion in their credit quality. In other words, good credits today tend to become somewhat worse credits over time and bad credits (conditional upon survival) tend to become better credits over time. Default studies on rated bonds by Moodys Investors Services support this assertion, as do other studies. We found further evidence for mean reversion in both our proprietary public firm default databases and in the Credit Research Database data on private firms. [85] The homogeneous Markov approach, focussing on the aspect of mean reversion, is in line with findings of Moody’s KMV. Mean reversion effects drive the forward or conditional PDs over time, respectively; see Figure 2.7. Therefore, we can hope to have at least a somewhat realistic model at disposal. However, if we compare PD term structures on a purely empirical base as published by S&P in [108] with the just calibrated term structure, we find as a disappointing surprise that curves do not match very well; see Figure 2.8. When we compare the homogeneous continuoustime Markov chain (HCTMCbased) credit curves with observed multiyear default frequencies from S&P we immediately get the impression that HCTMC modelbased PDs systematically overestimate empirically observed default frequencies (exception: AAA). Here are some remarks: • For good rating classes, AAA is the extreme case, the