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Preface v About the Authors ix A Brief Guide to the Book xi . From Single Credit Risks to Credit Portfolios 1 1.1 Modeling Single-Name Credit Risk ... . ...... 2 11 Ratings and Default Probabilities . . . . . 2 S1. 2 Credit Exposure ......... .... . . 10 S1.3 Loss Given Default . . . . . ... 14 1.2 Modeling Portfolio Credit Risk ........ . 17 1.2. Systematic and Idiosyncratic Credit Risk . 17 1.2.2 Loss Distribution of Credit Portfolios . .. 20 1.2.3 Practicability Versus Accuracy ......... 24 2 Default Baskets 27 2.1 Introductory Example: Duo Baskets . . . . .27 2.2 First- and Second-to-Default Modeling ...... 34 2.3 Derivation of PD Term Structures . ... . . . . 39 2.3.1 A Time-Homogeneous Markov Chain Approach . 40 2.3 2 A Non-Homogeneous Markov Chain Approach . 52 2.3 Extrapolation Problems for PD Term Structures 57 2.4 Duo Basket Evaluation for Multi-Year Horizons . . 59 .5 Dependent Default Tines .. . . 67 2 5 1 Default Times and PD Term Structures . . 67 2.5.2 Survival Function and Hazard Rate . . .. .68 2 5.3 Calculation of Default Time Densities and Haz- ard Rate Functions ............ 69 2.5a4 From Latent Variables to Default Times . . 78 2. 55 Dependence Modeling via Copula Functions . . 85 2 5.6 Copulas in Practice . ............ . 93 2.5.7 Visualization of Copula Differences and Mathe- matical Description by Dependence Meaures . 99 25.8 Impact of Copula Differences to he Du Baset 113 2,5.9 A Word of Caution .. .. ..... . 118 2.6 Nth-to Default Modeling ......... 20 2.6.1 Nth-to-Default Basket with the Gaussian Copula 121 2.6.2 Nth-to-Default Basket with the Student-t Copula 127 26.3 Nth-to-Default Basket with the Clayton Copula. 127 2.6.4 Nth-to-Default Simulation Study ..... 129 2.6.5 Evaluation of Cash Flows in Default Baskets . 136 2.66 Scenario Analysis ...... .. .... . 140 2.7 Example of a Basket Credit-Linked Noe (CLN . 147 3 Collateralized Debt and Synthetc Obligations 65 3.1 A General Perspective on CDO Modeling ....... 166 3.It A. Primer on CDOs .......... .. 167 3.12 Risk Transfer ...... .. . . 172 3.1.3 Spread and Rating Arbitrage ........... 178 3.1.4 FPunding Benefits ............ . 184 3.1.5 Regulatory Capital Relief ............ . 186 3.2 CDO Modeling Principles ............ . 190 3.3 CDO Modeling Approaches ............... 194 3.3.1 Introduction of a Sample CSO ...... .... 194 3.3.2 A First-Order Look at CSO Performance ... 199 3.3.3 Monte Carlo Simulation of he CSO .. 202 3.3,4 Implementing an Excess Cash Trap ....... 210 3.3.5 Muilti-Step and First Passage Time Models . 213 3.36 Analytic Semi-Anlyytic, and Comonotonic CDO Evaluation Approaches ............. 220 3.4 Single-Tranihe CDOs (STCDOs) ....... .... 250 3.4.1 Basics of Single-Tranche CDOs ... .. .... 250 34.2 CDS Indices as Reference Pool for STCDOs . 253 3.4.3 ITfaxx Europe Uranhed .......... 259 3.44 ITraxx Europe Index T'anches: Pricing, Delta Hedging, and ied Correlations ...... 271 35 Tranche Risk Measures .................. 287 3.5.1 Expected Shortfall Contributions ...... . 288 3.,.2 Tranche Hit Contributions of Single Names . . . 292 3.5.3 Applications: Asset Selection. Cost-to-Securitize 294 3.5.4 Remarks on Portfolios of CDOs . . . 299 4 Some Practical Remarks 303 5 Suggestions for Further Reading 307 0 Appendix 311 6.1 The Gamma Distribution . . . . .... . 311 6.2 The Chi-Square Distribution 3........... 312 6.3 The Student-t Distribution .. ... . .. .. 312 6 A. Natural Glayton-Like Copula Example . . . 314 6.5 Entropy-Based Rationale for Gaussian and Exponential Distributions as Natural Standard Choices . . . 315 6.6 Tail Orietation in Typical Latent Variable Credit Risk Models .. ............. .. 318 6.7 The Vasicek Limit Distribution .... ...... 320 6 8 One-Factor Versus Multi-Factor Models ... . 322 6.9 Description of the Sample Portfolio ......... 329 6.10 CDS Names in CDX.NA.IG and iTraxx Europe . 332 References 339