credit portfolio modeling handbook

Table of Contents

1. The past, present and future of credit risk 7

Introduction 7

Charting the course 9

Views of credit 9

2. The default/no-default world, and factor models 15

How are portfolios modeled? 15

Pictorial example 17

What is default probability? 18

What is default rate distribution? 19

Numerical example 20

Correlation numbers: A useful construct? 20

Portfolio analysis 21

Putting some structure in 22

Gaussian copula (quasi-Merton) model 22

Portfolio analysis with 1-factor Gaussian copula 25

Conclusions 26

3. Risk and optionalities 27

Nonlinear assets 27

Asset distributions 32

Correlation 38

Conclusions 39

4. Demystifying copulas 41

Modeling multivariate distributions 41

Copulas 44

Examples of copulas 45

Properties of copulas 49

Portfolio analysis with different copulas 54

Conclusions 56

5. Thinking unsystematically 57

Résumé 57

Independent random variables, pictorially 59

Independent random variables, mean-variance 64

Conclusions 64

The Quantitative Credit Strategist

4 29 October 2004

6. Characteristically elegant 65

Résumé 65

The characteristic function (Fourier transform) 65

Definition and properties 65

Examples of characteristic functions 66

Families 67

Inversion: some first thoughts 68

Central Limit Theorem 69

Numerical inversion 75

Conclusions 76

7. Posing on the saddle: the cowboys of portfolio theory 77

Résumé 77

The moment-generating function (MGF) 78

Review of Central Limit Theorem 80

Enter the saddle 82

Demonstration 83

Uniform approximation property 87

Second derivation of the saddle-point method 87

Alternative uses of saddle-point approximations 89

Conclusions 90

8. Getting the full picture 91

Résumé 91

Combining systematic and unsystematic risk 93

Example: Gaussian copula model 95

Granularity adjustment 95

Numerical example 1 95

Numerical example 2 98

Conclusions 98

Appendix: derivation of the granularity adjustment 100

9. Risk measures: how long is a risky piece of string? 103

Why think about risk measures? 103

Examples of risk measures 104

Artzner.s theory 104

Summary table 108

Subadditivity, convexity and risk sensitivity in more depth 109

Conclusion 110

10. Portfolio optimization: the importance of convexity 111

Risk and reward: the traditional view 111

Does this work with other risk measures? 113

Risk contributions 114

Convexity 115

An example of linear (non-convex) optimization 117

Conclusions 119

11. An advanced approach to correlation 121

Correlations between pairs of random variables 121

Time series and the differencing problem 126

Issues in credit-equity correlation 128

PR+2 Methodology . Part I 130

Back-testing of issuer-sector correlation 134

PR+2 Methodology . Part II 139

Back-testing 142

Appendix . Example of Bayes. theorem 145

12. Volatility, correlations, and the CAPM 147

Portfolio volatility 147

Risk contributions 149

Capital Asset Pricing Model (CAPM) 149

Convexity 151

Conclusions 152

13. Contributions to VaR and CVaR 153

Recap: from standard deviation to VaR 153

VaR contribution 156

From VaR to CVaR 163

Example 165

Conclusions 170

Appendix 171

Appendix Credit risk modeling 175

Overview of models 175

Reduced form models 176

Hybrid models 178

Structural models 180

Advanced structural models 183

CUSP: CSFB.s structural model 186

CUSP+: CSFB.s advanced structural credit risk model 188

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