Table of Contents 1. The past, present and future of credit risk 7 . The past, present and future of credit risk 7 Charting the course 9 Views of credit 9 2. The default/no-default world, and factor models 15 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 1 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 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 1 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? 103 103 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 0. Portfolio optimization: the importance of convexity 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 . An advanced approach to correlation 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 |