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Credit Suisse: Credit Portfolio Modeling
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Table of Contents
1. The past, present and future of credit risk 7
. The past, present and future of credit risk 7Introduction 7
Charting the course 9
Views of credit 9
2. The default/no-default world, and factor models 15
15How 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
1Modeling 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
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
1Résumé 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
103Why 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
0. Portfolio optimization: the importance of convexity 111Risk 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
. An advanced approach to correlation 121Correlations 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
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