Risk Management in Credit Portfolios (eBook)

Risk Management in Credit Portfolios 3
Foreword 5
Preface 7
Contents 9
List of Figures 13
List of Tables 15
Abbreviations 17
Chapter 1: Introduction 21
1.1 Problem Definition and Objectives of This Work 21
1.2 Course of Investigation 22
Chapter 2: Credit Risk Measurement in the Context of Basel II 25
2.1 Banking Supervision and Basel II 25
2.2 Measures of Risk in Credit Portfolios 28
2.2.1 Risk Parameters and Expected Loss 28
2.2.2 Value at Risk, Tail Conditional Expectation, and Expected Shortfall 31
2.2.3 Coherency of Risk Measures 36
2.2.4 Estimation and Statistical Errors of VaR and ES 42
2.3 The Unconditional Probability of Default Within the Asset Value Model of Merton 45
2.4 The Conditional Probability of Default Within the One-Factor Model of Vasicek 48
2.5 Measuring Credit Risk in Homogeneous Portfolios with the Vasicek Model 51
2.6 Measuring Credit Risk in Heterogeneous Portfolios with the ASRF Model of Gordy 55
2.7 Measuring Credit Risk Within the IRB Approach of Basel II 59
2.8 Appendix 63
2.8.1 Alternative Representation of the ES as an Indicator Function 63
2.8.2 Application of Itô´s Lemma 64
2.8.3 Application of Bayes´ Theorem for Continuous Distributions 65
2.8.4 Limit Distribution and Probability Density Function in the Vasicek Model 66
2.8.5 VaR and ES of the Limit Distribution in the Vasicek Model 68
2.8.6 Alternative Representation of the Bivariate Normal Distribution 69
2.8.7 Application of the Strong Law of Large Numbers 70
2.8.8 Application of Kronecker´s Lemma 72
2.8.9 Identity of the VaR in the ASRF Model 73
2.8.10 Identity of the ES in the ASRF Model 74
Chapter 3: Concentration Risk in Credit Portfolios and Its Treatment Under Basel II 77
3.1 Types of Concentration Risk 77
3.2 Incurrence and Relevance of Concentration Risk 79
3.3 Measurement and Management of Concentration Risk 82
3.4 Heuristic Approaches for the Measurement of Concentration Risk 87
3.5 Review of the Literature on Model-Based Approaches of Concentration Risk Measurement 90
Chapter 4: Model-Based Measurement of Name Concentration Risk in Credit Portfolios 93
4.1 Fundamentals and Research Questions on Name Concentration Risk 93
4.2 Measurement of Name Concentration Using the Risk Measure Value at Risk 95
4.2.1 Considering Name Concentration with the Granularity Adjustment 95
4.2.1.1 First-Order Granularity Adjustment for One-Factor Models 95
4.2.1.2 First-Order Granularity Adjustment for the Vasicek Model 100
4.2.1.3 Second-Order Granularity Adjustment for One-Factor Models 102
4.2.1.4 Second-Order Granularity Adjustment for the Vasicek Model 105
4.2.2 Numerical Analysis of the VaR-Based Granularity Adjustment 107
4.2.2.1 Impact on the Portfolio-Quantile 107
4.2.2.2 Size of Fine Grained Risk Buckets 110
4.2.2.3 Probing First-Order Granularity Adjustment 114
4.2.2.4 Probing Second-Order Granularity Adjustment 118
4.2.2.5 Probing Granularity for Inhomogeneous Portfolios 121
4.3 Measurement of Name Concentration Using the Risk Measure Expected Shortfall 123
4.3.1 Adjusting for Coherency by Parameterization of the Confidence Level 123
4.3.2 Considering Name Concentration with the Granularity Adjustment 128
4.3.2.1 First-Order Granularity Adjustment for One-Factor Models 128
4.3.2.2 First-Order Granularity Adjustment for the Vasicek Model 131
4.3.2.3 Second-Order Granularity Adjustment for One-Factor Models 132
4.3.2.4 Second-Order Granularity Adjustment for the Vasicek Model 133
4.3.3 Moment Matching Procedure for Stochastic LGDs 134
4.3.4 Numerical Analysis of the ES-Based Granularity Adjustment 141
4.3.4.1 Impact on the Portfolio-Quantile 141
4.3.4.2 Size of Fine Grained Risk Buckets 143
4.3.4.3 Probing First-Order Granularity Adjustment 146
4.3.4.4 Probing Second-Order Granularity Adjustment 150
4.3.4.5 Probing Granularity for Inhomogeneous Portfolios 153
4.4 Interim Result 154
4.5 Appendix 156
4.5.1 Alternative Derivation of the First-Order Granularity Adjustment 156
4.5.2 First and Second Derivative of VaR 163
4.5.2.1 First Derivative 164
4.5.2.2 Second Derivative 165
4.5.3 Probability Density Function of Transformed Random Variables 167
4.5.4 VaR-Based First-Order Granularity Adjustment for a Normally Distributed Systematic Factor 168
4.5.5 VaR-Based First-Order Granularity Adjustment for Homogeneous Portfolios 169
4.5.6 Arbitrary Derivatives of VaR 170
4.5.6.1 Mathematical Basics 170
4.5.6.1.1 Laplace Transform and Dirac´s Delta Function 170
4.5.6.1.2 Laurent Series, Singularities, and Complex Residues 171
4.5.6.1.3 Partitions 173
4.5.6.2 Determination of the Derivatives 173
4.5.6.2.1 Derivatives of the Distribution Function 173
4.5.6.2.2 Implicit Derivatives: Complex Residue Form 176
4.5.6.2.3 Implicit Derivatives: Combinatorial Form 177
4.5.6.2.4 Completion of the Derivation 181
4.5.7 Determination of the First Five Derivatives of VaR 183
4.5.8 Order of the Derivatives of VaR 188
4.5.9 VaR-Based Second-Order Granularity Adjustment for a Normally Distributed Systematic Factor 189
4.5.10 Third Conditional Moment of Losses 194
4.5.11 Difference Between the VaR Definitions 196
4.5.12 Identity of ES Within the Basel Framework 197
4.5.13 Arbitrary Derivatives of ES 198
4.5.14 Determination of the First Five Derivatives of ES 199
4.5.15 ES-Based Second-Order Granularity Adjustment for a Normally Distributed Systematic Factor 200
4.5.16 Probability Density Function of the Logit-Normal Distribution 201
Chapter 5: Model-Based Measurement of Sector Concentration Risk in Credit Portfolios 203
5.1 Fundamentals and Research Questions on Sector Concentration Risk 203
5.2 Incorporation of Sector Concentrations Using Multi-Factor Models 205
5.2.1 Structure of Multi-Factor Models and Basel II-Consistent Parameterization Through a Correlation Matching Procedure 205
5.2.2 Accounting for Sector Concentrations with the Model of Pykhtin 210
5.2.2.1 Derivation of the VaR-Based Multi-Factor Adjustment 210
5.2.2.2 Derivation and Implementation of the ES-Based Multi-Factor Adjustment 215
5.2.3 Accounting for Sector Concentrations with the Model of Cespedes, Herrero, Kreinin and Rosen 217
5.2.3.1 Design of the Diversification Factor 217
5.2.3.2 Computation of the Diversification Factor by Simulation 219
5.2.4 Accounting for Sector Concentrations with the Model of Düllmann 222
5.2.4.1 The Binomial Expansion Technique 222
5.2.4.2 The Infectious Defaults Model 224
5.2.4.3 Integrating Infectious Defaults into the BET-Model 227
Setup of the Model 227
Calibration and Implementation of the Model 229
5.3 Performance of Multi-Factor Models 232
5.3.1 Analysis for Deterministic Portfolios 232
5.3.2 Simulation Study for Homogeneous and Heterogeneous Portfolios 235
5.4 Interim Result 239
5.5 Appendix 240
5.5.1 Optimal Choice of the Single Correlation Factor 240
5.5.2 Conditional Correlation 243
5.5.3 Calculation of the Decomposed Variance 244
5.5.4 Derivatives of the Decomposed Variance Terms 248
5.5.5 Moment Matching in the BET-Model 253
5.5.5.1 Matching the First Moment 253
5.5.5.2 Matching the Second Moment 254
5.5.6 Interrelation of the Pairwise Default Correlation and the Asset Correlation 255
5.5.7 Expected Number of Defaults in the Infectious Defaults Model 256
Chapter 6: Conclusion 257
References 261