Rating Based Modeling of Credit Risk (eBook)

Front Cover 1
Rating Based Modeling of Credit Risk 4
Copyright Page 5
Table of Contents 8
Preface 12
Chapter 1. Introduction: Credit Risk Modeling, Ratings, and Migration Matrices 14
1.1 Motivation 14
1.2 Structural and Reduced Form Models 15
1.3 Basel II, Scoring Techniques, and Internal Rating Systems 16
1.4 Rating Based Modeling and the Pricing of Bonds 17
1.5 Stability of Transition Matrices, Conditional Migrations, and Dependence 18
1.6 Credit Derivative Pricing 19
1.7 Chapter Outline 20
Chapter 2. Rating and Scoring Techniques 24
2.1 Rating Agencies, Rating Processes, and Factors 24
2.1.1 The Rating Process 27
2.1.2 Credit Rating Factors 29
2.1.3 Types of Rating Systems 30
2.2 Scoring Systems 30
2.3 Discriminant Analysis 32
2.4 Logit and Probit Models 34
2.4.1 Logit Models 35
2.4.2 Probit Models 36
2.5 Model Evaluation: Methods and Difficulties 38
2.5.1 Model Performance and Benchmarking 38
2.5.2 Model Accuracy, Type I and II Errors 42
Chapter 3. The New Basel Capital Accord 44
3.1 Overview 44
3.1.1 The First Pillar—Minimum Capital Requirement 46
3.1.2 The Second Pillar—Supervisory Review Process 48
3.1.3 The Third Pillar—Market Discipline 48
3.2 The Standardized Approach 49
3.2.1 Risk Weights for Sovereigns and for Banks 49
3.2.2 Risk Weights for Corporates 52
3.2.3 Maturity 52
3.2.4 Credit Risk Mitigation 53
3.3 The Internal Ratings Based Approach 54
3.3.1 Key Elements and Risk Components 54
3.3.2 Derivation of the Benchmark Risk Weight Function 55
3.3.3 Asset Correlation 59
3.3.4 The Maturity Adjustment 61
3.3.5 Expected, Unexpected Losses and the Required Capital 63
3.4 Summary 63
Chapter 4. Rating Based Modeling 66
4.1 Introduction 66
4.2 Reduced Form and Intensity Models 67
4.2.1 The Model by Jarrow and Turnbull (1995) 72
4.2.2 The Model Suggested by Madan and ¨ Unal (1998) 73
4.2.3 The Model Suggested by Lando (1998) 74
4.2.4 The Model of Duffie and Singleton (1999) 76
4.3 The CreditMetrics Model 76
4.4 The CreditRisk+ Model 81
4.4.1 The First Modeling Approach 81
4.4.2 Modeling Severities 82
4.4.3 Shortcomings of the First Modeling Approach 84
4.4.4 Extensions in the CR+ Model 85
4.4.5 Allocating Obligors to One of Several Factors 85
4.4.6 The pgf for the Number of Defaults 86
4.4.7 The pgf for the Default Loss Distribution 88
4.4.8 Generalization of Obligor Allocation 88
4.4.9 The Default Loss Distribution 89
Chapter 5. Migration Matrices and the Markov Chain Approach 90
5.1 The Markov Chain Approach 90
5.1.1 Generator Matrices 91
5.2 Discrete Versus Continuous-Time Modeling 93
5.2.1 Some Conditions for the Existence of a Valid Generator 99
5.3 Approximation of Generator Matrices 101
5.3.1 The Method Proposed by Jarrow, Lando, and Turnbull (1997) 101
5.3.2 Methods Suggested by Israel, Rosenthal,and Wei (2000) 102
5.4 Simulating Credit Migrations 105
5.4.1 Time-Discrete Case 105
5.4.2 Time-Continuous Case 106
5.4.3 Nonparametric Approach 107
Chapter 6. Stability of Credit Migrations 110
6.1 Credit Migrations and the Business Cycle 110
6.2 The Markov Assumptions and Rating Drifts 115
6.2.1 Likelihood Ratio Tests 116
6.2.2 Rating Drift 117
6.2.3 An Empirical Study 118
6.3 Time Homogeneity of Migration Matrices 122
6.3.1 Tests Using the Chi-Square Distance 123
6.3.2 Eigenvalues and Eigenvectors 123
6.4 Migration Behavior and Effects on Credit VaR 126
Chapter 7. Measures for Comparison of Transition Matrices 142
7.1 Classical Matrix Norms 142
7.2 Indices Based on Eigenvaluesand Eigenvectors 144
7.3 Risk-Adjusted Difference Indices 146
7.3.1 The Direction of the Transition (DIR) 146
7.3.2 Transition to a Default or Nondefault State (TD) 147
7.3.3 The Probability Mass of the Cell (PM) 148
7.3.4 Migration Distance (MD) 149
7.3.5 Devising a Distance Measure 149
7.3.6 Difference Indices for the Exemplary Matrices 153
7.4 Summary 155
Chapter 8. Real-World and Risk-Neutral Transition Matrices 158
8.1 The JLT Model 158
8.2 Adjustments Based on the Discrete-Time Transition Matrix 161
8.3 Adjustments Based on the Generator Matrix 164
8.3.1 Modifying Default Intensities 165
8.3.2 Modifying the Rows of the Generator Matrix 166
8.3.3 Modifying Eigenvalues of the TransitionProbability Matrix 167
8.4 An Adjustment Technique Basedon Economic Theory 169
8.5 Risk-Neutral Migration Matrices and Pricing 170
Chapter 9. Conditional Credit Migrations: Adjustments and Forecasts 172
9.1 Overview 172
9.2 The CreditPortfolioView Approach 173
9.3 Adjustment Based on Factor Model Representations 178
9.3.1 Deriving an Index for the Credit Cycle 179
9.3.2 Conditioning of the Migration Matrix 180
9.3.3 A Multifactor Model Extension 184
9.4 Other Methods 186
9.5 An Empirical Study on Different Forecasting Methods 188
9.5.1 Forecasts Using the Factor Model Approach 189
9.5.2 Forecasts Using Numerical Adjustment Methods 191
9.5.3 Regression Models 192
9.5.4 In-Sample Results 193
9.5.5 Out-of-Sample Forecasts 197
Chapter 10. Dependence Modeling and Credit Migrations 200
10.1 Introduction 200
10.1.1 Independence 201
10.1.2 Dependence 202
10.2 Capturing the Structure of Dependence 204
10.2.1 Under General Multivariate Distributions 208
10.3 Copulas 209
10.3.1 Examples of Copulas 211
10.3.2 Properties of Copulas 212
10.3.3 Constructing Multivariate Distributions with Copulas 213
10.4 Modeling Dependent Defaults 214
10.5 Modeling Dependent Migrations 217
10.5.1 Dependence Based on a Credit Cycle Index 218
10.5.2 Dependence Based on Individual Transitions 219
10.5.3 Approaches Using Copulas 220
10.6 An Empirical Study on Dependent Migrations 222
10.6.1 Distribution of Defaults 222
10.6.2 The Distribution of Rating Changes 225
Chapter 11. Credit Derivatives 230
11.1 Introduction 230
11.1.1 Types of Credit Derivatives 232
11.1.2 Collateralized Debt Obligations (CDO) 235
11.2 Pricing Single-Named Credit Derivatives 237
11.3 Modeling and Pricing of Collateralized Debt Obligations and Basket Credit Derivatives 244
11.3.1 Estimation of Macroeconomic Risk Factors 248
11.3.2 Modeling of Conditional Migrations and Recovery Rates 250
11.3.3 Some Empirical Results 251
11.4 Pricing Step-Up Bonds 256
11.4.1 Step-Up Bonds 257
11.4.2 Pricing of Step-Up Bonds 257
Bibliography 262
Index 272