Rating Based Modeling of Credit Risk Theory and Application of Migration Matrices

Rating Based Modeling of Credit Risk Theory and Application of Migration Matrices

Preface xi 1 Introduction: Credit Risk Modeling, Ratings, and Migration Matrices 1 1.1 Motivation 1 1.2 Structural and Reduced Form Models 2 1.3 Basel II, Scoring Techniques, and Internal Rating Systems 3 1.4 Rating Based Modeling and the Pricing of Bonds 4 1.5 Stability of Transition Matrices, Conditional Migrations and Dependence 5 1.6 Credit Derivative Pricing 6 1.7 Chapter Outline 7 2 Rating and Scoring Techniques 11 2.1 Rating Agencies, Rating Processes, and Factors 11 2.1.1 The Rating Process 14 2.1.2 Credit Rating Factors 16 2.1.3 Types of Rating Systems 17 2.2 Scoring Systems 17 2.3 Discriminant Analysis 19 2.4 Logit and Probit Models 21 2.4.1 Logit Models 22 2.4.2 Probit Models 23 2.5 Model Evaluation: Methods and Difficulties 25 2.5.1 Model Performance and Benchmarking 25 2.5.2 Model Accuracy, Type I and II Errors 29 3 The New Basel Capital Accord 31 3.1 Overview 31 3.1.1 The First Pillar Minimum Capital Requirement 33 3.1.2 The Second Pillar Supervisory Review Process 35 3.1.3 The Third Pillar Market Discipline 35 3.2 The Standardized Approach 36 3.2.1 Risk Weights for Sovereigns and for Banks 36 3.2.2 Risk Weights for Corporates 39 3.2.3 Maturity 39 3.2.4 Credit Risk Mitigation 40 3.3 The Internal Ratings Based Approach 41 3.3.1 Key Elements and Risk Components 41 3.3.2 Derivation of the Benchmark Risk Weight Function 42 3.3.3 Asset Correlation 46 3.3.4 The Maturity Adjustment 48 3.3.5 Expected, Unexpected Losses and the Required Capital... 50 3.4 Summary 50

Rating Based Modeling 53 4.1 Introduction 53 4.2 Reduced Form and Intensity Models 54 4.2.1 The Model by Jarrow and Turnbull (1995) 59 4.2.2 The Model Suggested by Madan and Unal (1998) 60 4.2.3 The Model Suggested by Lando (1998) 61 4.2.4 The Model of Duffie and Singleton (1999) 63 4.3 The CreditMetrics Model 63 4.4 The CreditRisk+ Model 68 4.4.1 The First Modeling Approach 68 4.4.2 Modeling Severities 69 4.4.3 Shortcomings of the First Modeling Approach 71 4.4.4 Extensions in the CR+ Model 72 4.4.5 Allocating Obligors to One of Several Factors 72 4.4.6 The pgf for the Number of Defaults 73 4.4.7 The pgf for the Default Loss Distribution 75 4.4.8 Generalization of Obligor Allocation 75 4.4.9 The Default Loss Distribution 76 Migration Matrices and the Markov Chain Approach 77 5.1 The Markov Chain Approach 77 5.1.1 Generator Matrices 78 5.2 Discrete Versus Continuous-Time Modeling 80 5.2.1 Some Conditions for the Existence of a Valid Generator... 86 5.3 Approximation of Generator Matrices 88 5.3.1 The Method Proposed by Jarrow, Lando, and Turnbull (1997) 88 5.3.2 Methods Suggested by Israel, Rosenthal, and Wei (2000).. 89 5.4 Simulating Credit Migrations 92 5.4.1 Time-Discrete Case 92 5.4.2 Time-Continuous Case 93 5.4.3 Nonparametric Approach 94 Stability of Credit Migrations 97 6.1 Credit Migrations and the Business Cycle 97 6.2 The Markov Assumptions and Rating Drifts 102 6.2.1 Likelihood Ratio Tests 103 6.2.2 Rating Drift 104 6.2.3 An Empirical Study 105 6.3 Time Homogeneity of Migration Matrices 109 6.3.1 Tests Using the Chi-Square Distance 110 6.3.2 Eigenvalues and Eigenvectors 110 6.4 Migration Behavior and Effects on Credit VaR 113 6.5 Stability of Probability of Default Estimates 120 Measures for Comparison of Transition Matrices 129 7.1 Classical Matrix Norms 129 7.2 Indices Based on Eigenvalues and Eigenvectors 131 7.3 Risk-Adjusted Difference Indices 133 7.3.1 The Direction of the Transition (DIR) 133 7.3.2 Transition to a Default or Nondefault State (TD) 134 7.3.3 The Probability Mass of the Cell (PM) 135

ix 7.3.4 Migration Distance (MD) 136 7.3.5 Devising a Distance Measure 136 7.3.6 Difference Indices for the Exemplary Matrices 140 7.4 Summary 142 8 Real-World and Risk-Neutral Transition Matrices 145 8.1 The JLT Model 145 8.2 Adjustments Based on the Discrete-Time Transition Matrix... 148 8.3 Adjustments Based on the Generator Matrix 151 8.3.1 Modifying Default Intensities 152 8.3.2 Modifying the Rows of the Generator Matrix 153 8.3.3 Modifying Eigenvalues of the Transition Probability Matrix. 154 8.4 An Adjustment Technique Based on Economic Theory 156 8.5 Risk-Neutral Migration Matrices and Pricing 157 9 Conditional Credit Migrations: Adjustments and Forecasts 159 9.1 Overview 159 9.2 The CreditPortfolioView Approach 160 9.3 Adjustment Based on Factor Model Representations 165 9.3.1 Deriving an Index for the Credit Cycle 166 9.3.2 Conditioning of the Migration Matrix 167 9.3.3 A Multifactor Model Extension 171 9.4 Other Methods 173 9.5 An Empirical Study on Different Forecasting Methods 175 9.5.1 Forecasts Using the Factor Model Approach 176 9.5.2 Forecasts Using Numerical Adjustment Methods 178 9.5.3 Regression Models 179 9.5.4 In-Sample Results 180 9.5.5 Out-of-Sample Forecasts 184 10 Dependence Modeling and Credit Migrations 187 10.1 Introduction 187 10.1.1 Independence 188 10.1.2 Dependence 189 10.2 Capturing the Structure of Dependence 191 10.2.1 Under General Multivariate Distributions 195 10.3 Copulas 196 10.3.1 Examples of Copulas 198 10.3.2 Properties of Copulas 199 10.3.3 Constructing Multivariate Distributions with Copulas... 200 10.4 Modeling Dependent Defaults 201 10.5 Modeling Dependent Migrations 204 10.5.1 Dependence Based on a Credit Cycle Index 205 10.5.2 Dependence Based on Individual Transitions 206 10.5.3 Approaches Using Copulas 207 10.6 An Empirical Study on Dependent Migrations 209 10.6.1 Distribution of Defaults 209 10.6.2 The Distribution of Rating Changes 212 11 Credit Derivatives 217 11.1 Introduction 217 11.1.1 Types of Credit Derivatives 219 11.1.2 Collateralized Debt Obligations (CDO) 222

x Contents 11.2 Pricing Single-Named Credit Derivatives 224 11.3 Modeling and Pricing of Collateralized Debt Obligations and Basket Credit Derivatives 231 11.3.1 Estimation of Macroeconomic Risk Factors 235 11.3.2 Modeling of Conditional Migrations and Recovery Rates.. 237 11.3.3 Some Empirical Results 238 11.4 Pricing Step-Up Bonds 243 11.4.1 Step-Up Bonds 244 11.4.2 Pricing of Step-Up Bonds 244 Bibliography 249 Index 259