The Econometrics of Individual Risk (ebook)

Argumento de The Econometrics of Individual Risk (ebook)

"I don't know of any other book with this orientation. It promises to fill a gap in both the econometric and finance literature."--Torben G. Andersen, Kellogg School of Management, Northwestern University

"The Econometrics of Individual Risk gives a nice overview of a new area and manages to combine a good technical account with clarity. No other book to my knowledge has managed to fill this particular niche. It is well organized and well written, and the scholarship is excellent."--Kevin Dowd, Nottingham University Business School

"This book is simply outstanding. Its approach is powerful yet practical, and many of its results and insights are original. The combination of analytical power and applied sense is very, very rare. And the financial events modeled in the book are important and common in applied financial contexts."--Francis X. Diebold, University of Pennsylvania

Christian Gourieroux is Director of the Laboratory for Finance and Insurance at the Center for Research in Economics and Statistics (CREST) in Paris, and Professor at the University of Toronto. He is the coauthor of Statistics and Econometric Models, Simulation-Based Econometric Methods, and Time Series and Dynamic Models. Joann Jasiak is Associate Professor of Economics at York University, Toronto. She and Christian Gourieroux are the authors of Financial Econometrics (Princeton).0Preface xi

Chapter 1: Introduction 1
1.1 Market Risk and Individual Risk 1
1.2 Risk Variable 2
1.3 Scores 3
1.4 Organization of the Book 4
References 5

Chapter 2: Dichotomous Risk 7
2.1 Risk Prediction and Segmentation 7
2.1.1 Risk Prediction 8
2.1.2 Segmentation 11
2.2 Econometric Models 14
2.2.1 Discriminant Analysis 14
2.2.2 Dichotomous Qualitative Models 15
2.2.3 Comparison of Discriminant and Logit Models 18
2.3 Risk Heterogeneity 19
2.4 Concluding Remarks 20
2.5 Appendix: The Logistic Distribution 20
References 21

Chapter 3: Estimation 23
3.1 Estimation Methods 23
3.1.1 The Maximum Likelihood Approach 23
3.1.2 Maximum Likelihood Estimation of a Logit Model 25
3.1.3 Maximum Likelihood Estimation in Linear Discriminant Analysis 27
3.1.4 Test of the Linear Discriminant Hypothesis 28
3.2 Significance Tests 29
3.2.1 Likelihood-Based Testing Procedures 30
3.2.2 Application of the LM Test to the Logit Model 31
3.3 Implementation 32
3.3.1 Development of Score Methodology 33
3.3.2 Mortgage Score 35
3.4 Concluding Remarks 39
References 40

Chapter 4: Score Performance 43
4.1 Performance and Selection Curves 43
4.1.1 Definitions 43
4.1.2 Desirable Properties of a Score 46
4.1.3 Comparison of Scores 47
4.2 Discriminant Curves 49
4.2.1 Definitions 50
4.2.2 Linear Discriminant Analysis 52
4.3 Demand Monitoring, Credit Granting, and Scores 52
4.3.1 Time-Varying Quality of Credit Applicants 53
4.3.2 Analysis of Credit-Granting Decision 55
4.3.3 Performance Curves 58
4.4 Concluding Remarks 58
4.5 Appendix: Positive Dependence 59
References 60

Chapter 5: Count Data Models 61
5.1 Poisson Regression 62
5.1.1 The Model 62
5.1.2 Maximum Likelihood Estimator 63
5.1.3 Relationship with the Dichotomous Qualitative Model 64
5.2 The Negative-Binomial Regression 64
5.2.1 Model with Gamma Heterogeneity 64
5.2.2 The Bonus-Malus Scheme 66
5.3 Semi-Parametric Analysis 69
5.3.1 Mean and Variance Estimators 70
5.3.2 Estimation of the Heterogeneity Distribution 71
5.3.3 Determination of the Premium 72
5.4 Applications 73
5.4.1 Car Insurance 73
5.4.2 Presentation of Results 77
5.5 Concluding Remarks 82
References 83

Chapter 6: Durations 85
6.1 Duration Distributions 86
6.1.1 Characterizations of a Duration Distribution 86
6.1.2 Duration Dependence 88
6.1.3 Basic Duration Distributions 89
6.2 Duration Models 92
6.2.1 The Exponential Regression Model 93
6.2.2 The Exponential Model with Gamma Heterogeneity 94
6.2.3 Heterogeneity and Negative Duration Dependence 95
6.3 Semi-Parametric Models 98
6.3.1 Accelerated Hazard Model 98
6.3.2 Proportional Hazard Model 99
6.4 Applications 100
6.4.1 Pension Fund 100
6.4.2 Interest Rate Spreads 101
6.4.3 Prepayment Analysis 103
6.5 Concluding Remarks 107
6.6 Appendix 109
6.6.1 Expected Residual Lifetime 109
6.6.2 Computation of the Premium Rate for the Pension Contract 110
References 111

Chapter 7: Endogenous Selection and Partial Observability 113
7.1 Analysis of Dichotomous Risks from a Stratified Sample 113
7.1.1 Description of the Population and the Sample 113
7.1.2 Exogenous Stratification 115
7.1.3 Endogenous Stratification 115
7.1.4 The Role of Stratified Samples 117
7.2 Truncation and Censoring in Duration Models 117
7.2.1 Censoring 117
7.2.2 Truncation 118
7.2.3 Competing Risks 119
7.3 Bias Correction Using Rejected Credit Applications 120
7.3.1 Selectivity Bias 120
7.3.2 Boundaries for Risk Prediction 121
7.3.3 A Bivariate Model for Bias Correction 122
7.4 Concluding Remarks 126
7.5 Appendix: First-Order Expansion of the C.D.F. of a Bivariate Normal
Distribution 126
References 126

Chapter 8: Transition Models 129
8.1 Homogeneous Markov Chains 130
8.1.1 Distribution of the Markov Chain 130
8.1.2 Alternative Parametrizations of a Markov Chain 132
8.1.3 Two-State Space 134
8.1.4 Qualitative Representation of the Process 135
8.1.5 Estimation 136
8.2 Explanatory Variables 137
8.2.1 Specification of the Transition Probabilities 138
8.2.2 Specification of the Adjustment and Long-Run Parameters 138
8.2.3 Time-Dependent Markov Chain 139
8.3 Transitions between Score Categories 140
8.3.1 Revolving Consumer Credit 140
8.3.2 Corporate Rating Dynamics 143
8.4 Concluding Remarks 146
References 146

Chapter 9: Multiple Scores 149
9.1 Examples 150
9.1.1 Default Risk and Preselection 150
9.1.2 Term Structure of Default 151
9.1.3 Differentiated Incident Severity 152
9.1.4 Default and Prepayment 154
9.1.5 Default and Credit Promotion 156
9.1.6 Polytomous Logit Model 157
9.1.7 The Hypothesis of Irrelevant Alternatives 158
9.2 Profit- (Utility-) Optimizing Decisions 159
9.2.1 Promotional Mailing Decisions 159
9.2.2 Time-to-Default 161
9.2.3 Utility-Maximizing Behavior 162
9.3 Multi-Score Reduction Technique 163
9.3.1 Basic Notions 163
9.3.2 Singular Value Decomposition (SVD) 164
9.3.3 Statistical Inference 165
9.4 Household Portfolio Allocation 166
9.4.1 Description of the Data Set 166
9.4.2 Model Estimation 169
9.4.3 Reduction of the Number of Scores 176
9.5 Concluding Remarks 178
References 179

Chapter 10: Serial Dependence in Longitudinal Data 181
10.1 Poisson and Compound Poisson Processes 182
10.1.1 Poisson Process 182
10.1.2 Compound Poisson Process 184
10.1.3 From Discrete Time to Continuous Time 185
10.2 Models with Serial Dependence 186
10.2.1 Autoregressive Models 188
10.2.2 Time-Dependent Heterogeneity 192
10.3 Applications 195
10.3.1 Cost Sensitivity with Respect to Transitory Shocks 195
10.3.2 Learning in Revolving Credit 197
10.4 Concluding Remarks 205
10.5 Appendix: Distributions of the Duration and Count Variables 205
10.5.1 Distribution of the First Duration 205
10.5.2 Independence of Durations 206
10.5.3 Distribution of the Count Variable 206
References 206

Chapter 11: Management of Credit Risk 209
11.1 Measures of Risk and Capital Requirement 209
11.1.1 Value-at-Risk 210
11.1.2 Properties of a Risk Measure 212
11.2 Credit Portfolio 213
11.2.1 The P&L Distribution for a Credit Portfolio When the Horizon Is
Equal To the Maturity 214
11.2.2 The P&L Distribution for a Credit Portfolio When the Horizon Is
Shorter Than the Maturity 216
11.3 Corporate Bond Portfolio 223
11.3.1 Informational Content of Bond Prices 223
11.3.2 Default Correlation 224
11.3.3 Stochastic Transition Model 230
11.4 Concluding Remarks 235

References 235
Index 239