Credit-Risk Modelling (eBook)

David Jamieson Bolder is currently head of the World Bank Group’s (WBG) model-risk function. Prior to this appointment, he provided analytic support to the Bank for International Settlements’ (BIS) treasury and asset-management functions and worked in quantitative roles at the Bank of Canada, the World Bank Treasury, and the European Bank for Reconstruction and Development. He has authored numerous papers, articles, and chapters in books on financial modelling, stochastic simulation, and optimization. He has also published a comprehensive book on fixed-income portfolio analytics. His career has focused on the application of mathematical techniques towards informing decision-making in the areas of sovereign-debt, pension-fund, portfolio-risk, and foreign-reserve management.

Foreword 6
Preface 8
My Motivation 8
Transparency and Accessibility 9
Concreteness 9
Multiplicity of Perspective 10
Some Important Caveats 11
References 12
Acknowledgements 13
Contents 14
List of Figures 21
List of Tables 25
List of Algorithms 28
1 Getting Started 31
1.1 Alternative Perspectives 33
1.1.1 Pricing or Risk-Management? 33
1.1.2 Minding our P's and Q's 36
1.1.3 Instruments or Portfolios? 37
1.1.4 The Time Dimension 39
1.1.5 Type of Credit-Risk Model 40
1.1.6 Clarifying Our Perspective 41
1.2 A Useful Dichotomy 41
1.2.1 Modelling Implications 43
1.2.2 Rare Events 45
1.3 Seeing the Forest 48
1.3.1 Modelling Frameworks 49
1.3.2 Diagnostic Tools 51
1.3.3 Estimation Techniques 53
1.3.4 The Punchline 54
1.4 Prerequisites 54
1.5 Our Sample Portfolio 56
1.6 A Quick Pre-Screening 57
1.6.1 A Closer Look at Our Portfolio 57
1.6.2 The Default-Loss Distribution 59
1.6.3 Tail Probabilities and Risk Measures 60
1.6.4 Decomposing Risk 63
1.6.5 Summing Up 67
1.7 Final Thoughts 67
References 68
Part I Modelling Frameworks 69
Reference 70
2 A Natural First Step 71
2.1 Motivating a Default Model 72
A Bit of Structure 73
2.1.1 Two Instruments 75
2.1.2 Multiple Instruments 76
2.1.3 Dependence 80
2.2 Adding Formality 80
2.2.1 An Important Aside 83
2.2.2 A Numerical Solution 85
2.2.2.1 Bernoulli Trials 86
2.2.2.2 Practical Details 88
2.2.2.3 Some Results 90
2.2.3 An Analytical Approach 92
2.2.3.1 Putting It into Action 94
2.2.3.2 Comparing Key Assumptions 96
2.3 Convergence Properties 97
Convergence in Probability 98
Almost-Sure Convergence 99
Cutting to the Chase 102
2.4 Another Entry Point 103
A Numerical Implementation 106
The Analytic Model 107
The Law of Rare Events 110
2.5 Final Thoughts 112
References 112
3 Mixture or Actuarial Models 114
3.1 Binomial-Mixture Models 115
Conditional Independence 116
Default-Correlation Coefficient 117
The Distribution of DN 118
Convergence Properties 120
3.1.1 The Beta-Binomial Mixture Model 121
3.1.1.1 Beta-Parameter Calibration 124
3.1.1.2 Back to Our Example 125
3.1.1.3 Non-homogeneous Exposures 127
3.1.2 The Logit- and Probit-Normal Mixture Models 130
3.1.2.1 Deriving the Mixture Distributions 132
3.1.2.2 Numerical Integration 135
3.1.2.3 Logit- and Probit-Normal Calibration 136
3.1.2.4 Logit- and Probit-Normal Results 139
3.2 Poisson-Mixture Models 142
3.2.1 The Poisson-Gamma Approach 144
3.2.1.1 Calibrating the Poisson-Gamma Mixture Model 146
3.2.1.2 A Quick and Dirty Calibration 149
3.2.1.3 Poisson-Gamma Results 151
3.2.2 Other Poisson-Mixture Approaches 154
3.2.2.1 A Calibration Comparison 156
3.2.3 Poisson-Mixture Comparison 158
3.3 CreditRisk+ 160
3.3.1 A One-Factor Implementation 160
3.3.2 A Multi-Factor CreditRisk+ Example 170
3.4 Final Thoughts 176
References 177
4 Threshold Models 178
4.1 The Gaussian Model 179
4.1.1 The Latent Variable 179
4.1.2 Introducing Dependence 181
4.1.3 The Default Trigger 183
4.1.4 Conditionality 184
4.1.5 Default Correlation 187
4.1.6 Calibration 190
4.1.7 Gaussian Model Results 191
4.2 The Limit-Loss Distribution 194
4.2.1 The Limit-Loss Density 199
4.2.2 Analytic Gaussian Results 201
4.3 Tail Dependence 204
4.3.1 The Tail-Dependence Coefficient 205
4.3.2 Gaussian Copula Tail-Dependence 208
4.3.3 t-Copula Tail-Dependence 209
4.4 The t-Distributed Approach 211
4.4.1 A Revised Latent-Variable Definition 211
4.4.2 Back to Default Correlation 215
4.4.3 The Calibration Question 217
4.4.4 Implementing the t-Threshold Model 219
4.4.5 Pausing for a Breather 222
4.5 Normal-Variance Mixture Models 222
4.5.1 Computing Default Correlation 226
4.5.2 Higher Moments 227
4.5.3 Two Concrete Cases 229
4.5.4 The Variance-Gamma Model 230
4.5.5 The Generalized Hyperbolic Case 231
4.5.6 A Fly in the Ointment 233
4.5.7 Concrete Normal-Variance Results 235
4.6 The Canonical Multi-Factor Setting 240
4.6.1 The Gaussian Approach 240
4.6.2 The Normal-Variance-Mixture Set-Up 243
4.7 A Practical Multi-Factor Example 247
4.7.1 Understanding the Nested State-Variable Definition 248
4.7.2 Selecting Model Parameters 250
4.7.3 Multivariate Risk Measures 253
4.8 Final Thoughts 254
References 255
5 The Genesis of Credit-Risk Modelling 257
5.1 Merton's Idea 258
5.1.1 Introducing Asset Dynamics 261
5.1.2 Distance to Default 264
5.1.3 Incorporating Equity Information 266
5.2 Exploring Geometric Brownian Motion 268
5.3 Multiple Obligors 273
5.3.1 Two Choices 274
5.4 The Indirect Approach 275
5.4.1 A Surprising Simplification 277
5.4.2 Inferring Key Inputs 279
5.4.3 Simulating the Indirect Approach 280
5.5 The Direct Approach 283
5.5.1 Expected Value of An,T 285
5.5.2 Variance and Volatility of An,T 287
5.5.3 Covariance and Correlation of An,T and Am,T 289
5.5.4 Default Correlation Between Firms n and m 291
5.5.5 Collecting the Results 293
5.5.6 The Task of Calibration 293
5.5.7 A Direct-Approach Inventory 298
5.5.8 A Small Practical Example 298
5.6 Final Thoughts 308
References 310
Part II Diagnostic Tools 312
References 313
6 A Regulatory Perspective 314
6.1 The Basel Accords 315
6.1.1 Basel IRB 317
6.1.2 The Basic Structure 319
6.1.3 A Number of Important Details 322
6.1.4 The Full Story 327
6.2 IRB in Action 330
6.2.1 Some Foreshadowing 333
6.3 The Granularity Adjustment 336
6.3.1 A First Try 338
6.3.2 A Complicated Add-On 339
6.3.3 The Granularity Adjustment 344
6.3.4 The One-Factor Gaussian Case 345
6.3.5 Getting a Bit More Concrete 352
6.3.6 The CreditRisk+ Case 355
6.3.6.1 Some Loose Ends 364
6.3.7 A Final Experiment 367
6.3.8 Final Thoughts 373
References 375
7 Risk Attribution 377
7.1 The Main Idea 378
7.2 A Surprising Relationship 381
7.2.1 The Justification 383
7.2.2 A Direct Algorithm 387
7.2.3 Some Illustrative Results 390
7.2.4 A Shrewd Suggestion 392
7.3 The Normal Approximation 394
7.4 Introducing the Saddlepoint Approximation 398
7.4.1 The Intuition 399
7.4.2 The Density Approximation 402
7.4.3 The Tail Probability Approximation 404
7.4.4 Expected Shortfall 407
7.4.5 A Bit of Organization 408
7.5 Concrete Saddlepoint Details 410
7.5.1 The Saddlepoint Density 414
7.5.2 Tail Probabilities and Shortfall Integralls 417
7.5.3 A Quick Aside 418
7.5.4 Illustrative Results 419
7.6 Obligor-Level Risk Contributions 420
7.6.1 The VaR Contributions 421
7.6.2 Shortfall Contributions 426
7.7 The Conditionally Independent Saddlepoint Approximation 432
7.7.1 Implementation 438
7.7.2 A Multi-Model Example 441
7.7.3 Computational Burden 446
7.8 An Interesting Connection 447
7.9 Final Thoughts 452
References 452
8 Monte Carlo Methods 454
8.1 Brains or Brawn? 455
8.2 A Silly, But Informative Problem 456
8.3 The Monte Carlo Method 464
8.3.1 Monte Carlo in Finance 465
8.3.2 Dealing with Slowness 469
8.4 Interval Estimation 470
8.4.1 A Rough, But Workable Solution 470
8.4.2 An Example of Convergence Analysis 472
8.4.3 Taking Stock 475
8.5 Variance-Reduction Techniques 475
8.5.1 Introducing Importance Sampling 476
8.5.2 Setting Up the Problem 478
8.5.3 The Esscher Transform 482
8.5.4 Finding ? 485
8.5.5 Implementing the Twist 487
8.5.6 Shifting the Mean 492
8.5.7 Yet Another Twist 499
8.5.8 Tying Up Loose Ends 501
8.5.9 Does It Work? 504
8.6 Final Thoughts 510
References 511
Part III Parameter Estimation 513
9 Default Probabilities 514
9.1 Some Preliminary Motivation 515
9.1.1 A More Nuanced Perspective 516
9.2 Estimation 521
9.2.1 A Useful Mathematical Object 522
9.2.2 Applying This Idea 529
9.2.3 Cohort Approach 531
9.2.4 Hazard-Rate Approach 534
9.2.5 Getting More Practical 535
9.2.6 Generating Markov-Chain Outcomes 536
9.2.7 Point Estimates and Transition Statistics 541
9.2.8 Describing Uncertainty 546
9.2.8.1 Likelihood Theory 546
9.2.8.2 The Bootstrap Technique 553
9.2.9 Interval Estimates 561
9.2.10 Risk-Metric Implications 564
9.3 Risk-Neutral Default Probabilities 567
9.3.1 Basic Cash-Flow Analysis 567
9.3.2 Introducing Default Risk 570
9.3.3 Incorporating Default Risk 576
9.3.4 Inferring Hazard Rates 580
9.3.5 A Concrete Example 584
9.4 Back to Our P's and Q's 590
9.5 Final Thoughts 594
References 594
10 Default and Asset Correlation 597
10.1 Revisiting Default Correlation 598
10.2 Simulating a Dataset 602
10.2.1 A Familiar Setting 603
10.2.2 The Actual Results 609
10.3 The Method of Moments 611
10.3.1 The Threshold Case 615
10.4 Likelihood Approach 617
10.4.1 The Basic Insight 618
10.4.2 A One-Parameter Example 620
10.4.3 Another Example 624
10.4.4 A More Complicated Situation 628
10.5 Transition Likelihood Approach 637
10.5.1 The Elliptical Copula 638
10.5.2 The Log-Likelihood Kernel 644
10.5.3 Inferring the State Variables 648
10.5.4 A Final Example 650
10.6 Final Thoughts 655
References 656
A The t-Distribution 658
A.1 The Chi-Squared Distribution 659
A.2 Toward the t-Distribution 660
A.3 Simulating Correlated t Variates 664
A.4 A Quick Example 668
B The Black-Scholes Formula 670
B.1 Changing Probability Measures 670
B.2 Solving the Stochastic Differential Equation 674
B.3 Evaluating the Integral 676
B.4 The Final Result 679
C Markov Chains 680
C.1 Some Background 680
C.2 Some Useful Results 682
C.3 Ergodicity 684
D The Python Code Library 688
D.1 Explaining Some Choices 689
D.2 The Library Structure 690
D.3 An Example 692
D.4 Sample Exercises 693
References 694
Index 696
Author Index 701