Quantitative Risk Management: Concepts, Techniques, and Tools

Preface xiii

CHAPTER 1: Risk in Perspective 1
1.1 Risk 1
1.1.1 Risk and Randomness 1
1.1.2 Financial Risk 2
1.1.3 Measurement and Management 3
1.2 A Brief History of Risk Management 5
1.2.1 From Babylon to Wall Street 5
1.2.2 The Road to Regulation 8
1.3 The New Regulatory Framework 10
1.3.1 Basel II 10
1.3.2 Solvency 2 13
1.4 Why Manage Financial Risk? 15
1.4.1 A Societal View 15
1.4.2 The Shareholder's View 16
1.4.3 Economic Capital 18
1.5 Quantitative Risk Management 19
1.5.1 The Nature of the Challenge 19
1.5.2 QRM for the Future 22

CHAPTER 2: Basic Concepts in Risk Management 25
2.1 Risk Factors and Loss Distributions 25
2.1.1 General Definitions 25
2.1.2 Conditional and Unconditional Loss Distribution 28
2.1.3 Mapping of Risks:Some Examples 29
2.2 Risk Measurement 34
2.2.1 Approaches to Risk Measurement 34
2.2.2 Value-at-Risk 37
2.2.3 Further Comments on VaR 40
2.2.4 Other Risk Measures Based on Loss Distributions 43
2.3 Standard Methods for Market Risks 48
2.3.1 Variance -Covariance Method 48
2.3.2 Historical Simulation 50
2.3.3 Monte Carlo 52
2.3.4 Losses over Several Periods and Scaling 53
2.3.5 Backtesting 55
2.3.6 An Illustrative Example 55

CHAPTER 3: Multivariate Models 61
3.1 Basics of Multivariate Modelling 61
3.1.1 Random Vectors and Their Distributions 62
3.1.2 Standard Estimators of Covariance and Correlation 64
3.1.3 The Multivariate Normal Distribution 66
3.1.4 Testing Normality and Multivariate Normality 68
3.2 Normal Mixture Distributions 73
3.2.1 Normal Variance Mixtures 73
3.2.2 Normal Mean-Variance Mixtures 77
3.2.3 Generalized Hyperbolic Distributions 78
3.2.4 Fitting Generalized Hyperbolic Distributions to Data 81
3.2.5 Empirical Examples 84
3.3 Spherical and Elliptical Distributions 89
3.3.1 Spherical Distributions 89
3.3.2 Elliptical Distributions 93
3.3.3 Properties of Elliptical Distributions 95
3.3.4 Estimating Dispersion and Correlation 96
3.3.5 Testing for Elliptical Symmetry 99
3.4 Dimension Reduction Techniques 103
3.4.1 Factor Models 103
3.4.2 Statistical Calibration Strategies 105
3.4.3 Regression Analysis of Factor Models 106
3.4.4 Principal Component Analysis 109

CHAPTER 4: Financial Time Series 116
4.1 Empirical Analyses of Financial Time Series 117
4.1.1 Stylized Facts 117
4.1.2 Multivariate Stylized Facts 123
4.2 Fundamentals of Time Series Analysis 125
4.2.1 Basic Definitions 125
4.2.2 ARMA Processes 128
4.2.3 Analysis in the Time Domain 132
4.2.4 Statistical Analysis of Time Series 134
4.2.5 Prediction 136
4.3 GARCH Models for Changing Volatility 139
4.3.1 ARCH Processes 139
4.3.2 GARCH Processes 145
4.3.3 Simple Extensions of the GARCH Model 148
4.3.4 Fitting GARCH Models to Data 150
4.4 Volatility Models and Risk Estimation 158
4.4.1 Volatility Forecasting 158
4.4.2 Conditional Risk Measurement 160
4.4.3 Backtesting 162
4.5 Fundamentals of Multivariate Time Series 164
4.5.1 Basic Definitions 164
4.5.2 Analysis in the Time Domain 166
4.5.3 Multivariate ARMA Processes 168
4.6 Multivariate GARCH Processes 170
4.6.1 General Structure of Models 170
4.6.2 Models for Conditional Correlation 172
4.6.3 Models for Conditional Covariance 175
4.6.4 Fitting Multivariate GARCH Models 178
4.6.5 Dimension Reduction in MGARCH 179
4.6.6 MGARCH and Conditional Risk Measurement 182

CHAPTER 5: Copulas and Dependence 184
5.1 Copulas 184
5.1.1 Basic Properties 185
5.1.2 Examples of Copulas 189
5.1.3 Meta Distributions 192
5.1.4 Simulation of Copulas and Meta Distributions 193
5.1.5 Further Properties of Copulas 195
5.1.6 Perfect Dependence 199
5.2 Dependence Measures 201
5.2.1 Linear Correlation 201
5.2.2 Rank Correlation 206
5.2.3 Coefficients of Tail Dependence 208
5.3 Normal Mixture Copulas 210
5.3.1 Tail Dependence 210
5.3.2 Rank Correlations 215
5.3.3 Skewed Normal Mixture Copulas 217
5.3.4 Grouped Normal Mixture Copulas 218
5.4 Archimedean Copulas 220
5.4.1 Bivariate Archimedean Copulas 220
5.4.2 Multivariate Archimedean Copulas 222
5.4.3 Non-exchangeable Archimedean Copulas 224
5.5 Fitting Copulas to Data 228
5.5.1 Method-of-Moments using Rank Correlation 229
5.5.2 Forming a Pseudo-Sample from the Copula 232
5.5.3 Maximum Likelihood Estimation 234

CHAPTER 6: Aggregate Risk 238
6.1 Coherent Measures of Risk 238
6.1.1 The Axioms of Coherence 238
6.1.2 Value-at-Risk 241
6.1.3 Coherent Risk Measures Based on Loss Distributions 243
6.1.4 Coherent Risk Measures as Generalized Scenarios 244
6.1.5 Mean-VaR Portfolio Optimization 246
6.2 Bounds for Aggregate Risks 248
6.2.1 The General Fréchet Problem 248
6.2.2 The Case of VaR 250
6.3 Capital Allocation 256
6.3.1 The Allocation Problem 256
6.3.2 The Euler Principle and Examples 257
6.3.3 Economic Justification of the Euler Principle 261

CHAPTER 7: Extreme Value Theory 264
7.1 Maxima 264
7.1.1 Generalized Extreme Value Distribution 265
7.1.2 Maximum Domains of Attraction 267
7.1.3 Maxima of Strictly Stationary Time Series 270
7.1.4 The Block Maxima Method 271
7.2 Threshold Exceedances 275
7.2.1 Generalized Pareto Distribution 275
7.2.2 Modelling Excess Losses 278
7.2.3 Modelling Tails and Measures of Tail Risk 282
7.2.4 The Hill Method 286
7.2.5 Simulation Study of EVT Quantile Estimators 289
7.2.6 Conditional EVT for Financial Time Series 291
7.3 Tails of Specific Models 293
7.3.1 Domain of Attraction of Fréchet Distribution 293
7.3.2 Domain of Attraction of Gumbel Distribution 294
7.3.3 Mixture Models 295
7.4 Point Process Models 298
7.4.1 Threshold Exceedances for Strict White Noise 299
7.4.2 The POT Model 301
7.4.3 Self-Exciting Processes 306
7.4.4 A Self-Exciting POT Model 307
7.5 Multivariate Maxima 311
7.5.1 Multivariate Extreme Value Copulas 311
7.5.2 Copulas for Multivariate Minima 314
7.5.3 Copula Domains of Attraction 314
7.5.4 Modelling Multivariate Block Maxima 317
7.6 Multivariate Threshold Exceedances 319
7.6.1 Threshold Models Using EV Copulas 319
7.6.2 Fitting a Multivariate Tail Model 320
7.6.3 Threshold Copulas and Their Limits 322

CHAPTER 8: Credit Risk Management 327
8.1 Introduction to Credit Risk Modelling 327
8.1.1 Credit Risk Models 327
8.1.2 The Nature of the Challenge 329
8.2 Structural Models of Default 331
8.2.1 The Merton Model 331
8.2.2 Pricing in Merton's Model 332
8.2.3 The KMV Model 336
8.2.4 Models Based on Credit Migration 338
8.2.5 Multivariate Firm-Value Models 342
8.3 Threshold Models 343
8.3.1 Notation for One-Period Portfolio Models 344
8.3.2 Threshold Models and Copulas 345
8.3.3 Industry Examples 347
8.3.4 Models Based on Alternative Copulas 348
8.3.5 Model Risk Issues 350
8.4 The Mixture Model Approach 352
8.4.1 One-Factor Bernoulli Mixture Models 353
8.4.2 CreditRisk +356
8.4.3 Asymptotics for Large Portfolios 357
8.4.4 Threshold Models as Mixture Models 359
8.4.5 Model-Theoretic Aspects of Basel II 362
8.4.6 Model Risk Issues 364
8.5 Monte Carlo Methods 367
8.5.1 Basics of Importance Sampling 367
8.5.2 Application to Bernoulli-Mixture Models 370
8.6 Statistical Inference for Mixture Models 374
8.6.1 Motivation 374
8.6.2 Exchangeable Bernoulli-Mixture Models 375
8.6.3 Mixture Models as GLMMs 377
8.6.4 One-Factor Model with Rating Effect 381

CHAPTER 9: Dynamic Credit Risk Models 385
9.1 Credit Derivatives 386
9.1.1 Overview 386
9.1.2 Single-Name Credit Derivatives 387
9.1.3 Portfolio Credit Derivatives 389
9.2 Mathematical Tools 392
9.2.1 Random Times and Hazard Rates 393
9.2.2 Modelling Additional Information 395
9.2.3 Doubly Stochastic Random Times 397
9.3 Financial and Actuarial Pricing of Credit Risk 400
9.3.1 Physical and Risk-Neutral Probability Measure 401
9.3.2 Risk-Neutral Pricing and Market Completeness 405
9.3.3 Martingale Modelling 408
9.3.4 The Actuarial Approach to Credit Risk Pricing 411
9.4 Pricing with Doubly Stochastic Default Times 414
9.4.1 Recovery Payments of Corporate Bonds 414
9.4.2 The Model 415
9.4.3 Pricing Formulas 416
9.4.4 Applications 418
9.5 Affine Models 421
9.5.1 Basic Results 422
9.5.2 The CIR Square-Root Diffusion 423
9.5.3 Extensions 425
9.6 Conditionally Independent Defaults 429
9.6.1 Reduced-Form Models for Portfolio Credit Risk 429
9.6.2 Conditionally Independent Default Times 431
9.6.3 Examples and Applications 435
9.7 Copula Models 440
9.7.1 Definition and General Properties 440
9.7.2 Factor Copula Models 444
9.8 Default Contagion in Reduced-Form Models 448
9.8.1 Default Contagion and Default Dependence 448
9.8.2 Information-Based Default Contagion 453
9.8.3 Interacting Intensities 456

CHAPTER 10: Operational Risk and Insurance Analytics 463
10.1 Operational Risk in Perspective 463
10.1.1 A New Risk Class 463
10.1.2 The Elementary Approaches 465
10.1.3 Advanced Measurement Approaches 466
10.1.4 Operational Loss Data 468
10.2 Elements of Insurance Analytics 471
10.2.1 The Case for Actuarial Methodology 471
10.2.2 The Total Loss Amount 472
10.2.3 Approximations and Panjer Recursion 476
10.2.4 Poisson Mixtures 482
10.2.5 Tails of Aggregate Loss Distributions 484
10.2.6 The Homogeneous Poisson Process 484
10.2.7 Processes Related to the Poisson Process 487

Appendix 494
A.1 Miscellaneous Definitions and Results 494
A.1.1 Type of Distribution 494
A.1.2 Generalized Inverses and Quantiles 494
A.1.3 Karamata's Theorem 495
A.2 Probability Distributions 496
A.2.1 Beta 496
A.2.2 Exponential 496
A.2.3 F 496
A.2.4 Gamma 496
A.2.5 Generalized Inverse Gaussian 497
A.2.6 Inverse Gamma 497
A.2.7 Negative Binomial 498
A.2.8 Pareto 498
A.2.9 Stable 498
A.3 Likelihood Inference 499
A.3.1 Maximum Likelihood Estimators 499
A.3.2 Asymptotic Results:Scalar Parameter 499
A.3.3 Asymptotic Results:Vector of Parameters 500
A.3.4 Wald Test and Confidence Intervals 501
A.3.5 Likelihood Ratio Test and Confidence Intervals 501
A.3.6 Akaike Information Criterion 502

References 503
Index 529