Fundamental Aspects of Operational Risk and Insurance Analytics

Marcelo G. Cruz, PhD, is Adjunct Professor at New York University and a world-renowned consultant on operational risk modeling and measurement. He has written and edited several books in operational risk, and is Founder and Editor-in-Chief of The Journal of Operational Risk. Gareth W. Peters, PhD, is Assistant Professor in the Department of Statistical Science, Principle Investigator in Computational Statistics and Machine Learning, and Academic Member of the UK PhD Centre of Financial Computing at University College London. He is also Adjunct Scientist in the Commonwealth Scientific and Industrial Research Organisation, Australia; Associate Member Oxford-Man Institute at th Oxford University; and Associate Member in the Systemic Risk Centre at the London School of Economics. Pavel V. Shevchenko, PhD, is Senior Principal Research Scientist in the Commonwealth Scientific and Industrial Research Organisation, Australia, as well as Adjunct Professor at the University of New South Wales and the University of Technology, Sydney. He is also Associate Editor of The Journal of Operationa Risk. He works on research and consulting projects in the area of financial risk and the development of relevant numerical methods and software, has published extensively in academic journals, consults for major financial institutions, and frequently presents at industry and academic conferences.

Preface xvii

Acronyms xix

List of Distributions xxi

1 OpRisk in Perspective 1

1.1 Brief History 1

1.2 Risk-Based Capital Ratios for Banks 5

1.3 The Basic Indicator and Standardized Approaches for OpRisk 9

1.4 The Advanced Measurement Approach 10

1.4.1 Internal Measurement Approach 11

1.4.2 Score Card Approach 11

1.4.3 Loss Distribution Approach 12

1.4.4 Requirements for AMA 13

1.5 General Remarks and Book Structure 16

2 OpRisk Data and Governance 17

2.1 Introduction 17

2.2 OpRisk Taxonomy 17

2.2.1 Execution, Delivery, and Process Management 19

2.2.2 Clients, Products, and Business Practices 21

2.2.3 Business Disruption and System Failures 22

2.2.4 External Frauds 23

2.2.5 Internal Fraud 23

2.2.6 Employment Practices and Workplace Safety 24

2.2.7 Damage to Physical Assets 25

2.3 The Elements of the OpRisk Framework 25

2.3.1 Internal Loss Data 26

2.3.2 Setting a Collection Threshold and Possible Impacts 26

2.3.3 Completeness of Database (Under-reporting Events) 27

2.3.4 Recoveries and Near Misses 27

2.3.5 Time Period for Resolution of Operational Losses 28

2.3.6 Adding Costs to Losses 28

2.3.7 Provisioning Treatment of Expected Operational Losses 28

2.4 Business Environment and Internal Control Environment Factors (BEICFs) 29

2.4.1 Risk Control Self-Assessment (RCSA) 29

2.4.2 Key Risk Indicators 31

2.5 External Databases 33

2.6 Scenario Analysis 34

2.7 OpRisk Profile in Different Financial Sectors 37

2.7.1 Trading and Sales 37

2.7.2 Corporate Finance 38

2.7.3 Retail Banking 38

2.7.4 Insurance 39

2.7.5 Asset Management 40

2.7.6 Retail Brokerage 42

2.8 Risk Organization and Governance 43

2.8.1 Organization of Risk Departments 44

2.8.2 Structuring a Firm Wide Policy: Example of an OpRisk Policy 46

2.8.3 Governance 47

3 Using OpRisk Data for Business Analysis 48

3.1 Cost Reduction Programs in Financial Firms 49

3.2 Using OpRisk Data to Perform Business Analysis 53

3.2.1 The Risk of Losing Key Talents: OpRisk in Human Resources 53

3.2.2 OpRisk in Systems Development and Transaction Processing 54

3.3 Conclusions 58

4 Stress-Testing OpRisk Capital and the Comprehensive Capital Analysis and Review (CCAR) 59

4.1 The Need for Stressing OpRisk Capital Even Beyond 99.9% 59

4.2 Comprehensive Capital Review and Analysis (CCAR) 60

4.3 OpRisk and Stress Tests 68

4.4 OpRisk in CCAR in Practice 70

4.5 Reverse Stress Test 75

4.6 Stressing OpRisk Multivariate Models—Understanding the Relationship Among Internal Control Factors and Their Impact on Operational Losses 76

5 Basic Probability Concepts in Loss Distribution Approach 79

5.1 Loss Distribution Approach 79

5.2 Quantiles and Moments 85

5.3 Frequency Distributions 88

5.4 Severity Distributions 89

5.4.1 Simple Parametric Distributions 90

5.4.2 Truncated Distributions 92

5.4.3 Mixture and Spliced Distributions 93

5.5 Convolutions and Characteristic Functions 94

5.6 Extreme Value Theory 97

5.6.1 EVT—Block Maxima 98

5.6.2 EVT—Random Number of Losses 99

5.6.3 EVT—Threshold Exceedances 100

6 Risk Measures and Capital Allocation 102

6.1 Development of Capital Accords Base I, II and III 103

6.2 Measures of Risk 106

6.2.1 Coherent and Convex Risk Measures 107

6.2.2 Comonotonic Additive Risk Measures 109

6.2.3 Value-at-Risk 109

6.2.4 Expected Shortfall 114

6.2.5 Spectral Risk Measure 120

6.2.6 Higher-Order Risk Measures 122

6.2.7 Distortion Risk Measures 125

6.2.8 Elicitable Risk Measures 126

6.2.9 Risk Measure Accounting for Parameter Uncertainty 130

6.3 Capital Allocation 133

6.3.1 Coherent Capital Allocation 134

6.3.2 Euler Allocation 136

6.3.3 Standard Deviation 138

6.3.4 Expected Shortfall 139

6.3.5 Value-at-Risk 140

6.3.6 Allocation by Marginal Contributions 142

6.3.7 Numerical Example 143

7 Estimation of Frequency and Severity Models 146

7.1 Frequentist Estimation 146

7.1.1 Parameteric Maximum Likelihood Method 149

7.1.2 Maximum Likelihood Method for Truncated and Censored Data 151

7.1.3 Expectation Maximization and Parameter Estimation 152

7.1.4 Bootstrap for Estimation of Parameter Accuracy 156

7.1.5 Indirect Inference–Based Likelihood Estimation 157

7.2 Bayesian Inference Approach 159

7.2.1 Conjugate Prior Distributions 161

7.2.2 Gaussian Approximation for Posterior (Laplace Type) 161

7.2.3 Posterior Point Estimators 162

7.2.4 Restricted Parameters 163

7.2.5 Noninformative Prior 163

7.3 Mean Square Error of Prediction 164

7.4 Standard Markov Chain Monte Carlo (MCMC) Methods 166

7.4.1 Motivation for Markov Chain Methods 167

7.4.2 Metropolis–Hastings Algorithm 177

7.4.3 Gibbs Sampler 178

7.4.4 Random Walk Metropolis–Hastings within Gibbs 179

7.5 Standard MCMC Guidelines for Implementation 180

7.5.1 Tuning, Burn-in, and Sampling Stages 180

7.5.2 Numerical Error 185

7.5.3 MCMC Extensions: Reducing Sample Autocorrelation 187

7.6 Advanced MCMC Methods 188

7.6.1 Auxiliary Variable MCMC Methods: Slice Sampling 189

7.6.2 Generic Univariate Auxiliary Variable Gibbs Sampler: Slice Sampler 189

7.6.3 Adaptive MCMC 192

7.6.4 Riemann–Manifold Hamiltonian Monte Carlo Sampler (Automated Local Adaption) 196

7.7 Sequential Monte Carlo (SMC) Samplers and Importance Sampling 201

7.7.1 Motivating OpRisk Applications for SMC Samplers 202

7.7.2 SMC Sampler Methodology and Components 210

7.7.3 Incorporating Partial Rejection Control into SMC Samplers 216

7.7.4 Finite Sample (Nonasymptotic) Accuracy for Particle Integration 219

7.8 Approximate Bayesian Computation (ABC) Methods 220

7.9 OpRisk Estimation and Modeling for Truncated Data 223

7.9.1 Constant Threshold - Poisson Process 224

7.9.2 Negative Binomial and Binomial Frequencies 227

7.9.3 Ignoring Data Truncation 228

7.9.4 Threshold Varying in Time 232

7.9.5 Unknown and Stochastic Truncation Level 236

8 Model Selection and Goodness-of-Fit Testing for Frequency and Severity Models 238

8.1 Qualitative Model Diagnostic Tools 238

8.2 Tail Diagnostics 240

8.3 Information Criterion for Model Selection 242

8.3.1 Akaike Information Criterion for LDA Model Selection 242

8.3.2 Deviance Information Criterion 245

8.4 Goodness-of-Fit Testing for Model Choice (How to Account for Heavy Tails!) 246

8.4.1 Convergence Results of the Empirical Process for GOF Testing 247

8.4.2 Overview of Generic GOF Tests—Omnibus Distributional Tests 256

8.4.3 Kolmogorov–Smirnov Goodness-of-Fit Test and Weighted Variants: Testing in the Presence of Heavy Tails 260

8.4.4 Cramer-von-Mises Goodness-of-Fit Tests and Weighted Variants: Testing in the Presence of Heavy Tails 271

8.5 Bayesian Model Selection 283

8.5.1 Reciprocal Importance Sampling Estimator 284

8.5.2 Chib Estimator for Model Evidence 285

8.6 SMC Sampler Estimators of Model Evidence 286

8.7 Multiple Risk Dependence Structure Model Selection: Copula Choice 287

8.7.1 Approaches to Goodness-of-Fit Testing for Dependence Structures 293

8.7.2 Double Parameteric Bootstrap for Copula GOF 297

9 Flexible Parametric Severity Models: Basics 300

9.1 Motivation for Flexible Parametric Severity Loss Models 300

9.2 Context of Flexible Heavy-Tailed Loss Models in OpRisk and Insurance LDA Models 301

9.3 Empirical Analysis Justifying Heavy-Tailed Loss Models in OpRisk 303

9.4 Quantile Function Heavy-Tailed Severity Models 305

9.4.1 g-and-h Severity Model Family in OpRisk 311

9.4.2 Tail Properties of the g-and-h, g, h, and h–h Severity in OpRisk 321

9.4.3 Parameter Estimation for the g-and-h Severity in OpRisk 324

9.4.4 Bayesian Models for the g-and-h Severity in OpRisk 328

9.5 Generalized Beta Family of Heavy-Tailed Severity Models 333

9.5.1 Generalized Beta Family Type II Severity Models in OpRisk 333

9.5.2 Sub families of the Generalized Beta Family Type II Severity Models 336

9.5.3 Mixture Representations of the Generalized Beta Family Type II Severity Models 337

9.5.4 Estimation in the Generalized Beta Family Type II Severity Models 339

9.6 Generalized Hyperbolic Families of Heavy-Tailed Severity Models 340

9.6.1 Tail Properties and Infinite Divisibility of the Generalized Hyperbolic Severity Models 342

9.6.2 Subfamilies of the Generalized Hyperbolic Severity Models 344

9.6.3 Normal Inverse Gaussian Family of Heavy-Tailed Severity Models 346

9.7 Halphen Family of Flexible Severity Models: GIG and Hyperbolic 350

9.7.1 Halphen Type A: Generalized Inverse Gaussian Family of Flexible Severity Models 355

9.7.2 Halphen Type B and IB Families of Flexible Severity Models 361

10 Dependence Concepts 365

10.1 Introduction to Concepts in Dependence for OpRisk and Insurance 365

10.2 Dependence Modeling Within and Between LDA Model Structures 366

10.2.1 Where Can One Introduce Dependence Between LDA Model Structures? 368

10.2.2 Understanding Basic Impacts of Dependence Modeling Between LDA Components in Multiple Risks 369

10.3 General Notions of Dependence 372

10.4 Dependence Measures 387

10.4.1 Linear Correlation 390

10.4.2 Rank Correlation Measures 393

10.5 Tail Dependence Parameters, Functions, and Tail Order Functions 398

10.5.1 Tail Dependence Coefficients 398

10.5.2 Tail Dependence Functions and Orders 407

10.5.3 A Link Between Orthant Extreme Dependence and Spectral Measures: Tail Dependence 410

11 Dependence Models 414

11.1 Introduction to Parametric Dependence Modeling Through a Copula 414

11.2 Copula Model Families for OpRisk 422

11.2.1 Gaussian Copula 428

11.2.2 t-Copula 430

11.2.3 Archimedean Copulas 435

11.2.4 Archimedean Copula Generators and the Laplace Transform of a Non-Negative Random Variable 439

11.2.5 Archimedean Copula Generators, l1-Norm Symmetric Distributions and the Williamson Transform 441

11.2.6 Hierarchical and Nested Archimedean Copulae 452

11.2.7 Mixtures of Archimedean Copulae 454

11.2.8 Multivariate Archimedean Copula Tail Dependence 456

11.3 Copula Parameter Estimation in Two Stages: Inference for the Margins 457

11.3.1 MPLE: Copula Parameter Estimation 458

11.3.2 Inference Functions for Margins (IFM): Copula Parameter Estimation 459

12 Examples of LDA Dependence Models 462

12.1 Multiple Risk LDA Compound Poisson Processes and Lévy Copula 462

12.2 Multiple Risk LDA: Dependence Between Frequencies via Copula 468

12.3 Multiple Risk LDA: Dependence Between the k-th Event Times/Losses 468

12.3.1 Common Shock Processes 469

12.3.2 Max-Stable and Self-Chaining Copula Models 470

12.4 Multiple Risk LDA: Dependence Between Aggregated Losses via Copula 474

12.5 Multiple Risk LDA: Structural Model with Common Factors 477

12.6 Multiple Risk LDA: Stochastic and Dependent Risk Profiles 478

12.7 Multiple Risk LDA: Dependence and Combining Different Data Sources 482

12.7.1 Bayesian Inference Using MCMC 484

12.7.2 Numerical Example 485

12.7.3 Predictive Distribution 487

12.8 A Note on Negative Diversification and Dependence Modeling 489

13 Loss Aggregation 492

13.1 Analytic Solution 492

13.1.1 Analytic Solution via Convolutions 493

13.1.2 Analytic Solution via Characteristic Functions 494

13.1.3 Moments of Compound Distribution 496

13.1.4 Value-at-Risk and Expected Shortfall 499

13.2 Monte Carlo Method 499

13.2.1 Quantile Estimate 500

13.2.2 Expected Shortfall Estimate 502

13.3 Panjer Recursion 503

13.4 Panjer Extensions 509

13.5 Fast Fourier Transform 511

13.6 Closed-Form Approximation 514

13.7 Capital Charge Under Parameter Uncertainty 519

13.7.1 Predictive Distributions 520

13.7.2 Calculation of Predictive Distributions 521

13.8 Special Advanced Topics on Loss Aggregation 523

13.8.1 Discretisation Errors and Extrapolation Methods 524

13.8.2 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails 527

13.8.3 Recursions for Convolutions (Partial Sums) with Discretised Severity Distributions (Fixed n) 535

13.8.4 Alternatives to Panjer Recursions: Recursions for Compound Distributions with Discretised Severity Distributions 543

13.8.5 Higher Order Recursions for Discretised Severity Distributions in Compound LDA Models 545

13.8.6 Recursions for Discretised Severity Distributions in Compound Mixed Poisson LDA Models 547

13.8.7 Continuous Versions of the Panjer Recursion 550

14 Scenario Analysis 556

14.1 Introduction 556

14.2 Examples of Expert Judgments 559

14.3 Pure Bayesian Approach (Estimating Prior) 561

14.4 Expert Distribution and Scenario Elicitation: Learning from Bayesian Methods 563

14.5 Building Models for Elicited Opinions: Hierarchical Dirichlet Models 566

14.6 Worst-Case Scenario Framework 568

14.7 Stress Test Scenario Analysis 571

14.8 Bow-Tie Diagram 574

14.9 Bayesian Networks 576

14.9.1 Definition and Examples 577

14.9.2 Constructing and Simulating a Bayesian Net 580

14.9.3 Combining Expert Opinion and Data in a Bayesian Net 581

14.9.4 Bayesian Net and Operational Risk 582

14.10 Discussion 584

15 Combining Different Data Sources 585

15.1 Minimum Variance Principle 586

15.2 Bayesian Method to Combine Two Data Sources 588

15.2.1 Estimating Prior: Pure Bayesian Approach 590

15.2.2 Estimating Prior: Empirical Bayesian Approach 592

15.2.3 Poisson Frequency 593

15.2.4 The LogNormal Severity 597

15.2.5 Pareto Severity 601

15.3 Estimation of the Prior Using Data 606

15.3.1 The Maximum Likelihood Estimator 606

15.3.2 Poisson Frequencies 607

15.4 Combining Expert Opinions with External and Internal Data 609

15.4.1 Conjugate Prior Extension 610

15.4.2 Modeling Frequency: Poisson Model 611

15.4.3 LogNormal Model for Severities 618

15.4.4 Pareto Model 620

15.5 Combining Data Sources Using Credibility Theory 625

15.5.1 Bühlmann–Straub Model 626

15.5.2 Modeling Frequency 628

15.5.3 Modeling Severity 631

15.5.4 Numerical Example 633

15.5.5 Remarks and Interpretation 634

15.6 Nonparametric Bayesian Approach via Dirichlet Process 635

15.7 Combining Using Dempster–Shafer Structures and p-Boxes 638

15.7.1 Dempster–Shafer Structures and p-Boxes 639

15.7.2 Dempster’s Rule 641

15.7.3 Intersection Method 643

15.7.4 Envelope Method 644

15.7.5 Bounds for the Empirical Data Distribution 645

15.8 General Remarks 647

16 Multifactor Modeling and Regression for Loss Processes 649

16.1 Generalized Linear Model Regressions and the Exponential Family 649

16.1.1 Basic Components of a Generalized Linear Model Regression in the Exponential Family 650

16.1.2 Basis Function Regression 654

16.2 Maximum Likelihood Estimation for Generalized Linear Models 655

16.2.1 Iterated Weighted Least Squares Maximum Likelihood for Generalised Linear Models 655

16.2.2 Model Selection via the Deviance in a GLM Regression 657

16.3 Bayesian Generalized Linear Model Regressions and Regularization Priors 659

16.3.1 Bayesian Model Selection for Regularlized GLM Regression 665

16.4 Bayesian Estimation and Model Selection via SMC Samplers 666

16.4.1 Proposed SMC Sampler Solution 667

16.5 Illustrations of SMC Samplers Model Estimation and Selection for Bayesian GLM Regressions 668

16.5.1 Normal Regression Model 668

16.5.2 Poisson Regression Model 669

16.6 Introduction to Quantile Regression Methods for OpRisk 672

16.6.1 Nonparametric Quantile Regression Models 674

16.6.2 Parametric Quantile Regression Models 675

16.7 Factor Modeling for Industry Data 681

16.8 Multifactor Modeling under EVT Approach 683

17 Insurance and Risk Transfer: Products and Modeling 685

17.1 Motivation for Insurance and Risk Transfer in OpRisk 685

17.2 Fundamentals of Insurance Product Structures for OpRisk 688

17.3 Single Peril Policy Products for OpRisk 692

17.4 Generic Insurance Product Structures for OpRisk 694

17.4.1 Generic Deterministic Policy Structures 694

17.4.2 Generic Stochastic Policy Structures: Accounting for Coverage Uncertainty 700

17.5 Closed-Form LDA Models with Insurance Mitigations 705

17.5.1 Insurance Mitigation Under the Poisson-Inverse-Gaussian Closed-Form LDA Models 705

17.5.2 Insurance Mitigation and Poisson-α-Stable Closed-Form LDA Models 712

17.5.3 Large Claim Number Loss Processes: Generic Closed-Form LDA Models with Insurance Mitigation 719

17.5.4 Generic Closed-Form Approximations for Insured LDA Models 734

18 Insurance and Risk Transfer: Pricing Insurance-Linked Derivatives, Reinsurance, and CAT Bonds for OpRisk 750

18.1 Insurance-Linked Securities and CAT Bonds for OpRisk 751

18.1.1 Background on Insurance-Linked Derivatives and CAT Bonds for Extreme Risk Transfer 755

18.1.2 Triggers for CAT Bonds and Their Impact on Risk Transfer 760

18.1.3 Recent Trends in CAT Bonds 763

18.1.4 Management Strategies for Utilization of Insurance-Linked Derivatives and CAT Bonds in OpRisk 763

18.2 Basics of Valuation of ILS and CAT Bonds for OpRisk 765

18.2.1 Probabilistic Pricing Frameworks: Complete and Incomplete Markets, Real-World Pricing, Benchmark Approach, and Actuarial Valuation 771

18.2.2 Risk Assessment for Reinsurance: ILS and CAT Bonds 794

18.3 Applications of Pricing ILS and CAT Bonds 796

18.3.1 Probabilistic Framework for CAT Bond Market 796

18.3.2 Framework 1: Assuming Complete Market and Arbitrage-Free Pricing 798

18.3.3 Framework 2: Assuming Incomplete Arbitrage-Free Pricing 809

18.4 Sidecars, Multiple Peril Baskets, and Umbrellas for OpRisk 815

18.4.1 Umbrella Insurance 816

18.4.2 OpRisk Loss Processes Comprised of Multiple Perils 817

18.5 Optimal Insurance Purchase Strategies for OpRisk Insurance via Multiple Optimal Stopping Times 823

18.5.1 Examples of Basic Insurance Policies 826

18.5.2 Objective Functions for Rational and Boundedly Rational Insurees 828

18.5.3 Closed-Form Multiple Optimal Stopping Rules for Multiple Insurance Purchase Decisions 830

18.5.4 Aski-Polynomial Orthogonal Series Approximations 835

A Miscellaneous Definitions and List of Distributions 842

A.1 Indicator Function 842

A.2 Gamma Function 842

A.3 Discrete Distributions 842

A.3.1 Poisson Distribution 842

A.3.2 Binomial Distribution 843

A.3.3 Negative Binomial Distribution 843

A.3.4 Doubly Stochastic Poisson Process (Cox Process) 844

A.4 Continuous Distributions 844

A.4.1 Uniform Distribution 844

A.4.2 Normal (Gaussian) Distribution 844

A.4.3 Inverse Gaussian Distribution 845

A.4.4 LogNormal Distribution 845

A.4.5 Student’s t Distribution 846

A.4.6 Gamma Distribution 846

A.4.7 Weibull Distribution 846

A.4.8 Inverse Chi-Squared Distribution 847

A.4.9 Pareto Distribution (One-Parameter) 847

A.4.10 Pareto Distribution (Two-Parameter) 847

A.4.11 Generalized Pareto Distribution 848

A.4.12 Beta Distribution 848

A.4.13 Generalized Inverse Gaussian Distribution 849

A.4.14 d-variate Normal Distribution 849

A.4.15 d-variate t-Distribution 850

Bibliography 851

Index 892