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 xviiAcronyms xixList of Distributions xxi1 OpRisk in Perspective 11.1 Brief History 11.2 Risk-Based Capital Ratios for Banks 51.3 The Basic Indicator and Standardized Approaches for OpRisk 91.4 The Advanced Measurement Approach 101.4.1 Internal Measurement Approach 111.4.2 Score Card Approach 111.4.3 Loss Distribution Approach 121.4.4 Requirements for AMA 131.5 General Remarks and Book Structure 162 OpRisk Data and Governance 172.1 Introduction 172.2 OpRisk Taxonomy 172.2.1 Execution, Delivery, and Process Management 192.2.2 Clients, Products, and Business Practices 212.2.3 Business Disruption and System Failures 222.2.4 External Frauds 232.2.5 Internal Fraud 232.2.6 Employment Practices and Workplace Safety 242.2.7 Damage to Physical Assets 252.3 The Elements of the OpRisk Framework 252.3.1 Internal Loss Data 262.3.2 Setting a Collection Threshold and Possible Impacts 262.3.3 Completeness of Database (Under-reporting Events) 272.3.4 Recoveries and Near Misses 272.3.5 Time Period for Resolution of Operational Losses 282.3.6 Adding Costs to Losses 282.3.7 Provisioning Treatment of Expected Operational Losses 282.4 Business Environment and Internal Control Environment Factors (BEICFs) 292.4.1 Risk Control Self-Assessment (RCSA) 292.4.2 Key Risk Indicators 312.5 External Databases 332.6 Scenario Analysis 342.7 OpRisk Profile in Different Financial Sectors 372.7.1 Trading and Sales 372.7.2 Corporate Finance 382.7.3 Retail Banking 382.7.4 Insurance 392.7.5 Asset Management 402.7.6 Retail Brokerage 422.8 Risk Organization and Governance 432.8.1 Organization of Risk Departments 442.8.2 Structuring a Firm Wide Policy: Example of an OpRisk Policy 462.8.3 Governance 473 Using OpRisk Data for Business Analysis 483.1 Cost Reduction Programs in Financial Firms 493.2 Using OpRisk Data to Perform Business Analysis 533.2.1 The Risk of Losing Key Talents: OpRisk in Human Resources 533.2.2 OpRisk in Systems Development and Transaction Processing 543.3 Conclusions 584 Stress-Testing OpRisk Capital and the Comprehensive Capital Analysis and Review (CCAR) 594.1 The Need for Stressing OpRisk Capital Even Beyond 99.9% 594.2 Comprehensive Capital Review and Analysis (CCAR) 604.3 OpRisk and Stress Tests 684.4 OpRisk in CCAR in Practice 704.5 Reverse Stress Test 754.6 Stressing OpRisk Multivariate Models—Understanding the Relationship Among Internal Control Factors and Their Impact on Operational Losses 765 Basic Probability Concepts in Loss Distribution Approach 795.1 Loss Distribution Approach 795.2 Quantiles and Moments 855.3 Frequency Distributions 885.4 Severity Distributions 895.4.1 Simple Parametric Distributions 905.4.2 Truncated Distributions 925.4.3 Mixture and Spliced Distributions 935.5 Convolutions and Characteristic Functions 945.6 Extreme Value Theory 975.6.1 EVT—Block Maxima 985.6.2 EVT—Random Number of Losses 995.6.3 EVT—Threshold Exceedances 1006 Risk Measures and Capital Allocation 1026.1 Development of Capital Accords Base I, II and III 1036.2 Measures of Risk 1066.2.1 Coherent and Convex Risk Measures 1076.2.2 Comonotonic Additive Risk Measures 1096.2.3 Value-at-Risk 1096.2.4 Expected Shortfall 1146.2.5 Spectral Risk Measure 1206.2.6 Higher-Order Risk Measures 1226.2.7 Distortion Risk Measures 1256.2.8 Elicitable Risk Measures 1266.2.9 Risk Measure Accounting for Parameter Uncertainty 1306.3 Capital Allocation 1336.3.1 Coherent Capital Allocation 1346.3.2 Euler Allocation 1366.3.3 Standard Deviation 1386.3.4 Expected Shortfall 1396.3.5 Value-at-Risk 1406.3.6 Allocation by Marginal Contributions 1426.3.7 Numerical Example 1437 Estimation of Frequency and Severity Models 1467.1 Frequentist Estimation 1467.1.1 Parameteric Maximum Likelihood Method 1497.1.2 Maximum Likelihood Method for Truncated and Censored Data 1517.1.3 Expectation Maximization and Parameter Estimation 1527.1.4 Bootstrap for Estimation of Parameter Accuracy 1567.1.5 Indirect Inference–Based Likelihood Estimation 1577.2 Bayesian Inference Approach 1597.2.1 Conjugate Prior Distributions 1617.2.2 Gaussian Approximation for Posterior (Laplace Type) 1617.2.3 Posterior Point Estimators 1627.2.4 Restricted Parameters 1637.2.5 Noninformative Prior 1637.3 Mean Square Error of Prediction 1647.4 Standard Markov Chain Monte Carlo (MCMC) Methods 1667.4.1 Motivation for Markov Chain Methods 1677.4.2 Metropolis–Hastings Algorithm 1777.4.3 Gibbs Sampler 1787.4.4 Random Walk Metropolis–Hastings within Gibbs 1797.5 Standard MCMC Guidelines for Implementation 1807.5.1 Tuning, Burn-in, and Sampling Stages 1807.5.2 Numerical Error 1857.5.3 MCMC Extensions: Reducing Sample Autocorrelation 1877.6 Advanced MCMC Methods 1887.6.1 Auxiliary Variable MCMC Methods: Slice Sampling 1897.6.2 Generic Univariate Auxiliary Variable Gibbs Sampler: Slice Sampler 1897.6.3 Adaptive MCMC 1927.6.4 Riemann–Manifold Hamiltonian Monte Carlo Sampler (Automated Local Adaption) 1967.7 Sequential Monte Carlo (SMC) Samplers and Importance Sampling 2017.7.1 Motivating OpRisk Applications for SMC Samplers 2027.7.2 SMC Sampler Methodology and Components 2107.7.3 Incorporating Partial Rejection Control into SMC Samplers 2167.7.4 Finite Sample (Nonasymptotic) Accuracy for Particle Integration 2197.8 Approximate Bayesian Computation (ABC) Methods 2207.9 OpRisk Estimation and Modeling for Truncated Data 2237.9.1 Constant Threshold - Poisson Process 2247.9.2 Negative Binomial and Binomial Frequencies 2277.9.3 Ignoring Data Truncation 2287.9.4 Threshold Varying in Time 2327.9.5 Unknown and Stochastic Truncation Level 2368 Model Selection and Goodness-of-Fit Testing for Frequency and Severity Models 2388.1 Qualitative Model Diagnostic Tools 2388.2 Tail Diagnostics 2408.3 Information Criterion for Model Selection 2428.3.1 Akaike Information Criterion for LDA Model Selection 2428.3.2 Deviance Information Criterion 2458.4 Goodness-of-Fit Testing for Model Choice (How to Account for Heavy Tails!) 2468.4.1 Convergence Results of the Empirical Process for GOF Testing 2478.4.2 Overview of Generic GOF Tests—Omnibus Distributional Tests 2568.4.3 Kolmogorov–Smirnov Goodness-of-Fit Test and Weighted Variants: Testing in the Presence of Heavy Tails 2608.4.4 Cramer-von-Mises Goodness-of-Fit Tests and Weighted Variants: Testing in the Presence of Heavy Tails 2718.5 Bayesian Model Selection 2838.5.1 Reciprocal Importance Sampling Estimator 2848.5.2 Chib Estimator for Model Evidence 2858.6 SMC Sampler Estimators of Model Evidence 2868.7 Multiple Risk Dependence Structure Model Selection: Copula Choice 2878.7.1 Approaches to Goodness-of-Fit Testing for Dependence Structures 2938.7.2 Double Parameteric Bootstrap for Copula GOF 2979 Flexible Parametric Severity Models: Basics 3009.1 Motivation for Flexible Parametric Severity Loss Models 3009.2 Context of Flexible Heavy-Tailed Loss Models in OpRisk and Insurance LDA Models 3019.3 Empirical Analysis Justifying Heavy-Tailed Loss Models in OpRisk 3039.4 Quantile Function Heavy-Tailed Severity Models 3059.4.1 g-and-h Severity Model Family in OpRisk 3119.4.2 Tail Properties of the g-and-h, g, h, and h–h Severity in OpRisk 3219.4.3 Parameter Estimation for the g-and-h Severity in OpRisk 3249.4.4 Bayesian Models for the g-and-h Severity in OpRisk 3289.5 Generalized Beta Family of Heavy-Tailed Severity Models 3339.5.1 Generalized Beta Family Type II Severity Models in OpRisk 3339.5.2 Sub families of the Generalized Beta Family Type II Severity Models 3369.5.3 Mixture Representations of the Generalized Beta Family Type II Severity Models 3379.5.4 Estimation in the Generalized Beta Family Type II Severity Models 3399.6 Generalized Hyperbolic Families of Heavy-Tailed Severity Models 3409.6.1 Tail Properties and Infinite Divisibility of the Generalized Hyperbolic Severity Models 3429.6.2 Subfamilies of the Generalized Hyperbolic Severity Models 3449.6.3 Normal Inverse Gaussian Family of Heavy-Tailed Severity Models 3469.7 Halphen Family of Flexible Severity Models: GIG and Hyperbolic 3509.7.1 Halphen Type A: Generalized Inverse Gaussian Family of Flexible Severity Models 3559.7.2 Halphen Type B and IB Families of Flexible Severity Models 36110 Dependence Concepts 36510.1 Introduction to Concepts in Dependence for OpRisk and Insurance 36510.2 Dependence Modeling Within and Between LDA Model Structures 36610.2.1 Where Can One Introduce Dependence Between LDA Model Structures? 36810.2.2 Understanding Basic Impacts of Dependence Modeling Between LDA Components in Multiple Risks 36910.3 General Notions of Dependence 37210.4 Dependence Measures 38710.4.1 Linear Correlation 39010.4.2 Rank Correlation Measures 39310.5 Tail Dependence Parameters, Functions, and Tail Order Functions 39810.5.1 Tail Dependence Coefficients 39810.5.2 Tail Dependence Functions and Orders 40710.5.3 A Link Between Orthant Extreme Dependence and Spectral Measures: Tail Dependence 41011 Dependence Models 41411.1 Introduction to Parametric Dependence Modeling Through a Copula 41411.2 Copula Model Families for OpRisk 42211.2.1 Gaussian Copula 42811.2.2 t-Copula 43011.2.3 Archimedean Copulas 43511.2.4 Archimedean Copula Generators and the Laplace Transform of a Non-Negative Random Variable 43911.2.5 Archimedean Copula Generators, l1-Norm Symmetric Distributions and the Williamson Transform 44111.2.6 Hierarchical and Nested Archimedean Copulae 45211.2.7 Mixtures of Archimedean Copulae 45411.2.8 Multivariate Archimedean Copula Tail Dependence 45611.3 Copula Parameter Estimation in Two Stages: Inference for the Margins 45711.3.1 MPLE: Copula Parameter Estimation 45811.3.2 Inference Functions for Margins (IFM): Copula Parameter Estimation 45912 Examples of LDA Dependence Models 46212.1 Multiple Risk LDA Compound Poisson Processes and Lévy Copula 46212.2 Multiple Risk LDA: Dependence Between Frequencies via Copula 46812.3 Multiple Risk LDA: Dependence Between the k-th Event Times/Losses 46812.3.1 Common Shock Processes 46912.3.2 Max-Stable and Self-Chaining Copula Models 47012.4 Multiple Risk LDA: Dependence Between Aggregated Losses via Copula 47412.5 Multiple Risk LDA: Structural Model with Common Factors 47712.6 Multiple Risk LDA: Stochastic and Dependent Risk Profiles 47812.7 Multiple Risk LDA: Dependence and Combining Different Data Sources 48212.7.1 Bayesian Inference Using MCMC 48412.7.2 Numerical Example 48512.7.3 Predictive Distribution 48712.8 A Note on Negative Diversification and Dependence Modeling 48913 Loss Aggregation 49213.1 Analytic Solution 49213.1.1 Analytic Solution via Convolutions 49313.1.2 Analytic Solution via Characteristic Functions 49413.1.3 Moments of Compound Distribution 49613.1.4 Value-at-Risk and Expected Shortfall 49913.2 Monte Carlo Method 49913.2.1 Quantile Estimate 50013.2.2 Expected Shortfall Estimate 50213.3 Panjer Recursion 50313.4 Panjer Extensions 50913.5 Fast Fourier Transform 51113.6 Closed-Form Approximation 51413.7 Capital Charge Under Parameter Uncertainty 51913.7.1 Predictive Distributions 52013.7.2 Calculation of Predictive Distributions 52113.8 Special Advanced Topics on Loss Aggregation 52313.8.1 Discretisation Errors and Extrapolation Methods 52413.8.2 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails 52713.8.3 Recursions for Convolutions (Partial Sums) with Discretised Severity Distributions (Fixed n) 53513.8.4 Alternatives to Panjer Recursions: Recursions for Compound Distributions with Discretised Severity Distributions 54313.8.5 Higher Order Recursions for Discretised Severity Distributions in Compound LDA Models 54513.8.6 Recursions for Discretised Severity Distributions in Compound Mixed Poisson LDA Models 54713.8.7 Continuous Versions of the Panjer Recursion 55014 Scenario Analysis 55614.1 Introduction 55614.2 Examples of Expert Judgments 55914.3 Pure Bayesian Approach (Estimating Prior) 56114.4 Expert Distribution and Scenario Elicitation: Learning from Bayesian Methods 56314.5 Building Models for Elicited Opinions: Hierarchical Dirichlet Models 56614.6 Worst-Case Scenario Framework 56814.7 Stress Test Scenario Analysis 57114.8 Bow-Tie Diagram 57414.9 Bayesian Networks 57614.9.1 Definition and Examples 57714.9.2 Constructing and Simulating a Bayesian Net 58014.9.3 Combining Expert Opinion and Data in a Bayesian Net 58114.9.4 Bayesian Net and Operational Risk 58214.10 Discussion 58415 Combining Different Data Sources 58515.1 Minimum Variance Principle 58615.2 Bayesian Method to Combine Two Data Sources 58815.2.1 Estimating Prior: Pure Bayesian Approach 59015.2.2 Estimating Prior: Empirical Bayesian Approach 59215.2.3 Poisson Frequency 59315.2.4 The LogNormal Severity 59715.2.5 Pareto Severity 60115.3 Estimation of the Prior Using Data 60615.3.1 The Maximum Likelihood Estimator 60615.3.2 Poisson Frequencies 60715.4 Combining Expert Opinions with External and Internal Data 60915.4.1 Conjugate Prior Extension 61015.4.2 Modeling Frequency: Poisson Model 61115.4.3 LogNormal Model for Severities 61815.4.4 Pareto Model 62015.5 Combining Data Sources Using Credibility Theory 62515.5.1 Bühlmann–Straub Model 62615.5.2 Modeling Frequency 62815.5.3 Modeling Severity 63115.5.4 Numerical Example 63315.5.5 Remarks and Interpretation 63415.6 Nonparametric Bayesian Approach via Dirichlet Process 63515.7 Combining Using Dempster–Shafer Structures and p-Boxes 63815.7.1 Dempster–Shafer Structures and p-Boxes 63915.7.2 Dempster’s Rule 64115.7.3 Intersection Method 64315.7.4 Envelope Method 64415.7.5 Bounds for the Empirical Data Distribution 64515.8 General Remarks 64716 Multifactor Modeling and Regression for Loss Processes 64916.1 Generalized Linear Model Regressions and the Exponential Family 64916.1.1 Basic Components of a Generalized Linear Model Regression in the Exponential Family 65016.1.2 Basis Function Regression 65416.2 Maximum Likelihood Estimation for Generalized Linear Models 65516.2.1 Iterated Weighted Least Squares Maximum Likelihood for Generalised Linear Models 65516.2.2 Model Selection via the Deviance in a GLM Regression 65716.3 Bayesian Generalized Linear Model Regressions and Regularization Priors 65916.3.1 Bayesian Model Selection for Regularlized GLM Regression 66516.4 Bayesian Estimation and Model Selection via SMC Samplers 66616.4.1 Proposed SMC Sampler Solution 66716.5 Illustrations of SMC Samplers Model Estimation and Selection for Bayesian GLM Regressions 66816.5.1 Normal Regression Model 66816.5.2 Poisson Regression Model 66916.6 Introduction to Quantile Regression Methods for OpRisk 67216.6.1 Nonparametric Quantile Regression Models 67416.6.2 Parametric Quantile Regression Models 67516.7 Factor Modeling for Industry Data 68116.8 Multifactor Modeling under EVT Approach 68317 Insurance and Risk Transfer: Products and Modeling 68517.1 Motivation for Insurance and Risk Transfer in OpRisk 68517.2 Fundamentals of Insurance Product Structures for OpRisk 68817.3 Single Peril Policy Products for OpRisk 69217.4 Generic Insurance Product Structures for OpRisk 69417.4.1 Generic Deterministic Policy Structures 69417.4.2 Generic Stochastic Policy Structures: Accounting for Coverage Uncertainty 70017.5 Closed-Form LDA Models with Insurance Mitigations 70517.5.1 Insurance Mitigation Under the Poisson-Inverse-Gaussian Closed-Form LDA Models 70517.5.2 Insurance Mitigation and Poisson-α-Stable Closed-Form LDA Models 71217.5.3 Large Claim Number Loss Processes: Generic Closed-Form LDA Models with Insurance Mitigation 71917.5.4 Generic Closed-Form Approximations for Insured LDA Models 73418 Insurance and Risk Transfer: Pricing Insurance-Linked Derivatives, Reinsurance, and CAT Bonds for OpRisk 75018.1 Insurance-Linked Securities and CAT Bonds for OpRisk 75118.1.1 Background on Insurance-Linked Derivatives and CAT Bonds for Extreme Risk Transfer 75518.1.2 Triggers for CAT Bonds and Their Impact on Risk Transfer 76018.1.3 Recent Trends in CAT Bonds 76318.1.4 Management Strategies for Utilization of Insurance-Linked Derivatives and CAT Bonds in OpRisk 76318.2 Basics of Valuation of ILS and CAT Bonds for OpRisk 76518.2.1 Probabilistic Pricing Frameworks: Complete and Incomplete Markets, Real-World Pricing, Benchmark Approach, and Actuarial Valuation 77118.2.2 Risk Assessment for Reinsurance: ILS and CAT Bonds 79418.3 Applications of Pricing ILS and CAT Bonds 79618.3.1 Probabilistic Framework for CAT Bond Market 79618.3.2 Framework 1: Assuming Complete Market and Arbitrage-Free Pricing 79818.3.3 Framework 2: Assuming Incomplete Arbitrage-Free Pricing 80918.4 Sidecars, Multiple Peril Baskets, and Umbrellas for OpRisk 81518.4.1 Umbrella Insurance 81618.4.2 OpRisk Loss Processes Comprised of Multiple Perils 81718.5 Optimal Insurance Purchase Strategies for OpRisk Insurance via Multiple Optimal Stopping Times 82318.5.1 Examples of Basic Insurance Policies 82618.5.2 Objective Functions for Rational and Boundedly Rational Insurees 82818.5.3 Closed-Form Multiple Optimal Stopping Rules for Multiple Insurance Purchase Decisions 83018.5.4 Aski-Polynomial Orthogonal Series Approximations 835A Miscellaneous Definitions and List of Distributions 842A.1 Indicator Function 842A.2 Gamma Function 842A.3 Discrete Distributions 842A.3.1 Poisson Distribution 842A.3.2 Binomial Distribution 843A.3.3 Negative Binomial Distribution 843A.3.4 Doubly Stochastic Poisson Process (Cox Process) 844A.4 Continuous Distributions 844A.4.1 Uniform Distribution 844A.4.2 Normal (Gaussian) Distribution 844A.4.3 Inverse Gaussian Distribution 845A.4.4 LogNormal Distribution 845A.4.5 Student’s t Distribution 846A.4.6 Gamma Distribution 846A.4.7 Weibull Distribution 846A.4.8 Inverse Chi-Squared Distribution 847A.4.9 Pareto Distribution (One-Parameter) 847A.4.10 Pareto Distribution (Two-Parameter) 847A.4.11 Generalized Pareto Distribution 848A.4.12 Beta Distribution 848A.4.13 Generalized Inverse Gaussian Distribution 849A.4.14 d-variate Normal Distribution 849A.4.15 d-variate t-Distribution 850Bibliography 851Index 892

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 xviiAcronyms xixList of Distributions xxi1 OpRisk in Perspective 11.1 Brief History 11.2 Risk-Based Capital Ratios for Banks 51.3 The Basic Indicator and Standardized Approaches for OpRisk 91.4 The Advanced Measurement Approach 101.4.1 Internal Measurement Approach 111.4.2 Score Card Approach 111.4.3 Loss Distribution Approach 121.4.4 Requirements for AMA 131.5 General Remarks and Book Structure 162 OpRisk Data and Governance 172.1 Introduction 172.2 OpRisk Taxonomy 172.2.1 Execution, Delivery, and Process Management 192.2.2 Clients, Products, and Business Practices 212.2.3 Business Disruption and System Failures 222.2.4 External Frauds 232.2.5 Internal Fraud 232.2.6 Employment Practices and Workplace Safety 242.2.7 Damage to Physical Assets 252.3 The Elements of the OpRisk Framework 252.3.1 Internal Loss Data 262.3.2 Setting a Collection Threshold and Possible Impacts 262.3.3 Completeness of Database (Under-reporting Events) 272.3.4 Recoveries and Near Misses 272.3.5 Time Period for Resolution of Operational Losses 282.3.6 Adding Costs to Losses 282.3.7 Provisioning Treatment of Expected Operational Losses 282.4 Business Environment and Internal Control Environment Factors (BEICFs) 292.4.1 Risk Control Self-Assessment (RCSA) 292.4.2 Key Risk Indicators 312.5 External Databases 332.6 Scenario Analysis 342.7 OpRisk Profile in Different Financial Sectors 372.7.1 Trading and Sales 372.7.2 Corporate Finance 382.7.3 Retail Banking 382.7.4 Insurance 392.7.5 Asset Management 402.7.6 Retail Brokerage 422.8 Risk Organization and Governance 432.8.1 Organization of Risk Departments 442.8.2 Structuring a Firm Wide Policy: Example of an OpRisk Policy 462.8.3 Governance 473 Using OpRisk Data for Business Analysis 483.1 Cost Reduction Programs in Financial Firms 493.2 Using OpRisk Data to Perform Business Analysis 533.2.1 The Risk of Losing Key Talents: OpRisk in Human Resources 533.2.2 OpRisk in Systems Development and Transaction Processing 543.3 Conclusions 584 Stress-Testing OpRisk Capital and the Comprehensive Capital Analysis and Review (CCAR) 594.1 The Need for Stressing OpRisk Capital Even Beyond 99.9% 594.2 Comprehensive Capital Review and Analysis (CCAR) 604.3 OpRisk and Stress Tests 684.4 OpRisk in CCAR in Practice 704.5 Reverse Stress Test 754.6 Stressing OpRisk Multivariate Models—Understanding the Relationship Among Internal Control Factors and Their Impact on Operational Losses 765 Basic Probability Concepts in Loss Distribution Approach 795.1 Loss Distribution Approach 795.2 Quantiles and Moments 855.3 Frequency Distributions 885.4 Severity Distributions 895.4.1 Simple Parametric Distributions 905.4.2 Truncated Distributions 925.4.3 Mixture and Spliced Distributions 935.5 Convolutions and Characteristic Functions 945.6 Extreme Value Theory 975.6.1 EVT—Block Maxima 985.6.2 EVT—Random Number of Losses 995.6.3 EVT—Threshold Exceedances 1006 Risk Measures and Capital Allocation 1026.1 Development of Capital Accords Base I, II and III 1036.2 Measures of Risk 1066.2.1 Coherent and Convex Risk Measures 1076.2.2 Comonotonic Additive Risk Measures 1096.2.3 Value-at-Risk 1096.2.4 Expected Shortfall 1146.2.5 Spectral Risk Measure 1206.2.6 Higher-Order Risk Measures 1226.2.7 Distortion Risk Measures 1256.2.8 Elicitable Risk Measures 1266.2.9 Risk Measure Accounting for Parameter Uncertainty 1306.3 Capital Allocation 1336.3.1 Coherent Capital Allocation 1346.3.2 Euler Allocation 1366.3.3 Standard Deviation 1386.3.4 Expected Shortfall 1396.3.5 Value-at-Risk 1406.3.6 Allocation by Marginal Contributions 1426.3.7 Numerical Example 1437 Estimation of Frequency and Severity Models 1467.1 Frequentist Estimation 1467.1.1 Parameteric Maximum Likelihood Method 1497.1.2 Maximum Likelihood Method for Truncated and Censored Data 1517.1.3 Expectation Maximization and Parameter Estimation 1527.1.4 Bootstrap for Estimation of Parameter Accuracy 1567.1.5 Indirect Inference–Based Likelihood Estimation 1577.2 Bayesian Inference Approach 1597.2.1 Conjugate Prior Distributions 1617.2.2 Gaussian Approximation for Posterior (Laplace Type) 1617.2.3 Posterior Point Estimators 1627.2.4 Restricted Parameters 1637.2.5 Noninformative Prior 1637.3 Mean Square Error of Prediction 1647.4 Standard Markov Chain Monte Carlo (MCMC) Methods 1667.4.1 Motivation for Markov Chain Methods 1677.4.2 Metropolis–Hastings Algorithm 1777.4.3 Gibbs Sampler 1787.4.4 Random Walk Metropolis–Hastings within Gibbs 1797.5 Standard MCMC Guidelines for Implementation 1807.5.1 Tuning, Burn-in, and Sampling Stages 1807.5.2 Numerical Error 1857.5.3 MCMC Extensions: Reducing Sample Autocorrelation 1877.6 Advanced MCMC Methods 1887.6.1 Auxiliary Variable MCMC Methods: Slice Sampling 1897.6.2 Generic Univariate Auxiliary Variable Gibbs Sampler: Slice Sampler 1897.6.3 Adaptive MCMC 1927.6.4 Riemann–Manifold Hamiltonian Monte Carlo Sampler (Automated Local Adaption) 1967.7 Sequential Monte Carlo (SMC) Samplers and Importance Sampling 2017.7.1 Motivating OpRisk Applications for SMC Samplers 2027.7.2 SMC Sampler Methodology and Components 2107.7.3 Incorporating Partial Rejection Control into SMC Samplers 2167.7.4 Finite Sample (Nonasymptotic) Accuracy for Particle Integration 2197.8 Approximate Bayesian Computation (ABC) Methods 2207.9 OpRisk Estimation and Modeling for Truncated Data 2237.9.1 Constant Threshold - Poisson Process 2247.9.2 Negative Binomial and Binomial Frequencies 2277.9.3 Ignoring Data Truncation 2287.9.4 Threshold Varying in Time 2327.9.5 Unknown and Stochastic Truncation Level 2368 Model Selection and Goodness-of-Fit Testing for Frequency and Severity Models 2388.1 Qualitative Model Diagnostic Tools 2388.2 Tail Diagnostics 2408.3 Information Criterion for Model Selection 2428.3.1 Akaike Information Criterion for LDA Model Selection 2428.3.2 Deviance Information Criterion 2458.4 Goodness-of-Fit Testing for Model Choice (How to Account for Heavy Tails!) 2468.4.1 Convergence Results of the Empirical Process for GOF Testing 2478.4.2 Overview of Generic GOF Tests—Omnibus Distributional Tests 2568.4.3 Kolmogorov–Smirnov Goodness-of-Fit Test and Weighted Variants: Testing in the Presence of Heavy Tails 2608.4.4 Cramer-von-Mises Goodness-of-Fit Tests and Weighted Variants: Testing in the Presence of Heavy Tails 2718.5 Bayesian Model Selection 2838.5.1 Reciprocal Importance Sampling Estimator 2848.5.2 Chib Estimator for Model Evidence 2858.6 SMC Sampler Estimators of Model Evidence 2868.7 Multiple Risk Dependence Structure Model Selection: Copula Choice 2878.7.1 Approaches to Goodness-of-Fit Testing for Dependence Structures 2938.7.2 Double Parameteric Bootstrap for Copula GOF 2979 Flexible Parametric Severity Models: Basics 3009.1 Motivation for Flexible Parametric Severity Loss Models 3009.2 Context of Flexible Heavy-Tailed Loss Models in OpRisk and Insurance LDA Models 3019.3 Empirical Analysis Justifying Heavy-Tailed Loss Models in OpRisk 3039.4 Quantile Function Heavy-Tailed Severity Models 3059.4.1 g-and-h Severity Model Family in OpRisk 3119.4.2 Tail Properties of the g-and-h, g, h, and h–h Severity in OpRisk 3219.4.3 Parameter Estimation for the g-and-h Severity in OpRisk 3249.4.4 Bayesian Models for the g-and-h Severity in OpRisk 3289.5 Generalized Beta Family of Heavy-Tailed Severity Models 3339.5.1 Generalized Beta Family Type II Severity Models in OpRisk 3339.5.2 Sub families of the Generalized Beta Family Type II Severity Models 3369.5.3 Mixture Representations of the Generalized Beta Family Type II Severity Models 3379.5.4 Estimation in the Generalized Beta Family Type II Severity Models 3399.6 Generalized Hyperbolic Families of Heavy-Tailed Severity Models 3409.6.1 Tail Properties and Infinite Divisibility of the Generalized Hyperbolic Severity Models 3429.6.2 Subfamilies of the Generalized Hyperbolic Severity Models 3449.6.3 Normal Inverse Gaussian Family of Heavy-Tailed Severity Models 3469.7 Halphen Family of Flexible Severity Models: GIG and Hyperbolic 3509.7.1 Halphen Type A: Generalized Inverse Gaussian Family of Flexible Severity Models 3559.7.2 Halphen Type B and IB Families of Flexible Severity Models 36110 Dependence Concepts 36510.1 Introduction to Concepts in Dependence for OpRisk and Insurance 36510.2 Dependence Modeling Within and Between LDA Model Structures 36610.2.1 Where Can One Introduce Dependence Between LDA Model Structures? 36810.2.2 Understanding Basic Impacts of Dependence Modeling Between LDA Components in Multiple Risks 36910.3 General Notions of Dependence 37210.4 Dependence Measures 38710.4.1 Linear Correlation 39010.4.2 Rank Correlation Measures 39310.5 Tail Dependence Parameters, Functions, and Tail Order Functions 39810.5.1 Tail Dependence Coefficients 39810.5.2 Tail Dependence Functions and Orders 40710.5.3 A Link Between Orthant Extreme Dependence and Spectral Measures: Tail Dependence 41011 Dependence Models 41411.1 Introduction to Parametric Dependence Modeling Through a Copula 41411.2 Copula Model Families for OpRisk 42211.2.1 Gaussian Copula 42811.2.2 t-Copula 43011.2.3 Archimedean Copulas 43511.2.4 Archimedean Copula Generators and the Laplace Transform of a Non-Negative Random Variable 43911.2.5 Archimedean Copula Generators, l1-Norm Symmetric Distributions and the Williamson Transform 44111.2.6 Hierarchical and Nested Archimedean Copulae 45211.2.7 Mixtures of Archimedean Copulae 45411.2.8 Multivariate Archimedean Copula Tail Dependence 45611.3 Copula Parameter Estimation in Two Stages: Inference for the Margins 45711.3.1 MPLE: Copula Parameter Estimation 45811.3.2 Inference Functions for Margins (IFM): Copula Parameter Estimation 45912 Examples of LDA Dependence Models 46212.1 Multiple Risk LDA Compound Poisson Processes and Lévy Copula 46212.2 Multiple Risk LDA: Dependence Between Frequencies via Copula 46812.3 Multiple Risk LDA: Dependence Between the k-th Event Times/Losses 46812.3.1 Common Shock Processes 46912.3.2 Max-Stable and Self-Chaining Copula Models 47012.4 Multiple Risk LDA: Dependence Between Aggregated Losses via Copula 47412.5 Multiple Risk LDA: Structural Model with Common Factors 47712.6 Multiple Risk LDA: Stochastic and Dependent Risk Profiles 47812.7 Multiple Risk LDA: Dependence and Combining Different Data Sources 48212.7.1 Bayesian Inference Using MCMC 48412.7.2 Numerical Example 48512.7.3 Predictive Distribution 48712.8 A Note on Negative Diversification and Dependence Modeling 48913 Loss Aggregation 49213.1 Analytic Solution 49213.1.1 Analytic Solution via Convolutions 49313.1.2 Analytic Solution via Characteristic Functions 49413.1.3 Moments of Compound Distribution 49613.1.4 Value-at-Risk and Expected Shortfall 49913.2 Monte Carlo Method 49913.2.1 Quantile Estimate 50013.2.2 Expected Shortfall Estimate 50213.3 Panjer Recursion 50313.4 Panjer Extensions 50913.5 Fast Fourier Transform 51113.6 Closed-Form Approximation 51413.7 Capital Charge Under Parameter Uncertainty 51913.7.1 Predictive Distributions 52013.7.2 Calculation of Predictive Distributions 52113.8 Special Advanced Topics on Loss Aggregation 52313.8.1 Discretisation Errors and Extrapolation Methods 52413.8.2 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails 52713.8.3 Recursions for Convolutions (Partial Sums) with Discretised Severity Distributions (Fixed n) 53513.8.4 Alternatives to Panjer Recursions: Recursions for Compound Distributions with Discretised Severity Distributions 54313.8.5 Higher Order Recursions for Discretised Severity Distributions in Compound LDA Models 54513.8.6 Recursions for Discretised Severity Distributions in Compound Mixed Poisson LDA Models 54713.8.7 Continuous Versions of the Panjer Recursion 55014 Scenario Analysis 55614.1 Introduction 55614.2 Examples of Expert Judgments 55914.3 Pure Bayesian Approach (Estimating Prior) 56114.4 Expert Distribution and Scenario Elicitation: Learning from Bayesian Methods 56314.5 Building Models for Elicited Opinions: Hierarchical Dirichlet Models 56614.6 Worst-Case Scenario Framework 56814.7 Stress Test Scenario Analysis 57114.8 Bow-Tie Diagram 57414.9 Bayesian Networks 57614.9.1 Definition and Examples 57714.9.2 Constructing and Simulating a Bayesian Net 58014.9.3 Combining Expert Opinion and Data in a Bayesian Net 58114.9.4 Bayesian Net and Operational Risk 58214.10 Discussion 58415 Combining Different Data Sources 58515.1 Minimum Variance Principle 58615.2 Bayesian Method to Combine Two Data Sources 58815.2.1 Estimating Prior: Pure Bayesian Approach 59015.2.2 Estimating Prior: Empirical Bayesian Approach 59215.2.3 Poisson Frequency 59315.2.4 The LogNormal Severity 59715.2.5 Pareto Severity 60115.3 Estimation of the Prior Using Data 60615.3.1 The Maximum Likelihood Estimator 60615.3.2 Poisson Frequencies 60715.4 Combining Expert Opinions with External and Internal Data 60915.4.1 Conjugate Prior Extension 61015.4.2 Modeling Frequency: Poisson Model 61115.4.3 LogNormal Model for Severities 61815.4.4 Pareto Model 62015.5 Combining Data Sources Using Credibility Theory 62515.5.1 Bühlmann–Straub Model 62615.5.2 Modeling Frequency 62815.5.3 Modeling Severity 63115.5.4 Numerical Example 63315.5.5 Remarks and Interpretation 63415.6 Nonparametric Bayesian Approach via Dirichlet Process 63515.7 Combining Using Dempster–Shafer Structures and p-Boxes 63815.7.1 Dempster–Shafer Structures and p-Boxes 63915.7.2 Dempster’s Rule 64115.7.3 Intersection Method 64315.7.4 Envelope Method 64415.7.5 Bounds for the Empirical Data Distribution 64515.8 General Remarks 64716 Multifactor Modeling and Regression for Loss Processes 64916.1 Generalized Linear Model Regressions and the Exponential Family 64916.1.1 Basic Components of a Generalized Linear Model Regression in the Exponential Family 65016.1.2 Basis Function Regression 65416.2 Maximum Likelihood Estimation for Generalized Linear Models 65516.2.1 Iterated Weighted Least Squares Maximum Likelihood for Generalised Linear Models 65516.2.2 Model Selection via the Deviance in a GLM Regression 65716.3 Bayesian Generalized Linear Model Regressions and Regularization Priors 65916.3.1 Bayesian Model Selection for Regularlized GLM Regression 66516.4 Bayesian Estimation and Model Selection via SMC Samplers 66616.4.1 Proposed SMC Sampler Solution 66716.5 Illustrations of SMC Samplers Model Estimation and Selection for Bayesian GLM Regressions 66816.5.1 Normal Regression Model 66816.5.2 Poisson Regression Model 66916.6 Introduction to Quantile Regression Methods for OpRisk 67216.6.1 Nonparametric Quantile Regression Models 67416.6.2 Parametric Quantile Regression Models 67516.7 Factor Modeling for Industry Data 68116.8 Multifactor Modeling under EVT Approach 68317 Insurance and Risk Transfer: Products and Modeling 68517.1 Motivation for Insurance and Risk Transfer in OpRisk 68517.2 Fundamentals of Insurance Product Structures for OpRisk 68817.3 Single Peril Policy Products for OpRisk 69217.4 Generic Insurance Product Structures for OpRisk 69417.4.1 Generic Deterministic Policy Structures 69417.4.2 Generic Stochastic Policy Structures: Accounting for Coverage Uncertainty 70017.5 Closed-Form LDA Models with Insurance Mitigations 70517.5.1 Insurance Mitigation Under the Poisson-Inverse-Gaussian Closed-Form LDA Models 70517.5.2 Insurance Mitigation and Poisson-α-Stable Closed-Form LDA Models 71217.5.3 Large Claim Number Loss Processes: Generic Closed-Form LDA Models with Insurance Mitigation 71917.5.4 Generic Closed-Form Approximations for Insured LDA Models 73418 Insurance and Risk Transfer: Pricing Insurance-Linked Derivatives, Reinsurance, and CAT Bonds for OpRisk 75018.1 Insurance-Linked Securities and CAT Bonds for OpRisk 75118.1.1 Background on Insurance-Linked Derivatives and CAT Bonds for Extreme Risk Transfer 75518.1.2 Triggers for CAT Bonds and Their Impact on Risk Transfer 76018.1.3 Recent Trends in CAT Bonds 76318.1.4 Management Strategies for Utilization of Insurance-Linked Derivatives and CAT Bonds in OpRisk 76318.2 Basics of Valuation of ILS and CAT Bonds for OpRisk 76518.2.1 Probabilistic Pricing Frameworks: Complete and Incomplete Markets, Real-World Pricing, Benchmark Approach, and Actuarial Valuation 77118.2.2 Risk Assessment for Reinsurance: ILS and CAT Bonds 79418.3 Applications of Pricing ILS and CAT Bonds 79618.3.1 Probabilistic Framework for CAT Bond Market 79618.3.2 Framework 1: Assuming Complete Market and Arbitrage-Free Pricing 79818.3.3 Framework 2: Assuming Incomplete Arbitrage-Free Pricing 80918.4 Sidecars, Multiple Peril Baskets, and Umbrellas for OpRisk 81518.4.1 Umbrella Insurance 81618.4.2 OpRisk Loss Processes Comprised of Multiple Perils 81718.5 Optimal Insurance Purchase Strategies for OpRisk Insurance via Multiple Optimal Stopping Times 82318.5.1 Examples of Basic Insurance Policies 82618.5.2 Objective Functions for Rational and Boundedly Rational Insurees 82818.5.3 Closed-Form Multiple Optimal Stopping Rules for Multiple Insurance Purchase Decisions 83018.5.4 Aski-Polynomial Orthogonal Series Approximations 835A Miscellaneous Definitions and List of Distributions 842A.1 Indicator Function 842A.2 Gamma Function 842A.3 Discrete Distributions 842A.3.1 Poisson Distribution 842A.3.2 Binomial Distribution 843A.3.3 Negative Binomial Distribution 843A.3.4 Doubly Stochastic Poisson Process (Cox Process) 844A.4 Continuous Distributions 844A.4.1 Uniform Distribution 844A.4.2 Normal (Gaussian) Distribution 844A.4.3 Inverse Gaussian Distribution 845A.4.4 LogNormal Distribution 845A.4.5 Student’s t Distribution 846A.4.6 Gamma Distribution 846A.4.7 Weibull Distribution 846A.4.8 Inverse Chi-Squared Distribution 847A.4.9 Pareto Distribution (One-Parameter) 847A.4.10 Pareto Distribution (Two-Parameter) 847A.4.11 Generalized Pareto Distribution 848A.4.12 Beta Distribution 848A.4.13 Generalized Inverse Gaussian Distribution 849A.4.14 d-variate Normal Distribution 849A.4.15 d-variate t-Distribution 850Bibliography 851Index 892

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 xviiAcronyms xixList of Distributions xxi1 OpRisk in Perspective 11.1 Brief History 11.2 Risk-Based Capital Ratios for Banks 51.3 The Basic Indicator and Standardized Approaches for OpRisk 91.4 The Advanced Measurement Approach 101.4.1 Internal Measurement Approach 111.4.2 Score Card Approach 111.4.3 Loss Distribution Approach 121.4.4 Requirements for AMA 131.5 General Remarks and Book Structure 162 OpRisk Data and Governance 172.1 Introduction 172.2 OpRisk Taxonomy 172.2.1 Execution, Delivery, and Process Management 192.2.2 Clients, Products, and Business Practices 212.2.3 Business Disruption and System Failures 222.2.4 External Frauds 232.2.5 Internal Fraud 232.2.6 Employment Practices and Workplace Safety 242.2.7 Damage to Physical Assets 252.3 The Elements of the OpRisk Framework 252.3.1 Internal Loss Data 262.3.2 Setting a Collection Threshold and Possible Impacts 262.3.3 Completeness of Database (Under-reporting Events) 272.3.4 Recoveries and Near Misses 272.3.5 Time Period for Resolution of Operational Losses 282.3.6 Adding Costs to Losses 282.3.7 Provisioning Treatment of Expected Operational Losses 282.4 Business Environment and Internal Control Environment Factors (BEICFs) 292.4.1 Risk Control Self-Assessment (RCSA) 292.4.2 Key Risk Indicators 312.5 External Databases 332.6 Scenario Analysis 342.7 OpRisk Profile in Different Financial Sectors 372.7.1 Trading and Sales 372.7.2 Corporate Finance 382.7.3 Retail Banking 382.7.4 Insurance 392.7.5 Asset Management 402.7.6 Retail Brokerage 422.8 Risk Organization and Governance 432.8.1 Organization of Risk Departments 442.8.2 Structuring a Firm Wide Policy: Example of an OpRisk Policy 462.8.3 Governance 473 Using OpRisk Data for Business Analysis 483.1 Cost Reduction Programs in Financial Firms 493.2 Using OpRisk Data to Perform Business Analysis 533.2.1 The Risk of Losing Key Talents: OpRisk in Human Resources 533.2.2 OpRisk in Systems Development and Transaction Processing 543.3 Conclusions 584 Stress-Testing OpRisk Capital and the Comprehensive Capital Analysis and Review (CCAR) 594.1 The Need for Stressing OpRisk Capital Even Beyond 99.9% 594.2 Comprehensive Capital Review and Analysis (CCAR) 604.3 OpRisk and Stress Tests 684.4 OpRisk in CCAR in Practice 704.5 Reverse Stress Test 754.6 Stressing OpRisk Multivariate Models—Understanding the Relationship Among Internal Control Factors and Their Impact on Operational Losses 765 Basic Probability Concepts in Loss Distribution Approach 795.1 Loss Distribution Approach 795.2 Quantiles and Moments 855.3 Frequency Distributions 885.4 Severity Distributions 895.4.1 Simple Parametric Distributions 905.4.2 Truncated Distributions 925.4.3 Mixture and Spliced Distributions 935.5 Convolutions and Characteristic Functions 945.6 Extreme Value Theory 975.6.1 EVT—Block Maxima 985.6.2 EVT—Random Number of Losses 995.6.3 EVT—Threshold Exceedances 1006 Risk Measures and Capital Allocation 1026.1 Development of Capital Accords Base I, II and III 1036.2 Measures of Risk 1066.2.1 Coherent and Convex Risk Measures 1076.2.2 Comonotonic Additive Risk Measures 1096.2.3 Value-at-Risk 1096.2.4 Expected Shortfall 1146.2.5 Spectral Risk Measure 1206.2.6 Higher-Order Risk Measures 1226.2.7 Distortion Risk Measures 1256.2.8 Elicitable Risk Measures 1266.2.9 Risk Measure Accounting for Parameter Uncertainty 1306.3 Capital Allocation 1336.3.1 Coherent Capital Allocation 1346.3.2 Euler Allocation 1366.3.3 Standard Deviation 1386.3.4 Expected Shortfall 1396.3.5 Value-at-Risk 1406.3.6 Allocation by Marginal Contributions 1426.3.7 Numerical Example 1437 Estimation of Frequency and Severity Models 1467.1 Frequentist Estimation 1467.1.1 Parameteric Maximum Likelihood Method 1497.1.2 Maximum Likelihood Method for Truncated and Censored Data 1517.1.3 Expectation Maximization and Parameter Estimation 1527.1.4 Bootstrap for Estimation of Parameter Accuracy 1567.1.5 Indirect Inference–Based Likelihood Estimation 1577.2 Bayesian Inference Approach 1597.2.1 Conjugate Prior Distributions 1617.2.2 Gaussian Approximation for Posterior (Laplace Type) 1617.2.3 Posterior Point Estimators 1627.2.4 Restricted Parameters 1637.2.5 Noninformative Prior 1637.3 Mean Square Error of Prediction 1647.4 Standard Markov Chain Monte Carlo (MCMC) Methods 1667.4.1 Motivation for Markov Chain Methods 1677.4.2 Metropolis–Hastings Algorithm 1777.4.3 Gibbs Sampler 1787.4.4 Random Walk Metropolis–Hastings within Gibbs 1797.5 Standard MCMC Guidelines for Implementation 1807.5.1 Tuning, Burn-in, and Sampling Stages 1807.5.2 Numerical Error 1857.5.3 MCMC Extensions: Reducing Sample Autocorrelation 1877.6 Advanced MCMC Methods 1887.6.1 Auxiliary Variable MCMC Methods: Slice Sampling 1897.6.2 Generic Univariate Auxiliary Variable Gibbs Sampler: Slice Sampler 1897.6.3 Adaptive MCMC 1927.6.4 Riemann–Manifold Hamiltonian Monte Carlo Sampler (Automated Local Adaption) 1967.7 Sequential Monte Carlo (SMC) Samplers and Importance Sampling 2017.7.1 Motivating OpRisk Applications for SMC Samplers 2027.7.2 SMC Sampler Methodology and Components 2107.7.3 Incorporating Partial Rejection Control into SMC Samplers 2167.7.4 Finite Sample (Nonasymptotic) Accuracy for Particle Integration 2197.8 Approximate Bayesian Computation (ABC) Methods 2207.9 OpRisk Estimation and Modeling for Truncated Data 2237.9.1 Constant Threshold - Poisson Process 2247.9.2 Negative Binomial and Binomial Frequencies 2277.9.3 Ignoring Data Truncation 2287.9.4 Threshold Varying in Time 2327.9.5 Unknown and Stochastic Truncation Level 2368 Model Selection and Goodness-of-Fit Testing for Frequency and Severity Models 2388.1 Qualitative Model Diagnostic Tools 2388.2 Tail Diagnostics 2408.3 Information Criterion for Model Selection 2428.3.1 Akaike Information Criterion for LDA Model Selection 2428.3.2 Deviance Information Criterion 2458.4 Goodness-of-Fit Testing for Model Choice (How to Account for Heavy Tails!) 2468.4.1 Convergence Results of the Empirical Process for GOF Testing 2478.4.2 Overview of Generic GOF Tests—Omnibus Distributional Tests 2568.4.3 Kolmogorov–Smirnov Goodness-of-Fit Test and Weighted Variants: Testing in the Presence of Heavy Tails 2608.4.4 Cramer-von-Mises Goodness-of-Fit Tests and Weighted Variants: Testing in the Presence of Heavy Tails 2718.5 Bayesian Model Selection 2838.5.1 Reciprocal Importance Sampling Estimator 2848.5.2 Chib Estimator for Model Evidence 2858.6 SMC Sampler Estimators of Model Evidence 2868.7 Multiple Risk Dependence Structure Model Selection: Copula Choice 2878.7.1 Approaches to Goodness-of-Fit Testing for Dependence Structures 2938.7.2 Double Parameteric Bootstrap for Copula GOF 2979 Flexible Parametric Severity Models: Basics 3009.1 Motivation for Flexible Parametric Severity Loss Models 3009.2 Context of Flexible Heavy-Tailed Loss Models in OpRisk and Insurance LDA Models 3019.3 Empirical Analysis Justifying Heavy-Tailed Loss Models in OpRisk 3039.4 Quantile Function Heavy-Tailed Severity Models 3059.4.1 g-and-h Severity Model Family in OpRisk 3119.4.2 Tail Properties of the g-and-h, g, h, and h–h Severity in OpRisk 3219.4.3 Parameter Estimation for the g-and-h Severity in OpRisk 3249.4.4 Bayesian Models for the g-and-h Severity in OpRisk 3289.5 Generalized Beta Family of Heavy-Tailed Severity Models 3339.5.1 Generalized Beta Family Type II Severity Models in OpRisk 3339.5.2 Sub families of the Generalized Beta Family Type II Severity Models 3369.5.3 Mixture Representations of the Generalized Beta Family Type II Severity Models 3379.5.4 Estimation in the Generalized Beta Family Type II Severity Models 3399.6 Generalized Hyperbolic Families of Heavy-Tailed Severity Models 3409.6.1 Tail Properties and Infinite Divisibility of the Generalized Hyperbolic Severity Models 3429.6.2 Subfamilies of the Generalized Hyperbolic Severity Models 3449.6.3 Normal Inverse Gaussian Family of Heavy-Tailed Severity Models 3469.7 Halphen Family of Flexible Severity Models: GIG and Hyperbolic 3509.7.1 Halphen Type A: Generalized Inverse Gaussian Family of Flexible Severity Models 3559.7.2 Halphen Type B and IB Families of Flexible Severity Models 36110 Dependence Concepts 36510.1 Introduction to Concepts in Dependence for OpRisk and Insurance 36510.2 Dependence Modeling Within and Between LDA Model Structures 36610.2.1 Where Can One Introduce Dependence Between LDA Model Structures? 36810.2.2 Understanding Basic Impacts of Dependence Modeling Between LDA Components in Multiple Risks 36910.3 General Notions of Dependence 37210.4 Dependence Measures 38710.4.1 Linear Correlation 39010.4.2 Rank Correlation Measures 39310.5 Tail Dependence Parameters, Functions, and Tail Order Functions 39810.5.1 Tail Dependence Coefficients 39810.5.2 Tail Dependence Functions and Orders 40710.5.3 A Link Between Orthant Extreme Dependence and Spectral Measures: Tail Dependence 41011 Dependence Models 41411.1 Introduction to Parametric Dependence Modeling Through a Copula 41411.2 Copula Model Families for OpRisk 42211.2.1 Gaussian Copula 42811.2.2 t-Copula 43011.2.3 Archimedean Copulas 43511.2.4 Archimedean Copula Generators and the Laplace Transform of a Non-Negative Random Variable 43911.2.5 Archimedean Copula Generators, l1-Norm Symmetric Distributions and the Williamson Transform 44111.2.6 Hierarchical and Nested Archimedean Copulae 45211.2.7 Mixtures of Archimedean Copulae 45411.2.8 Multivariate Archimedean Copula Tail Dependence 45611.3 Copula Parameter Estimation in Two Stages: Inference for the Margins 45711.3.1 MPLE: Copula Parameter Estimation 45811.3.2 Inference Functions for Margins (IFM): Copula Parameter Estimation 45912 Examples of LDA Dependence Models 46212.1 Multiple Risk LDA Compound Poisson Processes and Lévy Copula 46212.2 Multiple Risk LDA: Dependence Between Frequencies via Copula 46812.3 Multiple Risk LDA: Dependence Between the k-th Event Times/Losses 46812.3.1 Common Shock Processes 46912.3.2 Max-Stable and Self-Chaining Copula Models 47012.4 Multiple Risk LDA: Dependence Between Aggregated Losses via Copula 47412.5 Multiple Risk LDA: Structural Model with Common Factors 47712.6 Multiple Risk LDA: Stochastic and Dependent Risk Profiles 47812.7 Multiple Risk LDA: Dependence and Combining Different Data Sources 48212.7.1 Bayesian Inference Using MCMC 48412.7.2 Numerical Example 48512.7.3 Predictive Distribution 48712.8 A Note on Negative Diversification and Dependence Modeling 48913 Loss Aggregation 49213.1 Analytic Solution 49213.1.1 Analytic Solution via Convolutions 49313.1.2 Analytic Solution via Characteristic Functions 49413.1.3 Moments of Compound Distribution 49613.1.4 Value-at-Risk and Expected Shortfall 49913.2 Monte Carlo Method 49913.2.1 Quantile Estimate 50013.2.2 Expected Shortfall Estimate 50213.3 Panjer Recursion 50313.4 Panjer Extensions 50913.5 Fast Fourier Transform 51113.6 Closed-Form Approximation 51413.7 Capital Charge Under Parameter Uncertainty 51913.7.1 Predictive Distributions 52013.7.2 Calculation of Predictive Distributions 52113.8 Special Advanced Topics on Loss Aggregation 52313.8.1 Discretisation Errors and Extrapolation Methods 52413.8.2 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails 52713.8.3 Recursions for Convolutions (Partial Sums) with Discretised Severity Distributions (Fixed n) 53513.8.4 Alternatives to Panjer Recursions: Recursions for Compound Distributions with Discretised Severity Distributions 54313.8.5 Higher Order Recursions for Discretised Severity Distributions in Compound LDA Models 54513.8.6 Recursions for Discretised Severity Distributions in Compound Mixed Poisson LDA Models 54713.8.7 Continuous Versions of the Panjer Recursion 55014 Scenario Analysis 55614.1 Introduction 55614.2 Examples of Expert Judgments 55914.3 Pure Bayesian Approach (Estimating Prior) 56114.4 Expert Distribution and Scenario Elicitation: Learning from Bayesian Methods 56314.5 Building Models for Elicited Opinions: Hierarchical Dirichlet Models 56614.6 Worst-Case Scenario Framework 56814.7 Stress Test Scenario Analysis 57114.8 Bow-Tie Diagram 57414.9 Bayesian Networks 57614.9.1 Definition and Examples 57714.9.2 Constructing and Simulating a Bayesian Net 58014.9.3 Combining Expert Opinion and Data in a Bayesian Net 58114.9.4 Bayesian Net and Operational Risk 58214.10 Discussion 58415 Combining Different Data Sources 58515.1 Minimum Variance Principle 58615.2 Bayesian Method to Combine Two Data Sources 58815.2.1 Estimating Prior: Pure Bayesian Approach 59015.2.2 Estimating Prior: Empirical Bayesian Approach 59215.2.3 Poisson Frequency 59315.2.4 The LogNormal Severity 59715.2.5 Pareto Severity 60115.3 Estimation of the Prior Using Data 60615.3.1 The Maximum Likelihood Estimator 60615.3.2 Poisson Frequencies 60715.4 Combining Expert Opinions with External and Internal Data 60915.4.1 Conjugate Prior Extension 61015.4.2 Modeling Frequency: Poisson Model 61115.4.3 LogNormal Model for Severities 61815.4.4 Pareto Model 62015.5 Combining Data Sources Using Credibility Theory 62515.5.1 Bühlmann–Straub Model 62615.5.2 Modeling Frequency 62815.5.3 Modeling Severity 63115.5.4 Numerical Example 63315.5.5 Remarks and Interpretation 63415.6 Nonparametric Bayesian Approach via Dirichlet Process 63515.7 Combining Using Dempster–Shafer Structures and p-Boxes 63815.7.1 Dempster–Shafer Structures and p-Boxes 63915.7.2 Dempster’s Rule 64115.7.3 Intersection Method 64315.7.4 Envelope Method 64415.7.5 Bounds for the Empirical Data Distribution 64515.8 General Remarks 64716 Multifactor Modeling and Regression for Loss Processes 64916.1 Generalized Linear Model Regressions and the Exponential Family 64916.1.1 Basic Components of a Generalized Linear Model Regression in the Exponential Family 65016.1.2 Basis Function Regression 65416.2 Maximum Likelihood Estimation for Generalized Linear Models 65516.2.1 Iterated Weighted Least Squares Maximum Likelihood for Generalised Linear Models 65516.2.2 Model Selection via the Deviance in a GLM Regression 65716.3 Bayesian Generalized Linear Model Regressions and Regularization Priors 65916.3.1 Bayesian Model Selection for Regularlized GLM Regression 66516.4 Bayesian Estimation and Model Selection via SMC Samplers 66616.4.1 Proposed SMC Sampler Solution 66716.5 Illustrations of SMC Samplers Model Estimation and Selection for Bayesian GLM Regressions 66816.5.1 Normal Regression Model 66816.5.2 Poisson Regression Model 66916.6 Introduction to Quantile Regression Methods for OpRisk 67216.6.1 Nonparametric Quantile Regression Models 67416.6.2 Parametric Quantile Regression Models 67516.7 Factor Modeling for Industry Data 68116.8 Multifactor Modeling under EVT Approach 68317 Insurance and Risk Transfer: Products and Modeling 68517.1 Motivation for Insurance and Risk Transfer in OpRisk 68517.2 Fundamentals of Insurance Product Structures for OpRisk 68817.3 Single Peril Policy Products for OpRisk 69217.4 Generic Insurance Product Structures for OpRisk 69417.4.1 Generic Deterministic Policy Structures 69417.4.2 Generic Stochastic Policy Structures: Accounting for Coverage Uncertainty 70017.5 Closed-Form LDA Models with Insurance Mitigations 70517.5.1 Insurance Mitigation Under the Poisson-Inverse-Gaussian Closed-Form LDA Models 70517.5.2 Insurance Mitigation and Poisson-α-Stable Closed-Form LDA Models 71217.5.3 Large Claim Number Loss Processes: Generic Closed-Form LDA Models with Insurance Mitigation 71917.5.4 Generic Closed-Form Approximations for Insured LDA Models 73418 Insurance and Risk Transfer: Pricing Insurance-Linked Derivatives, Reinsurance, and CAT Bonds for OpRisk 75018.1 Insurance-Linked Securities and CAT Bonds for OpRisk 75118.1.1 Background on Insurance-Linked Derivatives and CAT Bonds for Extreme Risk Transfer 75518.1.2 Triggers for CAT Bonds and Their Impact on Risk Transfer 76018.1.3 Recent Trends in CAT Bonds 76318.1.4 Management Strategies for Utilization of Insurance-Linked Derivatives and CAT Bonds in OpRisk 76318.2 Basics of Valuation of ILS and CAT Bonds for OpRisk 76518.2.1 Probabilistic Pricing Frameworks: Complete and Incomplete Markets, Real-World Pricing, Benchmark Approach, and Actuarial Valuation 77118.2.2 Risk Assessment for Reinsurance: ILS and CAT Bonds 79418.3 Applications of Pricing ILS and CAT Bonds 79618.3.1 Probabilistic Framework for CAT Bond Market 79618.3.2 Framework 1: Assuming Complete Market and Arbitrage-Free Pricing 79818.3.3 Framework 2: Assuming Incomplete Arbitrage-Free Pricing 80918.4 Sidecars, Multiple Peril Baskets, and Umbrellas for OpRisk 81518.4.1 Umbrella Insurance 81618.4.2 OpRisk Loss Processes Comprised of Multiple Perils 81718.5 Optimal Insurance Purchase Strategies for OpRisk Insurance via Multiple Optimal Stopping Times 82318.5.1 Examples of Basic Insurance Policies 82618.5.2 Objective Functions for Rational and Boundedly Rational Insurees 82818.5.3 Closed-Form Multiple Optimal Stopping Rules for Multiple Insurance Purchase Decisions 83018.5.4 Aski-Polynomial Orthogonal Series Approximations 835A Miscellaneous Definitions and List of Distributions 842A.1 Indicator Function 842A.2 Gamma Function 842A.3 Discrete Distributions 842A.3.1 Poisson Distribution 842A.3.2 Binomial Distribution 843A.3.3 Negative Binomial Distribution 843A.3.4 Doubly Stochastic Poisson Process (Cox Process) 844A.4 Continuous Distributions 844A.4.1 Uniform Distribution 844A.4.2 Normal (Gaussian) Distribution 844A.4.3 Inverse Gaussian Distribution 845A.4.4 LogNormal Distribution 845A.4.5 Student’s t Distribution 846A.4.6 Gamma Distribution 846A.4.7 Weibull Distribution 846A.4.8 Inverse Chi-Squared Distribution 847A.4.9 Pareto Distribution (One-Parameter) 847A.4.10 Pareto Distribution (Two-Parameter) 847A.4.11 Generalized Pareto Distribution 848A.4.12 Beta Distribution 848A.4.13 Generalized Inverse Gaussian Distribution 849A.4.14 d-variate Normal Distribution 849A.4.15 d-variate t-Distribution 850Bibliography 851Index 892