Advances in Heavy Tailed Risk Modeling

Preface xix Acronyms xxi Symbols xxiii List of Distributions xxv 1 Motivation for Heavy-Tailed Models 1 1.1 Structure of the Book 1 1.2 Dominance of the Heaviest Tail Risks 3 1.3 Empirical Analysis Justifying Heavy-Tailed Loss Models in OpRisk 6 1.4 Motivating Parametric, Spliced and Non-Parametric Severity Models 9 1.5 Creating Flexible Heavy-Tailed Models via Splicing 11 2 Fundamentals of Extreme Value Theory for OpRisk 17 2.1 Introduction 17 2.2 Historical Perspective on EVT and Risk 18 2.3 Theoretical Properties of Univariate EVT-Block Maxima and the GEV Family 20 2.4 Generalized Extreme Value Loss Distributional Approach (GEV-LDA) 40 2.4.1 Statistical Considerations for Applicability of the GEV Model 40 2.4.2 Various Statistical Estimation Procedures for the GEV Model Parameters in OpRisk Settings 42 2.4.3 GEV Sub-Family Approaches in OpRisk LDA Modeling 54 2.4.4 Properties of the Frechet-Pareto Family of Severity Models 54 2.4.5 Single Risk LDA Poisson-Generalized Pareto Family 55 2.4.6 Single Risk LDA Poisson-Burr Family 60 2.4.7 Properties of the Gumbel family of Severity Models 65 2.4.8 Single Risk LDA Poisson-LogNormal Family 65 2.4.9 Single Risk LDA Poisson-Benktander II Models 68 2.5 Theoretical Properties of Univariate EVT-Threshold Exceedances 72 2.5.1 Understanding the Distribution of Threshold Exceedances 74 2.6 Estimation Under the Peaks Over Threshold Approach via the Generalized Pareto Distribution 85 2.6.1 Maximum-Likelihood Estimation Under the GPD Model 87 2.6.2 Comments on Probability-Weighted Method of Moments Estimation Under the GPD Model 93 2.6.3 Robust Estimators of the GPD Model Parameters 95 2.6.4 EVT--Random Number of Losses 101 3 Heavy-Tailed Model Class Characterizations for LDA 105 3.1 Landau Notations for OpRisk Asymptotics: Big and Little 'Oh' 106 3.2 Introduction to the Sub-Exponential Family of Heavy-Tailed Models 113 3.3 Introduction to the Regular and Slow Variation Families of Heavy-Tailed Models 121 3.4 Alternative Classifications of Heavy-Tailed Models and Tail Variation 129 3.5 Extended Regular Variation and Matuszewska Indices for Heavy-Tailed Models 135 4 Flexible Heavy-Tailed Severity Models: alpha-Stable Family 139 4.1 Infinitely Divisible and Self-Decomposable Loss Random Variables 140 4.1.1 Basic Properties of Characteristic Functions 140 4.1.2 Divisibility and Self-Decomposability of Loss Random Variables 143 4.2 Characterizing Heavy-Tailed alpha-Stable Severity Models 148 4.2.1 Characterisations of alpha-Stable Severity Models via the Domain of Attraction 152 4.3 Deriving the Properties and Characterizations of the alpha-Stable Severity Models 156 4.3.1 Unimodality of alpha-Stable Severity Models 158 4.3.2 Relationship between L Class and alpha-Stable Distributions 160 4.3.3 Fundamentals of Obtaining the alpha-Stable Characteristic Function 163 4.3.4 From Lévy-Khinchin's Canonical Representation to the alpha-Stable Characteristic Function Parameterizations 167 4.4 Popular Parameterizations of the alpha-Stable Severity Model Characteristic Functions 171 4.4.1 Univariate alpha-Stable Parameterizations of Zolotarev A, M, B,W, C and E Types 172 4.4.2 Univariate alpha-Stable Parameterizations of Nolan S0 and S1 178 4.5 Density Representations of alpha-Stable Severity Models 181 4.5.1 Basics of Moving from a Characteristic Function to a Distribution or Density 182 4.5.2 Density Approximation Approach 1: Quadrature Integration via Transformation and Clenshaw-Curtis Discrete Cosine Transform Quadrature 187 4.5.3 Density Approximation Approach 2: Adaptive Quadrature Integration via Fast Fourier Transform (Midpoint Rule) and Bergstrom Series Tail Expansion 189 4.5.4 Density Approximation Approach 3: Truncated Polynomial Series Expansions 196 4.5.5 Density Approximation Approach 4: Reparameterization 198 4.5.6 Density Approximation Approach 5: Infinite Series Expansion Density and Distribution Representations 200 4.6 Distribution Representations of alpha-Stable Severity Models 207 4.6.1 Quadrature Approximations for Distribution Representations of alpha-Stable Severity Models 208 4.6.2 Convergent Series Representations of the Distribution for alpha-Stable Severity Models 209 4.7 Quantile Function Representations and Loss Simulation for alpha-Stable Severity Models 210 4.7.1 Approximating the Quantile Function of Stable Loss Random Variables 210 4.7.2 Sampling Realizations of Stable Loss Random Variables 214 4.8 Parameter Estimation in an alpha-Stable Severity Model 215 4.8.1 McCulloch's Quantile-Based alpha-Stable Severity Model Estimators 216 4.8.2 Zolotarev's Transformation to W-Class-Based alpha-stable Severity Model Estimators 217 4.8.3 Press's Method-of-Moments-Based alpha-stable Severity Model Estimators 218 4.9 Location of the Most Probable Loss Amount for Stable Severity Models 219 4.10 Asymptotic Tail Properties of alpha-Stable Severity Models and Rates of Convergence to Paretian Laws 220 5 Flexible Heavy-Tailed Severity Models: Tempered Stable and Quantile Transforms 227 5.1 Tempered and Generalized Tempered Stable Severity Models 227 5.1.1 Understanding the Concept of Tempering Stable Severity Models 228 5.1.2 Families and Representations of Tempering in Stable Severity Models 231 5.1.3 Density of the Tempered Stable Severity Model 241 5.1.4 Properties of Tempered Stable Severity Models 243 5.1.5 Parameter Estimation of Loss Random Variables from a Tempered Stable Severity Model 246 5.1.6 Simulation of Loss Random Variables from a Tempered Stable Severity Model 248 5.1.7 Tail Behaviour of the Tempered Stable Severity Model 252 5.2 Quantile Function Heavy-Tailed Severity Models 253 5.2.1 g-and-h Severity Model Family in OpRisk 257 5.2.2 Tail Properties of the g-and-h, g, h and h-h Severity in OpRisk 268 5.2.3 Parameter Estimation for the g-and-h Severity in OpRisk 270 5.2.4 Bayesian Models for the g-and-h Severity in OpRisk 273 6 Families of Closed-Form Single Risk LDA Models 279 6.1 Motivating the Consideration of Closed-Form Models in LDA Frameworks 279 6.2 Formal Characterization of Closed-Form LDA Models: Convolutional Semi-Groups and Doubly Infinitely Divisible Processes 281 6.2.1 Basic Properties of Convolution Operators and Semi-Groups for Distribution and Density Functions 282 6.2.2 Domain of Attraction of Lévy Processes: Stable and Tweedie Convergence 303 6.3 Practical Closed-Form Characterization of Families of LDA Models for Light-Tailed Severities 309 6.3.1 General Properties of Exponential Dispersion and Poisson-Tweedie Models for LDA Structures 309 6.4 Sub-Exponential Families of LDA Models 321 6.4.1 Properties of Discrete Exponential Dispersion Models 322 6.4.2 Closed-Form LDA Models for Large Loss Number Processes 326 6.4.3 Closed-Form LDA Models for the alpha-Stable Severity Family 333 6.4.4 Closed-Form LDA Models for the Tempered alpha-Stable Severity Family 349 7 Single Risk Closed-Form Approximations of Asymptotic Tail Behaviour 353 7.1 Tail Asymptotics for Partial Sums and Heavy-Tailed SeverityModels 356 7.1.1 Partial Sum Tail Asymptotics with Heavy-Tailed Severity Models: Finite Number of Annual Losses N = n 357 7.1.2 Partial Sum Tail Asymptotics with Heavy-Tailed Severity Models: Large Numbers of Loss Events 362 7.2 Asymptotics for LDA Models: Compound Processes 367 7.2.1 Asymptotics for LDA Models Light Frequency and Light Severity Tails: SaddlePoint Tail Approximations 368 7.3 Asymptotics for LDA Models Dominated by Frequency Distribution Tails 372 7.3.1 Heavy-Tailed Frequency Distribution and LDA Tail Asymptotics (Frechet Domain of Attraction) 374 7.3.2 Heavy-Tailed Frequency Distribution and LDA Tail Asymptotics (Gumbel Domain of Attraction) 375 7.4 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses 376 7.4.1 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: General Sub-exponential Severity Model Results 377 7.4.2 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Regular and O-Regularly Varying Severity Model Results 380 7.4.3 Remainder Analysis: First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models 385 7.4.4 Summary: First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models 388 7.5 Refinements and Second-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses 389 7.6 Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Dependent Losses 393 7.6.1 Severity Dependence Structures that Do Not Affect LDA Model Tail Asymptotics: Stochastic Bounds 402 7.6.2 Severity Dependence Structures that Do Not Affect LDA Model Tail Asymptotics: Sub-exponential, Partial Sums and Compound Processes 405 7.6.3 Severity Dependence Structures that Do Not Affect LDA Model Tail Asymptotics: Consistent Variation 410 7.6.4 Dependent Severity Models: Partial Sums and Compound Process Second-Order Tail Asymptotics 412 7.7 Third-order and Higher Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses 414 7.7.1 Background Understanding on Higher Order Tail Decomposition Approaches 414 7.7.2 Decomposition Approach 1: Higher Order Tail Approximation Variants 415 7.7.3 Decomposition Approach 2: Higher Order Tail Approximations 426 7.7.4 Explicit Expressions for Higher Order Recursive Tail Decompositions Under Different Assumptions on Severity Distribution Behaviour 430 8 Single Loss Closed-Form Approximations of Risk Measures 433 8.1 Summary of Chapter Key Results on Single-Loss Risk Measure Approximation (SLA) 433 8.2 Development of Capital Accords and the Motivation for SLAs 436 8.3 Examples of Closed-Form Quantile and Conditional Tail Expectation Functions for OpRisk Severity Models 440 8.3.1 Exponential Dispersion Family Loss Models 441 8.3.2 g-and-h Distribution Family Loss Models 445 8.3.3 Extended GPD: the Asymmetric Power Family Loss Models 446 8.4 Non-Parametric Estimators for Quantile and Conditional Tail Expectation Functions 448 8.5 First- and Second-Order SLA of the VaR for OpRisk LDA Models 451 8.5.1 Second-Order Refinements of the SLA VaR for Heavy-Tailed LDA Models 457 8.6 EVT-Based Penultimate SLA 468 8.7 Motivation for Expected Shortfall and Spectral Risk Measures 475 8.8 First- and Second-Order Approximation of Expected Shortfall and Spectral Risk Measure 478 8.8.1 Understanding the First-Order SLA for ES for Regularly Varying Loss Models 481 8.8.2 Second-Order SLA for Expected Shortfall for Regularly Varying Loss Models 485 8.8.3 Empirical Process and EVT Approximations of Expected Shortfall 488 8.8.4 SLA for Spectral Risk Measures 492 8.9 Assessing the Accuracy and Sensitivity of the Univariate SLA 496 8.9.1 Understanding the Impact of Parameter Estimation Error on a SLA 498 8.9.2 Understanding the SLA Error 502 8.10 Infinite Mean-Tempered Tail Conditional Expectation Risk Measure Approximations 503 9 Recursions for Distributions of LDA Models 517 9.1 Introduction 517 9.2 Discretization Methods for Severity Distribution 519 9.2.1 Discretization Method 1: Rounding 520 9.2.2 Discretization Method 2: Localized Moment Matching 522 9.2.3 Discretization Method 3: Lloyd's Algorithm 524 9.2.4 Discretization Method 4: Minimizing Kolmogorov Statistic 524 9.3 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails 525 9.4 Discretization Errors and Extrapolation Methods 533 9.5 Recursions for Convolutions (Partial Sums) with Discretized Severity Distributions (Fixed n) 535 9.5.1 De Pril Transforms for n-Fold Convolutions (Partial Sums) with Discretized Severity Distributions 537 9.5.2 De Pril's First Method 538 9.5.3 De Pril's Second Method 539 9.5.4 De Pril Transforms and Convolutions of Infinitely Divisible Distributions 540 9.5.5 Recursions for n-Fold Convolutions (Partial Sum) Distribution Tails with Discretized Severity 542 9.6 Estimating Higher Order Tail Approximations for Convolutions with Continuous Severity Distributions (Fixed n) 544 9.6.1 Approximation Stages to be Studied 547 9.7 Sequential Monte Carlo Sampler Methodology and Components 550 9.7.1 Choice of Mutation Kernel and Backward Kernel 553 9.7.2 Incorporating Partial Rejection Control into SMC Samplers 556 9.8 Multi-Level Sequential Monte Carlo Samplers for Higher Order Tail Expansions and Continuous Severity Distributions (Fixed n) 560 9.8.1 Key Components of Multi-Level SMC Samplers 562 9.9 Recursions for Compound Process Distributions and Tails with Discretized Severity Distribution (Random N) 565 9.9.1 Panjer Recursions for Compound Distributions with Discretized Severity Distributions 566 9.9.2 Alternatives to Panjer Recursions: Recursions for Compound Distributions with Discretized Severity Distributions 573 9.9.3 Higher Order Recursions for Discretized Severity Distributions in Compound LDA Models 575 9.9.4 Recursions for Discretized Severity Distributions in Compound Mixed Poisson LDA Models 577 9.10 Continuous Versions of the Panjer Recursion 581 9.10.1 The Panjer Recursion via Volterra Integral Equations of the Second Kind 581 9.10.2 Importance Sampling Solutions to the Continuous Panjer Recursion 583 A Miscellaneous Definitions and List of Distributions 587 A.1 Indicator Function, 587 A.2 Gamma Function, 587 A.3 Discrete Distributions, 587 A.3.1 Poisson Distribution, 587 A.3.2 Binomial Distribution, 588 A.3.3 Negative Binomial Distribution, 588 A.3.4 Doubly Stochastic Poisson Process (Cox Process), 589 A.4 Continuous Distributions, 589 A.4.1 Uniform Distribution, 589 A.4.2 Normal (Gaussian) Distribution, 590 A.4.3 Inverse Gaussian Distribution, 590 A.4.4 LogNormal Distribution, 591 A.4.5 Student's t-Distribution, 591 A.4.6 Gamma Distribution, 591 A.4.7 Weibull Distribution, 592 A.4.8 Inverse Chi-Squared Distribution, 592 A.4.9 Pareto Distribution (One Parameter), 592 A.4.10 Pareto Distribution (Two Parameter), 593 A.4.11 Generalized Pareto Distribution, 593 A.4.12 Beta Distribution, 594 A.4.13 Generalized Inverse Gaussian Distribution, 594 A.4.14 d-Variate Normal Distribution, 595 A.4.15 d-Variate t-Distribution, 595 References 597 Index 623