Gareth W. Peters, Pavel V. Shevchenko

Advances in Heavy Tailed Risk Modeling (eBook, PDF)

A Handbook of Operational Risk

• Format: PDF

Produktbeschreibung

ADVANCES IN HEAVY TAILED RISK MODELING A cutting-edge guide for the theories, applications, and statistical methodologies essential to heavy tailed risk modeling Focusing on the quantitative aspects of heavy tailed loss processes in operational risk and relevant insurance analytics, Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk presents comprehensive coverage of the latest research on the theories and applications in risk measurement and modeling techniques. Featuring a unique balance of mathematical and statistical perspectives, the handbook begins by introducing the motivation for heavy tailed risk processes. A companion with Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, the handbook provides a complete framework for all aspects of operational risk management and includes: * Clear coverage on advanced topics such as splice loss models, extreme value theory, heavy tailed closed form loss distribution approach models, flexible heavy tailed risk models, risk measures, and higher order asymptotic approximations of risk measures for capital estimation * An exploration of the characterization and estimation of risk and insurance modeling, which includes sub-exponential models, alpha-stable models, and tempered alpha stable models * An extended discussion of the core concepts of risk measurement and capital estimation as well as the details on numerical approaches to evaluation of heavy tailed loss process model capital estimates * Numerous detailed examples of real-world methods and practices of operational risk modeling used by both financial and non-financial institutions Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk is an excellent reference for risk management practitioners, quantitative analysts, financial engineers, and risk managers. The handbook is also useful for graduate-level courses on heavy tailed processes, advanced risk management, and actuarial science.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 656
• Erscheinungstermin: 5. Mai 2015
• Englisch
• ISBN-13: 9781118909553
• Artikelnr.: 42833131

Autorenporträt

Gareth W. Peters, PhD, is Assistant Professor in the Department of Statistical Science, Principal 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 the Oxford University; and Associate Member in the Systemic Risk Centre at the London School of Economics. Dr. Peters is also a visiting professor at the Institute of Statistical Mathematics, Tokyo, Japan. Pavel V. Shevchenko, PhD, is Senior Principal Research Scientist in the Division of Computational Informatics at 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 Operational 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.

Inhaltsangabe

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