Advances in Heavy Tailed Risk Modeling : A Handbook of Operational Risk.Series: Wiley Handbooks in Financial Engineering and Econometrics SerPublisher: New York : John Wiley & Sons, Incorporated, 2015Copyright date: ©2015Edition: 1st edDescription: 1 online resource (720 pages)Content type:
- online resource
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Cover -- Title Page -- Copyright -- Dedication -- Contents in Brief -- Contents -- Preface -- Acronyms -- Symbols -- List of Distributions -- Chapter 1 Motivation for Heavy-Tailed Models -- 1.1 Structure of the Book -- 1.2 Dominance of the Heaviest Tail Risks -- 1.3 Empirical Analysis Justifying Heavy-Tailed Loss Models-in OpRisk -- 1.4 Motivating Parametric, Spliced and Non-Parametric Severity Models -- 1.5 Creating Flexible Heavy-Tailed Models via Splicing -- Chapter 2 Fundamentals of Extreme Value Theory for OpRisk -- 2.1 Introduction -- 2.2 Historical Perspective on EVT and Risk -- 2.3 Theoretical Properties of Univariate EVT-Block Maxima and the GEV Family -- 2.4 Generalized Extreme Value Loss Distributional Approach (GEV-LDA) -- 2.4.1 Statistical Considerations for Applicability of the GEV Model -- 2.4.2 Various Statistical Estimation Procedures for the GEV Model Parameters in OpRisk Settings -- 2.4.3 GEV Sub-Family Approaches in OpRisk LDA Modeling -- 2.4.4 Properties of the Frechet-Pareto Family of Severity Models -- 2.4.5 Single Risk LDA Poisson-Generalized Pareto Family -- 2.4.6 Single Risk LDA Poisson-Burr Family -- 2.4.7 Properties of the Gumbel family of Severity Models -- 2.4.8 Single Risk LDA Poisson-LogNormal Family -- 2.4.9 Single Risk LDA Poisson-Benktander II Models -- 2.5 Theoretical Properties of Univariate EVT-Threshold Exceedances -- 2.5.1 Understanding the Distribution of Threshold Exceedances -- 2.6 Estimation Under the Peaks Over Threshold Approach via the Generalized Pareto Distribution -- 2.6.1 Maximum-Likelihood Estimation Under the GPD Model -- 2.6.2 Comments on Probability-Weighted Method of Moments Estimation Under the GPD Model -- 2.6.3 Robust Estimators of the GPD Model Parameters -- 2.6.4 EVT-Random Number of Losses -- Chapter 3 Heavy-Tailed Model Class Characterizations for LDA.
3.1 Landau Notations for OpRisk Asymptotics: Big and Little `Oh' -- 3.2 Introduction to the Sub-Exponential Family of Heavy-Tailed Models -- 3.3 Introduction to the Regular and Slow Variation Families-of Heavy-Tailed Models -- 3.4 Alternative Classifications of Heavy-Tailed Models and Tail Variation -- 3.5 Extended Regular Variation and Matuszewska Indices for Heavy-Tailed Models -- Chapter 4 Flexible Heavy-Tailed Severity Models: α-Stable Family -- 4.1 Infinitely Divisible and Self-Decomposable Loss Random Variables -- 4.1.1 Basic Properties of Characteristic Functions -- 4.1.2 Divisibility and Self-Decomposability of Loss Random Variables -- 4.2 Characterizing Heavy-Tailed α-Stable Severity Models -- 4.2.1 Characterisations of α-Stable Severity Models via the Domain of Attraction -- 4.3 Deriving the Properties and Characterizations of the α-Stable Severity Models -- 4.3.1 Unimodality of α-Stable Severity Models -- 4.3.2 Relationship between L Class and α-Stable Distributions -- 4.3.3 Fundamentals of Obtaining the α-Stable Characteristic Function -- 4.3.4 From Lévy-Khinchin's Canonical Representation to the α-Stable Characteristic Function Parameterizations -- 4.4 Popular Parameterizations of the α-Stable Severity Model Characteristic Functions -- 4.4.1 Univariate α-Stable Parameterizations of Zolotarev A, M, B, W, C and E Types -- 4.4.2 Univariate α-Stable Parameterizations of Nolan S0 and S1 -- 4.5 Density Representations of α-Stable Severity Models -- 4.5.1 Basics of Moving from a Characteristic Function to a Distribution or Density -- 4.5.2 Density Approximation Approach 1: Quadrature Integration via Transformation and Clenshaw-Curtis Discrete Cosine Transform Quadrature -- 4.5.3 Density Approximation Approach 2: Adaptive Quadrature Integration via Fast Fourier Transform (Midpoint Rule) and Bergstrom Series Tail Expansion.
4.5.4 Density Approximation Approach 3: Truncated Polynomial Series Expansions -- 4.5.5 Density Approximation Approach 4: Reparameterization -- 4.5.6 Density Approximation Approach 5: Infinite Series Expansion Density and Distribution Representations -- 4.6 Distribution Representations of α-Stable Severity Models -- 4.6.1 Quadrature Approximations for Distribution Representations of α-Stable Severity Models -- 4.6.2 Convergent Series Representations of the Distribution for α-Stable Severity Models -- 4.7 Quantile Function Representations and Loss Simulation for α-Stable Severity Models -- 4.7.1 Approximating the Quantile Function of Stable Loss Random Variables -- 4.7.2 Sampling Realizations of Stable Loss Random Variables -- 4.8 Parameter Estimation in an α-Stable Severity Model -- 4.8.1 McCulloch's Quantile-Based α-Stable Severity Model Estimators -- 4.8.2 Zolotarev's Transformation to W-Class-Based α-stable Severity Model Estimators -- 4.8.3 Press's Method-of-Moments-Based α-stable Severity Model Estimators -- 4.9 Location of the Most Probable Loss Amount for Stable Severity Models -- 4.10 Asymptotic Tail Properties of α-Stable Severity Models and Rates of Convergence to Paretian Laws -- Chapter 5 Flexible Heavy-Tailed Severity Models: Tempered Stable and Quantile Transforms -- 5.1 Tempered and Generalized Tempered Stable Severity Models -- 5.1.1 Understanding the Concept of Tempering Stable Severity Models -- 5.1.2 Families and Representations of Tempering in Stable Severity Models -- 5.1.3 Density of the Tempered Stable Severity Model -- 5.1.4 Properties of Tempered Stable Severity Models -- 5.1.5 Parameter Estimation of Loss Random Variables from a Tempered Stable Severity Model -- 5.1.6 Simulation of Loss Random Variables from a Tempered Stable Severity Model -- 5.1.7 Tail Behaviour of the Tempered Stable Severity Model.
5.2 Quantile Function Heavy-Tailed Severity Models -- 5.2.1 g-and-h Severity Model Family in OpRisk -- 5.2.2 Tail Properties of the g-and-h, g, h and h-h Severity in OpRisk -- 5.2.3 Parameter Estimation for the g-and-h Severity in OpRisk -- 5.2.4 Bayesian Models for the g-and-h Severity in OpRisk -- Chapter 6 Families of Closed-Form Single Risk LDA Models -- 6.1 Motivating the Consideration of Closed-Form Models in LDA Frameworks -- 6.2 Formal Characterization of Closed-Form LDA Models: Convolutional Semi-Groups and Doubly Infinitely Divisible Processes -- 6.2.1 Basic Properties of Convolution Operators and Semi-Groups for Distribution and Density Functions -- 6.2.2 Domain of Attraction of Lévy Processes: Stable and Tweedie Convergence -- 6.3 Practical Closed-Form Characterization of Families of LDA Models for Light-Tailed Severities -- 6.3.1 General Properties of Exponential Dispersion and Poisson-Tweedie Models for LDA Structures -- 6.4 Sub-Exponential Families of LDA Models -- 6.4.1 Properties of Discrete Exponential Dispersion Models -- 6.4.2 Closed-Form LDA Models for Large Loss Number Processes -- 6.4.3 Closed-Form LDA Models for the α-Stable Severity Family -- 6.4.4 Closed-Form LDA Models for the Tempered α-Stable Severity Family -- Chapter 7 Single Risk Closed-Form Approximations-of Asymptotic Tail Behaviour -- 7.1 Tail Asymptotics for Partial Sums and Heavy-Tailed Severity Models -- 7.1.1 Partial Sum Tail Asymptotics with Heavy-Tailed Severity Models: Finite Number of Annual Losses N=n -- 7.1.2 Partial Sum Tail Asymptotics with Heavy-Tailed Severity Models: Large Numbers of Loss Events -- 7.2 Asymptotics for LDA Models: Compound Processes -- 7.2.1 Asymptotics for LDA Models Light Frequency and Light Severity Tails: SaddlePoint Tail Approximations -- 7.3 Asymptotics for LDA Models Dominated by Frequency Distribution Tails.
7.3.1 Heavy-Tailed Frequency Distribution and LDA Tail Asymptotics (Frechet Domain of Attraction) -- 7.3.2 Heavy-Tailed Frequency Distribution and LDA Tail Asymptotics (Gumbel Domain of Attraction) -- 7.4 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses -- 7.4.1 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: General Sub-exponential Severity Model Results -- 7.4.2 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Regular and O-Regularly Varying Severity Model Results -- 7.4.3 Remainder Analysis: First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models -- 7.4.4 Summary: First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models -- 7.5 Refinements and Second-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses -- 7.6 Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Dependent Losses -- 7.6.1 Severity Dependence Structures that Do Not Affect LDA Model Tail Asymptotics: Stochastic Bounds -- 7.6.2 Severity Dependence Structures that Do Not Affect LDA Model Tail Asymptotics: Sub-exponential, Partial Sums-and Compound Processes -- 7.6.3 Severity Dependence Structures that Do Not Affect LDA Model Tail Asymptotics: Consistent Variation -- 7.6.4 Dependent Severity Models: Partial Sums and Compound Process Second-Order Tail Asymptotics -- 7.7 Third-order and Higher Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses -- 7.7.1 Background Understanding on Higher Order Tail Decomposition Approaches -- 7.7.2 Decomposition Approach 1: Higher Order Tail Approximation Variants -- 7.7.3 Decomposition Approach 2: Higher Order Tail Approximations.
7.7.4 Explicit Expressions for Higher Order Recursive Tail Decompositions Under Different Assumptions on Severity Distribution Behaviour.
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 in high consequence low frequency loss modeling. With a companion, Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, the book 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 distributional 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 modelling, 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 book is also a useful handbook for graduate-level courses on heavy tailed processes, advanced risk management, and actuarial science.
Description based on publisher supplied metadata and other sources.
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.