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Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk

Book Description

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.

    Table of Contents

    1. Cover Page
    2. Title Page
    3. Copyright
    4. Dedication
    5. Contents in Brief
    6. Contents
    7. Preface
      1. Acknowledgments
    8. Acronyms
    9. Symbols
    10. List of Distributions
    11. CHAPTER ONE: Motivation for Heavy-Tailed Models
      1. 1.1 Structure of the Book
      2. 1.2 Dominance of the Heaviest Tail Risks
      3. 1.3 Empirical Analysis Justifying Heavy-Tailed Loss Models in OpRisk
      4. 1.4 Motivating Parametric, Spliced and Non-Parametric Severity Models
      5. 1.5 Creating Flexible Heavy-Tailed Models via Splicing
    12. CHAPTER TWO: Fundamentals of Extreme Value Theory for OpRisk
      1. 2.1 Introduction
      2. 2.2 Historical Perspective on EVT and Risk
      3. 2.3 Theoretical Properties of Univariate EVT–Block Maxima and the GEV Family
      4. 2.4 Generalized Extreme Value Loss Distributional Approach (GEV-LDA)
      5. 2.5 Theoretical Properties of Univariate EVT–Threshold Exceedances
      6. 2.6 Estimation Under the Peaks Over Threshold Approach via the Generalized Pareto Distribution
    13. CHAPTER THREE: Heavy-Tailed Model Class Characterizations for LDA
      1. 3.1 Landau Notations for OpRisk Asymptotics: Big and Little ‘Oh’
      2. 3.2 Introduction to the Sub-Exponential Family of Heavy-Tailed Models
      3. 3.3 Introduction to the Regular and Slow Variation Families of Heavy-Tailed Models
      4. 3.4 Alternative Classifications of Heavy-Tailed Models and Tail Variation
      5. 3.5 Extended Regular Variation and Matuszewska Indices for Heavy-Tailed Models
    14. CHAPTER FOUR: Flexible Heavy-Tailed Severity Models: α -Stable Family
      1. 4.1 Infinitely Divisible and Self-Decomposable Loss Random Variables
      2. 4.2 Characterizing Heavy-Tailed α -Stable Severity Models
      3. 4.3 Deriving the Properties and Characterizations of the α -Stable Severity Models
      4. 4.4 Popular Parameterizations of the α -Stable Severity Model Characteristic Functions
      5. 4.5 Density Representations of α -Stable Severity Models
      6. 4.6 Distribution Representations of α -Stable Severity Models
      7. 4.7 Quantile Function Representations and Loss Simulation for α -Stable Severity Models
      8. 4.8 Parameter Estimation in an α -Stable Severity Model
      9. 4.9 Location of the Most Probable Loss Amount for Stable Severity Models
      10. 4.10 Asymptotic Tail Properties of α -Stable Severity Models and Rates of Convergence to Paretian Laws
    15. CHAPTER FIVE: Flexible Heavy-Tailed Severity Models: Tempered Stable and Quantile Transforms
      1. 5.1 Tempered and Generalized Tempered Stable Severity Models
      2. 5.2 Quantile Function Heavy-Tailed Severity Models
    16. CHAPTER SIX: Families of Closed-Form Single Risk LDA Models
      1. 6.1 Motivating the Consideration of Closed-Form Models in LDA Frameworks
      2. 6.2 Formal Characterization of Closed-Form LDA Models: Convolutional Semi-Groups and Doubly Infinitely Divisible Processes
      3. 6.3 Practical Closed-Form Characterization of Families of LDA Models for Light-Tailed Severities
      4. 6.4 Sub-Exponential Families of LDA Models
    17. CHAPTER SEVEN: Single Risk Closed-Form Approximations of Asymptotic Tail Behaviour
      1. 7.1 Tail Asymptotics for Partial Sums and Heavy-Tailed Severity Models
      2. 7.2 Asymptotics for LDA Models: Compound Processes
      3. 7.3 Asymptotics for LDA Models Dominated by Frequency Distribution Tails
      4. 7.4 First-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses
      5. 7.5 Refinements and Second-Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses
      6. 7.6 Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Dependent Losses
      7. 7.7 Third-order and Higher Order Single Risk Loss Process Asymptotics for Heavy-Tailed LDA Models: Independent Losses
    18. CHAPTER EIGHT: Single Loss Closed-Form Approximations of Risk Measures
      1. 8.1 Summary of Chapter Key Results on Single-Loss Risk Measure Approximation (SLA)
      2. 8.2 Development of Capital Accords and the Motivation for SLAs
      3. 8.3 Examples of Closed-Form Quantile and Conditional Tail Expectation Functions for OpRisk Severity Models
      4. 8.4 Non-Parametric Estimators for Quantile and Conditional Tail Expectation Functions
      5. 8.5 First- and Second-Order SLA of the VaR for OpRisk LDA Models
      6. 8.6 EVT-Based Penultimate SLA
      7. 8.7 Motivation for Expected Shortfall and Spectral Risk Measures
      8. 8.8 First- and Second-Order Approximation of Expected Shortfall and Spectral Risk Measure
      9. 8.9 Assessing the Accuracy and Sensitivity of the Univariate SLA
      10. 8.10 Infinite Mean-Tempered Tail Conditional Expectation Risk Measure Approximations
    19. CHAPTER NINE: Recursions for Distributions of LDA Models
      1. 9.1 Introduction
      2. 9.2 Discretization Methods for Severity Distribution
      3. 9.3 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails
      4. 9.4 Discretization Errors and Extrapolation Methods
      5. 9.5 Recursions for Convolutions (Partial Sums) with Discretized Severity Distributions (Fixed n )
      6. 9.6 Estimating Higher Order Tail Approximations for Convolutions with Continuous Severity Distributions (Fixed n )
      7. 9.7 Sequential Monte Carlo Sampler Methodology and Components
      8. 9.8 Multi-Level Sequential Monte Carlo Samplers for Higher Order Tail Expansions and Continuous Severity Distributions (Fixed n )
      9. 9.9 Recursions for Compound Process Distributions and Tails with Discretized Severity Distribution (Random N )
      10. 9.10 Continuous Versions of the Panjer Recursion
    20. APPENDIX A: Miscellaneous Definitions and List of Distributions
      1. A.1 Indicator Function
      2. A.2 Gamma Function
      3. A.3 Discrete Distributions
      4. A.4 Continuous Distributions
    21. References
    22. Index