Synopses & Reviews
A cutting-edge guide for the theories, applications, and statistical methodologies essential to heavy tailed risk modelingFocusing 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.
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 Computational Informatics at the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Australia; Associate Member Oxford-Man Institute at the Oxford University; and Associate Member in the Systemic Risk Centre at the London School of Economics. In addition, he is Visiting Professor at The Institute of Statistical Mathematics, Japan.
Pavel V. Shevchenko, PhD, is Senior Principal Research Scientist in the Division of Computational Informatics at the Commonwealth Scientific and Industrial Research Organisation (CSIRO) 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.
Synopsis
A companion book to Fundamental Aspects of Operational Risk Modeling and Insurance Analytics: A Handbook of Operational Risk (2014), this book covers key mathematical and statistical aspects of the quantitative modelling of heavy tailed loss processes in operational risk and insurance settings. This book can add value to the industry by providing clear and detailed coverage of modelling for heavy tailed operational risk losses from both a rigorous mathematical as well as a statistical perspective. Few books cover the range of details provided both the mathematical and statistical features of such models, directly targeting practitioners. The book focuses on providing a sound understanding of how one would mathematically and statistically model, estimate, simulate and validate heavy tailed loss process models in operational risk. Coverage includes advanced topics on risk modelling in high consequence low frequency loss processes. This features splice loss models and motivation for heavy tailed risk processes models. The key aspects of extreme value theory and their development in loss distributional approach modelling is considered. Classification and understanding of different classes of heavy tailed risk process models is discussed, this leads into topics on heavy tailed closed form loss distributional approach models and flexible heavy tailed risk models such as a-stable and tempered stable models. The remainder of the chapters covers advanced topics on risk measures and asymptotics for heavy tailed compound process models. The finishing chapter covers advanced topics including forming links between actuarial compound process recursions and monte carlo numerical solutions for capital and risk measure estimations.
About the Author
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.
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 Computational Informatics at the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Australia; Associate Member Oxford-Man Institute at the Oxford University; and Associate Member in the Systemic Risk Centre at the London School of Economics. In addition, he is Visiting Professor at The Institute of Statistical Mathematics, Japan.
Pavel V. Shevchenko, PhD, is Senior Principal Research Scientist in the Division of Computational Informatics at the Commonwealth Scientific and Industrial Research Organisation (CSIRO) 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.
Table of Contents
Preface xvii
Acronyms xxi
List of Symbols xxiii
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 (GEVLDA) 40
2.5 Theoretical Properties of Univariate EVT - Threshold Exceedances 72
2.6 Estimation Under the Peaks Over Threshold Approach via the Generalized Pareto Distribution 86
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 111
3.3 Introduction to the Regular and Slow Variation Families of Heavy Tailed Models 119
3.4 Alternative Classifications of Heavy Tailed Models and Tail Variation 127
3.5 Extended Regular Variation and Matuszewska Indices for Heavy Tailed Models 133
4 Flexible Heavy Tailed Severity Models: α-Stable Family 137
4.1 Infinitely Divisible and Self Decomposable Loss Random Variables 138
4.2 Characterizing Heavy Tailed α-Stable Severity Models 145
4.3 Deriving the Properties and Characterizations of the α-Stable Severity Models 154
4.4 Popular Parameterizations of the α-Stable Severity Model Characteristic Functions 168
4.5 Density Representations of α-Stable Severity Models 178
4.6 Distribution Representations of α-Stable Severity Models 204
4.7 Quantile Function Representations and Loss Simulation for α-Stable Severity Models 207
4.8 Parameter Estimation in an α-Stable Severity Model 211
4.9 Location of the Most Probable Loss Amount for Stable Severity Models 215
4.10 Asymptotic Tail Properties of α-Stable Severity Models and Rates of Convergence to Paretian Laws 217
5 Flexible Heavy Tailed Severity Models: Tempered Stable and Quantile Transforms 223
5.1 Tempered and Generalized Tempered Stable Severity Models 223
5.2 Quantile Function Heavy Tailed Severity Models 247
6 Families of Closed Form Single Risk LDA Models 275
6.1 Motivating the Consideration of Closed Form Models in LDA Frameworks 275
6.2 Formal Characterization of Closed Form LDA Models: Convolutional Semi-Groups and Doubly Infinitely Divisible Processes 277
6.3 Practical Closed Form Characterization of Families of LDA Models for Light Tailed Severities 304
6.4 Sub-Exponential Families of LDA Models 316
7 Single Risk Closed Form Approximations of Asymptotic Tail Behaviour 345
7.1 Tail Asymptotics for Partial Sums and Heavy Tailed Severity Models 348
7.2 Asymptotics for LDA Models: Compound Processes 359
7.3 Asymptotics for LDA Models Dominated by Frequency Distribution Tails 363
7.4 First Order Single Risk Loss Process Asymptotics for heavy tailed LDA Models: Independent Losses 367
7.5 Refinements and Second Order Single Risk Loss Process Asymptotics for heavy tailed LDA Models: Independent Losses 379
7.6 Single Risk Loss Process Asymptotics for Heavy Tailed LDA Models: Dependent Losses 384
7.7 Third and Higher Order Single Risk Loss Process Asymptotics for heavy tailed LDA Models: Independent Losses 404
8 Single Risk Closed Form Approximations of Risk Measures 423
8.1 Summary of Chapter Key Results on Single Loss Approximations 423
8.2 Development of Capital Accords and the Motivation for SLA’s 426
8.3 Examples of Closed form Quantile and Conditional Tail Expectation Functions for OpRisk Severity Models 430
8.4 Non-parametric Estimators for Quantile and Conditional Tail Expectation Functions 438
8.5 First and Second Order Single Loss Approximation of the VaR for OpRisk LDA Models (VaR-SLA) 441
8.6 EVT Based Penultimate Single Loss Approximations for (EVT-SLA-VaR) 458
8.7 Motivation for Expected Shortfall and Spectral Risk Measures 464
8.8 First and Second Order Approximation of Expected Shortfall and Spectral Risk 467
8.9 Assessing the Accuracy and Sensitivity of the Univariate Single Loss Approximations (SLA-VaR) 486
8.10 Infinite Mean Tempered Tail Conditional Expectation Risk Measure Approximations 494
9 Recursions for Distributions of LDA Models 507
9.1 Discretisation Methods for the Severity Distribution 509
9.2 Classes of Discrete Distributions: Discrete Infinite Divisibility and Discrete Heavy Tails 515
9.3 Discretisation Errors and Extrapolation Methods 522
9.4 Recursions for Convolutions (Partial Sums) with Discretised Severity Distributions (Fixed
n) 525
9.5 Estimating Higher Order Tail Approximations for Convolutions with Continuous Severity Distributions (Fixed
n) 533
9.6 Sequential Monte Carlo Sampler Methodology and Components 539
9.7 Multi-Level Sequential Monte Carlo Samplers for Higher Order Tail Expansions and Continuous Severity Distributions (Fixed
n) 549
9.8 Recursions for Compound Process Distributions and Tails with Discretized Severity Distribution (Random
N) 553
9.9 Continuous Versions of the Panjer Recursion 568
A. Miscellaneous Definitions and List of Distributions 573
A.1 Indicator Function 573
A.2 Gamma Function 573
A.3 Discrete Distributions 574
A.4 Continuous Distributions 575
Index 617