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"Covers key mathematical and statistical aspects of the quantitative modelling of heavy tailed loss processes in operational risk and insurance settings. Includes advanced topics on risk modelling in high consequence low frequency loss processes, key aspects of extreme value theory, and classification of different classes of heavy tailed risk process models. Primarily developed for advanced risk management practitioners and quantitative analysts. Suitable as a core reference for an advanced mathematical or statistical risk management masters course or a PhD research course on risk management and asymptotics"--
List of contents
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 Characteri
About the author
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.
Summary
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.
