Contents

PREFACE

ACRONYMS

SYMBOLS

LIST OF DISTRIBUTIONS

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

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 ...

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