Operational Risk Modeling Analytics phần 1 ppsx

1.2 Operational risk in insurance 1.3 The analysis of operational risk 1.4 The model-based approach 1.4.1 The modeling process 1.5 Organization of this book Basel 11 - Operational ri

This Page Intentionally Left Blank

Operational Risk

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A SHEWHART and SAMUEL S WILKS

Editors: David J Balding, Noel A C Cressie, Nicholas I Fisher,

Iain M Johnstone, J B Kudune, Geert Molenberghs, Louise M Ryan,

David W Scott, Adrian F M Smith

Editors Emeriti: Vic Barnett, J Stuart Hunter, David G Kendall, Jozef L Teugels

A complete list of the titles in this series appears at the end of this volume

Copyright 0 2006 by John Wiley & Sons, Inc All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form

or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should

be addressed to the Permissions Department, John Wiley & Sons, Inc., 1 11 River Street, Hoboken, NJ

07030, (201) 748-601 I, fax (201) 748-6008, or online at http://www.wiley.com/go/permission

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of

merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic format For information about Wiley products, visit our web site at www.wiley.com

Library of Congress Cutu~oging-in-Publication Data:

1.2 Operational risk in insurance

1.3 The analysis of operational risk

1.4 The model-based approach

1.4.1 The modeling process

1.5 Organization of this book

Basel 11 - Operational risk

2 Basic probability concepts

Part 11 Probabilistic tools for operational risk modeling

4 Models for the size of losses: Continuous distributions 57 4.2 A n inventory of continuous distributions 58 4.2.1 One-parameter distributions 58

4.1 Introduction 57

4.2.2 Two-parameter distributions 59 4.2.3 Three-parameter distributions 64 4.2.4 Four-parameter distributions 67 4.2.5 Distributions with finite support 68 4.3 Selected distributions and their relationships 68 4.3 I Introduction 68 4.3.2 Two important parametric families 69

4.4 Limiting distributions 70 4.5 The role of parameters 73 4.5 I Parametric and scale distributions 74

4.5.2 Finite mixture distributions 75 4.5.3 Data-dependent distributions 78

4.6 Tails of distributions 80 4.6.1 Classification based on moments 80 4.6.2 Classification based on tail behavior 81 4.6.3 Classification based on hazard rate

function 82

4.7 Creating new distributions 84 4.7.1 Introduction 84 4.7.2 Multiplication by a constant 84 4.7.3 Transformation by raising to a power 85

4.7.5 Continuous mixture of distributions 88

CONTENTS vii

4.7.6 Frailty models 90 4.8 TVaR for continuous distributions 93 4.8.1 Continuous elliptical distributions 94 4.8.2 Continuous exponential dispersion

distributions 97 4.9 Exercises 102 4.7.7 Splicing pieces of distributions 92

5 Models for the number of losses: Counting distributions 107 5.1 Introduction 107 5.2 The Poisson distribution 108 5.3 The negative binomial distribution 110 5.4 The binomial distribution 114 5.5 The (a,b,O) class f 14 5.7 Compound frequency models 122

5.8 Recursive calculation of compound probabilities 126 5.9 An inventory of discrete distributions 130 5.9.1 The (a,b,O) class 130 5.9.2 The (a, b, 1) class 132 5.9.3 The zero-truncated subclass 132 5.9.5 The compound class 135 5.10 A hierarchy of discrete distributions 136 5.11 Further properties of the compound Poisson class 137 5.12 Mixed frequency models 142 5.13 Poisson mixtures 144 5.14 Effect of exposure on loss counts 149 5.15 TVaR for discrete distributions 150

5.9.4 The zero-modified subclass 1 34

5.15.1 T VaR for discrete exponential dispersion

distributions 151 5.16 Exercises 156

6 Aggregate loss models 161 6.1 Introduction 161 6.2 Model choices 162 6.3 The compound model for aggregate losses 163 6.4 Some analytic results 168 6.5 Evaluation of the aggregate loss distribution 171

viii CONTENTS

6.6 The recursive method

6.6.1 Compound frequency models

6.6.2 Underflow/overjlow problems

6.6.3 Numerical stability

6.6.4 Continuous severity

6.6.5 Constructing arithmetic distributions

6.7 Fast Fourier transform methods

6.8 Using approximating severity distributions

6.10.3 T V a R for some severity distributions

6.10.4 Summary

6.11 Exercises

7 Extreme value theory: The study of jumbo losses

7.1 Introduction

7.2 Extreme value distributions

7.3 Distribution of the maximum

7.3.1 From a fixed number of losses

7.3.2 From a random number of losses

Stability of the maximum of the extreme value

distribution

7.4

7.5 The Fisher- Tippett theorem

7.6 Maximum domain of attraction

7.7 Generalized Pareto distributions

7.8 The frequency of exceedences

7.8.1 From a fixed number of losses

7.8.2 From a random number of losses

Stability of excesses of the generalized Pareto

7.9

7.10 Mean excess function

7.11 Limiting distributions of excesses

7.12 T V a R for extreme value distributions

Part 1 III Statistical methods for calibrating models of operational

9 Review of mathematical statistics

10.2 Method of moments and percentile matching

10.3 Maximum likelihood estimation

10.3.1 Introduction

10.3.2 Complete, individual data

10.3.3 Complete, grouped data

10.3.4 Truncated or censored data

10.4 Variance and interval estimation

10.5 Bayesian estimation

10.5.1 Definitions and Bayes ’ theorem

10.5.2 Inference and prediction

12.2 Representations of the data and model

12.3 Graphical comparison of the density and

distribution functions

12.4 Hypothesis tests

22.4.1 Kolmogorov-Smirnov test

12.4.2 Anderson-Darling test

12.4.3 Chi-square goodness-of-fit test

12.44 Likelihood ratio test

Pareto distribution 13.2.3 Estimating the Pareto shape parameter

13.2.4 Estimating extreme probabilities

13.3.1 Mean excess plots

CONTENTS X I

14 Fitting copula models

14.1 Introduction

14.2 Maximum likelihood estimation

14.3 Semiparametric estimation of the copula

14.5 Goodness-of-fit testing

14.6 An example

Appendix A Gamma and related functions

Appendix B Discretization of the severity distribution

B,l The method of rounding

B 2 Mean preserving

B.3 Undiscretization of a discretixed distribution

Appendix C Nelder-Mead simplex method

This Page Intentionally Left Blank

Preface

This book is is designed for the risk analyst who wishes to better understand the mathematical models and methods used in the management of operational risk in the banking and insurance sectors Many of the techniques in this book are more generally applicable to a wide range of risks However, each sector has its unique characteristics, its own data sources, and its own risk migation and management strategies Other major risk classes in the banking sector include credit risk and market risk In addition to these, the insurance sector also assumes the risk in the insurance contracts that it sells The product risk

in the insurance sector may dominate all other risk classes

This book is organized around the principle that much the analysis of opera- tional risk consists of the collection of data and the building of mathematical models to describe risk I have not assumed that the reader has any substan- tial knowledge of operational risk terminology or of mathematical statistics However, the book is more challenging technically than some other books on the topic of operational risk but less challenging than others that focus on risk mathematics This is intentional The purpose of the book is to provide detailed analytical tools for the practicing risk analyst as well as serving as a text for a university course

This book could serve as a text at the senior undergraduate or first-year graduate level for a course of one semester for students with a reasonable background in statistics, because many sections of the book can be covered rapidly Without a moderate background in statistics, students will require two semesters to cover the material in this book For chapters involving nu- merical computations, there are many exercises for students to practice and

xiv PREFACE

reinforce concepts in the text

Many of the concepts in this book have been developed in the insurance field, where the modeling and management of risk is a core activity This book is built on previous books by this author along with co-authors, in particular

Loss Distributions [53], Insurance Risk Models [93], and two editions of Loss

Models: From Data to Decisions [SS]

H H PAXJER

I am also indebted to two students, Yixi Shi and Shuyin Mai who assisted

in numerous technical aspects of producing this book And finally, thanks to

my wife Joanne Coyle, who tolerated my many weekends and evenings at the ofice

H.H.P

xv

This Page Intentionally Left Blank

Part I

Introduction to

This Page Intentionally Left Blank

Operational risk

Anything that can go wrong will go wrong

-Murphy

1.1 INTRODUCTION

Operational risk has only in recent years been identified as something that

should be actively measured and managed by a company in order to meet its objectives for stakeholders, including shareholders, customers, and manage- ment These objectives include future survival of the company, avoidance

of downgrades by rating agencies and remaining solvent for many years to come Operational risk is becoming a major part of corporate governance

of companies, especially in the financial services industry This industry in- cludes both banks and insurance companies, although they have somewhat different historical cultures in most countries More recently in other fields such as energy, where trading and hedging activity mirrors similar activity in the financial services industry, operational risk is being recognized as a vital part of a broader enterprise risk management framework

The definition of operational risk has not yet been universally agreed upon

In very general terms, operational risk refers to “risk” associated with the

“operations” of an organization “Risk” is not defined very specifically, nor is

“operations.’) Generally, the term “risk” refers to the possibility of things go- ing wrong, or the chances of things going wrong, or the possible consequences

of things that can go wrong “Operations” refers to the various functions of

3

4 OPERATlONAL RISK

the organization (usually a company such as a bank or insurance company)

in conducting its business It does not refer specifically to the products or services provided by the company In banking, operational risk does not in- clude the risk of losing money as a result of normal banking activities such

as investing, trading, or lending except to the extent that operational activ- ities affect those normal activities An example of such an operational risk

in banking is fraudulent activity, such as unauthorized lending where a loan officer ignores rules, or rogue trading in which a trader is involved in trading activity beyond limits of authorization The well-known classic example of a rogue trader is Nick Leeson, whose activities resulted in the failure of Barings Bank, leading to its takeover by the ING financial services conglomerate operational risk is generic in nature The operational risk concept applies

to organizations of all types However, the specifics of operational risk will vary from company to company depending on the individual characteristics

of the company For example, a manufacturer will be exposed to somewhat different operational risks than a bank or an insurance company, but many are the same The risk of shutdown of the operations of a company because

of IT failure, flooding, or an earthquake exists for any company While the principles of operational risk modeling and management apply to all types of organization, in this book we will look at operational risk from the vantage point of a financial institution, such as a bank or insurance company

Measurement and modeling of risk associated with operations for the finan- cial sector began in the banking industry Operational risk is one of several categories of risk used in enterprise risk management (ERM) ERM involves all types of risk faced by a company Operational risk is one part only Many financial institutions have incorporated ERM into a new governance paradigm in which risk exposure is better understood and managed The responsibility for the risk management function in a company often falls under the title of chief risk officer (CRO), a title first held by James Lam in the 1990s [72] The CRO is responsible for the entire ERM process of the company in all its business units Within the ERM process are processes for each risk category Within the operational risk category, the responsibilities include:

Developing operational risk policies and internal standards

Controlling the operational risk self-assessment in each business unit Describing and modeling all internal processes

Testing all processes for possible weaknesses

Developing operational risk technology

Developing key risk indicators

Planning the management of major business disruptions

Evaluating the risk associated with outsourcing operations

Maintaining a database of operational risk incidents

Developing metrics for operational risk exposure

Developing metrics for effectiveness of risk controls

Modeling losses using frequency and severity