Quantitative Risk Management: Concepts, Techniques, and Tools

by Alexander J. McNeil, Rüdiger Frey, Paul Embrechts

Overview

The implementation of sound quantitative risk models is a vital concern for all financial institutions, and this trend has accelerated in recent years with regulatory processes such as Basel II. This book provides a comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management and equips readers—whether financial risk analysts, actuaries, regulators, or students of quantitative finance—with practical tools to solve real-world problems. The authors cover methods for market, credit, and operational risk modelling; place standard industry approaches on a more formal footing; and describe recent developments that go beyond, and address main deficiencies of, current practice.

The book's methodology draws on diverse quantitative disciplines, from mathematical finance through statistics and econometrics to actuarial mathematics. Main concepts discussed include loss distributions, risk measures, and risk aggregation and allocation principles. A main theme is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. The techniques required derive from multivariate statistical analysis, financial time series modelling, copulas, and extreme value theory. A more technical chapter addresses credit derivatives. Based on courses taught to masters students and professionals, this book is a unique and fundamental reference that is set to become a standard in the field.

Product Details

ISBN-13:	9780691122557
Publisher:	Princeton University Press
Publication date:	10/16/2005
Series:	Princeton Series in Finance
Edition description:	Older Edition
Pages:	544
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Alexander J. McNeil is Professor of Mathematics at the Swiss Federal Institute of Technology (ETH) in Zurich. Rüdiger Frey is Professor of Financial Mathematics at the University of Leipzig. Paul Embrechts, Professor of Insurance Mathematics at the Swiss Federal Institute of Technology (ETH) in Zurich, is the coauthor of Modelling Extremal Events for Insurance and Finance.

Table of Contents

Preface xiii

CHAPTER 1: Risk in Perspective 1
1.1 Risk 1
1.1.1 Risk and Randomness 1
1.1.2 Financial Risk 2
1.1.3 Measurement and Management 3
1.2 A Brief History of Risk Management 5
1.2.1 From Babylon to Wall Street 5
1.2.2 The Road to Regulation 8
1.3 The New Regulatory Framework 10
1.3.1 Basel II 10
1.3.2 Solvency 2 13
1.4 Why Manage Financial Risk? 15
1.4.1 A Societal View 15
1.4.2 The Shareholder's View 16
1.4.3 Economic Capital 18
1.5 Quantitative Risk Management 19
1.5.1 The Nature of the Challenge 19
1.5.2 QRM for the Future 22

CHAPTER 2: Basic Concepts in Risk Management 25
2.1 Risk Factors and Loss Distributions 25
2.1.1 General Definitions 25
2.1.2 Conditional and Unconditional Loss Distribution 28
2.1.3 Mapping of Risks:Some Examples 29
2.2 Risk Measurement 34
2.2.1 Approaches to Risk Measurement 34
2.2.2 Value-at-Risk 37
2.2.3 Further Comments on VaR 40
2.2.4 Other Risk Measures Based on Loss Distributions 43
2.3 Standard Methods for Market Risks 48
2.3.1 Variance -Covariance Method 48
2.3.2 Historical Simulation 50
2.3.3 Monte Carlo 52
2.3.4 Losses over Several Periods and Scaling 53
2.3.5 Backtesting 55
2.3.6 An Illustrative Example 55

CHAPTER 3: Multivariate Models 61
3.1 Basics of Multivariate Modelling 61
3.1.1 Random Vectors and Their Distributions 62
3.1.2 Standard Estimators of Covariance and Correlation 64
3.1.3 The Multivariate Normal Distribution 66
3.1.4 Testing Normality and Multivariate Normality 68
3.2 Normal Mixture Distributions 73
3.2.1 Normal Variance Mixtures 73
3.2.2 Normal Mean-Variance Mixtures 77
3.2.3 Generalized Hyperbolic Distributions 78
3.2.4 Fitting Generalized Hyperbolic Distributions to Data 81
3.2.5 Empirical Examples 84
3.3 Spherical and Elliptical Distributions 89
3.3.1 Spherical Distributions 89
3.3.2 Elliptical Distributions 93
3.3.3 Properties of Elliptical Distributions 95
3.3.4 Estimating Dispersion and Correlation 96
3.3.5 Testing for Elliptical Symmetry 99
3.4 Dimension Reduction Techniques 103
3.4.1 Factor Models 103
3.4.2 Statistical Calibration Strategies 105
3.4.3 Regression Analysis of Factor Models 106
3.4.4 Principal Component Analysis 109

CHAPTER 4: Financial Time Series 116
4.1 Empirical Analyses of Financial Time Series 117
4.1.1 Stylized Facts 117
4.1.2 Multivariate Stylized Facts 123
4.2 Fundamentals of Time Series Analysis 125
4.2.1 Basic Definitions 125
4.2.2 ARMA Processes 128
4.2.3 Analysis in the Time Domain 132
4.2.4 Statistical Analysis of Time Series 134
4.2.5 Prediction 136
4.3 GARCH Models for Changing Volatility 139
4.3.1 ARCH Processes 139
4.3.2 GARCH Processes 145
4.3.3 Simple Extensions of the GARCH Model 148
4.3.4 Fitting GARCH Models to Data 150
4.4 Volatility Models and Risk Estimation 158
4.4.1 Volatility Forecasting 158
4.4.2 Conditional Risk Measurement 160
4.4.3 Backtesting 162
4.5 Fundamentals of Multivariate Time Series 164
4.5.1 Basic Definitions 164
4.5.2 Analysis in the Time Domain 166
4.5.3 Multivariate ARMA Processes 168
4.6 Multivariate GARCH Processes 170
4.6.1 General Structure of Models 170
4.6.2 Models for Conditional Correlation 172
4.6.3 Models for Conditional Covariance 175
4.6.4 Fitting Multivariate GARCH Models 178
4.6.5 Dimension Reduction in MGARCH 179
4.6.6 MGARCH and Conditional Risk Measurement 182

CHAPTER 5: Copulas and Dependence 184
5.1 Copulas 184
5.1.1 Basic Properties 185
5.1.2 Examples of Copulas 189
5.1.3 Meta Distributions 192
5.1.4 Simulation of Copulas and Meta Distributions 193
5.1.5 Further Properties of Copulas 195
5.1.6 Perfect Dependence 199
5.2 Dependence Measures 201
5.2.1 Linear Correlation 201
5.2.2 Rank Correlation 206
5.2.3 Coefficients of Tail Dependence 208
5.3 Normal Mixture Copulas 210
5.3.1 Tail Dependence 210
5.3.2 Rank Correlations 215
5.3.3 Skewed Normal Mixture Copulas 217
5.3.4 Grouped Normal Mixture Copulas 218
5.4 Archimedean Copulas 220
5.4.1 Bivariate Archimedean Copulas 220
5.4.2 Multivariate Archimedean Copulas 222
5.4.3 Non-exchangeable Archimedean Copulas 224
5.5 Fitting Copulas to Data 228
5.5.1 Method-of-Moments using Rank Correlation 229
5.5.2 Forming a Pseudo-Sample from the Copula 232
5.5.3 Maximum Likelihood Estimation 234

CHAPTER 6: Aggregate Risk 238
6.1 Coherent Measures of Risk 238
6.1.1 The Axioms of Coherence 238
6.1.2 Value-at-Risk 241
6.1.3 Coherent Risk Measures Based on Loss Distributions 243
6.1.4 Coherent Risk Measures as Generalized Scenarios 244
6.1.5 Mean-VaR Portfolio Optimization 246
6.2 Bounds for Aggregate Risks 248
6.2.1 The General Fr chet Problem 248
6.2.2 The Case of VaR 250
6.3 Capital Allocation 256
6.3.1 The Allocation Problem 256
6.3.2 The Euler Principle and Examples 257
6.3.3 Economic Justification of the Euler Principle 261

CHAPTER 7: Extreme Value Theory 264
7.1 Maxima 264
7.1.1 Generalized Extreme Value Distribution 265
7.1.2 Maximum Domains of Attraction 267
7.1.3 Maxima of Strictly Stationary Time Series 270
7.1.4 The Block Maxima Method 271
7.2 Threshold Exceedances 275
7.2.1 Generalized Pareto Distribution 275
7.2.2 Modelling Excess Losses 278
7.2.3 Modelling Tails and Measures of Tail Risk 282
7.2.4 The Hill Method 286
7.2.5 Simulation Study of EVT Quantile Estimators 289
7.2.6 Conditional EVT for Financial Time Series 291
7.3 Tails of Specific Models 293
7.3.1 Domain of Attraction of Fr chet Distribution 293
7.3.2 Domain of Attraction of Gumbel Distribution 294
7.3.3 Mixture Models 295
7.4 Point Process Models 298
7.4.1 Threshold Exceedances for Strict White Noise 299
7.4.2 The POT Model 301
7.4.3 Self-Exciting Processes 306
7.4.4 A Self-Exciting POT Model 307
7.5 Multivariate Maxima 311
7.5.1 Multivariate Extreme Value Copulas 311
7.5.2 Copulas for Multivariate Minima 314
7.5.3 Copula Domains of Attraction 314
7.5.4 Modelling Multivariate Block Maxima 317
7.6 Multivariate Threshold Exceedances 319
7.6.1 Threshold Models Using EV Copulas 319
7.6.2 Fitting a Multivariate Tail Model 320
7.6.3 Threshold Copulas and Their Limits 322

CHAPTER 8: Credit Risk Management 327
8.1 Introduction to Credit Risk Modelling 327
8.1.1 Credit Risk Models 327
8.1.2 The Nature of the Challenge 329
8.2 Structural Models of Default 331
8.2.1 The Merton Model 331
8.2.2 Pricing in Merton's Model 332
8.2.3 The KMV Model 336
8.2.4 Models Based on Credit Migration 338
8.2.5 Multivariate Firm-Value Models 342
8.3 Threshold Models 343
8.3.1 Notation for One-Period Portfolio Models 344
8.3.2 Threshold Models and Copulas 345
8.3.3 Industry Examples 347
8.3.4 Models Based on Alternative Copulas 348
8.3.5 Model Risk Issues 350
8.4 The Mixture Model Approach 352
8.4.1 One-Factor Bernoulli Mixture Models 353
8.4.2 CreditRisk +356
8.4.3 Asymptotics for Large Portfolios 357
8.4.4 Threshold Models as Mixture Models 359
8.4.5 Model-Theoretic Aspects of Basel II 362
8.4.6 Model Risk Issues 364
8.5 Monte Carlo Methods 367
8.5.1 Basics of Importance Sampling 367
8.5.2 Application to Bernoulli-Mixture Models 370
8.6 Statistical Inference for Mixture Models 374
8.6.1 Motivation 374
8.6.2 Exchangeable Bernoulli-Mixture Models 375
8.6.3 Mixture Models as GLMMs 377
8.6.4 One-Factor Model with Rating Effect 381

CHAPTER 9: Dynamic Credit Risk Models 385
9.1 Credit Derivatives 386
9.1.1 Overview 386
9.1.2 Single-Name Credit Derivatives 387
9.1.3 Portfolio Credit Derivatives 389
9.2 Mathematical Tools 392
9.2.1 Random Times and Hazard Rates 393
9.2.2 Modelling Additional Information 395
9.2.3 Doubly Stochastic Random Times 397
9.3 Financial and Actuarial Pricing of Credit Risk 400
9.3.1 Physical and Risk-Neutral Probability Measure 401
9.3.2 Risk-Neutral Pricing and Market Completeness 405
9.3.3 Martingale Modelling 408
9.3.4 The Actuarial Approach to Credit Risk Pricing 411
9.4 Pricing with Doubly Stochastic Default Times 414
9.4.1 Recovery Payments of Corporate Bonds 414
9.4.2 The Model 415
9.4.3 Pricing Formulas 416
9.4.4 Applications 418
9.5 Affine Models 421
9.5.1 Basic Results 422
9.5.2 The CIR Square-Root Diffusion 423
9.5.3 Extensions 425
9.6 Conditionally Independent Defaults 429
9.6.1 Reduced-Form Models for Portfolio Credit Risk 429
9.6.2 Conditionally Independent Default Times 431
9.6.3 Examples and Applications 435
9.7 Copula Models 440
9.7.1 Definition and General Properties 440
9.7.2 Factor Copula Models 444
9.8 Default Contagion in Reduced-Form Models 448
9.8.1 Default Contagion and Default Dependence 448
9.8.2 Information-Based Default Contagion 453
9.8.3 Interacting Intensities 456

CHAPTER 10: Operational Risk and Insurance Analytics 463
10.1 Operational Risk in Perspective 463
10.1.1 A New Risk Class 463
10.1.2 The Elementary Approaches 465
10.1.3 Advanced Measurement Approaches 466
10.1.4 Operational Loss Data 468
10.2 Elements of Insurance Analytics 471
10.2.1 The Case for Actuarial Methodology 471
10.2.2 The Total Loss Amount 472
10.2.3 Approximations and Panjer Recursion 476
10.2.4 Poisson Mixtures 482
10.2.5 Tails of Aggregate Loss Distributions 484
10.2.6 The Homogeneous Poisson Process 484
10.2.7 Processes Related to the Poisson Process 487

Appendix 494
A.1 Miscellaneous Definitions and Results 494
A.1.1 Type of Distribution 494
A.1.2 Generalized Inverses and Quantiles 494
A.1.3 Karamata's Theorem 495
A.2 Probability Distributions 496
A.2.1 Beta 496
A.2.2 Exponential 496
A.2.3 F 496
A.2.4 Gamma 496
A.2.5 Generalized Inverse Gaussian 497
A.2.6 Inverse Gamma 497
A.2.7 Negative Binomial 498
A.2.8 Pareto 498
A.2.9 Stable 498
A.3 Likelihood Inference 499
A.3.1 Maximum Likelihood Estimators 499
A.3.2 Asymptotic Results:Scalar Parameter 499
A.3.3 Asymptotic Results:Vector of Parameters 500
A.3.4 Wald Test and Confidence Intervals 501
A.3.5 Likelihood Ratio Test and Confidence Intervals 501
A.3.6 Akaike Information Criterion 502

References 503
Index 529

What People are Saying About This

Michael Gordy

McNeil, Frey, and Embrechts present a wide-ranging yet remarkably clear and coherent introduction to the modelling of financial risk. Unlike most finance texts, where the focus is on pricing individual instruments, the primary focus in this book is the statistical behavior of portfolios of risky instruments, which is, after all, the primary concern of risk management. This ought to be a core text in every risk manager's training, and a useful reference for experienced professionals.

Martin Baxter

This book provides a framework and a useful toolkit for analysis a wide variety of risk management problems. Common pitfalls are pointed out, and mathematical sophistication is used in pursuit of useful and usable solutions. Every financial institution has a risk management department that looks at aggregated portfolio-wide risks on longer time scales, and at risk exposure to large, or extreme, market movements. Risk managers are always on the lookout for good techniques to help them do their jobs. This very good book provides these techniques and addresses an important, and under-developed, area of practical research.
— Martin Baxter, Nomura International

Riccardo Rebonato

There is no book that provides the type of rigorous and detailed coverage of risk management topics that this book does. This could become the book on quantitative risk management.
— Riccardo Rebonato, Royal Bank of Scotland, author of "Modern Pricing of Interest-Rate Derivatives"

Engle III

This book is a compendium of the statistical arrows that should be in any quantitative risk manager's quiver. It includes extensive discussion of dynamic volatility models, extreme value theory, copulas, and credit risk. Academics, Ph.D. students, and quantitative practitioners will find many new and useful results in this important volume.
— Robert F. Engle III, 2003 Nobel Laureate in Economic Sciences, Michael Armellino Professor in the Management of Financial Services at New York University's Stern School of Business

From the Publisher

"This book is a compendium of the statistical arrows that should be in any quantitative risk manager's quiver. It includes extensive discussion of dynamic volatility models, extreme value theory, copulas, and credit risk. Academics, Ph.D. students, and quantitative practitioners will find many new and useful results in this important volume."—Robert F. Engle III, 2003 Nobel Laureate in Economic Sciences, Michael Armellino Professor in the Management of Financial Services at New York University's Stern School of Business

"This book provides a framework and a useful toolkit for analysis a wide variety of risk management problems. Common pitfalls are pointed out, and mathematical sophistication is used in pursuit of useful and usable solutions. Every financial institution has a risk management department that looks at aggregated portfolio-wide risks on longer time scales, and at risk exposure to large, or extreme, market movements. Risk managers are always on the lookout for good techniques to help them do their jobs. This very good book provides these techniques and addresses an important, and under-developed, area of practical research."—Martin Baxter, Nomura International

"McNeil, Frey, and Embrechts present a wide-ranging yet remarkably clear and coherent introduction to the modelling of financial risk. Unlike most finance texts, where the focus is on pricing individual instruments, the primary focus in this book is the statistical behavior of portfolios of risky instruments, which is, after all, the primary concern of risk management. This ought to be a core text in every risk manager's training, and a useful reference for experienced professionals."—Michael Gordy

"There is no book that provides the type of rigorous and detailed coverage of risk management topics that this book does. This could become the book on quantitative risk management."—Riccardo Rebonato, Royal Bank of Scotland, author of Modern Pricing of Interest-Rate Derivatives

Editorial Reviews

"A great summary of the latest techniques available within quantitative risk measurement. . . . [I]t is an excellent text to have on the shelf as a reference when your day job covers the whole spectrum of quantitative techniques in risk management."

Financial Engineering News

One of the Top 10 Technical Books on Financial Engineering by Financial Engineering News for 2006

"Quantitative Risk Managment can be highly recommended to anyone looking for an excellent survey of the most important techniques and tools used in this rapidly growing field."—-Holger Drees, Risk

"This book provides a state-of-the-art discussion of the three main categories of risk in financial markets, market risk, . . . credit risk . . . and operational risk. . . . This is a high level, but well-written treatment, rigorous (sometimes succinct), complete with theorems and proofs."—-D.L. McLeish, Short Book Reviews of the International Statistical Institute

"Quantitative Risk Management is highly recommended for financial regulators. The statistical and mathematical tools facilitate a better understanding of the strengths and weaknesses of a useful range of advanced risk-management concepts and models, while the focus on aggregate risk enhances the publication's value to banking and insurance supervisors."—-Hans Blommestein, The Financial Regulator

From the Publisher

Alexander McNeil, Rudiger Frey and Paul Embrechts have written a beautiful book. . . . [T]here is no book that can provide the type of rigorous, detailed, well balanced and relevant coverage of quantitative risk management topics that Quantitative Risk Management: Concepts, Techniques, and Tools offers. . . . I believe that this work may become the book on quantitative risk management. . . . [N]o book that I know of can provide better guidance.

Global Association of Risk Professionals (GARP) Review - Dr. Riccardo Rebonato

Quantitative Risk Management is highly recommended for financial regulators. The statistical and mathematical tools facilitate a better understanding of the strengths and weaknesses of a useful range of advanced risk-management concepts and models, while the focus on aggregate risk enhances the publication's value to banking and insurance supervisors.

The Financial Regulator - Hans Blommestein

This book provides a state-of-the-art discussion of the three main categories of risk in financial markets, market risk, . . . credit risk . . . and operational risk. . . . This is a high level, but well-written treatment, rigorous (sometimes succinct), complete with theorems and proofs.

Short Book Reviews of the International Statistical Institute - D.L. McLeish

Quantitative Risk Managment can be highly recommended to anyone looking for an excellent survey of the most important techniques and tools used in this rapidly growing field.

Risk - Holger Drees

This is a very impressive book on a rapidly growing field. It certainly helps to discover the forest in an area where a lot of trees are popping up daily.

SIAM Review - Hans Buhlmann

Alexander McNeil, Rudiger Frey and Paul Embrechts have written a beautiful book. . . . [T]here is no book that can provide the type of rigorous, detailed, well balanced and relevant coverage of quantitative risk management topics that Quantitative Risk Management: Concepts, Techniques, and Tools offers. . . . I believe that this work may become the book on quantitative risk management. . . . [N]o book that I know of can provide better guidance.

Doctor; Global Association of Risk Professionals (GARP) Review - Riccardo Rebonato

This is a very impressive book on a rapidly growing field. It certainly helps to discover the forest in an area where a lot of trees are popping up daily.
— Hans Bühlmann

SIAM Review

Alexander McNeil, Rudiger Frey and Paul Embrechts have written a beautiful book. . . . [T]here is no book that can provide the type of rigorous, detailed, well balanced and relevant coverage of quantitative risk management topics that Quantitative Risk Management: Concepts, Techniques, and Tools offers. . . . I believe that this work may become the book on quantitative risk management. . . . [N]o book that I know of can provide better guidance.
— Dr. Riccardo Rebonato

Global Association of Risk Professionals (GARP) Review

Quantitative Risk Management is highly recommended for financial regulators. The statistical and mathematical tools facilitate a better understanding of the strengths and weaknesses of a useful range of advanced risk-management concepts and models, while the focus on aggregate risk enhances the publication's value to banking and insurance supervisors.
— Hans Blommestein

The Financial Regulator

This book provides a state-of-the-art discussion of the three main categories of risk in financial markets, market risk, . . . credit risk . . . and operational risk. . . . This is a high level, but well-written treatment, rigorous (sometimes succinct), complete with theorems and proofs.
— D.L. McLeish

Short Book Reviews of the International Statistical Institute

Quantitative Risk Managment can be highly recommended to anyone looking for an excellent survey of the most important techniques and tools used in this rapidly growing field.
— Holger Drees

Risk