Market Risk Analysis, Value at Risk Models

Written by leading market risk academic, Professor Carol Alexander, Value-at-Risk Models forms part four of the Market Risk Analysis four volume set. Building on the three previous volumes this book provides by far the most comprehensive, rigorous and detailed treatment of market VaR models. It rests on the basic knowledge of financial mathematics and statistics gained from Volume I, of factor models, principal component analysis, statistical models of volatility and correlation and copulas from Volume II and, from Volume III, knowledge of pricing and hedging financial instruments and of mapping portfolios of similar instruments to risk factors. A unifying characteristic of the series is the pedagogical approach to practical examples that are relevant to market risk analysis in practice.

All together, the Market Risk Analysis four volume set illustrates virtually every concept or formula with a practical, numerical example or a longer, empirical case study. Across all four volumes there are approximately 300 numerical and empirical examples, 400 graphs and figures and 30 case studies many of which are contained in interactive Excel spreadsheets available from the the accompanying CD-ROM . Empirical examples and case studies specific to this volume include:

• Parametric linear value at risk (VaR)models: normal, Student t and normal mixture and their expected tail loss (ETL);
• New formulae for VaR based on autocorrelated returns;
• Historical simulation VaR models: how to scale historical VaR and volatility adjusted historical VaR;
• Monte Carlo simulation VaR models based on multivariate normal and Student t distributions, and based on copulas;
• Examples and case studies of numerous applications to interest rate sensitive, equity, commodity and international portfolios;
• Decomposition of systematic VaR of large portfolios into standard alone and marginal VaR components;
• Backtesting and the assessment of risk model risk;
• Hypothetical factor push and historical stress tests, and stress testing based on VaR and ETL.

List of Figures xiii

List of Tables xvi

List of Examples xxi

Foreword xxv

Preface to Volume IV xxix

IV.1 Value at Risk and Other Risk Metrics 1

IV.1.1 Introduction 1

IV.1.2 An Overview of Market Risk Assessment 4

IV.1.3 Downside and Quantile Risk Metrics 9

IV.1.4 Defining Value at Risk 13

IV.1.5 Foundations of Value-at-Risk Measurement 17

IV.1.6 Risk Factor Value at Risk 25

IV.1.7 Decomposition of Value at Risk 30

IV.1.8 Risk Metrics Associated with Value at Risk 33

IV.1.9 Introduction to Value-at-Risk Models 41

IV.1.10 Summary and Conclusions 47

IV.2 Parametric Linear VaR Models 53

IV.2.1 Introduction 53

IV.2.2 Foundations of Normal Linear Value at Risk 56

IV.2.3 Normal Linear Value at Risk for Cash-Flow Maps 67

IV.2.4 Case Study: PC Value at Risk of a UK Fixed Income Portfolio 79

IV.2.5 Normal Linear Value at Risk for Stock Portfolios 85

IV.2.6 Systematic Value-at-Risk Decomposition for Stock Portfolios 93

IV.2.7 Case Study: Normal Linear Value at Risk for Commodity Futures 103

IV.2.8 Student t Distributed Linear Value at Risk 106

IV.2.9 Linear Value at Risk with Mixture Distributions 111

IV.2.10 Exponential Weighting with Parametric Linear Value at Risk 121

IV.2.11 Expected Tail Loss (Conditional VaR) 128

IV.2.12 Case Study: Credit Spread Parametric Linear Value at Risk and ETL 135

IV.2.13 Summary and Conclusions 138

IV.3 Historical Simulation 141

IV.3.1 Introduction 141

IV.3.2 Properties of Historical Value at Risk 144

IV.3.3 Improving the Accuracy of Historical Value at Risk 152

IV.3.4 Precision of Historical Value at Risk at Extreme Quantiles 165

IV.3.5 Historical Value at Risk for Linear Portfolios 175

IV.3.6 Estimating Expected Tail Loss in the Historical Value-at-Risk Model 195

IV.3.7 Summary and Conclusions 198

IV.4 Monte Carlo VaR 201

IV.4.1 Introduction 201

IV.4.2 Basic Concepts 203

IV.4.3 Modelling Dynamic Properties in Risk Factor Returns 215

IV.4.4 Modelling Risk Factor Dependence 225

IV.4.5 Monte Carlo Value at Risk for Linear Portfolios 233

IV.4.6 Summary and Conclusions 244

IV.5 Value at Risk for Option Portfolios 247

IV.5.1 Introduction 247

IV.5.2 Risk Characteristics of Option Portfolios 250

IV.5.3 Analytic Value-at-Risk Approximations 257

IV.5.4 Historical Value at Risk for Option Portfolios 262

IV.5.5 Monte Carlo Value at Risk for Option Portfolios 282

IV.5.6 Summary and Conclusions 307

IV.6 Risk Model Risk 311

IV.6.1 Introduction 311

IV.6.2 Sources of Risk Model Risk 313

IV.6.3 Estimation Risk 324

IV.6.4 Model Validation 332

IV.6.5 Summary and Conclusions 353

IV.7 Scenario Analysis and Stress Testing 357

IV.7.1 Introduction 357

IV.7.2 Scenarios on Financial Risk Factors 359

IV.7.3 Scenario Value at Risk and Expected Tail Loss 367

IV.7.4 Introduction to Stress Testing 378

IV.7.5 A Coherent Framework for Stress Testing 384

IV.7.6 Summary and Conclusions 398

IV.8 Capital Allocation 401

IV.8.1 Introduction 401

IV.8.2 Minimum Market Risk Capital Requirements for Banks 403

IV.8.3 Economic Capital Allocation 416

IV.8.4 Summary and Conclusions 433

References 437

Index 441

Carol Alexander is a Professor of Risk Management at the ICMA Centre, University of Reading, and Chair of the Academic Advisory Council of the Professional Risk Manager’s International Association (PRMIA). She is the author of Market Models: A Guide to Financial Data Analysis (John Wiley & Sons Ltd, 2001) and has been editor and contributor of a very large number of books in finance and mathematics, including the multi-volume Professional Risk Manager’s Handbook (McGraw-Hill, 2008 and PRMIA Publications). Carol has published nearly 100 academic journal articles, book chapters and books, the majority of which focus on financial risk management and mathematical finance.

Professor Alexander is one of the world’s leading authorities on market risk analysis. For further details, see www.carolalexander.org

MARKET RISK ANALYSIS is a series of four volumes:

VOLUME I: Quantitative Methods in Finance

VOLUME II: Practical Financial Econometrics

VOLUME III: Pricing, Hedging and Trading Financial Instruments

VOLUME IV: Value-at-Risk Models

Although the four volumes are very much interlinked, each containing numerous cross-references to other volumes, they are written as self-contained texts.

VOLUME I covers the essential mathematical and financial background for subsequent volumes. There are six comprehensive chapters covering all the calculus, linear algebra, probability and statistics, numerical methods and portfolio mathematics that are necessary for market risk analysis. It is a complete and pedagogical introduction to quantitative methods applied to finance.

VOLUME II provides a detailed understanding of financial econometrics, with a unique focus on applications to asset pricing, fund management and market risk analysis. It covers equity factor models, including a detailed analysis of the Barra model and tracking error, principal component analysis, volatility and correlation, GARCH, cointegration, copulas, Markov switching, quantile regression, discrete choice models, non-linear regression, forecasting and model evaluation.

VOLUME III has five extensive chapters on the pricing, hedging and trading of bonds and swaps, futures and forwards, options and volatility, and detailed descriptions of mapping portfolios of these financial instruments to their risk factors. There are numerous examples, all coded in interactive Excel spreadsheets, including many pricing formulae for exotic options but excluding the calibration of stochastic volatility models, for which Matlab code is provided.

VOLUME IV builds on the three previous volumes to provide a comprehensive and detailed treatment of market VaR models. The exposition starts at an elementary level but, as in all the other volumes, the pedagogical approach accompanied by numerous interactive Excel spreadsheets allows readers to experience the application of parametric linear, historical simulation and Monte Carlo VaR models to increasingly complex portfolios. Starting with simple positions, readers are soon applying risk models to large international securities portfolios, commodity futures, path dependent options and much else. This rigorous treatment includes many new results as well as applications to regulatory and economic capital allocation, measurement of VaR model risk and stress testing.

Each volume is accompanied by a website which features numerous interactive Excel spreadsheets that illustrate the vast majority of the problems and case studies in these texts. For further information see the accompanying website.

Written by leading market risk academic, Professor Carol Alexander, VALUE-AT-RISK MODELS forms part four of the MARKET RISK ANALYSIS four volume set. Building on the three previous volumes this book provides by far the most comprehensive, rigorous and detailed treatment of market VaR models. It rests on the basic knowledge of financial mathematics and statistics gained from Volume I, of factor models, principal component analysis, statistical models of volatility and correlation and copulas from Volume II and, from Volume III, knowledge of pricing and hedging financial instruments and of mapping portfolios of similar instruments to risk factors. A unifying characteristic of the series is the pedagogical approach to practical examples that are relevant to market risk analysis in practice.

All together, the MARKET RISK ANALYSIS four volume set illustrates virtually every concept or formula with a practical, numerical example or a longer, empirical case study. Across all four volumes there are approximately 300 numerical and empirical examples, 400 graphs and figures and 30 case studies many of which are contained in interactive Excel spreadsheets available from the accompanying website. Excel examples and case studies provided with this volume include:

• Parametric linear value at risk (VaR) models: normal, Student t and normal mixture and their expected tail loss (ETL);
• New formulae for VaR based on autocorrelated returns;
• Historical simulation VaR models: how to scale historical VaR and volatility adjusted historical VaR;
• Monte Carlo simulation VaR models based on multivariate normal and Student t distributions, and based on copulas;
• Examples and case studies of numerous applications to interest rate sensitive, equity, commodity and international portfolios;
• Decomposition of systematic VaR of large portfolios into stand-alone and marginal VaR components;
• Backtesting and the assessment of risk model risk;
• Hypothetical and historical stress tests, liquidity adjustment to VaR, and stress testing based on VaR and ETL.