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