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.

Table of Contents:
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