Kevin Dowd is Professor of Financial Risk Management at Nottingham University. Kevin is an Adjunct Scholar at the Cato Institute in Washington, D.C., and a Fellow of the Pensions Institute at Birkbeck College.

• Preface to the Second Edition xiiiAcknowledgements xix1 The Rise of Value at Risk 11.1 The emergence of financial risk management 21.2 Market risk measurement 41.3 Risk measurement before VaR 51.3.1 Gap analysis 51.3.2 Duration analysis 51.3.3 Scenario analysis 61.3.4 Portfolio theory 71.3.5 Derivatives risk measures 81.4 Value at risk 91.4.1 The origin and development of VaR 91.4.2 Attractions of VaR 111.4.3 Criticisms of VaR 13Appendix: Types of Market Risk 152 Measures of Financial Risk 192.1 The mean–variance framework for measuring financial risk 202.2 Value at risk 272.2.1 Basics of VaR 272.2.2 Determination of the VaR parameters 292.2.3 Limitations of VaR as a risk measure 312.3 Coherent risk measures 322.3.1 The coherence axioms and their implications 322.3.2 The expected shortfall 352.3.3 Spectral risk measures 372.3.4 Scenarios as coherent risk measures 422.4 Conclusions 44Appendix 1: Probability Functions 45Appendix 2: Regulatory Uses of VaR 523 Estimating Market Risk Measures: An Introduction and Overview 533.1 Data 533.1.1 Profit/loss data 533.1.2 Loss/profit data 543.1.3 Arithmetic return data 543.1.4 Geometric return data 543.2 Estimating historical simulation VaR 563.3 Estimating parametric VaR 573.3.1 Estimating VaR with normally distributed profits/losses 573.3.2 Estimating VaR with normally distributed arithmetic returns 593.3.3 Estimating lognormal VaR 613.4 Estimating coherent risk measures 643.4.1 Estimating expected shortfall 643.4.2 Estimating coherent risk measures 643.5 Estimating the standard errors of risk measure estimators 693.5.1 Standard errors of quantile estimators 693.5.2 Standard errors in estimators of coherent risk measures 723.6 The core issues: an overview 73Appendix 1: Preliminary Data Analysis 75Appendix 2: Numerical Integration Methods 804 Non-parametric Approaches 834.1 Compiling historical simulation data 844.2 Estimation of historical simulation VaR and ES 844.2.1 Basic historical simulation 844.2.2 Bootstrapped historical simulation 854.2.3 Historical simulation using non-parametric density estimation 864.2.4 Estimating curves and surfaces for VAR and ES 884.3 Estimating confidence intervals for historical simulation VaR and ES 894.3.1 An order-statistics approach to the estimation of confidence intervals for HS VaR and ES 894.3.2 A bootstrap approach to the estimation of confidence intervals for HS VaR and ES 904.4 Weighted historical simulation 924.4.1 Age-weighted historical simulation 934.4.2 Volatility-weighted historical simulation 944.4.3 Correlation-weighted historical simulation 954.4.4 Filtered historical simulation 964.5 Advantages and disadvantages of non-parametric methods 994.5.1 Advantages 994.5.2 Disadvantages 1004.6 Conclusions 101Appendix 1: Estimating Risk Measures with Order Statistics 102Appendix 2: The Bootstrap 105Appendix 3: Non-parametric Density Estimation 111Appendix 4: Principal Components Analysis and Factor Analysis 1185 Forecasting Volatilities, Covariances and Correlations 1275.1 Forecasting volatilities 1275.1.1 Defining volatility 1275.1.2 Historical volatility forecasts 1285.1.3 Exponentially weighted moving average volatility 1295.1.4 GARCH models 1315.1.5 Implied volatilities 1365.2 Forecasting covariances and correlations 1375.2.1 Defining covariances and correlations 1375.2.2 Historical covariances and correlations 1385.2.3 Exponentially weighted moving average covariances 1405.2.4 GARCH covariances 1405.2.5 Implied covariances and correlations 1415.2.6 Some pitfalls with correlation estimation 1415.3 Forecasting covariance matrices 1425.3.1 Positive definiteness and positive semi-definiteness 1425.3.2 Historical variance–covariance estimation 1425.3.3 Multivariate EWMA 1425.3.4 Multivariate GARCH 1425.3.5 Computational problems with covariance and correlation matrices 143Appendix: Modelling Dependence: Correlations and Copulas 1456 Parametric Approaches (I) 1516.1 Conditional vs unconditional distributions 1526.2 Normal VaR and ES 1546.3 The t-distribution 1596.4 The lognormal distribution 1616.5 Miscellaneous parametric approaches 1656.5.1 Lévy approaches 1656.5.2 Elliptical and hyperbolic approaches 1676.5.3 Normal mixture approaches 1676.5.4 Jump diffusion 1686.5.5 Stochastic volatility approaches 1696.5.6 The Cornish–Fisher approximation 1716.6 The multivariate normal variance–covariance approach 1736.7 Non-normal variance–covariance approaches 1766.7.1 Multivariate t-distributions 1766.7.2 Multivariate elliptical distributions 1776.7.3 The Hull–White transformation-into-normality approach 1776.8 Handling multivariate return distributions with copulas 1786.8.1 Motivation 1786.8.2 Estimating VaR with copulas 1796.9 Conclusions 182Appendix: Forecasting Longer-term Risk Measures 1847 Parametric Approaches (II): Extreme Value 1897.1 Generalised extreme-value theory 1907.1.1 Theory 1907.1.2 A short-cut EV method 1947.1.3 Estimation of EV parameters 1957.2 The peaks-over-threshold approach: the generalised Pareto distribution 2017.2.1 Theory 2017.2.2 Estimation 2037.2.3 GEV vs POT 2047.3 Refinements to EV approaches 2047.3.1 Conditional EV 2047.3.2 Dealing with dependent (or non-iid) data 2057.3.3 Multivariate EVT 2067.4 Conclusions 2068 Monte Carlo Simulation Methods 2098.1 Uses of Monte carlo simulation 2108.2 Monte Carlo simulation with a single risk factor 2138.3 Monte Carlo simulation with multiple risk factors 2158.4 Variance-reduction methods 2178.4.1 Antithetic variables 2188.4.2 Control variates 2188.4.3 Importance sampling 2198.4.4 Stratified sampling 2208.4.5 Moment matching 2238.5 Advantages and disadvantages of Monte Carlo simulation 2258.5.1 Advantages 2258.5.2 Disadvantages 2258.6 Conclusions 2259 Applications of Stochastic Risk Measurement Methods 2279.1 Selecting stochastic processes 2279.2 Dealing with multivariate stochastic processes 2309.2.1 Principal components simulation 2309.2.2 Scenario simulation 2329.3 Dynamic risks 2349.4 Fixed-income risks 2369.4.1 Distinctive features of fixed-income problems 2379.4.2 Estimating fixed-income risk measures 2379.5 Credit-related risks 2389.6 Insurance risks 2409.6.1 General insurance risks 2419.6.2 Life insurance risks 2429.7 Measuring pensions risks 2449.7.1 Estimating risks of defined-benefit pension plans 2459.7.2 Estimating risks of defined-contribution pension plans 2469.8 Conclusions 24810 Estimating Options Risk Measures 24910.1 Analytical and algorithmic solutions for options VaR 24910.2 Simulation approaches 25310.3 Delta–gamma and related approaches 25610.3.1 Delta–normal approaches 25710.3.2 Delta–gamma approaches 25810.4 Conclusions 26411 Incremental and Component Risks 26511.1 Incremental VaR 26511.1.1 Interpreting Incremental VaR 26511.1.2 Estimating IVaR by brute force: the ‘before and after’ approach 26611.1.3 Estimating IVaR using analytical solutions 26711.2 Component VaR 27111.2.1 Properties of component VaR 27111.2.2 Uses of component VaR 27411.3 Decomposition of coherent risk measures 27712 Mapping Positions to Risk Factors 27912.1 Selecting core instruments 28012.2 Mapping positions and VaR estimation 28112.2.1 Basic building blocks 28112.2.2 More complex positions 28713 Stress Testing 29113.1 Benefits and difficulties of stress testing 29313.1.1 Benefits of stress testing 29313.1.2 Difficulties with stress tests 29513.2 Scenario analysis 29713.2.1 Choosing scenarios 29713.2.2 Evaluating the effects of scenarios 30013.3 Mechanical stress testing 30313.3.1 Factor push analysis 30313.3.2 Maximum loss optimisation 30513.3.3 CrashMetrics 30513.4 Conclusions 30614 Estimating Liquidity Risks 30914.1 Liquidity and liquidity risks 30914.2 Estimating liquidity-adjusted VaR 31014.2.1 The constant spread approach 31114.2.2 The exogenous spread approach 31214.2.3 Endogenous-price approaches 31414.2.4 The liquidity discount approach 31514.3 Estimating liquidity at risk (LaR) 31614.4 Estimating liquidity in crises 31915 Backtesting Market Risk Models 32115.1 Preliminary data issues 32115.2 Backtests based on frequency tests 32315.2.1 The basic frequency backtest 32415.2.2 The conditional testing (Christoffersen) backtest 32915.3 Backtests based on tests of distribution equality 33115.3.1 Tests based on the Rosenblatt transformation 33115.3.2 Tests using the Berkowitz transformation 33315.3.3 Overlapping forecast periods 33515.4 Comparing alternative models 33615.5 Backtesting with alternative positions and data 33915.5.1 Backtesting with alternative positions 34015.5.2 Backtesting with alternative data 34015.6 Assessing the precision of backtest results 34015.7 Summary and conclusions 342Appendix: Testing Whether Two Distributions are Different 34316 Model Risk 35116.1 Models and model risk 35116.2 Sources of model risk 35316.2.1 Incorrect model specification 35316.2.2 Incorrect model application 35416.2.3 Implementation risk 35416.2.4 Other sources of model risk 35516.3 Quantifying model risk 35716.4 Managing model risk 35916.4.1 Managing model risk: some guidelines for risk practitioners 35916.4.2 Managing model risk: some guidelines for senior managers 36016.4.3 Institutional methods to manage model risk 36116.5 Conclusions 363Bibliography 365Index 379

Kevin Dowd is Professor of Financial Risk Management at Nottingham University. Kevin is an Adjunct Scholar at the Cato Institute in Washington, D.C., and a Fellow of the Pensions Institute at Birkbeck College.

• Preface to the Second Edition xiiiAcknowledgements xix1 The Rise of Value at Risk 11.1 The emergence of financial risk management 21.2 Market risk measurement 41.3 Risk measurement before VaR 51.3.1 Gap analysis 51.3.2 Duration analysis 51.3.3 Scenario analysis 61.3.4 Portfolio theory 71.3.5 Derivatives risk measures 81.4 Value at risk 91.4.1 The origin and development of VaR 91.4.2 Attractions of VaR 111.4.3 Criticisms of VaR 13Appendix: Types of Market Risk 152 Measures of Financial Risk 192.1 The mean–variance framework for measuring financial risk 202.2 Value at risk 272.2.1 Basics of VaR 272.2.2 Determination of the VaR parameters 292.2.3 Limitations of VaR as a risk measure 312.3 Coherent risk measures 322.3.1 The coherence axioms and their implications 322.3.2 The expected shortfall 352.3.3 Spectral risk measures 372.3.4 Scenarios as coherent risk measures 422.4 Conclusions 44Appendix 1: Probability Functions 45Appendix 2: Regulatory Uses of VaR 523 Estimating Market Risk Measures: An Introduction and Overview 533.1 Data 533.1.1 Profit/loss data 533.1.2 Loss/profit data 543.1.3 Arithmetic return data 543.1.4 Geometric return data 543.2 Estimating historical simulation VaR 563.3 Estimating parametric VaR 573.3.1 Estimating VaR with normally distributed profits/losses 573.3.2 Estimating VaR with normally distributed arithmetic returns 593.3.3 Estimating lognormal VaR 613.4 Estimating coherent risk measures 643.4.1 Estimating expected shortfall 643.4.2 Estimating coherent risk measures 643.5 Estimating the standard errors of risk measure estimators 693.5.1 Standard errors of quantile estimators 693.5.2 Standard errors in estimators of coherent risk measures 723.6 The core issues: an overview 73Appendix 1: Preliminary Data Analysis 75Appendix 2: Numerical Integration Methods 804 Non-parametric Approaches 834.1 Compiling historical simulation data 844.2 Estimation of historical simulation VaR and ES 844.2.1 Basic historical simulation 844.2.2 Bootstrapped historical simulation 854.2.3 Historical simulation using non-parametric density estimation 864.2.4 Estimating curves and surfaces for VAR and ES 884.3 Estimating confidence intervals for historical simulation VaR and ES 894.3.1 An order-statistics approach to the estimation of confidence intervals for HS VaR and ES 894.3.2 A bootstrap approach to the estimation of confidence intervals for HS VaR and ES 904.4 Weighted historical simulation 924.4.1 Age-weighted historical simulation 934.4.2 Volatility-weighted historical simulation 944.4.3 Correlation-weighted historical simulation 954.4.4 Filtered historical simulation 964.5 Advantages and disadvantages of non-parametric methods 994.5.1 Advantages 994.5.2 Disadvantages 1004.6 Conclusions 101Appendix 1: Estimating Risk Measures with Order Statistics 102Appendix 2: The Bootstrap 105Appendix 3: Non-parametric Density Estimation 111Appendix 4: Principal Components Analysis and Factor Analysis 1185 Forecasting Volatilities, Covariances and Correlations 1275.1 Forecasting volatilities 1275.1.1 Defining volatility 1275.1.2 Historical volatility forecasts 1285.1.3 Exponentially weighted moving average volatility 1295.1.4 GARCH models 1315.1.5 Implied volatilities 1365.2 Forecasting covariances and correlations 1375.2.1 Defining covariances and correlations 1375.2.2 Historical covariances and correlations 1385.2.3 Exponentially weighted moving average covariances 1405.2.4 GARCH covariances 1405.2.5 Implied covariances and correlations 1415.2.6 Some pitfalls with correlation estimation 1415.3 Forecasting covariance matrices 1425.3.1 Positive definiteness and positive semi-definiteness 1425.3.2 Historical variance–covariance estimation 1425.3.3 Multivariate EWMA 1425.3.4 Multivariate GARCH 1425.3.5 Computational problems with covariance and correlation matrices 143Appendix: Modelling Dependence: Correlations and Copulas 1456 Parametric Approaches (I) 1516.1 Conditional vs unconditional distributions 1526.2 Normal VaR and ES 1546.3 The t-distribution 1596.4 The lognormal distribution 1616.5 Miscellaneous parametric approaches 1656.5.1 Lévy approaches 1656.5.2 Elliptical and hyperbolic approaches 1676.5.3 Normal mixture approaches 1676.5.4 Jump diffusion 1686.5.5 Stochastic volatility approaches 1696.5.6 The Cornish–Fisher approximation 1716.6 The multivariate normal variance–covariance approach 1736.7 Non-normal variance–covariance approaches 1766.7.1 Multivariate t-distributions 1766.7.2 Multivariate elliptical distributions 1776.7.3 The Hull–White transformation-into-normality approach 1776.8 Handling multivariate return distributions with copulas 1786.8.1 Motivation 1786.8.2 Estimating VaR with copulas 1796.9 Conclusions 182Appendix: Forecasting Longer-term Risk Measures 1847 Parametric Approaches (II): Extreme Value 1897.1 Generalised extreme-value theory 1907.1.1 Theory 1907.1.2 A short-cut EV method 1947.1.3 Estimation of EV parameters 1957.2 The peaks-over-threshold approach: the generalised Pareto distribution 2017.2.1 Theory 2017.2.2 Estimation 2037.2.3 GEV vs POT 2047.3 Refinements to EV approaches 2047.3.1 Conditional EV 2047.3.2 Dealing with dependent (or non-iid) data 2057.3.3 Multivariate EVT 2067.4 Conclusions 2068 Monte Carlo Simulation Methods 2098.1 Uses of Monte carlo simulation 2108.2 Monte Carlo simulation with a single risk factor 2138.3 Monte Carlo simulation with multiple risk factors 2158.4 Variance-reduction methods 2178.4.1 Antithetic variables 2188.4.2 Control variates 2188.4.3 Importance sampling 2198.4.4 Stratified sampling 2208.4.5 Moment matching 2238.5 Advantages and disadvantages of Monte Carlo simulation 2258.5.1 Advantages 2258.5.2 Disadvantages 2258.6 Conclusions 2259 Applications of Stochastic Risk Measurement Methods 2279.1 Selecting stochastic processes 2279.2 Dealing with multivariate stochastic processes 2309.2.1 Principal components simulation 2309.2.2 Scenario simulation 2329.3 Dynamic risks 2349.4 Fixed-income risks 2369.4.1 Distinctive features of fixed-income problems 2379.4.2 Estimating fixed-income risk measures 2379.5 Credit-related risks 2389.6 Insurance risks 2409.6.1 General insurance risks 2419.6.2 Life insurance risks 2429.7 Measuring pensions risks 2449.7.1 Estimating risks of defined-benefit pension plans 2459.7.2 Estimating risks of defined-contribution pension plans 2469.8 Conclusions 24810 Estimating Options Risk Measures 24910.1 Analytical and algorithmic solutions for options VaR 24910.2 Simulation approaches 25310.3 Delta–gamma and related approaches 25610.3.1 Delta–normal approaches 25710.3.2 Delta–gamma approaches 25810.4 Conclusions 26411 Incremental and Component Risks 26511.1 Incremental VaR 26511.1.1 Interpreting Incremental VaR 26511.1.2 Estimating IVaR by brute force: the ‘before and after’ approach 26611.1.3 Estimating IVaR using analytical solutions 26711.2 Component VaR 27111.2.1 Properties of component VaR 27111.2.2 Uses of component VaR 27411.3 Decomposition of coherent risk measures 27712 Mapping Positions to Risk Factors 27912.1 Selecting core instruments 28012.2 Mapping positions and VaR estimation 28112.2.1 Basic building blocks 28112.2.2 More complex positions 28713 Stress Testing 29113.1 Benefits and difficulties of stress testing 29313.1.1 Benefits of stress testing 29313.1.2 Difficulties with stress tests 29513.2 Scenario analysis 29713.2.1 Choosing scenarios 29713.2.2 Evaluating the effects of scenarios 30013.3 Mechanical stress testing 30313.3.1 Factor push analysis 30313.3.2 Maximum loss optimisation 30513.3.3 CrashMetrics 30513.4 Conclusions 30614 Estimating Liquidity Risks 30914.1 Liquidity and liquidity risks 30914.2 Estimating liquidity-adjusted VaR 31014.2.1 The constant spread approach 31114.2.2 The exogenous spread approach 31214.2.3 Endogenous-price approaches 31414.2.4 The liquidity discount approach 31514.3 Estimating liquidity at risk (LaR) 31614.4 Estimating liquidity in crises 31915 Backtesting Market Risk Models 32115.1 Preliminary data issues 32115.2 Backtests based on frequency tests 32315.2.1 The basic frequency backtest 32415.2.2 The conditional testing (Christoffersen) backtest 32915.3 Backtests based on tests of distribution equality 33115.3.1 Tests based on the Rosenblatt transformation 33115.3.2 Tests using the Berkowitz transformation 33315.3.3 Overlapping forecast periods 33515.4 Comparing alternative models 33615.5 Backtesting with alternative positions and data 33915.5.1 Backtesting with alternative positions 34015.5.2 Backtesting with alternative data 34015.6 Assessing the precision of backtest results 34015.7 Summary and conclusions 342Appendix: Testing Whether Two Distributions are Different 34316 Model Risk 35116.1 Models and model risk 35116.2 Sources of model risk 35316.2.1 Incorrect model specification 35316.2.2 Incorrect model application 35416.2.3 Implementation risk 35416.2.4 Other sources of model risk 35516.3 Quantifying model risk 35716.4 Managing model risk 35916.4.1 Managing model risk: some guidelines for risk practitioners 35916.4.2 Managing model risk: some guidelines for senior managers 36016.4.3 Institutional methods to manage model risk 36116.5 Conclusions 363Bibliography 365Index 379