Contents
IV.1 Value at Risk and Other Risk Metrics
IV.1.2 An Overview of Market Risk Assessment
IV.1.2.1 Risk Measurement in Banks
IV.1.2.2 Risk Measurement in Portfolio Management
IV.1.2.3 Risk Measurement in Large Corporations
IV.1.3 Downside and Quantile Risk Metrics
IV.1.3.1 Semi-Standard Deviation and Second Order Lower Partial Moment
IV.1.3.2 Other Lower Partial Moments
IV.1.3.3 Quantile Risk Metrics
IV.1.4.1 Confidence Level and Risk Horizon
IV.1.4.3 Mathematical Definition of VaR
IV.1.5 Foundations of Value-at-Risk Measurement
IV.1.5.1 Normal Linear VaR Formula: Portfolio Level
IV.1.5.4 Discounting and the Expected Return
IV.1.6 Risk Factor Value at Risk
IV.1.6.2 Normal Linear Equity VaR
IV.1.6.3 Normal Linear Interest Rate VaR
IV.1.7 Decomposition of Value at Risk
IV.1.7.1 Systematic and Specific VaR
IV.1.7.3 Marginal and Incremental VaR
IV.1.8 Risk Metrics Associated with Value at Risk
IV.1.8.2 Conditional VaR: Expected Tail Loss and Expected Shortfall
IV.1.8.3 Coherent Risk Metrics
IV.1.9 Introduction to Value-at-Risk Models
IV.1.9.2 Historical Simulation
IV.1.9.3 Monte Carlo Simulation
IV.1.9.4 Case Study: VaR of the S&P 500 Index
IV.1.10 Summary and Conclusions
IV.2 Parametric Linear VaR Models
IV.2.2 Foundations of Normal Linear Value at Risk
IV.2.2.1 Understanding the Normal Linear VaR Formula
IV.2.2.2 Analytic Formula for Normal VaR when Returns are Autocorrelated
IV.2.2.3 Systematic Normal Linear VaR
IV.2.2.4 Stand-Alone Normal Linear VaR
IV.2.2.5 Marginal and Incremental Normal Linear VaR
IV.2.3 Normal Linear Value at Risk for Cash-Flow Maps
IV.2.3.1 Normal Linear Interest Rate VaR
IV.2.3.3 Approximating Marginal and Incremental VaR
IV.2.3.4 Disaggregating Normal Linear Interest Rate VaR
IV.2.3.5 Normal Linear Credit Spread VaR
IV.2.4 Case Study: PC Value at Risk of a UK Fixed Income Portfolio
IV.2.4.1 Calculating the Volatility and VaR of the Portfolio
IV.2.4.2 Combining Cash-Flow Mapping with PCA
IV.2.4.3 Advantages of Using PC Factors for Interest Rate VaR
IV.2.5 Normal Linear Value at Risk for Stock Portfolios
IV.2.5.1 Cash Positions on a Few Stocks
IV.2.5.2 Systematic and Specific VaR for Domestic Stock Portfolios
IV.2.5.3 Empirical Estimation of Specific VaR
IV.2.5.4 EWMA Estimates of Specific VaR
IV.2.6 Systematic Value-at-Risk Decomposition for Stock Portfolios
IV.2.6.1 Portfolios Exposed to One Foreign Currency
IV.2.6.2 Portfolios Exposed to Several Foreign Currencies
IV.2.6.3 Interest Rate VaR of Equity Portfolios
IV.2.6.4 Hedging the Risks of International Equity Portfolios
IV.2.7 Case Study: Normal Linear Value at Risk for Commodity Futures
IV.2.8 Student t Distributed Linear Value at Risk
IV.2.8.1 Effect of Leptokurtosis and Skewness on VaR
IV.2.8.2 Student t Linear VaR Formula
IV.2.8.3 Empirical Examples of Student t Linear VaR
IV.2.9 Linear Value at Risk with Mixture Distributions
IV.2.9.1 Mixture Distributions
IV.2.9.2 Mixture Linear VaR Formula
IV.2.9.3 Mixture Parameter Estimation
IV.2.9.4 Examples of Mixture Linear VaR
IV.2.9.5 Normal Mixture Risk Factor VaR
IV.2.10 Exponential Weighting with Parametric Linear Value at Risk
IV.2.10.1 Exponentially Weighted Moving Averages
IV.2.10.2 EWMA VaR at the Portfolio Level
IV.2.10.3 RiskMetrics™ VaR Methodology
IV.2.11 Expected Tail Loss (Conditional VaR)
IV.2.11.1 ETL in the Normal Linear VaR Model
IV.2.11.2 ETL in the Student t Linear VaR Model
IV.2.11.3 ETL in the Normal Mixture Linear VaR Model
IV.2.11.4 ETL under a Mixture of Student t Distributions
IV.2.12 Case Study: Credit Spread Parametric Linear Value at Risk and ETL
IV.2.12.1 The iTraxx Europe Index
IV.2.13 Summary and Conclusions
IV.3.2 Properties of Historical Value at Risk
IV.3.2.1 Definition of Historical VaR
IV.3.2.2 Sample Size and Data Frequency
IV.3.2.3 Power Law Scale Exponents
IV.3.2.4 Case Study: Scale Exponents for Major Risk Factors
IV.3.2.5 Scaling Historical VaR for Linear Portfolios
IV.3.2.6 Errors from Square-Root Scaling of Historical VaR
IV.3.2.7 Overlapping Data and Multi-Step Historical Simulation
IV.3.3 Improving the Accuracy of Historical Value at Risk
IV.3.3.1 Case Study: Equally Weighted Historical and Linear VaR
IV.3.3.2 Exponential Weighting of Return Distributions
IV.3.3.3 Volatility Adjustment
IV.3.3.4 Filtered Historical Simulation
IV.3.4 Precision of Historical Value at Risk at Extreme Quantiles
IV.3.4.2 Extreme Value Distributions
IV.3.4.3 Cornish–Fisher Approximation
IV.3.4.4 Johnson Distributions
IV.3.5 Historical Value at Risk for Linear Portfolios
IV.3.5.1 Historical VaR for Cash Flows
IV.3.5.2 Total, Systematic and Specific VaR of a Stock Portfolio
IV.3.5.3 Equity and Forex VaR of an International Stock Portfolio
IV.3.5.4 Interest Rate and Forex VaR of an International Bond Position
IV.3.5.5 Case Study: Historical VaR for a Crack Spread Trader
IV.3.6 Estimating Expected Tail Loss in the Historical Value-at-Risk Model
IV.3.6.1 Parametric Historical ETL
IV.3.6.2 Empirical Results on Historical ETL
IV.3.6.3 Disaggregation of Historical ETL
IV.3.7 Summary and Conclusions
IV.4.2.1 Pseudo-Random Number Generation
IV.4.2.2 Low Discrepancy Sequences
IV.4.2.4 Sampling from Univariate Distributions
IV.4.2.5 Sampling from Multivariate Distributions
IV.4.2.6 Introduction to Monte Carlo VaR
IV.4.3 Modelling Dynamic Properties in Risk Factor Returns
IV.4.3.1 Multi-Step Monte Carlo
IV.4.3.2 Volatility Clustering and Mean Reversion
IV.4.3.3 Regime Switching Models
IV.4.4 Modelling Risk Factor Dependence
IV.4.4.1 Multivariate Distributions for i.i.d. Returns
IV.4.4.2 Principal Component Analysis
IV.4.4.4 Case Study: Modelling the Price – Volatility Relationship
IV.4.5 Monte Carlo Value at Risk for Linear Portfolios
IV.4.5.1 Algorithms for VaR and ETL
IV.4.5.2 Cash-Flow Portfolios: Copula VaR and PC VaR
IV.4.5.3 Equity Portfolios: ‘Crash’ Scenario VaR
IV.4.5.4 Currency Portfolios: VaR with Volatility Clustering
IV.4.6 Summary and Conclusions
IV.5 Value at Risk for Option Portfolios
IV.5.2 Risk Characteristics of Option Portfolios
IV.5.2.2 Delta and Vega Effects
IV.5.2.3 Theta and Rho Effects
IV.5.2.4 Static and Dynamic VaR Estimates
IV.5.3 Analytic Value-at-Risk Approximations
IV.5.3.1 Delta Approximation and Delta–Normal VaR
IV.5.3.2 P&L Distributions for Option Portfolios
IV.5.4 Historical Value at Risk for Option Portfolios
IV.5.4.1 VaR and ETL with Exact Revaluation
IV.5.4.2 Dynamically Hedged Option Portfolios
IV.5.4.4 Historical VaR for Path-Dependent Options
IV.5.4.5 Case Study: Historical VaR for an Energy Options Trading Book
IV.5.5 Monte Carlo Value at Risk for Option Portfolios
IV.5.5.1 Monte Carlo VaR and ETL with Exact Revaluation
IV.5.5.2 Risk Factor Models for Simulating Options VaR
IV.5.5.3 Capturing Non-normality and Non-linearity
IV.5.5.4 Capturing Gamma, Vega and Theta Effects
IV.5.5.6 Option Portfolios with a Single Underlying
IV.5.5.7 Option Portfolios with Several Underlyings
IV.5.5.8 Case Study: Monte Carlo VaR for an Energy Options Trading Book
IV.5.6 Summary and Conclusions
IV.6.2 Sources of Risk Model Risk
IV.6.2.2 Risk Factor or Asset Returns Model
IV.6.2.3 VaR Resolution Method
IV.6.3.1 Distribution of VaR Estimators in Parametric Linear Models
IV.6.3.2 Distribution of VaR Estimators in Simulation Models
IV.6.4.1 Backtesting Methodology
IV.6.4.2 Guidelines for Backtesting from Banking Regulators
IV.6.4.4 Backtests Based on Regression
IV.6.4.5 Backtesting ETL Forecasts
IV.6.4.6 Bias Statistics for Normal Linear VaR
IV.6.4.7 Distribution Forecasts
IV.6.4.8 Some Backtesting Results
IV.6.5 Summary and Conclusions
IV.7 Scenario Analysis and Stress Testing
IV.7.2 Scenarios on Financial Risk Factors
IV.7.2.1 Broad Categorization of Scenarios
IV.7.2.3 Hypothetical Scenarios
IV.7.2.4 Distribution Scenario Design
IV.7.3 Scenario Value at Risk and Expected Tail Loss
IV.7.3.1 Normal Distribution Scenarios
IV.7.3.2 Compound Distribution Scenario VaR
IV.7.4 Introduction to Stress Testing
IV.7.4.1 Regulatory Guidelines
IV.7.4.3 Stress Tests Based on Worst Case Loss
IV.7.5 A Coherent Framework for Stress Testing
IV.7.5.1 VaR Based on Stressed Covariance Matrices
IV.7.5.2 Generating Hypothetical Covariance Matrices
IV.7.5.3 Stress Tests Based on Principal Component Analysis
IV.7.5.4 Modelling Liquidity Risk
IV.7.5.5 Incorporating Volatility Clustering
IV.7.6 Summary and Conclusions
IV.8.2 Minimum Market Risk Capital Requirements for Banks
IV.8.2.2 Banking and Trading Book Accounting
IV.8.2.3 Regulatory Framework for Market Risk
IV.8.2.6 Incremental Risk Charge
IV.8.3 Economic Capital Allocation
IV.8.3.1 Measurement of Economic Capital
IV.8.3.2 Banking Applications of Economic Capital
IV.8.3.4 Risk Adjusted Performance Measures
IV.8.3.5 Optimal Allocation of Economic Capital