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Alexander Carol. Value-at-Risk Models

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Alexander Carol. Value-at-Risk Models
John Wiley & Sons, 2008. — 494 p. — (Market Risk Analysis, Vol. IV).
Value at Risk and Other Risk Metrics.
Introduction.
An Overview of Market Risk Assessment.
Risk Measurement in Banks.
Risk Measurement in Portfolio Management.
Risk Measurement in Large Corporations.
Downside and Quantile Risk Metrics.
Semi-Standard Deviation and Second Order Lower.
Partial Moment.
Other Lower Partial Moments.
Quantile Risk Metrics.
Defining Value at Risk.
Confidence Level and Risk Horizon.
Discounted P&L.
Mathematical Definition of VaR.
Foundations of Value-at-Risk Measurement.
Normal Linear VaR Formula: Portfolio Level.
Static Portfolios.
Scaling VaR.
Discounting and the Expected Return.
Risk Factor Value at Risk.
Motation.
Normal Linear Equity VaR.
Normal Linear Interest Rate VaR.
Decomposition of Value at Risk.
Systematic and Specific VaR.
Stand-alone VaR.
Marginal and Incremental VaR.
Risk Metrics Associated with Value at Risk.
Benchmark VaR.
Conditional VaR: Expected Tail Loss and Expected.
Shortfall.
Coherent Risk Metrics.
Introduction to Value-at-Risk Models.
Normal Linear VaR.
Historical Simulation.
Monte Carlo Simulation.
Case Study: VaR of the S&P Index.
Summary and Conclusions.
. Parametric Linear VaR Models.
Introduction.
Foundations of Normal Linear Value at Risk.
Understanding the Normal Linear VaR Formula.
Analytic Formula for Normal VaR when Returns are.
Autocorrelated.
Systematic Normal Linear VaR.
Stand-Alone Normal Linear VaR.
Marginal and Incremental Normal Linear VaR.
Normal Linear Value at Risk for Cash-Flow Maps.
Normal Linear Interest Rate VaR.
Calculating PV.
Approximating Marginal and Incremental VaR.
Disaggregating Normal Linear Interest Rate VaR.
Normal Linear Credit Spread VaR.
Case Study: PC Value at Risk of a UK Fixed Income Portfolio.
Calculating the Volatility and VaR of the Portfolio.
Combining Cash-Flow Mapping with PCA.
Advantages of Using PC Factors for Interest Rate VaR.
Normal Linear Value at Risk for Stock Portfolios.
Cash Positions on a Few Stocks.
Systematic and Specific VaR for Domestic Stock.
Portfolios.
Empirical Estimation of Specific VaR.
EWMA Estimates of Specific VaR.
Systematic Value-at-Risk Decomposition for Stock Portfolios.
Portfolios Exposed to One Foreign Currency.
Portfolios Exposed to Several Foreign Currencies.
Interest Rate VaR of Equity Portfolios.
Hedging the Risks of International Equity Portfolios.
Case Study: Normal Linear Value at Risk for Commodity Futures.
Contents ix.
Student t Distributed Linear Value at Risk.
Effect of Leptokurtosis and Skewness on VaR.
Student t Linear VaR Formula.
Empirical Examples of Student t Linear VaR.
Linear Value at Risk with Mixture Distributions.
Mixture Distributions.
Mixture Linear VaR Formula.
Mixture Parameter Estimation.
Examples of Mixture Linear VaR.
Normal Mixture Risk Factor VaR.
Exponential Weighting with Parametric Linear Value at Risk.
Exponentially Weighted Moving Averages.
EWMA VaR at the Portfolio Level.
RiskMetrics™ VaR Methodology.
Expected Tail Loss (Conditional VaR).
ETL in the Normal Linear VaR Model.
ETL in the Student t Linear VaR Model.
ETL in the Normal Mixture Linear VaR Model.
ETL under a Mixture of Student t Distributions.
Case Study: Credit Spread Parametric Linear Value at Risk and ETL.
The iTraxx Europe Index.
VaR Estimates.
Summary and Conclusions.
. Historical Simulation.
Introduction.
Properties of Historical Value at Risk.
Definition of Historical VaR.
Sample Size and Data Frequency.
Power Law Scale Exponents.
Case Study: Scale Exponents for Major Risk Factors.
Scaling Historical VaR for Linear Portfolios.
Errors from Square-Root Scaling of Historical VaR.
Overlapping Data and Multi-Step Historical Simulation.
Improving the Accuracy of Historical Value at Risk.
Case Study: Equally Weighted Historical and Linear VaR.
Exponential Weighting of Return Distributions.
Volatility Adjustment.
Filtered Historical Simulation.
Precision of Historical Value at Risk at Extreme Quantiles.
Kernel Fitting.
Extreme Value Distributions.
Cornish–Fisher Approximation.
Johnson Distributions.
Historical Value at Risk for Linear Portfolios.
Historical VaR for Cash Flows.
Total, Systematic and Specific VaR of a Stock Portfolio.
x Contents.
Equity and Forex VaR of an International Stock Portfolio.
Interest Rate and Forex VaR of an International Bond.
Position.
Case Study: Historical VaR for a Crack Spread Trader.
Estimating Expected Tail Loss in the Historical Value-at-Risk Model.
Parametric Historical ETL.
Empirical Results on Historical ETL.
Disaggregation of Historical ETL.
Summary and Conclusions.
. Monte Carlo VaR.
Introduction.
Basic Concepts.
Pseudo-Random Number Generation.
Low Discrepancy Sequences.
Variance Reduction.
Sampling from Unariate Distributions.
Sampling from Multariate Distributions.
Introduction to Monte Carlo VaR.
Modelling Dynamic Properties in Risk Factor Returns.
Multi-Step Monte Carlo.
Volatility Clustering and Mean Reversion.
Regime Switching Models.
Modelling Risk Factor Dependence.
Multariate Distributions for i.i.d. Returns.
Principal Component Analysis.
Behavioural Models.
Case Study: Modelling the Price – Volatility.
Relationship.
Monte Carlo Value at Risk for Linear Portfolios.
Algorithms for VaR and ETL.
Cash-Flow Portfolios: Copula VaR and PC VaR.
Equity Portfolios: ‘Crash’ Scenario VaR.
Currency Portfolios: VaR with Volatility Clustering.
Summary and Conclusions.
. Value at Risk for Option Portfolios.
Introduction.
Risk Characteristics of Option Portfolios.
Gamma Effects.
Delta and Vega Effects.
Theta and Rho Effects.
Static and Dynamic VaR Estimates.
Analytic Value-at-Risk Approximations.
Delta Approximation and Delta–Normal VaR.
P&L Distributions for Option Portfolios.
Delta–Gamma VaR.
Contents xi.
Historical Value at Risk for Option Portfolios.
VaR and ETL with Exact Revaluation.
Dynamically Hedged Option Portfolios.
Greeks Approximation.
Historical VaR for Path-Dependent Options.
Case Study: Historical VaR for an Energy Options.
Trading Book.
Monte Carlo Value at Risk for Option Portfolios.
Monte Carlo VaR and ETL with Exact Revaluation.
Risk Factor Models for Simulating Options VaR.
Capturing Non-normality and Non-linearity.
Capturing Gamma, Vega and Theta Effects.
Path Dependency.
Option Portfolios with a Single Underlying.
Option Portfolios with Several Underlyings.
Case Study: Monte Carlo VaR for an Energy Options.
Trading Book.
Summary and Conclusions.
. Risk Model Risk.
Introduction.
Sources of Risk Model Risk.
Risk Factor Mapping.
Risk Factor or Asset Returns Model.
VaR Resolution Method.
Scaling.
Estimation Risk.
Distribution of VaR Estimators in Parametric Linear.
Models.
Distribution of VaR Estimators in Simulation Models.
Model Validation.
Backtesting Methodology.
Guidelines for Backtesting from Banking Regulators.
Coverage Tests.
Backtests Based on Regression.
Backtesting ETL Forecasts.
Bias Statistics for Normal Linear VaR.
Distribution Forecasts.
Some Backtesting Results.
Summary and Conclusions.
. Scenario Analysis and Stress Testing.
Introduction.
Scenarios on Financial Risk Factors.
Broad Categorization of Scenarios.
Historical Scenarios.
Hypothetical Scenarios.
Distribution Scenario Design.
xii Contents.
Scenario Value at Risk and Expected Tail Loss.
Normal Distribution Scenarios.
Compound Distribution Scenario VaR.
Bayesian VaR.
Introduction to Stress Testing.
Regulatory Guidelines.
Systemic Risk.
Stress Tests Based on Worst Case Loss.
A Coherent Framework for Stress Testing.
VaR Based on Stressed Covariance Matrices.
Generating Hypothetical Covariance Matrices.
Stress Tests Based on Principal Component Analysis.
Modelling Liquidity Risk.
Incorporating Volatility Clustering.
Summary and Conclusions.
. Capital Allocation.
Introduction.
Minimum Market Risk Capital Requirements for Banks.
Basel Accords.
Banking and Trading Book Accounting.
Regulatory Framework for Market Risk.
Internal Models.
Standardized Rules.
Incremental Risk Charge.
Economic Capital Allocation.
Measurement of Economic Capital.
Banking Applications of Economic Capital.
Aggregation Risk.
Risk Adjusted Performance Measures.
Optimal Allocation of Economic Capital.
Summary and Conclusions.
References.
Index.
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