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1 List of Figures List of Tables Preface List of Abbreviations xiv xvi xviii xx 1 Introduction to Value at Risk (VaR) Economics underlying VaR measurement What is VaR? Calculating VaR The assumptions behind VaR calculations Inputs into VaR calculations Diversification and VaR Factors affecting portfolio diversification Decomposing volatility into systematic and idiosyncratic risk Diversification: Words of caution the case of long-term capital management (LTCM) 18 2 Quantifying Volatility in VaR Models The Stochastic Behavior of Returns Revisiting the assumptions The distribution of interest rate changes Fat tails Explaining fat tails Effects of volatility changes Can (conditional) normality be salvaged? Normality cannot be salvaged 34

2 x UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK 2.2 VaR Estimation Approaches Cyclical volatility Historical standard deviation Implementation considerations Exponential smoothing RiskMetrics volatility The optimal smoother lambda Adaptive volatility estimation The empirical performance of RiskMetrics GARCH Nonparametric volatility forecasting Historical simulation Multivariate density estimation A comparison of methods The hybrid approach Return Aggregation and VaR Implied Volatility as a Predictor of Future Volatility Long Horizon Volatility and VaR Mean Reversion and Long Horizon Volatility Correlation Measurement Summary 74 Appendix 2.1 Backtesting Methodology and Results 74 3 Putting VaR to Work The VaR of Derivatives Preliminaries Linear derivatives Nonlinear derivatives Approximating the VaR of derivatives Fixed income securities with embedded optionality Delta normal vs. full-revaluation Structured Monte Carlo, Stress Testing, and Scenario Analysis Motivation Structured Monte Carlo Scenario analysis Correlation breakdown Generating reasonable stress 103

3 xi Stress testing in practice Stress testing and historical simulation Asset concentration Worst Case Scenario (WCS) WCS vs. VaR A comparison of VaR to WCS Extensions Summary 113 Appendix 3.1 Duration Extending the VaR Approach to Non-tradable Loans Traditional Approaches to Credit Risk Measurement Expert systems Rating systems Credit scoring models Theoretical Underpinnings: Two Approaches Options-theoretic structural models of credit risk measurement Reduced form or intensity-based models of credit risk measurement Proprietary VaR models of credit risk measurement CreditMetrics The distribution of an individual loan s value The value distribution for a portfolio of loans Calculating the correlation between equity returns and industry indices for each borrower in the loan portfolio Calculating the correlation between borrower equity returns Solving for joint migration probabilities Valuing each loan across the entire credit migration spectrum Calculating the mean and standard deviation of the normal portfolio value distribution 149

4 xii UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK 4.4 Algorithmics Mark-to-Future Summary 153 Appendix 4.1 CreditMetrics: Calculating Credit VaR Using the Actual Distribution Extending the VaR Approach to Operational Risks Top-Down Approaches to Operational Risk Measurement Top-down vs. bottom-up models Data requirements Top-down models Multi-factor models Income-based models Expense-based models Operating leverage models Scenario analysis Risk profiling models Bottom-Up Approaches to Operational Risk Measurement Process approaches Causal networks or scorecards Connectivity models Reliability models Actuarial approaches Empirical loss distributions Parametric loss distributions Extreme value theory Proprietary operational risk models Hedging Operational Risk Insurance Self-insurance Hedging using derivatives Catastrophe options Cat bonds Limitations to operational risk hedging Summary 196 Appendix 5.1 Copula Functions Applying VaR to Regulatory Models BIS Regulatory Models of Market Risk 203

5 xiii The standardized framework for market risk Measuring interest rate risk Measuring foreign exchange rate risk Measuring equity price risk Internal models of market risk BIS Regulatory Models of Credit Risk The Standardized Model for credit risk The Internal Ratings-Based Models for credit risk The Foundation IRB Approach The Advanced IRB Approach BIS regulatory models of off-balance sheet credit risk Assessment of the BIS regulatory models of credit risk BIS Regulatory Models of Operational Risk The Basic Indicator Approach The Standardized Approach The Advanced Measurement Approach The internal measurement approach The loss distribution approach The scorecard approach Summary VaR: Outstanding Research Data Availability Model Integration Dynamic Modeling 235 Notes 236 References 257 Index 270

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