Multivariate Nonparametric Estimation of Value at Risk and Expected Shortfall for Nonlinear Returns Using Extreme Value Theory

The catastrophic failures of risk management systems in 2008 bring to the forefront the need for accurate and flexible estimators of market risk. Despite advances in the theory and practice of evaluating risk, existing measures are notoriously poor predictors of loss in high-quantile events. To extend the research concerned with modeling extreme value events, we utilize extreme value theory (EVT) to propose a multivariate estimation procedure for value-at-risk (VaR) and expected shortfall (ES) for conditional distributions of a time series of returns on a financial asset. Our approach extends the local linear estimator of conditional mean and volatility used in the conditional heteroskedastic autoregressive nonlinear (CHARN) model proposed by Martins-Filho and Yao (2006) by incorporating an exogenous time series resembling returns on the S&P 500 from January 1950 through September 2011. In combination with EVT, this model estimates the quantiles of the conditional distribution and subsequently the one-day forecasted VaR and ES. We examine the fi nite sample properties of our method and contrast them with the popular Gaussian GARCH estimator in an extensive Monte Carlo simulation. The method we propose generally outperforms the Gaussian GARCH estimator, particularly in samples greater than 1,000. Our results provide evidence of the e ffect of the curse of dimensionality, which arises because we include a second regressor.