  The Value-at-Risk (VaR) is a widely used instrument in financial risk
management. The question of estimating the VaR of loss return distributions at
extreme levels is an important question in financial applications, both from
operational and regulatory perspectives; in particular, the dynamic estimation
of extreme VaR given the recent past has received substantial attention. We
propose here a two-step bias-reduced estimation methodology called GARCH-UGH
(Unbiased Gomes-de Haan), whereby financial returns are first filtered using an
AR-GARCH model, and then a bias-reduced estimator of extreme quantiles is
applied to the standardized residuals to estimate one-step ahead dynamic
extreme VaR. Our results indicate that the GARCH-UGH estimates are more
accurate than those obtained by combining conventional AR-GARCH filtering and
extreme value estimates from the perspective of in-sample and out-of-sample
backtestings of historical daily returns on several financial time series.
