Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach

We propose a method for estimating VaR and related risk measures describing the tail of the conditional distribution of a heteroscedastic nancial return series. Our approach combines quasi maximum likelihood tting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting we show that our procedure gives better estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our tted models and a simulation approach we estimate the conditional quantiles of returns over multiple day horizons and nd evidence of a power scaling law, where the power depends in a natural way on the current volatility level. J.E.L. Subject Classi cation: C.22, G.10, G.21