Bayesian Estimation of Extreme Quantiles and the Distribution of Exceedances for Measuring Tail Risk

Estimating extreme quantiles and the number of future exceedances is an important task in financial risk management. More important than estimating the quantile itself is to insure zero coverage error, which implies the quantile estimate should, on average, reflect the desired probability of exceedance. In this research, we show that for unconditional distributions isomorphic to the exponential, a Bayesian quantile estimate results in zero coverage error. This compares to the traditional maximum likelihood method, where the coverage error can be significant under small sample sizes even though the quantile estimate is unbiased. More generally, we prove a sufficient condition for an unbiased quantile estimator to result in coverage error and we show our result holds by virtue of using a Jeffreys prior for the unknown parameters and is independent of the true prior. We derive a new, predictive distribution, and the moments, for the number of quantile exceedances, and highlight its superior performance. We extend our results to the conditional tail of distributions with asymptotic Paretian tails and, in particular, those in the Fréchet maximum domain of attraction which are typically encountered in finance. We illustrate our results using simulations for a variety of light and heavy-tailed distributions.