Quantifying Tail Risk with Bad Data

Abstract

We document substantial practitioner interest in measures of the downside tail risk of hedge funds, such as maximum drawdown (MDD) and worst one-period loss, together with a general sentiment that volatility does not convey enough information about tail risk. We show that past observed extremes are inappropriate estimators of tail risk, and propose a better, parametric estimator that is simple to implement and needs only short return histories as input. In addition, we characterize the statistical properties of downside risk measures and show that they depend linearly on volatility. Together with evidence that tail shape does not change much across funds, this explains why extreme downside performance measures rank funds similarly to the Sharpe ratio. Finally, we note that using sample standard deviation to estimate volatility when returns have fat tails is problematic. We show that the same technique employed in the paper can be used to improve estimation of the Sharpe ratio and other measures based on volatility.