Tales of Tails: Quantifying Extreme Downside Risk*

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 “classical” performance measures such as the Sharpe Ratio do not convey enough information about tail risk. We characterize the nite-sample distribution of these measures and show that it depends linearly on variance. Sample variance, appropriately transformed, as well as Bayesian estimates of the entire distribution of returns, are shown to yield better forward-looking estimates of extrememeasures than in-sample extremes themselves. We then show that worst oneperiod loss is “manipulation-proof ” in the sense of Ingersoll et al (). Moreover, MDD has a degree of robustness against return smoothing in the spirit of Getmansky et al. (). We prove that one cannot have it bothways: “manipulation-proof ”measures are not “smoothing-proof ” and vice-versa. Finally, we show that the expectation of manipulation-proof measures is an increasing function of sample size, and for worst one-period loss we provide an explicit adjustment for this problem. JEL classi cation: C, G, G