Distribution and statistics of the Sharpe Ratio.

Summary Because of the frequent usage of the Sharpe ratio in asset management to compare and benchmark funds and asset managers, it is relevant to derive the distribution and some statistics of the Sharpe ratio. In this paper, we show that under the assumption of independent normally distributed returns, it is possible to derive the exact distribution of the Sharpe ratio. In particular, we prove that up to a rescaling factor, the Sharpe ratio is a non centered Student distribution whose characteristics have been widely studied by statisticians. For a large number of observations, we can derive the asymtptotic distribution and find back the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general formula in this case of heteroscedasticity and autocorrelation.