  Sharpe ratio is widely used in asset management to compare and benchmark
funds and asset managers. It computes the ratio of the excess return over the
strategy standard deviation. However, the elements to compute the Sharpe ratio,
namely, the expected returns and the volatilities are unknown numbers and need
to be estimated statistically. This means that the Sharpe ratio used by funds
is subject to be error prone because of statistical estimation error. Lo
(2002), Mertens (2002) derive explicit expressions for the statistical
distribution of the Sharpe ratio using standard asymptotic theory under several
sets of assumptions (independent normally distributed - and identically
distributed returns). In this paper, we provide the exact distribution of the
Sharpe ratio for independent normally distributed return. In this case, the
Sharpe ratio statistic is up to a rescaling factor a non centered Student
distribution whose characteristics have been widely studied by statisticians.
The asymptotic behavior of our distribution provide 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.
