# The Sharpe Ratio

The Sharpe Ratio is perhaps the most widely used statistic for summarizing the achieved (or backtested!) past performance of some asset--mutual fund, hedge fund, trading strategy, etc. Defined as the mean return divided by the standard deviation (or $$\frac{\mu}{\sigma}$$), the Sharpe Ratio roughly measures the expected return per unit risk, with the idea that an investor will tailor their investment to a maximum level of risk. Expressed in 'annual' units (which is, 'per square root year'), a value of 1 or so should be considered very good for a mutual fund, while a value of 2 would be astounding. These interpretations are my own, but the Sharpe Ratio can be used to bound the probability of a loss; indeed for a mutual fund with an annual Sharpe of 1, a year over year loss is a "1 sigma event", destined to happen approximately 16 percent of the time. For a Sharpe of 2, however, that probability is around 2 percent.

The Sharpe Ratio is connected to the statistician's $$t$$ statistic, defined roughly as $$\sqrt{n}\frac{\mu}{\sigma}$$. Statisticians have spent decades thinking about the $$t$$ test, how it responds to violations of model assumptions (independence, heteroskedasticity, autocorrelation, non-normality, etc.), and how to make it robust to those assumptions. Much of those findings can be translated into facts about the Sharpe Ratio. We find more connections when we generalize to higher dimensions or take into account conditioning information: connections between the Markowitz Portfolio and the multivariate analogue of the $$t$$ statistic, Hotelling's $$T^2$$, and so on.