# 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.

If you want to learn more about the Sharpe Ratio, about its use, its distribution as a sample statistic, I recommend you:

- Read
*A Short Sharpe Course*, a free self-contained set of notes on the Sharpe Ratio produced by the author of this site. This work is only one third complete, still lacking the sections on market timing and the portfolio problem. - Read up on some of the historical research on the Sharpe Ratio.
- Check out our blog.