Confidence Intervals for Sharpe Ratios

A Sharpe ratio of 1.0 measured over three years has a confidence interval wide enough to include zero, which most summaries omit.

A confidence interval for a Sharpe ratio answers whether a measured edge is distinguishable from zero, and for a Sharpe ratio near zero the standard error of the estimate is approximately the square root of one divided by the number of years of data used to measure it. Most reported Sharpe ratios are quoted without this interval, and once it is attached, a headline number that looked decisive often turns out to be statistically indistinguishable from no edge at all.

For a Sharpe ratio near zero, the standard error of an estimate built from T years of return data is approximately the square root of one divided by T. The more complete version of the formula is the square root of the quantity one plus one half times the Sharpe ratio squared, all divided by T, and for a Sharpe ratio close to zero that extra term inside the numerator is small enough to ignore, which is why the simpler approximation is the one practitioners reach for first.

Because the standard error shrinks with the square root of the sample size rather than with the sample size itself, cutting the width of a confidence interval in half requires roughly four times as many years of data, not twice as many, a fact that makes a short backtest a much weaker basis for confidence than it appears.

A strategy measures an annual Sharpe ratio of 1.0 over 3 years of data. The standard error is approximately the square root of one third, which is about 0.577. A 95 percent confidence interval is the point estimate plus or minus 1.96 standard errors, giving roughly 1.0 minus 1.13 to 1.0 plus 1.13, or about negative 0.13 to 2.13. That interval includes zero, meaning the 3 years of data cannot rule out that the true, underlying edge is nothing at all, despite a measured Sharpe ratio that would read as attractive on a one-page summary.

The same measured Sharpe ratio of 1.0 over 10 years tells a different story. The standard error there is approximately the square root of one tenth, about 0.316, giving a 95 percent interval of roughly 1.0 minus 0.62 to 1.0 plus 0.62, or about 0.38 to 1.62, which excludes zero entirely. The headline number did not change between the two examples. The amount of data behind it did, and that difference is the entire basis for whether the number means anything.