Exercise: Sharpe-Ratio Confidence Interval from the CLT

Prerequisites: Central Limit Theorem, Expectation and Variance

Problem

1. Show that the sample Sharpe $\widehat{\text{SR}}$ is, by the CLT, approximately normal with mean $\text{SR}$ and standard error $\sqrt{(1 + \text{SR}^2/2)/n_{\text{years}}}$, where $n_{\text{years}} = n/252$ — under the i.i.d. Gaussian-returns assumption (Lo, 2002).

2. Suppose a manager reports $\widehat{\text{SR}} = 1.0$ from a one-year backtest ($n_{\text{years}} = 1$). Compute the 95% confidence interval for the true Sharpe. Does the interval exclude zero?

3. How many years of backtest are needed to conclude at the 95% level that the true Sharpe is strictly greater than zero, given a point estimate of $\widehat{\text{SR}} = 1.0$? Given $\widehat{\text{SR}} = 0.5$?

4. Conceptual. Why does the standard error grow with $\text{SR}$? Give a one-sentence intuition.

Hint

For part 1, don't re-derive the Lo (2002) result — state the standard error and use it. The key structural fact is that the sampling variance of $\widehat{\text{SR}}$ depends on the Sharpe ratio itself.