Solution: Sharpe-Ratio Confidence Interval from the CLT

Exercise: Sharpe-Ratio Confidence Interval from the CLT

Part 1

Under i.i.d. Gaussian returns, Lo (2002, FAJ) shows via the delta method applied to the CLT that

The variance has two pieces: the $1/n_{\text{years}}$ baseline (the CLT telling you the sample mean has variance $\sigma^2/n$), plus the $\tfrac{1}{2}\text{SR}^2/n_{\text{years}}$ correction from the jointly-estimated standard deviation. Under non-Gaussian returns a skew/kurtosis correction appears; we drop it here.

Part 2

For $\widehat{\text{SR}} = 1.0$, $n_{\text{years}} = 1$:

Part 3

Required: $\widehat{\text{SR}} - 1.96\cdot \text{SE} > 0$, i.e. $n_{\text{years}} > (1.96/\widehat{\text{SR}})^2 \cdot (1 + \tfrac{1}{2}\widehat{\text{SR}}^2)$.

• $\widehat{\text{SR}} = 1.0$: $n_{\text{years}} > (1.96)^2\cdot 1.5 \approx 5.76$ — about 6 years.
• $\widehat{\text{SR}} = 0.5$: $n_{\text{years}} > (1.96/0.5)^2\cdot 1.125 \approx 17.3$ — about 18 years.

At a Sharpe of $0.5$ you need roughly three times the backtest of a Sharpe-$1.0$ strategy for the same level of significance. This is why low-Sharpe systematic strategies are so hard to validate — you almost never have enough data.

Part 4

The standard error grows with Sharpe because a larger Sharpe is achieved either through larger $\mu$ or smaller $\sigma$, and the sample standard deviation $\hat\sigma$ itself is noisy; the fractional error in $\hat\sigma$ propagates through division and amplifies the fractional error in the ratio. Intuitively, a high-Sharpe strategy has a tight volatility estimate that contributes most of the uncertainty.

Takeaways

• Sharpe ratios from short backtests are almost meaningless. A one-year Sharpe of $1.0$ is consistent at the 95% level with true Sharpe of $-1.4$. This is why prop desks demand years of out-of-sample track record before sizing up.
• The $\sqrt n$-rate of the CLT sets the scale of validation. To halve a confidence interval you need to quadruple the backtest length.
• Higher point estimates do not trivially mean higher significance. Standard error grows with $\text{SR}$, so the $t$-statistic $\widehat{\text{SR}}/\text{SE} = \widehat{\text{SR}}\sqrt{n_{\text{years}}/(1 + \text{SR}^2/2)}$ is weakly concave in $\text{SR}$ — a strategy with Sharpe $2.0$ is not twice as significant as one with Sharpe $1.0$ at the same $n$.
• Lo (2002) is the canonical reference. Every backtester should know that formula.