Compare two correlated Sharpe ratios using matched returns, the corrected asymptotic variance, and a worked example that keeps return frequency and assumptions explicit.
Jobson–Korkie–Memmel test
The Jobson–Korkie–Memmel test is an asymptotic test of whether two population Sharpe ratios are equal, using their matched return observations. It distinguishes an estimated performance gap from evidence of a persistent difference. Its familiar closed-form standard error assumes independent, identically distributed return pairs from a joint normal distribution.
“Independent” applies across dates. The two strategies can be correlated within each date; that shared variation is central to the calculation. Comparing two separate Sharpe error bars discards it.
Keep the return pairs together
Let Xit=Rit−Rft be strategy i's excess return over the same cash benchmark for period t. Observe T aligned pairs (X1t,X2t) on the same dates and at the same frequency. Use comparable cost conventions. Do not correlate unrelated date ranges or fill missing returns with zeros just to obtain pairs.
Write the population ratios as Si=μi/σi, with σi>0, and the estimates as S^i=Xˉi/si. Here si uses the usual sample-variance divisor T−1; ρ^ is the sample correlation of the paired excess returns.
For the null hypothesis H0:S1−S2=0, the corrected, untransformed statistic is
The outer divisor is T, the number of paired observations. Changing the sample-variance convention changes finite-sample estimates; it does not turn this large-sample approximation into an exact test. Under the null and the model assumptions, z is asymptotically standard normal. A two-sided test uses p≈2{1−Φ(∣z∣)}. If the estimated difference variance is zero, this statistic is undefined.
Memmel's correction, especially equations (10)–(13), removes an error in the original transformed statistic and gives the simpler Sharpe-difference form above. It does not relax the Gaussian or time-independence assumptions.
Why correlation enters twice
Both the means and the standard deviations are estimated. Under the Gaussian model, the leading covariance between the two estimated ratios is
Cov(S^1,S^2)≈Tρ+21S1S2ρ2.
Combining this with Var(S^i)≈(1+Si2/2)/T yields the displayed difference variance. This is a delta-method calculation for correlated ratio estimates, not a test on the mean of X1−X2.
A 0.10 gap is not enough by itself
Take hypothetical same-period estimates S^1=0.20, S^2=0.10, ρ^=0.50 and T=120. Assuming iid jointly normal pairs,
V=0.0085,V=0.09220,z=1.08465.
The two-sided p-value is approximately 0.278. An approximate 95% confidence interval for S1−S2 is 0.10±1.96(0.09220), or [−0.0807,0.2807]. This does not reject equality at 5%; it does not establish that the strategies are equivalent.
At T 120 and same-period Sharpe estimates 0.20 and 0.10, approximate 95% difference intervals are minus 0.155 to 0.355 for return correlation zero, minus 0.081 to 0.281 for correlation 0.5, and 0.018 to 0.182 for correlation 0.9.
Original analytical sensitivity: only the paired-return correlation changes. The estimated gap stays 0.10; interval endpoints use the iid Gaussian approximation.
All ratios inside the formula must use the observation period. For monthly observations, use monthly Sharpes and the number of months. Substituting annualised ratios while retaining monthly T changes the calculation incorrectly. Under iid square-root annualisation, an annual-scale difference and its standard error both multiply by 12, leaving z unchanged. Decimal versus percentage return units also leave each Sharpe unchanged.
When the closed form stops answering the question
Skewness, heavy tails, serial dependence and volatility clustering change the estimator covariance. Uncorrelated returns alone are insufficient: squared returns can remain dependent. A longer record does not repair a misspecified variance formula.
Ledoit and Wolf, Sections 2–3, develop a general moment-based variance with HAC estimation and a studentized time-series bootstrap. The latter resamples blocks of return pairs, preserving contemporaneous association while approximating temporal dependence. Such methods still require suitable moment, stationarity and dependence conditions, plus bandwidth or block-length choices. Shuffling dates independently, bootstrapping each strategy separately, or labelling any resampling “robust” is insufficient.
A pairwise test also does not account for selecting the winner from many backtests. That requires a research design addressing multiple testing and selection; neither Memmel's correction nor resampling alone supplies it.
For the economic choice of performance measure, see Risk-Adjusted Performance Measures (public preview; full lesson requires membership), which compares Sharpe, Treynor, Jensen and active-risk measures.
