Sharpe ratio standard error, illustrative track-record threshold and serial-correlation scaling

Separate sampling uncertainty from evidence of investment skill

The standard error below is an asymptotic IID-normal approximation for a per-period Sharpe estimate. A Wald ratio is not an exact Student t statistic or a test of manager skill. Serial dependence, nonnormal returns, selection and repeated backtest searches require further analysis.

An absolute Wald ratio above 1.96 is an approximate two-sided 5% threshold against zero Sharpe under the assumptions; a negative ratio can also exceed that threshold in magnitude. Failure to cross it does not prove zero performance.

The sample-size calculation holds the assumed positive Sharpe fixed and inverts the displayed approximation. It is not a guarantee of future significance, a power analysis or an instruction to stop collecting observations once a threshold is crossed.

Use monthly Sharpe with monthly observation count. The separate annual-scaling example assumes stationary AR(1) correlations rho(k)=rho^k; it does not correct the IID standard error above.

Lo (2002) Sharpe Ratio Standard Error SE(SR)

B2 is the nonannualized monthly Sharpe estimate. B3 is a whole observation count of at least two; small samples can still make the asymptotic approximation unreliable.

=IF(COUNT(B2:B3)<>2,"Invalid inputs",IF(OR(B3<2,B3<>INT(B3)),"Invalid inputs",ROUND(SQRT((1 + 0.5 * (B2 ^ 2)) / B3), 4)))

Approximate Wald ratio against zero Sharpe

Divide by the unrounded standard error. The rounded display should not be used to make threshold decisions. This is not an exact t-test or a diagnosis of investment skill.

=IF(COUNT(B2:B3)<>2,"Invalid inputs",IF(OR(B3<2,B3<>INT(B3)),"Invalid inputs",ROUND(B2 / SQRT((1 + 0.5 * (B2 ^ 2)) / B3), 2)))

Illustrative positive-Sharpe observation threshold

B4 is a positive normal critical value, with 1.96 used here. Only a positive assumed Sharpe is eligible for this positive-performance threshold.

=IF(OR(COUNT(B2)<>1,COUNT(B4)<>1),"Invalid inputs",IF(B4<=0,"Invalid inputs",IF(B2<=0,"No positive threshold",CEILING((1 + 0.5 * (B2 ^ 2)) * ((B4 / B2) ^ 2), 1))))

Annual scaling under an AR(1) correlation model

For twelve periods, use sqrt(12) divided by sqrt(1+2 times the sum of (1-k/12)rho^k for lags 1 through 11). B5 is an assumed stationary AR(1) coefficient strictly between -1 and 1. Zero gives the IID multiplier. This scaling describes aggregation of returns under the model, not compounded wealth or a fitted correlation model.

=IF(OR(COUNT(B2)<>1,COUNT(B5)<>1),"Invalid inputs",IF(ABS(B5)>=1,"Invalid inputs",ROUND(B2*SQRT(12)/SQRT(1+2*((1-1/12)*B5^1+(1-2/12)*B5^2+(1-3/12)*B5^3+(1-4/12)*B5^4+(1-5/12)*B5^5+(1-6/12)*B5^6+(1-7/12)*B5^7+(1-8/12)*B5^8+(1-9/12)*B5^9+(1-10/12)*B5^10+(1-11/12)*B5^11)),4)))

Further reading

• Microsoft Support — CEILING function
• Andrew W. Lo (2002) — The Statistics of Sharpe Ratios, full paper