# p-value of Sharpe Ratio Differences

I am trying to understand what was done in this study by Research Affiliates on the small cap anomaly.

Looking at Table 1, how are the authors actually calculating the p-value?

I have read that the p-value can be derived by way of calculating the t-value from the

Sharpe Ratio * the sqrt(N) with N being the number of observations


Using R, I have tried to back into the p-value with the below, but have not had much success here. I am assuming that the data is monthly, which is what you will see below, but changing for daily does not change the result significantly.

#US P Value
# 46 years * 12 months
#.01 as difference in Small vs. Large cap Sharpe
46*12
.01*sqrt(552)
2*pt(-abs(.01*sqrt(552)),df=552-1)

#US P value - post Banz
# 31 years * 12 months
#.06 as difference in Small vs. Large cap Sharpe
31*12
.06*sqrt(372)
2*pt(-abs(.06*sqrt(372)),df=372-1)


Results:

> #US P Value
> # 46 years * 12 months
> 46*12
[1] 552
> .01*sqrt(552)
[1] 0.2349468
> 2*pt(-abs(.01*sqrt(552)),df=552-1)
[1] 0.8143373
>
> #US P value - post Banz
> # 31 years * 12 months
> 31*12
[1] 372
> .06*sqrt(372)
[1] 1.157238
> 2*pt(-abs(.06*sqrt(372)),df=372-1)
[1] 0.2479196


Please let me know if there is additional information I can provide / any questions.

• Your formulas are correct, I tried using years, months and days as well and do not match their numbers. – pyCthon Oct 25 '17 at 23:18
• Out of curiousity do you have a source for the formula you use ? Doesnt it assume equal variance for both Sharpe ratios, maybe this would explain the difference (ie slightly different variance). By the way I am not even sure how you would compute/define the variance of the Sharpe Ratio... – NegativeJo Oct 26 '17 at 19:26
• @NegativeJo - the formula of the Sharpe Ratio difference I am inferring from their notes in Table 1 "The difference in the P-values of Sharpe ratios represents the probability that the Sharpe ratio of the small-cap and large-cap portfolios are from the same distribution..." which I am inferring means subtracting one from the other. Additional source for the Sharpe ratio transformation into T-stat is in footnote 9 of the following link - stat.berkeley.edu/~aldous/157/Papers/harvey.pdf – steicher Oct 27 '17 at 2:49

The 'difference of Sharpe ratios' is likely via the test of Leung and Wong, which is based on the Central Limit Theorem and Delta Method. This is also described in section 4.3.1 of the Short Sharpe Course, and implemented in the R package SharpeR, c.f. Sharpe equality test.