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