Testing Differences Between Sharpe Ratios
Summary
The document asks how a study assessed whether small-cap and large-cap Sharpe ratios differed, after a simple calculation using the Sharpe difference multiplied by the square root of the observation count produced large p-values. The question presents monthly sample assumptions and calculations for the full period and a later subperiod, illustrating why the author doubts that this shortcut reproduces the study’s table.
The answer suggests that the study likely used the Leung and Wong test for equality of Sharpe ratios. That approach uses the Central Limit Theorem and the Delta Method to obtain inference for the difference, rather than treating the Sharpe difference alone as a t-statistic. The response points to a textbook treatment and an R implementation, but does not show the test formula, inspect the original study’s methods, or establish that this was definitely the authors’ procedure. Replication would require checking the study’s assumptions and inputs.
Key ideas
- Multiplying a Sharpe ratio difference by the square root of the sample size may not reproduce a study’s p-value.
- The answer identifies the Leung and Wong equality test as a likely method.
- The proposed test uses the Central Limit Theorem and Delta Method.
- The document does not verify the original study’s exact procedure or provide enough detail to reproduce it.
Tags
Full text
# p-value of Sharpe Ratio Differences # 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. ## Answer by steveo'america (score 1) https://quant.stackexchange.com/a/37819 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.