Skip to content
All library documents

Testing Whether Two Funds Have Different Sharpe Ratios

Article Quant Q&A · Author: Christian
Read the original

Summary

The document discusses testing whether two funds have different Sharpe ratios, prompted by an R routine that offers HAC-based asymptotic inference and circular bootstrap inference. It raises a practical input question—whether to provide prices, returns, or excess returns—and the response notes that Sharpe calculations use arithmetic returns, which can be derived from prices or supplied directly. The appropriate data and testing approach depend on how the statistic is defined and on the analyst's purpose.

For a conventional t-test framing, the output's t-statistic measures the observed difference relative to its uncertainty, while the p-value indicates how surprising a result at least that extreme would be under the null of equal Sharpe ratios. A chosen confidence level determines the rejection threshold. The response also suggests confidence intervals or standard hypothesis-testing methods for Sharpe estimates and points to related research, but it does not provide a detailed comparison of HAC and bootstrap assumptions, resolve the exact R package output, or establish which method is preferable for the user's data. Interpretation therefore depends on the calculation and its assumptions.

Key ideas

  • Sharpe ratios are calculated from arithmetic returns, which may be obtained from prices or supplied as returns.
  • The choice between HAC inference and a circular bootstrap depends on the analysis goals and assumptions.
  • A t-statistic scales the estimated Sharpe difference by its uncertainty.
  • A p-value evaluates how extreme the test statistic would be under the null that the Sharpe ratios are equal.
  • The interpretation of a software test depends on how its statistic and standard errors are calculated.

Tags

StatisticsPortfolio constructionRisk management
Full text
# Sharpe testing in R


# Sharpe testing in R












My goal test: The statistical significance of the difference in Sharpe ratio between funds A and B.

My data: I have daily prices from January 23 2008 until 10th of April 2019 (n = 2818 observations). I upload an excel sheet to r with prices of fund A in column 1 and prices of fund B in column 2.

R Code: I run the following code

```
## Run Sharpe testing (asymptotic hac)
x = SR_for_r[,1]
y = SR_for_r[,2]
ctr = list(type = 1, hac = TRUE)
out = sharpeTesting(x, y, control = ctr)
print(out)

## Run Sharpe testing (circular bootstrap)
x = SR_for_r[,1]
y = SR_for_r[,2]
set.seed(1234)
ctr = list(type = 2, nBoot = 1000, bBoot = 3)
out = sharpeTesting(x, y, control = ctr)
print(out)
```

My questions

1) Should I have fund prices, rate of return, or excess returns in columns 1 and 2 in the datasheet I import to R?

2) Should I use HAC standard errors only or use circular bootstrap to test the statistical significance of the Sharpe ratio difference?

3) How can I interpret the output from the test?

4) Does anyone know of an article of someone who have tested the statistical significance of the Sharpe ratio difference between two funds?

My Sources: The R code comes from here: https://rdrr.io/cran/PeerPerformance/man/sharpeTesting.html

## Answer by Chris (score 1)

https://quant.stackexchange.com/a/45336

Unless you're doing this as a purely educational exercise, it looks like you may be overcomplicating things.

Sharpe ratios follow a student's t distribution. You can thus use standard approaches to test hypotheses or create confidence intervals for each Sharpe estimate.

A quick google turned up this paper that addresses topics similar to what you're asking about.

To answer your specific questions:

(1) That depends entirely on how the calculation is being done. Sharpe ratios are calculated using arithmetic returns (which can be calculated from either prices or returns themselves).

(2) Depends on your intentions.

(3) Again, depends on what calculation is being done. If it's a standard t-test, pval represents the likelihood you'd see a tstat that extreme if the sample estimates weren't different (ie, type 1 error) under the hypothesis Sharpe1=Sharpe2. You typically pick some confidence level, a (eg, 95 or 90%), and reject the null (eg, Sharpe1 = Sharpe2) if pval < 1 - a. tstat is the statistic that's calculated to make this determination.

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.