Interpreting Degrees of Freedom, t-Statistics, and p-Values for Sharpe Estimates

Summary

The document explains how the Sharpe ratio relates to a t-statistic for testing whether mean returns equal the risk-free rate. Under the relationship given in the response, the t-statistic is the Sharpe ratio multiplied by the square root of the number of returns used. Reporting both quantities can therefore help distinguish the estimated risk-adjusted return from the statistic used for inference.

The response says to use the t-statistic with a Student t distribution and the appropriate degrees of freedom to obtain a one-tailed p-value under the null hypothesis. It describes the degrees of freedom as the number of monthly returns. This is a brief introductory explanation rather than a full treatment of inference: it gives no example and does not discuss adjustments for serial correlation, non-normal returns, or alternative degrees-of-freedom conventions. Readers should interpret the stated relationship in the context of the assumptions and estimation method used.

Key ideas

• The Sharpe ratio and the t-statistic for testing excess returns are related but are not identical.
• The stated relationship scales the Sharpe ratio by the square root of the return count to obtain the t-statistic.
• A Student t distribution and degrees of freedom are used to calculate a p-value under the null.
• The explanation does not address complications such as serial dependence or non-normal returns.

Tags

# What are "df", "t", and "p" in these sharpe ratio related estimates?


# What are "df", "t", and "p" in these sharpe ratio related estimates?












I am looking at some sharpe ratio related estimates and have not seen Sharpe stats broken down this way before. I don't know what is meant by df, t, and p. Can someone explain that to me? Thank you!

## Answer by nbbo2 (score 3)

https://quant.stackexchange.com/a/25566

The Sharpe Ratio and the T-Statistic for the hypothesis that returns are equal to the risk free rate, are closely related (occasionally some people mistakenly think they are the same).

In fact: "The t-statistic will equal the Sharpe Ratio times the square root of N (the number of returns used for the calculation)." 1

So it makes sense to show both. Then, by looking up the T-statistic t in a Student-t table with the right degrees of freedom you can come up with p the probability in a one-tailed test under the null. The df (degrees of freedom) is simply the number of (monthly) returns.

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.