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What is the proper way to test the significance of Sharpe Ratios or Information Ratios? The Sharpe Ratios will be based on various equity indices and may have variable look-back periods.

One solution that I have seen described simply applies a Student t-test, with the df set to the length of the look-back period.

I am hesitant to apply the above method due to the following concerns:

  1. I believe that the t-test is sensitive to skewness, however equity returns are generally negatively skewed.
  2. The mean return calculated using log returns is less than a mean return calculated using simple returns. I assume that this would make it more likely for a simple return based Sharpe Ratio to register as being significant compared to a log return based Sharpe Ratio, yet the underlying asset returns are technically the same.
  3. If the look-back period is small (i.e. sample size is small), the t-test might be appropriate, but at what threshold would it make sense to use a different test?

My first inclination is to avoid using the Student-t distribution and instead create a test based on the Asymmetric Power Distribution, which I have read has been shown to be a very close approximation of equity market returns, allowing for control over kurtosis and skewness.

My second inclination is to look at non-parametric tests, but having limited experience in their usage I'm not sure where to start and what pitfalls to avoid.

Am I overthinking this problem, are my concerns irrelevant?

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  • $\begingroup$ what would the t-test be with respect to? sharpe = 0? $\endgroup$
    – Trajan
    May 10, 2020 at 16:32

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Bailey and Marcos López de Prado designed a method do exactly that. They use the fact that Sharpe Ratio's are asymptotically normal distributed, even if the returns are not.

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here gamme_3 and gamma_4 are the skewness and kurtosis of the returns. They use this expression to derive the Probabilistic Sharpe Ratio.

enter image description here

SR^* is the value of the sharpe ratio under the null hypothesis, at 5% significance level Sharpe Ratio is significantly greater than SR* if the estimated PSR is larger than 0.95.

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    $\begingroup$ Thank you Shenkie, this solution addresses most of my questions. For those who are interested the paper referenced by Shenkie is "The Sharpe Ratio Efficient Frontier" by Bailey and Lopez de Prado. It not only describes a method to test Sharpe Ratios, but also provides a formula to identify how long of a look back period is required in order to have statistical confidence that a certain Sharpe is above a given threshold. The only thing that I am still scratching my head about is log vs. simple returns. $\endgroup$
    – cty.trader
    Jun 18, 2015 at 17:52
  • $\begingroup$ @cty.trader Use simple proportion/percent change returns or log actual returns. Don't combine them obviously. $\endgroup$
    – SARose
    May 10, 2016 at 13:27
  • $\begingroup$ @SARose - The issue I am trying to address arises when comparing the Sharpe or IR ratios computed using simple vs. log returns. Lets say I calculate the Sharpe for a hypothetical mutual fund; I use the simple(log) returns for the numerator and the simple(log) for the denominator, so there is no mixing of logs and simple returns. In most cases the Simple Sharpe will be greater than the Log Sharpe. This implies that if I do a hypothesis test on a Simple Sharpe, it is more likely to be significant than a test on the log Sharpe. Which results do I trust? $\endgroup$
    – cty.trader
    Aug 9, 2016 at 13:04
  • $\begingroup$ @cty.trader Yeah most of the time it will be greater but not significantly so. If you want a more intuitive answer you can use Bayesian techniques instead of a frequentist one. $\endgroup$
    – SARose
    Aug 9, 2016 at 13:34

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