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Connecting Sharpe ratio and Student t-statistic, and beyond

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  • Eric Benhamou

Abstract

Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions (independent normally distributed - and identically distributed returns). In this paper, we provide the exact distribution of the Sharpe ratio for independent normally distributed return. In this case, the Sharpe ratio statistic is up to a rescaling factor a non centered Student distribution whose characteristics have been widely studied by statisticians. The asymptotic behavior of our distribution provide the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general formula in this case of heteroscedasticity and autocorrelation.

Suggested Citation

  • Eric Benhamou, 2018. "Connecting Sharpe ratio and Student t-statistic, and beyond," Papers 1808.04233, arXiv.org, revised May 2019.
  • Handle: RePEc:arx:papers:1808.04233
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    References listed on IDEAS

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    1. J. Qi & M. Rekkas & A. Wong, 2018. "Highly Accurate Inference on the Sharpe Ratio for Autocorrelated Return Data," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 7(1), pages 1-2.
    2. William N. Goetzmann & Jonathan E. Ingersoll, Jr. & Matthew I. Spiegel & Ivo Welch, 2002. "Sharpening Sharpe Ratios," Yale School of Management Working Papers ysm29, Yale School of Management.
    3. Miller, Robert E. & Gehr, Adam K., 1978. "Sample Size Bias and Sharpe's Performance Measure: A Note," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 13(5), pages 943-946, December.
    4. Jobson, J D & Korkie, Bob M, 1981. "Performance Hypothesis Testing with the Sharpe and Treynor Measures," Journal of Finance, American Finance Association, vol. 36(4), pages 889-908, September.
    5. Qi, Feng & Mortici, Cristinel, 2015. "Some best approximation formulas and inequalities for the Wallis ratio," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 363-368.
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    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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