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Distribution and statistics of the Sharpe Ratio

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  • Eric Benhamou

    (MILES - Machine Intelligence and Learning Systems - LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique, LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

Abstract

Because of the frequent usage of the Sharpe ratio in asset management to compare and benchmark funds and asset managers, it is relevant to derive the distribution and some statistics of the Sharpe ratio. In this paper, we show that under the assumption of independent normally distributed returns, it is possible to derive the exact distribution of the Sharpe ratio. In particular, we prove that up to a rescaling factor, the Sharpe ratio is a non centered Student distribution whose characteristics have been widely studied by statisticians. For a large number of observations, we can derive the asymtptotic distribution and find back 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, 2021. "Distribution and statistics of the Sharpe Ratio," Working Papers hal-03207169, HAL.
    • Handle: RePEc:hal:wpaper:hal-03207169
    Note: View the original document on HAL open archive server: https://hal.science/hal-03207169
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    References listed on IDEAS

    as
    1. Eric Benhamou & David Saltiel & Beatrice Guez & Nicolas Paris, 2019. "Testing Sharpe ratio: luck or skill?," Papers 1905.08042, arXiv.org, revised May 2019.
    2. Eric Benhamou, 2018. "Trend without hiccups: a Kalman filter approach," Papers 1808.03297, arXiv.org.
    3. William Goetzmann & Jonathan Ingersoll & Matthew I. Spiegel & Ivo Welch, 2002. "Sharpening Sharpe Ratios," NBER Working Papers 9116, National Bureau of Economic Research, Inc.
    4. 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.
    5. Ying Liu & Marie Rekkas & Augustine Wong, 2012. "Inference for the Sharpe Ratio Using a Likelihood-Based Approach," Journal of Probability and Statistics, Hindawi, vol. 2012, pages 1-24, October.
    6. 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.
    7. 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.
    8. John Douglas (J.D.) Opdyke, 2007. "Comparing Sharpe ratios: So where are the p-values?," Journal of Asset Management, Palgrave Macmillan, vol. 8(5), pages 308-336, December.
    9. 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.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    JEL classification: C12; G11 Sharpe ratio; Student distribution; compounding effect on Sharpe; AR(1); Cramer Rao bound;
    All these keywords.

    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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