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The large-sample distribution of the maximum Sharpe ratio with and without short sales

Author

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  • Maller, Ross
  • Roberts, Steven
  • Tourky, Rabee

Abstract

In the Markowitz paradigm the portfolio having maximum Sharpe ratio is optimal. Previously the large sample distribution of this statistic has been calculated when short sales are allowed and sample returns and covariance matrix are asymptotically normally distributed. This paper considers the more complex situation when short sales are not allowed, and provides conditions under which the maximum Sharpe ratio is asymptotically normal. This is not always the case, as we show, in particular when the returns have zero mean. For this situation we obtain upper and lower asymptotic bounds (in distribution) on the possible values of the maximum Sharpe ratio which coincide when the returns are asymptotically uncorrelated. We indicate how the asymptotic theory, developed for the case of no short sales, can be extended to handle a more general class of portfolio constraints defined in terms of convex polytopes. Via simulations we examine the rapidity of approach to the limit distributions under various assumptions.

Suggested Citation

  • Maller, Ross & Roberts, Steven & Tourky, Rabee, 2016. "The large-sample distribution of the maximum Sharpe ratio with and without short sales," Journal of Econometrics, Elsevier, vol. 194(1), pages 138-152.
  • Handle: RePEc:eee:econom:v:194:y:2016:i:1:p:138-152
    DOI: 10.1016/j.jeconom.2016.04.003
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    References listed on IDEAS

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    Cited by:

    1. Hanke, Michael & Penev, Spiridon, 2018. "Comparing large-sample maximum Sharpe ratios and incremental variable testing," European Journal of Operational Research, Elsevier, vol. 265(2), pages 571-579.
    2. Mehmet Caner & Marcelo Medeiros & Gabriel Vasconcelos, 2020. "Residual-Based Nodewise Regression in Factor Models with Ultra-High Dimensions: Analysis of Mean-Variance Portfolio Efficiency and Estimation of Out-of-Sample and Constrained Maximum Sharpe Ratios," Papers 2002.01800, arXiv.org, revised Jun 2021.

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    More about this item

    Keywords

    Optimal portfolio; Maximum Sharpe ratio; Asymptotic distribution; Asymptotic normality; Short sales;
    All these keywords.

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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