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Robust portfolio optimization with Value-at-Risk-adjusted Sharpe ratios

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Abstract

We propose a robust portfolio optimization approach based on Value-at-Risk (VaR)-adjusted Sharpe ratios. Traditional Sharpe ratio estimates using a limited series of historical returns are subject to estimation errors. Portfolio optimization based on traditional Sharpe ratios ignores this uncertainty and, as a result, is not robust. In this article, we propose a robust portfolio optimization model that selects the portfolio with the largest worse-case-scenario Sharpe ratio within a given confidence interval. We show that this framework is equivalent to maximizing the Sharpe ratio reduced by a quantity proportional to the standard deviation in the Sharpe ratio estimator. We highlight the relationship between the VaR-adjusted Sharpe ratios and other modified Sharpe ratios proposed in the literature. In addition, we present both numerical and empirical results comparing optimal portfolios generated by the approach advocated here with those generated by both traditional and alternative optimization approaches.

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Notes

  1. Although the underlying return distribution is not normal, the distribution of Sharpe ratio estimators follows an asymptotically normal distribution.

  2. If there exists short-selling constraints on the n securities, l w is 0 and u w is 1. Relaxing such constraints allows for l w <0 and u w >1.

  3. VaRSR serves as a risk-adjusted Sharpe ratio. The level of VaRSR is strictly less than the Sharpe ratio. Occasionally, we may observe that VaRSR is negative while the Sharpe ratio is positive, especially when the risk-adjustment component is large.

  4. The accuracy of the Sharpe ratio standard deviation estimator increases as the sample size increases. Empirical studies show that the normality result of the Central Limit Theorem is generally a good approximation for sample sizes greater than 30 (Hogg and Tanis, 2009). For a more in-depth discussion concerning the convergence in the limit of large sample sizes, see Greene (2002).

  5. For more details please refer to the MATLAB function pearsrnd.

  6. We do not explicitly convert our model into a second-order cone program as many papers in robust optimization do because our inner minimization problem is relatively simple to solve even without this conversion.

  7. For a given value of γ, the optimal portfolio is the point on the Sharpe ratio efficient frontier with derivative equal to γ. As a result, varying the parameter γ and determining the optimal portfolio will provide the set of portfolios that comprise the Sharpe ratio efficient frontier.

  8. For details, please see http://www.hedgeindex.com/. As a proxy for the risk-free rate of interest, we use 1-month LIBOR rates.

  9. In the section ‘Simulation results’, we assumed the assets are uncorrelated. In this section, we implicitly use the historically accurate correlations between the various hedge fund indexes.

  10. It is, in principle, possible that the optimization procedure advocated here would have non-zero weight on an asset with a negative mean excess return if such an allocation would have diversification benefits to sufficiently alter the return distribution to increase the VaRSR.

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Correspondence to Geng Deng.

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Deng, G., Dulaney, T., McCann, C. et al. Robust portfolio optimization with Value-at-Risk-adjusted Sharpe ratios. J Asset Manag 14, 293–305 (2013). https://doi.org/10.1057/jam.2013.21

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  • DOI: https://doi.org/10.1057/jam.2013.21

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