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Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

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Abstract

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction.

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Notes

  1. While our proof is similar to the proof in Chen et al. (2011) in essence, their proof is faulty because their argument for exchanging max and min relies on a result of Zhu and Fukushima (2009) which is valid for discrete distributions. Our setting here, like that of Chen et al. (2011) is not confined to discrete distributions. Hence, a different justification is needed for exchanging max and min.

  2. It is a simple exercise to show that in the absence of the minimum mean return constraint the portfolio position in the ith risky asset tends to \(\pm \infty\) depending on the sign of the ith component of \(\tilde{\Upgamma}^{-1} \tilde{\mu}. \)

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Acknowledgments

The revised version of the paper benefited greatly from the comments of an anonymous referee.

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Authors and Affiliations

  1. Deparment of Industrial Engineering, Bilkent University, 06800, Ankara, Turkey

    A. Burak Paç & Mustafa Ç. Pınar

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  1. A. Burak Paç
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  2. Mustafa Ç. Pınar
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Correspondence to Mustafa Ç. Pınar.

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Paç, A.B., Pınar, M.Ç. Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity. TOP 22, 875–891 (2014). https://doi.org/10.1007/s11750-013-0303-y

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