Journal of Risk

Min-Max robust and CVaR robust mean-variance portfolios

Lei Zhu, Thomas F. Coleman, Yuying Li


This paper investigates robust optimization methods for mean-variance portfolio selection problems under the estimation risk in mean returns. We show that with an ellipsoidal uncertainty set based on the statistics of the sample mean estimates, the portfolio from the min-max robust mean-variance model equals the portfolio from the standard mean-variance model based on the nominal mean estimates but with a larger risk aversion parameter. We demonstrate that the min-max robust portfolios can vary significantly with the initial data used to generate uncertainty sets. In addition, min-max robust portfolios can be too conservative and unable to achieve a high return. Adjustment of the conservatism in the min-max robust model can be achieved only by excluding poor mean-return scenarios, which runs counter to the principle of min-max robustness. We propose a conditional value-at-risk (CVaR) robust portfolio optimization model to address estimation risk.We show that using CVaR to quantify the estimation risk in mean return, the conservatism level of the portfolios can be more naturally adjusted by gradually including better scenarios; the confidence level β can be interpreted as an estimation risk aversion parameter. We compare min-max robust portfolios with an interval uncertainty set and CVaR robust portfolios in terms of actual frontier variation, efficiency and asset diversification. We illustrate that the maximum worst-case mean return portfolio from the min-max robust model typically consists of a single asset, no matter how an interval uncertainty set is selected. In contrast, the maximum CVaR mean return portfolio typically consists of multiple assets. In addition, we illustrate that for the CVaR robust model, the distance between the actual mean-variance frontiers and the true efficient frontier is relatively insensitive for different confidence levels, as well as different sampling techniques.

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