A New Cone Programming Approach for Robust Portfolio Selection

The robust portfolio selection problems have recently been studied by several researchers (e.g., see [15, 14, 17, 25]). In their work, the “separable” uncertainty sets of the problem parameters (e.g., mean and covariance of the random returns) were considered. These uncertainty sets share two common drawbacks: i) the actual confidence level of the uncertainty set is unknown, and it can be much higher than the desired one; and ii) the uncertainty set is fully or partially box-type. The consequence of these drawbacks is that the resulting robust portfolio can be too conservative and moreover, it is usually highly non-diversified as observed in computational experiments. To combat these drawbacks, we consider a factor model for the random asset returns. For this model, we introduce a “joint” ellipsoidal uncertainty set for the model parameters and show that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters for any desired confidence level. We further show that the robust maximum risk-adjusted return problem with this uncertainty set can be reformulated and solved as a cone programming problem. Some computational experiments are performed to compare the performances of the robust portfolios corresponding to our “joint” uncertainty set and Goldfarb and Iyengar’s “separable” uncertainty set [15]. We observe that our robust portfolio has much better performance than Goldfarb and Iyengar’s in terms of wealth growth rate and transaction cost, and moreover, our robust portfolio is fairly diversified, but Goldfarb and Iyengar’s is highly non-diversified.