Utility-Deviation-Risk Portfolio Selection

We here provide a comprehensive study of the utility-deviation-risk portfolio selectionproblem. By considering the first-order condition for the corresponding objective function, we firstderive the necessary condition that the optimal terminal wealth satisfying two mild regularity con-ditions solves for a primitive static problem, called Nonlinear Moment Problem. We then illustratethe application of this general necessity result by revisiting the non-existence of the optimal solutionfor the mean-semivariance problem. Secondly, we establish an alternative version of the verificationtheorem serving as the sufficient condition that the solution, which satisfies another mild conditiondifferent from that for necessity, of the Nonlinear Moment Problem is the optimal terminal wealth ofthe original utility-deviation-risk portfolio selection problem. We then apply this general sufficiencyresult to revisit the various well-posed mean-risk problems already known in the literature, and toalso establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems under the assumption that the underlying utility satisfies the Inada Condition.To the best of our knowledge, the positive answers to the latter two problems have long been absentin the literature. In particular, the existence result in the utility-downside-risk problem is in contrastto the well-known non-existence of an optimal solution for the mean-downside-risk problem. As acorollary, the existence result in utility-semivariance problem allows us to utilize the semivariance asa proper risk measure in the classical portfolio management paradigm.