On the solution uniqueness in portfolio optimization and risk analysis

We consider the issue of solution uniqueness of the mean-deviation portfolio optimization problem and its inverse for asset returns distributed over a finite number of scenarios. Due to the asymmetry of returns, the risk is assessed by a general deviation measure introduced by [Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515–540] instead of the standard deviation as in the classical Markowitz optimization problem. We demonstrate that, in general, one cannot expect the uniqueness of Pareto-optimal profit sharing in cooperative investment and the uniqueness of solutions in the mean-deviation Black-Litterman asset allocation model. For a large class of deviation measures, we provide a resolution of the above non-uniqueness issues based on the principle of law-invariance. We provide several examples illustrating the non-uniqueness and the law-invariant solution.