Tight bounds for some risk measures, with applications to robust portfolio selection

In this paper we develop tight bounds on the expected values of several risk measures that are of interest to us. This work is motivated by the robust optimization models arising from portfolio selection problems. Indeed, the whole paper is centered around robust portfolio models and solutions. The basic setting is to find a portfolio that maximizes (respectively, minimizes) the expected utility (respectively, disutility) values in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations. First, we show that the single-stage portfolio selection problem within this framework, whenever the disutility function is in the form of lower partial moments (LPM), or conditional value-at-risk (CVaR), or value-at-risk (VaR), can be solved analytically. The results lead to the solutions for single-stage robust portfolio selection models. Furthermore, the results also lead to a multistage adjustable robust optimization (ARO) solution when the disutility function is the second-order LPM. Exploring beyond the confines of convex optimization, we also consider the so-called S-shaped value function, which plays a key role in the prospect theory of Kahneman and Tversky. The nonrobust version of the problem is shown to be NP-hard in general. However, we present an efficient procedure for solving the robust counterpart of the same portfolio selection problem. In this particular case, the consideration of the robustness actually helps to reduce the computational complexity. Finally, we consider the situation whereby we have some additional information about the chance that a quadratic function of the random distribution reaches a certain threshold. That information helps to further reduce the ambiguity in the robust model. We show that the robust optimization problem in that case can be solved by means of semidefinite programming (SDP), if no more than two additional chance inequalities are to be incorporated.