Portfolio optimization with two quasiconvex risk measures

Abstract

We study a static portfolio optimization problem with two risk measures: a principle risk measure in theobjective function and a secondary risk measure whose value is controlled in the constraints. This problem is of interestwhen it is necessary to consider the risk preferences of two parties, such as a portfolio manager and a regulator, at thesame time. A special case of this problem where the risk measures are assumed to be coherent (positively homogeneous)is studied recently in a joint work of the author. The present paper extends the analysis to a more general setting byassuming that the two risk measures are only quasiconvex. First, we study the case where the principal risk measure isconvex. We introduce a dual problem, show that there is zero duality gap between the portfolio optimization problem andthe dual problem, and finally identify a condition under which the Lagrange multiplier associated to the dual problemat optimality gives an optimal portfolio. Next, we study the general case without the convexity assumption and showthat an approximately optimal solution with prescribed optimality gap can be found by using the well-known bisectionalgorithm combined with a duality result that we prove.