Portfolio optimization with two quasiconvex risk measures

TL;DR

This paper extends portfolio optimization models to include two quasiconvex risk measures, analyzing convex and non-convex cases, and introduces duality and approximation methods for solving these problems.

Contribution

It generalizes previous models by allowing quasiconvex risk measures and develops duality and approximation techniques for the resulting optimization problems.

Findings

Zero duality gap under convex risk measures

Optimal portfolios identified via Lagrange multipliers

Approximate solutions achievable with bisection algorithm

Abstract

We study a static portfolio optimization problem with two risk measures: a principle risk measure in the objective function and a secondary risk measure whose value is controlled in the constraints. This problem is of interest when it is necessary to consider the risk preferences of two parties, such as a portfolio manager and a regulator, at the same 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 by assuming that the two risk measures are only quasiconvex. First, we study the case where the principal risk measure is convex. We introduce a dual problem, show that there is zero duality gap between the portfolio optimization problem and the dual problem, and finally identify a condition under which the Lagrange multiplier associated to the dual problem at optimality gives an optimal portfolio. Next, we study the general case without the convexity assumption and show that an approximately optimal solution with prescribed optimality gap can be achieved by using the well-known bisection algorithm combined with a duality result that we prove.