Dynamic mean-variance problem: recovering time-consistency

Abstract

As the foundation of modern portfolio theory, Markowitz’s mean-variance port-folio optimization problem is one of the fundamental problems of financial math-ematics. The dynamic version of this problem in which a positive linear com-bination of the mean and variance objectives is minimized is known to be time-inconsistent, hence the classical dynamic programming approach is not applicable. Following the dynamic utility approach in the literature, we consider a less re-strictive notion of time-consistency, where the weights of the mean and variance are allowed to change over time. Precisely speaking, rather than considering a fixed weight vector throughout the investment period, we consider an adapted weight process. Initially, we start by extending the well-known equivalence be-tween the dynamic mean-variance and the dynamic mean-second moment prob-lems in a general setting. Thereby, we utilize this equivalence to give a complete characterization of a time-consistent weight process, that is, a weight process which recovers the time-consistency of the mean-variance problem according to our definition. We formulate the mean-second moment problem as a biobjective optimization problem and develop a set-valued dynamic programming principle for the biobjective setup. Finally, we retrieve back to the dynamic mean-variance problem under the equivalence results that we establish and propose a backward-forward dynamic programming scheme by the methods of vector optimization. Consequently, we compute both the associated time-consistent weight process and the optimal solutions of the dynamic mean-variance problem.