Numerical methods for mean-risk portfolio optimization

Abstract

We consider various aspects of the dynamic mean-risk portfolio optimization problem, with specific focus on the numerical methods to solve this problem in the case where the risk measure is either the variance of wealth outcomes or the quadratic variation of the wealth process. In the case where the risk measure is the variance of wealth outcomes, the mean-risk problem is known as the dynamic mean-variance (MV) portfolio optimization problem. Since variance is not separable in the sense of dynamic programming, the MV objective violates Bellman's principle of optimality. As a result, the possibility of the time-inconsistency of the associated MV-optimal investment strategies leads to a number of competing approaches to formulating and solving the dynamic MV problem. We primarily focus on the two main approaches, namely the time-consistent MV (TCMV) and the pre-commitment MV (PCMV) optimization problems. In addition, this investigation also leads us to consider related approaches which are not necessarily MV optimal in some sense, such as the Mean-Quadratic Variation (MQV) portfolio optimization problem.