Abstract
We investigate the mean-quadratic variation (MQV) portfolio optimization problem and its relationship to the time-consistent mean-variance (TCMV) portfolio optimization problem. In the case of jumps in the risky asset process and no investment constraints, we derive analytical solutions for the TCMV and MQV problems. We study conditions under which the two problems are (i) identical with respect to MV trade-offs, and (ii) equivalent, i.e., have the same value function and optimal control. We provide a rigorous and intuitive explanation of the abstract equivalence result between the TCMV and MQV problems developed in [T. Bjork and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, working paper, 2010] for continuous rebalancing and no-jumps in risky asset processes. We extend this equivalence result to jump-diffusion processes (both discrete and continuous rebalancings). In order to compare the MQV and TCMV problems in a more realistic setting which involves investment constraints and modeling assumptions for which analytical solutions are not known to exist, using an impulse control approach we develop an efficient partial integro-differential equation (PIDE) method for determining the optimal control for the MQV problem. We also prove convergence of the proposed numerical method to the viscosity solution of the corresponding PIDE. We find that the MQV investor achieves essentially the same results concerning terminal wealth as the TCMV investor, but the MQV-optimal investment process has more desirable risk characteristics from the perspective of long-term investors with fixed investment time horizons. As a result, we conclude that MQV portfolio optimization is a potentially desirable alternative to TCMV.
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