Mean–Semivariance Optimal Portfolios In Discrete Time Using A Game-Theoretic Approach

This paper introduces a novel recursive scheme for optimal asset allocation based on a mean–semivariance reward functional and a game-theoretic approach in a discrete-time setting. Unlike established frameworks that can handle variance as a risk measure, this study shifts focus to semivariance, which cannot be handled by existing theory due to aspects of its definition, including the use of an indicator function. To address this problem and the corresponding challenges of time inconsistency in multi-period investment decisions, we propose an extended Bellman equation to find a Nash equilibrium. The main contribution of this paper is a computational framework and a numerical investigation of a semivariance-based allocation strategy, based on an extended Bellman equation. Our analysis is restricted to the two-asset case — one risky and one risk-free asset — as a proof of concept, leaving multi-asset extensions for future work. The results of the numerical study indicate that our proposed method shows potential in achieving favorable investment outcomes.