Strictly monotone mean-variance preferences with applications to portfolio selection

The monotone mean-variance (MMV) preference proposed by Maccheroni, et al. (Math. Finance 19(3): 487-521, 2009) fails to differentiate strictly dominant payoffs, which may cause inconsistency in portfolio decision-making. This paper introduces a broader class of strictly monotone mean-variance (SMMV) preferences and demonstrates its applications to portfolio selection problems. For the single-period portfolio problem under the SMMV preference, we derive the gradient condition for the optimal strategy, and investigate its association with the optimal mean-variance (MV) static strategy. We reduce the problem to solving a set of linear equations by analyzing the saddle point of some minimax problem. And results show that the optimal SMMV, MMV and MV strategies differ significantly in the single-period problem. Furthermore, we conduct numerical experiments and compare our results with those of Maccheroni, et al. (Math. Finance 19(3): 487-521, 2009). The findings indicate that our SMMV preferences provide a more rational basis for assessing given prospects. For the continuous-time portfolio problem under the SMMV preference, we consider continuous price processes with random coefficients, and establish a novel approach based on a general convex duality analysis to derive the optimal strategy. Interestingly, we find that the optimal strategies for SMMV, MMV and MV preferences coincide under a certain condition, and provide a classical microeconomic interpretation for this condition. We also characterize the optimal SMMV portfolio strategies relying on stochastic control techniques to facilitate potential extensions and refinements in future research.