Yusha Chen | Searcharxiv

arXiv2025

Time-consistent portfolio selection with monotone mean-variance preferences

We investigate time-inconsistent portfolio problems under a broader class of monotone mean-variance (MMV) preferences. Since the optimal strategies for MMV and mean-variance (MV) preferences coincide, the MMV optimal strategies at different initial times are necessarily time-inconsistent. To address this time-inconsistency, we consider Nash equilibrium controls of both open-loop and closed-loop types, and characterize them within a random parameter setting. The two control problems reduce to solving a flow of forward-backward stochastic differential equations and a system of extended Hamilton-Jacobi-Bellman equations, respectively. In particular, we derive semi-closed-form solutions for both types of equilibria under a deterministic parameter setting, and both solutions share the same representation, which is independent of the wealth state and the random path. We show that the investment amount under the MMV equilibrium exceeds that under the MV equilibrium, and the gap narrows over time. Furthermore, under a constant parameter setting, we find that the derived closed-loop Nash equilibrium control is a strong equilibrium strategy only when the interest rate is sufficiently large, whereas the derived open-loop Nash equilibrium control is necessarily a strong equilibrium strategy.

math.OC↗

arXiv2024

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.

q-fin.MF↗