Time-consistent portfolio selection with monotone mean-variance preferences

TL;DR

This paper studies time-inconsistent portfolio optimization under monotone mean-variance preferences, deriving equilibrium strategies via stochastic differential equations and HJB equations, with explicit solutions under certain settings.

Contribution

It introduces a Nash equilibrium framework for MMV preferences, providing semi-closed-form solutions and analyzing conditions for strong equilibrium strategies.

Findings

MMV optimal strategies coincide with MV strategies but are time-inconsistent.

Equilibrium investment exceeds MV investment, with the gap decreasing over time.

Closed-loop equilibrium is strong only when interest rates are high.

Abstract

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.