Strictly monotone mean-variance preferences with applications to portfolio selection

Paper: arXiv 2412.13523

Authors: Yike Wang, Yusha Chen, Jingzhen Liu, Zhenyu Cui

Abstract

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical extension to monotone mean-variance preferences using Fenchel conjugates, convex duality, and stochastic control techniques to address strict statewise dominance. However, empirical rigor is limited primarily to theoretical analysis and simple numerical toy examples rather than extensive real-world backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Resolve MMV preference failure in differentiating strictly dominant payoffs] --> B[Inputs: Single-Period Model & Continuous-Time Price Processes with Random Coefficients]
    B --> C[Methodology: Develop Strictly Monotone Mean-Variance Preferences]
    C --> D[Single-Period Computation: Minimax Saddle Point Analysis & Solving Linear Equations]
    C --> E[Continuous-Time Computation: General Convex Duality & Stochastic Control Techniques]
    D --> F[Key Outcomes: SMMV provides rational prospect assessment; Single-period SMMV, MMV, and MV strategies differ significantly]
    E --> G[Key Outcomes: Derived optimal strategies; Identified conditions where SMMV, MMV, and MV strategies coincide]

Cited References (35)

1. Convex Analysis and Monotone Operator Theory in H ilbert Spaces (2017) — link
2. Geometrical Methods in Variational Problems (1999) — link
3. Semimartingale theory of monotone mean-variance portfolio allocation (2020) — link
4. Dynamically optimal portfolios for monotone mean–variance preferences (2026)
5. Completion of a L 'evy market by power-jump assets (2005) — link
6. On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility (2012) — link
7. Mean-variance theory in complete markets (1982) — link
8. The Mathematics of Arbitrage (2006) — link

1 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.