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Strictly monotone mean-variance preferences with applications to portfolio selection

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.

Strictly monotone mean-variance preferences with applications to portfolio selection

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.

Paper Structure

This paper contains 34 sections, 18 theorems, 199 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Strictly monotone mean-variance preferences
  3. MMV preference revisited: lack of strict monotonicity
  4. SMMV preferences
  5. Properties and equivalent expressions of SMMV preference
  6. Single-period static portfolio selection
  7. Model and problem formulation
  8. Properties of solution
  9. Some examples
  10. Continuous-time dynamic portfolio management
  11. Model and problem formulation
  12. Microeconomics analysis for Assumption 4.1: single-period binomial framework
  13. Characterizations for the solution
  14. BSPDE characterization
  15. Duality characterization
  16. ...and 19 more sections

Key Result

Theorem 2.4

${V}_{ \theta, \zeta } (g) \le {V}_{ \theta, \zeta } (f) - \mathbb{E} [ ( f - g ) \zeta ]$ for $\zeta \in \mathbb{L}^{2}_{0} ( \Omega )$ and any $f,g \in \mathbb{L}^{2} ( \Omega )$ with $g \le f$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Comparison of MV, MMV, and SMMV preferences: ${U}_{2} ( {X}_{x} )$, ${V}_{2} ( {X}_{x} )$ and ${V}_{ 2, \frac{1}{2} } ( {X}_{x} )$ based on the setting described in \ref{['exp: illustrate MV MMV SMMV']}.

Theorems & Definitions (27)

  • Example 2.1
  • Example 2.2
  • Definition 2.3: SMMV preferences
  • Theorem 2.4
  • Remark 2.5: reasons for $\zeta \in \mathbb{L}^{2}_{0} ( \Omega )$
  • Example 2.6
  • Example 2.7
  • Lemma 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 17 more