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Time-consistent portfolio selection with strictly monotone mean-variance preference

Yike Wang, Yusha Chen

TL;DR

This work addresses time-inconsistent portfolio optimization under strictly monotone mean-variance (SMMV) preferences by formulating open-loop and closed-loop Nash equilibrium controls (ONEC and CNEC) to capture self-control dynamics across time. It develops a dual characterization: ONEC via a flow of forward-backward SDEs and CNEC via an extended HJB (BSPDE) system, with semi-closed-form solutions in deterministic-parameter settings. Notably, the path-independent ONEC and state-independent CNEC coincide under deterministic parameters, yielding an equilibrium that scales the conventional MV strategy by a factor $eta_t>1$ and is independent of wealth and path. The paper also analyzes strong-equilibrium properties, showing that open-loop equilibria are always strong, while closed-loop equilibria become strong only when the interest rate $r$ is sufficiently large, and it confirms these results through a numerical example illustrating increased equity exposure under SMMV relative to MV/MMV benchmarks.

Abstract

This paper is devoted to time-consistent control problems of portfolio selection with strictly monotone mean-variance preferences. These preferences are variational modifications of the conventional mean-variance preferences, and remain time-inconsistent as in mean-variance optimization problems. To tackle the time-inconsistency, we study the Nash equilibrium controls of both the open-loop type and the closed-loop type, and characterize them within a random parameter setting. The problem is reduced to solving a flow of forward-backward stochastic differential equations for open-loop equilibria, and to solving extended Hamilton-Jacobi-Bellman equations for closed-loop equilibria. In particular, we derive semi-closed-form solutions for these two types of equilibria under a deterministic parameter setting. Both solutions are represented by the same function, which is independent of wealth state and random path. This function can be expressed as the conventional time-consistent mean-variance portfolio strategy multiplied by a factor greater than one. Furthermore, we find that the state-independent closed-loop Nash equilibrium control is a strong equilibrium strategy in a constant parameter setting only when the interest rate is sufficiently large.

Time-consistent portfolio selection with strictly monotone mean-variance preference

TL;DR

This work addresses time-inconsistent portfolio optimization under strictly monotone mean-variance (SMMV) preferences by formulating open-loop and closed-loop Nash equilibrium controls (ONEC and CNEC) to capture self-control dynamics across time. It develops a dual characterization: ONEC via a flow of forward-backward SDEs and CNEC via an extended HJB (BSPDE) system, with semi-closed-form solutions in deterministic-parameter settings. Notably, the path-independent ONEC and state-independent CNEC coincide under deterministic parameters, yielding an equilibrium that scales the conventional MV strategy by a factor and is independent of wealth and path. The paper also analyzes strong-equilibrium properties, showing that open-loop equilibria are always strong, while closed-loop equilibria become strong only when the interest rate is sufficiently large, and it confirms these results through a numerical example illustrating increased equity exposure under SMMV relative to MV/MMV benchmarks.

Abstract

This paper is devoted to time-consistent control problems of portfolio selection with strictly monotone mean-variance preferences. These preferences are variational modifications of the conventional mean-variance preferences, and remain time-inconsistent as in mean-variance optimization problems. To tackle the time-inconsistency, we study the Nash equilibrium controls of both the open-loop type and the closed-loop type, and characterize them within a random parameter setting. The problem is reduced to solving a flow of forward-backward stochastic differential equations for open-loop equilibria, and to solving extended Hamilton-Jacobi-Bellman equations for closed-loop equilibria. In particular, we derive semi-closed-form solutions for these two types of equilibria under a deterministic parameter setting. Both solutions are represented by the same function, which is independent of wealth state and random path. This function can be expressed as the conventional time-consistent mean-variance portfolio strategy multiplied by a factor greater than one. Furthermore, we find that the state-independent closed-loop Nash equilibrium control is a strong equilibrium strategy in a constant parameter setting only when the interest rate is sufficiently large.

Paper Structure

This paper contains 16 sections, 9 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and problem formulation
  3. Characterization for ONEC
  4. Characterization for CNEC
  5. Numerical example
  6. Concluding remark
  7. Proof of lemmas and theorems
  8. Proof of \ref{['lem: Gateaux differentiability of g']}
  9. Proof of \ref{['lem: boundedness of gap']}
  10. Proof of \ref{['lem: estimate for gap']}
  11. Proof of \ref{['thm: verification theorem for ONEC']}
  12. Proof of \ref{['thm: path-independent ONEC']}
  13. Proof of \ref{['thm: verification theorem for CNEC']}
  14. Proof of \ref{['thm: state-independent CNEC']}
  15. Proof of \ref{['thm: identical trivial NECs']}
  16. ...and 1 more sections

Key Result

Lemma 3.1

For any $p,q \in \mathcal{P}_{0}$, where $\nabla g : \mathcal{P}_{0} \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is given by Thus, $g: \mathcal{P}_{0} \to \mathbb{R}$ is continuously Gâteaux differentiable, and $d g ( p, \cdot )$ as the Gâteaux differential of $g$ at $p$ is a continuous linear functional determined by $\nabla g ( p, \cdot, \cdot )$, which satisfies $\int_{ \mathbb{R} \ti

Figures (2)

  • Figure 1: Equilibrium investment amount in the risky asset with different preferences. The parameters are assumed to be deterministic, with $r = 0.012$, $\sigma = 0.158$, $\vartheta = 0.343$, $T = 25$ and $\theta = 5$. $\zeta$ varies from 0 to 0.7.
  • Figure 2: Equilibrium investment amount in the risky asset under the SMMV preference w.r.t. different values of risk premium. The parameters are assumed to be deterministic, with $r = 0.012$, $\sigma = 0.158$, $\zeta = 0.5$, $T = 25$ and $\theta = 5$. The value of the risk premium $\vartheta$ varies from 0.15 to 0.35.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Remark 3.7
  • Theorem 4.1
  • ...and 5 more