Portfolio Time Consistency and Utility Weighted Discount Rates

ArXiv ID: 2402.05113 “View on arXiv”

Authors: Unknown

Abstract

Merton portfolio management problem is studied in this paper within a stochastic volatility, non constant time discount rate, and power utility framework. This problem is time inconsistent and the way out of this predicament is to consider the subgame perfect strategies. The later are characterized through an extended Hamilton Jacobi Bellman (HJB) equation. A fixed point iteration is employed to solve the extended HJB equation. This is done in a two stage approach: in a first step the utility weighted discount rate is introduced and characterized as the fixed point of a certain operator; in the second step the value function is determined through a linear parabolic partial differential equation. Numerical experiments explore the effect of the time discount rate on the subgame perfect and precommitment strategies.

Keywords: Merton Portfolio Problem, Time Inconsistency, Stochastic Volatility, Subgame Perfect Equilibrium, Hamilton-Jacobi-Bellman (HJB) Equation, Asset Allocation

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats
  • Why: The paper employs advanced stochastic control theory, including an extended Hamilton-Jacobi-Bellman (HJB) equation, fixed-point iterations, and linear parabolic PDEs, indicating high mathematical density. However, the empirical rigor is limited as it focuses on theoretical characterization and numerical experiments without backtesting, real data, or implementation-heavy validation.
  flowchart TD
    A["Research Goal<br>Portfolio Time Consistency with<br>Stochastic Volatility & Non-Constant Discount"] --> B["Methodology: Subgame Perfect Equilibrium<br>via Extended HJB Equation"]
    
    B --> C{"Two-Stage Solution Approach"}
    C --> D["Stage 1: Fixed Point Iteration<br>Solve Utility Weighted Discount Rate"]
    C --> E["Stage 2: Linear Parabolic PDE<br>Compute Value Function"]
    
    D & E --> F["Data & Inputs<br>Power Utility, Volatility Process<br>Time Discount Parameters"]
    F --> G["Computational Process<br>Fixed Point Algorithm + Numerical PDE Solver"]
    G --> H["Key Findings/Outcomes<br>Subgame Perfect vs. Precommitment Strategies<br>Impact of Time Discount on Asset Allocation"]