Optimal consumption under loss-averse multiplicative habit-formation preferences

ArXiv ID: 2406.20063 “View on arXiv”

Authors: Unknown

Abstract

This paper studies a loss-averse version of the multiplicative habit formation preference and the corresponding optimal investment and consumption strategies over an infinite horizon. The agent’s consumption preference is depicted by a general S-shaped utility function of her consumption-to-habit ratio. By considering the concave envelope of the S-shaped utility and the associated dual value function, we provide a thorough analysis of the HJB equation for the concavified problem via studying a related nonlinear free boundary problem. Based on established properties of the solution to this free boundary problem, we obtain the optimal consumption and investment policies in feedback form. Some new and technical verification arguments are developed to cope with generality of the utility function. The equivalence between the original problem and the concavified problem readily follows from the structure of the feedback controls. We also discuss some quantitative properties of the optimal policies, complemented by illustrative numerical examples and their financial implications.

Keywords: Loss-Averse Utility, Multiplicative Habit Formation, HJB Equation, Optimal Investment, Consumption Strategy

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mathematical tools like Hamilton-Jacobi-Bellman equations, nonlinear free boundary problems, and concave envelope analysis, indicating high mathematical density. However, it lacks any empirical backtesting, datasets, or implementation details, focusing entirely on theoretical derivation and numerical examples for illustration rather than real-world application.

  flowchart TD
    A["Research Goal: Find optimal investment and consumption strategy under multiplicative habit formation with loss-averse, S-shaped utility"] --> B["Methodology: Reformulate via the concave envelope of utility and dual value function"]
    B --> C["Methodology: Solve the HJB equation via a nonlinear free boundary problem"]
    C --> D["Computational Process: Establish solution properties and feedback control policies"]
    D --> E["Verification: Prove equivalence between original and concavified problems"]
    E --> F["Outcome: Obtain optimal consumption and investment policies in feedback form"]
    F --> G["Outcome: Demonstrate quantitative properties and financial implications via numerical examples"]