Optimal consumption under loss-averse multiplicative habit-formation preferences

TL;DR

This paper analyzes optimal investment and consumption strategies for a loss-averse agent with multiplicative habit formation preferences, using a concavification approach and solving a free boundary problem to derive feedback policies.

Contribution

It introduces a novel framework for loss-averse habit formation preferences with S-shaped utility, providing explicit feedback strategies through a free boundary problem analysis.

Findings

Derived explicit feedback optimal policies for consumption and investment.

Established the equivalence between original and concavified problems.

Provided numerical examples illustrating the policies and their financial implications.

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.