Consumption and Portfolio Rules for Time-Inconsistent Investors

TL;DR

This paper extends classical continuous-time consumption and portfolio models to include time-inconsistent preferences, deriving solutions for naive and sophisticated agents and highlighting how utility functions influence optimal strategies.

Contribution

It introduces a modified HJB equation to solve for sophisticated agents and compares outcomes across different utility functions within a unified framework.

Findings

Optimal portfolio rules for CRRA within HARA are independent of discount rate.

Exponential utility leads to different portfolio behavior compared to CRRA.

Sophisticated agents' strategies differ from naive agents due to time inconsistency.

Abstract

This paper extends the classical consumption and portfolio rules model in continuous time (Merton 1969, 1971) to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton-Jacobi-Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and potential cases) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.