Time consistent portfolio strategies for a general utility function

TL;DR

This paper addresses the time inconsistency in portfolio optimization with a general utility function and non-constant discount rates, proposing subgame perfect strategies that align with optimal strategies under certain conditions.

Contribution

It introduces a method to derive time-consistent strategies in a general utility framework with variable discounting, using a fixed point approach for the utility-weighted discount rate.

Findings

Subgame perfect strategies coincide with optimal strategies under asymptotic utility assumptions.

A fixed point iteration effectively computes the utility-weighted discount rate.

Feedback forms for consumption and investment ratios are derived from the value function.

Abstract

We study the Merton portfolio management problem within a complete market, non constant time discount rate and general utility framework. The non constant discount rate introduces time inconsistency which can be solved by introducing sub game perfect strategies. Under some asymptotic assumptions on the utility function, we show that the subgame perfect strategy is the same as the optimal strategy, provided the discount rate is replaced by the utility weighted discount rate $\rho(t,x)$ that depends on the time $t$ and wealth level $x$. A fixed point iteration is used to find $\rho$. The consumption to wealth ratio and the investment to wealth ratio are given in feedback form as functions of the value function.