Optimal Portfolio Choice with Comfortable Consumption

Abstract

In this paper we investigate a Merton-type portfolio optimization problem with a minimum comfortable consumption constraint, utilizing a stochastic control approach. By translating the Hamilton-Jacobi-Bellman (HJB) equations into second-order ordinary differential equations (ODEs) through a novel method, we precisely characterize the set of candidate value functions. We then identify the optimal consumption rate, investment strategy and the value function explicitly by extending the recent stochastic perturbation method presented in Herdegen, Hobson and Jerome (2021). This approach can be applied to derive explicit solutions for other portfolio choice problems under constraints, with detailed studies of the corresponding HJB equations. In addition, we have extended the model when inflation is considered. We also discuss some applications, such as retirement funds, pension funds, endowment portfolios and the AK model for economic growth.

Keywords: Portfolio optimization, Minimum consumption, CRRA utility, Fenchel-Legendre transformation, Stochastic perturbation

JEL Classification: G01, G12, G14, G20