Optimal Life Insurance Purchase, Consumption and Investment on a financial market with multi-dimensional diffusive terms

TL;DR

This paper extends Merton's model to include a wage earner with a random lifetime, allowing for optimal consumption, investment, and life insurance purchase strategies in a multi-asset financial market, with explicit solutions for certain utility functions.

Contribution

It introduces a comprehensive model incorporating life insurance, random lifetime, and multi-dimensional risky assets, providing explicit solutions and new analytical insights.

Findings

Explicit solutions for optimal strategies under certain utility functions.

Analysis of the impact of life insurance on consumption and investment decisions.

New analytical results with economic interpretations.

Abstract

We introduce an extension to Merton's famous continuous time model of optimal consumption and investment, in the spirit of previous works by Pliska and Ye, to allow for a wage earner to have a random lifetime and to use a portion of the income to purchase life insurance in order to provide for his estate, while investing his savings in a financial market comprised of one risk-free security and an arbitrary number of risky securities driven by multi-dimensional Brownian motion. We then provide a detailed analysis of the optimal consumption, investment, and insurance purchase strategies for the wage earner whose goal is to maximize the expected utility obtained from his family consumption, from the size of the estate in the event of premature death, and from the size of the estate at the time of retirement. We use dynamic programming methods to obtain explicit solutions for the case of discounted constant relative risk aversion utility functions and describe new analytical results which are presented together with the corresponding economic interpretations.