An optimal life insurance policy in the investment-consumption problem in an incomplete market

TL;DR

This paper develops a novel model for optimal life insurance, investment, and consumption in an incomplete market with mortality risk, providing explicit solutions and numerical illustrations for exponential utility functions.

Contribution

It introduces a new model integrating life insurance, investment, and consumption in an incomplete market with mortality risk, with explicit solutions for exponential utilities.

Findings

Explicit solution for exponential utility case.

Numerical experiments illustrating optimal strategies.

Model captures real-life market incompleteness.

Abstract

This paper considers an optimal life insurance for a householder subject to mortality risk. The household receives a wage income continuously, which is terminated by unexpected (premature) loss of earning power or (planned and intended) retirement, whichever happens first. In order to hedge the risk of losing income stream by householder's unpredictable event, the household enters a life insurance contract by paying a premium to an insurance company. The household may also invest their wealth into a financial market. The problem is to determine an optimal insurance/investment/consumption strategy in order to maximize the expected total, discounted utility from consumption and terminal wealth. To reflect a real-life situation better, we consider an incomplete market where the householder cannot trade insurance contracts continuously. To our best knowledge, such a model is new in the insurance and finance literature. The case of exponential utilities is considered in detail to derive an explicit solution. We also provide numerical experiments for that particular case to illustrate our results.