Achieving a Given Financial Goal with Optimal Deferred Term Insurance Purchasing Policy

TL;DR

This paper develops optimal strategies for purchasing deferred term insurance to maximize the probability of achieving specific financial goals, considering both deterministic and stochastic financial environments with income, consumption, and risky investments.

Contribution

It introduces new control methods and solves complex differential equations to determine optimal insurance and investment strategies in a stochastic framework with time cutoffs.

Findings

Optimal strategies depend on time cutoffs m and n.

Deferred term insurance is preferable for those with limited financial resources.

Results extend previous models to include stochastic income and investment.

Abstract

This paper researches the problem of purchasing deferred term insurance in the context of financial planning to maximize the probability of achieving a personal financial goal. Specifically, our study starts from the perspective of hedging death risk and longevity risk, and considers the purchase of deferred term life insurance and deferred term pure endowment to achieve a given financial goal for the first time in both deterministic and stochastic framework. In particular, we consider income, consumption and risky investment in the stochastic framework, extending previous results in \cite{Bayraktar2016}. The time cutoff m and n make the work more difficult. However, by establishing new controls,``\emph{quasi-ideal value}" and``\emph{ideal value}", we solve the corresponding ordinary differential equations or stochastic differential equations, and give the specific expressions for the maximum probability. Then we provide the optimal life insurance purchasing strategies and the optimal risk investment strategies. In general, when m \geqslant 0, n>0, deferred term insurance or term life insurance is a better choice for those who want to achieve their financial or bequest goals but are not financially sound. In particular, if m >0, n \rightarrow \infty, our viewpoint also sheds light on reaching a bequest goal by purchasing deferred whole life insurance. It is worth noting that when m=0, n \rightarrow \infty, our problem is equivalent to achieving the just mentioned bequest goal by purchasing whole life insurance, at which point the maximum probability and the life insurance purchasing strategies we provide are consistent with those in \cite{Bayraktar2014, Bayraktar2016}.