Paper: arXiv 2301.04118

Authors: Yuqi Li, Lihua Zhang

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}.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using ODEs and SDEs to solve stochastic control problems in personal financial planning and insurance purchasing. However, it is purely theoretical with minimal empirical validation or backtesting using real market data.

Research Flowchart

  flowchart TD
    A[Goal: Maximize Probability of Reaching Financial/Bequest Goal via Deferred Term Insurance] --> B[Inputs: Income, Consumption, Risky Investment, Death & Longevity Risks]
    B --> C[Frameworks: Deterministic & Stochastic Systems with Time Cutoffs m, n]
    C --> D[Methodology: Establish 'Quasi-Ideal Value' & 'Ideal Value' Controls to Solve ODEs/SDEs]
    D --> E[Outcomes: Closed-Form Expressions for Max Probability, Optimal Insurance & Investment Strategies]
    E --> F[Key Insight: Deferred Term Insurance is Optimal for Non-Financially Sound Individuals]