An Overview of Optimal Life Insurance Purchase, Consumption and Investment Problems

We provide 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 consisting of one risk-free security and an arbitrary number of risky securities whose diffusive terms are driven by a multi-dimensional Brownian motion. The wage earner's problem is to find the optimal consumption, investment, and insurance purchase decisions in order to maximize expected utility of consumption and of the size of the estate in the event of premature death, and of the size of the estate at the time of retirement. Dynamic programming methods are used to obtain explicit solutions for the case of constant relative risk aversion utility functions, and new results are presented together with the corresponding economic interpretations.