Optimal Life Insurance, Consumption and Investment
We consider the problem of a wage earner who wants to make optimal, lifetime financial planning decisions for his family. With a random lifetime and given a specified income stream, he wants to purchase life insurance to protect his family against his death before retirement. He also invests a portion of his salary in a riskless asset as well as in a number of risky assets, and the balance of his income is consumed. The wage earner's problem is to find the optimal consumption, investment, and insurance purchase decisions in order to maximize the expected utility of (1) consumption, (2) the size of his estate in the event of premature death, and (3) the size of the estate at the time of retirement if he lives that long. With the risky securities modeled as multidimensional geometric Brownian motion, dynamic programming methods are used to obtain explicit solutions in the case of constant relative risk aversion utility functions, and some new results are presented together with the corresponding economic interpretations.