Optimal Life Insurance Purchase, Consumption and Portfolio Under an Uncertain Life

Abstract

In this dissertation, we set up a new quantitative model to investigate the problem of optimal life insurance purchase, consumption and portfolio investment strategies for a wage earner under an uncertain lifetime. We use two approaches, namely, the dynamic programming technique and the martingale technique, to analyze our model. First, by using the dynamic programming technique, we derive HJB equations for our problem and obtain explicit solutions for CRRA utility functions with or without subsistence requirements. Then we study the demand of life insurance by examining our explicit solutions and doing numerical experiments. Since explicit solutions to HJB equations are very rare, we develop a numerical method which is Makov Chain Approximation combining with logarithmic transformations. This numerical method is stable and easy-to-implement for our problem. Second, by using the martingale technique, we set up the mathematical framework for our model in a very general setting. A series of fundamental results are found, for example, the bankruptcy condition, the characteristic of admissible consumption, insurance premium and portfolio strategies, existence of optimality, etc.