Optimal Investment and Consumption Strategies Under Risk, an Uncertain Lifetime, and Insurance

Abstract

sumption in period j, such that either the risk aversion index -u(x)/u'(x), or the risk aversion index -xu'(x)/u'(x), is a positive constant for all x > 0. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual's horizon was infinite (or known with certainty). In this paper, we consider the same basic model with three modifications. First, we postulate that the individual's, lifetime is a random variable with a known probability distribution. Second, we introduce a utility function intended to represent the individual's bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made, all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only under special conditions. In Section 2, the various components of the decision process are constructed. In the earlier model, the individual's objective was assumed to be the maximization of expected utility from consumption over time. Here, we postulate, more generally, that his objective is to maximize expected utility from consumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position (which may be negative) and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a fair' rate, the individual faces both financial opportunities (borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time-shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the no-easy-money condition. While no limit is placed on borrowing, the individual is required to be solvent at the time of his death with