Lifetime Portfolio Selection in Continuous Time for a Multiplicative Class of Utility Functions

In a recent paper, Richard Meyer has studied the lifetime portfolio problem in continuous time for a multiplicative class of utility functions. Though obtaining a number of general characteristics of the optimal policy, he is able to obtain an analytic solution only for a special limiting case that corresponds to the additive family. The solution for this additive family is known from the independent work of Robert Merton, who, for continuous time, attacked this case directlv. The continuous time case requires the assumption that the distribution of returns follow an infinitely divisible normal process. The discrete time case does niot require this assumnption. The additive family in discrete time has been studied by Edmund Phelps, Nils Hakansson, David Levhari and T. N. Srinivasan, and Paul Samuelson. The purpose of this note is to point out that an analytic solution can be obtained for a subclass of the multiplicative family studied by Meyer. For discrete but not continuous time, I have given this solution in my 1972 paper. This solution differs significantly from that for the additive case in either discrete or continuous time. In the additive case, for a stationary distribution of returns, the proportion of wealth invested in risky securities doesn't change with age. This is not true for the solution in the multiplicative case. The proportion of wealth invested in risky securities increases or decreases with age as risk aversion is greater or less than that of the logarithmic utility function. The measure of risk aversion is the index of relative risk aversion developed by Kenneth Arrow and by John Pratt. If the logarithm may be thought of as the dividing line between optimists and pessimists, this result may be interpreted as follows. Optimists tend to gamble less as they grow older as they have less to gain, whereas pessimists gamble more as they have less to lose. Unlike the additive case, risk aversion in portfolio selection also depends on impatience in the multiplicative solution. Risk aversion in portfolio selection increases with impatience for optimists, while the reverse holds for pessimists. As the propensity to consume also depends on impatience, this means risk aversion and the propensity to consume will be correlated through their mutual dependence on impatience. For optimists, the correlation will be positive, whereas for the pessimists, it will be negative. Consider an investor at time t with a remaining lifespan of T-t. Following Mever, the utility of his remaining consumption stream will be taken to be the following: