Optimum Consumption and Portfolio Rules in a Continuous-Time Model

Abstract

A common hypothesis about the behavior of limited liability asset prices in perfect markets is the random walk of returns or in its continuous-time form the geometric Brownian motion hypothesis, which implies that asset prices are stationary and log-normally distributed. A number of investigators of the behavior of stock and commodity prices have questioned the accuracy of the hypothesis. In an earlier study described in the chapter, it was examined that the continuous-time consumption-portfolio problem for an individual whose income is generated by capital gains on investments in assets with prices assumed to satisfy the “geometric Brownian motion” hypothesis. Under the additional assumption of a constant relative or constant absolute risk-aversion utility function, explicit solutions for the optimal consumption and portfolio rules were derived. The changes in these optimal rules with respect to shifts in various parameters such as expected return, interest rates, and risk were examined by the technique of comparative statics. This chapter presents an extension of these results for more general utility functions, price behavior assumptions, and income generated also from noncapital gains sources. If the geometric Brownian motion hypothesis is accepted, then a general separation or mutual fund theorem can be proved such that, in this model, the classical Tobin mean-variance rules hold without the objectionable assumptions of quadratic utility or of normality of distributions for prices. Hence, when asset prices are generated by a geometric Brownian motion, the two-asset case can be worked on without loss of generality.