Journal Article A General Equilibrium Model of the Term Structure under Transaction Costs Get access Gordon Pye Gordon Pye University of California, Berkeley Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 38, Issue 4, October 1971, Pages 447–455, https://doi.org/10.2307/2296689 Published: 01 October 1971

It is widely assumed in portfolio theory that investors are risk-averse expected-utility maximizers. There is a good theoretical reason for assuming expected-utility maximization. Such behavior is well known to be consistent with several quite plausible postulates of rationality [5]. On the other hand, the main empirical foundation for such behavior in portfolio selection appears to be the observation of diversification. Risk-averse, expected-utility maximization implies diversification in portfolio selection, and investors are observed to diversify.

I. Introduction and summary, 226. — II. Valuation, 228. — III. Optimal dividend policy, 229. — IV. An empirical test, 234. — V. Comparative statics, 235. — VI. Capital budgeting, 237. — VII. Policy implications, 241.

Portfolio selection, 563. — Alternative capital gains tax, 567. — Borrowing, 568. — Uncertainty, 570. — Wealth and price effects, 571. — Market equilibrium, 573. — Empirical observations, 577.

Introduction, 60. — A Markov expectations model, 61. — Markov forecasts of the future interest rates, 66. — Observed yield curves, 69.

Following the lead of Markowitz,' the portfolio selection problem has usually taken the following form. An investor's utility for his portfolio is assumed to depend only on the mean and variance of its return over the following period. The investor has in mind means and variances for the returns on all the available securities. The problem is to determine the optimal proportions of these securities in his portfolio. Markowitz, himself, was largely concerned with calculating the dominant set of portfolios (i.e., those with minimum variance for given mean and maximum mean for given variance). Later Tobin 2 studied for two securities, one risky and the other riskless, how the optimal proportion of risky security would change with the mean and variance of its return. He also showed that the mean-variance approach of Markowitz would be consistent with the expected utility principle if the investor had a quadratic utility function or if he considered only a two parameter family of probability distributions. Recently Arrow 3 has reformulated the problem studied by Tobin. He lets the investor decide how to allocate a given amount of wealth between a risky and a riskless security so as to maximize his expected utility. Among other things he shows that the amount invested in the risky security will rise or fall with increasing wealth as minus U/U' (U being the investor's utility function) falls or rises with increasing wealth. In particular, for a quadratic U the amount invested in the risky security will necessarily decrease which is unrealistic. Thus, the mean-variance approach appears unduly restrictive for the portfolio problem, unless restrictions are placed on the form of the probability distribution. The restrictions require that the returns on the securities have a multivariate distribution such that a linear combination of the returns has a two parameter distribution. The multivariate normal is an important example of such a multivariate probability distribution. However, it may not be easy to find others when the returns are not independent. In this paper, the problem of portfolio selection is formulated so that it becomes formally identical to the traditional consumer theory for certain commodities. This permits all of the well-known results of this theory to be applied to the portfolio problem. It also permits the direct use of the traditional models for the price determination of certain commodities for determining the prices of securities. Investors are assumed to follow the expected utility principle and to form their own probability beliefs.

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: