Profit as the Risk-Taker's Surplus: A Probabilistic Theory

Abstract

7ROM among various recent contributions to the problem of risky investment decisions and portfolio diversification, the work of Harry Markowitz and that of James Tobin deserve particular attention.2 Both these authors interpret the as an individual who is attracted by certain characteristics of assets, and is repelled by other characteristics. A high mean of the expected frequency distribution of yields is viewed as an attractive feature; high dispersion (say, high variance) about the mean is considered a repelling feature for the investor; 3 and the question of how much of the attractive feature the individual is willing to trade for how much avoidance of the repelling feature is then said to depend on his preference system. At any rate, the with these tastes should abstain from acquiring portfolios (security-mixes) which have smaller yield and greater expected variance than other portfolios. Analytical frameworks of this kind suggest the use of indifference-functions. In particular the individual's gains from deciding how much of various assets to hold express themselves readily in a rise to a higher preference level, since he is essentially comparing objective marginal rates of transformation, which are provided by market opportunities, with subjective marginal rates of substitution between expected yield and avoidance of variance. For large numbers of securities the analysis becomes involved, but the basic principles remain the same and I will not go into further detail here. However, I would like to take my departure from the comments which Markowitz and Tobin have made on the probabilistic background of their analysis. Both authors feel that a set of axioms expressing the principles of operational utility and numerical subjective probability underlies their approach. My own position in this regard may be summarized in the following three points. (i) If we postulate strictly probabilistic behavior that is, if we interpret the decisionmaker as being guided by subjective degrees of belief that are consistent with one another by the specific standards of probability theory and as maximizing his utility-expectations then it is clearly desirable to develop the analysis in terms of alternative surpluses expressed in cardinal utility. In the present article I shall make such an attempt. A high degree of generality may be claimed for the validity of the results of such analysis, as long as one accepts strictly probabilistic basic assumptions. (2) I must, however, say that I do not regard it as generally fruitful to interpret the decisionmaking process in strictly probabilistic terms, say, in terms of L. J. Savage's axioms alone. One reason for this was expressed in articles by Daniel Ellsberg and by myself on an earlier occasion.4 In these two articles the reader will 'I am grateful to my colleague, John W. Hooper, for having read the manuscript of this article and for having made valuable suggestions. 2 Harry M. Markowitz, Journal of Finance, vii (March I952), and Portfolio Selection, Efficient Diversification of Investments, Cowles Foundation Monograph No. i6, New York: John Wiley & Sons, I959; James Tobin, Liquidity Preference as Behavior towards Risk, Review of Economic Studies, xxv (2) (February I958). 'The typical investor in this sense is a person with decreasing marginal utility of wealth. Such an will find a given mathematical expectation of money gains more attractive if it is associated with little dispersion than if the dispersion is great. However, acceptance of the variance as the uniquely relevant measure of dispersion implies specific (additional) constraints. It implies a quadratic aggregate utility function and/or a frequency distribution of expected returns which can be fully described by the mean and the second moment about it. See particularly Tobin, op. cit., and Marcel K. Richter, Cardinal Utility, Portfolio Selection and Taxation, Review of Economic Studies, XXVII (3) (June I960). See also p. I82 ff. These specific constraints on the utility functions and/or on the frequency distributions play a role in theories relating to the desirable degree of whenever the among which the diversifies have different probabilistic properties, but not so if these bets have the identical properties. In the latter case diversification will always diminish dispersion in the relevant sense, and in this latter case diversification will always be desirable to an with monotonically decreasing marginal utility. See section on limits of diversification, i8i if. 4See the symposium on Decisions under Uncertainty in