The fournal of finance VoL. XIX SEPTEMBER 1964 No. 3 CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* WILLIAM F. SHARPET ONE OF THE PROBLEMS which has plagued those attempting to predict the behavior of capital markets is the absence of a body of positive micro economic theory dealing with conditions of risk. Although many useful insights can be obtained from the traditional models of investment under conditions of certainty, the pervasive influence of risk in financial trans. actions has forced those working in this area to adopt models of price behavior which are little more than assertions. a typical classroom ex planation of the determination of capital asset prices, for example, usually begins with a careful and relatively rigorous description of the process through which individual preferences and physical relationships interact to determine an equilibrium pure interest rate. This is generally followed by the assertion that somehow a market risk-premium is also determined, with the prices of assets adjusting accordingly to account for differences in their risk A useful representation of the view of the capital market implied in such discussions is illustrated in Figure 1. In equilibrium, capital asset prices have adjusted so that the investor, if he follows rational procedures (primarily diversification), is able to attain any desired point along a capital market line He may obtain a higher expected rate of return on his holdings only by incurring additional risk. In effect, the market presents him with two prices: the price of time, or the pure interest rate (shown by the intersection of the line with the horizontal axis) and the price of risk, the additional expected return per unit of risk borne (the reciprocal of the slope of the line) s a great many people provided comments oad dearly versions of this Angeles, and to Professors Yoram Barzel, George Brabb, Bruce Johnson, Walter Oi and R. Haney Scott of the University of Washington t Associate Professor of Operations Research, University of Washington 1. Although some discussions are also consistent with a non-linear (but monotonic)curve
426 The Journal of finance At present there is no theory describing the manner in which the pric f risk results from the basic influences of investor preferences, the physi- al attributes of capital assets, etc. Moreover, lacking such a theory, it is difficult to give any real meaning to the relationship between the price of a single asset and its risk. Through diversification, some of the risk inherent in an asset can be avoided so that its total risk is obviously not the relevant influence on its price; unfortunately little has been said concerning the particular risk component which is relevant Capital Market Line Expected Rate of Return Pure Interest Rate FIGURE 1 In the last ten y rears a number of economists have developed normative models dealing with asset choice under conditions of risk. Markowitz, 2 following Von Neumann and Morgenstern, developed an analysis based on the expected utility maxim and proposed a general solution for the portfolio selection problem. Tobin showed that under certain conditions Markowitz's model implies that the process of investment choice can be broken down into two phases: first, the choice of a unique optimum combination of risky assets; and second, a separate choice concerning the allocation of funds between such a combination and a single riskless 2. Harry M. Markowitz, Portfolio Selection, Eficient Diversification of Investmen New York: John Wiley and Inc, 1959). The major elements of the theory firs appeared in his article "Portfolio Selection, "The Journal of Finance, XII (March 1952 James Tobin,"Liquidity Preference as Behavior Towards Risk, "The Review of economic dies, XX ruary,1958),65-86
asset. Recently, Hicks has used a model similar to that proposed by Tobin to derive corresponding conclusions about individual investor behavior, dealing somewhat more explicitly with the nature of the condi tions under which the process of investment choice can be dichotomized An even more detailed discussion of this process, including a rigorous proof in the context of a choice among lotteries has been presented by Gordon and Gangolli. Although all the authors cited use virtually the same model of investor behavior, none has yet attempted to extend it to construct a market equilibrium theory of asset prices under conditions of risk. T We will show that such an extension provides a theory with implications consistent with the assertions of traditional financial theory described above. Moreover, it sheds considerable light on the relationship between the price of asset and the various components of its overall risk. For these reasons it warrants consideration as a model of the determination of capital asset prices. ditions of risk. In Part III the equilibrium conditions for the capital market are considered and the capital market line derived. The implica tions for the relationship between the prices of individual capital assets and the various components of risk are described in Part Iv II. OPTIMAl INVESTMENT POLICY FOR THE INDIVIDUAL The Investor's Preference Function Assume that an individual views the outcome of any investment in probabilistic terms; that is, he thinks of the possible results in terms of some probability distribution. In assessing the desirability of a particular investment, however, he is willing to act on the basis of only two para 4. John R, Hicks, "Liquidity, The Economic Journal, lxXII (December, 1962),787 s.M. J. Gordon and Ramesh gangolli,"Choice Among and Scale of Play on Lottery Alternatives, " College of Business Administration, University of roch 1962 For another discussion of this relationship see W. F. Sharpe,"A Simplified Model for Portfolio Analysis, Management Science, Vol. 9, No. 2 (January 1963),277-293.A xuvorat oin inass an the Theory of Investment, "The 4merican Economic Review, 6. Recently Hirshleifer has suggested that the mean-variance approach used in the as a special case of a more general formulation due to Arrow. See Hirshleifer's "Investment Decision Under Uncertainty, "Papers and Proceedings eventy-Sisth Annual Meeting of the Ame ican Economic or Arrow's "Le Role des Valeurs Boursiers pour la Repartition la Meilleure des Risques, International Colloquium on Econometrics, 1952 A 7. After preparing this paper the author learned that Mr. Jack L. described here. Unfortunately Mr. Treynor's excellent work on thi unpublished
428 The Journal of finance meters of this distribution--its expected value and standard deviation This can be represented by a total utility function of the form where Ew indicates expected future wealth and ow the predicted standard deviation of the possible divergence of actual future wealth from Ew Investors are assumed to prefer a higher expected future wealth to a lower value, ceteris paribus(dU/dEw>0). Moreover, they exhibit risk-aversion, choosing an investment offering a lower value of wto one with a greater level, given the level of Ew(dU/dow<O) sumptions imply that indifference curves relating Ew and rard-slopin To simplify the analysis, we assume that an investor has decided to commit a given amount(Wi)of his present wealth to investment. Letting Wt be his terminal wealth and R the rate of return on his investment R Wt=R Wi+ Wi This relationship makes it possible to express the investor's utility in terms of R, since terminal wealth is directly related to the rate of return U Figure 2 summarizes the model of investor preferences in a family of indifference curves; successive curves indicate higher levels of utility as one moves down and/or to the right. 8. Under certain conditions the mean-vaniance unsatisfactory predictions of behavior. Markowitz semi-variance(the average of the squared deviations be preferable; in light of the formidable computational problems, variance and standard deviation 9. While only these characteristics are required for the analysis, it is generally assumed that the curves have the property of diminishing marginal rates of substitution between e and o, as do those in our diagrams. 10. Such indifference to maximize expected utility and that his total utility ed by a Inction of R with decreasing marginal utility. Both Markowit pre a derivation. A similar approach is used by Donald E. Farrar in The lnvestment Inder Uncertainty ( Prentice-Hall, 1962). Unfortunately Farrar makes an error derivation; he appeals to the von- Neumann-Morgenstern cardinal utility axioms to trans- form a function of the form E(U=a+ bER -cEr2-ca22 into one of the form E(U)=k1EB一k2012 That such a transformation is not consistent with the axioms can readily be seen in this ves in the er, on2 pla no lie on both a line and a non-linear curve (with a monotonic derivative). Thus the two functions must imply different orderings among alternative choices in at least some instance
Capital Asset Prices 429 FIGURE 2 The Investment Opportunity Curve The model of investor behavior considers the investor as choosing from a set of investment opportunities that one which maximizes his utility Every investment plan available to him may be represented by a point in the ER, On plane. If all such plans involve some risk, the ar of such points will have an appearance similar to that shown in Figure 2 The investor will choose from among all possible plans the one placing him on the indifference curve representing the highest level of utility (point F). The decision can be made in two stages: first, find the set of efficient investment plans and, second choose one from among this set. a plan is said to be efficient if (and only if) there is no alternative with either(1)the same En and a lower Or,(2)the same on and a higher E or(3)a higher ER and a lower oR. Thus investment z is inefficient since investments B,C, and D(among others)dominate it. The only plans which would be chosen must lie along the lower right-hand boundary AFBDCX)-the investment opportunity curve To understand the nature of this curve, consider two investment plans -A and B, each including one or more assets. Their predicted expected values and standard deviations of rate of return are shown in Figure 3
The Journal of finance If the proportion a of the individual's wealth is placed in plan A and the remainder(1-a)in B, the expected rate of return of the combination will lie between the expected returns of the two plar ERe aERa+(1-a) Erb The predicted standard deviation of return of the combination is: √a2oR2+(1-a)2m2+ note that this relationship includes rab, the correlation coefficient between the predicted rates of return of the two investment plans. a value of +1 would indicate an investor's belief that there is a precise positive relation ship between the outcomes of the two investments. A zero value would indicate a belief that the outcomes of the two investments are completely independent and-1 that the investor feels that there is a precise inverse relationship between them. In the usual case rab will have a value between 0 and +1. Figure 3 shows the possible values of Ere and aRe obtainable with different combinations of a and B under two different assumptions about Rb FIGURE 3
Capital Asset Price the value of rab. If the two investments are perfectly correlated, the combinations will lie along a straight line between the two points, since in this case both ere and oRe will be linearly related to the proportions invested in the two plans. 1 If they are less than perfectly positively cor- elated, the standard deviation of any combination must be less than that obtained with perfect correlation (since rab will be less); thus the combi- nations must lie along a curve below the line aB. azb shows such a curve for the case of complete independence (rab=0); with negative correlation the locus is even more U-shapec The manner in which the investment opportunity curve is formed is relatively simple conceptually, although exact solutions are usually quite difficult. 14 One first traces curves indicating Er, or values available with imple combinations of individual assets, then considers combinations of combinations of assets. The lower right-hand boundary must be either linear or increasing at an increasing rate(d2 or/dER>0). As suggested earlier, the complexity of the relationship between the characteristics of individual assets and the location of the investment opportunity curve makes it difficult to provide a simple rule for assessing the desirability of individual assets, since the effect of an asset on an investor s over-all investment opportunity curve depends not only on its expected rate of return (Eri)and risk (ori), but also on its correlations with the other by the equilibrium conditions for the model, as we will show in parteo available opportunities (ru, I2,...., rin). However, such a rule is implie The Pure rate of interest We have not yet dealt with riskless assets let p be such an asset; its risk is zero (ORp=0) and its expected rate of return, ERp, is equal(by definition) to the pure interest rate. If an investor places a of his wealth Ere =aERa t(1-a)er.= ERb+ ER,a but rab =1, therefore the expression under the square root sign can be factored qe=√/[aa+(1-a)ox2 a+(1一a)ckb = ORb+(oBa-oRb)a 12. This curvature is, in es 13. When rab =0, the slope of the curve at point a is Erh-E., at point B it is - When Iab=-1, the curve degenerates to two straight lines to 14. Markowitz has shown that this is a problem in parametric quadratic programmin (March and June, 1956), 111-133. A solution method for a special case is given in the author's"A Simplified Model for Portfolio Analysis, "op cit