CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK*

Abstract

One of the problems which has plagued those attempting to predict the behavior of capital markets is the absence of a body of positive microeconomic 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 transactions has forced those working in this area to adopt models of price behavior which are little more than assertions. A typical classroom explanation 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.11 Although some discussions are also consistent with a non-linear (but monotonic) curve. 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). At present there is no theory describing the manner in which the price of risk results from the basic influences of investor preferences, the physical 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. In the last ten years a number of economists have developed normative models dealing with asset choice under conditions of risk. Markowltz,22 Harry M. Markowitz, Portfolio Selection, Efficient Diversification of Investments (New York: John Wiley and Sons, Inc., 1959). The major elements of the theory first appeared in his article “Portfolio Selection,” The Journal of Finance, XII (March 1952), 77–91. following Von Neumann and Morgenstern, developed an analysis based on the expected utility maxim and proposed a general solution for the portfolio selection problem. Tobin33 James Tobin, “Liquidity Preference as Behavior Towards Risk,” The Review of Economic Studies, XXV (February, 1958), 65–86. 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 asset. Recently, Hicks44 John R. Hicks, “Liquidity,” The Economic Journal, LXXII (December, 1962), 787–802. 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 conditions 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.55 M. J. Gordon and Ramesh Gangolli, “Choice Among and Scale of Play on Lottery Type Alternatives,” College of Business Administration, University of Rochester, 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 related discussion can be found in F. Modigliani and M. H. Miller, “The Cost of Capital, Corporation Finance, and the Theory of Investment,” The American Economic Review, XLVIII (June 1958), 261–297. Although all the authors cited use virtually the same model of investor behavior,66 Recently Hirshleifer has suggested that the mean-variance approach used in the articles cited is best regarded as a special case of a more general formulation due to Arrow. See Hirshleifer's “Investment Decision Under Uncertainty,” Papers and Proceedings of the Seventy-Sixth Annual Meeting of the American Economic Association, Dec. 1963, or Arrow's “Le Role des Valeurs Boursieres pour la Repartition la Meilleure des Risques,” International Colloquium on Econometrics, 1952. none has yet attempted to extend it to construct a market equilibrium theory of asset prices under conditions of risk.77 After preparing this paper the author learned that Mr. Jack L. Treynor, of Arthur D. Little, Inc., had independently developed a model similar in many respects to the one described here. Unfortunately Mr. Treynor's excellent work on this subject is, at present, unpublished. 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 an 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. Part II provides the model of individual investor behavior under conditions of risk. In Part III the equilibrium conditions for the capital market are considered and the capital market line derived. The implications for the relationship between the prices of individual capital assets and the various components of risk are described in Part IV. Investors are assumed to prefer a higher expected future wealth to a lower value, ceteris paribus ( dU/dE w > 0 ) . Moreover, they exhibit risk-aversion, choosing an investment offering a lower value of σw to one with a greater level, given the level of E w ( dU / d σ w < 0 ) . These assumptions imply that indifference curves relating Ew and σw will be upward-sloping.99 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 Ew and σw, as do those in our diagrams. 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.1010 Such indifference curves can also be derived by assuming that the investor wishes to maximize expected utility and that his total utility can be represented by a quadratic function of R with decreasing marginal utility. Both Markowitz and Tobin present such a derivation. A similar approach is used by Donald E. Farrar in The Investment Decision Under Uncertainty (Prentice-Hall, 1962). Unfortunately Farrar makes an error in his derivation; he appeals to the Von-Neumann-Morgenstern cardinal utility axioms to transform a function of the form: E ( U ) = a + bE R − cE R 2 − c σ R 2 into one of the form: E ( U ) = k 1 E R − k 2 σ R 2 . That such a transformation is not consistent with the axioms can readily be seen in this form, since the first equation implies non-linear indifference curves in the ER, σ R 2 plane while the second implies a linear relationship. Obviously no three (different) points can 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. 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, σR plane. If all such plans involve some risk, the area composed 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 ER and a lower σR, (2) the same σR and a higher ER or (3) a higher ER and a lower σR. 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. 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 relationship 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 ERc and σRc obtainable with different combinations of A and B under two different assumptions about 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 ERc and σRc will be linearly related to the proportions invested in the two plans.1111 E Rc = α E Ra + ( 1 − α ) E R b = E Rb + ( E R a − E R b ) α σ R c = α 2 σ R a 2 + ( 1 − α ) 2 σ Rb 2 + 2 r ab α ( 1 − α ) σ Ra σ Rb but r ab = 1 , therefore the expression under the square root sign can be factored: σ Rc = [ α σ Ra + ( 1 − α ) σ R b ] 2 = α σ Ra + ( 1 − α ) σ Rb = σ Rb + ( σ Ra − σ Rb ) α If they are less than perfectly positively correlated, the standard deviation of any combination must be less than that obtained with perfect correlation (since rab will be less); thus the combinations must lie along a curve below the line AB.1212 This curvature is, in essence, the rationale for diversification. AZB shows such a curve for the case of complete independence ( r ab = 0 ) ; with negative correlation the locus is even more U-shaped.1313 When r ab = 0 , the slope of the curve at point A is − σ Ra E Rb − E Ra , at point B it is σ Rb E Rb − E Ra . When r ab = − 1 , the curve degenerates to two straight lines to a point on the horizontal axis. The manner in which the investment opportunity curve is formed is relatively simple conceptually, although exact solutions are usually quite difficult.1414 Markowitz has shown that this is a problem in parametric quadratic programming. An efficient solution technique is described in his article, “The Optimization of a Quadratic Function Subject to Linear Constraints,” Naval Research Logistics Quarterly, Vol. 3 (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. One first traces curves indicating ER, σR values available with simple 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 ( d 2 σ R / dE 2 R > 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 σRi), but also on its correlations with the other available opportunities (ri1, ri2, …., rin). However, such a rule is implied by the equilibrium conditions for the model, as we will show in part IV. This implies that all combinations involving any risky asset or combination of assets plus the riskless asset must have values of ERc and σRc which lie along a straight line between the points representing the two components. Thus in Figure 4 all combinations of ER and σR lying along the line PA are attainable if some money is loaned at the pure rate and some placed in A. Similarly, by lending at the pure rate and investing in B, combinations along PB can be attained. Of all such possibilities, however, one will dominate: that investment plan lying at the point of the original investment opportunity curve where a ray from point P is tangent to the curve. In Figure 4 all investments lying along the original curve from X to ϕ are dominated by some combination of investment in ϕ and lending at the pure interest rate. Consider next the possibility of borrowing. If the investor can borrow at the pure rate of interest, this is equivalent to disinvesting in P. The effect of borrowing to purchase more of any given investment than is possible with the given amount of wealth can be found simply by letting a take on negative values in the equations derived for the case of lending. This will obviously give points lying along the extension of line PA if borrowing is used to purchase more of A; points lying along the extension of PB if the funds are used to purchase B, etc. As in the case of lending, however, one investment plan will dominate all others when borrowing is possible. When the rate at which funds can be borrowed equals the lending rate, this plan will be the same one which is dominant if lending is to take place. Under these conditions, the investment opportunity curve a line in Figure Moreover, if the original investment opportunity curve is not linear at point the process of investment choice can be as first the optimum combination of risky assets (point and second borrow or to obtain the particular point on at which an indifference curve is tangent to the This proof first presented by Tobin for the case in which the pure rate of interest is zero considers the lending under conditions but not borrowing. Both authors present their analysis subject to as analysis independence and thus that the solution will no negative holdings of risky assets; the general thus his solution would generally negative holdings of some assets. The discussion in this paper is based on which includes on the holdings of all assets. with the analysis, it may be useful to alternative assumptions under which only a combination of assets lying at the point of between the original investment opportunity curve and a ray from P can be if borrowing is the investor will choose ϕ (and if his him to a point below ϕ on the line a number of choose to some of their funds in relatively this is not an if borrowing is possible but only to some the choice of ϕ would be made by all but those to considerable risk. These alternative to the thus the of borrowing or lending at the pure interest rate less than it to In to derive conditions for equilibrium in the capital market we two we a pure rate of interest, with all able to borrow or funds on we of investor A suggested by one of the are assumed to on the of various expected standard and correlation described in Part to these are and However, since the of a theory is not the of its assumptions but the of its and since these assumptions imply equilibrium conditions which a major part of financial it is from that this formulation be in view of the of alternative models to similar Under these given some set of capital asset prices, investor will view his in the same For one set of prices the as shown in Figure In this an investor with the preferences by indifference curves through would to some of his funds at the pure interest rate and to the in the combination of assets shown by point since this would give him the over-all An investor with the preferences by curves through would to all his funds in combination while an investor with indifference curves through would all his funds plus additional funds in combination ϕ in to his In any all would to purchase only those risky assets which combination The by to purchase the assets in combination ϕ and their of interest in assets not in combination ϕ of to a of prices. The prices of assets in ϕ will and, since an expected return future to present their expected will This will the of combinations which such assets; thus point ϕ (among others) will to the of its If the of future to present both ER and σR will under these conditions the point representing an asset would along a ray through the as its price the other the prices of assets not in ϕ will an in their expected and a of points representing combinations which them. Such price will to a of some combination or combinations will to different and thus to in prices. As the process the investment opportunity curve will to more with points such as ϕ to the and inefficient points as and to the asset prices of to a set of prices is for which asset at least one combination lying on the capital market Figure such an equilibrium The area in Figure representing ER, σR values with only risky assets has been at some from the horizontal for is that a more representation would it to the axis. in the area can be with combinations of risky assets, while points lying along the line can be by borrowing or lending at the pure rate plus an investment in some combination of risky assets. lying along from point A to point can be obtained in either For example, the ER, σR values shown by point A can be obtained by some combination of risky assets; the point can be by a combination of lending and investing in combination of risky assets. is to that in the shown in Figure many alternative combinations of risky assets are efficient lie along line and thus the theory not imply that all will the same This that there will be a unique combination of risky assets. proof of a unique optimum can be shown to be for the case of perfect correlation of efficient risky investment plans if the line their ER, σR points would through point P. In the on of this article the locus in this from a family of into one of straight lines to the thus the other all such combinations must be perfectly correlated, since they lie along a linear of the ER, σR ER, σR values given by combinations of any two combinations must lie the and a straight line the In this case they below such a straight since only in the case of perfect correlation will they along a straight the two combinations must be perfectly As shown in Part this not imply that the individual they are perfectly This provides a to the relationship between the prices of capital assets and different of risk. We have that in equilibrium there will be a simple linear relationship between the expected return and standard deviation of return for efficient combinations of risky assets. Thus has been said about such a relationship for individual assets. the ER, σR values with single assets will lie the capital market the of Moreover, such points may be the with no consistent relationship between their expected return and total risk However, there will be a consistent relationship between their expected and best be risk, as we will Figure the typical relationship between a single capital asset (point and an efficient combination of assets (point of which it is a The curve all ER, σR values which can be obtained with combinations of asset and combination As we such a combination in of a α of asset and ( 1 α ) of combination A value of α = 1 would indicate pure investment in asset while α = 0 would imply investment in combination however, that α = implies a total investment of more than the funds in asset since would be invested in and the other used to purchase combination which also includes some of asset This that a combination in which asset not at all must be represented by some negative value of such a In Figure the curve has been tangent to the capital market line at point This is no such curves must be tangent to the capital market line in equilibrium, since (1) they must it at the point representing the efficient combination and (2) they are at that if r = − 1 will the curve be the in Under these conditions a of would imply that the curve then some combination of assets would lie to the of the capital market an since the capital market line the efficient boundary of values of ER and σR. The that curves such as be tangent to the capital market line can be shown to to a relatively simple which the expected rate of return to various elements of risk for all assets which are in combination The standard deviation of a combination of and will σ = α 2 σ 2 + ( 1 − α ) 2 σ 2 + 2 r α ( 1 − α ) σ σ at α = 0 d σ d α = − 1 σ [ σ 2 − r σ σ ] but σ = σ at α = 0 . d σ d α = − [ σ − r σ ] The expected return of a combination will E = α E + ( 1 − α ) E at all values of dE d α = − [ E − E ] and, at α = 0 d σ dE = σ − r σ E − E . the equation of the capital market line σ R = ( E R − P ) where P is the pure interest rate. is tangent to the line when α = 0 , and since on the σ − r σ E − E = σ E − P r σ σ = − [ P E − P ] + [ 1 E − P ] E . meaning can best be seen if the relationship between the return of asset and that of combination is in a manner similar to that used in This model has been the model since its portfolio analysis solution can be by the so that the The method is described in the author's article, cited that we given a number of of the return of the two investments. The points as shown in The of the their will is, of of the total risk of the part of the is due to an relationship with the return on combination shown by the slope of the The of to in in account for of the in is this component of the total risk which we the risk. The the standard with is the This formulation of the relationship between and can be as a the predicted of to in given (the predicted risk of the of the predicted risk of asset can be This to the relationship derived from the of curves such as with the capital market line in the shown in Figure assets efficient combination must have and values lying on the line r = B 2 σ 2 σ 2 = B σ σ B = r σ σ . The expression on the is the expression on the of the last equation in B = − [ P E − P ] + [ 1 E − P ] E . will so that assets which are more to in will have higher expected than those which are less This with Obviously the part of an risk which is due to its correlation with the return on a combination be when the asset is to the the of this of risk it be related to expected The relationship illustrated in Figure provides a to the concerning the relationship between an risk and its expected thus we have only that the relationship for the assets which some particular efficient combination another combination been a different linear relationship would have been derived. this is As shown in the Consider the two assets and the in efficient combination and the in combination As shown B = − [ P E − P ] + [ 1 E − P ] E B = − [ P E − P ] + [ 1 E − P ] E . and are perfectly r = r B σ σ = B σ σ B = B [ σ σ ] . both and lie on a line which the at σ σ = E − P E − P B = B [ E − P E − P ] − [ P E − P ] + [ 1 E − P ] E = B [ E − P E − P ] from which we have the desired relationship between and B = − [ P E − P ] + [ 1 E − P ] E must therefore on the same line as we may any one of the efficient then the predicted of rate of return to that of the combination and these will be related to the expected rates of return of the assets in the manner in Figure The that rates of return from all efficient combinations will be perfectly provides the for any one of them. we may choose any perfectly with the rate of return of such The in Figure would then indicate alternative levels of a coefficient the of the rate of return of a capital asset to in the This possibility both a explanation for the that all efficient combinations will be perfectly and a useful of the relationship between an individual expected return and its risk. Although the theory implies only that rates of return from efficient combinations will be perfectly correlated, we that this would be due to their on the over-all level of If the investor to all but the risk from in of risk even in efficient since all other can be avoided by diversification, only the of an rate of return to the level of is relevant in assessing its risk. will there is a linear relationship between the of such and expected which are by in will return the pure interest those which with will higher expected rates of This discussion provides an to the second of the two in this In Part III it shown that with to equilibrium conditions in the capital market as a the theory to results consistent with the capital market line). We have shown that with to capital assets considered it also implications consistent with traditional it is for investment to a lower expected return from which little to in the than they from exhibit As suggested earlier, the of the implications not be considered a The of a for some of the major elements of traditional financial theory be a useful in its

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