{"doi":"10.1111/j.1540-6261.1964.tb02865.x","title":"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","journal":"The Journal of Finance","year":1964,"id":118,"datarank":36.432156797766524,"base_score":9.761923988214342,"endowment":9.761923988214342,"self_citation_contribution":1.4642885982321514,"citation_network_contribution":34.96786819953437,"self_endowment_contribution":1.4642885982321514,"citer_contribution":34.96786819953437,"corpus_percentile":99.3,"corpus_rank":2371,"citation_count":17359,"citer_count":200,"citers_with_citation_signal":200,"citers_with_endowment":200,"datacite_reuse_total":0,"is_dataset":false,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"1964-09-01","authors":[{"id":837,"name":"William F. 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