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    American Finance Association

    Capital Asset Prices: A Theory of Market Equilibrium under Conditions of RiskAuthor(s): William F. SharpeReviewed work(s):Source: The Journal of Finance, Vol. 19, No. 3 (Sep., 1964), pp. 425-442

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    h e j o u r n l o FIN NCEVOL. XIX SEPTEMBER 964 No. 3

    CAPITAL ASSET PRICES: A THEORY OF MARKETEQUILIBRIUM UNDER CONDITIONS OF RISK*WILLIAM F. SHARPEt

    I. INTRODUCTIONONE OF THE PROBLEMSwhich has plagued those attempting to predict thebehavior of capital markets is the absence of a body of positive micro-economic theory dealing with conditions of risk. Although many usefulinsights can be obtained from the traditionalmodels of investment underconditions of certainty, the pervasive influence of risk in financial trans-actions has forced those working in this area to adopt models of pricebehavior 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 theprocess through which individual preferences and physical relationshipsinteract to determine an equilibrium pure interest rate. This is generallyfollowed by the assertion that somehow a market risk-premiumis alsodetermined,with the prices of assets adjusting accordingly to account fordifferences n their risk.A useful representationof the view of the capital market implied insuch discussions is illustrated in Figure 1. In equilibrium,capital assetprices have adjusted so that the investor, if he follows rationalprocedures(primarily diversification), is able to attain any desired point along acapital market line.' He may obtain a higher expected rate of return onhis holdings only by incurring additional risk. In effect, the marketpresents 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 theprice of risk, the additional expected return per unit of risk borne (thereciprocalof the slope of the line).

    * A great many people provided commentson early versions of this paper which ledto major improvements n the exposition. In addition to the referees,who were mosthelpful, the author wishes to expresshis appreciationto Dr. Harry Markowitz of theRAND Corporation,Professor Jack Hirshleiferof the University of Californiaat LosAngeles,and to ProfessorsYoram Barzel, George Brabb, Bruce Johnson, Walter Oi andR. Haney Scott of the Universityof Washington.t AssociateProfessorof OperationsResearch,Universityof Washington.1. Althoughsome discussionsare also consistentwith a non-linear(but monotonic) curve.

    425

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    426 The Journalof FinanceAt present there is no theory describingthe manner in which the priceof risk results from the basic influencesof investor preferences,the physi-cal attributes of capital assets, etc. Moreover, lacking such a theory, it isdifficult to give any real meaning to the relationship between the price

    of a single asset and its risk. Through diversification, some of the riskinherent in an asset can be avoided so that its total risk is obviously notthe relevant influence on its price; unfortunately little has been saidconcerningthe particularrisk componentwhich is relevant.Risk

    Capital Market Line

    0 Expected Rate of ReturnPure Interest'Rate FIGURE 1

    In the last ten years a numberof economistshave developednormativemodels dealing with asset choice under conditions of risk. Markowitz,2following Von Neumann and Morgenstern,developed an analysis basedon the expected utility maxim and proposed a general solution for theportfolio selection problem. Tobin' showed that under certain conditionsMarkowitz's model implies that the process of investment choice can bebroken down into two phases: first, the choice of a unique optimumcombinationof risky assets; and second, a separatechoice concerningtheallocation of funds between such a combination and a single riskless

    2. Harry M. Markowitz, Portfolio Selection, Efficient Diversification of Investments(New York: John Wiley and Sons, Inc., 1959). The major elementsof the theory firstappeared n his article PortfolioSelection, The Journal of Finance,XII (March 1952),77-91.3. James Tobin, Liquidity Preferenceas Behavior Towards Risk, The Review ofEconomic Studies, XXV (February, 1958), 65-86.

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    CapitalAssetPrices 427asset. Recently, Hicks4 has used a model similar to that proposed byTobin to derive corresponding conclusions about individual investorbehavior, dealing somewhatmore 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 rigorousproof in the context of a choice among lotteries has been presented byGordonand Gangolli.5

    Althoughall the authorscited use virtually the same model of investorbehavior,6none has yet attempted to extend it to construct a marketequilibriumtheory of asset prices underconditionsof risk. We will showthat such an extensionprovides a theory with implicationsconsistent withthe assertions of traditional financial theory described above. Moreover,it sheds considerable light on the relationship between the price of anasset and the various components of its overall risk. For these reasonsit warrantsconsiderationas a model of the determinationof capital assetprices.Part II provides the model of individual investor behavior under con-ditions of risk. In Part III the equilibrium conditions for the capitalmarket are consideredand 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 FunctionAssume that an individual views the outcome of any investment inprobabilistic terms; that is, he thinks of the possible results in terms ofsome probability distribution.In assessing the desirability of a particularinvestment, however, he is willing to act on the basis of only two para-

    4. John R. Hicks, Liquidity, The Economic Journal,LXXII (December,1962), 787-802.5. M. J. Gordonand Ramesh Gangolli, ChoiceAmong and Scale of Play on LotteryType Alternatives, College of Business Administration,University of Rochester, 1962.For another discussionof this relationshipsee W. F. Sharpe, A SimplifiedModel forPortfolio Analysis, ManagementScience, Vol. 9, No. 2 (January 1963), 277-293. Arelateddiscussioncan be found in F. Modigliani and M. H. Miller, The Cost of Capital,CorporationFinance, and the Theory of Investment, The AmericanEconomic Review,XLVIII (June 1958), 261-297.6. Recently Hirshleiferhas suggested that the mean-varianceapproach used in thearticles cited is best regardedas a special case of a more general formulation due to-Arrow.See Hirshleifer's InvestmentDecision Under Uncertainty, Papersand Proceedingsof the Seventy-SixthAnnual Meeting of the AmericanEconomic Association, Dec. 1963,or Arrow's Le Role des ValeursBoursierespour la Repartition a Meilleuredes Risques,InternationalColloquiumon Econometrics,1952.7. After preparing his paper the author learnedthat Mr. Jack L. Treynor, of ArthurD. Little, Inc., had independentlydevelopeda model similar in many respectsto the onedescribedhere. UnfortunatelyMr. Treynor'sexcellent work on this subject is, at present,unpublished.

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    428 The Journal of Financemeters of this distribution-its expected value and standard deviation.8This can be representedby a total utility function of the form:

    U = f(E,, a,)where Ewindicates expected future wealth and cwthe predicted standarddeviation of the possible divergenceof actual future wealth from Ew.Investors are assumed to prefer a higher expected future wealth to alower value, ceteris paribus (dU/dEw > 0). Moreover, they exhibitrisk-aversion, choosing an investment offering a lower value of aw toone with a greater level, given the level of Ew (dU/dow < 0). These as-sumptions imply that indifference curves relating Ew and co will beupward-sloping.9

    To simplify the analysis, we assume that an investor has decided tocommit a given amount (WI) of his present wealth to investment. LettingWt be his terminal wealth and R the rate of return on his investment:R Wt WiWIwe have Wt R WI+ Wi.

    This relationship makes it possible to express the investor's utility interms of R, since terminalwealth is directly related to the rate of return:U = g(ER,OR) .Figure 2 summarizes the model of investor preferences in a family ofindifference curves; successive curves indicate higher levels of utility asone moves down and/or to the right.10

    8. Under certain conditions the mean-varianceapproach can be shown to lead tounsatisfactorypredictionsof behavior. Markowitz suggests that a model based on thesemi-variance the averageof the squareddeviationsbelow the mean) would be preferable;in light of the formidablecomputationalproblems,however,he bases his analysis on thevarianceand standard deviation.9. While only these characteristics re requiredfor the analysis, it is generallyassumedthat the curves have the propertyof diminishingmarginalrates of substitutionbetween

    EW and aw, as do those in our diagrams.10. Such indifferencecurves can also be derived by assumingthat the investor wishesto maximizeexpectedutility and that his total utility can be representedby a quadraticfunction of R with decreasingmarginalutility. Both Markowitz and Tobin present sucha derivation.A similar approach s used by Donald E. Farrarin The InvestmentDecisionUnder Uncertainty (Prentice-Hall,1962). UnfortunatelyFarrar makes an error in hisderivation; he appealsto the Von-Neumann-Morgensternardinal utility axioms to trans-form a function of the form:E(U) = a+ bER - cER2 -CR2into one of the form: E (U) = k1E -k2aR2.That such a transformations not consistentwith the axioms can readily be seen in thisform, since the first equation implies non-linearindifference urves in the ER' aR2 planewhile the second implies a linear relationship.Obviously no three (different) points canlie on both a line and a non-linearcurve (with a monotonic derivative). Thus the twofunctions must imply different orderings among alternative choices in at least someinstance.

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    Capital Asset Prices 429

    CYR

    The~.neten poruiy uv

    /he..

    or

    // -7.....

    Thsetofinvestmentopportuniti ureshtoewihmsm shsuiiyEvery investment plan available to him may be representedby a point inthe ER, OR plane. If all such plans involve some risk, the area composedof such points will have an appearancesimilar to that shown in Figure 2.The investor will choose from among all possible plans the one placinghim on the indifference curve representing the highest level of utility(point F). The decision can be made in two stages: first, find the set ofefficient investment plans and, second choose one from among this set.A plan is said to be efficientif (and only if) there is no alternative witheither (1) the same ER and a lower CR, (2) the same OR and a higher EBor (3) a higher ERand a lower CR.Thus investment Z is inefficientsinceinvestments B, C, and D (among others) dominate it. The only planswhich would be chosen must lie along the lower right-hand boundary(AFBDCX)- the investment opportunitycurve.To understandthe nature of this curve, consider two investment plans-A and B, each including one or more assets. Their predicted expectedvalues and standard deviations of rate of return are shown in Figure 3.

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    430 The Journal of FinanceIf the proportion a of the individual's wealth is placed in plan A and theremainder(1-a) in B, the expected rate of return of the combination willlie between the expected returns of the two plans:

    ER= aERa + (1 a) ERbThe predicted standarddeviation of return of the combinationis:

    RC Va2Ra 2 + (1 a)2 Rb2+ 2rab a(1 - a) CRaORbNote that this relationship ncludes rab, the correlation coefficientbetweenthe predicted rates of return of the two investment plans. A value of +1would indicate an investor's belief that there is a precise positive relation-ship between 'the outcomes of the two investments. A zero value wouldindicate a belief that the outcomes of the two investmentsare completelyindependentand -1 that the investor feels that there is a precise inverserelationshipbetweenthem. In the usual case rabwill have a value betweeno and +1.Figure 3 shows the possible values of ERcand ORCobtainable withdifferentcombinationsof A and B under two differentassumptionsabout

    OR

    aRb -B

    CRa-

    ERa ERb ERFIGURE

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    CapitalAssetPrices 431the value of rab. If the two investments are perfectly correlated, thecombinations will lie along a straight line between the two points, sincein this case both ERC nd oRc will be linearly related to the proportionsinvested in the two plans.11If they are less than perfectly positively cor-related, the standarddeviation of any combination must be less than thatobtained with perfect correlation (since rabwill be less); thus the combi-nations must lie along a curve below the line AB. 2AZB shows such acurve for the case of complete independence (rab - 0); with negativecorrelation the locus is even more U-shaped.13

    The manner in which the investment opportunity curve is formed isrelatively simple conceptually, although exact solutions are usually quitedifficult.14One first traces curves indicating ER,ORvalues available withsimple combinations of individual assets, then considers combinations ofcombinations of assets. The lower right-hand boundary must be eitherlinear or increasing at an increasing rate (d2CR/dE2R> 0). As suggestedearlier, the complexity of the relationship between the characteristics ofindividual assets and the location of the investment opportunity curvemakes it difficult to provide a simple rule for assessing the desirabilityof individual assets, since the effect of an asset on an investor's over-allinvestment opportunity curve depends not only on its expected rate ofreturn (ERI) and risk (CR1), but also on its correlations with the otheravailable opportunities (rii, rI2 ...., rin). However, such a rule is impliedby the equilibriumconditions for the model, as we will show in part IV.The Pure Rate of Interest

    We have not yet dealt with riskless assets. Let P be such an asset; itsrisk is zero (oRp = 0) and its expected rate of return, ERR, is equal (bydefinition) to the pure interest rate. If an investor places a of his wealth11. ERC = aERa + (1 -a) ERb = ERb + (ERa ERb)

    ORC= V/a2Ra2 + (1- a)2 cvRb2+ 2rab a(1 - a) rRaaRbbutab 1, therefore he expression nderthe squareroot signcan be factored:

    YRc = \/[aaRa + (1 GO)cRb]2- a YRa + (1 - a) aRb- 0Rb + (OYRa YRb) a12. This curvature is, in essence, the rationale for diversification.

    13. When rab = 0, the slope of the curve at point A is - , at point B it isERb - ERaYRb . When rab =-1, the curve degeneratesto two straight lines to a pointERb- ERaon the horizontalaxis.

    14. Markowitzhas shown that this is a problemin parametricquadratic programming.An efficientsolutiontechnique s described n his article, The Optimizationof a QuadraticFunction 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 theauthor's A SimplifiedModel for Portfolio Analysis, op. cit.

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    432 The Journal of Financein P and the remainder n somerisky asset A, he would obtain an expectedrate of return:

    ERC= aERP+ (1 - a) ER1aThe standarddeviation of such a combinationwould be:

    0Rc - V\a202Rp + ( -a)2aua2 + 2rpaa(1-a) (}RplRabut since ORp = 0, this reduces to:

    CrR = (1 a) (Ra.This implies that all combinations involving any risky asset or combi-nation of assets plus the riskless asset must have values of ERCand OCRwhich lie along a straight line between the points representing the twocomponents.Thus in Figure 4 all combinationsof ERand ORying alongOaR

    P' 'vFIGURiE 4

    the line PA are attainable 'if some money is loaned at the pure rate andsome pBlacedn A. Similarly,by lending at the pure rate and investing inB, combinationsalong PB can be attained. Of all such possibilities, how-ever, one will dominate: that investment plan lying at the point of theoriginalinvestmentopprortunityurvewhere a ray frompoint P is tangentto the curve. In Figure 4 all investments lying along the original curve

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    Capital Asset Prices 433from X to cPare dominatedby some combinationof investment in 4 andlending at the pure interest rate.

    Considernext the possibility of borrowing. If the investor can borrowat the pure rate of interest, this is equivalent to disinvesting in P. Theeffect of borrowing to purchase more of any given investment than ispossible with the given amount of wealth can be found simply by lettinga take on negative values in the equationsderived for the case of lending.This will obviously give points lying along the extension of line PA ifborrowing s used to purchasemore of A; points lying along the extensionof PB if the fundsare used to purchaseB, etc.

    As in the case of lending, however, one investment plan will dominateall others when borrowing is possible. When the rate at which funds canbe borrowedequals the lending rate, this plan will be the same one whichis dominantif lending is to take place. Under these conditions,the invest-ment opportunity curve becomes a line (POZ in Figure 4). Moreover,if the original investment opportunitycurve is not linear at point c, theprocess of investment choice can be dichotomized as follows: first selectthe (unique) optimumcombination of risky assets (point c), and secondborrowor lend to obtain the particularpoint on PZ at which an indiffer-ence curve is tangent to the line.'5Before proceedingwith the analysis, it may be useful to consideralter-native assumptionsunderwhich only a combination of assets lying at thepoint of tangency between the original investment opportunitycurve anda ray from P can be efficient.Even if borrowing s impossible,the investorwill choose 4 (and lending) if his risk-aversion leads him to a pointbelow 4)on the line Pq).Since a large numberof investors choose to placesome of their funds in relatively risk-free investments, this is not an un-likely possibility. Alternatively, if borrowing is possible but only up tosome limit, the choice of 4) would be made by all but those investorswilling to undertakeconsiderable risk. These alternativepaths lead to themain conclusion, thus making the assumption of borrowing or lendingat the pure interest rate less onerous than it might initially appear to be.

    III. EQUILIBRIUMN THE CAPITALMARKETIn order to derive conditions for equilibriumin the capital market weinvoke two assumptions.First, we assume a commonpure rate of interest,with all investors able to borrow or lend funds on equal terms. Second,

    we assume homogeneityof investor expectations:16 investors are assumed15. This proof was first presentedby Tobin for the case in which the pure rate ofinterest is zero (cash). Hicks considersthe lending situation under comparableconditionsbut does not allow borrowing. Both authors present their analysis using maximizationsubject to constraintsexpressedas equalities.Hicks' analysis assumes independenceandthus insures that the solution will include no negative holdings of risky assets; Tobin'scovers the general case, thus his solution would generally include negative holdings ofsome assets. The discussion in this paper is based on Markowitz' formulation, whichincludesnon-negativityconstraints n the holdingsof all assets.16. A term suggestedby one of the referees.

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    434 The Journal of Financeto agree on the prospects of various investments-the expected values,standard deviations and correlation coefficients described in Part II.Needless to say, these are highly restrictive and undoubtedly unrealisticassumptions.However,since the proper test of a theory is not the realismof its assumptionsbut the acceptabilityof its implications,and since theseassumptions imply equilibrium conditions which form a major partof classical financial doctrine, it is far from clear that this formulationshould be rejected-especially in view of the dearth of alternativemodelsleading to similarresults.

    Under these assumptions,given some set of capital asset prices, eachinvestor will view his alternatives in the same manner. For one set ofprices the alternatives might appear as shown in Figure 5. In this situa-aR

    ClC2

    - -C3C1/

    /

    B~ B~2

    1/ /3

    P ERFIGURE 5

    tion an investor with the preferencesindicated by indifferencecurves A1throughA4would seek to lend some of his funds at the pure interest rateand to invest the remainder n the combinationof assets shown by point4,0 ince this wouldgive him the preferred over-allpositionA*. An investorwith the preferences ndicatedby curves B1 throughB4 would seek to in-vest all his funds in combination 4, while an investor with indifferencecurvesC1throughC4wouldinvest all his fundsplus additional (borrowed)

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    Capital Asset Prices 435funds in combination4 in order to reach his preferredposition (C*). Inany event, all would attempt to purchase only those risky assets whichenter combination b.

    The attempts by investors to purchase the assets in combination4 andtheir lack of interest in holding assets not in combination ' would, ofcourse, lead to a revisionof prices. The prices of assets in 4 will rise and,since an asset's expected return relates future income to present price,their expectedreturnswill fall. This will reducethe attractivenessof com-binations which include such assets; thus point ' (among others) willmove to the left of its initial position.'7On the other hand, the prices ofassets not in ' will fall, causing an increase in their expected returnsanda rightwardmovement of points representingcombinationswhich includethem. Suchprice changeswill lead to a revision of investors'actions; somenew combination or combinations will become attractive, leading to dif-ferent demandsand thus to furtherrevisions in prices. As the process con-tinues, the investment opportunity curve will tend to become more linear,with points such as ' moving to the left and formerly inefficientpoints(such as F and G) moving to the right.

    Capital asset prices must, of course, continue to change until a set ofprices is attained for which every asset enters at least one combinationlying on the capital market line. Figure 6 illustrates such an equilibriumcondition.'8All possibilities in the shaded area can be attained with com-binations of risky assets, while points lying along the line PZ can be at-tained by borrowingor lending at the pure rate plus an investment insome combinationof risky assets. Certainpossibilities (those lying alongPZ from point A to point B) can be obtained in either manner. For ex-ample, the ER, R values shownby point A can be obtainedsolely by somecombination of risky assets; alternatively, the point can be reachedby acombinationof lending and investing in combinationC of risky assets.

    It is important to recognize that in the situation shown in Figure 6many alternative combinationsof risky assets are efficient (i.e., lie alongline PZ), and thus the theory does not imply that all investors will holdthe same combination. 9On the other hand, all such combinationsmustbe perfectly (positively) correlated,since they lie along a linearborderof17. If investors consider the variability of future dollar returns unrelated to presentprice, both ER and cR will fall; under these conditionsthe point representingan assetwould move along a ray throughthe originas its price changes.18. The area in Figure 6 representingER' aR values attained with only risky assetshas been drawn at some distancefrom the horizontalaxis for emphasis.It is likely thata more accuraterepresentationwould place it very close to the axis.19. This statement contradictsTobin's conclusionthat there will be a unique optimalcombinationof risky assets. Tobin's proof of a unique optimum can be shown to beincorrect for the case of perfect correlationof efficient risky investment plans if theline connecting heir ER, aR points would pass throughpoint P. In the graph on page 83of this article (op. cit.) the constant-risk ocus would, in this case, degeneratefrom afamily of ellipsesinto one of straight lines parallelto the constant-returnoci, thus givingmultiple optima.

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    436 The Journalof Financethe ER, oR region.20This provides a key to the relationship between theprices of capital assets and different types of risk.aRl

    B

    p ERFIGURE

    IV. THE PRICESOF CAPITALASSETSWe have argued that in equilibriumthere will be a simple linear rela-tionshipbetweenthe expectedreturnand standarddeviationof returnforefficient combinations of risky assets. Thus far nothing has been saidaboutsucha relationship or individualassets. Typically the ER,ORvaluesassociatedwith single assets will lie above the capital market line, reflect-ing the inefficiencyof undiversifiedholdings. Moreover, such points maybe scattered throughoutthe feasible region, with no consistent relation-ship between their expected return and total risk (OR). However, there

    will be a consistent relationshipbetween their expected returnsand whatmightbest be called systematic risk, as we will now show.Figure 7 illustrates the typical relationship between a single capital20. ER, 0R values given by combinationsof any two combinationsmust lie withinthe regionand cannotplot above a straight ine joining the points. In this case they cannotplot below such a straightline. But since only in the case of perfect correlationwill theyplot along a straight line, the two combinationsmust be perfectly correlated.As shownin Part IV, this does not necessarilyimply that the individual securities they containare perfectlycorrelated.

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    Capital Asset Prices 437asset (point i) and an efficientcombinationof assets (point g) of whichit is a part.The curve igg' indicatesall ERJ,R values which can be obtainedwith feasible combinations of asset i and combination g. As before, wedenote such a combination in terms of a proportion a. of asset i and(1 a) of combinationg. A value of a 1 would indicate pure invest-

    ?R _

    P ERFiGuRE7

    ment in asset i while a = 0 would imply investment in combination g.Note, however, that a = .5 implies a total investment of more than halfthe funds in asset i, since half would be invested in i itself and the otherhalf used to purchasecombinationg, which also includes some of asset i.This means that a combination n which asset i does not appearat all mustbe represented by some negative value of a. Point g' indicates such acombination.In Figure 7 the curveigg' has been drawntangent to the capital marketline (PZ) at point g. This is no accident. All such curves must be tangentto the capital market line in equilibrium,since (1) they must touch it atthe point representing the efficient combination and (2) they are con-tinuous at that point.2' Under these conditions a lack of tangency would

    21. Only if rig= -1 will the curve be discontinuousover the range in question.

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    438 The Journal of Financeimply that the curve intersects PZ. But then some feasible combinationofassets would lie to the right of the capital market line, an obvious impos-sibility since the capital market line represents the efficientboundary offeasible values of ERand OR.The requirement that curves such as igg' be tangent to the capitalmarket line can be shown to lead to a relatively simple formula whichrelates the expected rate of return to various elements of risk for all as-sets which are included in combinationg.22Its economicmeaningcan bestbe seen if the relationshipbetween the return of asset i and that of com-bination g is viewed in a manner similar to that used in regressionanaly-sis.28Imagine that we were given a number of (ex post) observationsofthe return of the two investments. The points might plot as shown in Fig.8. The scatter of the R, observationsaround their mean (which will ap-proximateERi) is, of course, evidence of the total risk of the asset - CRi.But part of the scatter is due to an underlyingrelationshipwith the returnon combinationg, shown by Big,the slope of the regressionline. The re-sponse of R, to changes in Rg (and variations in Rg itself) account for

    22. The standarddeviation of a combinationof g and i will be:or= V/a2aRi2 + (1 - a)2 ORg2 + 2rig a(l a) cRiaRg

    at a = 0:do 1daL = d- [0Rg2 - rjgcvRjaTglbut or= Rg at a = 0. Thus:dada - [cvRg rigaRilThe expectedreturnof a combinationwill be:

    E = aERi + (1 - ca) ERgThus, at all values of a:

    dEdL = _ [ERg - ERJ]

    and, at a = 0:dcv cvRg rigcfRidE ERg-ERi

    Let the equationof the capitalmarket ine be:CvR= S(ER - P)

    whereP is the pure interestrate. Since igg' is tangent to the line when c = 0, and since(ERg, CvRg)ies on the line:

    qp ~- igq.,i dR,Rg jgj RgERgERi ERg Por:

    E l+ ERi.rgRE [ERg-P LERg -P]

    23. This model has been called the diagonalmodel since its portfolio analysis solutioncan be facilitatedby re-arranginghe data so that the variance-covariancematrix becomesdiagonal. The method is described in the author's article, cited earlier.

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    CapitalAssetPrices 439Return on Asset i (Ri)

    ,. ~~~ig

    -4..ERi

    Return on Combination g (Rg)FIGuRE8

    much of the variation in Ri. It is this componentof the asset's total riskwhich we term the systematic risk. The remainder,24 eing uncorrelatedwith Rg,is the unsystematic component.This formulationof the relation-shipbetweenR, and Rgcan be employedex ante as a predictivemodel. Bigbecomes the predicted response of Ri to changes in Rg. Then, given ORg(the predicted risk of Rg), the systematic portion of the predicted riskof each asset can be determined.This interpretation allows us to state the relationship derived fromthe tangency of curves such as igg' with the capital market line in theform shown in Figure 9. All assets entering efficient combinationg musthave (predicted) Big and ER,values lying on the line PQ.25Prices will

    24. ex post, the standard error.25.

    /Big20 Rg2 BigcRgg - aRi2 ORiand:

    -rigcvRiBig =- 4oRgThe expressionon the right is the expressionon the left-hand side of the last equationinfootnote 22. Thus:Big = E- Rg P + [EgiP] ERI.

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    440 The Journal of Financeadjust so that assets which are more responsiveto changes in Rg will havehigher expected returns than those which are less responsive. This accordswith common sense. Obviously the part of an asset's risk which is due toits correlationwith the return on a combinationcannot be diversifiedawaywhen the asset is added to the combination.Since Big indicates the magni-tude of this type of risk it should be directly related to expected return.The relationship llustratedin Figure 9 provides a partial answer to thequestion posed earlier concerning the relationshipbetween an asset's risk

    Big Q

    0 J______________->f -P ~~~~~~~~~~Pure Rate of Interest

    FIGUREand its expected return. But thus far we have argued only that the rela-tionship holds for the assets which enter some particular efficient com-bination (g). Had another combination been selected, a different linearrelationshipwould have been derived. Fortunately this limitation is easilyovercome. As shown in the footnote,26 we may arbitrarilyselect any one

    26. Considerthe two assets i and i*, the former included in efficientcombinationgand the latter in combinationg*. As shown above:Big = -[E1,g'P] + [E P] ERI

    and:

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    Capital Asset Prices 441of the efficient combinations, then measure the predicted responsivenessof every asset's rate of return to that of the combination selected; andthese coefficients will be related to the expected rates of return of theassets in exactly the mannerpicturedin Figure 9.The fact that rates of return from all efficient combinations will beperfectly correlatedprovides the justification for arbitrarilyselecting anyone of them. Alternatively we may choose instead any variable perfectlycorrelatedwith the rate of return of such combinations.The vertical axisin Figure 9 would then indicate alternative levels of a coefficientmeasur-ing the sensitivity of the rate of return of a capital asset to changes in thevariable chosen.This possibility suggests both a plausible explanation for the implica-tion that all efficientcombinationswill be perfectly correlatedand a use-ful interpretation of the relationship between an individual asset's ex-pected return and its risk. Although the theory itself implies only thatrates of return from efficient combinations will be perfectly correlated,we might expect that this would be due to their common dependenceonthe over-all level of economic activity. If so, diversificationenables theinvestor to escape all but the risk resulting from swings in economic ac-tivity-this type of risk remainseven in efficientcombinations.And, sinceall other types can be avoided by diversification,only the responsivenessof an asset's rate of return to the level of economic activity is relevant in

    BJ*g* = [ERg* P + [E *-P] ER11*Since Rg and Rg* are perfectly correlated:

    rj*g* = rj*gThus:Bi*g*oRg* Bi*gcrRg

    and:[ ORg. = Bi*g[ Crjg

    Since both g and g* lie on a line which intercepts the E-axis at P:YRg ERg-P

    (yRg* ERg* Pand:Bj*g* = Bi*g P

    Thus:

    - [ ]ER:P + [E g p] ERi* = Bi*g E Pz]from which we have the desired relationship between R,* and g:

    Bj*g[] ErP+ [El IP]Bi*g must therefore plot on the same line as does Big.

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    442 The Journal of Financeassessing its risk. Prices will adjust until there is a linear relationshipbetween the magnitude of such responsiveness and expected return. As-sets which are unaffectedby changes in economic activity will return thepure interest rate; those which move with economicactivity will promiseappropriatelyhigher expected rates of return.This discussionprovides an answer to the second of the two questionsposed in this paper. In Part III it was shown that with respect to equi-librium conditions in the capital market as a whole, the theory leads toresults consistent with classical doctrine (i.e., the capital market line).We have now shown that with regard to capital assets considered in-dividually, it also yields implicationsconsistent with traditionalconcepts:it is commonpractice for investmentcounselorsto accept a lower expectedreturnfrom defensive securities (those which respondlittle to changes inthe economy) than they requirefrom aggressive securities (which exhibitsignificantresponse). As suggested earlier, the familiarity of the implica-tions need not be considereda drawback.The provisionof a logical frame-work for producing some of the major elements of traditional financialtheory shouldbe a useful contribution n its own right.