《投资学原理 Investments》课程教学资源（阅读资料）The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets

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THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS* John Lintner Introduction and Preview of Som lusions titive markets when utility functions are quad- HE effects of risk and uncertainty upon ratic or rates of return are multivariate normal asset prices, upon rational decision rules We then note that the same conclusion follows for individuals and institutions to use in selecting from an earlier theorem of Roy's |19) without security portfolios, and upon the proper selection dependence on quadratic utilities or normalit of projects to include in corporate capital bud- The second section shows that if short sales are gets, have increasingly engaged the attention of permitted, the best portfolio-mix of risk assets professional economists and other students of the can be determined by the solution of a single capital markets and of business finance in recent simple set of simultaneous equations without years. The essential purpose of the present paper recourse to programming methods, and when is to push back the frontiers of our knowledge of covariances are zero, a still simpler ratio scheme the logical structure of these related issues, albeit gives the optimum, whether or not short sales under idealized conditions. The immediately are permitted. When covariances are not all following text describes the contents of the paper zero and short sales are excluded, a single quad- and summarizes some of the principal results. ratic programming solution is required,but The first two sections of this paper deal with sufficient folios by risk-averse investors who have the al- work, we concentrate on the set of risk assets ternative of investing in risk-free securities with held in risk averters'portfolios. In section III we a positive return (or borrowing at the same rate develop various significant equilibrium proper of interest)and who can sell short if they wish. ties weithin the risk asset portfolio. In particular, The first gives alternative and hopefully more we establish conditions under which stocks will transparent proofs(under these more general be held long(short)in optimal portfolios even market conditions)for Tobin's important "sep- when "risk premiums"are negative(positive) aration theorem”that We also develop expressions for different combi ate composition of the non-cash assets is inde- nations of expected rate of return on a given endent of their aggregate share of the invest- security, and its standard deviation, variance ment balance (and hence of the optimal and/or covariances which will result in the same holding of cash)for risk averters in purely compe- relative holding of a stock, ceteris paribus. These indifference functions" provide direct evidence *This is another in a series of interrelated theoretical on the moot issue of the appropriate functional relationships between"required rates of ret ion, and more recently the Ford Foundation, to the Harvard and relevant risk parameter(s)-and on the Buasietess ahoolowlede generhes support for th is work is most related issue of how"risk classes"of securitie ab his colleagues Professors Bishop, Christenson, Kahr, Raiffa, may best be delineated (if they are to be used,- (especially) Schlaifer, for extensive discussion and com mentary on an earlier draft of this. Tobin [2I, especially pp. 82-851. Tobin assumed that but responsibility for funds rfections rema y his own allocated only over"monetary assets"(risk ee cash and default-free bonds of Market Equilibrium Under Conditions of Risk"(Journal of low. Other approaches are reviewed in Farrar [38] Finance, September I964) appeared after this paper was in It should be noted that the classic paper by Modigl he printers. My first scction, and Miller [i6] was silent on these issues. Corporations were ich parallels the first half of his paper(with corresponding assumed to be divided into homogeneous classes having the conclusions), sets the algebraic framework for sections II, property that all shares of all corporations in any given class Ill and (which have no counterpart in his paper) and for differed (at most) by a"scale factor, "and hence ection IV on the equilibrium prices of risk assets, concerning fectly correlated with each other and(6)were perfect substi- hich our results differ significantly for reasons which will be tutes for each other in perfect markets(p. 266). No comment plored elsewhere. Sharpe does not take up the capital was made on the measure of risk or dgeting problem developed in section V below.] attributes)relevant to the identification of different"equiva- [13]

THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among uncertainty per se(as distinct from the effects of y the standard deviation (or coefficient of implications of such uncertainty. In particulg economists that relative risks are best measured diverse expectations), and to derive further variation)of the rate of return, but in the simp- the aggregate market value of any companys t cases considered- specifically when all equity is equal to the capitalization at the risk covariances are considered to be invariant (or free interest rate of a uniquely defined certainty zero)-the indifference functions are shown to equivalent of the probability distribution of the be linear between expected rates of return and aggregate dollar returns to all holders of its stock their variance, not standard deviation. 4(With For each company, this certainty equivalent is variances fixed, the indifference function between the expected value of these uncertain returns less the ith expected rate of return and its pooled an adjustment term which is proportional to covariance with other stocks is hyperbolic. their aggregate risk. The factor of proportion There is no simple relation between the expected ality is the same for all companies in equilibirum rate of return required to maintain an investor's and may be regarded as a market price of dollar relative holding of a stock and its standard devia- risk. The relevant risk of each company's stock tion.Specifically, when covariances are non- is measured, moreover, not by the standard de zero and variable the indifference functions are viation of its dollar returns, but by the sum of the complex and non-linear even if it is assumed that variance of its own aggregate dollar returns and the correlations between rates of return on differ- their lotal covariance with those of all other stocks ent securities are invariant The next section considers some of the impli To this point we follow Tobin[21] and Marko- cations of these results for the normative aspects witz 14] in assuming that current security prices of the capital budgeting decisions of a company are given, and that each investor acts on his own whose stock is traded in the market. For sim perhaps unique)probability distribution over plicity, we impose further assumptions required rates of return given these market prices. In the to make capital budgeting decisions independent rest of the paper, we assume that investors' of decisions on how the budget is financed. 6 The joint probability distributions pertain to dollar capital budgeting problem becomes a quadratic returns rather than rates of return and for programming problem analogous to that intro- simplicity we assume that all investors assign duced earlier for the individual investor. This identical sets of means, variances, and covari- capital budgeting-portfolio problem is formula- ances to the distribution of these dollar returns. ted, its solution is given and some of its more be, it enables us, in section IV, to derive a set of the minimum expected return(in dollars of ex- (stable) equilibrium market prices which at pected present value) required to justify the least fully and explicitly reflect the presence of allocation of funds to a given risky project is shown to be an increasing function of each of the lent return"classes. Both Propositions I(market value of firm following factors: (i) the risk-free rate of return between the expected return on equity shares and the debt(ii) the"market price of(dollar )risk";(iii)the equity ratio for firms within a given class) are derived from variance in the project's own presentvalue return porate bonds are riskless securities); they involve no inter- iv) the project,s aggregate present value re- class comparisons, ". nor do they involve any assertion as turn-covariance with assets already held by the to what is anea e ruste compe ns, tio to investors for assuming company, and (o) its total covariance with other aThis is. for instance, the presumption of Hirschleifer projects concurrently included in the capital 18, p. 1131, although he was careful not to commit himself to budget. All five factors are involved explicitly his measure alone in a paper prim ir i fator st he stahe rd in the corresponding (derived)formula for the Gordon [s, especially pp 6g and 76r. See also Dorfman in investment project. In this model, all means he constant 6We also ass ommon stock portfolios are not term will be larger, and the er, the higher the( fixed) "inferior goods level of covariances of the gi ocks with invariant, and any effect of changes in capital budgets on the od is th the cash covariances between the values of different companies'stocks dividend and the increase in market price du

VALUATION OF RISK ASSETS and (co)variances of present values must be cept in the final section, we assume that the calculated at the riskless rate r*. We also show interest rate paid on such loans is the same as he that there can be no"risk-discountrale to be used would have received had he invested in risk-free in computing present values to accept or reject savings accounts, and that there is no limit on the individual projects. In particular, the"cost of amount he can borrow at this rate. Finally(5) capital"as defined (for uncertainty)anywhere he makes all purchases and sales of securities and in the literature is not the appropriate rate to use all deposits and loans at discrete points in time in these decisions even if all new projects have the so that in selecting his portfolio at any"trans same“risk” as existing assets action point, each investor will consider only The final section of the paper briefly examines (i) the cash throw-off (typically interest pay- the complications introduced by institutional ments and dividends received)within the period limits on amounts which either individuals or to the next transaction point and (ii) changes in corporations may borrow at given rates, by rising the market prices of stocks during this same costs of borrowed funds, and certain other "real period. The return on any common stock is de world"complications. It is emphasized that fined to be the sum of the cash dividends received the results of this paper are not being presented plus the change in its market price. The retur as directly applicable to practical decisions, be- on any portfolio is measured in exactly the sar cause many of the factors which matter very way, including interest received or paid ctice have had to be or assumed away. The function of these sim- Assumptions Regarding Investors plifying assumptions has been to permit a (1)Since we posit the existence of assets rigorous development of theoretical relationships yielding posilive risk-free returns, we assume that and theorems which reorient much current each investor has already decided the fraction of theory (especially on capital budgeting) and pro- his total capital he wishes to hold in cash and vide a basis for further work. 7 More detailed non-interest bearing deposits for reasons of conclusions will be found emphasized at numerous liquidity or transactions requirements. 10Hence points in the text. forth, we will speak of an investor's capital as the stock of funds he has available for profitable I-Portfolio Selection for an Individual Investor: investment after optimal cash holdings have been deducted. We also assume that(2)each investor Market Assumptions will have assigned a joint probability distribution We assume that(1)each individual investor incorporating his best judgments regarding the can invest any part of his capital in certain risk- have specified an expected value and variance to returns on all individual stocks, or at least will free assets (e. g deposits in insured savings ac- every return and a covariance or correlation to counts)all of which pay interest at a common positive rate, exogeneously determined: and that every pair of returns. All expected values of (2) he can invest any fraction of his capital in any returns are finite, all variances are non-zero and or all of a given finite set of risky securities which finite, and all correlations of returns are less than are(3) traded in a single purely competitive one in absolute value(i.e the covariance matrix market, free of transactions costs and taxes, at is positive-definite). The investor computes the depend on his investments or transactions. We on any possible portfolio, or mix of any specified also assume that (4)any investor may, if he amounts of any or all of the individual stocks, by wishes, borrow funds to invest in risk assets. Ex- forming the appropriately weighted average or TThe relation between the results of this paper and the sum of these components expected returns models which were used in (rr] and [r2] is indicated at the end variances and covariances 10These latter decisions are independent of the decisions Government bonds of appropriate maturity regarding the another important example when their"yield"is urn a-aining funds between risk-free ' Solely for conveniet shall usually refer ominate those with no return once Investments as common though the analysis is of liquidity and requirements are satisfied at the

THE REVIEW OF ECONOMICS AND STATISTICS With respect to an investor's criterion for optimal mix of risk assets conditional on a given choices among different attainable combinations gross investment in this portfolio, and then for of assets, we assume that(3)if any two mixtures mally proving the critical invariance property of assets have the same expected return, the inves- stated in the theorem. Tobin used more restric tor will prefer the one having the smaller ra7〃ce tive assumptions that we do regarding the avail of return, and if any two mixtures of assets have able investment opportunities and he permitted the same variance of returns, he will prefer the no borrowing l1 Under our somewhat broadened one having the greater expected value. Tobin [21, assumptions in these respects, the problem fits pp. 75-76 has shown that such preferences are neatly into a traditional Fisher framework, with implied by maximization of the expected value different available combinations of expected of a von Neumann-Morgenstern utility function values and standard deviations of return on al if either(a) the investors utility function is con- ternative stock portfolios taking the place of cave and quadratic or(b)the investor's utility the original "production opportunity''set and function is concave, and he has assigned probabil- with the alternative investment choices being y distributions such that the returns on all pos- concurrent rather than between time periods sible portfolios differ at most by a location and scale Within this framework, alternative and more parameter, (which will be the case if the joint dis- transparent proofs of the separation theorem tribution of all individual stocks is multivariate are available which do not involve the actual calculation of the best allocation in stocks over individual stock issues. As did Fisher, we shall Alternative Proofs of the Separation Theorem present a simple algebraic proof, set out the Since the interest rates on riskless savings logic of the argument leading to the theorem, and bank deposits("loans to the bank")and on bor- depict the essential geometry of the problem As a preliminary step, we need to establish the rowed funds are being assumed to be the same, relation between the investor's total investment G the gross amount invested in stocks, (i)the stocks, his total net return from all his invest- fraction of this amount invested in each indivi. ments (including riskless assets and any borrow dual stock, and(i) the net amount invested in ing), and the risk parameters of his investment loans(a negative value showing that the investor position. Let the interest rate on riskless assets has borrowed rather than lent). But since the or borrowing ber*, and the uncertain return(divi total nel investment(the algebraic sum of stocks dends plus price appreciation) per dollar invested plus loans)is a given amount, the problem sim- in the given porfolio of stocks be r. Let u rep- ply requires finding the jointly optimal values resent the ratio of gross investment in stocks to for(1)the ratio of the gross investment in stocks aTobin considered the special case where cash with no to the total net investment, and(2) the ratio of required that all assets be held in non-l the gross investment in each individual stock to (thereby ruling out short sales), and that the total value of risk the total gross investment in stocks. It turns out that although the solution of(1)depends upon constraints were not introduced into his formal solution of the that of (2), in our context the latter is indepen- optimal investment mix, which in turn was used in proving the dent of the former. Specifically, the separation Mfgis independent of the programming constraints neglec- e property stated in the theorem. Our proof of the theorem asserts that ted in Tobins proof. Later in this section we show that when Given the assumptions about borrowing, short sales are properly and explicitly introduced into the set lending, and investor preferences stated earlier in olio mix are identical to those derived by Tobin, but that this section, the optimal proportionate composition insistence on no short sales results in a somewhat more complex of the stock (risk-asset) portfolio(i.e. the solution programming problem(when covariances are non-zero),which ralio of the gross investment in stocks to the total net available. be readily handled with computer programs now to sub-problem 2 above)is independent of the may howev 1An alternative algebraic proof using utility functions invesiment ock wood Rainha Tobin proved this important separation theo- and presented a similar proof of the theorem in an unpublished em by deriving the detailed solution for the

VALUATION OF RISK ASSETS 17 total net investment (stock plus riskless assets value of w). Since any expected return y can be minus borrowing). Then the investors net obtained from any stock mix, an investor adher- return per dollar of total net investment will be ing to our choice criterion will minimize the (1)y=(1-v)r*+er =r*+u(f-r*);0$w1 indicates that the investor borrows to buy variance associated with any y (and hence any w stocks on margin and pays interest amounting value) the investor may prefer, and consequent the absolute value of(1-r*. From(1)we is independent of y and w. This establishes the determine the mean and variance of the net re- separation theorem 4, once we note that our turn per dollar of total net investment to be: assumptions regarding available portfolios in- (2a) y=r*+w(r-r*),and sure the existence of a maximum e It is equally apparent that after determining the optimal stock portfolio(mix) by ma Finally, after eliminating w between these two 0, the investor can complete his choice of an equations, we find that the direct relation be- over-all investment position by substituting tween the expected value of the investor's net the e of this optimal mix in(3)and decide which return per dollar of his total net investment and over-all investment position by substituting the risk parameters of his investment position is: of the available(, ay) pairs he prefers by refer y=r*+Boy, where ring to his own utility function. Substitution of this best y value in(2a) determines a unique In terms of any arbitrarily selected stock port best value of the ratio we of gross investment in folio, therefore, the investor's net expected rate the optimal stock portfolio of return on his total net investment is related investment, and hence, the optimal amount of linearly to the risk of return on his total net investments in riskless savings deposits or the investment as measured by the standard deviation optimal amount of borrowing as well of his return. Given any selected stock portfolio, This separation theorem thus has four immedi this linear function corresponds to Fisher's ate corrolaries which can be stated market opportunity line"; its intercept is the risk-free rater"and its slope is given by 8, which (i) Given the assumptions about borrowing is determined by the parameters i and ar of the and lending stated above, any investor whose choices maximize the expectation of any pa ticu- also see from(2a) that, by a suitable choice of w, lar utility function consistent with these condi the investor can use any stock mix(and its asso- tions will make identical decisions regarding the iated"market opportunity line")to obtain an proportionate composition of his stock (risk-asset) because of (26 )and(36), as he increases his in- utility functions whose expectation he maximizes vestment w in the (tentatively chosen)mix, the,(ii) Under these assumptions, only a singl standard deviation y (and hence the variance a'y)of the return on his total investment also relevant to the investors decision regarding his becomes proportionately greater nvestments in risk assets. '7(The next section ow consider all possible stock portfo See also the appendix, note I for a different form of proof. Those portfolios having the same 0 value will ie on the same"market opportunity line, "but on individual stocks are finite, that all variances are positive those having different o values will ofer differ- and finite, and that the variance-covariance matrixispositive nt"market opportunity lines"(between expected detinite return and risk) for the investor to 16 When probability assessments are multivariate normal the utility function may be Polynomial, exponential, eto vestors problem is to choose which stock port- Even in the "non-normal"case when utility functions are folio-mix(or market opportunity line or 0 value) quadratic, they may vary in its parameters. See also the to use and how intensively to use it (the proper When the above conditions hold (see also final para-

THE REVIEW OF ECONOMICS AND STATISTICS shows this point can be obtained directly without has been determined, the investor completes the calculating the remainder of the efficient set. optimization of his total investment position Given the same assumptions, (iii) the para- by selecting the point on the ray through M meters of the investors particular utility within which is tangent to a utility contour in the he relevant set determine only the ratio of his standard manner. If his utility contours are as total gross investment in stocks to his total net in the Ui set in chart 1, he uses savings accounts investment(including riskless assets and borrow- and does not borrow. If his utility contours are g); and(iv)the investor's wealth is also, conse- as in U; set, he borrows in order to have a gross uently, relevant to determining the absolute size investment in his best stock mix greater than his of his investment in individual stocks, but not to net investment balance the relative distribution of his gross investment in stocks among individual issues. Risk aversion, Normality and the separation The Geometry of the Separation Theorem and Its The above analysis has been based on the Corrolaries The algebraic derivations given above assumptions regarding markets and investors stated at the beginning of this section. One epresented graphically as in chart 1.Any given available stock portfolio is characterized crucial premise was investor risk-aversion in the form of preference for expected return and prefer by a pair of values(or, F) which can be repre- ence against relurn-variance, ceteris paribus. We sented as a point in a plane with axes ay and y. noted that tobin has shown that either concave Our assumptions insure that the pointsrepresent quadratic utility functions or multivariate nor- ing all available stock mixes lie in a finite region, mality(of probability assessments) and any con- lI parts of which lie to the right of the vertical axis, and that this region is bounded by a closed cave utility were suficient conditions to validate nis premise, but they curve. 18 The contours of the investor's utility to be necessary conditions. This is probably for- function are concave upward, and any movement in a north and or west direction denotes con-(or wealth, function, in spite of its popularity in tours of greater utility. Equation(3)shows that theoretical work, has several undesirably restric borrowing, or lending with any particular stock tive and implausible properties, 20 and, despite portfolio lie on y from the point (0, r*)row ear. The optimal set of produc- though the point corresponding to the stock mix tion opportunities available is found by moving along the en question. Each possible stock portfolio thus higher present value lines to the highest attainable. This best Given the properties of the utility function, it is particular utility function which determines only whether he bvious that shifts from one possible mix to either case)to reach hi best over-all position.The another which rotate the associated market op- erences between this case and ours lie in the concurrent nature portunity line counter colckwnse will move the inves- of the comparisons(instead of inter-period), and the ro rotation pivot the line he had tentatively chosen. The slope of the riskless return (instead of parallel shifts in present value his market-opportunity line given by(3)is a, section Ia and the limit of the favorable rotation is given negative marginal utilities of income or wealth much"too simply ly does the quad by the maximum attainable e, which identifies in empirical work unless the risk-aversion parameter is very the optimal mix M. t9 Once this best mix, M, small-in which case it cannot account for the degree of risk a, common stocks, like potatoes ange of Markowitz Efficient Set suggested by Baumol [2] is in Ireland, are"inferior"goods. Offering more return at the than needed by a factor strictly proportionate to the he risk would so sate investors that they would reduce thei portfolios he retains in his truncated set! This is cause they were more attractive. (Thereb the relevant set is a single portfolio under these con- as Tobin [2I] noted, denying the negatively sloped demand ard doctrine in“ liqui See Markowitz [I4] as cited in the appendix, note I I The analogy with the standard Fisher two-period pro- tally, be avoided by "limit arguments on quadratic utilities duction-opportunity case in perfect markets with equal bor- such as he used, once borrowing and leverage are admitted

VALUATION OF RISK ASSETS its mathematical convenience, multivariate nor- of 0- is thus rigorously appropriate in the non mality is doubtless also suspect, especially per- multivariate normal case for Safety-Firsters who haps in considering common stocks minimax the stated upper bound of the chance It is, consequently, very relevant to note that of doing less well on portfolios including risk by using the Bienayme-Tchebycheff inequality, assets than they can do on riskless investments, Roy [19] has shown that investors operating on just as it is for concave-expected utility maxi. his"Safety First" principle (i.e. make risky in- mizers in the "normal"case. On the basis vestments so as to minimize the upper bound of of the same probability judgments, these Safety- the probability that the realized outcome will fall Firsters will use the same proximate criterion below a pre-assigned "disaster level"")should function(max 0)and will choose proportionately maximize the ratio of the excess expected port- the same risk asset portfolios as the more folio return (over the disaster level)to the orothodox"utility maximizers""we have hitherto standard deviation of the return on the port- considered folio21- which is precisely our criterion of max 0 when his disaster level is equated to the risk- II-Portfolio Selection: The Optimal Stock Mix free rate r*. This result, of course, does not depend on multivariate normality, an na uses a Before finding the optimal stock mix different argument and form of utility function mix which maximizes 0 in( 3b) above-it necessary to express the return on any arbitrary The Separalion Theorem, and its Corrolaries mix in terms of the returns on individual stocks i)and (ii) above-and all the rest of our follo ing analysis which depends on the maximization included in the portfolio. Although short sale are excluded by assumption in most of the writings on portfolio optimization, this restric tive assumption is arbitrary for some purposes at least, and we therefore broaden the analysis in this paper to include short sales whenever they Computation of Returns on a Stock Mix, When We assume that there are m different stocks in short sales as negative purchases. We shall use the following basic notation The ratio of th the itn stock(the market value of the amount bought or sold) to the gross FIGURE I investment in all stocks. A positive value of hi indicates a purchase, while This function also implausibly imp Pratt [I7I and a negative value indicates a short sale Arrow [r]have noted that the ins i, -The return per dollar invested in a e would be willing to pay to hedge rise pr sult, see Hicks purchase of the i stock(ca dends plus price appreciation Roy also notes that when judgmental distributions are As above, the return per dollar inves multi the probability of"disaster"(failure to do better in stocks than d in a particular mix or portfolio of savings deposits or government bonds held to maturity). It hould be noted, however, minimization of the probability of Consider now a gross investment in the entire strictly equivalent to expected utility maximization under all mix, so that the actual investment in the il risk-averters'utility functions. The equivalence is not re- stock is equal to h. The returns on purchases aster occurs, one if it doesn't ), as caimed by roy (g, p. 43] and short sales need to be considered separately and Markowitz [I4, P. 293 and following. I First, we see that if hi is invested in a pur

THE REVIEW OF ECONOMICS AND STATISTICS chase(hi>0), the return will be simply hi. (5) i=EiIhiF-r*)+ hiIr* For reasons which will be clear immediately how- er. we write this in the form because >i hi=I by the definition of hil Now suppose that Ih: I is invested in a short a, The expectation and variance of the return on (F;-r*)+|h ly stock mix is consequently sale(hthi t, is the negative of any price appreciation during ()12(x)12(2;hh,)12 this period. In addition, the short seller will Since h may be either positive or negative, receive interest at the riskless rate r*on the equation (6a) shows that a portfolio with sales price placed in escrow, and he may or may i>r*and hence with 0>o exists if there is not also receive interest at the same rate on his one or more stocks with i not exactly equal to cash remittance to the lender of the stock. To r*. We assume throughout the rest of the paper facilitate the formal analysis, we assume that that such a portfolio exists both interest components are aleays received by the short seller, and that margin requirements are Determination of the Optimal stock portfolio Ioo%. In this case, the short seller's return per As shown in the proof of the Separation dollar of his gross investment will be(2r*-r), Theorem above, the optimal stock portfolio is and if he invests hil in the short sale (h;< o), the one which maximizes 6 as defined in equa its contribution to his portfolio return will be: tion(8). We, of course, wish to maximize this (4b)h1|( Since the right-hand sides of (4a)and(46)are (9) E: hiI=I identical, the total return per dollar invested in which follows from the definition of h:|. But any stock mix can be written as we observe from equation(8)that e is a homog eneous function of order zero in the hi: the value the short seller to waive interest on his deposit with the lender of e is unchanged by any proportionate change in arket parlance, for the borrowers of all hi. Our problem thus reduces to the simpler a vector of stock is large relative to the supply available for this purpose, unconstrained maximum of 8 in equation(8) tock. See Sidney m. Robbins, [18, pp. s8-sgl. It will be after which we may scale these initial solution developed belochanging the varia che expected return of short values to satisfy the constraint oted that these practice rmit the identification of the appropriate Crocks for short sale assuming the expected return is(2r-ii. The Oplimum Portfolio When Short Sales these stocks were to be borrowed"fat or a premium paid, are permitted it would be simply necessary to iterate ihe soltion after replacing We first examine the partial derivatives of(8) shoud be substituted(where fas is paid, the term (+pi) with respect to the h; and find w“3m)t()ah1=(),-M+3) imagine req rests aifl r fot stuegte ting the uase o a b solute where, values in analyzing short sales (I1)A=x/σ2=2h/2;2hh,元

VALUATION OF RISK ASSETS 21 The necessary and sufficient condilions on the the sum of their absolute values. A comparison relative values of the h, for a stationary and the of equations(I6) and(II)shows further that unique (global)maximums are obtained by (18) 2:,01=X0=2o0x 2 setting the derivatives in (Io)equal to zero, i.e. the sum of the absolute values of the z 0 which give the set of equations yields, as a byproduct, the value of the ratio of (12) ziti+ 2 ti=ti,i=I,2,.., m; the expected excess rate of return on the optimal rite portfolio to the variance of the return on this best portfolio It is also of interest to note that if we form the It will be noted the set of equations (I2) corresponding A-ratio of the expected excess which are identical to those Tobin derived by a return to its variance for each ith stock, we have different route24- are linear, in the own-wari. at the optimum ances, pooled covariances, and excess returns of the respective securities; and since the covariance (1g) h,=(i/)-2ixih, ii/awhere matrix x is positive definite and hence non ngular, this system of equations has a unique The optimal fraction of each security in the best solution (I4)x0=∑元 covariance with other securities to its own vari where aii represents the ijth element of ( 2-1, ance. Consequently, if the investor were to act the inverse of the covariance matrix. Using on the assumption that all covariances were his solution may also be zero, he could pick his optimal portfolio mix written in terms of the primary variables of the very simply by determining the \; ratio of the problem in the form expected excess return i=fi -r* of each (r5)h:0=(A0)-1x,P(F;-r+),alli stock to its variance iii =Pii, and setting each Moreover, since(I3)implie h;=入/Σλ; for with no covariances,25A;= (16)E;|z;|=A2;|h a=to/oro With this simplifying assumption he ratios of each stock suffice to determine Ao may readily be evaluated, after introducing the optimal mix by simple arithmetic; 26 in the the constraint(9)as more general case with non-zero covariances, a single set7 of linear equations must be solved in quently be scaled to the optimal proportions of programming is required and no more than one aaIt is clear from a comparison of equations(8)and(Ir), we are working re relevant to the maximization of e 8 as given in(8)and all its first partials shown The Optimum Portfolio When Short sales a minimum )when a >o because 0 is a convex function with a cate the above analysis if the investor is willing positive-definite quadratic torm in its denomina For the to act on an assumption of no correlations b 9=e ason, any maximum of e is a unique(global )maximum between the returns on different stocks. In th ver, formally required no short selling or borrowing, implying case, he finds his best portfolio of "long" holding long as there is a single riskless asset (pp. 84-85); but the onstraints were ignored in his derivation. We have shown 26 With no covariances, the set of equations(I2)reduces that this set of equations is valid when short sales are properly to Mh =ii/ii=A, and after summing over al included in the portfolio a perfect 2... m, and using the constraint(o), we have imme 油成 markets in unlimited amounts. The alternative set of equi- that x|=E:IMl, and xo> ofor max e(instead of mi brium conditions required when short sales are ruled out is 24 Using a more restricted market setting, Hicks [6, p. or] ren immediately below. The complications introduced by has also reached an equivalent result when covariances are paper wing restrictions are examined in the final section of the zero(as he assumed throughout bove 27 See, however, footnote 2

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