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$w< o, variance of his over-all return a'y associated with where a value of v <1 indicates that the investor any expected return he may choose by confining holds some of his capital in riskless assets and all his investment in stocks to the mix with the interest amounting to(1-v0r* while largest 0 value. This portfolio minimizes the a >1 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(h< o, this al r*+Σh1(F;-r*)=r*+zh price received must be deposited in escrow, and in where Fi represents the variance arif when addition, an amount equal to margin require- i=j, and covariances when i#j. The notation ments on the current price of the stock sold must has been further simplified in the right-hand remitted or loaned to the actual owner of the expressions by defining securities borrowed to effect the short sale.)(7 know that the short seller must pay to the and making appropriate substitutions. in the person who lends him the stock any dividends(3b)can thus be written hence borrowed), and his capital gain (or loss)( 8) 0=7l=7=->thi 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