VALUATION OF RISK ASSETS 21 The necessary and sufficient conditions on thethe sum of their absolute values.A comparison relative values of the h:for a stationary and the of equations (16)and (II)shows further that: unique (global)maximum23 are obtained by (18):=0=0/2; setting the derivatives in (Io)equal to zero, i.e.the sum of the absolute values of the which give the set of equations yields,as a byproduct,the value of the ratio of (12）8成十2=元，i=1,2,···,m; the expected excess rate of return on the optimal where we write portfolio to the variance of the return on this (I3)名=λh. best portfolio. It is also of interest to note that if we form the It will be noted the set of equations (12)- 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-vari- at the optimum: ances,pooled covariances,and excess returns of the respective securities;and since the covariance (Ig）heo=(入/入)-2h,e/where matrix is positive definite and hence non- 入i=无/ci. singular,this system of equations has a unique The optimal fraction of each security in the best solution portfolio is equal to the ratio of its A;to that of the entire portfolio,less the ratio of its pooled (I4）80=2元； covariance with other securities to its own vari- where represents the ijth element of (ance.Consequently,if the investor were to act the inverse of the covariance matrix.Using on the assumption that all covariances were (13),(7),and (66),this 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 A:ratio of the problem in the form expected excess return元：=r:-r*of each (15)h0=(入)-12产f5-r*,alli. stock to its variance=,and setting each Moreover,since (13)implies h:=入/；for with no covariances,25zλt= (t6):z:|=λ2：lh:, A=2.With this simplifying assumption, the A: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 (I7）2:z:0|=入2：h,0|=λ0 single set27 of linear equations must be solved in The optimal relative investments z:can conse- the usual way,but no (linear or non-linear) quently be scaled to the optimal proportions of programming is required and no more than one the stock portfolio h,by dividing each zr by point on the "efficient frontier"need ever be computed,given the assumptions under which 23It is clear from a comparison of equations (8)and (I), we are working. showing that sgn 0 sgn A,that only the vectors of values corresponding to>oare relevant to the maximization of 0. Moreover,since 6 as given in (8)and all its first partials shown The Optimum Portfolio When Short Sales in (Io)are continuous functions of the /it follows that when are not Permitted short sales are permitted,any maximum of 6 must be a station- The exclusion of short sales does not compli- ary value,and any stationary value is a maximum(rather than 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 denominator.For the to act on an assumption of no correlations same reason,any maximum of 6 is a unique(global)maximum. between the returns on different stocks.In this 2See Tobin,[21],equation (3.22),p.83.Tobin had,how- ever,formally required no short selling or borrowing,implying case,he finds his best portfolio of "long"holding that this set of equations is valid under these constraints [so by merely eliminating all securities whose A; long as there is a single riskless asset (pp.84-85);but the constraints were ignored in his derivation.We have shown 2s With no covariances,the set of equations (I2)reduces that this set of equations is valid when short sales are properly toλa=/ia=入i,and after summing over all i=I, included in the portfolio and borrowing is available in perfect 2...m,and using the constraint (9),we have immediately markets in unlimited amounts.The alternative set of equi- that|λo|=:|入，andλo>ofor max(instead of min) librium conditions required when short sales are ruled out is 24 Using a more restricted market setting,Hicks [6,p.8or] given immediately below.The complications introduced by has also reached an equivalent result when covariances are borrowing restrictions are examined in the final section of the zero (as he assumed throughout). paper. 27 See,however,footnote 22,above. This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/termsVALUATION OF RISK ASSETS 21 The necessary and sufficient conditions on the relative values of the hi for a stationary and the unique (global) maximum23 are obtained by setting the derivatives in (io) equal to zero, which give the set of equations (I2) Zitij + jZjaxij = i, i = I, 2, . . ,m; where we write (I3) zi = Xhi. It will be noted the set of equations (I 2)- which are identical to those Tobin derived by a different route24 -are linear in the own-vari- ances, pooled covariances, and excess returns of the respective securities; and since the covariance matrix xc is positive definite and hence non- singular, this system of equations has a unique solution (I4) zi? = 2jj where xii represents the i0th element of (x)-l the inverse of the covariance matrix. Using (I3), (7), and (6b), this solution may also be written in terms of the primary variables of the problem in the form (I5) hi? = (XO)-12jrii(j - r*), all i. Moreover, since (I3) implies (i6) 2zi I = X2h i, XO may readily be evaluated, after introducing the constraint (g) as (I 7) 2;i Izi? t i Ihi? I = ?o (~~) ~ = h0 The optimal relative investments zi? can conse- quently be scaled to the optimal proportions of the stock portfolio hi?, by dividing each zi? by the sum of their absolute values. A comparison of equations (i6) and (ii) shows further that: (i8) 2i -zi? I = ? = X?/a o2; i.e. the sum of the absolute values of the zi0 yields, as a byproduct, the value of the ratio of the expected excess rate of return on the optimal portfolio to the variance of the return on this best portfolio. It is also of interest to note that if we form the corresponding X-ratio of the expected excess return to its variance for each ith stock, we have at the optimum: (i9) hi? = (Xi/XO) - 2;j?xijiii where Xi = The optimal fraction of each security in the best portfolio is equal to the ratio of its Xi to that of the entire portfolio, less the ratio of its pooled covariance with other securities to its own vari- ance. Consequently, if the investor were to act on the assumption that all covariances were zero, he could pick his optimal portfolio mix very simply by determining the Xi ratio of the expected excess return xi = i- r* of each stock to its variance xij = rii, and setting each hi= Xi/2Xi; for with no covariances,25 2Ai = XO = 0/1ae2. With this simplifying assumption, the Xi ratios of each stock suffice to determine the optimal mix by simple arithmetic;26 in the more general case with non-zero covariances, a single set27 of linear equations must be solved in the usual way, but no (linear or non-linear) programming is required and no more than one point on the "efficient frontier" need ever be computed, given the assumptions under which we are working. The Optimum Portfolio When Short Sales are not Permitted The exclusion of short sales does not compli- cate the above analysis if the investor is willing to act on an assumption of no correlations between the returns on different stocks. In this case, he finds his best portfolio of "long" holding by merely eliminating all securities whose Xi- 231t is clear from a comparison of equations (8) and (ii), showing that sgn 0 = sgn X, that only the vectors of hi values corresponding to X > o are relevant to the maximization of 0. Moreover, since 0 as given in (8) and all its first partials shown in (io) are continuous functions of the hi, it follows that when short sales are permitted, any maximum of 0 must be a station- ary value, and any stationary value is a maximum (rather than a minimum) when X > o because 0 is a convex function with a positive-definite quadratic torm in its denominator. For the same reason, any maximum of 0 is a unique (global) maximum. 24See Tobin, [2I], equation (3.22), p. 83. Tobin had, how- ever, formally required no short selling or borrowing, implying that this set of equations is valid under these constraints [so long as there is a single riskless asset (pp. 84-85)]; but the constraints were ignored in his derivation. We have shown that this set of equations is valid when short sales are properly included in the portfolio and borrowing is available in perfect markets in unlimited amounts. The alternative set of equi- librium conditions required when short sales are ruled out is given immediately below. The complications introduced by borrowing restrictions are examined in the final section of the paper. 25 With no covariances, the set of equations (I2) reduceS to Xhs = = Xi, and after summing over all i =I, 2 ... m, and using the constraint (9), we have immediately that I X 0 = Ii I Xi 1, and X > o for max 0 (instead of min 0). 26 Using a more restricted market setting, Hicks [6, p. 8oi] has also reached an equivalent result when covariances are zero (as he assumed throughout). 27 See, however, footnote 22, above. This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms