22 THE REVIEW OF ECONOMICS AND STATISTICS ratio is negative,and investing in the remaining (22a)=i=I,2,...m; issues in the proportionsk:=λ：/λ；in accord- where ance with the preceding paragraph. (22b-d)z0≥o,v:0≥o,z,,0-o. But in the more generally realistic cases when covariances are nonzero and short sales are not This system of equations can be expeditiously admitted,the solution of a single bilinear or solved by the Wilson Simplicial Algorithm [23] Now let m'denote the number of stocks with quadratic programming problem is required to determine the optimal portfolio.(All other strictly positive holdings z>o in (226),and points on the“efficient frontier,.”of course, renumber the entire set of stocks so that the continue to be irrelevant so long as there is a subset satisfying this strict inequality [and, riskless asset and a"perfect"borrowing market.) hence also,by (22d)v:=o]are denoted I, The optimal portfolio mix is now given by the 2,...,m'.Within this m'subset of stocks set of which maximize 0 in equation (8) found to belong in the optimal portfolio with posi- subject to the constraint that allo.As tive holdings,we consequently have,using the before,the (further)constraint that the sum of constraint (I9), the h;be unity (equation o)may be ignored in （红7a）2-1mz:0=λ02-1m'h,0=λ0 the initial solution for the relalive values of the so that the fraction of the optimal portfolio in- k:[because 0 in (8)is homogeneous of order vested in the ith stock (where i =I,2...m)is zerol.To find this optimum,we form the (23）h:0=8/入0=z,/Σ-1mz0. Lagrangian function Once again,using (I7a)and (II),the sum of the (2o)(九W）=0+24h zo within this set of stocks held yields as a by- which is to be maximized subject to k;o and product the ratio of the expected excess rate of 4;2o.Using (II),we have immediately return on the optimal portfolio to the variance 8中 of the return on this best portfolio: (2I) ah:≥o口-(ha+h,) (18a）2-1mz,0=λ0=0/g2. +a4,之o. Moreover,since z:>o in (22a and 226)strictly As in the previous cases,we also must have implies v,=o by virtue of(22c),equation (22a) A>o for a maximum (rather than a minimum) for the subset of positively held stocks i=I,2 of中，and we shall write z:=λh:andy:=au .m'is formally identical to equation (12). The necssary and sufficient conditions for the We can,consequently,use these equations to vector of relalive holdings z:which maximizes bring out certain significant properties of the 6 in (20)are consequently,28 using the Kuhn- security portfolios which will be held by risk- Tucker theorem [ol, averse investors trading in perfect markets.20In Equation(22a-22d)can readily be shown to satisfy the the rest of this paper,all statements with respect to six necessary and two further sufficient conditions of the "other stocks"will refer to other stocks included Kuhn-Tucker theorem.Apart from the constraintso within the porlfolio. and o which are automatically satisfied by the com- puting algorithm [conditions (22band 22c)]the four necessary conditions are: III Risk Premiums and Other Properties This condition is satisfied as a sri )「0φ7 of Stocks Held Long or Short in Optimal Portfolios equalily in our solutions by virtue of equation (22a)[See equation (2)].This strict equality also shows that, Since the covariances between most pairs of 2)k:0「8φ7 o=o,the first complementary slackness stocks will be positive,it is clear from equation condition is also satisfied. (Io)that stocks held long (>o)in a port- 3）[9e]●≥o.This coition ssisied because from folio will generally be those whose expected equation (20), The two additional suficiency conditions are of course oby virtue of equation ()This satisfied because the variance-covariance matrix is positive Ld」 definite,making中（飞，a concave function on and same equation shows that the second complementary 中(，w)a convex function of. slackness condition, 29 More precisely,the properties of portfolios when both may be writtenowhich is 4）aJ the investors and the markets satisfy the conditions stated at the outset of section I or,alternatively,when investors also satisfied because of equation (22c)since o. satisfy Roy's premises as noted previously. 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/terms22 THE REVIEW OF ECONOMICS AND STATISTICS ratio is negative, and investing in the remaining issues in the proportions hi = Xi/2Xi in accord- ance with the preceding paragraph. But in the more generally realistic cases when covariances are nonzero and short sales are not admitted, the solution of a single bilinear or quadratic programming problem is required to determine the optimal portfolio. (All other points on the "efficient frontier," of course, continue to be irrelevant so long as there is a riskless asset and a "perfect" borrowing market.) The optimal portfolio mix is now given by the set of hiz which maximize 0 in equation (8) subject to the constraint that all hi > o. As before, the (further) constraint that the sum of the hi be unity (equation 9) may be ignored in the initial solution for the relative values of the hi [because 0 in (8) is homogeneous of order zero]. To find this optimum, we form the Lagrangian function (20) 4(h, u) = 6 + 2iuihi which is to be maximized subject to hi > o and ui > o. Using (ii), we have immediately (2aI) il0 <+ - Xi(hitii + Ijkjti) + au i 0o. As in the previous cases, we also must have X > o for a maximum (rather than a minimum) of p, and we shall write zi = Xhi and vi = aui. The necssary and sufficient conditions for the vector of relative holdings zi0 which maximizes 0 in (20) are consequently,28 using the Kuhn- Tucker theorem [9], (22a) zi0tij + 2jzj -vij- = vi, i = I, 2, . . .m; where (22b-d) zi? > o, vi0 > o, z.0v.0 = a. This system of equations can be expeditiously solved by the Wilson Simplicial Algorithm [23]. Now let m' denote the number of stocks with strictly positive holdings zi? > o in (22b), and renumber the entire set of stocks so that the subset satisfying this strict inequality [and, hence also, by (22d) vi? = o] are denoted i, 2, . . ., i'. Within this m' subset of stocks found to belong in the optimal portfolio with posi- tive holdings, we consequently have, using the constraint (i9), (I7a) li=m'Zi? = o.1i=im'hio = ,o so that the fraction of the optimal portfolio in- vested in the ith stock (where i = I, 2 . .. im') is (23) hi? = zi-/x? = z 0/2i=jm,z1o . Once again, using (I7a) and (ii), the sum of the zio within this set of stocks held yields as a by- product the ratio of the expected excess rate of return on the optimal portfolio to the variance of the return on this best portfolio: (I8a) z ilm'Z 0 = X0 = X0/o Moreover, since zi? > o in (22a and 22b) strictly implies vi? = o by virtue of (22c), equation (22a) for the subset of positively held stocks i = I, 2 . . . m' is formally identical to equation (I2). We can, consequently, use these equations to bring out certain significant properties of the security portfolios which will be held by risk- averse investors trading in perfect markets.29 In the rest of this paper, all statements with respect to "other stocks" will refer to other stocks included within the portfolio. III Risk Premiums and Other Properties of Stocks Held Long or Short in Optimal Portfolios Since the covariances between most pairs of stocks will be positive, it is clear from equation (I9) that stocks held long (hi? > o) in a port- folio will generally be those whose expected 28 Equation (22a-22d) can readily be shown to satisfy the six necessary and two further sufficient conditions of the Kuhn-Tucker theorem. Apart from the constraints h _ o and u > o which are automatically satisfied by the com- puting algorithm [conditions (22b and 22c)] the four necessary conditions are: )o [?] 0 -o. This condition is satisfied as a strict Lhi equality in our solutions by virtue of equation (22a) [See equation (2i)]. This strict equality also shows that, L) h hi ? = -, the first complementary slackness condition is also satisfied. 3) 0 ] > o. This condition is satisfied because from equation (20), [aI = hi > o by virtue of equation (22b). This same equation shows that the second complementary slackness condition, 4) ui [ a]0 = a, may be written ui0 hi0 = o which is als iie also satisfied because of equation (22C) since a /- a. The two additional sufficiency conditions are of course satisfied because the variance-covariance matrix x is positive definite, making 4 (h, u0) a concave function on h and 0 (h0, u) a convex function of u. 29 More precisely, the properties of portfolios when both the investors and the markets satisfy the conditions stated at the outset of section I or, alternatively, when investors satisfy Roy's premises as noted previously. 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