5136437447

able seŁ In ttus case ttiere is a security wcucn aoes not en ter mto any effident portfolio. (2) Two securities have tbe same ji_{. In this case the isomean lines are paraUcl to a boundary linę. It may happen that the effident portfolio with marimum Eis a. diversified portfolio. (3) A case wherein oniy one portfolio is effident.

The effident set in the 4 security case is, as in the 3 security and also the A⁷ security case, a series of connected linę segments. At one end of the effident set is the point of minimum variance; at the other end is a point of majomum expectcd return¹⁰ (see Fig. 4).

No w that -we have seen the naturę of the set of effident portfolios, it is not difficult to see the naturę of the set of effident (E, V) combma-tions. In the three security case E = a₀ + <hXi + <hXt is a piane; V — h + hXi + hX₂ + boXA + buX\ + is a paraboloid.¹ As shown in Figurę 5, the section of the E-plane over the effident portfolio set is a series of connected linę segments. The section of the F-parab-oloid over the effident portfolio set is a series of connected parabola segments. If we plotted V against E for effident portfolios we would again get a series of connected parabola segments (see Fig. 6). This re-sult obtains for any number of securities.

Various reasons recommend the use of the expected retum-variance of return rule, both as a hypothesis to explain well-established invest-ment behavior and as a marim to guide one’s own action. The rule serves better, we will see, as an explanation of, and guide to, “invest-ment” as distinguished from “speculative” behavior.

4

10.    Just as we used the eąuation ^ Xt «■ 1 to reduce the dimensionality in the three

*-i

security case, we can use it to represent tbe four security case in 3 dimensional space. Eluninating Xt we get E * E{X_t, Xt, Xi), V — V(X,, Xj, Xx). The attainable set is rep-resented, in three-space, by the tetrahedron with vertices (0,0,0), (0,0,1), (0,1,0), (1,0,0), representing portfolios with, respcctively, X_t « 1, X% -» 1, X_t = 1, Xi ~ 1.

Lct Sm be the subspace conasting of all points with X* — 0. Similarly we can denne są,oa to be the subspace consisting of all points with Xt «* 0, * j* a_lt ..., aa. For e3ch subspace s_a i,... ,aa we can define a criluallinc lat,... aa. This linę is the Jocus of points P wbere P nunirnizesF for all points mSoi,...,aa with the same E as P. Ii a point is in Soi, . - . ,oa and is cfficicnt it mtist be on loi,...» aa. The effident set may be traced outby stazting at the point of minimum availablc vari3nce, moving continuously along various la,,..., aa according to delinite raks, ending in a point which gives manmum E. As in the two dimensional case the point with minimum available v ariance may be in the interior of the ayailable set ot on one of its boundaries. Typically we proceed along a givcn critical linę until either this linę intcrsccts one of a larger subspace or mects a boundary (and simultaneously the critical linc of a lower dimensional subspace). In either of thcsc cascs the effident linę turns and continucs along the ncw linę. The effident linę terminates when a point with marimum E is reached.

1

   See footnote 8.