theN security case, a series of connected line segments. At one end of
the efficient set is the point of minimum variance; at the other end is
a point of maximum expected returnlo (see Fig. 4). Now that we have seen the nature of the set of efficient portfolios,
it is not difficult to see the nature of the set of efficient (E, V) combinations.
In the three security case E = a0 +alXl+a2X2 is a plane; V =
bo+blX1 +hX2 +b12XlX2 +b1lx:+~BX;is a paraboloid.ll As shown in Figure 5, the section of the E-plane over the efficient portfolio
set is a series of connected line segments. The section of the V-paraboloid
over the efficient portfolio set is a series of connected parabola
segments. If we plotted V against E for efficient portfolios we would
again get a series of connected parabola segments (see Fig. 6). This result
obtains for any number of securities.
Various reasons recommend the use of the expected return-variance
of return rule, both as a hypothesis to explain well-established investment
behavior and as a maxim to guide one's own action. The rule
serves better, we will see, as an explanation of, and guide to, \
as distinguished from ('speculative\10. Just as we used the equation 5Xi = I to reduce the dimensionality in the three i=1
security case, we can use it to represent the four security case in 3 dimensional space.
Eliminating X, we get E = E(X1, Xz, Xs), V = V(X1, Xz, Xs). The attainable set is represented,
in three-space, by the tetrahedron with vertices (O,0, O), (0,0, I), (0,1, O), (1,0, O),
representing portfolios with, respectively, X4 = 1, Xs= 1, Xz= 1, XI = 1.
Let sisa be the subspace consisting of all points with
X4 = 0. Similarly we can define
sol, . . . ,aa to be the subspace consisting of all points with Xi = 0, i # a ~ ,. . . ,aa. For
each subspace sol, . . . ,aa we can define a critical lilzelal, . . . aa. This line is the locus of points P where P minimizes V for all points in sol, .. . ,aa with the same E as P. If a point is in s,l, .. . ,aa and is efficient it must be on lal, . . .,aa. The efficient set may be traced out by starting at the point of minimum available variance, moving continuously along
variouslal, . . . ,aa according to definite rules, ending in a point which gives maximum E.
As in the two dimensional case the point with minimum available variance may be in the
interior of the available set or on one of its boundaries. Typically we proceed along a given critical line until either this line intersects one of a larger subspace or meets a boundary
(and simultaneously the critical line of a lower dimensional subspace). In either of these
cases the efficient line turns and continues along the
new line. The efficient line terminates when a point with maximum E is reached. 11. See footnote 8.
v efficient E, V eombinofionr E Portfolio Selection 89
Earlier we rejected the expected returns rule on the grounds that it
never implied the superiority of diversification. The expected returnvariance
of return rule, on the other hand, implies diversification for a
wide range of pi, aij.This does not mean that the E-V rule never implies
the superiority of an undiversified portfolio. It is conceivable that
one security might have an extremely higher yield and lower variance
than all other securities; so much so that one particular undiversified
portfolio would give maximum E and minimum V. Rut for a large,
presumably representative range of pi, aijthe E-V rule leads to efficient
portfolios almost all of which are diversified. Not only does the E-V hypothesis imply diversification, it implies
the \reason.'' The adequacy
of diversification is not thought by investors to depend solely on the
number of different securities held. A portfolio with sixty different railway
securities, for example, would not be as well diversified as the same
size portfolio with some railroad, some public utility, mining, various
sort of manufacturing, etc. The reason is that it is generally more
likely for firms within the same industry to do poorly at the same time
than for firms in dissimilar industries.