文档介绍：Portfolio Selectiorc 83
Two conditions-at least-must be satisfied before it would be prac-
tical to use efficient surfaces in the manner described above. First, the
investor must desire to act according to the E-V maxim. Second, we
must be able to arrive at reasonable pi and cij. We will return to these
matters later.
Let us consider the case of three securities. In the three security case
our model reduces to
4) Xi>O for i=l,2,3.
From (3) we get
3') X,= 1-X1-Xz
If we substitute (3') in equation (1) and (2) we get E and V as functions
of X1 and Xz. For example we find
1') = CL~+ XI (PI - CLB) + X2 (CLZ - pa)
The exact formulas are not too important here (that of V is given be-
low).* We can simply write
a) E = E (XI, X2)
b) V = V (Xi, X2)
c) X~>O,X~>O,~-X~-X~>O
By using relations (a), (b), (c), we can work with two dimensional
geometry.
The attainable set of portfolios consists of all portfolios which
satisfy constraints (c) and (3') (or equivalently (3) and (4)). The at-
binations of XI, X2 are represented by the triangle abc in
Figure 2. Any point to the left of the Xz axis is not attainable because
it violates the condition that X1 > 0. Any point below the XI axis is
not attainable because it violates the condition that Xz > 0. Any
84 The Journal of Finance
point above the line (1 - XI - XZ = 0) is not attainable because it
violates the condition that X3 = 1 - XI - X2 )/ 0.
We define an isomean curve to be the set of all points (portfolios)
with a given expected return. Similarly an isovariance line is defined to
be the set of all points (portfolios) with a given variance of return.
An examination of the formulae for E and V tells us the shapes of the
isomean and isovariance curves. Specifically they tell us that typicallyg
the isomean curves are a system of parallel straight lines; the isovari-
ance curves are a system of concentric ellipses (see Fig. 2). For example,