likely than A, or both equally likely. If the investor were consistent in his opinions on such
matters, he would possess a system of probability beliefs. We cannot expect the investor
to be consistent in every detail. We can, however, expect his probability beliefs to be
roughly consistent on important matters that have been carefully considered. We should
also expect that he will base his actions upon these probability beliefs-even though they be in part subjective.
This paper does not consider the difficult question of how investors do (or should) form their probability beliefs. 82 The Journal of Finance For fixed probability beliefs (pi, oij) the investor has a choice of various
combinations of E and V depending on his choice of portfolio
XI, . . . ,XN.Suppose that the set of all obtainable
(E, V) combinations
were as in Figure 1.The E-V rule states that the
investor would
(or should) want to select one of those portfolios which give rise to the
(E, V) combinations indicated as efficient in the figure; i.e., those with
minimum V for given E or more and maximum E for given V or less.
There are techniques by which we can compute the set of efficient
portfolios and efficient (E, V) combinations associated with given pi attainable E, V combinations
andoij. We will not present these techniques here. We will, however,
illustrate geometrically the nature of the efficient surfaces for cases
in which N (the number of available securities) is small.
The calculation of efficient surfaces might possibly be of practical
use. Perhaps there are ways, by combining statistical
techniques and
the judgment of experts, to form reasonable probability beliefs (pi,
aij).We could use these beliefs to compute the
attainable efficient
combinations of (E, V). The investor, being informed of what (E, V)
combinations were attainable, could state which he desired. We could
then find the portfolio which gave this desired combination.
Portfolio Selection 83
Two conditions-at least-must be satisfied before it would be practical
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 uij. 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') Xs= 1-XI--Xz
Ifwe substitute (3') in equation (1)and (2) we get E and V as functions
of X1 and Xz. For example we find 1') E' =~3 +x1(111 -~ 3 +) x2 (112 -113) The exact formulas are not too important here (that of V is given below). 8 We can simply write
a) E =E (XI, Xd b) V = V (Xi, Xz)
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 attainable
combinations of XI, X2 are represented by the triangle
abcin
Figure 2. Any point to the left of the Xz axis is not attainable because
it violates the condition that X1 3 0. Any point below the X1 axis is
not attainable because it violates the condition that Xz3 0. Any
84 The Journal of Finance point above the line (1 - X1 - Xz= 0) is not attainable because it
violates the condition that X3 = 1 - XI - Xz>0. We define an isomeancurve to be the set of all points (portfolios)
with a given expected return. Similarly an
isovarianceline 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
theisomean curves are a system of parallel straight