马克维茨投资组合选择(4)

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

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