of rigor and
generality. The chief limitations from this source are (1) we do not
derive our results analytically for the n-security case; instead, we
present them geometrically for the 3 and 4 security cases; (2) we assume
static probability beliefs. In a general presentation we must recognize
that the probability distribution of yields of the various securities is a
function of time. The writer intends to present, in the future, the general,
mathematical treatment which removes these limitations.
We will need the following elementary concepts and results of
mathematical statistics:
Let Y be a random variable, i.e., a variable whose value is decided by
chance. Suppose, for simplicity of exposition, that Y can take on a
finite number of values yl, yz, . . . ,y,~L. et the probability that Y =
5. U'illiams, op. cit., pp. 68, 69. 80 The Journal of Finance yl, be pl; that Y = y2 be pzetc. The expected value (or mean) of Y is defined to be
The variance of Y is defined to be
V is the average squared deviation of Y from its expected value. V is a
commonly used measure of dispersion. Other measures of dispersion,
closely related to V are the standard deviation, u = .\\/V and the coefficient of variation, a/E.
Suppose we have a number of random variables: R1, . . . ,R,. If R is
a weighted sum (linear combination) of the Ri then R is also a random variable. (For example R1, may be the number
which turns up on one die; R2, that of another die, and R the sum of
these numbers. In this case n = 2, a1 = a2 = 1). It will be important for us to know how the expected value and
variance of the weighted sum (R) are related to the probability distribution
of the R1, . . . ,R,. We state these relations below; we refer
the reader to any standard text for proof.6
The expected value of a weighted sum is the weighted sum of the
expected values. I.e., E(R) = alE(R1) +aZE(R2) + . . . + a,E(R,)
The variance of a weighted sum is not as simple. To express it we must
define \i.e., the expected value of [(the deviation of R1 from its mean) times
(the deviation of R2 from its mean)]. In general we define the covariance betweenRi and R as
~ i=jE( [Ri-E (Ri) I [Ri-E (Rj)I f uij may be expressed in terms of the familiar
correlation coefficient
(pij). The covariance between Ri and Rj is equal to [(their correlation)
times (the standard deviation of Ri) times (the standard deviation of Rj)l:
U i j = P i j U i U j
6. E.g.,J. V. Uspensky, Introduction to mathematical Probability (New York: McGraw-
Hill, 1937), chapter 9, pp. 161-81.
Portfolio Selection The variance of a weighted sum is
If we use the fact that the variance of Ri is uii then Let Ri be the return on the iN\security. Let pi be the expected vaIue
ofRi; uij, be the covariance between Ri and Rj (thus uii is the variance
ofRi). Let Xi be the percentage of the investor's assets which are allocated
to the ithsecurity. The yield (R) on the portfolio as a whole is
The Ri (and consequently R) are considered to be random
variables.'
The Xi are not random variables, but are fixed by the investor. Since
the Xi are percentages we have ZXi= 1. In our analysis we will exclude
negative values of the Xi (i.e., short sales); therefore Xi >0 for alli. The return (R) on the portfolio as a whole is a weighted sum of random
variables (where the investor can choose the weights). From our
discussion of such weighted sums we see that the expected return E from the portfolio as a whole is and the variance is
7. I.e., we assume that the investor does (and should) act as if he had probability beliefs
concerning these variables. I n general we ~vouldexpect that the investor could tell us, for
any two events (A and B), whether he personally considered A more likely than B, B more