bre44380_ch07_162-191.indd 179 09/02/15 04:11 PM
Chapter 7 Introduction to Risk and Return 179invested. Each of the other boxes contains the covariance between that pair of securities,
weighted by the product of the proportions invested.^31(^31) The formal equivalent to “add up all the boxes” is
Portfolio variance = (^) ∑
i=1
N
(^) ∑
j=1
N
xixjσij
Notice that when i = j, σij is just the variance of stock i.
◗ FIGURE 7.13
To find the variance of an
N-stock portfolio, we must
add the entries in a matrix like
this. The diagonal cells con-
tain variance terms (x^2 σ^2 ) and
the off-diagonal cells contain
covariance terms (xixjσij).
Stock
Stock
N
N
1 1 2 3 4 5 6 7
23 45 67
Did you notice in Figure 7.13 how much more important the covariances become as we add
more securities to the portfolio? When there are just two securities, there are equal num-
bers of variance boxes and of covariance boxes. When there are many securities, the number
of covariances is much larger than the number of variances. Thus the variability of a well-
diversified portfolio reflects mainly the covariances.
Suppose we are dealing with portfolios in which equal investments are made in each of N
stocks. The proportion invested in each stock is, therefore, 1/N. So in each variance box we
have (1/N)^2 times the variance, and in each covariance box we have (1/N)^2 times the covari-
ance. There are N variance boxes and N^2 – N covariance boxes. Therefore,
Portfolio variance = N (^) ( __^1
N
(^) )^
2
× average variance
- (N^2 − N) (^) ( ^1
N
(^) )
2
× average covariance
= ^1
N
× average variance + (^) ( 1 − __^1
N
(^) ) × average covariance
EXAMPLE 7.1 ● Limits to Diversification
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