Section 7.1 introduced the concept of diversification and the limits to the benefits of diver-
sification resulting from systematic risk. Given the tools we have developed, we can recon-
sider this intuition more rigorously and at the same time sharpen our insight regarding the
C H A P T E R
7
Optimal Risky Portfolios
227
Recall from Equation 7.16, restated here, that the general formula for the variance of a
portfolio is
s
p
2
5 a
n
i
51
a
n
j
51
w
i
w
j
Cov(r
i
, r
j
)
(7.16)
Consider now the naive diversification strategy in which an
equally weighted portfolio is
constructed, meaning that w
i
5 1/ n for each security. In this case Equation 7.16 may be
rewritten as follows, where we break out the terms for which i 5 j into a separate sum,
noting that Cov(r
i
, r
i
)
5 s
i
2
:
s
p
2
5
1
n a
n
i
51
1
n
s
i
2
1 a
n
j
51
j
2i
a
n
i
51
1
n
2
Cov(r
i
, r
j
)
(7.17)
Note that there are
n variance terms and
n (
n 2 1) covariance terms in Equation 7.17.
If we define the average variance and average covariance of the securities as
s
2
5
1
n a
n
i
51
s
i
2
(7.18)
Cov
5
1
n(n
2 1) a
n
j
51
j
2i
a
n
i
51
Cov(r
i
, r
j
)
(7.19)
we can express portfolio variance as
s
p
2
5
1
n
s
2
1
n
2 1
n
Cov
(7.20)
Now examine the effect of diversification. When the average covariance among security
returns is zero, as it is when all risk is firm-specific, portfolio variance can be driven to zero.
We see this from Equation 7.20. The second term on the right-hand side will be zero in this
scenario, while the first term approaches zero as n becomes larger. Hence when security
returns are uncorrelated, the power of diversification to reduce portfolio risk is unlimited.
However, the more important case is the one in which economywide risk factors impart
positive correlation among stock returns. In this case, as the portfolio becomes more highly
diversified ( n increases), portfolio variance remains positive. Although firm-specific risk,
represented by the first term in Equation 7.20, is still diversified away, the second term
simply approaches Cov as n becomes greater. [Note that ( n 2 1)/ n 5 1 2 1/ n, which
approaches 1 for large n. ] Thus the irreducible risk of a diversified portfolio depends on
the covariance of the returns of the component securities, which in turn is a function of the
importance of systematic factors in the economy.
To see further the fundamental relationship between systematic risk and security corre-
lations, suppose for simplicity that all securities have a common standard deviation, s , and
all security pairs have a common correlation coefficient, r . Then the covariance between
all pairs of securities is r s
2
, and Equation 7.20 becomes
s
p
2
5
1
n
s
2
1
n
2 1
n
rs
2
(7.21)
The effect of correlation is now explicit. When r 5 0, we again obtain the insurance
principle, where portfolio variance approaches zero as n becomes greater. For r . 0, how-
ever, portfolio variance remains positive. In fact, for r 5 1, portfolio variance equals s
2
regardless of
n, demonstrating that diversification is of no benefit: In the case of perfect
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228
P A R T I I
Portfolio Theory and Practice
correlation, all risk is systematic. More generally, as n becomes greater, Equation 7.21
shows that systematic risk becomes r s
2
.
Table 7.4 presents portfolio standard deviation as we include ever-greater numbers of
securities in the portfolio for two cases, r 5 0 and r 5 .40. The table takes s to be 50%. As
one would expect, portfolio risk is greater when r 5 .40. More surprising, perhaps, is that
portfolio risk diminishes far less rapidly as n increases in the positive correlation case. The
correlation among security returns limits the power of diversification.
Note that for a 100-security portfolio, the standard deviation is 5% in the uncorrelated
case—still significant compared to the potential of zero standard deviation. For r 5 .40,
the standard deviation is high, 31.86%, yet it is very close to undiversifiable systematic
risk in the infinite-sized security universe,
"rs
2
5 ".4 3 50
2
5 31.62%. At this point,
further diversification is of little value.
Perhaps the most important insight from the exercise is this: When we hold diversified
portfolios, the contribution to portfolio risk of a particular security will depend on the
covariance of that security’s return with those of other securities, and not on the security’s
variance. As we shall see in Chapter 9, this implies that fair risk premiums also should
depend on covariances rather than total variability of returns.
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