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Well-Diversified Portfolios
Consider the risk of a portfolio of stocks in a single-factor market. We first show that if a
portfolio is well diversified, its firm-specific or nonfactor risk becomes negligible, so that
only factor (or systematic) risk remains. The excess return on an n -stock portfolio with
weights w
i
, S w
i
5 1, is
R
P
5 E(R
P
)
1 b
P
F
1 e
P
(10.3)
where
b
P
5 g w
i
b
i
; E(R
P
)
5 gw
i
E(R
i
)
are the weighted averages of the b
i
and risk premiums of the n securities. The portfolio
nonsystematic component (which is uncorrelated with F ) is e
P
5 S w
i
e
i
, which similarly is
a weighted average of the e
i
of the n securities.
We can divide the variance of this portfolio into systematic and nonsystematic sources:
s
P
2
5 b
P
2
s
F
2
1 s
2
(e
P
)
where s
F
2
is the variance of the factor F and s
2
( e
P
) is the nonsystematic risk of the portfo-
lio, which is given by
s
2
(e
P
)
5 Variance(gw
i
e
i
)
5 gw
i
2
s
2
(e
i
)
Note that in deriving the nonsystematic variance of the portfolio, we depend on the fact
that the firm-specific e
i
s are uncorrelated and hence that the variance of the “portfolio” of
nonsystematic e
i
s is the weighted sum of the individual nonsystematic variances with the
square of the investment proportions as weights.
If the portfolio were equally weighted, w
i
5 1/ n, then the nonsystematic variance would be
s
2
(e
P
)
5 Variance (gw
i
e
i
)
5 a a
1
n b
2
s
2
(e
i
)
5
1
n
a
s
2
(e
i
)
n
5
1
n
s
2
(e
i
)
where the last term is the average value of nonsystematic variance across securities. In
words, the nonsystematic variance of the portfolio equals the average nonsystematic vari-
ance divided by n. Therefore, when n is large, nonsystematic variance approaches zero.
This is the effect of diversification.
This property is true of portfolios other than the equally weighted one. Any portfolio for
which each w
i
becomes consistently smaller as n gets large (more precisely, for which each
w
i
2
approaches zero as n increases) will satisfy the condition that the portfolio nonsystem-
atic risk will approach zero. This property motivates us to define a well-diversified portfolio
as one with each weight, w
i
, small enough that for practical purposes the nonsystematic
variance, s
2
( e
P
), is negligible.
a. A portfolio is invested in a very large number of shares ( n is large). However, one-half of the portfolio
is invested in stock 1, and the rest of the portfolio is equally divided among the other n 2 1 shares. Is
this portfolio well diversified?
b. Another portfolio also is invested in the same n shares, where n is very large. Instead of equally
weighting with portfolio weights of 1/ n in each stock, the weights in half the securities are 1.5/ n
while the weights in the other shares are .5/ n. Is this portfolio well diversified?
CONCEPT CHECK
10.2
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P A R T I I I
Equilibrium in Capital Markets
Because the expected value of e
P
for any well-diversified portfolio is zero, and its
variance also is effectively zero, we can conclude that any realized value of e
P
will be
virtually zero. Rewriting Equation 10.1, we conclude that, for a well-diversified portfolio,
for all practical purposes
R
P
5 E(R
P
)
1 b
P
F
The solid line in Figure 10.1, panel A plots the excess return of a well-diversified port-
folio A with E ( R
A
) 5 10% and b
A
5 1 for various realizations of the systematic factor. The
expected return of portfolio A is 10%; this is where the solid line crosses the vertical axis.
At this point the systematic factor is zero, implying no macro surprises. If the macro factor
is positive, the portfolio’s return exceeds its expected value; if it is negative, the portfolio’s
return falls short of its mean. The excess return on the portfolio is therefore
E(R
A
)
1 b
A
F
5 10% 1 1.0 3 F
Compare panel A in Figure 10.1 with panel B, which is a similar graph for a single stock ( S )
with b
s
5 1. The undiversified stock is subject to nonsystematic risk, which is seen in
a scatter of points around the line. The well-diversified portfolio’s return, in contrast, is
determined completely by the systematic factor.
In a single-factor world, all pairs of well-diversified portfolios are perfectly correlated:
Their risk is fully determined by the same systematic factor. Consider a second well-
diversified portfolio, Portfolio Q, with R
Q
5 E ( R
Q
) 1 b
Q
F. We can compute the standard
deviations of P and Q, as well as the covariance and correlation between them:
s
P
5 b
P
s
F
; s
Q
5 b
Q
s
F
Cov(R
P
, R
Q
)
5 Cov(b
P
F, b
Q
F)
5 b
P
b
Q
s
F
2
r
PQ
5
Cov(R
P
, R
Q
)
s
P
s
Q
5 1
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