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Solutions to Study Problems, Set 4
Chapter 8
6.
a.The standard deviation of each individual stock is given by:
2
/
1
i
2
2
M
2
i
i
)]
e
(
[
σ
+
σ
β
=
σ
Since
β
A
= 0.8,
β
B
= 1.2,
σ
(e
A
) = 30%,
σ
(e
B
) = 40%, and
σ
M
= 22%, we get:
σ
A
= (0.8
2
×
22
2
+ 30
2
)
1/2
= 34.78%
σ
B
= (1.2
2
×
22
2
+ 40
2
)
1/2
= 47.93%
b.
The expected rate of return on a portfolio is the weighted average of the
expected returns of the individual securities:
E(r
P
) = w
A
E(r
A
) + w
B
E(r
B
) + w
f
r
f
where w
A
, w
B
, and w
f
are the portfolio weights for Stock A, Stock B, and
Tbills, respectively.
Substituting in the formula we get:
E(r
P
) = (0.30
×
13) + (0.45
×
18) + (0.25
×
8) = 14%
The beta of a portfolio is similarly a weighted average of the betas of the
individual securities:
β
P
= w
A
β
A
+ w
B
β
B
+ w
f
β
f
The beta for Tbills (
β
f
) is zero.
The beta for the portfolio is therefore:
β
P
= (0.30
×
0.8) + (0.45
×
1.2) + 0
=
0.78
The variance of this portfolio is:
)
e
(
P
2
2
M
2
P
2
P
σ
+
σ
β
=
σ
where
2
M
2
P
σ
β
is the systematic component and
)
e
(
P
2
σ
is the nonsystematic
component.
Since the residuals (e
i
) are uncorrelated, the nonsystematic
variance is:
)
e
(
w
)
e
(
w
)
e
(
w
)
e
(
f
2
2
f
B
2
2
B
A
2
2
A
P
2
σ
+
σ
+
σ
=
σ
= (0.30
2
×
30
2
) + (0.45
2
×
40
2
) + (0.25
2
×
0) = 405
where
σ
2
(e
A
) and
σ
2
(e
B
) are the firmspecific (nonsystematic) variances of
Stocks A and B, and
σ
2
(e
f
), the nonsystematic variance of Tbills, is zero.
The residual standard deviation of the portfolio is thus:
σ
(e
P
) = (405)
1/2
= 20.12%
The total variance of the portfolio is then:
47
.
699
405
)
22
78
.
0
(
2
2
2
P
=
+
×
=
σ
1
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View Full DocumentThe standard deviation is 26.45%.
CFA Problem 2:
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 Spring '09
 TongYao
 Management

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