1
U
NIVERSITY OF
N
ORTH
C
AROLINA
K
ENAN
-F
LAGLER
B
USINESS
S
CHOOL
BUSI 580: INVESTMENTS
P
ART
I:
P
ORTFOLIO
T
HEORY AND
A
SSET
P
RICING
Prof. Günter Strobl
Spring 2010
Solution to Problem Set 2
A.
The Capital Asset Pricing Model
1.
If the stock’s correlation coefficient with the market portfolio doubles (with all other
variables such as variances unchanged), then beta, and therefore the risk premium,
will also double. The current risk premium is 14% - 6% = 8%. The new risk premium
will be 16% and the new discount rate for the security will be 16% + 6% = 22%.
If the stock pays a constant perpetual dividend, then we know that the dividend (D)
must satisfy the equation for the present value of a perpetuity:
Price = Dividend / Discount rate
→
D =
$50 × 14% = $7
At the new discount rate of 22%, the stock will be worth $7 / 0.22 = $31.82.
2.
(a)
σ
p
2
= 0.75
2
× 0.4
2
+ 0.25
2
× 0.3
2
+ 2 × 0.75 × 0.25 × 0.02 = 0.103
(b)
β
1
=
σ
1,M
/
σ
M
2
= 0.064 / 0.04 = 1.6
β
2
=
σ
2,M
/
σ
M
2
= 0.032 / 0.04 = 0.8
β
p
= 0.75 ×
β
1,M
+ 0.25 ×
β
2,M
= 1.4
(c) R
1
2
=
β
1
2
σ
M
2
/
σ
1
2
= 1.6
2
× 0.04 / 0.16 = 0.64
R
2
2
=
β
2
2
σ
M
2
/
σ
2
2
= 0.8
2
× 0.04 / 0.09 = 0.284
R
p
2
=
β
p
2
σ
M
2
/
σ
p
2
= 1.4
2
× 0.04 / 0.103 = 0.761
3.
In a world with only two risky securities, we know that the weight of stock A in the
optimal risky portfolio (tangency portfolio) is given by:
1
1
See the lecture notes on portfolio theory.

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