E 120 : Problem Set 10 Solutions  Fall 2010
Problem 1
In the Markowitz model, the minimum standard deviation for a desired expected return is
given by the Capital Market Line. In this case, we have
ρ
= 0
.
10,
r
0
= 0
.
05,
ρ
M
= 0
.
15, and
σ
M
=
σ
(
ρ
M
) = 0
.
18. The equation of the Capital Market Line is given by
ρ
=
r
0
+
ρ
M

r
0
σ
M
σ
(
ρ
)
=
⇒
σ
(
ρ
) =
σ
M
ρ

r
0
ρ
M

r
0
= 0
.
18
×
0
.
10

0
.
05
0
.
15

0
.
05
= 0
.
09
which is the minimum standard deviation you can achieve.
Problem 2
In this case, we have
ρ
= 0
.
20,
r
0
= 0
.
05,
ρ
M
= 0
.
15, and
σ
M
=
σ
(
ρ
M
) = 0
.
18. Using the
Capital Market Line, the minimum standard deviation that can be achieved is given by
σ
(
ρ
) =
σ
M
ρ

r
0
ρ
M

r
0
= 0
.
18
×
0
.
20

0
.
05
0
.
15

0
.
05
= 0
.
27 or 27%
>
20%
Hence, it is not possible to achieve an expected return of 20% and a standard deviation of
20%
Problem 3
(a) We have
r
0
= 0
.
04,
ρ
= 0
.
20 and
r
=
0
.
15
0
.
20
=
⇒
˜
ρ
=
ρ

r
0
= 0
.
16 and ˜
r
=
r

r
0
e
=
0
.
11
0
.
16
Also,
V
=
0
.
01
0
0
0
.
0025
It follows that
w
(
ρ
) =
˜
ρ
˜
r
T
V

1
˜
r
V

1
˜
r
=
0
.
1537
0
.
8943
So the minimum variance portfolio with an expected return of 0
.
20 is given by
w
A
=
0
.
1537,
w
B
= 0
.
8943 and
w
0
= 1

e
T
w
(
ρ
) =

0
.
048.
Hence, Jack should borrow
$1000
×
0
.
048 = $48 from the bank, and invest $1000
×
0
.
1537 = $153
.
70 in Stock A
and $1000
×
0
.
8943 = $894
.
30 in Stock B.
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 Fall '08
 ILAN
 Standard Deviation, Capital Asset Pricing Model, Modern portfolio theory, ρm

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