Solution to Problem Set 3
Investments
Prof. PierreOlivier Weill
1.
(a) The return on the risk free asset is given as 8%. The standard deviation
of that return is 0 by definition, since the asset is risk free.
(b) Expected return is given by:
E
(
R
p
)
=
w
M
E
(
R
M
) +
w
f
E
(
R
f
)
=
(
.
5)(
.
16) + (
.
5)(
.
08) =
.
12
Because the standard deviation of the return on the risk free asset is 0,
the standard deviation of the portfolio is:
σ
p
=
w
M
σ
M
= (
.
5)(
.
10) =
.
05 = 5%
(c) The standard deviation of return will be equal to:
σ
p
=
w
M
σ
M
= (1
.
25)(
.
10) = 0
.
125 = 12
.
5%
Expected return will be equal to:
E
(
R
p
) =
w
M
E
(
R
M
) + (1

w
M
)
R
f
= 1
.
25(
.
16) + (

.
25)(
.
08) =
.
18
This result can also be obtained using:
E
(
R
p
) =
R
f
+
E
(
R
M
)

R
f
σ
M
σ
p
=
.
08 +
.
16

.
08
.
10
.
125 =
.
18
(d) From above we have:
σ
p
=
w
M
σ
M
for the risk of the portfolio. The question asks for
w
M
and
w
f
that
produces
σ
p
= 2
σ
M
. Substituting 2
σ
p
for
σ
M
into the equation gives:
2
σ
M
=
w
M
σ
M
This implies
w
M
= 2
1
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We also know that
w
f
= 1

w
M
= 1

2 =

1
This says the following in words: To produce a portfolio that is twice as
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 Spring '08
 Miyakawa
 Standard Deviation, Russell Fund, wM σM

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