H
AAS
S
CHOOL
OF
B
USINESS
U
NIVERSITY
OF
C
ALIFORNIA
AT
B
ERKELEY
UGBA 103
A
VINASH
V
ERMA
S
OLUTION
TO
H
OMEWORK
10
1.
Assume that the CAPM holds.
E
p
R
f
σ
p
M
(Market Portfolio)
Security i
Portfolio a
Portfolio q
Security j
Securityk
[4 points each for a total of 44 points for (a) through (k) in 1.1 and 1.2]
1.1.
Indicate whether the following statements are true or false based on what you can infer from the
graph above
without scaling it
.
(a).
Portfolio q
has no diversifiable risk.
True.
Portfolio
q
is on the CML. Therefore, it is a portfolio of the risk free asset
and the market portfolio. The risk free asset has no risk, and the market
portfolio has no diversifiable risk. Therefore, portfolio q has no diversifiable risk.
(b).
1
=
aM
β
.
False.
If
β
aM
were equal to one, then by CAPM,
E
a
would have to EQUAL
E
M
.
However we can see from the graph that
E
a
>
E
M
.
(c).
Portfolio q
is a portfolio in which we have invested positive amounts both in the risk free asset
and in the market portfolio.
True.
Portfolio q is on the CML. Therefore, (
E
q
–
R
f
) /
σ
q
= (
E
M
–
R
f
) /
σ
M
. Since
Portfolio
q
is a portfolio of the risk free asset and the market portfolio,
σ
q
=
x
M
*
σ
M
. We can see from the graph that
σ
q
<
σ
M
x
M
*
σ
M
<
σ
M
0<
x
M
< 1,
and since the two portfolio weights sum up to one, 0<
x
Rf
< 1.
(d).
Security i
has no systematic risk.
True.
We can see from the graph that
E
i
, the expected return on Security
i
,
equals the risk free rate. Given CAPM and
E
i
=
R
f
,
β
iM
, = 0. By definition,
systematic risk is given by
β
iM
2
*
σ
M
2
, which is zero given that
β
iM
, = 0.
(e).
M
a
aM
σ
σ
σ
*
=
.
True.
Since Portfolio
a
is on the CML, (
E
a
–
R
f
) /
σ
a
= (
E
M
–
R
f
) /
σ
M
(
E
a
–
R
f
) = (
σ
a
/
σ
M
)*(
E
M
–
R
f
). Also by CAPM, (
E
a
–
R
f
) =
β
aM
*(
E
M
–
R
f
).
Therefore, (
σ
a
/
σ
M
) =
β
aM
. By using the definition of beta,
β
aM
=
σ
aM
/
σ
M
2
.
Therefore, (
σ
a
/
σ
M
) =
β
aM
=
σ
aM
/
σ
M
2
. Crossmultiplying, we get
σ
aM
=
σ
a
*
σ
M
.
(f).
2
M
kM
σ
σ
False.
We can see from the graph that
E
k
, the expected return on Security
k
,
equals
E
M
, the expected return on the market portfolio. Given CAPM, and the
fact that
E
k
=
E
M
,
β
kM
, = 1. By using the definition of beta,
β
kM
=
σ
kM
/
σ
M
2
= 1
σ
kM
=
σ
M
2
.
(g).
0<
ρ
qM
<1
False.
Since Portfolio
q
is on the CML, (
E
q
–
R
f
) /
σ
q
= (
E
M
–
R
f
) /
σ
M
(
E
q
–
R
f
) = (
σ
q
/
σ
M
)*(
E
M
–
R
f
). Also by CAPM, (
E
q
–
R
f
) =
β
qM
*(
E
M
–
R
f
).
Therefore, (
σ
q
/
σ
M
) =
β
qM
. By using the definition of beta,
β
qM
=
σ
qM
/
σ
M
2
.
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 Spring '10
 CHOW
 Capital Asset Pricing Model, Financial Markets, Market Portfolio

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