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Winter 10 Actsc 372 Assignment 4 Solution
5 Marks for each part. (total 60 marks)
1 a.
Let:
X
1
= the proportion of Security 1 in the portfolio and
X
2
= the proportion of Security 2 in the portfolio
and note that since the weights must sum to 1.0,
X
1
= 1 – X
2
Recall that the beta for a portfolio (or in this case the beta for a factor) is the
weighted average of the security betas, so
β
P1
= X
1
β
11
+ X
2
β
21
β
P1
= X
1
β
11
+ (1–X
1
)
β
21
Now, apply the condition given in the hint that the return of the portfolio does not
depend on F
1
. This means that the portfolio beta for that factor will be 0, so:
β
P1
= X
1
β
11
+ X
2
β
21
β
P1
= 0 = X
1
(1.0) + (1 – X
1
)(0.5)
and solving for X
1
and X
2
:
X
1
= – 1, X
2
= 2
Thus, sell short Security 1 and buy Security 2.
To find the expected return on that portfolio, use
R
P
= X
1
R
1
+ X
2
R
2
so applying the above:
E(R
P
) = –1(20%) + 2(20%) = 20%
β
P1
= –1(1) + 2(0.5) = 0
b.
Following the same logic as in part a, we have
β
P2
= 0 =
β
P1
= X
3
β
31
+ (1–X
3
)
β
41
β
P2
= 0 = X
3
(1) + (1 – X
3
)(1.5)
and
X
3
= 3, X
4
= –2
Thus, sell short Security 4 and buy Security 3. Then,
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View Full DocumentE(R
P2
) = 3(10%) + (–2)(10%) = 10%
β
P2
= 3(0.5) – 2(0.75) = 0
Note that since both β
P1
and β
P2
are 0, this is a risk free portfolio!
c.
The portfolio in part b provides a risk free return of 10%, which is higher than
the 5% return provided by the risk free security. To take advantage of this
opportunity, borrow at the risk free rate of 5% and invest the funds in a portfolio
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 Spring '10
 Tan,KenS

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