UNIVERSITY OF TORONTO
Joseph L. Rotman School of Management
April 30, 2007
Davydenko/Derrien/
MGT337Y
FINAL EXAMINATION
Florence/Lu/Nevard
SOLUTIONS
1. (a) Let
S
be the price of the security, we have
S
=
100
1 +
r
+
101
(1 +
r
)
2
+
· · ·
+
199
(1 +
r
)
100
.
(1)
Multiply both sides of (1) by (1 +
r
), we obtain
(1 +
r
)
S
= 100 +
101
1 +
r
+
· · ·
+
199
(1 +
r
)
99
.
(2)
Subtract (1) from (2), we have
rS
= 100 +
1
1 +
r
+
1
(1 +
r
)
2
+
· · ·
+
1
(1 +
r
)
99

199
(1 +
r
)
100
⇒
rS
= 100 +
A
99
r

199
(1 +
r
)
100
⇒
S
=
100 +
A
99
r
r

199
r
(1 +
r
)
100
.
The second equality follows because the terms in the middle on the right hand side
are the present value of an annuity of $1 for 99 years.
Putting
r
= 0
.
08, we have
S
= $1405
.
04.
(b) The monthly interest rate is
r
m
= 0
.
12
/
12 = 0
.
01. Let
x
be the monthly payment,
the present value of the twelve payments must be equal to $10000. This implies
xA
12
r
m
= $10000
⇒
x
=
$10000
A
12
0
.
01
=
$10000
11
.
2551
= $888
.
49
.
(3)
Therefore, the monthly payment is $888.49.
(c) (i) Since this loan requires you to pay $900/month whereas the loan in part (b)
only requires you to pay $888.49/month.
You obviously would prefer the 12% loan
that is compounded on a monthly basis.
1
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(ii) For the last monthly payment, you will pay an interest of $10.26, so the principal
repayment is $900

$10
.
26 = $889
.
74. On a monthly basis, the interest rate is
r
m
=
10
.
26
889
.
74
= 0
.
01153
.
Therefore, the effective annual interest rate is
r
= (1 +
r
m
)
12

1 = 0
.
1475
,
which is much higher than the quoted 8%/year. This explains why you prefer the loan
in part in (b).
2. (a) The decomposition of the total risk of Robotoy is
σ
2
R
=
β
2
R
σ
2
M
+
σ
2
,
(4)
where the first term on the right hand side is the market risk of Robotoy and the
second term is the unique risk of Robotoy. It follows that the market risk of Robotoy
is (1
.
39)
2
×
(0
.
1802)
2
= 0
.
06274 and the unique risk of Robotoy is
σ
2
R

β
2
R
σ
2
M
=
(0
.
4021)
2

0
.
06274 = 0
.
09895.
(b) If you hold just one stock, Robotoy is indeed riskier than Mobitoy. However, if you
include Robotoy in a well diversified portfolio (say the market portfolio), then only
its systematic risk (beta) matters.
In this case, Robotoy is less risky than Mobitoy
because it has a lower beta. This explains why the expected return of Robotoy is lower
than that of Mobitoy.
(c) For a portfolio with 55% in Robotoy and 45% in Mobitoty, its expected return and
standard deviation are given by
μ
p
=
0
.
55
×
0
.
1601 + 0
.
45
×
0
.
2092 = 0
.
1822
,
σ
p
=
(0
.
55)
2
(0
.
4021)
2
+ (0
.
45)
2
(0
.
3422)
2
+ 2(0
.
55)(0
.
45)(0
.
6)(0
.
4021)(0
.
3422)
1
2
=
0
.
3369
.
(d) In order to find out the expected return of the portfolio, we need to know three
things: its beta, the riskfree rate, and the risk premium on the market portfolio. We
first figure out the composition of the portfolio.
Let
x
M
be the percentage that is
invested in the market portfolio, we have
σ
p
=
x
M
σ
M
Since the portfolio has a standard deviation of 20% and
σ
M
= 18
.
02%, we find that
x
M
= 20
/
18
.
02 = 1
.
11. As the beta of the portfolio is a weighted average of the beta
of the individual assets in the portfolio, we have
β
p
=
x
M
β
M
= 1
.
11
×
1 = 1
.
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 Spring '11
 SabrinaButti
 Dividend, share price, Market Portfolio

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