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UNIVERSITY OF TORONTO
Joseph L. Rotman School of Management
RSM332
Tutorial #8 Problem Set
Nov.10/Nov.11 2011
1. There are two stocks,
A
and
B
. The table below gives next year’s returns of the two
stocks,
R
A
and
R
B
, depending on the state of the world, and the probability of each
state of the world. You also know that
E
(
R
A
) = 8
.
5% and
E
(
R
B
) = 6
.
9%. Finally,
you can borrow and lend at the riskfree interest rate
r
f
= 3%.
Probability
R
A
R
B
r
f
Expansion
0.2
20%
3%
Normal
0.5
12%
7%
3%
Recession
6%
3%
(a) Fill in the blanks in the table. (5 marks)
(b) Compute the variances of
R
A
and
R
B
and the correlation between them. (5 marks)
(c) What is the standard deviation of a portfolio that combines
A
and
B
(but not the
riskfree asset) and has the expected return of 7.3%? (5 marks)
(d) What is the correlation between
R
A
and returns on an equallyweighted portfolio of
A
and the riskfree asset (i.e., portfolio weights are 0.5 for each of the two)? (5 marks)
Solution:
(a) Probabilities need to sum up to one, so the missing probability is 10.20.5=0.3.
We should then use the formula for the expected value to compute the missing returns:
E
(
R
A
) = 0
.
2
×
0
.
2 + 0
.
5
×
0
.
12 + 0
.
3
×
x
= 0
.
085
⇒
x
=

0
.
05
E
(
R
B
) = 0
.
2
×
y
+ 0
.
5
×
0
.
07 + 0
.
3
×
0
.
06 = 0
.
069
⇒
y
= 0
.
08
where
x
and
y
are missing returns on A and B, respectively.
Probability
R
A
R
B
r
f
Expansion
0.2
20%
8%
3%
Normal
0.5
12%
7%
3%
Recession
0.3
5%
6%
3%
1
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View Full Document (b)
σ
2
A
= 0
.
2
×
(0
.
2

0
.
085)
2
+ 0
.
5
×
(0
.
12

0
.
085)
2
+ 0
.
3
×
(

0
.
05

0
.
085)
2
= 0
.
008725
σ
2
B
= 0
.
2
×
(0
.
08

0
.
069)
2
+ 0
.
5
×
(0
.
07

0
.
069)
2
+ 0
.
3
×
(0
.
06

0
.
069)
2
= 0
.
000049
Cov[
R
A
,R
B
] = 0
.
2
×
(0
.
2

0
.
085)(0
.
08

0
.
069) + 0
.
5
×
(0
.
12

0
.
085)(0
.
07

0
.
069) +
+ 0
.
3
×
(

0
.
05

0
.
085)(0
.
06

0
.
069) = 0
.
000635
ρ
A,B
=
Cov[
R
A
,R
B
]
σ
A
σ
b
= 0
.
97
.
(c) We ﬁrst need to calculate the weights of the portfolio that has expected return of
7.3%:
0
.
073 = 0
.
085
w
A
+ 0
.
069
w
B
1 =
w
A
+
w
B
,
which yields
w
A
= 0
.
25,
w
B
= 0
.
75. Now we can use the portfolio variance formula:
σ
2
P
=
w
2
A
σ
2
A
+
w
B
σ
2
B
+ 2
w
A
w
B
cov
(
R
A
,R
B
) = 0
.
000811
,
⇒
σ
P
= 0
.
0285
.
(d) Since the variance of the riskfree rate is zero and the covariance between the risk
free rate and anything else is zero, the variance of the equallyweighted portfolio and
the covariance of the portfolio with
R
A
are
σ
2
P
= 0
.
5
2
σ
2
A
+ 0
.
5
2
×
0 + 2
×
0
.
5
×
0
.
5
×
0 = 0
.
5
2
σ
2
A
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This note was uploaded on 12/20/2011 for the course RSM 332 taught by Professor Raymondkan during the Fall '08 term at University of Toronto Toronto.
 Fall '08
 RAYMONDKAN

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