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Midterm

RSM332H1 Study Guide - Midterm Guide: Capital Market, Savings Account, Nsb Di 2


Department
Rotman Commerce
Course Code
RSM332H1
Professor
William Huggins
Study Guide
Midterm

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UNIVERSITY OF TORONTO
Joseph L. Rotman School of Management
Oct. 21, 2008 Ezer/Kan/Florence
RSM332 MID-TERM EXAMINATION Pomorski/Zhou
SOLUTIONS
1. (a) For Mr. Oh, his consumption at time 0 and time 1 are given by C0= 1000 I0
and C1= 30I
1
2
0. Therefore, we can write his utility as
UO=C
1
2
0C
1
2
1= (1000 I0)1
2(30I
1
2
0)1
2=30(1000 I0)1
2I
1
4
0=30(1000I
1
2
0I
3
2
0)1
2.
Differentiating UOwith respect to I0, we obtain
dUO
dI0
=1
230(1000I
1
2
0I
3
2
0)1
21000
2I1
2
03
2I
1
2
0.
Setting the derivative equal to zero, we have
1000
2I1
2
03
2I
1
2
0= 0 I0= 333.33.
Therefore, the optimal investment by Mr. Oh is I
0= 333.33.
(b) If Mrs. Ner follows the investment decision of Mr. Oh, her consumption at time 0
and time 1 will be C0= 1000 333.33 = 666.67 and C1= 30333.33 = 547.72. It
follows that her utility will be
UN(C0, C1) = C
1
4
0C
3
4
1= (666.67)1
4(547.72)3
4= 575.30.
(c) The utility of Mrs. Ner is given by
UN=C
1
4
0C
3
4
1= (1000 I0)1
4(30I
1
2
0)3
4= 303
4(1000 I0)1
4I
3
8
0= 303
4(1000I
3
2
0I
5
2
0)1
4.
Differentiating UNwith respect to I0, we obtain
dUN
dI0
=1
4303
4(1000I
3
2
0I
5
2
0)3
43000
2I
1
2
05
2I
3
2
0.
Setting the derivative equal to zero, we have
3000
2I
1
2
05
2I
3
2
0= 0 I0= 600.
1

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Therefore, the optimal investment by Mrs. Ner is I
0= 600. At the optimal level of
investment, the utility of Mrs. Neh is given by
UN= (1000 I
0)1
4(30I
0
1
2)3
4= 631.19.
(d) When Mrs. Ner invests 333.33, her consumption at time 0 and 1 are given by
C0= 666.67 and C1= 547.72. In order for Mrs. Ner to have a utility of 631.19, she
needs a consumption at time 0 to satisfy
C
1
4
0(547.72)3
4= 631.19 C0= 965.98.
This means that she needs an additional consumption of 965.98 666.67 = 299.31 at
time 0 for her to be indifferent.
(e) Since there is a capital market, Fisher’s separation works and the investment de-
cision does not depend on the utility function. Mr. Oh’s and Mrs. Ner’s optimal
investment plans are identical. Hence, there is no need to compensate Mrs. Ner for
having to separate ownership and control: she’s perfectly happy to let Mr. Oh make
the investment decision.
2. (a) For a fair comparison, we need to compare the effective annual interest rate of the
two saving accounts. The effective annual rate for Account A is
rA
e=1 + 0.12
22
1 = (1.06)21=0.1236.
The effective annual rate for Account B is
rB
e=e0.1175 1 = 0.1247.
Therefore, Account B should be the preferred one because it offers a higher effective
annual rate.
When the money is deposited in Account B, the present value of the six deposits is
PV = 5000A6
0.1247 =5000
0.1247 "11
(1.1247)6#= 5000 ×4.0575 = 20287.33.
It follows that the future value of the deposits in 20 years is
FV20 = PV(1 + 0.1247)20 = 20287.33 ×10.4856 = 212724.19.
(b) (i) The future value of his student loan at t= 5 is
FV = 25000(1.1)4+ 20000(1.1)3+ 20000(1.1)2+ 30000(1.1) = 120422.5.
2
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