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# 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 âˆ’I0
and C1= 30I
1
2
0. Therefore, we can write his utility as
UO=C
1
2
0C
1
2
1= (1000 âˆ’I0)1
2(30I
1
2
0)1
2=âˆš30(1000 âˆ’I0)1
2I
1
4
0=âˆš30(1000I
1
2
0âˆ’I
3
2
0)1
2.
Diï¬€erentiating UOwith respect to I0, we obtain
dUO
dI0
=1
2âˆš30(1000I
1
2
0âˆ’I
3
2
0)âˆ’1
2î€’1000
2Iâˆ’1
2
0âˆ’3
2I
1
2
0î€“.
Setting the derivative equal to zero, we have
1000
2Iâˆ’1
2
0âˆ’3
2I
1
2
0= 0 â‡’I0= 333.33.
Therefore, the optimal investment by Mr. Oh is Iâˆ—
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 âˆ’333.33 = 666.67 and C1= 30âˆš333.33 = 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 âˆ’I0)1
4(30I
1
2
0)3
4= 303
4(1000 âˆ’I0)1
4I
3
8
0= 303
4(1000I
3
2
0âˆ’I
5
2
0)1
4.
Diï¬€erentiating UNwith respect to I0, we obtain
dUN
dI0
=1
4303
4(1000I
3
2
0âˆ’I
5
2
0)âˆ’3
4î€’3000
2I
1
2
0âˆ’5
2I
3
2
0î€“.
Setting the derivative equal to zero, we have
3000
2I
1
2
0âˆ’5
2I
3
2
0= 0 â‡’I0= 600.
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Therefore, the optimal investment by Mrs. Ner is Iâˆ—
0= 600. At the optimal level of
investment, the utility of Mrs. Neh is given by
UN= (1000 âˆ’Iâˆ—
0)1
4(30Iâˆ—
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 â‡’C0= 965.98.
This means that she needs an additional consumption of 965.98 âˆ’666.67 = 299.31 at
time 0 for her to be indiï¬€erent.
(e) Since there is a capital market, Fisherâ€™s separation works and the investment de-
cision does not depend on the utility function. Mr. Ohâ€™s and Mrs. Nerâ€™s optimal
investment plans are identical. Hence, there is no need to compensate Mrs. Ner for
having to separate ownership and control: sheâ€™s perfectly happy to let Mr. Oh make
the investment decision.
2. (a) For a fair comparison, we need to compare the eï¬€ective annual interest rate of the
two saving accounts. The eï¬€ective annual rate for Account A is
rA
e=î€’1 + 0.12
2î€“2
âˆ’1 = (1.06)2âˆ’1=0.1236.
The eï¬€ective annual rate for Account B is
rB
e=e0.1175 âˆ’1 = 0.1247.
Therefore, Account B should be the preferred one because it oï¬€ers a higher eï¬€ective
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 "1âˆ’1
(1.1247)6#= 5000 Ã—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 Ã—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.
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