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RSM332H1 (10)

William Huggins (2)

Midterm

Department

Rotman CommerceCourse Code

RSM332H1Professor

William HugginsStudy Guide

MidtermThis

**preview**shows pages 1-2. to view the full**6 pages of the document.**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.

1

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

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.

2

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