TM 661
Chapter 2
Solutions 1
# 9) Suppose you wanted to become a millionaire at retirement. If an annual compound
interest rate of 8% could be sustained over a 40-year period, how much would have to
be deposited yearly in the fund in order to accumulate $1 million.
F = 1,000,000
Soln:
0
1
2
3
4
.
.
A A A A
.
40
A
A = F(A/F, I, 40)
= 1,000,000 (A/F, 8, 40)
= 1,000,000 (0.00386)
= 3,860
A = 1,000,000 (A/F, 10, 40)
= 1,000,000 (0.00226)
= 2,259
A = 1,000,000 (A/f, 12, 40)
= 1,000,000 (0.00130)
= 1,303
Chapter 2
Solutions 1
TM 661
Chapter 2
Solutions 2
14. Kim deposits $1,000 in a savings account; 4 years after the deposit, half of the account
balance is withdrawn. $2,000 is deposited annually for an 8-year period, with the first
deposit occurring 2 years after the withdrawal. The total balance is withdrawn 15
years after the initial deposit. If the account earned interest of 8% compounded
annually over the 15 year period, how much was withdrawn at each withdrawal point?
F
Soln:
500(1.08)4
0
1
2
3
4
5
1,000
6
7
13
2,000 2,000
2,000
15
At year 4, value in the account is given as
F = 1,000(F/P, 8, 4)
= 1,000(1.08)4
= 1,360
Leaving half in the account leaves 680 in period 4. Then in year 15, the total amount
in the account is given by
= 680(F/P, 8, 11) + 2,000(F/A, 8, 8)(F/P, 8, 2)
= 680(1.08)11 + 2,000(10.6366)(1.08)2
= $ 26,399
Chapter 2
Solutions 2
TM 661
Chapter 2
Solutions 3
27) Dr. Shieh deposits $3,000 in a money market fund. The fund pays interest at a rate of
12% compounded annually. Just 3 years after making the single deposit, he
withdraws one third the accumulated money in his account. Then 5 years after the
initial deposit, he withdraws all of the accumulated money remaining in the account.
How much does he withdraw 5 years after his initial deposit?
Soln:
F
1/3 accum.
1
2
3
4
5
3,000
Amount accumulated by year 3.
F = 3,000(F/P, 12, 3)
= 3,000 (1.12)3
= 4,215
Withdrawing 1/3 leaves $2,810 which then earns interest for two more years
F5 = 2,810 (F/P, 12, 2)
= 2,810 (1.12)2
= $3,525
Chapter 2
Solutions 3
TM 661
Chapter 2
Solutions 4
38. Lynne borrows $15,000 at 1.5% per month. She desires to repay the money using equal
monthly payments over 36 months. Lynne makes four such payments and decides to
pay off the remaining debt with one lump sum payment at the time for the 5th
payment. What should be the size of the payment if interest is truly compounded at a
rate of 1.5% per month.
Soln:
15,000
1
2
3
36
A A
A
A
A = 15,000 (A/P, 1.5, 36)
= 15,000 (.0362)
= 543
After making 4 payments of 543, Lynne wishes to pay off the loan in the 5th payment.
15,000
1
543
2
3
4
5
X
15,000 = 543 (P/A, 1.5, 4) + X (1.015)-5
15,000 = 543 (3.8544) + X (.9283)
X = $ 13,905
Chapter 2
Solutions 4
TM 661
Chapter 2
Solutions 5
47) A person borrows $10,000 and wishes to pay it back witn 9 equal annual payments.
What will the payments be if the interest used is 12% compounded (a) annually, (b)
semi-annually, and (c) continuously?
15,000
Soln:
1
2
3
.
.
.
9
A
a) ieff = 12%
A = 10,000(A/P, 12, 9)
= 10,000 (0.1877)
= $1,877
b) ieff = (1+.06)2 - 1 = .1236
A = 10,000(A/P, 12.36, 9)
= 10,000 (0.1903)
= $1,903
c) ieff = e.12 - 1 = .1275
A = 10,000(A/P, 12.75, 9)
= 10,000 (0.1931)
= $1,931
or
= 10,000(A/P, 12, 9)cont
= 10,000(0.1931)
= $1,931
formula p. 61 or table Appendix B, p. 450
Chapter 2
Solutions 5
TM 661
Chapter 2
Solutions 6
54) A firm buys a new computer that costs $100,000. It may either pay cash now or pay
$20,000 down and $30,000 per year for 3 years. If the firm can earn 15% on
investments, which option should the firm choose?
Soln:
or
100
20
1
2
3
30
30
30
for alternative b) compute Present Worth
= 20 + 30 (P/A, 15, 3)
= 20 + 30 (2.2832)
= 88.497
Based on the firm’s time-value-of-money, alternative b) is cheaper.
Chapter 2
Solutions 6
TM 661
Chapter 2
Solutions 7
62) Consider the following cash flow
t
At
0
-10,000
1
6,500
2
6,000
3
5,500
4
5,000
5
4,500
6
4,000
7
3,500
8
3,000
At 10% annual compound interest, compute the equivalent uniform annual cash flow.
Soln: We are given a gradient cash flow decreasing at 500 per period. The present worth
consists of the present worth of the annuity (6,500) less the present worth of the
gradient (-500).
P
= -10,000 + 6,500 (P/A, 10, 8) - 500(P/G, 10, 8)
= -10,000 +6,500 (5.3349) - 500 (16.0287)
= 16,662
A = 16,662 (A/P, 10, 8)
= 16,662 (0.1874)
= $3,123
Alternatively,
A = -10,000 (A/P, 10, 8) + 6,500 - 500 (A/G, 10, 8)
= -10,000 (0.1874) + 6,500 -500 (3.0045)
= $3,124
Chapter 2
Solutions 7
TM 661
Chapter 2
Solutions 8
67) Ms. Torro-Tamos borrows $7,000 and repays the loan with 4 quarterly payments of
$600 during the first year and 4 quarterly payments of $1,500 during the second year.
Determine the effective interest rate for this transaction.
7,000
Soln:
1
2
3
4
5
6
7
8
600
1,500
7,000 = 600 (P/A, i, 4) + 1,500 (P/A,i, 4)(P/F, i, 4)
i
1.0%
2.0%
3.0%
4.0%
3.5%
3.6%
NPV(i)
(966.00)
(561.00)
(184.00)
167.00
(5.16)
29.90
The quarterly interest rate is roughly 3.5% per quarter.
ieff = (1 + .035)4 - 1
= .1475 = 14.75%
Chapter 2
Solutions 8