Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges
Подождите немного. Документ загружается.
Recall from Chapter 14 that the following procedure is used to create an amortization
schedule.
474 Part 6 Corporate and Special Applications
Creating a Loan Amortization Schedule
6
Learning Objective
Create a loan amortization schedule.
to Create an Amortization Schedule
For each row except the last:
1. Interest payment 5 Unpaid balance 3 Monthly interest rate
2. Principal payment 5 Monthly payment 2 Interest payment
3. New unpaid balance 5 Old unpaid balance 2 Principal payment
For the last row:
1. Interest payment 5 Unpaid balance 3 Monthly interest rate
2. (Then ADD) Monthly payment 5 Unpaid balance 1 Interest payment
3. Principal payment 5 Unpaid balance
STEPS
EXAMPLE K
Mike Ward borrows $2,400. The loan is amortized over 4 months at 12% compounded
monthly (1% per month on the unpaid balance). The PVAF (Table 23-2) is 3.90197 and
the first three monthly payments will be $615.07. Create the amortization schedule for
the loan. 1% interest is charged on the unpaid balance each month.
Unpaid InterestPrincipal Total New
Month
Balance Payment Payment Payment Balance
1 $2,400.00 $24.00 $591.07 $615.07 $1,808.93
2 1,808.93 18.09 596.98 615.07 1,211.95
3 1,211.95 12.12 602.95 615.07 609.00
4 609.00 6.09 609.00 615.09 0
STEP 3STEP 2STEP 1
In example K, note that each month’s beginning unpaid balance is multiplied by the
monthly interest rate (1%) and rounded to the nearest cent.
© RYAN MCVAY/PHOTODISC/GETTYIMAGES
Month 1 Month 2 Month 3
$2,400.00 0.01 $24.00 $1,808.93 0.01 $18.09 $1,211.95 0.01 $12.12
$615.07 $24.00 $591.07 $615.07 $18.09 $596.98 $615.07 $12.12 $602.95
$2,400.00 $591.07 $1,808.93 $1,808.93 $596.98 $1,211.95 $1,211.95 $602.95 $609.00
Month 4
$609.00 0.01 $6.09
$609.00 $6.09 $615.09
$609.00 $609.00 $0.00
Chapter 23 Annuities 475
Amortize a $1,500 purchase over 3 months at an annual rate of 12%. First, use
Table 23-2 or a calculator to calculate the first two monthly payments.Then show
the calculations to construct a 3-month amortization schedule.
The periodic interest rate is 12% 4 12 5 1%, and the number of periods is 3.
The present value annuity factor from row 3 of the 1.00% column of Table 23-2
is 2.94099.The first two loan payments are $1,500 4 2.94099 5 $510.03.
Month 1 Month 2 Month 3
Unpaid balance: Purchase price $1,500.00 From month 1 $1,004.97 From month 2 $504.99
Interest payment: $1,500.00 3 0.01 5 $15.00 $1,004.97 3 0.01 5 $10.05 $504.99 3 0.01 5 $5.05
Total payment: From above $510.03 From above $510.03 $504.99 1 $5.05 5 $510.04
Principal payment: $510.03 2 $15.00 5 $495.03 $510.03 2 $10.05 5 $499.98 (Unpaid balance) $504.99
New balance: $1,500 2 $495.03 5 $1,004.97 $1,004.97 2 $499.98 5 $504.99 $504.99 2 $504.99 5 $0.00
COMPLETE ASSIGNMENT 23.2.
✔
CONCEPT CHECK 23.6
m 5 12
i 550.01
n 5 3
PVAF 5
5 2.94098521
1 2 (1 1 0.01)
2 3
0.01
0.12
12
annuity
future value of an annuity
future value of an annuity factor
(FVAF)
ordinary annuity
present value of an annuity
present value of an annuity factor
(PVAF)
sinking fund
time line
Chapter Terms for Review
Try Microsoft
®
Excel
Try working the problems using the Microsoft Excel templates found on your student
CD. Solutions for the problems are also shown on the CD.
476 Part 6 Corporate and Special Applications
Notes
THE BOTTOM LINE
Chapter 23 Annuities 477
Summary of chapter learning objectives:
Learning Objective
23.1
Compute the future value of an annuity.
Example
1. Compute the future value of $900 invested every month for 2.5 years,
with interest at 6% compounded monthly.
Answers: 1. $29,052.02 2. $299.82 3. $44,619.46 4. $918.43 5. $913.69
6. $10,000; $100.00; $621.24; $721.24; $9,378.76
23.2
Compute the regular payments of an annuity from the
future value.
2. Compute the regular annuity payment that is required to accumulate
$6,000 after 17 quarterly payments at an interest rate of 8% com-
pounded quarterly.
23.3
Compute the present value of an annuity.
3. Compute the present value of $4,750 withdrawn every half-year for
years, with interest at 10% compounded semiannually.6
1
2
23.4
Compute the regular payments of an annuity from the
present value.
4. An account starts at $4,000. Compute the regular, equal annual
withdrawal that is required to empty the account in 6 years if the
interest is 10% compounded annually.
23.5
Compute the loan payment required to amortize a loan.
5. Compute the loan payment that is required to amortize a $20,000 loan
in 24 monthly payments, with an interest rate of 9% compounded
monthly.
23.6
Create a loan amortization schedule.
6. A $10,000 loan at a rate of 12% compounded monthly is amortized in
15 monthly payments of $721.24. Compute the entries for the first
row of the amortization schedule.
SELF-CHECK
Review Problems for Chapter 23
478 Part 6 Corporate and Special Applications
1 For each annuity, find either the future value (compound amount) or the payment, as indicated. Then compute the
compound interest earned by the annuity. (Use Table 23-1)
Payment Rate Period Time Future Value Interest
$13,600 12% monthly 10 mo a. b.
$ 5,000 8% semiannually 20 yr c. d.
Future Value Rate Period Time Payment Interest
$30,000 5% annually 18 yr e. f.
$28,000 6% quarterly 6 yr g. h.
2 For each annuity, find either the present value or the payment, as indicated. Then compute the compound interest
earned by the annuity. (Use Table 23-2)
Payment Rate Period Time Present Value Interest
$1,750 12% semiannually 2 yr a. b.
$8,400 16% quarterly 3 yr c. d.
Present Value Rate Period Time Payment Interest
$24,000 10% annually 17 yr e. f.
$60,000 9% monthly 2.5 yr g. h.
3 Sharon Wilder planned to save money for retirement. She put $750 every month in an investment that paid a return
of 6% compounded monthly. How much would Sharon have in her account after 4 years?
4 Med-West wanted to set up a sinking fund to have $15,000,000 in 20 years. The company would make annual pay-
ments that would pay a return of 6% per year. What size should the payments be?
5 Nancy Duncan received a payment of $75,000 from a life insurance company. She put it in an account that would
pay 6% compounded quarterly. Nancy wanted to make equal quarterly withdrawals from the account for 10 years,
when the account would be empty. What size withdrawals can Nancy make?
6 Wayne Runn read about an investment opportunity on the Internet. The Website explained that Wayne would re-
ceive payments of $1,000 every 6 months for 14 years. If the returns are based on 8% compounded semiannually,
what is the present value of this investment opportunity?
Answers to the Self-Check can be found in Appendix B at the back of the text.
Chapter 23 Annuities 479
Assignment 23.1: Annuities—Future Value
Name
Date Score
1
Learning Objectives
2
9% 4 12 5 0.75% $2,200 3 26.18847 5 $57,614.63
12 3 2 5 24 payments
6% 4 4 5 1.5% $24,000 4 34.48148 5 $696.03
4 3 7 5 28 payments
8% 4 2 5 4% $3,500 3 29.77808 5 $104,223.28
2 3 10 5 20 payments
8% 4 1 5 8% $200,000 4 73.10594 5 $2,735.76
1 3 25 5 25 payments
15% 4 12 5 1.25% $500 3 45.11551 5 $22,557.76
12 3 3 5 36 payments
5% 4 4 5 1.25% $30,000 4 51.48956 5 $582.64
4 3 10 5 40 payments
10% 4 2 5 5% $1,000 3 75.29883 5 $75,298.83
2 3 16 5 32 payments
A (28 points) For each of the following annuities, find the future value or the amount of the periodic
payment. Round answers to the nearest cent. (4 points for each correct answer)
Payment Payment Length Future
Amount Periods Interest Rate of Annuity Value
1. $2,200 monthly 9% compounded monthly 2 years
2. quarterly 6% compounded quarterly 7 years $24,000
3. $3,500 semiannually 8% compounded semiannually 10 years
4. annually 8% compounded annually 25 years $200,000
5. $500 monthly 15% compounded monthly 3 years
6. quarterly 5% compounded quarterly 10 years $30,000
7. $1,000 semiannually 10% compounded semiannually 16 years
Score for A (28)
$75,298.83
$582.64
$22,557.76
$2,735.76
$104,223.28
$696.03
$57,614.63
B (32 points) For each of the following annuities, find the future value, the amount of the periodic
payment, or the total amount of interest paid. Round answers to the nearest cent. (4 points for each
correct answer)
8.
Calculate the future value of a 25-year annuity with payments of $3,000 each year and an interest rate of
5% compounded annually.
9. How much total interest is earned on an annuity with payments of $300 per month for 4 years and an
interest rate of 6% compounded monthly?
10. An annuity consists of quarterly payments of $1,600 each for 10 years at an interest rate of 6%
compounded quarterly. Compute the future value of the annuity.
11. A 7-year annuity has semiannual payments of $8,000 each and an interest rate of 8% compounded
semiannually. What will be the total amount of interest earned?
12. A sinking fund has 10 annual payments, has an interest rate of 10% compounded annually, and has a
future value of $75,000. Compute the amount of each annual payment.
13. An 8-year annuity with quarterly payments and an interest rate of 5% compounded quarterly has a
future value of $45,000. How much total interest does the annuity earn?
14. Calculate the amount of each monthly payment in a 1-year annuity that has a future value of $5,000 and
an interest rate of 9% compounded monthly.
$399.76
$8,124.48
$4,705.91
$34,335.28
$86,828.62
$1,829.35
$143,181.30
480 Part 6 Corporate and Special Applications
Assignment 23.1 Continued
5% 4 1 5 5%
1 3 25 5 25 payments $3,000 3 47.72710 5 $143,181.30
6% 4 12 5 0.5% $300 3 54.09783 5 $16,229.35
12 3 4 5 48 payments $16,229.35 2 ($300 3 48) 5 $1,829.35
6% 4 4 5 1.5%
4 3 10 5 40 payments $1,600 3 54.26789 5 $86,828.62
8% 4 2 5 4% $8,000 3 18.29191 5 $146,335.28
2 3 7 5 14 payments $146,335.28 2 ($8,000 3 14) 5 $34,335.28
10% 4 1 5 10% $75,000 4 15.93742 5 $4,705.91
1 3 10 5 10 payments
5% 4 4 5 1.25% $45,000 4 39.05044 5 $1,152.36
4 3 8 5 32 payments $45,000 2 ($1,152.36 3 32) 5 $8,124.48
9% 4 12 5 0.75%
12 3 1 5 12 payments $5,000 4 12.50759 5 $399.76
15. Determine the total interest earned by an annuity with semiannual payments for 18 years, an interest rate
of 10% compounded semiannually, and a future value of $25,000.
Score for B (32)
C (40 points) In each of the following applications, find the future value of the annuity, the amount of the
periodic payment, or the total amount of interest earned. Round answers to the nearest cent. (4 points
for each correct answer)
16.
Jim Walter decides to make semiannual deposits in his credit union account because it is guaranteeing a
rate of 8% compounded semiannually for the next 5 years. How much will Jim have after making equal
semiannual deposits of $2,500 for 5 years?
17. Calvin White is planning for his daughter’s college education. An investment advisor recommends an
investment whose prospectus claims it will return 9% compounded monthly. If the investment does
return 9% compounded monthly, how much must Calvin invest each month for 4 years if he wants to
have a total of $50,000 after the last deposit?
18. Maxfield International is raising $25,000,000 by selling bonds that will mature in 20 years. Maxfield
plans to make equal annual payments in a sinking fund to repay the bonds. If Maxfield can earn 6%
per year, what amount should it deposit each year in order to have $25,000,000 at the end of 20 years?
19. Luis Garza is quite certain he will need to replace some construction equipment in 3 years. He decides
to set up a sinking fund now to help buy the equipment. Luis estimates that he can deposit $1,400 each
month for 3 years in a sinking fund that will pay 6% compounded monthly. How much will Luis’ sinking
fund be worth after the last deposit?
20. Bill Starnes is planning that his twin daughters could get married in 6 years. He thinks that he should
start saving now to try to accumulate $30,000 by the end of the 6 years. Assuming that Bill can find an
investment that will pay 8% compounded quarterly, what amount must Bill deposit each quarter to have
the necessary $30,000 at the end of the 6 years?
$986.13
$55,070.54
$679,613.95
$869.25
$30,015.28
$15,609.04
Chapter 23 Annuities 481
Assignment 23.1 Continued
10% 4 2 5 5% $25,000 4 95.83632 5 $260.86
2 3 18 5 36 payments $25,000 2 ($260.86 3 36) 5 $15,609.04
8% 4 2 5 4%
2 3 5 5 10 payments $2,500 3 12.00611 5 $30,015.28
9% 4 12 5 0.75%
12 3 4 5 48 payments $50,000 4 57.52071 5 $869.25
6% 4 1 5 6%
1 3 20 5 20 payments $25,000,000 4 36.78559 5 $679,613.95
6% 4 12 5 0.5%
12 3 3 5 36 payments $1,400 3 39.33610 5 $55,070.54
8% 4 4 5 2%
4 3 6 5 24 payments $30,000 4 30.42186 5 $986.13
21. Joseph Woo imports patio furniture from various countries. He prefers to have cash available when he
goes on buying trips. Suppose that Joseph makes equal monthly deposits into a risky investment that
promises to pay 12% compounded monthly. If he deposits enough each month to accumulate $60,000
by the end of 2 years, and if the investment pays as promised, how much of the $60,000 will the bank
have paid in interest?
22. Jeanne Knowles will graduate from high school in a few months. She has found a part-time job and is
trying to determine how much money she can save in 6 years. Calculate the future value after 6 years if
Jeanne makes semiannual deposits of $600 each into an investment account that promises a return of 8%
compounded semiannually.
23. Musical Instrument Manufacturing, Inc., (MIMI) just sold $40,000,000 in bonds. The bonds will mature
in 20 years. MIMI will make equal semiannual payments into a sinking fund that will earn 10% com-
pounded semiannually. If MIMI has the $40,000,000 after 20 years, what amount of the total was earned
from the interest?
24. Every three months, Katie Webb sends $1,600 to her granddaughter, Jenny. To encourage Jenny to save
more money, Jenny’s father promises to give her interest of 12% compounded quarterly on everything
that she saves. If Jenny always saves the entire $1,600 each quarter, and receives these payments every
quarter for 5 years, how much money will Jenny’s father have to pay her in interest?
25. This year, Doug McCombs charged all his family’s Christmas gifts on a credit card, and the result was a
minor financial disaster. Planning for next year, Doug decides to save money each month from February
through November and put it into an account that will pay 12% compounded monthly. He plans to
make 10 equal deposits, and he wants to have accumulated $3,500 once he makes the tenth deposit.
Calculate the size of each deposit.
Score for C (40)
$334.54
$10,992.59
$26,754,941.60
$9,015.49
$6,614.16
482 Part 6 Corporate and Special Applications
Assignment 23.1 Continued
12% 4 12 5 1% $60,000 4 26.97346 5 $2,224.41
12 3 2 5 24 payments $60,000 2 ($2,224.41 3 24) 5 $6,614.16
8% 4 2 5 4%
2 3 6 5 12 payments $600 3 15.02581 5 $9,015.49
10% 4 2 5 5% $40,000,000 4 120.79977 5 $331,126.46
2 3 20 5 40 payments $40,000,000 2 ($331,126.46 3 40) 5 $26,754,941.60
12% 4 4 5 3% $1,600 3 26.87037 5 $42,992.59
4 3 5 5 20 payments $42,992.59 2 ($1,600 3 20) 5 $10,992.59
12% 4 12 5 1% $3,500 4 10.46221 5 $334.54
10 months 5 10 payments
Chapter 23 Annuities 483
Assignment 23.2: Annuities—Present Value
Name
Date Score
A (28 points) For each of the following annuities, find the present value or the amount of the periodic
payment. Round answers to the nearest cent. (4 points for each correct answer)
Payment Payment Length Present
Amount Periods Interest Rate of Annuity Value
1. $1,500 semiannually 8% compounded semiannually 9 years
2. quarterly 6% compounded quarterly 12 years $35,000
3. $800 monthly 6% compounded monthly 4 years
4. annually 10% compounded annually 27 years $86,000
5. $2,500 quarterly 8% compounded quarterly 8 years
6. semiannually 6% compounded semiannually 25 years $100,000
7. $750 monthly 9% compounded monthly 3 years
Score for A (28)
$23,585.11
$3,886.55
$58,670.83
$9,310.16
$34,064.26
$1,028.13
$18,988.95
3
Learning Objectives
4 5 6
8% 4 2 5 4% $1,500 3 12.65930 5 $18,988.95
2 3 9 5 18 payments
6% 4 4 5 1.5%
4 3 12 5 48 payments $35,000 4 34.04255 5 $1,028.13
6% 4 12 5 0.5%
12 3 4 5 48 payments $800 3 42.58032 5 $34,064.26
10% 4 1 5 10%
1 3 27 5 27 payments $86,000 4 9.23722 5 $9,310.16
8% 4 4 5 2%
4 3 8 5 32 payments $2,500 3 23.46833 5 $58,670.83
6% 4 2 5 3%
2 3 25 5 50 payments $100,000 4 25.72976 5 $3,886.55
9% 4 12 5 0.75%
12 3 3 5 36 payments $750 3 31.44681 5 $23,585.11