Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges

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324 Part 4 Interest Applications

16.6 Some students who use calcula-

tors to find PVFs might be interested to

see that they can calculate (1 1 i )

and then press the

⁄x key,or even that

(1 1 i )

is the same as 1 4 (1 1 i )

PRESENT VALUE FORMULA

If you use the present value factors (Table 16-2 or a calculator), you use a different

formula, the present value formula,

Present value 5 Future value 3 Present value factor (from Table 16-2 or a calculator)

or, PV 5 FV 3 PVF

EXAMPLE K

Rework example J using Table 16-2 and the present value formula. How much must

Edison Motors invest today to have $20,000 in 2 years if the interest rate is 6%

compounded monthly?

$20,000 is the future value, for which Edison wants to know the present value.

There are 12 compounding periods in 1 year (monthly).

Periodic rate 5 6% 4 12 5 0.5%

Number of compounding periods 5 12 3 2 years 5 24

The present value factor in the 0.5% column and row 24 of Table 16-2 is

0.88719.

Substitute these values into the present value formula.

Present value 5 Future value 3 Present value factor (from Table 16-2)

5 FV 3 PVF 5 $20,000 3 0.88719 5 $17,743.80

The answer to example J was $17,743.71. The discrepancy between that result and

$17,743.80 in example K is due to rounding. If the two tables had more decimal places

instead of just five, this discrepancy would disappear. In fact, using the calculator PVF

from the margin, we get PV 5 $20,000 3 0.88718567 5 $17,743.7134, which is identical

to the nearest cent.

EXAMPLE L

Solve the present value problem from example H. If Polly Layer can earn 5% compounded

annually, how much should she deposit today in investments for her son and daughter

so that the investments will be worth $60,000 and $70,000 in 6 and 8 years, respectively?

Son Daughter

Future value: $60,000 $70,000

Term: 6 years 8 years

Rate: 5% compounded 5% compounded

annually annually

Periods per year: 1 (annual) 1 (annual)

Periodic rate: 5% 4 1 5 5% 5% 4 1 5 5%

Compounding periods: 1 3 6 years 5 61 3 8 years 5 8

PV factor (Table 16-2): 0.74622 0.67684

Present value: $60,000 3 0.74622 5 $70,000 3 0.67684 5

$44,773.20 $47,378.80

STEP iii

STEP ii

STEP i

STEP iii

STEP ii

STEP i

With a calculator,

PVF 5

FVF

(1 1 i)

PV 5 FV 3 PVF

5 $20,000 3 0.88718567

5 $17,743.71

5 $70,000 3 0.67683936

5 $47,378.76

5 0.88718567

PVF 5

(1 1 0.005)

n 5 12 3 2 5 24

i 5

0.06

5 0.005

m 5 12

Son

5 $60,000 3 0.74621540

5 $44,772.92

Daughter

5 0.67683936

PVF 5

(1 1 0.05)

n 5 1 3 8 5 8

i 5

0.05

5 0.05

m 5 1

5 0.74621540

PVF 5

(1 1 0.05)

n 5 1 3 6 5 6

i 5

0.05

5 0.05

m 5 1

The reason for two formulas and two tables is historical, predating handheld calcula-

tors. Without a calculator, a multiplication problem is typically easier than a long divi-

sion problem with the same two large numbers.

Theoretically, we need only one formula and one table. The second present value for-

mula and the table of present value factors permit us to solve present value problems by

using multiplication instead of division. Look at example J. To solve the problem requires

that we divide $20,000 by 1.12716, which is extremely time-consuming to do without a

calculator. (The answer is $17,743.71.) Using Table 16-2, we can solve the same problem

by multiplying $20,000 by 0.88719, a relatively easy calculation even without a calculator.

(The answer is $17,743.80; the difference is due to rounding in the creation of the table.)

NOTES ABOUT THE FUTURE VALUE

AND PRESENT VALUE TABLES

Chapter 16 Compound Interest 325

16.7 Finance courses and financial de-

cision-making normally involve more

work with present values than with fu-

ture values.Before the age of calcula-

tors,a traditional finance textbook

might not have used a future value

table very often.

16.8 Emphasize again that the present

value formula and table were devel-

oped to make an “easier”multiplication

problem compared to a very tedious

long division problem in “precalculator

days”—less than 35 years ago.

The numbers in the future value table

(Table 16-1) are actually just the future

value of $1.00 at a specific interest rate

and for a specific period of time. For

example, suppose that you invest $1.00 for

2 years at 6% compounded annually. This

is the same problem as example A, except

that the principal is only $1.00 instead of

$2,000.00

The calculations shown at the right

have not been rounded off. The answer,

which is $1.1236, is the future value of the

$1.00 investment. Now, find row 2 and the

6.00% column of Table 16-1. The future

value factor is 1.12360—exactly the same

as $1.1236, without the dollar sign and

with five decimal places.

$1.00 Original principal

0.06 Interest rate

$0.0600 First-year interest

1.0000 First-year principal

$1.0600 Second-year principal

3 0.06 Interest rate

$0.0636 Second-year interest

1.06

00 Second-year principal

$1.1236 Final compound amount

Each number in the present value table (Table 16-2) can be calculated directly from

the corresponding number in the future value table. The corresponding numbers are

reciprocals of each other. The reciprocal of a number is found by dividing the number

into 1.

Look back at examples J and K, which

showed two different ways to solve the

same problem. In example J we used a fu-

ture value factor, which was 1.12716. In

example K we used a present value factor,

which was 0.88719. Each factor is in row

24 and the 0.50% column of its respective

table. With your calculator, divide 1 by

1.12716 to get 0.88718549, which, rounded

to five places, is 0.88719. And dividing 1

by 0.88719 gives 1.12715427.

1 4 1.12716 5 0.88718549, or 0.88719

1 4 0.88719 5 1.12715427, or 1.12716

Examine your calculator. You may have a reciprocal key, labeled “1/x.” If you have this

key, enter 1.12716 and press the . The calculator will display 0.88718549. Press the

again and the calculator will display 1.12716, or perhaps 1.12716000.

326 Part 4 Interest Applications

a. What present value (principal) invested for 3 years at 10% compounded

semiannually will result in a total future value of $4,000? (Use Table 16-2

or a calculator.)

Semiannually means 2 periods per year.

Periodic rate 5 10% 4 2 5 5% per half-year

Number of periods 5 2 3 3 years 5 6 periods

The present value factor from row 6 of the 5.00% column in Table 16-2 is 0.74622.

PV 5 FV 3 PVF 5 $4,000 3 0.74622 5 $2,984.88

b. Four years ago, a woman invested money at 9% compounded monthly. If the invest-

ment is now worth $12,000, how much compound interest did she earn in the

4 years? (Use Table 16-2 or a calculator.)

Monthly means 12 periods per year.

Periodic rate 5 9% 4 12 5 0.75% per month

Number of periods 5 12 3 4 years 5 48 periods

The present value factor from row 48 of the 0.75% column in Table 16-2 is

0.69861.

PV 5 FV 3 PVF 5 $12,000 3 0.69861 5 $8,383.32

Compound interest 5 Future value 2 Present value

5 $12,000 2 $8,383.32 5 $3,616.68

COMPLETE ASSIGNMENT 16.2.

✔

CONCEPT CHECK 16.3

compound amount

compound amount factors

compound interest

exponent

future value

future value factor

number of compounding periods

period (compounding period)

periodic interest rate

power

present value

present value factor

Chapter Terms for Review

5 0.74621540

PVF 5

(1 1 0.05)

n 5 2 3 3 5 6

i 5

0.10

5 0.05

m 5 2

5 0.69861414

PVF 5

(1 1 0.0075)

n 5 12 3 4 5 48

i 5

0.09

5 0.0075

m 5 12

5 $4,000 3 0.74621540

5 $2,984.86

5 $12,000 3 0.69861414

5 $8,383.37

THE BOTTOM LINE

Chapter 16 Compound Interest 327

Summary of chapter learning objectives:

Learning Objective

16.1

Compute future values from tables and formulas.

Example

1. Compute the future value of $8,000 invested at 9% compounded

monthly for 3 years.

2. Compute the compound interest earned on $5,000 invested at 6%

compounded quarterly for 5 years.

Answers: 1. $10,469.20 2. $1,734.30 3. $4,454.69 4. $1,957.08 5. $634.95

16.2

Compute present values from future value tables.

3. Compute the present value that has to be invested at 10% com-

pounded semiannually for 6 years to result in $8,000.

16.3

Compute using present value tables and formulas.

4. If $6,000 is the future value after 13 years at 9% compounded

annually, compute the principal (present value).

5. An investment made 16 months ago is worth $5,634.95 today. If the

interest rate was 9% compounded monthly, what was the amount of

compound interest?

SELF-CHECK

Review Problems for Chapter 16

328 Part 4 Interest Applications

1 Calculate the future value (compound amount) and compound interest. (Use Table 16-1 or a calculator.)

Principal Rate Time Future Value Interest

$ 4,000 6% compounded monthly 3 yr a. b.

$12,000 8% compounded quarterly 7 yr c. d.

$30,000 10% compounded annually 16 yr e. f.

$ 8,000 10% compounded semiannually 10 yr g. h.

2 Calculate the present value (principal) and compound interest. (Use Table 16-2 or a calculator.)

Future

Value Rate Time Present Value Interest

$25,000 6% compounded annually 9 yr a. b.

$ 6,000 8% compounded semiannually 12 yr c. d.

$15,000 9% compounded monthly 4 yr e. f.

$40,000 6% compounded quarterly 5 yr g. h.

3 Vernon Lee received a $6,000 bonus from his employer. He can invest it safely in his credit union at 4% com-

pounded quarterly. What will be the value of the investment in 7 years?

4 Kathy Shutter inherited $26,760. She invested it immediately in an investment fund paying 8% compounded

semiannually. How much interest would Kathy earn if she left principal and interest invested for 9 years?

5 Sandy Hopkins was planning to buy a new car in 3 years. She has some money today that she can invest for 3

years in an account that will pay 6% compounded quarterly. How much of it would she need to deposit today

so that she will have $8,000 in her account in 3 years?

6 Doug Jurgensen will need to buy a $25,000 wood lathe in 2 years. He can deposit excess profits from this

year in an investment that should pay 9% compounded monthly. If Doug earns the $25,000 in 2 years, how

much will he earn in interest?

Answers to the Self-Check can be found in Appendix B at the back of the text.

Chapter 16 Compound Interest 329

Assignment 16.1: Future Value (Compound Amount)

Name

Date Score

A (28 points) Find the future value (compound amount) and the compound interest, as indicated, for

each of the following investments. Round answers to the nearest cent. Use Table 16-1 or a calculator.

(2 points for each correct answer)

Future Compound

Principal Rate Term Value Interest

1. $6,000 6% compounded monthly 4 years

$1,622.94$7,622.94

Learning Objective

6% 4 12 5 0.5% $6,000 3 1.27049 5 $7,622.94

12 3 4 5 48 periods $7,622.94 2 $6,000 5 $1,622.94

8% 4 2 5 4% $750 3 2.77247 5 $2,079.35

2 313 5 26 periods $2,079.35 2 $750 5 $1,329.35

8% 4 4 5 2% $20,000 3 1.88454 5 $37,690.80

4 3 8 5 32 periods $37,690.80 2 $20,000 5 $17,690.80

12% 4 1 5 12% $12,500 3 29.95992 5 $374,499.00

1 3 30 5 30 periods $374,499 2 $12,500 5 $361,999.00

9% 4 12 5 0.75% $5,000 3 1.14396 5 $5,719.80

18 months 5 18 periods $5,719.80 2 $5,000 5 $719.80

6% 4 4 5 1.5% $14,450 3 1.26899 5 $18,336.91

4 3 4 5 16 periods $18,336.91 2 $14,450 5 $3,886.91

4% 4 2 5 2% $4,000 3 1.42825 5 $5,713.00

2 3 9 5 18 periods $5,713 2 $4,000 5 $1,713.00

2. $750 8% compounded semiannually 13 years

$1,329.35$2,079.35

3. $20,000 8% compounded quarterly 8 years

$17,690.80$37,690.80

4. $12,500 12% compounded annually 30 years

$361,999.00$374,499.00

5. $5,000 9% compounded monthly 18 months

$719.80$5,719.80

6. $14,450 6% compounded quarterly 4 years

$3,886.91$18,336.91

7. $4,000 4% compounded semiannually 9 years

$1,713.00$5,713.00

Score for A (28)

Note to teachers: Answers were computed using Table 16-1. Calculator answers may vary slightly.

B (32 points) Find the future value (compound amount) or the compound interest, as indicated, for each

of the following investments or loans. Round answers to the nearest cent. Use Table 16-1 or a calculator.

(4 points for each correct answer)

Compute the future value (compound amount) of $4,500 invested for 10 years at 5% compounded

quarterly.

$7,396.29

330 Part 4 Interest Applications

Assignment 16.1 Continued

5% 4 4 51.25% $4,500 3 1.64362 5 $7,396.29

4 310 5 40 periods

15% 4 12 5 1.25% $25,000 3 1.17526 5 $29,381.50

13 months 5 13 periods $29,381.50 2 $25,000 5 $4,381.50

8% 4 1 5 8% $38,260 3 2.71962 5 $104,052.66

1 3 13 5 13 periods

12% 4 2 5 6% $7,900 3 6.84059 5 $54,040.66

2 3 16.5 5 33 periods $54,040.66 2 $7,900 5 $46,140.66

9% 4 12 5 0.75% $15,000 3 1.25127 5 $18,769.05

12 3530 periods2

16% 4 2 5 8% $1,780 3 2.15892 5 $3,842.88

2 3 5 5 10 periods $3,842.88 2 $1,780 5 $2,062.88

9% 4 1 5 9% $10,000 3 4.32763 5 $43,276.30

1 3 17 5 17 periods $43,276.30 2 $10,000 5 $33,276.30

5% 4 4 5 1.25% $18,000 3 1.25058 5 $22,510.44

4 3 4.5 5 18 periods

9. How much compound interest will you pay if you borrow $25,000 for 13 months at 15% compounded

monthly?

$4,381.50

10. Calculate the future value (compound amount) on a loan of $38,260 at 8% compounded annually for

13 years.

$104,052.66

11. How much compound interest will you earn if you invest $7,900 for 16.5 years at 12% compounded

semiannually?

$46,140.66

12. What total amount (principal and interest) must be repaid in years on a loan of $15,000 at 9% com-

pounded monthly?

$18,769.05

13. Determine the total compound interest that you will have to pay if you borrow $1,780 at 16% com-

pounded semiannually and don’t pay it back for 5 years.

$2,062.88

14. How much compound interest will you earn if you invest $10,000 for 17 years at 9% compounded

annually?

$33,276.30

15. Compute the future value (compound amount) of $18,000 invested for 4.5 years at 5% compounded

quarterly.

$22,510.44

Score for B (32)

C (40 points) Business Applications. Find the future value (compound amount) or the compound

interest, as indicated. Round answers to the nearest cent. Use Table 16-1 or a calculator. (4 points

for each correct answer)

16.

Karen Wilson thinks that she needs to borrow $7,600 for 2 years. She doesn’t have a very good credit

rating, so most finance companies want to charge her a high interest rate. She finally finds a lender that

will loan her the money at 12% compounded monthly. How much interest will Karen have to pay to this

particular lender?

$2,049.95

Chapter 16 Compound Interest 331

Assignment 16.1 Continued

12% 4 12 5 1% $7,600 3 1.26973 5 $9,649.95

12 3 2 5 24 periods $9,649.95 2 $7,600 5 $2,049.95

16% 4 2 5 8% $5,000 3 2.51817 5 $12,590.85

2 3 6 5 12 periods $12,590.85 2 $5,000 5 $7,590.85

5% 4 4 5 1.25% $11,200 3 1.18995 5 $13,327.44

4 3514 periods3

8% 4 1 5 8% $7,150 3 1.99900 5 $14,292.85

1 3 9 5 9 periods $14,292.85 2 $7,150 5 $7,142.85

9% 4 12 5 0.75% $15,000 3 1.06956 5 $16,043.40

9 months 5 9 periods

17. Mary Sousa receives a telephone call from a salesperson who describes “an incredible investment

opportunity.” The investment promises a return of 16% compounded semiannually for investments

of $5,000 or more. One disadvantage is that no money will be paid out for a long time. Another disad-

vantage is that the investment is very risky. Mary doesn’t think that she will need the money for 6 years,

so she decides to invest $5,000. If the investment pays what it promises, how much interest will Mary

earn in the 6 years?

$7,590.85

18. William Wang wants to borrow money from his father to buy a car. William’s father is trying to teach

him how to manage money, so he agrees to loan him the money, but at 5% compounded quarterly.

William borrows $11,200 and repays everything—principal plus all of the interest—in years.

How much does William pay back to his father?

$13,327.44

19. Don Hildebrand is trying to decide whether to invest money in a bank or in something a little riskier

that will pay a higher return. One very simple investment promises to pay a minimum of 8% com-

pounded annually, but he must leave all of the money and interest invested for 9 years. How much inter-

est will Don earn during the 9 years if he invests $7,150 and the investment pays the minimum?

$7,142.85

20. Marcia Juarez and her brother-in-law have a successful business with several employees. They decide

to borrow $15,000 to pay their quarterly deposits for payroll taxes and federal income tax. They get

the money at 9% compounded monthly and repay all interest and principal after 9 months. How much

do they repay?

$16,043.40

21. Sammie Crass inherited $16,780. She wants to invest it in something relatively safe so that she can trans-

fer all the money to her children’s college fund in about 8 years. One investment brochure (called a

prospectus) states that it will pay a return of 8% compounded quarterly. How much will Sammie have

total, principal plus interest, after 8 years?

$31,622.58

332 Part 4 Interest Applications

Assignment 16.1 Continued

8% 4 4 5 2% $16,780 3 1.88454 5 $31,622.58

4 3 8 5 32 periods

4% 4 1 5 4% $24,500 3 1.26532 5 $31,000.34

1 3 6 5 6 periods $31,000.34 2 $24,500 5 $6,500.34

8% 4 2 5 4% $4,750 3 1.36857 5 $6,500.71

2 3 4 5 8 periods $6,500.71 2 $4,750 5 $1,750.71

6% 4 12 5 0.5% $32,000 3 1.13846 5 $36,430.72

26 months 5 26 periods

8% 4 4 5 2% $1,800 3 1.17166 5 $2,108.99

4 3 2 5 8 periods $2,108.99 2 $1,800 5 $308.99

22. To help his daughter and son-in-law purchase their first new car, Robert Chiu loans them $24,500. They

agree on an interest rate of 4% compounded annually, and Mr. Chiu tells them that they can pay it all

back, the $24,500 plus the interest, in 6 years. How much interest will Mr. Chiu receive from them?

$6,500.34

23. Sandee Millet owns and operates an art supply store in a suburban shopping center. Sandee learns about

an investment that claims to pay a return of 8% compounded semiannually for 4 years. Sandee decides to

invest $4,750. Compute the amount of interest that she will earn in the 4 years.

$1,750.71

24. Ken Ortman is a student at medical school. He borrowed $32,000 for 26 months at the rate of 6% com-

pounded monthly. How much total, principal plus compound interest, must Ken repay at the end of the

26 months?

$36,430.72

25. The County Employees Credit Union pays an interest rate of 8% compounded quarterly on savings

accounts of $1,000 or more, with the requirement that the money be deposited for at least 6 months.

How much interest will Marilyn Bunnell earn if she deposits $1,800 and leaves it in the credit union for

2 years?

$308.99

Score for C (40)

Chapter 16 Compound Interest 333

Assignment 16.2: Present Value

Name

Date Score

A (28 points) Find the present value (principal) and the compound interest, as indicated, for each of the

following investments. (Hint: Subtract the present value from the future value to find the compound

interest.) Use Table 16-1, Table 16-2, or a calculator. Round answers to the nearest cent. (2 points for

each correct answer)

Future Present Compound

Value Rate Term Value Interest

1. $3,900 6% compounded semiannually 3 years

$633.83$3,266.17

Learning Objectives

2 3

6% 4 2 5 3% $3,900 4 1.19405 5 $3,266.19 or $3,900 3 0.83748 5 $3,266.17

2 3 3 5 6 periods $3,900 2 $3,266.17 5 $633.83

8% 4 4 5 2% $15,000 4 1.74102 5 $8,615.64 or $15,000 3 0.57437 5 $8,615.55

4 3 7 5 28 periods $15,000 2 $8,615.55 5 $6,384.45

6% 4 1 5 6% $23,000 4 1.89830 5 $12,116.10 or $23,000 3 0.52679 5 $12,116.17

1 3 11 5 11 periods $23,000 2 $12,116.17 5 $10,883.83

9% 4 12 5 0.75% $6,800 4 1.43141 5 $4,750.56 or $6,800 3 0.69861 5 $4,750.55

12 3 4 5 48 periods $6,800 2 $4,750.55 5 $2,049.45

6% 4 4 5 1.5% $10,000 4 1.81402 5 $5,512.62 or $10,000 3 0.55126 5 $5,512.60

4 3 10 5 40 periods $10,000 2 $5,512.60 5 $4,487.40

8% 4 2 5 4% $50,000 4 1.60103 5 $31,229.90 or $50,000 3 0.62460 5 $31,230.00

2 3 6 5 12 periods $50,000 2 $31,230 5 $18,770.00

6% 4 12 5 0.5% $2,500 4 1.09393 5 $2,285.34 or $2,500 3 0.91414 5 $2,285.35

18 months 5 18 periods $2,500 2 $2,285.35 5 $214.65

Note to teachers: Problems were solved using both Table 16-1 and Table 16-2. The given answers come from the

Table 16-2 solutions. Answers from the solutions using Table 16-1 and calculator solutions may vary slightly.

2. $15,000 8% compounded quarterly 7 years

$6,384.45$8,615.55

3. $23,000 6% compounded annually 11 years

$10,883.83$12,116.17

4. $6,800 9% compounded monthly 4 years

$2,049.45$4,750.55

5. $10,000 6% compounded quarterly 10 years

$4,487.40$5,512.60

6. $50,000 8% compounded semiannually 6 years

$18,770.00$31,230.00

7. $2,500 6% compounded monthly 18 months

$214.65$2,285.35

Score for A (28)