Chapter TwelveCOMPOUND INTEREST AND PRESENT VALUE

Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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1. Compare simple interest with compound interest.2. Calculate the compound amount and interest

manually and by table lookup.3. Explain and compute the effective rate (APY).

LU 12-1 Compound Interest (Future Value) – The Big Picture

LEARNING UNIT OBJECTIVES

LU 12-2 Present Value -- The Big Picture

1. Compare present value (PV) with compound interest (FV).

2. Compute present value by table lookup.3. Check the present value answer by compounding.

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COMPOUND INTEREST

• So far we have discussed only simple interest, which is interest on the principal alone. Simple interest is either paid at the end of the loan period or deducted in advance.

• In this chapter we look at the power of compounding—interest paid on earned interest.

• Compounding involves the calculation of interest periodically over the life of the loan (or investment). After each calculation, the interest is added to the principal. Future calculations are on the adjusted principal (old principal plus interest). Compound interest, then, is the interest on the principal plus the interest of prior periods. Future value (FV), or the compound amount, is the final amount of the loan or investment at the end of the last period.

• Before you learn how to calculate compound interest and compare it to simple interest, you must understand the terms that follow.

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COMPOUNDING INTEREST (FUTURE VALUE)

Compound Interest –

The interest on the principal plus the interest of prior

periods

Compounding –

Involves the calculation of interest periodically over the life

of the loan or investment

Present Value –

The value of a loan or investment today

Future Value (compound amount) –

The final amount of the loan or investment at the end of the

last period

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COMPOUNDING TERMS

Compounding Periods Interest Calculated

Compounding Annually

Compounding Semiannually

Compounding Quarterly

Compounding Monthly

Compounding Daily

Once a year

Every 6 months

Every 3 months

Every month

Every day

• Compounding changes the interest rate.• Rate for each period is annual interest rate divided by

the number of times the interest is compounded per year.

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FUTURE VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE 12.1)

$0.00$0.50$1.00$1.50

$2.00$2.50$3.00$3.50$4.00$4.50$5.00

0 1 2 3 4Number of periods

Compounding goes from present value to future value

Present

value

After 1

period, $1 is worth $1.08

After 2 periods,

$1 is worth $1.17

After 3 periods,

$1 is worth $1.26

Future

Value

After 4 periods, $1 is worth $1.36

$1.00

$1.08 $1.1664 $1.2597 $1.3605

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FUTURE VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE

12.1)

Year 1 Year 2 Year 3 Year 41.00$ 1.08$ 1.17$ 1.26$ 0.08 x .10 x .10 x .10

Interest 0.08$ 0.09$ 0.09$ 0.10$ Beg. Bal 1.00 1.08 1.17 1.26 End of year 1.08$ 1.17$ 1.26$ 1.36$

Manual Calculation

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TOOLS FOR CALCULATING COMPOUND INTEREST

Number of periods (N) Number of years

multiplied by the number of times the interest is compounded per year

Rate for each period (R) Annual interest rate

divided by the number of times the interest is

compounded per year

If you compounded $100 for 4 years at 8% annually,

semiannually, or quarterly, what is N and R?

Annually:

Semiannually:

Quarterly:

Periods Rate

4 x 1 = 4

4 x 4 = 16

4 x 2 = 8

8% / 1 = 8%Annually:

Semiannually:

Quarterly: 8% / 4 = 2%

8% / 2 = 4%

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SIMPLE VERSUS COMPOUND INTEREST

Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest and maturity value?

I = P x R x T

I = $80 x .08 x 4

I = $25.60

MV = $80 + $25.60

MV = $105.60

I = P x R x T

I = $80 x .08 x 4

I = $25.60

MV = $80 + $25.60

MV = $105.60

Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded amount?

Year 1 Year 2 Year 3 Year 480.00$ 86.40$ 93.31$ 100.77$ x .08 x .08 x .08 x .08

Interest 6.40$ 6.91$ 7.46$ 8.06$ Beg. bal 80.00 86.40 93.31 100.77 End of year 86.40$ 93.31$ 100.77$ 108.83$

Interest: $108.83 -- $80.00 = $28.83

Simple Compounded

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CALCULATING COMPOUND AMOUNT

BY TABLE LOOKUPStep 1. Find the periods: Years multiplied by number of

times interest is compounded in 1 year.

Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.

Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor for the compound amount of $1.

Step 4. Multiply the table factor by the amount of the loan. This gives the compound amount.

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Period 1% 1.50% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 1.0100 1.0150 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000

2 1.0201 1.0302 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100

3 1.0300 1.0457 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310

4 1.0406 1.0614 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641

5 1.0510 1.0773 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105

6 1.0615 1.0934 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716

7 1.0721 1.1098 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487

8 1.0829 1.1265 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436

9 1.0937 1.1434 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579

10 1.1046 1.1605 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937

11 1.1157 1.1780 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531

12 1.1260 1.1960 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384

13 1.1381 1.2135 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523

14 1.1495 1.2318 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975

15 1.1610 1.2502 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1772

Future value of $1 at compound interest (Partial)

FUTURE VALUE OF $1 AT COMPOUND INTEREST (TABLE 12.1)

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CALCULATING COMPOUND AMOUNT BY TABLE LOOKUP

Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the

balance of her account at the end of 5 years?

Periods (N) = 4 x 5 = 20

Rate (R) = 6%/4 = 1.5%Table Factor = 1.3469Compounded Amount:$8,000 x 1.3469 = $10,775.20

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NOMINAL AND EFFECTIVE RATES (APY) OF INTEREST

Truth in

Savings Law

Annual

PercentageYield

Effective rate (APY) = Interest for 1 year

Principal

Nominal Rate (stated rate) –

The rate on which the bank calculates interest

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CALCULATING EFFECTIVE RATE APY

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NOMINAL AND EFFECTIVE RATES (APY) OF INTEREST COMPARED

(FIGURE 12.3)

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Period 6.00% 6.50% 7.00% 7.50% 8.00% 8.50% 9.00% 9.50% 10.00%

1 1.0618 1.0672 1.0725 1.0779 1.0833 1.0887 1.0942 1.0996 1.1052

2 1.1275 1.1388 1.1503 1.1618 1.1735 1.1853 1.1972 1.2092 1.2214

3 1.1972 1.2153 1.2337 1.2523 1.2712 1.2904 1.3099 1.3297 1.3498

4 1.2712 1.2969 1.3231 1.3498 1.3771 1.4049 1.4333 1.4622 1.4917

5 1.3498 1.3840 1.4190 1.4549 1.4917 1.5295 1.5862 1.6079 1.6486

6 1.4333 1.4769 1.5219 1.5682 1.6160 1.6652 1.7159 1.7681 1.8220

7 1.5219 1.5761 1.6322 1.6904 1.7506 1.8129 1.8775 1.9443 2.0136

8 1.6160 1.6819 1.7506 1.8220 1.8963 1.9737 2.0543 2.1381 2.2253

9 1.7159 1.7949 1.8775 1.9639 2.0543 2.1488 2.2477 2.3511 2.4593

10 1.8220 1.9154 2.0136 2.1168 2.2253 2.3394 2.4593 2.5854 2.7179

15 2.4594 2.6509 2.8574 3.0799 3.3197 3.5782 3.8568 4.1571 4.4808

20 3.3198 3.6689 4.0546 4.4810 4.9522 5.4728 6.0482 6.6842 7.3870

25 4.4811 5.0777 5.7536 6.5195 7.3874 8.3708 9.4851 10.7477 12.1782

30 6.0487 7.0275 8.1645 9.4855 11.0202 12.8032 14.8747 17.2813 20.0772

Interest on a 1% deposit compounded daily--360 day basis

COMPOUNDING INTEREST DAILY (TABLE 12.2)

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COMPOUNDING INTEREST DAILY

Use Table 12.2 to calculate what $1,500 compounded daily for

5 years will grow to at 7%.

N = 5

R = 7%

Factor, 1.4190

$1,500 x 1.4190 = $2,128.50

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PRESENT VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE

12.4)

$0.00$0.10$0.20$0.30$0.40$0.50$0.60$0.70$0.80$0.90$1.00$1.10$1.20

0 1 2 3 4

Number of periods

Present value goes from the future value to the present value

Present value

$.7350$.7938

$.8573

$.9259

$1.0000Future Value

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CALCULATING PRESENT VALUE BY TABLE LOOKUP

Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year.

Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.

Step 3. Go down the Period column of the table to the number desired; look across the row to find the rate. At the intersection of the two columns is the table factor for the compound value of $1.

Step 4. Multiply the table factor by the future value. This is the present value.

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Period 1% 1.50% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.9901 0.9852 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 0.9803 0.9707 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264

3 0.9706 0.9563 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513

4 0.9610 0.9422 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830

5 0.9515 0.9283 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209

6 0.9420 0.9145 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645

7 0.9327 0.9010 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132

8 0.9235 0.8877 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665

9 0.9143 0.8746 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241

10 0.9053 0.8617 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855

11 0.8963 0.8489 0.8043 0.7224 0.6496 0.5847 0.5268 0.4751 0.4289 0.3875 0.3505

12 0.8874 0.8364 0.7885 0.7014 0.6246 0.5568 0.4970 0.4440 0.3971 0.3555 0.3186

13 0.8787 0.8240 0.7730 0.6810 0.6006 0.5303 0.4688 0.4150 0.3677 0.3262 0.2897

14 0.8700 0.8119 0.7579 0.6611 0.5775 0.5051 0.4423 0.3878 0.3405 0.2992 0.2633

15 0.8613 0.7999 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394

Present value of $1 at end period (partial)

PRESENT VALUE OF $1 AT END PERIOD (TABLE 12.3)

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COMPARING COMPOUND INTEREST (FV)

(TABLE 12.1) WITH PRESENT VALUE (PV) (TABLE 12.3)

Compound value Table 12.1Table Present Future12.1 Value Value1.3605 x $80 = $108.84

(N = 4, R = 8%)

We know the present dollar

amount and find what the dollar amount is worth

in the future.

We know the future dollar amount and find what the

dollar amount is worth in the

present.

Present value Table 12.3Table Future Present12.3 Value Value

.7350 x $108.84 = $80.00

(N = 4, R = 8%)

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CALCULATING PRESENT VALUE AMOUNT

BY TABLE LOOKUPRene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at

the beginning of the year to have $20,000 in 4 years?

Periods (N) = 4 x 4 = 16Rate (R) = 8%/4 = 2%Table Factor = .7284

Compounded Amount:$20,000 x .7284 = $14,568

Invest Today

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PRACTICE QUIZ-1

1-Complete the following without a table (round each calculation to the nearest cent as needed):

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PRACTICE QUIZ-1

2-Solve the previous problem by using compound value (FV) in Table 12.1.3-Lionel Rodgers deposits $6,000 in Victory Bank, which pays 3% interest compounded semiannually. How much will Lionel have in his account at the end of 8 years?4-Find the effective rate (APY) for the year: principal, $7,000; interest rate, 12%; and compounded quarterly.5-Calculate by Table 12.2 what $1,500 compounded daily for 5 years will grow to at 7%.

For step by step solution watch the video for LU 12-1 ( Go to: McGraw-Hill’s Connect; Assignment # 4; Question 1; Click the eBook & resources options drop down menu; scroll down to LU12-1 and click

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PRACTICE QUIZ-2

Use the present value Table 12.3 to complete:

3-Bill Blum needs $20,000 6 years from today to attend V.P.R. Tech. How much must Bill put in the bank today (12% quarterly) to reach his goal?4-Bob Fry wants to buy his grandson a Ford Taurus in 4 years. The cost of a car will be $24,000. Assuming a bank rate of 8% compounded quarterly, how much must Bob put in the bank today?

For step by step solution watch the video for LU 12-1 ( Go to: McGraw-Hill’s Connect; Assignment # 4; Question 5; Click the eBook & resources options drop down menu; scroll down to LU12-2 and click

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PROBLEM 12-13

Solution:

8 years x 2 = 16 periods 6% 2 = 3% $40,000 x 1.6047 = $64,188

Lynn Ally, owner of a local Subway shop, loaned $40,000 to Pete Hall to help him open a Subway franchise. Pete plans to repay Lynn at the end of 8 years with 6% interest compounded semiannually. How much will Lynn receive at the end of 8 years? LU 12-1(2)

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PROBLEM 12-15

Solution:

Mystic

4 years x 2 = 8 periods

10% 2

= 5%

$10,000 x 1.4775 = $14,775 -- 10,000 $ 4,775

Four Rivers

4 years x 4 = 16 periods

8% 4

= 2%

$10,000 X 1.3728 = $13,728 -- 10,000 $ 3,728

Melvin Indecision has difficulty deciding whether to put his savings in Mystic Bank or Four Rivers Bank. Mystic offers 10% interest compounded semiannually. Four Rivers offers 8% interest compounded quarterly. Melvin has $10,000 to invest. He expects to withdraw the money at the end of 4 years. Which bank gives Melvin the better deal? Check your answer. LU 12-1(3)

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PROBLEM 12-16

Solution:

3 years x 2 = 6 periods9% 2 = 4.5%

$59,534.50 x 1.3023 = $77,531.78

Lee Holmes deposited $15,000 in a new savings account at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $40,000 at 9% interest compounded semiannually. At the end of 6 years, what is the balance in Lee’s account? LU 12-1(2)

$15,000 x 1.3023 = $19,534.50 + 40,000.00 $ 59,534.50

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PROBLEM 12-27

Solution:

8 years x 2 = 16 periods

6% 2 = 3%

$6,000 x .6232 = $3,739.20

Paul Havlik promised his grandson Jamie that he would give him $6,000 8 years from today for graduating from high school. Assume money is worth 6% interest compounded semiannually. What is the present value of this $6,000?LU 12-2(2)

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DRILL PROBLEMS

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