Chapter 12

Compound Interest and Compound Interest and Present ValuePresent Value

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

12-2

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

Compound Interest and Present Value#12#12Learning Unit ObjectivesCompound Interest (Future Value) – The Big Picture

LU12.1LU12.1

12-3

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

2. Compute present value by table lookup

3. Check the present value answer by compounding

Compound Interest and Present Value#12#12Learning Unit ObjectivesPresent Value -- The Big PictureLU12.2LU12.2

12-4

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) - is the final amount of the loan or investment at the end of the last

period

12-5

Compounding Terms

Compounding Periods Interested Calculated

Compounding Annually Once a year

Compounding Semiannually Every 6 months

Compounding Quarterly Every 3 months

Compounding Monthly Every month

Compounding Daily Every day

12-6

Figure 12.1 Future Value of $1 at 8% for Four Periods

$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 4

Number 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

12-7

Figure 12.1 Future Value of $1 at 8% for Four Periods

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 1.26End of year 1.08$ 1.17$ 1.26$ 1.36$

Manual Calculation

12-8

Tools for Calculating Compound Interest

Number of periods (N) Number of years

multiplied 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: 4 x 1 = 4Semiannually: 4 x 2 = 8Quarterly: 4 x 4 = 16

Annually: 8% / 1 = 8%Semiannually: 8% / 2 = 4%Quarterly: 8% / 4 = 2%

Periods Rate

12-9

Simple Versus Compound Interest

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

Simple CompoundedCompounded

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 93.31 100.77End of year 86.40$ 93.31$ 100.77$ 108.83$

Interest: $108.83 - $80.00 = $28.83

SimpleSimple

12-10

Calculating Compound Amount 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 is the table factor

Step 4. Multiply the table factor by the amount of the loan.

12-11

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)

Table 12.1 - Future Value of $1 at Compound Interest

12-12

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?

N = 4 x 5 = 20

R = 6% = 1.5% 1

Table Factor = 1.3469

Compounded Amount:

$8,000 x 1.3469 = $10,775.20

12-13

Nominal and Effective Rates (APY) of Interest

Truth in

Savings

Law

Annual

Percentage

Yield Effective Rate = Interest for 1 year (APY) Principal

Nominal Rate (Stated Rate) - The rate on which the bank calculates interest.

12-14

Calculating Effective Rate APY

Blue, 8% compounded quarterlyPeriods = 4 (4 x 1)Percent = 8% = 2% 4Principal = $8,000Table 12.1 lookup: 4 periods, 2%

1.0824 x $8,000Less $8,659.20

$8,000.00 659.20

APY 659.20 = .0824 $8,000

= 8.24%

Sun, 8% compounded semiannuallyPeriods = 2 (2 x 1)Percent = 8% = 4% 2Principal = $8,000Table 12.1 lookup: 2 periods, 4%

1.0816 x $8,000Less $8,652.80

$8,000.00 652.80

APY 652.80 = .0816 $8,000

= 8.16%

12-15

Figure 12.3 - Nominal and Effective Rates (APY) of Interest Compared

Annual

Semiannual

Quarterly

Daily

$1,060.00

$1,060.90

$1,061.40

$1,061.80

6.00

6.09%

6.14%

6.18%

$1,000 + 6%

Beginning Nominal rate Compounding End Effective rate balance of interest period balance (APY) of interest

12-16

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

Table 12.2 - Compounding Interest Daily

12-17

Compounding Interest Daily

Calculate by Table 12.2 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

12-18

Figure 12.4 Present Value of $1 at 8% for Four Periods

$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

$1.0000

Future Value

12-19

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 is the table factor.

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

12-20

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)

Table 12.3 - Present Value of $1 at End Period

12-21

Comparing Compound Interest (FV) Table 12.1 with Present Value (PV) Table 12.3

Compound value Table 12.1 Present value Table 12.3Table Present Future Table Future Present12.1 Value Value 12.3 Value Value

1.3605 x $80 = $108.84 .7350 x $108.84 = $80.00

(N = 4, R = 8) (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

12-22

Calculating Present Value Amount by Table Lookup

Rene 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?

N = 4 x 4 = 16

R = 8% = 2% 4

Table Factor = .7284

Compounded Amount:

$20,000 x .7284 = $14,568

Invest Today

12-23

Problem 12-13:

Solution:

8 years x 2 = 16 periods 6% 2

= 3%

$40,000.00 x 1.6047 = $64,188.

12-24

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

12-25

Problem 12-16:

Solution:

3 years x 2 = 6 periods $15,000 x 1.3023 = $19,534.50 +40,000.00 $59,534.50

9% 2

= 4.5%

$59,534.50 x 1.3023 = $77,531.78

12-26

Problem 12-27:

Solution:

8 years x 2 = 16 periods 6% 2

= 3%

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