Lecture Practical business math procedures (10/e): Chapter 12 - Jeffrey Slater

Chapter 12 - Compound interest and present value. After you have mastered the material in this chapter, you will be able to: Compare simple interest with compound interest, calculate the compound amount and interest manually and by table lookup, explain and compute the effective rate (APY), compare present value (PV) with compound interest (FV), compute present value by table lookup, check the present value answer by compounding.

Compound Interest and 

Present Value

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

Learning Unit Objectives

Compound Interest (Future Value) –  The Big Picture

LU12.1

1 Compare present value (PV) with 

compound interest (FV)

2 Compute present value by table lookup

3 Check the present value answer by 

Compound Interest and Present Value

#12

Learning Unit Objectives

Present Value ­­ The Big Picture

LU12.2

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

Compounding Terms

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

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

$1.259

7 $1.3605

for Four Periods

Year 1 Year 2 Year 3 Year 4

1.00

$ $ 1.08 $ 1.17 $ 1.26

Interest $ 0.08 $ 0.09 $ 0.09 $ 0.10

End of year $ 1.08 $ 1.17 $ 1.26 $ 1.36

Manual Calculation

Number of periods  (N) 

Number of years 

multiplied the number 

of times the interest is 

compounded per year

Rate for each period  

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?

Semiannually: 4 x 2 = 8

Semiannually: 8% / 2 = 4%

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

I = P x R x T

I = $80 x .08 x 4

I = $25.60

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?

80.00

$ $ 86.40 $ 93.31 $ 100.77

Interest $ 6.40 $ 6.91 $ 7.46 $ 8.06 Simple

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. 

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:

 Truth in 

Savings 

Law Annual Percentage

Yield Effective Rate =   Interest for 1 year

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

Blue, 8% compounded quarterly

Periods = 4 (4 x 1)

Percent = 8%  = 2%

       4

Principal = $8,000

Table 12.1 lookup: 4 periods, 2%

         1.0824

      x $8,000

Less      $8,659.20

$8,000.00      659.20 APY       659.20   = .0824

Sun, 8% compounded semiannually Periods = 2 (2 x 1)

Percent = 8%  = 4%

       2 Principal = $8,000 Table 12.1 lookup: 2 periods, 4%

         1.0816

      x $8,000 Less      $8,652.80

$8,000.00      652.80 APY       652.80   = .0816

(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

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

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

Number of periods

Present value goes from the future value to the present value

Present  value

$.7350

$.7938 $.8573

$.9259 $1.0000

Future  Value

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

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

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

We know the 

present dollar 

amount and find 

We know the  future dollar  amount and find 

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