The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

Finding the Total Interest Earned

Our annuity formulas do not provide any easy, direct way to find the total amount of interest

earned. However, we can easily find this out with a slightly indirect approach:

Example 4.2.7 How much total interest did Carrie (from Example 4.2.6) earn?

To solve this, we fi rst observe that the money in her account comes from two sources: the

money she deposits, and the interest she earns. It is not hard to determine the total of her

deposits. She made monthly deposits for 5 years, for a total of n 
 60 deposits. Each one

was for $250, so in total she deposited (60)($250) 
 $15,000.

If $15,000 of her future value came from her deposits, the rest must have come from

interest. So

Carrie’s total interest 
 $16,786.39  $15,000 
 $1,786.39.

The Future Value of an Annuity Due

With an annuity due, payments are made at the beginning of each period rather than the

end. Each payment is made earlier, so it stands to reason that an annuity due would have

a larger future value than an ordinary annuity, since the payments have longer to earn

interest.

To see how much more, let’s revisit the 5 year, $1,200 per year annuity with 7.2%

interest, this time, though, as an annuity due. The first payment would earn interest

from the start of the first year until the end of the fifth year, for a total of 5 years. The

second payment would earn interest for 4 years, the third payment for 3 years, and so

on. Thus, following the bucket approach just as we did before, we get that the future

value would be:

Payment from Year Payment Amount Years of Interest Future Value

1 $1,200 5 $1,698.85

2 $1,200 4 $1,584.75

3 $1,200 3 $1,478.31

4 $1,200 2 $1,379.02

5 $1,200 1 $1,286.40

Grand total $7,427.33

Notice that this is the same table we used for the ordinary annuity, except that every pay-

ment is being credited with 1 additional year of interest. Since crediting 1 year’s extra

interest is equivalent to multiplying everything through by 1.072, we get our future value

by multiplying the ordinary annuity’s future value by 1.072. Trying it out, we see that

$6,928.48(1.072) does indeed equal $7,427.33.

This leads us to a formula for annuities due:

FORMULA 4.2.3

Future Value for an Annuity Due

FV 
 PMT s _n|i (1  i)

where

FV represents the FUTURE VALUE of the annuity

PMT is the amount of each PAYMENT

i is the INTEREST RATE per period

and

s _n (^) | (^) i is the FUTURE VALUE ANNUITY FACTOR (as defi ned in Defi nition 4.2.1)
4.2 Future Values of Annuities 155