The Mathematics of Money

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

4.2 Future Values of Annuities 147

Approach 1: The Chronological Approach

A natural way to approach this problem is to build up the account value year by year, credit-

ing interest as it comes due and adding new payments as they are made.

At the end of the first year, $1,200 is deposited. No interest would be paid, since there

was no money in the account prior to this deposit.

At the end of the second year, 1 year’s worth of interest would be paid on the $1,200,

raising the account value to $1,200(1.072) 
 $1,286.40. In addition, $1,200 comes in from

the second-year deposit, bringing to total to $2,486.40.

We could continue along this way until the end of the 5 years, the results of which can

be summarized in the table below:

Year

Starting

Balance

Interest

Earned Deposit

Ending

Balance

1 $0.00 $0.00 $1,200.00 $1,200.00

2 $1,200.00 $86.40 $1,200.00 $2,486.40

3 $2,486.40 $179.02 $1,200.00 $3,865.42

4 $3,865.42 $278.31 $1,200.00 $5,343.73

5 $5,343.73 $384.75 $1,200.00 $6,928.48

This approach is logical, does not involve any new formulas, and it nicely reflects what goes

on in the account as time goes by. But it is obvious that we wouldn’t want to use it to find

the future value of 40 years of weekly payments. This chronological approach may be fine

in cases where the total number of payments is small, but clearly we have reason to find an

alternative.

Approach 2: The Bucket Approach

Suppose now that you opened up a new account for each year’s deposit, instead of making

them all to the same account. Assuming that all of your accounts paid the same interest rate,

and assuming you didn’t mind the inconvenience of keeping track of multiple accounts,

would this have made any difference to the end result?

The bucket approach arises from the observation that keeping separate accounts would

make no difference whatever in the total end result. It doesn’t make any difference whether

you have a dollar or 20 nickels, and it doesn’t make any difference whether you have

one large account or 20 smaller ones. Since everything would come out the same if each

deposit (and its interest) were kept in a separate “bucket,” we pretend that this is actually

what happens. That’s probably not what actually does happen, but since the result would be

the same if it did, pretending this happens won’t change our final answer.

The first payment of $1,200 is placed in the first bucket. This money is kept on deposit

from the end of year 1 until the end of year 5, a total of 4 years. So, at the end of the fifth

year, its bucket will contain a total of $1,200(1.072)^4 
 $1,584.75.

We do the same for each of the other five payments, and then put them all back together

again at the end for the total:

Payment from Year Payment Amount Years of Interest Future Value

1 $1,200 4 $1,584.75

2 $1,200 3 $1,478.31

3 $1,200 2 $1,379.02

4 $1,200 1 $1,286.40

5 $1,200 0 $1,200.00

Grand total $6,928.48

Notice that we arrive at the same total as with the chronological approach, as we would

have hoped. Unfortunately, the bucket approach would still be very tedious to use if there

were a lot of payments instead of just five. As things stand, we still have not found our goal