The Mathematics of Money

166 Chapter 4 Annuities

Example 4.3.5 For the scenario of the previous example, determine how much total

interest Joe would be earning.

Since he is making 1,080 deposits of $67.01 each, his total deposits add up to (1,080)($67.01)


 $72,370.80. So the total interest earned is:

To tal interest 
 $1,000,000  $72,370.80 
 $927,629.20

So most of Joe’s million comes from the result of compound growth over time. He is planning

on earning a fairly high rate, and while his account balance probably won’t be much in the early

years, what there is will be earning compound interest for a very long period of time, and, as we

have repeatedly seen, compound interest over a long period of time is powerful stuff indeed.

Whether or not Joe’s assumptions (that he should have $1,000,000 and that he can

earn 9%) are reasonable is a matter that will be addressed in other parts of this book, in

Chapters 6 and 7. However, whether they are too high, too low, or a mix, the inescapable

mathematical conclusion here is that compound interest working over long periods of time

can accumulate a large sum from surprisingly small deposits. The following example will

illustrate the importance of time.

Example 4.3.6 How much would Joe’s semimonthly deposits need to be to

accumulate $1,000,000 at age 70, again assuming a 9% growth rate, assuming that

he starts at (a) age 65, (b) age 50, (c) age 35, (d) age 25 (already done), (e) age 18,

and (f) at age 2 (obviously assuming someone started the deposits for his benefi t)? For

each starting age, determine how much of the $1,000,000 comes from his deposits,

and how much comes from interest on the deposits.

Each of the calculations will be essentially the same as the one we did above, so we will not

show them here. For practice, though, you should verify the calculations for at least one or

two of the rows. The results are displayed in the table below:

Starting

Age

Number of

Deposits Annuity Factor

Each

Deposit

To tal

Deposits Total Interest

2 1,632 119,650.08513797 $8.36 $13,643.52 $986,356.48

18 1,248 28,221.56380297 $35.43 $44,216.64 $955,783.36

25 1,080 14,923.81979405 $67.01 $72,370.80 $927,629.20

35 840 5,919.72900240 $168.93 $141,901.20 $858,098.80

50 480 1,341.15067989 $745.63 $357,902.40 $642,097.60

65 120 151.19807368 $6,613.84 $793,660.80 $206,339.20

It should come as no surprise that the sooner the deposits start, the smaller they can be, but

it may still be surprising just how much of an impact this has.

EXERCISES 4.3

A. Sinking Funds

1. A community college is planning to add a new academic building 5 years from now. The college’s administration wants to
   set aside some money from its annual budget in each of the next 5 years in order to accumulate a fund of $1,200,000 to
   use toward the project. If the college can earn 4.55% interest, how much should it set aside each year to meet this goal?


2. Brad has nothing in his savings account right now, but realizes it would be a very good idea to build up a savings
   balance. His account pays 4% interest. How much should he deposit each month if he wants to have $10,000 in this
   account in 2 years.