The Mathematics of Money

88 Chapter 3 Compound Interest

interest on the $5,400. In other words, you want to receive interest on your accumulated

interest!

That is precisely the point of compound interest. With compound interest, interest is

paid on both the original principal and on any interest that accumulates along the way.

When interest is paid on interest, we say that it compounds. To see how this works, let’s

revisit the $5,000 bank account, this time calculating compound interest.

Compound Interest

Consistent with our first view of this account, suppose that we look at the account balance

year by year, assuming that interest is credited to your account annually. Then in the first

year you would earn interest on your original $5,000, just as before:

First Year

I 
 PRT

I 
 ($5,000)(0.08)(1)

I 
 $400.00

Things change though in the second year. Since we are now using compound interest,

when we calculate interest for the second year we will treat the entire $5,400.00 as the

principal:

Second Year

I 
 PRT

I 
 ($5,400.00)(0.08)(1)

I 
 $432.00

Not surprisingly, in the second year you earn more interest. The extra $32 is due entirely to

“interest on interest.” You can verify for yourself that $32 is 8% interest on $400 for 1 year.

In the third year, the interest grows even larger, since interest will now be paid on the

accumulated balance of $5,400  $432 
 $5,832.

Third Year

I 
 PRT

I 
 ($5,832.00)(0.08)(1)

I 
 $466.56

As the balance grows year by year, so does the interest. It is worthwhile to compare the

growth of the account year by year under compound interest versus its growth under simple

interest.

SIMPLE INTEREST COMPOUND INTEREST

Difference in

Year Start Interest End Start Interest End End Balance

1 $5,000.00 $400.00 $5,400.00 $5,000.00 $400.00 $5,400.00 $0.00

2 $5,400.00 $400.00 $5,800.00 $5,400.00 $432.00 $5,832.00 $32.00

3 $5,800.00 $400.00 $6,200.00 $5,832.00 $466.56 $6,298.56 $98.56

4 $6,200.00 $400.00 $6,600.00 $6,298.00 $503.88 $6,802.44 $202.44

5 $6,600.00 $400.00 $7,000.00 $6,802.44 $544.20 $7,346.64 $346.64

Note that, as time goes by and the account balance grows, so does the impact of

compounding. Over the first couple of years, the difference between simple and com-

pound interest is not all that significant. But as the principal used for compound interest

grows larger and larger, the interest earned on that principal also grows and grows. Even

more interestingly, while the pace of growth under simple interest remains constant, the