The Mathematics of Money

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

To begin with, let’s consider a $5,000 loan for 5 years at 8% simple interest. From our

previous work we can easily calculate the interest:

I 
 PRT

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

I 
 $2,000

and so the maturity value would be $5,000  $2,000 
 $7,000.

What we have not yet looked at is the way the loan progresses toward this goal as time

passes through the 5-year term. This calculation takes us from principal to maturity value

without any thought about how the amount of interest grows over the loan’s term. Suppose,

then, that instead of just jumping from the $5,000 principal to the $7,000 maturity value

5 years later we take a look at the loan year by year along the way.

In the first year, the interest would be

I 
 PRT

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

I 
 $400

In the second year, the interest would be the same. The principal is still $5,000, the interest

rate is still 8%, and the second year is 1 year long, just like the first one. Likewise, the inter-

est in the third, fourth, and fifth years would also be the same. Thus under simple interest

the loan is growing at a constant rate of $400 per year:

Year Interest Balance

Start N/A $5,000.00

Year 1 $400.00 $5,400.00

Year 2 $400.00 $5,800.00

Year 3 $400.00 $6,200.00

Year 4 $400.00 $6,600.00

Year 5 $400.00 $7,000.00

We see that at this pace we arrive at an ending balance of $7,000.00 at maturity, just as

expected.

This is all well and good, but now imagine that this is a deposit that you have made in

a bank certificate. At the end of the first year, you would have $5,400.00 in your account.

That money belongs to you. By leaving it on deposit at the bank for the second year, you

are in effect loaning the bank your $5,400.00. Yet, according to our simple interest calcula-

tion, you are being paid interest only on your original $5,000.00. Even though your account

balance grows and grows because of the accumulation of interest, you continue to be paid

interest only on that original $5,000.00.

That is how simple interest works. No matter how long the loan continues, under simple

interest the borrower pays (and lender receives) interest only on the original principal,

not on any interest that accumulates along the way. Thus, in year 2 you get interest on the

$5,000, since that was original principal, but you are not entitled to any interest on the

$400. Even though that money is yours, and even though you are letting your bank have

use of it, it is not considered principal and it does not earn interest. The same thing happens

in years 3, 4, and 5. It doesn’t matter how big your balance gets; only the original $5,000

earns interest. Even if you left the account open at the same rate for 10,000 years (in which

time your balance would grow to more than $4 million), you would still continue to earn

interest at the same plodding rate of $400 per year.

This doesn’t seem quite fair. It seems reasonable that you should receive interest

on the entire amount of your account balance. If the bank has the use of $5,400 of

your money in year 2, then you have every reason to expect that it should pay you

3.1 Compound Interest: The Basics 87