The Mathematics of Money

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

1.1 Simple Interest and the Time Value of Money 3

Despite this, though, you probably wouldn’t be willing to let me live there for the year

for free. Even though you’ll get the house back at the end just as it was at the start, you’d

still expect to be paid something for a year’s use of your house. After all, though you

wouldn’t actually give up any of your property by lending it to me, you nonetheless would

be giving up something: the opportunity to live in your house during the year that I am

there. It is only fair that you should be paid for the property’s temporary use. In other,

ordinary terms, you’d expect to be paid some rent. There is nothing surprising in this. We

are all familiar with the idea of paying rent for a house or apartment. And the same idea

applies for other types of property as well; we can rent cars, or party tents, or construction

equipment, and many other things as well.

Now let’s suppose that I need to borrow $20, and you agree to lend it to me. If I offered

to pay you back the full $20 one year from today, would you agree to the loan under those

terms? You would be getting your full $20 back, but it hardly seems fair that you wouldn’t

get any other compensation. Just as in the example of the house, even though you will

eventually get your property back, over the course of the year you won’t be able to use it.

Once again it only seems fair that you should get some benefit for giving up the privilege

of having the use of what belongs to you.

We ordinarily call the payment for the temporary use of property such as houses, apart-

ments, equipment, or vehicles rent. In the case of money, though, we don’t normally use

that term. Instead we call that payment interest.

Definition 1.1.1

Interest is what a borrower pays a lender for the temporary use of the lender’s money.

Or, in other words:

Definition 1.1.2

Interest is the “rent” that a borrower pays a lender to use the lender’s money.

Interest is paid in addition to the repayment of the amount borrowed. In some cases, the

amount of interest is spelled out explicitly. If we need to determine the total amount to be

repaid, we can simply add the interest on to the amount borrowed.

Example 1.1.1 Sam loans Danielle $500 for 100 days. Danielle agrees to pay her

$80 interest for the loan. How much will Danielle pay Sam in total?

Interest is added onto the amount borrowed. $500 
 $80  $580. Therefore Danielle will

pay Sam a total of $580 at the end of the 100 days.

In other cases, the borrower and lender may agree on the amount borrowed and the amount

to be repaid without explicitly stating the amount of interest. In those cases, we can deter-

mine the amount of interest by finding the difference between the two amounts (in other

words, by subtracting.)

Example 1.1.2 To m loans Larry $200, agreeing to repay the loan by giving Larry

$250 in 1 year. How much interest will Larry pay?

The interest is the difference between what Tom borrows and what he repays. $250 – $200 

$50. So Larry will pay a total of $50 in interest.

It is awkward to have to keep saying “the amount borrowed” over and over again, and so

we give this amount a specific name.

Definition 1.1.3

The principal of a loan is the amount borrowed.

So in Example 1.1.1 the principal is $500. In Example 1.1.2 we would say that the principal

is $200 and the interest is $50.

There are a few other special terms that are used with loans as well.