The Mathematics of Money

232 Chapter 5 Spreadsheets

to pay less. It is fi ne for Ted and Kirsti (from Example 5.3.3) to pay any amount they like

in any given month, but it must be at least $845.76. While it is mathematically possible to

determine how long it would take to pay off their loan if they paid just $800 a month, in

reality their lender would probably not allow this.

But sometimes a lender might allow, or even encourage, a borrower to pay less than the

payment required to pay off the loan in the expected amount of time. For example, a lender

might allow smaller payment for someone who has a temporary fi nancial strain, such as

loss of a job, military deployment, or completing a graduate degree. There are also more

sinister possibilities: a lender might permit an initially low payment to make a borrower

feel like the amount he is borrowing is not as large as it really is, or might encourage lower

payments so that more interest will pile up on a debt. Regardless of the reasons for such a

situation, though, the fi nancial consequences can be serious, as the following example will

illustrate.

Example 5.3.4 Jennifer owes $2,538 on a personal loan with a 12½% interest rate.

She was originally supposed to make payments of at least $100 a month, but the

lender has told her that from now on she can pay as little as $20 a month if she wants.

If she decides to reduce her payment to just $20 a month, how long will it take her to

pay off the remaining balance?

Looking at an amortization table for this loan with a $20 payment, we get:

1 Rate: 12.50% Initial Balance: $2,538.00

2

3

A B C D

Month Payment To Principal Ending Balance

1 $20.00 -$6.44 $2,544.44

To Interest

$26.44

4 2 $20.00 $26.50 -$6.50 $2,550.94

E

Notice that the amount going to principal is a negative number, and that the ending balance

is increasing. Because $20.00 is not enough to cover the fi rst month’s interest, the $6.44

that it falls short must be added to the balance. In the second month, the interest is a bit

higher, so the $20.00 payment is a bit further short, and so the balance grows by a little

bit more. Looking out to further months, we can see that the situation just continues to get

worse and worse; after 20 years, for example, Jennifer’s debt will have grown to $9,352.10

at this rate. And there is no opportunity for things to ever get better.

We can conclude from this that at $20.00 a month, Jennifer will never manage to pay off

this balance.

When the amount of the payment falls below the interest charges, the shortfall causes the

balance to increase instead of decreasing. This sort of situation is referred to as negative

amortization.

Example 5.3.5 What is the minimum monthly payment Jennifer (from Example

5.3.4) needs to schedule in order to avoid negative amortization?

From the amortization table, we can see that if her payment is less than $26.44, we will

have negative amortization. If she pays exactly $26.44 she won’t encounter negative amor-

tization, but her balance will never go down either, since that would be just enough to pay

the interest, and nothing more. If she pays $26.45 (or more) she will be reducing interest

by a little bit with each payment. The $26.45 a month won’t accomplish much with each

payment, but at least it will accomplish some reduction in the balance.

Her minimum to avoid negative amortization then is $26.45.

Of course, if she only pays a penny over the minimum, it will take a very long time to pay

off the debt. But at that rate it will at least eventually get paid off. As the cliché goes, little

things mean a lot. Slight changes in a payment schedule can make an enormous difference.

Example 5.3.6 If Jennifer pays $25 a month, how much will she owe in 20 years? If

she pays $27.50, how much will she owe in 20 years? What if she pays $30.00?