The Mathematics of Money

184 Chapter 4 Annuities

Example 4.5.2 Suppose Pat and Tracy took out a 15-year mortgage instead, also

for $158,000 and at a 7.2% rate. Their monthly payment would then be $1,437.87.

Construct an amortization table for their fi rst 6 monthly payments.

For the fi rst month, interest would be paid on the full $158,000, so the interest would be

I 
 ($158,000)(0.072)(1/12) 
 $948.00, just as in the 30-year example. This leaves

$1,437.87  $948.00 
 $489.87 to go toward reducing principal, and so after this pay-

ment they will owe $158,000  $489.87 
 $157,510.13. Note that, because of the larger

payment, more progress is made toward reducing the balance than with the fi rst payment

of the 30-year example.

The calculations for the next month are similar, except that instead of using $158,000 we

instead use the slightly smaller $157,510.13, applying the same logic as before.

Payment

Number

Payment

Amount

Interest

Amount

Principal

Amount

Remaining

Balance

1 $1,437.87 $948.00 $489.87 $157,510.13

2 $1,437.87 $947.25 $490.62 $157,019.51

3 $1,437.87 $946.50 $491.37 $156,528.14

4 $1,437.87 $945.75 $492.12 $156,036.02

5 $1,437.87 $944.99 $492.88 $155,543.14

6 $1,437.87 $933.26 $504.61 $155,038.53

7 $1,437.87 $930.23 $507.64 $154,530.89

8 $1,437.87 $927.19 $510.68 $154,020.21

9 $1,437.87 $924.12 $513.75 $153,506.46

10 $1,437.87 $921.04 $516.83 $152,989.63

11 $1,437.87 $917.94 $519.93 $152,469.70

12 $1,437.87 $914.82 $523.05 $151,946.65

Note that while the amount of interest in the first month is the same for both the 15-year

and 30-year loans, in the second month this is no longer true. This is because the first pay-

ment does more to drop the balance of the 15-year loan, and so its smaller balance leads to

less interest. This trend continues in the ensuing months, and in fact the difference becomes

more pronounced as time goes by. The greater progress toward killing off the balance with

the 15-year loan comes both from the larger payment and also from the interest savings of

having a smaller balance. While the interest savings are small early on, they grow over time

and become more and more significant as time goes by.

In these examples we compared 30-year and 15-year mortgage loans. While we assumed

that the interest rates were the same, we did consider these as two separate potential loans.

That did not necessarily have to be the case, however. In most circumstances, a borrower

can choose to pay more than the required monthly payment on a loan and enjoy the interest

saving benefits of doing so. The ability to do this is a consequence of an approach to inter-

est on a loan sometimes referred to as the United States Rule. For most financial situations

in the United States, for each time period you pay the interest only on the amount that you

actually owe. If you pay extra to reduce the balance, this reduces the amount of interest

you pay.

Some loans, however, will have what are known as prepayment penalties. When

a loan has such a penalty, paying extra to reduce the term of the loan may mean that

extra fees will be charged to the borrower. Once common, prepayment penalties are now

fairly rare.

Also, some loans, or other financial transactions that closely resemble but may not tech-

nically speaking actually be loans (such as leases or “rent-to-own” plans), do not work

in this way, and either forbid extra payments or apply them in some other way (such as

treating any extra you pay this month as an early payment of the next month’s payment.)

Again, this is not the norm, but it should be noted that that this sort of thing does sometimes

happen. As with anything in the financial world, read the fine print!

cf