Copyright © 2008, The McGraw-Hill Companies, Inc.
We will use the term amortized loan to refer to any loan whose payments are calculated
by using the annuity factors and whose progress toward payoff therefore follows what
is illustrated in an amortization table. We will move forward assuming that all loans are
amortized unless stated otherwise.The Remaining Balance of a Loan
One very depressing fact about longer term loans is that, early on, very little progress is made
toward paying off the debt. Most of the early payments go toward interest, with little left
over to reduce the balance. In Example 4.5.1, after 6 months of payments, Pat and Tracy had
reduced their debt from $158,000 down to $157,241.77. If that slow progress is not depress-
ing enough in itself, we can note that they have paid a total of 6($1,072.49) $6,434.94 to
reduce their debt by a whopping $158,000 $157,241,77 $758.23. After 12 months, the
progress was not much more impressive.
At some point, though, this has to change. Little by little, the principal is dropping,
and with each passing month more and more of the payment goes toward killing it off.
In the next 6 months, we can see from the amortization table that Pat and Tracy pay off
$157,241.77 $156,455.83 $785.94. This is more than $758.23, and in the following
6 months it stands to reason they will pay off still more, and so on and so on. How far along
will they be after, say, 5 years? Or 10 years? Or 20 years?
We could answer these questions by carrying our amortization table farther. This would
certainly work, but it wouldn’t be much fun. Even if we programmed a computer to do the
calculation, an amortization that carried out monthly payments for 5, 10, or 20 years would
be quite an undertaking.
We can determine the remaining balance of a loan at any point by giving a sly answer
to the question “What do they still owe?” The answer we want to this question is of course
the lump sum dollar amount that is owed, but another way of answering the question would
be to observe that they owe the remaining payments on the loan! This seems like a smart-
alecky and not very helpful answer until we realize that whatever the remaining debt, it
must be equivalent to the remaining payments. To illustrate:Remaining paymentsPV of remaining
payments =The remaining payments on the loan form an annuity, and the remaining balance is a single
sum at the start of this annuity that is equivalent to it. In other words: the amount owed is
the present value of the remaining payments. Even though this is not a formula per se, it
seems significant enough to deserve special notice:“FORMULA” 4.4.5
The Remaining Balance of a LoanAt any point in the term of an amortized loan, the amount owed is equal to the
present value of the remaining payments.We can test this out by using it to find the amount that Pat and Tracy would owe after
6 months of their 30-year mortgage. We know from the amortization table that this amount
is $157,241.77. Using this new approach, we would observe that the original loan called
for 360 payments, and after the sixth payment there would be 360 6 354 left. And so:PV PMT a _n (^) | (^) i
PV $1,072.49a ___ 354 | (^) .006
PV $1,072.49(146.614395231)
PV $157,241.77
The fact that the answers come out the same both ways should gives us extra confidence
in this approach.
4.5 Amortization Tables 185