Chapter 5 Time Value of Money 1515-18 AMORTIZED LOANS
14
An important application of compound interest involves loans that are paid off in
installments over time. Included are automobile loans, home mortgage loans, stu-
dent loans, and many business loans. A loan that is to be repaid in equal amounts
on a monthly, quarterly, or annual basis is called an amortized loan.^15
Table 5-4 illustrates the amortization process. A homeowner borrows $100,000
on a mortgage loan, and the loan is to be repaid in! ve equal payments at the end
of each of the next 5 years.^16 The lender charges 6% on the balance at the beginning
of each year.
Our! rst task is to determine the payment the homeowner must make each
year. Here’s a picture of the situation:
(^0) I = 6% 3 4
$100,000 PMT PMT PMT PMT PMT
1 2 5
The payments must be such that the sum of their PVs equals $100,000:
$100,000! __(1.06)PMT 1 " __(1.06)PMT 2 " __(1.06)PMT 3 " __(1.06)PMT 4 " __(1.06)PMT 5! ∑
t! 1
5__(1.06)PMT (^) t
We could insert values into a calculator as follows to get the required payments,
$23,739.64:^17
N I/YR PV PMT FV
5 6 100000 0
–23,739.64
Amortized Loan
A loan that is repaid in
equal payments over its
life.
Amortized Loan
A loan that is repaid in
equal payments over its
life.
(^14) Amortized loans are important, but this section can be omitted without loss of continuity.
(^15) The word amortized comes from the Latin mors, meaning “death”; so an amortized loan is one that is “killed o# ”
over time.
(^16) Most mortgage loans call for monthly payments over 10 to 30 years, but we use a shorter period to reduce the
calculations.
(^17) You could also factor out the PMT term;! nd the value of the remaining summation term (4.212364); and divide
it into the $100,000 to! nd the payment, $23,739.64.
Amount borrowed: $100,000
Years: 5
Rate: 6%
PMT: $$23,739.64
Year
Beginning
Amount
(1)
Payment
(2)
Interesta
(3)
Repayment of
Principalb
(4)
Ending
Balance
(5)
1 $100,000.00 $23,739.64 $6,000.00 $17,739.64 $82,260.36
2 82,260.36 23,739.64 4,935.62 18,804.02 63,456.34
3 63,456.34 23,739.64 3,807.38 19,932.26 43,524.08
4 43,524.08 23,739.64 2,611.44 21,128.20 22,395.89
5 22,395.89 23,739.64 1,343.75 22,395.89 0.00
a Interest in each period is calculated by multiplying the loan balance at the beginning of the year by the
interest rate. Therefore, interest in Year 1 is $100,000.00(0.06)! $6,000; in Year 2, it is $4,935.62; and so forth.
bRepayment of principal is equal to the payment of $23,739.64 minus the interest charge for the year.
Tabl e 5 - 4 Loan Amortization Schedule, $100,000 at 6% for 5 Years