# Amortization

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```					Amortization
Definition
• A loan with a fixed rate is said to be
amortized
– when both principal and interest are paid by a
sequence of equal payments made over equal
time intervals.

• The idea is that the loan is paid off, or
"killed”, at the end of the term of the loan.
– The French word Mort means death!
The fundamental question:

• How much should each periodic loan
payment be so that a loan is amortized at
the end of its the term?

• Let’s consider our first example.
Example #1
Monica purchased a condo for \$260,000.
The condo was financed by making a 10%
down payment and signing a 30-year
mortgage at 6.5% compounded monthly.

a) What will be her monthly payments?

b) What will be the total interest paid?
• Monica will pay \$1,479.04 every month.
• The total interest paid is equal to
–   Total payment made minus amount borrowed

–   360(1479.04)-234000

–   Hence, the total interest paid is \$298,454.4
Example #2
Monica purchased a condo for \$260,000. The
condo was financed by making a 10% down
payment but signing a 15-year mortgage
instead of a 30-year mortgage at 6.5%
compounded monthly.

a) What will be her monthly payments?

b) What will be the total interest paid?
• Monica will pay \$2,038.39 every month.
• But the total interest paid is equal to
–   Total payment made minus amount borrowed

–   180(2038.39)-234000

–   Hence, the total interest paid is \$132,910.20
• So on a 15-year mortgage, the monthly
payment is \$2,038.39, which is about
37.8% (\$559) higher than the monthly
payment on the 30-year mortgage.
• But the interest paid on the 15-year
mortgage (\$ 132,910.20) is significantly
lower than that on the 30-year mortgage
(\$298,454.4), not even half of it.
• You may argue that people should only
take short term loans
• but there is a price to pay:
– higher installments

• If you can't afford high periodic payments,
then you have no choice but pay a lot of
interest.
Example #3
•   John bought a townhouse on September 31, 1995 by
taking out a 30-year, \$112,475 mortgage at 9%.
a) Calculate the unpaid balance of the loan on September
31, 2005, just after making the 120th payment.
b) How much equity did John have on his home if it was
appraised at \$ 300,000 on September 31,2005?
c) How much interest was paid during the first 10 years of
the loan?
d) How much interest will be paid during the last 10 years
of the loan?
John will pay \$905 every month.
a) After 120 payments have been made, the unpaid
balance of the loan is the amount that the
remaining 240 payments of \$905 will amortize.

•   This is the “present value” of the loan that will
be paid over the remaining 20 years.

•   So we need to find that PV.
•   John will still owe \$100,586.18 on the
townhouse.
b) The equity on a home is the difference
between the appraised value of the house
and the unpaid balance on the home
–   Since John’s town home was appraised at
\$300,000 in 2005, his equity was
•   300000-100586.18

–   Thus John’s equity was \$199,413.82 in 2005
c) Over the first 10 years of the loan, the
principal decreased from \$112,475 to
\$100,586.18
–   But 112,475-100,586.18= \$11,888.92
–   That is, only \$11,888.92 of the payments was
applied toward the principal.

•   Thus the interest paid over the first 10
years is
–   120(905)- 11,888.92 =\$96,711.18.
d) We find the amount of the loan that the last 120
payments will amortize.

•   This is the “present value” of the loan that will
be paid over the remaining 10 years.
•   So we need to find that PV.
•   The balance that John will be paying over the
last 10 years is \$71,442.23.
•   The interest paid over the last 10 years of the
mortgage is 120(905)-71442.23 = \$37,157.77
•    This example clearly illustrates that
1. Initially the larger portion of the payments
goes toward payment of interest charges.
2. But, as time passes, the larger portion of the
payments goes toward payment of the
principal

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