Amortization
MATH 102
Contemporary Math
S. Rook
Overview
• Section 9.5 in the textbook:
– Amortized loans
– Amortization schedules
– Finding the unpaid balance on a loan
Amortized Loans
Amortized Loans
• Amortized loan: a special type of loan that is paid
off by making a series of regular & equal payments
– Part of each payment goes towards paying off the
simple interest from the unpaid balance while the
rest goes towards paying off the principal of the loan
– This differs from installment loans where the interest
over the lifetime of the loan is computed at purchase
• Interest for an amortized loan is computed on the
unpaid balance
Size of Payment for an Amortized
Loan
• To find the regular payment per month for an
 r 
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amortized loan:
 1    1 
nt
nt
– Use the formula:

 r
P 1    R  
 n



n
r
n

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• DO NOT be
intimidated by this formula!!!
– The left side is the compound interest formula
(Section 9.2) while the right side is the formula to
compute an annuity
Size of Payment for an Amortized
Loan (Continued)
– Theory for the formula can be found on page 431
of the textbook:
• Essentially, each payment can be thought of going
into a sink fund
• The amortized loan is paid off when the value of
the sink fund (right side) equals or exceeds that of
the compounded original principal computed by
the lender (left side)
• Again, DO NOT be intimidated by the formula
– Calculate in steps
• For some, the calculation may be further simplified
by using a TI-xx calculator
Amortized Loans (Example)
Ex 1: Find the monthly payment required for
each amortized loan:
a) Amount, $5,000; rate, 10%; time, 4 years
b) Amount, $8,000; rate, 7.5%; time, 6 years
Amortized Loans (Example)
Ex 2: Wilfredo bought a new boat for $13,500.
He paid a $2,000 down payment and financed
the rest for 4 years at an interest rate of 7.2%.
a) Find his monthly payment
b) Calculate the total amount of interest he
will pay off over the lifetime of the loan
Amortization Schedules
Amortization Schedules
• An amortization schedule (table) is a breakdown of
how payments are used to pay interest and principal
– Each row of the table represents a payment
• Calculate the payment per month for the amortized
loan (see previous slides)
– e.g. Find the payments for a $1200 loan at 9.6% for 5
years
• For each payment:
– Calculate the interest owed for the month by using
the simple interest formula
• e.g. Calculate the interest owed for the 1st payment
Amortization Schedules
(Continued)
– Subtract the interest from the payment
• The rest is applied to the unpaid balance
• e.g. How much goes towards the principal?
– Subtract the remaining payment from the unpaid
balance
• Represents the new unpaid balance after the payment
is applied
• e.g. What is the new unpaid balance?
• Amortization tables show that later payments
mostly go towards the principal while a good
amount goes towards interest for early payments
Amortization Schedules (Example)
Ex 3: Complete an amortization schedule for
the first three payments of the given loan:
a) Amount, $12,500; rate, 8.25%; time, 4
years
b) Amount, $1900; rate, 8%; time, 18 months
Amortization Schedules (Example)
Ex 4: Assume that you have taken out a 30-year
mortgage for $100,000 at an annual rate of
7%.
a) Construct an amortization table for the first
three payments
b) Repeat part a) if you decide to pay an extra
$100 per month to pay off the mortgage more
quickly
Finding the Unpaid Balance of a
Loan
Finding the Unpaid Balance of a
Loan
• Suppose we wish to terminate a loan
prematurely before the last payment
– Obviously, we still owe some amount because the
loan will not be repaid until we reach the last
payment
• i.e. The lender’s compounded principal will exceed
how much is in the sink fund
– Unpaid balance:
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
 1    1 
nt
• For theory, see
 n
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U

P
1

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R
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page 435 in the
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r
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textbook
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n
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Finding the Unpaid Balance of a
Loan (Continued)
• Either we wish to pay the remaining balance
of the loan or we wish to refinance
• To refinance means to take out a second loan
at a lower interest rate to pay the unpaid
balance of the first loan
– Sometimes there is a refinancing free expressed as
a percentage
• Anytime we wish to terminate a loan, we must
know its unpaid balance
Finding the Unpaid Balance of a
Loan (Example)
Ex 5: You have taken an amortized loan at 8.5%
for 5 years to pay off your new car which cost
$12,000. How much would you pay if after 3
years, you decided to pay off the loan?
Finding the Unpaid Balance of a
Loan (Example)
Ex 6: Suppose that you have taken a 20 year mortgage
on a home for $100,000 at an annual interest rate of
8%. After 5 years, you decide to refinance the
unpaid balance at an annual interest rate of 6%.
a) What is your payment under the original
mortgage?
b) What is your unpaid balance when you decide to
refinance?
c) What is your payment per month after you
refinance?
Summary
• After studying these slides, you should know
how to do the following:
– Calculate the regular payment for an amortized loan
– Construct an amortization schedule for any number of
payments
– Find the unpaid balance of a loan before it is fully repaid
• Additional Practice:
– See problems in Section 9.5
• Next Lesson:
– Lines, Angles, & Circles (Section 10.1)