Amortization

MATH 102Contemporary 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 amortized loan:– Use the formula:• 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

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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:• For theory, see

page 435 in the textbook

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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)