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Chapter 5. Section 5.4 Amortized Loans. Amortized Loans

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An amortized loan is a type of investment (for the loaner) in which the amount of the loan, plus the interest is paid off in a series of fixed regular payments. A simple interest amortized loan is an ordinary annuity whose future value is equal to the loans future value and the payments coincide with the compounding periods in the investment. This enables us to get a formula right away for a loan.

Simple Interest Amortized Loan

Value of Payments = Value of Lump Sum

Future Value of Annuity = Future Value of Loan Amount

where pymt represents the amount of the payments, P is the amount of the loan, n is the number of payments and i is the periodic rate that corresponds to the compounding periods.

Most often loans are paid on monthly so they will have monthly compounding periods. Except for certain special types of loans they are considered monthly.

A person wants to buy a new car that costs $22,400.00. They have been able to save $2,500.00 to put down. They decide to finance the rest at 7.2% making monthly payments for 4 years.

a) How much are the monthly payments?

For this loan we have the following:

P = 22,400-2,500 = 19,900

n = 12·4 = 48

b) How much will they pay in interest over the life of the loan?

They will make 48 payments of $478.38 each. This gives 48·478.38 = 22962.20

The interest is the difference between what they pay and what the amount of the loan was for.

22,962.20 – 19,900 = 3,062.20

Most housing lenders require at least a 20% down payment on a house to avoid a charge called Private Mortgage Insurance (or PMI).

Example

A couple wants to buy a house that sells for $175,000. They agree to put 20% down to avoid the PMI and finance the rest at 6% for 30 years.

a) How much will their monthly payments be?

P=175,000-(.2)·175,000=175,000-35,000=140,000

n = 12·30 = 360

b) How much will they pay in interest over the life of the loan?

They will make 360 payments of $839.37 each.

This gives 360·839.37 = 302,173.20

The interest is the difference between what they pay and what the amount of the loan was for.

302,173.20 - 140,000 =162,173.20

Most banks will only make a home loan if the amount of the payment does not exceed 38% of the borrower's monthly income. If a couple brings home a joint income of $1500 what is the maximum 30 year home loan they can qualify for at an interest rate of 4.8%?

The maximum payment they could make is 38% of $1500.

pymt = .38·1500 = 570.00

n = 12·30 = 360

Amortization Schedule for a Loan

When paying off an amortized loan part of each payment goes to paying off the loan and part is the interest on the loan. An amortization schedule for a loan is a table that shows how much of each payment goes to interest, how much goes to paying off the principal and what the current balance is on the loan.

The amount of interest paid each period is the periodic rate (i) times the outstanding balance. The rest of the payment goes to pay down the principal.

A company borrows $10,000 dollars and agrees to pay it back in 4 payments at 6.8% compounded quarterly.

a) Find the amount of each payment.

P = 10,000

n = 4

b) Fill in the amortization schedule for this loan below. (note i = .017)

2437.15

170.00

10,000.00-2437.15 = 7562.85

128.57

7562.85-2478.58 = 5084.27

2478.58

2520.72

86.43

5084.27-2520.72 = 2563.55

2563.55-2563.57 = -.02

2563.57

43.58

1. Payment 1 interest :10,000·(.017) = 170, so principal is 2607.15-170 = 2437.15

2. Payment 2 interest : 7562.85·(.017) = 128.57, so principal is 2607.15-128.57 = 2478.58

3. Payment 3 interest : 5084.27·(.017) = 86.43, so principal is 2607.15-86.43 = 2520.72

4. Payment 4 interest : 2563.55·(.017) = 43.58, so principal is 2607.15-43.58 = 2478.58

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