Chapter 5

Section 5.4

Amortized Loans

Amortized Loans

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

nn

iPi

ipymt

1

11

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.

Example

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 12

072.i

n = 12·4 = 48

38.478435003.55

939405.26518

939405.26518435003.55

1900,1911

111

48

12072.

12072.

48

12072.

pymt

pymt

pymt

iPi

ipymt n

n

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

Private Mortgage Insurance

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

ExampleA 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

12

06.in = 12·30 = 360

37.839515042.1004

529717.843160

529717.843160515042.1004

1000,14011

111

360

1206.

1206.

360

1206.

pymt

pymt

pymt

iPi

ipymt n

n

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

Example

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

12

048.in = 12·30 = 360

68.640,108208590.4

064385.457224

208590.4064385.457224

111

570

111

360

12048.

12048.

360

12048.

P

P

P

iPi

ipymt n

n

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 = 10,000 017.4

068.i n = 4

15.2607103161.4

537355.10697

537355.10697103161.4

1000,1011 4

4068.

4068.

4

4068.

pymt

pymt

pymt

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

Payment

Number

Principal

Portion

Interest

Portion

Total

Payment

Balance

0 - - - 10,000.00

1 2607.15

2 2607.15

3 2607.15

4 2607.15

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

2437.15 170.00

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

10,000.00-2437.15 = 7562.85

2478.58

2520.72

2563.57

128.57

86.43

43.58

7562.85-2478.58 = 5084.27

5084.27-2520.72 = 2563.55

2563.55-2563.57 = -.02

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