tudied in Section B the so-called add-on interest loan, in which the interest charged each payment period is based on the original amount borrowed. These loans are more commonly entered into for moderately priced consumer items. For much larger purchases - such as in buying homes or new cars, for example - the simple interest amortized loan is the
We use the following notation in our calculations with simple interest amortized loan
p = payment amount r = annual interest rate
n = # of payments per year R = r/n = periodic interest rate
t = # of years P = original principal
N = n · t = total # of payments
The formula for the amount of each payment on the loan is
equation
In the vast majority of home mortgages, payments are made on a monthly basis. For such loans the number of payments per year is n = 12, while the periodic interest rate is the annual interest rate divided by 12, or R = r/12. The formula for the monthly payment then becomes
equation
example 1
Lucky and Lucille
Lucky and Lucille borrowed $20,000 to buy a car, getting a simple interest amortized loan with a 12% annual interest rate, and with payments to extend over 3 years. We calculate their monthly payment, the sum of all their payments, and the total interest they will pay.
The principal is P = $20,000, the periodic interest rate is R = 12%/12 = 1% = .01, and the number of payments is N = 12 · t = 12 · 3 = 36. The monthly payment formula gives
equation
As they will make 36 payments of this amount, the total sum of money they will pay to the lender over the life of the loan is
36 · $664.29 = $23,914.44 .
The total interest they will pay is the difference in what they pay back and the principal they borrowed,
$23,914.44 − $20,000.00 = $3,914.44 .
Now we prepare a so-called amortization schedule for the first three payments of the loan in the previous example. The periodic, or monthly, interest rate on the loan is R = 12%/12 = 1% = .01; thus Lucky and Lucille must each month pay 1% interest on the outstanding balance on the loan. Using the simple interest formula, we calculate the interest for the first month as
I = P · R · t = $20,000 · .01 · 1 = $200 .
This amount comes out of their first payment, and the remainder,
$664.29 − $200.00 = $464.29 ,
is applied to pay off the principal. Thus, after the first month the outstanding principal, or balance, is
$20,000.00 − $464.29 = $19,535.71 .
For the second month, the interest on the outstanding principal is
I = P · R · t = $19,535.71 · .01 · 1 = $195.36 .
This amount comes out of the second payment, and the remainder,
$664.29 − $195.36 = $468.93 ,
is applied to the principal; the new balance on the principal is the difference
$19,535.71 − $468.93 = $19,066.78 .
Finally, for the third month the interest is
I = P · R · t = $19,066.78 · .01 · 1 = $190.67 ,
the amount applied to the principal is
$664.29 − $190.67 = $473.62 ,
and the outstanding balance after the third month is
$19,066.78 − $473.62 = $18,593.16 .
We summarize the above calculations in an amortization table:
Payment
Number
Total
Payment
Amount to
Interest
Amount to
Principal
Balance
$20,000.00
1$664.29$200.00$464.29$19,535.71
2$664.29$195.36$468.93$19,066.78
3$664.29$190.67$473.62$18,593.16
A complete amortization table would continue through all 36 payments, until after the last payment the balance reduces to zero.
example 2
Enchanted Lake
The Nakashima family will borrow $240,000 to buy a house in Kailua on Enchanted Lake. From American Savings they can get a loan at a 7% annual interest rate, with monthly payments stretched over 30 years. We will calculate the
monthly payment,
total amount of all payments,
total interest to be paid on
Having found the home of their dreams, the Howe family is looking into financing. They must borrow $200,000 to purchase their new house. Calculate their monthly payment under the following plans:
15 year loan, 6% annual interest rate,
20 year loan, 6.3% annual interest rate,
25 year loan, 6.6% annual interest rate,
30 year loan, 7.2% annual interest rate.
Also, under each plan calculate the sum of all their payments, as well as the total interest they will pay.
Mr. Brewster with salesman
Mr. Brewster is pleased to have just purchased a brand new car, financing it with a $30,000 loan at a 12% annual interest rate, with payments to be made over 5 years. Calculate
his monthly payment,
the total of all his payments,
the total interest he will pay.
Also, construct an amortization table for the first
Step-by-step explanation:
hope it helps
