Use PMT = [tex]$\frac{P(\frac{r}{n})}{\left[1-(1+\frac{r}{n})^{-nt}\right]}$[/tex] to determine the regular payment amount, rounded to the nearest cent. The cost of a home is financed with a $140,000 30-year fixed-rate mortgage at 2.5%.
a. Find the monthly payments and the total interest for the loan.
b. Prepare a loan amortization schedule for the first three months of the mortgage.
a. The monthly payment is $553.17.
(Do not round until the final answer. Then round to the nearest cent as needed.)
The total interest for the loan is $ [ ]
(Use the answer from part a to find this answer. Round to the nearest cent as needed.)
Answer (1)
Calculate the total amount paid by multiplying the monthly payment by the total number of payments: T o t a l P ai d = 553.17 \t 360 = 199141.20 .
Calculate the total interest paid by subtracting the principal from the total amount paid: T o t a l I n t eres t = 199141.20 − 140000 = 59141.20 .
Calculate the interest and principal paid each month and update the remaining balance for the first three months.
The total interest for the loan is 59141.20 .
Explanation
Problem Analysis We are given the formula for calculating the monthly payment (PMT) of a loan, and we are asked to find the total interest paid over the life of a home loan and to prepare an amortization schedule for the first three months. The principal amount (P) is $140,000, the annual interest rate (r) is 2.5% or 0.025, and the loan term (t) is 30 years with monthly payments (n = 12). The monthly payment is given as $553.17.
Calculating Total Amount Paid First, we need to calculate the total amount paid over the 30-year loan term. This is done by multiplying the monthly payment by the total number of payments, which is the number of years times the number of payments per year.
Total Payment Calculation The total number of payments is n \t = 12 \t 30 = 360 . Therefore, the total amount paid is: T o t a l \t P ai d = PMT \t \t n \t t = 553.17 \t \t 12 \t 30 = 553.17 \t 360 = 199141.20
Calculating Total Interest Next, we calculate the total interest paid over the life of the loan. This is the difference between the total amount paid and the original principal amount.
Total Interest Calculation The total interest paid is: T o t a l \t I n t eres t = T o t a l \t P ai d − P = 199141.20 − 140000 = 59141.20
Amortization Schedule Introduction Now, let's prepare the loan amortization schedule for the first three months. We need to calculate the interest paid, principal paid, and remaining balance for each month.
Month 1 Calculations For month 1:
The initial principal is $140,000.
The monthly interest rate is r / n = 0.025/12 = 0.00208333 .
The interest paid is I n t eres t 1 = P \t ( r / n ) = 140000 \t 0.00208333 = 291.67 (rounded to the nearest cent).
The principal paid is P r in c i p a l 1 = PMT − I n t eres t 1 = 553.17 − 291.67 = 261.50 .
The remaining balance is B a l an c e 1 = P − P r in c i p a l 1 = 140000 − 261.50 = 139738.50 .
Month 2 Calculations For month 2:
The initial principal is $139,738.50.
The interest paid is I n t eres t 2 = B a l an c e 1 \t ( r / n ) = 139738.50 \t 0.00208333 = 291.12 (rounded to the nearest cent).
The principal paid is P r in c i p a l 2 = PMT − I n t eres t 2 = 553.17 − 291.12 = 262.05 .
The remaining balance is B a l an c e 2 = B a l an c e 1 − P r in c i p a l 2 = 139738.50 − 262.05 = 139476.45 .
Month 3 Calculations For month 3:
The initial principal is $139,476.45.
The interest paid is I n t eres t 3 = B a l an c e 2 \t ( r / n ) = 139476.45 \t 0.00208333 = 290.58 (rounded to the nearest cent).
The principal paid is P r in c i p a l 3 = PMT − I n t eres t 3 = 553.17 − 290.58 = 262.59 .
The remaining balance is B a l an c e 3 = B a l an c e 2 − P r in c i p a l 3 = 139476.45 − 262.59 = 139213.86 .
Final Answer The total interest for the loan is $59,141.20.
Examples
Understanding loan amortization is crucial in personal finance. For instance, when buying a car or a house, knowing how much of each payment goes toward interest and principal helps in financial planning. It allows you to see the long-term cost of borrowing and make informed decisions about early repayments or refinancing. This understanding extends to business contexts, where companies use amortization schedules to manage debt and assets, ensuring transparency and sound financial management.
