PMT = [tex]$\frac{p(\frac{r}{n})}{[\square]}[/tex] to determine the regular payment amount, rounded to the nearest cent. The cost of a home is financed with a [tex]$140,000[/tex] [tex]$\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right][/tex] year fixed-rate mortgage at 2.5%. Find the monthly payments and the total interest for the loan. Prepare a loan amortization schedule for the first three months of the mortgage. The monthly payment is $[tex]$\square[/tex]. (Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (2)
Calculate the monthly interest rate and the total number of payments: i = 12 0.025 , N = 12 × 30 = 360 .
Use the PMT formula to find the monthly payment: PMT = 1 − ( 1 + i ) − N P × i = 553.17 .
Calculate the total interest paid: T o t a l I n t eres t = ( PMT × N ) − P = 59141.20 .
Prepare the amortization schedule for the first three months, showing the breakdown of principal and interest payments.
The monthly payment is $553.17 .
Explanation
Problem Analysis We are given a problem about calculating the monthly payments, total interest, and creating an amortization schedule for a home loan. Here's how we'll approach it:
Calculate the monthly payment: We'll use the PMT formula to find the regular payment amount.
Calculate the total interest: We'll find the total amount paid over the loan term and subtract the principal to find the total interest.
Create an amortization schedule: We'll show the breakdown of principal and interest for the first three months of the loan.
Identify Given Values First, let's identify the given values:
Principal (P): $140 , 000
Annual interest rate (r): 2.5% or 0.025
Loan term (t): 30 years
Number of times interest is compounded per year (n): 12 (monthly)
Calculate Monthly Interest Rate and Total Payments Now, we calculate the monthly interest rate (i) and the total number of payments (N):
Monthly interest rate: i = n r = 12 0.025 = 0.00208333...
Total number of payments: N = n × t = 12 × 30 = 360
Calculate Monthly Payment Next, we use the PMT formula to calculate the monthly payment:
PMT = 1 − ( 1 + i ) − N P × i
PMT = 1 − ( 1 + 0.00208333 ) − 360 140000 × 0.00208333
PMT = 1 − ( 1.00208333 ) − 360 291.666666
PMT = 1 − 0.414162 291.666666
PMT = 0.585838 291.666666
PMT = 553.169258
Rounding to the nearest cent, the monthly payment is $553.17 .
Calculate Total Interest Now, we calculate the total amount paid over the loan term:
T o t a lP ai d = PMT × N = 553.17 × 360 = 199141.20
And the total interest paid:
T o t a l I n t eres t = T o t a lP ai d − P = 199141.20 − 140000 = 59141.20
So, the total interest paid over the 30-year loan term is $59 , 141.20 .
Create Amortization Schedule Now, let's prepare the amortization schedule for the first three months:
Month 1:
Beginning Balance: $140 , 000.00
Interest Paid: $140 , 000.00 × 0.00208333 = $291.67
Principal Paid: $553.17 − $291.67 = $261.50
Ending Balance: $140 , 000.00 − $261.50 = $139 , 738.50
Month 2:
Beginning Balance: $139 , 738.50
Interest Paid: $139 , 738.50 × 0.00208333 = $291.12
Principal Paid: $553.17 − $291.12 = $262.05
Ending Balance: $139 , 738.50 − $262.05 = $139 , 476.45
Month 3:
Beginning Balance: $139 , 476.45
Interest Paid: $139 , 476.45 × 0.00208333 = $290.58
Principal Paid: $553.17 − $290.58 = $262.59
Ending Balance: $139 , 476.45 − $262.59 = $139 , 213.86
Final Answer In summary:
The monthly payment is $553.17 .
The total interest paid is $59 , 141.20 .
The amortization schedule for the first three months is shown above.
Examples
Understanding loan amortization is crucial in personal finance. For instance, when buying a car or a house, knowing the monthly payment and how much of it goes towards interest versus principal helps in budgeting and financial planning. The amortization schedule provides a clear breakdown, allowing you to see how the loan balance decreases over time. This knowledge can also assist in making informed decisions about early loan repayment or refinancing.
The monthly payment for the mortgage is $553.17, and the total interest paid over the loan term is $59,141.20. An amortization schedule for the first three months shows how payments are distributed between interest and principal reduction.
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