Answer:
(a) The size of the monthly payment for the mortgage is approximately $717.47.
(b) The balance at the end of the seven-year term is approximately $68,607.47.
(c) The size of the monthly payment for the renewal term at 7% interest is approximately $825.84.
Step-by-step explanation:
Calculating the Monthly Payment
To calculate the size of the monthly payment, we can use the formula for the monthly payment of an amortized loan:
Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Payments))
In this case, the loan amount is $93,000, the interest rate is 5.1% compounded semi-annually, and the loan term is 15 years (180 months).
First, we need to convert the annual interest rate to a monthly interest rate. Since the interest is compounded semi-annually, the effective interest rate per compounding period is half of the annual interest rate. Therefore, the monthly interest rate is (5.1% / 2) / 100 = 0.0255.
Plugging in the values into the formula:
Monthly Payment = (93000 * 0.0255) / (1 - (1 + 0.0255)^(-180))
Using a calculator, the monthly payment is approximately $717.47.
Determining the Balance at the End of the Seven-Year Term
To determine the balance at the end of the seven-year term, we can use the formula for the remaining balance of an amortized loan:
Balance = Loan Amount * (1 + Monthly Interest Rate)^(-Number of Payments)
In this case, the loan amount is $93,000, the interest rate is 5.1% compounded semi-annually, and the number of payments is 7 years (84 months).
Using the formula:
Balance = 93000 * (1 + 0.0255)^(-84)
Using a calculator, the balance at the end of the seven-year term is approximately $68,607.47.
Calculating the Monthly Payment for the Renewal Term
To calculate the size of the monthly payment for the renewal term, we can use the same formula as before, but with the new interest rate. In this case, the interest rate is 7% compounded semi-annually.
Using the formula:
Monthly Payment = (93000 * (7% / 2) / 100) / (1 - (1 + (7% / 2) / 100)^(-180))
Using a calculator, the size of the monthly payment for the renewal term is approximately $825.84.