Question: Use \[ P \left( \frac{1}{n} \right) \] to determine the regular payment amount, rounded to the…

Use \[ P \left( \frac{1}{n} \right) \] to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $155,000 mortgage. Which mortgage loan has the larger total cost (closing costs + the amount paid for points + total cost of interest)? By how much? Mortgage A: 30-year fixed at 7.25% with closing costs of $2200 and 1 point. Mortgage B: 30-year fixed at 6.75% with closing costs of $2200 and 2 points. Choose the correct answer below, and fill in the answer box to complete your choice. (Do not round until the final answer. Then round to the nearest dollar as needed.) A. Mortgage B has a larger total cost than mortgage A by $ B. Mortgage A has a larger total cost than mortgage B by $

Solution

To solve this problem, we will compute the total cost of each mortgage, considering both the closing costs and the total cost of interest over the life of the loan. We’ll use the formula given for the monthly payment (usually the formula for an annuity): \[ P = \frac{rPV}{1 - (1 + r)^{-n}} \] where: - \( P \) is the monthly payment, - \( r \) is the monthly interest rate, - \( PV \) is the present value (loan amount), - \( n \) is the total number of payments. Mortgage A: - Loan amount: $155,000 - Interest rate: 7.25% annually or \(\frac{7.25}{12}\%\) monthly - Closing costs: $2200 - Points: 1 point = 1% of $155,000 = $1550 First, calculate the monthly interest rate: \[ r = \frac{7.25}{100 \times 12} = \frac{0.0725}{12} \] Calculate \( n \): \[ n = 30 \times 12 = 360 \] Calculate the monthly payment \( P \): \[ P = \frac{r \times 155,000}{1 - (1 + r)^{-360}} \] Calculate the total payment over the life of the loan: \[ P \times 360 \] Calculate the total cost for Mortgage A: \[ \text{Total cost of Mortgage A} = (\text{Total payment} + \text{closing costs} + \text{points}) \] Mortgage B: - Loan amount: $155,000 - Interest rate: 6.75% annually or \(\frac{6.75}{12}\%\) monthly - Closing costs: $2200 - Points: 2 points = 2% of $155,000 = $3100 First, calculate the monthly interest rate: \[ r = \frac{6.75}{100 \times 12} = \frac{0.0675}{12} \] Calculate \( n \): \[ n = 30 \times 12 = 360 \] Calculate the monthly payment \( P \): \[ P = \frac{r \times 155,000}{1 - (1 + r)^{-360}} \] Calculate the total payment over the life of the loan: \[ P \times 360 \] Calculate the total cost for Mortgage B: \[ \text{Total cost of Mortgage B} = (\text{Total payment} + \text{closing costs} + \text{points}) \] Compare the total costs and determine the differences: Calculate the difference in total cost between the two mortgages, and choose the option with the correct statement from the given options \( A \) or \( B \).