The price of a home is \( \$ 230,000 \). The bank requires a \( 15 \% \) down payment. The buyer is offered two mortgage options: 15 -year fixed at \( 6.5 \% \) or 30 -year fixed at \( 6.5 \% \). Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 15 -year option? Use the following formula to determine the regular payment amount. PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) Find the monthly payment for the 15-year option. \( \$ \square \) (Round to the nearest dollar as needed.)
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To start off, let's calculate the down payment: Down payment = \( \$ 230,000 \times 0.15 = \$ 34,500 \) Now, the loan amount after the down payment will be: Loan amount = \( \$ 230,000 - \$ 34,500 = \$ 195,500 \) Next, for the 15-year mortgage with a fixed rate of \( 6.5\% \): 1. Convert the annual interest rate to a monthly rate and the number of payments: - Monthly interest rate, \( r = \frac{0.065}{12} = 0.00541667 \) - Total number of payments for 15 years, \( n \times t = 12 \times 15 = 180 \) 2. Substitute these values into the formula \( PMT = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} \): \[ PMT = \frac{195500 \left(0.00541667\right)}{1 - (1 + 0.00541667)^{-180}} \] - Using a calculator, you’ll find that: \[ PMT \approx \frac{1063.75}{0.611} \approx 1740 \] So, the monthly payment for the 15-year option is approximately **$1,740**. To find the total interest paid: For the 15-year mortgage: Total payment over 15 years = \( 1,740 \times 180 = 313,200 \) Total interest paid = Total payment - Loan amount = \( 313,200 - 195,500 = 117,700 \) For the 30-year mortgage with the same rate: 1. Total payments for 30 years: \( 12 \times 30 = 360 \) 2. Monthly payment calculation (substituting \( n \times t = 360 \)) gives: \[ PMT \approx \frac{195500(0.00541667)}{1 - (1 + 0.00541667)^{-360}} \approx 1230 \] Total payment over 30 years = \( 1,230 \times 360 = 442,800 \) Total interest paid = Total payment - Loan amount = \( 442,800 - 195,500 = 247,300 \) Now, the buyer saves in interest by choosing the 15-year option: Interest savings = \( 247,300 - 117,700 = 129,600 \) To wrap it up: - Monthly Payment for 15-Year Option: **$1,740** - Total Interest Savings: **$129,600**
