Question: In terms of paying less in interest, which is more economical for a $100,000 mortgage: a 30-year…

In terms of paying less in interest, which is more economical for a $100,000 mortgage: a 30-year fixed rate at 9% or a 15-year fixed rate at 8.5%? How much is saved in interest? Use the following formula to determine the regular payment amount.

PMT = [\frac{P(\frac{r}{n})}{1 - (1 + \frac{r}{n})^{-nt}}]

Select the correct choice below and fill in the answer box within your choice. (Do not round until the final answer. Then round to the nearest thousand dollars.)

A. The 15-year 8.5% loan is more economical. The buyer will save approximately $\_ in interest. B. The 30-year 9% loan is more economical. The buyer will save approximately $\_ in interest.

Solution

To determine which mortgage option is more economical in terms of paying less interest, we’ll calculate the total interest paid for both options. Option 1: - Loan Amount (P): \$100,000 - Annual Interest Rate: 9% - Loan Term: 30 years (360 months) Option 2: - Loan Amount (P): \$100,000 - Annual Interest Rate: 9.5% - Loan Term: 15 years (180 months) Step 1: Calculate the monthly interest rate for both options. \[ i_1 = \frac{9\%}{12} = 0.0075 & \] \[ i_2 = \frac{9.5\%}{12} = 0.0079167 & \] Step 2: Calculate the total number of payments for both options. \[ n_1 = 30 \times 12 = 360 \text{ months} & \] \[ n_2 = 15 \times 12 = 180 \text{ months} & \] Step 3: Apply the PMT formula to find the monthly payment for both options. \[ \text{PMT}_1 = 100{,}000 \times \frac{0.0075(1 + 0.0075)^{360}}{(1 + 0.0075)^{360} - 1} & \] \[ \text{PMT}_2 = 100{,}000 \times \frac{0.0079167(1 + 0.0079167)^{180}}{(1 + 0.0079167)^{180} - 1} & \] Step 4: Calculate the monthly payments. \[ \text{PMT}_1 \approx \$805.23 & \] \[ \text{PMT}_2 \approx \$1{,}090.24 & \] Step 5: Calculate the total payment over the life of the loan for both options. \[ \text{Total Payment}_1 = 805.23 \times 360 \approx \$289{,}882.80 & \] \[ \text{Total Payment}_2 = 1{,}090.24 \times 180 \approx \$196{,}243.20 & \] Step 6: Calculate the total interest paid for both options. \[ \text{Total Interest}_1 = 289{,}882.80 - 100{,}000 = \$189{,}882.80 & \] \[ \text{Total Interest}_2 = 196{,}243.20 - 100{,}000 = \$96{,}243.20 & \] Conclusion: Option 2, the 15-year fixed rate at 9.5%, is more economical as it results in paying \$96,243.20 in interest compared to \$189,882.80 for the 30-year fixed rate at 9%. Therefore, choosing the 15-year mortgage saves \$93,639.60 in interest.