Question: In terms of paying less in interest, which is more economical for a $140,000 mortgage: a 30-year…

In terms of paying less in interest, which is more economical for a $140,000 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.

PMI = \frac{P\left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-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 loan option is more economical, we need to calculate the total interest paid for both mortgage options using the given formula for the regular payment amount (PMT): \[ \text{PMT} = \frac{P \cdot \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-n \cdot t}} \] Where: - \( P = 140,000 \) (loan principal) - \( r \) is the annual interest rate - \( n = 12 \) (number of payments per year) - \( t \) is the term of the loan in years We’ll calculate the PMT for both options and then find the total payment and total interest for each. Option 1: 30-year loan at 9% interest 1. \( r = 0.09 \) 2. \( t = 30 \) 3. Calculate monthly interest rate: \(\frac{r}{n} = \frac{0.09}{12} = 0.0075\) 4. Substitute into PMT formula: \[ \text{PMT} = \frac{140,000 \times 0.0075}{1 - (1 + 0.0075)^{-360}} \] 5. Calculate PMT: \[ \text{PMT} = \frac{1,050}{1 - (1.0075)^{-360}} \approx 1,126.16 \] 6. Calculate total payment over 30 years: \[ \text{Total payment} = 1,126.16 \times 360 = 405,417.60 \] 7. Calculate total interest: \[ \text{Total interest} = 405,417.60 - 140,000 = 265,417.60 \] Option 2: 15-year loan at 8.5% interest 1. \( r = 0.085 \) 2. \( t = 15 \) 3. Calculate monthly interest rate: \(\frac{r}{n} = \frac{0.085}{12} = 0.0070833\) 4. Substitute into PMT formula: \[ \text{PMT} = \frac{140,000 \times 0.0070833}{1 - (1 + 0.0070833)^{-180}} \] 5. Calculate PMT: \[ \text{PMT} = \frac{991.66}{1 - (1.0070833)^{-180}} \approx 1,378.03 \] 6. Calculate total payment over 15 years: \[ \text{Total payment} = 1,378.03 \times 180 = 248,045.40 \] 7. Calculate total interest: \[ \text{Total interest} = 248,045.40 - 140,000 = 108,045.40 \] Comparison: - Total interest for 30-year loan: $265,417.60 - Total interest for 15-year loan: $108,045.40 The 15-year loan at 8.5% is more economical. The buyer will save approximately: \[ 265,417.60 - 108,045.40 = 157,372 \] Therefore, option A is correct. The buyer will save approximately $157,000 in interest.