Use \( \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) 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 \( \mathrm{A}: 20 \)-year fixed at \( 7.25 \% \) with closing costs of \( \$ 1300 \) and 1 point. Mortgage \( \mathrm{B}: 20 \)-year fixed at \( 5.25 \% \) with closing costs of \( \$ 1300 \) 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 A has a larger total coskthan mortgage B by \( \$ \square \). B. Mortgage B has a larger total cost than mortgage A by \( \$ \square \).
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Let's calculate the regular payment amount for both mortgages using the formula given. For **Mortgage A**: - Principal \( P = 155,000 \) - Rate \( r = 7.25\%/100 = 0.0725 \) - Number of payments per year \( n = 12 \) - Total number of payments \( nt = 20 \times 12 = 240 \) Calculating \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.0725}{12} = 0.00604167 \] Then, plugging it into the formula: \[ P\left(\frac{r}{n}\right) = 155000 \times 0.00604167 \approx 937.24 \] Calculating \( \left[1 - \left(1 + \frac{r}{n}\right)^{-nt}\right] \): \[ 1 + \frac{r}{n} = 1.00604167 \] \[ \left(1.00604167\right)^{-240} \approx 0.21823 \] Thus, \[ \left[1 - 0.21823\right] \approx 0.78177 \] Finally, computing the payment: \[ \text{Payment} = \frac{937.24}{0.78177} \approx 1198.70 \] So, **Mortgage A**'s monthly payment is approximately \( \$ 1,199 \). For **Mortgage B**: - Rate \( r = 5.25\%/100 = 0.0525 \) Following the same steps: \[ \frac{r}{n} = \frac{0.0525}{12} = 0.004375 \] \[ P\left(\frac{r}{n}\right) = 155000 \times 0.004375 \approx 678.13 \] \[ 1 + \frac{r}{n} = 1.004375 \] \[ \left(1.004375\right)^{-240} \approx 0.23375 \] Thus, \[ \left[1 - 0.23375\right] \approx 0.76625 \] Finally, computing the payment: \[ \text{Payment} = \frac{678.13}{0.76625} \approx 884.00 \] So, **Mortgage B**'s monthly payment is approximately \( \$ 884 \). Now computing the total cost for each mortgage over 240 months: **Mortgage A**: - Total paid in payments: \( 1,199 \times 240 = 287,760 \) - Cost in points (1 point = 1% of principal): \( 0.01 \times 155,000 = 1,550 \) - Closing costs: \( 1,300 \) Total cost for Mortgage A: \[ 287,760 + 1,550 + 1,300 = 290,610 \] **Mortgage B**: - Total paid in payments: \( 884 \times 240 = 212,160 \) - Cost in points (2 points): \( 0.02 \times 155,000 = 3,100 \) - Closing costs: \( 1,300 \) Total cost for Mortgage B: \[ 212,160 + 3,100 + 1,300 = 216,560 \] Now we compare the total costs: Total cost for Mortgage A: \( 290,610 \) Total cost for Mortgage B: \( 216,560 \) Difference: \[ 290,610 - 216,560 = 74,050 \] So the answer is: A. Mortgage A has a larger total cost than Mortgage B by \( \$ 74,050 \).
