A mortgage loan of $100,000 at 6% compounded monthly is amortized by equal monthly payments over 25 years. What is the total amount of interest that would be paid during the first year?

Question

Quizplus · Accepted Answer

To calculate the total amount of interest paid during the first year, we first need to determine the monthly payment amount and then calculate the interest portion of those payments. The monthly payment can be calculated using the formula for a fixed-rate mortgage, which is:$$ M = P \frac{r(1+r)^n}{(1+r)^n - 1} $$where:- $M$ is the total monthly mortgage payment,- $P$ is the principal loan amount ($100,000),- $r$ is the monthly interest rate (annual rate divided by 12, so 0.06/12 = 0.005),- $n$ is the number of payments (25 years * 12 months = 300 payments).Plugging in the numbers:$$ M = 100,000 \frac{0.005(1+0.005)^{300}}{(1+0.005)^{300} - 1} $$$$ M \approx 644.31 $$The total monthly payment is approximately $644.31. The interest paid in the first month is the entire loan amount times the monthly interest rate:$$ \text{First month interest} = 100,000 \times 0.005 = 500 $$Since the payments are fixed, the portion of each payment that goes towards interest decreases over time, while the portion that goes towards the principal increases. However, for the first year, we can approximate the total interest paid by recognizing that it will be slightly less each month than the first month's interest.To calculate the total interest paid in the first year accurately, we need to subtract the principal portion from each monthly payment and sum the interest portions. However, given the options provided and understanding the nature of amortization, we can estimate and check which option is closest to our rough calculation.The first month's interest is $500, and it will slightly decrease each month. Multiplying $500 by 12 gives $6,000 as a rough estimate for the first year, but since the interest portion decreases each month, the actual total will be less.Option C, $5,952, is the closest to our rough estimate and accounts for the decrease in interest portion over the 12 months.

A Mortgage Loan of $100,000 at 6% Compounded Monthly Is