Question

the table shows the terms of a fixed-rate mortgage.
component | cost
(p) principal | $200,000
(r) monthly interest rate |.4%
(n) total number of monthly payments (30-year term) | 360
(m) monthly payment | $1059.85
which formula should be used to correctly calculate the monthly mortgage payment?
\\( \bigcirc \\ m = p\frac{\leftr(1 - r)^{n}\
ight}{(1 + r)^{n}} \\)
\\( \bigcirc \\ m = p\frac{\leftr(1 + r)^{n}\
ight}{\left(1 + r)^{n} - 1\
ight} \\)
\\( \bigcirc \\ m = p\frac{r}{\left(1 + r)^{n} - 1\
ight} \\)
\\( \bigcirc \\ m = p\frac{\leftr(1 + r)^{n}\
ight}{(n + r)} \\)

Explanation:

The formula for calculating the monthly payment (M) of a fixed - rate mortgage is derived from the concept of an amortizing loan. The general formula for the monthly mortgage payment is $M = P\frac{R(1 + R)^{n}}{(1 + R)^{n}-1}$, where:

• $P$ is the principal amount of the loan.
• $R$ is the monthly interest rate (annual interest rate divided by 12, and in decimal form).
• $n$ is the total number of monthly payments (loan term in years multiplied by 12).

Let's analyze each option:

• Option 1: $M = P\frac{[R(1 - R)^{n}]}{(1 + R)^{n}}$: The numerator has $(1 - R)^{n}$, which is incorrect for mortgage payment calculations as we deal with compounding interest (growth, so $(1 + R)^{n}$), not discounting in this way.
• Option 2: $M = P\frac{[R(1 + R)^{n}]}{[(1 + R)^{n}-1]}$: This matches the standard fixed - rate mortgage monthly payment formula.
• Option 3: $M = P\frac{R}{[(1 + R)^{n}-1]}$: The numerator is missing the $(1 + R)^{n}$ term. This formula would be incorrect as it does not account for the compounding of interest over the loan term.
• Option 4: $M = P\frac{[R(1 + R)^{n}]}{(N + R)}$: The denominator $(N + R)$ is incorrect. The correct denominator for the mortgage payment formula is $(1 + R)^{n}-1$.

Answer:

The formula $M = P\frac{[R(1 + R)^{n}]}{[(1 + R)^{n}-1]}$ (the second option) should be used to calculate the monthly mortgage payment.