Question

What is the formula that i can use to solve the problem below.
Derek borrows $265,351.00 to buy a house. He has a 30-year mortgage with a rate of 4.10%. The monthly mortgage payment is $________.

Answer 1

To calculate the monthly mortgage payment for a loan, you can use the formula for a fixed-rate mortgage payment, often referred to as the mortgage payment formula:

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

Where:
- \( M \) = monthly payment
- \( P \) = loan principal (amount borrowed)
- \( r \) = monthly interest rate (annual rate divided by 12)
- \( n \) = number of payments (loan term in months)

### Given:
- Principal \( P = 265,351.00 \)
- Annual interest rate = 4.10% = 0.0410
- Monthly interest rate \( r = \frac{0.0410}{12} \)
- Loan term = 30 years \( = 30 \times 12 = 360 \) months

### Steps to Solve:
1. Calculate the monthly interest rate \( r \):
\[
r = \frac{0.0410}{12} \approx 0.00341667
\]

2. Determine the total number of payments \( n \):
\[
n = 30 \times 12 = 360
\]

3. Substitute \( P \), \( r \), and \( n \) into the mortgage payment formula:
\[
M = 265,351 \frac{0.00341667(1 + 0.00341667)^{360}}{(1 + 0.00341667)^{360} - 1}
\]

4. Calculate \( (1 + r)^n \):
\[
(1 + 0.00341667)^{360} \approx 3.4406 \quad \text{ (using a calculator)}
\]

5. Substitute back to find \( M \):
\[
M = 265,351 \frac{0.00341667 \times 3.4406}{3.4406 - 1}
\]
\[
M = 265,351 \frac{0.01175699}{2.4406} \approx 265,351 \times 0.00481 \approx 1,275.27
\]

So, the monthly mortgage payment is approximately $1,275.27.