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 $________.

1 answer

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.