To calculate Derek's monthly mortgage payment, we can use the formula for a fixed-rate mortgage payment, which is given by:
\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) = monthly payment
- \( P \) = principal loan amount (the amount borrowed)
- \( r \) = monthly interest rate (annual rate divided by 12)
- \( n \) = number of payments (loan term in months)
Given:
- \( P = 278,971.00 \)
- Annual interest rate = 4.34% = 0.0434
- Monthly interest rate \( r = \frac{0.0434}{12} = 0.00361667 \)
- Loan term = 30 years = \( 30 \times 12 = 360 \) months
Now we can plug these values into the formula:
\[ M = 278,971 \frac{0.00361667(1 + 0.00361667)^{360}}{(1 + 0.00361667)^{360} - 1} \]
Calculating \( (1 + r)^{360} \):
\[ (1 + 0.00361667)^{360} \approx 3.4748851 \]
Now, substituting this back into the formula:
\[ M = 278,971 \frac{0.00361667 \times 3.4748851}{3.4748851 - 1} \]
Calculating the numerator:
\[ 0.00361667 \times 3.4748851 \approx 0.0125582 \]
And the denominator:
\[ 3.4748851 - 1 \approx 2.4748851 \]
So:
\[ M = 278,971 \frac{0.0125582}{2.4748851} \approx 278,971 \times 0.0050757 \approx 1,415.41 \]
Thus, Derek's monthly mortgage payment is approximately $1,415.41.