Derek borrows $311,156.00 to buy a house. He has a 30-year mortgage with a rate of 5.98%. After making 114.00 payments, how much does he owe on the mortgage?

1 answer

To find out how much Derek owes on his mortgage after 114 payments, we need to calculate the remaining balance on the mortgage. The formula for the remaining balance \( B \) on a fixed-rate mortgage after \( n \) payments is given by:

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

where:

  • \( P \) is the principal (initial loan amount).
  • \( r \) is the monthly interest rate (annual rate divided by 12).
  • \( N \) is the total number of payments.
  • \( n \) is the number of payments made so far.

Given values:

  • Principal \( P = 311,156.00 \)
  • Annual interest rate = 5.98%, so the monthly interest rate \( r = \frac{5.98%}{12} = \frac{0.0598}{12} \approx 0.00498333 \)
  • Total number of payments \( N = 30 \times 12 = 360 \)
  • Payments made \( n = 114 \)

Step 1: Calculate \( (1 + r)^N \)

\[ (1 + r)^N = (1 + 0.00498333)^{360} \approx 5.029356 \]

Step 2: Calculate \( (1 + r)^n \)

\[ (1 + r)^n = (1 + 0.00498333)^{114} \approx 1.648611 \]

Step 3: Calculate the remaining balance \( B \)

\[ B = 311,156 \frac{(5.029356 - 1.648611)}{(5.029356 - 1)} \]

Calculating \( 5.029356 - 1.648611 \): \[ 5.029356 - 1.648611 \approx 3.380745 \]

Now, calculating \( 5.029356 - 1 \): \[ 5.029356 - 1 \approx 4.029356 \]

Now substituting these values into the formula for \( B \): \[ B = 311,156 \frac{3.380745}{4.029356} \]

Calculating the fraction: \[ \frac{3.380745}{4.029356} \approx 0.839803 \]

Now, calculating \( B \): \[ B \approx 311,156 \times 0.839803 \approx 261,205.07 \]

Conclusion

After making 114 payments, Derek owes approximately $261,205.07 on the mortgage.