Answer:
Step-by-step explanation:
To calculate the annual mortgage payment, we can use the formula for the fixed-rate mortgage payment:
\[ P = \dfrac{P_{\text{loan}} \times r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( P \) = Mortgage payment
- \( P_{\text{loan}} \) = Principal loan amount (Cost of the house - Down payment)
- \( r \) = Monthly interest rate (Annual interest rate / 12)
- \( n \) = Total number of payments (Years of the mortgage * 12)
Given:
- Cost of the house = $195,000
- Down payment = $40,000
- Annual interest rate = 4.2%
- Mortgage term = 30 years
1. Calculate the principal loan amount:
\[ P_{\text{loan}} = \text{Cost of the house} - \text{Down payment} \]
\[ P_{\text{loan}} = \$195,000 - \$40,000 = \$155,000 \]
2. Convert the annual interest rate to a monthly interest rate:
\[ r = \dfrac{\text{Annual interest rate}}{12 \times 100} \]
\[ r = \dfrac{4.2}{12 \times 100} = 0.0035 \]
3. Calculate the total number of payments:
\[ n = \text{Years of the mortgage} \times 12 \]
\[ n = 30 \times 12 = 360 \]
4. Plug the values into the formula to find the annual mortgage payment:
\[ P = \dfrac{\$155,000 \times 0.0035(1 + 0.0035)^{360}}{(1 + 0.0035)^{360} - 1} \]
Use this formula to calculate the mortgage payment for each year and the total interest paid over the loan term. Then, determine the remaining loan balance after 5, 10, 15, and 30 years. Finally, subtract the remaining loan balance from the total cost of the loan to find the total interest paid.
