To calculate Jimmy's monthly mortgage payment, we can use the formula for a fixed-rate mortgage, 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 financed)
- \( r \) = monthly interest rate (annual rate / 12 months)
- \( n \) = number of payments (loan term in months)
Step 1: Determine the Principal Loan Amount
Jimmy's house costs $232,000, and he makes a down payment of $20,000. Therefore, the principal loan amount (\( P \)) is:
\[ P = 232,000 - 20,000 = 212,000 \]
Step 2: Calculate the Monthly Interest Rate
The annual interest rate is 4.6%, so we need to convert this to a monthly rate:
\[ r = \frac{4.6%}{100} \div 12 = \frac{0.046}{12} \approx 0.00383333 \]
Step 3: Determine the Number of Payments
For a 15-year loan, the total number of payments (\( n \)) is:
\[ n = 15 \times 12 = 180 \text{ months} \]
Step 4: Calculate the Monthly Payment
Now we can substitute these values into the mortgage payment formula:
\[ M = 212,000 \frac{0.00383333(1 + 0.00383333)^{180}}{(1 + 0.00383333)^{180} - 1} \]
First, we calculate \( (1 + r)^{n} \):
\[ (1 + 0.00383333)^{180} \approx 1.898322 \]
Now, plug that back into the formula:
\[ M = 212,000 \frac{0.00383333 \times 1.898322}{1.898322 - 1} \]
Calculating the numerator and denominator:
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Numerator: \[ 0.00383333 \times 1.898322 \approx 0.007267 \]
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Denominator: \[ 1.898322 - 1 \approx 0.898322 \]
So now,
\[ M = 212,000 \frac{0.007267}{0.898322} \approx 212,000 \times 0.008078 \] \[ M \approx 1717.53 \]
Final Result
Jimmy's estimated monthly mortgage payment is approximately $1,717.53.
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