Question

Jimmy buys a house for $232,000. He makes a down payment of $20,000 and finances the balance. How much are his monthly payments if the current rate is 4.6% and it is a fifteen year loan?

Answer 1

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:

- **Numerator**:
\[
0.00383333 \times 1.898322 \approx 0.007267
\]

- **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**.