To determine which expression correctly calculates the monthly payment for a 30-year loan of \[tex]$190,000 at an annual interest rate of 11.4%, compounded monthly, let's go through the key steps in the loan payment calculation.
1. Determine the monthly interest rate:
The annual interest rate is 11.4%, which when compounded monthly, yields a monthly interest rate:
\[
\text{Monthly Interest Rate} = \frac{11.4\%}{12} = \frac{0.114}{12} \approx 0.0095
\]
2. Determine the number of payments:
For a 30-year loan with monthly payments, the total number of payments is:
\[
\text{Total Payments} = 30 \times 12 = 360
\]
3. Use the loan payment formula:
The monthly payment \( P \) for a loan amount \( L \), with a monthly interest rate \( r \), over \( n \) total payments, is given by the formula:
\[
P = \frac{L \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Plugging in the values we have:
\[
L = 190,000
\]
\[
r = 0.0095
\]
\[
n = 360
\]
Therefore, the expression we need to use is:
\[
P = \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1}
\]
Now, comparing this to the given options:
- Option A: \(\frac{\$[/tex] 190.000 \cdot 0.0095 (1 - 0.0095)^{300}}{(1 - 0.0095)^{360} - 1}\)
- This formula has terms with [tex]\(1 - 0.0095\)[/tex] which is incorrect when calculating compound interest.
- Option B: [tex]\(\frac{\$ 190.000 \cdot 0.0095 (1 - 0.0095)^{300}}{(1 - 0.0095)^{300} + 1}\)[/tex]
- This also uses [tex]\(1 - 0.0095\)[/tex] which is incorrect.
- Option C: [tex]\(\frac{\$ 190.000 \cdot 0.0095 (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} + 1}\)[/tex]
- This uses [tex]\(+ 1\)[/tex] in the denominator, which is also incorrect.
- Option D: [tex]\(\frac{\$ 190.000 \cdot 0.0095 (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1}\)[/tex]
- This matches our derived formula perfectly.
Therefore, the correct expression is:
[tex]\[
\boxed{\frac{\$ 190.000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}}
\][/tex]
Using this expression, the resulting monthly payment is calculated to be approximately \$1867.07.
