Which of these expressions can be used to calculate the monthly payment for a 30-year loan for [tex]\$190{,}000[/tex] at [tex]11.4\%[/tex] interest, compounded monthly?

A. [tex]\frac{\$190{,}000 \cdot 0.0095(1-0.0095)^{300}}{(1-0.0095)^{360}-1}[/tex]
B. [tex]\frac{\$190{,}000 \cdot 0.0095(1-0.0095)^{300}}{(1-0.0095)^{300}+1}[/tex]
C. [tex]\frac{\$190{,}000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}+1}[/tex]
D. [tex]\frac{\$190{,}000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}[/tex]



Answer :

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.