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

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

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

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

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



Answer :

Given the problem, we need to identify which of the provided expressions correctly represents the formula for calculating the monthly payment on a 30-year loan for \[tex]$190,000 at an annual interest rate of 11.4%, compounded monthly. To start, let's outline the standard formula for calculating the monthly payment \( M \) of a fixed-rate mortgage loan: \[ M = \frac{P \cdot r(1 + r)^n}{(1 + r)^n - 1} \] where: - \( P \) is the loan principal (the amount borrowed) - \( r \) is the monthly interest rate - \( n \) is the total number of payments (months) The given parameters are: - Principal \( P = \$[/tex]190,000 \)
- Annual interest rate [tex]\( \text{annual\_interest\_rate} = 11.4\% \)[/tex] (or 0.114)
- Loan period [tex]\( \text{years} = 30 \)[/tex]

### Step-by-step solution:

1. Convert the annual interest rate to a monthly interest rate:
[tex]\[ \text{monthly\_interest\_rate} = \frac{11.4\%}{12} = \frac{0.114}{12} = 0.0095 \][/tex]

2. Calculate the number of months (payments):
[tex]\[ \text{months} = 30 \times 12 = 360 \][/tex]

3. Identify the correct formula by comparing it with the standard formula:
- The monthly interest rate [tex]\( r = 0.0095 \)[/tex]
- The number of payments [tex]\( n = 360 \)[/tex]
- The principal [tex]\( P = \$190,000 \)[/tex]

Substituting these values into the standard formula, we get:

[tex]\[ M = \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1} \][/tex]

By analyzing the given choices:

\- Option A: [tex]\(\frac{\$ 190,000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{360}-1}\)[/tex] contains errors involving incorrect interest rate adjustments.

\- Option B: [tex]\(\frac{\$ 190.000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{300}+1}\)[/tex] incorrectly adjusts the exponent on the denominator.

\- Option C: [tex]\(\frac{\$ 190.000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{300}+1}\)[/tex] also contains incorrect interest rate adjustments and exponents.

\- Option D: [tex]\(\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}\)[/tex] correctly uses the loan parameters and the formula structure.

Thus, the correct expression that can be used to calculate the monthly payment for the specified loan is:

[tex]\[ \boxed{\text{D. } \frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}} \][/tex]