To determine the monthly payment for the installment loan, we need to use the formula for calculating the monthly payment of an installment loan:
[tex]\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment.
- [tex]\( P \)[/tex] is the principal loan amount (the amount financed).
- [tex]\( r \)[/tex] is the monthly interest rate (annual percentage rate divided by the number of payments per year).
- [tex]\( n \)[/tex] is the total number of payments over the term of the loan (number of payments per year multiplied by the number of years).
Given:
- Amount Financed, [tex]\( P = 13000 \)[/tex] dollars
- Annual Interest Rate, [tex]\( r_{\text{annual}} = 0.05 \)[/tex] (expressed as decimal)
- Number of Payments per Year, [tex]\( n_{\text{year}} = 12 \)[/tex]
- Time in Years, [tex]\( t = 4 \)[/tex] years
First, calculate the monthly interest rate:
[tex]\[ r = \frac{r_{\text{annual}}}{n_{\text{year}}} = \frac{0.05}{12} \approx 0.004167 \][/tex]
Next, calculate the total number of payments:
[tex]\[ n = n_{\text{year}} \times t = 12 \times 4 = 48 \][/tex]
Now we can plug these values into the formula to get the monthly payment:
[tex]\[ M = 13000 \times \frac{0.004167(1 + 0.004167)^{48}}{(1 + 0.004167)^{48} - 1} \][/tex]
Through careful calculation, we get:
[tex]\[ M \approx 299.38 \][/tex]
So, the monthly payment is approximately \$299.38.
