Suppose you want to buy a car. The dealer offers a financing package consisting of a 3.6% APR compounded monthly for a term of four years. Suppose that you want your monthly payments to be at most $500. What is the maximum amount that you should finance? Round your answer to the nearest dollar.
To find the maximum amount that should be financed, we can use the formula for monthly payments:
\[ P = \frac{r(PV)}{1 - (1 + r)^{-n}} \]
Where:
- \( P \) = monthly payment
- \( r \) = monthly interest rate (APR divided by 12)
- \( PV \) = present value or loan amount
- \( n \) = number of payments
Given:
- Monthly payment \( P = $500 \)
- \( r = \frac{3.6\%}{12} = 0.003 \)
- \( n = 4 \times 12 = 48 \) months
Substitute these values into the formula:
\[ 500 = \frac{0.003(PV)}{1 - (1 + 0.003)^{-48}} \]
\[ 500 = \frac{0.003(PV)}{1 - (1.003^{-48})} \]
\[ 500 = \frac{0.003(PV)}{1 - 0.859123} \]
\[ 500 = \frac{0.003(PV)}{0.140877} \]
\[ 500 \times 0.140877 = 0.003(PV) \]
\[ PV = \frac{500 \times 0.140877}{0.003} \]
\[ PV \approx 23302 \]
Therefore, the maximum amount that should be financed is approximately $23,302.