Question

Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most $600. If the prevailing interest rate is 3.6%/year compounded monthly for a 48-month loan, what is the most expensive car that Dan can afford, assuming that he will receive $7000 for his trade-in? (Round your answer to the nearest cent.)

Answer 1

To calculate the most expensive car Dan can afford, we use the present value of an annuity formula considering his maximum monthly payment, the interest rate, and the loan term. Adding the trade-in value of his current car to the calculated amount gives us the final price of the car he can afford.

Step-by-step explanation:

Dan is looking at options for acquiring a new car and needs to understand what is the most expensive car he can afford based on his financial situation. To determine this, we need to consider the monthly payment he can make, the interest rate on the loan, and the duration of the payment period, which is 48 months in this scenario. We also take into account the trade-in value of his current car, which is $7,000.

We can use the present value of an annuity formula:
PV = PMT × ((1 - (1 + r)^-n) / r)
where PV is the present value (the amount that Dan can finance), PMT is the monthly payment ($600), r is the monthly interest rate (3.6% per year compounded monthly is 0.003), and n is the total number of payments (48).

First, we calculate the monthly interest rate:
Monthly interest rate = Annual rate / 12
0.036 / 12 = 0.003 or 0.3%

Then, we plug the values into the formula:
PV = $600 × ((1 - (1 + 0.003)^-48) / 0.003)

After calculating the present value, we add the trade-in value of $7,000 to find out the most expensive car Dan can afford.

If the calculations are done correctly, the final result will give us not only the most expensive car Dan can afford but also the overall cost of the loan.