Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most $400. 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 $5000 for his trade-in? (Round your answer to the nearest cent.)
To calculate the maximum price of the car Dan can afford, we need to consider the present value of the loan of $400 per month for 48 months at an interest rate of 3.6% compounded monthly.
First, calculate the present value of the annuity using the formula:
PV = PMT * (1 - (1 + r)^-n) / r
Where:
PV = Present Value
PMT = Payment per period ($400)
r = Interest rate per period (3.6%/12 months = 0.003 per month)
n = Number of periods (48 months)
PV = $400 * (1 - (1 + 0.003)^-48) / 0.003
PV = $400 * (1 - 0.726508) / 0.003
PV = $400 * 0.273492 / 0.003
PV = $36,938.08
Now, add the trade-in value of $5000 to the present value of the loan to find the maximum price of the car Dan can afford:
Maximum Price = $36,938.08 + $5000
Maximum Price = $41,938.08
Therefore, Dan can afford a car that costs at most $41,938.08.