Question

You are taking out a car loan and will make payments of $400 each month, for a total of 60 monthly payments. You also have $6500 for a down payment. The APR is 10.8% with quarterly compounding. How much does the car cost?

Answer 1

Annual interest rate = 10.80%
Quarterly interest rate = 2.70%

Effective annual rate = (1 + Quarterly interest rate)^4 - 1
Effective annual rate = (1 + 0.0270)^4 - 1
Effective annual rate = 1.11245 - 1
Effective annual rate = 0.11245

Monthly interest rate = (1 + Effective annual rate)^(1/12) - 1
Monthly interest rate = (1 + 0.11245)^(1/12) - 1
Monthly interest rate = 1.00892 - 1
Monthly interest rate = 0.00892 or 0.892%

Monthly payment = $400
Number of payments = 60

Present value of monthly payments = $400/1.00892 + $400/1.00892^2 + … + $400/1.00892^59 + $400/1.00892^60
Present value of monthly payments = $400 * (1 - (1/1.00892)^60) / 0.00892
Present value of monthly payments = $400 * 46.306808
Present value of monthly payments = $18,522.72

Cost of car = Down payment + Present value of monthly payments
Cost of car = $6,500 + $18,522.72
Cost of car = $25,022.72