Question

Derek plans to buy a $30,508.00 car. The dealership offers zero percent financing for 52.00 months with the first payment due at signing (today). Derek would be willing to pay for the car in full today if the dealership offers him $____ cash back. He can borrow money from his bank at an interest rate of 4.65%.

Answer 1

Monthly Installment = 30508/52 = $586.6923

Amount to be paid at the BEGINNING of each month = PV of Annuity = P*[1-{(1+i)^-n}]/i

Note: For the purpose of calculation (so that formula can be applied), it will be considered that amount will be paid for 51 months at the end of each month starting from 1 month from now, and we will also add an additional installment that will be paid today. Effectively, we have a total of PV of next 51 installments and today’s installment.

Where, P = Annuity = 586.6923, i = Interest Rate = 0.0465/12 = 0.003875, n = Number of Periods = 52-1 = 51

Therefore, Present Value = PV of next 51 Installments + Today’s Installment

= [586.6923*{1-((1+0.003875)^-51)}/0.003875]+586.6923 = 27102.75632+586.6923 = $27689.45

Cash Back = Listed Price-PV = 30508-27689.45 = $2818.55