Question

Automobile dealerships often advertise promotions such as “no payments for 90 days!” or similar deals that sound too good to be true. The goal, of course, is to entice buyers with the ever-tantalizing prospect of getting something now without having to worry about how to pay for it until later. Suppose that you are looking to buy a car priced at $35,000 and are offered a loan with a down payment of $5,000 and an APR of 6% over 60 months.

(a) Calculate the monthly payment for this loan

(b) Now suppose that the dealer offers you “no money down, no interest, and no payments for 90 days!” and you naively interpret this to imply that the remaining payments are unchanged. In present value terms, what price do you think that you are paying for the car?

(c) Somewhat coming to your senses, you realize that your monthly payment will go up to offset the missing initial payments, but still believe that you will see some savings because of the time value of avoiding interest for three months. What monthly payment do you calculate and what do price do you think you are paying? (d) (5) Finally, you realize that if the dealer had any intention of offering you a discount, he would have let you know how much you’d be saving, and that “no interest” simply means that any unpaid interest is added to the principal. What is your actual monthly payment?

Answer 1

(a) Monthly payment for this loan

Price of the car -----------------------------------------35000

Less: Down payment in cash--------------------------   5000

Balance to be paid in instalments(35000-5000)---- 30000

at an APR of 6% ,ie. 0.06/12=0.005 % per month, for 60 months

Using PV of annuity formula,

PV of loan=Mthly.installment*(1-(1+mthly.interest %)^-)No.of mthly.instalments))/Mthly.interest %

ie. 30000=Mthly.Pmt.*(1-(1+0.005)^-60)/0.005

solving online, we get the monthly payment towards the loan as,

580

b.ie. The above payment of $ 580 start only after 90 days,ie. 3 months--

$ 580 paid for (60-3)=57 months, from the start of 4th month

So, PV of the payments =PV at start of Yr.4/(1+0.005)^3

PV=(580*(1-(1+0.005)^-57)/0.005)/(1+0.005)^3

PV=28278

So, the price he thinks he is paying for the car=

28278

c. So, now recalculating with the original price

but with no down payment & payment starting after 3 months

35000=(Mthly.Pmt.*(1-(1+0.005)^-57)/0.005)/(1+0.005)^3

Solving the above, the monthly payment will be

718

So, actual monthly payment

with down payment -- $ 580

without any down payment--- $ 718