Question

Karim Soltan is shopping for a new vehicle, and has noticed that many vehicle manufacturers are offering special deals to sell off the current year’s vehicles before the new models arrive. Karim’s local Ford dealership is advertising 3.9% financing for a full 48 months (i.e., 3.9% compounded monthly) or up to $4000 cash back on selected vehicles.

The vehicle that Karim wants to purchase costs $24 600 including taxes, delivery, licence, and dealer preparation. This vehicle qualifies for $1800 cash back if Karim pays cash for the vehicle. Karim has a good credit rating and knows that he could arrange a vehicle loan at his bank for the full price of any vehicle he chooses. His other option is to take the dealer financing offered at 3.9% for 48 months.

Karim wants to know which option requires the lower monthly payment. He knows he can use annuity formulas to calculate the monthly payments.

Questions
a. Suppose Karim buys the vehicle on July 1. What monthly payment must Karim make if he chooses the dealer’s 3.9% financing option and pays off the loan over 48 months? (Assume he makes each monthly payment at the end of the month and his first payment is due on July 31.)
b. Suppose the bank offers Karim a 48-month loan with the interest compounded monthly and the payments due at the end of each month. If Karim accepts the bank loan, he can get $1800 cash back on this vehicle.

Help Karim work out a method to calculate the bank rate of interest required to make bank financing the same cost as dealer financing. First, calculate the monthly rate of interest that would make the monthly bank payments equal to the monthly dealer payments. Then calculate the effective rate of interest represented by the monthly compounded rate. If the financing from the bank is at a lower rate of interest compounded monthly, choose the bank financing. The reason is that the monthly payments for the bank’s financing would be lower than the monthly payments for the dealer’s 3.9% financing.
(i) How much money would Karim have to borrow from the bank to pay cash for this vehicle?
(ii) Using the method above, calculate the effective annual rate of interest and the nominal annual rate of interest required to make the monthly payments for bank financing exactly the same as for dealer financing.
c. Suppose Karim decides to explore the costs of financing a more expensive vehicle. The more expensive vehicle costs $34 900 in total and qualifies for the 3.9% dealer financing for 48 months or $2500 cash back. What is the highest effective annual rate of interest at which Karim should borrow from the bank instead of using the dealer’s 3.9% financing?

Answer 1

1a 3. q Given ioloist ayment Pesiods find 48 H Compounded amdul the Let oul Compound amaun oan amounl- Heui = Annual rali 4 inlae Number ofpyme peiod g Ahember of paymeutpewi oels 24 6 00 (1.9 Caumpound amaul 24.600 685) Q?745 Total Compouaded Num be Manthly amaw aymed Paymet ad 29745 S 98.854 2 () i)The amout be wili $24,600 -1,800 92,800

C) Monthly dealu payment fs98 8.8542 Here monthly payment for honk financ dial irancing es Same as Total amonn 28,745 Compo unded monthly S8. 8542 x U8 2874 oul 28 7u5 2 12t 22 loo 28,74 2+1 22,800 12 1-2607 124 4.&39 Total annual in Yali Annual Yali : F22 600 29 x 10D 22,800

Total annual inoe22 boo (1 2 20 Given at inle Yali + 3.9% Joan 34 90D amoul Compounded camoud pa Total dbaleu annum 2- 34,900 36,285 Tlee fore Compound iniseu fla aprum 36 285-349od st'suf The highet eeetive val leiett karim shouhd boowo the al ohich hank insleaddeales 9 han finanting ),385 x100 3,94% F4,900 4.27 x 100 34900- 26 ftere 0o u Cash back amont