You are considering buying a car worth $30,000. The dealer, who is anxious to sell the car, offers you an attractive financing package. You have to make a down-payment of $3,500, and pay the rest over 5 years with annual payments. The dealer will charge you interest at a constant annual interest rate of 2%, which may be different from the market interest rate.

The dealer offers you a second option: you pay cash, but get a $2,500 rebate. Assume that the market annual interest rate is constant at 5%. Which of the following statements is true?

• You should pay cash since this option costs $430 less
• You should pay cash since this option costs $340 less
• You should borrow since this option costs $340 less
• You should borrow since this option costs $430 less
• Both options cost the same

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.

Solution: B

Let the annual payment [math]=C[/math]. The PV of all my payments, discounted at the dealer's rate, must equal to the price, i.e.,

[[math]] \begin{aligned} & 3500+\frac{C}{0.02}\left(1-\frac{1}{1.02^5}\right)=30000 \\ & 3500+4.71346 C=30000 \\ & C=\$ 5,622.20 \end{aligned} [[/math]]

Since I can save at a higher rate, the cost of the financing plan in (a) is only 3500 + C/0.05 (1 − 11.05⁵ ) = $27, 841.17.

The cost of the second option is 30, 000 − 2, 500 = $27, 500. I should pay cash.

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.