You are considering the purchase of a new car, the reborn VW Beetle, and you have been offered two different deals from two different dealers. Dealer A offers to sell you the car for $20,000, but allows you to put down $2,000 and pay back $18,000 over 36 months (fixed payment each month) at a rate of 8% compounded monthly. Dealer B offers to sell you the car for $19,500 but requires a down payment of $4,000 with repayment of the remaining $15,500 over 36 months at 10% compounded monthly.

Which of the following statements is true?

• If the annual discount rate, compounded monthly, is above 8.5%, then deal A is better than deal B
• If the annual discount rate, compounded monthly, is above 9.5%, then deal A is better than deal B
• If the annual discount rate, compounded monthly, is below 9%, then deal A is better than deal B
• Both deals are equally good regardless of the discount rate
• Deal A is always better than deal B

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

First use the annuity formula to determine the monthly payments [math]C_a[/math] and [math]C_b[/math] for dealerships A and B, respectively, ignoring the initial down payments:

(a) Dealership A: [math]\mathrm{PV}_a=\$ 18,000, r_a=0.08 / 12[/math], and [math]t=36[/math] months [math]\Rightarrow C_a=\$ 564.05[/math].

(b) Dealership B: [math]\mathrm{PV}_b=\$ 15,500, r_b=0.10 / 12[/math], and [math]t=36[/math] months [math]\Rightarrow C_b=\$ 500.14[/math].

If the monthly discount rate is currently r, then the net present values of the two packages are

[[math]] \begin{aligned} \mathrm{NPV}_a & =2,000+C_a\left[\frac{1}{r}-\frac{1}{r(1+r)^{36}}\right] \\ \mathrm{NPV}_b & =4,000+C_b\left[\frac{1}{r}-\frac{1}{r(1+r)^{36}}\right] .\end{aligned} [[/math]]

It is clearly more advantageous to accept dealership A's offer if and only if [math]\mathrm{NPV}_a\lt[/math] [math]\mathrm{NPV}_b[/math]. Substituting the expressions from above and simplifying, we have that [math]\mathrm{NPV}_a\lt[/math] [math]\mathrm{NPV}_b[/math] if and only if

[[math]]\left[\frac{1}{r}-\frac{1}{r(1+r)^{36}}\right]\lt31.29 [[/math]]

By trial and error, the cross-over point is at r = 0.00778. The conclusion is that if the current annual interest rate for a 36-month period (compounded monthly) is above 9.34%, you should choose dealership A. If the current annual interest rate for a 36-month period (compounded monthly) is below 9.34%, you should choose dealership B.

References

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