The automotive dealership industry often offers two alternatives for new car buyers. The so called “cash back” deal allows buyers to pay a lower price for a car than the sticker price. The “special financing” deal allows the buyer to take advantage of a low interest rate over the life of the loan to buy the car. Often, buyers assume the cash back deal is better because it results in a lower price paid for the car. However, a little financial math can reveal the true better deal.

Suppose that a car dealership offers to sell a buyer a new car for $15,000. The buyer may choose between $1,000 cash back or a low 2% APR interest rate for the three year life of a loan to buy the car. Monthly payments on the car with the special financing option can be calculated using the present value of an annuity formula. In this case, the annuity is the monthly payments and the present value is the price of the car.

PVA = CF * [((1 + r)^{n} – 1) / (r * (1 + r)^{n}]

15000 = CF * [((1 + 0.001667)^{36} – 1) / (0.001667 * (1 + 0.001667)^{36}]

CF = $429.64

So, the monthly payments by taking the 2% APR special financing deal at a price of $15,000 are $429.64.

Suppose that if the cash back option is taken, the buyer can borrow $14,000 ($15,000 – $1000) from the bank at a rate of 8% APR. Using the same formula above, the monthly payments for this option are:

14000 = CF * [((1 + 0.006667)^{36} – 1) / (0.006667 * (1 + 0.006667)^{36}]

CF = $438.71

So, the monthly payments by taking the 8% APR from the bank and paying $14,000 for the car are $438.71.

To find out which is the better deal, the buyer needs to compare the monthly payments under both offers. Since the monthly payments under the special financing deal ($429.64) are less then the cash back deal ($438.71), the special financing deal is the better choice. The difference of the two payments is $9.07 per month. However, keep in mind that this comparison is only possible because the life of the loan (3 years) and the number of payments (36 months) are the same for both deals. If one or both of these changes, this comparison does not reveal the true better deal.

Remembering the principle of the Time Value of Money, wise investors know interest rates compound in a way that distort the value of money. Most people would assume that since the cash back deal lowers the price of the car from $15,000 to $14,000 that the total amount paid will be less than if the special financing deal is chosen. The principle to remember is that the present value of an asset is only one way to calculate the asset’s worth and that interest rates have a significant effect on the value of an asset.