# 1. A car company is offering a choice of deals. You can receive $2,000 cash back on the purchase... ## Question: A car company is offering a choice of deals. You can receive$2,000 cash back on the purchase or a 2.6 percent APR, 3-year loan. The price of the car is \$22,000 and you could obtain a 3-year loan from your credit union, at 6.6 percent APR.

Which deal is cheaper?

## Dealership Incentives:

Dealership often provides incentives to buyer in one of two forms: a cash discount or a lower financing rate. For consumers to determine which one is better, they need to compare the loan payments under each scenario.

The second deal is cheaper.

To see which one is cheaper, we can compare the monthly payments under each case. If you receive the cash back, then your loan amount is 22,000 - 2,000 = 20,000, and the term of the loan is 3-year, 6.6% APR. We can use the following formula to compute the monthly payment for a loan with principal {eq}P {/eq}, monthly interest rate {eq}r{/eq} and number of monthly payments {eq}T{/eq}:

• {eq}\displaystyle \frac{Pr}{1 - (1 + r)^{-T}} {/eq}

Applying the formula, with cash back, the monthly payment is:

• {eq}\displaystyle \frac{20,000*2.6\%/12}{1 - (1 + 6.6\%/12)^{-36}} = 613.89 {/eq}

With the other option, the amount borrowed is 22,000, the term on the loan is 3-year, 2.6% APR, applying the formula, the monthly payment is:

• {eq}\displaystyle \frac{22,000*2.6\%/12}{1 - (1 + 2.6\%/12)^{-36}} = 578.11 {/eq}

Since monthly payment is lower with the second option, it is the cheaper one.