0% Financing vs. a Cash Rebate
A student is buying a used car with a sticker price of $14,200. Dealer A offers 0% APR financing for 48 months with no rebate. Dealer B offers a $2,000 manufacturer rebate plus financing at 5.9% APR compounded monthly for 48 months. Both deals require no down payment, and the student can afford either monthly payment. The student's uncle insists that '0% is always the best deal because you pay no interest.' The student also has $2,000 in a savings account earning 3.5% APR compounded monthly, which could be used as a down payment, and plans to keep the car for at least six years. The student wants to compare the two financing offers and decide whether to use the savings. The car has no other fees.
1. Compute the monthly payment and total amount paid over 48 months under each dealer's offer. Which offer costs less in total?
Read the worked answer
Dealer A: payment = 14,200/48 ≈ $295.83; total = $14,200.
Dealer B: amount financed = 14,200 − 2,000 = $12,200. The monthly rate is r = 0.059/12 ≈ 0.0049167 and n = 48. The payment is P = 12,200·r/[1 − (1 + r)^−48]. Compute (1 + r)^48 ≈ 1.26545, so the denominator is 1 − 1/1.26545 ≈ 0.20976. Payment ≈ 12,200(0.0049167)/0.20976 ≈ 59.98/0.20976 ≈ $285.96. Total ≈ 285.96 × 48 ≈ $13,726. Dealer B costs about $474 less, so the rebate offer is cheaper despite the interest.
2. If Dealer B's APR were 9% instead of 5.9%, would the conclusion change? Compute the new payment and total, and explain what this shows about the uncle's claim.
Read the worked answer
At 9% APR, r = 0.09/12 = 0.0075. (1.0075)^48 ≈ 1.4314, denominator ≈ 1 − 1/1.4314 ≈ 0.30137. Payment ≈ 12,200(0.0075)/0.30137 ≈ 91.50/0.30137 ≈ $303.61. Total ≈ 303.61 × 48 ≈ $14,573. That is about $373 more than Dealer A's $14,200. So the 0% offer wins at 9% APR. The uncle's '0% is always best' is not a logical rule; the rebate offer wins when the interest cost on the smaller financed amount is less than the rebate. The comparison depends on the APR, the loan term, and the rebate amount.
3. Should the student use the $2,000 savings as a down payment? Compare the relevant rates and give a reasoned recommendation that considers liquidity.
Read the worked answer
The savings earns 3.5% APR compounded monthly, while Dealer B's loan costs 5.9% APR. Using $2,000 to reduce the loan principal saves interest at 5.9%, which is more than the 3.5% the savings earns, so under Dealer B the student comes out ahead by using the savings, provided an emergency fund remains. The monthly payment would fall: principal $10,200 at 5.9% for 48 months gives about 10,200(0.0049167)/0.20976 ≈ $239.10, total ≈ $11,477 plus the $2,000 used, or about $13,477 all-in. Under Dealer A's 0% loan, applying the $2,000 does not save interest; it only lowers the monthly payment and reduces liquidity, so keeping the savings may be preferable. The recommendation should therefore be conditional on which financing is chosen and on the need for cash reserves.