A buyer of a 2003 Protege S Hatchback has a choice of $0\%$ financing for $60$ months or a $\$3600$ rebate. He plans to make no down payment. The buyer is able to qualify for $7\%$ annual effective financing through his credit union and thereby take advantage of the rebate. Let $Y$ denote his negotiated price for the Protege S Hatchback. How large must $Y$ be in order for the $0\%$ dealer financing to be preferable?
For the credit union financing, I think the payment would be $$\dfrac{Y}{ \biggl( \dfrac{ 1-(1 + .06785/12)^{-60}}{.06785/12}\biggr)}.$$ How do I account for the rebate? How do I compare this with the $0\%$ financing option. The present value of the $0\%$ finance option is $0$ correct? How do we make the comparison?
No, the present value of the $0\%$ option is not $0$. The $0\%$ option means that they don't charge interest, but they still charge money. Under the $0\%$ option, he will make $60$ monthly payments of $\frac Y{60}$, and you need to take the present value of these payments at $7\%$ effective. For the rebate, the present value of the credit union payments is $Y$, and you have to subtract $3600$ to account for the rebate. The break-even point comes when these two values are equal.