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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?

minger
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2 Answers2

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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.

saulspatz
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  • then $\dfrac{Y}{60}a_{60|.0678/12}=Y- \bigl(3600(1.07^{-5})\bigr)$ and $Y=16670.21$ ? – minger Mar 19 '20 at 22:13
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    No, I was wrong. I was thinking the rebate comes at the end of the payments, but of course it comes at the beginning. I've corrected my answer. – saulspatz Mar 19 '20 at 23:26
  • so $Y=23380.84$ which agrees with my textbook's answer. thanks – minger Mar 19 '20 at 23:36
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A simple approach is to compare the payments each way, because he will own the car in $60$ months after paying either loan. If he takes the dealer loan, he will pay $\frac Y{60}$ every month. For the credit union, you should replace $Y$ with $Y-3600$ in the formula because that is how much he has to borrow after the rebate. You now have a linear equation in $Y$ to solve. Why is it $0.06785$ in your formula instead of $0.07$?

Ross Millikan
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  • Would the Y-3600 be considered a down payment? Don't we have to convert from effective into nominal? – minger Mar 19 '20 at 03:35
  • No, the check he has to hand the dealer is $Y-3600$ because of the rebate. If he doesn't make a down payment, that is what he has to borrow from the credit union. A better way to look at the dealer deal is he is borrowing $Y-3600$ and paying $3600$ in interest. If he pays less than $3600$ interest to the credit union, the credit union is a better deal. I can't help with nominal/effective interest. – Ross Millikan Mar 19 '20 at 03:41