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Dealer A will sell me a car for \$20,000 with \$2,000 down and the remaining $18,000 paid back over 36 months at a rate of 8% compounded monthly.

Dealer B will sell me the same car for \$19,500 with \$4,000 down and the remaining $15,500 paid back over 36 months at a rate of 10% compounded monthly.

(I assume these interest rates are APRs, since having them be monthly rates results in you paying more in interest than the price of the car.)

I'm asked to find the discount rate at which these deals are equally attractive. I assume this means the rate at which if I were to forgo buying a car and deposit amounts equal to the payments instead, I would end up having the same present value for both deals.

But when I try to calculate this I run into problems. I calculated the monthly payments for both deals, but then when I try to equate their present values and solve for the discount rate, I get that there is no positive discount rate for which these two deals are equally attractive. I've checked my math over and over and I'm sure it's correct so I must be thinking about the problem incorrectly. Anyone have any ideas?

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  • $\begingroup$ Can you please post your equations? Not necessarily all of them, but the beginning - in particular the one that defines the discount rate you are looking for. $\endgroup$
    – Oliv
    Sep 19 '16 at 8:56
  • $\begingroup$ For a particular dealer, are monthly payments the same each month or are they interest plus $\frac1{36}$ of the original loan? $\endgroup$
    – Henry
    Sep 19 '16 at 12:09
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Presuming the question is about inter-temporal choices and corresponding discount rate (not central banking). It is rather easy to prove existence of such discount rate.

At one extreme, when discount rate is 100% (carpe diem mode), the person in question (N) will only look at the immediate value and expense. Dealer A effectively sells the car for \$2,000 (in present terms), while Dealer B sells it for \$4,000. So N will choose A.

At the other extreme, when discount rate is 0% (immortal person without any propensity for investment, perhaps?), N will just sum up all the value and expense streams. So A effectively sells the car for \$20,000 (both in present terms and in nominal terms), while B for \$19,500. So N will choose B.

With discounting function being continuous, it follows that somewhere between 100% and 0% there is a discount rate where A and B have the same present value.

PS: I am not sure if I should have posted this as a comment, but it's too long, and actually proves existence of the answer. I may expand this to include the actual solution if the question is updated to specify whether equal payments or equal principal payments or some other amortization schedule is used.

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  • $\begingroup$ Your third paragraph ("At the other extreme...") seems to ignore the interest payments on the loans $\endgroup$
    – Henry
    Sep 28 '16 at 23:55
  • $\begingroup$ You are right, I got distracted by not being sure what kind of schedule to use, and ultimately forgot about interest. $\endgroup$ Sep 29 '16 at 3:12
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I would have thought that you could work out the amounts paid, and then compare the differences

So for Dealer A, I would have thought the payments were $\$2000$ immediately followed by $36$ equal monthly payments of around $\$591.122$, though it does depend on how you translate $10\%$ into a monthly interest payment

Similarly for Dealer B, I would have thought the payments were $\$4000$ immediately followed by $36$ equal monthly payments of around $\$522.275$, with the same caveat

So the difference between them is $\$2000$ less paid to Dealer A immediately than to Dealer B, but offset by about $\$68.847$ paid more each month for $36$ months. I make those equal at a monthly discount rate of about $0.92677\%$ which compounds up to an annual discount rate of about $11.706\%$

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