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I'm solving the following problem:

Two dealers compete to sell you a new Hummer with a list price of \$45,000. Dealer C offers to sell it for \$40,000 cash. Dealer F offers “0-percent financing:” 48 monthly payments of \$937.50. (48x937.50=45,000)

(a) You can finance purchase by withdrawals from a money market fund yielding 2% per year. Which deal is better?

So I need to calculate the present value of the financing option with a yearly rate of $r=0.02$, and monthly cashflows of $C=\\\$937.50$. My logic for this is that we need to first convert the annual interest rate to the monthly rate so that $r\to r/12$. Moreover, we need to ensure that our discounting is consistent, in that $(1+r)^T$ represents $T$ years of the time horizon. Therefore, the exact expression for the present value is

$$ PV = C \sum_{n=1}^{48} \left(\frac{1}{\left(1+\frac{r}{12}\right)^{1/12}}\right)^n. $$

However, the official solution to the problem states that

$$ PV = C \sum_{n=1}^{48} \left(\frac{1}{\left(1+r \right)^{1/12}}\right)^n. $$

So the only difference is that the official solution compounds the yearly interest for each payment, and mine compounds on a monthly basis. So my question is which one is the correct solution.

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1 Answer 1

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The official solution is correct. Consider the case where $r = 0.02$. The monthly rate in this case is $1.02^{1/12}$ or about $1.0016516$.

By the way, $(1 + 0.02/12)^{1/12}$ is about $1.0001388$. That is not going to annualize to 2%.

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  • $\begingroup$ Thanks for your response, and I like that as a sanity check. I get what you are saying that after 12 years we need to compound so that we accrue 1.02 in interest. I guess my confusion is that in every other problem I have solved (not that many tbf), when we change the compounding period we need to also need to account for the number of payment periods in the interest rate. I guess, logically at least, i'm failing to see a relevant distinction even though what your answer says makes perfect sense. $\endgroup$
    – Wolfgang
    Apr 8, 2023 at 19:29
  • $\begingroup$ If you look at the term having $n=12$ you will see that the official solution is correct: the discounting factor is $\frac{1}{1+r}$ as we would expect for discounting for one year at an annual rate $r$. $\endgroup$
    – nbbo2
    Apr 8, 2023 at 19:35

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