What difficulties can be met while teaching the "less than yearly compounded interest" concept?

Based on my own (learning) experience, an objection arose when I was presented with the formula:

$$A = A_0 \left(1+\frac{r}{n}\right)^{nt}.$$

The objection is : "why should the rate be diminished (that is, divided by $n$) on the sole ground that the period is shorter?"

  • Is this objection frequent?

  • In which confusion does this objection originate?

  • How to teach this concept in order to avoid this miscomprehension?

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    $\begingroup$ Can you clarify your question? If you're asking why a six-month investment would return less than the annual percentage rate, the answer that you would deserve less then 12 months of interest seems pretty obvious. $\endgroup$ – Matthew Daly Feb 26 at 13:40
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    $\begingroup$ @MatthewDaly The question asks why the rate would be diminished, not the value of the account. I thought this question was asking about the motivation for taking yearly interest and compounding it, say, monthly. However, I don’t think it’s merely because the period is shorter, but I would leave it to a financial historian to fill in those details. $\endgroup$ – Nick C Feb 26 at 14:44
  • $\begingroup$ @NickC Yeah, a specific question that students have trouble appreciating would definitely clear up this confusion. $\endgroup$ – Matthew Daly Feb 26 at 15:16
  • $\begingroup$ I'd say, $t$ may be the time taken (in some fixed unit such as years), but $nt$ is the number of periods in which compounding is applied; accordingly, since $1+\text{something}$ is taken to this power, it should be the rate factor per period and not per year (which at least motivates why it wouldn't be $1+r$ lest interest become infinite in the limit; now I used to object to (arithmetically) dividing by $n$ given that the compounding is "geometric", but that's another question). $\endgroup$ – Vandermonde Feb 26 at 20:20
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    $\begingroup$ The objection I would raise: why do people call this the "less than yearly compounded interest"? If the yearly interest is $r$, the monthly one should be $(1+r)^{1/12}-1$ not $r/12$, to amount to the same after a year. Or maybe I misunderstand the problem. $\endgroup$ – Michael Bächtold Feb 27 at 0:33

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