# How to explain the “less than yearly compounded interest” concept?

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?

• 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. – Matthew Daly Feb 26 '20 at 13:40
• @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. – Nick C Feb 26 '20 at 14:44
• @NickC Yeah, a specific question that students have trouble appreciating would definitely clear up this confusion. – Matthew Daly Feb 26 '20 at 15:16
• 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). – Vandermonde Feb 26 '20 at 20:20
• 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. – Michael Bächtold Feb 27 '20 at 0:33