The equation for compound interest can be derived by considering the recurrence relation $a_{n+1}=a_n+sa_n=(1+r)a_n$ where $s>0$ is the growth rate, $a_n$ is the current balance, and $a_{n+1}$ is the new balance after one "compounding cycle." This gives $$a_n=(1+s)^n a_0$$ where $n$ is the number of compounding cycles that have elapsed. Of course, this can be equation can be rewritten as $a_n=(1+s)^{mt}$ where $t$ is years and $m$ is number of compounding cycles per year. Typically, the compound interest equation is expressed as $$a_n=\left(1+\frac{r}{m}\right)^{mt}a_0$$ where $r/m=s$, the compounding rate. Does $r$ have any meaning other than the compounding rate scaled by $m$? What I mean is....I find calling $r$ the "annual interest rate" very misleading. I would expect the annual interest rate to be the interest rate such that, were I to be compounding annually, the growth would match my original $a_n$ equation. As it stands, if I were to compound annually with the "annual interest rate" for $a_n$ I would get a completely different growth than $a_n$; that is, $a_n=\left(1+\frac{r}{m}\right)^{mt} \neq (1+r)^t$
How can this be reconciled? I suppose an answer would address why is $r$ called an annual rate and whether $r$ can be given any meaning other than its seemingly arbitrary definition.
