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A dollar return with interest $i$ invested for $T$ years with compounding interest frequency of $m$ times each year is:

$$1*(1+\frac{i}{m})^{mt}.$$

My Question

  1. Why do we divide $i$ by $m$? Is this because $i$ represents annual interest rate, but it is compounded $m$ times a year, so we need to compute the effective interest rate at each compounding periods?

  2. How do we analytically derive this formula?

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  • $\begingroup$ @AlexC Great. Can you make your response to an answer. Thanks! $\endgroup$
    – Frank Swanton
    Oct 1, 2019 at 0:46

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This is about interest rate conventions and terminology. When people say "i percent a year compounded m times a year" it means the following: $i$ is called the quoted rate, which is not directly used in the calculation. Instead the first step is to calculate $\frac{i}{m}$ which is called the periodic rate and then apply this rate to every period. If there are t years, there are $mt$ periods and therefore the formula above follows.

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    $\begingroup$ It is also a natural way to express the relationship, as $lim_{m\to\infty}(1+\frac{i}{m})^{mt}=e^{it}$. (It converges to the continuous compounding expression in the limit.) $\endgroup$
    – Drew
    Oct 1, 2019 at 2:33

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