# Derivation of the formula for $m$ compounding periods per year: $(1+\frac{i}{m})^{mt}$

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?

• @AlexC Great. Can you make your response to an answer. Thanks! – Frank Swanton Oct 1 '19 at 0:46

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.
• 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.) – Drew Oct 1 '19 at 2:33