From an anual growth rate to a monthly compunded

Question

Let's say I have an amount that grew in 1 year of 5%. From 100, to 105. How do I calulcate the average monthly growth?

Using the CAGR, I could write:

$(105/100)^{1/12}-1 = 0.41\%$

Is this formula correct, is there any other way without using Begin value (100) and End Value (105) ?

Answer 1

Let us take the reverse of the process you used to calculate the compounded amount. Suppose the growth rate is $(\frac{105}{100})^{\frac{1}{12}} - 1$. If we compound the growth rate $12$ times from $100$, we should get $105$. Ignoring the $(-1)$ in your answer (to convert a growth rate to a percentage), we have:

$$(100){(\frac{105}{100})^{\frac{1}{12}}}^{12}$$ $$=(100)(\frac{105}{100})$$ $$=105$$

Therefore your answer is correct.

This formula in its essence, takes the root of $\frac{1}{n}$ of the growth rate, where $n$ is the compounding period, so that when multiplied $n$ times it returns the original value. Therefore, you only need the change between the start value and end value, since this problem describes rates of change.