# From an anual growth rate to a monthly compunded

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) ?

• Do you want to accept my answer? – Toby Mak Jun 3 '17 at 13:12
• well my questions was slightly different, i.e. if there was another formula without having to use begin value and end value (e.g. just using the growth 5% and the periods 12), but oh well accepted – giò Jun 5 '17 at 7:01
• There is no other formula, because the start and end values must be included to find the percentage increase. – Toby Mak Jun 5 '17 at 8:18

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