CAGR and Why are Square Roots Taken?

Hi,

I am working on a couple of calucalations for a colleague that wants to arrive at a Compounded Annual Growth Rate (CAGR) for a time-series of investment data:

Total Return (adding 1 to each period's % return and then multiplying them all together to get a Total Return 1.xxxx and the final CAGR calculation which takes the nth root of the Total Return (nth root being the number of % return periods) and subtracting 1.

I understand mechanically how to do the problem, but I don't understand for Total Return what the formula means: why do I add 1 to each return and then multiply them sequentially - what does that do/mean as "Total Return"?

Then, for the CAGR calculation, why is the nth root being taken? In other words, and not only for this problem, why is the nth root or square root used - what does it ulimately do to the output of any problem?

Thanks for helping with my conceptual understanding!

Total Return / CAGR example

Here is an example I found:

hp 12c (platinum) - Calculating a Compound Annual Growth Rate : HP Calculator : Educalc.net

Start with $1,000. In year one you get a 20% return ($1,200 at year end). In year two you go up another 10% ($1,320 at end of year two), down 15% in year three ($1,122), and up 30% in year four ($1,458.60 ending amount).

Multiply the returns for each year to get the total return.

1.20 * 1.10 * 0.85 * 1.30 = 1.4586 (or 45.86%)

Now, all we need is the CAGR. For two years we took the square root. For three years, you would take the cube root. For four years, that's the... quad root or something? I just use my trusty spreadsheet to do these calculations. Spreadsheets and some calculators take roots by using the inverse of the root as an exponent, so a square root is 1/2, a cube root is 1/3, etc. In this example, that's:

1.4586^(1/4), or 1.4586^0.25, which equals 1.099.

1.099 - 1= 0.099, or 9.9%.