The compound annual growth rate over a number of periods is the single growth rate that, if applied year-to-year, results in the absolute growth from the value in the first to the last period. It is a way to see the underlying overall growth from year to year that may be obscured by the high growth and low growth years.
Why can’t you take the average of each of the year-to-year growth rates? You can get this without even knowing the absolute numbers (e.g., sales in a given year). This will get you close to compound annual growth rate, but without connecting the growth rate to the actual absolute growth, the growth rate does not get properly weighted by the value of that growth. Compound growth rates must be calculated from the actual values, or they will not reflect the extent to which the growth from one year to the next is compounded by the actual values.
The compound growth rate reflects the singular growth rate which, if applied every year, will result in the growth from the first to the final year. The average growth rate will not do this. See the sample data below:
To the extent that there is uneven growth (in the example the growth rates that vary considerably from year-to-year), the average annual growth rate and the compound annual growth rate (CAGR) will differ. If there is even growth, as in the special case when the growth rate per year is exactly the same every year (say, 12.86%), then the CAGR will be exactly the same as the average growth rate.
Please note that the compound annual growth rate imputes but does not reveal the variation from year to year in growth rates, only the impact from the first to the last year in the forecast.
So, here is the actual calculation of compound annual growth from Year 1 (“y1”) to Year X (“yx”), where y1 is the first year and yx is the final year:
Let’s break it down: If you wanted to simply know the growth from y1 to yx (let’s say 2014 to 2024), then you simply take the sales in year x and divide it by the sales in year 1. That would be shown as (Salesy1/Salesyx)-1. So, if sales in year 1 are $108.1 and year x (in this case year 10) are $357.0, then the absolute growth from y1 to yx is (357.0/108.1)-1, or 2.3, written as a percent as 230%, meaning the value in yx is 2.3 times higher than the value in y1. To break down this absolute growth to the compounded growth, you raise the growth rate to the inverse of the number of years to be compounded, or the growth raised to the power of 1/10.
Again, the compound growth rate is the rate that, if applied each year in the forecast period, will reflect the value in the final year. As a test that this is indeed the actual compound growth rate (shown in green), we apply this growth rate of 12.69% each year and derive the same value in year 10 that was the result of all the variable growth.
If you instead applied the simple average growth rate of 12.86% year-to-year (calculated earlier), the result would be 362.6, which is 5.6 too high. It’s a rough estimate, but not really close enough when the numbers being calculated are in millions or billions.