If a company's shipments grew 100% from 100 to 200 and then in the next period fell 50% back 100, then the average annual growth rate would be 25%. But this growth rate is incorrect as you're back at your principal. A compounded annual growth rate calculation would yield the correct 0% annual growth rate.
Why does AAGR overestimate while CAGR yields the correct growth rate? I understand that compounding may play a role but I can't quite wrap my head as to how.
On
Yes, this is because of compounding and the fact that the first year’s percentage and the second year’s percentage are percentages of different things.
This is an example of the arithmetic-geometric mean inequality (https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means) which is a famous mathematical theorem with many proofs and many applications.
The arithmetic mean is when you take $n$ numbers, add them up, and divide by $n$. In your case $2.00$ (first year) and $0.50$ (second year) give you $1.25$.
The geometric mean is when you take $n$ numbers, multiply them together, and take the $n$th root. In your case this gives $1.00$.
The arithmetic mean is always greater than or equal to the geometric mean, and it is equal only if all the $n$ numbers in your list are identical.
You can't average growth rates because the growth is in terms of the principal. The doubling in the first year started with a small number while in the second year, you are halving a larger number. Since each year has a different principal, the actual increases and decreases are different than the percentage changes.
In your calculation, $100\%$ growth in year one means that after year one, your principal is multiplied by $(\text{original }100\% +\text{growth }100\%)=2$. Therefore, if $P$ is your original shipment amount, then after one year, your shipment amount is $2P$.
Then, the loss of $50\%$ in the second year means that after year two, your principal is multiplied by $(\text{original }100\% - \text{loss }50\%)=\frac{1}{2}$. The principal at the beginning of year two is $2P$, so the results after year $2$ is $P$.
The main difference is that the second time, your principal has changed. You aren't computing from the first principal, but the principal after the growth of the first year. This makes a $50\%$ reduction in the second year the same as a $100\%$ increase in the first year since the first year growth corresponds to doubling output while the second year loss corresponds to halving the output.