# Correct way to calculate the S&P500 average CAGR of earnings over 10 years rolling periods

Let's say we have only the following data for the earnings (e) and we calculated for each 10 years period the total earnings growth (G) and its Compounded Annual Growth Rate (AG)

$$\begin{equation} \\G_n = {e_n/e_{n-1}}-1 \end{equation}$$

$$\begin{equation} \\AG_n = \sqrt{1+G_n}-1 \end{equation}$$ What's the correct way to calculate the average CAGR of the earnings over these 10 years periods?

Method 1) We could do the average of G and than calculate the CAGR of such average $$\begin{equation} \\CAGR1 = \sqrt{1+\frac{\sum{G_n}}{n}}-1 \end{equation}$$

Method 2) We could simply do the average of the AG $$\begin{equation} \\CAGR2 = \frac{\sum{AG_n}}{n} \end{equation}$$

In this simple example there is a little difference in the result (1.86% vs 1.79%), but still there is a difference in the result.

I can understand the results are different because they are algebraically different, but I can't intuitively understand how they can be different. Which is the one that really represents the correct average?

I tried to calculate the same thing using S&P500 earnings over rolling 10 years periods (since March/1957 to September/2019) and the result are very different:

If I use method (1) I get an average total earning growth over 10 years of 107,59% and its CAGR is 7.6%

If I use method (2) I get an average CAGR of 6.4%

Which one is the correct method?

If they are both correct and I have being asked by someone - "What has been the historical average annual compounded growth rate of the earnings of the S&P500 over 10 years?" - what should I answer - 7.6% or 6.4%?

• it occurs to me a better question is why you need an average of rolling 10y CAGR? there's not really a need to calculate an average given the question you asked above...the compound annual growth rate (which is itself an 'average' annual return) can be calculated directly Feb 5, 2020 at 0:18
• @Chris each 10 year period has a different CAGR of the earnings, I want to know what was the average CAGR. Feb 5, 2020 at 8:58
• I realize that, but it isn't clear why you'd need that. At a minimum, in taking an average over your rolling CAGR, you're double counting annual returns. That's a weird thing to do. Feb 5, 2020 at 19:45
• @Chris I need it to answer the question "What has been the historical average annual compounded growth rate of the earnings of the S&P500 over 10 years periods?" Feb 6, 2020 at 17:01