Averaging Rolling Earnings Growth Rates and Annualized CAGRs
Summary
The document compares two ways to summarize earnings growth across overlapping ten-year periods. One approach averages total growth across periods and then converts that average into an annualized compounded rate. The other calculates each period’s annualized rate first and then takes the arithmetic mean of those rates. The two procedures answer different statistical questions, so they need not produce the same result.
Using historical S&P 500 earnings data, the author reports substantially different figures from the two calculations and asks which should describe historical average annual compounded growth. The document does not give a resolution or discuss the effects of overlapping windows, changing earnings bases, or the distribution of period returns. The appropriate summary depends on the intended interpretation: an annualized rate derived from mean total growth versus the mean of period-specific annualized rates. Those quantities should be labeled clearly rather than treated as interchangeable.
Key ideas
- Averaging total growth before annualizing differs from averaging each period’s annualized growth rate.
- The two methods summarize different properties of the rolling periods.
- The author’s historical S&P 500 calculation produces different averages under the two methods.
- The intended interpretation should determine which measure is reported, with its calculation stated explicitly.
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Full text
# Correct way to calculate the S&P500 average CAGR of earnings over 10 years rolling periods
# 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[10]{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[10]{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%?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.