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How much has the Stock Market returned per year?

 The Stock Market (S&P 500) has returned on average 10.4% per year.  This data goes back to 1871.  This comprises of an average price appreciation of 6.2% and a historical dividend yield of 4.2%.

Distribution of Returns

Although it is tempting to use the average returns for forward assumptions, there is one big caveat: the distribution of returns has been large.  The worst year returned -38.3% while the best year returned 53.1%.

S&P 500 Distribution of Annual Returns

Percentile Range

Another useful way to use the data is to look at the 25th and 75th percentile, which represents the 25th worst return and the 75th best return.  25th = -1.3% and 75th = 20.6%.  This is called the interquartile range and provides a sense of range.  There is also the quartile deviation that is half the value of the interquartile range to provide a measure of dispersion so in this case 11.0%.  With the mean value which is the same as 50th percentile we can get a sense of the middle, so in this case 11.0%.  In this data set, it looks like historically 50% of the value are between -1.3% and 20.6% having a middle value of 11.0% and have the quartile deviation at 10%

Normal Distribution

The more popular approach is to assume the numbers are normally distributed following the central limit theorem so getting the average and standard deviation is all you need to get a sense of the range of values.  In this case, the average is 10.4% with a standard deviation of 16.8%.   If the distribution of returns in normally distributed, then 68% of the values fall within 1 standard deviation and 95% fall within 2 standard deviations.  In our data, this means 68% of the numbers are between [10.4%-16.8%, 10.4%+16.8%] or [-6.4%, 27.2%]. With 95% of the numbers between [23.2%, 44%]

 

Summary

Although historic S&P 500 returns have returned on average 10.4%, it is also useful to understand how using percentile range and normal distributions approach can be used to help quantify the potential uncertainty of outcomes.
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