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Regression to the mean Michel Pireu     | Business Day Wednesday, April 10, 2019
From Peter Bernstein in Against the Gods: The Remarkable story of Risk

Mathematics tells us that the variance – a measure of how observations tend to distribute themselves around their average level – of a series of random numbers should increase precisely as the length of the series grows.

Observations over three-year periods should show triple the variance of observations over one year, and observations over a decade should show ten times the variance of annual observations. If, on the other hand, the numbers are not random, because regression to the mean is at work, the ratio of the change variance to the time period will be less than one.

Baylor University Professors, William Reichenstein and Dovalee Dorsett, studied the S&P 500 from 1926 to 1993 and found that the variance of three-year returns was only 2.7 times the variance of annual returns; the variance of eight-year returns was only 5.6 times the variance of annual returns. When they assembled realistic portfolios containing a mixture of stocks and bonds, the ratios of variance to time period were even smaller than for portfolios consisting only of stocks.

Clearly, long run volatility in the stock market is less than it would be if the extremes had any chance of taking over.

This finding has profound implications for long-term investors, because it means the uncertainty about rates of return over the long run is much smaller than in the short run.

Reichenstein and Dorsett’s principal findings: For a one-year holding period, there is a five percent chance that investors in the stock market will lose at least 25% of their money, and a five percent chance that they will make more than 40%. Over thirty years, there is a five percent chance that they will make less than 20%, and a five percent chance that they could end up over fifty times richer.
 
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