Mean Reversion  Can It Be Proved?
IntroductionMean reversion refers to stock market "memory" as opposed to randomness of returns. In other words, the random market hypothesis basically says that forward stock returns are independent of what has preceded them. This would mean that a run of higher than average returns would be just as likely to be followed by more years of higher returns as it would lower returns and vice versa. Mean reversion in simple terms means that a run of high returns is more likely to be followed by lower lower returns and vice versa. It turns out that while most knowledgeable stock market experts believe that mean reversion exists it is very difficult to prove statistically. The studies that I'm aware of that have looked at this had usually used the S&P 500 or a total market index for analysis. Intuitively, the concept of mean reversion has much appeal. After all, over the very longterm stock prices should at least loosely track earnings and dividends. If for example earnings and dividends are increasing 10% per year but the market has been returning 20% per year it should be due for a fall. The random walk hypothesis if strictly followed would have us believe that in such a case the market would not necessarily be due for a correction because it is so efficient that the price at any given time reflects proper valuation, given all that is known at the moment. So how does one quantify mean reversion? One way is to look at the standard deviation of returns. This is discussed on the Enhanced Monte Carlo Simulation  Part I page. The problem with this is that it is difficult to apply statistical analysis to the results. The generally preferred method is to perform autocorrelations between adjacent time periods using historical data. Using a fiveyear timeframe as an example, if mean reversion exists one would expect that the five years following a fiveyear period of outperformance would be followed by underperformance in the next timeframe. Thus, if high returns are followed by low returns than they are negatively correlated. On the other hand, if the opposite occurs then one would have a positive momentum effect. The problem with statistical analysis in this setting is the relative paucity of data. We have only a few decades of reliable domestic market data dating back to about 1926. If one were looking at fiveyear autocorrelations then generally he or she would take, say, the 1926 to 1930 period and compare it to the 1931 to 1935 period, then 19271931 compared to 19321936, 19281932 compared to 19331937, etc. This would be then carried out until one runs out of data. This would amount to the construction of something like 71 rolling fiveyear period correlations if taken to the end of year 2002. The 71 data pairs would then be analyzed to generate a correlation coefficient for the rolling fiveyear periods. This seems like a fair amount of data but it really isn't because of the considerable overlap involved. The data overlap means we cannot use conventional descriptive statistics to analyze the results. So what we do? I believe that the way to attack the problem is through simulation techniques. For my simulation experiment I decided not to limit myself to the Standard & Poor's 500 index as most others have done. Instead, I chose to look at all subsets within the market since some groups of stocks behave quite differently from others  for example large cap versus microcap and value versus growth. Fortunately, there is an outstanding source of data with which to study. This comes from the work of researchers Eugene Fama and Kenneth French. They have taken data since 1926 and divided the market and divided the market into segments based on market cap and book value. They have several sets of data and I chose the 25 Portfolios Formed on Size and BooktoMarket (5 x 5) to use for my work. This data set includes five different market cap quintiles which are each subdivided into five booktomarket quintiles for a total of 25 portfolios. These range from tiny value stocks to megacap growth stocks. I chose to use real (inflationadjusted) returns in order to negate the nonmean reverting random effects introduced by inflation. What I did then was take each of the 25 portfolios and run a separate analysis. First, autocorrelations were performed for rolling 2 to 20year time periods (19 separate autocorrelations). I chose this time frame because some previous studies have shown that mean reversion is most likely to occur at four or fiveyear intervals but I wanted to be sure I included plenty of data on either side. These autocorrelations were plotted on a graph. Then 10,000 simulations were run with the same yearly returns for the index but shuffled in random order so that the entire sequence would have no "memory", thus having no predilection for mean reversion or momentum. For each shuffled data series autocorrelations were performed for each 2 to 20 year rolling periods, the same as was done for the actual historical sequence. The autocorrelations were then sorted and each data point from the actual historical series was evaluated relative to the 10,000 simulated autocorrelations. It was then decided to construct upper and lower boundaries on a graph that would include all 19 data points from each simulated data series approximately 95.5% of the time. I chose this number since it would include the median autocorrelation plus or minus two standard deviations which is a typical test for statistical significance (p < 0.045). It follows that of the 10,000 simulated runs about 450 would have at least one data point falling outside one of the two boundaries. It turns out that the probability of any single point falling out is about 0.23% which is very close to what would be predicted from binomial probability tables (i.e. 0.23% probability X 19 trials = cumulative 4.5%). FindingsPlease see the following graph using the Standard & Poor's 500 index data:
Each of the 19 red dots represents the actual correlation coefficient for rolling 2 to 20 year periods (19272002). As you can see, each dot falls within the upper and lower boundaries which implies a lack of statistically significant mean reversion. This pretty much bears out what other researchers have found in that there is no real statistical evidence for mean reversion for this particular index. So what happens if we look at the 25 Fama and French portfolios? Here is a summary of what I found: Table 1
Table 1 shows the results for each of the 25 portfolios. The top row refers to size quintiles with 1Q being the smallest and 5Q being the largest capitalization groups. The labels in the lefthand column refer to booktomarket quintiles. Note that high booktomarket ratios are equivalent to low pricetobook ratios. The squares that are green and contain numbers are those which had a least one rolling period that fell below the lower boundary for statistical significance. The numbers are the length of the rolling periods which were outside the boundary. It appears that mean reversion was most likely to occur at around the year four and occasionally year five which is pretty much in line with what others have found when studying different markets. Each of the 25 graphs is available here. Note that on these 25 graphs any data point (correlation coefficient) that fell outside the boundary lines is colored green while the rest are colored red. ObservationsLooking at table 1 we can see that there are eight green squares out of of 25 total. This means that slightly less than one third of the total show what I would consider to be statistically significant mean reversion at the four and/or fiveyear intervals. If these indexes were composed of random numbers than we would expect one, or at most two or three, green squares and they would not be congregated around years 4 and 5. Eight green squares would be an almost one in one million occurrence if the numbers had no memory. However, we must be careful here since all 25 portfolios have varying degrees of positive correlation with each other. This would not happen with random numbers. That said, however, I believe that the evidence is quite clear that there is statistically significant mean reversion, at least with regard to some of the portfolios. If you look closely at the 25 Mean Reversion Charts you will see that many of the portfolios showed negative autocorrelations which were very nearly significant around year four. In fact, all 25 showed negative autocorrelations below the median value at years three and four. This would be highly unlikely in the absence of some element of mean reversion. Note also from table 1 that the best evidence of mean reversion comes from the value weighted portfolios in the small and midcap sectors. It is the larger growth oriented portfolios that show less evidence of mean reversion. One can only speculate why. My feeling is that value stocks tend to adhere closer to underlying fundamentals which is the essence of mean reversion tendency. It is the growth stocks that tend to become disconnected with reality for months or years as was witnessed during the 1990s largecap growth bubble. In fact, if you look at some of the charts from the larger size quintiles you'll notice that there are positive autocorrelations at various times within many of them. I believe that this may be the result of momentum investing but this is only a guess. If my supposition is true that would seem that value stocks would be more suited to contrarian strategies and growth stocks would be more suited to momentum strategies since the former depends on mean reversion and the latter upon positive rather than negative autocorrelations. Conclusions
Last edited: 04/21/2005

