A common argument in favour of equity investments is that, even though they go down, and go down hard, they will bounce right back up again. Effectively, the argument goes, equity returns will revert to a mean. But mean reversion is a strange concept - like the soul, there are more people saying it does or doesn’t exist than there are people saying what they think it means. My plan is to resolve the debate, once and for all; and do so in the next few hundred words.

There is, as mentioned, no single accepted definition of mean reversion^{1} ; since the issue is quite subtle, we break down the problem into two separate ideas: independence of returns and “active” mean reversion.

Independence of returns means simply that extreme results are typically followed by less extreme results. If one rolls a 5, then rolls another die, one expects a lower score. Active mean reversion is more difficult to define; we use it to mean a stronger trend to smoothing. That is, rather than assuming a set of independent data points, we assume an underlying force “pulling” data back to the mean.

To clarify what we mean by this distinction, consider a two-dimensional drunkard’s walk. At each point, the drunkard moves left or right with equal probability. Now consider successive (not rolling) 10 step intervals. If he moves all ten steps to the right, you still expect him to make half of his next ten steps left and half right - i.e. his future movements do not depend on his past movements, so his movements (“returns”) are independent. However if, instead, after each step he were twice as likely to move back to where he came from, we would say there was active mean reversion.

Armed with our set of definitions, we now look at some data - specifically, Professor Shiller’s monthly US equity data from 1870^{2}. For independence of returns, this is easy. Throughout the period, the year-on-year correlation of excess returns is observed to be 5.1%. This is equivalent to a p-value of 55% - that is, if year-on-year excess returns are independent, it is more likely than not that the results would be at least as extreme as this; so it seems that returns are independent year-on-year.

This suggests active mean reversion isn’t happening, however, it merits a deeper look. The method we chose to look at was standard deviation^{3} - the idea being that any active pull to parity should result in lower volatilities for long-term holding periods. We then weight this figure by the square root of the time held, since we’re comparing annualized returns.

However, because there is not that much data, we found the values obtained varied depending on the starting point. Particularly for the longer periods, the standard deviation of excess returns will be different if you take periods from (say) 1870-1880, 1880-1890, and so on, or if you take periods 1871-1881, 1881-1891, etc.

To compensate, we took the average over each such “run”; that is, for ten year periods, we defined run 1 as: 1870-1880, 1880-1890..., and run 2 as: 1871-1881, 1881-1891... etc. We then took the average standard deviation over all the “runs” for each time period. We did this in an attempt to make as much use of the data as possible while limiting the bias that comes from re-using data (if we compared, say, excess returns from 1870-1880 with those from 1871-1881, then we should not be surprised if they were similar).

*Source - Professor Robert Shiller; Calculations - Redington*

*there is no significant risk reduction from holding equities for longer periods.*

So we see good reasons to believe that returns are independent and very little grounds for believing that any form of active mean reversion happens

^{4}. This leads us to the seemingly perverse view that the reason equities have mean-reverted historically is precisely because they are random. Or, in plain English, our somewhat less spectacular conclusion is that recent equity returns are no guide either way to equity returns in the future.

This highlights a significant flaw in the conventional wisdom, namely that equities are safer over the long-term. The idea, called “time diversification”, is that over sufficiently long periods (typically 20-30 years), your returns will revert to a long-term mean. This is contradicted by our findings.

Moreover, without any form of active mean reversion the whole argument collapses: if your portfolio falls in value, and has active mean reversion, then you can “ride out the storm”; if returns are independent, then there is no storm to ride out, and a large loss might well be followed by another large loss. Any other view is akin to the gambler’s fallacy. In reality, without active mean reversion, holding equities for longer periods may be trading the apparent security of a seemingly lower probability of loss for the more sinister increase in the risk of much larger losses.

If holding equities for longer makes them riskier

^{5}, then how should investors think about earning a risk premium from equities? The obvious answer is by thinking in terms of risk; in particular, by thinking about risk limits over a shorter period. If you’re not prepared to take a gamble once, you shouldn’t take it 30 times; so if you’re not prepared to take more than a certain level of risk in one year (however you choose to measure your risk), then you shouldn’t take more risk every year for 30 years. If equities always bounce straight back, then time diversification makes sense; our research suggests they don’t, and that it doesn’t. Since equity returns do not seem to revert to a mean, having a disciplined risk management framework for the equity part of a portfolio is really the only viable approach - it is certainly far superior to blindly trusting the flawed logic of time diversification.

[1] “Mean Reversion”- (Exley, Mehta, Smith) gives a good outline of the difficulties with the definition

[2] From Professor Shiller’s website, http://www.econ.yale.edu/~shiller/

[3] We also looked at autocorrelation, and reached the same conclusions

[4] See also http://www.investmenteurope.net/digital_assets/6305/2013_yearbook_final_web.pdf

[5] Other studies support this conclusion, and suggest that returns are far more dependent on timing than on length of time held; for example, even over 20 year periods an investor can experience substantial real losses

See: http://www.nytimes.com/interactive/2011/01/02/business/20110102-metrics-graphic.html?ref=business&_r=1&

*Please note that all opinions expressed in this blog are the author’s own and do not constitute*

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