Efficient Frontier
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William J. Bernstein

Almost Normal:
Of Fat Tails, Mean Reversion, and Survival


If there is one belief professed by almost all successful investors, it is a faith in mean reversion. This creed can be expressed very simply: what goes up must come down, and vice versa. Or it can be put more formally: prolonged high realized returns must of necessity lower expected returns, and vice versa. And, if one is a showoff: long-period autocorrelations with lag = 1 tend to be negative.

But whether you say tomato or to-mah-to, it all means the same thing—unless you can throw money into stocks when they’re in the toilet and avoid buying into bubbles, your best chance at a successful retirement is the demise of a wealthy relative.

Although the evidence for mean reversion is weak, most financial economists believe that it exists. The lack of supportive data likely stems from the statistical nature of mean reversion; since mean reversion is a long-term phenomenon, we cannot collect enough independent observations to prove the case.

Two recent working papers speak to this issue, and their conclusions are not reassuring. The first, an offering from Philippe Jorion at U.C. Irvine, expands upon previous work with William Goetzmann on twentieth-century equity returns. Jorion probes the same database with a variety of techniques—variance ratios, loss probability, and value-at-risk—and shows that while mean reversion seemed to occur in the most successful nations, it did not occur in others, and that when the entire sample was examined, there was no evidence for it whatsoever.

Worse, nations that experienced "interruptions" of their capital markets demonstrated mean aversion, that is, in unstable nations, one terrible period was likely to be followed by another. The only good news was that a diversified global portfolio demonstrated less risk than that of any single nation.

The second paper, by Xavier Gabaix and his colleagues at MIT, explains why life in the financial markets has a fat tail and why the highly improbable seems to occur so often. Take, for example, the stock-market crash of October 19, 1987, during which the S&P 500 lost 23% in a single day. Since the daily standard deviation (SD) of this index is almost exactly 1%, it was a 23 SD event. How improbable is that? Don’t even try to calculate the value—you’ll get a migraine from all the zeros.

The reason, according to Gabaix et. al., is that security returns are not really normally distributed; they only look that way. To illustrate this phenomenon, I’ve plotted the frequency of monthly returns of the S&P 500 since 1926. The mean value is 0.974% per month, and the SD of monthly returns is 5.63%. Next, I plotted the actual frequency of returns at 1 SD intervals against a normal distribution (as it’s known in the trade, "i.i.d.," independent, identical distribution), as shown below:

Looks like a pretty good fit doesn’t it? Actually, it ain’t. It only appears that way because we’re plotting frequency on the y-axis using an arithmetic scale. This method doesn’t do a very good job of showing what’s happening at low probabilities. For example, there was one monthly return in excess of 7 SD above the mean and two between 6 SD and 7 SD—roughly a 0.1% and 0.2% incidence, respectively, whereas a normal distribution would have predicted an incidence of about 0.0000000001% and 0.0000001%, respectively. Clearly, something is wrong with the normal assumption.

The problem, and the solution, shows up clearly when we plot incidence and probability logarithmically, as shown below:

At the extremes, the probability seems to fall off along a straight line—that is, geometrically when using a semilog plot—not normally, which would yield a much steeper falloff. The authors studied this phenomena using the returns of thousands of securities at intervals as short as fifteen minutes, yielding millions of observations. Their data are breathtaking. With the author’s permission, I’ve reproduced a graph of daily returns for the U.S., Japanese, and Hong Kong Markets:

Note, first, how the distribution of returns is nearly identical across nations. Second, see that in the high probability (low SD) region, there is a "flat spot" of returns, above which they bend over linearly using a double-logarithmic plot. The slope of this plot has a power of almost exactly three—i.e. incidence falls off as the cube of increasing SD. Since there is a "flat region" at low SDs, it’s not quite as simple as a –4 SD event being one eighth as probable as a –2 SD event. But the key thing is that using their formulation, events such as the 1987 crash fall into the realm of possibility, even probability. When the authors looked at fifteen-minute returns, they observed events at the +/- 70 SD level (that’s right, seventy standard deviations).

Even more remarkable, the authors present a case for other "power laws" as well. In addition to the "cubic law" of equity returns, there is a cubic law for the incidence of the number of trades, a first power law for the number of investors as a function of size, and a "half-cubic" law relating the number of trades to volume. If you’re good at canonical math, the authors will even supply you with an impressive theoretical model that explains their findings.

What ties together the work of Jorion and Gabaix et. al. is the notion that it’s a wild world out there and, as the nice folks at Long Term Capital Management found out a few years back, you can’t always depend on mean reversion and price convergence to save your bacon.

A medical analogy will suffice. Patients with acute illnesses can wind up in one of three places in the hospital: the routine medical or surgical ward, where uncomplicated cases are managed, the intensive care unit, where the most serious cases are treated and the outcome is not so certain, and the morgue. If you confine your analysis to the ward, you would conclude that human health mean reverts—the sick tend to get well. If you visit the intensive care unit, you might not be so sure. And if you spend most of your time in the morgue, it would be obvious that human health mean averts.

The coin depicted at the head of the article, an Athenian stater from around 293 B.C., supplies a metaphor for the same process at a national level. By that date, the decline of that great ancient city-state was well advanced, and in a desperate, unsuccessful bid to hold off the Macedonian hordes at its gates, the Athenians stripped gold from the sacred statue of Athena in the Parthenon and minted it into these coins to pay for the city’s defense.

For the past seventy years, financial economists have spent most of their time "on the ward" with the healthiest cases—nations like the U.S. and the U.K. As long as these nations remain stable, their stock markets will appear to mean revert. If there is any good news here (beyond the diversification benefit demonstrated in Jorion’s current and previous work), it is that the world’s major economies and securities markets should remain healthy for the foreseeable future, and thus continue to bless us with what appears to be mean reversion.

And if they do not, and their markets "mean avert," our portfolios will be the least of our worries.

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Copyright © 2004, William J. Bernstein. All rights reserved.

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