Beware
theLong
Tail
Economic modelsof risk don’t add up,
cadre of researchers
caution By Rachel Ehrenberg

22 | SCIENCE NEWS | November 5, 2011

econophysics.indd 22

Stanley had been exploring the notionthat extreme economic events, the bubbles and crashes of financial markets,might be described by a mathematicallaw — a tidy law, like acceleration due
to gravity. And he isn’t the only outsider
who has had an eye on the markets. Scientists from a range of fields have beenporing over financial data, finding somecurious patterns in the process.
These patterns suggest that standard
economic models based on the notionof equilibrium — markets will fluctuate
but then settle down like the surface ofa still pond — may not capture the wholestory. Freak events may be a normal partof long-term economic behavior. If that’strue, then the mathematical methods
guiding Wall Street’s estimation of risk
are seriously flawed, offering a dangerous false sense of security.
“You have to understand that the bad
events can be really, really bad,” says

J. Doyne Farmer, who is trained in physics and does research spanning several
disciplines at the Santa Fe Institute inNew Mexico. “And there’s a significant
chance that over a five-year period wewill get hit by a really big event. That’s
where the rubber really hits the road.”

Gaussian-colored glasses
Discounting extreme events as improbable is a long-held tradition in economics,
notes Stanley. Many mathematical models assume that financial data, such aschanges in the price of a stock, fit what isknown as a Gaussian, or normal, distribution — the good old bell curve. Most datacluster around an average. Move to eitherside of the average, and the data points
become increasingly scarce, tapering off
in a predictable way. A blizzard in July or
the Dow Jones dropping 20 percent in
one day are considered so rare that theymight as well be impossible.

ILLUSTRATION: JANEL KILEY, BACKGROUND: MALIKETH/ISTOCKPHOTOW

hen H. Eugene Stanley
heard that Lehman
Brothers had filed forbankruptcy, a small partof him was thrilled.
Of course, the news was distressing.
The firm’s seismic collapse had disastrous consequences, not only for theglobal economy but also for Stanley’s
daughter-in-law, who became instantly
unemployed. But Lehman’s downfall
was exactly the kind of rare event that
Stanley, a physicist at Boston University,
had been expecting.
“Many economists will tell you thatthe chances of something really big andbad happening are really, really small,”
Stanley says. But when viewed through adifferent lens, he contends, catastrophic
events — such as Lehman filing for bankruptcy in 2008 — aren’t exceptional butinevitable.
At the time of Lehman’s collapse,

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The Gaussian bell’s roots in financego back to work by French mathematician Louis Bachelier, who modeledchanges in share prices in the early 1900s.
Bachelier recognized that some of his
model’s assumptions were flawed, including the premise that the probability ofextreme events is vanishingly small (hereportedly called such events “contaminators”). Yet these assumptions werepreserved in later models, including theBlack-Scholes formula, which underlies
much of Wall Street’s estimation of risk.
In some respects, the long reign of thisGaussian approach isn’t that surprising. Many things measured in the realworld fit the Gaussian mold, says Mark
Newman of the Center for the Study ofComplex Systems at the University ofMichigan in Ann Arbor. Take the height ofadult American males: It generally hovers
around 6 feet, or about 180 centimeters.
Plot the number of men with heights
lower and higher, and the data points oneither side taper off quickly. “You don’t
get a mile-high human,” Newman says.
With truly Gaussian distributions,
measurements that appear extraordinary, such as a person a mile tall, areprobably flukes; perhaps the measurer
didn’t know how to use a ruler or madea mistake in writing down the number.
Termed “outliers,” these data points areoften thrown out of the analysis.
But when it comes to financial data, agrowing body of research suggests thatoutliers can be more like babies thanbathwater. Such events may still be very
rare; Stanley says that the probability
that stocks would crash as they did on
Black Monday in 1987 was “as close as
you can come to never.” Yet Black Monday still happened. And while much offinance does behave within the boundsof a normal distribution, ignoring therare, large events doesn’t capture reality.
When rare extremes are included in
the picture — if there really are a handful of mile-tall men — then one of the bell
curve’s sloping sides doesn’t come tidily
to a close. It splays out in what’s known
as a long (or fat or heavy) tail. Researchers
analyzing financial data are finding thesetails over and over and over again.
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econophysics.indd 23

In the 1990s — when Stanley coined
the term “econophysics” to describe suchresearch — two of Stanley’s graduate students spotted a signature long tail in U.S.
market data. The team analyzed everytransaction for 1,000 stocks in the major
markets, looking at how much the pricesof those stocks changed and how often.
The more than 200 million data points
included a handful of extremes, causing
the graph to splay outward.
Instead of dismissing such tails because
they don’t fit the models, researchers
might need to rework the models because
they don’t fit the data, Stanley and others argue. “The model should really be
driven by the data,” he says. “For a physicist, there are no outliers. If I saw a glassof water float up in the air, we’d have tore-examine the law of gravity.”
The late mathematician Benoît
Mandelbrot, father of fractals, made asimilar observation in the 1960s after
examining variation in cotton prices. He
later called the Gaussian distribution “amodel child,” one “which is commonly
called ‘normal,’ but in fact deserves lessand less to be considered as such.”

Power upLong tails are a mathematical clue that adifferent kind of behavior may be at play,
one that physicists have long been fascinated by. When data follow what is calleda power law distribution, the outlandish
data points that generate the tail aren’t
aberrant freaks; they fit right in.
A commonly cited power law, often
referred to as Zipf’s law, represents populations of towns and cities. While themajority of places people live are fair- tomiddling-sized, a handful, such as New
York City, have populations that are crazylarge, so large that they are hard to fit
with all the other towns and cities on a
typical graph. To better visualize such
distributions, researchers can analyze
their data logarithmically — a mathematical sleight of hand that compresses thedistance between the numbers. When apower law is at play, the plotted logarithms of the data don’t skew awkwardly
but fall on an elegant straight line.
A classic case of the power law in economics (although it has recently beendisputed) is the distribution of wealth,described by Italian Vilfredo Pareto in
1896. The wealth of the richest people isorders of magnitude greater than that ofthe ordinary. Forbes put Bill Gates’ net
worth as of September at $59 billion; try
fitting that on a traditional chart witheverybody else’s.
A new analysis of transactions on theS&P 500 share index and the German
DAX Future stock market also uncovered power laws at work. As prices start
to rise or fall, larger and larger chunks ofstock are sold with greater and greater
frequency, Stanley, Tobias Preis ofBoston University and Artemis Capital Asset Management, and Johannes
Schneider of Johannes Gutenberg University Mainz in Germany reported inMay in the Proceedings of the National
Academy of Sciences. The trade volume
and the time between successive trades
both exhibit power law behavior.
The researchers also observed apeculiar trait of power laws: They are

Spotting a tail Some models assume that financial data follow a typical bell curve (black), withdata points clustered around an average and then tapering off quickly to either side. But a handfulof extreme events can cause the curve to splay out in what is known as a long or fat tail (red).

November 5, 2011 | SCIENCE NEWS | 23

10/19/11 12:00 PMFEATURE | BEWARE THE LONG TAIL

24 | SCIENCE NEWS | November 5, 2011

econophysics.indd 24

A different distribution
When displayed on a typical graph, data that follow a power law distribution forma long tail (below, left graph). But graph the same data on a log scale and they fall
onto a straight line (right). By applying several statistical tests, a recent analysis
identified data sets that probably show power law behavior (bottom).
Typical graph.003
.002
.001
002x105
4x105
Magnitude

Log scale

10-2
Percent of sample.004Percent of sample

10-3
10-4
10-5
10-6
10-7
10-8
104

105
106
Magnitude

107

Power law behavior

The number ofsightings of birdsof different species in the North
American Breeding Bird Survey
in 2003

The intensityof wars 1816–
1980, measured
as number ofbattle deaths
and adjusted for
population size

Frequency ofoccurrence of
unique words in
Herman Melville’s
Moby Dick

The number ofadherents toreligious bodies
and sects, as
published onadherents.com

SOURCES: M. NEWMAN/ARXIV.ORG 2006; A. CLAUSET, C.R. SHALIZI AND M. NEWMAN/ARXIV.ORG 2009crash.” His team’s simulations suggestthat adding leverage to a market tips
the distribution of price changes froma Gaussian to a power law distribution.
And when banks cut off many borrowers to control risk, the situation can getworse, the team reported in 2009 in aSanta Fe Institute working paper.
Power laws in one area of the economy may lead to others, says economist
Xavier Gabaix of New York University’s
Stern School of Business. The power law
distribution of CEO pay may arise fromthe interplay between a phenomenon
known as the economics of superstars
and a power law that exists for firm size,
he and colleague Augustin Landier, whois now at the Toulouse School of Economics in France, reported in 2008 in
the Quarterly Journal of Economics. Even
though there are only slight differences
in talent within the cream of the crop,

firms want the best CEO. Competition
among large firms for really good CEOs
can lead to huge differences in income,
especially when some firms are supersized, the researchers contend.
Another consequence of firm size’s
power law is that major events, such as
an employee strike or the invention ofthe smartphone, don’t dissipate gently
through the market. When superlarge
firms, such as Nokia, have huge successes
or failures, these events can steamroll
their way across an entire economy. Infact, such idiosyncratic shocks to the
100 largest U.S. firms may account for
about one-third of the volatility seenin the whole market, Gabaix reported
in May in Econometrica.

An eye on outliersSome researchers argue that understanding the whole economic picture

BIRD: KENCANNING/ISTOCKPHOTO; WAR: ARCHIVES.GOV/STILL PICTURE BRANCH (NNSP); HANDS: BRADLEYM/ISTOCKPHOTO“scale-free.” Say, for example, thatcomputer files 2 kilobytes in size are
one-fourth as common as files of 1 kilobyte. Under a power law regime, file sizesof 2 megabytes would be one-fourth
as common as those of 1 megabyte, andso on. Like Mandelbrot’s fractal geometry of a coastline or cauliflower whorl,
whether you view from afar or zoom
in with a microscope, the proportions
remain the same.
These data suggest to Stanley that
global financial crashes and the bubbles
that precede them aren’t outliers. The
same mechanisms that cause the smaller
blips occurring in markets daily may
also be generating bigger crashes.
Knowing that such extreme events
will happen doesn’t mean researchers can predict when, Stanley says. But
acknowledging power law behavior may
help investors and regulators pin theright number on risk. Having a powerlaw distribution changes how often
you’d expect to see an event sitting far
from the data’s average, a distance measured in “standard deviations.” With aGaussian model, an event that’s 100
standard deviations out — so far out
it’s considered impossible — has aprobability of about 1 in 10350. With apower law distribution, that likelihood
shoots up to 1 in 108, Stanley notes.
Farmer, who made a small fortune
working in the financial sector throughout the 1990s, says knowing how often big
events may hit is crucial for estimating
risk: “You have to understand your tail.”
Recent work by Farmer, Stefan Thurnerof the Medical University of Vienna andYale economist John Geanakoplos suggests that some investment strategies
can actually create a power law long tail.
Say you see an underpriced stock. You
buy it, which normally would push theprice up a bit. But if you’re using leverage(borrowed money) to try to amplify yourreturns, and then the bank cuts you off,you might be forced to sell prematurely orsell off other assets. This selling can push
prices down and then other outfits maysell too, because they see the price sliding.
“Heavy-tailed events can be caused byleverage,” Farmer says. “It can create awww.sciencenews.org

10/19/11 1:26 PM1 in 10

1 in 1001 in 1,000

1 in 10,000

Know the risk A change in stock price expected

in 1 of 100 trading days has a similar size under apower law regime (red circles) and a Gaussian distribution (shaded circles). But for a 1-in-10,000 event, thepower law predicts a much bigger price change.

will mean incorporating more than justpower laws. Occasionally crashes are solarge that they are outliers even fromthe power law distribution, says Didier
Sornette of ETH Zurich. Their specialness makes them predictable, says
Sornette, who calls the standout events“dragon-kings.”“Dragons are not like the ordinary animals you meet in the zoo,” Sornette says.
“They require new mechanisms, newbiology to explain them.” The “kings”
part of the name refers to the fortunesof royal families, which after centuries
of concentrating wealth have so muchthat they no longer even fit in Pareto’s
distribution of wealth.
“We believe that there are events thatare in a class of their own,” Sornette says.
“While the power law distribution is agood characterization of the distribution of returns, it actually misses theelephant in the room, the dragon-king.”
Sornette and his colleagues argue
that understanding dragon-kings mayhelp economists spot markets teetering

toward crashes. Out-of-control growth
can be one sign of an approachingdragon-king. When herding behavior
among investors amps up, a stock’s orindex’s growth rate can increase fasterthan exponentially, leading to moreherding, Sornette says. This positive
feedback among investors, the same
sort of feedback that concentrates thewealth of kings, brings the system to atipping point. About two-thirds of thetime, a crash results, Sornette wrote ina 2009 paper online at arXiv.org.
Working out of the Financial Crisis
Observatory at ETH Zurich, Sornette
and his colleagues are now trying to use
this aggressive growth as a signature
to identify crashes before they happen(often encrypting the data so as not toinfluence the markets).
The researchers seem to be on tosomething. While other market watchers remained enthusiastic about theoutlandish growth of the Shanghai Stock
Exchange Composite Index into the summer of 2009, Sornette and his colleagues

Event prediction A research team in Zurich was able to successfully predict two downturns
in a Chinese market before they happened. Gray lines in the graph below show the expected
range of dates predicted for the downturn; red lines show when the predictions were announced.

Shanghai Stock Exchange
Composite Index

BOTTOM: Z.-Q. JIANG ET AL/JOURNAL OF ECONOMIC BEHAVIOR & ORGANIZATION 2010

SOURCE: J. FARMER AND J. GEANAKOPLOS/POWER LAWS IN ECONOMICS AND ELSEWHERE 2008

6.0 E34.0 E32.0 E32005

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econophysics.indd 25

announced on July 10 that a downturn was coming. They predicted thatthe bubble burst would begin between
July 17 and 27. It popped on July 29.
Though great strides are being madein understanding outliers, how to reconcile the newfound importance ofseemingly freak events with traditionalmodels based on stability and equilibrium isn’t yet clear.
Many economists agree that current
models grossly oversimplify things:“Almost no economists think that theGaussian is a very good approximationof reality,” Gabaix says.
But power law math is much messierthan Gaussian math. Even figuring out
where a power law distribution begins
can be tough. Pareto’s classic case ofincome probably follows a power law
only in its tail, for example, with thewealth of the majority of the population
based on labor for pay.
To keep things simple, models leaveout a lot, Gabaix says. The key, and avery difficult thing, is making sure thatthe most important ingredients areincluded. “Power laws,” he says, “areone of those intriguing facts that force
people to write new theories that hopefully will explain them.” s

Explore more2006

2007

Year2008

2009

2010

s For Sornette et al’s prediction papers:
www.er.ethz.ch/fco
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