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A S T E R I NG U NCE R T A I NT Y / W I L D U N C E R T A IN T Y

A focus on the exception

hat prove the ruleitional risk management tools focus on what is normal and consider extreme events as ancillaries. In a wacterised by volatility and uncertainty, Benoit Mandelbrot and Nassim Taleb argue that this approach uided, and propose an alternative methodology where large deviations dominate the analysis

Conventional studies of uncer-

tainty, whether in statistics,

economics, finance or social

science, have largely stayedclose to the so-called bellcurve, a symmetrical graph

epresents a probability distribution.

o great effect to describe errors inomical measurement by the 19th-cen-

mathematician Carl Friedrich Gauss,l curve, or Gaussian model, has since

ed our business and scientific culture,rms like sigma, variance, standard

on, correlation, R-square and Sharpe

re all directly linked to it.u read a mutual fund prospectus, or a

funds exposure, the odds are that itpply you, among other information,

me quantitative summary claiming to

re risk. That measure will be basedof the above buzzwords that derive

he bell curve and its kin.measures of future uncertainty sat-

ur ingrained desire to simplify bying into one single number mattersre too rich to be described by it. In

n, they cater to psychological biasesr tendency to understate uncertainty

r to provide an illusion of understand-world.

bell curve has been presented as nor-

or almost two centuries, despite itseing obvious to any practitioner with

cal sense. Granted, it has been tink-with using such methods as comple-

ry jumps, stress testing, regime

ng or the elaborate methods knownRCH, but while they represent a good

they fail to address the bell curvesmental flaws.

problem is that measures of uncer-

using the bell curve simply disregardsibility of sharp jumps or discontinui-

d, therefore, have no meaning or conse-. Using them is like focusing on the

nd missing out on the (gigantic) trees.while the occasional and unpredicta-

ge deviations are rare, they cannot be

ed as outliers because, cumulatively,mpact in the long term is so dramatic.

traditional Gaussian way of looking atrld begins by focusing on the ordi-

and then deals with exceptions or so-

outliers as ancillaries. But there is

also a second way, which takes the so-called

exceptional as a starting point and deals with

the ordinary in a subordinate manner simply

because that ordinary is less consequential.These two models correspond to two mutu-ally exclusive types of randomness: mild or

Gaussian on the one hand, and wild, fractal

or scalable power laws on the other. Meas-urements that exhibit mild randomness are

suitable for treatment by the bell curve orGaussian models, whereas those that are

susceptible to wild randomness can only beexpressed accurately using a fractal scale.

The good news, especially for practitioners,

is that the fractal model is both intuitively

and computationally simpler than the Gaus-sian, which makes us wonder why it was not

implemented before.Let us first turn to an illustration of mild

randomness. Assume that you round up

1,000 people at random among the generalpopulation and bring them into a stadium.

Then, add the heaviest person you can thinkof to that sample. Even assuming he weighs

300kg, more than three times the average, hewill rarely represent more than a very small

fraction of the entire population (say, 0.5 per

cent). Similarly, in the car insurance busi-ness, no single accident will put a dent on a

companys annual income. These two exam-ples both follow the Law of Large Num-

bers, which implies that the average of a

random sample is likely to be close to themean of the whole population.

In a population that follows a mild type ofrandomness, one single observation, such as

a very heavy person, may seem impressive

by itself but will not disproportionatelyimpact the aggregate or total. A randomness

that disappears under averaging is trivialand harmless. You can diversify it away by

having a large sample.There are specific measurements where

the bell curve approach works very well,

such as weight, height, calories consumed,death b heart attacks or erformance of a

gambler at a casino. An individual that is a

few million miles tall is not biologically pos-

sible, but an exception of equivalent scale

cannot be ruled out with a different sort ofvariable, as we will see next.

Wild randomnessWhat is wild randomness? Simply put, it isan environment in which a single observa-

tion or a particular number can impact thetotal in a disproportionate way. The bell

curve has thin tails in the sense that largeevents are considered possible but far too

rare to be consequential. But many funda-

mental quantities follow distributions thathave fat tails namely, a higher probabil-

ity of extreme values that can have a signifi-cant impact on the total.

One can safely disregard the odds of run-

ning into someone several miles tall, orsomeone who weighs several million kilo-

grammes, but similar excessive observationscan never be ruled out in other areas of life.

Having already considered the weight of1,000 people assembled for the previous exper-iment, let us instead consider wealth. Add to

the crowd of 1,000 the wealthiest person to befound on the planet Bill Gates, the founder

of Microsoft. Assuming that his net worth isclose to $80bn, how much would he represent

of the total wealth? 99.9 per cent? Indeed, all

the others would represent no more than thevariation of his personal portfolio over the

past few seconds. For someones weight torepresent such a share, he would need to

weigh 30m kg.

Try it again with, say, book sales. Line upa collection of 1,000 authors. Then, add the

most read person alive, JK Rowling, theauthor of the Harry Potter series. With sales

of several hundred million books, she would

dwarf the remaining 1,000 authors whowould collectively have only a few hundred

thousand readers.So, while weight, height and calorie con-

sumption are Gaussian, wealth is not. Norare income, market returns, size of hedge

funds, returns in the financial markets,

number of deaths in wars or casualties interrorist attacks. Almost all man-made varia-

bles are wild. Furthermore, physi

continues to discover more and m

ples of wild uncertainty, such as

sity of earthquakes, hurricanes orEconomic life displays numerousof wild uncertainty. For example,

1920s, the German currency m

three to a dollar to 4bn to the dollayears. And veteran currency tr

remember when, as late as the 19term interest rates jumped by sev

sand per cent.We live in a world of extreme con

where the winner takes all. Co

example, how Google grabs much traffic, how Microsoft represents t

PC software sales, how 1 per centpopulation earns close to 90 times t

20 per cent or how half the capita

the market (at least 10,000 listed is concentrated in less than 100 cor

Taken together, these facts enough to demonstrate that it

called outlier and not the regulneed to model. For instance, a vnumber of days accounts for the b

stock market changes: just ten trarepresent 63 per cent of the retu

past 50 years.Let us now return to the Gaus

closer look at its tails. The

defined as a standard deviation the average, which could be arou

per cent in a stock market or 8 toheight. The probabilities of exceed

ples of sigma are obtained by

mathematical formula. Using thione finds the following values:

Probability of exceeding:0 sigmas: 1 in 2 times

1 sigmas: 1 in 6.3 times2 sigmas: 1 in 44 times

3 sigmas: 1 in 740 times4 sigmas: 1 in 32,000 times

5 sigmas: 1 in 3,500,000 times

6 sigmas: 1 in 1,000,000,000 times7 sigmas: 1 in 780,000,000,000 tim

8 sigmas: 1 in 1,600,000,000,000,09 sigmas: 1 in 8,900,000,000,000,0

times

10 sigmas: 1 in 130,000,000,000,00000 times

and, skipping a bit:

20 sigmas: 1 in 36,000,000,000,000

000,000,000,000,000,000,000,000,00

000,000,000,000,000,000,000,000,00000 times

Soon, after about 22 sigmas, o

googol, which is 1 with 100 zeroesWith measurements such as h

weight, this probability seems reas

it would require a deviation from thof more than 2m. The same cann

We live in a world of

extreme concentration

where the winner takes all.Consider how Google grabs

much of internet traffic or

how Microsoft represents

the bulk of software sales

Rob Mitchell

EDITOR Ravi Mattu

Stephen Stafford

LLUSTRATION Brett Ryder

S DEVELOPMENT MANAGER Kate Harris

NERSHIP PUBLICATIONSOne Southwark Bridge London SE1 9HL

7873 3000

R Richard Foster/masteringuncertainty

onsors of this publication, Ernst & Young had no input in the

its contributors or editorial content 1950 2006

3000

2500

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1000

500 S&P 500

S&P 500without the ten biggestone-day moves

The difference ten days makeBy removing the ten biggest one-day moves from the S&P 500over the past 50 years, we see a huge difference in returns. Andyet conventional finance treats these one-day jumps as mereanomalies.