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MASTERING UNCERTAINTY
/WILD UNCERTAINTY 3
of variables such as financial markets. For
example, a level desribed as a 22 sigma has
been exceeded with the stock market crashes
of 1987 and the interest rate moves of 1992.
The key here is to note how the frequen
cies in the preceding list drop very rapidly,
in an accelerating way. The ratio is not
invariant with respect to scale.
Let us now look more closely at a fractal,
or scalable, distribution using the example of
wealth. We find that the odds of encountering
a millionaire in Europe are as follows:
Richer than 1 million: 1 in 62.5
Richer than 2 million: 1 in 250
Richer than 4 million: 1 in 1,000
Richer than 8 million: 1 in 4,000
Richer than 16 million: 1 in 16,000
Richer than 32 million: 1 in 64,000
Richer than 320 million: 1 in 6,400,000
This is simply a fractal law with a “tail
exponent”, or “alpha”, of two, which means
that when the number is doubled, the inci
dence goes down by the square of that
number – in this case four. If you look at the
ratio of the moves, you will notice that this
ratio is invariant with respect to scale.
If the “alpha” were one, the incidence
doubled. This would produce a “flatter” dis
tribution (fatter tails), whereby a greater
contribution to the total comes from the low
probability events.
Richer than 1 million: 1 in 62.5
Richer than 2 million: 1 in 125
Richer than 4 million: 1 in 250
Richer than 8 million: 1 in 500
Richer than 16 million: 1 in 1,000
We have used the example of wealth here,
but the same “fractal” scale can be used for
stock market returns and many other varia
bles. Indeed, this fractal approach can prove to
be an extremely robust method to identify a
portfolio’s vulnerability to severe risks. Tradi
tional “stress testing” is usually done by select
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 Spring '09
 Taleb,NassimN

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