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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