Used to great effect to describe errors in astronomical measurement by the 19th-century mathematician Carl Friedrich Gauss, the bell curve, or Gaussian model, has since pervaded our business and scientific culture, and terms such as sigma, variance, standard deviation, correlation, R-square and the Sharpe ratio are all directly linked to it. That measure will be based on one of the above buzz words that derive from the bell curve and its kin.

In addition, they cater to psychological biases and our tendency to understate uncertainty in order to provide an illusion of understanding the world. The problem is that measures of uncertainty using the bell curve simply disregard the possibility of sharp jumps or discontinuities and, therefore, have no meaning or consequence.

Using them is like focusing on the grass and missing out on the gigantic trees.

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The traditional Gaussian way of looking at the world begins by focusing on the ordinary, and then deals with exceptions or so-called outliers as ancillaries. These two models correspond to two mutually exclusive types of randomness: Measurements that exhibit mild randomness are suitable for treatment by the bell curve or Gaussian models, whereas those that are susceptible to wild randomness can only be expressed accurately using a fractal scale.

The good news, especially for practitioners, is that the fractal model is both intuitively and computationally simpler than the Gaussian, 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, people at random among the general population and bring them into a stadium. Then, add the heaviest person you can think of to that sample. Even assuming he weighs kg, more than three times the average, he will rarely represent more than a very small fraction of the entire population say, 0.

In a population that follows a mild type of randomness, one single observation, such as a very heavy person, may seem impressive by itself but will not disproportionately impact the aggregate or total. A randomness that disappears under averaging is trivial and 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 by heart attacks or performance of a gambler at a casino.

An individual that is a few million miles tall is not biologically possible, but an exception of equivalent scale cannot be ruled out with a different sort of variable, as we will see next.

What is wild randomness? Simply put, it is an environment in which a single observation or a particular number can impact the total in a disproportionate way. One can safely disregard the odds of running into someone several miles tall, or someone who weighs several million kilogrammes, but similar excessive observations can never be ruled out in other areas of life.

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Having already considered the weight of 1, people assembled for the previous experiment, let us instead consider wealth. Add to the crowd of 1, the wealthiest person to be found on the planet — Bill Gates, the founder of Microsoft.

Indeed, all the others would represent no more than the variation of his personal portfolio over the past few seconds. Try it again with book sales.

Line up a collection of 1, authors. Then, add the most read person alive, JK Rowling, the author of the Harry Potter series. With sales of several hundred million books, she would dwarf the remaining 1, authors who would collectively have only a few hundred thousand readers.

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So, while weight, height and calorie consumption are Gaussian, wealth is not. Nor are income, market returns, size of hedge funds, returns in the financial markets, number of deaths in wars or casualties in terrorist attacks. Almost all man-made variables are wild. Furthermore, physical science continues to discover more and more examples of wild uncertainty, such as the intensity of earthquakes, hurricanes or tsunamis.

Economic life displays numerous examples of wild uncertainty. For example, during the s, the German currency moved from three to a dollar to 4bn to the dollar in a few years. And veteran currency traders still remember when, as late as the s, short-term interest rates jumped by several thousand per cent.

We live in a world of extreme concentration where the winner takes all. Consider, for example, how Google grabs much of internet traffic, how Microsoft represents the bulk of PC software sales, how 1 per cent of the US population earns close to 90 times the bottom 20 per cent or how half the capitalisation of the market at least 10, listed companies is concentrated in less than corporations. For instance, a very small number of days accounts for the bulk of the stock market changes: Let us now return to the Gaussian for a closer look at its tails.

The probabilities of exceeding multiples of sigma are obtained by a complex mathematical formula. Using this formula, one finds the following values:. With measurements such as height and weight, this remote probability makes sense, as it would require a deviation from the average of more than 2m. The same cannot be said of variables such as financial markets. For example, a level described as a 22 sigma has been exceeded with the stock market crashes of or the interest rate moves of The key here is to note how the frequencies 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:.

If you look at the ratio of the moves, you will notice that this ratio is invariant with respect to scale. It assumes that whenever one has seen in the past a large move of, say, 10 per cent, one can conclude that a fluctuation of this magnitude would be the worst one can expect for the future. This method forgets that crashes happen without antecedents.

Before the crash of , stress testing would not have allowed for a 22 per cent move. Using a fractal method, it is easy to extrapolate multiple projected scenarios. Using this model, a —15 per cent move would happen every 16 years, and so forth. This will give you a much clearer idea of your risks by expressing them as a series of possibilities. You can also change the alpha to generate additional scenarios — lowering it means increasing the probabilities of large deviations and increasing it means reducing them.

What would such a method reveal? It can also show how some portfolios can benefit inordinately from wild uncertainty.

Despite the shortcomings of the bell curve, reliance on it is accelerating, and widening the gap between reality and standard tools of measurement.

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The consensus seems to be that any number is better than no number — even if it is wrong. We live in a world primarily driven by random jumps, and tools designed for random walks address the wrong problem. While scalable laws do not yet yield precise recipes, they have become an alternative way to view the world, and a methodology where large deviation and stressful events dominate the analysis instead of the other way around.

We do not know of a more robust manner for decision-making in an uncertain world. Benoit Mandelbrot is Sterling professor emeritus of mathematical sciences at Yale University. Copyright The Financial Times Limited You may share using our article tools.

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Print a single copy of this article for personal use. Contact us if you wish to print more to distribute to others. Privacy policy Terms Copyright. Register Subscribe Sign in. Subscribe You are signed in. Wild randomness What is wild randomness? Using this formula, one finds the following values: We find that the odds of encountering a millionaire in Europe are as follows: Richer than 1 million: Quick links FT Live How to spend it The FT Property Listings Social Media hub The Banker The Banker Database Global Risk Regulator fDi Intelligence fDi Markets fDi Benchmark Professional Wealth Management This is Africa Investors Chronicle MandateWire FTChinese.