Before the onset of the current liquidity crisis, Benoit Mandelbrot, who died in October, was probably most famous for fractal geometry. His name will forever be associated with the colourful posters based on the Mandelbrot set, although the mathematics behind the artwork will be of little interest to non-mathematicians.

However, for many years he warned that the principles underlying these psychedelic posters also provided a better model for market returns than the theories based on the normal distribution. In particular, he believed that investors basing their decisions on the normal distribution did not understand the magnitude of the risks they were running.

These ideas were set out in Mandelbrot’s book *The (Mis)Behaviour of Markets*, reviewed in the October 2009 edition of *The Actuary*. However, although this book was reviewed by this magazine in the middle of the current financial crisis, it had been published some time earlier in 2005, by which time Mandelbrot had been expressing his reservations with modern finance theory for nearly half a century.

At the heart of Mandelbrot’s argument was his view that returns were fractal in nature. This means that the broad structure of returns looks similar no matter what timescale is considered. The patterns seen in returns over a few minutes of trading could appear identical to those witnessed over many years, suggesting that if prices can dip sharply for an afternoon, they can do the same for a decade. This gives mathematical form to Keynes’ statement that markets can remain irrational longer than you can remain solvent.

The distribution of fractal returns follows a power law — the probability of observing a return of 2x is 1/2^n times the probability of observing a return of x. Such ‘power law’ distributions have much fatter tails than the normal distribution, the level of fatness increasing with *n*. This means that extreme positive or negative returns are much more likely under the assumption of fractal returns than under the normal distribution.

Mandelbrot’s fractal model appears to fit observed market behaviour well, certainly better than modern portfolio theory. Attempts have been made to bring models based on the normal distribution into line with observations. However, Mandelbrot regarded many of the solutions — such as generalised autoregressive conditional heteroscedasticity (GARCH) models for varying volatility — as attempts to shore up a framework that was fundamentally unsound. His view was that fractals offer a simpler model that explains reality better.

So why are fractal models not used more widely? Well, for one thing they are not well suited to indicate whether one strategy might be better or worse than another given a particular set of liabilities. It can also be difficult to calibrate fractal models when particular risk measures are required for statutory purposes. Be that as it may, even if fractal models are not used, accepting that the fractal view of the world is even partly valid suggests that certain steps should be taken.

The level of uncertainty implied by fractal models means that it is important to avoid trying to describe risk in terms of a single number such as the standard deviation. Taken in isolation, such measures can give the misleading impression that the true underlying level of risk is known; it is, in fact, unknowable. This is not to say that such measures are useless — although the standard deviation is a poor measure of the frequency or size of extreme events — but it is important to consider a range of information. This information should then be used to inform decisions rather than being regarded as ‘the answer’.

It is also important to consider as long a history as possible when analysing possible future events. While the nature of a market might change considerably over time, the nature of markets in general has been the same for millennia. Abandoning the models with which we are familiar might be a step too far — there is a limit to the uses to which fractal models can be put. Nevertheless, they do offer additional insights into the risks present in portfolios, and as such the possibility that returns might be fractal should be borne in mind when investment decisions are being made.

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*Paul Sweeting is professor of actuarial science at the University of Kent*