Serious Money: Mathematician takes a hammer to the investmen decision-making edifice, writes Chris Johns

There are two ways to invest: the route followed by the purveyors of snake-oil and the proper way. The methodology adopted by the charlatans is always a variant on the toss of a coin. Guesses about the direction of asset prices are plucked from the air but always based on an edifice of spurious mumbo-jumbo that seems to add scientific weight to conclusions reached.

The correct way involves the assimilation of a lot of increasingly complex best-practice techniques and ensuring you have extremely timely access to all necessary information and data.

Even then, we need a dose of humility and need to recognise that there is still an awful lot we don't know, particularly about the shorter term ups and downs of market prices.

Much of the formidable quantitative edifice that we can build in support of our investment decisions works best over long periods. The short term has essentially been given up to the black arts of the traders, who most of the time make bets on the bets that others are making. But our fundamental models come into their own over the long haul: we believe in regression to the mean, which in this context means that all asset prices will eventually drift back to "fair value".

Many professionals believe that we are quite good at judging the proper value of an asset and pretty poor at guessing how long markets will take to reach it. The cornerstone of modern financial theory is the "efficient markets hypothesis" which, among other things, says that all asset prices are completely unforecastable.

Practitioners don't believe this for a second, arguing that forecasting is extremely difficult but not impossible. Of course, there are competing models and anyone who claims to have invented the perfect mousetrap is usually greeted with derision.

Nearly all of our models rely on the bell curve, familiar to anyone who has studied Leaving Cert maths. The "normal distribution", to give it its proper name, underlies everything from modern portfolio theory to the pricing of share options.

The bell curve occurs frequently in nature, as well as the text book: the most common example is the distribution of people's height. A chart of the number of people who reach all possible heights looks like a bell, with the peak of the curve representing the average height of the population. This normal curve has some wonderful statistical properties that lend themselves to all sorts of financial applications.

Benoit Mandelbrot, a professor at Yale, is a mathematician who has been arguing for years that all of this stuff about the bell curve is nonsense. The world of smooth curves and elegant geometry is utterly bogus in Mandelbrot's view.

His vision is of a much rougher world, where virtually everything from natural phenomena such as floods and coastlines to the underlying behaviour of stock prices can only be described in terms of jagged lines rather than simple curves.

His new book, The (Mis)Behaviour of Markets, is an attempt to make his speciality, the formidable mathematics of fractal geometry, more accessible to a wider audience.

Although the maths is hard, the underlying ideas are straightforward. It's a subtle point, but the normal distribution makes firm predictions about the nature of price changes, but is mute on the level of the price itself. That means we can say all sorts of useful things about the riskiness of investments. We can construct proper portfolios for pension funds or individuals and it also allows us to price executive share options. The "value at risk" models employed by banks and regulators to keep banks from blowing up their balance sheets often have a normal distribution lurking in the background.

Mandelbrot argues that the bell curve assumptions underlying much of the finance industry are simply wrong. If price changes did follow a normal distribution there would be far fewer extreme events like stock market crashes or Russian debt crises. Moreover, another key assumption underlying the bell curve appears to be violated in the real world: asset prices seem to be affected by the past. There is a "memory" effect at work in many stock and bond prices. If markets were efficient this simply could not happen.

One key conclusion that arises from all of this is that markets are often much riskier than orthodox finance would have us believe. Mandelbrot goes on to argue that the "random walk" prediction is also wrong: many asset prices will follow a "fractal" shape.

A fractal often looks like a jagged edge, an apparently completely random variable that on closer examination often conforms to a complex repeating pattern. The coastline of the west of Ireland looks like a randomly formed, extremely rough line. But it can be modelled using Mandelbrot's fractals.

But, as the author freely admits, modelling is not the same as forecasting. Few of Mandelbrot's techniques can be easily adapted for making market forecasts, although some are trying. I know of at least one UK hedge fund making explicit use of Hurst exponents (don't ask) in investment strategy.

One particular conclusion reached by Mandelbrot is, I think, extremely powerful. Perhaps the biggest puzzle in modern finance is why the historic relationship between equities and bonds reveals stocks outperforming their fixed income counterparts by up to eight times the level predicted by the theory.

The historic "equity risk premium" is simply too big: the return from equities relative to bonds has been way too high. But Mandelbrot's reworking of the measurement of risk suggests why this has been the case: equities really are that risky, particularly in terms of extreme events.

Mainstream finance is fighting back. If options were as mis-priced as Mandelbrot suggests, most sellers of these derivatives would have gone out of business long ago. Options existed long before Black and Scholes invented their famous formula, and the industry has recognised some of Mandelbrot's points with extensions and modifications to the bell curve assumptions. But Mandelbrot sees these amendments to conventional wisdom as mere sticking plasters on a corpse.

Mandelbrot has no time for "fundamental" analysis - the study of things like economic growth, inflation and all the other factors that determine asset prices. He says that is simply too hard. But that's finance and economics. His models of market prices are therefore explicitly "black box": he knows how asset prices will move but cheerfully admits he has no idea why.

As a fundamental analyst I find all of this encouraging. Mandelbrot, I believe, is filling in some of the gaps in our knowledge about short-term price determination. Over the long haul, only us fundamentalists can shed any light on why - and where - asset prices will move.