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Monday, 20 July 2009

Mandelbrot's Mad Markets

Haunted By Statistics

As we've wandered down the echoing corridors of behavioural finance we seem to be haunted by a troublesome spectre, which refuses to go away no matter how much we prove to it that it’s a figment of economists’ imaginations. Discovered by Francis Galton, appropriated by Harry Markowitz and embedded in risk management models ever since, our ghoulish apparition is a mathematical construct, the Gaussian distribution, aka the bell curve.

The Gaussian distribution keeps on reappearing throughout economic theories as a rule-of-thumb description of how markets behave. With which there is just the smallest problem – markets don’t behave as it would predict. We’ve known this for over forty years, ever since Benoît Mandelbrot showed that cotton prices bound around in a decidedly peculiar way. Markets behave madly far more often than the standard models predict so why anyone should be surprised that they fail catastrophically every so often is a bit of a mystery, really.

Modelling Random Behaviour

The original idea that market returns follow a Gaussian pattern was a reasonable one. For a long time we thought that most human behaviour followed the familiar bell curve with lots of results clustering around the central, mean value and fewer and fewer as you get further out. Classically, human height is based on a Gaussian distribution. Markets seemed to be just another facet of this structure.

Modelling market behaviour in this way implies that share prices fluctuate randomly but do so in a way which maps to the Gaussian curve – lots of fluctuations around the average value and then less and less as you move away from it. Basically small changes around the central value are much more common than large ones. This fits with a common sense view of how share prices move around their fundamental value – you’ll get a lot of price movements around that value and occasional movements that are much further out, where the share is significantly over or undervalued.

Back at the start of the twentieth century when this model was first developed it was a reasonable stab at describing the way markets behave. However, it wasn’t based on any underlying theory of what was actually going on – it’s purely a way of generating outputs that look like a real market. This is the way that most models work – a flight simulator program doesn’t really include a full scale model of weather, it simply uses a set of rules to try and mimic it while you press the space bar frantically in an attempt to move some illusory flaps and avoid some kamikaze pixels.

Roughly this is how life insurers work, too – they have a model of people’s lifespans and life insurance premiums are calculated on this basis in order to allow them to make their profit margin. Life insurers don’t model exactly how people age and die – they simply use a statistical distribution to calculate the probability of death occurring based on age, gender and lifestyle. Hence the fact that people keep coming up with stupid ways of killing themselves isn't a problem, merely a statistical insignificance.

Losing Alpha Centurii

So to develop models of market risk we need a statistical distribution that tells us what the probability is of any given return and the Gaussian curve seemed to fit the bill. Unfortunately it turns out that markets don’t follow a Gaussian distribution so that any models based on this assumption will be wrong. It’s like physicists discovering quantum mechanics but then carrying on modelling the universe without taking its findings into account and then acting all surprised when Alpha Centurii keeps turning up in the wrong place.

Actually, the analogy with quantum mechanics is better than that. Physicists know that their current model of Everything is wrong but don’t know exactly why – something’s adrift in Einstein’s universe: it keeps misbehaving, refusing to produce results in accordance with the theory leading to speculation that 90% of the damn thing has gone into hiding under some kind of celestial witness protection scheme. Similarly economists know that they don’t understand the way markets fluctuate but can’t find a description that fits the empirical data.

What’s really dangerous about modelling market risk using a Gaussian distribution is that it fails to capture the extreme behaviour that’s exhibited from time to time. In essence at the very time risk management is really needed the models fall over. Yet Benoît Mandelbrot showed back in the 1960’s that at its extremes the market comes over all fractal.

Going Fractal

Fractal behaviour looks nothing like a Gaussian distribution. What it gives you is lots of small changes occasionally interrupted by giant ones – known as a Lévy distribution with no mean to regress to in sight. A Lévy distribution is “scale invariant” or, to put it another way, there’s no characteristic change in value – the changes don’t cluster around the average (because there isn't one, duh) – and it predicts that we’ll see many more extreme events than the classical models would suggest. Which, of course, is exactly what we see in real markets.

The obvious answer, you’d think, is to replace the use of Gaussian distributions with Lévy ones. Unfortunately it’s not that simple because markets don’t actually follow a Lévy distribution either. In fact they seem to do a bit of a mix of the two with something else thrown in – which we also can’t model. There’s something really odd underlying market behaviour and the best we can do is guess that it’s something to do with how mainly irrational people behave under conditions of uncertainty.

Roughly, most of the time market behaviour approximates to a Gaussian distribution. During these periods the risk models work pretty well and organisations can make small amounts of money over and over again by exploiting predictable human errors. At extremes, however, the Lévy flight kicks in and the risk models go to hell in a jet propelled handcart. Nassim Taleb memorably refers to organisations as “picking up pennies in front of a steamroller”: most of the time the steamroller trundles along and the pennies can be gathered safely but occasionally the afterburners kick in and the penny gatherers get squished.

Not Going Fractal

If we peer closely at some of the risk management programs around today we still find the Gaussian distribution at their heart. The Black-Scholes option pricing model uses it. So does the broken-hearted copula model that was used to model counterparty risk for sub-prime mortgage assets. Why do we keep on using models that we know are faulty?

Well, for one reason, a badly functioning model of risk is better than no model at all as long as you recognise the limitations. It’s better to be safe 60% of the time than none. Secondly, lots of attempts have been made to develop better models but so far market behaviour has defied these. There’s something about the way that raw capitalism works that means it doesn’t follow any path we can predict.

Regardless, if we’re using faulty models we need to be aware of this and to treat them with the caution they deserve. That we haven’t has been perfectly evident from the problems we’ve repeatedly suffered from over the past few years. Imperfect models of risk ought to be better than no models at all but the evidence so far is that this isn’t the case – people abrogate their responsibilities and start putting unjustified faith in the models rather than thinking for themselves.

Chaos

Something, somewhere, at the heart of capitalist markets defies modelling. It may be that the inherent instability of capitalism is due to the interaction of irrational humans or even, possibly, that there’s something more fundamental than that going on. Fractal behaviour is typical of chaotic systems – systems which are forever poised on a knife edge and in which the slightest perturbation can dramatically change the final outcome. If markets are genuinely chaotic then we won’t understand them until physics figures out how the universe is really put together. We could be waiting a long time.

In the meantime, though, a little more humility in the face of the unknown would be in order from the people charged with managing the world’s finances. Some leaders who understand elementary statistics would be a start.


Related Articles: Regression To The Mean: Of Nazis and Investment Analysts, Correlation Is Not Causality (And Is Often Spurious), Markowitz's Portfolio Theory And The Efficient Frontier

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