Today is the 20th anniversary of Black Monday, the largest one-day fall in the Dow Jones index in history. Which raises the question: could it happen again?

If you believe returns are normally distributed, the answer - for practical purposes - is: no.

The 22.6% drop in the Dow then was (assuming annualized volatility of 20%) an 18 standard deviation event. And on a bell curve, the odds are against such events happening even once in the entire history of the universe.

But do returns follow a bell curve? What could we say to someone who says: "Returns are normally distributed - we were just very unlucky to see it in our lifetime."?

We could reply that lower standard deviation events are also more likely than a bell curve predicts; I estimated before the summer's falls that three standard deviation events are four times more likely than a bell curve predicts.

"Aha" our normalist might reply. "That's irrelevant. You can't infer the likelihood of 18 standard deviation events from three standard deviation ones. Indeed, probabilities must sum to one, so the fact that somethings happen more often than they should means that other things must happen less often. So perhaps 18SD events are even less likely than a bell curve predicts."

We could then try the wisdom of crowds argument. Options consistently price in a higher chance of a big fall than the bell curve says they should.

"This isn't wisdom" replies the normalist. "It's Charles Mackay's madness of crowds. It's a common cognitive bias to over-react to easily-remembered events. The memory of 1987 means deep out-of-the-money put options are over-priced. Look how many hedge funds have made money selling them down the years."

We could then point the normalist to this paper, which gives evidence that share returns are distributed as a power law with an exponent of 3. This predicts that a 10% daily fall should happen roughly once every 11 years and a 20% fall every 100 years.

"Come off it" replies the normalist. "Any fool can fit a curve to a sample. It doesn't follow that the curve will continue to fit the future data. That's just blind extrapolation."

I reckon, then, that all we're left with are Bayesian priors. It's not the data that tell us a crash could happen again, but common sense. We know that investors can herd, that there can be common mode failures in which disasters come not as single soldiers but as whole batallions, and that, well, shit happens.

This, though, means that "probability" is not something that exists independently, and is given by the data. Instead, it's something subjective that we impose onto the data.

Which in turn means that risk management is not - or at least should not be - narrow number-crunching. That's not science, but rather the appearance of science. Riccardo Rebonato shows why in the most readable book on financial risk you're likely to find.

To "assume" that the Bell Curve applies is on a par with the mumbo-jumbo of the modelling of Global Warming.

Posted by: dearieme | October 19, 2007 at 11:42 AM

"This, though, means that "probability" is not something that exists independently, and is given by the data. Instead, it's something subjective that we impose onto the data."

This may be a bit picky, but if probability does not exist independently, what is it that you are trying to figure out? Isn't it more accurate to write that while probability may exist independently, our ability to apprehend it is poor, and calculations based on data may do a worse job of describing probability than subjective evaluations.

Posted by: Luis Enrique | October 19, 2007 at 01:15 PM

IOW probability is socially constructed. Probability is a measurement, and like all measurements it is only "true" at the point it is taken.

Posted by: Matt Munro | October 19, 2007 at 02:56 PM

I wonder whether oceanographers have such debates about the normal distribution of waves and the 100 s.d. tsunamis that occasionally occur.Or perhaps they realise that the tsunamis are caused by a different process than everyday waves.

Posted by: james c | October 19, 2007 at 03:42 PM

If you believe returns are normally distributed, the answer - for practical purposes - is: no.

Why on earth would I believe that? I might just about be able to invoke the central limit theorem if I looked at yearly averages, and even that would be a bit of a stretch (even then, the CLT doesn't strictly apply, but I could probably construct a plausible argument that would suggest that one would expect the distribution to tend towards the normal anyway.)

But daily volitility? Why on earth should I expect that to be normally distributed? I see no reason for it to be, and every reason for it not to be.

Posted by: Sam | October 19, 2007 at 07:47 PM

How about Extreme Value Theory ?

Posted by: Dipper | October 19, 2007 at 08:00 PM

The normal (Gaussian) distribution and Central Limit Theorem do not fit with stock prices. Their use in this context is an example of the lamppost effect - the convenient mathematics of the Gaussian is attractive in the way the drunk looks for his lost keys, not in the place where he dropped them, but under the lamppost where there is light enough to see.

Posted by: steve_roberts | October 21, 2007 at 11:47 PM