## by Benoit Mandelbrot

According to Modern Portfolio Theory, the basis for most of modern stock market analysis, the crash of October 1987 shouldn’t have happened in our lifetimes. In fact, it shouldn’t have happened in the lifetime of the country. It should have happened, perhaps, once in the lifetime of the universe. It was just our good fortune to have lived through it. Now we can relax.

In fact, even as Robert Merton and Myron Scholes were accepting the Nobel Prize for their contributions, the markets were preparing to given the Royal Swedish Academy second thoughts.

So what went wrong? Why did MPT fail so catastrophically? What assumptions failed? Why do academics and researchers continue to base their research on those assumptions? And is there anything out there that might be better?

Benoit Mandelbrot, Father of Fractals, Creator of Chaos Theory, Maker of the Mandelbrot Set, has some ideas, and The (Mis)Behavior of Markets tries to bring them to the public.

I fell in love with fractals and chaos theory in college, when both it and I were young. I didn’t pursue a career in physics, and left them behind, I thought for good. Ah, but life is strange, and just a couple of years before I decided to turn towards finance, so did Mandelbrot and fractals, and there there are, my old friends, waving me over to the table for drinks.

Actually Mandelbrot started out as a market analyst, but got sidetracked for a couple of decades, and is now turning his attention back to the problem that got him started: why historical cotton prices don’t look anything like a normal distribution.

Mandelbrot spends the first half of the book explaining the roots of Modern Portfolio Theory, and why the simplifying assumptions simplify it out of relevance to the real world. He doesn’t suffer fools gladly, and while he doesn’t actually come out and call anyone a fool, he does let the read draw his own conclusions.

I’ve covered the problems with MPT is previous reviews and comments, so I’ll just touch on them here. MPT is based on the early 20th Century work by M. Bachelier, who took his cue from Brownian motion, the random motion of particles suspended in water. That motion is normally distributed. Well, if dust motes, why not stock prices? Bachelier posited that stock prices moved some small amount each day, that the movement was normally distributed, and that there was no memory from one day to the next.

Why normally distributed? Well, like the guy who loses his keys over there in the dark, but looks for them over here under the streetlamp because the light’s better, it’s the only distribution that we can really find closed-form solutions for.

Modern Portfolio Theorists follow Merton in assuming that time is continuous. On the way down, or on the way up, the stock price hits every tick, (this used to be 1/8 of a dollar, now it’s a penny), so you can always get the share price when you put in your order,

These assumptions don’t hold. They’re made largely because they’re the only way to get to a closed-form solution to the math. What’s more, anyone with eyes in his head ought to be able to see that they’re not even usually true.

Mandelbrot starts from his distribution of cotton prices, and works from there. First, prices are more likely to follow what’s called a Cauchy distribution. More peaked in the middle, and with much, much fatter tails, the Cauchy distribution is very ill-behaved.

From there, readers familiar with fractals will find themselves skipping large sections. Mandelbrot reintroduces self-similarity, scaling, fractal dimension, and so on. But he also introduces the idea of time-warping, warping along the x-axis as well as the y-axis. Do this, the produces some very realistic-looking charts.

And that’s pretty much where the argument ends. Mandelbrot has produced charts that look right, and returns distributions that mimic reality, but he admits that it’s purely descriptive. There’s no mechanism, only behavioral speculation, and there’s no real way, yet, to make money other than by selling books about it.

What are the implications for finance? Well first, with those fat tails, the Cauchy distribution is much, much riskier than the normal distribution, with the likelihood of ruin much higher. Second, the thing about the realistic-looking charts, that are indistinguishable from real charts, is that they really are random. Technical analysts, beware! Mandelbrot has some sympathy for the illusory patterns you see in the charts, but illusory they are. If I can produce something randomly that looks like the real thing, and you think you see predictive patterns in it that aren’t there, what does that say about the predictive power of the patterns you see in real charts?

Mandelbrot also reminds us that he started to come up with this stuff just before the advent of MPT, but that in the trendy world of finance, his uncertainly was overshadowed by MPT’s promise of managing risk. He reminds us more than several times, and it does get a bit tiresome. There’s nothing the matter with enjoying being right, but even the disinterested observer, who’s only looking for something better, get weary of hearing “I told them so.”

He make up for it, though, but devoting the last chapter of his book to an uncritical discussion of current research based on his work. He’s boosting these guys, with no expectation of recompense, in the best academic tradition, and he deserves credit for that. If he can use his name and popularity to create a community pursuing these ideas, more power to him.

We are reminded that this is a field still in its infancy, not really allowed to grow by an industry that thought it had the magic key. Retrenching and pursuing this line of thought, which currently offers little if any profit path, is hard. But it’s better than building on sand, on a coastline whose length you can’t even measure.

Tags: Fractals, Markets