The (mis)Behavior of Markets: A Fractal View of Financial Turbulence – Mandelbrot and Hudson

2004, Basic Books, 328 pages

Back Blurb

The (mis)Behavior of Markets offers a revolutionary reevaluation of the tools and models of modern financial theory. From the gyrations of the Dow to the dollar-euro exchange rate, mathematical superstar and inventor of fractal geometry shows us how to understand the volatility of markets in far more accurate terms than the failed theories that have brought the financial system to the brink of disaster. Updated with a new preface on the financial crisis of 2008, Mandelbrot’s insights are more valuable than ever.

Author Blurbs

Benoit Mandelbrot is Sterling Professor Emeritus of Mathematical Sciences at Yale University and a Fellow Emeritus at IBM’s Thomas J. Watson Laboratory. The inventor of fractal geometry, he has received the Wolf Prize in Physics, the Japan Prize in science and technology, and awards from the U.S. National Academy of Sciences, the IEEE, and numerous universities in the United States and abroad. His many books include Fractals: Form, Chance and Dimension, which later expanded into the classic The Fractal Geometry of Nature. He lives in Cambridge, Massachusetts.

Richard L. Hudson was managing editor of the Wall Street Journal’s European edition for six years, and a Journal reporter and editor for more than two decades. He is a graduate of Harvard University and a former Knight Fellow of MIT. Now the CEO and editor of Science Publishing Ltd., he lives in Brussels, Belgium.

A Fractal View of Financial Turbulence?

Fractal (from Latin frango “to break” e.g. fracture, fraction, fragment, etc.,) geometry was invented by Mandelbrot. It is a real-world, anti-Euclidean geometry in that it is the geometry of rough surfaces as opposed to the straight lines and perfect planes of Euclid. You can use fractal geometry to model structures where similar patterns recur in smaller and larger scales: for example cauliflower heads (a small head is a smaller version of a larger head) or coastlines (little nooks and crannies are scaled down versions of fjords). The immediate question than is: what do fractals have to do with financial turbulence? Well, the answer is that, rises and declines in stock prices also recur in similar and recurring patterns in smaller and larger time scales. For intuitive proof, compare one day, one month, one year, and one decade stock charts. Would you be able to tell, were the dates removed, which was which? Wall Street pros can’t. It’s just a bunch of wiggly lines. Wiggly lines that go back and forth like the coastline. And, like the coastline that exhibits self-similarity under 1x, 10x, and 1000x magnification, the stock chart exhibits self-similarity in one day, one month, or one year intervals.

This was Mandelbrot’s key insight, and a momentous one. It’s momentous because the implication is that, even if stock and commodity prices don’t go up and down randomly (they react to news, world events, and investor sentiment), their fluctuations can be modelled by the rules of probability as though they were random.

Late Work

One of the appealing things about The mis(Behavior) of Markets is that it is a late work. He turned 80 in 2004, the year the book came out. Long time readers of this blog will know that I’ve been a fan of late works for a long time: Beethoven’s Opus 111, Bach’s The Art of the Fugue, Mozart’s Requiem (unfinished at the time of death and played at his funeral), Nietzsche’s Ecce Homo, Sophocles’ Oedipus at Colonus (he successfully defended himself in court against charges of senility by citing his play), and Goethe’s second part of Faust. Directness of theme, abandonment of artifice, a brutal sense of honesty, a heartfelt and personal expression, a sense of possibility, and a glimpse of the bigger picture characterize the best late works. There’s an excellent book that talks about late style by scholar Edward W. Said entitled On Late Style: Music and Literature against the Grain. It’s fitting that that work is itself a late work.

Mandelbrot himself was keenly aware that he was himself producing a late work in writing The (mis)Behavior of Markets. Here’s a telling quote from the book (in the prelude written by coauthor Hudson):

In 2004, in his eightieth year, Mandelbrot continues making trouble. He works the same full schedule–including weekends–as he always has. He continues publishing new research papers and books, lecturing at Yale, and traveling the world of scientific conferences to advance his views. Why not? After all, as he points out, Racine’s most enduring play, Athalie; Verdi’s greatest opera, Falstaff; Wagner’s Ring Cycle–all were written in the twilight of life, when the artist, after years of experience and experimentation, was at the height of his powers.

Prejudice against the Speculative Markets

Mandelbrot devotes a chapter of the book to mathematician Louis Bachelier. Bachelier had dared to write base his dissertation on the volatility of bonds at The Bourse at the Paris exchange. His idea was that, although you could never know where future bond prices would end up, you could mathematically evaluate the odds of the fluctuations because bond prices would follow a ‘random walk’. The random walk is based on the random path of pollen grains suspended in water. And just as the path of pollen grains could be plotted on the bell curve, so could bond prices.

Unfortunately for Bachelier, academia deemed The Bourse to be to degraded of a place for true mathematicians. So, instead of graduating with a ‘trés honorable’ honour, he received a ‘mention honorable’. This consigned him to a life of obscurity. It wasn’t until the 1950s, a decade after his death, that his star picked up. Some are born posthumously.

Now it seems that the prejudice against true mathematicians working in finance remains to us today. Since my childhood, I’ve loved reading science books. Inevitably, each one will mention Mandelbrot and how fractal geometry is the best thing since sliced bread. But, you know, I don’t think any of them talked about Mandelbrot’s pioneering work in the 1960s examining price volatility in cotton markets (in the 60s, historic data on cotton pricing was complete, readily available, and accurate). Since then, Mandelbrot has devoted a lot of time and published quite a bit on how markets work. Heck, Eugene Fama was one of his students (he supervised his dissertation). But it wasn’t until I stumbled on this book (probably through a Marketwatch or Bloomberg article) that I had any idea that Mandelbrot had anything to say about the markets. In fact, I was so surprised when I found out, I googled to see if this was the Mandelbrot or another fellow with the same name.

So, You Think You Know What Risk Is…

Risk can be many things. Risk can be loss. Risk can be when something happens that you didn’t expect would happen. These sorts of risk are hard to quantify. But, if risk is volatility, it can be quantified. Take the 52 week high and low of Apple stock. The range between the high and low is the volatility. This sort of volatility can be expressed mathematically, using the laws of probability–that’s why the economists like it. They put in into formulas and win Nobel Prizes.

Beginning with Bachelier, it was thought that, if one graphed the daily movements of a stock, the price data would arrange itself into the standard distribution of a bell curve. The mean price would fill out the familiar bulge in the centre of the curve, and the larger price swings would be captured in the ‘tails’ of the curve. The larger the price swing, the less probable it is to happen. The bell curve is popular because it fits many natural phenomena. Human height, for example, fits a bell curve: 68% of American men are between 68-72 inches tall; 95% are between 66-74 inches tall; 98% are between 64-76 inches tall. The bell curve doesn’t rule out a 10′ giant. But the tail at this extreme is so flat that you would never expect to see one. IQ scores and the returns on betting on a series of coin tosses also fit a bell curve.

The idea of using the standard distribution of the bell curve to represent market risk was so prevalent that when Bachelier was rediscovered in the 1960s, the standard tools of finance all took it up. As a result, the standard tools MBA students learn to model the market are all based on the mild and predictable sort of risk the bell curve predicts. These tools are: modern portfolio theory or MPT by Harry Makowitz, the capital asset pricing model or CAPM by William Sharpe, and the Black-Scholes formula by Fischer Black, Myron Scholes, and Robert Merton. Markowitz, Sharpe, Scholes, and Merton all received Nobel Prizes for their work. Black would have as well, if he had lived another two years (the Nobel is not awarded posthumously).

The question Mandelbrot poses is: what if the bell curve is wrong? What if the odds of catastrophic ‘tail’ events in the market such as the 29.2% decline on Black Monday (October 19, 1987) are a thousand or a million times more likely than what the standard model posits? And, what is more, what if the stock market has a memory?–the standard model is based on a random walk. Like how each flip of the coin, the daily movements of the stock market are independent of one another. But, what if, in the real world, volatility cascades? Cascades in that a 3% drop one day increases the odds that it will continue to fall in the following days? Mandelbrot’s answer? If the bell curve is wrong, then we are like shipbuilders who think gales are rare and hurricanes are myths. We sail into doom. And we encourage others to sail into the storm with the comfort of dead wrong economic models. It’s like if we planned a mission to go to Mars based on old Ptolemaic models of the solar system.

To show how the bell curve is a poor measure of risk, Mandelbrot provides examples from the cotton, commodity, and stock markets: the data doesn’t fit the curve. ‘Impossible’ tail events happen in reality far more frequently than the bell curve allows.

The Solution

This was the most confusing part for me. Mandelbrot himself says that the math isn’t complete. Just as Bachelier had to wait a good 60 years for the math to catch up to his ideas, Mandelbrot’s ideas of fractal turbulence may have to wait another generation or two. He himself says the math is very hard. This book is more a call to arms that something has to be done. He does offer some suggestions, though.

In addition to the standard, bell curve distribution, there appear to be other probability distributions. There’s the Cauchy distribution. And there’s also a whole family of L-stable or ‘Levy’ distributions. These other distributions, from what I gather, have fatter tails. But they too, don’t capture the how real world risk works. It may be that they overstate the odds of catastrophic tail events. And it does not appear possible to insert other types of probability distributions into the standard models of finance (e.g., Black-Scholes, MPT, and CAPM). All the standard model, for some reason that a mathematician would understand, use the bell curve because it fits into the equation. Here I could be wrong, but that’s what I’m gathering.

In the future, the multifractal model of financial turbulence might be able to create a ‘fingerprint’ of a stock’s volatility. Right now, one of our best models of risk is the VaR or ‘Value at Risk’ model (also based on the bell curve). You start off by deciding how safe you want to be. Let’s say the maximum loss you are willing to take in a year is 10%. You then find a stock using the VaR model where, 95% of the time, the losses will be 10% or less. How is this safe, asks Mandelbrot?–the point is that 5% of the time, the losses can be more than 10% and up to 100%. It is only the illusion of safety. But let’s say someone uses the multifractal model to create a fingerprint of a stock’s volatility. This, to me, would still be based on historic price data. If the company hasn’t gone out of business, would the fingerprint capture the possibility of the stock going bankrupt, or, in other words, going down 100%?

Mandelbrot says many times that it’s not possible to make money (yet) from this multifractal view of stock market volatility. That may be true, but I wonder…if all the standard models underestimate volatility (because they use the bell curve), then wouldn’t that mean that the market is underpricing volatility? There should be some way of betting on irrational gains and losses and making money. Let’s say the market is saying that the odds of a 10% daily collapse is 1:1,000,000. But the odds are actually higher, 1:100,000 or something like that. There must be some financial instrument you could use to short the market so that when the 10% daily collapse happens, you could clean up since you know the ‘true’ odds and the market, which uses the incorrect model, has mispriced that eventuality.

Another idea to make money: if volatility is greater than commonly thought, would that be an argument for buying an equal weight index rather than a market weight index? With an equal weight index, you would have the same constituent stocks as an index investor (in a market weight index such as the S&P 500). But since stocks bounce up and down from their ‘average’ or ‘mean’ price, each time you invest fresh money, you would buy whichever stock had fallen the most. If you used a buy and hold strategy, because volatility is greater, you should be able to pick up a couple of points over the market weight buy and hold investor. Or?

Of course, these ideas aren’t based on the multifractal model, but rather, on volatility itself. Perhaps the way to make money on the multifractal model would be to market it to a data company such as Thomson Reuters. Thomson Reuters would use the mathematical model to project future growth, volatility, and other parameters of a stock. It might even use the model to draw future stock charts and run them through Monte Carlo simulators. Investors would, in turn, use this information in putting together resilient and efficient portfolios which maximize return and minimize risk.

Betting on Volatility and Turbulence

I should mention that I’ve put money on a sort of equal weight index. In March 2015, I picked 20 small companies in my play portfolio. They were picked somewhat randomly, but not completely at random. Since, for the most part, I’m an index investor, and the Canadian TSX Composite is dominated by financials (banks and insurance), oil & gas, and materials (mining), one rule was that none of these 20 companies could be from these sectors. The idea was that I wanted the small-cap portfolio to zig when the TSX Composite was zagging. So I ended up picking some industrials, consumer staples, healthcare, and technology stocks. Why small-cap? Well, I wanted to capture volatility and small-caps tend to move up and down violently than their more stable large cap brethren. In this small-cap portfolio, the idea is never to sell. But whenever I added money to the portfolio, I would top up the stock that had been most hammered (to keep the portfolio equal weight). This way, when things turn around (which they will do unless your pick goes bankrupt), you’ll pick up a little extra because you’ve said yes to volatility. Why 20 stocks? Well, you have to have enough stocks to have some winners and losers. And you can’t have too many stocks that you’re just replicating the index. Between 20-30 seems like a good number where the losers will hit you, but not too hard and the winners will help you, but also not too much. If you have too few stocks, and just happen to pick the losers, you’re going to get really hurt. But if you have too many stocks, the winners aren’t really going to impact the portfolio that much. It’s a question of concentration.

It seems that some Paris firms are doing something similar and calling this a multifractal strategy. Mandelbrot dismisses such attempts in his book as being far from multifractal: to him, it is just betting on stocks ‘reverting to the mean’. He’s absolutely right. But I can’t help but think that if volatility is so great, and if volatility is the measure of how much a stock deviates from its mean price, then shouldn’t it be easy to pick up a few extra points by a continual buying and holding equal weight strategy? Here were my picks from three and a half years ago and the performance:

AG Growth International (AFN) +17.8%

AGT Food and Ingredients (AGT) -28.0%

Alcanna (CLIQ) -21.4%

Boston Pizza Royalty Income Fund (BPF.UN) -21.5%

Boyd Group (BP.UN) +133.5%

Capstone Infrastructure (CSE) taken private at a gain of +37.6%, used proceeds to buy Cipher Pharmaceuticals

Chemtrade Logistics (CHE.UN) -27.6%

Cipher Pharmaceuticals (CPH) -49.6%

Clearwater Seafoods (CLR) -39.0%

Descartes Systems (DSG)  +126.2%

Great Canadian Gaming (GC) +74.7%

Highliner Seafoods (HLF) -59.3%

Innergex Renewable Energy (INE) +17.4%

Intertape Polymer (ITP) -7.3%

K-Bro Linen (KBL) -20.2%

Morneau Shepell (MSI) +56.2%

NFI Group (NFI) +257.3%

Northwest Company (NWC) +13.7%

Park Lawn (PLC) +4.4%

Premium Brands (PBH) +246.5%

Student Transportation (STB) taken private at a gain of 38.9%, used proceeds to buy Park Lawn

Western One (WEQ) -44.2%

If you look at the returns, I think you’ll agree there’s no shortage of volatility: the best performer was NFI (a maker of buses and motorhomes) at +257% and the laggard was Highliner Seafoods (a maker of fishsticks) at -59%. In the portfolio, two stock more than tripled (NFI and PBH), two other stocks more than doubled (BYD and DSG), and three stocks lost more than 40% (CPH, HLF, and WEQ). The volatility is there.

But the question is, how did the portfolio do? In three and a half years, with dividends, it’s up 26.3%. That equates to a rate of return of 6.9% each year. Compare this to the S&P TSX Small Cap Index (market weight). Total return in the last three and a half years is 20.1% for an annualized return of 5.7%. The little equal weight portfolio has done well compared to the market weight index. But of course the results are statistically meaningless as the two portfolios hold different stocks. You’d have to compare a market cap to a equal weight portfolio tracking the same index to draw meaningful conclusions. Perhaps a topic for a future blog?

One insight, does, however, emerge from this small cap portfolio: the business model really gives you little idea of how a stock will perform. Who knew that a bus and motorhome manufacturer (New Flyer) would triple? Who knew that Premium Brand Holdings, a company that makes Starbucks breakfast sandwiches and the sliced meats you find at grocery stores would be a top performer?–geez, they just make black forest ham! Who knew that a worldwide lentil distributor would be down a third? Aren’t people supposed to be eating more lentils? Who knew that Highliner Seafoods would be down over half? Isn’t seafood consumption up and growing? And why is K-Bro Linen down a fifth?–don’t they have the lock on hospital linen cleaning contracts in all the major cities?

If you had asked ten experts three and a half years ago to predict where these 20 stocks were going, I don’t think any of them would even be remotely close. You would have to have known that India would have frozen out Canadian lentils (AGT). You would have to have known that the government would have stripped CLR of part of their arctic clam license. You would have had to have predicted that oil would go down to twenty dollars a barrel (WEQ). You would have had to have predicted Valeant’s business model would explode, dragging down the whole pharmaceutical industry (CPH). How could anyone have known? And you know, it’s going to be like this going forwards. The things that will affect this small cap portfolio are the things we don’t know yet. Until then, I’ll keep picking up a few points on volatility. That I do know will be there. Funny, the only certain thing is that things are uncertain.

A Mystery

Mandelbrot spends a bit of time talking about power laws. Instead of a bell curve distribution where the tails are imperceptible, cotton prices, wheat prices, interest rates, and some stocks follow a power law distribution which allows for large price swings. Gravity and earthquakes also follow a power law distribution: double the distance or power, and the force of attraction or the frequency is four times less. In a section of the book, Mandelbrot tells the story of Harold Edwin Hurst, a hydrologist who cracked the code of how high to build a dam to tame the Nile.

The problem with calculating how high to build the dam was that the Nile would not only experience really dry and really wet years, but also that the wet and dry years would cluster together in an unpredictable pattern. In his attempt to understand flooding, Hurst looked through any reliable, long-running records on climate he could find, from tree ring growth, sunspot patterns, discharges from Lake Huron, and annual water levels at Lake Dalalven in Sweden. People thought he was a crack, since how could such varied phenomena be related? He looked through 51 different phenomena, and found that everywhere he looked, the range obeyed a three-fourths (0.73) power law. It was as though this three-fourths power law is a constant of nature. Now this is interesting. It makes me want to learn math so that I can figure out why this is. This is one of these questions that you could spend your life looking into.

I have to agree with Taleb that this is “the deepest and most realistic finance book ever published.” I read it three times. And, if I didn’t have a stack of other deserving titles, I would have read it a fourth time.

Until next time, I’m Edwin Wong, and I’m doing Melpomene’s work.

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