My views on the market, tech, and everything else

Benoit Mandelbrot’s Ideas about Investing and Markets (Made as Simple as Possible, but not Simpler).


Benoit Mandelbrot was a Polish-born, French and American “mathematician with broad interests in the practical sciences.” I met him only once at a lecture at Microsoft Research before he passed away in 2010. Mandelbrot and Richard Hudson are the authors of the influential book The Misbehavior of Markets: A Fractal View of Financial Turbulence. I suggest you read this book for yourself (even though it is not an easy book to read). My comments on Mandelbrot in this post are shorter than usual, since Mandelbrot’s own words convey his ideas well. I am not going to dig deeply into fractals in this post since most of you will stop reading, but I will explain generally what they are. You might want to think about recent movements in the market price of Bitcoin as you read this post and ask yourself how Mandelbrot’s ideas may help explain them. 

The intent of this post is to explain Mandelbrot’s influence to someone who is not an expert. The other objective is to ask whether people who say they understand Mandelbrot’s ideas actually invest or implement policies in ways that reflect that understanding. Is what they actually do consistent with what they claim to believe? This issue reminds me of a quotation often mistakenly attributed to Mark Twain: “Everybody talks about the weather, but how many people do anything about it?” Stated differently, “Everybody talks about Mandelbrot’s ideas, but how many people do anything about it?”

1.“The idea of chance in markets is difficult to grasp, perhaps because… the millions of people who buy and sell securities are real individuals, complex and familiar. But to say the record of their transactions, the price chart, can be described by random processes is not to say the chart is irrational or haphazard; rather, it is to say it is unpredictable.” 

Making short-term predictions about how a price chart reflecting the actions of millions of people will fluctuate is more than just hard. The word Mandelbrot uses is “unpredictable” rather than difficult. Again: not predictable. Mandelbrot here is talking about an aggregate or macro phenomenon. Will a blind squirrel find a nut once in a while in making a macro predictions that times a market? Sure. But not more than twice in a row. Mandelbrot is not saying that investors should throw their hands in the air and quit, but rather that they should use the tools of probability in a more refined and nuanced way. Investors who like to know what they do not know or cannot know will benefit from Mandelbrot’s ideas. Risk comes from now knowing what you are doing and avoiding those areas is a very good thing. If you knew exactly where you will die, the intelligent response is to never go there. The same principle applies in investing.

The good news is that there are other approaches to investing which work around the problem Mandelbrot identifies. Why would you try to predict the unpredictable when there are other investing methods that avoid this problem? Why swim across a section of a swiftly moving river containing a deadly whirlpool when shallow relatively easy to cross sections of the same river are nearby? This is especially true since no points are awarded for degree of difficulty in investing. n approach similar to Mandelbrot’s is also advocated by the investor Howard Marks when he says you cannot predict. but you can prepare.  Mandelbrot’s version of that advice is:

“These techniques do not come closer to forecasting a price drop or rise on a specific day on the basis of past records. But they provide estimates of the probability of what the market might do and allow one to prepare for inevitable sea changes.”

2.  “Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it.” “The brain highlights what it imagines as patterns; it disregards contradictory information. “So limited is our knowledge that we resort, not to science, but to shamans.”

The human desire for predictability and order will result in people sometimes seeing patterns that they believe will deliver certain outcomes. This human desire is often encouraged by promoters who know how to turn dysfunctional thinking into cash – their cash. Unfortunately, our brains can conspire against us when it comes to successful investing.  Much of the work a successful investor must do is related to avoiding emotional or psychological mistakes. Even a random number generator will spit out results that a human brain will see as a pattern. There are some traders who try to claim that Mandelbrot borrowed ideas from the creator of Elliot Wave Theory. Mandelbrot himself responded to that claim by saying: “The idea is ancient, but his use and mine stand in absolute contrast.” Elliot claimed to have spotted a pattern that can predict price moves. Mandelbrot is saying that the reverse is true. A side bar about complex adaptive systems might be useful here, but must wait for a future post. Until then there is my blog post on Murray Gell-Mann if you are interested. When I asked a friend about this attempt to link Elliot’s ideas to Mandelbrot he said: “Mandelbrot was heavily empirical. He was interested in describing how markets work and much less interested in the mechanisms. But he absolutely understood complexity.” Another friend said: “Mandelbrot described complex systems correctly as discrete/quantum like systems and not ‘waves.’”

3. “If you can find some market properties that remain consistent over time or place, you can … make sounder financial decisions.” 

Five examples of approaches to investing which can be an investor’s friend because they are what Mandelbrot calls “consistent over time and place” are:

  1. Low fees and expenses- this approach should never go out of style.
  2. “Diversify as broadly as you can.”- this approach is the only free lunch in investing.
  3. Don’t try to time markets- with this approach the investor instead prepares for unpredictable outcomes.
  4.  Focus on microeconomic factors impacting specific understandable businesses.
  5. “The multifractal model successfully predicts what the data show: that at short time frames prices vary wildly, and at longer time frames they start to settle down.”

These five approaches are the equivalent of places in a river where the water level is low and crossing is relatively easy.  An investing system that is consistent with some of all of these five approaches has a greater probability of being successful.

4. “If you are going to use probability to model a financial market, then you had better use the right kind of probability. 

  • Wild randomness is like the gaseous phase of matter: high energies, no structure, no volume. No telling what it can do, where it will go. The fluctuation from one value to the next is limitless and frightening.
  • Mild randomness, then, is like the solid phase of matter: low energies, stable structures, well-defined volume. It stays where you put it. 
  • Slow randomness is intermediate between the others, the liquid state. 

Real markets are wild. Their price fluctuations can be hair-raising – far greater and more damaging than the mild variations of orthodox finance. That means that individual stocks and currencies are riskier than normally assumed. It means that stock portfolios are being put together incorrectly; far from managing risk, they may be magnifying it. It means that some trading strategies are misguided, and options mis-priced.” “Every so often, not so rarely, prices change dramatically, and today prices move much more quickly and these changes are much more important. But it has always been like that. There are stories in the Merchant of Venice by Shakespeare, and even much older books than that, which talked about the existence of a category of people, bankers, who knew very well from experience that ships sometimes went safely on a long trip and sometimes didn’t. And when they didn’t return, it was a big loss to their business. A single loss could very well sink a big company.”

When Mandelbrot said: “there is no telling where wild randomness will go” he was making a statement about the fundamental unpredictability of changes in short-term prices. He was not talking about, for example, using certain statistical factors to make long term predictions about an index. He was not talking about making predictions about the long term prices of a specific business using microeconomics principles. Mandelbrot was pointing out that the magnitude of fluctuations will be far greater than many people people imagine. Charlie Munger makes a similar point: “When people talk about sigmas in terms of disaster probabilities in markets, they’re crazy. They think probabilities in markets are Gaussian distributions because it’s easy to compute and teach, but if you think Gaussian distributions apply to markets, then you must believe in the tooth fairy. It reminds me of when I asked a doctor at a medical school why he was still teaching an outdated procedure, and he replied, ‘It’s easier to teach.'” Mandelbrot put it this way: “Anywhere the bell-curve assumption enters the financial calculations, an error can come out.” Portfolio theory “regards large market shifts as too unlikely to matter or as impossible to take into account. Typhoons, in effect, are defined out of existence.”

In Mandelbrot’s view markets are “fractal,” a name created by Mandelbrot in 1975 to describe repeating or self-similar mathematical patterns. Mandelbrot elaborates here:

“A fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole. In finance, this concept is not a rootless abstraction but a theoretical reformulation of a down-to-earth bit of market folklore— namely, that movements of a stock or currency all look alike when a market chart is enlarged or reduced so that it fits the same time and price scale. An observer then cannot tell which of the data concern prices that change from week to week, day to day or hour to hour.”

Robert Hagstrom, in his wonderful book Investing the Last Liberal Art, elaborates:


Fractal mathematics cannot predict outcomes that results from complex adaptive systems but they can tell us that such outcomes will inevitably happen sometime. All one can do then is prepare for their arrival and wait.


In a review of the book by Mandelbrot and Hudson Nassim Taleb writes:

People are so conditioned by advice offering charlatans in business books that anything remotely away from it seems, as I was told, quite “odd”. BM’s, of course, does not give you a recipe. It was therefore amusing to see the book reviews complaining about the “now what?”

The answer to “now what?” is to internalize that there is no “recipe” when it comes to the behavior of complex systems. People want a recipe so badly that getting over this desire arguably requires a process similar to the famous stages of the Kubler-Ross model: denial, anger, bargaining, depression and acceptance. Wisdom in no small part depends on accepting what you do not know or cannot know.

There are times to be aggressive as an investor and times to be very cautious. Knowing the difference is critical.

5. “Wild price swings, business failures, windfall trading profits – these are key phenomena. In all their drama and power, they should matter most to bankers, regulators and investors.”

A small number of outcomes can drive the lion’s share of financial returns. The FT captures in this paragraph an important point made by Mandelbrot:

“Mandelbrot distinguished between ‘Joseph’ effects and ‘Noah’ effects. Joseph effects – seven fat years here, seven lean years there – occurred when markets were evolving gradually and continuously. Noah effects were cataclysms – the Flood, or the week of September 11 2001, when the New York Stock Exchange closed for five days and dropped 7.5 per cent on re-opening. Because Joseph effects rule the market most of the time, they are what models measure. But Noah effects make and unmake investors.”

As just one example, Professor Bessembinder writes:

“Four out of every seven common stocks that have appeared in the CRSP database since 1926 have lifetime buy-and-hold returns less than one-month Treasuries. When stated in terms of lifetime dollar wealth creation, the best-performing four percent of listed companies explain the net gain for the entire U.S. stock market since 1926, as other stocks collectively matched Treasury bills. These results highlight the important role of positive skewness in the distribution of individual stock returns, attributable both to skewness in monthly returns and to the effects of compounding. The results help to explain why poorly-diversified active strategies most often underperform market averages.”

6. “In finance, I believe the conventional models and their more recent ‘fixes’ violate the Hippocratic Oath to ‘do no harm.’ They are not merely wrong; they are dangerously wrong.” “People think that risk means that if you invest $10, you may get back $11 if you’re lucky, perhaps $10.30, but somewhere close to $10. In fact, if you look at the actual data of trading, not for every price, but for the important prices on the market, large price changes are observed often enough to matter a lot.” “If one does not take account of the possibility of a price going up very suddenly, or going down very suddenly, one takes a risk that is higher than anyone wants.”

It is common now for people to say, “Well, I understand long tails since I have studied events like the failure of Long-Term Capital Management and I have now included the impact of these fat tails risks in my financial models.” As I noted above, what some people say they believe is often not what they actually do.  Mandelbrot addressed this overconfident view when he said: “Conventional models … are not merely wrong; they are dangerously wrong. They are like a shipbuilder who assumes that gales are rare and hurricanes myth; so he builds his vessel for speed, capacity and comfort — giving little thought to stability and strength.” Mandelbrot co-wrote an article with Nassim Taleb in which they say:

Your mutual fund’s annual report, for example, may contain a measure of risk (usually something called beta). It would indeed be useful to know just how risky your fund is, but this number won’t tell you. Nor will any of the other quantities spewed out by the pseudoscience of finance: standard deviation, the Sharpe ratio, variance, correlation, alpha, value at risk, even the Black-Scholes option-pricing model.

The problem with all these measures is that they are built upon the statistical device known as the bell curve. This means they disregard big market moves: They focus on the grass and miss out on the (gigantic) trees. Rare and unpredictably large deviations like the collapse of Enron’s stock price in 2001 or the spectacular rise of Cisco’s in the 1990s have a dramatic impact on long-term returns –but “risk” and “variance” disregard them.

7. “In a networked world, mayhem in one market spreads instantaneously to all others—and we have only the vaguest of notions how this happens, or how to regulate it.” 

Mandelbrot is pointing out that an increasingly networked world is a much more turbulent and unpredictable world. Billions of people with supercomputers in their pockets interconnected by networks creates the potential for mayhem in addition to valuable new products and services. When I hear people talk about a Great Stagnation I want to laugh out loud. The pace of actual change and the potential for more massive new shifts has never been greater. As Charlie Munger says: “The great lesson in microeconomics is to discriminate between when technology is going to help you and when it’s going to kill you…there are all kinds of wonderful new inventions that give you nothing as owners except the opportunity to spend a lot more money in a business that’s still going to be lousy. The money still won’t come to you. All of the advantages from great improvements are going to flow through to the customers.”

The types of feedback loops that can drive societal, economic and technical change have been put on steroids in this hyper connected digital era. If you think you understand all the implications of this increase in “inter-connectedness” you do not understand the problem. The old bumper sticker that said “Shit happens” must now say: “Shit happens faster and at far greater scale.” Bitcoins, which are themselves digital, are another accelerator of the phenomenon Mandelbrot wrote and talked about. 

8-12 are best understood by reading Mandelbrot in the original. When you read this it is a good idea to keep in mind this point made by Paul Samuelson: “Benoit Mandelbrot has been an incorrigibly original mind, and economists have been blessed by his insights. On the scroll of great non-economists who have advanced economics by quantum leaps, next to John von Neumann we read the name of Benoit Mandelbrot” (Paul Samuelson, Gaussian Self-Affinity). If you can’t read this screen shot due to print quality, I suggest you buy the book. If you can read it, I suggest you buy the book anyway!




Video interviews:

 FT interview: https://www.youtube.com/watch?v=vxbxXBrOPS8

Microsoft Research talk (sorry for the sound quality) https://www.youtube.com/watch?v=AQgXmjUwsMo

Inequality and finance; differences between Bachelier and Mandelbrot https://www.youtube.com/watch?v=6iVWFsxkdyo

Physics and Finance https://www.youtube.com/watch?v=vLiItUfqdS0

Stories: https://www.webofstories.com/play/benoit.mandelbrot/127

Mandelbrot’s finance book: https://www.amazon.com/Misbehavior-Markets-Fractal-Financial-Turbulence/dp/0465043577

Wikipedia bio: https://en.wikipedia.org/wiki/Benoit_Mandelbrot











Hagstrom book: https://www.amazon.com/Investing-Liberal-Robert-G-Hagstrom/dp/1587991381

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