# Mandelbrot, Chaos Theory, and The Misbehavior of Markets

Stock market returns are not just random—they are randomly random. That is, the returns of the market are **stochastic**.

While I love the word stochastic, it and **Chaos Theory** are not well understood by ordinary people. This is too bad because stochasticism explains much of what goes on around us daily. Like, weather, or are you exposed to a virus today just going about your business? And, if exposed, will you get infected; or symptomatic or not? These are all stochastic processes.

The real implication of Chaos Theory on the stock market, however, is that **market returns do not have a normal distribution**. That is, there is no Bell Curve of stock market returns. Returns are randomly random, or fractal.

While chaos theory can seem far removed from the stock market, **Mandelbrot** brings theory and practice together in an important yet mostly neglected book. Let’s explore chaos, fractals, and Mandelbrot to figure out how to improve DIY investing.

## What is Chaos Theory, and What does it have to do with the Stock Market?

Chaos Theory reveals symmetries that occur in complex adaptive non-linear systems. Whilst that is a mouthful, the general idea is that initial starting conditions (a butterfly flapping its wings on a different continent) can set off massive consequences later (such as a hurricane resulting from the butterfly flap). Though one cannot predict the outcome, patterns emerge, which are fractal.

It is easy to see fractals in stock market returns. Tick-by-tick returns for a single stock mirror the patterns of returns for the market over the decades. If you take the x-axis (time) off any return chart, no one will be able to tell you if the returns occurred over seconds or years. Thus, stock market returns are fractal.

Above, you can see a fractal. The patterns all look the same if you zoom way in or pull way back out. **The small patterns are inherent in and make up the large patterns.**

What does chaos theory have to do with the stock market? Well, market returns do not follow a normal distribution (bell) curve. They are not even a random walk down wall street. They are actually stochastic returns that can be modeled via the fractals of chaos theory.

All stock market models are based upon the idea that the returns fit a normal distribution curve. Thus, fundamentally,** all historical models of the stock market are wrong**.

What? The model used to describe the stock market is wrong? Yup.

Let’s explore Mandelbrot’s book and see what’s up with that.

## Mandelbrot and Investing

Benoit Mandelbrot is best known for coining the term “fractal” and his work in the early field of chaos theory. He summarizes how chaos theory can be applied to the stock market in his book:

The Misbehavior of Markets: A Fractal View of Financial Turbulence

“The brain highlights what it imagines as patterns; it disregards contradictory information. Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it.”

― Benoît B. Mandelbrot, The (Mis)Behavior of Markets

In the interest of your time, don’t pick up the book unless you have an interest in fractals and investing. Or, note that Nassim Nicholas Taleb remarked that Mandelbrot’s book is: “The deepest and most realistic finance book ever published.”^{}

The underlying idea, regardless, is important to note. And here it is: the returns of the market are not represented by a normal distribution (bell-shaped) curve. There are many technical ways to describe this, but if you want to keep it simple, think of it this way: volatility clusters.

That is, when there is volatility, there is a lot of it. When the market is calm, there is very little volatility. That seems self-evident, but it gets you so far away from the normal distribution of returns that you realize you cannot use standard deviation as the marker for volatility.

Mandelbrot’s subtitle is “a fractal view of financial turbulence.” You now understand that turbulence (or volatility) is fractal. He points out that we have to view risk, rewards, and extreme risk (ruin) through the lens of fractals to have a deep understanding of them.

Before we get to some deeper points, let’s go through the book’s three sections.

## The Misbehavior of Markets: A Fractal View of Financial Turbulence

### First Part: Models Fail to Describe the Market

The first part of the book is rather straightforward. Current models fail to adequately describe the financial system. Mostly this part is a history lesson of how we got to the current models and why they fail to truly represent the nature of the stock market.

### Second Part: Chaos Theory

The next part develops the MultiFractal theory and discusses the true nature of markets. This is worth the price of admission. I’m not going to talk much more about the multifractal theory, as you will need to read the book to understand this concept cohesively.

If you want a simpler, less market-specific overview of the topic, pick up James Gleick’s Chaos. This book is one of three that changed my life and describes Chaos Theory in much more general terms.

### Third Part: Future Paths

As far as future paths, Mandelbrot does a decent job describing the problem, but he fails to provide solutions. This is not a criticism; it results from working with a complex non-linear system.

In summary, this book is not about making a killing on Wall Street; it is a book that explains why others fail to do so. The simple fact: most are using the wrong model.

But what to replace it with: now that is the question. It begins with fractals.

## Fractals in The Stock Market

Since fractals are paramount to understanding how we can apply them to the stock market, let’s spend a little more time unpacking fractals.

What are Fractals?

Fractals are self-similar. They start with basic rules and wind up amazingly complex. Yet smaller parts inherently reflect the whole.

Fractals are how we develop complexity from simplicity. It is seen in the shapes of nature (waves, weather, sand, landscape, trees, etc.).

Since it is difficult to comprehend fractals, often we resort to pictures or fractal geometry to provide an example.

Above, you can see another geographic representation of a fractal. Consider this a three-dimension picture of stock market returns regardless of the time frame.

Or:

How can you get 5-10 “100-year floods” in a short amount of time? Because weather (and climate) doesn’t distribute normally, it is fractal.

This is why there are multiple recessions in a short period of time or long stretches of boom times. The market has no memory, and tomorrow’s returns are not just random, they are randomly random. Small inputs can change everything in complex adaptive systems with unknown (and indeed unknowable) positive and negative feedback loops.

Said another way, there are “fat tails” to worry about. And moreover, since volatility clusters, **fat tails are contagious**.

So, while a normal distribution curve suggests that tail events are rare, fractals predict that tail events are startlingly common. Thus the “100-year flood” comes twice in one year. This feature of the market, rather than being a bug, helps change our understanding of both risk and reward and fat tails (extreme risk… AKA ruin).

## A Fractal View of Risk, Reward, and Ruin

How can a DIY investor profit from understanding chaos theory? By understanding how fractals affect risk, return, and ruin (which is extreme risk or black swans).

### A Fractal View of Risk

So what is risk? This is a profound question and the source of much misunderstanding on wall street.

Market folks who want to sell their snake oil want you to think that risk is volatility, and you can hedge risk.

Mandelbrot argues that markets are much riskier than most realize. In fact, as the theory it is based upon is inadequate and inaccurate, the entire financial system is on thin ground. His solution smacks of market timing, but, in general, discussions of risk are tricky and complex.

### A Fractal View of Reward

Since the market returns are not “normal” or random, much of the return comes in big jumps. Since volatility clusters, take advantage of the times of low volatility.

### A Fractal View of Ruin

Mandelbrot’s most interesting ideas come from his discussion of ruin. So much so, that Taleb’s The Black Swan is dedicated to Mandelbrot.

Black swan events are much more common in the real world than using normal distribution models would have you predict. These are the clustered fat tails that can lead to ruin.

Mandelbrot says: “Conventional models … are not merely wrong; they are dangerously wrong”

Additionally, markets deceive. **Humans are pattern seeking animals. Yet, there are no non-fractal patterns in the market.** We, therefore, see things where there are not, and this commonly leads to mistakes.

“If you are going to use probability to model a financial market, then you had better use the right kind of probability. 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. Anywhere the bell-curve assumption enters the financial calculations, an error can come out.”

## Lesson to Learn from Mandelbrot on the Stock Market

“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.”

Yet Mandelbrot believe there are lessons to take despite the randomness. Here is a list:

- Low fees and expenses are paramount
- Broad diversification is a free lunch
- Know that unpredictable outcomes are predictable
- The data settles down with longer time frames, stay invested
- Avoid those overconfident from conventional models of the market

Many of these points also can be considered icons of behavioral finance.

The last point reminds me of a joke: microeconomics is what economics specifically gets wrong and macroeconomics is what they get wrong in general.

## Summary: Chaos Theory in Stock Market

Sometimes it is tough to break out of the echo chamber of investment blogs and wall street advice. Mandelbrot and his Misbehaviors of Markets deserve more credit than it gets and will help break you of the notion that some people actually understand how the whole thing works.

Remember that Wall Street works most efficiently at separating you from your money. Their models and products may change, but at the end of the day, it is all “complexity with a side of fees.” Discover how to be an excellent DIY Physician Investor instead.

Break free of the echo chamber; read this free thinker, consider what risk actually is, and inform your investing by learning about chaos theory. And note, due to less overconfidence, women are naturally better investors!

I’ve wanted to write about chaos theory for a while but haven’t found the vehicle. Mandelbrot and his pretty fractals are the ticket.

For you, what about a go at understanding fractals, time, and the Stock Market? As Mandelbrot says “the very heart of finance is fractal.”

This book is an introduction to fractals in economics. After decimating current models, it ends with suggestions for future methods that may be fruitful.

In the end, however, no answers are provided by this book. But you might ask better questions.

Understand how the common model is flawed so deeply that there are risks beyond what it predicts. Risk is not standard deviation; it is not volatility. It, too, is fractal.

Good work as always David. If I understand this, doesn’t it imply that SWR calculations as well as paying someone large amounts of money to manage your portfolio to ‘beat the market’ are both inherently flawed and a waste of time and money?

I think it implies that most things people do to study the market is a waste of time and money!