Fractals: The True Nature of Stock Market Returns – Part 2

Last Updated on

While discussing the distribution of wealth in India, I had referred to the idea of self-similarity with the example of the Pareto principle. Today I would like to discuss how the stock market behaves in a similar way!

The first part of this series is here: Fat Tails: The True Nature of Stock Market Returns – Part 1

To recap, we say that 20% of the population hold 80% wealth. If we zoom in on the 20%, then: 20% of 20% population hold 80% of 80% wealth.

If we zoom in on the 20%, then: 20% of 20% (4%) population hold 80% of 80% wealth (64%)

If we zoom in on the 4%, then: 20% of 4% (or 20% of 20% of 20%) or 0.8% of the population hold 80% of 64% (or 80% of 80% of 80%) or 51% wealth.


20% holds 80% –> 4% holds 64% –> 0.8% –> 51% –>

Each segment obeys the Pareto principle. The segments make up a whole and the whole also obeys the Pareto principle.

Wealth distribution is self similar!

This is known as self-similarity or fractal behaviour: parts of a whole behave like the whole. Parts of the parts behave like the parts and also behave like the whole.

Reade more about this: The 80/20 rule: Making sense of richest 1% Indians owning 58% wealth!

I had mentioned that nature is fractal in nature. Before we head to the stock market, another fascinating example.

The Coastline Paradox

Suppose you wanted to measure the length of the coastline of a country (say Britan – because this problem originated there).

The Coastline Paradox (Wikipedia)

The answer you get depends on the yardstick used. The answer is nice and small for a long stick. Use a smaller stick, then the distance is bigger! This is the paradox – the answer depends on your measuring stick.

The ups and downs of the coastline over a zoom level of say 100 Km x 100 Km is similar to the ups and downs over a tighter zoom of 10 Km x 10 Km. This, in turn, is similar to the features observed over 1 Km x 1 Km.

Coastlines are self-similar!

The small ups and downs that you see over one kilometre are reproduced over 10s and 100s of kilometres.

Now look at the coastline of Great Britan and see if you can spot the similarity with the stock market. The ups and downs of the coastline resemble stock price movements.

The ups and downs in price observed daily are similar to the ups and downs over a week, over a month and even over a year.

The stock market is self-similar!

What is similar to all three situations? Parts of a whole, behave like the whole. Parts of the parts, behave like the parts and also like the whole.

Snowflakes are self-similar!

More on this and other examples later.

Let us get back to what we were discussing yesterday – the S & P 500.

We had looked a return distribution like this.

In his ground-breaking 1963 paper, Benoit Mandelbrot showed that cotton prices were self-similar. I will attempt a simplistic version of his research with the S & P 500.

The horizontal axis of the above graph represents return bins The vertical axis, the frequency with which daily returns occurred in each bin. Suppose I study the rate at which distribution falls off, this is what I get:

The horizontal axis represents return bins in log scale the vertical axis the frequency with which data falls in each bin (also in log scale).

Mandelbrot showed that rate at which daily returns fell, is similar to the rate which weekly returns fell and the rate at which monthly returns fell. (Yearly is a bit different).

This is the self-similarity. The movement over a day is similar to the movement over a week and over a month – just like a coastline under various zoom levels.

Note: Above data is only for outliers (not so frequent returns).

He went on to show that one can go on to construct a stock market-like behaviour over say months and years starting from a small variation (say over the course of a day).

Here is a picture from his famous Scientific Amercian article. The below image is a direct link from the article used under the fair use doctrine for commentary only.

This is a direct link to the Scientific American article mentioned above. No copyright infringement is intended. Clicking on the image will take you the scientific Amercian page.


Notice that the last panel represents stock market like behaviour and it has been generated from a simple construction like this: /\/ (top panel).

Naturally, this will not exactly reproduce the market behaviour, but gives you an idea of how it behaves.


Reason 1: This kind of self-similarity can account for extreme market events much better than a bell curve. So we get a better understanding of risk. More on this later.

Reason 2: Man has this desire to predict market movements. Using fractals may (repeat, may) get better results.

Traders are familiar with this behaviour via Elliot Wave Theory. Arguably, T S Elliot (founder of Elliot waves ) knew about fractals decades before Mandelbrot did.  The idea of self-similarity is common to both theories, but I need to dig deeper before I can comment on the efficacy of Elliot waves.


The (Mis)Behaviour of Markets: A fractal view of risk, ruin and reward by Benoit B Mandelbrot (available at

The variation of certain speculative prices, The Journal of Business, Vol. 36, No. 4 (Oct 1963), pp. 394-419, by Benoit B Mandelbrot. Available here

PUNE DIY Investor Meet Feb 26th 2017

The second Pune DIY meet will be held on 26th Fb 2017. You can register via this link

You Can Be Rich Too With Goal Based Investing

Your first investment should be buying this book

The calculators are really awesome and will give you all possible insights

Thank you, readers, for your generous support and patronage.

Amazon Hardcover Rs. 267. 33% OFF

Kindle at (Rs. 244.30)

Google Play Store (Rs. 244.30)

Now just Rs. 280  with additional 10% discount with code: Republic10 at Infibeam

If you use mobikwik wallet, and purchase via infibeam, you can get up to 100% cashback!!

  • Ask the right questions about money
  • get simple solutions
  • Define your goals clearly with worksheets
  • Calculate the correct asset allocation for each goal.
  • Find out how much insurance cover you need, and how much you need to invest with nine online calculator modules
  • Learn to choose mutual funds qualitatively and quantitatively.

More information is available here: A Beginner’s Guide To Make Your Money Dreams Come True!

What Readers Say

Also Available At

Bookadda Rs. 371. Flipkart Rs. 359 ($ 3.70 or Rs. 267)

Google Play Store (Rs. 244.30)

Do share if you found this useful

About the Author M Pattabiraman author of freefincal.comM. Pattabiraman(PhD) is the author and owner of  He is an associate professor at the Indian Institute of Technology, Madras since Aug 2006. Pattu” as he is popularly known, has co-authored two print-books, You can be rich too with goal based investing (CNBC TV18) and Gamechanger and seven other free e-books on various topics of money management.  He is a patron and co-founder of “Fee-only India” an organisation to promote unbiased, commission-free investment advice. Pattu publishes unbiased, promotion-free research, analysis and holistic money management advice. Freefincal serves more than one million readers a year (2.5 million page views) with numbers based analysis on topical issues and has more than a 100 free calculators on different aspects of insurance and investment analysis. He conducts free money management sessions for corporates  and associations(see details below). Previous engagements include World Bank, RBI, BHEL, Asian Paints, TamilNadu Investors Association etc. Contact information: freefincal {at} Gmail {dot} com (sponsored posts or paid collaborations will not be entertained)
Want to conduct a sales-free "basics of money management" session in your office?
I conduct free seminars to employees or societies. Only the very basics and getting-started steps are discussed (no scary math):For example: How to define financial goals, how to save tax with a clear goal in mind; How to use a credit card for maximum benefit; When to buy a house; How to start investing; where to invest; how to invest for and after retirement etc. depending on the audience. If you are interested, you can contact me: freefincal [at] Gmail [dot] com. I can do the talk via conferencing software, so there is no cost for your company. If you want me to travel, you need to cover my airfare (I live in Chennai)

Connect with us on social media

Content Policy

Freefincal has original unbiased, conflict-of-interest-free,  topical reports, reviews, commentary and analysis on all aspects of personal finance like mutual funds, stocks, insurance etc. All guest authors and contributors to the site also do not have any conflict of interest. If you find the content useful, please consider supporting us by (1) sharing our articles and (2) disabling ad-blockers for our site if you are using one. No promotional content We do not accept sponsored posts and link exchange requests from content writers and agencies. This is our privacy policy Our website is non-profit in nature. The revenue from the advertisement will only be used for hosting charges, domain registration charges, specific plugins necessary for traffic growth and analytics services for search engine optimisation.

Do check out my books

You Can Be Rich Too with Goal-Based Investing

You can be rich too with goal based investingMy first book is meant to help you ask the right questions, seek the right answers and since it comes with nine online calculators, you can also create custom solutions for your lifestyle! Get it now.  It is also available in Kindle format.
Gamechanger: Forget Startups, Join Corporate & Still Live the Rich Life You WantGamechanger: Forget Start-ups, Join Corporate and Still Live the Rich Life you wantMy second book is meant for young earners to get their basics right from day one! It will also help you travel to exotic places at low cost! Get it or gift it to a young earner

The ultimate guide to travel by Pranav Surya

Travel-Training-Kit-Cover This is a deep dive analysis into vacation planning, finding cheap flights, budget accommodation, what to do when travelling, how travelling slowly is better financially and psychologically with links to the web pages and hand-holding at every step.  Get the pdf for ₹199 (instant download)

Free Apps for your Android Phone

All calculators from our book, “You can be Rich Too” are now available on Google Play!
Install Financial Freedom App! (Google Play Store)
Install Freefincal Retirement Planner App! (Google Play Store)
Find out if you have enough to say "FU" to your employer (Google Play Store)

Blog Comment Policy

Your thoughts are vital to the health of this blog and are the driving force behind the analysis and calculators that you see here. We welcome criticism and differing opinions. I will do my very best to respond to all comments asap. Please do not include hyperlinks or email ids in the comment body. Such comments will be moderated and I reserve the right to delete the entire comment or remove the links before approving them.


  1. Dear Pattu,

    Interesting learning. Thank you.

    Is this not the principle on which most technical analysts work.

    So what role does fundamentals play. Or do they also eventually fit into this model.

    Curious to learn.

    1. Thank you. The pure fractal approach is to determine the nature of the underlying distribution and scaling factors. The problem is, such data set will not exactly fit into any known distribution. This is the reason we cannot predict. Technical analysts construct Elliot waves from a given data set. I don’t think they worry about the nature of the distribution. Both parties do not worry about fundamentals 🙂
      The aim of the above model is primarily to model risk with extreme events accounted for better.

Leave a Reply

Your email address will not be published. Required fields are marked *