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Fat Tails: The True Nature of Stock Market Returns – Part 1

In a two-part post, I discuss the nature of stocks market returns, first by pointing out the influence of extreme market events and then by considering their fractal or self-similar nature. Readers may recall that last week, we considered self-similarity in wealth distribution where I had mentioned their universal nature.

Both posts in this series shall deal with the same data set: daily, weekly and monthly returns of the S&P 500 from Jan 3rd, 1950 to Jan 21st, 2017 obtained from Yahoo Finance.

Before we begin a quick announcement: I have added SEBI registered fee-only planner Piyush Khatri to my List of Fee-only Financial Planners in India. Piyush is based in Hyderabad but can travel to Bangalore if necessary.

101: The nature of stock market returns

The most common representation of stock market returns is the bell curve.

The horizontal axis is the returns (daily/weekly/monthly/yearly) and the vertical axis is how frequently a particular value of return was observed.

The most frequent return is in the centre - aka the average or the (arithmetic) mean.

The standard deviation or departure from the mean (a measure of risk) is represented by the width of the curve.

One of the most important properties of this "distribution" is how rapidly the frequency (or probability) of observing returns well above or well below the average decreases.

For example, if the average return is +10%, the probability of observing a return of +100% or -100% is practically zero, as per this curve.

I had earlier written about this and how to understanding the idea of compounding here: Understanding the Nature of Stock Market Returns

However, extreme market events are rare, but not improbable. For example, the Sensex rose by an astounding +270% due to price fixing by Harshad Mehta. When the scam was exposed, the markets corrected (only!) by -40%.

These returns cannot be explained by the bell curve. The alternatives, however, are not easy to handle. Therefore to help us get an idea about how much returns can swing, I had ignored these extreme events and had used a modified bell curve to estimate: The Return I Expect From Equity Over the Long-term? (Part 2).

However, I believe it is our duty to understand what we are up against as completely as can and hence this series will be called The "true nature" of stock market returns.

In this post, I shall focus on the deviation from a bell curve. To understand this, let us look at an ideal bell curve with the vertical axis plotted in log scale. We will use this as a reference.

The Bell Curve In Log Scale

Two key features:

1: The slope off from the central peak is gentle.

2: The slope is smooth. The probability decreases constantly away from the centre.

Both these features will be absent when we consider real data.

Daily Returns of the S & P 500 (log scale)

Weekly Returns of the S & P 500 (log scale)

Monthly Returns of the S & P 500 (log scale)

The fat tails are those spikes that stand out (or down) from the rest of the distribution. Many have written about this, the first being Benoit B Mandelbrot. What is often overlooked is the fact that real-life distributions are much thinner than a bell curve. The fall from the peak is rapid (this, of course, decreases as we go from daily to monthly returns).

 Daily returns vs time for a bell curve (Ideal)

If the daily returns are plotted against date (time) then it should look like this: range bound and no abnormal spikes.

White Noise from a Gaussian Signal (Wikipedia)

In other words, if stock returns followed a bell curve, the market would be a zero-sum game. It is not.

Daily Returns vs Time (Real data)

Notice that huge upward and downward spikes. Those are the fat tail or black swan events.

If we had a used a bell curve model (and most of us still do), those events would be more than impossible!! Yet those returns are quite possible and occur with reasonable regularity, but not frequently.

The danger with modelling risk based on a bell curve is that we overlook these wide swings. Doing so could change our life's forever.

Handling fat tail events mathematically is not easy simply because of the number of parameters involved. So a bell curve approximation is a reasonable foothold, provided we understand the serious limitations.

Unfortunately, a model if used long enough appears like a law of nature and enough people do not "bother to look" (from "the Big Short") deeper.

Take my own learning curve. Being a physicist, I saw the bell curve is a law of nature. However natural process follows a set of rules, written in stones. If those rules are disobeyed, there would no life on Earth.

So I made the mistake of assuming the same for the stock market. It is when I started learning about departures from the bell-curve, I moved away from the associated greek alphabets - alpha, beta, omega, the Sharpe, Sortino and Treynor ratios to just downside and upside capture ratios.

My goals for this year: develop tools for analysing fat tails and self-similarity (to be covered in part 2)

So, how do we account for extreme events? First, we understand their nature better. We shall do this in part 2.


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

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10 thoughts on “Fat Tails: The True Nature of Stock Market Returns – Part 1

  1. Shan

    Aha, glad to see you're coming over to the real world from the academic world. Economics and markets are not 'science' and follow no rules. I'm glad you're seeing this finally 🙂

    Some other tips:
    1. Read The Black Swan by Nassim Nicholas Taleb
    2. I hope I never ever read the sentence "Volatility is a measure of risk" on your blog ever again 🙂
    3. Fat tails work both ways, you can lose a lot and you can gain a lot too. There is no limit, no law which says that you have to perform like nifty or some other human constructed index. So any comparison of any fund with the index is comparing one artificial thing with another artificial thing. This quickly devolves into financial porn.
    4. Ultimately, bell curves, returns, etc are all in the past. This financial history is potentially useful for academics but limited use if you want to get rich because the returns you seek are all in the future.

    Welcome to the real world. 🙂

    1. freefincal

      On the contrary, market behaviour can be understood by a well defined set of rules. The adherence is not perfect like science, but that should not stop us from trying measure risk. Then there are other who use it to predict.
      Volatility IS a measure of risk. It is only a question of how the volatility is measured. Anything that does not use bell curve parameters is not intuitive.
      Fat tails have nothing to do with benchmarking.
      Lot of traders get rich each business day using financial "history".
      When I have access to the work of his idol, Mandelbrot, Taleb can wait.

  2. ravi

    Awesome. I always wondered why the capture ratios, are a better measure of risk as compared to the alpha, beta ratios. Fat tails are more common volatility paradigm. Curious to know how you will factor in these outliers. Machine Learning or Deep Learning is what I am thinking

    1. freefincal

      Solutions are not easy. Markets do not follow set patterns (even among non-normal distributions). Any metric that does not used the std dev is going to be tough to understand.

  3. Dilip

    Nice approach in studying the market behavior (i.e.Human behavior) .

    I think, if you can take up Indian market (though it's not so old) behavior example would be nice.

    Even great if you can put forward the all above and future charts spreadsheet links.

    Thanks !

  4. kalyan

    I remember my early days from the job when I first started looking at market. I saw the plots and told my self - easy, Markov modelling. Once my proffessor at IISc Prof:Sundar Rajan told us that if a very practical and popular problem is not solved for long time it is better not to approach it trivially as many intelligent people would have already tried to solve it with no success.

    I still think that all this academic research is correct but we are looking at a non stationary random process and so we have only one sample to look at and so the margin for error is huge. All these approaches try to make it stationary random variable and try to project the data as large set of sample points but what we have is one time series which is just one sample of the random process - if that makes sense.

  5. kalyan

    To add to the above comment my conclusions about mutual funds study is (which I am sure as wrong as any other) in an ideal world the fund picking does not matter. Even in present situation it doesnt matter as long as you take care of few things which doesnt have anything to do with performance. Process is more important than manager or past performance. As long as you avoid funds with bad process (like LIC which is forced to buy few stocks by govt which they may not be interested to buy if given independence) it should be fine. But as this is very subjective, may be look at last 10 years performance and if the mutual fund is in top quartile or above average pick it. It really doesnt matter which fund and which fund house after that. I might sound contradicting here but here performance is trying to check the process indirectly - performance itself is not criterion.


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