Are Stock Returns Normally Distributed?

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According to “ Fama & French Forum: “ Distributions of daily and monthly stock returns are rather symmetric about their means, but the tails are fatter (i.e., there are more outliers) than would be expected with normal distributions. (This topic takes up half of Eugene F. Fama's 1964 PhD thesis. Eugene Fama is the 2013 Nobel laureate in economic sciences

In the old literature on this issue, the popular alternatives to the normal distributions were non-normal symmetric stable distributions (which are fat-tailed relative to the normal) and t-distributions with low degrees of freedom (which are also fat-tailed). The message for investors is: expect extreme returns, negative as well as positive.

Did you see the patterns or characteristics in the below charts?

Of course, this is not the “patterns “ which I have described in a separate blog under the blog title of  “ Patterns, Patterns, Patterns! “  

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Yes! The stocks market return is not in the form of  “perfect normal ( aka Gaussian ) distribution “.

1      1)      It  “skewed “ towards +ve return in the long run

2      2)      Black swan ( Crisis ) tend to follow by White Swan ( Opportunities )

image credit to the business

Concept explained : Gaussian ( Normal ) Distribution by Wikipedia & Investopedia

The Normal (Bell Curve) Distribution

In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.

The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. 

Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

The normal distribution is sometimes informally called the bell curve

Data Sets (like the height of 100 humans, marks obtained by 45 pupils in a class, etc.) tend to have many values at the same data point or within the same range. This distribution of data points is called the normal or bell curve distribution. For example, in a group of 100 individuals, 10 may be below 5 feet tall, 65 may stand between 5 and 5.5 feet and 25 may be above 5.5 feet. This range-bound distribution can be plotted as follows:

Similarly, data points plotted in graphs for any given data set may resemble different types of distributions. Three of the most common are left-aligned, right-aligned and jumbled distributions:

Note the red trendline in each of these graphs. This roughly indicates the data distribution trend. The first, “LEFT Aligned Distribution,” indicates that a majority of the data points falls in the lower range. In second “RIGHT Aligned Distribution” graph, the majority of data points fall in the higher end of the range, while they last, “Jumbled Distribution,” represents a mixed data set without any clear trend.

There are a lot of cases where the distribution of data points tends to be around a central value and that graph shows a perfect normal distribution, equally balanced on both sides with the highest number of data points concentrated in the centre.
Here is a perfect, normally distributed data set.

A lot of real-life examples fit the bell curve distribution:

  • Toss a fair coin many times (say 100 times or more) and you will get balanced normal distribution of heads and tails.
  • Roll a pair of fair dice many times (say 100 times or more) and the result will be a balanced, normal distribution centred around the number 7 and uniformly tapering towards extreme-end values of 2 and 12.
  • The height of individuals in a group of considerable size and marks obtained by people in a class, both follow normal patterns of distribution.
  • In finance, changes in the log values of Forex rates, price indices, and stock prices are assumed to be normally distributed

Searching  Abnormality in Normal Distribution 

With the above, we have a better understanding of these two characteristics that :

1)      Market skewed towards +ve return, in the long run, investing in much longer horizon tend to produce +ve return. Below chart  may give us some clue on this  :

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2)       Black Swan ( Crisis ) tend to follow by White Swan ( opportunities ), we may capitalize or take advantage of any crisis as the most quoted phrase by Warren Buffett  “Be Fearful When Others Are Greedy and Greedy When Others Are Fearful”.

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Although the market will eventually recover from any crisis and Black swan event shall follow by a White swan, but the importance of holding strong fundamental stocks will ensure that one survive post-crisis. 

Always remember that speculative and penny stock hit badly during the crisis , as such, having a diversified portfolio with good fundamental ( value ) stocks is important for us to “sail through “ the perfect storms during a financial crisis.


Quote Of The Day :

“Many investors today focus on earnings, but I focus on assets and don’t try to predict next months’ earnings, which is a much more difficult approach to investing.” By Walter Schloss

PS :

Anyone having a chart of STI’s return in “ bell curve distribution table “ to share with us will be much appreciated ! as the data shown in this blog is just only for the US market.


  1. STE, I always fond of the subject of math/statistical modeling and its application on financial and portfolio investment.

    So I'm really enjoying your write up on this. Thanks!

    1. Hi Yaruzi,
      Thanks for the comments , yah ! statistic modeling in finance is always an interesting subject to explore ,, but as we know well , there is "pitfall " in using such modeling , stocks market still 90% psychological driven ( behavioral finance ) .. hahaha
      Cheers !!

  2. I read a book 《华尔街的物理学》, or English title:"The Physics of Wall Street". there has explain this Distributions, butterfly effect, prediction of crisis and etc..

  3. Hi Ali ,
    Yes , I have the book in my book list as well , is a great book by James Owen Weatherall, it is about how "quants" is using physics modeling to beat the market ... interesting read .
    Cheers !!


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