### Are Stock Returns Normally Distributed?

According to “ Fama &
French Forum : “

**Distributions of daily and monthly stock returns are rather symmetric**about their means, but the**(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 2013 Nobel laureate in economic sciences)***tails are fatter*
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 below charts ?**

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

**“ Patterns , Patterns , Patterns ! “**

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 )

**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**

**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 the 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 center.

###

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

*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 centered 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 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 :

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”.

Although

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

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

###
**“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 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 only for US market.

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

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

Hi Yaruzi,

DeleteThanks 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 !!

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..

ReplyDeleteHi Ali ,

ReplyDeleteYes , 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 !!