This article was
first published in Moneylife on March 11th, 2013: Co-Author- Khemchand H. Sakaldeepi
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In
reality, we have experienced 5, 6, 7 or even more than that, sigma events more
frequently than what the normal distribution suggests and we dare to accept.
Between
1998-2013, out of a total of 3,785 days, movement in the CNX 500 was outside 3
sigma on 60 occasions, that is 1.59% of the total. By normal distribution, less
than 0.03% observations should fall outside the 3 sigma
In
the world of investments, returns are measured by the first moment of prices
(mean) and the risks are measured by the second moment (standard deviation or
sigma). Most of the classical theories of finance are based on the assumption
that the returns are normally distributed. In the probability theory, the
normal distribution is a bell shaped curve of probability values for various
natural events—hence the word ‘normal’. This distribution assumes that the
tails or the ends are flatter and extreme events are rare. For example, this
means that the probability of returns moving more than three standard
deviations beyond the mean is 0.03%, or virtually nil. But what is ‘normal’
in markets?
In
the Indian context, taking daily CNX 500 data from 1 January 1998 to 28
February 2013 (more than 15 years), 99.73% of the daily returns should ideally
fall within -4.97% and 5.09%. Or less than 0.03% observations should fall
outside the 3 sigma.
Out
of a total of 3,785 daily observations during the period of analysis, 60 times
the returns were outside 3 sigma in the case of CNX 500, that is 1.59% of the
total observations. Clearly much more than we bargain for. The rule book says
that if we are looking at daily events, a 5 sigma event would occur once in
4,776 years. A 6 sigma event would occur once in 1.388 million years and after
that, the numbers are, let's just say too big to bother.
On
17 May 2004, the financial market experienced a more than 7 standard deviation
fall, when markets crashed due to political uncertainty. Markets fell more than
5 to 6 standard deviations many times in 2007 and 2008, owing to global melt
down. Similarly, the market posted a more than 9 standard deviation gain, once
again due to the political scenario in the country at that time.
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This
is true globally, not just in India. For instance, Goldman Sachs, Citigroup,
UBS, Merrill Lynch, all experienced large (as large as 25) sigma events on
multiple days in 2007 and 2008. There was the South East Asian crisis, the 11
September 2001 attacks on the World Trade Centre, the Euro crisis, all in the
past two decades.
It
is not just that these events occur more frequently, these events have greater
impact, as well. The impact is, in fact, higher due to the surprise element
attached to them. It hits one at the place where it hurts the most and makes it
very difficult to recover.
Our
observations suggest that the distribution is more leptokurtic in nature, with
fatter tails. This means that more observations are concentrated around the
mean and tails are fatter, or have greater number of observations than
suggested by the normal distribution.
So
what we must do is first, acknowledge the limitation of our knowledge that we
cannot explain everything and second, we must believe that such events occur
more frequently than we had thought. This must call for better risk management
systems. Perhaps these events indicate that we must prepare for more
incorrigible things that will happen.
What
this also points to is that the assumption of normal distribution does not
hold. Hence, financial mathematicians must look at distributions with fatter
tails for building their theories and models.
Additionally,
Daniel Kahneman’s prospect theory says that humans are more likely to act to
avoid loss than to achieve a gain, articulated very well in his book “Thinking
fast and slow”. If we accept this to be true, then it becomes all the more
important for the theorists and professional money managers to rethink the way
they build models or the appropriateness of the models which they use.
As
for the investors, it would be wise to question their financial advisor on the
soundness of their advice during a large sigma event!
