Fat tails have a tale to tell
Nupur Pavan Bang  and  Khemchand H Sakaldeepi 11 March 2013

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.

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.

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!

(Nupur Pavan Bang is a senior researcher at the Centre for Investment, Indian School of Business, Hyderabad. Khemchand H Sakaldeepi is a researcher at the Centre for Investment, Indian School of Business, Hyderabad.)