Terms explained - Normal Distribution, Bayes Formula

Normal Distribution Explained from the Indian Stock Market Perspective

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.

Source - Moneylife Website

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Bayes Formula

Bayes Formula relates inverse representations of probabilities concerning two events.

Calculate and interpret an updated probability using Bayes Formula

P(A/B) = P(AB)/P(B) 

(Note - P(A/B) means Probability of A given B has occurred)

P(B/A) = P(BA)/P(A)

Hence,

P(AB) = P(A/B) * P(B)   P(BA) = P(B/A) * P(A)

Since, P(AB) = P(BA)

Hence, P(A/B) * P(B) = P(B/A) * P(A)

P(B/A) = {P(A/B) * P(B)}/P(A)

Total Probability Rule to find P(A) 
That is P(A) = P(A/B)*P(B) + P(A/B(compliment))*P(B(compliment))


Replace B with Event
Replace A with Information

P(Event/Information) = {P(Information/Event) * P(Event)}/P(Information)

To solve problems such as, 

In a school, there are 54% boys and 46% girls. The girl students wear Blue sweaters and red sweaters in equal numbers, while all the boys wear blue sweaters. An observer sees a random student wearing a blue sweater. What is the probability that the student is a girl?

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