Normal distribution in share market

In earlier consideration, Nifty’s potential trading range was determined by its annualised volatility. We ascertained an upper and lower limit, suggesting that Nifty will stay within the established limits.

What level of confidence can we have regarding this? Is there a chance Nifty will not stay in the given parameters? If so, what is the likelihood it would go past the range and how likely is it that it would remain within it? If it does exceed the range, what are its boundaries?

Answering these questions is immensely valuable for various reasons, as it serves as a fundamental basis for examining markets in quantitative terms, unlike the typical processes of fundamental and technical analysis.

Let’s go a bit further and find our answers.

– Random Walk

This conversation is critical to the matter before us, and it should be quite engaging.

Galton Board which has pins affixed to its surface. Compartments lie underneath each pin where the items that drop from above can be collected.

The intention is to release a tiny ball from a higher position than the pins. Immediately it’s dropped, its trajectory converges with a pin, after which it could proceed either direction before meeting another one. This continues until it trickles down and lands in one of the bins below.

Once you drop the ball from the top, its path is out of your control. The Random Walk, as it is known, comprises an utterly natural trajectory that cannot be predetermined or manipulated.

What would be the result of dropping multiple balls consecutively? Each one will likely have their own unique trajectory before they end up in one of the bins. But what do you think the overall pattern of distribution would look like?

• Will the balls land up in the same bin?

• Will they all get distributed equally across the bins? or

• Will they randomly fall across the various bins?

Those unfamiliar with this experiment may assume that the balls will be scattered randomly across the bins, instead of forming any sort of order. However, there appears to be more structure here than first meets the eye.

It appears that when dropping multiple objects onto a Galton Board with each taking a random walk, the resulting pattern of distribution is quite specific.

• The majority of the balls tend to land in the central bin

• As you progress away from the main bin, there will be a decreasing number of balls in each one. Whether you look left or right, you should expect to see fewer balls as you move further out.

• The bins located at the extreme ends contain only a few balls.

This distribution is commonly referred to as the “Normal Distribution” or the bell curve, as you may recall from your school days. Regardless of how many times you repeat this experiment, the balls will consistently form a normal distribution.

This is the Galton Board experiment, a highly regarded experiment. Watch the following video to get a better understanding of the concept: 

So why do you think we are discussing the Galton Board experiment and the Normal Distribution?

Many phenomena in real life exhibit this natural pattern. For instance –

• Gather a group of adults and measure their respective weights, then sort them into bins, for example 40 to 50kgs, 50 to 60kgs, etc. This will generate a normal distribution if you count the number of people in each bin.

• Do the same with people’s heights, and you’re likely to get a normal distribution.

• People’s shoe sizes follow a normal distribution.

• Weight of fruits, vegetables

• Commute time on a given route

• Lifetime of batteries

This list is seemingly endless, but one variable worthy of note is the daily returns of a stock, which usually follows a normal distribution.

Daily returns of stocks or indices cannot be predicted – it’s like the random walk of a ball. However, by collecting daily returns across a certain period and looking at the distribution, one can observe a normal distribution or what’s known as bell curve.

To demonstrate this clearly, I have presented the distribution of the day-by-day returns for these stocks/indices.

• Nifty (index)

• Bank Nifty (index)

• TCS (large cap)

• Cipla (large cap)

• Kitex Garments (small cap)

• Astral Poly (small cap)

It is evident that the daily returns for stocks and indices exhibit a normal distribution.

Fair enough, now you must be wondering why is it related to Volatility. Let’s dive deep. 

 – Normal Distribution

Exploring normal distribution for the first time can be overwhelming, so I’ll explain the concept, connect it to the Galton board experiment and extrapolate it to stock markets. 

Data can be dispersed in several ways. The Normal Distribution is the most studied, but other patterns such as binomial distribution, uniform distribution, poisson distribution, and chi square distribution also exist. While further research needs to take place in this field, the Normal Distribution is probably the best understood presently.

The properties of the normal distribution provide insights into the dataset. The shape of the curve is defined by two values: the mean and standard deviation.

The mean is the central value which most values are concentrated around. This tends to be the average of a given distribution. For example, in the Galton board experiment, this would be the bin with the highest number of balls in it.

I can number the bins from left to right, starting with one and going up to nine. The fifth bin is indicated by a red arrow, and this is our ‘average’ bin. From here, we can see how the data spreads out on either side of our reference value – this spread is known as dispersion, or volatility in the stock market.

You should be aware that ‘Standard Deviation’ (SD) by default refers to the first SD. Likewise, there are a 2nd SD, 3rd SD and further ones. When I’m talking about SD then I mean simply the standard deviation value; 2SD is twice this value, 3SD is three times it, and so on.

For instance, let’s consider the Galton Board experiment with a standard deviation (SD) of 1 and an average of 5. In this case:

• 1 SD would cover the bins between the 4th bin (5 – 1) and the 6th bin (5 + 1), representing one bin to the left and one bin to the right of the average bin.
• 2 SD would cover the bins between the 3rd bin (5 – 2*1) and the 7th bin (5 + 2*1).
• 3 SD would encompass the bins between the 2nd bin (5 – 3*1) and the 8th bin (5 + 3*1).

Taking the above example into account, let’s discuss the general theory surrounding the normal distribution that you should be familiar with.

• Within the 1st standard deviation one can observe 68% of the data

• Within the 2nd standard deviation one can observe 95% of the data

• Within the 3rd standard deviation one can observe 99.7% of the data

Applying this to the Galton board experiment –

• Within the 1st standard deviation i.e between 4th and 6th bin we can observe that 68% of balls are collected

• Between the 2nd standard deviation, which includes the bins from the 3rd to the 7th bin, we can observe that approximately 95% of the balls are collected.

• Within the 3rd standard deviation i.e between 2nd and 8th bin we can observe that 99.7% of balls are collected

You – I’m about to drop a ball on the Galton board. Can you guess which bin it will land in?

Me – I cannot accurately predict the specific bin where the ball will land since its trajectory follows a random path. However, I can estimate the range of bins it is likely to end up in.

You – Can you provide an estimate of the range?

Me – It is probable that the ball will land somewhere between the 4th and 6th bins.

You – How confident are you about this estimation?

Me – I am quite confident that the ball will fall within the range of the fourth through sixth bins.

You – A 68% accuracy seems relatively low. Can you provide a more precise range?

Me – With a 95% level of confidence, I can guarantee that the ball will fall between the 3rd and 7th bins. For an even more precise prediction, I am 99.5% certain that it will be within the range of the 2nd to 8th bin.

You – So, there is no chance for the ball to land in the 1st or 10th bin?

Me – While the probability is very low, there is still a chance for the ball to land in bins outside the range of the 3rd standard deviation. 

You – How low is the probability?

Me – The likelihood of such an occurrence, often referred to as a ‘Black Swan’ event, is less than 0.5%.

You – Can you explain more about the concept of a Black Swan?

Me – Certainly. ‘Black Swan’ events are unexpected and rare occurrences with a low probability of happening. It is challenging to predict when and how such events will take place, but it is important to acknowledge their possibility. 

The image provided illustrates the scattered distribution of numerous balls, with only a few congregating at either end.

– Normal Distribution and stock returns

Above, we discussed the normal distribution and why it is relevant to stock returns. Knowing the mean and standard deviation of the stock return gives us more insight into how they behave or how much they differ from each other. As an example, let’s consider the Nifty index and conduct further analysis.

We can observe a normal distribution of daily returns. To calculate the average and standard deviation, we need to use log daily returns. If you need assistance with this, please refer to the previous chapter.

Daily Average / Mean = 0.02%

Daily Standard Deviation / Volatility = 0.75%

Let’s assume the current market price of Nifty = 12000

Now, let’s calculate the following:

1. Nifty’s likely trading range for the next 1 year:

Average = 0.02%

SD = 0.75%

Converting to annualized numbers:

Average = 0.02% * 252 = 5.04%

SD = 0.75% * Sqrt(252) = 11.92%

With 68% confidence, the range is:

Upper Range = Average + 1 SD = 5.04% + 11.92% = 16.96%

Lower Range = Average – 1 SD = 5.04% – 11.92% = -6.88%

Converting back to regular percentages:

Upper Range = 12000 * exponential(16.96%) = 14482

Lower Range = 12000 * exponential(-6.88%) = 10460

This calculation suggests that Nifty is likely to trade between 10460 and 14482 with 68% confidence for the next 1 year.

Increasing the confidence level to 95% (2nd standard deviation):

Upper Range = Average + 2 SD = 5.04% + 2 * 11.92% = 29.88%

Lower Range = Average – 2 SD = 5.04% – 2 * 11.92% = -17.84%

Converting back to regular percentages:

Upper Range = 12000 * exponential(29.88%) = 25133

Lower Range = 12000 * exponential(-17.84%) = 6013

With 95% confidence, Nifty is likely to trade between 6013 and 25133 for the next 1 year.

2. Nifty’s likely trading range for the next 30 days:

Average = 0.02%

SD = 0.75%

Converting to the desired time period:

Average = 0.02% * 30 = 0.6%

SD = 0.75% * Sqrt(30) = 3.27%

With 68% confidence, the range is:

Upper Range = Average + 1 SD = 0.6% + 3.27% = 3.87%

Lower Range = Average – 1 SD = 0.6% – 3.27% = -2.67%

Converting back to regular percentages:

Upper Range = 12000 * exponential(3.87%) = 12359

Lower Range = 12000 * exponential(-2.67%) = 11573

With 68% confidence, Nifty is likely to trade between 11573 and 12359 over the next 30 days.

Increasing the confidence level to 95% (2nd standard deviation):

Upper Range = Average + 2 SD = 0.6% + 2 * 3.27% = 7.14%

Lower Range = Average – 2 SD = 0.6% – 2 * 3.27% = -5.94%

Converting back to regular percentages:

Upper Range = 12000 * exponential(7.14%) = 13676

Lower Range = 12000 * exponential(-5.94%) = 10347

With 95% confidence, Nifty is likely to trade between 10347 and 13676 over the next 30 days.