## Amazing! with the help of volatility and normal distribution pattern we can find the top and bottom of the stock; get the details here (Statistical Analysis)

##### Wed, Sep 5, 2018 3:43 PM on Beginner's Guide, Exclusive, Latest, Recommended, Stock Market,

*-Krishna Khatiwada*

Random Walk Theory says that the stock returns are random and predictability of movement by past performance or any other information is very tough but amazingly, in our research, the daily returns of the NEPSE index were normally distributed.

*(-1, +1 represents the 1 ^{st} positive and negative SD and others numbers correspondingly represents the respective SD)*

In the normal distribution pattern, most of the values or event lie around the mean value and other values are equally distributed from the mean value. The ideal normally distributed figure is given above.

We have researched on major global indices and amazingly, their returns are also normally distributed. Random things are not always random; there is a certain pattern they follow and in most of the cases, they follow the normal distribution pattern. We will first understand the volatility, then we will come back to normal distribution pattern. We are sure that the information given below will be very efficient and effective for your investment career so please stick with us.

**What is volatility?**

Volatility measures the fluctuation in the stock price. If the average annual returns of the stock "A" is 3% and let's say the annualized volatility is 25%. Then this means that the stock price will fluctuate in the range of Exponential power of 28% on the upper range and Exponential power of -22% on the lower range. In simple words, the volatility is the standard deviation of returns of the stock or index.

For example, if the current market of the price of stock "A" is Rs 1,500 then according to the above assumption, the price of the stock will fluctuate between the ranges of Rs 1,984 to Rs 1,203. Thus, with the idea of volatility, we will know the probable lows and highs of the stock for any period of time (days, weeks, month). In case of Nepal, the information on volatility and average returns are not easily available. So forth, we need to calculate the volatility and average annual returns on our own.

**How to calculate the volatility and average annual returns?**

First, we need to find the daily average returns of stock of at least two years. For that, we have to collect the data of daily closing price and we can get that data from the page https://www.sharesansar.com/price-history

For index history data we can use https://www.sharesansar.com/index-history-data

First we have to calculate the average returns by the formula Log_{e }(today's closed price/yesterday's closed price). In this way we can find the average daily returns for the long period of time. We have calculated the same in excel file.

We have calculated the daily returns of one year (235 trading days). Now, we need to find the average daily returns by dividing the sum of daily returns by the number of period. Then we need to find the standard deviation. Standard deviation simply expresses the average deviation of the numbers from the mean value. To find the Standard deviation, we need to find the average variance and then we need to square root the variance. In excel sheet we can find S.D by using the direct formula. In our calculation, we got the daily standard deviation or volatility of 1.13% and daily returns of -0.13%.

Now, to convert the daily volatility into the yearly volatility. We have to find the average number of trading days in one year. Here in case, we have total 235 trading days in a considered year.

Now,

Annualized volatility= daily volatility * square root of 235 (daily volatility *square root of a number of trading days in one year)

Annualized return = {(daily returns + 1) ^235 -1}*100

We have calculated the same in excel sheet and we got an annualized volatility of 17.34% and annualized returns of 0.31%.

By using this information, we can calculate the most probable highest and lowest point of NEPSE index for next 12 months and can enter and exit the market at the bottom and at the top.

To bring more accuracy in finding the top and bottom, we can use the normal distribution pattern of NEPSE index.

The graph below shows the frequency of daily returns of NEPSE index over a year. As we can see in the graph, most numbers of returns are revolving around the mean value. Most of the daily returns are in the range of -1.53% to 1.59%. The graph is not a perfect normal distributed pattern.

To get the better idea we have simplified the graph with the use of S.D (S.D represents the volatility)

As we can see from above graph that 181(77%) daily returns out of 235 days lies in the 1^{st} standard deviation and 217(92.3%) daily returns out of 235 days lies between the 2^{nd} standard deviation. In an ideal normal distributed pattern, 68% of the events lies between the 1^{s }standard deviation, 95% of the event lies between the 2^{nd} standard deviation and 99.7% of the events lies between the 3^{rd} standard deviation.

Now, we can calculate the probable low and high of the index for next year.

The upper range for 1SD is "average return +1SD" and the lower range is "average return -1SD".

*(Annual SD value is considered in our calculation and we can find the range of any time period by using SD of the same time period)*

From the above data

Upper range= -0.31+17.34%= 17.03%

Lower range = -0.31%- 17.34%= -17.65%

This % are the log percentage thus to find lower and upper value, we need to multiply index current value with the exponential power of these percentage. NEPSE is currently trading at 1190 points.

So, the Upper range = **1190*e^ ^{ (17.03%) }= 1410 **

Lower range = **1190*e^ ^{ (-17.65%)} = 997**

Now we can say that, we are 77% confident that index lowest value will not exceed 997 points and upper value will not exceeds 1410 points in next one year.

We can increase our confidence level by considering 2^{nd} standard deviation. Now we are increasing our confidence to 92.3%.

Upper range= -0.31% + 2*1SD = 34.37%

Lower range = 34.99%

With the similar calculation, we can say that we are 92.3% sure that NEPSE will trade between the range of 1678-830 points, next year.

If you want more confidence then, we can include 3^{rd} Standard deviation in our calculation.

*(Don’t treat these value as absolute)*

**How to utilize the above information?**

If NEPSE is trading in the range of 997 points then it's time to buy as we are 77% confident that price won't go beyond that but still if it goes below 997 points then we buy heavily if it is in trading at the level of 830 points as we are 92.3% confident that price will not go beyond that. Note that we can never find the absolute top and bottom of any market and values given above are the most probable values.

ShareSansar's newly developed web-based software SS Pro has calculated the VaR (Value-at-Risk) through the concept of volatility. VaR is given in the section of Tearsheet of SS Pro software.

As you can see in above picture, 1 week VaR of Agriculture Development Bank at 5% is 13.283 which means we are 95% confident that the ABDL stock will not lose more than Rs 13.283 in next week. If we want more confident, then we can look at the VaR of 1% which signifies the confidence level of 99%.

**Key things to remember:**

- Standard Deviation is the volatility
- Returns of NEPSE index follow the Normal distribution pattern.
- In the normal distribution pattern, most of the returns lie in the 1
^{st}standard deviation. - As we increase the Standard deviation, our confidence on the value increases.