Crystal Ball for Nifty Using Transition Probability Matrix

None | Jan. 23, 2019, 3:35 p.m.

There has always been a lot of uncertainty regarding monsoon in India.It seems like Indra the rain God - reads the latest weather forecast, and acts exactly opposite just to screw up the data.

But generally, if it has rained today, will it rain tomorrow as well? It will depend on the season, of course. In monsoon, the chance of another rainy day may be higher, whereas in winter, it may be 50-50. If it is an average summer, the chance of second rainy day will be very less, as most days are expected to be sunny.

After this British style of opening up a discussion with chats of weather, how does this relate to Nifty?

Every time the market heats up or crashes down, an investor wonders about tomorrow. Nifty has risen 40% in 52 weeks. Where does that lead us from now? Will the stride continue or are the times a-changin'? The next year’s estimates by broker houses depend on a lot of factors, including the analyst’s mood that day. But what does history say when such events occur?

For this, we will first define our ‘seasons’, where we can expect different price movements. We have a price chart of Nifty along with its trend limits. We will have 6 seasons as defined by the placement of price in a particular limit.

Thus Seasons go from costliest to cheapest, from 1 to 6.But is it necessary to have 6 different seasons? Does the price really behave differently in different seasons, or can we use the whole period to study at one go? The following seemingly complicated chart holds the answer.

This chart tells the story of all the seasons. Each panel represents the season (marked by the title of every panel), and the chart shows the distribution of 52-week returns in that season. We can clearly see that the distribution of returns is different in every season. As we go from season 1 to season 6 that is go from Upper limit to lower limit, the return distribution keeps on shifting towards right, so that more and larger wins appear with lesser and smaller losses.

Hence it makes sense to study every season separately, rather than studying whole period together.

Now that the six seasons are defined, we will first see the general tendency of price to belong in each season. Using the weekly data, following is the distribution of seasons along with the vital statistics of every season.

This table gives detailed insight about every season. We can see the number of weeks (or the probability of times) that each season lasted, with the average, minimum, maximum and median returns. Lastly, we can see the probability of positive & negative returns in next52 weeks for every season.

We can see that as the seasons change, the average annual returns for coming period turn progressively higher. It shows the harder the season today, the better the next year’s harvest will be. And it makes sense to study each season separately, rather than studying the whole period as one.

The extreme seasons- Season 1 &Season 6 - are very rare. The probability of price lying in these seasons is extremely low, about one in a hundred times. The profits and losses in these cases are sure events.

Season 1 results surely in loss (with Probability of negative returns 1.00 with -46% average loss).

Season 6 results surely in profit (with Probability of positive returns 1.00 with 91% average profit).

Coming slightly nearer to the trend, seasons 2 & 5 are still rare, but not as much as season 1 & 6. They occur 8 to 17 times out of 100. The returns follow similar pattern.

Season 2 results most probably in loss (with probability of negative returns 0.98 with -27% average loss)

Season 5 results likely in profit (with probability of positive returns 0.61 with 29% average profit).

Closest to the trend, the seasons 3 & 4 are the most frequent seasons, with season 3 taking almost half of the trading horizon. 48 times out of 100, the price lies in season 3.

Season 3 has average returns of 10%, with positive returns in 63 times out of 100.

Season 4 has average returns of 27%, with positive returns in 77 times out of 100.

Since Season3 is spread over a long time, we will study these further to get better understanding of movement of price.

A question to proceed: Given that Nifty is in Season 3 AND it is up 40% in past 52 weeks. What is the expected return in next year?

This is a real-time question. The Nifty Index right now is actually in season 3, and has returned 40% from in past 52 weeks till the end of August 2014. What to expect in next 52 weeks?

The question is really in 3 parts, first two giving information, and last one seeking information:

•  Part I: Current season

•  Part II: Latest 52-Week returns

•  Part III (actual question): Next 52-week returns?

We already have information about the current season that the price lies in, as well as the latest returns.The problem thus gets nearer to its solution.

We will now use ‘Transition Probability Matrix’. Considering our example at the beginning, there are two states a day can be in, either rainy or sunny (ignoring any other state). Then given that it is a sunny day today, tomorrow it can either transition to a rainy day or remain sunny with some probabilities. The table that summarizes these probabilities of transitions is called ‘Transition probability matrix.’

P1: Transition Probability of Sunny day to Rainy | P2: Transition probability from Rainy Day to Sunny

Since it can be either rainy or sunny day, the horizontal sum of all rows will be 1.

Given that you have information of the season as well as the current state of the latest monthly returns, the following table will give you the future roadmap of the price given its current state.

Transition Probability Matrix for Season 3 to read this puzzle?!

It takes a little focus to read the findings; this is not an ‘In-Your-Face’ information table. The Transition Probability Matrix gives the probabilities of transition to various states given today’s state.

First, make sure you are reading correct table. Cross check whether it is really season 3 for today’s price. (It is.) Then, in the first column, find out the current state of the price, as per latest annual return. As it is 40% for Aug 14, choose the row ‘Above 30%’ in first column.

I will fade the other rows now, so that we can see the desired row only.

Transition Probability Matrix for Season 3

Let us read the row. The column ‘Weeks’ tells the number of times this phenomenon has occurred, when the price was in Season 3 and had a return of 30% or more. There have been 230 such weeks in history of Nifty.

The next set of colorful columns depicts the returns in next 52 weeks with their probability. The colors show the ‘heat’ of the series.The darker the reds(or greens), the lowest (or highest) the probability of that return. As can be seen, positive returns are more probable (green) than losses (colored in red).

So to answer our question: given that the Nifty is in season 3 and the 52-week return is more than 30%, these are the possibilities for next 52-week return:

1.  Less than -20% return with probability 0.10

2.  Between -20% and -10% return with probability 0.05

3.  Between -10% and 0% return with probability 0.10

4.  Between 0% and 10% return with probability 0.05

5.  Between 10% and 20% return with probability 0.04

6.  Between 20% and 30% return with probability 0.06

7.  Above 30% return with probability 0.60

This is how one can read a Transition Probability Matrix. This is just a starting step in an exciting possible study under Time Series Analysis, but for that, we need to cross check a number of assumptions before using advanced techniques.

The changing of seasons is not a sudden phenomenon (except the extremes), so you don’t need to worry going through different tables every time. Currently Nifty is in season 3, and unless a drastic change occurs, it will likely stay in that state for a good amount of time.

This tool gives an additional edge to an analyst to confirm/reassess his short and long term views about the markets. Since most stocks have a large beta, a good idea about the future movement of the index is a valuable tool in the pursuit of perfection.

Even though Nifty is not currently in Season 4, I am putting its TPM because it is the second most recurring season.

Transition Probability Matrix for Season 4

For season 4, clearly profit is highly probable than loss in future irrespective of the current return.

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