Application of Markov Chain

Following to the article of “EMFPS: Efficient Portfolio of Assets (The Optimization for Risk, Return and Probability)” on the link: http://emfps.blogspot.com/2011_10_09_archive.html

I would like to remind you about Step (7) – Part (A) where we fix Rp and change probabilities into limited range. But how can we fix the probabilities distribution into a limited range? It means that we should anticipate the probabilities distribution for the future. Here I have used from homogeneous Markov Chain as a tool to foresee the probabilities distribution.

Now, let me start again the problem with the example mentioned on above article as follows:

We had the data for the forecasted returns of assets A, B, C, D, E, and F from 2012 to 2017 as follows:

				Assets Return (%)
Year		A	B	C	D	E	F
2012		7	19	7	25	8	17
2013		9	16	11	21	10	15
2014		11	14	13	19	12	13
2015		14	12	16	15	14	11
2016		18	10	20	12	16	9
2017		21	8	23	9	18	6

I consider the people in city or country or the location (M) who are the owners of assets A, B, C, D, E, and F. I assume the owners of each asset as a particular area or state are dealing as follows:

Owners of Assets                 Particular Area (State)

A                                                             S1

B                                                             S2

C                                                             S3

D                                                            S4

E                                                             S5

F                                                             S6

At the first, we need to obtain recent data of the total transactions for each particular area for instance, how many people who are the owners of asset “A “would like to deal only into area “A” or liquid their asset to go area B, C, D, E or F (in the period of exact time). The approach will be certainly based on the balance of Risk and Expected Return rate. Therefore, we have to highlight them that the conditions included Risk and Expected Return is equal for all six assets.

How can we find statistics data for the whole of the transactions?

It is clear; there are two types of the data to collect:

1) Primary data: By using of distributing the survey and questionnaire among the people to know their interest.

2) Secondary data:   Refer to historical data collected from internet and so many finance websites for instance, the Volume of transacted shares in a Stock Index during the period of the last time.

Definitely all people do not fill our questionnaire accurately. Therefore, we should have a cross –section of primary and secondary data. On the other hand, we should have a good PEST analysis and then a good Industry analysis for each asset to confirm the combination of primary and secondary data in which we should perceive all Economic, Politic, Society -Cultural and Technology indicators which are affecting on all transactions.

Now, assume we have made our final collected data as follows:

-In Area S1: 30% of people are interested in dealing into area S1, 12% of people are interested in liquating their asset and deal into area S2, 18% deals into S3, 20% deals into S4, 15% deals into S5, 0.05% deals into S6

- In Area S2: 10% of people are interested in dealing into area S1, 20% of people are interested in liquating their asset and go into area S2, 30% go to S3, 0.05% go to S4, 17% go to S5, 0.18% go to S6

-In Area S3: 20% of people are interested in dealing into area S1, 10% of people are interested in liquating their asset and go into area S2, 12% go to S3, 21% go to S4, 18% go to S5, 21% go to S6

-In Area S4: 10% of people are interested in dealing into area S1, 18% of people are interested in liquating their asset and go into area S2, 15% go to S3, 25% go to S4, 10% go to S5, 20% go to S6

-In Area S5: 10% of people are interested in dealing into area S1, 13% of people are interested in liquating their asset and go into area S2, 25% go to S3, 16% go to S4, 14% go to S5, 25% go to S6

-In Area S6: 20% of people are interested in dealing into area S1, 0.09% of people are interested in liquating their asset and go into area S2, 13% go to S3, 17% go to S4, 25% go to S5, 15% go to S6

To simplify above information, we can use from a matrix which is named as transition or movement matrix (S) as follows:

Matrix (S) =

	S1	S2	S3	S4	S5	S6
S1	0.3	0.12	0.18	0.2	0.15	0.05
S2	0.1	0.2	0.3	0.05	0.17	0.18
S3	0.2	0.1	0.12	0.21	0.18	0.21
S4	0.1	0.18	0.15	0.25	0.1	0.2
S5	0.1	0.13	0.25	0.16	0.14	0.25
S6	0.2	0.09	0.13	0.17	0.25	0.15

We have the starting of the movement (deals) on rows and the ending of the movement (transactions) on columns in the period of the exact time.

Let me model the problem as a Markov Chain (homogeneous) to reach the fixed probabilities distributions in the future (at least until 2017 year in this example).

Of course, we should know that there are three conditions (properties) for each problem to be considered as a Markov Chain as follows:

-Each one of the deals done in this system stays a Risk and Expected Return exactly equal after the transaction for all assets in the period of the distinct time.

-The total percentage transactions made by the people into each area must be equal to 1.

- The primary probabilities distribution is not changed over the distinct time (Matrix (S)).

I as well as know that my assumptions are not exactly accurate. But to solve any problem, we should be able to simplify a complicated problem into boundaries conditions in which we need to sure if our assumptions are reasonable. In fact, there is the fundamental difference between the accurate and the reasonable.

If we consider that Matrix (S) is for the first transaction, we will be able to anticipate the probabilities for the second transaction as follows:

S^2 =

0.1794	0.138	0.1856	0.1863	0.1513	0.1594
0.1597	0.1293	0.1874	0.1633	0.1768	0.1835
0.1675	0.1337	0.1806	0.1832	0.1643	0.1707
0.16	0.1414	0.1821	0.177	0.1646	0.1749
0.1605	0.1289	0.1731	0.1779	0.1757	0.1839
0.1648	0.1328	0.1879	0.1818	0.1597	0.173

After the third transaction:

S^3 =

0.167449	0.135237	0.182456	0.179637	0.16344	0.171781
0.165079	0.132657	0.182574	0.178067	0.166625	0.174998
0.16546	0.134598	0.182678	0.179218	0.164359	0.173687
0.164409	0.134689	0.183509	0.17763	0.165285	0.174478
0.164464	0.133764	0.182849	0.178746	0.165509	0.174668
0.165867	0.134181	0.181737	0.179471	0.164906	0.173838

After the fifth transaction:

S^5 =

0.165483	0.134269	0.182655	0.178807	0.164941	0.173846
0.165455	0.13424	0.182622	0.17884	0.16496	0.173882
0.165456	0.134254	0.182647	0.178809	0.16496	0.173875
0.165446	0.134232	0.182638	0.178808	0.164981	0.173895
0.165443	0.134243	0.182634	0.178818	0.164971	0.173891
0.165457	0.134257	0.182643	0.178814	0.164955	0.173874

After the eighth transaction:

S^8 =

0.165457	0.134249	0.182641	0.178815	0.164962	0.173877
0.165456	0.134249	0.182641	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877

After the thirteenth transaction:

S^13 =

0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877
0.165456	0.134249	0.18264	0.178815	0.164962	0.173877

As we can see, after 13 deals and the ending of the period of distinct time, the probabilities distribution is equal for all areas of S1 to S7 as follows:

Areas	S1	S2	S3	S4	S5	S6
Assets	A	B	C	D	E	F
Pr	16.55%	13.42%	18.26%	17.88%	16.50%	17.39%

It has been presented us that asset “C” will be the best option to invest in the future. And so we can take above fixed probabilities distribution for our analysis in said article.

Another application of this Markov Chain (homogeneous) is to track the number of the petitions and deals divided into each area after ending time of all transactions.

Assume we have 100 petitions to deal for each area in starting time of the transactions. We replace the number of petitions into a row vector (transposed vector) which is named “X” as follows:

S1          S2        S3       S4        S5      S6 

100       100      100     100       100    100

X = [100,100,100,100,100,100]

^{After the first deal, we will have below situation for the number of deals:}

^{X * S = [98, 82, 113, 104, 99, 104]}

^{After the thirteenth transaction, we will have below situation for the number of deals:}

^{X * (S^13) = [99, 81, 110, 107, 99,104]}

^{As we can see, the number of deals into each area in the starting time will be approximately equal to the ending time of the transactions.}

Note:  “All spreadsheets and calculation notes are available. The people, who are interested in having my spreadsheets of this method as a template for further practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call me on my cellphone: +989109250225.   Please be informed these spreadsheets are not free of charge.”

 

^{To be continued ………} 

The Constant Issues, Universal Laws and Boundary Conditions in Physics Theory

How can an Industrial Revolution be raised?

During the period of several centuries ago, the Human discovered some of the constant natural numbers such as Pi, e, G and so on that each one of them had the influence on development of the world. In fact, the constant numbers were the origin of the Universal laws as the constant points for instance; the Newton’s law of universal gravitation was inferred from “G” (Universal Constant). Therefore, the answer to above question is to find out the constant numbers in which they give us the vita, the life, the energy and Dynamic. For example, please consider the sun and other stars which are the constant points and give us the dynamic. Another example is about someone who stays in the job or the location for several years (the constant) and gain the great credit to launch new business and dynamic. In the financial analysis, if we find any accurate constant data other than the abruption data as our assumption, we can expand a new dynamical idea by using of the cause – effect system and measure the impact of the constant data by a Balance Scorecard framework. These are the constant points.

Now, the question is about the total impact of a constant number. It depends on the period of the sustainability and the stability time for constant data. For instance, if the data or number is the constant only for several years, definitely the total influence of data on the systems dynamic is limited. Of course, the debate can be very challenging because we assume that all data or numbers are the stable on the period of limited time. Actually, there is not any constant natural numbers but we have the boundaries for them such as “e” and diagram of y = 1 / x

The process of dynamic for a constant natural number or point can be started by a wave then we will have high frequency wave and after a resonance, probably it will go toward a very, very high frequency wave and finally we will not have any wave but it is a perpendicular line on the time axis just like to a spark. This area is the same boundaries because the properties of the points on the line are variable.

It means if we discover a constant natural number or point, we can generate a dynamic system or we can say the behind of any dynamic system, we have the constant number or point. A constant natural number or point can be a new idea or a new strategy in which the engineers or strategists can use them to produce a new dynamic system.

According to above mentioned, I stated two statuses: 1) The starting point by using of the constant number 2) The ending line (reach to boundaries)

Third status is very important and it is to move on the boundaries. Even though the movement on the boundaries is very risky and dangerous, it can be gained to the change into a new system in which we have again a starting point and the huge new opportunities.

Let me bring you three examples as follows:

-Please consider you are driving on highway. How can you go on speed line? Firstly you should reach on the boundary of the speed line then you should move on the boundary finally you will enter on speed line (new system)

-In the climate change, firstly the clouds are moving toward the boundary then we will have a spark because of the cloud’s movement on the boundary finally we will have a climate change or new climatology system (rainy weather).

-In puzzle game, assume you have several choices (locations) for a puzzle. Please close your eyes and go on the boundary of the puzzle by your finger then move on the boundary. Finally you will discover which location is the matched to your puzzle.

In the result, let us review again all three statuses as follows:

1) The starting point by using of the constant number: The good example is Newton’s laws of Motion. In this case, we can see a new idea or the constant point had been generated by cross –section of Art (Leonardo da Vinci 1452-1519), Philosophy (Newton 1642–1727) in which the case had been changed to the most important Universal law in the world.

2) The ending line (reach to boundaries) or reaching to the critical point: The good example is the relative theory by Albert Einstein in which the base of this theory is to assume the boundary for the light’s velocity (V = c). In this case, we cannot still move on the boundaries equal to the light’s velocity. Therefore, we are limited the applications and gains only by using of the boundaries conditions because we do not still know all real dimensions on this space. It is why we have so many models of quantum mechanics in the world. Sometimes we have to find new or free boundaries and utilize from the Scenario analysis and Simulation method to solve the problem. For instance, in the Steam Generators (Boilers), we can only design a boundary for pressure and temperature during the period of the time by using of Simulation method. Have you ever seen a Boiler with pressure capacity 20,000 Bar?

 3) The movement on the boundaries: ……………

Perhaps, the best way is to generate new constant number and to obtain the applications of it in the real life. It means to return back to Status (1).