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 ……… }