Monday, November 14, 2011

Efficient Portfolio of Assets: Markov Chain & the Constant Eigenvector

Following to my previous article of EMFPS: Efficient Portfolio of Assets (CON): Application of Markov Chain” posted on the link of “http://emfps.blogspot.com/2011/11/emfps-efficient-portfolio-of-assets-con.html“, I am willing to continue my debate about Zero –Risk (Risk free). How can we find the zero- risk on portfolio of assets?
At the first, we should bear in mind that there is the fundamental issue of Markov Chain which is as follows:
“If Matrix (A) is the result of Markov chain, Vector [ai1] will be the constant Eigenvector of Matrix (A), Matrix (A') and Matrix (A^k) in which all elements of vector [ai1] are the same or vector [ai1] is the Scalar Multiplication of unit vector. It means that: A* c* [ai1] = (lambda)*c*[ai1] where:  ai1 = 1, Eigen value (lambda) = 1, Scalar Multiplication = c”
For example, assume we have:
 Matrix (A) =

0.3
0.12
0.18
0.2
0.15
0.05
0.1
0.2
0.3
0.05
0.17
0.18
0.2
0.1
0.12
0.21
0.18
0.21
0.1
0.18
0.15
0.25
0.1
0.2
0.1
0.13
0.25
0.16
0.14
0.25
0.2
0.09
0.13
0.17
0.25
0.15


Matrix (A) ^2 =

0.179
0.138
0.186
0.186
0.151
0.159
0.16
0.1293
0.187
0.163
0.177
0.184
0.168
0.1337
0.181
0.183
0.164
0.171
0.16
0.1414
0.182
0.177
0.165
0.175
0.161
0.1289
0.173
0.178
0.176
0.184
0.165
0.1328
0.188
0.182
0.16
0.173


Unit Vector (a) =
1
1
1
1
1
1
And, C =12
[Matrix^2] * C * Unit vector (a) = (lambda) * C * Unit vector (a)
12
12
12
12
12
12
Where: lambda = 1
Now, let me turn back to the article of “EMFPS: Efficient Portfolio of Assets (The Optimization for Risk, Return and Probability)” on link: http://emfps.blogspot.com/2011_10_09_archive.html
In that article, we wanted to decrease the risk of portfolio assets in given expected return rate.
If we assume the matrix of assets sorted by annual return and the time is completely compatible with Matrix of Markov chain, we are able to reach the zero – risk (Risk Free).
 As I already stated that it is clear, our 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.
I think the assumption of total sum return rate of all assets for each given time just equal to 100% could be considered the reasonable because we can increase varieties of the assets into our portfolio where the total return rate will be equal to 100%. This is the most important assumption to apply Markov chain in this article.
Therefore, the steps of reaching to Zero – Risk are as follows:
-Total sum return rate of all assets is obtained just equal to 100% for the given time for instance one year.
-Referring to above mentioned, Matrix (X) which is included the proportions of each asset will be equal to Matrix (P) which is Expected Portfolio Return Annually (Rp). In this case, we should not only consider the number of years that we are anticipating but also we should make a square matrix. In the result, the elements of both matrixes will be equal to:
(100 / number of years) %
-The most important step is to be equal all related probabilities distribution for outcomes. In other word, it is to be fixed Expected Portfolio Return Annually (Rp) during the period of one year. The access to the constant return rate is very challenging. As I mentioned in my previous article of “Efficient Portfolio of Assets: Application of Markov Chain (CON)” on below link: http://emfps.blogspot.com/
“This is a game and maybe the application of the Game Theory will help us to find the best analysis. Application of the Game Theory after PEST, Industry and SWOT analysis will guide us to find how much percentage of the shares and which ones should be purchased or sold in which the total action will be affected on Pr and Pe.”
Here is an example:
Assume we have the portfolio of assets as follows:

Year
A
B
C
D
E
F
2012
0.15
0.19
0.16
0.25
0.08
0.17
2013
0.18
0.16
0.20
0.21
0.10
0.15
2014
0.16
0.14
0.16
0.19
0.17
0.18
2015
0.18
0.20
0.16
0.15
0.14
0.17
2016
0.18
0.18
0.20
0.12
0.16
0.16
2017
0.21
0.18
0.23
0.14
0.18
0.06


The elements of matrixes of (X) and (P) will be equal to:
100 / 6 = 16.7 %
In the result, we have the Standard Deviation of expected portfolio returns (Qr) and Coefficient of Variation (CV) approximately equal to zero. Of course, the easiest status (without any challenge) to minimize the risk is to assume a low return rate as expected return or expected value. As you can see, the return rate on a U.S. Treasury bill is the free risk.
Now, let me expand this application for another field. I would like to refer you to my article of “Where money goes? Where power comes from?” On below link:
http://emfps.blogspot.com/2011_02_20_archive.html
Where I stated: “If a function of w=f(x,y,z,t) wants to move far from a point (p), the change of amounts for this function depends on its direction”
Assume:   Grad “f’ = ai + bj+ ck
If we have: a+b+c = 1
Then, the directional derivative on function “f” in which the point P(x1, y1, z1) in the space is moving on direction of unit vector always will be equal 1.

Now, we consider that we have so many functions such as “f, g, w, h…
If we include all directional derivatives of the functions into matrix (X) as follows:
Matrix (X) =
Grad “f”
Grad “g”
Grad “w”
Grad “h”
There is a Eigenvector equal to unit vector which is responding to Eigen value equal to 1 (lambda = 1)



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 ………