As we know, the
finance could be divided into two fields: 1) Asset Pricing 2) Corporate finance
1) Asset
Pricing: In this case,
the investors are willing to know how they should cope with their assets. What
is the price of the assets? Should they sell their assets or purchase new ones?
Is there any business model to liquidate the assets immediately?
In this article,
I will not have any debate on asset pricing models but I can refer you to my
research proposal of “Application of Game Theory as a New Strategy Implementation in the
Residential Real Estate Market” where the
combination of Markov
Chain and Monte Carlo Simulation Model plus the Constant Eigenvector approach (mentioned on link: http://emfps.blogspot.com/2011/11/efficient-portfolio-of-assets-markov.html) have
been applied as research methodology. The final result will be about how the investors
can liquidate their residential properties in the stagnated market.
2) Corporate finance: It is about the firm’s concern and finally decision
making in finance such as capital structure and leverage, dividend policy and
so on. There are many applications of the Game theory in corporate finance. In
this article, I am willing to discuss about Dividend Signalling theory which is
a type of Dividend Relevance Theory. My example will be the case of Gainesboro
Machine Tools Corporation on below link: http://emfps.blogspot.com/2012/03/case-analysis-of-gainesboro-machine.html
By the constant
or increasing dividends, the firm can show the positive signals about the
future prospects of the company consequently an increase in share price and
vice versa, absence of dividends or decreasing dividends presents negative
signal resulting in decline in share price. Of course, this is still a traditional
theory. In the real world, due to the uncertainty in business and financial
risks, nowadays the firms are following the Game theory in which the
repurchasing (buy back) of stocks by the company could also be a positive signal
to increase of the share prices.
Before going to
the application of the game theory on this type of dividend policy, let me have
an overview on the statistics as follows:
Permutations & Combinations
What is the
total number of outcomes when you have “n” possibilities and you choose “r’ of
these possibilities? Definitely it depends on the type of combination:
-If the replacement
does not matter, it is a combination. For instance, 1234 = 4312
-If the replacement
does matter, it is a permutation where 1234 is different with 4312
On the other
hand, we have permutations & combinations with and without repetition.
Example (1): In Backgammon
game, we have two
dices. What is the probability of each status after throwing dices?
Firstly, we
should find the number of total combinations. Which one of above mentioned
should we use?
-Permutation
with repetition
-Permutation
without repetition
-Combination
with repetition
-Combination
without repetition
Since the
replacement does not matter (because 3 &4 is equal 4&3) and also we can
have repetition status such 6 & 6, we should apply the formula for the
Combination with repetition as follows:
C (n + r -1, r)
= [(n + r – 1)!] / [r! (n – 1)!]
Where we have:
n = 6 total possibilities
r = 2 our choice
C (n + r -1, r) =
7! /
2!(6-1)! = 21 number of outcomes
X = each outcome
P (X) = probability of each outcome
P (X) = 1 / 21 = 4.8%
Example (2): In the case of Combination without repetition, I assume
we have the set of possibilities A = [1, 2, 3, 4, 5, 6, 7, 8] and we have 3
choice. What is the total number of combinations without repetition?
We can use from Binomial formula as follows:
C (n, r) = n! / [r!(n – r)!]
Where:
n = 8
r = 3
C (n, r) = 8! / [3!(8 -3)!] = 56
We can also use from Pascal’s Triangular to get all
outcomes:
0
|
1
|
||||||||
1
|
1
|
1
|
|||||||
2
|
1
|
2
|
1
|
||||||
3
|
1
|
3
|
3
|
1
|
|||||
4
|
1
|
4
|
6
|
4
|
1
|
||||
5
|
1
|
5
|
10
|
10
|
5
|
1
|
|||
6
|
1
|
6
|
15
|
20
|
15
|
6
|
1
|
||
7
|
1
|
7
|
21
|
35
|
35
|
21
|
7
|
1
|
|
8
|
1
|
8
|
28
|
56
|
70
|
56
|
28
|
8
|
1
|
As you can see, the number of rows is 9 but n = 8.
The best way to find all combinations and permutations is to utilize sensitivity analysis by using of one way - data table on excel spreadsheet just like below sheet:
The best way to find all combinations and permutations is to utilize sensitivity analysis by using of one way - data table on excel spreadsheet just like below sheet:
In the case of corporate financial strategy (in
this article), I assume that there is a normal probability distribution then I utilize
from Combination without repetition (using of Pascal’s Triangular) to obtain
all outcomes in which I use Binomial formula. Before that, please review
examples (3) and (4).
Example (3): Now, let me bring an example in the field of
Strategic Management. I consider two competitors “A” and “B” and I also assume
that there are 9 driving forces (please see link: http://emfps.blogspot.com/2012/02/fuzzy-delphi-method-to-design-strategic.html)
which are affecting on new SWOT matrix of “A” or “B” as follows:
DF1, DF2, DF3 …DF10
How can we calculate the total number
of outcomes made by the combination of driving forces?
We can use from below formula:
X = the number of driving forces
n = X -1, n = 8
Total number of outcomes = SUM C (8, r)
Where: r = 0, 1, 2, 3…8
Total number of outcomes = C (8, 0) + C
(8, 1) + C (8, 2) + C (8, 3) + C (8, 4) + C (8, 5) + C (8, 6) + C (8, 7) + C
(8, 8) = 256
You can also use from Pascal’s Triangular as follows:
n = 8, Total number of outcomes = 1 + 8 + 28
+ 56 + 70 + 56 + 28 + 8 +1 = 256
Another formula can be below cited:
Total number of outcomes = 2 ^ (X – 1)
= 2 ^ 8 = 256
Example (4): If we assume that firm “A” gains 4 key success
factors form all driving forces and the firm “B” gains 5 key success factors from
all driving forces, how can we calculate the probability of competitive
advantage for the firm “A”?
Total number of outcomes for fourth key success
factors = C (8, 0) + C (8, 1) + C (8, 2) + C (8, 3) + C (8, 4) = 163
But the best solution is to use from Pascal’s
Triangular as follows:
Total number of outcomes for 4 key success factors
= 1 + 8 + 28 + 56 + 70 = 163
Total number of outcomes = 256
Probability of competitive advantage for the firm “A”
= 163 / 256 = 64%
The Case of Gainesboro Machine Tools Corporation:
Dividend Policy
We assume that
Gainesboro will apply dividend signalling theory as follows:
According to
Exhibit 8, the assumption of dividend-payout ratio is 40% of net income. If the
company consider 20% (for instance) of net income as dividend-payout ratio and
simultaneously repurchase its stocks, the number of the shareholders will be
very important to obtain positive signalling. Why?
Here, I am
willing to use again from example (4) where I have already make 1000 trials
(rows) of Pascal’s
Triangular by excel spreadsheet.
The approach of the shareholders is two different
choices:
A = Buyers of the stocks (number)
B = Sellers of the stocks (number)
An increase on the number of buyers accompanied by
dividend payout will have the positive signalling on demand and consequently an
increase on stock prices. But repurchasing of the shares by the company is limited
to source of net income because we have:
Total amount of money to repurchase the stocks =
total number of the buyers * the number of shares purchased by each buyer
Therefore, an increase on total number of the
buyers will decrease the number of stocks purchased by each buyer
Now, I assume below conditions:
-A = 51% of total number of the shareholders
-B = 49% of total number of the shareholders
If total number of the shareholders is variable as
follows, by using of Pascal’s Triangular, we are able to calculate all
outcomes:
Total Number of Shareholders Total Outcomes
100
|
6.34E+29
|
|||
300
|
1.02E+90
|
|||
500
|
1.6E+150
|
|||
750
|
3E+225
|
|||
1000
|
5.4E+300
|
Total
number of Shareholders
|
Total
outcomes of sellers
|
Total
outcomes of buyers
|
||
100
|
3.16913E+29
|
4.15826E+29
|
||
300
|
4.1612E+89
|
6.90715E+89
|
||
500
|
5.8944E+149
|
1.1529E+150
|
||
750
|
9.7878E+224
|
2.205E+225
|
||
1000
|
1.5243E+300
|
4.0545E+300
|
In the result, we will have a diagram
for the probability of outcomes as follows (I have also included 50% buyer and
seller):
As you can see, by increasing of total
number of shareholders, if the percentage of buyers goes down to 49% of total
number of shareholders, the probability of stocks’ demand will decrease and
vice verse.
This is a general diagram in
which we can utilize it even for the people's voting.
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.”
