Application of Pascal’s Triangular in Corporate Financial Strategy

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:

 

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.”