The generating New probability theorems (Con)

Following to previous article posted on link:  http://emfps.blogspot.com/2016/09/the-generating-new-probability-theorems.html, you can review second theorem as follows:

 Theorem (2): Rule of sixty plus (60 %+)

I start theorem (2) just like theorem (1) by using three dices but I change the function to a Transcendental function.

As we know, there are two types of functions in mathematics (Calculus). The first functions are named Algebraic function. These ones are the functions which are applied in below equation:

  0  =  (Pn (x)y^n  +...P1(x)y  + P0 (x        

                  ......... (Where P0 (x), P1(x
 Pn (x) are the polynomials of "x
  

I used Algebraic function for theorem (1).

Second functions are named Transcendental function including two group: 1) Trigonometric function 2) Exponential function. These functions are not compatible with above equation.

In theorem (2), I will apply a type of trigonometric function.

Assume we again set three dices on the table in one row from left to right. We will have:

Dice1      Dice2     Dice3

Now, consider three dices as three variables as follows:

Dice1 = x,    Dice2 = y, Dice3 = z

I am willing to define function f (x, y) below cited:

z = f (x, y) = x + COS y

We know the domain of "x" and "y" is the same the range of "z" equal to 1, 2, 3, 4, 5 and 6.

I am willing to know, what is below probability:

P (z <= x + COS y) =?

We can calculate the probability which is exactly equal to 50.00 %.

P (z <= x + COS y) = 50.00 %

Now, I will pull out another dice from basket and add forth dice on the table right side of dice 3. We will have Dice 4 and I will assign variable "w" to this dice:

Dice 4 = w

Suppose the function will be:

w = f (x, y, z) = x + COS (y + z)

What is below probability?

P (w <= x + COS (y + z)) =?

We have 6^4 = 1296   permutations with repetition.

The probability is again calculated exactly equal to 50.00 %.

P (w <= x + COS (y + z)) = 50. 00%

Then, I will pull out another dice from basket and add fifth dice on the table right side of dice 4. We will have Dice 5 and I will assign variable "t" to this dice:

Dice 5 = t

Suppose the function will be:

t = f (x, y, z, w) = x + COS (y + z + w)

It is clear, permutations with repetition is equal to 6^5 = 7776 and the probability will again be 50%.   

P (t <= x + COS (y + z + w)) = 50%

Finally, I will pull out another dice from basket and add sixth dice on the table right side of dice 5. We will have Dice 6 and I will assign variable "r" to this dice:

Dice 6 = r

Suppose the function will be:

r = f (x, y, z, w, t) = x + COS (y + z + w + t)

What is below probability?

P (r <= x + COS (y + z + w + t)) =?

Permutations with repetition is equal to 6^6 = 46656 and the probability is again 50%.

 P (w <= x + COS (y + z + w + t)) = 50%

Therefore, we can reach to a general theorem in which there are "n" dices:

If we have "n" dices with below function:

xn = f (x1, x2, x3, ……..xn-1) = x1 + COS ( x2 + x3+ ….. xn-1) 

Then, below Probability will be 50%:

 P(xn <= x1 + COS (x2 + x3+ ….. xn-1)) = 50% 

What is the application of theorem (2)?

Let me again refer to my article of "A Template for Financial Section of a Business Plan (Con)" 

posted on link:

http://emfps.blogspot.com/2016/08/a-template-for-financial-section-of_25.html

If you want to test these theorems and see real application of these theorems in the field of financial management

you should select 10, 20, 50 or more companies and apply these theorems on growth rate of sales and costs or amount of sales and costs in the sequential years, quarters and so on.

Here, I have bought an example for 20 company that I have covered the names. You can follow me as follows:

Step 1: Select 20 companies in different industries

Step 2: Go to Google finance and search name of each company

Step 3: For each company click on Financials (left side of page)

Step 4: Copy total revenue and cost of revenue for five sequential period of 13 weeks and paste on excel spreadsheet

Step 5: Calculate growth rate for total revenue and cost of revenue

Step 6: Calculate the function of theorem (2) for two sequential period of 13 weeks and compare it with third sequential period of 13 weeks

Step 7: Calculate the probability

You can see Figure (1) which is my example:

( 1) Figure

  In my example, I am predicting the periods of 13 weeks ending 2016- 03 – 26 and 13 weeks ending 2016 – 06 – 25. You can see, for function of z = x + cos y, the probability is exactly 100% for both periods (for revenue and cost). It means that the predictions are 100% true. But these predictions are not useful because they show us very high upper stream line. It is just like, I say to you that the sales will not increase more than 150 or 200% for the next period. Therefore, there are two important questions

1. Why is there the significant difference between the probability of theorem (2) and real probability which is 100% instead of 50%?

2. How can we decrease upper stream line where the predictions will be useful for us?

Answer to question (1):

As we can see, the growth rates usually are between -100% to + 100% (-1 to +1).

 Therefore, let me consider all dices contains infinity numbers which are members of Real Number between -1 to 1 as follows 

D1 = 

 {x │ x ϵ R, -1< x <1}

     =  D2 

{y │ y ϵ R, -1< y <1}

=   D3

{z │ z ϵ R, -1< z <1}

Now, if we use above conditions and apply Monte Carlo method for our analysis, we will find that the probability for theorem (2) will increase to more than 85% with Average approximately 85% by standard deviation of 0.001. Please see Figure 2

 Figure (2)

Answer to question (2):

Only way to decrease upper stream line is, to divide function of theorem (2) to some numbers. For instance, I apply the Monte Carlo analysis for below functions:

z = f (x, y) = (x + COS (y)) / 4

According to Figure (1), the probabilities are 95% and 85% for my example. In the reference with Monte Carlo analysis, we can say:

P (z <= (x + COS y) /4) > 65%   with Average > 64 % and standard deviation approximately 0.0015

z = f (x, y) = (x + COS (y)) / 6

Figure (1) shows us, the probabilities are 95% and 72.5% for my example. Regarding to Monte Carlo analysis, we can say:

P (z <= (x + COS y) /6) > 60%   with Average approximately 60 % and standard deviation 0.0018

Since theorem (2) divided to 6 is more useful for us, I can say that the Rule of sixty plus (60 %+) is as follows:

P (z <= (x + COS y) /6) > 60%

The Generating New Probability Theorems

The purpose of this article is to generate new theorems of probability and to find out some applications of these theorems. In this case, suppose that we have a covered basket that contains many dices. In many blind tests, we will reach in and pull out a dice and set it on the table on one row from left to right. It is clear, each dice has six events (choices) including 1, 2, 3, 4, 5, and 6.

Theorem (1): Rule of fifty plus (50 %+)

I start this theorem by using three dices. At the first, I pull out one dice from basket then second dice and finally third dice and set them on the table on one row from left to right.

 The question is: What is the probability for number of third dice less than or equal to average of numbers first and second dices?

Let us have the dices as follows:

D1 = first dice

D2 = second dice

D3 = third dice

D1    D2    D3

I am willing to know:

P (D3 ≤ ((D2 + D1) / 2)) =?

Definitely, we have 6^3 = 216   permutations with repetition.

We can calculate the probability equal to 54. 1667%.

 P (D3 ≤ ((D2 + D1) / 2)) = 54.1667%

Or, what is the probability for number of third dice more than or equal to average of numbers first and second dices? 

The answer is the same:

 P (D3 ≥ ((D2 + D1) / 2)) = 54.1667%

Now, another question is: What is the probability number of third dice less than or equal to twice number second dice minus first dice?

P (D3 ≤ ((2*D2) - D1)) =?

The calculation shows us that the probability is the same equal to 54. 1667%

P (D3 ≤ ((2*D2) - D1)) = 54.1667%

Therefore, we can say:

P (D3 ≤ ((D2 + D1) / 2)) = P (D3 ≥ ((D2 + D1) / 2)) = P (D3 ≤ ((2*D2) - D1))   

Let me expand this idea as follows:

If we assume all dices contains infinity numbers which are members of Real Number as follows:

D1 and D2 and D3 are subsets of Real Number

In this case, each dice is included a set of Real Number in which the numbers of sets will be different, then we will have:

P (D3 ≤ ((2*D2) - D1)) > 50%

I name this theorem: The rule of fifty plus (50 %+)

 What are the applications?

Here I have brought an example of financial management.

Let me again refer you to my article of "A Template for Financial Section of a Business Plan (Con)"
posted on link: 
 http://emfps.blogspot.com/2016/08/a-template-for-financial-section-of_25.html

We can apply this theorem to predict assumptions when we have the final reports of previous years 
by probability of more than fifty percent (50 %+) for instance, I utilize this theorem for growth rate of sales as follows:

If we replace the dices by years of sales (Income statement) and use from Monte Carlo method to iterate calculations, we will reach to the rule of 50 %+. (Please see below pic)

On above spreadsheet, we have sales for year 3, 4 and 5. Then I use different probability distribution and different growth rate for CUT OFF. By one way data table, you can see an average of P (x) is equal to 53.9% by standard deviation of 0.000499.

 P (x) = P (year5 ≤ ((2*year4) – year3)) = 53.9%

It means that there is a probability more than 50% in which sales in year 5 will be less than or equal to twice sales in year 4 minus sales in year 3.

You can check this theorem (50 %+) for all income statements, annual reports and so on.

Even though we have found out this theorem, there is still more than 40 % risk to use this theorem. But, how can we decrease the risk of projection for assumptions or increase the probability prediction of our assumptions?

For deducting this risks, we have to generate other theorems by increasing the number of dices or changing the functions.

A Modified Template for Financial Section of a Business Plan

Following to article of "A Template for Financial Section of a Business Plan" posted on link: http://emfps.blogspot.com/2016/08/a-template-for-financial-section-of.htm,

the purpose of this article is to develop and improve previous template. So, I am willing to introduce you a new ratio (S) extracted from some new theorems in mathematics in which by using this ratio, we will be able to focus on only one assumption and eliminate other assumptions for our analysis. In fact, by changing other assumptions, this ratio (S) always stays the constant.   

Developing of Previous Template

In previous template, we had only one column for our assumptions. It means, by changing a variable (assumption), the changes for all years are the same. For instance, if we change the growth of sales from 4% t0 6%, the NPV shows us a growth of 6% for year 1, year 2, year 3 and so on. Please see 

below pic

In reality, there is not the same growth for all years. Therefore, I added other cells on my spreadsheet where we will have a set of assumptions for each year. This gives us the opportunity for better 

analysis by using the Monte Carlo method. Please see blow pic:

The Improve of Previous Template

In new template, I added new assumption which is the Tax rate.  In previous template, we have a big problem when EBTI is negative because we have already added tax payment to net income. In this case, there is the concept which is named NOL.

What is NOL? It is Net Operating Loss in which we can deduct our NOL from the taxes we paid in prior years and get a refund, or we can apply it to future years to lower our tax bill.

This will help us to recover some net operating losses but usually not all. 

The concept of NOL is the complicated and you cannot easily claim your NOL. Some people believe to deduct entire NOL from future taxes (Tax loss carry forwards) or to add tax payment to net income when EBTI is negative (my previous template). Let us see below example:

 Assume Company X has net sales of $2,000,000 but expenses of $2,200,000. Its net operating loss is $2,000,000 - $2,200,000 = -$200,000.

Company X will probably not have to pay taxes that year, because it has negative taxable income. But let's assume that next year, Company X makes more sales and records $700,000 of taxable income. Company X pays a corporate tax rate of 40%.

Normally, the company would need to pay $700,000 x 40% = $280,000 in taxes. But because it had a tax loss carry forward from last year, it can apply last year's loss to this year's tax bill, reducing it significantly (or even to $0, depending on the jurisdiction Company X is in).

Let's assume that Company X can apply the entire -$200,000 tax loss carryforward to this year's tax bill. Instead of owing $700,000 x 40% = $280,000 in taxes, Company X now owes only ($700,000 - $200,000) x 40% = $200,000 in taxes.

But other people say another story as follows:

"It's rare to see a company acquired for the purpose of NOLs today (at least through a direct acquisition). This is because of 382 limitations on the usability of NOLs in the case of a change in control of a company's equity rendering your NOLs almost worthless on a present value basis. NOLs are currently limited to 3.98% of the value of the company during a change of control. This number is determined by the IRS monthly and (along with the value of the company for 382 purposes) will be fixed at the time of the change of control. Based on your business and applicable tax laws, you may also have to distinguish between cash taxes and accounting taxes. The actual formula for NOI after taxes is simply: NOI - taxes. This is equivalent to (1-taxes) * NOI if your taxes are positive, but should be just NOI since your taxes are zero if your NOI is negative."

Therefore, we have the pessimistic and optimistic comments. Anyway, I choose the pessimistic situation because here is a good opportunity for our analysis as follows:

For obtaining zero taxes on my spreadsheet when NOI is negative, I use a simple trick in which I add one row under item of tax payment (Adjusted tax payment) by below formula on all months and years of cash flow statement:

=IF (B36>0; B36; 0)-B36

Please see below pics:

In this case, if your tax payment is negative, adjusted tax payment will be positive where the total sum is zero. But if your tax payment is positive, adjusted tax payment will be zero.

Then, on spreadsheet of Income statement, I add tax payment and adjusted tax payment.

But what is good opportunity?

Above trick gives us a good opportunity on our analysis. If we change the tax rate for each year but NPV, IRR and also Enterprise value do not change, it means that the combination of other assumptions presents net operating loss in its year. For example: you can see when I change tax rate in year 4 and year 5, NPV does not change. Therefore, we have net losses in year 4 and year 5. (Please see below pics)

A New Ratio (S)

Now, let me introduce you a new ratio that if you apply it on each assumption, by changing other assumptions this ratio always stay the constant.

Example:

I consider Cost of Capital as base of assumption to generate this new ratio. Of course, we can choose any assumption as the base of calculation this ratio.

- First, I use form a sensitivity analysis for the Cost of capital and NPV. (see below pic)

 - Then, the ratio of "S" is equal to NPV 3 minus NPV 2 divide NPV 4 – NPV 1

S = (NPV3 – NPV 2) / (NPV 4 – NPV 1)

We can use from IRR and also Enterprise value instead NPV.

In below pic, you can see the ratio of "S" has been calculated by using above formula for all assumptions:

Now, I change all assumptions except the Cost of Capital in below pic:

As you can see, the ratio of "S" will stay the constant for all changes of assumptions.

Indeed, what is application of the ratio of "S" for our analysis?

For answering to above question, at the first we should familiar to a new theorem of mathematics which is the rule of 0.333333….

Then we can start our analysis by using the Monte Carlo Method.
Of course, there is another ratio which is named ratio of "P". This ratio has also the property just like ratio of "S" where by changing all assumptions ratio "P" stays the constant. The formula for ratio "P" is as follows:

P = (NPV 2 – NPV1) / (NPV3 – NPV2) or

P = (EV2 – EV1) / (EV3 – EV2) or

P = (IRR2 – IRR1) / (IRR3 – IRR2)

Please see below pic:

Now, I change all assumptions and you can see ratio "S" and "P" stay the constant. Please see below pic:

Are there other ratios which have the same property?