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Sunday, October 2, 2011

EMFPS: How Can We Mix Several Types of Aggregates to Find Out the Best Size Gradation?


As you know, one of the most important tests to qualify the aggregates used in construction jobs such as Concrete, Pavement (Asphalt), Embankment at Roads and Yards and so on is sieve analysis which determines the particle-size distribution of an aggregate. We have so many tests to qualify the aggregates for instance, Alkali-Aggregate Reaction, Stripping (for chemical properties) and Gradation and size, Toughness and abrasion resistance (Los Angeles Abrasion Test), Durability and soundness, Particle shape and surface texture, Specific gravity, Cleanliness and deleterious materials (for physical properties).
What is different between grading test of aggregate and other tests?
When other tests do not obtain the appropriate results, we have to reject the aggregate. But if we have not a good sorting of sieve analysis which is not compatible with the standards, we will be able to improve the gradation of aggregates by mixing several types of them. As the matter of fact, in the real job, we cannot usually find the aggregate generated in river mining (especially fine aggregate for concrete) in which the particle size distribution of the aggregate (sorting) will be completely compatible with the standards. What can we do?
In this article, I have brought a simple calculation accompanied by an example to solve this problem. Why do I bring you this article? Because I am willing to establish a link between Engineering fields and Financial Management so that in my next article you will see to determine the risk minimizing combination (Risk Diversification), we can use the same method.
Here, I explain this method with an example in the field of Concrete Technology.
Assume we are willing to produce the fresh concrete at the site. One of the most important components of the concrete to gain high workability and economic savings is good sorting of fine aggregate. Two important factors are controlling the fine aggregate of the concrete:

1) Sandy Equivalent (SE) > 75%

2) Fineness Modulus (FM) that should be:  2.4 < FM < 3.2

If Fineness Modulus increases more than 3.2, we have to use more cement to reach our specific compressive strength of concrete accompanied by high workability (75mm < slump < 100mm) in which we will not have any economic savings.


The formula of FM is as follows:
FM = SUM (cumulative percentage on specified sieves) / 100

Where:

F.M =fineness modulus


As we can see, F.M is completely referred to sieve analysis of the fine aggregate.
specified sieves
=
0.150 mm (No. 100), 0.30 mm (No. 50), 0.60 mm (No. 30), 1.18 mm (No. 16), 2.36 mm (No. 8), 4.75 mm (No. 4),  9.5 mm (0.375-in.)
Now, let me bring an example of an available sample of fine aggregate (A) that the sieve analysis is as follows:
Size of sieves    3/8”  No.4    No.8    No.16    No.30     No.50     No.100
Passing (%)       100   85         63         47         28           12            7
We can calculate:      F.M = 3.58
According to ASTM C 33, fine - aggregate grating limits is below cited:
Size of sieves   3/8”    No.4     No.8      No.16     No.30    No.50    No.100
Passing (%)     100     95-100   80-100    50-85     25-60      5-30       0-10           
We can see that fine aggregate (A) is not compatible with ASTM C 33 and so F.M is outside of the limit.
Now, we want to mix two other fine aggregates of (B) and (C) with fine aggregate of (A) so that we will have a new fine aggregate which will cover all standard requirements. The sieve analysis of these two fine aggregates is as follows:
Sieve Analysis of Fine- aggregate (B):
Size of sieves    3/8”      No.4      No.8       No.16      No.30      No.50      No.100
Passing (%)        100        93          85           70            43           23             12
Sieve Analysis of Fine- aggregate (C):
Size of sieves    3/8”      No.4      No.8       No.16      No.30      No.50      No.100
Passing (%)        100      100         97           83           65            43              22
What will be the volume percentages of A, B and C for our mixed aggregate?
Now, we have three variables which are “x” as the volume percentage of A, “y” as the volume percentage of B and “z” as the volume percentage of C.
It is clear, we can easily find these variables by using of the solving aMatrix Inverse as follows:
Assume we have Matrixes of S, X and P as follows:
S = Matrix (m*n)   Where: m = n
X = Matrix (m*1)
P = Matrix (m*1)
S.X = P
S’ * S * X = S’ * P
S’ * S = I
I * X = S’ * P
I * X = X
X  = S’ * P
You can solve above function by useing of Excel (Please see my spreadsheet of Excel).
There are some conditions to solve the matrix extracted from our example (mixed aggregates) below cited:
 1) The number of rows, which are the  the size of sieves, should be equal to the number of the variables (x, y, z) or columns in matrix S. In fact,we have: m = n
The number of rows in matrix X and P are equal to matrix S.
The number of columns in matrix X and P are equal to 1.
2) In this example, “x” is related to aggregate type A and respectively “y” for B and “z” for C.
3) Elements of matrixes (aij) are the same the percentages of passing
4)We should  have enough date issued by the standards such as ASTM, AASHTO and so on.
5) We should use from the best data issued by the standards as Index Data which are compatible with our requirements to solve this matrix. What is the meaning? What are the  Index Data? Here, the Index Data is to choose the size of sieves. One the the most important conditions is to choose the best size of sieves to decrease the variances (Diversification) that it could  be also used to reduce the risk of assets in the field of Financial Management. I will depict it in my next article.
For the better perception, let me continue the example as follows:
According to above mentioned, we have three variables and we should choose three size of sieves included in below table:

                                                       Passing (%)
Size of Sieves      A (x)         B (y)     C (z)       Average of Limits
3/8”                           100            100             100            100
No.4                          85              93               100            97.5
No.8                          63              85               97              90
No.16                        47              70               83              67.5
No.30                        28              43               65              42.5
No.50                        12              23               43              17.5
No.100                       7               12               22              5
     
How can we choose the best Size of Sieves?
Definitely we should diversify and decrease the variances. I suggest you below method as follows:
V1 = [(85- 97.5) +(93 - 97.5) +(100- 97.5)] / 3
V2= ----------------------------------------------
V3 = -----------------------------------------------
V4 = -----------------------------------------------
V5 = -----------------------------------------------
V6 = ----------------------------------------------                               
 Then we should find the least amount of SUM (Vi) which is reaching to zero.
(In formula of SUM (Vi), we should know that n = 3 because we have only three variables and should choose three Size of sieves among six Size of Sieves).   
Of course, it will be very hard and we should obtain all permutations and combinations. I will explain this method completely in my next article about Risk Diversification of Assets in the field of Financial Management.
Now, the size of sieves: No.8, No.16 and No.100 have the least amount of
SUM (V) but when we try to find the variables, we find below percentages:
x = 141%
y = -113%
z = 72% 
F.M = 3.3
As we can see, the results are not logic. Therefore, we should use from next SUM (V) which give us the size of sieves: No.8, No.16 and No.50
The results are as follows:
x = 96%
y = 4%
z = 0%
F.M = 3.5
It means that we should use 96% of aggregate type A and 4% aggregate type B. We do not need to use from aggregate type C.

In fact, to find the appropriate results, you can utilize from try and error method by using of my Excel spreadsheet.




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.”



Wednesday, September 28, 2011

EMFPS: How Can We Obtain “Beta” to Analyze CAPM? (CON)




Referring to my previous article of "EMFPS: How Can We Obtain “Beta” to Analyze CAPM?" on link: http://emfps.blogspot.com/2011/09/emfps-how-can-we-obtain-beta-to-analyze.htmllet me explain you the next steps by an example as follows:

1) Without any dividends:
Assume the data of Company “M” from Jan 3, 2005 to Dec 1, 2010 is below cited:

Date
Open
High
Low
Close
Avg Vol
ADj Close*
Jan 3, 2005
4.74
4.92
4.7
4.88
792,600
    4.21
--------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------
 Dec 1,           2010
 5.74
  6.52
 5.74
 6.23
4,193,000
    6.23

The data of the related Stock Index of the company is as follows:


Date
Open
High
Low
Close
Avg Vol
Adj Close*
Jan 3, 2005
907.02
940.94
897.13
916.27
54,441,400
916.27
--------------------------------------------------------------------------------------------------------
---------------------------------------------------------------------------------------------------------
---------------------------------------------------------------------------------------------------------
Dec 1, 2010
1,482.69
1,529.95
1,477.57
1,518.91
141,888,300
1,518.91

-To calculate the return rate for both Company and Stock Index, we can use below formula because there is not any dividends:
Ra = (Pt – Pt-1) / Pt-1
Ra = the return rate for Stock Index and Company
P t-1 = Close price on Jan 3, 2005
Pt = Close price in the next month
-On excel spreadsheet, calculate one cell by using of above formula and copy on all cells related to next months until Dec 1, 2010.
-Now, you have all rate of return for each month (Company and Stock Index). Calculate the average for Stock Index only
-Multiply the average of Stock Index by 12 to obtain the rate of return annually (Rm) which is utilized to analyze CAPM
-Use from excel formula to calculate covariance of the return rate for both Company and Stock Index:    =COVAR (Array1, Array2)
 Where:
Array 1 = all return rate of Company
Array 2 = all return rate of Stock Index
-Use from excel formula to variance of return rate for Stock Index:  
= VAR (number1, [number2]…..)
-Calculate the Beta of Company: COVAR (Ri,Rm) / VAR (Rm)
According to my spreadsheet, I obtained the Beta to be equal 1.501137


2) With dividends:
Now, we assume that the company had paid the dividends as follows:
Date
    Dividends

17-Jan-07
                                                          0.05
27-Jul-
07
0.05
29-Oct-08
0.25
29-Jul-09
0.05
28-Jul-10
0.11

We should look at the date of dividend and improve the company’s return rate accordingly on our spreadsheet by using of below formula:
Ra = [D + (Pt – Pt-1)] / Pt-1
Where:
Ra = the return rate of Company
D = dividend
Pt = Close price for month after dividend date
Pt-1 = Close price for month before dividend date
I calculate the Beta of the Company which is equal to 1.499519


Why are they approximately the same (without and with dividend)?
Because when the company paid the dividend accordingly its share price fell down which shows us the dividend policy of the company. I can say to you this dividend policy is not wrong because the Beta of with dividend is not more than the beta of without dividend.

Conclusion
As the result of this article, I would like to compare the market return of this company with historical market return of the world (see sheet 4 of my spreadsheet).
I calculated annually return rate of Company “M” by adding all monthly return rate (with dividends) then I used from historical returns mentioned on below link:



To analyze the Beta, I got the gradient as follows:

Beta = Delta (company’s return) / Delta (Market return)

Here is my data:
Year
Rc
Rm
2005
-7.67%
4.83%
2006
64.12%
15.61%
2007
-3.31%
5.48%
2008
-49.42%
-36.58%
2009
39.61%
25.92%
2010
41.53%
14.86%

Where:
Rc = the return rate of Company
Rm= the return rate the whole of the market


    Year
                  Beta
2006
    6.659839
2007
6.656976
2008
1.096235
2009
1.424453
2010
-0.17352
B (av)
3.132797


As we can see, in this case, the average of Beta is not important but the trend
of the Beta is very contemplative. I remember that I wrote an article
of “ New Economic System” on below link:


Maybe this method to analyze the Beta will lead us toward the better 
perception and analysis of economic systems.

 
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.”

 









Monday, September 26, 2011

EMFPS: How Can We Obtain the “Beta” to Analyze CAPM?


We need to CAPM analysis not only to evaluate the risk of company’s assets portfolio but also to assess the feasibility of the projects by using of capital budgeting method. As you know, the formula of Capital Asset Pricing Model (CAPM) is as follow:
CAPM = Rf + [B * (Rm – Rf)]
In this article, I am willing to expand my debate on “B”.
At the first, we should divide the companies in two groups:
                      ·                      Private companies: Where the expected return of the company has been calculated in accordance with EBITDA multiplier’s method.
                      ·                      Public companies: Where the expected return of the company has been calculated in accordance with the share price.
Here, I am willing to depict the Beta of public companies. Then I will bring you an example as the practice and finally I will compare the market return of a Stock Index with historic market return of the world in which the result will be referred to the Economic Systems.
Before everything, we should clear our target as follows:
1) Do we want to anticipate the Beta for next several years?
2) Do we want to calculate the Beta by using of historical information? In this case, we should classify the companies in accordance with their dividend policy as follows:
Ø  The company has paid the dividend in the period of our limited time.
Ø  The company has not paid any dividend in the period of our limited time.
If we are expected to anticipate “B” for next several years, we should use from behavioral approach for instance, simulation method such as the Monte Carlo simulation programs in which we have to define several possible alternative outcomes just like to Scenario Analysis and then we should try to determine probability distribution and random numbers to estimate the percentage of probability which is matched to each required return and market return for each one of the outcomes. For example, we can extract some economic data forecasts from IMF (International Monetary Fund) such as GDP Growth (Constant Prices, National Currency) for revenue growth, Inflation (Average Consumer Price Change %) for COGS, Unemployment Rate (% of Labour Force) and government interest rate for Selling and Administrative Expenses and so on. Then we need to build an Excel spreadsheet such as Kimi Ford’s sensitivity analysis included in Exhibit (2) of Case analysis of Nike, Inc: Cost of Capital (see the link of http://emfps.blogspot.com/2011/06/case-analysis-of-nike-inc-cost-of.html). Finally we will obtain something like below example by using of the Monte Carlo simulation program:


Assume we want to calculate the Beta of Company “M” for the period of next ten years. According to above mentioned, we have obtained below data:
Outcomes of the Economy  Probability  Market Return  Company’s Return
  Stagnant                                  15%                    7%                        9%
  Slow growth                            25%                    11%                     13%
  Medium growth                       30%                    14%                     18%
  Rapid growth                          30 %                    21%                     27%
Now, we are able to calculate the expected return of Market and Company by multiplying the percentage of probability by them as follows:
Expected return of Market = SUM [(probability)*(Market Return)] = 14.30 %
Expected return of Company = SUM [(probability)*(Company’s Return)] = 18.10 %
Regarding to the formula of the Beta, we have:
B = Cov (ra, rm) / Qm^2
Where:
Cov (ra, rm) = SUM [(probability) * (Market return - Expected return of Market) * (Company's return -Expected return of Company)]
Qm^2 = variance of the return on the market portfolio = SUM [(probability)* ((Market return - Expected return of Market)^2)]
Cov (ra, rm) = 0.003207
Qm^2 = 0.00242
B = 0.003207 / 0.00242 = 1.325
Now, let us focus on historical data to analyze the Beta of Company “M”. In this case, we do not need to have any probability distribution because all events have been already occurred and we can consider the related probabilities to be equal. Therefore, the expected return is the same the average of Company’s returns during the period of our chosen time.
The method of the Beta analysis will be done step by step as follows:
-Go to one of the financial websites such as: http://finance.yahoo.com
-On “GET QUOTES” search the name of your chosen Company
-Click on “Historical prices”
-I usually search for monthly but you can also search daily or weekly (you should remember that finally the return rate should be changed to annually).
-Copy and Paste all data on your Excel spreadsheet
- On “GET QUOTES” search the related Stock Index of the company to obtain the market return
-Click on “Historical prices”
-Search your chosen time just like to the time extracted for the company
-Copy and Paste all data on your Excel spreadsheet
To be continued …….

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.”

Monday, September 5, 2011

Case Analysis of Nike, Inc.: Cost of Capital (CON)


Referring to my previous article (http://emfps.blogspot.com/2011/06/case-analysis-of-nike-inc-cost-of.html), someone who had received my spreadsheet, had the problem to calculate the Cost of Debt by using of IRR method (Method 2). As the matter of fact, this method can be utilized to obtain the required return rate on Bonds in which you can use it even in your own business.
How can we use from this spreadsheet to calculate the required return rate of Bonds?
Here I am willing to explain it step by step as follows:
Please look at the spreadsheet.
-Enter your new Coupon rate (annually) on Cell C9 and Price of Bond based on par value equal $100 on Cell C10
-You should use the try and error method by changing on required return rate speculated by you on Cell C21 and C22 in which finally Cell C23 will be equal to Cell C17 (C23 = C17).
How can you guess the required return rate?
I have already depicted it on below link:
Where I stated: “Which method is to calculate cost of debt better than others?  It is important to find the relationship between the required return and the coupon interest rate. When the required return is greater than the coupon interest rate, the bond value ……”
Let me bring you an example as follows:
Assume the company has issued the $500,000 nominal amount of 8% rate of Bonds from 1999 to 2009 was issued by a subsidiary at $95.068 per $100 par value. How can we calculate the required return rate by using of my spreadsheet?

-Enter on Cell C6: 1999
-Enter on Cell C6: 2009
-Enter on Cell C9: 8%
-Enter on Cell C10: 95.068
-Since the price of Bond is less than Par value, our speculate on required return rate should be more than Coupon rate (annually)
-Click on Cell C21where we can see this formula: = - PV (rate, number of years, Coupon payment). Of course, the previous numbers on Cell C21 are: = - PV (0.1415, 25,135)
-Replace the amount for required return more than Coupon rate, for instance, 8.5%. Where we have: = - PV (0.085, 10, 80) or = - PV (0.085, C19, C18)
-Click on Cell C22 where we can see this formula: = Par Value / (1+required return) ^n. Of course, the previous numbers on Cell C22 are: = 1000/ ((1+0.1415) ^ 25)
-Replace the same amount of required return (8.5%) and power equal to 10. Where we have: = 1000 / ((1+0.085) ^ 10 or = C20 / ((1+0.085) ^ C19

-We can see that Cell C23 is more than Cell C17 (967.19 > 950.68). Therefore, we should increase again the required return rate to 8.8%
-All previous steps should be repeated. Now, we have: C23 = 948.20 which is less than C17= 950.68
-In this case, we should decrease the required return for instance to 8.76%. Finally we will have C23 approximately equal to C17 (C23 = 950.71 and C17 = 950.68)

In the result, the required return rate will be equal to 8.76%.

 

Note:  “All spreadsheets and calculation notes are available. The people, who are interested in having my spreadsheets of this case analysis 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.”