Fuzzy Method for Decision Making: A Case of Asset Pricing Model

There are always many ways to solve the problem but we should only choose the best way among them to take an action. How can we find the optimized choice? The process of optimization the ways for solving of the problem is defined as decision making in which the output of this process will be an action. Nowadays, we need to use the process of decision making in all sciences such as Strategic Management, Financial Management, Engineering, Medicine and especially Political science. When we refer to this process, we can often see that there is the lack of information or vague values for the parameters which are affecting on the case something like uncertainty on data and so on that it is named as Fuzzy environment. Fuzzy Set theory gives us the opportunity to grade all alternatives (ways) where we should abandon the approach of 0, 1 in our mind. In fact, this method is one the best tools to solve the paradoxes.

The purpose of this article is to apply intersection of fuzzy sets to obtain the results (alternatives) in the frame of a new fuzzy set as the process of decision making where the alternatives are a crisp set and maximum membership function of this new fuzzy set gives us the best alternative (best way) to take an action. Then a new method of fuzzy number intersections has been demonstrated. A case study of asset pricing model is given to perceive purposed method. Finally, some laws and a new formula about distance measurement among fuzzy numbers have been listed.

Methodology

According to Bellman and Zadeh (1970), if two fuzzy sets “Д” and “И” have the intersection, there is a new fuzzy set “Ж” with membership function µ Ж (x) in which “Ж” is named the decision as follows:

Ж = Д ∩ И = {(x, µ Ж (x) |x ϵ [a1, a2], µ Ж (x) ϵ [0, α ≤ 1]}

Where interval [a1, a2] is a crisp set which is named the alternatives consequently for fuzzy set “Ж”, we have:

µ Ж (x) = min (µ Д (x), µ И (x))      and      x ϵ [a1, a2]         

Here is a schematic figure of above definition:

Fig (1)

We definitely apply the maximum degree of membership function of set “Ж” that it gives us the maximum “x” among alternatives. Therefore, we have:

x max  = {x |x ϵ [a1, a2], µ Ж (x) = max (min (µ Д (x), µ И (x)))}

Above schematic diagram shows us the same left and right sides of continuous piecewise – quadratic fuzzy number. We can also use triangular and trapezoidal fuzzy numbers to demonstrate the concept of this method as follows:

Fig (2)

Case Study: Asset Pricing Model

Mr. X (or company X) is willing to launch a finance plan as follows:

According to his economic study, the parameters such as money supply, monetary base, money multiplier, interest rate, inflation rate, Demand – Supply curves and so on show a fuzzy environment. In fact, there is a stagnated market to transact some assets in which you cannot easily liquidate them to get cash money while the attractive markets including stock markets, corporate bonds have a high volume of transactions. He says: “Where money goes?” He starts his finance plan by the schedule in the period of 6 months. According to his projection, If he sells his second car to invest in corporate bonds and some stock markets, the internal rate of return (IRR) will be about 26% after 6 months whereas the increase valuation of his second car plus the costs of transportation and so on will show his IRR equal to 18% during the period of next 6 months. Therefore, the sale of his second car and investment on stock markets and corporate bonds will be ok. But how can he sell his second car in the stagnated market? He has to advertise his second car on websites or newspaper which will have the cost. How many times should he advertise his second car? It is clear, he should pay the cost for each time advertisement. Several times of advertisement will have several times of making payment. How can he optimize his cost of advertisement? He has to find the best price (the best way or the best alternative) for advertising where the increase of calls by buyers will proceed to make a deal. In a fuzzy environment, he has considered to apply fuzzy method to find the best price for advertising step by step as follows:

Ø  He review second car prices which are the same model with his car and they have advertised on websites or newspaper during the period of the last week.

Ø  To calculate the average of the prices collected from Internet and newspaper.

Ø  He will look for some agents who will certainly buy his second car by cash but very chipper than the prices advertised on Internet or newspaper.

Ø  To calculate the average of the prices quoted by the agents.

Ø  To get the maximum price advertised on Internet or newspaper

He arranges the set of alternatives (crisp set) for right and left side of fuzzy number as follows:

Right side of fuzzy number:

Average of prices quoted by the agents = $ 5000

Maximum price advertised on Internet or newspaper = $ 10000

Referring to Figure (2), we have:

Д = {(x, µ Д (x)) | (x ϵ [0, 5000], µ Д (x) = 1), (x ϵ [5000, 10000], 0 ≤ µ Д (x) ≤ 1)}

Left side of fuzzy number:

The average of the prices collected from Internet and newspaper = $ 6400

Maximum price advertised on Internet or newspaper = $ 10000

In the reference with Figure (2), we have:

И = {(x, µ И (x)) | (x ϵ [6400, 10000], 0 ≤ µ И (x) ≤ 1), (x ≥ 10000, µ И (x) = 1)}

Finally, we will have fuzzy set “Ж” as follows:

Ж = Д ∩ И = {(x, µ Ж (x) |x ϵ [6400, 10000], µ Ж (x) ϵ [0, α ≤ 1]}

Therefore, crisp set of alternatives is interval [6400, 10000]. Figure (3) shows us that the best price of the first try to advertise is equal to $7900 with confidence level of 0.42 (α = 0.42).

Fig (3)

 

New Method of Fuzzy Numbers Intersection

The most crucial problem to use above method is to take the mistake about minimum prices. The most of time, we have doubt about minimum prices quoted by buyers. On the other hand, considering the average of the prices collected from Internet or newspaper as minimum price of left side fuzzy number is not really logical and accurate price in which this is also the same for right side fuzzy number. What can we do?

One of the best ways is, to define the range for minimum prices instead of a fixed average price. Then we will apply two ways sensitivity analysis for the range of minimum prices and “x” which is the price variations from 0 to maximum prices. In this case, we will have an area of intersections by fuzzy numbers on our diagram. The best price of first try to advertise will be calculated by two methods as follows:

Ø  Applying Finite Element Analysis (FEA) on spreadsheet of excel

Ø  Calculating the Gravity Centre of the area

In this article, I will utilize the second method which is to calculate the gravity centre of the intersection area.

Note (1): “The professional people who are interested in applying Finite Element Analysis (FEA) on spreadsheet of excel, can use below link which is an article of “Mechanical and Electromechanical Systems

Simple Thermal Heat Transfer Finite Element Analysis in Excel 2007” by Tom Scarince on April 1st, 2010:

http://www.thsengineering.com/2010/04/thermal-heat-transfer-finite-element-analysis-excel/”

Now, let me start this new method by two examples as follows:

Example (1)

Mr. X (or company X) believes that the range of minimum prices quoted by buyers is into interval [$5000, $10000]. He also considers the range of the prices collected from Internet or newspaper into interval [$5000, $10000]. The results are as follows:

 Right side of fuzzy number:

The range of minimum prices quoted by the agents = [$5000, $10000]

Maximum price advertised on Internet or newspaper = $ 10000

Left side of fuzzy number:

The range of the prices collected from Internet and newspaper = [$5000, $10000)]

Maximum price advertised on Internet or newspaper = $ 10000

According to the method defined for fuzzy number intersection, the final results can be presented in Figure (4):

Fig (4)

If we calculate the gravity centre of area (ABCD), the best price of the first try to advertise will be equal to $8883.33 with confidence level of 0.52 (α = 0.52).

Example (2)

Mr. X (or company X) discovers that the range of minimum prices quoted by buyers is into interval [$4000, $6000]. He also gets the range of the prices collected from Internet or newspaper between $6000 to $8000 and the maximum price is still equal to $10000. The results are as follows:

 Right side of fuzzy number:

The range of minimum prices quoted by the agents = [$4000, $6000]

Maximum price advertised on Internet or newspaper = $ 10000

Left side of fuzzy number:

The range of the prices collected from Internet and newspaper = [$6000, $8000)]

Maximum price advertised on Internet or newspaper = $ 10000

According to the method defined for fuzzy number intersection, the final results can be presented in Figure (5):

Fig (5)

If we calculate the gravity centre of area (ABCD), the best price of the first try to advertise will be equal to $8183.33 with confidence level of 0.4 (α = 0.4).

This method can be also used in business, finance, strategic management, political science, engineering, and medicine and so on. For instance, in financial management, we can replace dividend, time series analysis of share prices, IRR and NPV of the projects instead of second car price.

Case Questions:

Ø  What is the impact of sale a second car by using of above fuzzy method on overall market? Can we say this transaction will be considered as the index for new situation of market and new minimum price for this fuzzy model where new fuzzy numbers will go ahead?  

Ø  Assume, Mr. X (or company X) is willing to sell 50 second cars simultaneously. What is the impact of sale simultaneously 50 second cars by using of above fuzzy method on overall market? Can we say these transactions will be considered as the index for new situation of market and new minimum prices for this fuzzy model where new fuzzy numbers will go ahead? 

Ø  How many second cars should be sold to move overall market? Should we add the application of Game theory to this fuzzy model to answer this question?

Ø  If Mr. X (or company X) sells his second car by $8000, he has to buy the same model car by $11500 after 6 months. Will his investment cover his lost?

 For answering to above questions, we should try to learn new definitions and laws of distance measurement among fuzzy numbers. Let us review some basic rules in this case as follows:

If we have triangular fuzzy numbers Д, И and Ж:

(1)   d (Д, Д) = 0

(2)   d (Д, И) = d (И, Д)

(3)   If d (Д, И) < d (Д, Ж), then И is closer to Д.

(4)   The New Inequality Theorem in Fuzzy Numbers (refer to link: http://emfps.blogspot.nl/2012/02/fuzzy-delphi-method-to-design-strategic_06.html)

Here, I am willing to demonstrate two new laws or formulas as follows:

Ø  If we have triangular fuzzy number Д = (a, b, c) and interval [x1, x2] where:

a ≤ x1≤ b  and   b ≤ x2 ≤ c and  ∆x = x2 - x1

By using of geometrical theorems, we can easily prove below formula:

 µ = 1 – [∆x / (c – a)]                   0 ≤ µ≤ 1

In the result, we have:

Ø  In the reference with above formula, if we have two triangular fuzzy numbers

 Д = (a1, b1, c1) with membership function µ Д (x) and И = (a2, b1, c2) with membership function µ И (x) and below conditions:

a1 ≤ a2 ≤ b1   and     b1 ≤ c2 ≤ c1  

Then, we can get below formula:

(c2 - a2) / (c1 - a1) = 1 – µ Д (x)

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……

References:

- Bojadziev, George., & Bojadziev, Maria (2007). FUZZY LOGIC FOR BUSINESS, FINANCE, AND MANAGEMENT (2^nd ed.). London:  World Scientific Publishing Co. Pte. Ltd.

- Fu, Guangtao. (2006). A fuzzy optimization method for multicriteria decision making:

An application to reservoir flood control operation. Expert Systems with Applications, 34, 145- 149.

 

 

 

  

Challenging in Project Management Today: What Challenges Managers Face Today in Projects Due to the New Management System, Development and So on?

“I keep six honest serving men (they taught me all I knew): Their names are What and Why and When and How and Where and Who.”  Rudyard Kipling

There are so many challenges that the project managers are encountered during the period of the project life cycling. I have brought here the most important ones that I had experienced while I was working as project manager in EPC projects.

Before to proceed toward the challenges which the project managers face today, let me start by using guidelines of PMBOK book (Project Management Body of Knowledge) as the Bible of the Project Management field. Here is the Project Management framework in accordance with PMBOK.

                                                Project Management Framework

Regarding to above chart, I am willing to divide all challenges to three parts which are Stakeholders’ needs and expectations, 9 Knowledge Areas and Tools and Techniques. In fact, I can consider the challenges as External and Internal. You can see that we are again speaking about External and Internal analysis. Therefore, let me tell you that the first step to practice the Project Management by managers is to learn the basic of Strategic Management as follows:

Why should the project managers learn about the Comprehensive Strategic – Management Model?  The reasons are as follows:

• To issue the large tenders
• To evaluate the bids and the quotations, purchase orders
• To predict the strategies, mission and vision of the main contractors, vendors
• Bid meetings, negotiations

In fact, the SWOT analysis of each main contractor or vendor will give good information to the project managers to save the cost and time accompanied by quality of the project.

At the first let me have the debate about external challenges as follows:

• The first step is that the project manager as well as knows the Stakeholders of project to familiar their needs and expectations.

Who are the Stakeholders of the project? They are Customers, Consumers, Owners, Partners (Joint ventures), Funder, Main contractor, Subcontractors, Society and Government, Internal Personnel, Shareholders and Clients.

If you don’t know your stakeholders, you won’t perceive what they expect. Then you are not able to have a good communication among them. If Stakeholders have the conflicts all together you cannot coordinate and integrate them and consequently your project will be failed. In the other word, the project manger should know the real concepts of the Expected Return rate and Risk.

• Your client has uncertainty about the scope and specifications of project and the client doesn’t really know what they want. One of the most important problems that I had in my first project was the approval of the Layout of the project in which the client should confirm their needs. It took so long time for confirmation.
• Your client has delay with your communications.
• Please consider that you have been chosen as the project manager and the client ask you to handle the project which had been already determined the Cost, the Time and the Scope. What will be your action? Do you agree with the quotation obtained by the client without to take your analysis? Here will be the origin of the conflict between you and the client. Because if there is not any balance among the scope the time and the cost, you have to decrease the quality of the project in all sections such as procurement, low standard in engineering, low quality in construction, the limited communication, limited vendor list, high risk project and so on. On the other hand, you see that the client inject some unskilled people in your team project. Definitely if you do not accept the analysis of the client, you should leave your job. This is the big challenge form the project managers today.
• Another challenge is a huge fluctuation on economic indicators such as inflation rate, exchange currency and so on during the period of the project life cycle. Sometimes the changes are affected on the cost and risk of the project so that you have to stop the project because the project is not feasible any more.
• One of the most important external challenges for the project managers is the authority. What kind of authority (if any) does the project manager has or exercises?

In most cases the project manager has no authority at all on the resources, and he's under the mercy of the functional manager. However, the project manager through diplomacy, communications, and by setting himself/herself as an example for the resources, he/she can impose his/her authority (even without having formal authority). Of course, maybe it is logical issue because all responsibilities which are associated to the penalty and lost in the project are assigned the owners. There are two kinds of authority. The authority on the people and staff that it doesn’t matter very much because even the CEO uses his/her influences to push the skilled people to do their obligations. The second kind of authority is decision – making authority. This is a major problem for many project managers, especially when it comes to decisions to spend money. When you are a project manager, you are not allowed to spend the money more than for instance $50 without any approval. What will be the outcomes? Definitely the outcomes will be approached to increase the time of the project in which the balance of the scope, the cost and the quality consequently will be changed. It is possible; the project will be failed because of too much delay. For instance, sometimes the project manager has to cross – check some documents, data sheets, DWG’s or invoices of subcontractors. He/ she should spend money to get the consultant services outside of the organization. To prevent of delay, the project manager should pass some regulations rapidly. Therefore, he/she will be charged the additional cost. If each issue should be approved by functional manager, the project manager certainly needs to new overall time schedule every day.  I don’t understand while the client has project budgets of hundreds of thousands of dollars; do not trust the project manager.

Now, let me bring some examples of internal challenges in which the approach is 9 knowledge areas and tools and techniques in project management.

• Project Integration Management

- Approach of the project management should be always help and cover the executive agents such as Consulting Engineers, Main and Subcontractors, Project team in addition to coordinate among them to reach his target. The project manager should know that only someone does not take a mistake who does not work. Therefore, sometimes he should shut one’s eyes on some negligible issues. In other words, he should be flexible. Of course, the project manger should have the concern about balancing of Time, the Cost, the Scope and Quality during the period of project life cycle.

- The project manager should obtain enough information from his sub-projects before to put out to the bid among the subcontractors for instance, to estimate time and cost of the sub- projects (Items price list or break down price list) in which there is a high return rate accompanied by low risk. Therefore, he needs to have basic and details engineering and details time schedule before the bid by coordinating Human resources such as engineering department, commercial department and so on.

• In Procurement (Sales contract)

- It is suggested to the project manager that details time schedule presented by main contractor is enclosed as an article of shipment documents included in Sales contract so that main contractor could not supply any machinery at the site before commencement date of civil work or erection work. Because main contractors are not usually interested in doing their obligations in Construction contract (“C” in EPC) and they will take the most their profits by Procurement of equipment and Engineering (“P&E” in EPC) that these margins are often more than P.B.G (Performance Bank Guarantee) presented by main contractor.

-Break down price lists should be checked exactly because it is possible that shipment of a machine or equipment only absorbs 50% of L/C. In the result, the main contractor, which has gained so much profit by shipping only one or two equipment at the site, is not interested to doing his obligations included in the construction contract. Because the civil and erection works are always accompanied by high risk level.

-Details time schedule should truly be designed so that it should be established a logical relation among shipments of the equipment (procurement) and progress of civil and erection works.

-When  a new revision of the document ( contract, appendix, DWG, data sheet and so on) is issued by Contractor, the  project manager should check it out not only the cases are depended on decisions of the meeting but also other parts of the document should be controlled word to word and number to number. For instance, in one of the projects, I remember that we had discussed about mechanical specifications of machinery (App 3 from Sales contract) at the meeting. But I saw that it had been deleted some civil work specifications in another part of new revision APP (3) by Contractor so that total difference cost was more than one million Euro. And So I remember that the basic elevation code (BM) had been changed in new revision of DWG related to site plan and  had send us for our approval  while the real debate  on the site plan map was about the layout no BM code so that it was caused a subject of claim. The project manger should teach to his team that they should compare all numbers and words on the documents between previous and new document issued by the contractor.

• Communication Management

     The most important challenge of the communication is the negotiation between the project manager and main contractor at the meetings. Here I have brought some examples for good negotiation as follows:

      - Each time we become nervous, we have to pay the cost of it. In fact, it will be the goodwill for the opponent (main contractor).      

      - At the meeting, when we are obliged to think, we will be at the extreme of   weakness and we have to surrender and give up (in fact, this is the important principal of our life. When we are thinking to solve of the problem, we are at the extreme of weakness and the problem is at the extreme of the power). Therefore, we should quickly say to the correspondent (at the meeting): “Let me check it and reply to you ASAP” or “What’s your mean, I don’t understand it” and so on. 

     - The best consequence of the meeting is to reach a Win-Win result. What are the superiority conditions to this outcome?

Before the staring of the meeting, the both parties should have a PEST, Industry and SOWT analysis as the initial information from each other and the project. Because there are two states at the meeting: 1) our opponent bluff     2) our opponent shows us that he is very stupid. The second one is very dangerous for us. Each time we bear in mind that the people do not understand and are stupid, it is the starting time of losing our opportunities and benefits. In fact, win – win approach is to reach the boundaries conditions.  Who are we able to reach the boundaries conditions? Definitely it will be done with the increase of our information.

• Human resources Management

In the field of Human resource management, I am willing to depict the challenges on two groups of the project team which are engineering department and  control project department as follows:

Ø  In Engineering

-The most important thing to present a new design in the field of Engineering is to collect and select the exact basic data. In fact, the calculation note and engineering software help us to increase the rate of analysis so that the feedbacks show us the confirmation or non-confirmation of basic data. Therefore, we should know that engineering tools (software and calculations) do two works for us: 1) to analyze of basic data 2) to proceed toward the limits (for saving of cost and time).

    - The project manager should have the concern about a new design because it brings for the project the new science production and the innovation. Therefore, the project manger should cross - check the design of engineering department by using of locally information, previous executed designs, references, returning analysis, and using of other consultant engineers and vice versa. It means, when engineers proceed to design and calculate of some details in the fields of civil, mechanic, process, electric and so on, they should guess the details of final design completely before starting of the calculation by using of locally information, other projects just like to this project, returning analysis, references, their experiences and etc.

An optimum design will guide us toward the exact estimation and saving of money and time. For example: I remember that I rejected basic and details engineering of Electrical Substation Building in one of the projects so that the new revision had been changed from 2500 m2( Area) to 750 m2 and total saving money was about 800000 Euro accompanied by 6 months saving time. In another project, I remember that I warned to engineering department about some basic data for designing of utilities system (Cooling Towers, Steam Generators, Electrical system etc) in which engineering department had to changed the basic data logically in accordance with the real consumptions and matched  with the contemporary percentages. And so, I rejected basic engineering of sewage water (Domestic + foul water) and raw water treatment because of inlet volume (basic data) so that finally it had been changed from 6 m3/day to 1.5 m3/day. Therefore, when a project manager receives a new design from engineering department, he should spend money to obtain other designs to be planned by other consultant engineers outside of his company for comparing with this new design because he is expected to find out the best way to reach the target of project (saving the cost and the time). Of course, a project manager should care about locally information, previous executed designs, references, returning analysis too.

- Before final approving of details engineering included DWGs, data sheets and so on, the executive engineers and site manager should check them because a designing and calculating engineer usually works in 2Dimentioal system but an executive engineer always works in 3Dimentioal space. I mean that the site engineers should also sign all of these documents before final approval.

Ø  In Project Control:

-       Since preparing of a details time schedule (CPM) is the most important item in the projects, if we increase the number of activities (WBS) accompanied by relationships among them, we will be able to lead the project smoothly. In fact, a significant CPM should present us details activities. Therefore, basic data are very important just like to engineering field.

-       A good project controller should be able to have a good friendship with engineers in engineering department, engineers at the site, financial department, commercial department, and coordination among them. Because he/she is someone who is supposed to establish a logical cash flow compatible with engineering activities, civil and erection activities, procurement ( shipments) by tracking of critical paths.

-       I think that it will be valuable that a project manager spend money to get several details time schedule from different sources.

-       A good project controller should track compatibility between primary cost estimating and cash flow in addition to other tracking every day (for presenting warning to project manager).

-       Engineering department, site manager, financial department, and commercial department should sign the details time schedule (CPM) before final approval.

Nowadays, one of the most crucial things to consider is to monitor the project by using of new technologies. In fact, the project manager should familiar to new technologies of monitoring which will significantly help to the project manager for a better control of the project.

Finally, I would like to suggest you a very good reference book in the field of the Project Management which is: “Plant Project Engineering Guidebook for Mechanical and Civil Engineers” by Morley H. Selver, P. Eng.

EMFPS: How Can We Get the Power Set of a Set by Using of Excel? (The Case Analysis of Big Data)

Following to my previous article posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html, let me tell you one of the applications of this method in Risk Management.  In this article, I am willing to solve a case in which the template is referred to my article of “EMFPS: Efficient Portfolio of Assets (The Optimization for Risk, Return and Probability)” posted on link: 

http://emfps.blogspot.com/2011/10/emfps-efficient-portfolio-of-assets.html.

The Case: To Find Minimum Risk for Portfolio of Assets in Variety of the Outcomes

If we have had a range of probability for our outcomes where there is not the same probability assigned to the outcomes, how can we calculate minimum risk for the portfolio of assets? What are the probability distributions assigned to the expected portfolio return in which we will have minimum risk?

Let me start the solution of this case by using of the example stated in article of “EMFPS: Efficient Portfolio of Assets (The Optimization for Risk, Return and Probability)” posted on link: 
http://emfps.blogspot.com/2011/10/emfps-efficient-portfolio-of-assets.html as follows:

Referring to step (3), I define the range of probability for six outcomes equal to 10% to 21%. 

10% ≤ Pr ≤ 21%

The solution is step by step below cited:

Ø  On the spreadsheet of excel, we should type:

A1 = p

A2 = 6

A3 = 0.1, A4, = 0.11, A5 = 0.12, A6 = 0.13, A7 = 0.14, A8 = 0.15, A9 = 0.16, A10 = 0.17

A11 = 0.18, A12 = 0.19, A13 = 0.2, A14 = 0.21

Ø  Using from VB code written by Myrna Larson mentioned in article of  “EMFPS: How Can We Get the Power Set of a Set by Using of Excel?” posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html

Ø  We will have 665280 cells filled with different permutations (please check it by formula = PERMUT (12,6) on excel)

               Ø  Due to each subsets is into one cell, to split the elements of subsets into other cells, we should utilize      Data Tab – Data Tools – Text to Columns

Ø  Using from formula = =IF(SUM(A2:F2)=1,A2,"") on excel to find all combinations of probabilities which are equal to 1or 100%. Copy and paste this formula for all 665280 cells.

Ø  Using from formula: Rv= SUM [Rp(i) * P(i)] to find the expected value of portfolio returns. Copy and paste this formula for all 665280 cells.

Ø  Using from formula: Qr = {SUM [(Rp(i) – Rv)^2) * P(i)]}^0.5 to find Standard Deviation of the expected portfolio returns. Copy and paste this formula for all 665280 cells.

Ø   Using from formula: CV = Qr / Rv to find Coefficient of Variation. Copy and paste this formula for all 665280 cells.

Now, we will be able to find the minimum Qr and CV for the range of 10% ≤ Pr ≤ 21% as follows:

Qr(min)	0.1993867
CV(min)	0.0144341

There are 12 combinations of the probability distributions (in the range of 10% to 21%) assigned to the outcomes which show us the minimum risk as follows:

	C1	C2	C3	C4	C5	C6
Outcome (1)	0.21	0.21	0.21	0.21	0.21	0.21
Outcome (2)	0.18	0.18	0.18	0.18	0.19	0.19
Outcome (3)	0.19	0.19	0.2	0.2	0.18	0.18
Outcome (4)	0.2	0.2	0.19	0.19	0.2	0.2
Outcome (5)	0.1	0.12	0.1	0.12	0.1	0.12
Outcome (6)	0.12	0.1	0.12	0.1	0.12	0.1

C7     C8    C9         C10      C11     C12 Outcome (1)                   0.21 0.21 0.21 0.21 0.21 0.21 Outcome (2)                0.19 0.19 0.2 0.2 0.2 0.2 Outcome (3)                    0.2 0.2 0.18 0.18 0.19 0.19 Outcome (4)                0.18 0.18 0.19 0.19 0.18 0.18 Outcome (5)                  0.1 0.12 0.1 0.12 0.1 0.12 Outcome (6)   0.12 0.1 0.12 0.1 0.12 0.1

(C1 to C12 are all 12 combinations)

What do you think about using of my previous simulation model mentioned in article of “Application of Pascal’s Triangular plus Monte Carlo Analysis to Find the Least Squares Fitting for a Limited Area” posted on link: http://emfps.blogspot.com/2012/05/application-of-pascals-triangular-plus_23.html?

I have already calculated the minimum risk by using of this simulation model. The final results are as follows:

CV(min)	0.014946785
Qr(min)	0.206693106

As you can see, the error is approximately equal to 3.6%.

How can we decrease the error by using of this simulation model?

Definitely we will be able to decrease the error by increasing the calculation iteration.

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.”