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Contents
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2.A. What is it?

1. The Input Data has Uncertainties that can be quantified;
2. The "answer", or Output, must accurately represent the Input Data;
3. The calculated uncertainty in the "answer", or Output, must accurately reflect the Uncertainty in the Input data; and
4. The calculated uncertainty in the "answer", or Output, must be an accurate measure of the validity of the model.

1. We know there is uncertainty in our Input since a pro forma is by definition, an estimation of performance.
2. We know that the Monte Carlo will calculate "answers" that accurately represent the Input Data.
3. The Calculated Uncertainty of the "answers" will accurately reflect the Uncertainty of the Input Data.
4. Further, since we may be asked to invest in this project, the actual uncertainty in the final results of the pro forma are an accurate measure of the validity of the model. This is very important, if we want to make an informed investment decision!

2.B. Uncertainty in Business Plans & "TABU"

Technical Axiom of Business Uncertainty - The less you know about a Project, the higher your uncertainty regarding the Project.

• All "surprises" will most certainly raise costs. In fact the bigger the surprise, the larger the cost increase (or, profit decrease).
• Those that treat a pro forma as "fact" are telling you how little they know about the Project!
• Cost estimations will most likely be high, rather than low.
• Income estimations will most likely to be low, rather than high.
• The further out in time your estimations, the larger the uncertainty.
• Any Technology that has yet to be scaled to the size required in a Business Plan has two big sources of Uncertainty. One is the cost of R&D and the other is the cost of scaling.
• The costs of Technologies will nearly always have to be repaid at the end of the project from increased sales or profits.
• Technologies seldom lower required capital or construction costs.
• Savings that are expected from operations in a less developed country will seldom be as high as predicted. Unexpected operating factors, long distances, and governmental graft will certainly raise costs.
  
  

2.C. Errors, Uncertainty & Monte Carlo

• Errors are the difference between a "measured" (or determined!) value of a parameter and its "true" value. They include both Random Errors and Uncertainty.
  
  
• Random Errors are an effect of chance and a function only of the physical system being analyzed. Further, these Random Errors of a system are not reducible through either further study or by further measurement.
  
  
• Uncertainties of any system are due simply to the assessor's lack of knowledge about the system being studied. It is considered to subjective in nature since it may be redued by either further measurements or study.
  

1. Analytical calculation using the equation A = L x W. This is the way most of the "general population" would perform the calculation. This is also the way the calculation would be performed if the error or uncertainty were not deemed to be important. In this case, we would simply ignore any potential error or uncertainty and just use the "best", or "most likely", values of the Length and Width to calculate the Area.
   
   A = L x W = 750 ft x 999 ft = 749,250 ft²
   
   
2. If it were deemed to be important to account for any "deviation" in the Length and Width we could use standard differential calculus techniques to "propagate" the Standard Deviation from the Length and Width to the Area. This is what is taught and used in all university science and engineering laboratories. The Area would still be calculated as above.
   
   A = L x W = 749,250 ft
   
   In addition, we would "analyze" the data we have for the Length and Width and determine their "Standard Deviation". Notice that we have not really addressed error to this point. The fact of the matter is that we know that a certain fraction of our uncertainty is due to error, but we do not have to make that assessment at this time. Our main goal is to always reduce uncertainty. If we reduce uncertainty, we automatically reduce our risk! We can now calculate the Standard Deviation (or, we could just as easily use the Variance which is the square of the Standard Deviation) in the Area (SD_A) as propagated from the SD_L and SD_W. In our case, the Standard Deviation for the Length is SD_L = 46.9 ft and the Standard Deviation for the Width is SD_W = 64.6 ft. If we use the standard differential calculus techniques, we get the propagated Standard Deviation of the Area to be:
   
   SD_A = [(W x SD_L)² + (L x SD_W)²]^1/2 = 73,499 ft
   
   However, it must be noted that the partial derivative required for this calculation is sometimes difficult, if not impossible, to determine in a closed form. Therefore, Monte Carlo will even become the "method of choice" to propagate the Standard Deviation!
   
   
3. Monte Carlo Calculation. We performed this calculation above in Part A, and found we could obtain a large set of numbers that provide an added measure of validity to the final calculated Area.
   • The "Most Likely" Area (The same as Methods 1 and 2, above)
   • The Standard Deviation of the Calculated Area (As in Method 1, above)
   • The Probability that the Area will be at least some Value.
   • The Probability that the Area will be less than some Value
   
   
   

Let's suppose that two different people want to determine the area of a "triangular-shaped" piece of land that is defined by "Metes and Bounds". This method is archaic and means that the property line is describe by its meeting or crossing certain physical landmarks. In our example, the property line starts at a "spring" on Farmer Jones' property and proceeds "approximately" north for 3500 rods to tree near Farmer Brown's Barn. The Property Line then proceeds southwest for 1600 rods to a large rock on the banks of Little Horn Creek. From this large rock, the property line proceeds back to the starting point. To make matters worse, the barn on Farmer Brown's property was torn down years ago and there is now a forest of trees near where the barn was once supposedly located. To add to the confusion, there are now two springs on Farmer Brown's property. As you can see, there are several sources of uncertainty in this project!

The first person (Mr. Black) might "guess" where the Tree was, where the "right" spring is located, and which big rock by the Creek is the one in the property description. He would then use geometry to calculate his area. However, since Mr. Black is an honest person, he would say that the property has an area of "approximately AAA acres".

The 2nd person, Ms. Harris, is an expert in Statistics and Monte Carlo simulation techniques and she therefore did the following:

1. She determined "general areas" where the corners are supposed to be located.
2. She determined the "absolutely longest possible" distances for each of the three sides.
3. She also determined the "absolutely shortest possible" distances for each side.
4. She then examined the local courthouse records and walked the property line so she could determine the "most likely" distance for each side. (Mr. Black may well have used this, or a similar method to determine the length of the 3 sides of the triangle.)
5. With this information, she constructed a Distribution Function to represent the length of each side.
6. Finally, she performed a Monte Carlo calculation using 5000 iterations to calculate a "distribution" of answers.

The difference between the answers that Mr. Black and Ms. Harris would give is quite significant. Mr. Black would give you a number but he has no idea of its numerical uncertainty. On the other hand, Ms. Harris would give some like "It has an area of XXX Acres with an uncertainty of x acres. Further, there is a 90% probability that the size is greater than YYY acres and a 50% Probability that the size is greater than ZZZ acres.