Navigation Bar
Place mouse on line with 
symbol to expand menu.
Purchase Dr. Wright's timely Report titled "The How & Why of Technical Due Diligence" for US$25.
Read and Save Dr. Wright's Free pdf article titled "Technical Due Diligence, Errors & Uncertainty".
Read about Dr. Wright's unique Program that saves both Time and Money - Lowers Risk!
Read Dr. Wright's article published in the March 1, 2001, issue of "Expert-Zine.com".
|
RISK ANALYSIS USING
MONTE CARLO SIMULATION
If you are interested in Monte Carlo Simulation of scientific problems
rather than investments, please click on this green button:  .
Monte Carlo Simulation is a quick and cost-effective method to quantify your Risk in a New Project or Investment. NASA, the US nuclear weapons program, and important scientific endeavors throughout the world have used this technique extensively since the advent of computers. It is just now finding general acceptance in the field of Finance.
Contents
Click on the Section to
immediately Navigate to that Section
2.E. Monte Carlo Theory Review
Since Monte Carlo is a Simulation technique, let's first define exactly what we mean by Simulation.
A true Simulation will merely describe a system, not optimize it!
(However, it should be noted that a true simulation may be modified in a manner
such that it can be used to significantly enhance the efficiency of a system.)
Therefore, our goal in Simulation is to build an experimental model that will simply describe a real system.
However, the breadth and extent of Simulation models is extensive! This can be illustrated by considering the three general "classifications" of Simulation Models, below. And in each of these "classifications", I have defined two possible "characteristics".
- 1. Functional Classification
- Deterministic Characteristic - These are "exact" models that will produce the same outcome each time they are run.
- Stochastic Characteristic - These models include some "randomness" that may produce a different outcome each time it is run. This randomness forces us to make a large number of runs to develop a "trend" in our "collection" of outcomes. Further, the exact number of how many "runs" you must make to obtain the "right trend" is simply a matter of statistics.
- 2. Time Dependence
- Static Characteristic - These models are not time-dependent. This even includes calculations for return on investments after a fixed period of time.
- Dynamic Characteristic - These models depict change in a system over many time intervals during the calculation process.
- 3. Input Data
- Discrete Characteristic - The input data are collections of "distributions". These "distributions" are series of discrete numbers.
- Continuous Characteristic - The input data can be described by an equation.
We can now say that my Monte Carlo Risk Analysis is a "True Simulation" as it describes how uncertainty is passed from input variables to the output answer. We could further define it as a "Stochastic, Static Simulation that uses Continuous Input". I stated that I used "Continuous Input" because I fit the "input data" to an equation before I start the Monte Carlo process. If a prediction were required, then "every possible" option would have to be considered and this is where the well-known "Variance Reduction Methods" (antithetic variables, etc.) would be used to reduce the number of iterations required in the simulation.
The various types of Distribution Functions were discussed in Part 2.D of this Monte Carlo Section.
|
Figure 1 is a "typical" Distribution Function for a variable "X". The horizontal axis is the "value of the variable, X" and the vertical axis is the "frequency of X". By examining Figure 1, it can be seen that the highest frequency occurs for a single value of X. This value of X for which the frequency is largest is defined as the "most likely value" value of X. The minimum and maximum values of X (XMin and XMax) are at the "end-points" of the curve. In most "real-world" systems of measured values, the XMin and XMax values also will have "minimum" frequencies.
|
This particular Distribution Function has the general "bell shape" that represents the statistics of many natural, or measured, phenomena. Next, let's "normalize" this Distribution Function. This means that we will multiply the frequencies of the Distribution Function by a constant such that the area under the "new" Normalized Distribution Function is equal to "1". This is very important, because it allows us to state that a fraction of the area under "new" Normalized Distribution Function curve between XMin and another number, Xi, is a unique number between 0 and 1 (as long as our Xi is between XMin and XMax).
- We now have a method to use random numbers between the value of "0" and "1" to always "randomly select" a value of Xi, for all i.
- Further, this set of Xi will always fall within our Frequency Distribution.
This is the basis of the Monte Carlo Simulation Process!
I believe that the best way to understand how the Monte Carlo simulation process works is to simply go through a test calculation from start to finish.
-
The Problem
We want to calculate the Area of a rectangle, A, that has a Width, W, and a Length, L. We know that the Length is approximately 1000 ft and the Width is approximately 750 ft.
The Distribution Functions
However, when the Width and Length were measured 10 different times with three different meter sticks by 10 different people, we obtained a series of answers (300) for both the Width and Length. The data from these measured Lengths and Widths were sorted and place into two histograms (one for Width and one for Length) with evenly sized bins (let's say 1 ft in width). These two histograms are called "frequency distributions and they give us the number of measurements that fall in each 1-ft wide bin. If we made enough measurements, we would see a "bell-shaped" curve where the "bin" with the most members (highest frequency) is called the "most likely" value. In fact, the most likely value for the Width was determined to be 750 ft but values ranged from 710 ft to 785 ft. For the Length, the most likely value was determined to be 999 ft, the minimum value was 981 ft and the maximum value measured was 1027 ft. Based on these data we determined the following parameters for the Length and Width.
Side
|
Absolute Minimum Value
|
Most Likely Value
|
Absolute Maximum Value
|
|
Width
|
700 ft
|
750 ft
|
795 ft
|
|
Length
|
971 ft
|
999 ft
|
1,040 ft
|
Note that the Absolute Maximum and Absolute Minimum are respectively more and less than the maximum and minimum values we first determined. This is because we want there to be a zero probability that any value less that the "Absolute Minimum" and any value more than the "Absolute Maximum" will ever occur.
Next, we use the set of values in the above table to calculate a Normalized Distribution Function for the Length and a Normalized Distribution Function for the Width. Finally, we are ready to start our iterative calculation.
The Iterative Calculation
The steps of the iterative calculation are as follows:
- Start the "ith" iteration
- Use a Random Number between the 0 and 1 and the Normalized Distribution Function of the Length to determine a single value, Li, to be used in this iteration.
- Use a Random Number between the 0 and 1 and the Normalized Distribution Function of the Width to determine a single value, Wi, to be used in this iteration.
- Calculate the Area for the "ith" iteration from the equation
Ai=Wi x Li
- Place the value of Ai in the appropriate "bin" of a histogram. This histogram will become a Frequency Distribution for Ai when all of the iterations are complete.
- Go to Step 1 and start the process over if the Frequency Distribution Histogram in Step 5, is not complete.
The Results
The results are summarized below.
- The Amost likely = Lmost likely x Wmost likely. This is what we would expect! This is also what would be calculated if we used the "most likely" values of the Width and Length in a simple Area calculation.
- More importantly, we can now use our Area Histogram to calculate such things as:
- The Probability that the Area will be at least some Value.
- The Probability that the Area will be less than some Value
- The Standard Deviation of the Calculated Area
Now, we not only have the "most likely value of the Area, but we also have several measures of the Probability that the Area will be more or less than a certan value. These Probabilities are actually a measure of the validity, or Uncertainty, of the Area.
Monte Carlo Simulation methods are primarily used in situations where:
- The Input Data has Uncertainties;
- The "answer", or Output, must accurately represent the Input Data;
- The calculated uncertainty in the "answer", or Output, must accurately reflect the Uncertainty in the Input data; and
- The calculated Uncertainty in the "answer", or Output, must be an accurate measure of the validity of the model.
This is why Monte Carlo Simulation is the perfect tool for evaluating pro forma of Investment Opportunities.
- We know there is Uncertainty in our Input since a pro forma is by definition, an estimation of performance.
- We know that the Monte Carlo will calculate "answers" that accurately represent the Input Data.
- The Calculated Uncertainty of the "answers" will accurately reflect the Uncertainty of the Input Data.
- 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!
A more complete development of how both "discrete" and "continuous real-valued" Distribution Functions are used in actual Monte Carlo calculations is included in my Report titled "The How & Why of Technical Due Diligence".
This report can be purchased
for US$25. You will be taken to the Credit Card Payment Department by clicking on the
green button below.
|
Go to Top of Page
|
