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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
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2.D. Probability Distribution Functions
The selection of the Probability Distribution Functions (commonly called the "p.d.f.", or "pdf") to be used in a Monte Carlo Simulation is extremely important. Remember that the "Garbage In - Garbage Out" axiom defines the single most important source of error in the Monte Carlo Simulation technique. The Distribution Functions that are used in any simulation process must reflect reality and be normalized over the appropriate domain! Below are brief descriptions of some p.d.f.'s that can be used in different situations.
- Gaussian p.d.f. - Use a "Gaussian-type" or normal distribution for most variables that have been scientifically measured or derived, or represent classifications of plants, animals or humans. Significant disadvantages to the Gaussian distribution function are that it symmetric about its "mean" and spans the domain of, and is normalized over, all real numbers (from - infinity to + infinity).
- Poisson p.d.f. - Use a "Poisson-type" distribution for the failure of manufacturing or "process" components, or any rare phenomena. It serves as a useful probability model for events occuring randomly over time or space when all that is known is the average number of occurences per unit time or space. The only input required is the average rate of occurences per unit time or space. The "variance" is always equal to the "mean" of a Poisson Distribution.
- Flat p.d.f. - Use a constant value (a flat distribution curve) if there is thought to be little or no variation in the distribution.
- Triangular p.d.f. - Use a triangular distribution if you know very little about the distribution. The Triangular Distribution Function can be completely defined by knowing the absolute minimum value, the most likely value, and the absolute maximum value. It can also be easily skewed to match reality. When people estimate "most likely" values of parameters they tend to favor the answers they want. In other words, most people will underestimate cost projections and overestimate income projections. Therefore, the absolute minimums and maximums should be skewed to compensate for this "natural" bias.
- Other "Angular" p.d.f.'s - Use an "angular" distribution that is derived to be a good fit to the data represented by the more conventional Distribution Functions (Gaussian, Poisson, etc.) These angular Distribution Functions are constructed by a series of straight-line approximations to the more complex continuous distribution functions. This significantly reduces the computation time required in Monte Carlo calculations while retaining the validity of the simulation. Further, these angular distribution functions allow for the more "realistic" upper and lower bounds that are not plus and minus infinity nor must they be symmetric about the most likely value. I call the "angular" distribution function that is constructed to fit Gaussian-type data the "Gaussangular DistributionTM" Function. There is also a "Poissangular" Distribution Function to represent Poisson-type data. In fact, this methodology can be applied to fit any type of data to any "realistic" distribution function! These angular Distribution Functions can be created by specifying the absolute minimum value, the most likely value, and the absolute maximum value.
- Reaction, or cross-section, p.d.f.'s - These are used in physics and chemistry problems. They are tabulated for both chemical and nuclear reactions.
As can be seen from the examples listed above, the p.d.f. can have many forms. However, each p.d.f. must realistically describe each variable in the model for the Monte Carlo Simulation to be valid!
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