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Monte Carlo Simulation Business

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									Monte Carlo Simulation
  and Risk Analysis


   James F. Wright, Ph.D.
   Monte Carlo Simulation
            Scientific Uses


Complex Systems Where Experiments
    are not Possible or Practical

 • Space Program
 • Nuclear and Thermonuclear Weapons
 • Catastrophic Mechanical Systems Design
 • Chemical & Nuclear Reactions
Monte Carlo Simulation
     Economic Systems



  It is Fiduciary Insurance!

 • Verify Economic Predictions
 • Investment Risk Analysis
 • Portfolio Management
        Mathematical Simulation
Let’s consider Conventional Mathematical Simulation, or Modeling, of Real Systems.


           In the Study of Mathematics we
               Represent Real Systems
              with an Equation, or Metric.

  The y, is the answer that examine how Monte Carlo Simulation the Mathematical
                   Now let’s                           The f (xi) is
                                  y = f (x )
  represents the ”state” of the “regular” Mathematical function that uses the Input
                   differs from                 i       Simulation
  Real System we are                                   Variables, xi, to produce the
  modeling.                                            answer, y.
        Monte Carlo Simulation

• We recognize that the values of Input Variables for
  our Metrics are seldom, if ever, exactly known.
• Therefore, we use Realistic Distribution
  Functions to Represent the Values of these Input
  Variables.
• By Sampling these Distribution Functions in a
  Random Manner, our Answer is a Discrete
  Distribution Function that accurately and
  precisely represents both the Metric and the Input
  Distribution Functions.
          Distribution Function

• It is a Frequency Distribution that is Normalized
  (The Area under the Curve is equal to one). A
  common example is the “Bell Curve.”
• It can be Represented by either its pdf (Probability
  Density Function) or cdf (Cumulative Distribution
  Function).
• It may be either Continuous or Discrete

            (See my Book on the Subject)
          Monte Carlo Simulation

            In the Monte Carlo Method we
               Represent Real Systems
              with an Equation, or Metric.
                    Now let’s summarize our Monte Carlo Model.


The Answer to our Metric,      yk = f [g(xi)]        Each Input Variable is now
                                                     represented by a realistic
yk, is a discrete
Distribution Function                                Distribution Function, g(xi),
rather than a single value!                          rather than single-valued
                                                     variables.
                  Monte Carlo Simulation
It is important to note that in order for the yk to be realistic, both of the following must be true.
 The Monte Carlo method uses Distribution Functions, gi(x), to represent precisely represent
1. Each of the individual input distribution functions must accurately andeach Input Variable
     the Metric used
 inthe input data. to Simulate the Real System. The Monte Carlo solution to the Metric is a
 Discrete Distribution, yk , that is representative of both the Metric, and the Input
2. The metric, or equation, must realistically represent the process being modeled.Distribution
 Functions. (See my Book on the Subject)




 Rather than use a “Best” single Value for the Input variables of the metric, we represent each
 with a Distribution Function. This Distribution Function includes the unique Absolute
 Minimum Value, Absolute Maximum Value, and all points in between including the Best (or
 Most Likely) Value. The calculated answer is represented by a Discrete Probability
 Distribution that Accurately and Precisely reflects the cumulative effects of each of the Input
 Variables and the Metric.
This Book applies the general technique of Monte Carlo Simulation to the evaluation of
Business Prospects. However as shown in this brief presentation, the technique can be
applied to any mathematical function you are using to model your real physical system.
Monte Carlo Simulation


   James F. Wright, Ph.D.
        432-367-1542
    Drjfw@drjfwright.com

								
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