1. Monte Carlo Simulation and Risk Analysis James F. Wright, Ph.D.

2. Monte Carlo Simulation Scientific Uses Space Program Nuclear and Thermonuclear Weapons Catastrophic Mechanical Systems Design Chemical & Nuclear Reactions Complex Systems Where Experiments are not Possible or Practical

3. Monte Carlo Simulation Economic Systems Verify Economic Predictions Investment Risk Analysis Portfolio Management It is Fiduciary Insurance!

4. Mathematical Simulation In the Study of Mathematics we Represent Real Systems with an Equation, or Metric. y = f (x i ) The y, is the answer that represents the ”state” of the Real System we are modeling. The f (x i ) is the Mathematical function that uses the Input Variables, x i, to produce the answer, y. Now let’s examine how Monte Carlo Simulation differs from “regular” Mathematical Simulation Let’s consider Conventional Mathematical Simulation, or Modeling, of Real Systems.

5. 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.

6. 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)

7. Monte Carlo Simulation In the Monte Carlo Method we Represent Real Systems with an Equation, or Metric. y k = f [g(x i )] Each Input Variable is now represented by a realistic Distribution Function, g(x i ), rather than single-valued variables. The Answer to our Metric, y k, is a discrete Distribution Function rather than a single value! Now let’s summarize our Monte Carlo Model.

8. Monte Carlo Simulation The Monte Carlo method uses Distribution Functions, g i (x), to represent each Input Variable in the Metric used to Simulate the Real System. The Monte Carlo solution to the Metric is a Discrete Distribution, y k, that is representative of both the Metric, and the Input 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. It is important to note that in order for the y k to be realistic, both of the following must be true. 1.Each of the individual input distribution functions must accurately and precisely represent the input data. 2.The metric, or equation, must realistically represent the process being modeled. It is important to note that in order for the y k to be realistic, both of the following must be true. 1.Each of the individual input distribution functions must accurately and precisely represent the input data. 2.The metric, or equation, must realistically represent the process being modeled.

9. 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.

10. Monte Carlo Simulation James F. Wright, Ph.D. 432-367-1542 Drjfw@drjfwright.com