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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
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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