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Monte Carlo Simulation

A technique that helps modelers
examine the consequences of
continuous risk


Most risks in real world generate hundreds
of possible outcomes
Provides fuller picture of the risk in an
asset or investment by considering
Different input assumptions & scenarios
 Likelihood of inputs & scenarios occurring

5 Steps of Monte Carlo
Simulation





Build a spreadsheet model that has dynamic
relationships between input assumptions and key
outputs
Perform sensitivity analysis to identify the key
uncertain inputs that have the most potential impact on
the key outputs
Quantify possible values for the key uncertain inputs
by specifying probability distributions
Simulate numerous scenarios from the input
probability distributions and record output results
Summarize recorded output results to measure risks
and likelihood of different outcomes
Random Number Generator
(RNG)

=Rand() function in Excel

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Randomly generates a number between 0 and 1
Used to represent a cumulative probability P(X) between 0%
and 100%
Will be used to identify the input value X such that the
probability that value X or a lower value occurs is equal to
P(X)
For example,
 Rand()=.5


Used to find input assumption value X where 50% of the input
assumption values are smaller and 50% of possible input assumption
values are larger than X.
Rand()=.9

Used to find input assumption value X where 90% of the input
assumption values are smaller than X.
Normal (u, σ)
P(X≤u)=.5
P(u- σ ≤x ≤u+ σ)=.65
σ
Prob
u- σ
u
u+ σ
x
Normal Input Distributions
=Norminv(rand(), mean, standard
deviation)
 For a specified mean and standard
deviation, this formula looks up the value
for the input distribution that results in
rand()% of the assumption values being
smaller than the returned value.
 The Normal distribution is a continuous
distribution

Continuous –vs- Discrete Distributions
In discrete distributions, the values
generated for a random variable must be
from a finite distinct set of individual
values.
 In continuous distributions, the values
generated for a random variable are
specified from a set of uninterrupted
values over a range; an infinite number
of values is possible

Uniform (a, b)
Prob
P(X ≤ u)=. 5
a
u=a+(b-a)
2
b
X
Uniform Distribution
=a + (b-a)*rand()
 Where a is the smallest value that could
occur, b is the largest value
 Values between a and b are assumed to
be equally likely to occur
 Values are assumed to be continuous
and not discrete

Probability Distributions in
Analytic Solver
Select cell that represents an uncertain
input assumption
 On the Analytic Solver Platform ribbon,
click Distributions in the Simulation
Model group
 Pick a Common continuous distribution
or a Discrete distribution
 Enter values for its parameters in dialog
box and click Save

Generating Values for Random
Variables

Analytic Solver will insert:
=PsiDistribution(parameters)
function into the input assumption cell from
which values will be randomly selected
 Pressing [F9] Recalc key will simulate next set
of values in model
 Double-clicking on the input cell will open the
dialog box for further editing or review
 This function can be copied to other cells
Specifying Input Distributions

Practice in Class Exercise 2!
Tracking Output Measures
(KPI’s) in Analytic Solver

Select cell that represents an output measure

On the Analytic Solver Platform ribbon, click
Results in the Simulation Model group

Click Output and In Cell option

This will append the PsiOutput() function to
the formula in the cell which triggers ASP to
record the cell’s value during the simulation
Tracking Output Statistics in
Worksheet with Analytic Solver

Select cell that represents an output measure

On the Analytic Solver Platform ribbon, click
Results in the Simulation Model group

Click Statistic and desired statistic option (e.g.
Mean)

Click on the blank cell where you want this
statistic entered

=PsiStatistic(OutputMeasure) function will
be entered in the cell and calculated after the
simulation is run
Simulation Settings


Select number of trials (e.g. runs,
scenarios) to be performed:

On the Analytic Solver Platform ribbon, click
Options in the Options group

Set Trials per Simulation to desired number
on Simulation tab.
You can also check Use Correlations if
you want to correlate input distributions
Run Simulation

On the Analytic Solver Platform ribbon, click
Interactive in the Simulate button for the Solve
Action group

The simulation will be run and a dialog box will
pop up with the distributions for your output
measures. The requested statistics will also
become available in worksheet. You can click
on any chart to see statistics and interactive
graph.
Correlated Input Variables


On the Analytic Solver Platform ribbon, click
Correlations in the Simulation Model group
Select Matrices and move random variables
that you want to correlate to right side of task
bar.
 Click on the square that has a scatter plot for
two random variables and move the
correlation bar to define the relationship
between the two variables. Save in blank cells
in worksheet and then run simulation.