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Monte Carlo Simulation Spreadsheet Modeling

Why Simulate

- When we use the word simulation, we refer to any

analytical method meant to imitate a real-life

system, especially when other analyses are too

mathematically complex or too difficult to

reproduce. - Without the aid of simulation, a spreadsheet

model will only reveal a single outcome,

generally the most likely or average scenario.

Spreadsheet risk analysis uses both a spreadsheet

model and simulation to automatically analyze the

effect of varying inputs on outputs of the

modeled system.

Why Monte Carlo

- Monte Carlo simulation was named for Monte Carlo,

Monaco, where the primary attractions are casinos

containing games of chance. Games of chance such

as roulette wheels, dice, and slot machines,

exhibit random behavior.

- The random behavior in games of chance is similar

to how Monte Carlo simulation selects variable

values at random to simulate a model. When you

roll a die, you know that either a 1, 2, 3, 4, 5,

or 6 will come up, but you dont know which for

any particular roll. Its the same with the

variables that have a known range of values but

an uncertain value for any particular time or

event (e.g. interest rates, staffing needs, stock

prices, inventory, phone calls per minute).

Modeling Uncertainty

- What do you do with uncertain variables in your

spreadsheet - For each uncertain variable (one that has a range

of possible values), you define the possible

values with a probability distribution. The type

of distribution you select is based on the

conditions surrounding that variable. - Distribution types include
- Discreet
- Bernouilli (Success, Failure in one trial)
- Binomial (Number of Success/Failure in n trials
- Multi-Nominal (number of times we observe a

particular outcome in n trials.) - Poisson (number of occurrences of a particular

event during a time period 0 to time period t) - Continuous Variables
- Exponential Distribution
- Normal Distribution

Excel Functions RAND( ) a random number

between (0,1) RANDBETWEEN (a,b) a random number

between values of (a, b)

What happens during a simulation

- A simulation calculates multiple scenarios of a

model by repeatedly sampling values from the

probability distributions for the uncertain

variables and using those values for the cell.

Two ways to simulate multiple scenarios. SIMTABLE

and DATA TABLES - What do we do with the output of the simulation-

Measure Risk and optimize managerial choices,

capacity, quality, markets to enter, product to

invest RD. - Summary statistics
- Averages, Standard Deviations, Median
- Plot distribution of the values of the

performance measure and the relative frequencies,

using the Frequency function. - To use the frequency function 1. put in the bin

values to a column, 2. highlight the empty column

next to bin column, as many rows as the number of

bin rows. - FREQUENCY (bin array (highlight the cells

containing the bin values)), Data Array

(highlight the column that contains the data) )

then CTRLSHIFTENTER

Simtable vs. Data Tables

- Simulation Table from Simtools
- By leaving an empty column to the left , Copy /

Refer to the cell which contains the random

formula. - highlight as many rows as the number of

simulations you want have for the random formula. - Go to tools/ simtools/simtable

- Simulation by Data Tables
- Use one-way table.
- Set-up a one - way table referring to the

following. - Do not input any values on the left column and

refer to an empty cell for the input .

Empty Column

Random formula

Excel will replicate the formula as many times as

the number of rows, each time (each row) randomly

evaluating the formula

You need to copy and paste special/values after

the table is calculated

Another common technique is just to copy the

random formula several times to replicate the

simulation.

Simulating a Discreet Variable

1. A Bernoulli Random Variable generate a single

random variable between (0,1)

Boy

1/2

Baby

IF ( RAND( ) lt1/2, boy,girl)

1/2

Girl

2. A Binomial Random Variable with n trials

generate n Bernoulli variables and count the

number of times you have success in n trials. Say

success is defined as having a boy, in this case

to simulate the number of boys in n births,

Repeat (1) n times in a column and count the

number of times, the outcome is a boy. You can

use the function COUNTIF( datarange, condition)

Generate RAND( )

Or use BINOMINV(RAND(.),n,p)

Given our assumption about the probability

distribution of our random variable (normal,

poisson, binomial distribution), we can generate

a draw from the distribution by using the

following general technique.

Cumulative and inverse cumulative probability

distributions

Probability distribution of the random variable X

This is by assumptions of the model Assumption

should be justifies based on some historical data

, information about the variable..

1

yP(Xltx)

a

Rand()b

0

x

x

possible value of our random variable

Generate a random number between 0,1 and use

INVERSE cumulative functions available from

excel to map the random number between 0,1 to an

X value

NORMDIST(x,m,s,0) Probability distribution NORMD

IST(x,m,s,1) Cumulative distribution NORMINV(a,m

,s) Inverse cumulative distribution function

0ltalt1

Other Inverse Cumulative Functions

- Binomial Random Variable
- BinomDist(s,n,p,0) calculates the probability

of having s successes out of n trials when the

probability of success in each trial is p. - BinomDist(s,n,p,1) calculates the probability

of having Less Than and Equal To s successes out

of n trials when the probability of success in

each trial is p. - BinomInv(Rand(),n,p) generates a random number

of successes out of n trials. - Poisson Random Variable
- Poison(x,m,0) calculates the probability of x

occurrences of an event in a given time period

when the mean occurrence rate of an event is m. - Poison(x,m,1) calculates the probability of

number occurrences of an event in a given time

period is less than and equal to x, when the

mean occurrence rate of an event during that time

period is m. - PoisonInv(Rand (),m)generates a random number of

occurences for an event during a time period,

when the mean (expected) occurrence rate of that

event during that time period is m. - Exponential Random Variable
- Denotes the random time between occurences of two

identical events. If customer arrivals are

random, then the time between two arrivals can be

modeled as exponential variable with the expected

time between two arrivals m. - P(Xx) 1/mExp(x/m) exp() is the exponential

function in excel. - P(Xltx)1-Exp(x/m)
- ExpoInv(rand(),m) generates a random time

between the two occurrences of a particular event

(e.g. customer arrivals).

How should we choose the probability distribution

for our random variables

- Nature of the Variable Data (fit distributions)

or Theory - Law of Large Numbers works if we are interested

in the average of a sum of random variables with

arbitrary distribution. - Relationship between Distribution Functions

Approximations Binomial-Poisson-Normal

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