Monte Carlo Simulation

A technique that helps modelers examine the consequences of continuous riskMost risks in real world generate hundreds

of possible outcomes

Provides fuller picture of the risk in an asset or investment by consideringDifferent input assumptions & scenariosLikelihood 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 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.

Prob

Normal (u, σ)P(X≤u)=.5

P(u- σ ≤x ≤u+ σ)=.65

σ

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)

a u=a+(b-a) b X 2

Prob

P(X ≤ u)=. 5

Uniform Distribution

=a + (b-a)*rand()Where a is the smallest value that could

occur, b is the largest valueValues between a and b are assumed to

be equally likely to occurValues are assumed to be continuous

and not discrete

Triangular Distribution

For symmetrical triangles:=a + (b-a)*(rand()+rand())/2

Where a is the smallest value that could occur, b is the largest value

m is the most likely value to occur and is assumed to be halfway between a and b for symmetrical triangles

Values are assumed to be continuous and not discrete in both symmetrical and nonsymmetrical triangular distributions

Discrete Distribution

P(x)X P(x)10 .211 .312 .413 .1

10 11 12 13 x

.2

1.0/0

.9

.5

1110

13

12

Cumulative Probability Distribution

X P(x)10 .211 .312 .413 .1

P(X≤x).2.5.91.0

Discrete Distributions

Set up a table where the first two columns contain the cumulative probability range for a value and the third column contains the respective discrete values. Make sure the first column starts at 0

Lower Prob

Upper Prob

Discrete Value

0 .2 10

.2 .5 11

.5 .9 12

.9 1.0 13

Vlookup formula for the Discrete Distribution

Use the Excel function:=vlookup(rand(),table,3)

This function will look up the rand() number in column 1 of the table, identify the row that represents the correct cumulative probability, and look up the value associated with that probability in column 3.

See Vlookup Excel Tutorials in MyLMUConnect for logic of the Vlookup formula

Specifiying Input Distributions

Practice in Class Exercise 2!