## Monte Carlo

## Monte Carlo

## Simulation

## Simulation

Natalia A. Humphreys Natalia A. Humphreys April 6, 2012 April 6, 2012University of Texas at Dallas

### Aknowledgement

### Aknowledgement

Wayne L. Winston,Wayne L. Winston, “Microsoft Excel Data Analysis“Microsoft Excel Data Analysis

and Business Modeling”

### Aknowledgement

### Aknowledgement

Wayne L. Winston,Wayne L. Winston, “Microsoft Excel Data Analysis“Microsoft Excel Data Analysis

and Business Modeling”

### Overview

### Overview

Part IPart I

Questions answered with the help of Questions answered with the help of MCSMCS

HistoryHistory

Typical simulationsTypical simulations

Part II: Part II: Simulation examplesSimulation examples

Part III: AdvanPart III: Advantages of tages of MCS over MCS over deterministicdeterministic

analysis

### Challenges

### Challenges

We are constantly faced with uncertainty, ambiguity,We are constantly faced with uncertainty, ambiguity,

and

and variabilityvariability..

Risk analysis is part of every decision we make.Risk analysis is part of every decision we make.

### We’d like to accurately predict (estimate) the

### We’d like to accurately predict (estimate) the

probabilitie

probabilities s of uncertain events.of uncertain events.

Monte Carlo simulation enables us to modelMonte Carlo simulation enables us to model

situations that present uncertainty and play them situations that present uncertainty and play them out thousands of times on a computer.

### Questions answered

### Questions answered

### with the help of MCS

### with the help of MCS

How should a greeting card How should a greeting card company determinecompany determine

how many cards to produce?

how many cards to produce?

How should a car How should a car dealership determine how manydealership determine how many

cars to order?

cars to order?

### What is the probability that

### What is the probability that

### a new product’s cash

### a new product’s cash

flows will have a positive net present

flows will have a positive net present value (NPV)?value (NPV)?

### Modeling with MCS

Monte Carlo Simulation (MCS) lets you see all the

possible outcomes of your decisions and assess the impact of risk, allowing for better decision making under uncertainty.

### MCS: Where did the

### Name Come From?

During the 1930s and 1940s, many computer

simulations were performed to estimate the probability that the chain reaction needed for the atom bomb would work successfully.

The Monte Carlo method was coined then by the

physicists John von Neumann, Stanislaw Ulam and

Nicholas Metropolis, while they were working on this and other nuclear weapon projects (Manhattan Project) in the Los Alamos National Laboratory.

It was named in homage to the Monte Carlo Casino, a

famous casino in the Monaco resort Monte Carlo where Ulam's uncle would often gamble away his money.

### Who Uses MCS?

General Motors (GM) Procter and Gamble (P&G) Eli Lilly

Wall Street firms

Sears

Financial planners

### MCS Use

General Motors (GM), Procter and Gamble (P&G),

and Eli Lilly use simulation to estimate both the average return and the riskiness of new products.

### MCS Use: GM

Forecast net income for the corporation Predict structural costs and purchasing costs Determine its susceptibility to different risks:

Interest rate changes

### MCS Use: Lilly

Determine the optimal plant capacity that should be

### MCS Use: Wall Street

Price complex financial derivatives Determine the Value at Risk (VaR) of investment

portfolios.

By definition, Value at Risk at security level p for a

random variable X is the number VaR_p(X) such that

Pr(X<VaR_p(X))=p

In practice, p is selected to be close to 1: 95%, 99%, 99.5%

### MCS Use: Procter &

### Gamble

### MCS Use: Sears

How many units of each product line should be

### MCS Use: Financial

### Planners

Determine optimal investment strategies for their

### MCS Use: Others

### Value “real options”:

Value of an option to expand, contract, or postpone a

### MCS Applications

Physical Sciences Engineering Computational Biology Applied Statistics Games Design and visuals

Finance and business (Actuarial Science) Telecommunications

### Part II

### We’ll now discuss how Monte Carlo simulation

### =RAND() function

When you enter the formula =RAND() in a cell, you

get a number that is equally likely to assume any value between 0 and 1.

Get a number less than or equal to 0.25 around 25% of

the time

Get a number that is at least 0.9 around 10% of the

### xamp e : scre e

### Random Variable

### Simulation

Demand for a calendar is governed by the following

discrete r.v.: DEMAND PROBABILITY 10,000 0.10 20,000 0.35 40,000 0.30 60,000 .25

### Discrete r.v.

### Simulation(cont.)

How can we have Excel play out, or simulate, this

demand for calendars many times?

We associate each possible value of the RAND

### Discr r.v. Sim (cont.)

The following assignment ensures that a demand of

10,000 will occur 10 percent of the time, and so on.

DEMAND RANDOM NUMBER ASSIGNED

10,000 Less than 0.10

20,000 Greater than or equal to 0.10 and less than

0.45

40,000 Greater than or equal to 0.45 and less than 0.75

### Discr r.v. Sim (cont.)

Creating the following cutoff table, we then use it to

### look up the values “assigned” to each random

number: CUTOFF DEMAND 0 10,000 0.1 20,000 0.45 40,000 0.75 60,000

TRIAL RAND SIM

DEMAND 1 0.823097422 60,000 2 0.076074298 10,000 3 0.364201634 20,000 4 0.698116365 40,000

### Discr r.v. Sim (cont.)

The function used to create the values in the thirdcolumn of the second table is called the VLOOKUP function.

Its syntax in Excel is:

VLOOKUP( lookup_value, table_array,

### Discr r.v. Sim (cont.)

Thus, the VLOOKUP(0.823097422, LOOKUP, 2,

1)=60,000

TRUE=1, FALSE=0

If VLOOKUP can't find lookup value, and range

lookup is TRUE, it uses the largest value that is less than or equal to lookup value.

### Discr r.v. Sim (cont.)

If we simulate 400 values of calendar demand and

then calculate the fraction of time each demand

### appears in the simulation, we’ll get a table similar to

the following:
DEMAND FRACTION
OF TIME
10,000 _{0.10250}
20,000 _{0.35500}
40,000 _{0.29250}
60,000 _{0.25000}
DEMAND PROBABILI
TY
10,000 0.10
20,000 0.35
40,000 0.30
60,000 0.25

### xamp e : orma

### Random Variable

### Simulation

Suppose we want to simulate 400 trials or iterations

for a normal r.v. with a mean

### μ

=40,000 and standard deviation### σ

=10,000 What is a normal random variable?

Let us first define the standard normal random

### Standard Normal

### Random Variable

### Its distribution has a form of a “bell” curve around

the zero.

Standard Normal Distribution Table is a table that

shows probability that a standard normal random variable Z is less than a number z:

### Φ(z

)=Pr(Z<z)### Connection between

### any Normal r.v. and a

### Standard Normal r.v.

If

### Z is N(0, 1) and is Y is N(μ, σ^2), then

### Normal Random

### Variable Simulation

Suppose we want to simulate 400 trials or iterations

for a normal r.v. with a mean

### μ

=40,000 and standard deviation### σ

=10,000 The formula NORMINV(RAND(),

### μ

,### σ

) will generatea simulated value of a normal r.v. having a mean

### Normal r.v. Sim (cont.)

33,518.16 = NORMINV(0.258433031, 40,000, 10,000)

This value could also be looked up using the

Standard Normal Distribution table.

TRIAL RAND NORMAL RV

1 0.258433031 33,518.16 2 0.344835199 36,006.98 3 0.927522163 54,575.82 4 0.248403053 33,204.76

### Example 3: How Many

### Cards to Produce?

### Suppose the demand for a Valentine’s Day card is

governed by the following discrete r.v.:

DEMAND PROBABILITY

10,000 0.10 20,000 0.35 40,000 0.30 60,000 .25

### Cards to Produce?

### (cont.)

The greeting card sells for $4.00

The variable cost of producing each card is $1.50 Leftover cards will be disposed at $0.20 per card

### How many cards should be printed to get

### the highest profit?

### Cards to Produce?

### (cont.)

We simulate each possible production quantity

(10,000, 20,000, 40,000 or 60000) many times (e.g. 1,000 iterations)

Then we determine which order quantity yields the

### Cards to Produce?

### (cont.)

1 produced 10,000 2 rand 0.400927091

3 demandcard 20,000

4 unit prod cost $1.50 5 unit price $4.00 6 unit disp cost $0.20 7 revenue $40,000.00 8 total var cost $15,000.00 9 total disposing cost

### Cards to Produce?

### (cont.)

Our sales and cost parameters are in 4, 5, and 6 Enter a trial production quantity in 1

Create a random number in 2 with =RAND() Simulate demand for the card in 3 with

VLOOKUP(rand, lookup, 2)

The number of unites sold is

### Cards to Produce?

### (cont.)

Revenue in 7: MIN (Produced, Demand)*unit price Total production cost in 8: produced*unit production

cost

If we produce more cards than are demanded, the

number of units left over equals production minus demand

### Cards to Produce?

### (cont.)

Disposal cost in 9:

unit disposal cost*MAX(produced-demand, 0)

Total profit in 10:

### Cards to Produce?

### (cont.)

We would like an efficient way to calculate profit for

each production quantity

### We’ll use a two

-way data tablemean (ave profit) 24,985 45,984 57,311 44,218 st dev (risk) - 12,321.19 48,346.89 73,622.44 25,000 10,000 20,000 40,000 60,000 1 25000 50000 16000 -60000 2 25000 50000 100000 66000 3 25000 50000 16000 66000 4 25000 50000 100000 150000 5 25000 50000 100000 -18000

### Cards to Produce?

### (cont.)

Enter 1-1000 on the left corresponding to our 1,000

trials

Enter possible production quantities (third row)

We want to calculate profit for each trial number and

each production quantity

Refer to the formula for profit in the upper left cell of

our data table by entering =B11

We are now ready to trick Excel into simulating

1,000 iterations of demand for each production quantity.

### Cards to Produce?

### (cont.)

Select the table range and then click Table on the

Data menu.

Click on any blank cell (e.g. I14) as the column

input cell and choose production quantity (cell B1) as the row input cell.

We calculate the average simulated profit for each

production quantity

We calculate the standard deviation of simulated

### Cards to Produce?

### Conclusion

Producing 40,000 cards always yields the largest

expected profit

However, it also appear to have a large standard

### The Impact of Risk in

### Our Decision

Producing 20,000 cards instead of 40,000, the

expected profits drop by about 22%, but the risk drops almost 73%.

Therefore, if we are extremely risk averse,

producing 20,000 cards might be the right decision.

Note that producing 10,000 cards always has a

std.dev. of zero cards because if we produce 10,000 cards we will always sell all of them and have none left over.

### Confidence Interval for

### Mean Profit

Into what interval are we 95% sure the true mean

will fall?

This interval is called the 95% confidence interval

for mean profit .

### It’s computed by the following formula:

Mean Profit ±(1.96*profit std.dev

### .)/√(number iterations)

### Problems

1 A GMC dealer believes that demand for 2005

Envoys will normally be distributed with a mean of 200 and standard deviation of 30. His cost of

receiving an Envoy is $25,000, and he sells an

Envoy for $40,000. Half of all leftover Envoys can be sold for $30,000. His is considering ordering 200, 220, 240, 260, 280, and 300 Envoys. How many should he order?

### Problems (cont.)

2 A small supermarket is trying to determine how

many copies of Newsweek magazine they should order each week. They believe their demand for Newsweek is governed by the following discrete random variable DEMAND PROBABILITY 15 0.10 20 0.20 25 0.30 30 0.25 35 0.15

### Problems (cont.)

2 The supermarket pays $1.00 for each copy ofNewsweek and sells each copy for $1.95. They can return each unsold copy of Newsweek for $0.50.

How many copies of Newsweek should the store order to maximize its profit?

### Part III: Advantages of

### MCS

### In conclusion, we’ll discuss some advantages of

MCS over deterministic,

### or “single

-### point estimate”

analysis.### Advantages of MCS

MCS provides a number of advantages over

deterministic,

### or “single

-### point estimate” analysis:

Probabilistic Results Graphical Results Sensitivity Analysis Scenario Analysis

### Probabilistic Results

Results show not only what could happen, but how

### Graphical Results

Because of the data a Monte Carlo simulation### generates, it’s easy to create graphs of different

outcomes and their chances of occurrence.

This is important for communicating findings to

### Sensitivity Analysis

With just a few cases, deterministic analysis makes

it difficult to see which variables impact the outcome the most.

### In Monte Carlo simulation, it’s easy to see which

### Scenario Analysis

### In deterministic models, it’s very difficult to model

different combinations of values for different inputs to see the effects of truly different scenarios.

Using Monte Carlo simulation, analysts can see

exactly which inputs had which values together when certain outcomes occurred.

### Correlation of Inputs

### In Monte Carlo simulation, it’s possible to model

interdependent relationships between input variables.

### It’s important for accuracy to represent how, in

reality, when some factors go up, others go up or down accordingly.