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

Aknowledgement Wayne L. Winston, “Microsoft Excel Data Analysis and Business Modeling”, 2004

Overview Part I Part II: Simulation examples Questions answered with the help of MCS History Typical simulations Part II: Simulation examples Part III: Advantages of MCS over deterministic analysis

Challenges We are constantly faced with uncertainty, ambiguity, and variability. Risk analysis is part of every decision we make. We’d like to accurately predict (estimate) the probabilities of uncertain events. Monte Carlo simulation enables us to model situations that present uncertainty and play them out thousands of times on a computer.

Questions answered with the help of MCS How should a greeting card company determine how many cards to produce? How should a car dealership determine how many cars to order? What is the probability that a new product’s cash flows will have a positive net present value (NPV)? What is the riskiness of an investment portfolio?

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 Other companies, organizations and individuals

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 Exchange rate fluctuations

MCS Use: Lilly Determine the optimal plant capacity that should be built for each drug

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 Model and optimally hedge foreign exchange risk

MCS Use: Sears How many units of each product line should be ordered from suppliers

MCS Use: Financial Planners Determine optimal investment strategies for their clients’ retirement.

MCS Use: Others Value “real options”: Value of an option to expand, contract, or postpone a project

MCS Applications Physical Sciences Engineering Computational Biology Applied Statistics Games Design and visuals Finance and business (Actuarial Science) Telecommunications Mathematics

Part II We’ll now discuss how Monte Carlo simulation works by looking at a few simulation examples

=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 time What happens when I enter =RAND() in a cell?

Example 1: Discrete 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 function with a possible demand for calendars.

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 60,000 Greater than or equal to 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: TRIAL RAND SIM DEMAND 1 0.823097422 60,000 2 0.076074298 10,000 3 0.364201634 20,000 4 0.698116365 40,000 CUTOFF DEMAND 10,000 0.1 20,000 0.45 40,000 0.75 60,000

Discr r.v. Sim (cont.) The function used to create the values in the third column of the second table is called the VLOOKUP function. Its syntax in Excel is: VLOOKUP( lookup_value, table_array, col_index_num, range_lookup )

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 PROBABILITY 10,000 0.10 20,000 0.35 40,000 0.30 60,000 0.25 DEMAND FRACTION OF TIME 10,000 0.10250 20,000 0.35500 40,000 0.29250 60,000 0.25000

Example 2: 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 What is a normal random variable? Let us first define the standard normal random variable.

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) A standard normal r.v. Z is a r.v. with μ=0 and σ=1

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 Y=σZ+μ

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 generate a simulated value of a normal r.v. having a mean μand standard deviation σ.

Normal r.v. Sim (cont.) 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 33,518.16 = NORMINV(0.258433031, 40,000, 10,000) This value could also be looked up using the Standard Normal Distribution table.

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 In this section we’ll show how Monte Carlo simulation can be used as a tool to help businesses make better decisions.

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? In this section we’ll show how Monte Carlo simulation can be used as a tool to help businesses make better decisions.

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 maximum average profit over the 1,000 iterations In this section we’ll show how Monte Carlo simulation can be used as a tool to help businesses make better decisions.

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 $- 10 profit $25,000.00 In this section we’ll show how Monte Carlo simulation can be used as a tool to help businesses make better decisions.

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 MIN (Production Quantity, Demand) In this section we’ll show how Monte Carlo simulation can be used as a tool to help businesses make better decisions.

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 In this section we’ll show how Monte Carlo simulation can be used as a tool to help businesses make better decisions.

Cards to Produce? (cont.) Disposal cost in 9: unit disposal cost*MAX(produced-demand, 0) Total profit in 10: Revenue – total var cost – total disposing cost

Cards to Produce? (cont.) We would like an efficient way to calculate profit for each production quantity We’ll use a two-way data table mean (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 100000 66000 3 4 150000 5 -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 profits for each production quantity

Cards to Produce? Conclusion Producing 40,000 cards always yields the largest expected profit However, it also appear to have a large standard deviation (risk)

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) In our example: (53,650.46 59,628.26 )

Problems 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.) 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.) The supermarket pays $1.00 for each copy of Newsweek 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 Correlation of Inputs

Probabilistic Results Results show not only what could happen, but how likely each outcome is.

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 other stakeholders.

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 inputs had the biggest effect on bottom-line results.

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.  This is invaluable for pursuing further analysis.

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.

References Wayne L. Winston, “Microsoft Excel Data Analysis and Business Modeling”, 2004 http://office.microsoft.com/en-us/excel- help/introduction-to-monte-carlo-simulation- HA001111893.aspx Monte Carlo Simulation http://www.palisade.com/risk/monte_carlo_simulati on.asp