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

2. 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.

3. 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?

4. 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.

5. 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.

6. Who Uses MCS?  General Motors (GM)  Procter and Gamble (P&G)  Eli Lilly  Wall Street firms  Sears  Financial planners  Other companies, organizations and individuals

7. 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.

8. 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

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

10. 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%

11. MCS Use: Procter & Gamble  Model and optimally hedge foreign exchange risk

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

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

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

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

16. Part III: Advantages of MCS  In conclusion, we’ll discuss some advantages of MCS over deterministic, or “single-point estimate” analysis.

17. 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

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

19. 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.

20. 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.

21. 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.

22. 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.