Monte Carlo Analysis

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##### Monte Carlo Analysis

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1. Monte Carlo Analysis A Technique for Combining Distributions

2. Purpose of lecture • Introduce Monte Carlo Analysis as a tool for managing uncertainty • To demonstrate how it can be used in the policy setting • To discuss its uses and shortcomings, and how they are relevant to policy making processes. FIN 591: Financial Modeling, Spring 2004

3. What is Monte Carlo Analysis? • It is a tool for combining distributions, and thereby propagating more than just summary statistics • It uses a random number generation, rather than analytic calculations • It is increasingly popular due to high speed personal computers. FIN 591: Financial Modeling, Spring 2004

4. Background/History • “Monte Carlo” from the gambling town of the same name (no surprise) • Limited use because time consuming • Much more common since late 80’s • Too easy now? FIN 591: Financial Modeling, Spring 2004

5. Why do Monte Carlo Analysis? • Combining distributions • With more than two distributions, solving analytically is very difficult • Simple calculations lose information • Mean  mean = mean • 95% %ile  95%ile  95%ile! • Gets “worse” with 3 or more distributions. FIN 591: Financial Modeling, Spring 2004

6. Monte Carlo Analysis • Takes an equation • Example: Risk = probability  consequence • Draws randomly from defined distributions • Multiplies, stores • Repeats this over and over and over… • Results displayed as a new, combined distribution. FIN 591: Financial Modeling, Spring 2004

7. Simple Example • Skin cream additive is an irritant • Many samples of cream provide information on concentration: • mean 0.02 mg chemical/application • standard dev. 0.005 mg chemical/application • Two tests show probability of irritation given application • low p(effect per mg exposure)=0.05 / mg • high p(effect per mg exposure)=0.10 / mg. FIN 591: Financial Modeling, Spring 2004

8. Skin cream additive data FIN 591: Financial Modeling, Spring 2004

9. Analytical Results • Risk = Exposure  potency • Mean risk = 0.02 mg  0.075 / mg = 0.0015 or 0.15% probability that someone using the cream will be irritated. FIN 591: Financial Modeling, Spring 2004

10. Analytical results • “Conservative estimate” • Use upper 95th %ile Risk = 0.03 mg  0.0975 / mg = 0.0029 or p(irritation|application) = 0.29%. FIN 591: Financial Modeling, Spring 2004

11. Monte Carlo: Visual example Exposure = normal (mean 0.02 mg, s.d. = 0.005 mg) Potency = uniform (range 0.05 / mg to 0.10 / mg) FIN 591: Financial Modeling, Spring 2004

12. Random Draw One p(irritate) = 0.0165 mg × 0.063 / mg = 0.0010 FIN 591: Financial Modeling, Spring 2004

13. Random Draw Two p(irritate) = 0.0175 mg × 0.089 / mg = 0.0016 Summary: {0.0010, 0.0016} FIN 591: Financial Modeling, Spring 2004

14. Random Draw Three p(irritate) = 0.152 mg × 0.057 / mg = 0.0087 Summary: {0.0010, 0.0016, 0.00087} FIN 591: Financial Modeling, Spring 2004

15. Random Draw Four p(irritate) = 0.0238 mg × 0.085 / mg = 0.0020 Summary: {0.0010, 0.0016, 0.00087, 0.0020} FIN 591: Financial Modeling, Spring 2004

16. After Ten Random Draws Summary {0.0010, 0.0016, 0.00087, 0.0020, 0.0011, 0.0018, 0.0024, 0.0016, 0.0015, 0.00062} Mean = 0.0014 Standard deviation = (0.00055). FIN 591: Financial Modeling, Spring 2004

17. Using software • Could write this program using a random number generator • But, several software packages exist • I use @Risk • User friendly • Customizable • RNG good up to about 10,000 iterations. FIN 591: Financial Modeling, Spring 2004

18. 100 iterations (less than two seconds) • Monte Carlo results • Mean 0.00161 • Standard Deviation 0.00048 • Compare to analytical results • Mean 0.0015 • standard deviation n/a. FIN 591: Financial Modeling, Spring 2004

19. Summary chart - 100 trials FIN 591: Financial Modeling, Spring 2004

20. Summary - 10,000 Trials • Monte Carlo results • Mean 0.00150 • Standard Deviation 0.000472 • Compare to analytical results • Mean 0.00150 • standard deviation n/a. FIN 591: Financial Modeling, Spring 2004

21. Summary chart - 10,000 trials FIN 591: Financial Modeling, Spring 2004

22. Issues: Sensitivity Analysis • Which input distributions have the greatest effect on the eventual distribution • Which parameters can both be influenced by policy and reduce risks • When better data can be most valuable (information isn’t free…nor even cheap). FIN 591: Financial Modeling, Spring 2004

23. Issues: Correlation • Two distributions are correlated when a change in one is associated with a change in another • Example: People who eat lots of peas may eat less broccoli (or may eat more…) • Usually doesn’t have much effect unless significant correlation (||>0.75). FIN 591: Financial Modeling, Spring 2004

24. Generating Distributions • Invalid distributions create invalid results, which leads to inappropriate policies • Two options • Empirical • Theoretical. FIN 591: Financial Modeling, Spring 2004

25. Empirical Distributions • Most appropriate when developed for the issue at hand. • Example: local fish consumption • Survey individuals or otherwise estimate • Data from individuals elsewhere may be very misleading • A number of very large data sets have been developed and published. FIN 591: Financial Modeling, Spring 2004

26. Empirical Distributions • Challenge: when there’s very little data • Example of two data points • Uniform distribution? • Triangular distribution? • Not a hypothetical issue…is an ongoing debate in the literature • Key is to state clearly your assumptions • Better yet…do it both ways! FIN 591: Financial Modeling, Spring 2004

27. Which Distribution? FIN 591: Financial Modeling, Spring 2004

28. Random number generation • Shouldn’t be an issue…@Risk is good to at least 10,000 iterations • 10,000 iterations is typically enough, even with many input distributions. FIN 591: Financial Modeling, Spring 2004

29. Theoretical Distributions • Appropriate when there’s some mechanistic or probabilistic basis • Example: small sample (say 50 test animals) establishes a binomial distribution • Lognormal distributions show up often in nature, particular economics/business. FIN 591: Financial Modeling, Spring 2004

30. Some Caveats • Beware believing that you’ve really “understood” uncertainty • Central tendencies are NOT “real risk” • Distributions are only PART of uncertainty • Beware misapplication • Ignorance at best • Fraudulent at worst. FIN 591: Financial Modeling, Spring 2004

31. Example (after Finkel 1995) Alar “versus” aflatoxin Exposure has two elements Juice consumption Alar/UDMH residue Peanut butter consumption aflatoxin residue Potency has one element aflatoxin potency UDMH potency Risk = (consumption  residue  potency)/body weight FIN 591: Financial Modeling, Spring 2004

32. Inputs for Alar & aflatoxin FIN 591: Financial Modeling, Spring 2004

33. Alar and Aflatoxin Point Estimates • Aflatoxin estimates: • Mean = 0.028 • Alar (UDMH) estimates: • Mean = 0.046. FIN 591: Financial Modeling, Spring 2004

34. Alar and Aflatoxin Monte Carlo • 10,000 runs • Generate distributions • (don’t allow 0) • Don’t expect correlation. FIN 591: Financial Modeling, Spring 2004

35. Aflatoxin and Alar Monte Carlo Results (Point Values) FIN 591: Financial Modeling, Spring 2004

36. FIN 591: Financial Modeling, Spring 2004

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40. End FIN 591: Financial Modeling, Spring 2004