1. Monte Carlo Analysis A Technique for Combining Distributions

2. FIN 591: Financial Modeling, Spring 2004 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.

3. 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.

4. 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?

5. 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.

6. 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.

7. 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.

8. FIN 591: Financial Modeling, Spring 2004 8 Skin cream additive data PotencyExposure Information type {low, high}Mean, deviation Data{0.05, 0.10}0.02 mg, 0.005 mg Distribution?Uniform? Triangular? Normal? Lognormal?

9. 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.

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

11. 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)

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

13. 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}

14. 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}

15. 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}

16. 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).

17. 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.

18. FIN 591: Financial Modeling, Spring 2004 18 100 iterations (less than two seconds) Monte Carlo results Mean0.00161 Standard Deviation0.00048 Compare to analytical results Mean 0.0015 standard deviationn/a.

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

20. FIN 591: Financial Modeling, Spring 2004 20 Summary - 10,000 Trials Monte Carlo results Mean0.00150 Standard Deviation0.000472 Compare to analytical results Mean 0.00150 standard deviationn/a.

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

22. 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).

23. 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).

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

25. 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.

26. 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!

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

28. 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.

29. 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.

30. 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.

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

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

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

34. 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.

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

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40. FIN 591: Financial Modeling, Spring 2004 40 End