Monte Carlo Analysis A Technique for Combining Distributions

FIN 591: Financial Modeling, Spring 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 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 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 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 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 Simple Example Skin cream additive is an irritant Many samples of cream provide information on concentration: mean 0.02 mg chemical/application standard dev 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 Skin cream additive data PotencyExposure Information type {low, high}Mean, deviation Data{0.05, 0.10}0.02 mg, mg Distribution?Uniform? Triangular? Normal? Lognormal?

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

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

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

FIN 591: Financial Modeling, Spring Random Draw One p(irritate) = mg × / mg =

FIN 591: Financial Modeling, Spring Random Draw Two p(irritate) = mg × / mg = Summary: {0.0010, }

FIN 591: Financial Modeling, Spring Random Draw Three p(irritate) = mg × / mg = Summary: {0.0010, , }

FIN 591: Financial Modeling, Spring Random Draw Four p(irritate) = mg × / mg = Summary: {0.0010, , , }

FIN 591: Financial Modeling, Spring After Ten Random Draws Summary {0.0010, , , , , , , , , } Mean = Standard deviation = ( ).

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

FIN 591: Financial Modeling, Spring iterations (less than two seconds) Monte Carlo results Mean Standard Deviation Compare to analytical results Mean standard deviationn/a.

FIN 591: Financial Modeling, Spring Summary chart trials

FIN 591: Financial Modeling, Spring Summary - 10,000 Trials Monte Carlo results Mean Standard Deviation Compare to analytical results Mean standard deviationn/a.

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

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

FIN 591: Financial Modeling, Spring 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 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 Which Distribution?

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

FIN 591: Financial Modeling, Spring 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 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 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

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

FIN 591: Financial Modeling, Spring Alar and Aflatoxin Point Estimates Aflatoxin estimates: Mean = Alar (UDMH) estimates: Mean =

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

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

FIN 591: Financial Modeling, Spring

FIN 591: Financial Modeling, Spring

FIN 591: Financial Modeling, Spring

FIN 591: Financial Modeling, Spring

FIN 591: Financial Modeling, Spring End