Chapter 6 Chapter 16 Sections , 4.0, Lecture 16 GRKS.XLSX Lecture 16 Low Prob Extremes.XLSX Lecture 16 Uncertain Emp Dist.XLSX Lecture 16 Combined Distribution.xlsm Simulating Uncertainty Lecture 16

Risk is when we have random variability from a known (or certain) probability distribution Uncertainty is when we have random variability from unknown (or uncertain ) distributions Known distribution can be a parametric or non-parametric distribution –Normal, Empirical, Beta, etc. Risk vs. Uncertainty

We have random variables coming from unknown or uncertain distributions May be based on history or on purely random events or reactions by people in the market place Could be a hybrid distribution as –Part Normal and part Empirical –Part Beta and part Gamma We are uncertain and must test alternative Dist. Uncertainty

Conceptualize a hybrid distribution Part Normal and part Empirical –Simulate a USD as USD = uniform(0,1) –If USD <0.2 then Ỹ = Ŷ * (1+EMP(S i, F(x))) –IF USD>=0.2 and USD<=0.8 then Ỹ = NORM(Ŷ, Std Dev) –If USD > 0.8 then Ỹ = Ŷ * (1+EMP(S’ i, F’(x))) Where S’ i are sorted “large” values for Y and S i are sorted “small” values for Y Uncertainty

Hybrid Distribution

This is how we will model low probability, high impact events, i.e., Black Swans The event may have a 1 or 2% chance but it would mean havoc for your business Low risk events must be included in the business model or you will under estimate the potential risk for the business decision This is a subjective risk augmentation to the historical distribution Uncertainty

When you have little or no historical data for a random variable assume a distribution such as: –GRKS (Gray, Richardson, Klose, and Schumann) –Or EMP I prefer GRKS because Triangle never returns min or max and we usually ask manager for the min and max that is observed 1 in 10 years, i.e., a 10% chance of occurring GRKS Distribution for Uncertainty

GRKS parameters are –Min, Middle or Mode, and Max Define Min as the value where you have a 97.5% chance of seeing greater values Define Max as the value where there is a 97.5% chance of seeing lower values –In other words, we are bracketing the distribution with ±2 standard deviations GRKS has a 50% chance of seeing values less than the middle Once estimated the parameters can be adjusted GRKS Distribution

Parameters for GRKS are Min, Middle, Max Simulate it as =GRKS(Min, Middle, Max) Note: not necessarily equal distance from middle =GRKS(12, 20, 50) GRKS Distribution minmiddlemax 1.0 minmiddlemax

Easy to modify the GRKS distribution to represent any subjective risk or random variable. This makes the dist. very flexible From the Simetar Toolbar click on GRKS Distribution and fill in the menu Edit table of deviates for Xs and F(Xs) to change the distribution shape to conform to your subjective expectations Simulate distribution using =EMP(S i, F(x)) GRKS Distribution

The GRKS menu asks for –Minimum –Middle –Maximum –No. of intervals in Std Deviations beyond the min and max. I like 4 intervals to give more flexibility to customizing the distribution. –Always request a chart so you can see what your distribution looks like after you make changes in the X’s or Prob(x)’s GRKS Distribution

The GRKS menu generates the following table and CDF chart: Prob(X i ) is the Y axis and X i is the X axis Has 13 equal distant intervals for X’s; so we have parameters for EMP 50% observations below Mode 2.275% below the Minimum 2.275% above the Maximum Modeling Uncertainty with GRKS

Actually it is easy to model uncertainty with an EMP distribution We estimate the parameters for an EMP using the EMP Simetar icon for the historical data –Select the option to estimate deviates as a percent of the mean or trend Next we modify the probabilities and Xs based on your expectations or knowledge about the risk in the system Modeling Uncertainty with EMP

Below is the input data and the EMP parameters as fractions of the trend forecasts Note price can fall a maximum of 25.96% from Ŷ Price can be a max of 20.54% greater than Ŷ Modeling Uncertainty with EMP

The changes I made are in Bold. Then calculated the Expected Min and Max. F(X) is used for all three random variables. You may not want to do this. You may want a different F(x) for each variable.

Results from simulating the modified distribution for Price Note probabilities of extreme prices Modeling Uncertainty with EMP

Do not assume historical data has all the possible risk that can affect your business Use yours or an expert’s experience to incorporate extreme events which could adversely affect the business Modify the “historical distribution” based on expected probabilities of rare events See the next side for an example. Summary Modeling Uncertainty

Assume you buy an input and there is a small chance (2%) that price could be 150% greater than your Ŷ Historical risk from EMP function showed the maximum increase over Ŷ is 59% with a 1.73% I would make the changes to the right in bold and simulate the modified distribution as an =EMP() Simulation results are provided on the right Modeling Low Probability Extremes