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## Materials for Lecture 11

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**Materials for Lecture 11**• Chapter 6 • Chapter 16 Sections 3.2 - 3.7.3, 4.0, • Lecture 11 GRKS.XLSX • Lecture 11 Low Prob Extremes.XLSX • Lecture 11 Uncertain Emp Dist.XLSX**Risk vs. Uncertainty**• 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 • Know distribution can be a parametric or non-parametric distribution • Normal, Empirical, Beta, etc.**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 as • Part Normal and part Empirical • Simulate a USD as USD = uniform(0,1) • If USD <0.2 then Ỹ = Ŷ * (1+EMP(Si, F(x))) • IF USD>=0.2 and USD<=0.8 then Ỹ = NORM(Ŷ , StdDev) • If USD > 0.8 then Ỹ = Ŷ * (1+EMP(S’i, F’(x)))**Uncertainty**• This is where 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**GRKS Distribution for 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**• 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 + and – 2 standard deviations • GRKS has a 50% chance of seeing values less than the middle**1.0**min middle max min middle max GRKS Distribution • Parameters for GRKS are Min, Middle, Max • Simulate it as =GRKS(Min, Middle, Max) Note it does not have to be equal size =GRKS(12, 20, 50)**GRKS Distribution**• Easy to modify the GRKS distribution to represent any subjective risk or random variable • From the Simetar Toolbar click on GRKS Distribution and fill in the menu • Edit table of deviates forXs and F(Xs) to change the distribution shape to conform to your subjective expectations • Simulate it as an =EMP(Si , 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 better now • 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**Modeling Uncertainty with GRKS**The GRKS menu generates the following table and chart: • Prob(Xi) is the Y axis and Xi 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 EMP**• 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.**Modeling Uncertainty with EMP**• Results from simulating the modified distribution for Price • Note probabilities of extreme prices**Summary Modeling Uncertainty**• 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 that could adversely affect your business • Modify the “historical distribution” based on expected probabilities of rare events • See the next side for an example.**Modeling Low Probability Extremes**• 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