Materials for Lecture 10 • Chapter 6 • Chapter 16 Sections 3.2 - 3.7.3, 4.0, • Lecture 10 GRKS.XLSX • Lecture 10 Low Prob Extremes.XLSX • Lecture 10 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 • Known 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 for Xs and F(Xs) to change the distribution shape to conform to your subjective expectations • Simulate distributionas =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 intervals • 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