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• Chapter 6• Chapter 16 Sections 3.2 - 3.7.3, 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
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• 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
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• 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
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• 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(Si, 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 Si are sorted “small” values for Y
Uncertainty
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Hybrid Distribution
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Hybrid Distribution
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• 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
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• 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
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• 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
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• 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
min middle max
1.0
min middle max
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• 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(Si , F(x))
GRKS Distribution
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• 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
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The GRKS menu generates the following table and CDF 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 GRKS
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Modeling Uncertainty with GRKS
3. If you want to modify the distribution, edit the values in the table.To demonstrate this I repeated Step 2 and then modified the Xi values in the table below.The assumption is that I think Y should be 50 about 35% of the time.The values in Prob(Xi) and Xi that I changed are in Bold.
GRKS Distribution With the Following Parameters:Minimum Mode Maximum
25 65 100Interval Prob(Xi) Xi
Pseudo Min 1 0.00 0.002 0.01 50.00
Minimum 3 0.02 50.004 0.07 50.005 0.16 50.006 0.35 50.00
Mode 7 0.50 65.008 0.69 73.759 0.84 82.50
10 0.93 91.25Maximum 11 0.98 100.00
12 0.99 108.75Pseudo Max 13 1.00 117.50
0.000.100.200.300.400.500.600.700.800.901.00
40.00 50.00 60.00 70.00 80.00 90.00 100.00 110.00 120.00
GRKS Distribution
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• 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
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• 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
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Modeling Uncertainty with EMPThe 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.
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• Results from simulating the modified distribution for Price
• Note probabilities of extreme prices
Modeling Uncertainty with EMP
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Modeling Uncertainty with EMP
00.10.20.30.40.50.60.70.80.9
1
3 5 7 9 11 13 15
Prob
Comparison of CDFs for Original and Modified Price Distributions
Price 1 Price 2
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• 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
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• 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