• 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

• 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(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

Hybrid Distribution

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

min middle max

1.0

min middle max

• 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

• 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(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

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

• 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

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.

• Results from simulating the modified distribution for Price

• Note probabilities of extreme prices

Modeling Uncertainty with EMP

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

• 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