1. Materials for Lecture 20
Explain the use of Scenarios and Sensitivity Analysis in a simulation model
Utility based risk ranking methods, setting the stage
Chapter 10 pages 1-3
Chapter 16 Sections 7, 8 and 9
Lecture 20 Scenario Example.xlsx
Lecture 20 Sensitivity Elasticity.xlsx
Lecture 20 Whole Farm Model Scenarios.xlsx

2. Applying a Simulation Model
Assume we have a working simulation model
Model has the following parts
Input section where the user enters all input values that are management control variables and exogenous policy or time series data
Stochastic variables that have been validated
Equations to calculate all dependent variables
Equations to calculate the KOVs
A KOV table to send to the simulation engine
We are ready to run scenarios on control variables and make recommendations

3. Scenario and Sensitivity Analysis
Simetar simulation engine controls
Number of scenarios
Sensitivity analysis
Sensitivity elasticities

4. Scenario Analysis
Base scenario – complete simulation of the model for 500 or more iterations with all variables set at their initial or base values
Alternative scenario – complete simulation of the model for 500 or more iterations with one or more of the control variables changed from the Base
Every scenario must use the same random USDs
Scenario loop
Iteration loop
IS = 1, M
Change management variables (X) from one scenario to the next
IT = 1, N
Next scenario
Use the same random USDs for all random variables, so identical risk for each scenario

5. Scenario Analysis
All values in the model are held constant and you systematically change one or more variables
Number of scenarios determined by analyst
Random number seed is held constant and this forces Simetar to use the same random USDs to generate random values for the stochastic variables for every scenario (benefit of Pseudo Random Numbers)
Use =SCENARIO() a Simetar function to increment each of the scenario (manager) control variables

6. Example of a Scenario Table
5 Scenarios for the risk and variable costs
Purpose is to look at the impacts of different management scenarios on net returns

7. Results of the Scenario Analysis

8. Example of a Scenario Table
Create as big a scenario table as needed
Add all control variables into the table

9. Example Scenario Table of Controls

10. Sensitivity Analysis
Sensitivity analysis seeks to determine how sensitive the KOVs are to small changes in one particular variable
Does net return change a little or a lot when you change variable X by 10%?
Does NPV change greatly if the assumed fixed cost changes by 10%?
Simulate the model numerous times changing the “change” variable for each simulation
Must ensure that the same random values are used for each simulation
Simetar has a sensitivity option that insures the same random values used for each run

11. Sensitivity Analysis
Simetar uses the Simulation Engine to specify the change variable and the percentage changes to test
Specify as many KOVs as you want
Specify only ONE sensitivity variable
Simulate the model and 7 scenarios are run

12. Demonstrate Sensitivity Simulation
Change the Price per unit as follows
+ or – 5%
+ or – 10%
+ or – 15%
Simulates the model 7 times
The initial value initially in the change cell
Two runs for + and – 5% for the control variable
Two runs for + and – 10% for the control variable
Two runs for + and – 15% for the control variable
Collect the statistics for only a few KOVs
For demonstration purposes collect results for the variable doing the sensitivity test on
Could collect the results for several KOVs

13. Sensitivity Results
Test the sensitivity of price received for the product on Net Cash Income
Note that we get 7 sets of results in SimData
Labels indicate the % difference from the initial value of the change variable

14. Display Sensitivity Results in a Chart

15. Sensitivity Elasticities (SE)
Sensitivity of a KOV with respect to (wrt) multiple variables in the model can be estimated and displayed in terms of elasticities, calculated as:
SEij = % Change KOVi
% Change Variablej
Calculate SE’s for a KOVi wrt change variablesj at each iteration and then calculate the average and standard deviation of the SE
SEij’s can be calculated for small changes in Control Variablesj, say, 1% to 5%
Necessary to simulate base with all values set initially
Simulate model for an x% change in Vj
Simulate model for an x% change in Vj+1

16. Sensitivity Elasticities
The more sensitive the KOV is to a variable, Vj, the larger the SEij
Display the SEij’s in a table and chart

17. Sensitivity Elasticities

18. Sensitivity Elasticities

19. Utility Based Risk Ranking Procedures
Utility and risk are often stated as a lottery
Assume you own a lottery ticket that will pay $10 or $0, with a probability of 50%
Risk neutral DM will sell the ticket for $5
Risk averse DM will sell ticket for a “certain (non-risky)” payment less than $5, say $4
Risk loving DM will only sell the ticket if paid a “certain” amount greater than $5, say $7
Amount of the “certain” payment to sell the ticket is the DM’s “Certainty Equivalent” or CE
Risk premium (RP) is the difference between the CE and the expected value
RP = E(Value) – CE
RP = 5 – 4

20. Utility Based Risk Ranking Procedures
CE is used everyday when we make risky decisions
We implicitly calculate a CE for each risky alternative
“Deal or No Deal” game show is a good example
Player has 4 unopened boxes with amounts of:
$5, $50,000, $250,000, and $0
Offered a “certain payment” (say, $65,000) to exit the game, the certain payment is always less than the expected value (E(x) =$75, in this example)
If a contestant takes the Deal, then the “Certain Payment” offer exceeded their implicit CE for that particular gamble
Their CE is based on their risk aversion level

21. Utility Based Risk Ranking Procedures
Black Utility Function is for Normal Risk Averse Person
Red line is for a Risk Lover
Maroon line is for a very risk averse person
Utility
E($) =$5 $0
$10
Income
CE($) Risk Averse DM

22. Ranking Risky Alternatives Using Utility
GIven a simple assumption, “the DM prefers more to less,” then we can rank risky alternatives with CE
DM will always prefer the risky alternative with the greater CE
To calculate a CE, “all we have to do” is assume a utility function and that the DM is rational and consistent, calculate their risk aversion coefficient, and then calculate the DM’s utility for a risky choice

23. Ranking Risky Alternatives Using Utility
Utility based risk ranking tools in Simetar
Stochastic dominance with respect to a function (SDRF)
Certainty equivalents (CE)
Stochastic efficiency with respect to a function (SERF)
Risk Premiums (RP)
All of these procedures require estimating the DM’s risk aversion coefficient (RAC) as it is the parameter for the Utility Function

24. Suggestions on Setting the RACs
Anderson and Dillon (1992) proposed a relative risk aversion (RRAC) schedule of
0.0 risk neutral
0.5 hardly risk averse
1.0 normal or somewhat risk averse
2.0 moderately risk averse
3.0 very risk averse
4.0 extremely risk averse (4.01 is a maximum)
Rule for setting RRAC and ARAC range is:
Utility Function
Lower RAC
Upper RAC
Neg Exponential Utility
ARAC
4/Wealth
Power Utility
RRAC

25. Assuming a Utility Function for the DM
Power utility function
Use this function when assuming the DM exhibits relative risk aversion RRAC
DM willing to take on more risk as wealth increases
Poor person buys a $1 lottery ticket
A rich person buys 1,000 $1 lottery tickets
Both may feel the same amount of risk relative to wealth
Use when ranking risky scenarios with a KOV that is calculated over multiple years, as:
Net Present Value (NPV)
Present Value of Ending Net Worth (PVENW)

26. Assuming a Utility Function for the DM
Negative Exponential utility function
Use this function when assuming DM exhibits constant absolute risk aversion ARAC
DM will not take on more risk as wealth increases
Poor person buys one $1 lottery ticket
A rich person buys one $1 lottery ticket
Both feel that same amount of risk relative to wealth
Use when ranking risky scenarios using KOVs for single year, such as:
Annual net cash income or return on investments
You get the same rankings with Power and Negative Exponential utility functions, if you use correct the RACs

27. Summarize Validation Tests
The Following Slides are a summary of distribution validation
They are very important so I summarized them here in one place.
Good test material

28. Summarize Validation Tests
Validation of simulated distributions is critical to building good simulation models
Selection of the appropriate statistical tests to validate the simulated random variables is essential
Appropriate statistical tests changes as we change the method for estimating the parameters

29. Summarize Univariate Validation Tests
If the data are stationary and you want to simulate using the historical mean
Distributions such as:
Normal as =NORM(Ῡ, σY) or
Empirical as =EMP(Historical Ys)
Validation Tests for Univariate distribution
Compare Two Series tab in Simetar
Student-t test of means as H0: ῩHist = ῩSim
F test of variances as H0: σ2Hist = σ2Sim
You want both tests to Fail to Reject the null H0

30. Summarize Univariate Validation Tests
If the data are stationary and you want to simulate using a mean that is not equal to the historical mean
Distribution
Use Empirical as a fraction of the mean so the Si = Sorted((Yi - Ῡ)/Ῡ) are deviates and simulate using the formula:
Ỹ = Ῡ(new mean) * ( 1 + EMP(Si, F(Si), [CUSDi] ))
Validation Tests for Univariate distribution
Test Parameters
Student-t test of means as H0: ῩNew Mean = ῩSim
Chi-Square test of Std Dev as H0: σHist = σSim
You want both tests to Fail to Reject the null H0

31. Summarize Univariate Validation Tests
If the data are non-stationary and you use OLS, Trend, or time series to project Ŷ
Distribution
Use =NORM(Ŷ , Standard Deviation of Residuals) OR
Use Empirical and the residuals as fractions of Ŷ calculated for Si = Sorted((Yi - Ŷj)/Ŷ) deviates and simulate each variable using:
Ỹi = Ŷi * (1+ EMP(Si, F(Si) ))
Validation Tests for Univariate distribution
Test Parameters
Student-t test of means as H0: ŶNew Mean = ῩSim
Chi-Square test of Std Dev as H0: σê = σSim
You want both tests to Fail to Reject the null H0

32. Summarize Univariate Validation Tests
If the data have a cycle, seasonal, or structural pattern and you use OLS or any econometric forecasting method to project Ŷ
Distribution
Use =NORM(Ŷ, σê of the residuals)
Use Empirical and the residuals as fractions of Ŷ calculated for Si = Sorted((Yi - Ŷ)/Ŷ) and simulate using the formula
Ỹ = Ŷ * (1 + EMP(Si, F(Si) ))
Validation Tests for Univariate distribution
Test Parameters tab
Student-t test of means as H0: ŶNew Mean = ῩSim
Chi-Square test of Std Dev as H0: σê = σSim
You want both tests to Fail to Reject the null H0

33. Summarize Multivariate Validation Tests
If the data are stationary and you want to simulate using the historical means and variance
Distribution
Use Normal =MVNORM(Ῡ vector, ∑ matrix) or
Empirical =MVEMP(Historical Ys,,,, Ῡ vector, 0)
Validation Tests for Multivariate distributions
Compare Two Series for 10 or fewer variables
Hotelling T2 test of mean vectors as H0: ῩHist = ῩSim
Box’s M Test of Covariances as H0: ∑Hist = ∑Sim
Complete Homogeneity Test of mean vectors and covariance simultaneously
You want all three tests to Fail to Reject the null H0
Check Correlation
Performs a Student-t test on each correlation coefficient in the correlation matrix: H0: ρHist = ρSim
You want all calculated t statistics to be less than the Critical Value t statistic so you fail to reject each t test (Not Bold)

34. Summarize Multivariate Validation Tests
If you want to simulate using projected means such that Ŷt ≠ Ῡhistory
Distribution
Use Normal as = MVNORM(Ŷ Vector, ∑matrix) or
Empirical as = MVEMP(Historical Ys ,,,, Ŷ vector, 2)
Validation Tests for Multivariate distribution
Check Correlation
Performs a Student-t test on each correlation coefficient in the matrix: H0: ρHist = ρSim
You want all calculated t statistics to be less than the Critical Value t statistic so you fail to reject each t test
Test Parameters, for each j variable
Student-t test of means as H0: ŶProjected j = ῩSim j
Chi-Square test of Std Dev as H0: σê j = σSim j