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

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

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

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

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

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

Results of the Scenario Analysis

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

Example Scenario Table of Controls

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

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

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

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

Display Sensitivity Results in a Chart

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

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

Sensitivity Elasticities

Sensitivity Elasticities

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

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) =$75,001.25 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

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

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

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

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 4.000001

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)

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

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

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

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

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

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

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

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)

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