1. Brief Review –Forecasting for 3 weeks –Simulation Motivation for building simulation models Steps for developing simulation models Stochastic variables and why they are included in models What financial simulation model is used for Parametric Distributions (N, U, Bernoulli) Test Results –Mean –Std Dev Welcome Back From Spring Break

2. Chapter 6 Chapter 16 Sections 3.2 - 3.7.3, 4.0, Lecture 10 Demo Distributions.xlsx Lecture 10 Empirical Distributions.xlsx Materials for Lecture 9

3. These are Non-Parametric Distributions –Discrete Uniform –Empirical –GRKS –Triangle Parametric Distributions –Fixed form, such as Uniform, Normal, Beta, Gamma, etc. and are estimated by UPES Empirical Probability Distribution

4. Discrete Uniform Empirical distribution used where only fixed values can occur –Each value has an equal probability of being drawn –No interpolation between observed values Function might be used for things such as, –Discrete number of labors who show up to work –Number of steers on a truck –Simulating a fair die –Letter grades Discrete Uniform Empirical

5. 4X PDF for DE(3, 4, 6, 7) 367 CDF for DE(3, 4, 6, 7) 4X367 0.25.5.75 1 PDF and CDF for a Discrete Uniform Distribution. Discrete Uniform Empirical Distribution -Discrete Empirical means that each observed value of X i, has an equal probability of being observed Row 110 212 320=DEMPIRICAL (A1:A5) 415 513 BCA - Parameters for a DE(x 1, x 2, x 3, …, x n ) based on history

6. Simulate this type of random variable two ways in Simetar –Discrete empirical with equal probabilities =DEMPIRICAL(A1:A5) =RANDSORT(A1:A5) Discrete Uniform Empirical

7. =RANDSORT(I1:I5) Random shuffle of names; highlight 5 cells and =RANDSORT(I1:I5, [Option]) then hit Ctrl Shift Enter Option can be set to: 0 causes it to draw a sample every time press F9 1 causes Simetar to make only one draw, so get one sample Discrete Empirical -- Alphanumeric

8. An empirical distribution is defined totally by the observations for the data, no distributional form is assumed Parameters to simulate an empirical distribution –Forecasted values: means (Ῡ) or forecasts (Ŷ) –Calculate the deviation from the mean or forecast –Sort the deviations from the mean or forecast from low to high –Assign a cumulative probability to each data point (usually equal probability). Cumulative probabilities go from zero to one –Assume the distribution is continuous, so interpolate between the observed points Use the Inverse Transform formula to simulate the distribution This requires simulation of a USD for use in interpolation Use Emp icon to estimate parameters Empirical Distribution

9. Empirical distribution should be used if –Random variable is continuous over its range, –You have < 20 observations for the variable, and/or –You cannot easily estimate parameters for the true PDF Simulate crop yields as an Empirical distribution when you have only 10 historical values –In this situation we know: Yield can be any positive value We don’t have enough observations to test for normality We know the 10 random values were observed with a probability of 1/10, or one observation each year Using the Empirical Distribution

10. PDF and CDF for an Empirical Dist. F(x) Probability Density FunctionCumulative Distribution Function minmax X f(x) min max X 0.0 1.0 We interpolate the Dark Black line in the CDF based on the discrete CDF and use it as the approximation for a continuous distribution

11. Inverse Transform for Simulating an Empirical Distribution F(x) 0.0 1.0 Y1Y1 Y2Y2 Y3Y3 Y4Y4 Y5Y5 Y6Y6 Y7Y7 U(0,1) = 0.45 StochasticDerived by linear interpolation ỸiỸi Start with a random USD Interpolate the Ỹ axis using the USD value

12. Empirical distribution is usually simulated as percent deviations from mean or trend: percent deviates from mean = ( Y t – Ῡ t )/Ῡ t Parameters are: –Mean of the data is either Ῡ t or Ŷ t –Sorted deviations from mean or forecasted Ŷ are S t = Sort [(Y t – Ῡ t )/Ῡ t ] or S t = Sort [(Y t – Ŷ t )/ Ŷ t ] –Probabilities for S t ’s, are called F(S t ) or F(x) values and MUST range from 0.0 to 1.0 Use the parameters to simulate random variable Ỹ: Ỹ = Ῡ t * (1 + EMP(S t, F(S t ), [USD]) ) Simulating Empirical Distributions

13. Empirical Distribution -- No Trend Given a random variable, Ỹ, with 11 observations Develop the parameters if simulating variable using the mean to forecast the deterministic component: Parameter for deterministic component is the mean or the second column Calculate the stochastic component or ê as: ê i = Y i – Ῡ Convert the residual to fractional deviation of forecast mean value: Dev i = ê i / Ῡ Sort the Dev i values from low to high (S i ) and assign the probabilities of S i or F(Si) Simulate Ỹ in two steps: Stoch Dev i = EMP(Sort Dev, Prob Dev, USD) Stoch Ỹ T+i = Ῡ T+i * (1 + Stoch Dev i ) Recall : Dev i = (Y i - Ῡ ) / Ῡ rearrange terms or ( Ῡ * Dev i ) = Y i – Ῡ so Y i = Ῡ + ( Ῡ * Dev i )

14. Empirical Dist. -- With Trend Parameters for EMP() if deterministic component is the trend forecast Calculate the stochastic component or ê as: ê i = Y i – Ŷ i Convert residual to fractional deviate of forecast value: Dev i = ê i / Ŷ i Sort the Dev i values from low to high (S i ) and calculate the probabilities of S i or F(Si) Simulate Ỹ as follows: Stoch Dev i = EMP(S i, F(S i ), USD ) Ỹ T+i = Ŷ T+i * (1 + Stoch Dev i ) Derived from: Stoch Dev i = (Y i - Ŷ i ) / Ŷ i or Y i – Ŷ i = (Ŷ i * Stoch Dev i ) or Y Stoch i = Ŷ i + (Ŷ i * Stoch Dev i ) Ỹ T+I Could have been developed from a structural or time series equation, then ê i are the residuals from the regression

15. Let: S i be in B1:B10 and F(S i ) in A1:A10 If S i expressed as actual values =EMP(S i ) or =EMP(B1:B10) If S i expressed as residuals from mean = Ῡ + EMP(B1:B10, A1:A10) If S i expressed as fractional deviates from trend or trend: S i = (ẽ / Ŷ) = Ŷ * (1 + EMP(B1:B10, A1:A10)) Simulate Emp Distribution with Simetar

16. Advantages of Emp Distribution –It lets the data define the shape of the distribution –Does not force an assumed distribution shape on the variable –The larger the number of observations in the sample, the closer Emp will approximate the true distribution Disadvantages of Emp Distribution –It has finite min and max values –It does not adhere to known probabilities and parameters –Parameters can be difficult to estimate w/o Simetar Simulating an Emp Distribution

17. Advantages of specifying the S i ’s as a fraction of forecasted values –Guarantees the “relative risk” for a random variable is the same as the historical period Coefficient of variation for the sample data is constant over time CV t = (σ / Ῡ t ) * 100 –Allows you to use any mean (Ŷ or Ῡ) for the simulated planning horizon and it will have the same CV as the historical period Historical Ῡ can be 100 and the mean for the forecast period Ŷ can be 150 and the Ỹ values will have the same CV as the historical data. Simulating an Emp Distribution