Welcome Back From Spring Break

1. Welcome Back From Spring Break • 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)

2. Test Results Mean 80.4, StdDev13.1, Range 45-100

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

4. Non-Parametric and Parametric Distributions • Non-Parametric Distributions – not a fixed form that is parameter dependent • Discrete Uniform • Empirical • GRKS • Triangle • Parametric Distributions (covered last lecture) • Fixed form, shape dependent on parameters • Uniform, Normal, Beta, Gamma

5. Discrete (Uniform) Empirical • Discrete Empirical distribution used where only fixed values can occur • Each value has an equal probability of being drawn • No interpolation between observed values • Examples of Discrete Empirical distributions • Discrete number of labors who show up to work • Number of steers on a cattle truck • Simulating a fair die: 1, 2, 3, 4, 5, 6 • Letter grades: A, B, C, D, F

6. Discrete (Uniform) Empirical Distribution PDF for DE(3, 4, 6, 7) CDF for DE(3, 4, 6, 7) 1 A B C Row 1 10 2 12 .75 3 20 =DEMPIRICAL (A1:A5) 4 15 5 13 .5 .25 0 3 4 6 7 X 3 4 6 7 X PDF and CDF for a Discrete Uniform Distribution. - Parameters for a DE(x1, x2, x3, …, xn) based on history - Discrete Empirical means that each observed value of Xi, has an equal probability of being observed

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

8. Discrete Empirical -- Alphanumeric • =RANDSORT(I1:I5) • Random shuffle of names; highlight 5 cells and Type =RANDSORT(I1:I5) then press and hold Ctrl Shift Enter

9. Empirical Distribution • 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 percentage deviation from the mean or forecast = (Yi- Ŷi) / Ŷi • Sort the deviations from the mean or forecast from low to high • Assign a cumulative probability to each sorted deviates (usually assume equal probability for each data point). • Cumulative probabilities go from 0.0 to 1.0; named F(x) • 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

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

11. Using the Empirical Distribution • 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 less than 20historical values • Assume we have 10 observed yields: • Yield can be any positive value, not discrete values • 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 • So F(x) goes from 0.0 to 1.0 in equal increments

12. Simulating Empirical Distributions • Empirical distribution is usually simulated as percent deviations from mean or trend: percent deviates from mean = (Yt –Ῡt )/Ῡt • Parameters are: • Mean of the data is either Ῡtor Ŷt • Sorted deviations from mean or forecasted Ŷ are St = Sort [(Yt –Ῡt )/Ῡt ] or St = Sort [(Yt –Ŷt)/ Ŷt ] • Probabilities for St’s, are called F(St) or F(x) values and MUST range from 0.0 to 1.0 • Use the parameters to simulate random variable Ỹ: Ỹ = Ῡt * (1 + EMP(St, F(St), [USD]) )

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: êi = Yi – Ῡ • Convert the residual to fractional deviation of forecast mean value: Devi = êi / Ῡ • Sort the Devi values from low to high (Si) and assign the probabilities of Si or F(Si) • Simulate Ỹ in two steps: Stoch Devi = EMP(Sort Dev, Prob Dev, USD) • Stoch ỸT+i = ῩT+i * (1 + Stoch Devi) • Recall : Devi = (Yi- Ῡi) / Ῡirearrange terms or (Ῡ * Devi) =Yi – Ῡ so Yi = Ῡ + (Ῡ * Devi)

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

15. Simulate Emp Distribution with Simetar • Let: Si be in B1:B10 and F(Si) in A1:A10 • If Si expressed as actual values =EMP(Si ) or =EMP(B1:B10) • If Si expressed as residuals mean or OLS = Ῡ + EMP(B1:B10, A1:A10) • If Si expressed as fractional deviates from trend or trend: Si = (ẽ / Ŷ) = Ŷ * (1 + EMP(B1:B10, A1:A10)) Memorize these formulas. They are important.

16. Simulating an Emp Distribution • 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

17. Simulating an Emp Distribution • Advantages of specifying the Si’s as fractional deviates 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 CVt = (σ / Ῡt) * 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.

18. Ỹi Stochastic Derived by linear interpolation Inverse Transform for Simulating an Empirical Distribution 1.0 F(x) Start with a random USD U(0,1) = 0.45 Interpolate the Ỹ axis using the USD value 0.0 Y1 Y2 Y6 Y3 Y4 Y5 Y7