Materials for Lecture 13 Chapter 2 pages 6-12, Chapter 6, Chapter 16 Section 3.1 and 4 Read Richardson & Mapp article Lecture 13 Probability of Revenue.xlsx Lecture 13 Flow Chart.xlsx Lecture 13 Farm Simulator.xlsx Lecture 13 Uniform.xlsx

What is a Simulation Model? A simulation Model is a mathematical representation of a system of equations When you think through the many steps to solve a problem you are constructing a model When you think or plan your way through a complex situation you are making a virtual model Computer games are models Econometric equations can be part of a model We build models so we do not have to experiment on the actual economic system Will the business be successful if we change management practices, etc.?

Outline for the Lecture Organization of a model in an Excel Workbook Steps for model development Parts of a simulation model Generating random variables from uniform distributions Estimating parameters for other distributions Parameters are the numbers that define the center and the dispersion about the center for a random variable For a Normally distributed random variable, the parameters are the Mean & Std Dev For Empirical it is a little more complicated ….

Organization of Models in Excel Input Data: Costs, inflation & interest rates Production functions Assets & liabilities Scenarios to analyze, etc. Historical Data for Stochastic Variables: Prices Production levels Other variables not controlled by management Equations to calculate variables: Production, Receipts, Costs, Amortize Loans, Update Asset values, Taxes, etc. Tables to report financial results: Income statement, cash flow, balance sheet, financial ratios KOV Table List all output variables of interest Model Outputs: Statistics for KOVs Probability charts Decision summary Final report tables

Organization of Models in Excel Sheet 1 (Model) Assumptions and all Input Data Control variables for managing the system Logical flow of all calculations Table of intermediate results Pro Forma financial tables of results Key Output Variables (KOVs) Table to send to SimData Sheet 2 (Stoch) Historical data for all random variables Calculations to estimate the parameters for all random variables Simulate all random values to be used in the Model Sheets 3-N (SimData, Stoplite, SERF, STODOM, etc) Simulation results tables and charts

Equations and Calculations to Model Design Steps KOVs Intermediate Results Tables and Reports Equations and Calculations to Get Values for Reports Stochastic Variables Exogenous and Control Variables Design Build Model development is like building a pyramid Design the model from the top down Build from the bottom up

Steps for Model Development Determine the purpose of the model and KOVs Draw a sketch of how data will be used to calculate the KOVs Determine all of the variables necessary to calculate the KOVs For example to calculate Net Present Value (NPV): NPV = -Beg NW +∑(PV Net Returns) +PV Ending NW Annual net cash withdrawals (money that leaves the business) which are a function of net returns Ending net worth which is a function of assets and liabilities This means you need a balance sheet and a cash flow statement to calculate annual cash reserves An annual income statement is needed as input into a cash flow Annual net returns are calculated from an income statement

Flow Chart for Simulating NPV

Steps for Model Development Write out the equations by hand or at least in Word This organizes your thoughts and the model’s structure Avoids problem of forgetting important sections Example of equations to simulate receipts: Output/hour = a stochastic variable Hours Operated = management control value (scenario variable) Production = Output/hour * Hours Operated Price = forecast mean each year with a risk component Receipts = Price * Production Define input variables Exogenous variables are not controlled by management and are deterministic; usually policy driven Stochastic variables management can not control and are random in nature: weather, input & output prices, interest rates Control variables the manager can manipulate and are usually used for sensitivity and/or scenario analyses

Steps for Model Development Stochastic variables (most time is spent here) Identify key random variables that affect the system Estimate parameters for the assumed distributions Normality – means and standard deviations Empirical – sorted deviates and probabilities Other distributions should be tested Use the best possible econometric model to forecast deterministic part of stochastic variables to reduce risk Model validation starts here Use statistical tests of the simulated stochastic variables to insure that random variables are simulated correctly Correlation tests, means tests, variance tests CDF and PDF charts to compare history to simulated values Key to validating model are statistical tests

More About Stochastic Variables What are Stochastic Variables? Random variables we can not control, such as: Prices, yields, interest rates, rates of inflation, sickness, etc. Represented by the residuals from regression equations -- this is the part of the variable we did not predict Why include stochastic variables? To get a more robust simulation answer Draw random values from a PDF rather than a single or deterministic value The result is that we can assign probabilities to KOVs We can incorporate risk in our decisions of selecting between scenarios

More About Stochastic Variables Production of agricultural products is stochastic due to many factors Weather, producers’ response to prices (acres planted, inputs used in production, etc.) Output Y Input X1

Prices are Stochastic Due to Demand Being Stochastic Supply and Demand Model You learned there is one Demand and one Supply But there are many, due to risk in the market Qx = a + b1Px +b2Y + b3Py gives a single line for Demand Qx = a + b1Px +b2Y + b3Py + ẽ gives infinite Demands After harvest Supply is a constant, so we get an infinite number of Prices as we draw ẽ values at random Price/U Supply Demand is stochastic so we can have an infinite number of Demand functions passing through the QD distribution Demand Quantity/UT

The Basic Business Model Profit is generally our Most Important KOV 𝜋 = Total Receipts – Variable Cost – Fixed Cost 𝜋 = ∑(Pi * Ỹi * Qi ) - ∑(VCi * Qi ) – FC Where Pi is the stochastic price for product i, as $/bu. Ỹi is stochastic production level as yield or bu./acre VCi is variable cost per unit of production for i, or $/bu. Qi is the level of resources committed to i, as acres ~ ~

Ending Cash is our Second Most Important KOV All businesses want to avoid a negative ending cash balance Scenario analyses used to predict P(cash < 0) Ending cash reserves calculated as Net Cash Income = Total Receipts – Total Cash Costs Total Cash Available = Beginning Cash + Net Cash Income + Interest Earned Total Cash Outflow = Principal Payments + Income Taxes + Machinery Down Payments + Cash Withdrawals & Dividends + Repay Cash Flow Deficitst-1 Ending Cash = Total Cash Available – Total Cash Outflow

Univariate Random Variables More than 50 Univariate Distributions in Simetar Uniform Distribution Normal and Truncated Normal Distribution Empirical, Discrete Empirical Distribution GRKS Distribution Triangle Distribution Bernoulli Distribution Conditional Distribution We will focus on learning to use these but there are many more in Simetar See Chapter 16, Sections 3.1 and 4

Uniform Distribution A continuous distribution where each range has an equal probability of being observed 20% chance of seeing a value between 0 and 0.2 or between 0.8 and 1.0 Parameters for the uniform are minimum and maximum values and the domain includes all real number’s =UNIFORM(minimum, maximum) The mean and variance of this distribution are:

PDF and CDF for a Uniform Dist. Probability Density Function Cumulative Distribution Function f(x) F(x) 1.0 0.0 min max min max X X

When to Use the Uniform Distribution Use the uniform distribution when every range of length “n” between the minimum and maximum values has an equal chance of occurrence Use this distribution when you have no idea what type of distribution to use Uniform distribution is used to simulate all random variables via the Inverse Transform procedure and USD An example of how USD is used to simulate a Standard Normal Distribution Uniform Deviate Std. Normal Dev. - + 3 0.5 1.0 0.8 0.6 0.4 0.2 SNDi USDi Inverse Transform for Generating a SND from a USD

Uniform Standard Deviate (USD) In Simetar we simulate the USD as: =UNIFORM(0,1) or =UNIFORM() Produces a Uniform Standard Deviate (USD) 0 to 1 Special case of the Uniform distribution USD is the building block for all random number generation using the Inverse Transformation method for simulation. Inverse Transform uses a USD to simulate a Uniform distribution as: X = Min + (Max-Min) * USD

Simulate a Uniform Distribution Alternative ways to program the Uniform( ) distribution function = Uniform(Min, Max,[USD]) = Uniform(10,20) Not recommended method = Uniform(A1,A2) This is the preferred method = Uniform(A1,A2,A3) where a USD is calculated in cell A3

Uses for a Uniform Standard Deviate USD can be used in all random variable formulas in Simetar to facilitate correlating random variables For example in Simetar we can add USDs: =NORM(mean, std dev, [USD1]) =GRKS(min, middle, max, [USD2]) = EMP( Si, F(Si), [USD3]) =EMP(values , , [USD4]) NOTE: every variable has its own unique USD. Do not use a USD more than ONCE! Note the [ ] means that USD is optional

Generating Random Numbers Generate a Uniform Standard Deviate (USD) =UNIFORM(0,1) Simetar defaults to simulate 500 values (can be changed to 1,000s) These are called iterations or draws Iterations are separate, uncorrelated draws of random variables Equal chance of observing a number in each of the intervals; both charts are for the same output

USD Output in SimData Simetar saves the 500 samples in SimData and calculates summary statistics The mean of a Uniform (0,1) distribution is 0.5 The minimum is 0.0 The maximum is 1.00 See how close the results are for 500 iterations!

Inverse Transform Use the 500 USDs to simulate random variables for your Ŷ variable This involves translating the USDs from a 0 to 1 scale to the scale for your random variable This is done using the Inverse Transform method shown on the next slide. NOTE: you must have a separate USD for every random variable Y

Inverse Transform The 500 USDs converted from the 0 to 1 scale on the Y axis by direct interpolation Each random USD is associated with a unique “random” Y value to get 500 Ỹs

Inverse Transform Results of 500 iterations for Y using Inverse Transform in an Empirical Dist. USDs and their resulting Ỹs

Simulate the Normal Distribution Parameters for a Normal Distribution Mean or Ŷ from OLS Std Dev or σ of residuals Simulated using the formula for a Normal Ỹ = Ŷ + σ * SND Where the SND is a “standard normal deviate” We generate 500 SNDs and thus simulate (calculate) 500 random Y’s

Simulate the Standard Normal Deviate (SND) SND is a random value between ±∞ SND has a mean of zero and a standard deviation of one SND is simulated by =NORM(0,1) SNDs are the “number of standard deviations from the mean” or the number of σ’s Ỹ is from the Ŷ or Ῡ Uniform Deviate Std. Normal Dev. - + 3 0.5 1.0 0.8 0.6 0.4 0.2 SNDi USDi Inverse Transform for Generating a SND from a USD

Simulate Normal Distribution Next apply the random SNDs to the Normal distribution formula Ỹ = Ŷ + σ * SND In Simetar all of these steps are done for you: = NORM(Ŷ, σ) or = NORM(Ŷ, σ, USD) Remember where to get Ŷ and σ ? In forecasting we estimated Ŷ = a + bX1 +bX2 σ = Standard Deviation of residuals

Normal Distribution: Simetar Code and Output The USD is used to calculate the SND The SND is used to simulate Ỹ Simetar gives same result in one step

Forecasting REVIEW Notes The following is a mathematical review of the forecasting techniques we have covered in class We will not cover these in class They are for your benefit as a summary of the math all in one place

Forecast Techniques - Moving Average Example of a 3 period MA model Calculate and simulate as Normally distributed, Simulate it for a future period, say, year 16 as Stochastic Comp Deterministic Component

Forecasting Techniques - Simple Exponential Smoothing Example is: Simulate for a future period, say, 25 as: Deterministic Component Stochastic Comp

Forecasting Techniques - Regression Models · Trend Regression · Multiple Regression · Non-Linear Trend Regression · Harmonic Regression Use the residuals to simulate the risk in the forecast For example, if we had used OLS to estimate a cycle

Forecast Techniques - Using a Seasonal Price Index for Forecasting: · Seasonal index for each month Ii, i = 1, 2, …, 12 · Annual forecast for year t is · Deterministic monthly forecast for month 6 · To make this stochastic need to add risk on the index I and on - Risk on annual forecasts component is from the residuals on the annual forecast ) or use a MVE for the Deviations from mean). - Risk on the monthly index is from the index for month i. · Stochastic monthly forecast for month 6 if assume residuals are normally dist.

Forecast Techniques -- Seasonal Forecast Finding the risk measure for the monthly index, or · Calculate the Seasonal Index Table to get index values Iij Calculate parameters for a Multivariate Empirical Distribution as a Fraction of the Mean – using the values in the Index table ( the 12 months and N years of prices or sales numbers) Correlation matrix using unsorted deviations from the mean Sorted deviations from the mean as a fraction Probabilities for the sorted deviates · Calculate CUSD’s using the correlation Matrix. Calculate a separate 12x1 vector for each year to forecast · Calculate the Stochastic Index Value for each year as: · Seasonal forecast value in year i, month j is: · See Lecture 16 Probabilistic Forecasting.XLS Worksheet Seasonal Forecast

Forecast Techniques - Times Series Stochastic Forecasts - AR and VAR models can be estimated and deterministic forecasts can be developed - The one period ahead forecast can be simulated stochastic by adding risk where is the std. dev. of the residuals for the AR( ) model - This is a stochastic application of the Chain Rule forecasting formula

Time Series Stochastic Forecasts - The second and third periods ahead stochastic forecasts from the AR Model become more complex as: and