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Materials for Lecture • Chapter 2 pages 6-12, Chapter 6, Chapter 16 Section 3.1 and 4 • Lecture 7 Probability of Revenue.xls • Lecture 7 Flow Chart.xls • Lecture 7 Farm Simulator.xls • Lecture 7 Uniform.xls • Lecture 7 Theta UPES.xls • Lecture 7 View Distributions.xls Simulation Models • A Model is a mathematical representation of an actual 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 a model • We build models so we do not have to experiment on actual system – Will the business be successful if we change management practices, etc.? Developing Simulation Models • • • • Organization of a model in an Excel Workbook Steps for model development Parts in a simulation model Generating random variables from uniform distributions • Estimating parameters for other distributions Organization of Models in Excel Input Data, such as – Costs, inflation & interest rates, Production functions Assets & liabilities Scenarios to analyze, etc. Historical Data for Random Variables, such as – Prices Production levels Other variables not controlled by management Equations to calculate variables – Production, Receipts, Costs, Amortize Loans, Update Asset values, etc. Tables to report financial results – Income statement, cash flow, balance sheet KOV Table – List all output variables of interest 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 Table of final results – the Key Output Variables (KOVs) • Sheet 2 (Stoch) – Historical data for random variables – Calculations to estimate the parameters for random variables – Simulate all random values • Sheets 3-N (SimData, Stoplite, CDF) – Simulation results and charts Model Design Steps KOVs Design Intermediate Results Tables and Reports Build Equations and Calculations to Get Values for Reports Stochastic Variables Exogenous and Control Variables • 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 or flowchart of how data will interact to calculate the KOVs • Determine the variables necessary to calculate the KOVs – For example to calculate Net Present Value (NPV) we need: • Annual net cash withdrawals 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 Example Flowchart of a Model Flow Chart for Simulating NPV Control Variables for Manager such as: Levels of Production, Debt Levels, Market Share Macro Data as inflation rates interest rates Sections and Equations for the Model Generate the Stochastic Values Use Projected Means and Historical Data for Random Variables Use the Stochastic Values in the Equations for the Model Equations for the System to model Production = f( scale of the farm and stochastic values) Price = f( stochastic values) Revenue = Price * Production for each enterprise Variable Costs by Enterprise = Production * Unit Cost Costs = Variable Costs for each enterprise + Fixed Costs Net Returns = Revenue - Costs Balance Sheet Information Asset Valuation Liabilities Net Worth Annual Projected Mean Prices Key Output Variables Net Present Value Probability of Net Returns > 0 Probability of NPV > 0 ( or Prob of Success) Probability of Increasing Real Net Worth Analyze KOVs Budgets for each of the Enterprises Stochastic Variables -- need the historical data to estimate parameters for random variables Steps for Model Development • Write out the equations by hand – This organizes your thoughts and the model’s structure – Avoids problem of forgetting important sections – Example of equations for a model at this point: • • • • • Output/hour = stochastic variable Hours Operated = management control value Production = Output/hour * Hours Operated Price = forecast mean each year with a risk component Receipts = Price * Production • Define input variables – Exogenous variables are out of the control of management and are constant; usually policy driven – Stochastic variables management can not control and are random in nature: weather or market driven – Control variables the manager can manipulate Steps for Model Development • Stochastic variables (40% of time is spent here) – Identify all random variables that affect the system – Estimate parameters for the assumed distributions • Normality – means and standard deviations • Empirical – sorted deviates and probabilities – Use the best model possible econometric 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 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 as this is the part of a variable we did not predict • Why include stochastic variables? – So we can get a more robust simulation answer – Rather than a single value output we get a PDF – We can assign probabilities of success – We can consider risk in our decisions Simple Economic Model • A supply and Demand Model – You learned there is one Demand and one Supply – But there are many, due to the risk on the equations Qx = a + b1Px +b2Y + b3Py gives a single line for Demand Qx = a + b1Px +b2Y + b3Py + ẽ gives infinite Demands – Now if Supply is a constant we get an infinite number of Prices as we draw ẽ values at random Price/U Supply Demand Quantity/UT Simple Business Model • Profit is generally the Key Output Variable of interest P = Total Receipts – Variable Cost – Fixed Cost P = ∑(P~i * Ỹi ) - ∑(VCi * Ỹi * Qi ) – FC Where P~i 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 Univariate Random Variables • More than 40 Univariate Distributions in Simetar – – – – – – – Uniform Distribution Normal and Truncated Normal Distribution Empirical, Discrete Empirical Distribution GRKS Distribution Triangle Distribution Bernoulli Distribution Conditional Distribution • Excel probability distributions have been made Simetar compatible, e.g., – Beta, Gamma, Exponential, Log Normal, Weibull Uniform Distribution • A continuous distribution where each range has an equal probability of being observed • Parameters for the uniform are minimum and maximum values and the domain includes all real number’s =UNIFORM(min,max) • The mean and variance of this distribution are: min max 2 max min 2 2 12 PDF and CDF for a Uniform Dist. Probability Density Function f(x) min max X Cumulative Distribution Function F(x) 1.0 0.0 min max 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 For example USD is used to simulate a Normal Distribution Uniform Deviate 1.0 USDi 0.8 0.6 0.5 0.4 0.2 - 3 0 SNDi + 3 Std. Normal Dev. 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) – Special case of the Uniform distribution • USD is 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) * Uniform(0,1) X = Min + (Max-Min) * USD Simulate a Uniform Distribution • Alternative ways to program the =Uniform( ) function = Uniform(Min, Max,[USD]) = Uniform(10,20) = Uniform(A1,A2) = Uniform(A1,A2,A3) where a USD is calculated in cell A3 Uses for a Uniform Standard Deviate • The uniform standard deviate (USD) is used in all of the random number formulas in Simetar to facilitate correlating random variables • For example in Simetar we can add USDs: =NORM(mean, std dev, [USD]) =TRIANGLE(min, middle, max, [USD]) = EMP( Si, F(Si), [USD]) • Add USDs in formulas so we can later correlate random variables with CUSDs Simulating Random Variables • Must assume a probability distribution shape – Normal, Beta, Empirical, etc. • Estimate parameters required for the assumed distribution • Here are the parameters for selected distributions – – – – Normal ( Mean, Std Deviation ) Beta ( Alpha, Beta, Min, Max ) Uniform ( Min, Max ) Empirical ( Si, F(Si) ) • Often times we assume several distribution forms, estimate their parameters, simulate them and pick the one which best fits the data Steps for Parameter Estimation • Step 1: Check for the presence of a trend, cycle or structural pattern – If present remove it & work with the residuals (ẽt) – If no trend or structural pattern, use actual data (X’s) • Step 2: Estimate parameters for several assumed distributions using the X’s or the residuals (ẽt) • Step 3: Simulate the different distributions • Step 4: Pick the best match based on – – – – Mean, Standard Deviation -- use validation tests Minimum and Maximum Shape of the CDF vs. historical series Penalty function =CDFDEV() to quantify differences Parameter Estimator in Simetar • Use Theta Icon in Simetar – Estimate parameters for 16 parametric distributions – Select MLE method of parameter estimation – Provides equations for simulating distributions Parameter Estimator in Simetar • Results for Theta Estimate parameters for 16 distributions – Selected MLE in this example – Provides equations for simulating distributions based on a common USD Univariate Parameter Estimation for Random Variable at 2/27/2010 4:23:57 PM MLEs Random Variables Distribution Parameters Parm. 1 Parm. 2 Distribution MLE Beta α, β ; A≤x≤B, α,β>0 1.195 1.324 Beta 1.683 Double Exponential α, β ; α≤x<∞, -∞<α<∞, 2.270β>0 0.286 Double Exponential 1.823 Exponentialμ, σ; -∞<x<∞, -∞<μ<∞, 1.500 σ>0 0.744 Exponential 1.582 Gamma α, β ; 0≤x<∞, α,β>0 36.568 0.061 Gamma 1.792 Logistic μ, σ; 0≤x<∞, -∞<μ<∞, 2.243σ>0 0.208 Logistic 1.796 Log-Log μ, σ; -∞<x<∞, -∞<μ<∞, 2.062 σ>0 0.352 Log-Log 1.776 Log-Logisticμ, σ; 0≤x<∞, -∞<μ<∞, 10.543σ>0 2.224 Log-Logistic 1.815 Lognormal μ, σ; 0≤x<∞, -∞<μ<∞, 0.794σ>0 0.167 Lognormal 1.794 Normal μ, σ; -∞<x<∞, -∞<μ<∞, 2.244 σ>0 0.367 Normal 1.784 Pareto α, β ; α≤x<∞, α,β>0 1.500 2.571 Pareto 1.566 Uniform a, b ; a≤x≤b 1.500 3.050 Uniform 1.662 Weibull α, β ; 0≤x<∞, α,β>0 6.555 310.531 Weibull 1.715 Binomial n, p ; x=0,1,2,...,n;4.000 0≤p≤1 0.438 Binomial 1 Geometric p ; x=1,2,...; 0≤p≤1 0.364 Geometric 1 Poisson λ ; x=0,1,...; 0≤λ<∞ 1.750 Poisson 0 Negative Binomial s, p ; x=1,2,...; 0≤p≤1 Negative Binomial Common USD 0.10 Which is the Best Distribution? • Use Simetar function =CDFDEV( history, sim data) – Perfect fit has a CDFDEV value of Zero – Pick the distribution with the lowest CDFDEV Distributions Beta Double Exponential Exponential Gamma Logistic Log-Log Log-Logistic Lognormal Normal Pareto Uniform Weibull Binomial Geometric Poisson CDFDEV Formula 0.02 =CDFDEV(Sheet1!$A$2:$A$21,SimData!B9:B108) 0.37 =CDFDEV(Sheet1!$A$2:$A$21,SimData!C9:C108) 2.66 =CDFDEV(Sheet1!$A$2:$A$21,SimData!D9:D108) 0.07 =CDFDEV(Sheet1!$A$2:$A$21,SimData!E9:E108) 0.16 =CDFDEV(Sheet1!$A$2:$A$21,SimData!F9:F108) 0.43 =CDFDEV(Sheet1!$A$2:$A$21,SimData!G9:G108) 0.34 =CDFDEV(Sheet1!$A$2:$A$21,SimData!H9:H108) 0.10 =CDFDEV(Sheet1!$A$2:$A$21,SimData!I9:I108) 0.05 =CDFDEV(Sheet1!$A$2:$A$21,SimData!J9:J108) 79.04 =CDFDEV(Sheet1!$A$2:$A$21,SimData!K9:K108) 0.03 =CDFDEV(Sheet1!$A$2:$A$21,SimData!L9:L108) 0.08 =CDFDEV(Sheet1!$A$2:$A$21,SimData!M9:M108) 1.04 =CDFDEV(Sheet1!$A$2:$A$21,SimData!N9:N108) 33.81 =CDFDEV(Sheet1!$A$2:$A$21,SimData!O9:O108) 4.19 =CDFDEV(Sheet1!$A$2:$A$21,SimData!P9:P108) Use the “View Distributions.xls” • For a random variable with 10 observations can estimate the parameters and view the shape of the distribution