Materials for Lecture 13• Chapter 2 pages 6-12, Chapter 6,

Chapter 16 Section 3.1 and 4• Lecture 13 Probability of

Revenue.xlsx• Lecture 13 Flow Chart.xlsx• Lecture 13 Farm Simulator.xlsx• Lecture 13 Uniform.xlsx• Lecture 13 Theta UPES.xlsx• Lecture 13 View Distributions.xlsx

Simulation Models• A Model is a mathematical representation

of any 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.?

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– Parameters are the numbers that define the

center and the dispersion about the center of the random variable

– For a Normally distributed random variable, the parameters are the Mean & Std Dev

– For Empirical ….

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

Model Outputs:

Statistics for KOVs

Probability charts

Decision summarys

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– 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

• Model development is like building a pyramid – Design the model from the top down– Build from the bottom up

KOVs

Intermediate ResultsTables and Reports

Equations and Calculations to

Get Values for ReportsStochastic Variables

Exogenous and Control Variables

Design

Build

Steps for Model Development

• Determine the purpose of the model and KOVs

• Draw a sketch 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

Flow Chart for Simulating NPV

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

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

Macro Data as inflation ratesinterest rates

Annual Projected Mean Prices

Budgets for each of the Enterprises

Stochastic Variables -- need the historical data to estimate parameters for random variables

Control Variables for Manager such as: Levels of Production, Debt Levels, Market Share

Analyze KOVs

• 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 deterministic; 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 (most 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 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

Steps for Model Development

• 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?– To get a more robust simulation answer– PDF rather than a single value– We can assign probabilities to KOVs– We can incorporate risk in our decisions

Stochastic Variables?

• A Supply and Demand Model – You learned there is one Demand and one

Supply– But there are many, due to the risk on the

equationsQx = a + b1Px +b2Y + b3Py gives a single line for

DemandQx = 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

Simple Economic Model

Supply

Demand Quantity/UT

Price/U

• Profit is generally our Key Output Variable of interest = Total Receipts – Variable Cost – Fixed Cost = ∑(Pi * Ỹi ) - ∑(VCi * Ỹi * 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

The Basic Business Model

~

~

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

• Excel probability distributions have been made Simetar compatible, e.g., – Beta, Gamma, Exponential, Log Normal, Weibull

• 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:

Uniform Distribution

min max

2

12

minmax 22

PDF and CDF for a Uniform Dist.

min max X

f(x)

min max X

F(x)

0.0

1.0Probability Density Function Cumulative Distribution Function

• 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

When to Use the Uniform Distribution

Uniform Deviate

Std. Normal Dev.- + 30

0.5

1.0

0.8

0.6

0.4

0.2

3 SNDi

USDi

Inverse Transform for Generating a SND from a USD

For example USD is used to simulate a Normal Distribution

• 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

Uniform Standard Deviate (USD)

Simulate a Uniform Distribution

• Alternative ways to program the Uniform( ) distribution 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

• USD can be used in all 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])

=EMP(values , , [USD]) • Note the [USD] means that USD is

optional

Generating Random Numbers

• Generate a Uniform Standard Deviate (USD)=UNIFORM(0,1)Simetar simulates 500 valuesThese are called iterationsThey are 500 samples or draws

• Equal chance of observing a number in each of the intervals; both charts are for the same output

0

0.02

0.04

0.06

0.08

0.1

0.12

0.00 0.13 0.25 0.38 0.50 0.62 0.75 0.87 1.00

USD = UNIFORM(0,1)

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

Prob

CDF for Uniform(0,1)

USD Output in SimData

• Simetar saves the 500 samples in SimData and calculates summary statisticsSimetar Simulation Results for 500 Iterations. 9:36:20 AM 2/17/2013 (1 sec.). © 2011.Variable Sheet1!B7Mean 0.499985StDev 0.288988CV 57.79939Min 0.000895Max 0.999165Iteration USD

1 0.5127932 0.3073163 0.5812774 0.7874955 0.942096 0.7359717 0.0489238 0.23733

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 are converted from 0 to 1 scale to the Y scale by direct interpolation

• Each random USD is associated with a unique “random” Y value to get 500 Ỹs

0.000.100.200.300.400.500.600.700.800.901.00

55.00 60.00 65.00 70.00 75.00

USD or F(x)CDF of a Random Variable

Inverse Transform

• Results of 500 iterations for Y using Inverse Transform

• USDs and their resulting ỸsSimetar Simulation Results for 500 Iterations. 10:15:19 AM 2/17/2013 (1 sec.). © 2011.Variable Sheet1!G33Sheet1!G34Mean 0.499985 65.19666StDev 0.288988 3.136123CV 57.79939 4.810251Min 0.000895 56.38011Max 0.999165 74.43161Iteration USD Y-Tilda

1 0.512793 65.226072 0.307316 63.615343 0.581277 65.79394 0.787495 67.724645 0.94209 70.203086 0.735971 67.178927 0.048923 60.036648 0.23733 62.918439 0.955568 70.68873

10 0.634662 66.23654

Simulate The Normal Distribution

• Parameters for a Normal Distribution – Mean or Ŷ from OLS– Std Dev or σ of residuals

• Simulated using the formulaỸ = Ŷ + σ * SNDWhere the SND is a “standard normal deviate” We generate 500 SNDs and thus simulate (calculate) 500 random Y’s

Generating SNDs• Generate 500 USDs and transform them to SNDs

using the Inverse Transform (=NORMSDIST(USD))

• SND’s have mean of Zero and range from ≈ ±∞• SNDs are the “number of standard deviations

from the mean” or the number of σ’s Ỹ is from the Ŷ or Ῡ

00.10.20.30.40.50.60.70.80.9

1

-3.2 -2.2 -1.2 -0.2 0.8 1.8 2.8

Draw 500 Random USDs

Convert 500 USDs to Unique SNDs

Simulate Normal Distribution

• Next apply the random SNDs in the Normal distribution formula

Ỹ = Ŷ + σ * SNDIn 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 = Std 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 Simetar Simulation Results for 500 Iterations. 7:56:32 PM 2/17/2013 (0 sec.). © 2011.Variable Sheet1!B47Sheet1!B48Sheet1!B49Sheet1!B50Mean 0.499985 -0.00015 65.48175 65.48175StDev 0.288988 1.001471 3.946465 3.946465CV 57.79939 -650265 6.026817 6.026817Min 0.000895 -3.12303 53.1755 53.1755Max 0.999165 3.143506 77.86988 77.86988Iteration USD SND Y Tilda Simetar

1 0.512793 0.032072 65.60874 65.608742 0.307316 -0.50347 63.49834 63.498343 0.581277 0.20516 66.29082 66.290824 0.787495 0.797758 68.62605 68.626055 0.94209 1.572561 71.6793 71.67936 0.735971 0.630975 67.96882 67.968827 0.048923 -1.65539 58.95901 58.95901

Simulating Random Variables• Must assume a probability distribution shape

– Normal, Beta, Empirical, etc.• Estimate parameters required to define 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 2/27/2012 9:01:56 PMRandom Variables

Distribution Parameters Parm. 1 Parm. 2 Distribution MOMBeta α, β ; A≤x≤B, α,β>01.896 2.117 Beta 200.607Double Exponential α, β ; α≤x<∞, -∞<α<∞, β>0187.812 5.025 Double Exponential207.451Exponential μ, σ ; -∞<x<∞, -∞<μ<∞, σ>0180.705 7.107 Exponential 213.405Gamma α, β ; 0≤x<∞, α,β>0698.392 0.269 Gamma 204.728Inverse Gaussian μ, σ ; 0≤x<∞, μ>0, σ>0187.812 0.003 Inverse Gaussian 205.160Logistic μ, σ ; -∞x<∞, -∞<μ<∞, σ>0187.812 3.918 Logistic 205.801Log-Log μ, σ ; -∞<x<∞, -∞<μ<∞, σ>0184.614 5.541 Log-Log 210.081Log-Logistic μ, σ ; 0≤x<∞, -∞<μ<∞, σ>047.975 187.678 Log-Logistic 206.526Lognormal μ, σ ; 0≤x<∞, -∞<μ<∞, σ>05.235 0.038 Lognormal 204.930Normal μ, σ ; -∞<x<∞, -∞<μ<∞, σ>0187.812 7.107 Normal 204.334Pareto α, β ; α≤x<∞, α,β>0180.969 27.446 Pareto 213.999Uniform A, B ; A≤x≤B175.503 200.122 Uniform 199.874Weibull α, β ; 0≤x<∞, α,β>034.000 ############## Weibull ############Binomial n, p ; x=0,1,2,...,n; 0≤p≤1256.000 0.729 Binomial 203Geometric p ; x=1,2,...; 0≤p≤10.005 Geometric 865Poisson λ ; x=0,1,...; 0≤λ<∞187.317 Poisson 220Negative Binomial s, p ; x=1,2,...; 0≤p≤1 Negative Binomial

Common USD0.99

MOM Estimates

Which is the Best Distribution?• Use Simetar function =CDFDEV(History,

SimData)– Perfect fit has a CDFDEV value of Zero– Pick the distribution with the lowest CDFDEVDistributions CDFDEV Formula

Beta 0.02 =CDFDEV(Sheet1!$A$2:$A$21,SimData!B9:B108)Double Exponential 0.37 =CDFDEV(Sheet1!$A$2:$A$21,SimData!C9:C108)Exponential 2.66 =CDFDEV(Sheet1!$A$2:$A$21,SimData!D9:D108)Gamma 0.07 =CDFDEV(Sheet1!$A$2:$A$21,SimData!E9:E108)Logistic 0.16 =CDFDEV(Sheet1!$A$2:$A$21,SimData!F9:F108)Log-Log 0.43 =CDFDEV(Sheet1!$A$2:$A$21,SimData!G9:G108)Log-Logistic 0.34 =CDFDEV(Sheet1!$A$2:$A$21,SimData!H9:H108)Lognormal 0.10 =CDFDEV(Sheet1!$A$2:$A$21,SimData!I9:I108)Normal 0.05 =CDFDEV(Sheet1!$A$2:$A$21,SimData!J9:J108)Pareto 79.04 =CDFDEV(Sheet1!$A$2:$A$21,SimData!K9:K108)Uniform 0.03 =CDFDEV(Sheet1!$A$2:$A$21,SimData!L9:L108)Weibull 0.08 =CDFDEV(Sheet1!$A$2:$A$21,SimData!M9:M108)Binomial 1.04 =CDFDEV(Sheet1!$A$2:$A$21,SimData!N9:N108)Geometric 33.81 =CDFDEV(Sheet1!$A$2:$A$21,SimData!O9:O108)Poisson 4.19 =CDFDEV(Sheet1!$A$2:$A$21,SimData!P9:P108)

Use the “View Distributions.xlsx”

• For a random variable with 10 observations can estimate the parameters and view the shape of the distribution