Probability Distributions and Stochastic Budgeting AEC 851 – Agribusiness Operations Management Spring, 2006

Recapping Mean-Variance Methods covered: –Mean-variance efficiency –Quadratic Programming variants Minimize Variance s.t. min. Exp Income Maximize Exp. Income s.t. max Variance E-V utility function (as proxy for constant absolute risk aversion) Assumptions required –Decision maker cares only about mean & variance –Outcome variable follows Normal distribution

Beyond Mean-Variance Skewed probability distributions

Stochastic Budgets Stochastic budgets are built around: 1) Mean (“typical”) values 2) Probability distributions for drawing random values of key input variables that affect outcome variable How to come up with probability distributions?

Common probability distributions, key parameters & shapes EmpiricalPrior data or Estimated values Form varies UniformMin, maxFlat NormalMean, varianceSymmetric TriangularMin, max, most likely value Skewed

When probability info missing Probability distributions needing least info: –Uniform –Triangular Estimating empirical probabilities (visual impact method) –Given some counters (e.g., 50), build histogram of believed outcomes –Most likely value? Cutoff value below/above which no more than 25%?

Triangular distribution: For eliciting subjective estimates Determined by Min, Max, Most likely value (MLV) Mean –(Min + MLV + Max)/3 Variance –(Min 2 +MLV 2 +Max 2 - Min*MLV-Min*Max- MLV*Max)/18 Min MLVMax x Pr(x)

Other distributions Beta, gamma, lognormal –For continuous variables (smooth curve); may be skewed; beta has min & max Bernoulli, binomial, neg. binomial –Binomial outcomes (Yes/No, On/Off) with and without equal probabilities Poisson –Discrete outcomes (e.g., number of persons arriving in line)

Correlated risks Most outcomes involve more than one uncertain process Is it reasonable to assume that random variables are independent?

 = -0.91

Factoring in correlated risk Empirical data available: –Estimate correlation coefficients uses rank correlation, rather than linear correlation) Empirical data not available: –Develop joint probability table using counters Pr(A & B) = Pr(A|B)*Pr(B) Where A is outcome variable influenced by B –Use Uniform or Triangular illustration

Effect of correlated price & quantity risk on mean outcome Formula for expected income if price and yield are correlated: What effect will this have on income? –Average income? –Variability of income?

@RISK spreadsheet generates random numbers from the Input Variable probability distributions that you specify Result is probability distribution(s) for the Output Variable(s)

Creating a stochastic budget or open an Excel version that is linked 2.Build a budget 3.Identify risky budget components 4.Specify probability distributions for those risky components based on available data

Analyzing a stochastic budget will recognize the cells functions as Input Variables for the risk analysis 2.Specify the Output Variable(s) 3.If certain components are correlated, specify rank correlation in “List Inputs” 4.If certain components should be held constant, lock them up them using Fix/Vary 5.Check that “Simulation Settings” OK 6.Run “Simulate”

Interpreting a stochastic budget analysis 1.“Statistics” screen shows summary statistics of all random variables 2.“Graph” will display histogram of highlighted variable 3.“Sensitivity” will evaluate sensitivity of Output to different Input variables a)“Hurricane” graphs display correlations 4.Scenario shows probability of being above or below key thresholds

Commands for Continuous Distributions in Excel RiskUniform(Min, Max) –Uniform distribution gives equal probability of any value in range from Min to Max RiskTriang(Min,MLV,Max) –Triangular distribution gives highest probability of Most Likely Value (MLV) within fixed range RiskNormal(Mean, Std Dev) –Normal “bell-shaped” distribution (no Min or Max)

Commands for Empirical Distributions in Excel RiskHistogrm(Min, Max, {p 1, p 2 … p n }) –Histogram distribution gives n specified probabilities (p i ) of n equal interval outcomes RiskCumul(Min,Max,{x 1,… x n },{cp 1,…cp n }) –Cumulative distribution gives n specified outcomes (increasing in size) and n associated cumulative probabilities of outcomes

Command for Discrete Distribution in Excel RiskDiscrete({x 1,… x n },{p 1,…p n }) –Discrete distribution gives n specified discrete outcomes and n associated probabilities –Outcomes can take only exact values of the x i –Examples: An event that will or will not occur Mutually exclusive outcomes