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Probability Distributions and Stochastic Budgeting

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

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Probability Distributions and Stochastic Budgeting

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

  2. 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

  3. Beyond Mean-Variance • Skewed probability distributions

  4. 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?

  5. Common probability distributions, key parameters & shapes

  6. 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%?

  7. Triangular distribution: For eliciting subjective estimates Pr(x) • Determined by Min, Max, Most likely value (MLV) • Mean • (Min + MLV + Max)/3 • Variance • (Min2+MLV2+Max2-Min*MLV-Min*Max-MLV*Max)/18 x Min MLV Max

  8. 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)

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

  10.  = -0.91

  11. Factoring in correlated risk • Empirical data available: • Estimate correlation coefficients (@RISK 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 distribution • @RISK illustration

  12. 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?

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

  14. Creating a stochastic budget in @RISK • Open @RISK or open an Excel version that is linked to @RISK • Build a budget • Identify risky budget components • Specify probability distributions for those risky components based on available data

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

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

  17. Basic @RISK 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)

  18. Basic @RISK Commands for Empirical Distributions in Excel • RiskHistogrm(Min, Max, {p1, p2 … pn}) • Histogram distribution gives n specified probabilities (pi) of n equal interval outcomes • RiskCumul(Min,Max,{x1,… xn},{cp1,…cpn}) • Cumulative distribution gives n specified outcomes (increasing in size) and n associated cumulative probabilities of outcomes

  19. Basic @RISK Command for Discrete Distribution in Excel • RiskDiscrete({x1,… xn},{p1,…pn}) • Discrete distribution gives n specified discrete outcomes and n associated probabilities • Outcomes can take only exact values of the xi • Examples: • An event that will or will not occur • Mutually exclusive outcomes

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