Probability Distributions and Stochastic Budgeting

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Probability Distributions and Stochastic Budgeting

• AEC 851 Agribusiness Operations Management
• Spring, 2006

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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

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Beyond Mean-Variance

• Skewed probability distributions

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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?

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Common probability distributions, key parameters
shapes
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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?

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Triangular distribution For eliciting
subjective estimates
Pr(x)

• Determined by Min, Max, Most likely value
  (MLV)
• Mean
• (Min MLV Max)/3
• Variance
• (Min2MLV2Max2-MinMLV-MinMax-MLVMax)/18

x
Min
MLV
Max
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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)

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Correlated risks

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

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? -0.91
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Factoring in correlated risk

• Empirical data available
• Estimate correlation coefficients (_at_RISK uses
  rank correlation, rather than linear correlation)
• Empirical data not available
• Develop joint probability table using counters
• Pr(A B) Pr(AB)Pr(B)
• Where A is outcome variable influenced by B
• Use Uniform or Triangular distribution
• _at_RISK illustration

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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?

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_at_RISK spreadsheet program

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

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Creating a stochastic budget in _at_RISK

• Open _at_RISK or open an Excel version that is
  linked to _at_RISK
• Build a budget
• Identify risky budget components
• Specify probability distributions for those risky
  components based on available data

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Analyzing a stochastic budget in _at_RISK

• _at_RISK will recognize the cells with _at_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

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Interpreting a stochastic budget analysis in _at_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

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Basic _at_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)

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Basic _at_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

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Basic _at_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