Prob. Distributions & Stochastic Budgeting: Mean-Variance Analysis & Stochastic Budgets, Slides of Business Management and Analysis

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Probability Distributions and

Stochastic Budgeting

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

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

Empirical Prior data or

Estimated values

Form

varies

Uniform Min, max Flat

Normal Mean, variance Symmetric

Triangular Min, max,

most likely value

Skewed

Triangular distribution:

For eliciting subjective estimates

• Determined by Min, Max, Most likely value (MLV)
• Mean
  • (Min + MLV + Max)/
• Variance
  • (Min 2 +MLV 2 +Max 2 - MinMLV-MinMax- MLV*Max)/ Min (^) MLV Max 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)

Tart Cherry Price & Michigan Yields, 1993-

0

10

20

30

40

50

60

0 2,000 4,000 6,000 8,000 10,000 12,

Michigan Yield (lb/ac)

Grower Price (cents/lb)

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

@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)

Creating a stochastic budget

in @RISK

1. Open @RISK or open an Excel

version that is linked to @RISK

2. Build a budget

3. Identify risky budget components

4. Specify probability distributions for

those risky components based on

available data

Interpreting a stochastic

budget analysis in @RISK

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

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)

Basic @RISK Command for

Discrete Distribution in Excel

• RiskDiscrete({x 1 ,… x (^) n },{p 1 ,…pn })
  • 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