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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
- 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
- “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 a) “Hurricane” graphs display correlations
- 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