Download 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
• 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
Probability of Tart Cherry Grower Price, 1970-2002
60%
Probability
50%
40%
30%
20%
10%
0%
Under 10
10-19
20-29
30-39
Grower Price Range
40-49
50-59
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
Form
Estimated values varies
Uniform
Min, max
Flat
Normal
Mean, variance
Symmetric
Triangular Min, max,
Skewed
most likely value
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
Pr(x)
– (Min + MLV + Max)/3
• Variance
– (Min2+MLV2+Max2Min*MLV-Min*MaxMLV*Max)/18
Min
MLV
Max
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?
Grower Price (cents/lb)
Tart Cherry Price & Michigan Yields, 1993-2002
60
50
40
 = -0.91
30
20
10
0
0
2,000
4,000
6,000
8,000
Michigan Yield (lb/ac)
10,000
12,000
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
Effect of correlated price &
quantity risk on mean outcome
• Formula for expected income if price
and yield are correlated:
E ( )  E ( p y ) E ( y )  Cov( p y , y )  p x x  FC
• What effect will this have on income?
– Average income?
– Variability of income?
@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
Analyzing a stochastic budget
in @RISK
1. @RISK will recognize the cells with @RISK
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 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 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
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