Decision and Risk Analysis

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Decision and Risk Analysis

• Financial Modelling
• Risk Analysis II
• Kiriakos Vlahos
• Spring 2000

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Session overview

• Probability distributions for Risk Analysis
• Subjective
• Regression and Forecasting models
• Historic data
• Resampling
• Distribution fitting
• Sampling distributions
• Using histograms
• The inversion method
• Correlated random variables
• Comparing uncertain outcomes
• Dynatron case

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Using regression models in risk analysis
Example Ferric regression model Cost 11.75
7.93 (1/Capacity) Standard Error (SE)
0.98 _at_RISK formula for cost Cost 11.75
7.93 (1/Capacity) RiskNormal(0,0.98)
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Using historic dataResampling
_at_RISK funcion RISKDUNIFORM(datarange)
At every iteration it picks one of the historic
values at random.
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Historic data - Distribution fitting
2. Histogram
1. Historic data
4. Fit theoretical distribution
3.Cumulative function
5. Then use theoretical distribution in _at_RISK
Use statistical packages for distribution fitting
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Cumulative functions of standard distributions
Cumulative function
Distribution function
Uniform
Triangular
Normal
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Random sampling

• Probabilistic simulation depends on creating
  samples of random variables
• In order to carry out random sampling we need
• a set of random numbers
• a distribution or cumulative function for each of
  the random variables
• a mechanism for converting random numbers into
  samples of the above distributions
• Tables of random numbers
• Pseudo random number generators
• e.g. Rj1 MOD(a Rj c, m)
• The initial R is the seed
• Excel RAND() function

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Inversion method
Pick random number between 0 and 1
Read sample
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Modelling correlated variables
Demand risknormal(100,20) Price
risknormal(100,20) Sales Demand Price
Min 2,000 Max 20,500 St.d. 2900
Assuming correlation of -0.8
Min 5,500 Max 13,500 St.d. 1300
Always try to model correlation between random
variables
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Expected value
Production 100 Demand risknormal(100,20) Sales
min(Production, Demand)
If we replace Demand with its expected value then
Sales equals 100. But the expected value of
Sales is less than 100. In general
i.e. replacing uncertain inputs with their
average values does not result in the expected
value of the output unless the function is linear.
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Dynatron

• Decide about
• The production level of Dynatron toys
• the split into super and standard

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Dynatron - Decision Alternatives
Field Sales Representatives
Production Manager
Gassman
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Cost Accounting
Additional production costs
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Base case model
Profit Revenue - Inventory cost - Investment
cost
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Dynatron - Demand uncertainty
Median demand 150 Minimum 50 and maximum 300 1 in
4 chance that demand is at least 190 3 in 4
chances that demand is at least 125
Cumulative function
RiskCumul(50,300,125,150,190,0.25,0.5,0.75)
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Standard/super split uncertainty
of supers Median 40 Minimum 30 and maximum
60 75 chance to be 45 or less 25 to be 36 or
less
Cumulative function
RiskCumul(0.3,0.6,0.36,0.4,0.45,0.25,0.5,0.75)
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Dynatron - Simulaton Results
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Comparing risky assets
Case 1
AgtgtB
B
A
Profit
Case 2
A
AgtgtB
B
Profit
Case 3
AgtgtB ?
B
A
Profit
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Risk-return tradeoff
Return
Efficient frontier
Dominated options
Risk
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Screening risky options
Cumulative Probability functions
1
AgtgtB
B
A
0
Return
1
if area (1) gt area (2)
(2)
then project A gtgt B
Requires risk aversion
B
A
(1)
0
Return
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Dynatron - Simulation Results
Cumulative probability distributions
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Dynatron - Simulation Results
Cumulative probability distributions
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Summary

• Integrating regression and forecasting models
  with risk analysis
• Using historic data in risk analysis
• Resampling
• Distribution fitting
• Sampling distributions
• The inversion method
• Model correlation between random variables!
• Comparing uncertain outcomes
• Screening options
• Risk return tradeoff
• Risk preferences