Transcript Measures_of_Risk_and_Utility - Analytica Wiki

Measures of Risk and Utility
Analytica Users Group
Gentle Intro to Modeling Uncertainty
Webinar Series
Session #4
20 May 2010
Lonnie Chrisman
Lumina Decision Systems
Copyright © 2010 Lumina Decision Systems, Inc.
Today’s Outline
• What is risk?
• (Expected) Utility
Risk neutrality, risk aversion
Utility of non-monetary outcomes
• Specific risk measures
• Uses of risk measures
Copyright © 2010 Lumina Decision Systems, Inc.
Course Syllabus
(tentative)
Over the coming weeks:
• What is uncertainty? Probability.
• Probability Distributions
• Monte Carlo Sampling
• Measures of Risk and Utility (Today)
• Risk analysis for portfolios
(risk management)
• Common parametric distributions
• Assessment of Uncertainty
• Hypothesis testing
Copyright © 2010 Lumina Decision Systems, Inc.
What is Risk?
• A state of uncertainty where some outcomes
are substantially undesirable.
Considerations that some (but not everyone)
see as inherent in the concept of risk:
Involves outcomes that can be avoided or
mitigated.
Concerns deviation from expected value.
Involves harm.
Asymmetric – concerns bad outcomes only
Concerns events not previously conceptualized as
possibilities.
Copyright © 2010 Lumina Decision Systems, Inc.
Risk-Return Tradeoffs
• Decisions often involve tradeoffs between
expected benefit and level of risk.
• This implies a metric for quantifying risk.
Copyright © 2010 Lumina Decision Systems, Inc.
Types of things that
might be at risk
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Money
Property
Lives (risk of death)
Shortening of lifespan
Physical well-being ( risk of injury, pain )
Emotional well-being
Reputation
Power or influence
Health of the planet (environment)
The society’s condition or values
• Discussion: What units of measurement might be
appropriate for each of the above?
Copyright © 2010 Lumina Decision Systems, Inc.
Deal or No Deal?
You are a contestant on a game show.
Hidden in one of two boxes is $1,000,000. The
other box is empty. You can open only one
box and keep its contents.
Or, you receive $400,000 if you leave now
without selecting either box.
• What do you choose? Why?
• Does this game involve “risk”?
How would you quantify the amount of risk?
• At what threshold amount paid for leaving
would you be indifferent?
Copyright © 2010 Lumina Decision Systems, Inc.
Regret
• One metric for risk is minimum regret.
• Does not use probability of outcome.
Outcome
Decision
Box 1
Play $1M
Regret: 0
Stop $400K
Regret: $600K
Box 2
0
Regret: $400K
$400K
Regret: 0
Potential regret
$400K
$600K
At Risk: $400K
Copyright © 2010 Lumina Decision Systems, Inc.
Deal or no Deal #2
A friend presents you with two boxes.
Hidden in one is $10, the other is
empty. You can select one box and
keep its contents.
Or, you will be given $4 if you stop now.
• Why is this decision any different than
the previous one?
Copyright © 2010 Lumina Decision Systems, Inc.
Utility Functions
• The utility of an outcome reflects a degree
of benefit for the decision maker.
• Twice the money doesn’t usually mean twice
the benefit.
• Daniel Bernoulli: Your utility is proportional
to Ln(wealth), the logarithm of your net
wealth.
• Exercise: Estimate your own net wealth. For
the $1M deal game:
What is your expected utility if you chose a box?
What is your utility if you leave with $400K?
At what threshold amount with they be the same?
Copyright © 2010 Lumina Decision Systems, Inc.
Risk Neutrality & Risk Aversion
Most of us
Lottery
player
Copyright © 2010 Lumina Decision Systems, Inc.
Non-Monetary Utility
• A philanthropic organization must decide
between two projects in Africa:
Malaria treatments:
Will save the lives of X children under the age of 10.
AIDS prevention
Will prevent Y new cases of AIDS (mostly young adults).
• Discussion: How could you define utility functions in
such a way that these could be meaningfully
compared?
Copyright © 2010 Lumina Decision Systems, Inc.
Exercise (financial risk)
Build a model of a potential 5 yr rental
property investment.
Purchase price: $250K
$50K down payment
(Mortgage: $200K at 5.5% 30yr fixed) – not needed for model
Total net income over 5 yrs:
Normal($-25K,$10K)
Appreciation in 5 yrs: Normal(12%,10%)
To be sold after 5 years.
Mortgage balance at that time: $185K
Compute Profit, Return-on-investment
View Mean, CDF results
Copyright © 2010 Lumina Decision Systems, Inc.
Some possible (single-number) risk
measures for previous example
• Expected profit (?)
Does this capture “risk”?
Mean(profit)
• Expected change in log-wealth utility.
Mean( Ln(profit+wealth) – Ln(wealth) )
• Probability of losing money
Probability(profit<0)
• Standard deviation (of profit) – aka “volatility”
SDeviation(Profit)
• 5% fractile (of profit)
GetFract(profit,5%)
• Exercise: Encode each of these in the Rental
Investment model.
Copyright © 2010 Lumina Decision Systems, Inc.
Value at Risk (VaR)
• Definition: The 5% five-year VaR is the 5%
percentile for the loss at the 5 year mark
(relative to the value now).
• Note: Also called the 95% five-year VaR.
In Analytica example:
GetFract( -Profit, 95% )
• Will be a positive number (the amount of
loss) if Probability(Profit<0) > 5%.
• 1% VaR is also commonly used.
Copyright © 2010 Lumina Decision Systems, Inc.
ChanceDist
• Given:
Index Outcome (possible outcomes)
Array P indexed by Outcome
(probabilities)
• ChanceDist(Probs,Outcome)
Encodes the discrete distribution.
Copyright © 2010 Lumina Decision Systems, Inc.
Exercise
0.1
0.9
Bear
+0.3%
0.1
Crash
-10%
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Bull
-0.2%
0.001
0.1
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0.899
Start
($1)
0.9
Model the above transitions & price changes over 100
days/transitions.
Compute the 100-day 5% VaR.
(Use SampleSize=1000 and Random Latin Hypercube)
•
Compute the worst loss among 1000 sampled runs.
Copyright © 2010 Lumina Decision Systems, Inc.
Expected Shortfall
• Also known as:
Conditional value at risk (CVaR)
Expected tail loss
• Definition: The expected loss when the loss
exceeds the VaR.
• Exercise: Compute the 100-day 5% expected
shortfall for the previous example.
Mean(loss, w:loss>=value_at_risk)
Copyright © 2010 Lumina Decision Systems, Inc.
Uses for a risk measure
• Decision making
As an objective.
As a constraint.
Explicit risk/reward trade-offs.
• Reporting / monitoring
Communicating level of risk being incurred (in a
portfolio, or by an organization).
Regulation (Basel II & Sarbanes-Oxley)
• Explaining
Behavior analysis
Copyright © 2010 Lumina Decision Systems, Inc.
Summary
• Several conceptions of “risk” exist.
• Utility allows:
Direct incorporation of risk attitudes into
decision making
Incorporation of non-monetary considerations.
• Some possible measures of risk:
Standard deviation (volatility)
Minimum regret
Probability of loss
Fractile levels
Value at risk (VaR)
Expected shortfall
Copyright © 2010 Lumina Decision Systems, Inc.