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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 • • • 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. 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 Decision • One metric for risk is minimum regret. • Does not use probability of outcome. Outcome Box 1 Box 2 Play $1 M 0 Regret: 0 Stop $400 K Regret: $600 K Regret: $400 K Regret: 0 Potential regret $400 K $600 K At Risk: $400 K 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 $1 M deal game: What is your expected utility if you chose a box? What is your utility if you leave with $400 K? 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: $250 K $50 K down payment (Mortgage: $200 K at 5. 5% 30 yr fixed) – not needed for model Total net income over 5 yrs: Normal($-25 K, $10 K) Appreciation in 5 yrs: Normal(12%, 10%) To be sold after 5 years. Mortgage balance at that time: $185 K 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) Get. Fract(profit, 5%) • Exercise: Encode each of these in the Rental Investment model. Copyright © 2010 Lumina Decision Systems, Inc.
Value at Risk (Va. R) • Definition: The 5% five-year Va. R is the 5% percentile for the loss at the 5 year mark (relative to the value now). • Note: Also called the 95% five-year Va. R. In Analytica example: Get. Fract( -Profit, 95% ) • Will be a positive number (the amount of loss) if Probability(Profit<0) > 5%. • 1% Va. R is also commonly used. Copyright © 2010 Lumina Decision Systems, Inc.
Chance. Dist • Given: Index Outcome (possible outcomes) Array P indexed by Outcome (probabilities) • Chance. Dist(Probs, Outcome) Encodes the discrete distribution. Copyright © 2010 Lumina Decision Systems, Inc.
Exercise 0. 1 0. 9 Bear +0. 3% 0. 1 Bull -0. 2% 0. 001 0. 1 Crash -10% • • 0. 899 Start ($1) 0. 9 Model the above transitions & price changes over 100 days/transitions. Compute the 100 -day 5% Va. R. (Use Sample. Size=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 (CVa. R) Expected tail loss • Definition: The expected loss when the loss exceeds the Va. R. • 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 (Va. R) Expected shortfall Copyright © 2010 Lumina Decision Systems, Inc.