Copyright © 2010 Lumina Decision Systems, Inc. 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 $1,000,000. 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 Box 1Box 2 Play$1M0 Stop$400K Regret: $600K Regret: $400KRegret: 0 Potential regret $400K $600K At Risk: $400K Decision

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 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 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. Bear +0.3% Bull -0.2% Crash -10% Start ($1)

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