Measures of Risk and Utility

1. Measures of Risk and Utility Analytica Users GroupGentle Intro to Modeling Uncertainty Webinar Series Session #4 20 May 2010 Lonnie ChrismanLumina Decision Systems

2. Today’s Outline • What is risk? • (Expected) Utility • Risk neutrality, risk aversion • Utility of non-monetary outcomes • Specific risk measures • Uses of risk measures

3. 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

4. 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.

5. Risk-Return Tradeoffs • Decisions often involve tradeoffs between expected benefit and level of risk. • This implies a metric for quantifying risk.

6. Types of things thatmight 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?

7. 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?

8. Regret • One metric for risk is minimum regret. • Does not use probability of outcome. Potential regret $400K Regret: 0 Regret: $400K Decision $600K Regret: $600K Regret: 0 At Risk: $400K

9. 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?

10. 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?

11. Risk Neutrality & Risk Aversion Most of us Lottery player

12. 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?

13. 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

14. 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.

15. 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.

16. ChanceDist • Given: • Index Outcome (possible outcomes) • Array P indexed by Outcome (probabilities) • ChanceDist(Probs,Outcome) • Encodes the discrete distribution.

17. Exercise 0.1 0.9 0.899 • 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% 0.1 0.001 0.1 Start($1) Crash-10% 0.9

18. 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)

19. 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

20. 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