1. Risk Measures IEF 217a: Lecture Section 3 Fall 2002

2. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

3. Histogram

4. Another Histogram

5. Histogram Full picture of risk subject to: –Underlying model is correct –Samples big enough Easy to get with simulations Problems –Complicated to Summarize Evaluate

6. Risk -> Single Number Many attempts

7. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

8. Variance Expected value Variance,

9. Variance Equal Sample Expected value Variance,

10. Why Variance? Easy to describe –People learn it in statistics classes Normal Distribution –Completely described by Expected Value Variance (dispersion/risk) –Central limit theorem Useful properties (adding)

11. Variances for Two: Covariance E(x) = expected value of x Covariance(x,y) = cov(x,y)

12. Useful Properties If x is independent of y then

13. Case for Variance A pictorial example of variance Matlab example: –varexam1.m

14. Case Against Variance Asymmetry is not captured well by variance For asymmetric distributions variance might not be a good measure

15. Mean Absolute Deviation Close Relative to Variance Not as easy to compute Not used as often –Less often reported with Normal distribution Less sensitive to outliers than variance

16. Mean Absolute Deviation Expected value MAD,

17. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

18. CAPM-Beta –r: individual stock return –a: constant for stock I –b: beta –e: idiosyncratic risk

19. More on Beta

20. The Importance of Portfolio Thinking (Markowitz) Previously –Risk of a stock = Variance( r ) Post CAPM –Risk of a stock = beta (b)

21. CAPM Example Matlab: betatest.m Portfolio problem –0.8 market, 0.2 individual stock Experiment –Measure contribution to portfolio std from Increase beta on individual stock Increase variance of e, idiosyncratic risk on stock

22. CAPM Example

23. CAPM Example Interpreting Results Beta risk contributes more to the overall risk of the portfolio then idiosynchratic Intuition: –Idiosyncratic risk is uncorrelated with the overall portfolio so it contributes less to overall risk –Market risk is more important since it is correlated with the rest of the portfolio

24. “Traditional” CAPM Mean/Variance world E(return) linear function of beta We wont dwell on this, why? –Outside of mean/variance world –Nonlinear dependence

25. Why do we still care about systematic (beta) risk? Risk depends on –What you are considering to add to your portfolio and, –Its relationship to whatever is in your portfolio Big picture risk analysis!! Example: (general factor models)

26. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

27. What is VaR? VaR = Value at Risk Percentile in left tail Maximum loss, leaving out low probability events matlab example: varpic.m

28. 5% VaR

29. Why VaR? Nonnormality Derivatives More relevant risk measure –Capital requirements –Firm wide risk reports

30. Close Relation: Expected Tail Loss (ETL) E( r ) given that r < 5th percentile Expected loss given that something really bad happens matlab example: etlex.m

31. Expected Tail Loss Expected Value in the Tail

32. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

33. Expected Utility Replace expected value with

34. What is u(x)? u(x) = utility function Important property E(u(x)) often decreasing in measures of x riskiness

35. Example u(x) = log(x) –x = [1; 2], probs = [0.5; 0.5] E(x) = 1.5 E(u(x)) = E(log(x)) = 0.35 –x = [0.5; 2.5], probs = [0.5; 0.5] E(x) = 1.5 E(u(x)) = E(log(x)) = 0.11

36. Why Does Expected Utility Fall? Function curvature –Moving -0.5 has bigger impact then –Moving +0.5 x Log(x) 1.5

37. Utility Functions Many different possibilities –Log, Power –Mean/Variance (Quadratic) –Specific functions (loss aversion, semivariance) Is this the right way to measure risk? –Yes: under restrictive assumptions –No: In general, but it is often used Easy to deal with in computer simulations

38. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

39. Time Future horizon –1 Day –1 Month –1 Year –10 Years Liquidity –Maximum sustainable loss

40. Common Investor Advice Long Horizon Investors (young) –More funds in equity (risky) Short Horizon Investors (old) –More funds in bonds (less risky) Does this make sense? –Montecarlo test We’ll check this many times

41. Portfolio Experiment Annual return: Normal, mean 0.05, std 0.15 $100 starting investment Two Horizons –5 years –20 years Look at histograms and tails matlab: timehorizon1.m

42. Liquidity Good strategy in the long run –High expected return May sustain large losses in the short run –Needs short term financial backing Might go bankrupt before big returns are realized

43. Liquidity Example FX Trader Daily Return = [-0.1; 0.1] prob = [0.4;0.6] Trade for 20 days Start with $100 Must stop when portfolio drops below $80 –(Pull the plug) new matlab commands: next slide

44. Liquidity Example Matlab commands –cumprod –Returns: r = [r1 r2 r3 r4] –cumprod(1+r) (1+r1) (1+r1)(1+r2) (1+r1)(1+r2)(1+r3) (1+r1)(1+r2)(1+r3)(1+r4) matlab: ruin.m

45. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

46. Historical Perspective Risk -> Number: Very old problem Different approaches –Economists/statisticians Expected utility Expected loss –Business/finance/investment CAPM/Risk neutral pricing –Psychologists Real people behave in strange ways with risk

47. Risk Measures Assigning a number to risk Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives

48. Risk Types (Jorion 1) Market risk Credit risk Operational risk –Model risk Legal risk