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Risk Measures

Risk Measures. IEF 217a: Lecture Section 3 Fall 2002. Risk Measures Assigning a number to risk. Histogram Variance Beta (CAPM) Value at Risk (VaR) Expected utility Time Historical Perspectives. Histogram. Another Histogram. Histogram. Full picture of risk subject to:

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Risk Measures

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

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

  8. Variance • Expected value • Variance,

  9. VarianceEqual 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 DeviationClose 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 MeasuresAssigning 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 ExampleInterpreting 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 MeasuresAssigning 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 LossExpected Value in the Tail

  32. Risk MeasuresAssigning 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 Log(x) 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 MeasuresAssigning 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 MeasuresAssigning 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 MeasuresAssigning 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

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