Analytics of Risk Management II: Statistical Measures of Risk

1. Risk Management Lecturer: Mr. Frank Lee Session 4 Analytics of Risk Management II: Statistical Measures of Risk

2. Overview Quantitative measures of risk - 3 main types: • Sensitivity – derivative based measures • Volatility & Statistical measures of risk: • Attitudes to risk • Relation with Finance Theory • Portfolio Theory, CAPM & APT • Post-modern Portfolio Theory • Downside Risk measures • Statistical underpinning • Value at Risk

3. Value = Expected Net Present Value n E(NPV) = ΣE(Rt) - E(Ct) (1 + R)t t = 1

4. A Set of Definitions • Risk - The outcome in a particular situation is unknown but the probability distribution from which the outcome is to drawn is known • Uncertainty - The outcome in a particular situation is unknown as is the probabilitydistribution from which the outcome is drawn.

5. Problems of which situations are risky and which are uncertain • Risky situations can form the basis for tractable financial analysis • Uncertain situations are considerably less analytically tractable

6. Examples of Risk and Uncertainty • Risk - Prices in Financial Markets Share Prices Interest Rates Exchange Rates • Uncertainty - Terrorist Attacks Wars Financial Crises

7. Statistics and Estimates • Mean - Measure of Central Tendency, like Median and Mode. • Variance - Measure of dispersion round the mean the squared term ensures that Variance is positive and gives extra weight to observations furthest from the mean. • Standard Deviation - Square Root of Variance. It is in same metric as Mean

8. Probability Distributions • We generally focus on normal distributions • Normal Distributions are entirely specified by Mean and Standard Deviation

9. Normal Distribution Frequency 34.13% 34.13% -s +s m Return

10. Risk and Return • Return - Average or Expected Return • Risk • For Normal Distributions Standard Deviation is totally satisfactory • For Non-normal Distributions there may be a diversity of statistical and psychological measures or risks

11. Attitudes to Risk • Utility Function - Defined over a probability distribution of returns. • Mean and Variance approach • Higher Moments • Time

12. Attitudes to Risk • Risk Averse - Like Return Dislike Risk • Risk Neutral - Like Return and totally unconcerned about Risk • Risk Loving - Like Return and Like Risk

13. Utility and Risk Utility U(B) U(*) U(A) W(A) W(*) W(B) Wealth

14. Risk Aversion and Utility W(A) + W(B) = W(*) 2 U(*) > U(A) + U(B) 2 Prefer W(*) to bet with 0.5 probability of W(A) and 0.5 probability of W(B) which has the same expected value of wealth of W(*)

15. Preferences for Risk and Return Return B A X C D Risk

16. For Risk Averse Investor • A Preferred to X • X Preferred to D • C and X no ordering • B and X no ordering

17. Indifference Curves for the Risk Averse Investor U1 Return U2 U3 x Risk

18. Indifference Curve Slope is a Measure of Risk Aversion Return a b c d ab > cd xy xy Risk x y

19. Measures of Risk • Standard Deviation • Combination of Moments • Value at Risk • Expected Tail Loss • Moments Relative to Benchmarks - Risk Free Rate, Zero Return, Capital Asset Pricing Model, Arbitrage Pricing Theory

20. Measuring Risk Variance - Average value of squared deviations from mean. A measure of volatility. 2 = Standard Deviation - Square root of variance (square root of average value of squared deviations from mean). A measure of volatility.

21. Standard Deviation • Square root of variance • Equates risk with uncertainty • Implies symmetric, normal return distribution • Upside volatility penalized same as downside volatility • Measures risk relative to the mean • Same risk for all goals

22. Moments ith Moment around a = E(R - a)i Measure of Skewness = E(R - E(R))3 (Minus value skewed to left, Positive Value skewed to right) Measure of Kurtosis = E(R - E(R))4 (Larger value flatter the distribution)

23. Modern Portfolio Theory

24. Portfolio Theory Assumptions • Investors Risk Averse • Investors only interested in the Mean and Standard Deviation of the Distribution of Returns on an Asset • Investors have knowledge of Mean and Standard Deviation of Returns • A Range of Risky Assets • At Least One Riskless Asset

25. Measuring Risk

26. Portfolio Risk Covariance = Correlation = (-1 < xy <1)

27. Variables Trend Together Y Figure 1 X

28. Variables Trend in Opposite Directions Y Figure 2 X

29. Correlation Values • One - Perfect linear relation • Between zero and one variables trend together • Zero - No relation between variables • Between zero and minus one variables trend in opposite directions • Minus one - variables have negative perfect linear relation

30. Portfolio Return & Risk

31. 1 2 3 4 5 6 N 1 2 3 4 5 6 N Portfolio Risk The shaded boxes contain variance terms; the remainder contain covariance terms. To calculate portfolio variance add up the boxes STOCK STOCK

32. Hedging with a Portfolio Return A B Time

33. Simple Portfolio Impacts

34. Correlation Coefficients Revisited

35. The Set of Risky Assets E(R) A SD(R)

36. Optimal Set of Risky Assets E(R) y x SD(R)

37. Adding the Riskless Asset E(R) Rf SD(R)

38. The Capital Market Line E(R) G The Capital Market Line E H D SD(R)

39. Investors Choice E(R) y x SD(R)

40. Portfolio Theory Conclusions • All Investors hold the same set of risky assets if they hold risky assets • All investors must hold the market portfolio • Their risk preferences determine whether they gear up or down by borrowing or lending

41. Security Market Line Return . Market Return = rm Efficient Portfolio Risk Free Return = rf BETA 1.0

42. Security Market Line Return SML rf BETA 1.0 SML Equation = rf + B ( rm - rf )

43. Beta and Unique Risk Covariance with the market Variance of the market

44. Downside Risk Measures

45. Expert Opinions • Markowitz (1992): Since an investor worries about underperformance rather than over-performance, semi-deviation is a more appropriate measure of investor's risk than variance. • Sharpe (1963): Under certain conditions the mean-variance approach leads to unsatisfactory predictions of investor behavior.

46. Post-Modern Portfolio Theory Two Fundamental Advances on MPT: • Downside risk replaces standard deviation • PMPT permits non-normal return distributions

47. ‘PMPT’ v MPT • Risk measure: • Downside Risk vs. Standard Deviation • Probability distribution: • Lognormal vs. Normal. • The same application: • Asset allocation/portfolio optimalisation • and performance measurement

48. Downside risk measures Shortfall probability Average shortfall Semi-variance

49. Downside Risk • Defined by below-target semideviation • Standard deviation of below-target returns • Differentiates between risk and uncertainty • Naturally incorporates skewness • Recognizes that upside volatility is better than downside volatility • Combines frequency and magnitude of bad outcomes • No single riskless asset

50. Value at Risk (VAR)