1. Portfolio Selection with Higher Moments Campbell R. Harvey Duke University, Durham, NC USA National Bureau of Economic Research, Cambridge, MA USA http://www.duke.edu/~charvey Inquire UK Autumn Seminar 22-24 September 2002 Royal Bath Hotel, Bournemouth

2. Campbell R. Harvey2 1. Objectives The asset allocation setting What is risk? Conditional versus unconditional risk The importance of higher moments Estimation error New research frontiers

3. Campbell R. Harvey3 2. Modes/Inputs of Asset Allocation Types of asset allocation –Strategic –Tactical Type of information –Unconditional –Conditional

4. Campbell R. Harvey4 Strategic Tactical Unconditional Conditional Slow evolving weights Dynamic weights Constant weights 2. Modes/Inputs of Asset Allocation

5. Campbell R. Harvey5 2. Modes/Inputs of Asset Allocation Conditioning information makes a difference

6. Campbell R. Harvey6 3. Performance Depends on Business Cycle Data through June 2002

7. Campbell R. Harvey7 3. Performance Depends on Business Cycle Data through June 2002

8. Campbell R. Harvey8 3. Performance Depends on Business Cycle Data through June 2002

9. Campbell R. Harvey9 3. Performance Depends on Business Cycle Data through June 2002

10. Campbell R. Harvey10 4. Conditioning Information and Portfolio Analysis Adding conditioning information is like adding extra assets to an optimization

11. Campbell R. Harvey11 4. Conditioning Information and Portfolio Analysis Er Vol Traditional fixed weight optimization (contrarian) in 2-dimensional setting

12. Campbell R. Harvey12 4. Conditioning Information and Portfolio Analysis Er Vol Add conditioning information and weights change through time. Frontier shifts.

13. Campbell R. Harvey13 5. What is Risk? Traditional models maximize expected returns for some level of volatility Is volatility a complete measure of risk?

14. Campbell R. Harvey14 5. What is Risk? Much interest in downside risk, asymmetric volatility, semi-variance, extreme value analysis, regime-switching, jump processes,...

15. Campbell R. Harvey15 6. Skewness... These are just terms that describe the skewness in returns distributions. Most asset allocation work operates in two dimensions: mean and variance -- but skew is important for investors. Examples:

16. Campbell R. Harvey16 6. Skewness 1. The $1 lottery ticket. The expected value is $0.45 (hence a -55%) expected return. –Why is price so high? –Lottery delivers positive skew, people like positive skew and are willing to pay a premium

17. Campbell R. Harvey17 6. Skewness 2. High implied vol in out of the money OEX put options. –Why is price so high? –Option limits downside (reduces negative skew). –Investors are willing to pay a premium for assets that reduce negative skew

18. Campbell R. Harvey18 6. Skewness 2. High implied vol in out of the money S&P index put options. –This example is particularly interesting because the volatility skew is found for the index and for some large capitalization stocks that track the index – not in every option –That is, one can diversify a portfolio of individual stocks – but the market index is harder to hedge. –Hint of systematic risk

19. Campbell R. Harvey19 6. Skewness 3. Some stocks that trade with seemingly “too high” P/E multiples –Why is price so high? –Enormous upside potential (some of which is not well understood) –Investors are willing to pay a premium for assets that produce positive skew –[Note: Expected returns could be small or negative!]

20. Campbell R. Harvey20 7. Skewness 3. Some stocks that trade with seemingly “too high” P/E multiples –Hence, traditional beta may not be that meaningful. Indeed, the traditional beta may be high and the expected return low if higher moments are important

21. Campbell R. Harvey21 7. Skewness

22. Campbell R. Harvey22 7. Skewness

23. Campbell R. Harvey23 7. Skewness

24. Campbell R. Harvey24 7. Skewness

25. Campbell R. Harvey25 7. Skewness

26. Campbell R. Harvey26 7. Higher Moments & Expected Returns CAPM with skewness invented in 1973 and 1976 by Rubinstein, Kraus and Litzerberger Same intuition as usual CAPM: what counts is the systematic (undiversifiable) part of skewness (called coskewness)

27. Campbell R. Harvey27 7. Higher Moments & Expected Returns Covariance is the contribution of the security to the variance of the well diversified portfolio Coskewness is the contribution of the security to the skewness of the well diversified portfolio

28. Campbell R. Harvey28 7. Higher Moments & Expected Returns Data through June 2002

29. Campbell R. Harvey29 7. Higher Moments & Expected Returns Data through June 2002

30. Campbell R. Harvey30 7. Higher Moments & Expected Returns Data through June 2002

31. Campbell R. Harvey31 7. Higher Moments & Expected Returns Data through June 2002

32. Campbell R. Harvey32 8. Factors 1. SR (systematic risk) is the beta,  i in the simple CAPM equation 2. TR (total risk) is the standard deviation of country return  i 3. IR (idiosyncratic risk) is the standard deviation of the residual in simple CAPM, e it Related to simple CAPM: R it – r ft =  i +  i [R mt – r ft ] + e it

33. Campbell R. Harvey33 8. Factors 4. Log market capitalization Related to size

34. Campbell R. Harvey34 8. Factors 5. Semi-Mean is the semi-standard deviation with B = average returns for the market 6. Semi-r f is the semi-standard deviation with B = U.S. risk free rate 7. Semi-0 is the semi-standard deviation with B = 0 Related to semi-standard deviation: Semi-B =, for all R t < B

35. Campbell R. Harvey35 8. Factors 8. Down-  iw is the  coefficient from market model using observations when country returns and world returns are simultaneously negative. 9. Down-  w is the  coefficient from market model using observations when world returns negative. Related to downside beta

36. Campbell R. Harvey36 8. Factors 10. VaR is a value at risk measure. It is the simple average of returns below the 5th percentile level. Related to value at risk

37. Campbell R. Harvey37 8. Factors 11. Skew is the unconditional skewness of returns. It is calculated by taking the Mean(e i 3 ) {Standard deviation of (e i )}^3 12. Skew5%: {(return at the 95 th percentile – mean return) - (return at 5 th percentile level – mean return)} - 1 Related to skewness

38. Campbell R. Harvey38 8. Factors 13. Coskew1 is: (  e i * e m 2 )/T {square root of (  (e i 2 )/T)) } * {(  e m 2 )/T)} 14. Coskew2 is: (  e i * e m 2 )/T {standard deviation of (e m )}^3 Related to coskewness

39. Campbell R. Harvey39 8. Factors 15. Kurt is the kurtosis of the return distribution Related to spread

40. Campbell R. Harvey40 8. Factors 16. ICRGC is the log of the average monthly International Country Risk Guide’s (ICRG) country risk composite 17. CCR is the log of the average semi-annual country risk rating published by Institutional Investor. 18. ICRGP is the log of the average monthly ICRG political risk ratings. Related to political risk

41. Campbell R. Harvey41 8. Factors 19. betahml - HML 20. betasmb - SMB Related to Fama-French 3-factor model

42. Campbell R. Harvey42 8. Factors 21. betaoil - Oil Price (Change in Brent index) 22. binfl - Weighted average of G7 inflation using GDP deflator. Related to commodity prices and inflation

43. Campbell R. Harvey43 8. Factors 23. betafx - The trade weighted FX to $ given by the Federal Reserve 24. betafx1- Simple average $ -Euro and $-Yen Related to FX risk

44. Campbell R. Harvey44 8. Factors 25. bintr - Real interest rate - Weighted average short-term interest rate/Weighted average of inflation 26. bterm - Weighted average difference between long and short rates Related to Interest Rates

45. Campbell R. Harvey45 8. Factors 27. betaip - OECD G7 industrial production Related to Economic Activity

46. Campbell R. Harvey46 9. Results

47. Campbell R. Harvey47 9. Results

48. Campbell R. Harvey48 9. Results

49. Campbell R. Harvey49 9. Results Harvey and Siddique (2000, Journal of Finance) “Conditional Skewness in Asset Pricing Tests” find that skewness is able to explain one of the most puzzling anomalies in asset pricing: momentum

50. Campbell R. Harvey50 9. Results 12-month momentum

51. Campbell R. Harvey51 10. Conditional Skewness Bakshi, Harvey and Siddique (2002) examine the fundamental determinants of volatility, covariance, skewness and coskewness

52. Campbell R. Harvey52 10. Conditional Skewness

53. Campbell R. Harvey53 10. Conditional Skewness Skewness can be especially important in hedge fund strategies where derivatives play an explicit role in trading strategies

54. Campbell R. Harvey54 10. Conditional Skewness Source: Lu and Mulvey (2001)

55. Campbell R. Harvey55 10. Conditional Skewness Source: Lu and Mulvey (2001)

56. Campbell R. Harvey56 11. Three-Dimensional Analysis

57. Campbell R. Harvey57 12. Estimation Error Goal is the maximize expected utility (find the point on the frontier that best matches our utility) However, all the moments are estimated with error Traditional analysis does not take this estimation error into account

58. Campbell R. Harvey58 12. Estimation Error Small movements along the frontier can cause radical swings in weights

59. Campbell R. Harvey59 12. Estimation Error Popular “solutions” involve the resampling of the efficient frontier Basically, the step are: –(1) Calculate the means, variances and covariances –(2) Simulate data based on (1) –(3) Solve for efficient weights –(4) Repeat (2) and (3) many times –(5) Average the weights for each asset to get the “resampled” frontier, call it w*

60. Campbell R. Harvey60 12. Estimation Error However, the average of a set of maximums is not the maximum of an average The expected utility for w* will be less than the maximum expected utility Hence, current techniques are suboptimal

61. Campbell R. Harvey61 12. Estimation Error Harvey, Liechty, Liechty and Müller (2002) “Portfolio Selection with Higher Moments” provide an alternative approach –(1) Generate samples of parameters (means, etc) using a Bayesian estimation procedure –(2) Estimate expected utility –(3) Find weights that maximize expected utility

62. Campbell R. Harvey62 12. Estimation Error Harvey, Liechty, Liechty and Müller (2002) “Portfolio Selection with Higher Moments” provide an alternative approach –(4) For two moments, use Normal distribution –(5) For three moments, use Skew Normal distribution

63. Campbell R. Harvey63 12. Estimation Error

64. Campbell R. Harvey64 12. Estimation Error

65. Campbell R. Harvey65 12. Estimation Error

66. Campbell R. Harvey66 13. Conclusions Both conditioning information and higher moments matter People make portfolio choices based on “predictive” distributions – not necessarily what has happened in the past Investors have clear preference over skewness which needs to be incorporated into our portfolio selection methods

67. Campbell R. Harvey67 Readings “Distributional Characteristics of Emerging Market Returns and Asset Allocation," with Geert Bekaert, Claude B. Erb and Tadas E. Viskanta, Journal of Portfolio Management (1998), Winter,102-116. “Autoregressive Conditional Skewness,” with Akhtar Siddique, Journal of Financial and Quantitative Analysis 34, 4, 1999, 465-488. “Conditional Skewness in Asset Pricing Tests,” with Akhtar Siddique, Journal of Finance 55, June 2000, 1263-1295. “Time-Varying Conditional Skewness and the Market Risk Premium,” with Akhtar Siddique, Research in Banking and Finance 2000, 1, 27-60. “The Drivers of Expected Returns in International Markets,” Emerging Markets Quarterly 2000, 32-49. “Portfolio Selection with Higher Moments,” with John Liechty, Merrill Liechty, and Peter Müller, Working paper. “Fundamental Risk,” with Gurdip Bakshi and Akhtar Siddique, Working paper. Nan Q. Lu and John M. Mulvey, “Analyses of Market Neutral Hedge Fund Returns” ORFE-01-1, Princeton University