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

2. 1. Objectives The asset allocation setting What is risk?
Conditional versus unconditional risk
The importance of higher moments
Estimation error
New research frontiers
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3. 2. Modes/Inputs of Asset Allocation
Types of asset allocation
Strategic
Tactical
Type of information
Unconditional
Conditional
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4. 2. Modes/Inputs of Asset Allocation
Constant
weights
Dynamic
weights
Slow evolving
weights
Strategic
Tactical
Unconditional
Conditional
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5. 2. Modes/Inputs of Asset Allocation
Conditioning information makes a difference
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6. 3. Performance Depends on Business Cycle
Data through June 2002
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7. 3. Performance Depends on Business Cycle
Data through June 2002
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8. 3. Performance Depends on Business Cycle
Data through June 2002
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9. 3. Performance Depends on Business Cycle
Data through June 2002
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10. 4. Conditioning Information and Portfolio Analysis
Adding conditioning information is like adding extra assets to an optimization
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11. 4. Conditioning Information and Portfolio AnalysisEr
Traditional fixed weight
optimization (contrarian)
in 2-dimensional setting
Vol
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12. 4. Conditioning Information and Portfolio AnalysisEr
Add conditioning
information and weights
change through time.
Frontier shifts.
Vol
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13. 5. What is Risk?
Traditional models maximize expected returns for some level of volatility
Is volatility a complete measure of risk?
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14. 5. What is Risk?
Much interest in downside risk, asymmetric volatility, semi-variance, extreme value analysis, regime-switching, jump processes, ...
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15. 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:
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16. 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
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17. 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
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18. 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
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19. 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!]
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20. 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
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21. 7. Skewness
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22. 7. Skewness
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23. 7. Skewness
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24. 7. Skewness
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25. 7. Skewness
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26. 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)
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27. 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
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28. 7. Higher Moments & Expected Returns
Data through June 2002
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29. 7. Higher Moments & Expected Returns
Data through June 2002
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30. 7. Higher Moments & Expected Returns
Data through June 2002
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31. 7. Higher Moments & Expected Returns
Data through June 2002
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32. 8. Factors
Related to simple CAPM: Rit – rft = ai + bi[Rmt – rft] + eit
1. SR (systematic risk) is the beta, bi in the simple CAPM equation
2. TR (total risk) is the standard deviation of country return si
3. IR (idiosyncratic risk) is the standard deviation of the residual in simple CAPM, eit
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33. 8. Factors 4. Log market capitalization Related to size
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34. 8. Factors
Related to semi-standard deviation:
Semi-B =, for all Rt < B
5. Semi-Mean is the semi-standard deviation with B = average returns for the market
6. Semi-rf is the semi-standard deviation with B = U.S. risk free rate
7. Semi-0 is the semi-standard deviation with B = 0
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35. 8. Factors
Related to downside beta
8. Down-biw is the b coefficient from market model using observations when country returns and world returns are simultaneously negative.
9. Down-bw is the b coefficient from market model using observations when world returns negative.
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36. 8. Factors
Related to value at risk
10. VaR is a value at risk measure. It is the simple average of returns below the 5th percentile level.
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37. 8. Factors
Related to skewness
11. Skew is the unconditional skewness of returns. It is calculated by taking the
Mean(ei3)
{Standard deviation of (ei)}^3
12. Skew5%:
{(return at the 95th percentile – mean return) -(return at 5th percentile level – mean return)} - 1
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38. 8. Factors 13. Coskew1 is: (S ei * em2)/T
Related to coskewness
13. Coskew1 is:
(S ei * em2)/T
{square root of (S(ei2 )/T)) } * {(S em2)/T)}
14. Coskew2 is:
{standard deviation of (em)}^3
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39. 8. Factors 15. Kurt is the kurtosis of the return distribution
Related to spread
15. Kurt is the kurtosis of the return distribution
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40. 8. Factors
Related to political risk
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.
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41. 8. Factors 19. betahml - HML 20. betasmb - SMB
Related to Fama-French 3-factor model
19. betahml - HML
20. betasmb - SMB
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42. 8. Factors 21. betaoil - Oil Price (Change in Brent index)
Related to commodity prices and inflation
21. betaoil - Oil Price (Change in Brent index)
22. binfl - Weighted average of G7 inflation using
GDP deflator.
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43. 8. Factors
Related to FX risk
23. betafx - The trade weighted FX to $ given by the Federal Reserve
24. betafx1- Simple average $ -Euro and $-Yen
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44. 8. Factors
Related to Interest Rates
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
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45. 8. Factors 27. betaip - OECD G7 industrial production
Related to Economic Activity
27. betaip - OECD G7 industrial production
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46. 9. Results
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47. 9. Results
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48. 9. Results
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49. 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
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50. 9. Results
12-month momentum
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51. 10. Conditional Skewness
Bakshi, Harvey and Siddique (2002) examine the fundamental determinants of volatility, covariance, skewness and coskewness
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52. 10. Conditional Skewness
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53. 10. Conditional Skewness
Skewness can be especially important in hedge fund strategies where derivatives play an explicit role in trading strategies
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54. 10. Conditional Skewness Source: Lu and Mulvey (2001)
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55. 10. Conditional Skewness Source: Lu and Mulvey (2001)
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56. 11. Three-Dimensional Analysis
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57. 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
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58. 12. Estimation Error
Small movements along the frontier can cause radical swings in weights
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59. 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*
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60. 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
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61. 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
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62. 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
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63. 12. Estimation Error
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64. 12. Estimation Error
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65. 12. Estimation Error
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66. 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
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67. 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,
“Autoregressive Conditional Skewness,” with Akhtar Siddique, Journal of Financial and Quantitative Analysis 34, 4, 1999,
“Conditional Skewness in Asset Pricing Tests,” with Akhtar Siddique, Journal of Finance 55, June 2000,
“Time-Varying Conditional Skewness and the Market Risk Premium,” with Akhtar Siddique, Research in Banking and Finance 2000, 1,
“The Drivers of Expected Returns in International Markets,” Emerging Markets Quarterly 2000,
“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
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