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 http://www.duke.edu/~charvey

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

2. Modes/Inputs of Asset Allocation • Types of asset allocation • Strategic • Tactical • Type of information • Unconditional • Conditional Campbell R. Harvey

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

2. Modes/Inputs of Asset Allocation • Conditioning information makes a difference Campbell R. Harvey

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

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

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

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

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

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

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: Campbell R. Harvey

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 Campbell R. Harvey

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 Campbell R. Harvey

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 Campbell R. Harvey

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!] Campbell R. Harvey

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 Campbell R. Harvey

7. Skewness Campbell R. Harvey

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) Campbell R. Harvey

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 Campbell R. Harvey

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

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 Campbell R. Harvey

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

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 Campbell R. Harvey

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. Campbell R. Harvey

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. Campbell R. Harvey

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 Campbell R. Harvey

8. Factors Related to coskewness 13. Coskew1 is: (S ei * em2)/T {square root of (S(ei2 )/T)) } * {(S em2)/T)} 14. Coskew2 is: (S ei * em2)/T {standard deviation of (em)}^3 Campbell R. Harvey

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

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. Campbell R. Harvey

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

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

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 Campbell R. Harvey

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 Campbell R. Harvey

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

9. Results Campbell R. Harvey

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 Campbell R. Harvey

9. Results 12-month momentum Campbell R. Harvey

Portfolio Selection with Higher Moments