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Managing Higher Moments in Hedge Fund Allocation

Boston College June 11, 2004. Managing Higher Moments in Hedge Fund Allocation. Campbell R. Harvey Duke University, Durham, NC USA National Bureau of Economic Research, Cambridge, MA USA http://www.duke.edu/~charvey. 1. Objectives. Framework The importance of higher moments

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Managing Higher Moments in Hedge Fund Allocation

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  1. Boston College June 11, 2004 Managing Higher Moments in Hedge Fund Allocation Campbell R. Harvey Duke University, Durham, NC USA National Bureau of Economic Research, Cambridge, MA USA http://www.duke.edu/~charvey

  2. 1. Objectives • Framework • The importance of higher moments • Rethinking risk • Characteristics of hedge fund returns • Rethinking optimization • Skewness and expected returns • Implementation • Conclusions Campbell R. Harvey

  3. 2. Framework Markowitz (1952) Stage 1: • “...starts with observation and experience and ends with beliefs about the future performances of available securities” Campbell R. Harvey

  4. 2. Framework Markowitz (1952) Stage 2: • “...starts with relevant beliefs and ends with the selection of a portfolio” • Markowitz only dealt with Stage 2 in context of the famous mean-variance framework Campbell R. Harvey

  5. 2. Framework Markowitz (1952) Important caveat, p.90-91: • If preferences depend on mean and variance, an investor “will never accept an actuarially fair bet.” Campbell R. Harvey

  6. 2. Framework Markowitz (1952) Important caveat, p.90-91: • If preferences also depend skewness, an investor “then there some fair bets which would be accepted.” Campbell R. Harvey

  7. 3. Motivation 50 years later, we have learned: • Investors have an obvious preference for skewness • Returns (or log returns) are non-normal Campbell R. Harvey

  8. 3. Motivation Campbell R. Harvey Source: Shadwick and Keating (2003)

  9. 3. Motivation Preferences: 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

  10. 3. Motivation Preferences: 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

  11. 3. Motivation Preferences: 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

  12. 3. Motivation Preferences: 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

  13. 3. Motivation Preferences: 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

  14. 3. Motivation Returns: • Crisis events such as August 1998 • Scholes (AER 2000, p.19) notes: • “This 20-basis point change was a move of 10 standard deviations in the swap spread.” Campbell R. Harvey

  15. 3. Motivation Returns: • 10 standard deviation move has a probability of 10-24 -- under a normal distribution Campbell R. Harvey

  16. 3. Motivation Returns: • 10 standard deviation move has a probability of 10-24 -- under a normal distribution • Roughly the probability of winning the Powerball Lottery ... Campbell R. Harvey

  17. 3. Motivation Returns: • 10 standard deviation move has a probability of 10-24 -- under a normal distribution • Roughly the probability of winning the Powerball Lottery ... 3 consecutive times! • (See Routledge and Zin (2003)) Campbell R. Harvey

  18. 3. Motivation Returns: • The most unlikely arena to see normally distributed returns is the hedge fund industry • Use of derivatives, derivative replicating strategies, and leverage make the returns non-normal Campbell R. Harvey

  19. 3. Motivation Returns: • Consider an excerpt from a presentation of one of the largest endowments in the U.S. from March 2004 Campbell R. Harvey

  20. The Evolution of Large Endowment Asset Mixes % of Total Portfolio 198819911994199720002003 • US Equity 45.6 45.9 40.1 39.4 32.4 24.8 • Non-US Equity 3.1 6.0 13.5 14.8 13.5 13.6 • Hedge Funds .7 2.0 6.4 8.8 11.7 24.0 • Non-Marketable 3.8 5.3 6.2 7.1 18.7 12.6 • Bonds 33.0 32.0 25.5 20.2 16.6 17.2 • Real Estate 2.9 3.2 3.3 5.4 4.7 6.2 Campbell R. Harvey

  21. Asset Mix-Large Endowments Versus the Average Fund June 2003 % of Portfolio Large Average EndowmentsEndowment • US Equity 24.8 49.0 • Non-US Equity 13.6 8.2 • Hedge Funds 24.0 6.1 • Non-Marketable 12.6 4.1 • Bonds 17.2 25.8 • Real Estate 6.2 2.8 • Cash 1.6 4.0 “Traditional” 43.6 78.8 (US stocks, bonds, cash) Campbell R. Harvey

  22. Selected Endowment Asset Mixes June 2003 % of Endowment HarvardYaleVirginia • US Equity 18.4 15.1 6.2 • Non-US Equity 19.6 14.8 5.8 • Hedge Funds 54.7 • Private Equity 8.6 15.2 13.1 • Equity and Related 46.6 45.1 79.8 • Real Estate 5.1 13.1 2.8 • Natural Resources 5.8 6.9 2.8 • Commodities 3.8 • TIPS 6.7 7.7 • Inflation hedges 21.4 20.0 13.3 • Absolute Return 12.2 25.2 6.3 • Bonds 24.7 7.5 0 • Cash -4.9 2.2 .6 • Total Fixed 19.8 9.7 .6 Campbell R. Harvey

  23. Endowment Returns by Size of Fund Periods ending 6/30/2003 1 year3 years5 years10 years • > $1 billion 4.1 -.7 6.9 11.5 • $501mm to $1b 2.9 -2.3 3.9 9.3 • $101mm to $500mm 2.7 -2.4 3.1 8.8 • $51mm to $100mm 2.7 -2.8 2.1 8.1 • $26mm to $50mm 3.1 -2.3 2.4 8.1 • Less than $25mm 3.5 -2.3 2.2 7.2 Campbell R. Harvey

  24. 3. Motivation Manager explained the following fact: • “If I use the same expected returns as in 1994 and add the hedge fund asset class, the optimized portfolio mix tilts to hedge funds. The Sharpe Ratio of my portfolio goes up.” Campbell R. Harvey

  25. 3. Motivation Manager’s “optimization” based on traditional Markowitz mean and variance. • Does this make sense? Campbell R. Harvey

  26. 3. Motivation Source: Naik (2003) Campbell R. Harvey

  27. 3. Motivation Source: Naik (2003) Campbell R. Harvey

  28. 3. Motivation Source: Naik (2003) Campbell R. Harvey

  29. 3. Motivation Source: Naik (2003) Campbell R. Harvey

  30. 4. Rethinking Risk • Much interest in downside risk, asymmetric volatility, semi-variance, extreme value analysis, regime-switching, jump processes, ... Campbell R. Harvey

  31. 4. Rethinking Risk • …all related to skewness • Harvey and Siddique, “Conditional Skewness in Asset Pricing Tests” Journal of Finance 2000. Campbell R. Harvey

  32. Average Returns: January 1995-April 2004 Campbell R. Harvey Source: HFR

  33. Volatility: January 1995-April 2004 Source: HFR Campbell R. Harvey

  34. Skewness: January 1995-April 2004 Source: HFR Campbell R. Harvey

  35. Kurtosis: January 1995-April 2004 Source: HFR Campbell R. Harvey

  36. Coskewness: January 1995-April 2004 Source: HFR Campbell R. Harvey

  37. Beta market: January 1995-April 2004 Source: HFR Campbell R. Harvey

  38. Beta market (August 1998): January 1995-April 2004 Source: HFR Campbell R. Harvey

  39. Beta chg. 10-yr: January 1995-April 2004 Source: HFR Campbell R. Harvey

  40. Beta chg. slope: January 1995-April 2004 Source: HFR Campbell R. Harvey

  41. Beta chg. spread: January 1995-April 2004 Source: HFR Campbell R. Harvey

  42. Beta SMB: January 1995-April 2004 Source: HFR Campbell R. Harvey

  43. Beta HML: January 1995-April 2004 Source: HFR Campbell R. Harvey

  44. 5. Rethinking Optimization • Move to three dimensions: mean-variance-skewness • Relatively new idea in equity management but old one in fixed income management Campbell R. Harvey

  45. 5. Rethinking Optimization Campbell R. Harvey

  46. 5. Rethinking Optimization Campbell R. Harvey

  47. 5. Rethinking Optimization Campbell R. Harvey

  48. 5. Rethinking Optimization Campbell R. Harvey

  49. 6. 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

  50. 6. 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

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