Managing Higher Moments in Hedge Fund Allocation Campbell R. Harvey Duke University, Durham, NC USA National Bureau of Economic Research, Cambridge, MA USA Boston College June 11, 2004

Campbell R. Harvey2 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. Harvey3 2. Framework Markowitz (1952) Stage 1: “...starts with observation and experience and ends with beliefs about the future performances of available securities”

Campbell R. Harvey4 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. Harvey5 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. Harvey6 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. Harvey7 3. Motivation 50 years later, we have learned: Investors have an obvious preference for skewness Returns (or log returns) are non-normal

Campbell R. Harvey8 3. Motivation Source: Shadwick and Keating (2003)

Campbell R. Harvey9 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. Harvey10 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. Harvey11 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. Harvey12 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. Harvey13 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. Harvey14 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. Harvey15 3. Motivation Returns: 10 standard deviation move has a probability of under a normal distribution

Campbell R. Harvey16 3. Motivation Returns: 10 standard deviation move has a probability of under a normal distribution Roughly the probability of winning the Powerball Lottery...

Campbell R. Harvey17 3. Motivation Returns: 10 standard deviation move has a probability of under a normal distribution Roughly the probability of winning the Powerball Lottery... 3 consecutive times! –(See Routledge and Zin (2003))

Campbell R. Harvey18 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. Harvey19 3. Motivation Returns: Consider an excerpt from a presentation of one of the largest endowments in the U.S. from March 2004

Campbell R. Harvey20 The Evolution of Large Endowment Asset Mixes % of Total Portfolio  US Equity  Non-US Equity  Hedge Funds  Non-Marketable  Bonds  Real Estate

Campbell R. Harvey21 Asset Mix-Large Endowments Versus the Average Fund June 2003 % of Portfolio LargeAverage Endowments Endowment  US Equity  Non-US Equity  Hedge Funds  Non-Marketable  Bonds  Real Estate  Cash “Traditional” (US stocks, bonds, cash)

Campbell R. Harvey22 Selected Endowment Asset Mixes June 2003 % of Endowment Harvard Yale Virginia  US Equity  Non-US Equity  Hedge Funds54.7  Private Equity  Equity and Related  Real Estate  Natural Resources  Commodities 3.8  TIPS  Inflation hedges  Absolute Return  Bonds  Cash  Total Fixed

Campbell R. Harvey23 Endowment Returns by Size of Fund Periods ending 6/30/ year3 years5 years10 years  > $1 billion  $501mm to $1b  $101mm to $500mm  $51mm to $100mm  $26mm to $50mm  Less than $25mm

Campbell R. Harvey24 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. Harvey25 3. Motivation Manager’s “optimization” based on traditional Markowitz mean and variance. Does this make sense?

Campbell R. Harvey26 3. Motivation Source: Naik (2003)

Campbell R. Harvey27 3. Motivation Source: Naik (2003)

Campbell R. Harvey28 3. Motivation Source: Naik (2003)

Campbell R. Harvey29 3. Motivation Source: Naik (2003)

Campbell R. Harvey30 4. Rethinking Risk Much interest in downside risk, asymmetric volatility, semi-variance, extreme value analysis, regime-switching, jump processes,...

Campbell R. Harvey31 4. Rethinking Risk …all related to skewness Harvey and Siddique, “Conditional Skewness in Asset Pricing Tests” Journal of Finance 2000.

Campbell R. Harvey32 Average Returns: January 1995-April 2004 Source: HFR

Campbell R. Harvey33 Volatility: January 1995-April 2004 Source: HFR

Campbell R. Harvey34 Skewness: January 1995-April 2004 Source: HFR

Campbell R. Harvey35 Kurtosis: January 1995-April 2004 Source: HFR

Campbell R. Harvey36 Coskewness: January 1995-April 2004 Source: HFR

Campbell R. Harvey37 Beta market: January 1995-April 2004 Source: HFR

Campbell R. Harvey38 Beta market (August 1998): January 1995-April 2004 Source: HFR

Campbell R. Harvey39 Beta chg. 10-yr: January 1995-April 2004 Source: HFR

Campbell R. Harvey40 Beta chg. slope: January 1995-April 2004 Source: HFR

Campbell R. Harvey41 Beta chg. spread: January 1995-April 2004 Source: HFR

Campbell R. Harvey42 Beta SMB: January 1995-April 2004 Source: HFR

Campbell R. Harvey43 Beta HML: January 1995-April 2004 Source: HFR

Campbell R. Harvey44 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. Harvey45 5. Rethinking Optimization

Campbell R. Harvey46 5. Rethinking Optimization

Campbell R. Harvey47 5. Rethinking Optimization

Campbell R. Harvey48 5. Rethinking Optimization

Campbell R. Harvey49 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. Harvey50 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. Harvey51 6. Higher Moments & Expected Returns

Campbell R. Harvey52 6. Higher Moments & Expected Returns

Campbell R. Harvey53 6. Higher Moments & Expected Returns

Campbell R. Harvey54 6. Higher Moments & Expected Returns

Campbell R. Harvey55 7. New Metrics Old Sharpe Ratio= Excess return/vol Alternative = Excess return/[vol-adj(skew)] Alternative = alpha from 3-moment CAPM

Campbell R. Harvey56 7. New Metrics Traditional Markowitz optimization over mean and variance New optimization over mean, variance and skewness

Campbell R. Harvey57 8. Implementation Harvey, Liechty, Liechty and Müller (2004) “Portfolio Selection with Higher Moments”

Campbell R. Harvey58 9. Conclusions Data not normal – especially hedge fund returns Investors have clear preference over skewness which needs to be incorporated into our portfolio selection methods – and performance evaluation Remember Markowitz’s “two stages”. Ex ante skewness is difficult to measure.

Campbell R. Harvey59 9. Conclusions While we have only talked about average risk, it is likely that the risk (including skewness) changes through time

Campbell R. Harvey60 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, 2004.