Managing Higher Moments in Hedge Fund Allocation

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Managing Higher Moments in Hedge Fund Allocation
Boston College June 11, 2004

• Campbell R. Harvey
• Duke University, Durham, NC USA
• National Bureau of Economic Research, Cambridge,
  MA USA
• http//www.duke.edu/charvey

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1. Objectives

• Framework
• The importance of higher moments
• Rethinking risk
• Characteristics of hedge fund returns
• Rethinking optimization
• Skewness and expected returns
• Implementation
• Conclusions

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2. Framework

• Markowitz (1952)
• Stage 1
• ...starts with observation and experience and
  ends with beliefs about the future performances
  of available securities

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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

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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.

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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.

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3. Motivation

• 50 years later, we have learned
• Investors have an obvious preference for skewness
• Returns (or log returns) are non-normal

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3. Motivation
Source Shadwick and Keating (2003)
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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

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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

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3. Motivation

• Preferences
• 2. High implied vol in out of the money SP 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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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!

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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

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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.

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3. Motivation

• Returns
• 10 standard deviation move has a probability of
  10-24 -- under a normal distribution

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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 ...

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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))

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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

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3. Motivation

• Returns
• Consider an excerpt from a presentation of one of
  the largest endowments in the U.S. from March
  2004

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• The Evolution of Large Endowment Asset Mixes
• of Total Portfolio
• 1988 1991 1994 1997 2000 2003
• 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

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• Asset Mix-Large Endowments Versus the Average
  Fund
• June 2003
• of Portfolio
• Large Average
• Endowments Endowment
• 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)

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• Selected Endowment Asset Mixes
• June 2003
• of Endowment
• Harvard Yale Virginia
• 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

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• Endowment Returns by Size of Fund
• Periods ending 6/30/2003
• 1 year 3 years 5 years 10 years
• gt 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

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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.

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3. Motivation

• Managers optimization based on traditional
  Markowitz mean and variance.
• Does this make sense?

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3. Motivation
Source Naik (2003)
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3. Motivation
Source Naik (2003)
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3. Motivation
Source Naik (2003)
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3. Motivation
Source Naik (2003)
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4. Rethinking Risk

• Much interest in downside risk, asymmetric
  volatility, semi-variance, extreme value
  analysis, regime-switching, jump processes, ...

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4. Rethinking Risk

• all related to skewness
• Harvey and Siddique, Conditional Skewness in
  Asset Pricing Tests Journal of Finance 2000.

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Average Returns January 1995-April 2004
Source HFR
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Volatility January 1995-April 2004
Source HFR
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Skewness January 1995-April 2004
Source HFR
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Kurtosis January 1995-April 2004
Source HFR
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Coskewness January 1995-April 2004
Source HFR
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Beta market January 1995-April 2004
Source HFR
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Beta market (August 1998) January 1995-April
2004
Source HFR
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Beta chg. 10-yr January 1995-April 2004
Source HFR
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Beta chg. slope January 1995-April 2004
Source HFR
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Beta chg. spread January 1995-April 2004
Source HFR
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Beta SMB January 1995-April 2004
Source HFR
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Beta HML January 1995-April 2004
Source HFR
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5. Rethinking Optimization

• Move to three dimensions mean-variance-skewness
• Relatively new idea in equity management but old
  one in fixed income management

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5. Rethinking Optimization
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5. Rethinking Optimization
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5. Rethinking Optimization
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5. Rethinking Optimization
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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)

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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

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6. Higher Moments Expected Returns
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6. Higher Moments Expected Returns
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6. Higher Moments Expected Returns
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6. Higher Moments Expected Returns
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7. New Metrics

• Old Sharpe Ratio Excess return/vol
• Alternative Excess return/vol-adj(skew)
• Alternative alpha from 3-moment CAPM

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7. New Metrics

• Traditional Markowitz optimization over mean and
  variance
• New optimization over mean, variance and skewness

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8. Implementation

• Harvey, Liechty, Liechty and Müller (2004)
  Portfolio Selection with Higher Moments

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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 Markowitzs two stages. Ex ante
  skewness is difficult to measure.

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9. Conclusions

• While we have only talked about average risk, it
  is likely that the risk (including skewness)
  changes through time

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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, 2004.