Portfolio Selection with Higher Moments

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Portfolio Selection with Higher Moments
Inquire UK Autumn Seminar 22-24 September
2002 Royal Bath Hotel, Bournemouth

• 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

• 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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2. Modes/Inputs of Asset Allocation

• Types of asset allocation
• Strategic
• Tactical
• Type of information
• Unconditional
• Conditional

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

• Conditioning information makes a difference

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3. Performance Depends on Business Cycle
Data through June 2002
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3. Performance Depends on Business Cycle
Data through June 2002
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3. Performance Depends on Business Cycle
Data through June 2002
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3. Performance Depends on Business Cycle
Data through June 2002
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4. Conditioning Information and Portfolio Analysis

• Adding conditioning information is like adding
  extra assets to an optimization

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4. Conditioning Information and Portfolio Analysis
Er
Traditional fixed weight optimization
(contrarian) in 2-dimensional setting
Vol
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4. Conditioning Information and Portfolio Analysis
Er
Add conditioning information and weights change
through time. Frontier shifts.
Vol
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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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5. What is Risk?

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

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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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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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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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6. Skewness

• 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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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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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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7. Skewness
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7. Skewness
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7. Skewness
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7. Skewness
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7. Skewness
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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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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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7. Higher Moments Expected Returns
Data through June 2002
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7. Higher Moments Expected Returns
Data through June 2002
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7. Higher Moments Expected Returns
Data through June 2002
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7. Higher Moments Expected Returns
Data through June 2002
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8. Factors
Related to simple CAPM Rit rft ai biRmt
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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8. Factors
Related to size

• 4. Log market capitalization

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8. Factors
Related to semi-standard deviation Semi-B ,
for all Rt lt 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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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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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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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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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

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8. Factors
Related to spread

• 15. Kurt is the kurtosis of the return
  distribution

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8. Factors
Related to political risk

• 16. ICRGC is the log of the average monthly
  International Country Risk Guides (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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8. Factors
Related to Fama-French 3-factor model

• 19. betahml - HML
• 20. betasmb - SMB

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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.

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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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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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8. Factors
Related to Economic Activity

• 27. betaip - OECD G7 industrial production

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9. Results
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9. Results
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9. Results
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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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9. Results
12-month momentum
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10. Conditional Skewness

• Bakshi, Harvey and Siddique (2002) examine the
  fundamental determinants of volatility,
  covariance, skewness and coskewness

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10. Conditional Skewness
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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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10. Conditional Skewness
Source Lu and Mulvey (2001)
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10. Conditional Skewness
Source Lu and Mulvey (2001)
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11. Three-Dimensional Analysis
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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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12. Estimation Error

• Small movements along the frontier can cause
  radical swings in weights

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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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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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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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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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12. Estimation Error
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12. Estimation Error
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12. Estimation Error
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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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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.
• 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