portfolio selection with higher moments campbell r. harvey duke university, durham, nc usa national...

Portfolio Selection with Higher Moments

Campbell R. HarveyDuke University, Durham, NC USA

National Bureau of Economic Research, Cambridge, MA USA

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

Inquire UK Autumn Seminar

22-24 September 2002 Royal Bath Hotel, Bournemouth

Campbell R. Harvey 2

1. Objectives

• The asset allocation setting

• What is risk?

• Conditional versus unconditional risk

• The importance of higher moments

• Estimation error

• New research frontiers


2. Modes/Inputs of Asset Allocation

• Types of asset allocation– Strategic– Tactical

• Type of information– Unconditional– Conditional


Strategic Tactical

Unconditional Conditional

Slow evolvingweights

Dynamicweights

Constantweights




• Conditioning information makes a difference


3. Performance Depends on Business Cycle

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Expansion geometric mean Recession geometric mean

Average Annual Returns During U.S. Business Cycle Phases

Data through June 2002



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Expansion std.dev. Recession std.dev.

Average Annual Volatility During U.S. Business Cycle Phases




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Expansion correlation with US Recession correlation with US

Correlations During U.S. Business Cycle Phases




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Expansion covariance with US Recession covariance with US

Covariances During U.S. Business Cycle Phases



4. Conditioning Information and Portfolio Analysis

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



Er

Vol

Traditional fixed weightoptimization (contrarian)in 2-dimensional setting



Er

Vol

Add conditioninginformation and weightschange through time. Frontier shifts.


5. What is Risk?

• Traditional models maximize expected returns for some level of volatility

• Is volatility a complete measure of risk?


5. What is Risk?

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


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:


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


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


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


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!]


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


7. Skewness

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7. Skewness

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7. Skewness

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Variance

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7. Skewness

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Variance

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Expected Return

RF


7. Skewness

05 10 15

Variance

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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)



• 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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Average Skewness in Developed Markets




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Compo

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Average Skewness in Emerging Markets





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Average Excess Kurtosis in Developed Markets




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Average Excess Kurtosis in Emerging Markets


8. Factors

1. SR (systematic risk) is the beta, i in the simple CAPM equation

2. TR (total risk) is the standard deviation of country return i

3. IR (idiosyncratic risk) is the standard deviation of the residual in simple CAPM, eit

Related to simple CAPM: Rit – rft = i + i[Rmt – rft] + eit


8. Factors

4. Log market capitalization

Related to size


8. Factors

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

Related to semi-standard deviation:Semi-B =, for all Rt < B

T

t t BRT1

2)()/1(


8. Factors

8. Down-iw is the coefficient from market model using observations when country returns and world returns are simultaneously negative.

9. Down-w is the coefficient from market model using observations when world returns negative.

Related to downside beta


8. Factors

10. VaR is a value at risk measure. It is the simple average of returns below the 5th percentile level.

Related to value at risk


8. Factors

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

Related to skewness


8. Factors

13. Coskew1 is: ( ei * em

2)/T {square root of ((ei

2 )/T)) } * {( em2)/T)}

14. Coskew2 is: ( ei * em

2)/T {standard deviation of (em)}^3

Related to coskewness


8. Factors

15. Kurt is the kurtosis of the return distribution

Related to spread


8. Factors

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.

Related to political risk


8. Factors

19. betahml - HML

20. betasmb - SMB

Related to Fama-French 3-factor model


8. Factors

21. betaoil - Oil Price (Change in Brent index)

22. binfl - Weighted average of G7 inflation using

GDP deflator.

Related to commodity prices and inflation


8. Factors

23. betafx - The trade weighted FX to $ given by the Federal Reserve

24. betafx1- Simple average $ -Euro and $-Yen

Related to FX risk


8. Factors

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

Related to Interest Rates


8. Factors

27. betaip - OECD G7 industrial production

Related to Economic Activity


9. Results

y = -0.8121x + 0.8964

R2 = 0.118

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etur

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9. Results

y = 0.1586x + 0.3226

R2 = 0.1673

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9. Results Multiple Regressions - All Markets

Risk1 / Risk2 c0 p-value c1 p-value c2 p-value R2

SR / TR -0.3051 0.2370 0.4638 0.0540 0.1054 0.0000 0.470SR / IR -0.2491 0.3280 0.6110 0.0080 0.0944 0.0000 0.459SR / Size 0.3817 0.2400 0.9780 0.0000 -0.0451 0.5150 0.278SR / Semi - Mean -0.2923 0.2900 0.4350 0.0970 0.1610 0.0010 0.427

SR / Semi - rf -0.1101 0.7130 0.6483 0.0200 0.1114 0.0460 0.335

SR / Semi-0 -0.0878 0.7640 0.6460 0.0210 0.1119 0.0450 0.335

SR / Down-iw 0.0555 0.8360 0.6375 0.0320 0.4776 0.0840 0.320

SR / Down-w 0.1791 0.4770 0.5493 0.1210 0.3252 0.1300 0.309

SR / VAR -0.0937 0.7440 0.5945 0.0360 -0.0371 0.0320 0.344SR / skew 0.0580 0.8020 0.8795 0.0000 0.5256 0.0010 0.427SR / skew5% 0.0881 0.7430 1.0000 0.0000 1.0737 0.1310 0.308SR / coskew1 0.0873 0.7460 0.9747 0.0000 -1.0720 0.1380 0.307SR / coskew2 -0.0099 0.9680 0.9638 0.0000 -0.8360 0.0040 0.396SR / kurt -0.3296 0.2950 0.8553 0.0000 0.1302 0.0080 0.381SR / ICRGC 7.5150 0.0640 0.9550 0.0000 -1.6895 0.0720 0.323SR / CCR 2.6242 0.0270 0.9561 0.0000 -0.5971 0.0400 0.339SR / ICRGP 2.5330 0.4660 0.9618 0.0000 -0.5372 0.5100 0.278


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


9. Results

y = -5.3067x + 24.869

R2 = 0.5934

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Skew

Mea

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12-month momentum


10. Conditional Skewness

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


10. Conditional SkewnessFor 1996

0. 7390. 686

0. 6340. 581

0. 5280. 475

0. 4220. 370

0. 3170. 264

0. 2110. 158

0. 1060. 053

0. 000

book_mkt

11. 4510. 59

9. 748. 88

8. 027. 16

6. 315. 45

4. 593. 73

2. 882. 02

1. 160. 31

- 0. 55

l ogs i ze

f 5s kew

- 7. 00

- 5. 44

- 3. 89

- 2. 33

- 0. 78

0. 78

2. 33

3. 89

5. 44

7. 00



• Skewness can be especially important in hedge fund strategies where derivatives play an explicit role in trading strategies



Co-Skewness Measure (Definition 2)(Total of 42 Funds, over Jan 1997 - Feb 2001)

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Coskewness

Mea

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(Geo

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ric)

Source: Lu and Mulvey (2001)



Co-Skewness Measure (Definition 2)(Total of 42 Funds, over Jan 1997 - Feb 2001)

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Coskewness

Mea

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etu

rns

(Ari

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etic

)

Source: Lu and Mulvey (2001)


11. Three-Dimensional Analysis


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



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



• 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*



• 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



• 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



• 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



Results Using Multivariate Normal

Max Expected Utility using

Normal

Expected Utility using Max Utility

weights from Normal

Expected Utility using Michaud

and Normal

0.00 0.154 0.154 0.150 0.03 0.009 0.009 0.008 0.06 -0.125 -0.125 -0.128

utility = – *2



Results Using Multivariate Skewed Normal

Max Expected Utility using

Skewed Normal

Expected Utility using Michaud

and Skewed Normal

Expected Utility using Max

Utility weights from Normal

Expected Utility using Michaud weights from

Normal 0.00 0.01 0.122 0.119 0.122 0.109 0.00 0.50 0.106 0.103 0.074 0.050 0.03 0.01 0.003 0.002 0.000 -0.004 0.03 0.50 0.004 -0.007 -0.001 -0.006 0.06 0.01 -0.132 -0.134 -0.135 -0.141 0.06 0.50 -0.132 -0.138 -0.136 -0.143

utility = – *2 + * I{ >0} + 2** I{ < 0}

is skewness and I{ } is an indicator function, I{ < 0} = 1, if < 0 and 0, if >= 0.


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


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

portfolio selection with higher moments campbell r. harvey duke university, durham, nc usa national...

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