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Managing Higher Moments in Hedge Fund Allocation
Campbell R. HarveyDuke University, Durham, NC USA
National Bureau of Economic Research, Cambridge, MA USA
http://www.duke.edu/~charvey
Boston College June 11, 2004
Campbell R. Harvey 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
Source: Shadwick and Keating (2003)
Campbell R. Harvey 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
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
Campbell R. Harvey 21
Asset Mix-Large Endowments Versus the Average FundJune 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)
Campbell R. Harvey 22
Selected Endowment Asset MixesJune 2003
% of EndowmentHarvard 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 Return12.2 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 FundPeriods ending 6/30/2003
1 year 3 years 5 years 10 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
-4-3-2-101234567
1 2 3 4 5
S&P Quintiles
S&P 500Global Macro
Source: Naik (2003)
Campbell R. Harvey 27
3. Motivation
-8
-6
-4
-2
0
2
4
6
8
1 2 3 4 5
S&P Quintiles
S&P 500Trend Followers
Source: Naik (2003)
Campbell R. Harvey 28
3. Motivation
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1 2 3 4 5
Credit spread quintiles
Delta(BAA-10yTBond)x10FI Arb
Source: Naik (2003)
Campbell R. Harvey 29
3. Motivation
Event Driven Index: Exposure to Russell 3000 Index (Period: Jan '90 to Dec '91)
-8
-6
-4
-2
0
2
4
6
-15 -10 -5 0 5 10
Russell 3000 Index Returns
Eve
nt D
riven
Inde
x R
etur
ns
LOWESS fit
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
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Average Returns: January 1995-April 2004
Source: HFR
Campbell R. Harvey 33
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
5.0%
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Volatility: January 1995-April 2004
Source: HFR
Campbell R. Harvey 34
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Skewness: January 1995-April 2004
Source: HFR
Campbell R. Harvey 35
0.0
5.0
10.0
15.0
20.0
25.0
30.0
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Kurtosis: January 1995-April 2004
Source: HFR
Campbell R. Harvey 36
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Coskewness: January 1995-April 2004
Source: HFR
Campbell R. Harvey 37
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Beta market: January 1995-April 2004
Source: HFR
Campbell R. Harvey 38
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Beta market (August 1998): January 1995-April 2004
Source: HFR
Campbell R. Harvey 39
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Beta chg. 10-yr: January 1995-April 2004
Source: HFR
Campbell R. Harvey 40
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Beta chg. slope: January 1995-April 2004
Source: HFR
Campbell R. Harvey 41
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Beta chg. spread: January 1995-April 2004
Source: HFR
Campbell R. Harvey 42
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
Beta SMB: January 1995-April 2004
Source: HFR
Campbell R. Harvey 43
-0.4
-0.3
-0.3
-0.2
-0.2
-0.1
-0.1
0.0
0.1
0.1
0.2
S&P500
Convert
ible
Arb
Distre
ssed
Emer
ging M
arkets
: Total
Emer
ging M
arkets
: Glo
bal
Equity H
edge
Equity M
arke
t Neu
tral
Mar
ket Neu
tral :
Stat
Arb
itrag
e
Equity N
on-H
edge
Event-D
riven
Fixed
Inco
me: Total
Fixed
Inco
me: Arb
itrag
e
Fixed
Inco
me: Con
verti
ble B
onds
Fixed
Inco
me: Dive
rsifie
d
Fixed
Inco
me: High
Yiel
d
Fixed
Inco
me: M
ortgag
e-Bac
ked
Mac
ro
Mer
ger Arb
itrag
e
Relativ
e Valu
e Arb
itrag
e
Fund W
eigted
Compos
ite
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
0
5
10
15
Variance
- 2
- 1
0
1
2
Skewness
5
7.5
10
12.5
Expected Return
0
5
10
15
Variance
Campbell R. Harvey 46
5. Rethinking Optimization
0
5
10
15
Variance
- 2
- 1
0
1
2
Skewness
5
7.5
10
12.5
Expected Return
RF
0
5
10
15
Variance
Campbell R. Harvey 47
5. Rethinking Optimization
0 5 10 15
Variance
- 2
- 1
0
1
2
Skewness
5
7.5
10
12.5
Expected Return
RF
0 5 10 15
Variance
- 2
- 1
0
1
2
Skewness
Campbell R. Harvey 48
5. Rethinking Optimization
05 10 15
Variance
- 2- 1012
Skewness
5
7.5
10
12.5
Expected Return
RF
05 10 15
Variance
5
7.5
10
12.5
Expected Return
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 51
6. Higher Moments & Expected Returns
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Australi
a
Austria
Belg
ium
Canad
a
Den
mar
k
Finlan
d
France
Ger
man
y
Hong K
ong
Irelan
d It
aly
Japan
Nether
lands
New
Zea
land
Norway
Portugal
Spain
Swed
en
Switzer
land UK US
World
World
ex-U
S
EAFE
Average Skewness in Developed Markets
Campbell R. Harvey 52
6. Higher Moments & Expected Returns
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Argen
tina
Bahrai
n
Brazil
Chile
China
Colombia
Czech
Rep
ublicEgy
pt
Greece
Hunga
ry
India
Indo
nesia
Israe
l
Jord
an
Korea
Mala
ysia
Mex
ico
Mor
occo
Nigeria
Oman
Pakist
an
Peru
Philipp
ines
Poland
Russia
Saudi
Arabia
Slovak
ia
South
Africa
Sri Lan
ka
Taiw
an
Thaila
nd
Turke
y
Venez
uela
Zimba
bwe
Compo
site
Average Skewness in Emerging Markets
Campbell R. Harvey 53
6. Higher Moments & Expected Returns
-1
0
1
2
3
4
5
6
Australi
a
Austria
Belg
ium
Canad
a
Den
mar
k
Finlan
d
France
Ger
man
y
Hong K
ong
Irelan
d It
aly
Japan
Nether
lands
New
Zea
land
Norway
Portugal
Spain
Swed
en
Switzer
land UK US
World
World
ex-U
S
EAFE
Average Excess Kurtosis in Developed Markets
Campbell R. Harvey 54
6. Higher Moments & Expected Returns
-1
0
1
2
3
4
5
6
Argen
tina
Bahrai
n
Brazil
Chile
China
Colombia
Czech
Rep
ublicEgy
pt
Greece
Hunga
ry
India
Indo
nesia
Israe
l
Jord
an
Korea
Mala
ysia
Mex
ico
Mor
occo
Nigeria
Oman
Pakist
an
Peru
Philipp
ines
Poland
Russia
Saudi
Arabia
Slovak
ia
South
Africa
Sri Lan
ka
Taiw
an
Thaila
nd
Turke
y
Venez
uela
Zimba
bwe
Compo
site
Average Excess Kurtosis in Emerging Markets
Campbell R. Harvey 55
7. New Metrics
• Old Sharpe Ratio= Excess return/vol
• Alternative = Excess return/[vol-adj(skew)]
• Alternative = alpha from 3-moment CAPM
Campbell R. Harvey 56
7. New Metrics
• Traditional Markowitz optimization over mean and variance
• New optimization over mean, variance and skewness
Campbell R. Harvey 57
8. Implementation
• Harvey, Liechty, Liechty and Müller (2004) “Portfolio Selection with Higher Moments”
Campbell R. Harvey 58
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. Harvey 59
9. Conclusions
• While we have only talked about average risk, it is likely that the risk (including skewness) changes through time
Campbell R. Harvey 60
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.
