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![Page 1: Portfolio Selection with Higher Moments Campbell R. Harvey Duke University, Durham, NC USA National Bureau of Economic Research, Cambridge, MA USA charvey.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d0c5503460f949e0d9f/html5/thumbnails/1.jpg)
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
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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
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Campbell R. Harvey 3
2. Modes/Inputs of Asset Allocation
• Types of asset allocation– Strategic– Tactical
• Type of information– Unconditional– Conditional
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Campbell R. Harvey 4
Strategic Tactical
Unconditional Conditional
Slow evolvingweights
Dynamicweights
Constantweights
2. Modes/Inputs of Asset Allocation
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Campbell R. Harvey 5
2. Modes/Inputs of Asset Allocation
• Conditioning information makes a difference
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Campbell R. Harvey 6
3. Performance Depends on Business Cycle
-30
-20
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Norway
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Swed
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Switzer
land UK US
World
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ex-U
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EAFE
Expansion geometric mean Recession geometric mean
Average Annual Returns During U.S. Business Cycle Phases
Data through June 2002
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Campbell R. Harvey 7
3. Performance Depends on Business Cycle
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Hong K
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Irelan
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Japan
Nether
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Norway
Portugal
Spain
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UK USW
orld
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Expansion std.dev. Recession std.dev.
Average Annual Volatility During U.S. Business Cycle Phases
Data through June 2002
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Campbell R. Harvey 8
3. Performance Depends on Business Cycle
-0.2
0
0.2
0.4
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0.8
1
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man
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Hong K
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d It
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Japan
Nether
lands
New
Zea
land
Norway
Portugal
Spain
Swed
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Switzer
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UK USW
orld
World
ex-U
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EAFE
Expansion correlation with US Recession correlation with US
Correlations During U.S. Business Cycle Phases
Data through June 2002
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Campbell R. Harvey 9
3. Performance Depends on Business Cycle
0
5
10
15
20
25
30
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40
45
Australi
a
Austria
Belg
ium
Canad
a
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mar
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Finlan
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France
Ger
man
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Hong K
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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
Expansion covariance with US Recession covariance with US
Covariances During U.S. Business Cycle Phases
Data through June 2002
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Campbell R. Harvey 10
4. Conditioning Information and Portfolio Analysis
• Adding conditioning information is like adding extra assets to an optimization
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Campbell R. Harvey 11
4. Conditioning Information and Portfolio Analysis
Er
Vol
Traditional fixed weightoptimization (contrarian)in 2-dimensional setting
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Campbell R. Harvey 12
4. Conditioning Information and Portfolio Analysis
Er
Vol
Add conditioninginformation and weightschange through time. Frontier shifts.
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Campbell R. Harvey 13
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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Campbell R. Harvey 14
5. What is Risk?
• Much interest in downside risk, asymmetric volatility, semi-variance, extreme value analysis, regime-switching, jump processes, ...
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Campbell R. Harvey 15
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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Campbell R. Harvey 16
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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Campbell R. Harvey 17
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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Campbell R. Harvey 18
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
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Campbell R. Harvey 19
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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Campbell R. Harvey 20
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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Campbell R. Harvey 21
7. Skewness
0
5
10
15
Variance
- 2
- 1
0
1
2
Skewness
5
7.5
10
12.5
Expected Return
0
5
10
15
Variance
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Campbell R. Harvey 22
7. Skewness
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
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Campbell R. Harvey 23
7. Skewness
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
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Campbell R. Harvey 24
7. Skewness
0
5
10
15
Variance
- 2
- 1
0
1
2
Skewness
57.51012.5
Expected Return
RF
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Campbell R. Harvey 25
7. Skewness
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
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Campbell R. Harvey 26
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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Campbell R. Harvey 27
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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Campbell R. Harvey 28
7. 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
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Irelan
d It
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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
Data through June 2002
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Campbell R. Harvey 29
7. 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
Data through June 2002
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Campbell R. Harvey 30
7. Higher Moments & Expected Returns
Data through June 2002
-1
0
1
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4
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Japan
Nether
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New
Zea
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Norway
Portugal
Spain
Swed
en
Switzer
land UK US
World
World
ex-U
S
EAFE
Average Excess Kurtosis in Developed Markets
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Campbell R. Harvey 31
7. Higher Moments & Expected Returns
Data through June 2002
-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
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Campbell R. Harvey 32
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
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Campbell R. Harvey 33
8. Factors
4. Log market capitalization
Related to size
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Campbell R. Harvey 34
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(
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Campbell R. Harvey 35
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
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Campbell R. Harvey 36
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
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Campbell R. Harvey 37
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
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Campbell R. Harvey 38
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
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Campbell R. Harvey 39
8. Factors
15. Kurt is the kurtosis of the return distribution
Related to spread
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Campbell R. Harvey 40
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
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Campbell R. Harvey 41
8. Factors
19. betahml - HML
20. betasmb - SMB
Related to Fama-French 3-factor model
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Campbell R. Harvey 42
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
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Campbell R. Harvey 43
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
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Campbell R. Harvey 44
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
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8. Factors
27. betaip - OECD G7 industrial production
Related to Economic Activity
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9. Results
y = -0.8121x + 0.8964
R2 = 0.118
-1
0
1
2
3
4
5
-1.50 -1.00 -0.50 0.00 0.50
Coskew2
Mea
n R
etur
ns
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9. Results
y = 0.1586x + 0.3226
R2 = 0.1673
-1
0
1
2
3
4
5
0.00 5.00 10.00 15.00 20.00
Kurtosis
Mea
n R
etur
ns
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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
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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
y = -5.3067x + 24.869
R2 = 0.5934
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
Skew
Mea
n
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 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
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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
Co-Skewness Measure (Definition 2)(Total of 42 Funds, over Jan 1997 - Feb 2001)
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
Coskewness
Mea
n R
etu
rns
(Geo
met
ric)
Source: Lu and Mulvey (2001)
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10. Conditional Skewness
Co-Skewness Measure (Definition 2)(Total of 42 Funds, over Jan 1997 - Feb 2001)
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
Coskewness
Mea
n R
etu
rns
(Ari
thm
etic
)
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
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
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12. Estimation Error
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.
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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