Discussion of
Portfolio Choice with Model Misspeci…cation
by Pesaran, Uppal, and Za¤aroni
Xiaoji Lin
Ohio State University
UBC Summer Finance Conference
July 31, 2015
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Context: Portfolio choice, APT, and misspeci…cation
The literature on APT
Ross ’76, ’77, Huberman ’82, Chamberlain and Rothschild ’83, Ingersoll ’84
Portfolio selection in the presence of "mispricing"
Dybvig and Ross ’85, Green ’86, Grinblatt and Titman ’89, Kosowski et al ’06
The implications of the no-arbitrage constraint imposed by APT for
the mean-variance portfolio estimation
Jobson and Korkie ’80, Black and Litterman ’90, Green and Holli…eld ’92, MacKinlay and
Pastor ’00, Jagannathan and Ma ’03,
Contribution: Extends the original APT to allow for large pricing
errors due to model misspeci…cation and decomposes the
mean-variance portfolio into an "alpha" portfolio and a beta portfolio.
2 / 17
Context: Portfolio choice, APT, and misspeci…cation
The literature on APT
Ross ’76, ’77, Huberman ’82, Chamberlain and Rothschild ’83, Ingersoll ’84
Portfolio selection in the presence of "mispricing"
Dybvig and Ross ’85, Green ’86, Grinblatt and Titman ’89, Kosowski et al ’06
The implications of the no-arbitrage constraint imposed by APT for
the mean-variance portfolio estimation
Jobson and Korkie ’80, Black and Litterman ’90, Green and Holli…eld ’92, MacKinlay and
Pastor ’00, Jagannathan and Ma ’03,
Contribution: Extends the original APT to allow for large pricing
errors due to model misspeci…cation and decomposes the
mean-variance portfolio into an "alpha" portfolio and a beta portfolio.
2 / 17
Outline
1
Main results in the paper
2
Tests of the theory implications in the data
3 / 17
The APT
In matrix form:
R = A + BF + "
E["jF ] = 0
E[""0 jF ] =
R is N 1 vector of asset returns
A is N 1 vector of intercepts
B = [b1 ; b2 ; :::bN ] is a N K matrix of factor sensitivities
" is N 1 vector of disturbance term.
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Notes
Given the structure, Ross (1976) shows that the absence of arbitrage in
large economies implies that
=( )
+B
K
where is Nx1 vector of expected returns and is the model zero-beta
parameter, and equals the risk-free rate if such a rate exists
This relationship holds exactly only as the number of assets !in…nity.
5 / 17
Key result 1
Extends the interpretation of the APT and shows that it can capture
large pricing errors that arise from mis-measured or missing factors
|{z}
pricing errors
=
A
|{z}
loadings
m
|{z}
latent/missing factors
+
a
|{z}
idio. pricing errors
The original APT interpretation
pricing errors are small
pricing errors are unrelated to factors
6 / 17
Key result 1
Extends the interpretation of the APT and shows that it can capture
large pricing errors that arise from mis-measured or missing factors
|{z}
pricing errors
=
A
|{z}
loadings
m
|{z}
latent/missing factors
+
a
|{z}
idio. pricing errors
The original APT interpretation
pricing errors are small
pricing errors are unrelated to factors
6 / 17
Key result 2
Under the extended APT
the optimal mean-variance portfolio
= an alpha portfolio +
{z
}
|
depends on pricing errors
a beta portfolio
{z
}
|
depends on factor risk premia/loadings
The original APT implies
the optimal mean-variance portfolio =
a beta portfolio
{z
}
|
depends on factor risk premia/loadings
Obtains similar decompositions for the global minimum-variance
portfolio and the e¢ cient-frontier portfolios.
7 / 17
Key result 2
Under the extended APT
the optimal mean-variance portfolio
= an alpha portfolio +
{z
}
|
depends on pricing errors
a beta portfolio
{z
}
|
depends on factor risk premia/loadings
The original APT implies
the optimal mean-variance portfolio =
a beta portfolio
{z
}
|
depends on factor risk premia/loadings
Obtains similar decompositions for the global minimum-variance
portfolio and the e¢ cient-frontier portfolios.
7 / 17
Key result 2
Under the extended APT
the optimal mean-variance portfolio
= an alpha portfolio +
{z
}
|
depends on pricing errors
a beta portfolio
{z
}
|
depends on factor risk premia/loadings
The original APT implies
the optimal mean-variance portfolio =
a beta portfolio
{z
}
|
depends on factor risk premia/loadings
Obtains similar decompositions for the global minimum-variance
portfolio and the e¢ cient-frontier portfolios.
7 / 17
Key result 3
Practical implications:
The decomposition results and the restriction arising from the
extended APT can be used to improve the estimation of portfolio
weights when the model misspeci…cation is present.
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My discussion: Test the theory results in the data
1
A single factor model: Market
2
A two factor model: Market + Equity issuance shock (Belo, Lin, and
Yang ’15)
3
A two factor model: Market + a latent factor without the APT
restriction on pricing errors ( 0
< 0)
4
A two factor model: Market + a latent factor with the APT
restrictions on pricing errors ( 0
< 0)
Test assets: 5 BM portfolios, 5 investment portfolio, 5 size portfolios
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Standard asset pricing tests
Single factor model: CAPM
Two factor model: Market factor + Equity issuance shock (ICS)
A
CAPM
B
16
16
14
14
Market+ ICS
SZ1
SZ1
IK5
10
SZ2
SZ4
SZ3
BM1
8
Predicted
Predicted
SZ2
12
SZ5 BM2
BM3BM4 IK1
IK4 IK3
IK2
6
6
8
C
10
12
BM5
BM3
16
IK1
IK5
SZ5
BM1
6
Latent factor (δ=∞)
8
D
BM2
IK2
IK4 IK3
10
12
14
16
Latent factor (δ= 1)
12
SZ1
IK5
BM1
SZ2
SZ4
SZ3
latent w/ restriction
16
14
Predicted
SZ4
BM4
10
6
14
BM5
SZ3
8
16
10
12
14
12
SZ1
10
IK5
SZ2
SZ3
SZ4
BM4
BM5
10 / 17
Model misspeci…cation with no restriction
Model: Market + latent factor
The APT restriction:
A
=1
CAPM
B
16
16
14
14
Market+ ICS
SZ1
Predicted
Predicted
SZ2
12
SZ1
IK5
10
SZ2
SZ4
SZ3
BM1
8
SZ5 BM2
BM3BM4 IK1
IK4 IK3
IK2
6
6
8
C
10
12
12
SZ4
BM4
10
BM3
SZ5
BM1
6
14
16
IK1
IK5
8
BM5
BM5
SZ3
6
8
BM2
IK2
IK4 IK3
10
12
14
16
Latent factor (δ=∞)
16
Predicted
14
12
10
SZ1
IK5
BM1
SZ5
8
6
6
8
SZ2
SZ4
SZ3
BM2BM3BM4 IK1
IK4 IK3
IK2
10
12
BM5
14
16
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Model misspeci…cation with restriction
Model: Market + latent factor
The APT restriction:
=1
A
CAPM
B
16
16
14
14
12
12
Market+ ICS
SZ1
SZ1
IK5
10
SZ2
SZ4
SZ3
BM1
8
Predicted
Predicted
SZ2
SZ5 BM2
BM3BM4 IK1
IK4 IK3
IK2
6
6
8
C
10
12
16
Market+latent w/ restriction
Predicted
SZ1
IK5
BM1
SZ5
6
8
SZ2
SZ4
SZ3
BM2BM3BM4 IK1
IK4 IK3
IK2
10
12
BM5
14
8
D
12
6
SZ5
BM1
6
Latent factor (δ=∞)
14
8
BM3
16
IK1
IK5
6
14
16
10
SZ4
BM4
10
8
BM5
BM5
SZ3
BM2
IK2
IK4 IK3
10
12
14
16
Latent factor (δ= 1)
16
14
12
SZ1
10
IK5
8
SZ2
SZ3
SZ4
BM4
BM2BM3 IK1
BM5
BM1
SZ5
IK4
IK2IK3
6
6
8
10
12
14
16
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Model misspeci…cation with tighter restriction
Model: Market + latent factor
The APT restriction:
= 1 and
= 0:01
B L a te n t fa c to r (δ=0 .0 1 )
16
15
15
14
14
13
13
Ma r ke t+la ten t w/ r e str ictio n
Ma r ke t+la ten t w/ r e str ictio n
A L ate n t fa c to r (δ=1 )
16
12
11
SZ1
10
SZ2
IK5
SZ3
SZ4
9
BM5
BM4
12
11
SZ1
10
SZ2
IK5
SZ3
SZ4
9
BM3 IK1
BM1
BM2
SZ5
8
BM3 IK1
BM1
BM2
SZ5
7
IK4 IK3
IK2
7
IK4 IK3
IK2
6
5
BM5
BM4
8
6
6
8
10
12
14
16
5
6
8
10
12
14
16
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Model misspeci…cation with restriction
The ICS factor and the latent factor are positively correlated
10
50
ICS-MI
Latent
Correlation = 0.27
0
- 10
1975
0
1980
1985
1990
1995
Year
2000
2005
2010
- 50
2015
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Model misspeci…cation with restriction
The HML and SMB factors and the latent factor are correlated
SMB
Latent
40
20
0
-20
-40
1975
Corr. = -0.13
1980
1985
1990
1995
Year
2000
2005
2010
40
2015
HML
Latent
20
0
-20
-40
1975
Corr. = 0.26
1980
1985
1990
1995
Year
2000
2005
2010
2015
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The mean-variance, alpha, and beta portfolios
The alpha portfolio drives most of the variations of the mean-variance
portfolio
4
Alpha Portfolio, SR. = 1.02
Beta Portfolio, SR. = 0.67
MV Portfolio, SR. = 1.13
3
Excess Return (%)
2
1
0
-1
-2
-3
1975
1980
1985
1990
1995
Year
2000
2005
2010
2015
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Conclusion
Very nice paper!
Understanding model misspeci…cation is important for asset pricing
test and portfolio choice
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