Financial Data Analysis
Professor S. Kou, Department of IEOR, Columbia University
Lecture 2. Estimation Risk and Portfolio Selection: Empirical Bayesian
Estimators and the BlackLitterman Bayesian model
Although the mean variance analysis is a major breakthrough in °nance,
relatively few people in practice use it directly. Some main problems are
(1) Sometimes it is hard to estimate the covariance matrix
C
, although
it is a big problem if the number of assets involved is small.
(2) The result is very sensitive to the mean vector
R
, which is very
di¢ cult to be estimated.
(3) The portfolio weights are sensitive to estimation errors in
C
and
R
.
(4) The whole theory is based on the quadratic utility function, which
itself is problematic.
The literature is quite rich on the estimation risk topic: Jobson and Ko
rkie (1980, 1981), Frost and Savarino (1986, 1988), Jorion (1986), Michaud
(1989, 2000), Best and Grauer (1991), Black and Litterman (1992), Scherer
(2004); see a survey paper by Brandt (2005) and the book by Meucci (2005).
There are two main approaches to handle the di¢ culties.
First, there
are empirical Bayesian estimation procedures, which are the ones that we
shall discuss next. Later in this lecture we shall consider a popular Bayesian
approach to do portfolio optimization, the BlackLitterman (1992) model,
which is used by Goodman Sachs, . A key di/erence between the two ap
proaches is that the empirical Bayesian approaches only depend on historical
data, while Bayesian approaches rely on both data and subjective views.
1
Problems with the empirical means and covari
ances
Suppose that we observe a vector of
N
stock returns
X
1
,
X
2
, ...,
X
n
during
n
periods. The mean vector is
R
1
°
N
, and covariance is
C
N
°
N
. One way to
estimate the mean vector
R
is to simply use the sample mean vector
^
R
;
^
R
=
1
n
n
X
i
=1
X
i
:
Now assuming that
X
1
,
X
2
, ...,
X
n
are i.i.d. random vector with the mul
tivariate normal distribution. Then
S
2
=
n
X
i
=1
(
X
i
°
^
R
)
>
(
X
i
°
^
R
)
has a Wishart distribution with
E
°
S
2
±
= (
n
°
1)
C
:
1
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To estimate
C
one can use the sample covariance
^
C
=
1
n
°
1
S
2
:
The commend in Splus for
^
C
is
var
.
Recall that in the presence of the riskfree rate, the vector of optimal
mean variance e¢ cient portfolio weights is given by
w
=
°
±
C
±
1
(
R
°
r
1
)
To get an unbiased estimator for the portfolio weights we shall use an
unbiased estimator
^
w
=
°
±
(
n
°
N
°
2)
²
S
2
³
±
1
(
^
R
°
r
1
) =
°
±
n
°
N
°
2
n
°
1
±
´
^
C
µ
±
1
(
^
R
°
r
1
)
:
To show this estimator is unbiased we use the fact that
S
2
and
^
R
are inde
pendent, and
²
S
2
³
±
1
has an inverse Wishart distribution with mean
C
±
1
n
°
N
°
2
:
Indeed,
E
[ ^
w
]
=
°
±
(
n
°
N
°
2)
±
E
h
²
S
2
³
±
1
i
±
E
(
^
R
°
r
1
)
=
°
±
(
n
°
N
°
2)
±
C
±
1
n
°
N
°
2
±
(
R
°
r
1
)
=
°
±
C
±
1
±
(
R
°
r
1
) =
w:
Kan and Zhou (2007) show that the above unbiased portfolios per
form better (highest expected outofsample performance) respect to the
one based on an unbiased estimate of the covariance matrix (i.e.
without
the term
n
±
N
±
2
n
±
1
), and that the MLE (obtained using
´
1
n
±
1
S
2
µ
±
1
to estimate
C
±
1
). However, it is still an open problem to compute
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 Fall '10
 StevenKou
 Normal Distribution, Variance, Financial Engineering, Estimation theory, Ri, Covariance matrix, Covariances

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