Risk Reduction in Large Portfolios: A Role for
Portfolio Weight Constraints1
Ravi Jagannathan2
Northwestern University and NBER
Tongshu Ma3
University of Utah
April 28, 2001
1 We
thank Torben Andersen, Louis Chan, Gregory Connor, Kent Daniel,
Ludger Hentschel, Andrew Lo, Jesper Lund, Robert McDonald, Steve Ross, Jay
Shanken and seminar participants at Copenhagen Business School, MIT, Norwegian School of Management, Hong Kong University of Science and Technology,
University of Rochester, University of Illinois at Urbana-Champaign, and University of Utah for helpful comments. We are responsible for any errors or omissions.
2 Department of Finance, Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208. [email protected], 847-491-8338.
3 Department of Finance, David Eccles School of Business, University of Utah,
Salt Lake City, UT84112. [email protected], 801-581-7506.
Abstract
Mean-variance efficient portfolios constructed using sample moments often
involve taking extreme long and short positions. Hence practitioners often
impose portfolio weight constraints when constructing efficient portfolios.
Green and Hollifield (1992) argue that the presence of a single dominant
factor in the covariance matrix of returns is why we observe extreme positive and negative weights. If this were the case then imposing the weight
constraint should hurt whereas the empirical evidence is often to the contrary. We reconcile this apparent contradiction. We show that constraining
portfolio weights to be nonnegative is equivalent to using the sample covariance matrix after reducing its large elements and then form the optimal
portfolio without any restrictions on portfolio weights. This shrinkage helps
reduce the risk in estimated optimal portfolios even when they have negative
weights in the population. Surprisingly, we also find that once the nonnegativity constraint is imposed, minimum variance and minimum tracking error
portfolios constructed using the sample covariance matrix perform as well as
those constructed using covariance matrices estimated using factor models
and shrinkage estimators.
1
Introduction
Markowitz’s (1952, 1959) portfolio theory is one of the most important theoretical developments in finance. Mean-variance efficient portfolios play an
important role in this theory. Such portfolios constructed using sample moments often involve taking large short positions in a number of assets and
large long positions in other assets. In practice most investors impose the
constraint that portfolio weights should be nonnegative and below an upper
bound while constructing mean-variance efficient portfolios. Green and Hollifield (1992) argue that it is difficult to dismiss the observed extreme negative
and positive weights as entirely due to imprecise estimation of the inputs used
to construct mean-variance efficient portfolios. If extreme negative weights
in efficient portfolios arise due to the presence of a single dominant factor
as Green and Hollifield (1992) argue then it would appear that imposing
the portfolio weight constraints would lead to a loss in efficiency. However
empirical findings in this area suggest that imposing these constraints on
portfolio weights improves the efficiency of optimal portfolios constructed
using sample moments1 . Does this mean that the evidence does not support
the Green and Hollifield (1992) view?
In this paper we provide an answer to this question. We show that imposing the nonnegativity constraint on portfolio weights can help even when
Green and Hollifield are right — i.e., the true covariance matrix is such that
efficient portfolios involve taking large negative positions in a number of assets. We show that each of the no-shortsales constraints is equivalent to
reducing the sample covariances of the corresponding asset with other assets
by a certain amount. Stocks that have high covariances with other stocks
tend to receive negative portfolio weights. Hence, to the extent high estimated covariances are more likely to be caused by estimation error, imposing
the nonnegativity constraint can reduce the sampling error. It follows from
1
See Frost and Savarino (1988) for an excellent discussion.
1
the theory of shrinkage estimators that imposing the no-shortsales constraint
can help even when the constraints do not hold in the population.
We also study the impact of upper bounds on portfolio weights. We
show that each upper bound constraint is equivalent to increasing the sample
covariances of the corresponding asset with other assets by a certain amount.
Since stocks that have low covariances with other stocks tend to get extreme
high portfolio weights, and these extreme low covariances are more likely to
be caused by estimation error, this would reduce sampling error and help the
out-of-sample performance of the optimal portfolios. Our empirical evidence
suggests that imposing upperbounds on portfolio weights does not lead to a
significant improvement in the out-of-sample performance of minimum risk
portfolios when no-shortsales restrictions are already in place. Therefore the
role of upperbounds is more to ensure that the minimum risk portfolios that
come out of the optimization exercise can be constructed in practice.
The most common estimator of a covariance matrix is the corresponding sample covariance matrix of historical returns. It has long been recognized that mean-variance efficient portfolios constructed using the sample
covariance matrix perform poorly out-of-sample.2 The primary reason is
that the large covariance matrices encountered in practice require estimating
too many parameters. For example, the covariance matrix of the returns of
500 stocks will have about 125,000 distinct parameters. However, monthly
returns for all traded stocks are only available for the past 450 months or so.
This gives less than 2 degrees of freedom per estimated parameter. Consequently the elements of the covariance matrix are estimated imprecisely.
Several solutions to this problem have been suggested. The first is to
impose more structure on the covariance matrix to reduce the number of
parameters that have to be estimated. This includes factor models and constant correlation models. The second approach is to use shrinkage estimators,
2
See Jobson and Korkie (1980, 1981), Frost and Savarino (1986, 1988), Jorion (1986),
Michaud (1989), Best and Grauer (1991), and Black and Litterman (1992).
2
shrinking the sample covariance matrix toward some target. An example of
such a target would be the identity matrix. The third approach is to use data
of higher frequency, for example, daily data in place of monthly data. These
three approaches are widely used both by practitioners and academics.
For any covariance matrix estimator, we can impose the usual portfolio
weight constraints. These constraints reduce sampling errors while at the
same time introduce specification errors. This is because, according to Green
and Hollifield, they are wrong in population. The gain from imposing these
constraints depends on the trade off between the reduction in sampling error
and the increase in specification error. For covariance matrix estimators that
have large sampling error, such as the sample covariance matrix, imposing
these constraints is likely to be helpful. However, for the factor models and
shrinkage estimators, imposing such constraint is likely to hurt. These are
indeed what we find empirically.
Another finding of this paper is that once the nonnegativity constraint is
imposed, portfolios constructed using the sample covariance matrices perform
as well as portfolios constructed using covariance matrices estimated using
factor models and shrinkage methods. Also, when covariance matrices are
estimated using daily data, corrections for microstructure effects that have
been suggested in the literature do not lead to superior performance. So far,
this has not been recognized in the literature.3
The reminder of the paper is organized as follows. First, we provide a
theoretical analysis of the effect of imposing the no-shortsales restriction
and upper bounds on portfolio weights while constructing minimum risk
portfolios. Then we show that these portfolio weight constraints have a
shrinkage interpretation. Finally, we show using Monte Carlo methods that
3
It is well recognized in the literature that the imposition of portfolio weight constraints
leads to superior out-of-sample performance of mean-variance efficient portfolios. However to our knowledge no one has noticed that the performance improvement is so large
that it is comparable to that attained using the other alternatives.
3
imposing the no-shortsales restriction can help even when imposing it would
hurt if the true covariance matrix of returns were known.
Imposing the portfolio weight constraints reduces sampling error but introduces specification error. Whether the net effect is positive or negative is
an empirical issue. In the next Section, we examine the effect of the usual
portfolio weight constraints for portfolio variance minimization and tracking
error minimization. We focus on minimum variance portfolios based on the
findings reported in the literature that it has as large an out-of-sample Sharpe
Ratio as other efficient portfolios4 . Our interest in minimum tracking error
portfolios arises from the observation that it may often be necessary in practice to construct portfolios using a subset of all available stocks that have low
transactions costs and high liquidity to track certain benchmark indices that
may contain assets that may not be actively traded. We find that for factor
models and shrinkage estimators, imposing the usual portfolio weight constraint reduces the portfolio’s efficiency slightly. On the other hand, when
the no-shortsales restriction is imposed, minimum variance and minimum
tracking error portfolios constructed using the sample covariance matrix perform as well as those constructed using more sophisticated estimates of the
covariance matrix. This is a useful result for institutional investors because
4
Using monthly stock index return data for G7 countries Jorion (1985) convincingly
argues that ‘... benefits from diversification are more likely to accrue from a reduction
in risk.’ In the data set he examined, the global minimum variance portfolio had the
best out of sample performance. It out performed classical tangent portfolio, the tangent
portfolio construced using the Bayes-Stein estimator for the vector of mean returns, and
the value weighted and equally weighted portfolios. Using simulation methods Jorion
(1986) showed that this conclusion is robust if the sample size is not large. Jorion
(1991) found that the minimum variance portfolio constructed using returns on seven
industry stock index portfolios performed as well as the CRSP equally weighted and value
weighted stock indices during the January 1926 to December 1987 period, in out of sample
tests. The performance was comparable to that of the tangent portfolio constructed using
the Bayes-Stein estimator for the mean.. Bloomfield, Leftwich and Long (1977) found
that portfolios constructed using mean-variance optimization did not dominate an equally
weighted portfolio..
4
most of them cannot short sell. We conclude the paper after that.
2
The Effect of Portfolio Weight Constraints
Portfolio weight constraints are usually imposed in portfolio optimization.
Previous researches have noticed their effects empirically (for example, Frost
and Savarino (1988)). In this section we examine the implications of the
usual portfolio weight constraints from a theoretical point of view. We will
first look at the effect of the no-shortsales constraint only. We then turn
to the case when both the no-shortsales constraint and an upper bound on
portfolio weights are imposed. The analysis is very similar in the two cases.
2.1
The Effect of the No-Shortsales Restriction
Given an estimated covariance matrix S, the portfolio variance minimization
problem with nonnegative weight constraint is:
minω ω 0 Sω
s.t.
X
(1)
ωi = 1
(2)
i
ω i ≥ 0, i = 1, 2, · · · , N.
(3)
The Kuhn-Tucker conditions (necessary and sufficient) are
X
j
Si,j ω j − λi = λ > 0, i = 1, 2, · · · , N.
λi ≥ 0,
and λi = 0 if ω i > 0, i = 1, 2, · · · , N.
(4)
(5)
Let us first examine why there are negative portfolio weights in the minimum variance portfolio. The (unconstrained) portfolio variance minimization
5
problem is (1) and (2) only. The first order conditions are (2) and
N
X
j=1
ω j Si,j = λ > 0, i = 1, 2, · · · , N.
(6)
The above condition says that at the optimum, stock i’s marginal contribution to the portfolio variance is the same as stock j’s, for any i and j.
Suppose stock i tends to have higher covariances with other stocks, i.e.,
the ith row of S tends to have larger elements than other rows. Then stock
P
i’s marginal contribution to the portfolio variance, 2 N
j=1 ω j Si,j , will tend
to be bigger than other stocks’ marginal contribution. Therefore, to achieve
optimality, we need to reduce stock i’s portfolio weight and increase other
stocks’ weights. Stock i may even have negative weight if its covariances
with other stocks weighted using the optimally chosen weight is sufficiently
high. Therefore, a stock tends to receive a negative portfolio weight in the
minimum variance portfolio if it has higher variance and higher covariances
with other stocks. In a one-factor structure, these are the high-beta stocks.
Let S be an estimated covariance matrix. Denote the solution to the
constrained variance minimization problem (1) - (3) as ω + (S). Now we can
construct a new covariance estimate S̃, such that its implied (unconstrained)
minimum variance portfolio is exactly ω + (S). Hence we can say that the
effect of imposing portfolio weight constraint is as if we are using S̃ instead
of S.
Let λi (i = 1, 2, ..., N) be the Lagrange multipliers for the nonnegativity
constraints (3) and λ the Lagrange multiplier for (2). Then we have the
following proposition.
Proposition 1 The global minimum variance portfolio of
S̃ = S − (λi + λj )i,j=1,···,N
is ω + (S). Furthermore, S̃ is symmetric and positive definite.
6
(7)
Proof: Suppose that (ω 1 , · · · , ω N , λ1 , · · · , λN , λ) is the solution to the constrained portfolio variance minimization problem (1) - (3). To show that
ω = (ω 1 , · · · , ω N ) is the (unconstrained) minimum variance portfolio of S̃, it
suffices to verify the first order condition:
S̃ω = Sω − (λi + λj )N
i,j=1 ω




= Sω − 






= Sω − 





= 



λ
λ
..
.
λ

P
λ1 j6=1 ω j
P
λ2 j6=2 ω j
..
.
λN
P
j6=N

λ1

.. 
. 
λN
ωj












.



The second equality follows from the fact that ω i λi = 0, the third equality
P
follows from the same fact and j wj = 1, and the last equality follows from
(4). The fact that S̃ω = λ · (1, 1, · · · , 1)0 shows that ω solves the (unconstrained) variance minimization problem for covariance matrix S̃.
S̃ is clearly symmetric. Its global minimum variance portfolio has a variance of ω 0 S̃ω = ω 0 · (λ, λ, · · · , λ)0 = λ > 0. Hence S̃ is positive definite.
QED
This result shows that constructing the constrained minimum variance
portfolio from S is equivalent to constructing the (unconstrained) minimum
variance portfolio from S̃ = S − (λi + λj )i,j=1,···,N . Notice that the effect
of imposing the constraint is that, whenever the nonnegativity constraint is
binding for stock i, its covariances with other stocks, Sij , j 6= i, are reduced
by λi + λj , a positive quantity, and its variance is reduced by 2λi . We saw
7
before that a stock will receive a negative portfolio weight in the minimum
variance portfolio if its covariances are relatively too high. Therefore, the
new covariance matrix estimate, S̃, is constructed by shrinking the large
covariances toward the average covariances. Since the largest covariance
estimates are more likely caused by estimation error, this shrinking may
reduce the estimation error.
In general, given ω + (S), there are many covariance matrix estimates that
have ω + (S) as their (unconstrained) minimum variance portfolio. Is there
anything special about S̃? We do have an answer to this question when
returns are jointly normal and S is the MLE of the population covariance
matrix.
Let ht = (r1t, , r2t , · · · , rNt )0 be iid normal N (µ, Ω). Then the MLE of Ω is
P
S = Tt=1 (ht − h̄)(ht − h̄)0 /T. It is well known that with whatever estimate
of the covariance matrix, the MLE of the mean is always the sample mean.
(See, for example, Morrison (1990), p.98.) With this estimate of the mean,
the log-likelihood (as a function of the covariance matrix alone) becomes
l(Ω) = CONST −
T
T
ln |Ω| − tr(SΩ−1 ).
2
2
This can also be considered as the likelihood function of Ω−1 .
Now consider the constrained MLE of Ω, subject to the constraint that
the global minimum variance portfolio constructed from Ω has nonnegative
weights only. Let Ωi,j denote the (i, j)-th element of Ω and Ωi,j denote the
(i, j)-th element of Ω−1 , then the constraint is
X
j
Ωi,j ≥ 0, i = 1, 2, · · · , N.
(8)
So the constrained maximum likelihood problem is
max
−
i,j
Ω
, i≤j
T
T
ln |Ω| − tr(SΩ−1 )
2
2
subject to (8). We have the following proposition:
8
(9)
Proposition 2 Assume that returns are jointly iid normal N (µ, Ω). Let S
be the unconstrained MLE of Ω.
(i) Given S, let {λi , ω i }N
i=1 be the solution to the constrained portfolio variance minimization problem (1) - (3), and construct S̃ according to (7). Then
S̃ and {λi }N
i=1 jointly satisfy the first order conditions for the constrained ML
problem. (ii) Let S̃ and {λi }N
i=1 jointly satisfy the first order conditions for the
P
P P
constrained ML problem. For i = 1, · · · , N, define ω i = j S̃ i,j / k l S̃ k,l ,
the normalized row sums of S̃ −1 . Then {λi , ω i }N
i=1 is the solution to the constrained portfolio variance minimization problem (1) - (3), given S.
Proof: First, let’s write out the first order conditions for the constrained
ML problem. Let the constrained MLE be Ω̂. The Kuhn-Tucker conditions
for the constrained maximization problem are (Morrison (1990), p. 99):
1
1
∂l
Ω̂i,i − Si,i = −λi , all i
=
i,i
∂Ω
2
2
∂l
= Ω̂i,j − Si,j = −(λi + λj ), all i < j
∂Ωi,j
X i,j
λi ≥ 0, and λi = 0 if
Ω̂ > 0.
(10)
(11)
(12)
j
These conditions imply that the constrained MLE can be written as
Ω̂ = S − (λi + λj )N
i,j=1 .
Notice that Ω̂ has the same form as S̃.
We will only prove part (i). The proof of part (ii) is similar. Let {λi , ω i }N
i=1
be the solution to the constrained portfolio variance minimization problem
(1) - (3), given S, and construct S̃ according to (7). Then we can easily
verify that {λi }N
i=1 and S̃ satisfies (10), (11), and the first half of (12). Now
P
we need to verify the second half of (12), i.e., j S̃ i,j > 0 implies λi = 0.
By Proposition 1, ω is the unconstrained global minimum variance portfolio
P
P P
of S̃, so ω i = j S̃ i,j / j k S̃ j,k . From the second half of (5) we know that
P
QED
ω i > 0 implies λi = 0. This says that j S̃ i,j > 0 implies λi = 0.
9
This Proposition states that when S is the MLE, the constrained portfolio
variance minimization problem is equivalent to the constrained ML estimation problem. So if instead of imposing the constraint in the optimization
stage, we impose it in the estimation stage, we get exactly the same constrained optimal portfolio. Furthermore, the fact that S̃ is the constrained
MLE illustrates that it is in a sense the “closest” to S (which is the unconstrained MLE) among all covariance matrix estimates that have ω + (S) as
their global minimum variance portfolio.
2.2
The Joint Effect of an Upper Bound on Portfolio
Weights and the No-Shortales Restriction
Given an estimated covariance matrix S, the portfolio variance minimization
problem with both nonnegative weight constraint and an upper bound on
portfolio weights is:
minω ω 0 Sω
s.t.
X
(13)
ωi = 1
(14)
i
ω i ≥ 0, i = 1, 2, · · · , N.
(15)
ω i ≤ ω̄, i = 1, 2, · · · , N.
(16)
The Kuhn-Tucker conditions (necessary and sufficient) are
X
j
Si,j ω j − λi + δ i = λ > 0, i = 1, 2, · · · , N.
(17)
λi ≥ 0,
and λi = 0 if ω i > 0, i = 1, 2, · · · , N.
(18)
δi ≥ 0,
and δ i = 0 if ω i < ω̄, i = 1, 2, · · · , N.
(19)
Let us first examine why some portfolio weights may exceed the upper
bound. The following discussion is parallel to the discussion when only the
no-short-sales constraint is imposed. Suppose stock i tends to have lower
10
covariances with other stocks, i.e., the ith row of S tends to have smaller elements than other rows. Then stock i’s marginal contribution to the portfolio
P
variance, 2 N
j=1 ω j Si,j , will tend to be smaller than other stocks’ marginal
contribution. Therefore, to minimize portfolio variance, we need to increase
stock i’s portfolio weight and decrease other stocks’ weights. Stock i’s covariances with other stocks can be so much smaller than other stocks’ covariances that at optimal, it will receive a portfolio weight that exceeds the
upper bound. Therefore, a stock tends to receive a large positive portfolio
weight in the minimum variance portfolio if it has lower variance and lower
covariances with other stocks. In a single factor model, these are the low-beta
stocks.
Let S be an estimated covariance matrix. Denote the solution to the constrained portfolio variance minimization problem (13)-(16) as ω ++ (S). Now
we can construct a new covariance estimate S̃, such that its (unconstrained)
global minimum variance portfolio is exactly ω ++ (S). So if a fund manager
voluntarily imposes these kinds of weight constraints (i.e., these constraints
are not due to institutional restrictions), he/she in essence believes in S̃ instead of S.
Let λi (i = 1, 2, ..., N) be the Lagrange multipliers for the nonnegativity
constraints in (15), δ i (i = 1, 2, ..., N ) the multipliers for the constraints in
(16), and λ the multiplier for (14) as before. Then we have the following
proposition.
Proposition 3 The global minimum variance portfolio of
S̃ = S + (δ i + δ j − λi − λj )N
i,j=1
(20)
is ω ++ (S).
Proof: Suppose that (ω 1 , · · · , ω N , λ1 , · · · , λN , λ, δ 1 , · · · , δ N ) is the solution
to the constrained portfolio variance minimization problem (13)-(16). To
11
show that ω = (ω 1 , · · · , ω N )0 is the (unconstrained) minimum variance portfolio from S̃, it suffices to verify the first order condition:
N
S̃ω = Sω − (λi + λj )N
i,j=1 ω + (δ i + δ j )i,j=1 ω



λ
δ +
 1   1
 ..  
= Sω −  .  + 

λN

λN




δN +

δN

λ
 1  

 
= Sω −  ...  + 




= 



P

λ + ω̄ i δ i
P
λ + ω̄ i δ i
..
.
λ + ω̄
P
i δi

PN

j=1
δj ωj
..
.
PN
j=1
δj ωj






δ1
X 


.. 

+
ω̄
δi 
.

i


1

.. 
. 
1




.



The second and third equalities follow from the fact that λi ω i = 0 and
P
δ i (ω̄ − ω i ) = 0, and the last equality follows from (17). Since λ + ω̄ i δ i is
P
positive, the fact that S̃ω = (λ + ω̄ i δ i ) · (1, 1, · · · , 1)0 shows that ω is the
(unconstrained) global minimum variance portfolio for covariance matrix S̃.
QED
This result shows that constructing the constrained minimum variance
portfolio from S is equivalent to constructing the (unconstrained) minimum
variance portfolio from S̃ = S + (i + j − λi − λj )N
i,j=1 . We are familiar with
N
the effect of subtracting (λi + λj )i,j=1 . The effect of adding (i + j ) is the opposite: whenever the upper bound on portfolio weights is binding for stock
i, its covariances with other stocks, Sij , j 6= i, are raised by (i + j ), a positive
quantity, and its variance is raised by 2i . We saw before that a stock will receive a large positive portfolio weight in the minimum variance portfolio if its
covariances are relatively too low. Therefore, the new covariance estimate, S̃,
is constructed by shrinking both the small covariances and large covariances
toward the average. Since the extreme covariance estimates are more likely
12
caused by estimation error, this shrinking may reduce the estimation error.
Under normality and assuming that S is the MLE of the covariance matrix, then as in the previous subsection, we have the following result that
roughly says that S̃ = S + (δ i + δ j − λi − λj )N
i,j=1 is a constrained MLE of
the population covariance matrix.
Proposition 4 Assume that returns are jointly iid normal N (µ, Ω). Let S
be the unconstrained MLE of Ω.
(i) Given S, let {λi , δ i , ω i }N
i=1 be the solution to the constrained portfolio
variance minimization problem (13) - (16), and construct S̃ according to
(20). Then S̃ and {λi , (1 − ω̄) δ i }N
i=1 jointly satisfy the first order conditions
for the constrained ML problem with the constraints
X
j
Ωi,j ≥ 0, i = 1, 2, · · · , N,
j
Ωi,j ≤ ω̄
X
XX
k
l
Ωk,l , i = 1, 2, · · · , N.
(ii) Let S̃ and {λi , δ i }N
i=1 jointly satisfy the first order conditions for the
P
P P
constrained ML problem. For i = 1, · · · , N, define ω i = j S̃ i,j / k l S̃ k,l ,
the normalized row sums of S̃ −1 . Then {λi , δ i /(1 − ω̄), ω i }N
i=1 is the solution
to the constrained portfolio variance minimization problem (13) - (16), given
S.
The proof is similar to that of Proposition 2.
2.3
The Effects of Portfolio Weight Constraints: An
Example
This subsection looks at the shrinkage-like effects of portfolio weight constraints through an example. The covariance matrix estimate S is the sample
covariance matrix of a random sample of 30 stocks among all domestic common stocks that have all monthly returns from 1988 to 1997 (120 months).
13
Column 1 of Table 1 lists the row sums of S. We can see that the row sums
vary over a wide range: the lowest row sum is 0.28, the highest is 10.4. The
standard deviation of the row sums is 2.4 (see the bottom of the Table 1).
The second column lists the unconstrained global minimum variance portfolio weights. There is a general tendency that if the row sum is high (low),
the corresponding portfolio weight will be negative (positive). In fact the
correlation coefficient between row sums and portfolio weights is -0.725.
The Lagrange multipliers for the nonnegativity constraint are listed in
column 3. There is a tendency for the multipliers to be positive when the
row sums are high. In fact the correlation coefficient between the two is
0.766. The row sums of S̃ is listed in column 4. Column 5 gives the changes
from row sums of S to row sums of S̃.5
We can see that there is a tendency that when a row sum is high, it is
reduced by a larger amount. In fact, the correlation coefficient between row
sums of S and changes in row sums is -0.798. This confirmed our intuition
that the effect of the nonnegative constraint leads to shrinking the largest
covariance estimates toward zero. Column 5 tells us that the amount of
shrinking can be sizable. For example, the second highest row sum (10.23)
is reduced by 2.47.
Table 2 reports similar results when both nonnegativity and an upper
bound of 10% are imposed. The first two columns of Table 2 are the same as
those in Table 1. Columns 3 and 4 list the two sets of Lagrange multipliers.
We can see a pattern: high row sums implies positive Lagrange multiplier
for the nonnegativity constraint; low row sums implies positive Lagrange
5
The shrinking reduces all row sums. Since S is an unbiased estimator of the population
covariance matrix, S̃ will be a downward biased estimator of the population covariance
matrix (element by element). One way to correct for this bias is to multiply S̃ by a
constant that is larger than one. The other is to add an equal amount to all elements of
S̃. Both adjustments would result in the same global minimum variance portfolio as that
associated with S̃, so we will not concern about the bias for our purpose.
14
multipliers for the upper bound. The correlation between row sums of S and
changes of row sums is -0.819. This confirms that restricting the portfolio
weights to be positive shrink the lowest covariance estimates of S toward the
average value, and hence reduces sampling error.
2.4
Imposing the Wrong Constraint May Help: Evidence from a Simulation Study
People might think that the no-shortsales constraint helps because the constraint holds in population. Green and Hollifield (1992) argue correctly that
the true minimum variance portfolio will have extreme long and short positions. Their conclusion seems to imply that imposing the nonnegativity
constraint might be counter productive. The following simulation study illustrates that, although the constraints are violated in population, they can
still help. This is because of the shrinkage effect of the portfolio weight
constraints reduces sampling error.
The simulation study reported in Table 3 supports this conclusion. There
we estimated a covariance matrix using the sample covariance matrix of the
historical returns of a random sample of 30 domestic common stocks (the
same 30 stocks as those in Tables 1 and 2). Treating the covariance matrix as the true one, we simulate return data, and use the simulated return
data to estimate the covariance matrix and form the optimal portfolios. We
construct four global minimum variance portfolios: the unconstrained and
nonnegativity constrained ones from the true covariance matrix and the estimated covariance matrix. The standard errors of these portfolios are calculated using the true covariance matrix and the portfolio weights, and are
reported in the table. We see that the constrained portfolio from the true
covariance matrix does worse than the unconstrained one. This means that
the nonnegativity constraint is violated in population, consistent with Green
and Hollifield’s conclusion. Yet unless we have 180 months or more returns to
15
estimate the covariance matrix, the constraint helps the ex post performance
of the global minimum variance portfolios. This demonstrates that imposing
the wrong constraint may actually help.
3
The Effect of Portfolio Weight Constraints:
An Empirical Examination
3.1
Data and Methodology
In this section we examine empirically the effect of portfolio weight constraints. As we have said before, whether these weight constraints help or
hurt is an empirical issue and will depend on the specific covariance matrix estimator. For the estimators that have large sampling errors such as
the sample covariance matrix, the portfolio weight constraints are likely to
be helpful, as documented by Frost and Savarino (1988) and demonstrated
again in the simulation study in the last section. However, for the factor
models and shrinkage estimators, the portfolio weight constraints are likely
to be harmful.
We examine the effect of the portfolio weight constraints on the outof-sample performance of minimum variance and minimum tracking error
portfolios formed using a number of covariance matrix estimators. This is
done following the methodology in Chan, Lakonishok and Karcesky (1999).
At the end of each April 1968 to 1998, we randomly choose 500 stocks from
all common domestic stocks traded on the NYSE and the AMXE, with stock
price greater than $5, market capitalization more than the 80th percentile of
the size distribution of NYSE firms, and with monthly return data for all the
immediately preceding 60 months. We use return data for the preceding 60
months to estimate the covariance matrix of the returns of the 500 stocks.
For estimators that use daily data, the daily returns during the previous 60
16
months of the same 500 stocks are used. When a daily return is missing, the
equally-weighted market return of that day is used instead.
When variance minimization is the objective, we form three global minimum variance portfolios using each covariance matrix estimator — only two if
the covariance matrix estimate is singular. The first portfolio is constructed
without imposing any restrictions on portfolio weights, the second is subject to the constraint that portfolio weights should be nonnegative, and the
third, in addition, faces the restriction that no more than 2 percent of the
investment can be in any one stock. Each of these portfolios are held for one
year, their returns for each of the following 12 months are recorded, and at
the end of April of the next year, the same process is repeated. This gives at
most three minimum variance portfolios that have post formation monthly
returns from May 1968 to April 1999 for each covariance matrix estimator.
We use the standard deviation of the monthly return on these portfolios to
compare the different covariance matrix estimators.
For tracking error minimization, following Chan et al (1999) we assume
the investor is trying to track the return of the S&P 500 index. As in the
case of portfolio variance minimization, we construct three tracking error
minimizing portfolios for each covariance estimator. Notice that constructing
the minimum tracking error variance portfolio is the same constructing the
minimum variance portfolio using returns in excess of the benchmark subject
to the restriction that the portfolio weights sum to one.
3.2
Covariance Matrix Estimators
The first estimator is the sample covariance matrix:
SN =
T
1 X
(ht − h̄)(ht − h̄)0 ,
T − 1 t=1
where T is the sample size, ht is a N × 1 vector of stock returns in period t,
and h̄ is the average of these return vectors.
17
The second estimator assumes that returns are generated according to
Sharpe’s (1963) one-factor model given by:
rit = αi + β i rmt + ²it ,
where rmt is the period t return on the value-weighted portfolio of stocks
traded on the NYSE, AMEX and NASDAQ t. Then the covariance estimator
is,
(21)
S1 = s2m BB 0 + D.
Here B is the vector of β’s, s2m is the sample variance of rmt , and D has the
sample variances of the residuals along the diagnal and zeros elsewhere.
The third estimator is the optimal shrinkage estimator of Ledoit (1999).
It is a weighted average of the sample covariance matrix and the one-factor
model-based estimator:
SL =
α
α
S1 + (1 − )SN ,
T
T
where α is a parameter that determines the shrinkage intensity that is estimated using the data. Ledoit (1999) shows this estimator outperforms the
constant correlation model (Elton and Gruber (1973), Schwert and Seguin
(1990)), the single-factor model, the industry factor model, and the principal
component model with five principal components.6 For the fourth set of
stocks we consider the Fama and French (1993) three-factor model, and the
Connor and Korajczyk’s (1986, 1988) five-factor model and a three-factor
version that includes only the first three of the five factors. This gives three
additional covariance matrix estimators each corresponding to one of these
multifactor models7 .
6
For tracking error variance minimization, we also considered Ledoit’s (1996) estimator
that shrinks the sample covariance matrix toward the identity matrix. The results are
similar and we do not report them.
7
For tracking error variance minimization, the loadings on the first factor in these
multifactor models are set to zero for every stock.
18
Finally we consider several covariance matrix estimators that use daily
return data. These include the daily return sample covariance matrix, daily
one-factor model, daily Fama-French three-factor model, and daily ConnorKorajczyk five-factor and three-factor models. These models are similar to
the corresponding monthly models. However, we incorporate the corrections
for microstructure effects suggested by Scholes and Williams (1977), Dimson
(1979) and Cohen et al. (1983). We also develop a new estimator for the
covariance matrix of returns using daily return data that nests these three
estimators. Details on the estimation of the covariance matrix of monthly
returns using daily return data that allow for microstructure effects are provided in the Appendix.
3.3
Empirical Results
Table 4 gives the characteristics of minimum variance portfolios constructed
using various covariance matrix estimates. As expected, the sample covariance matrix estimated using monthly return data is singular. When there
are no restrictions on portfolio weights all factor models perform equally
well with about 40% of the portfolio weights being negative. The portfolios involve taking a substantial amount of short positions, and they sum
to more than 50 percent of the portfolio value. The shrinkage estimator
proposed by Ledoit (1999) improves the performance of the minimum variance portfolio — an 8 percent improvement when compared to the Sharpe
single-factor model and a 4 percent improvement when compared to the fivefactor Connor-Korajczyk model. Since the Ledoit estimator is a particular
weighted average of the one-factor model and the sample covariance matrix,
we examined whether a simple average of the two estimators would do equally
well. The random average of the one-factor model and the sample covariance
matrix has an annualized out-of-sample standard deviation of 10.34 percentage points (not reported in Table 4) which is not much different from the
19
10.76 percentage points for the Ledoit estimator. We thus suspect that the
sampling errors associated with the estimated optimal shrinkage weights in
the Ledoit estimator is rather large.
Next, let us turn to the case when the no-shortsales restriction is imposed. There is a unique solution to the portfolio variance minimization
problem using the sample covariance matrix although it is singular. Surprisingly the minimum variance portfolio constructed using the monthly sample
covariance matrix compares favorably with all the other covariance matrix
estimators. The out-of-sample annualized standard deviation is about 12
percentage points per year for all of the estimators, including the shrinkage
estimator of Ledoit. Imposing the no-shortsales restriction leads to a small
increase — between 8 and 14 percent — in the standard deviation of the minimum variance portfolios constructed using factor models. This decline in
the performance is consistent with the observation by Green and Hollifield
(1992) that shortsales restrictions probably do not hold in the population.
The number of assets in the portfolio varies from a low of 24 for the sample covariance matrix estimator to a high of 42 for the Connor-Korajczyk
five-factor model. In comparison, the equally weighted portfolio of the 500
stocks has an annualized standard deviation of 17 percentage points, the
value weighted portfolio of the 500 stocks has a standard deviation of 16
percentage points. A portfolio of randomly picked 25 stocks has a standard
deviation of 18 percentage points which is 40 percent more than that of the
optimal minimum variance portfolio constructed using the sample covariance
matrix subject to the no-shortsales restrictions. Imposing an upper bound
of two percentage points on portfolio weights in addition to a lower bound
of zero does not affect the out-of-sample variance of the resulting minimum
variance portfolios in any significant way.
The out-of-sample standard deviations of the minimum variance portfolios are on average about four times as large as the corresponding in-sample
standard deviations when portfolio weights are not constrained (except that
20
they must sum to unity). This has been observed by Frost and Savarino
(1988) and termed “in-sample optimism.” When lower and upper bound on
portfolio weights are imposed the out-of-sample standard deviations are less
than twice the corresponding in sample numbers. Therefore when portfolio
weight constraints are imposed, the in-sample standard deviations become
better predictors of the corresponding out-of-sample standard deviations.
When daily returns are used, the sample covariance matrix estimator
performs the best when there are no portfolio weight constraints. The
corrections for microstructure effects suggested in the literature do not lead
to superior performance. The short positions are smaller for the factor
models. They sum to approximately 120 percent of the portfolio value for
the sample covariance matrices and 50 percent of the portfolio value for factor
models. When portfolio weights are constrained to be nonnegative all the
models perform about equally. The number of stocks in the portfolio range
from a low of 39 for the one-factor model (new) to a high of 65 for the daily
sample covariance matrix.
Table 5 gives the corresponding numbers for the minimum tracking error
portfolios.
Table 6 reports the t-tests for the difference between the mean returns and
mean squared returns for the portfolios compared with the nonnegativityconstrained portfolio from the sample covariance matrix. Since the differences in mean returns are all insignificant, the t-test for the difference in
squared returns serve as a test for the difference in return variances, which
is the focus of our study. We can see that for both portfolio variance minimization and tracking error minimization, once the nonnegativity constraint
is imposed, the more sophisticated estimators do not in general give better
out-of-sample performance than the monthly return sample covariance matrix. However, there is evidence that using daily returns can achieve smaller
out-of-sample tracking error. Why the use of daily returns can help for the
21
tracking error minimization case and not the total risk minimization case is
an issue of future investigation.
4
Concluding Remarks
Investors often face or voluntarily impose no-shortsales restrictions in optimal
risk reduction. However, Green and Hollifield (1992) argue that the optimal
portfolios should involve extreme positive and negative weights. We reconcile this apparent contradiction. We show that imposing the no-shortsales
constraint on a stock is equivalent to reducing the sample covariances of the
corresponding asset with other assets by a certain amount and then constructing the optimal portfolio. Since the extreme negative portfolio weights
are associated with the extreme high row sums of the estimated covariance
matrix, and these extreme high row sums are more likely to be caused by
estimation error, reducing these row sums reduces sampling error. It follows
from the theory of shrinkage estimators that imposing the no-shortsales constraint can help even when the constraint does not hold in the population.
The upper bounds on portfolio weights have similar shrinking effects. Using
a Monte Carlo simulation we show that this is indeed what is happening in
optimal portfolio risk reduction.
Imposing the weight constraints also introduces specification error, since
these constraints are violated in population, as demonstrated by Green and
Hollifield. So the net effect depends on the tradeoff between the reduction
in sampling error and the increase in specification error, and whether the
net effect is positive or negative is an empirical issue. Frost and Savarino
(1988) and our simulation study show that for the sample covariance matrix, imposing the weight constraints helps. Our empirical results also show
that for factor models and shrinkage estimators, imposing the weight constraints hurts slightly. On the other hand, once the no-shortsales restriction
is imposed, portfolios constructed from the sample covariance matrix per22
form as well as portfolios constructed using covariance matrices, estimated
using factor models and shrinkage methods.
The effect of the portfolio weight constraints has been examined empirically. Our result on the shrinkage effect of such constraints is the first
“theoretical” result on this issue. One potential use of this result is to gauge
the tightness of a set of portfolio weight constraints by the change from S to
S̃. Suppose the change is quite significant. This means that the effect of the
constraints is quite strong and the constraints are quite tight. This measure
of tightness can help portfolio managers to decide the appropriate lower and
upper bounds on portfolio weights when constructing the optimal portfolios.
Our empirical findings raise the following questions that deserve further
examination. (a) Why the use of daily data can help achieve better tracking
error reduction but not portfolio variance reduction? A possible explanation
is that covariances caused by exposure to the dominant factor change much
more over time when compared to those that are caused by exposure to other
factors. While the use of daily return data helps to reduce the sampling error it does not take into account time variations in the covariance structure.
This may be less important once the dominant factor is removed as in the
case of tracking error minimization. (b) Why corrections for microstructure
effects do not help even though they provide more precise estimates of factor
betas? This could be due to the greater instability of microstructure effects across stocks over time. (c) Why do we find large in-sample-optimism
even for factor models and shrinkage estimators?. Past researchers appear
to attribute this to sampling error. Additional Monte Carlo simulations indicate that when returns have an exact factor structure, and the covariance
matrices are constructed using estimated factors, thel in-sample optimism is
attenuated. This means that sampling error alone is unlikely to provide a
complete answer. Part of the reason has to be that either returns do not have
an exact factor structure or that the the structure of the covariance matrix
is changing over time. Hence shrinkage methods and factor models, which
23
are intended to reduce sampling errors, can only achieve limited success.
5
Appendix: Covariance Matrix Estimators
that Use Daily Returns
This appendix describes how we estimate monthly covariance matrices using daily return data after taking into account serial correlations and crosscorrelations at various leads and lags induced by microstructure effects. We
consider the estimators proposed by Scholes and Williams (1977), Dimson
(1979), and Cohen, et al (1983, hereafter CHMSW). We point out that the
CHMSW estimator of the sample covariance matrix is usually not positive
semi-definite. We then propose a new methodology to estimate the covariance matrices of monthly returns using data on daily returns that are always
positive semi-definite. Our estimator uses the fact that the continuously
componded monthly return is the sum of continuously compounded daily returns. Our estimator can be readily modified based on restrictions imposed
by microstructure models.
The CHMSW estimator is based on the following relationship:
t
t
, rk,t
) = cov(rj,t , rk,t ) +
cov(rj,t
L
X
cov(rj,t , rk,t−n )
n=1
+
L
X
cov(rj,t−n , rk,t ),
(22)
n=1
for any pair of stocks j and k, j 6= k. Here rt denotes the true date t return and r denotes the observed return, t and t − n denote dates t and
t
t
t − n. Based on this relation we can estimate cov(rj,t
, rk,t
) using observed
daily returns. In practice, L is set to three and the variances are estimated
using sample variances without any adjustment (Cohen et al. (1983) and
Shanken (1987)). The Scholes-Williams estimator is the special case of the
24
above, with L set to one. The estimate of the full sample covariance matrix
using the CHMSW method is denoted as “Daily Sample Covariance Matrix
(CHMSW)” in Tables 4-6.
Equation (22) is also valid if either asset (or both) is a portfolio. For
a well-diversified portfolio, (22) is approximately valid for its variance also.
Based on this, we can estimate a stock’s beta as its covariance with the
market portfolio divided by the market portfolio’s variance. The daily onefactor models (Scholes-Williams and CHMSW) in Tables 4-6 are estimated
using this strategy.
There is a problem with these estimators. The estimated covariance matrix, constructed from individual covariances and variances, is not positive
semi-definite. This is problematic for portfolio optimization. We propose a
new estimator that does not have this problem.8 Notice that monthly log
returns are simply sums of daily log returns:
ri,τ =
τm
X
ri,t .
t=(τ −1)m+1
Here τ is month τ , t is day t, and m is the number of days in a month. Then
the monthly return covariance is
cov(ri,τ , rj,τ ) =
τm
X
τm
X
cov(ri,t , rj,s ).
t=(τ −1)m+1 s=(τ −1)m+1
Unlike the CHMSW estimator (22), the above is valid for any i and j, even
if either one (or both) is a portfolio. Assuming covariance stationarity as
usual, we can drop the time subscripts, and get
covM (ri , rj ) = m · covD (ri,t , rj,t )
8
There is a difference between our approach and the approach taken by Scholes and
Williams and CHMSW. While they want to estimate the “true” covariances and betas
using the daily returns, we want to use daily returns to estimate the covariances and betas
of monthly returns.
25
+ (m − 1) · (covD (ri , rj,t+1 ) + covD (ri,t+1 , rj,t ))
+ (m − 2) · (covD (ri , rj,t+2 ) + covD (ri,t+2 , rj,t ))
+ ···
+ (covD (ri,t , rj,t+m−1 ) + covD (ri,t+m−1 , rj,t )).
(23)
The superscripts M and D denote the covariance of the monthly returns and
the covariance of the daily returns, respectively. We set m = 21 for there are
about 21 trading days in one month.
An obvious approach is to use the sample counterpart of the RHS of
(23) to estimate the monthly return covariance. However, to guarantee that
the covariance matrix is positive semi-definite, we need to further adjust the
covariance estimates (23) slightly. Let
R0 = (rt,i − r.,i )t=1,...,T ; i=1,...,N ,
the matrix of demeaned returns. For j = 1, · · · , m − 1, let R−j be the same
size as R0 , with the first j rows set to zeros, and the remaining T − j rows
the same as R0 ’s first T − j rows (i.e., R−j is the matrix of lag-j demeaned
returns). Let
Ωj = R00 R−j /T,
S = m Ω0 +
m−1
X
j=1
(m − j)(Ωj + Ω0j ).
(24)
Then S is positive semi-definite (see Newy and West (1987) for the proof)
and S is a consistent estimator of the RHS of (23). The covariance matrix
estimated this way is denoted as “Daily Sample Covariance Matrix (New)”
in Tables 4 - 6.
If we assume a k−factor model, then the beta estimates are
b = (var(f
ˆ ))−1 cov(f,
ˆ
r).
Here b is k × N, var(f
ˆ ) is the estimated factor covariance matrix (of size
k × k) and is estimated according to (24), and cov(f,
ˆ
r) is estimated (similar
26
to (24)) by


m−1
X
1
0
cov(f,
ˆ
r) = m F00 R0 +
(m − j)(F00 R−j + F−j
R0 ) ,
T
j=1
with F−j defined similarly as R−j .
For factor models, the residual covariance matrix is assumed diagonal
and the residual variances are estimated by the sample variances of residuals calculated from observed stock returns, observed factor returns, and the
estimated betas. In Tables 4-6, the estimators denoted as “Daily 1 Factor
Model (New),” “Daily Connor-Korajczyk 3 Factor Model,” “Daily ConnorKorajczyk 5 Factor Model,” and “Daily Fama-French 3 Factor Model,” are
all estimated using this strategy.
We construct the daily Connor-Korajczyk factors and Fama-French factors that follow the same procedure as outlined in Connor and Korajczyk
(1988) (using daily returns instead of monthly returns) and Fama and French
(1993).
Finally, in Tables 4-6, the estimator “Daily Sample Covariance Matrix”
denotes the sample covariance of daily returns.
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31
Table 1: Row sums, unconstrained portfolio weights, Lagrangian multipliers, row sums
This table shows the effects of nonnegativity constraints on portfolio weights. The covariance
matrix is the sample covariance matrix of monthly returns of a random sample of 30 stocks
from 1988-1997 (120 months). The column "row sums" is the row sums of the sample
covariance matrix. "S_tilde" is the shrinked version of the sample covariance matrix, defined
in Proposition 1. All numbers are scaled by multiplying by 100.
row sums
3.476
5.429
3.531
3.796
10.417
6.994
1.290
3.757
5.090
7.389
2.696
0.284
2.444
3.611
5.361
10.234
3.388
7.291
1.905
4.706
2.395
2.286
4.030
2.858
4.126
3.608
5.124
7.635
4.171
6.074
unconstrained
portfolio weights
6.913
-0.258
6.624
5.045
-0.235
-6.289
15.038
1.099
-0.936
1.806
2.561
8.706
14.211
4.812
1.647
-2.457
4.067
0.294
12.931
1.401
8.887
10.457
3.153
4.638
2.705
-0.730
-1.604
-0.704
-0.816
-2.969
Lagrangian
multipliers
0
0
0
0
0.0454
0.0467
0
0
0.0171
0
0
0
0
0
0
0.0723
0
0.0229
0
0
0
0
0
0
0
0.0046
0.0265
0.0274
0.0045
0.0322
row sums of
S_tilde
3.176
5.129
3.231
3.496
8.754
5.293
0.990
3.457
4.277
7.089
2.396
-0.015
2.144
3.311
5.061
7.764
3.088
6.303
1.605
4.406
2.096
1.987
3.730
2.558
3.826
3.171
4.030
6.512
3.736
4.807
Corrcoef(row sums, unconstrained portfolio weights) = - 0.725.
Corrcoef(row sums, Lagrangian multipliers) = 0.766.
Corrcoef(row sums, changes in row sums) = - 0.798.
std(row sums) = 2.389.
std(row sums of S_tilde) = 1.977.
changes in
constrained
row sums
portfolio weights
-0.300
5.179
-0.300
0.133
-0.300
3.087
-0.300
3.948
-1.663
0.000
-1.701
0.000
-0.300
15.303
-0.300
1.496
-0.814
0.000
-0.300
0.400
-0.300
3.832
-0.300
10.136
-0.300
11.670
-0.300
3.345
-0.300
0.100
-2.470
0.000
-0.300
4.119
-0.988
0.000
-0.300
12.180
-0.300
1.092
-0.300
6.801
-0.300
7.726
-0.300
3.864
-0.300
3.683
-0.300
1.905
-0.438
0.000
-1.094
0.000
-1.123
0.000
-0.435
0.000
-1.267
0.000
Table 2: Row sums, unconstrained portfolio weights, Lagrangian multipliers, row
sums of the shrinked covariance matrix estimates, and constrained portfolio weights
This table shows the effects of both nonnegativity constraints and upper bounds on portfolio weights.
The covariance matrix is the sample covariance matrix of monthly return data of a random sample
of 30 stocks from 1988-1997 (120 months). The column "row sums" is the row sums of the sample
covariance matrix. "S_tilde" is the shrinked version of the sample covariance matrix, defined in
Proposition 3. All numbers are scaled by multiplying by 100.
Unconstrained
Lagrangian Multipliers Row sums Changes in Constrained
Row sums portfolio weights lower bounds upper bounds of S_tilde row sums portfolio weights
3.476
6.913
0
0
5.429
-0.258
0
0
3.531
6.624
0
0
3.796
5.045
0
0
10.417
-0.235
0.031
0
6.994
-6.289
0.0489
0
1.290
15.038
0
0.0194
3.757
1.099
0
0
5.090
-0.936
0.0158
0
7.389
1.806
0
0
2.696
2.561
0
0
0.284
8.706
0
0.0082
2.444
14.211
0
0.0078
3.611
4.812
0
0
5.361
1.647
0
0
10.234
-2.457
0.0763
0
3.388
4.067
0
0
7.291
0.294
0.0339
0
1.905
12.931
0
0.0095
4.706
1.401
0
0
2.395
8.887
0
0
2.286
10.457
0
0
4.030
3.153
0
0
2.858
4.638
0
0
4.126
2.705
0
0
3.608
-0.730
0
0
5.124
-1.604
0.0192
0
7.635
-0.704
0.023
0
4.171
-0.816
0.0035
0
6.074
-2.969
0.0303
0
Corrcoef(row sums, unconstrained portfolio weights) = -0.725.
Corrcoef(row sums, Lagrangian multipliers on lower bounds) = 0.766.
Corrcoef(row sums, Lagrangian multipliers on upper bounds) = -0.467.
Corrcoef(row sums, changes in row sums) = -0.819.
std(row sums) = 2.389.
std(row sums of S_tilde) = 1.938.
3.24
5.19
3.29
3.56
9.25
5.29
1.63
3.52
4.38
7.15
2.46
0.29
2.44
3.37
5.12
7.71
3.15
6.04
1.95
4.47
2.16
2.05
3.79
2.62
3.89
3.37
4.31
6.71
3.83
4.93
-0.24
-0.24
-0.24
-0.24
-1.17
-1.70
0.34
-0.24
-0.71
-0.24
-0.24
0.01
0.00
-0.24
-0.24
-2.53
-0.24
-1.26
0.05
-0.24
-0.24
-0.24
-0.24
-0.24
-0.24
-0.24
-0.81
-0.93
-0.34
-1.15
5.225
0.147
3.752
4.067
0.000
0.000
10.000
1.276
0.000
0.405
5.216
10.000
10.000
4.392
0.294
0.000
4.375
0.000
10.000
0.875
9.516
9.381
4.448
4.514
1.940
0.179
0.000
0.000
0.000
0.000
Table 3: A simulation study of the effect of portfolio weight constraints
The sample covariance matrix of the monthly returns of 30 randomly selected stocks from
1988-1997 are used as the true covariance matrix. Return data are simulated using the true covariance.
The table reports the ex post standard errors of four global minimum variance portfolios:
The unconstrained and nonnegativity constrained ones from the true covariance matrix and the
sample covariance matrix of the simulated returns. These standard errors are in percentage per year.
The results are averages over 500 independent replications.
Estimation
True Covariance Matrix
Window (months)
Unconstrained
Constrained
60
8.668
9.110
120
8.668
9.110
180
8.668
9.110
Sample Covariance Matrix
Unconstrained
Constrained
12.211
10.403
9.983
9.763
9.476
9.553
Table 4: Ex post mean, standard deviation and other characteristics of the minimum variance portfolios.
Mean and standard deviation are in percentage per year. Maximum and minimum portfolio weights and short interest are in percentages.
Short interest is the sum of negative portfolio weights. The C after each estimator indicate the nonnegativity constrained minimum
variance portfolio, the D indicates the minimum variance portfolio with both nonnegativity constraint and and upper bound of 2%.
For the equally-weighted and value-weighted portfolios of 25 stocks, the 25 stocks are a random sample of the 500 ones.
The in-sample estimate of standard deviations (column 4) are the average in-sample estimates of the portfolios' standard deviations.
Covariance Matrix Estiamtor
Monthly Sample Covariance Matrix, C
Monthly Sample Covariance Matrix, D
One Factor Model
One Factor Model C
One Factor Model D
Ledoit
Ledoit C
Ledoit D
Connor-Korajczyk 3 Factor
Connor-Korajczyk 3 Factor C
Connor-Korajczyk 3 Factor D
Connor-Korajczyk 5 Factor
Connor-Korajczyk 5 Factor C
Connor-Korajczyk 5 Factor D
Fama-French 3 Factor
Fama-French 3 Factor C
Fama-French 3 Factor D
Daily Sample Covariance Matrix
Daily Sample Covariance Matrix C
Daily Sample Covariance Matrix D
Daily Sample Covariance Matrix (CHMSW) C
Daily Sample Covariance Matrix (CHMSW) D
Daily Sample Covariance Matrix (New)
Daily Sample Covariance Matrix (New) C
Daily Sample Covariance Matrix (New) D
Daily 1 Factor Model (Scholes-Williams)
Daily 1 Factor Model (Scholes-Williams) C
Daily 1 Factor Model (Scholes-Williams) D
Daily 1 Factor Model (CHMSW)
Daily 1 Factor Model (CHMSW) C
Daily 1 Factor Model (CHMSW) D
Daily 1 Factor Model (New)
Daily 1 Factor Model (New) C
Daily 1 Factor Model (New) D
Daily Connor-Korajczyk 3 Factor
Daily Connor-Korajczyk 3 Factor C
Daily Connor-Korajczyk 3 Factor D
Daily Connor-Korajczyk 5 Factor
Daily Connor-Korajczyk 5 Factor C
Daily Connor-Korajczyk 5 Factor D
Daily Fama-French 3 Factor
Daily Fama-French 3 Factor C
Daily Fama-French 3 Factor D
Equally-weighted portfolio of the 500 stocks
Vaue-weighted portfolio of the 500 stocks
Equally-weighted portfolio of the 25 stocks
Vaue-weighted portfolio of the 25 stocks
Standard
In-sample
Maximum Minimum
Mean
Deviation estimate of std Weight
Weight
13.55
12.43
7.08
18.4
0
13.55
12.85
8.25
2.0
0
13.99
11.69
2.55
3.8
-0.9
12.68
12.62
5.99
11.3
0
13.51
12.50
6.92
2.0
0
13.09
10.76
2.96
4.9
-1.7
12.79
12.29
6.90
13.4
0
13.49
12.43
7.79
2.0
0
13.41
11.46
2.51
3.8
-1
12.71
12.76
5.73
11.1
0
13.25
12.52
6.61
2.0
0
12.89
11.21
2.62
4.1
-1.3
12.26
12.71
6.54
13.2
0
13.18
12.41
7.45
2.0
0
13.04
11.35
2.73
4.2
-1.4
12.65
12.38
6.59
12.2
0
13.34
12.53
7.49
2.0
0
14.06
10.64
3.42
6.3
-2.5
13.95
12.34
5.70
8.8
0
14.22
12.28
6.06
2.0
0
13.89
12.31
5.87
9.1
0
14.18
12.25
6.25
2.0
0
13.94
10.60
3.50
6.3
-2.6
13.81
12.26
5.96
9.0
0
14.12
12.21
6.33
2.0
0
15.03
11.77
2.92
3.4
-0.9
13.35
12.32
6.07
9
0
13.75
12.26
6.61
2
0
14.57
11.73
2.98
3.4
-0.9
13.10
12.18
6.33
8.7
0
13.46
12.28
6.91
2
0
13.86
11.80
2.52
3.6
-0.8
12.96
12.54
6.09
11.9
0
13.16
12.31
6.91
2
0
13.56
11.53
2.42
3.3
-0.9
12.51
12.39
5.86
10.6
0
12.83
12.37
6.50
2
0
13.23
11.38
2.62
3.8
-1.1
12.68
12.27
6.24
11.8
0
12.87
12.31
6.93
2
0
13.07
11.25
2.84
4
-1.4
13.03
12.03
6.73
12.7
0
13.09
12.15
7.51
2
0
14.52
17.48
17.83
0.2
0
13.39
15.60
7.3
0
15.16
17.78
4
0
14.25
17.79
30
0
Short No. of Posi. No. of Neg.
Interest
Weights
Weights
0
24.1
0
0
59.5
0
-50.7
268.8
229.9
0
39.3
0
0
63.1
0
-80.6
283.2
216.0
0
39.7
0
0
65.4
0
-50.8
281.8
216.7
0
42.6
0
0
64.9
0
-61.0
285.6
213.2
0
39.7
0
0
64.8
0
-63.5
284.1
214.8
0
40.1
0
0
64.0
0
-122.4
270.7
229
0
64.7
0
0
81.1
0
0
62.5
0
0
80.1
0
-128.5
269.5
230
0
62.9
0
0
79.5
0
-52.2
270
228.9
0
51.2
0
0
68.8
0
-52.9
264.2
234.7
0
47.2
0
0
65.1
0
-50.4
261.2
237.5
0
39.3
0
0
60.6
0
-49.1
287.6
211
0
51.4
0
0
69.3
0
-57.9
284.3
214.7
0
48.5
0
0
67.8
0
-66.9
276.8
222.1
0
40.7
0
0
63.6
0
0
500
0
0
500
0
0
25
0
0
25
0
Table 5: Ex post mean, standard deviation and other characteristics of the minimum tracking error variance portfolios.
Mean and standard deviation are those of the tracking errors when tracking the S&P 500 index. Both are in percentage per year.
Maximum and minimum portfolio weights and short interest are in percentages.
Short interest is the sum of negative portfolio weights. The C after each estimator indicate the nonnegativity constrained minimum
variance portfolio, the D indicates the minimum variance portfolio with both nonnegativity constraint and and upper bound of 2%.
For the equally-weighted and value-weighted portfolios of 25 stocks, the 25 stocks are a random sample of the 500 ones.
Covariance Matrix Estiamtor
Monthly Sample Covariance Matrix, C
Monthly Sample Covariance Matrix, D
One Factor Model
One Factor Model C
One Factor Model D
Ledoit
Ledoit C
Ledoit D
Connor-Korajczyk 3 Factor
Connor-Korajczyk 3 Factor C
Connor-Korajczyk 3 Factor D
Connor-Korajczyk 5 Factor
Connor-Korajczyk 5 Factor C
Connor-Korajczyk 5 Factor D
Fama-French 3 Factor
Fama-French 3 Factor C
Fama-French 3 Factor D
Daily Sample Covariance Matrix
Daily Sample Covariance Matrix C
Daily Sample Covariance Matrix D
Daily Sample Covariance Matrix (CHMSW) C
Daily Sample Covariance Matrix (CHMSW) D
Daily Sample Covariance Matrix (New)
Daily Sample Covariance Matrix (New) C
Daily Sample Covariance Matrix (New) D
Daily 1 Factor Model (Scholes-Williams)
Daily 1 Factor Model (Scholes-Williams) C
Daily 1 Factor Model (Scholes-Williams) D
Daily 1 Factor Model (CHMSW)
Daily 1 Factor Model (CHMSW) C
Daily 1 Factor Model (CHMSW) D
Daily 1 Factor Model (New)
Daily 1 Factor Model (New) C
Daily 1 Factor Model (New) D
Daily Connor-Korajczyk 3 Factor
Daily Connor-Korajczyk 3 Factor C
Daily Connor-Korajczyk 3 Factor D
Daily Connor-Korajczyk 5 Factor
Daily Connor-Korajczyk 5 Factor C
Daily Connor-Korajczyk 5 Factor D
Daily Fama-French 3 Factor
Daily Fama-French 3 Factor C
Daily Fama-French 3 Factor D
Equally-weighted portfolio of the 500 stocks
Vaue-weighted portfolio of the 500 stocks
Equally-weighted portfolio of the 25 stocks
Vaue-weighted portfolio of the 25 stocks
Standard
In-sample
Maximum Minimum
Mean
Deviation estimate of std Weight
Weight
4.28
3.36
0.000
2.4
0
4.23
3.40
0.032
1.8
0
4.91
5.04
1.050
0.9
0
4.91
5.04
1.050
0.9
0
4.91
5.04
1.050
0.9
0
4.18
3.48
1.243
1.8
-0.3
4.21
3.34
1.311
2.2
0
4.27
3.36
1.313
1.8
0
4.14
4.71
1.130
1.1
-0.1
4.08
4.70
1.141
1.1
0
4.08
4.70
1.141
1.1
0
4.06
4.52
1.121
1.1
-0.1
4.00
4.51
1.138
1.2
0
4.00
4.51
1.138
1.2
0
3.76
4.41
1.223
1.4
-0.2
3.71
4.39
1.247
1.5
0
3.71
4.39
1.247
1.5
0
3.69
2.94
1.525
5.3
-1
3.93
2.78
1.772
5.4
0
4.17
2.96
1.911
2
0
3.92
2.75
1.787
5.4
0
4.17
2.92
1.916
2
0
3.72
2.92
1.532
5.3
-1
3.93
2.73
1.784
5.4
0
4.17
2.89
1.916
2
0
4.88
5.18
1.265
0.7
0
4.88
5.18
1.265
0.7
0
4.88
5.18
1.265
0.7
0
4.88
5.07
1.034
0.7
0
4.88
5.07
1.034
0.7
0
4.88
5.07
1.212
0.7
0
4.97
5.06
1.034
0.9
0
4.97
5.06
1.034
0.9
0
4.97
5.06
1.034
0.9
0
4.18
4.54
1.166
1.1
-0.1
4.16
4.53
1.179
1.1
0
4.16
4.53
1.179
1.1
0
4.06
4.40
1.184
1.1
-0.1
3.99
4.38
1.200
1.2
0
3.99
4.38
1.200
1.2
0
3.82
4.50
1.241
1.3
-0.2
3.76
4.50
1.266
1.5
0
3.77
4.51
1.266
1.4
0
4.92
6.58
6.841
0.2
0
3.6
2.37
6.7
0
4.66
8.62
4
0
4.13
8.24
29.7
0
Short
Interest
0
0
0
0
0
-8.7
0
0
-1.5
0
0
-2.1
0
0
-3.3
0
0
-36.6
0
0
0
0
-36.8
0
0
0
0
0
0
0
0
0
0
0
-1.9
0
0
-2.6
0
0
-3.7
0
0
0
0
0
0
No. of Posi. No. of Neg.
Weights
Weights
200
0
194
0
499
0.7
499
0
499
0
391
107.1
314
0
312
0
467.5
31.1
449
0
449
0
457.7
40.8
430.9
0
430.9
0
433.1
64.3
394.6
0
394.5
0
322.4
176.3
231.3
0
227.6
0
231.4
0
226.3
0
322.2
176.5
228
0
224.2
0
498.8
1.1
498.8
0
498.8
0
499.9
0.1
499.9
0
499.9
0
499.6
0.3
499.6
0
499.6
0
454.9
43.6
431.7
0
431.7
0
445.4
52.8
415.1
0
415
0
427.8
70.1
384.1
0
383.8
0
500
0
500
0
25
0
25
0
Table 6: T-tests of equal mean returns and equal mean squared returns
This table reports the t-tests of equal mean returns and equal mean squared returns of the minimum variance and minimum
tracking error variance portfolios. For each such portfolio, we test whether its mean return and mean squared return are
statistically different from those of the no-short-sale constrained portfolio constructed from the sample covariance matrix
of monthly returns. The t-tests are calculated as mean(x1-x2)/std(x1-x2)*sqrt(No of Obs), where x1 is either the return
( in testing the equality of mean return) or squared return (in testing the equality of the mean squared return) of the
constrained portfolio constructed from the monthly return sample covariance matrix, x2 is either the return or squared
return of the portfolio constructed from another covariance matrix estimator.
Mininum Variance Portfolio
Equality in mean Equality in mean
return
squared return
Monthly Sample Covariance Matrix, D
One Factor Model
One Factor Model C
One Factor Model D
Ledoit
Ledoit C
Ledoit D
Connor-Korajczyk 3 Factor
Connor-Korajczyk 3 Factor C
Connor-Korajczyk 3 Factor D
Connor-Korajczyk 5 Factor
Connor-Korajczyk 5 Factor C
Connor-Korajczyk 5 Factor D
Fama-French 3 Factor
Fama-French 3 Factor C
Fama-French 3 Factor D
Daily Sample Covariance Matrix
Daily Sample Covariance Matrix C
Daily Sample Covariance Matrix D
Daily Sample Covariance Matrix (CHMSW) C
Daily Sample Covariance Matrix (CHMSW) D
Daily Sample Covariance Matrix (New)
Daily Sample Covariance Matrix (New) C
Daily Sample Covariance Matrix (New) D
Daily 1 Factor Model (Scholes-Williams)
Daily 1 Factor Model (Scholes-Williams) C
Daily 1 Factor Model (Scholes-Williams) D
Daily 1 Factor Model (CHMSW)
Daily 1 Factor Model (CHMSW) C
Daily 1 Factor Model (CHMSW) D
Daily 1 Factor Model (New)
Daily 1 Factor Model (New) C
Daily 1 Factor Model (New) D
Daily Connor-Korajczyk 3 Factor
Daily Connor-Korajczyk 3 Factor C
Daily Connor-Korajczyk 3 Factor D
Daily Connor-Korajczyk 5 Factor
Daily Connor-Korajczyk 5 Factor C
Daily Connor-Korajczyk 5 Factor D
Daily Fama-French 3 Factor
Daily Fama-French 3 Factor C
Daily Fama-French 3 Factor D
Equally-weighted portfolio of the 500 stocks
Vaue-weighted portfolio of the 500 stocks
Equally-weighted portfolio of the 25 stocks
Vaue-weighted portfolio of the 25 stocks
-0.01
0.26
-0.76
-0.05
-0.40
-1.05
-0.07
-0.10
-0.73
-0.26
-0.47
-1.41
-0.37
-0.38
-1.08
-0.22
0.38
0.35
0.65
0.29
0.62
0.29
0.23
0.57
0.88
-0.18
0.18
0.57
-0.37
-0.08
0.16
-0.45
-0.31
0.00
-0.85
-0.61
-0.21
-0.77
-0.60
-0.31
-0.51
-0.44
0.54
-0.10
0.82
0.32
1.36
-1.16
0.19
0.15
-2.99
-0.60
-0.02
-1.93
0.41
0.13
-2.45
0.33
-0.13
-2.34
-0.29
0.23
-3.86
-0.11
-0.22
-0.20
-0.31
-3.82
-0.35
-0.44
-0.91
-0.24
-0.35
-1.01
-0.53
-0.36
-0.95
0.10
-0.32
-1.28
-0.28
-0.28
-2.02
-0.55
-0.44
-2.41
-0.95
-0.78
6.79
5.25
6.94
7.07
Minimum Tracking Error
Equality in mean Equality in mean
return
squared return
-0.53
1.06
1.06
1.06
-0.32
-0.30
-0.03
-0.24
-0.36
-0.36
-0.42
-0.55
-0.55
-1.02
-1.15
-1.15
-1.30
-0.87
-0.28
-0.89
-0.29
-1.24
-0.87
-0.29
0.96
0.95
0.95
1.01
1.01
1.01
1.15
1.15
1.15
-0.19
-0.24
-0.24
-0.44
-0.61
-0.60
-0.82
-0.95
-0.93
0.73
-1.48
0.29
-0.10
1.49
5.35
5.34
5.34
1.06
-0.32
0.08
5.12
5.12
5.12
5.00
4.86
4.87
4.46
4.42
4.41
-3.29
-4.43
-3.26
-4.82
-3.72
-3.40
-5.09
-4.11
5.33
5.33
5.33
5.21
5.21
5.21
5.34
5.35
5.35
5.26
5.28
5.28
4.63
4.57
4.57
4.28
4.25
4.26
6.76
-7.33
9.48
9.88