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DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Piotr Fiszeder  
Nicolaus Copernicus University in Toruń 

Minimum Variance Portfolio Selection  
for Large Number of Stocks – Application  

of Time-Varying Covariance Matrices 

A b s t r a c t. An evaluation of the efficiency of different methods of the minimum variance 
portfolio selection was performed for seventy stocks from the Warsaw Stock Exchange. Eight 
specifications of multivariate GARCH models and six other methods were used. The application 
of all considered GARCH-class models was more efficient in stocks allocation than the imple-
mentation of the other analyzed methods. The simple specifications of multivariate GARCH 
models, whose parameters were estimated in two stages, like the DCC and CCC models were the 
best performing models. 
K e y w o r d s: multivariate GARCH models, time-varying covariance matrix, portfolio selection.  

Introduction  

 The selection of estimators of the population mean, variance and covariance 
of financial returns and closely connected with it the selection of forecasting 
methods of the population mean, variance and covariance of returns plays 
a vital role in the construction of efficient portfolios. There is a consensus 
among both financial market practitioners and scientists, that financial returns 
are difficult to forecast and the portfolio construction process according to the 
criteria proposed by Markowitz (1952, 1959) is very sensitive to the selection of 
an estimator of the expected value of returns (see Michaud, 1989; Best, Grauer, 
1991; Chopra, Ziemba, 1993). Small differences in estimates of the expected 
returns often lead to a meaningful portfolio reconstruction (see e.g. Jobson, 
Korkie, 1980). It is believed that the estimation of population variances and 
covariances of returns is easier than the estimation of population means (Merton 
1980; Nelson, 1992), however the selection of estimators of population vari-
ances and covariances of returns has also a significant impact on an asset alloca-
tion (see e.g. Litterman, Winkelmann, 1998; Chan, Karceski, Lakonishok, 
1999). Traditionally used estimators of the population mean, variance and co-



Piotr Fiszeder 88

variance, namely the arithmetic mean and empirical variance and covariance 
calculated on the basis of available historical data do not give the best results in 
efficient portfolios construction (see e.g. Jorion, 1991; Sheedy, Trevor, Wood, 
1999; Johannes, Polson, Stroud, 2002; Flavin, Wickens, 2006). 

 The classical approach to the selection of efficient portfolios is static, i.e. 
a chosen period or moment is considered, but not their sequence. Such an ap-
proach ignores therefore the variability of conditional variances and covariances 
of returns. A dynamic approach to the selection of efficient portfolios for ho-
mogeneous assets, i.e. stocks, based on the forecasts of variances and covari-
ances of returns constructed from multivariate GARCH models is presented in 
this paper. There is an extensive literature on time-varying conditional variances 
of returns but also on time-varying conditional covariances and correlation co-
efficients between financial series. The multivariate GARCH process allows to 
describe both time-varying conditional variances and covariances of returns. If 
variances and covariances of returns are not constant, then the forecasts based 
on multivariate GARCH models should give additional benefits in the selection 
of efficient portfolios. The literature on the application of GARCH models in 
construction of portfolios is poor, and the results of such analyses, due to the 
complexity of the problem, are still fragmentary. In most of the studies the uni-
variate GARCH model is applied or a very limited number of assets are used. 
Among the few investigations in which multivariate GARCH models were ap-
plied for large portfolios the following papers can be mentioned: Engle, Shep-
pard (2001), Engle, Colacito (2006), Osiewalski, Pajor (2010).    

 The primary purpose of this study is to evaluate the effectiveness of differ-
ent methods of the minimum variance portfolio selection, mainly with the ap-
plication of various multivariate specifications of GARCH models. The adopted 
approaches to the portfolio construction differ only by forecasting methods of 
the covariance matrix of returns. The paper is an extension of the author's earli-
er work presented in Fiszeder (2004, 2007). The plan for the rest of the paper is 
as follows. Section 1 outlines the way in which efficient portfolios are selected. 
In section 2 competing methods of the covariance matrix estimation are pre-
sented. Section 3 contains the effectiveness analysis of the minimum variance 
portfolio selection for seventy stocks quoted on the Warsaw Stock Exchange 
(WSE) and section 4 presents the conclusions.  

1.  Dynamic Process of Portfolio Selection with Application  
of GARCH Models 

 It is difficult to evaluate the influence of the choice of the covariance matrix 
estimator against the selection of the population mean estimator in construction 
of efficient portfolios. One of the ways to eliminate the influence of the choice 
of the mean estimator is the minimum variance portfolio selection. Shares of 



Minimum Variance Portfolio Selections for Large Number of Stocks – Application… 89

individual assets in the minimum variance portfolio depend solely on the covar-
iance matrix, that is why such procedure is used in the paper. 

 For a given t, based on all the available data from the period ],1[ t  parame-
ters of a GARCH model are estimated1. The forecast of the covariance matrix 
based on the estimated model is formulated at time t , where the   is the 
forecast horizon. The constructed forecast is used for the selection of efficient 

portfolio. Let )...,,,(' 21 pttNpttpttptt www   W , where pttiw   is the 

share of asset i in the portfolio at time t , ptt H is the forecast of the condi-

tional covariance matrix of returns at time t . The variance of portfolio re-

turns is then equal to pttpttptt   WHW' . In order to select the minimum 

variance portfolio (the global minimum) the following quadratic programming 
problem has to be solved: 

min,'  pttpttptt  WHW      (1) 

subject to the constraint: 

,1'  lW ptt    (2) 

where l  is the 1N  vector of ones.  

When short selling is not allowed the boundary conditions have to be addition-
ally imposed: 

0 pttiw      for ....,,2,1 Ni   (3) 

The whole procedure is repeated for successive periods (with the arrival of sub-
sequent data).  
If short selling is allowed, then shares of assets in the minimum variance portfo-
lio are defined by the following formula:  

,
1 1 lHW 




  ptt
ptt

ptt C 


   (4) 

where lHl 1' 
  pttpttC  . 

The variance of the minimum variance portfolio is then equal to 

pttptt CV    /1 . 

Other efficient portfolios can be selected by finding the minimum variance port-
folio subject to a minimum expected return.  

                                                 
1 If conditional expected values of returns are different from zero, then parameters of condi-

tional mean equations should also be estimated.  



Piotr Fiszeder 90

2. Specifications of Multivariate GARCH Models  

 Eight parameterizations of multivariate GARCH models were applied in the 
analysis: Scalar BEKK, Integrated, CCC, Orthogonal, DCC, Integrated DCC, 
DECO-DCC and additionally Scalar BEKK with Student-t innovations. The 
results obtained for the GARCH models were compared with the outcomes for 
six other methods: equal shares in all stocks, the unconditional covariance ma-
trix of returns, the rolling covariance matrix2, the 25-day rolling covariance 
matrix, the exponentially weighted moving average estimate of the covariance 
matrix (hereafter the EWMA covariance matrix)3, the EWMA covariance ma-
trix with a smoothing parameter set to 0.94 (hereafter the RiskMetrics4). 

 Only some basic information about the considered multivariate models is 
given below. More details can be found for example in Bauwens, Laurent, 
Rombouts (2006) or Silvennoinen, Teräsvirta (2009).  

 Let assume that tε  can be either returns with the mean zero or residuals 
from filtered time series: 

,),(~  1 ttt N H0ε       (5) 

where 1t  is the set of all information available at time t-1 and tH  is the 

NN   symmetric conditional covariance matrix. 

 The estimation of parameters of the general form of a multivariate GARCH 
model, VECH model5 (Kraft, Engle, 1983), is very difficult even for a small 
number of assets. For this reason simpler parameterizations of multivariate 
GARCH models were used in the study. Considered were only those specifica-
tions which ensure the positive definiteness of the covariance matrix. 

Baba, Engle, Kraft, Kroner (1990) introduced the following form the so-called 
BEKK(p,q) model6 (see Engle, Kroner, 1995): 

,''
1

  

1

'' 






p

j
jjtjiitit

q

i
it EHEDεεDCCH      (6) 

where C, iD  and jE  are NN   parameter matrices and C is an upper triangu-

lar matrix.  

                                                 
2 The value of a rolling window was chosen to minimize the variance of a portfolio in a pre-

sample. 
3 The value of a smoothing parameter was chosen to minimize the variance of a portfolio in 

a pre-sample. 
4 The RiskMetrics methodology was developed in the investment bank J. P. Morgan for 

measuring market risk with VaR. The value of a smoothing parameter in the EWMA model was 
often set by financial market practitioners to 0.94 for daily data. 

5 The name of the model comes from the application of the vech operator for the conditional 
covariance matrix. 

6 The name of the model is formed by the first letters of the authors surnames. 



Minimum Variance Portfolio Selections for Large Number of Stocks – Application… 91

 The BEKK model is also too complex for large portfolios and the imposi-
tion of restrictions on parameters in estimation is necessary. The so-called Sca-

lar BEKK model can be obtained by replacing the matrices iD  and jE  by the 

scalars 2/1
id  and 2/1

je . Additionally the variance targeting approach (see Engle, 

Mezrich, 1996) was used. For example for 1 pq  instead of the product 'CC  
the following formula can be substituted:  

,)1(' 11 SCC ed       (7) 

where S is the sample covariance matrix given as 



t

t
ttt 1

'1
εεS . Two specifica-

tions of the conditional distribution of tε  in (5) were considered for the Scalar 
BEKK model, namely multivariate normal and Student-t. 
A further simplification of the model can be obtained by the assumptions 

0'CC  and  
 


q

i

p

j
ji ed

1 1

1. This formulation is often called the Integrated 

multivariate model7. For 1 pq  the model has only one parameter. 
 In the Constant Conditional Correlations (CCC) model of Bollerslev (1990), 
which is outside the BEKK class, the time-varying conditional covariances are 
parameterized to be proportional to the product of the corresponding conditional 
standard deviations: 

, ttt DΓDH       (8) 

where tD  is a NN   diagonal matrix ),...,,(diag 2/12/1
2

2/1
1 Ntttt hhhD  with ith  

defined as any univariate GARCH model, and Γ  is a NN   matrix of the 
time–invariant conditional correlations. 
 Alexander and Chibumba (1996) introduced the Orthogonal GARCH model 
defined as: 

,2/1
tmtt fΛuεV       (9) 

,2/12/1 VVVH tt       (10) 

where ),...,,(diag 21 NvvvV , with iv  the population variance of it , mΛ  is 

a matrix of dimension mN   given by ),...,,diag( 2/12/1
2

1/2
1 mmm lllPΛ  , 

0...21  mlll  being the m largest eigenvalues of the population correlation 

matrix of tu , and mP  the mN   matrix of associated (mutually orthogonal) 

eigenvectors, )'...( 21 mtttt ffff  is a random process such that 0)(1  ttE f , the 

                                                 
7 The name of the model comes from an analogous parameterization of the univariate 

IGARCH model. 



Piotr Fiszeder 92

conditional covariance matrix of tf  is equal to ),...,,(diag 222
t 21 mttt fff Q , 

2
itf  is defined as a univariate GARCH model (for mi ,...,2,1 ) and the condi-

tional covariance matrix  of tu is equal to mtmt 'ΛQΛV  . 
 The Dynamic Conditional Correlation (DCC) model of Engle (2002) can be 
defined as: 

,tttt DRDH       (11) 

,1*1*  tttt QQQR      (12) 

,)'()1(
111 1
 






 


P

j
jtj

Q

i
ititi

Q

i

P

j
jit QzzSQ       (13) 

where ),...,,(diag 2/12/1
2

2/1
1 Ntttt hhhD , kth  can be defined as a univariate GARCH 

model (for Nk ,...,2,1 ), tz  is a vector of standardized values of kt , i.e. 

ktktkt hz / , tR  is the time-varying conditional correlation matrix of tz ,  

S is the sample covariance matrix of tz , *
tQ  is a diagonal matrix composed of 

the square root of the diagonal elements of tQ  and the parameters have to satis-

fy the condition 10
1 1
 
 


Q

i

P

j
ji  . 

 Besides the standard DCC model two other modifications were also consid-

ered. When the sum of the parameters is equal to one 1
1 1
 
 


Q

i

P

j
ji   the DCC 

model is called the Integrated DCC model. Engle and Kelly (2008) introduced 
the DECO-DCC (dynamic equicorrelation) model. In this parameterization of 
the DCC model the equality of all pairwise conditional correlations at each time 
is assumed. 

3.  Evaluation of Portfolio Performance for Seventy Stocks Quoted 
on the WSE 

 The presented approach for the minimum variance portfolio selection was 
evaluated for Polish stocks. The investigated period was November 17, 20008 to 
June 30, 2009 (2158 daily returns). All stocks quoted in the specified period on 
the Warsaw Stock Exchange were considered in the analysis. The companies 
for which the percent of non-trading days was higher than 5% were omitted in 
order to avoid the problem of non-synchronous trading. In total seventy stocks 
were analyzed, however all the models used in the paper can be applied for 

                                                 
8 Since the introduction of the new trading system Warset. 



Minimum Variance Portfolio Selections for Large Number of Stocks – Application… 93

a much larger number of companies. An evaluation of portfolio performance 
was based on data from January 2004 to June 2009 (1380 observations).  
The following five steps were performed: (1) the estimation of parameters of all 
considered models9 (at the beginning for data from November 17, 2000 to De-
cember 30, 2003)10, (2) the construction of one-day ahead forecasts of the con-
ditional covariance matrix, (3) the minimum variance portfolio selection (4) the 
ex post calculation of the portfolio variance as a square of the realized portfolio 
return, (5) the extension of the sample with one observation. All the steps were 
repeated 1380 times. Every time the estimation of parameters was performed 
for increasing sample size. For each model and method the mean of the portfo-
lio variances was calculated. The results are presented in Table 1 (a square root 
of the mean is given). 

Table 1.  Estimates of the standard deviations of returns for the minimum variance port-
folios  

Portfolio designation 
Standard deviation 

(× 10-2) 
Rank 

Integrated DCC 0.9257 1 
DCC 0.9316 2 
CCC 0.9326 3 

DECO-DCC 0.9843 4 
Scalar BEKK Student-t 1.0030 5 

Scalar BEKK 1.0065 6 
Orthogonal 70 factors 1.0854 7 

Integrated 1.1155 8 
Unconditional covariance matrix 1.1155 8 

Rolling covariance matrix 1.2634 10 
Equal shares in all stocks 1.3538 11 

RiskMetrics 1.6549 12 
EWMA covariance matrix 2.9846 13 

25-day rolling covariance matrix 29.7629 14 
 
 The best performing model in the selection of the minimum variance portfo-
lio was the Integrated DCC model. On the other hand the worst performing 
method was the 25-day rolling covariance matrix. 
 An application of all the GARCH-class models was more effective in allo-
cation of stocks than the implementation of the other analyzed methods. The 
first four positions in the ranking (see Table 1) were occupied by the models, 
whose parameters were estimated in two steps and in the first stage parameters 
of a univariate GARCH model were estimated.  
 The Integrated model had the worst rank among all the applied GARCH 
models, however in this case the model was reduced to the unconditional covar-

                                                 
9 Codes written by the author in the Gauss programming language were applied.  
10 Logarithmic returns were used in the study. Because daily data were used, the logarithmic 

return on a portfolio is very close to the weighted average of the logarithmic returns on the indi-
vidual assets.  



Piotr Fiszeder 94

iance matrix of returns11. The Orthogonal GARCH model, which assumes that 
returns depend on common independent factors extracted by principal compo-
nent analysis ranked the one before last among the GARCH models. This result 
follows, first of all, from the loss of important information.  
 The methods used by financial market practitioners, i.e. the rolling covari-
ance matrix and the exponentially weighted moving average estimate of the 
covariance matrix took a distant positions in the ranking, even behind the un-
conditional covariance matrix of returns. The very poor result of the rolling 
covariance matrix with a low value of the rolling window equal to 25 deserves 
to be highlighted. The value of the smoothing parameter, set often for daily data 
in the RiskMetrics methodology to 0.94 is not the optimal value for the Polish 
stock market (it is definitely too low). 
 For a large number of assets like in this study, probably none of the re-
strictions, assumed by the considered GARCH models, is met. However, one 
can attempt to indicate the restrictions, which have a stronger negative influence 
on the performance of portfolio, in the case when they are not met. For exam-
ple, the assumption that models describing conditional variances and covari-
ances have the same parameters for all series12 (like in the Scalar BEKK and 
Integrated models) is probably too strong. It also seems that the presupposition 
of the constancy of conditional correlation coefficients is less restrictive. 
 The statistical significance of the observed differences between the perfor-
mance of the different models was not verified, however some of them are rele-
vant from the economic point of view. For example, the difference between the 
first in the ranking, namely the Integrated DCC model and the unconditional 
covariance matrix of returns means a decrease of about 3 percentage points of 
the standard deviation per year.  
 Additionally, values of the Schwarz information criterion (SIC) were calcu-
lated, when it was possible to evaluate a joint likelihood function (see Table 2). 
The ranking of the models according to the SIC is similar to the one for the 
portfolio performance evaluation (Table 1). The exceptions are the Scalar 
BEKK model with Student-t innovations and the Integrated model which take 
better position in the SIC ranking. The better performance of GARCH models 
with Student-t innovations in the rankings based on the information criteria is 
common in other studies (see Fiszeder, 2009). The rankings of models con-
structed on the basis of the information criteria provide useful clues for the se-
lection of models for the minimum variance portfolio construction. It has to be 
remembered, however, that some characteristics are crucial in the evaluation of 
the general fit of the model in a sample (like for instance a type of conditional 
density), but their influence on the performance of portfolio is not so important. 

 

                                                 
11 Estimates of the parameter on the lagged conditional covariance matrix were equal to one. 
12 It means, that the volatility dynamics of all series is very similar. 



Minimum Variance Portfolio Selections for Large Number of Stocks – Application… 95

Table 2.  Ranking of the models based on the Schwarz information criterion  

Portfolio designation SIC Rank 
Integrated DCC -679060.02 1 

DCC -666933.43 2 
Scalar BEKK Student-t -661928.17 3 

DECO-DCC -658854.85 4 
Integrated -656773.56 5 

Scalar BEKK -645459.24 6 
RiskMetrics 2052174.1 7 

 
 It is interesting to compare the results of this analysis with the similar study 
for twenty stocks (Fiszeder, 2007). The GARCH models, whose parameters 
were estimated in one stage (like the Scalar BEKK and Integrated models) took 
further positions in the ranking. It seems that the assumption that models de-
scribing conditional variances have the same parameters for all series becomes 
more and more restrictive for a larger number of assets. Furthermore, the differ-
ences between evaluations of portfolios, obtained with the use of the applied 
methods, were greater for the seventy stocks. The reason for such results is 
probably the higher differentiation of the seventy stocks, among which are both 
huge but also very small companies. In the study for the twenty stocks only the 
biggest companies were considered. This comparison for different numbers of 
stocks clearly shows that the received results depend on the properties of finan-
cial time series. 

Conclusions 

 The dynamic approach to the selection of efficient portfolios for a large 
number of homogeneous assets, i.e. stocks, based on the forecasts of variances 
and covariances of returns constructed from multivariate GARCH models has 
been presented in this paper. An evaluation of the efficiency for different meth-
ods of the minimum variance portfolio selection was performed for the seventy 
stocks from the Warsaw Stock Exchange. The eight specifications of multivari-
ate GARCH models and the six other methods were used.  
 Capturing time-varying variances and covariances of stock returns does not 
always increase the efficiency of the asset allocation process. The application of 
all the considered GARCH-class models was more efficacious in the allocation 
of stocks than the implementation of the other analyzed methods, including the 
methods employed often by financial market practitioners. The simple specifi-
cations of multivariate GARCH models, whose parameters were estimated in 
two stages, like the DCC and CCC models, were the best performing models. 
  



Piotr Fiszeder 96

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Konstrukcja portfeli o minimalnej wariancji dla dużej liczby spółek –
zastosowanie zmiennych w czasie macierzy kowariancji 

Z a r y s  t r e ś c i. W pracy dokonano oceny efektywności różnych metod tworzenia portfeli 
o minimalnej wariancji, w tym przede wszystkim z wykorzystaniem różnych specyfikacji wielo-
równaniowych modeli GARCH. Badanie zostało przeprowadzone dla 70 spółek notowanych na 
GPW w Warszawie. Zastosowano osiem parametryzacji modelu GARCH: skalarny BEKK, zinte-
growany, CCC, ortogonalny dla 70 czynników, DCC, zintegrowany DCC, DECO-DCC, skalarny 
BEKK z warunkowym rozkładem t Studenta oraz  sześć innych metod: równe udziały dla 
wszystkich aktywów, bezwarunkowa macierz kowariancji stóp zwrotu, ruchoma macierz kowa-
riancji, ruchoma macierz kowariancji ze stałą wygładzania równą 25, metoda wyrównywania 
wykładniczego dla macierzy kowariancji oraz metoda wyrównywania wykładniczego dla macie-
rzy kowariancji z parametrem wygasania równym 0,94. 

S ł o w a  k l u c z o w e: wielorównaniowe modele GARCH, zmieniająca się w czasie macierz 
kowariancji, konstrukcja portfela.  

 

 

 

 

 

 

 

 



 

 

 

 

 

 

 

 

 


