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

Anna Pajor 
Cracow University of Economics 

Bayesian Optimal Portfolio Selection  
in the MSF-SBEKK Model†  

A b s t r a c t. The aim of this paper is to investigate the predictive properties of the MSF-Scalar 
BEKK(1,1) model in context of portfolio optimization. The MSF-SBEKK model has been pro-
posed as a feasible tool for analyzing multidimensional financial data (large n), but this research 
examines forecasting abilities of this model for n = 2, since for bivariate data we can obtain and 
compare predictive distributions of the portfolio in many other multivariate SV specifications. 
Also, approximate posterior results in the MSF-SBEKK model (based on preliminary estimates of 
nuisance matrix parameters) are compared with the exact ones.  

K e y w o r d s: portfolio analysis, MSV models, MSF-SBEKK model, forecasting.  

Introduction  

 It is well known that in portfolio selection (computing the weights of the 
assets in the portfolio) correlations among the assets are essential. The weights 
of the minimum variance portfolio depend on the conditional covariance matrix 
(see Aguilar, West, 2000, Pajor, 2009). Thus for active portfolio management 
multivariate time series forecasts should be applied.  

The aim of the paper is to examine the predictive properties of the MSF-
SBEKK model (being the hybrid of the Multiplicative Stochastic Factor and 
scalar BEKK specifications; hence the model is called MSF-SBEKK; see 
Osiewalski, Pajor, 2009) in context of the optimal portfolio selection problem. 
The multi-period minimum conditional variance portfolio is considered (as in 
Pajor, 2009). In the optimization process we use the predictive distributions of 
future returns and the predictive conditional covariance matrices obtained from 
the Bayesian MSF-SBEKK and other multivariate stochastic volatility (MSV) 
models. In order to compare predictive results in the MSF-SBEKK model with 

                                                 
† Research supported by a grant from the Cracow University of Economics. The author would 

like to thank Janusz Jaworski for language verification of the manuscript. 



Anna Pajor  42

those obtained in other MSV specifications, we consider only bivariate portfoli-
os. The bivariate stochastic volatility models are used to describe the daily ex-
change rate of the euro against the Polish zloty and the daily exchange rate of 
the US dollar against the Polish zloty. Based on these two currencies we con-
sider the Bayesian portfolio selection problem. In the next section we briefly 
present the MSF-SBEKK model. Section 3 is devoted to the optimal portfolio 
construction. In section 4 we present and discuss the empirical results. Some 
concluding remarks are presented in the last section. 

1. Bayesian MSF-SBEKK Model 

 Let xj,t denote the price of asset j (or the exchange rate as in our application) 
at time t for j = 1,2, ..., n and t = 1, 2, ..., T+s. The vector of growth rates  
yt =(y1,t, y2,t, ..., yn,t), where yj,t = 100 ln (xj,t/xj,t-1), is modelled using the basic 
VAR(1) framework: 

ttt ξRyδy  1 , t = 1, 2, ... ,T, T+1, ..., T+s, (1) 

where { tξ } is a process with time-varying volatility, T denotes the number of 

the observations used in estimation, and s is the forecast horizon, δ  is  
a n-dimensional vector, R is a nn matrix of parameters. 

 Following Osiewalski and Pajor (2009), for tξ we assume the so-called type 
I MSF-SBEKK(1,1) hybrid specification: 

tttt g εHξ 2/1 , (2) 

tgtt gg   1lnln ,    ),(~})','{( 1]1)1[(  nntt iiN I0ε  ,1 (3) 

  111 ')1(   tttt HξξAH  . (4) 

That is, tξ  is conditionally normal with mean vector 0 and covariance matrix 
gtHt, where gt is a latent process and Ht is a square matrix of order n that has the 
scalar BEKK(1,1) structure . Thus, the conditional distribution of yt, given its 
past and latent variables, is normal with mean 1 tt Ryδy  and covariance 
matrix gtHt. The model defined by (2)-(4) includes as special cases two simple 
basic structures. When 0g  and  = 0 we have the scalar BEKK(1,1) mod-

el, while β = 0 and γ = 0 lead to the MSF model (see Osiewalski, Pajor, 2009). 
Note that the model has one latent process which helps in explaining outlying 
observations, and time-varying conditional correlations as in the scalar 
BEKK(1,1) structure.  

                                                 
1 { )','( tt ε } is a sequence of independent and identically distributed normal random vectors 

with mean vector zero and covariance matrix In+1. 



Bayesian Optimal Portfolio Selection in the MSF-SBEKK Model 43

In (4) A is a free symmetric positive definite matrix of order n; for A-1 we as-
sume the Wishart prior with n degrees of freedom and mean In; β and γ are free 
scalar parameters, jointly uniformly distributed over the unit simplex. As re-
gards initial conditions for Ht, we take H0 = h0In and treat h0 > 0 as an additional 
parameter, a priori exponentially distributed with mean 1. For the parameters of 
the latent process we use the same priors as Osiewalski, Pajor (2009); for  : 
normal with mean 0 and variance 100, truncated to (-1, 1), for 2

g : exponential 

with mean 200; g0 is equal 1. The n(n+1) elements of )')'((0 Rδδ vec  are 
assumed to be a priori independent of remaining parameters, with the  
N(0, In(n+1)) prior truncated by the restriction that all eigenvalues of R lie inside 
the unit circle. 

In this paper we also want to check how the approximation proposed and ex-
plained in Osiewalski, Pajor (2009) influences the predictive distribution of 
future logarithmic returns and, in consequence, the optimal portfolio composi-
tion. Therefore we apply this approximation. That is, we use Ordinary Least 
Squares (OLS) for the VAR(1) parameters and replace A by the empirical co-
variance matrix of the OLS residuals from the VAR(1) part. The Bayesian anal-
ysis for the remaining parameters and future return rates is based on the condi-
tional posterior and predictive distributions given the particular values of vector 
δ0 and matrix A.  

All distributions are sampled using the Gibbs scheme with Metropolis-Hastings 
steps, as shown in detail in Osiewalski, Pajor (2009). 

2. Portfolio Selection Problem in the MSF-SBEKK Model 

 We denote by tΘ  the latent variable vector at time t, by θ  the parameter 
vector, and we assume that:  

a)  ttt εΣξ 2/1 , where ),(~}{ nt iiN I0ε ,  

b) tΣ  is a function of the latent variables Θ  for t , and of the past of tξ , 

i.e. );,( 1 tt    ξΘΣΣ , 

c) the vector tξ , conditional on );,( 1 t   ξΘ , is independent of 

);( t Θ . 

In Pajor (2009) it was assumed that );( tt  ΘΣΣ . Now we relax the as-

sumption, allowing tΣ  to depend on the past of tξ  as in the MSF-SBEKK 
model. The s-period portfolio at time T is defined by a vector 
wT+s|T = (w1,T+s|T, w2,T+s|T, ..., wn,T+s|T), where wi,T+s|T is the fraction of wealth 
invested in asset i (1  i  n). The return on the portfolio that places weight 
wi,T+s|T on asset i at time T is approximately a weighted average of the returns on 



Anna Pajor  44

the individual assets. The weight applied to each return is the fraction of the 
portfolio invested in that asset:  

TsTw

n

i
TsTiTsTiTsTw RzwR |,

1
|,|,|,

~



  , (5) 

where zi,T+s|T is the rate of return on the asset i from the period T to T+s, i.e. 





 

sT

Tt
tiTsTi yz

1
,|,  (i = 1, ..., n). If TsT |Σ  is the matrix of conditional covariances 

of zT+s|T = (z1,T+s|T, z2,T+s|T, ..., zn,T+s|T), then the conditional variance of return on 
the portfolio is  

),...,,|~( |, sTTTTsTwRVar  ΘΘ  = TsTTsTTsTTsTV |||
2

| '   wΣw , (6) 

where T is the -algebra generated by ε  and Θ  for T , i.e. 

);,( TT    Θε .  

The vector of the rates of return at time T+k (k > 0, k  s) satisfies: 

.
1

1

0









 

k

j
jT

jk
T

k
k

j

j
kT ξRyRδRy  (7) 

Based on equation (7) we have: 

,
1 011

1

0
|   











 

s

j
jT

js

i

i
s

k
T

k
s

k

k

j

j
TsT ξRyRδRz  (8) 

Since 0ΘΘξξ  ),...,,|'( sTTTjTiTE   for i  j, the conditional covariance 

matrix of zT+s|T in the MSF-SBEKK(1,1) model becomes: 

,)'()(
1 0

*

0
|  











 

s

j

js

i

i
jT

js

i

i
TsT RΣRΣ  (9) 

where  

.),...,,|'(*
sTTTjTjTjT E   ΘΘξξΣ   

Finally, the conditional variance of return on the portfolio is:  

),...,,|~( |, sTTTTsTwRVar  ΘΘ  = .)'()(' |
1 0

*

0
| TsT

s

j

js

i

i
jT

js

i

i
TsT 











   wRΣRw  

It is easy to show that in the MSF-SBEKK(1,1) model: 

,),...,,|'( 1111   TTsTTTTT gE HΘΘξξ   

,')1(),...,,|( 11 TTTsTTTTT E HξξAΘΘHH     

and for 2 < k  s: 



Bayesian Optimal Portfolio Selection in the MSF-SBEKK Model 45

)],,...,,|()()1[(

),...,,|'(

11 sTTTkTkTkT

sTTTkTkT

Egg

E








ΘΘHA

ΘΘξξ




 

).,...,,|()()1(

),...,,|(

11 sTTTkTkT

sTTTkT

Eg

E








ΘΘHA

ΘΘH




 

Consequently2: 

).()1(1

),...,,|'(

1

1
1

2

1 1





























 




k

j
jkTTkT

k

i

i

j
jkTkT

sTTTkTkT

gggg

E

HA

ΘΘξξ

 

The most popular approach assumes that the investor selects the portfolio with 
minimum variance (see Markowitz, 1959, Elton, Gruber, 1991). Here we as-
sume that the conditional variance of the portfolio is minimized and that short 
sales are allowed (wi,T+s|T < 0 reflects a short selling). Then the problem for the 
investor reduces to solving the quadratic programming problem: 

TsTTsTTsT
TsT

||| 'min
|




wΣw
w

  subject to w1,T+s|T + w2,T+s|T + ... +wn,T+s|T = 1. 

In this way we obtain so-called the minimum conditional variance portfolio (the 
portfolio that has the lowest risk of any feasible portfolio):  

,
' 1

|

1
|

|,
ιΣι

ιΣ
w 






 
TsT

TsT
TsTMV  (10) 

which has a return: 

,
'

'
1

|

|
1

|
|,

ιΣι

zΣι







 
TsT

TsTTsT
TsTMVR   (11) 

and the conditional variance at time T: 

,
'

1
),...,,|'(

1
|

2
|,|,

ιΣι
ΘΘyw 


 

TsT
TsTMVsTTTsTTsTMV VVar   (12) 

where ι  is an n1 vector of ones.  

Next we consider a s-period portfolio selection problem where the investor min-
imizes the conditional variance of the portfolio with a given level of return 

*
|,|,

~
TsTpTsTw RR   . This problem reduces to solving the quadratic programming 

problem: 

                                                 
2 A very similar result was obtained by Piotr De Silva in his unpublished master’s disserta-

tion. 



Anna Pajor  46

TsTTsTTsT
TsT

||| 'min
|




wΣw
w

 subject to 














.'

,1'
*

|,||

|

sTTpTsTTsT

TsT

Rzw

ιw
 

When *
|,|,

~
TsTpTsTw RR   , the solution for the s-period portfolio is: 

.
)'()')('(

))(''(
2

|
1

||
1

||
1

|

|
*

|,
1

||
1

|
1

||
1

|

|,*

TsTTsTTsTTsTTsTTsT

TsTTsTpTsTTsTTsTTsTTsTTsT

TsTMVR

R
p
























 




zΣιzΣzιΣι

zιΣιzΣΣιzΣ
w (13) 

It is important to stress that the classic portfolio choice scheme assumes the 
covariance matrix and expected returns at time T to be known. In our Bayesian 
models the minimum conditional variance portfolio ( TsTMV |, w ), and the mini-

mum conditional variance portfolio with a given level of return,  
(

TsTMVRp |,* 
w ) are random vectors as measurable functions of zT+s|T, and TsT |Σ . 

Hence, the predictive distributions of TsTMV |, w , and 
TsTMVRp |,* 

w  (also, of 

TsTMVV |,  , and 
TsTMVRp

V
|,*  ) are induced by the distribution of zT+s|T, and TsT |Σ . 

In practice, to compute the weights of the assets in the portfolio we must use 
some characteristic of these predictive distributions. As the predictive mean (for 

TsTMV |, w  or 
TsTMVRp |,* 

w ) may not exist, we consider the predictive medians of 

TsTiMV |,, w  and 
TsTiMVRp |,,* 

w , denoted by )',...,( |,,|,1,|,
op

TsTnMV
op

TsTMV
op

TsTMV ww  w  

and )',...,(
|,,|,|, ***

op

TsTnMVR

op

TsTMVR

op

TsTMVR ppp
ww


w , and defined respectively by 

conditions: 

5.0}|Pr{ |,,|,,   yww op
TsTiMVTsTiMV  and 5.0}|Pr{ |,,|,,   yww op

TsTiMVTsTiMV , 

5.0}|Pr{
|,,|,, ** 


yww op

TsTiMVRTsTiMVR pp
 and ,5.0}|Pr{

|,,|,, ** 


yww op

TsTiMVRTsTiMVR pp
 

for i = 1, ..., n-1, and 





 

1

1
|,,|,, 1

n

i

op
TsTiMV

op
TsTnMV ww , 







1

1
|,,|,, ** 1

n

i

op

TsTiMVR

op

TsTnMVR pp
ww . 

In multivariate stochastic variance models there is no analytical solution for the 
optimal portfolio selection problem even for n = 2 assets. To evaluate the quan-
tiles of the predictive distributions of TsTMV |, w  and 

TsTMVRp |,* 
w , and then find 

the portfolio, we use Markov chain Monte Carlo methods – the Gibbs sampler 
with the Metropolis-Hastings algorithm.  



Bayesian Optimal Portfolio Selection in the MSF-SBEKK Model 47

3. Empirical Results 

 As the dataset we use the same daily exchange rates as in Pajor (2009). 
Thus, we consider the daily exchange rate of the euro against the Polish zloty 
and the daily exchange rate of the US dollar against the Polish zloty from Janu-
ary 2, 2002 to June 29, 2007. The data were downloaded from the website of 
the National Bank of Poland. The dataset of the percentage daily logarithmic 
growth (return) rates, yt, consists of 1388 observations (for each series). As the 
first growth rates are used as initial conditions, T = 1387 remaining observations 
on yt are modelled.  

3.1. Bayesian Model Comparison  

 In Table 1 we rank the models by the increasing value of the decimal loga-
rithm of the Bayes factor of VAR(1)-SJSV against the alternative models.  
We see that for our dataset the models with three latent processes describe the 
time-varying conditional covariance matrix much better than the models with 
one or two latent processes. The VAR(1)-SJSV model wins our model compari-
son, being about 8.5 orders of magnitude better than the VAR(1)-TSVEUR_USD 
model. The decimal log of the Bayes factor of the VAR(1)-MSF-SBEKK model 
relative to the VAR(1)-SJSV model is 27.32. The presence of more latent pro-
cesses improves fit enormously, but seems infeasible for highly dimensional 
time series. Assuming equal prior model probabilities, the VAR(1)-MSF-
SBEKK model is about 20.73 orders of magnitude more probable a posterior 
than the VAR(1)-MSF model (with the constant conditional correlations), and 
about 32 orders of magnitude better than the VAR(1)-SBEKK model. Note that 
the VAR(1)-MSF-SBEKK model is about 6.6 orders of magnitude better than 
another hybrid model – the VAR(1)-MSF-DCC model, proposed by Osiewalski, 
Pajor (2007).  

Table 1.  Logs of Bayes factors in favour of VAR(1)-SJSV model  

Model Number of latent processes  Number of parameters Log10 (BSJSV,i) Rank 
VAR(1)-SJSV 3 18 0 1 
VAR(1)-TSVEUR_USD 3 18 8.51 2 
VAR(1)-TSVUSD_EUR 3 18 11.10 3 
VAR(1)-JSV 2 15 19.60 4 
VAR(1)-MSF-SBEKK 1 14 27.32 5 
VAR(1)-MSF-SBEKK 
with the approximation 

1 14 29.30 6 

VAR(1)-MSF-IDCC 1 18 32.00 7 
VAR(1)-MSF-DCC 1 20 33.88 8 
VAR(1)-MSF(SDF) 1 12 48.05 9 
VAR(1)-SBEKK 0 12 59.70 10 
VAR(1)-BMSV 2 14 158.51 11 

Note:  the decimal logarithm of the Bayes factors were calculated using the Newton and Raftery method (see 
Newton, Raftery, 1994). Only the results for the VAR(1)-MSF-SBEKK, VAR(1)-MSF and VAR(1)-SBEKK 
models are new; the remaining ones were obtained by Pajor (2009). 



Anna Pajor  48

In the bivariate case considered here it is possible to compare exact and approx-
imate Bayesian results relate to estimation of the VAR(1)-MSF-SBEKK model. 
Thus, in Tables 1 we present the decimal logarithm of the Bayes factor for both 
cases. Using the approximate Bayesian approach proposed by Osiewalski, Pajor 
(2009) leads to smaller values of the data density, but it seems that the fit does 
not significantly change.  

Of course, our model comparison relies on the prior distributions for the param-
eters of the models, but these prior distributions are not very informative. 

3.1. Predictive Properties of the MSF-SBEKK Models in Portfolio  
Selection 

 It is important to investigate the predictive properties of the MSF-SBEKK 
model in portfolio selection. In addition, we can examine how the exact and 
approximate posterior results may differ. Thus, in this section we report the 
results of building the optimal portfolios using the MSF-SBEKK model.  
We consider the hypothetical portfolios, which consist of two currencies: the 
US dollar and euro. We assume that there are no transaction costs and that we 
may reallocate zloty to long as well as to short positions across the currencies. 
Allocation decisions are made at time T based on the predictive distribution for 
yT+k  and kT Σ  for k =1, ..., 60. 

)|( |,1, yTsTMVwp   

SJSV TSVEUR_USD MSF-SBEKK 

MSF-SBEKK with app. 

 

MSF SBEKK 

Figure 1.  Quantiles of the predictive distributions of the minimum conditional vari-
ance portfolios (the fractions of wealth invested in the US dollar). The cen-
tral black lines represent the medians, and the grey lines represent the quan-
tiles of order 0.05, 0.25, 0.75, 0.95, respectively  

-2

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24



Bayesian Optimal Portfolio Selection in the MSF-SBEKK Model 49

In Figure 1 we show the quantiles of the predictive distributions of the mini-
mum conditional variance portfolio sTTMVw |,1,  (the fraction of wealth invested 

in the US dollar). If the medians of the marginal predictive distributions are 
treated as point forecasts, in model with time-varying conditional correlation 
coefficient the optimum weights to invest in the USD/PLN are negative, indi-
cating the short sale of the US dollar (the median of the marginal predictive 
distribution of sTTMVw |,1,  is equal to about -0.4 in the most probable a posterior 

model, and about -0.22 in the VAR(1)-MSF-SBEKK model). The short position 
on the US dollar is connected with corresponding long position on the euro.  
We see that in VAR(1)-MSV models with more than one latent process the 
predictive distributions are very widely dispersed and fat-tailed, thus leaving us 
with considerable uncertainty about the future returns of these portfolios. Sur-
prisingly, in the VAR(1)-MSV models with one latent process or in the 
VAR(1)-SBEKK model the minimum conditional variance portfolios are esti-
mated more precisely – the inter-quartile ranges are relatively small. It seams 
that the VAR(1)-MSF-SBEKK and VAR(1)-SBEKK models produce portfolios 
with lowest risk measured by the conditional variance (see Figure 2). Note that 
the predictive distributions of sTTMVw |,1,  for s = 1, ..., 60 produced by the 

VAR(1)-MSF-SBEKK model are located in areas of high predictive densities 
obtained in the best model (i.e. VAR(1)-SJSV). 

)|( |, yTsTMVVp   

SJSV TSVEUR_USD MSF-SBEKK 

 
MSF-SBEKK with app. 

 

MSF SBEKK 

 

Figure 2.  Quantiles of the predictive distributions of the conditional standard deviation 
of the minimum conditional variance portfolios. The central black lines rep-
resent the medians, and the grey lines represent the quantiles of order 0.05, 
0.25, 0.75, 0.95, respectively 

0

0.5

1

1.5

2

2.5

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3.5

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07

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24



Anna Pajor  50

As in Pajor (2009) we can see that the predictive distributions related to the 
portfolio with bound on return are more diffuse – the inter-quartile ranges are 
higher (see Figure 3 and 4). Comparing the minimum conditional variance port-
folio and the minimum conditional variance portfolio with the return equal to at 
least 5%, we can see that the distributions of the forecasted value of 

sTTMVRp
w

|,*  

and 
sTTMVRp

V
|,*  are more dispersed and have very thick tails. Thus uncertainty 

connected with the optimal portfolio with return at least 5% on annual base is 
huge. In all models the quantiles of the conditional standard deviation of the 
optimal portfolios (see Figure 4) indicate increasing volatility with the forecast 
horizon. 

)|(
|,1,* y
TsTpMVR

wp


 

SJSV TSVEUR_USD MSF-SBEKK 

 
MSF-SBEKK with app. 

 

MSF SBEKK 

 

Figure 3. Quantiles of the predictive distributions of the minimum conditional vari-
ance portfolios with the return equal to at least 5% on annual base (the frac-
tion of wealth invested in the US dollar). The central black lines represent 
the medians, and the grey lines represent the quantiles of order 0.05, 0.25, 
0.75, 0.95, respectively 

Finally, as in Pajor (2009) we use the medians of 
TsTMVRp

w
|,1,* 

 to construct hy-

pothetical portfolios for s = 1, 2, .., 60. Let WT = 10000 PLN be the initial 
wealth of the investor at time T (on June 29, 2007). If we assume that there are 
no transaction costs and the investor uses the median of the predictive distribu-

tion of 
TsTMVRp |,* 

w  (denoted by op

TsTMVRp
w

|,1,* 
) to construct optimal portfolio, then 

the investor’s wealth at time T+s is given by: 

-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
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07

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9-

24



Bayesian Optimal Portfolio Selection in the MSF-SBEKK Model 51

)]/()/([ ,2,2|,2,,1,1|,1,|, *** TsT
op

TsTMVRTsT
op

TsTMVRTTsTMVR
xxwxxwWW

ppp


 ,  

s = 1, 2, .., 60. 
 

)|(
|,* y
TsTpMVR

Vp


 

SJSV TSVEUR_USD MSF-SBEKK 

 

MSF-SBEKK with app. 

 

MSF SBEKK 

 

Figure 4.  Quantiles of the predictive distributions of the conditional standard deviation 
of the minimum conditional variance portfolio with the return equal to at 
least 5% on annual base. The central black lines represent the medians, and 
the grey lines represent the quantiles of order 0.05, 0.25, 0.75, 0.95, respec-
tively 

 
Figure 5.  Wealth of the investor at time T+s for s = 1, ..., 60 (the optimal portfolio is 

constructed on the medians of 
TsTpMVR

w
|,* 

) 

0

0.5

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1.5

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07

-0
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9850

10000

10150

10300

10450

20
07

-0
7-

02

20
07

-0
7-

05

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07

-0
7-

08

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11

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29

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01

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07

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07

-0
8-

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07

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28

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07

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31

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07

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07

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07

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07

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9-

21

20
07

-0
9-

24

JSV

SJSV

TSV_E_U

TSV_U_E

MSF-SBEKK

SBEKK

MSF

BMSV

BANK 
DEPOSIT

equally-weighted portfolio

app. MSF-SBEKK



Anna Pajor  52

In Figure 5, we present the plot of 
TsTMVRp

W
|,*   for s =1, 2, ..., 60, and compare 

them with a bank deposit with the interest rate equal to 4.7% on annual base 
(the quotation of the 3-month Warsaw Interbank Offered Rate on June, 29 
2007). The best results we obtain in the VAR(1)-JSV model – at a 2-month 
horizon the average return of the optimal portfolios is equal to 0.098%, which 
represents annual return of 24.58%. In the best model (i.e. VAR(1) – SJSV) the 
average return of the optimal portfolios is equal to 0.065%, which represents 
annual return of 16.34%, whereas in the VAR(1)-MSF-SBEKK and VAR(1)-
SBEKK models we have 0.048% and 0.044%, respectively (i.e. 12.02% and 
11.05% per annum, respectively). It is important to stress that in the VAR(1)-
MSF-SBEKK model the returns of the hypothetical investments are higher than 
those of the bank deposit, indicating good forecasting properties of the model. 
In the VAR(1)-MSF model (with constant conditional correlation) the average 
return of the portfolio is negative (we obtained -0.006% i.e. -0.16% per annum). 
Thus the SBEKK structure is very important in forecasting. In the approximated 
VAR(1)-MSF-SBEKK model the average return is equal to 0.04% (i.e. 9.43% 
per annum). Thus using approximation in the VAR(1)-MSF-SBEKK leads to 
worse predictive results. After two months the return of the optimal portfolio is 
lower than the interest rate of the bank deposit, but still it is positive. Note that 
the average return of equally-weighted portfolio is equal to -0.047, i.e. -11.80% 
per annum. 

Conclusions 

 The paper investigates the predictive abilities of the VAR(1)-MSF-SBEKK 
model in portfolio selection. The predictive distributions of the optimal portfo-
lios produced by the VAR(1)-MSF-SBEKK model are compared with those 
obtained in unparsimonious (but more probable a posterior) MSV specifica-
tions. The predictive distributions of the weights of the optimal portfolios pro-
duced by the VAR(1)-MSF-SBEKK model are located in areas of high predic-
tive densities obtained in the best MSV model (i.e. VAR(1)-SJSV). Unfortu-
nately, in all models the predictive distributions of the optimal portfolio are very 
spread and have heavy tails. Our main finding is that the VAR(1)-MSF-SBEKK 
model is useful (but not very impressive) for building the multi-period optimal 
minimum conditional variance portfolio. It seems that the approximation pro-
posed by Osiewalski, Pajor (2009) results in worse predictive properties of the 
VAR(1)-MSF-BEKK model, but for large portfolios this approximation  
is necessary. 

  



Bayesian Optimal Portfolio Selection in the MSF-SBEKK Model 53

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Bayesowska optymalizacja portfela w modelu MSF-SBEKK 

Z a r y s  t r e ś c i. Celem artykułu jest analiza prognostycznych własności bayesowskiego mode-
lu MSF-SBEKK w kontekście wyboru optymalnego portfela inwestycyjnego. Wykorzystywany 
w artykule wielowymiarowy proces MSF-SBEKK posiada elementy struktury skalarnego procesu 
BEKK oraz procesu MSF. Obecność, w jego definicji, odrębnego czynnika losowego pozwala 
lepiej opisywać zjawisko grubych ogonów, zaś w strukturze SBEKK uzależnia się warunkowe 
wariancje oraz warunkowe korelacje od przeszłych wartości procesu. Proces MSF-SBEKK posia-
da zatem nietrywialną strukturę i może być wykorzystany do opisu zależności miedzy stopami 
zwrotu kilkudziesięciu (a nawet kilkuset) instrumentów finansowych. W artykule dokonane zosta-
ło porównanie prognoz uzyskanych w dwuwymiarowym modelu MSF-SBEKK oraz w innych 
modelach z klasy MSV na przykładzie portfela walutowego, złożonego z kursu dolara amerykań-
skiego oraz euro. Uzyskane wyniki wskazują na dobre własności prognostyczne modelu MSF-
SBEKK, choć uproszczenia w sposobie jego estymacji mogą je pogarszać.  

S ł o w a  k l u c z o w e: model MSF-SBEKK, modele MSV, analiza portfelowa, prognoza.  

 

 

 

 

 

 



 

 

 

 


