Microsoft Word - 03_Pajor_A.docx 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 -1 0 1 20 0 7- 07 -0 2 20 0 7- 07 -0 9 20 0 7- 07 -1 6 20 0 7- 07 -2 3 20 0 7- 07 -3 0 20 0 7- 08 -0 6 20 0 7- 08 -1 3 20 0 7- 08 -2 0 20 0 7- 08 -2 7 20 0 7- 09 -0 3 20 0 7- 09 -1 0 20 0 7- 09 -1 7 20 0 7- 09 -2 4 -2 -1 0 1 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 -2 -1 0 1 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 9- 24 -2 -1 0 1 20 07 -0 7 -0 2 20 07 -0 7 -0 9 20 07 -0 7 -1 6 20 07 -0 7 -2 3 20 07 -0 7 -3 0 20 07 -0 8 -0 6 20 07 -0 8 -1 3 20 07 -0 8 -2 0 20 07 -0 8 -2 7 20 07 -0 9 -0 3 20 07 -0 9 -1 0 20 07 -0 9 -1 7 20 07 -0 9 -2 4 -2 -1 0 1 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 9- 24 -2 -1 0 1 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 9- 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 3 3.5 4 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 20 0 7 -0 7- 02 20 0 7 -0 7- 09 20 0 7 -0 7- 16 20 0 7 -0 7- 23 20 0 7 -0 7- 30 20 0 7 -0 8- 06 20 0 7 -0 8- 13 20 0 7 -0 8- 20 20 0 7 -0 8- 27 20 0 7 -0 9- 03 20 0 7 -0 9- 10 20 0 7 -0 9- 17 20 0 7 -0 9- 24 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 20 0 7- 0 7- 0 2 20 0 7- 0 7- 0 9 20 0 7- 0 7- 1 6 20 0 7- 0 7- 2 3 20 0 7- 0 7- 3 0 20 0 7- 0 8- 0 6 20 0 7- 0 8- 1 3 20 0 7- 0 8- 2 0 20 0 7- 0 8- 2 7 20 0 7- 0 9- 0 3 20 0 7- 0 9- 1 0 20 0 7- 0 9- 1 7 20 0 7- 0 9- 2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 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 6 7 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 9- 24 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 0 0 7 -0 7- 0 2 2 0 0 7 -0 7- 0 9 2 0 0 7 -0 7- 1 6 2 0 0 7 -0 7- 2 3 2 0 0 7 -0 7- 3 0 2 0 0 7 -0 8- 0 6 2 0 0 7 -0 8- 1 3 2 0 0 7 -0 8- 2 0 2 0 0 7 -0 8- 2 7 2 0 0 7 -0 9- 0 3 2 0 0 7 -0 9- 1 0 2 0 0 7 -0 9- 1 7 2 0 0 7 -0 9- 2 4 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 00 7 -0 7 -0 2 2 00 7 -0 7 -0 9 2 00 7 -0 7 -1 6 2 00 7 -0 7 -2 3 2 00 7 -0 7 -3 0 2 00 7 -0 8 -0 6 2 00 7 -0 8 -1 3 2 00 7 -0 8 -2 0 2 00 7 -0 8 -2 7 2 00 7 -0 9 -0 3 2 00 7 -0 9 -1 0 2 00 7 -0 9 -1 7 2 00 7 -0 9 -2 4 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 9- 24 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 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 1 1.5 2 2.5 3 3.5 4 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7- 0 7- 0 2 2 00 7- 0 7- 0 9 2 00 7- 0 7- 1 6 2 00 7- 0 7- 2 3 2 00 7- 0 7- 3 0 2 00 7- 0 8- 0 6 2 00 7- 0 8- 1 3 2 00 7- 0 8- 2 0 2 00 7- 0 8- 2 7 2 00 7- 0 9- 0 3 2 00 7- 0 9- 1 0 2 00 7- 0 9- 1 7 2 00 7- 0 9- 2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7 -0 7- 02 2 00 7 -0 7- 09 2 00 7 -0 7- 16 2 00 7 -0 7- 23 2 00 7 -0 7- 30 2 00 7 -0 8- 06 2 00 7 -0 8- 13 2 00 7 -0 8- 20 2 00 7 -0 8- 27 2 00 7 -0 9- 03 2 00 7 -0 9- 10 2 00 7 -0 9- 17 2 00 7 -0 9- 24 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7- 07 -0 2 2 00 7- 07 -0 9 2 00 7- 07 -1 6 2 00 7- 07 -2 3 2 00 7- 07 -3 0 2 00 7- 08 -0 6 2 00 7- 08 -1 3 2 00 7- 08 -2 0 2 00 7- 08 -2 7 2 00 7- 09 -0 3 2 00 7- 09 -1 0 2 00 7- 09 -1 7 2 00 7- 09 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 20 07 -0 7- 02 20 07 -0 7- 09 20 07 -0 7- 16 20 07 -0 7- 23 20 07 -0 7- 30 20 07 -0 8- 06 20 07 -0 8- 13 20 07 -0 8- 20 20 07 -0 8- 27 20 07 -0 9- 03 20 07 -0 9- 10 20 07 -0 9- 17 20 07 -0 9- 24 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7 -0 7 -0 2 2 00 7 -0 7 -0 9 2 00 7 -0 7 -1 6 2 00 7 -0 7 -2 3 2 00 7 -0 7 -3 0 2 00 7 -0 8 -0 6 2 00 7 -0 8 -1 3 2 00 7 -0 8 -2 0 2 00 7 -0 8 -2 7 2 00 7 -0 9 -0 3 2 00 7 -0 9 -1 0 2 00 7 -0 9 -1 7 2 00 7 -0 9 -2 4 9850 10000 10150 10300 10450 20 07 -0 7- 02 20 07 -0 7- 05 20 07 -0 7- 08 20 07 -0 7- 11 20 07 -0 7- 14 20 07 -0 7- 17 20 07 -0 7- 20 20 07 -0 7- 23 20 07 -0 7- 26 20 07 -0 7- 29 20 07 -0 8- 01 20 07 -0 8- 04 20 07 -0 8- 07 20 07 -0 8- 10 20 07 -0 8- 13 20 07 -0 8- 16 20 07 -0 8- 19 20 07 -0 8- 22 20 07 -0 8- 25 20 07 -0 8- 28 20 07 -0 8- 31 20 07 -0 9- 03 20 07 -0 9- 06 20 07 -0 9- 09 20 07 -0 9- 12 20 07 -0 9- 15 20 07 -0 9- 18 20 07 -0 9- 21 20 07 -0 9- 24 JSV SJSV TSV_E_U TSV_U_E MSF-SBEKK SBEKK MSF BMSV BANK DEPOSIT equally-weighted portf olio 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 References Aguilar, O., West, M. (2000), Bayesian Dynamic Factor Models and Portfolio Allocation, Journal of Business and Economic Statistics, 18, 338–357. Elton, J.E., Gruber, M.J. (1991), Modern Portfolio Theory and Investment Analysis, John Wiley & Sons, Inc, New York. Markowitz, H.M. (1959), Portfolio Selection: Efficient Diversification of Investments, New York, John Wiely & Sons, Inc. Newton, M.A., Raftery, A.E. (1994), Approximate Bayesian Inference by the Weighted Likeli- hood Bootstrap [with discussion], Journal of the Royal Statistical Society B, 56(1), 3–48 Osiewalski, J. (2009), New Hybrid Models of Multivariate Volatility (a Bayesian Perspective), Przegląd Statystyczny (Statistical Review), 56, z. 1, 15–22. Osiewalski, J., Pajor, A. (2007), Flexibility and Parsimony in Multivariate Financial Modelling: a Hybrid Bivariate DCC–SV Model, [in:] Financial Markets. Principles of Modeling, Forecasting and Decision-Making (FindEcon Monograph Series No.3), [ed.:] W. Milo, P. Wdowiński, Łódź University Press, Łódź, 11–26. Osiewalski, J., Pajor, A. (2009), Bayesian Analysis for Hybrid MSF–SBEKK Models of Multi- variate Volatility, Central European Journal of Economic Modelling and Econometrics, 1(2), 179–202. Pajor, A. (2009), Bayesian Portfolio Selection with MSV Models, Przegląd Statystyczny (Statisti- cal Review), 56, z. 1, 40–55. 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.