بهینه سازی پرتفوی در حضور بازده مالی وابسته همراه با حافظه بلندمدت : یک رویکرد مبتنی بر عضو رابط

In this paper, we seek to examine the effect of the presence of long memory on the dependence structure between financial returns and on portfolio optimization. First, we focus on the dependence structure using copulas. To select the best copula, in addition to the goodness of fit tests, we employ a graphical method based on visual comparison of the fitted copula density and the smoothed copula density estimated by wavelets. Moreover, we check the stability of the copula parameter. The empirical results show that the long memory affects the dependence structure. Second, we analyze the impact of this dependence structure on the optimal portfolio. We propose a new approach based on minimizing the Conditional Value at Risk and assuming that the dependence structure is modeled by the copula parameter. The empirical results show that our approach outperforms the traditional minimizing variance approach, where the dependence structure is represented by the linear correlation coefficient.

Modern Portfolio Theory argues that investors can benefit from diversification by investing in financial assets with lower correlations. It assumes that the financial returns are subject to joint-normal distribution. Therefore, the dependence between financial returns is fully described by the linear correlation coefficient and efficient portfolios are given by the traditional mean variance optimization program (Markowitz, 1952). However, empirical studies find that the assumption of normality of financial returns distribution is not hold. The departures from normality have been examined using the skewness (the third moment of distribution) and the kurtosis (the fourth moment of distribution). Some studies find evidence of negative skewness and excess kurtosis. Specifically, negative skewness indicates a higher probability of negative returns, that is, the market gives higher probability to decreases than increases in asset pricing while excess kurtosis makes extreme observations more likely than in the normal case, which means that the market gives higher probability to extreme observations than in normal distribution. Others studies find evidence of time-varying skewness (Harvey and Siddique, 1999) and time-varying kurtosis (Jondeau and Rockinger, 2003 and Brooks et al., 2005). Moreover, there is evidence of long memory in the mean and in the variance of financial returns (Crato, 1994, Lobato and Savin, 1998 and Sadique and Silvapulle, 2001). Consequently, the linear correlation coefficient is no longer suitable and can largely lead to misleading results (for more discussions about the bias in the linear correlation coefficient, see Embrechts et al., 1999, Embrechts et al., 2002 and Forbes and Rigobon, 2000). Furthermore, several authors find evidence of asymmetry in the dependence between stock market returns (Longin and Solnik, 2001, Ang and Bekaert, 2002, Ang and Chen, 2002, Das and Uppal, 2004, Patton, 2004 and Hong et al., 2007) and between exchange rate returns (Beine, 2004 and Li, 2011). In particular, stock market returns exhibit greater dependence during market downturns than market upturns and exchange rate returns seem to be more correlated in periods of depreciation than in periods of appreciation which increase the downside risk. To reproduce the asymmetric dependence between financial returns, some authors propose to rely on the concept of exceedance correlation1 (Ang and Chen, 2002) and the extreme value theory2 (Longin and Solnik, 2001, Hartmann et al., 2004, Poon et al., 2004 and Beine et al., 2010). An important limitation of these methods, is that they are subject to the problem of choosing the appropriate threshold and the use of the extreme value distribution such as Pareto distribution. Others authors consider multivariate Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model3 with skewness (Harvey and Siddique, 1999) and/or kurtosis (Jondeau and Rockinger, 2003 and Brooks et al., 2005). Few authors use regime-switching models (Ang and Bekaert, 2002 and Ang and Chen, 2002). Although these models allow for time-varying conditional correlations, they cannot reproduce asymmetries in asymptotic tail dependence. Tail dependence refers to dependence that arises between random variables from extreme observations. To overcome this shortcoming, we propose an alternative approach to modeling the dependence structure of multivariate data, without imposing any assumption in marginal distributions, based on copula theory. The advantage of copula lies in separating marginal distributions and dependence structure from joint distribution. In particular, we consider Archimedean copulas since they allow for asymmetry and dependence in the extreme tails. Archimedean copula functions have been applied by several authors to model the dependence structure between stock market returns (Costinot et al., 2000, Patton, 2004, Canela and Collazo, 2006, Kole et al., 2005, Jondeau and Rockinger, 2006, Hu, 2006, Rodriguez, 2007, de Melo Mendes and Kolev, 2008, Sun et al., 2009, Ignatievay and Platen, 2010, Aloui et al., 2011, Chollete et al., 2011 and Garcia and Tsafack, 2011) and the dependence structure between exchange rate returns (Liu, 2006 and Manner, 2007). A major implication of this asymmetric dependence between financial returns is that traditional mean variance optimization analysis portfolios are not efficient with respect to their effective risk profile. To take into account the asymmetric dependence structure between financial returns when determining the optimal portfolio, Fantazzini (2004) proposes an optimization program for a portfolio of stock market indexes, where the dependence structure is modeled by Kendall’s tau instead of the linear correlation coefficient. DiClemente and Romano (2004) propose an optimization program for a portfolio of credit risks, where the dependence structure between credit assets is modeled by copula functions. He and Gong (2009) propose also an optimization program for a portfolio of market and credit risks, where their dependence structure is modeled by copula functions. The objective of this paper is twofold. First, we focus on modeling the dependence structure between financial returns using copula functions. As financial returns, we consider one pair of daily stock market returns and one pair of daily exchange rate returns. In particular, we investigate whether the presence of long memory behavior affects the dependence structure between financial returns. Second, we study the influence of the dependence structure between financial returns on the optimal portfolio. To select the best copula, the authors compare the values of the log-likelihood, the information criterion and the distance of Cramér-von Mises between the fitted copula and the empirical copula. In this paper, we apply, in addition, an innovative graphical method based on visual comparison of the fitted density copula and the smoothed density copula. More precisely, the smoothed density copula is estimated by using a non-parametric estimator based on the wavelet decomposition as advanced by Genest et al. (2009b). Further, we check for stability of the copula parameter. To determine the optimal portfolio, we refer first to the classical mean risk framework proposed by Markowitz (1952). We consider that risk is measured by variance and that the dependence structure between financial returns is modeled by the linear correlation coefficient. Thus, the optimization problem involves on minimizing the variance for a given return. Then, we replace the linear correlation coefficient by the copula parameter to reproduce the true dependence structure between financial returns. It is well known that variance has been criticized because it is a symmetric measure and treats downside risk and upside risk in the same way while the investors assign greater importance to downside risk than to upside risk (Post and van Vliet, 2004 and Ang et al., 2006). In the same context, Tsafack (2009) shows that the use of symmetric dependence tends to underestimate extreme risk in the presence of asymmetric dependence. For the accuracy of risk measure, some authors introduced several downside risk measures in their optimization program. As downside risk measures, Markowitz (1959) considers the semi-variance. Using safety first principle,4Roy, 1952 and Arzac and Bawa, 1977 consider Value at Risk (VaR).5 To take into account the tail events, Susmel (2001) uses extreme value theory to reproduce the fact that the returns are fat-tailed. In this paper, we provide an alternative method for modeling the tail events based on copulas. In addition, VaR has been criticized for not being a coherent risk measure. Indeed, VaR is not sub-additive, i.e., the risk associated with a given portfolio can be larger than the sum of the stand alone risks of its components when measured by the VaR (for more details, see Artzner et al., 1999, McNeil et al., 2005 and Gilboa, 2009). For this reason, we retain the Conditional Value at Risk (CVaR)6 as downside risk measure instead of the VaR. Following DiClemente and Romano, 2004 and He and Gong, 2009, we develop an optimization program based on minimizing the CVaR and, where the dependence structure between financial returns is modeled by the copula parameter. Our findings have significant implications. First, understanding and measuring the interdependence between financial returns is important for international financial market integration and portfolio diversification. Second, the optimization program proposed here can help the investors to compute more accurately the extreme risk of the portfolio. The remainder of this paper is organized as follows. In the next section, we review briefly the main concepts of the copula theory and we present an empirical application to modeling the dependence structure between financial returns. In Section 3, we analyse the effect of this dependence structure on the optimization portfolio. In Section 4, we conclude the paper.

In this paper, we investigate the impact of the presence of long memory on the dependence structure between a pair of stock market returns and a pair of exchange rate returns and we check if this dependence structure affects the optimization portfolio. For that, we conduct our analysis in two steps. In a first step, we examine separately the dependence structure between raw returns and the dependence structure between filtered returns obtained from ARFIMA-FIGARCH process. The empirical results show evidence of asymmetric dependence between financial returns and reveal that the existence of long range dependence masks the real dependence structure between financial returns. More precisely, the true dependence between financial returns series is higher than the one observed. In a second step, we study how the dependence structure between raw returns or filtered returns affects portfolio optimization. We first consider the classical mean variance framework as advanced by Markowitz (1952), where dependence is measured by Gumbel copula parameter instead of the linear correlation coefficient. Then, we propose an optimization program, where risk is measured by CVaR. The empirical results show that the efficient frontier is influenced by the existence of a long memory behavior and the choice of the measure dependence. In spite of the obtained results, we conclude that the presence of long memory affects both the dependence structure between financial returns and the efficient frontier.