تجزیه و تحلیل خوشه ای برای بهینه سازی سبد سرمایه گذاری

We consider the problem of the statistical uncertainty of the correlation matrix in the optimization of a financial portfolio. By assuming idealized conditions of perfect forecast ability for the future return and volatility of stocks and short selling, we show that the use of clustering algorithms can improve the reliability of the portfolio in terms of the ratio between predicted and realized risk. Bootstrap analysis indicates that this improvement is obtained in a wide range of the parameters N (number of assets) and T (investment horizon). The predicted and realized risk level and the relative portfolio composition of the selected portfolio for a given value of the portfolio return are also investigated for each considered filtering method. We also show that several of the results obtained by assuming idealized conditions are still observed under the more realistic assumptions of no short selling and mean return and volatility forecasting based on historical data.

The problem of portfolio optimization is one of the most important issues in asset management (Elton and Gruber, 1995). Since the seminal work of Markowitz (1959), which solved the problem under a certain number of simplifying assumptions (see also Section 2), many other studies have been devoted to consider several aspects of portfolio optimization both from a theoretical and from an applied point of view. A huge number of studies considering key aspects of portfolio optimization theory are present in the finance literature. Here we refer to a few studies considering, for example, the real performance of portfolios constructed using sample moments (Jorion, 1985), the realized Sharpe ratio of the global minimum variance portfolio (Jagannathan and Ma, 2003), the role of constraints in portfolio optimization (Eichhorn et al., 1998 and Jagannathan and Ma, 2003), the shrinkage estimator of large dimensional covariance matrices (Ledoit and Wolf, 2004a and Ledoit and Wolf, 2004b). The aim of the present study is to focus on the role of the correlation coefficient matrix in portfolio optimization. The estimation of the correlation matrix is unavoidably associated with a statistical uncertainty, which is due to the finite length of the asset return time series. Recently, there have been several contributions in the econophysics literature devoted to quantify the degree of statistical uncertainty present in a correlation matrix. The results of these investigations have been obtained by using concepts and tools of random matrix theory (RMT) (Metha, 1990). The RMT quantification of the statistical uncertainty associated with the estimation of the correlation coefficient matrix of a finite multivariate time series has been recently used to devise a procedure to filter the information present in the correlation coefficient matrix which is robust with respect to the unavoidable statistical uncertainty (in the econophysics literature the term of noise dressing has been used) (Galluccio et al., 1998, Laloux et al., 1999, Laloux et al., 2000, Plerou et al., 1999, Plerou et al., 2002, Gopikrishnan et al., 2001, Drozdz et al., 2001, Rosenow et al., 2002, Pafka and Kondor, 2003, Pafka and Kondor, 2004, Rosenow et al., 2003, Guhr and Kalber, 2003, Malevergne and Sornette, 2004, Sharifi et al., 2004 and Burda and Jurkiewicz, 2004). The correlation matrices obtained by this filtering procedure has been used in portfolio optimization in some studies (Laloux et al., 2000 and Rosenow et al., 2002), which have shown that under the assumption of perfect forecasting of future returns and volatilities the distance between the predicted optimal portfolio and the realized one is smaller for the filtered correlation matrix than for the original one at a given level of the portfolio return. In recent years, other filtering procedures of the correlation coefficient matrix performed using correlation based clustering procedures have also been proposed in the econophysics literature (Mantegna, 1999, Kullmann et al., 2000, Kullmann et al., 2002, Bonanno et al., 2000, Bonanno et al., 2001, Bonanno et al., 2003, Bonanno et al., 2004, Giada and Marsili, 2001, Maslov, 2001, Bernaschi et al., 2002, Onnela et al., 2002, Mendes et al., 2003, Micciche et al., 2003, Maskawa, 2003, Di Matteo et al., 2004, Basalto et al., 2005 and Tumminello et al., 2005). These methods also select information of the correlation coefficient matrix which is representative of the entire matrix and it is often less affected by the statistical uncertainty and therefore more stable than the entire matrix during the time evolution of the system. In this paper we investigate how the portfolio optimization procedure is sensitive to different filtering procedures applied to the correlation coefficient matrix. Specifically, we consider filtering procedures based on RMT and on correlation based clustering procedures. We proceed in two steps. In the first step, similarly as in Laloux et al. (2000) and Rosenow et al. (2002), we assume perfect forecasting ability of future returns and volatilities. We also assume that short selling is allowed in the portfolio optimization procedure. These are quite idealized conditions. In the second step, we assume more realistic conditions. Specifically, we consider no short selling constraint and we assume imperfect forecasting of mean returns and volatilities. The forecast is obtained just by using estimation based on historical data. We limit the number of potential control parameters by investigating the global minimum variance portfolio. This is not a severe limitation because it has been shown that under weight constraints the ex-post Sharpe ratio of the global minimum variance portfolio is often no smaller than other efficient portfolios (Jagannathan and Ma, 2003). We verify that several of the results obtained under idealized conditions are still observed for the global minimum variance portfolio under much more realistic conditions. The paper is organized as follows. In Section 2 we describe briefly the mean variance optimization problem, we define the notation and we summarize the problem of the estimation of the correlation matrix. In Section 3 we review the approach recently introduced (Laloux et al., 2000 and Rosenow et al., 2002) which makes use of the RMT to improve the portfolio optimization in the presence of estimation errors due to the finiteness of sample data. In Section 4 we describe the clustering algorithms used to perform the portfolio optimization. These algorithms are the average linkage and the single linkage. In Section 5 we describe the portfolio optimization procedure performed with clustering algorithms and we compare the obtained results with the one of the RMT under the idealized assumptions of perfect forecasting and short selling allowed. The more realistic conditions of forecasting based on historical data and no short selling for the global minimum variance portfolios are investigated in Section 6. Finally in Section 7 we summarize our results and indicate future work extending and possibly improving our method.

In this paper we have performed portfolio optimization by using filtered correlation coefficient matrices. These matrices have been obtained by applying different filtering methods to the original correlation coefficient matrix. We have proposed two filtering methods based on the average linkage and single linkage clustering procedures. The optimal portfolios obtained with these two new methods have been compared with the one based on RMT recently proposed in Laloux et al. (2000) and Rosenow et al. (2002) both under idealized conditions and under more realistic conditions. In the idealized case, a large set of simulations have shown that clustering methods are outperforming RMT filtering when we consider the reliability of the estimation of the realized portfolio with respect to the predicted one for portfolios with a number of assets ≈50<N<≈500≈50<N<≈500. Hence, for relatively large portfolios the clustering filtering methods provide a more reliable estimation of the predicted risk-return profile both with respect to the Markowitz basic estimation and with respect to the determination of the correlation coefficient done with the RMT filtering The portfolios obtained with the average linkage show a predicted and realized risk-return profile which are characterized by high values of reliability and are often located within the corresponding profiles obtained both with the Markowitz basic estimation and after the RMT filtering. In the case of the single linkage clustering method the risk-return profile shows risk levels which are systematically higher than the ones obtained both with the Markowitz basic estimation and after the RMT filtering. Therefore with respect to the aspect of the level of risk associated with the selected portfolios the most successful methods are the average linkage and the RMT filtering. Another aspect investigated in our study refers to the composition of the portfolios selected. We have quantified the degree of homogeneity of the distribution of the wealth across the stocks of the portfolio through the ‘effective size’ of the portfolio. A small number of this parameter indicates an uneven distribution of the portfolio wealth suggesting that during portfolio re-balancing only a subset of stocks will be significantly involved. The investigation of the ‘effective size’ of the portfolio has shown that the average linkage and the RMT are characterized by not too different values of the ‘effective size’. In fact for small portfolios (e.g. N=50N=50) the RMT has for most values of T a smaller value of the ‘effective size’ whereas the pattern is reversed for medium (N=300N=300) and large portfolios (N=500N=500) both for the minimum and intermediate value of rprp. The pattern is clearly different for the case of the single linkage filtering. In this case the ‘effective size’ is always significantly less than the one observed in the cases of RMT filtering. The above discussion of the idealized case shows that the different filtering procedures provide different portfolio optimization results that are characterized by specific strengths or weaknesses. In other words, for each value of N and T, the most useful filtered correlation coefficient matrix can be different depending on the strongest constraint the investor has among the risk level of the portfolio, the reliability of the estimation and the portfolio ‘effective size’. We believe that the two clustering methods we have proposed here and the RMT are not exhaustive with respect to all potential aspects of portfolio optimization and, probably, other filtering methods could also provide very interesting results in specific regimes of the different control parameters. These findings obtained in the idealized case have been compared with the results obtained for the global minimum variance portfolio selected under more realistic conditions. We have verified that the reliability results of clustering portfolio optimization methods are largely still present under the realistic assumptions of mean return and volatility forecasts and no short selling. Results about portfolio realized risk and Sharpe ratio indicate that the average linkage clustering portfolio optimization method often outperforms Markowitz or RMT methods when one assumes perfect forecasting of returns and volatilities but no short selling conditions (Case III of Tables 3 and 4). Always considering the realized risk and Sharpe ratio, in the absence of perfect forecasting ability and no short selling (Case IV of Tables 3 and 4), we do not observe any method systematically outperforming the other ones. The last observation concerns the effective size of portfolios. It is worth noting that, under realistic conditions (Case IV of Table 5), different portfolios characterized by similar values of the Sharpe ratio can present a much lower value of N(eff)N(eff) when one uses the single linkage clustering portfolio optimization procedure. This result can be achieved at the cost of accepting a slightly higher level of the realized risk. The different results of the different filtering methods raise the scientific question of which is the reason for the difference between the various filtering procedures. A precise quantification of the information retained by the different filtered matrices would be very useful. This goal is left for future research.