انتخاب یک استراتژی سرمایه گذاری مطلوب: نقش قوی پرتفوی بر پایه جفت وسیله های اتصال

This paper is concerned with the efficient allocation of a set of financial assets and its successful management. Efficient diversification of investments is achieved by inputing robust pair-copulas based estimates of the expected return and covariances in the mean-variance analysis of Markowitz. Although the whole point of diversifying a portfolio is to avoid rebalancing, very often one needs to rebalance to restore the portfolio to its original balance or target. But when and why to rebalance is a critical issue, and this paper investigates several managers' strategies to keep the allocations optimal. Findings for an emerging market target return and minimum risk investments are highly significant and convincing. Although the best strategy depends on the investor risk profile, it is empirically shown that the proposed robust portfolios always outperform the classical versions based on the sample estimates, yielding higher gains in the long run and requiring a smaller number of updates. We found that the pair-copulas based robust minimum risk portfolio monitored by a manager which checks its composition twice a year provides the best long run investment.

Financial institutions and portfolio managers are primarily concerned with the efficient allocation and monitoring of sets of financial assets. Periodic portfolio rebalancing, aiming to restore the investment back to its desired target risk and return, is a crucial step in the process of controlling risk. Commonly asked questions are how often a portfolio should be rebalanced, and which would be the best indicators of changes in the global economy or in the balance among the component assets. Efficient diversification of investments based on the mean–variance analysis of Markowitz (1952) is widely used by institutional investors. Statistically, the resulting efficient frontier just relies on the estimates of the expected return and covariance matrix, and the sample estimates are the usual inputs. However, the statistical good properties of the sample estimates are attached to the highly improbable assumption of multivariate normality. A better characterization of the data underlying multivariate distribution will provide more reliable estimation of the efficient frontier. This means we must know not just the marginal univariate series behavior and their correlations, but their whole d-dimensional probability distribution. This may be accomplished by modeling the data through pair-copulas ( Berg and Aas, 2009, Fischer et al., 2008, Aas et al., 2007 and Min and Czado, 2008). A pair-copula construction is just a hierarchical decomposition of a multivariate copula into a cascade of bivariate copulas. Since an appropriate copula function can be found for any type of association – linear, nonlinear, ranging from perfect negative to perfect positive dependence – one can expect the model to truthfully represent the data at hand. Estimation takes place at the level of the two-dimensional data, therefore avoiding the famous curse of dimensionality. The analysis of financial data from emerging markets poses some specific challenges. Atypical points in transaction prices (from non-confirmed unexpected news, market manipulation, and so on) distort classical statistical inference, corrupting the inputs to the mean–variance algorithm. A distorted correlation matrix and inflated risk estimates will provide misleading allocations. To handle deviations from the true underlying distribution, robust methodologies are called for. We suggest to apply the robust estimates for pair-copulas models, initially proposed in Mendes et al. (2007). For each parametric copula family there exist a robust weighted minimum distance or a weighted maximum likelihood estimator providing accurate estimates under contaminations. The robust portfolios are obtained by inserting the robust pair-copulas based mean and covariance estimates in the mean–variance Markowitz procedure. Robust methods typically detect the pattern implied by the vast majority of the data, providing more stable estimates. Robust allocations are resistant to unjustified sudden fluctuations of the market, which are identified by the robust estimates as point contaminations. Therefore, robust portfolios are primarily designed for long run investments. We note that the notion of “long run” may vary across markets and to account for changes in the economy, some periodic rebalancing of the portfolio may still be needed. It is expected from a robust investment to yield higher gains in the long run and to require a smaller number of updates. There is no universally accepted best strategy for portfolio management. Best strategy will change with investor risk aversion, portfolio target return or standard deviation. Among many others, we consider the popular strategy followed by institutional investors that monitors a portfolios at an annual (or monthly) frequency and then rebalances only if the allocation to an asset shifts more than some threshold (5%, 1%). We do not consider additional factors when implementing the rebalancing strategies, such as trading costs or cost of time spent which would reduce the return of the portfolio. However we keep track of the rebalancing frequency of each manager and are able to draw some conclusions based on their number of rebalancings. Summarizing, in this paper we address both problems of composing and managing portfolios, given a set of financial instruments. We do not address the issue of choosing the component assets. We robustly estimate the data multivariate distribution using pair-copulas obtaining the inputs which will define the robust efficient frontier. The trajectory of a target return and the minimum risk portfolios will be managed by twelve managers during a 2-year period of out-of-sample investigation. We use data from emerging markets because of the higher volatility of these stock markets and their greater potential for interdependence with the major markets. More specifically, we use six-dimensional contemporaneous daily log-returns from the most traded Brazilian stocks, due to Brazil's important position among emerging equity markets. The robust portfolios are compared to their classical version based on the sample empirical estimates. The contributions of this paper are three fold: (i) we introduce and investigate the performance of pair-copulas based robust portfolios; (ii) we investigate 12 managing strategies aiming to keep (or restore) the portfolio target, to guarantee the same risk aversion level; (iii) we illustrate the ideas using Brazilian data. Findings in the paper are striking and convincing. We found that despite the investment type, the robust methodology always outperform the classical version. We are able to determine the best rule for restoring the portfolio to its original balance and keep the allocations optimal. We show that the best strategy depends on the investor risk profile, and that pair-copulas based robust minimum risk portfolios monitored by a manager which checks its composition twice a year provides the best long run investment. Additional exercises are not provided because the aim of the paper is to find best portfolio composition along with best long run managing strategy for a given set of financial instruments. The remaining of this paper is organized as follows. In Section 2 we briefly consider the classical mean–variance methodology for obtaining efficient portfolios, and briefly review the definitions of pair-copulas and robust estimates. In Section 3 we define 12 strategies for portfolio monitoring. In Section 4 we analyze two 6-dimensional data sets and assess the performance of classical and robust target and minimum risk portfolios. In Section 5 we discuss and summarize the results.

As usual, the conclusions drawn in this paper only apply to data possessing similar characteristics. However, the results and discussions may shed some light on the largely discussed topic of portfolio allocation and rebalancing strategies, and may be easily tested and extended to other investments based on different asset characteristics (expected return, volatility and correlations). More importantly, this work has shown that the robustly estimated pair-copulas based mean–variance inputs are more accurate thus increasing the chances of producing a financial instrument which will truthfully yield what was expected from it. From the analyzes carried on the superiority of the robust portfolio was clear. But even a robust portfolio must be properly managed. According to the empirical analysis of the first data set composed only by daily stock returns, for the Robust Minimum Risk portfolio the second best manager with respect to higher accumulated gains is Manager 9 (which uses the VaR). However, Manager 9 has done 9 rebalancings, whereas Manager 12 carried on only one after one year and a half. One may be tempted to conclude that the good performance of the winner may be just due to the changes in the economy which may have taken the portfolio away from the efficient frontier. It was not the case. Fig. 8 illustrates and shows the positions of the robust minimum risk portfolios from both managers (based on weights from last rebalancing and using the entire validation period data) along with the updated efficient frontier. We observe that the portfolios are very close and still close to the curve, with the point risk × return from Manager 9 (triangle in pink) even a little bit higher. The success of this procedure may be credited to the timing of Manager 12 combined with the good stability of the robust pair-copulas method of portfolio construction. Actually, for the second data set used based on less volatile assets, Manager 12 was also the best option for the Classical Minimum Risk portfolio. Manager 9 was also the second best option for the Classical Minimum Risk portfolio, and the best one for the Robust Target. Thus, one may say that controlling the Value-at-Risk is also an efficient strategy. Another issue is whether or not the target portfolio should be restored to the same target. We did not address this problem, but this consideration will certainly not change the result towards the excellent performance of the robust method. Many other managers' rules could be defined. Another strategy that seems promising is a variation of the rules followed by managers 10, or 11, or 12, where the threshold determining the need for rebalancing would not be fixed for all assets. Instead, it would vary according to each asset weight or importance in the composition. In addition, it deserves further investigation the actual value of the threshold. Other values beyond the assumed 5% and 10% may lead to better performances. Another important result drawn from this empirical analysis is concerned with costs. We found that despite portfolio type, robust portfolios typically demand a smaller number of updates lowering costs. Finally, we found that despite the rebalancing rule, the robust portfolios always outperform the classical versions. Fig. 9 shows the differences between the accumulated gains from the robust and the Classical Target Portfolios, having fixed the managing rule. Given the same manager and the same portfolio target, the robust method is always superior to the classical method. This is also true for the Minimum Risk portfolios, and the outstanding performance of the Robust Minimum Risk portfolio for all fixed managers can be seen in Fig. 10. In summary, observing that the dependence between assets go beyond the linear correlation, in this paper we proposed modeling log-returns data using robustly estimated pair-copula models. The method is appealing simple and able to handle contaminations that may occur when working with financial data. We illustrated the idea in the context of emerging stock markets using two data sets composed by the most liquid Brazilian stocks, a long-term inflation-indexed Brazilian treasury bonds index and a floating rate Brazilian Treasury bill index. The empirical analyzes carried on in this paper indicated that for any type of portfolio we are able to find the best manager strategy. Moreover, they indicated that the robustly estimated pair-copulas based portfolios always outperform the classical versions despite the managing strategy. Methodology seems promising for any risk profile investor. The interested reader (or investor) may easily tailor these ideas to his/her needs, repeating these exercises using other data sets and considering other investor risk aversion levels, and even defining new rules for managing. We are very confident that it will always be a combination of manager and a robust portfolio outperforming its classical version. The authors will be happy to compute the robust estimates and check the performance of any portfolio for any data set the reader shall have. Future research may include the investigation of different managers' rules for other types of portfolios. Indeed, we have a final recommendation. If the objective is a minimum risk long run investment the best one can do is to allocate the assets using robustly estimated pair-copulas estimates and inspect the portfolio each 6 months, rebalancing whenever allocations change by 10%. This will guarantee the investment characteristics and provide the cheapest managing strategy.