جدا کردن گندم از کاه: آشنایی با بازده پرتفوی در بازار نوظهور

In this paper we apply Random Matrix Theory (RMT) to study daily return correlations of 83 companies that are part of the Chilean stock market during the period 2000 to 2011. We find that using RMT to identify statistically significant correlations within our sample of stocks significantly improves the efficiency of a family of Markowitz Portfolios. Moreover, by using Vector Autoregressive analysis we identify global risk aversion as the main driver of the Chilean equity market returns followed in importance by shocks to the monthly rate of inflation and the country's monetary policy rate. By studying the effects of macroeconomic variables on the constructed portfolio returns we reach a better understanding of the true risks involved in an emerging market portfolio.

The Great Recession of 2008–2009 was a vivid reminder that financial correlations breakdown during periods of high volatility. It also highlighted the importance of identifying stable correlations in order to quantify the underlying risk of diversified portfolios. This paper contributes to current debates on portfolio optimization by examining the statistical significance of the correlations across the Chilean stock market. Two questions frame our investigation. First, we ask if all the return correlations across the Chilean stock market are equally statistically significant. Second, we ask which are the main macroeconomic drivers affecting the Chilean stock market returns. To answer these questions, we use Random Matrix Theory (RMT) to study the daily returns of 83 Chilean stocks that are part of the IPSA and IGPA indices from January 2000 to January 2011. The RMT helps us to separate the wheat from the chaff in the correlation matrix. Using Markowitz's Portfolio Theory (MPT), we then compare the efficiency of portfolios constructed under RMT with others constructed under the standard approach which considers all covariances in the correlation matrix equally significant. Finally, we use Vector Autorregresion approach (VAR) to determine the impact of a set of macroeconomic and financial variables on the optimal portfolios derived from our significant eigenvalues. We focus on the Chilean stock market as we think that it provides a good case study for other emerging markets due to a number of reasons. First, the Chilean equity market is one of the most developed within the emerging market world with a market capitalization of 120% of its GDP. Other markets such as the Brazilian one have much lower market capitalizations as percentage of their GDP (58%) while Mexico has a market capitalization of 39% of GDP. Second, Chile is a small open economy with almost no restrictions to the access of international investors. Therefore, both idiosyncratic and global factors are likely to be important determinants of the stock market returns. Assessing the relative importance of domestic versus international factors in explaining domestic market volatility is key for developing public policy and market regulation in other emerging economies that are following the path of financial liberalization. Our main findings can be summarized as follows: First, applying Random Matrix Theory to a portfolio composed of Chilean equities improves its efficiency compared to a portfolio constructed using a standard MPT approach by at least 48%.2 Second, using VAR analysis, we identify global risk aversion as the main macroeconomic determinant of the Chilean equity market returns followed in importance by shocks to the monthly rate of inflation and the country's monetary policy rate. Third, it is possible to diversify away some of the market portfolio risk by adding positions on the portfolios constructed by the second and third largest eigenvalues. Fourth, the three smallest eigenvalues produce portfolio returns that are mostly uncorrelated with macroeconomic shocks. These portfolios are also uncorrelated with the market portfolio. Finally, we show that the insights provided by the RMT approach can help us to improve some existing models of the MV-GARCH literature, with significant improvements in realized risk predictions. In general, the stability and statistical significance of empirical correlations are crucial for risk and portfolio management, since the probability of large losses in a portfolio is mainly driven by the way the correlations between the assets in it behave. For example, a position which is simultaneously long in stocks and short in bonds is riskier than one holding any of those assets exclusively. This is because bonds and stocks usually move in opposite directions during periods of crisis (Laloux et al., 1999). The use of correlation matrices has a long history in portfolio management and is one of the main ingredients of Markowitz's Portfolio Theory, which solves the following dual problem: given a set of financial assets characterized by their average return and risk; what is the optimal weight on each asset such that the overall portfolio provides the best return for a fixed level of risk, or equivalently, the lowest risk for a given return? The solution to this problem entails the resolution of a system of non-linear equations where the correlation matrix C has to be inverted. The final outcome of this exercise is a risk–return relation, the so-called “efficient frontier”. Despite the apparent success of this theory in explaining optimal portfolios, it has been widely criticized by its inability to provide good risk estimates. A possible explanation for this failure in the Markowitz's Portfolio Theory, has found room in the field of Statistical Physics, which states that the estimation of a correlation matrix may be difficult in case the length of the time series (henceforth, T) is not very large compared to the total number of assets (henceforth, N). In these cases we should expect the covariances to contain a lot of noise and therefore be, to a large extent, random. In particular, the smallest eigenvalues are precisely those that are the most sensitive to this noise and their associated eigenvectors the ones that determine the least risky portfolios ( Laloux et al., 1999 and Laloux et al., 2000). With all these problems in mind, it becomes apparent the need to seek for new methods to separate out “signals”, which contain economically relevant information, from “noise” which do not. From this perspective, it turns out interesting to compare the properties of an empirical correlation matrix C with respect to a “null hypothesis” purely random, that one could obtain from a time series of completely independent assets. In this context, deviations from this random counterpart could indicate evidence of relevant economic information. The Econophysics literature deals with this issue by using the Random Matrix Theory (RMT), a technique taken from Statistical Physics. RMT has been recently applied to the study of financial correlations (Laloux et al., 1999 and Laloux et al., 2000). A number of studies show the advantages of using the RMT approach to remove noise from correlation matrices, with significant improvements in portfolio choice (Gopikrishnan et al., 2001, Plerou et al., 1999, Plerou et al., 2000a, Plerou et al., 2000b, Plerou et al., 2001 and Plerou et al., 2002). Our paper contributes to the academic literature in two important ways. First, by applying RMT to the Chilean stock market we provide further evidence of the benefits of using this technique in building efficient portfolios. Second, to the best of our knowledge we are one of the first to explore the relationship between macroeconomic variables and portfolio returns constructed using statistically significant correlations. The rest of the paper is organized as follows. Section 2 introduces the relevant literature on Random Matrix Theory while in Section 3, we provide an eigenvalue analysis of the Chilean stock market. In Section 4, we apply the RMT notions in a Markowitz's portfolio framework to improve portfolio risk estimates. In Section 5, we look for the macroeconomic and financial drivers of stock returns. Finally, in Section 6, we conclude and propose lines for future research.

In this paper we apply Random Matrix Theory to study the correlations of 83 Chilean stocks that are part of the IPSA and IGPA indices during the period 2000 to 2011. We find that using Random Matrix Theory to identify statistically significant correlations within our sample of stocks significantly improves the efficiency of a family of Markowitz Portfolios. Moreover, by using Vector Autoregressive analysis we identify the global risk aversion as the main driver of the market returns followed in importance by shocks to the monthly rate of inflation and the country's monetary policy rate. With the use of statistically significant eigenvalues we can construct portfolios, which are uncorrelated to macroeconomic and financial factors such as global risk aversion. Despite the recent success of Random Matrix Theory in financial applications, challenges remain. Recent evidence has suggested that meaningful correlations can be measured in the bulk of the MP eigenvalue spectrum (Burda and Jurkiewicz, 2004, Burda et al., 2004, Kwapièn et al., 2006 and Malevergne and Sornette, 2004). One possible reason for this is the fact that we are assuming that our returns are normally distributed and under this premise we carry out our comparisons with the empirical correlation matrix. In reality, returns in financial markets are not normally distributed and exhibit a number of features, such as fat tails, volatility clustering and non-stationarity. Recent literature has been improving these matters by developing extensions of the MP distribution to account for these phenomena (Bouchaud and Potters, 2009 and Potters et al., 2005). However, still one of the main difficulties has been to generate predictions during critical periods, where there is a high collectivity between stocks, especially during drawdowns. In this context, an interesting line of future research is to extend this approach focusing on turbulent periods in financial markets. In the present paper, we have tried to deal with this by combining the RMT approach with some standard models of the MV-GARCH literature in order to capture to some extent the short run dynamics of financial correlations with significant improvements in correlation estimates. Another interesting line of future research would be extending our analysis to regional stocks such as the MILA or the MSCI LATAM. Also, it would be interesting to apply RMT to cross assets within the Chilean financial market including the return of equities, fixed income and currency. This would give us a more complete picture of the factors that account for the volatility of the Chilean financial market. All in all, the results obtained for the Chilean case provide important evidence of the broad applicability of the RMT approach to equity returns in emerging markets portfolios. The Chilean stock market is characterized by the existence of large institutional investors such as private pension funds that maintain important structural positions on this market. Applying RMT to their portfolio optimization could be useful in diversifying some of the systemic risk of the Chilean market.