حرکت مشترک بین المللی بازده بازار سهام: تجزیه و تحلیل موجی

The assessment of the comovement among international stock markets is of key interest, for example, for the international portfolio diversification literature. In this paper, we re-examine such comovement by resorting to a novel approach, wavelet analysis. Wavelet analysis allows one to measure the comovement in the time–frequency space. In this way, one can characterize how international stock returns relate in the time and frequency domains simultaneously, which allows one to provide a richer analysis of the comovement. We focus on Germany, Japan, UK and US and the analysis is done at both the aggregate and sectoral levels.

The analysis of the comovement of stock market returns is a key issue in finance as it has important practical implications in asset allocation and risk management. Since the seminal work of Grubel (1968) on the benefits of international portfolio diversification (see also, Levy and Sarnat (1970) and Agmon (1972)) this topic has received a lot of attention in international finance. In fact, a growing body of literature has emerged more recently on studying the comovement of international stock prices (see, for example, King et al. (1994), Lin et al. (1994), Longin and Solnik, 1995 and Longin and Solnik, 2001, Karolyi and Stulz (1996), Forbes and Rigobon (2002), Brooks and Del Negro, 2005 and Brooks and Del Negro, 2006). In particular, most of those studies have found that the comovement of stock returns is not constant over time. For instance, Brooks and Del Negro (2004) and Kizys and Pierdzioch (2009) found evidence of increasing international comovement of stock returns since the mid-90s among the major developed countries. It has been current practice to evaluate the comovement of stock returns through the correlation coefficient while the evolving properties have been investigated either through a rolling window correlation coefficient (see, for example, Brooks and Del Negro (2004)) or by considering non-overlapping sample periods (see, for example, King and Wadhwani (1990) and Lin et al. (1994)). However, the comovement analysis should also take into account the distinction between the short and long-term investor (see, for example, Candelon et al. (2008)). From a portfolio diversification view, the first kind of investor is naturally more interested in the comovement of stock returns at higher frequencies, that is, short-term fluctuations, whereas the latter focuses on the relationship at lower frequencies, that is, long-term fluctuations. Hence, one has to resort to the frequency domain analysis to obtain insights about the comovement at the frequency level (see, for example, A'Hearn and Woitek (2001) and Pakko (2004)). One should note that, despite its recognized interest, analysis in the frequency domain is much less found in the financial empirical literature (see, for example, Smith (2001)). In this paper, we re-examine the stock return comovement among the major developed economies through a novel approach, wavelet analysis. Wavelet analysis constitutes a very promising tool as it represents a refinement in terms of analysis in the sense that both time and frequency domains are taken into account. Although wavelets have been more popular in fields such as signal and image processing, meteorology, physics, among others, such analysis can also provide fruitful insights about several economic phenomena (see, for example, Ramsey and Zhang, 1996 and Ramsey and Zhang, 1997). The pioneer work of Ramsey and Lampart, 1998a and Ramsey and Lampart, 1998b draws on wavelets to study the relationship between several macroeconomic variables (see, for example, Crowley (2007) for a survey). In particular, wavelet analysis provides a unified framework to measure comovement in the time–frequency space. The study of the comovement of stock market returns is crucial for risk assessment of portfolios. A higher comovement among the assets of a given portfolio implies lower gains, in terms of risk management, stemming from portfolio diversification. Hence, the evaluation of the comovement is of striking importance to the investor so that he can best assess the risk of a portfolio. On one hand, it has been acknowledged that the comovement of stock returns varies over time. Hence, one has to be able to capture this time-varying feature as it implies an evolving risk exposure. On the other hand, the distinction between short and long-term investors should not be ignored as the first is more interested on short-run movements whereas the latter on long-run fluctuations. That is, if the degree of the comovement of stock returns varies across frequencies the risk for each type of investor will also be different. In contrast with time or frequency domain approaches which allow one to focus only on one of these issues, wavelet analysis encompasses both. In particular, through wavelets one can assess simultaneously the strength of the comovement at different frequencies and how such strength has evolved over time. In this way it is possible to identify regions in the time–frequency space where the comovement is higher and the benefits of portfolio diversification in terms of risk management are lower. In addition, we also extend such analysis to the sectoral level. That is, besides considering the aggregate stock returns, we also distinguish ten sectors for each country. For the international diversification of equity portfolios, the assessment of the comovement at the sectoral level also plays a role (see, for example, Roll (1992), Heston and Rouwenhorst (1994) and Griffin and Karolyi (1998)). For instance, it is important to assess if the evidence of greater interdependence of international stock markets is confined or not to a small set of sectors (see, for example, Berben and Jansen (2005)). Again, wavelet analysis can provide interesting insights on how such international comovement has evolved over time across frequencies for the different sectors. Hence, this paper provides a fresh new look into the characterisation of the comovement among international stock returns. We focus on the major developed economies, namely Germany, Japan, United Kingdom and United States over the last four decades. Moreover, by considering the decomposition of the aggregate index in ten sectors, we also provide insights at the sectoral level. This paper is organised as follows. In Section 2, the comovement measure in the wavelet domain is presented. In Section 3, a data overview is provided and in Section 4 the empirical results for the major developed economies are discussed. Finally, Section 5 concludes.

The assessment of the international comovement of stock returns is crucial so as to shed light, for example, on the potential benefits of international portfolio diversification. This paper provides a new look into the comovement measurement of stock returns by resorting to wavelet analysis. Wavelet analysis allows one to assess the time- and frequency-varying comovement within a unified framework. This analysis is of particular interest in the context of the study of stock returns comovement as it is by now a stylized fact that the degree of comovement has changed over time and because one should be able to take into consideration the distinction between the short- and long-term investor, that is, the frequency domain. In fact, with wavelet analysis one can take into account the time and frequency domains simultaneously.In this paper, we consider the stock returns for the major developed countries, namely Germany, Japan, United Kingdom and United States over the last four decades and besides the aggregate index we also consider its decomposition in ten main sectors, so as to provide insights at the sectoral level. A noteworthy finding of this paper is that the strength of the comovement of international stock returns depends on the frequency. In general, we find that comovement between markets is stronger at the lower frequencies suggesting that the benefits from international diversification may be relatively less important in the long-term than in the short-term. Therefore, the nature of the investor, in terms of short or long-term profile, should be taken into account when addressing the international portfolio diversification problem. We also found that the strength of the comovement in the time–frequency space varies across countries as well as across sectors. For instance, even though the Japanese stock market is generally weakly correlated with the other developed countries stock markets considered (as in Berben and Jansen, 2005), there are some sectors (technology and consumer goods) displaying strong comovements at particular frequencies and time periods. Finally, it was also found that the degree of comovement has changed over time, in line with the findings of Brooks and Del Negro (2004), among others. However, such changes are found to be confined, in several cases, to particular frequency ranges. Moreover, the detected changes are of different natures regarding their persistence in time. For example, the degree of comovement of the German market with the US and UK markets is characterized by some permanent changes over time: a gradual but steady increase of the comovement at the lower frequencies, and also a sudden increase after the end of the nineties for the other frequencies. On the other hand, the episodes of stronger comovement at higher frequencies between the US and UK markets around the 1987 crash and at the end of the century technological bubble are clearly of a distinct transitory nature. The first phenomena may be explained by the increased integration of financial markets whereas the latter may be associated with contagion. All these results highlight the importance of taking into consideration the time and frequency-varying properties of stock returns comovement in designing international portfolios as it may influence the benefits of international portfolio diversification in a non-negligible way.