Dispersion as Cross-Sectional Correlation

Transcription

1 Dispersion as Cross-Sectional Correlation Bruno Solnik Jacques Roulet Abstract This paper introduces the concept of cross-sectional dispersion of stock-market returns as an alternative approach to estimating the global correlation level of stock markets. Our objective is to derive a simple instantaneous measure of the general level of global market correlation. Our cross-sectional method of estimating global correlation is dynamic and gives instantaneous information on the trend of global correlation using cross-sectional data. The traditional time-series method requires a long period of observations, and overlapping data have to be used to study the change in correlation. However, both methods yield similar estimates for a long period. A combination of the cross-sectional and time-series approaches should be of practical use to global asset managers. *HEC- School of Management Jouy en Josas, Cedex, France Tel: , Fax: solnik@hec.fr

2 Introduction The case for international investing has been built on the rather low correlations across national equity markets. If global correlations are low, spreading investments across countries makes it possible to diversify the total risk of a portfolio. In addition, if global correlations are low, active managers have more opportunities to find assets and markets that will have a large return. So from both a risk and a return viewpoint, low global correlations are beneficial. The traditional approach to estimating the extent of markets' comovement is to conduct a time-series estimation 1 of market correlations over a fixed period of time, for example an estimation window of 5 years of monthly data. This method is poorly suited to studying the change in correlation over time, as a large number of observations are required to estimate just one correlation coefficient. To study time variation in correlation, one has to resort to overlapping observations, with a rolling estimation window. This is not a satisfactory method, because two successive correlation estimates are based on almost the same data set and differ only because one old observation is dropped and replaced by a new one. It takes a long time for a permanent change in the general level of global correlation to be reflected in the estimation. A temporary change hardly gets noticed, as it affects only a few observations in the estimation window. The question of changes in the level of global correlations has strong practical relevance. Investment managers optimize their asset allocations partly on the basis of the international covariance structure of market returns. The level of global correlation affects the degree of diversification needed in their portfolios while the size of the excess return of asset classes is 1 For example, Solnik, Boucrelle and Le Fur (1996) looked at the correlation of several markets with the U.S. stock market over the period They computed the moving correlation over the previous 36 months for all country pairs. A 36-month estimation window is quite short to get statistically-reliable correlation estimates. They found that the correlation fluctuates over time, but showed only a modest increase over the period. See also Erb, Harvey and Viskanta (1994). 1

3 determined by the dispersion. What they need is an indicator that will instantaneously track the time variation in global correlation. In the absence of such a quantitative indicator, we are left with emotional impressions that can be misleading and difficult to integrate into a structured quantitative asset allocation process. In this paper, we propose a new and simple way to measure the instantaneous changes in the general level of correlation, based on the cross-sectional dispersion of returns. It is a simple model linking cross-sectional dispersion to global correlation. It needs refinement, and its practical implementation in asset management is not straightforward. However, we suggest different ways to make the concept of crosssectional global correlation useful for practitioners. Clearly, a portfolio manager needs to have refreshed estimates of market correlations in periods when market volatility fluctuates widely over time. The globalization of investments and the instantaneous flow of information have rendered markets particularly prone to contagion whenever a regional crisis develops. Our dispersion measure and the global correlation measure that we derive from it are practical, albeit partial, answers to this need, as they make it possible to instantaneously detect changes in correlation in the international marketplace. The traditional time-series approach to measuring correlation The traditional approach is to measure the time-series correlation for each market pair over a fixed time period, typically 5 years of monthly data (the estimation window). A sufficient number of observations (e.g. 60 months of data) is required to derive statistically significant estimates 2. The distribution of returns is assumed to be multivariate-normal with constant parameters. Basically, correlation, volatility and expected returns are assumed constant over the five-year estimation window. One computes the mean five-year return for each market and 2

4 studies the deviations of monthly returns from their mean returns for a market pair. Several points should be stressed: Each pairwise correlation coefficient is computed separately. To answer our lead question on the general level of global correlation, we need to compute a crosssectional average of the correlations across all markets. This is an unconditional estimation, where correlation is assumed to be constant over the 60-month estimation window. It is assumed that return distributions remain the same over the whole period. Because correlation needs to be estimated over a timeseries of data, it is difficult to conclude anything about the change in correlation over time, at least in a statistically meaningful way. For example, a total period of 15 years yields only 3 independent observations of the 60-month correlation coefficient. While the moving correlation can be computed with overlapping estimation windows (by replacing one monthly observation each month), two successive correlation estimates are strongly, and somewhat spuriously, dependent. Refinements to the time-series method have been proposed. Pairwise correlations can be computed using weights that give more importance to recent periods. This is somewhat ad-hoc as the set of weights are chosen arbitrarily. An attempt to use simultaneously the information on all markets can be based on the multivariate GARCH approach with time-varying covariances 3. Unfortunately, there are severe limitations as discussed in Kroner and Ng (1998). The "larger" model used is the VECH model, where the covariance between two markets is estimated as a weighted average of past cross-products of the unexpected returns of two markets. The time-varying covariance between two markets is still only a function of 2 From now on, we will illustrate our discussion using one month as the observation period and 5 years as the estimation window. 3 See Longin and Solnik (1995), Kroner and Ng (1998) or Ramchand and Susmel (1998). 3

5 returns on these two markets ("diagonal" model) and there are two practical shortcomings: the number of parameters to be estimated is huge (630 for 20 countries) even in a diagonal model and it is difficult to ensure that the correlation matrix is indeed semi-definite-positive as it should be. To circumvent this non-positiveness problem a BEKK model has been proposed; unfortunately there are now 1010 parameters to be estimated for 20 countries and the model is still diagonal. Simpler models can be used, but their treatment of time variation in correlation is even more simplistic. A cross-sectional approach to measuring correlation: dispersion There is another way to look at the general questions of stock markets' comovement and global risk measurement. It has to do with the dispersion of returns on the various markets for any given observation period. This is a cross-sectional approach rather than a time-series one. Let us call world return the average return on all markets. If markets are moving together ( highly correlated ), all markets will provide a similar return in any given month. Although the world return will vary over time, the dispersion of all national market returns around this world return will be small in any given month. On the other hand, a large dispersion of national market returns around the world return in a given period would indicate that markets are not really moving together and that there is ample room for global risk diversification (or profit opportunities for active investors). Let us take 1997 as an example, when the return (in U.S. dollars, dividends reinvested) on the MSCI World Index was 16.3%. There was a large dispersion of returns in U.S. dollars among national markets: Switzerland (44.4%), Germany (43.8%) and Italy (35.8%) topped the list, France (13.1%) was somewhere in the middle, while the worst performers were Singapore (-30.2%) and Japan (-23.8%). This is clearly a large dispersion of returns. 4

6 Our approach is to calculate, for each period, the cross-sectional dispersion of country index returns around the world return, as measured by the standard deviation of national market returns for that period. All returns are measured in a common currency, and we use the U.S. dollar in our examples. The observation frequency has to be selected. Daily returns cannot be used because of the international time difference. Markets are not open at the same time, and there is often little or no overlap in trading in different regions. This non-simultaneity problem also pollutes estimations based on weekly returns (one day out of five), but is less serious for estimations based on monthly returns. Using monthly data in our examples, we get one independent measure of dispersion each month, and we can study its change over time. The inverse relation between cross-sectional dispersion and time-series correlation will be detailed later, but let us first look at the data and give an illustration of the concept of dispersion. Empirical investigation We use monthly returns in U.S. dollars of the 15 developed stock markets originally included 4 in the MSCI World Index, from January 1971 to September A list of these markets is given in Table 1. For each month t, we can estimate the dispersion of returns, as measured by the cross-sectional standard deviation of the 15 market returns, σ e (t). These figures are plotted on Figure 1. The monthly dispersion looks remarkably stable over time, and remains around an average value of 4.5% per month, or 15.6% annually 5. The dotted line in Figure 1 gives the average world return for each month. This world return fluctuates widely from one month to the next, but has little influence on the dispersion. For example, a large 4 Several countries were later included in the MSCI Index: Hong Kong, Singapore,...The MSCI World Index had 22 markets as of the end of To annualize the monthly standard deviation, one simply needs to multiply it by the square root of 12. 5

7 negative world return of -20% appears in October 1987, but the global dispersion is equal to 8.5% in that month. A similar dispersion was observed in January 1987, when the monthly return was a strong 7% and in July 1986 when the world return was zero. A simple time-series regression between the world return and monthly dispersions confirms the visual impression with an adjusted R-squared of only 1.0%. Dispersion was higher in the period and appeared lower in the more recent period. It increased again in The straight line in Figure 1 is the regression line between dispersion and time. Dispersion slowly declines with time, suggesting that markets are increasingly moving together, but the R-squared is only 5%. The fitted dispersion moved from 5.06% per month at the start of 1971 to 3.85% per month in September 1998 (however, the actual dispersion in 1998 was well above the fitted line), and the slope has a t-statistic of To summarize, the dispersion around the world return has remained quite stable: it (slowly) decreased in the first 27 years but jumped up again in In other words, the potential benefits of international diversification have not disappeared over the past twenty years. We now turn to a more formal discussion of the link between time-series correlation and cross-sectional dispersion. We introduce the concept of cross-sectional correlation and illustrate it by proposing a simple model for asset returns. A simple model of cross-sectional correlation Let us assume that the return on any national market i at time t, R it, is driven by the return on the world market, R wt, plus an independent term e it. R it = α i + R wt + e it (1) 6

8 The return on the world market is the average return of all markets, and for simplicity we assume an equally-weighted average. A market-capitalization-weighted average could be used instead, but it would require the knowledge of the market weights at each point in time. Equation (1) assumes that all national markets have a beta of one relative to the world index. This assumption is made to simplify the mathematical exposition and can be done without as shown later. We further assume that deviations from the world index are multivariate-normal and independent: e it ~ N(0, σ e (t)) (2) This equation implies that, in any month, the return of all national markets deviates from the world return by a tracking error that is drawn from the same normal distribution. However, the standard deviation of this tracking error, σ e (t), can fluctuate over time. As stated above, we call this standard deviation of returns on all markets around the world return dispersion. As the world market is the equally-weighted average of all markets, its volatility can change over time and we identify it as σ w (t). At each point in time, we have the correlation between market i and the world index, which is identified as ρ iw (t) and can be time-varying. Given equations (1) and (2), the instantaneous correlation ρ iw (t) is equal to: ρ iw σw () t 1 (t) = = σi() t 2 2 (1+ σ (t) / σ (t)) e w (3) To simplify the notations, we will drop the time subscript, but the reader should be aware that all variables are assumed to be time-varying: 7

9 ρ iw = 1 2 e / 2 w (1+ σ σ ) (4) Likewise, the correlation between any pair of national markets, i and j, is simply equal to: ρ ij 1 = 1+ σ 2 / σ 2 w e thus ρij = ρ 2 iw (5) Admittedly, our model of return is somewhat simplified, as it implies that all markets have the same correlation with the world index. This comes from the assumption that all markets have betas equal to one, with the same tracking error. However, this simple model makes it possible to compute and study a useful measure easily: the average level of global correlation. Further refinements can be introduced, but we keep the model simple to make it easier to understand our approach. Note that there is an inverse relationship between global correlation and dispersion. The higher the dispersion, the lower the correlation. A cross-sectional estimation of correlation Global dispersion can be used to estimate global correlation based on equation (3). We cannot directly measure the time variation in world market volatility, so we make the assumption that it is constant over time, and equal to its long-run average (a monthly standard deviation of 4.5%, or 15.6% annually). Other assumptions could easily be incorporated without much influence on the conclusion. The simplest is to use the standard times-series estimation based on a rolling window of 60 months. A more sophisticated approach would be to use a GARCH model. An univariate GARCH(1,1) is often used to model time variation in stock-market volatility. 8

10 However, as the correlation in equation (3) is a function of the dispersion (estimated crosssectionally, thus with a large variance) and the world market volatility (estimated in timeseries using overlapping data, thus with a small variance), most of the variance of the correlation will be due to the variance of the dispersion. Hence we estimate the monthly global correlation using equation (3), an estimate of the world market volatility and our estimate of monthly dispersion. We call it cross-sectional correlation, to differentiate it from the traditional time-series correlation estimation. This global correlation is plotted in Figure 2 from January 1971 to September The flat line is the unconditional correlation with world returns, equal to 0.69 for monthly returns 6. The unconditional correlation is simply obtained by computing the traditional time-series correlation coefficient for each market over the whole period and averaging these across all markets. Clearly, this traditional correlation is very close to the long-run average correlation computed with our cross-sectional methodology (they differ only in the third decimal). Our approach makes it possible to study the time variation in correlation and has the advantage of being immediately available on a monthly or even more frequent basis. It should be remembered that our monthly correlation estimates are independent from one month to the next, as they are based on the global dispersion of returns for that month: the estimation does not use historical data, and certainly not the past 60 months as the time-series approach usually does. A stable period of high (or low) correlation is not a spurious statistical artifact caused by overlapping data as in the case of time-series estimations with rolling windows. Clearly, correlation was quite low in the periods and (until October), while there was a period of high correlation in the mid-1990s. Note that correlation declined at the end of 1997 and into

11 Again, it appears that correlation has increased somewhat in the past 25 years. The solid line in Figure 2 is the regression line between correlation and time. Correlation slowly increases with time. The fitted correlation moved from 0.62 at the start of 1971 to 0.74 in September 1998, the slope has a t-statistic of 5.91 and the R-squared is equal to 9.1%. Differences between the two methods to estimate correlation Our cross-sectional method to estimate global correlation is dynamic and gives instantaneous information on the change in the level of global correlation using short-term data. The traditional time-series method requires a long period of observations, and overlapping data have to be used to study the change in correlation. A drawback of the dispersion-based methodology is that it requires a sufficiently large number of markets to obtain statistical significance, so it gives information only on movements in the general level of global correlation, not on individual correlations between countries or regions. The reason why the two methods could, in some instances, lead to different estimates should be stressed. Our cross-sectional method is conditional on the world return, while the traditional time-series method is unconditional, in the sense that for each country it studies the deviation of national returns from their long-term mean 7. One method looks at relative returns, the other at absolute returns. We feel that our approach is more appropriate for the current asset-management paradigm, where performance is measured relative to a benchmark rather than in absolute terms. Here we use the equally-weighted world index as a benchmark. But the approach could be extended seamlessly to use a market-capitalization world index or any other index as a benchmark. 6 The average unconditional monthly correlation of national markets with the world is 0.69, while the average pairwise correlation between two national markets is 0.49 (the square of 0.69). 7 By long term we mean over the window used to estimate correlation, typically 5 years of monthly data. 10

12 In some instances, the two methods could lead to somewhat different conclusions about the level of global correlation. The difference can be illustrated with a simple intuitive example. Let us consider three periods where national markets exhibit the same dispersion of 4% around the world index, but where the world return is -10%, +10% and 0%. The long-term 3-period mean return is 0%. In the first period, the returns on all national markets are negative (centered around -10%). In the second period, the returns on all national markets are positive (centered around +10%). In the third period, the returns are slightly positive or negative and centered around zero. The traditional time-series correlation would look at deviations of national returns from their long-term means (close to 0%) and consider that markets are very strongly correlated in period 1 and period 2 because all national markets exhibit together a much lower (or higher) return than their mean return. In period 3, they appear uncorrelated. However, in terms of relative returns, national markets exhibit the same dispersion around the world index in all three periods. 8 Clearly, the results reported above suggest that correlation does not increase dramatically in periods of bear markets and are in contrast with the often-cited quote: "diversification fails us when we need it most". It is not possible to derive a "mathematical" comparison of the two approaches as they are fundamentally different; their linkage depends on the time-dynamics postulated for the true correlation and volatility. A brief empirical comparison of the two indicators of global correlation can be performed. For each month we compute independently two different measures of global correlation: one derived from our dispersion-based approach and one 8 To avoid such a problem caused by a couple of extreme observations (small-sample bias), the time-series approach must use a large number of observations so that the empirical distribution of returns approaches the true distribution. Unfortunately, the time-series estimation also requires that the distribution remain unchanged over the whole estimation period; such an assumption is more likely to be violated over longer time periods. If the three observations used in our simple example were indeed representative of the true distribution of returns (rather than random occurrences in a small sample), it would imply that the stock markets are extremely volatile, and our cross-sectional method would also find a high correlation because σw would be much larger than σ e in equation (3). 11

13 derived from the traditional time-series approach taking the average time-series correlation for all markets (calculated over the past 36 months). As can be expected, the dispersion-based correlation is much more volatile than the time-series correlation (which uses the past 36 months). However, both measures are linked as their correlation over time is equal to 26.7%. Furthermore, a change in the dispersion-based correlation helps forecast a change in the timeseries correlation. Extension and practical use The simple model presented above assumed constant world market volatility in equation (3). An easy extension would be to use the classic times-series estimate based on historical returns up to t. Another possible extension would be to model time variation in world market volatility using a simple univariate GARCH(1,1) representation for σ w (t), as is often done by practitioners. Time variation in such a model depends only on the last innovation (shock) in the world return. So equation (3) would still yield a fresh estimate of correlation 9 based on returns observed at time t. We estimated the cross-sectional correlation with the three different models for world market volatility, and found results that are very similar to those given in Figure 1. We could also drop the assumptions that all markets have a unitary beta relative to the world market. This would require assuming that betas are stable. Equation (1) would become: R it = α i + β i R wt + e it (1a) Then the correlations would become: 9 Again note that equation (3) is consistent with the phenomenon observed in the literature, that correlation increases when market volatility increases. 12

14 ρ iw = e i 2 w (1+ σ / β σ ) (4a) Likewise, the correlation between any pair of national markets, i and j, would simply be: ρ ij = e i w e j 2 w (1+ σ / β σ ) (1+ σ / β σ ) (5a) These correlations could be estimated assuming that the betas are constant over time, but different across countries. Betas estimated over the whole 27-year period are reported in Table 1, with their standard errors and R-squared. The unconditional correlation of each country with the equally-weighted world index is given in the last column; the average is 0.69 as already mentioned. The betas are not very far from unity, and their low standard errors suggest that they are quite stable over time. The U.S. stock market has the lowest beta (0.687), and a high correlation with the world (0.8). In practical asset-allocation implementations, one could go a step further and combine the time-series and cross-sectional approaches with a periodic estimation of individual time-series correlation for each country pair and a dynamic updating of the general level of correlation via our dispersion method. For obvious structural, socioeconomic and sociopolitical reasons, correlations differ across countries. There are some closely linked regional blocs and some countries with loose economic ties. For example, the correlation between the German and Dutch stock markets is higher than the correlation between the U.S. and Hong Kong ones. However, in periods of considerable economic shocks, when global factors strongly influence all economies, the general level of correlation between all markets increases. While the structural relations, which explain differences in correlation across market pairs, are likely to remain quite stable in the short run, this is not the case for the general level of correlation. 13

15 Conclusions This paper introduces the concept of cross-sectional dispersion of stock-market returns as an alternative approach to estimate the global level of correlation of stock markets. Our objective is to derive an instantaneous measure of the general level of global market correlation. We first define the concept of dispersion and illustrate its changes from January 1971 to September 1998 for the original MSCI developed-market indices. We use monthly returns in U.S. dollars. We find that the dispersion is quite stable over the period, at an average of 4.5% per month (15.6% annually), but has a tendency to decrease with time, thus pointing toward more integrated equity markets. A simple model of global market correlation is then derived in an international setting, postulating that each country has a beta of one relative to the world market. Global correlation is linked, in a simple manner, to dispersion as we show that global correlation is inversely proportional to dispersion. On the basis of our model, we show how to estimate the global market correlation, using cross-sectional (or contemporaneous) short-term data. The usefulness of the model is then illustrated by studying whether developed markets have become increasingly correlated over time. We find that correlation has a positive trend, increasing from 0.62 at the beginning of 1971 to 0.74 in September 1998, but the slope of the regression is quite weak. This findings suggests that equity markets are getting more integrated, but at a slower pace than expected by practitioners. This is consistent with the findings in Solnik, Boucrelle and Le Fur (1996), who further stress that the growth of new markets partly offsets this trend. Our cross-sectional method to estimate global correlation is dynamic and gives instantaneous information on the changes of the level of global correlation using short-term 14

16 data. The traditional time-series method requires a long period of observations, and overlapping data have to be used to study the changes in correlations. However, both methods yield similar estimates for a long period (5 years or more). We also believe that it is an important concept in the current controversy between country and industry allocation in global asset management. Structuring asset allocation along industry lines, rather than country lines, is currently an act of faith when looking at all the published empirical evidence based on long-term historical data. 10 Such an industry approach would make sense if the dispersion of returns of global industry indices is larger than that of country indices. It would mean that country factors have become less distinctive (more homogeneous ). Clearly, the global asset allocation process should be structured along factors that maximize global dispersion, thus minimize global correlation. 10 For a review see Lombard, Roulet and Solnik (1999). 15

17 References Erb C.B., Harvey C.R. and Viskanta T.E. 1994, Forecasting International Correlation, Financial Analyst Journal, vol. 50, no. 6 (November/December): Kroner K. F. and Ng V. K. 1998, Modeling Asymmetric Comovements of Asset Returns, Review of Financial Studies, 11 (Winter): Lombard T., Roulet J. and Solnik B. 1999, Pricing of Domestic Versus Multinational Companies, Financial Analyst Journal, vol. 55, no. 2 (March/April): Longin F. and Solnik B. 1995, Is the Correlation in International Equity Returns Constant: ?, Journal of International Money and Finance, 14 (February):3-26. Ramchand, L. and Susmel R. 1998, Volatility and Cross Correlation across Major Stock Markets, Journal of Empirical Finance, 5 (October): Solnik B., Bourcrelle C. and Le Fur Y. 1996, International Market Correlation and Volatility, Financial Analyst Journal, vol. 52, no. 5 (September/October):

18 20% Figure 1: Dispersion of Monthly Returns 15 markets, in US$, Standard deviation in % per month 15% 10% 5% % pe r m onth 0% -5% % -15% -20% -25% standard deviation World return fitted line 17

19 Figure 2: International Correlation Monthly US$ returns 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% correlation fitted line unconditional mean 18

20 Table 1: Some Statistics on the Stock Markets Included in the Study Country Beta Standard error R**2 with World Correlation with World Belgium Denmark France Germany Italy Norway Netherlands Spain Sweden Switzerland U.K Australia Japan Canada U.S.A