Correlations and Diversification Benefits S. Mehmet Ozsoy∗ January 17, 2018
Abstract Return correlations are shown to be asymmetric: They are higher in bear markets than bull markets, and higher during recessions than during booms. Heightened correlations from bull to bear markets are associated with diminished diversification benefits, consistent with the common intuition. However, we show that this is not the case for counter-cyclical correlations: Both the correlations and diversification benefits increase from booms to recessions. Thus, we claim that the correlation coefficient is a misleading measure of diversification opportunities. We also analyze how different causes behind a correlation increase have different implications for diversification benefits.
Keywords: Stock return asymmetries, asymmetric correlations, diversification benefits
Ozyegin University, Faculty of Business, Nisantepe Mah. Orman Sok. No 34-36, Istanbul, Turkey 34794. Email: [email protected]
There is widespread evidence that the correlations of returns in financial markets are time-varying. One striking feature of time-varying correlations is the asymmetric nature of them: correlation are higher during bear markets than during bull markets, and higher during recessions than during booms. These heightened correlations are considered to be detrimental to the risk diversification aim of the investors. Given that investors might seek diversification opportunities especially for difficult times such as bear markets or recessions, it would be an unpleasant feature of markets if diversification benefits are indeed slim during those times due to the heightened correlations. However, the relationship between the correlations and diversification benefits is not as straightforward. In this paper we analyze the diversification benefits over the business cycle (boom versus recessions) as well as with respect to market trend (bull versus bear markets). The asymmetric nature of correlations can be seen in Table 1. Calculating the return correlations among five size-sorted portfolios we show that correlations are higher in bear markets than bull markets, and higher during recession than during booms. These findings are consistent with those of the vast literature on correlation asymmetry1 . One of our contributions is to distinguish the two types of asymmetric correlations. While the heightened correlation over the market trend is associated with diminished diversification benefits, it is not the case for asymmetric correlations over the business cycles. To be specific, both correlations and diversification benefits are higher during recessions than during booms. Our findings also support the arguments of Statman and Scheid (2005, 2008) who argue that the correlation coefficient is a misleading measure of diversification benefits and show that benefits of diversification can remain high although the correlations increased. Unlike Statman and Scheid (2005, 2008), we compare and contrast correlations and diversification benefits not only over the market trend but also over the business cycles. The claim that correlation is not a reliable measure of diversification benefits can be best supported by an example with simultaneously increasing correlations and diversification benefits. We provide that example: as the economy moves from booms to recessions, diversification opportunities increase while correlations increase as well. 1
Ang and Chen (2002), Erb, Harvey and Viskanta (1994), Hong, Tu and Zhou (2007), Longin and Solnik (2001).
Correlation and diversification benefits How much diversification opportunity is present when we want to invest in a set of assets? The amount of diversification benefits are commonly associated with return correlations among the assets to be invested. Yet return correlation is a poor measure of diversification benefits. Diversification benefits are defined as the diversifiable risk, the possible reduction in portfolio risk due to diversification. Campbell, Lettau, Malkiel and Xu (2001) for instance, found that the benefits of diversification among U.S. stocks increased during 1962-1997 because diversifiable risk increased during the period. We measure the diversification benefits as reduction in risk (volatility) due to forming a portfolio. For instance, in the case of two assets, if the average volatility of asset is denoted by σ and the volatility of portfolio is denoted by σp , the diversification benefits is equal to the reduction in volatility: ExcessV ol = σ − σp . Given that we have seen the increase in correlations from bull to bear markets, and from booms to recessions we next look at the change in diversification benefits. To understand the diversification opportunities in different periods, using daily return data we calculate monthly volatility levels for individual assets and a portfolio. Then we report the average values of these volatilities over different subsamples. Individual assets are five Fama-French size-sorted portfolios. In Table 2 we compare the diversification benefits during bull and bear markets, and during boom and recessions. Given that correlations are higher in bear markets than bull markets, common intuition about correlations would imply lower diversification benefits in bear markets than in bull markets. What we see in the left panel of Table 2 is consistent with the common intuition: diversification benefits are smaller in bear markets, in other words there is less opportunities for diversification. However, when we look at the diversification benefits over the business cycle periods we see a completely different result: The diversification opportunities do increase in recessions rather than decrease, contrary to the common intuition about the relationship between correlations and diversification benefits. The right panel of Table 2 displays the average risk reduction during booms and recessions. Although correlations are higher in recessions than booms, diversification benefits are higher as well. This is clearly at odds with the common intuition about the correlations and diversification benefits. Recessions are the times of very high correlations of portfolio returns, yet the diversification benefits are even larger than during booms let alone any deterioration. Before discussing the reasons behind these results we show the statistical significance of differences in diversification benefits with the help of panel regressions. To see the effect of bull and bear markets on diversification benefits we run the following 3
panel regression: ExcessV oli,t = δ0 + δ1 Beart + εi,t
where ExcessV oli,t is our measure of diversification benefits in month t. It is calculated for each pair of assets (size-sorted portfolios) for every month t in our sample. Beart is a dummy which takes the value of one for months during bear markets, when the market return is below its average, and zero for bull markets, when the market return is above its average. Therefore, the impact of bull market is captured by δ0 and the impact of bear markets is δ0 + δ1 . In other words δ1 captures the additional effect of bear markets and a statistically significant negative δ1 implies that diversification benefits are lower in bear markets. In a similar fashion we run the following regression for the impact of recession on risk reduction; ExcessV oli,t = θ0 + θ1 Recessiont + εi,t
where Recessiont is a dummy variable that takes the value of one in months during NBER determined recession periods and zero otherwise. Testing the significance of θ1 suffices for seeing whether diversification benefits are different in recessions compared to booms. The results of two sets of regressions are collected in Table 3. The column labeled I contains the results of regression for bull and bear market while the column labeled II is for booms and recessions. What we see in Table 3 is that the coefficients of bear markets and recessions have opposite signs. Consistent with the findings above, diversification benefits increase during recessions while they decrease during bear markets, and these effects are statistically significant. The last column of the table shows the results of a regression in which both dummy variables are included. We can conclude that bear markets and recessions have separate and opposite effects on diversification benefits. We next show that the interesting case of recessions, heightened correlations are accompanied by greater diversification benefits, is a quite general characteristic of U.S. economy rather than being specific to any certain period. In order to compare each recession and boom periods, we present the average excess volatility for each boom and recession period since 1963 in Table 4. Each row of the table corresponds to a different recession or boom period. It is clearly the case that average risk reduction (excess volatility) is, with one exception, higher during each recession period compared to the preceding and succeeding boom periods. Different recession periods do not necessarily share the same features and it could
be the case that not every recessionary period come with greater diversification benefits. Yet, the pattern visible in Table 4 shows that our earlier results are not driven by some recession periods but this pattern is a general phenomenon, at least for the U.S. stock markets we study. These impacts in opposite directions of bear markets and recessions are intriguing since both periods can be considered as “bad times” for investments in financial markets and they are both associated with heightened correlations. We next try to understand the causes behind this difference.
Different reasons, different consequences In this section we show that the same level of correlation can be associated with different levels of diversification benefits. Correlations can increase due to different reasons, and resulting diversification benefits can differ accordingly as well. With this in mind, we assume a CAPM structure for returns, i.e. ri = α + βi rm + σi2 , where rm is the aggregate market return, and βi is the market beta of asset i, and σi is the idiosyncratic volatility. Similarly the return of the other asset can be decomposed as, rj = α + βj rm + σj2 . Given this setup, correlation between these two returns can increase due to following reasons: increase in volatility of aggregate market (σm ), increase in market betas (βi and βj ), decrease in idiosyncratic volatilities (σi and σj ), and any combination of these three causes. We first show that the same amount change in correlation can imply different amount of diversification benefits to the investors. Given the CAPM structure, we generate different scenarios with two regimes, low and high correlation regimes. However, in each scenario the reason behind high correlation is different. We report these scenarios in Panel A of Table 5, together with risk reduction as a result of diversification. In the first two columns of Table 5, we see the low and high correlations, ρL and ρH respectively. In each row of the table correlations are the same, which is created this way on purpose. Each row corresponds to a different scenario for the same amount of increase in correlations, from ρL to ρH . Third column of the table specifies the cause of increase in correlation. While the fourth column reports the diversification benefits (excess volatility) during low correlation regime, the fifth column depicts the the diversification benefits during high correlation regime. We have one main message from panel A of the table. First, although the correlations are exactly the same under each scenario, corresponding risk reduction available to investors are different. To put another way, the same correlations are associated with different di-
versification benefits. Depending on the cause of increase in correlations, implications can be vastly different. The variation of diversification benefits across scenarios is quite significant, although correlations are exactly the same in every scenario. The highest available risk reduction is almost twice the smallest one (in the scenario with low idiosyncratic volatilities). We next try to engineer scenarios similar to empirically observed bull-to-bear market and boom-to-recession transitions. Similar to the empirical observation, we want correlations to increase from one regime to another. However, while heightened correlations yield lowered diversification benefits in one scenario, they are associated with heightened diversification benefits in the other one. These scenarios are collected in panel B of Table 5. The first scenario is created to mimic empirically observed counter-cyclical correlations and associated increase in diversification benefits. In the second row of panel B, the scenario for bull and bear markets is presented. The behavior of correlations in two scenarios are exactly the same: return correlation increase from 0.70 to 0.78, yet due to different reasons. In the first case, from booms to recessions, correlation increase is mostly due to increase in aggregate market volatility. In the second case, from bull to bear markets, market volatility plays a rather limited role in increasing the correlation. Importantly, for the first case, we could engineer a scenario in which higher correlations can be accompanied with larger diversification benefits. This is important in understanding when and why correlations would have implications inconsistent with the common interpretation of the correlation coefficient in relation to diversification opportunities.
Diversification opportunities in different crises We next try to understand the extent of potential diversification benefits in different crisis periods. Reinhart and Rogoff (2009) provide a taxonomy of crises as currency crisis, stock market crisis and banking crisis. For that purpose, we regress excess volatilities on dummy variables for different types of crisis. In Table 7 we summarize the results of regressions. We see that the diversification benefits increase during stock market crises and during banking crises, although statistically insignificant in the latter case. These results are similar to our earlier results regarding recessions in the sense that diversification opportunities improve during difficult times. The currency crisis seem to be tough times to diversify as we can see that the risk reduction available through diversification decreases during currency crisis. These results suggest that different type of crises have quite different implication for risk diversification.
Conclusion We show that during recessions both the correlations and diversification benefits are higher compared to booms. This finding supports the arguments that correlation coefficient is a misleading measure of diversification benefits. Although correlations are high,the heightened volatility during recessions causes greater diversification benefits by inducing a greater dispersion of asset returns. The same amount of variation in correlations, we show, can lead to different degrees of diversification benefits.
Figures and Tables
Table 1: Asymmetric and Counter-cyclical correlations
1 1 1 1 2 2 2 3 3 4
& & & & & & & & & &
2 3 4 5 3 4 5 4 5 5
Asymmetric Bull Bear 0.929 0.956 0.895 0.931 0.846 0.897 0.748 0.794 0.945 0.967 0.904 0.940 0.806 0.849 0.944 0.965 0.851 0.884 0.901 0.923
Counter-cyclical Boom Recession 0.938 0.964 0.907 0.939 0.863 0.910 0.761 0.821 0.952 0.975 0.916 0.950 0.818 0.873 0.951 0.973 0.860 0.905 0.907 0.939
The table reports the average pairwise return correlations among the five size-sorted portfolios, for different subsamples (Bull, Bear, Boom and Recession). Daily data spans the period from January, 1963 to December, 2015. For each month realized return correlations between size-portfolios are calculated (636 observations) and the average realized correlations over different subsamples are reported. Columns 2 and 3 present the correlation coefficients for bull and bear markets, while columns four and five present correlations during boom and recessions, where boom and recession periods are defined according to NBER determined dates. Return variables are in the excess of risk free rate which is approximated by the one-month US Treasury bill rate.
Table 2: Diversification Benefits and Correlation Asymmetry
1 1 1 1 2 2 2 3 3 4
& & & & & & & & & &
2 3 4 5 3 4 5 4 5 5
Asymmetric Bull Bear 0.049 0.042 0.078 0.068 0.116 0.102 0.214 0.206 0.034 0.028 0.071 0.058 0.165 0.152 0.035 0.027 0.115 0.107 0.072 0.067
Counter-cyclical Boom Recession 0.045 0.051 0.071 0.084 0.106 0.130 0.204 0.248 0.032 0.030 0.064 0.069 0.156 0.174 0.031 0.030 0.110 0.118 0.069 0.071
The table reports the average decrease in volatility (multiplied by 100) due to forming an equally weighted portfolio, during different subsamples (Bull, Bear, Boom and Recession). Daily data spans the period from January, 1963 to December, 2015. For each pair of assets (pair of two size-sorted portfolios), an equally weighted portfolio is formed at the beginning of each month in the sample and then the volatility of portfolio is subtracted from the average volatility of assets in the portfolio (636 observations). The decrease in volatility is our diversification benefits measure and its average value is calculated for different subsamples and reposted in the table.
Table 3: Diversification Benefits: Panel Regressions Dep var: ExcessVoli,t Bear
# of obs
* p < 0.1; ** p < 0.05 The table reports estimates from panel regressions, including coefficients estimates and t-statistics (in parentheses). The dependent variable is monthly excess volatility, calculated for each pair. Bear and Recession are dummy variables taking value of one for bear markets and recessions, respectively. There are ten pairs of returns and 636 months in the sample.
Table 4: Diversification benefits in recessions
1963.01 1970.01 1970.12 1973.12 1975.04 1980.02 1980.08 1981.08 1982.12 1990.08 1991.04 2001.04 2001.12 2008.01 2009.07
1969.12 1970.11 1973.11 1975.03 1980.01 1980.07 1981.07 1982.11 1990.07 1991.03 2001.03 2001.11 2007.12 2009.06 2015.12
Average Excess Volatility
0.058 0.062 0.051 0.095 0.059 0.106 0.068 0.094 0.105 0.115 0.133 0.164 0.085 0.098 0.082
Boom Recession Boom Recession Boom Recession Boom Recession Boom Recession Boom Recession Boom Recession Boom
84 11 36 16 58 6 12 16 92 8 120 8 73 18 78
The table reports the average excess volatility (multiplied by 100) calculated for each boom and recession period in the sample. The first column shows the beginning and end dates of each business cycle period, while the last column shows the duration of the period.
Table 5: Heightened correlations: Different causes and different consequences ρL
Cause of high correlation, ρH
Excess Volatility when ρL
Excess Volatility when ρH
Panel A: The Role of Different Causes 0.33 0.33 0.33 0.33 0.33 0.33
0.60 0.60 0.60 0.60 0.60 0.60
High σi & high β High σi & σm High β High σm High β & High σm Low σi
0.0477 0.0477 0.0477 0.0477 0.0477 0.0477
0.0372 0.0372 0.0354 0.0354 0.0354 0.0204
Panel B: Mimicking Bull/Bear and Boom/Recession 0.70 0.70
Boom to Recession Bull to Bear
The table reports the monetary compensations to naive investor as explained in Exercise 1 and Exercise 2. Exercise 1 sets the expected continuously compounded return on risky assets as µ = 0.07, and the volatility of the continuously compounded return increases from 15 percent in low volatility state to 18.3 percent in high volatility state. This increase in volatility causes the correlation of asset returns in low volatility state, ρL = 0.50, to increase to ρH = 0.60 in high volatility state. The constant risk-free rate is set as rf = 0.05, and the constant relative risk aversion as γ = 4. Exercise 2 uses the same parameters, but lets the betas and idiosyncratic volatilities of risky assets to differ between High and Low correlation regimes. return gap is forecasted return gap for high corr regime and required compensation is the average.
Table 6: Volatility increases especially in recessions
vol(rm ) vol(r1 ) vol(r2 ) vol(r3 ) vol(r4 ) vol(r5 )
Bull/Bear Bull Bear 2.63 3.24 2.45 3.11 2.78 3.45 2.67 3.33 2.63 3.28 2.73 3.31
Change in volatility
Boom/Recession Boom Recession 2.68 4.54 2.57 4.06 2.88 4.59 2.77 4.44 2.70 4.56 2.76 4.66 64 %
The table reports the average market volatility (multiplied by 100) during bull and bear markets, and during booms and recessions. Realized volatility for excess market return is calculated for each month and then averaged over bull and bear markets, and boom and recession periods.
Table 7: Excess Volatility during Different Types of Crises Dep var: ExcessVoli,t Currency Crisis
Stock Market Crisis
# of obs
* p < 0.1; ** p < 0.05 The table reports estimates from panel regressions, including coefficients estimates and t-statistics (in parentheses). The dependent variable is monthly excess volatility, calculated for each pair. Regressors are dummy variables that capture different types of crises according to the taxonomy of Reinhart and Rogoff (2009). Years of currency crisis are 1969, 1971, 1975, 2002 and 2003. Stock market crises are during years 1974, 2002 and 2008, while the banking crises are from 1984 to 1992, and from 2007 to 2010.
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