# Concentration of Trading in S&P 500 Stocks. Recent studies and news reports have noted an increase in the average correlation

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2 subsets of the market such as non-s&p 500 stocks, randomly chosen portfolios of 500 stocks and the top 500 stocks by market value. Our results show that the concentration index for the S&P 500 stocks has decreased since the mid-1970 s and that the distribution of trading volumes within the S&P 500 has become more evenly spread over time. This finding is consistent with the evidence that ETF and index traders have gained dominance over active investors, and that stocks within the S&P 500 that were previously ignored or neglected are now regularly traded simply because they are part of the index. For stocks that are not in an index, such as the non-s&p 500 stocks and randomly chosen 500 stocks portfolios, we find concentration indexes that have steadily increased over the last five decades, indicating that trading in these stocks has become more concentrated. Measuring Trading Concentration The power law has been used to model a wide range of economic phenomena such as firm size (e.g., Axtell, 2001), stock returns (Gabaix et al., 2003) and net income of firms (Okuyama et al., 1999) and stock trading volumes (Balakrishnan, Miller & Shankar 2008). A generalization of the power law, applied to any item that can be ranked by size (such as firm size, stock returns, net income, or trading volumes) can be stated as: Sizei) x (Size Ranki) q = constant (or) Log(SizeRank) = (1/q) * Log(constant) (1/q) * Log(Size) (1) The exponent q is the power law exponent, also known as Zipf s parameter (Zipf, 1949). Following Balakrishnan, Miller & Shankar (2008) and Naldi (2003) we use this

3 power law exponent as an index of market concentration, with larger exponent values indicating greater levels of concentration 1. As an illustration of the concentration index concept, consider the distribution of trading volumes, on two different days in a simulated market with six stocks, shown in Table 1. Table 1 Rank Trading Volume Day 1 Day 2 % of total Trading volume Volume % of total volume Stock A 1 63, % 14, % Stock B 2 97, % 26, % Stock C 3 159, % 45, % Stock X 4 255, % 114, % Stock Y 5 342, % 406, % Stock Z 6 447, % 757, % Total Volume 1,365,100 1,365,100 The total volumes of shares traded on the two days in Table 1 are identical, but there are distinct differences in the distribution. The distribution on Day 1 appears more evenly spread (or less concentrated) than on Day 2. For example, the two highest volumes on Day 1 account for 68% compared to the 85% for Day 2; at the other end, the two lowest volumes on Day 1 account for 12% compared to the 3% for Day 2. To confirm this intuitive understanding that the distribution on Day 1 is less concentrated, we compute the concentration indexes for the two days. Using the power law equation described above, we find that the power law exponent or the concentration index for Day 1 is 1.2, and that the 1 Balakrishnan, Miller & Shankar (2008) report that the power law exponent reflects changes in concentration better than the more commonly used Hirschman Herfindahl Index (HHI) measure of concentration. They find that the daily HHI measures are extremely elastic and that the huge fluctuations in the daily HHI measures obscure any trends in the concentration of daily trading.

4 index for Day 2 is 2.5, or almost double the concentration index for Day 1. These concentration index value values confirm the visual and intuitive evidence that trading is more concentrated (or less evenly spread) on Day 2 when compared to Day 1. The methodology used above can be applied to any distribution that can be ranked by size and the power law exponent can serve as an index of concentration in that distribution. As our example shows, the concentration index is lower when distributions are more even,. In the extreme case where all values in a distribution are equal, for e.g., when all stocks have the same trading volumes, the concentration index will be zero indicating a perfectly even distribution. Concentration in S&P 500 stocks The S&P 500 index, introduced in 1957, is well known as an indicator of stock market performance. Starting in the late 1970 s it also became the benchmark for index funds, a concept championed by Paul Samuelson and others (Bogle, 2011). Though the index funds concept initially encountered considerable skepticism from active investment managers, index funds have gained in popularity over the last three decades. In September 2012, the total value of assets that were in S&P 500 index funds amount to \$1.31 trillion in assets (S&P, 2012). Viewed another way, S&P 500 index funds alone hold about 11% of the total market capitalization of the 500 stocks in the index and account for a much larger fraction of the daily trading volume in these stocks. Prior to the introduction of index funds, investors actively chose individual stocks in the S&P 500 based on the stocks individual merits, or criteria other than membership in the index. In this environment, some of the index stocks would have received more attention while others may have received no attention at all; that is, the distribution of trading in the S&P 500 stocks would have been uneven. However, the advent of index

6 Figure 1 Concentration indexes, , based on dollar volume 3.0 Non S&P 500 stocks 2.0 S&P 500 stocks To confirm the divergence between the two series of concentration indexes, we plot their difference in Figure 2. As that chart shows, the difference between the concentration indexes for the S&P 500 and the non-s&p 500 stocks remained small until 1975 but started to increase thereafter. We confirm this by estimating the slope of the differences line in two periods, 1960 to 1975 and 1976 to For the period, the slope is statistically insignificant at 0.08 x 10-5 (t-value=0.4; R 2 =0). However, for the 1976 to 2011 period, the slope is a statistically significant 21.4 x 10-5 (t-value=310; R 2 = 0.91).

7 3.0 Figure 2: Difference between daily concentration indexes for non S&P 500 stocks and S&P 500 stocks Concentration indexes for other 500 stock portfolios The number of stocks traded in the non-s&p 500 segment (in the CRSP database) increased from 557 in January 1960 to 6554 in December To rule out the possibility that the increase in the concentration index for non-s&p 500 stocks arises mainly from the increase in the number of stocks that were traded, we compare the concentration indexes for the S&P 500 stocks with the indexes for other portfolios of 500 stocks. We first construct eight portfolios of 500 stocks drawn at random each day from the overall market and compute the daily concentration index for each of these eight portfolios. We average the indexes for these eight portfolios to arrive at the concentration index for portfolios of randomly chosen 500 stocks. In addition to this, we also choose the top 500 stocks by market value each day and compute the daily concentration index for these top 500 stocks.

8 In Figure 3, we present the concentration indexes of the randomly chosen 500 stock portfolios and the top 500 stocks portfolio alongside those for the S&P 500 stocks. Figure 3 Concentration indexes of S&P 500 versus other 500 stock portfolios 3.0 Randomly chosen 500 stk portfolios 2.0 S&P 500 stocks 1.0 Top 500 stocks We observe that the concentration indexes for the randomly chosen 500 stock portfolios in Figure 3 follow the same upward trend that we observed for non-s&p 500 stocks, indicating that the distribution of trading has been more uneven over time. On the other hand, the indexes for the top 500 stocks by market value are more or less flat, indicating that the distribution of trading among the top 500 stocks has been even over time. In summary, concentration indexes for S&P 500 appear different from the indexes for non-s&p 500 stocks as well as portfolios of 500 stocks that do not have an index investor

9 following such as the random 500 or the top 500. We confirm the statistical significance of these visual observations by regressing the indexes against calendar time for all the portfolios under discussion. For the regressions, we consider two time periods: 1960 to 1975 or the period prior to the introduction of index funds and 1976 to 2011, the period in which the index funds have been in operation. The summary results of these regressions are in Table 2. Table 2 Slope estimates for regression of concentration indexes vs. calendar time Market Sector Slope estimate 1960 to to 2010 t- value r- squared Slope estimate t- valu e r- squared Non-S&P 500 stocks 10.7 x x S&P 500 stocks 10.6 x x Random 500 stocks 14.0 x x Top 500 stocks 1.84x x Consistent with the trend lines in Figure 3, the slope estimates for the non-s&p 500 stocks time series are significantly positive and similar in value in both the and the time periods, as shown in Table 2. However, we see a structural change between the two periods for the S&P 500 stocks. In the period prior to the introduction of index funds, , the slope estimate for S&P stocks, x 10-5, is significantly positive and nearly identical to the Non-S&P 500 stocks estimate. However, in the period, the S&P 500 stocks slope estimate, x 10-5, is significantly

10 negative and substantially different from the positive estimate for the Non-S&P 500 stocks. The estimates for the randomly chosen 500 stocks portfolios are very similar to the estimates for the non-s&p 500 stocks and confirm the upward trend seen in Figure 3. In the case of the Top 500 stocks, there is no discernible trend over time, as evident from the relatively low estimates and goodness-of-fit measures. Comparison of Market Cap and Volume concentrations The regression results above suggest that the entry of index fund investors after 1975 set in motion a gradual change towards more even, uniform trading in S&P 500 index stocks. Over time, their trading appears to have gained dominance over the trading of active investors in individual S&P 500 stocks. To explore this notion further, we compute the concentration indexes based on the market capitalizations of the stocks in the S&P 500 and compare this with the concentration indexes for the dollar-weighted volumes of the S&P 500 stocks. If all the trading in S&P 500 stocks is done only by index investors, we would expect the two concentration indexes, the market cap based index and the volume based index, to be identical since index investors trade these stocks in proportion to the market cap. Alternatively, if active investors play a large role in daily trading, we would expect the indexes for the volume based index to be higher than the market cap based indexes, since active investors would not be trading in all the S&P 500 stocks. We show both these series in Figure 4 from 1975, the beginning of the index fund era, to In the early years of this period, the volume based concentration index is higher than the market cap based concentration index, suggesting that the volumes were more concentrated in some S&P stocks than the market weights would support. However, over time, the distance between the two series has decreased and the series more or less

11 converge after This convergence suggests that the volume of daily trading in S&P 500 stocks is almost proportional to the market cap of the stocks in the index and that index traders dominate the market for S&P 500 stocks. Figure 4 S&P 500 Market Cap indexes versus S&P 500 dollar volume indexes 2.0 S&P 500 Dollar Volume 1.5 S&P 500 Market Cap Conclusions This paper examines the distribution of daily trading volumes in US markets over the last five decades and introduces the concentration index, a measure of trading concentration. We find that concentration in stocks that are not in the S&P 500 index has steadily increased from 1960 to However, most importantly, we find that the trend in the trading concentration for S&P 500 stocks exhibits distinctly different behavior. After

12 increasing along with the non-s&p 500 stocks from 1960 to 1975, the trend for S&P 500 stocks starts diverging in 1976 and then steadily declines, indicating that trading among S&P 500 stocks has become more uniform over time. This divergence in the trend for S&P 500 stocks coincides with the introduction of S&P 500 index funds in 1976 and is consistent with the hypothesis that S&P 500 index fund investors, who buy the entire index, have gradually become the dominant traders in S&P 500 stocks over the last 35 years. In the process, they appear to have displaced the active investors who buy individual stocks in the S&P 500 index based on criteria other than membership in the S&P 500 index.

13 References Axtell, R.I., Zipf distribution of U.S. firm sizes. Science 293, P.V. Balakrishnan, J. M. Miller, S.G. Shankar, Power law and evolutionary trends in stocks markets. Economics Letters 98, Bogle, J., How The Index Fund Was Born, Wall Street Journal, September 3, 2011, p.15 Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H.E., A theory of power- law distributions in financial market fluctuations. Nature 423, Gabaix, X., Power Laws in Economics and Finance. Annual Review of Economics 1, Naldi, M., Concentration indices and Zipf s law. Economics Letters 78, Naldi, M., Salaris, C., Rank-size distribution of teletraffic and customers over a wide area network. European Transactions on Telecommunications 17, Okuyama, K., Takayasu, M., Takayasu, H., Zipf's law in income distribution of companies. Physica A 269, S&P, Retrieved September 10, 2012, from 500/en/us/?indexId=spusa-500-usduf--p-us-l-- Zipf, G.K., Human behavior and the principle of least effort, Addison-Wesley, Cambridge. 13

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing

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