A Liquidity-based Stock Network

Transcription

1 A Liquidity-based Stock Network Zhenyu Gao Wenxi Jiang y Da Tian z November 2016 Abstract Stocks are connected through common ownership of nancial institutions. Firm shocks can be transmitted and ampli ed through these interconnections, aggregating into market level uctuations. We formalize this intuition by estimating a parsimonious model using mutual fund holding data. The model allows us to quantify the aggregate outcome of propagation of rm level shocks through the network. Using earnings surprises as a proxy for rm shocks, we nd that our model s aggregation is signi cantly correlated with the aggregate market return. Also, the network structure estimated by our model can predict subsequent volatility of aggregate market returns and forecast the volatility of and correlation between individual stocks future returns. The Chinese University of Hong Kong. gaozhenyu@baf.cuhk.edu.hk. y The Chinese University of Hong Kong. wenxijiang@baf.cuhk.edu.hk. z University of Florida. davidtianpku@gmail.com. 1

2 1 Introduction The recent theoretic literature has recognized the important role of network structure in generating aggregate uctuations. Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012), for example, point out that idiosyncratic shocks at rm level can propagate over various interconnections among rms, with potentially signi cant implications for aggregate volatility. Also, several market-wide breakdowns, such the stock market crash in 1998 and the recent nancial crisis, have highlighted that the common ownership of distressed assets by nancial institutions can serve as a channel through which contagion is ampli ed. Based on these observations, regulators concern whether nancial network is a source of systemic risk (e.g., Yellen 2013). In this paper, we construct a stock network built on common holdings of - nancial institutions and develop formal tests to analyze the propagating e ect of this liquidity-based network. We nd empirical evidence that the network structure is an important factor that explains the uctuation of the aggregate stock market. Our ndings shed light on the discussion of nancial network and market instability. We start by developing a simple linear model that can be readily estimated using data on fund holdings. Our model takes the stock holdings of each mutual fund as given and aims to analyze how an idiosyncratic price shock to a stock transmits in the economy. Our analysis relies on two assumptions. The rst one is pro-cyclical trading of mutual funds. That is, institutional investors tend to buy (sell) assets in their portfolio following positive (negative) performance. In reality, this tendency can be driven by performance-based asset management and risk 2

3 management practice, such as Value-at-Risk rules (see, e.g., Shleifer and Vishny 1997, Lou 2012, and Adrian and Shin 2014). Thus, stock level shocks can generate uctuations in a fund s performance and further lead to purchases or sales of stocks the fund holds, not only stocks that experience shocks. The second assumption is liquidity of stocks traded in secondary market. That is, trading will generate price impact and the magnitude of impact is proportional to dollar size of a trade. We use a simple example to illustrate how we quantify the transmission process in network. Suppose that Fund A holds Stocks X and Z in its portfolio, and Stock X experiences a negative " shock while no news for Stock Z. The model rst tries to quantify the direct link e ect, i.e., how much the " shock will transmit to the price of stock Z via the common ownership of Fund A. The higher weight of Stock X in portfolio, the larger impact of the shock on Fund A s return. Changes in return simulate trading of Stock Z, and the price change in Stock Z is proportional to the amount of shares traded relative to the number of shares outstanding and the price impact parameter. Thus the magnitude equals the product of (1) Stock X s weight in Fund A portfolio, (2) Fund A s sensitivity parameter of trading to performance, (3) Fund A s ownership of Stock Z, and (4) Stock Z s price impact parameter. (1) and (3) can be easily calculated using public data, while (2) and (4) need to be estimated endogenously (as discussed in Section 2.3). If more than one fund holds both Stocks X and Z, the summation of the products is the total direct link e ect. We denote a an S S matrix, L, to represent the direct link e ect for every pair of stocks in the market (where S represents the number of stocks). The direct link e ect originated from Stock X can be further transmitted from Stock Z to stocks immediately linked with Z and eventually to all stocks (including 3

4 Stock X itself), which we labeled as the indirect e ect. The outcome of price uctuations corresponds to the total network e ect, which can be captured by L s Leontief Inverse, denoted as N. This matrix accounts for all possible direct link and indirect e ects of interconnections between any pair of rms. The N matrix provides a simple formula of how idiosyncratic shocks aggregate across the whole economy. To estimate the N matrix, we use the data on mutual funds quarterly holdings of common stocks in the U.S. Mutual funds, as active players in the stock market, are subject to performance-driven ows, which is consistent with our assumptions. Also, the data provides monthly information on mutual funds returns and assets under management, which enables us to estimate the sensitivity of trading to performance. At the beginning of each quarter, we estimate the N matrix and then apply it to market return variables. Speci cally, our framework can link the aggregation function of N to market level uctuations in both the rst and the second moments. First, let us denote an S1 vector,, as rm level shocks at a point of time, and an S1 vector, H, as the weight of stocks market capitalizations (i.e., value weighted) or as a vector of equal weights (i.e., equal weighted). Then, N measures the total e ect of each rm s shock after the propagation through network. Further, H 0 N aggregates the e ect into the market level and thus ought to be correlated with the contemporaneous market movement. Second, we can link N to the second moment of stocks returns. That is, H 0 N 0 N 0 H corresponds to the variance of aggregate market returns. We develop empirical tests to verify these two predictions. Our empirical results are summarized below. First, using unexpected earnings as a proxy for rm level shock, the aggregation 4

5 function of N exhibits signi cant correlation with market returns. In each month, we construct the vector,, whose s th element represents the s th rm s earnings surprise announced in the month. Then, we calculate the value of H 0 N for each month and nd that the market return exhibits signi cantly positive contemporaneous correlation with H 0 N (with t-statistics above 2.7) for both value-weighted and equal-weighted returns. This correlation is robust to controlling for contemporaneous and lagged Fama-French. In addition, the simple average of stocks earnings surprise, (i.e., H 0 ), without mapping through the network structure N, does not exhibit any signi cant correlation with market returns. Second, assuming stock-level shocks are independently and identically distributed over time, the aggregate e ect of network suggested by the model, H 0 NN 0 H, should predict future market volatility. The time-series variation in the network structure changes how rm level shocks are aggregated into the market level. When the value of H 0 NN 0 H is high at a quarter-end, it indicates more instability in the system and thereby the volatility of market returns in the subsequent quarter ought to be higher. We nd consistent evidence to this conjecture: H 0 NN 0 H signi cantly predicts the volatility of market index returns in the subsequent quarter, controlling for current market volatility and Fama-French factors. The univariate regression generates a R-squared of 18%. Third, to further boost the persuasiveness of our analyses, we test an auxiliary prediction of our model regarding the cross-section of individual stocks. In particular, the cross-sectional analysis has the advantage of high statistical power. As NN 0 corresponds to the variance-covariance matrix of individual stocks, assuming rm level shocks are i.i.d over time, o -diagonal (diagonal) elements of NN 0 ought to be able predict stocks price correlation (volatility) in the future. 5

6 We nd this is indeed the case in data. O -diagonal elements in NN 0 are positively correlated with two stocks correlation of residual returns (adjusted from Fama-French four factors), controlling for past return correlation and stocks size and industry. The predictability is both statistically signi cant and economically meaningful. Also, diagonal elements in NN 0 exhibit signi cant predictive power for a stock s future return volatility. The result is robust controlling for the stock s current volatility. As a robustness check, we directly control for the e ect of the direct link between a pair of stocks (i.e., LL 0 ) and nd that our network measure remains signi cant. Our paper is closely related to the theoretic literature on network e ect and market wide stability. Allen and Gale (2000), for example, argue that denser interconnections can mitigate systemic risk as idiosyncratic shocks average out in network. On the other hand, Gabaix (2011), Acemoglu et al. (2012, 2015), and Elliott, Golub and Jackson (2014), among others, argue that the average-out e ect unnecessarily dominates, and, depending on the structure of network, links in network can serve as the propagation mechanism of micro shocks throughout the economy. Following this vein of literature, we focus on a liquidity-based network and provide some empirical evidence to the debate. Our paper also contributes to the growing empirical literature on network. The networks studied in the previous papers are based on di erent linkages. Ahern (2013) and Ozdagli and Weber (2016) look at production-based networks in which sectors in an economy are connected by output-input relation. Ozsoylev et al. (2014) examine an investor network based on information ows. By comparison, we consider connections among stocks based on common holdings of mutual funds, complementing the literature by providing another framework to describe 6

7 the network of nancial assets. Our paper is inspired by some previous work on the common ownership and asset prices. Greenwood and Thesmar (2011) and Anton and Polk (2014), for example, show that the common ownership and trading by mutual funds generate more comovement of stocks. In particular, Greenwood, Landier, and Thesmar (2015) examine the network e ect of common holdings. They focus on a network of 49 European banks connected with debt holdings and identify which bank is the most vulnerable in network during sovereign bond crisis. Our paper constructs a network of all common stocks in the U.S. and pushes the agenda further by linking network structure to uctuations in the aggregate market. The paper proceeds as follows. Section 2 develops the model. Sections 3 and 4 introduce the data and presents empirical results, respectively. Section 5 concludes. 2 Model 2.1 Network Setup Consider a directed weighted network with S stocks and M institutions. If institution m holds stock s, two directed edges are built up between them. The edge l m;s represents the impact of stock s s price change on institution m s fund ows. This performance-driven-fund- ows channel has been studied in a large literature (e.g. Shleifer and Vishny (1997) and Lou (2012)). The strength of this link is proportional to institution m s portfolio share of stock s denoted as w ms and sensitivity of fund ow to performance, i.e., l m;s = w ms, meaning that a shock to a stock is more likely to a ect the fund performance and in turn its ows if this 7

8 stock weights more in its portfolio. The other directed edge l s;m from institution m to stock s measures the price pressure from the ow-induced trading (for example, Edelen and Warner (2001), Goetzmann and Massa (2003), and Coval and Sta ord (2007)). l s;m = o sm, where o sm is institution m s ownership of stock s, and is the price impact of liquidity shock to stocks. This means that the higher the ownership, the larger impact of the institution on the stock price. For all S stocks and M institutions, we have matrix L M;S representing stock to institution linkage: L M;S = W 0 ; (1) and matrix L S;M representing institution to stock linkage: L S;M = O; (2) where W SM is the portfolio share matrix (and W 0 is the transpose of W ) and O SM the ownership 2 matrix as follows: w 11 w 12 ::: w 1M o 11 o 12 ::: o 1M w 21 w 22 ::: w 2M o 21 o 22 ::: o 2M W = and O = ; the entry 6 ::: ::: ::: ::: 7 6 ::: ::: ::: ::: w S1 w S2 ::: w SM o S1 o S2 ::: o SM w ij is the value weights of stock i in institution j s portfolio and o ij is the fraction of shares of stock i held by institution j. Our model is parsimonious and we make several important assumptions: rst, we assume that fund ows induce a proportional trading across the institution s holdings; second, with a constant, we assume price impact due to liquidity shock 8

9 is the same across stocks; third, investors treat all institutions equally and therefore the sensitivity of fund ow to fund performance is assumed to be constant across institutions. Note that and can be time-varying and we estimate the product of these two parameters for each quarter. We consider this simple framework due to three reasons: rst, we are mainly interested in the variation of stock ownership and portfolio shares which are obtained from the data; second, the fund ow sensitivity to performance and price impact due to liquidity shocks are not directly observable and may su er measurement errors; third, this simple assumption makes our estimation straightforward and transparent as we will show in the next section. This network is bipartite in the sense that stocks are linked only through institutions and that institutions are linked only through stocks. The model assumes away any stock links not via institutions. Stocks can be also connected through various channels such as supply chains and social networks of rms board members but in our paper we focus on this liquidity-based connection. To simplify our following analysis, we construct the stock linkage matrix L as L = L S;M L M;S = OW 0 : (3) Note that this is still a directed weighted graph with the element l i;j = X k o ik w kj, indicating how much a shock to stock j is transmitted to stock i through institutions that connect them. This adjacency matrix L is thus asymmetric, meaning l i;j and l j;i are not equal and shock transmission from stock i to j could be di erent from that from j to i. The shock transmission can go beyond the direct link if we consider the feed- 9

10 back and chain e ects. A shock to a stock induces the liquidity shocks to other stocks through the direct linkage. These shocks can in turn trigger another round of shocks and so on. This "liquidity spiral" is related to Brunnermeier and Pedersen (2009) but the nature of network structure captures more than the feedback loop in their model as the connectedness between stocks enables contagion and ampli cation, which can further reinforce feedback e ects. Suppose that idiosyncratic shocks hit stocks, by examining multiple rounds, S1 we quantify the accumulated shocks to stocks as + L + L 2 + ::: (4) = (I L) 1 = N; here we de ne N (I L) 1 (5) as the Leontief inverse of the stock linkage matrix L. Therefore, N is the adjacency matrix of stock network. 2.2 Network and Shock Transmission We take stock holdings of nancial institutions as exogenously given when constructing this liquidity based network. We will focus on the shock transmission and ampli cation via network. Given t+1 S1 as the idiosyncratic shocks to stocks at time t + 1, through this 10

11 stock network, stock returns from time t to t + 1 are R t+1 = N t t+1 + " t+1 ; (6) where N t is the network matrix N at the end of t. " t+1 is an error term with expectation of zero and variance 2 ";t+1 at t and includes shocks from other resources independent of idiosyncratic shocks to stocks. Consequently the market return at t + 1 is MR t+1 = H 0 tr t+1 = H 0 tn t t+1 + H 0 t" t+1 (7) where H t is a S 1 vector of the weights of stock market capitalizations (for value weighted returns) or equal weights (for equal weighted returns) at t. The market variance is thus simply written as V ar market;t+1 = H 0 tn t N 0 th t 2 + H 0 th t 2 ";t+1; (8) where we assume stock-level shocks are independently and identically distributed with a variance 2 across stocks and over time. Given the return structure, we can also examine the second moments crosssectionally: the volatility of individual stocks and correlation between them, i.e., the stock comovement. The variance-covariance matrix of individual stock returns: t+1 SS = E t [R t+1 R 0 t+1] (9) = N t N 0 t 2 ;t+1 + H 0 th t ";t+1 : 11

12 The diagonal entries correspond to return variance and o -diagonal entries measure return covariance. 2.3 Estimation In this subsection, we develop the method to estimate the key parameters of our model. To construct the network, we only need to estimate parameter, de ned as the product of the fund ow sensitivity and price impact by matching the correlations of fund ows between the model and real data. Like shocks to stocks, the performance driven fund ows are also accumulated over multiple rounds, i.e., F low t+1 = L M;S;t t+1 + L M;S;t L t t+1 + ::: (10) = W 0 tn t t+1 ; The correlation of fund ows between institutions is cov t (F low t+1 ) = E[(W 0 tn t t+1 + e t+1 )(W 0 tn t t+1 + e t+1 ) T ] = 2 2 W 0 tn t N 0 tw t = 2 2 W 0 t(i S O t W 0 t) 1 (I S W t O 0 t) 1 W t ; where I S is the S S identical matrix. We de ne t () = W 0 t(i S O t W 0 t) 1 (I S W t O 0 t) 1 W t. 12

13 Then the matrix of correlation coe cients of fund ows among institutions in the model is Cor t () = (diag( t ())) 1 2 t ()(diag( t ())) 1 2 ; where diag( t ) is a S S diagonal matrix with the diagonal entries of t. We then estimate by minimizing the sum of the squares of di erences of correlation coe cients between real data and o -diagonal entries of Cor t (). We then compute the monthly fund ows following the literature F low i;m = AUM i;m AUM i;m 1 (1 + R i;m ); for fund i at month m where AUM i;m is the asset under management and R i;m is the fund return. Within each quarter, we calculate the correlation coe cients of fund ows between institutions. We then use ownerships and portfolio shares from institution holding data in the last quarter to construct Cor t (). Finally we estimate in quarter t by matching correlation coe cients between the data and the model with the least square method. Using the estimated, together with the portfolio share w, and the ownership o, we construct the network N at the end of each quarter following Equations (3) and (5). Next we will apply this network structure into mutual fund holdings. Mutual funds play an important role in the stock market and are subject to performancedriven ows. We will use the data of mutual fund ows to estimate key parameters in the model and empirically study the e ect of this network on aggregate outcomes and cross-sectional stock returns. 13

14 3 Data Description 3.1 Data Sources We focus on mutual funds in our estimation and empirical analyses. The mutual fund data include CRSP and Thomson Reuters Mutual Fund Holdings (S12). Our sample period is from 1991 to We obtain monthly fund returns and total net asset value from CRSP, and mutual funds quarterly holdings information from S12. If the mutual funds have multiple share classes, the total net assets (TNA) of di erent classes are summed together to generate the asset under management (AUM) of the fund. We use WFICN to link these datasets together. We apply several criteria for mutual funds to construct our sample. Following Lou (2012), we restrict mutual funds in the sample to be aggressive growth, growth, growth and income, balanced, unclassi ed, or missing. We drop funds with AUM less than 1 million dollars. Using the holding information, we calculate the value of holding for each fund at the end of every quarter, and calculate the ratio of the holding value to the AUM for each fund. Only funds with a moving average of ratios over current and the past three quarters that is between 0.75 and 1.2 are kept in the sample. We also restrict the funds ratio between end-of-quarter assets (reported in S12) and AUM (end-of-quarter assets divided by AUM) to be in the range from 0.5 to 2. When the fund s holding is missing, we assume that its portfolio in the current quarter remains the same with that in the previous quarter. In addition, we construct the measure of unexpected earning (UE) using stock 1 Although we can obtain stock information and fund holding data from an earlier period of time, the monthly data of fund returns and total net asset value from CRSP are only available since

15 earnings data and analysts forecast data obtained from IBES. It is calculated as the di erence between reported earnings and analyst consensus scaled by past price. 3.2 Summary Statistics Table 1 reports key statistics of the mutual fund sample from 1991 to The number in the table is as of the end of each year. The number of funds is relatively small in early years of the sample, increases signi cantly during the 1990s and reaches to a peak before the recent nancial crisis, and then starts decreasing afterwards. The number of stocks counts the stocks that are held by at least one mutual fund. For example, in 1997, the 1292 funds in the sample hold 7071 di erent stocks in total, the maximum over the twenty-four years of our sample period. On average, there are 5500 di erent stocks in all funds portfolios each year. We follow Anton and Polk (2014) to de ne large stocks as those with market capitalization greater than the median of NYSE. The number of large stocks in mutual funds position is relatively stable. The total AUM is the sum of each fund s asset under management at the end of each year. While the size of AUM generally increases every year, it experiences sharp drops during the burst of internet bubble in 2001 and the recent nancial crisis. The average AUM across the twenty-four years is about 2,256 billion dollars. Market Capitalizations of stocks are the total market value of all the stocks or large stocks in the sample, and are calculated based on the price and shares outstanding of the last trading day each year for each security. Two crises during our sample period also impose a signi cant impact on the market value of stocks. Besides, the numbers show that large stocks on average account 15

16 for a dominant portion of 79% of the total market value. Table 2 summarizes the statistics of variables in di erent speci cations. Panel A shows statistics for variables in Table 4. The monthly value weighted and equal weighted market returns come from CRSP Index / Stock File Indexes. They represent stocks traded in NYSE, AMEX and NASDAQ. Network_E ect represents the value of H 0 N measured at the end of each month. It can be value weighted or equal weighted depending on whether H, the weighting scheme vector, is the weights of stocks last month market capitalization (VW) or equal weights. Average_UE is the average of every stock s UE in each month, valued-weighted (VW) or equal-weighted (EW). SMB, HML and MOM represent size, value, and momentum factors, respectively. They come from French s website. Our sample includes 285-month observations. The average value weighted market return is 0.88% monthly or 10.56% annually. The equal weighted market return is slightly higher 1.18% monthly or 14.2% annually. The average of monthly network e ects is 0.01% for both the value weighted and the equal weighted. Panel B of Table 2 includes statistics for variables in Table 5. Market_Volatility is the standard deviation of daily market index returns. The market indexes are the same with those in panel A. Network_E ect represents the square root of H 0 NN 0 H measured at the end of each quarter. The average aggregate market volatility each quarter is about 0.99% for value-weighted market returns and 0.81% for equal weighted market returns. The average of stock network e ect across quarters is 2.96% for the value-weighted e ect and 2.21% for the equal-weighed e ect. SMB, HML, MOM and MKT are the quarterly mean of daily returns for the size, value, momentum and market factors, respectively. Panel C of Table 2 reports statistics for variables in Table 6. We focus on large 16

17 stocks in cross-sectional analyses. Stock_Volatility is the standard deviation of daily returns for a stock in that quarter. Network_E ect represents the square root of diagonal entries of NN 0 at quarter end. Direct_Link is the square root of diagonal entries of LL 0 measured at the end of quarter for each stock. Stock_Cap is the log of a stock s quarter-end market capitalization. The average quarterly stock price volatility is about 0.05%. The average of quarterly stock network e ect is 0.22% and the average of the direct link e ect is 0.13%. Panel D of Table 2 presents statistics for variables in Table 7. Correlation is the correlation coe cient of daily adjusted returns between two stocks each quarter. Network_E ect is the value of the o -diagonal entry of NN 0 for a pair of stocks, scaled by the product of square roots of the two stocks diagonal entries of NN 0. Direct_Link is the value of the o -diagonal entry of LL 0 for a pair of stocks, scaled by the product of square roots of the two stocks diagonal entries of LL 0. SIC2 and SIC3 are dummy variables that equal one if the rst two and three digit of the SIC code of two stocks are the same, zero otherwise, respectively. Stock_Cap_Di is the absolute value of the di erence in market capitalization between two stocks. The average correlation of daily returns between two stocks is The average of quarterly stock network e ect is and the average of the direct link e ect is Table 3 reports the correlation among variables. Panel A is the correlation between monthly market return, stock network e ect and other variables in Table 4. Panel B is the correlation between the volatility of market return, stock network e ect in each quarter and other variables in Table 5. Panel C reports the correlation between quarterly stock volatility, stock network e ect and other variables in Table 6. Panel D summarizes the correlation between the quarterly stock correlation, 17

18 stock network e ect and other variables in Table 7. Generally, the stock network e ects do not have a strong correlation with other variables. 4 Empirical Analyses Using the mutual fund data, we follow the methodology in Subsection 2.3 to estimate the parameter and construct the network matrices N at the end of each quarter. In this section we link the network to aggregate market uctuations as well as the volatility of and comovement between individual stocks. We rst conduct time series analyses at aggregate level and then cross-sectional tests for individual stocks. 4.1 Aggregate Market Returns and Volatility We rst examine the importance of our stock network to aggregate uctuations and systemic risks. We conduct time series tests on aggregate market returns (the rst moment) and volatility (the second moment) Aggregate Market Returns Our stock network links individual stock shocks to aggregate market returns as shown in Equation (7). We use unexpected earnings as a proxy for stocks fundamental shocks. Earning surprises m S1 are de ned as the di erence between reported earnings and analyst consensus scaled by past price. If there is no announcement in month m, it equals zero. Using the holding data in the previous quarter end, 18

19 we construct the stock network N t and conduct the following time series test: SS MR m = H 0 mn t m + 2 Controls m + e m ; (11) where H m is the weights of stock market capitalizations (for value weighted returns) S1 or equal weights (for equal weighted returns) at the end of month m. We control the average of every stock s unexpected earnings H 0 m m and the market return in the previous month MR m 1. We also control for Fama and French (1993) SMB and HML factors, and the MOM factor. We report Newey-West (1987) standard errors with 11-month lag. Table 4 reports coe cient estimates. Columns (1) to (4) present the value weighted market returns and Columns (5) to (8) show the equal weighted market returns. Column (1) shows a positive and statistically signi cant contemporaneous correlation between the network e ect HmN 0 t m, i.e., the shock attributed to earnings surprises through stock network, and market return in the univariate regression. This result is also economically meaningful: an increase of one standard deviation in our network measure is associated with a one percent increase in market returns, compared with average market returns of 0:88% per month. The coef- cient remains almost unchanged if we include SMB, HML, and MOM factors (Column (2)), and further lagged market return and factors (Column (3)). Most importantly, after we add Hm 0 m, the aggregate shock directly from earnings surprises, as a control in Column (4), the network e ect is still positive and signi cant with the magnitude almost the same while the direct e ect is insigni cant. This con rms the importance of stock network to aggregate uctuations. Without the 19

20 ampli cation mechanism provided by network, the aggregate e ect of unexpected earnings is negligible due to diversi cation. When we turn to the equal weighted market returns, coe cient estimates of these four speci cations are slightly smaller but still statistically signi cant as shown in Columns (5) to (8). It is again important to note that the direct impact of earnings surprises on aggregate market returns is insigni cant, which suggests that shocks to individual stocks get ampli ed rather than diversi ed through stock network Aggregate Market Volatility In this subsection, we study the e ect of stock network on aggregate market volatility. We are interested whether stock network can contribute to market level variations. Furthermore, Equation (8) from our model suggests that the second moment of stock network p HtN 0 t NtH 0 t can predict the market volatility at t+1. Speci cally V ol market;t+1 = p H 0 t N t N 0 th t + 2 Controls t + e t+1 ; (12) where we extract the holding information and stock market capitalizations at the end of quarter t and construct both value-weighted and equal-weighted network e ect. We control the current quarter market volatility as volatility is persistent over time. We then add factors including MKT, Fama and French (1993) SMB and HML, and MOM. Also we calculate Newey-West (1987) standard errors with three-quarter lags. Table 5 reports regression results. Market volatility in the rst three columns is constructed by the value weighted returns and that in the rest of columns is 20

21 calculated by the equal weighted returns. Column (1) presents the coe cient by regressing the market volatility in Quarter t + 1 on p H 0 tn t N 0 th t estimated at Quarter t, i.e., the contribution from stock network in a univariate regression. Stock network can positively predict the future volatility at the 1% signi cance level (t-statistic of 3.87). The univariate regression generates a R-squard of 18%. The prediction power remains strong after we control current market volatility (Column (2)) and additionally four factors (Column (3)). As shown in Column (4) to (6), network can still signi cantly predict the equal weighted market volatility with a smaller magnitude in all three speci cations. In summary, we present both the rst and second moment evidence on the important role of stock network in amplifying idiosyncratic shocks of individual stocks and driving aggregate returns and volatility. Particularly, it is noteworthy that stock network makes diversi cation unsuccessful as we discussed in the rst moment test. This echoes a series of work on the network origins of macro uctuations and failures of diversi cation including Acemoglu et al. (2012), Elliott, Golub, and Jackson (2014), among others. 4.2 Cross-sectional Returns In this section, we investigate the relation between stock network and cross sectional returns. There are two important features of stock network: rst, the feedback e ect in that how individual shocks get intensi ed through multiple loops; second, whether shocks can be contagious across stocks through networks beyond the direct links, i.e., the chain e ect. Note that these two e ects can be reinforced by each other. For this purpose, we examine the e ect of stock network on stock 21

22 volatility and comovement. Equation (9) of our model directly builds up the relation between the variancecovariance matrix of individual stock returns and our stock network. The diagonal entries of matrix correspond to the variance of individual stocks while the o -diagonal ones are covariance. We have the following speci cations for return volatility and for return correlation, respectively: V ol i;t+1 = q[n t N 0 t] (i;i) + 2 Contr t + e i;t+1 (13) and Cor i;j;t+1 = [N t N 0 t] (i;j) p [Nt N 0 t] (i;i) p [Nt N 0 t] (j;j) (14) + 2 Contr t + e i;j;t+1 ; where [] (i;j) denotes the entry (i; j) of the matrix. Following Anton and Polk (2014), we focus on large stocks de ned as those with market capitalization greater than the median value of NYSE for the crosssectional analyses. To implement the data to our prediction test, we rst adjust the stock returns by considering four-factor residuals. We also add the current quarter volatility (correlation) to control the persistence pattern. In addition, we control for the current quarter market capitalization for the volatility prediction and the absolute value of the di erence in market capitalizations of two stocks in the current quarter for the comovement prediction. Table 6 and 7 present the coe cient estimates of 22

23 Fama MacBeth regressions. We then calculate Newey-West (1987) standard errors with three-quarter lags. We report regression results of stock volatility in Table 6. Column (1) shows that stock network can predict stock volatility in the univariate regression and the coe cient estimate is positive and highly signi cant with a t-statistic of The coe cient becomes smaller but the signi cance level remains very high (tstatistic of 5.99) after we control for market capitalizations and current quarter stock volatility in Column (2). To highlight the network e ect beyond the direct link, we need to consider the prediction power from L t. Column (3) presents the regression result after controlling p [L t L 0 t] (i;j), the contribution from the direct link to stock volatility suggested in the model. The network e ect still exists with a statistically signi cant coe - cient. The coe cient becomes smaller than that in Column (2), consistent with the fact that as the rst round impact, the direct link absorbs a part of contribution of network e ect. Interestingly, the network e ect even dominates the direct link e ect as the coe cient of the former is almost three times as large as that of the latter, which suggests that the network e ect is important to stock volatility and it is necessary to consider the feedback and contagion loops beyond the impact via the direct link. Next, we study how stock network a ects stock comovement since these stocks get connected through this network. Table 7 presents whether network predicts the return correlation of stock pairs. Column (1) reports the result of regressing correlation coe cients of pair-wise stock returns next quarter on stock network in current quarter without any controls. The coe cient estimate is positive and highly signi cant (t-statistic of 11.64), suggesting network s predictability on stock 23

24 comovement. We then follow the literature to add several variables that potentially contribute to return comovement: we control the return correlation in the current quarter because comovement is persistent; we also consider the industry shocks as a source of comovement and include two dummy variables SIC2 and SIC3 that indicate if they belong to the same industry (the rst two and three digits of SIC codes). Last, we add the absolute value of market value di erences to control the size e ect. As shown in Column (2), the magnitude of coe cient of network decreases but it remains at a high signi cance level. In Column (3), we again add the impact from the direct link as a control. Network can still predict the future return correlation at the 1% signi cance level in addition to the direct link. To summarize, our cross-sectional tests con rm that stock network is considerably involved with return uctuations and comovement. Network might help the individual shock to stocks generate large and contagious uctuations and this e ect is not fully driven by the direct link between stocks, thus di erentiating our study from the previous literature on common ownership and stock comovement such as Anton and Polk (2014). 5 Conclusion In one of her keynote speeches, Yellen (2013) opened the discussion on how complex links in nancial market in uences the systemic risk: Complex links among nancial market participants and institutions are a hallmark of the modern global nancial system.... [T]here is little doubt that some degree of interconnectedness is vital to the functioning of our nancial system. 24

25 ... [I]terconnections among nancial intermediaries are not an unalloyed good. Complex interactions among market actors may serve to amplify existing market frictions,... In the paper, we try to shed some light on this ongoing debate by looking at one speci c type of nancial interconnections. That is, one asset s price is correlated with another s through common ownership of nancial institutions. We construct this network and develop a simple linear model that can be readily estimated with mutual fund holding data to analyze the propagation e ect of network. Our empirical results suggest that the structure of this liquidity-based stock network is an important factor that explains not only the movements of the aggregate stock market but also the volatility of and correlation between individual stocks. 25

26 References Acemoglu, Daron; Vasco M Carvalho; Asuman Ozdaglar and Alireza Tahbaz Salehi "The Network Origins of Aggregate Fluctuations." Econometrica, 80(5), Acemoglu, Daron; Asuman Ozdaglar and Alireza Tahbaz-Salehi "Systemic Risk and Stability in Financial Networks." The American Economic Review, 105(2), Adrian, Tobias and Hyun Song Shin "Procyclical Leverage and Value-at-Risk." Review of Financial Studies, 27(2), Ahern, Kenneth R "Network Centrality and the Cross Section of Stock Returns." Available at SSRN Allen, Franklin and Douglas Gale "Financial Contagion." Journal of political economy, 108(1), Anton, Miguel and Christopher Polk "Connected Stocks." The Journal of Finance, 69(3), Brunnermeier, Markus K and Lasse Heje Pedersen "Market Liquidity and Funding Liquidity." Review of Financial Studies, 22(6), Coval, Joshua and Erik Stafford "Asset Fire Sales (and Purchases) in Equity Markets." Journal of Financial Economics, 86(2), Edelen, Roger M and Jerold B Warner "Aggregate Price Effects of Institutional Trading: A Study of Mutual Fund Flow and Market Returns." Journal of Financial Economics, 59(2), Elliott, Matthew; Benjamin Golub and Matthew O Jackson "Financial Networks and Contagion." The American Economic Review, 104(10), Fama, Eugene F and Kenneth R French "Common Risk Factors in the Returns on Stocks and Bonds." Journal of Financial Economics, 33(1), Gabaix, Xavier "The Granular Origins of Aggregate Fluctuations." Econometrica, 79(3), Goetzmann, William N and Massimo Massa "Index Funds and Stock Market Growth," Journal of Business, 76(1), Greenwood, Robin; Augustin Landier and David Thesmar "Vulnerable Banks." Journal of Financial Economics, 115(3),

27 Greenwood, Robin and David Thesmar "Stock Price Fragility." Journal of Financial Economics, 102(3), Lou, Dong "A Flow-Based Explanation for Return Predictability." Review of Financial Studies, 25(12), Newey, Whitney K and Kenneth D West "A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix." Econometrica, 55(3), Ozsoylev, Han N; Johan Walden; M Deniz Yavuz and Recep Bildik "Investor Networks in the Stock Market." Review of Financial Studies, 27(5), Shleifer, Andrei and Robert W Vishny "The Limits of Arbitrage." The Journal of Finance, 52(1), Weber, Michael and Ali Ozdagli "Monetary Policy through Production Networks: Evidence from the Stock Market," 2016 Meeting Papers. Society for Economic Dynamics. Yellen Janet "Interconnectedness and systemic risk: Lessons from the financial crisis and policy implications". Speech at the American Economic Association/American Finance Association 27

28 Table 1. Number of Equity Funds and Stocks This table reports statistics of the mutual fund and stock sample at the end of each year. The sample starts from 1991 to The sample includes all equity funds and stocks that are held by at least one fund at each year-end. Large stocks are common stocks with market capitalization greater than the median of NYSE. Total AUM is the sum of assets under management of all funds (in billion $). The last two columns report the total market capitalization of all stocks and large stocks (in billion $). Year # of Funds # of Stocks # of Large Stocks Total AUM Market Cap of All Stocks Market Cap of Larges Stocks ($B) ($B) ($B)

29 Table 2. Summary Statistics This table represents summary statistics of main variables. Note that some variables may be defined differently in different panels. N is the stock network matrix as defined in Equation (5). L is the stock linkage matrix as defined in Equation (3). H, the weighting scheme vector, is the weights of stocks last month market capitalization (VW) or equal weights (EW). MKT refers to the market factor, and SMB, HML, MOM are Fama-French size, value, and momentum factors, respectively. Panel A reports statistics of variables used in Table 4. Market_Return is monthly stock market return, weighted by stocks past market capitalization (VW) or weighted equally (EW). Network_Effect represents the value of H Nη measured at the end of each month. η is a vector of stocks unexpected earnings (UE) announced during each month. UE is calculated as the difference between reported earnings and analyst consensus scaled by past price. Average_UE is the average of every stock s UE in each month, valued-weighted (VW) or equal-weighted (EW). Panel B reports statistics of variables used in Table 5. Market_Volatility, the standard deviation of daily stock market returns during quarter t+1. It is calculated using value-weighted returns (VW) and equal-weighted returns (EW). Network_Effect represents the square root of H NN H measured at the end of quarter t. Panel C reports statistics of variables used in Table 5. Stock_Volatility is the standard deviation of a stock daily adjusted returns during quarter t+1. Daily returns are adjusted with Fama-French 4-factor model. Network_Effect represents the square root of diagonal entries of NN at quarter end. Direct_Link is the square root of diagonal entries of LL measured at the end of quarter t for each stock. Stock_Cap is the log of a stock s quarter-end market capitalization. Panel D reports statistics of variables used in Table 7. Correlation is the correlation coefficient of daily adjusted returns of stocks i and j during quarter t+1. Network_Effect is the value of the off-diagonal entry of NN for a pair of stocks, scaled by the product of square roots of the two stocks diagonal entries of NN. Direct_Link is the value of the off-diagonal entry of LL for a pair of stocks, scaled by the product of square roots of the two stocks diagonal entries of LL. SIC2 and SIC3 are dummy variables that equal one if the first two and three digit of the SIC code of two stocks are the same, zero otherwise, respectively. Stock_Cap_Diff is the absolute value of the difference in market capitalization between two stocks. Panel C and D use the sample of large stocks, defined as stocks whose market capitalization is larger than the median market capitalization in NYSE. Panel A: Monthly Market Return and Stock Network Mean SD P10 P25 P50 P75 P90 Obs. Market_Return (VW) Market_Return (EW) Network_Effect (VW) Average_UE (VW) Network_Effect (EW) Average_UE (EW) SMB HML MOM Panel B: Quarterly Market Volatility and Stock Network Mean SD P10 P25 P50 P75 P90 Obs. Market_Volatility (VW) Market_Volatility (EW) Network_Effect (VW) Network_Effect (EW) SMB HML MOM MKT

30 Panel C: Quarterly Stock Volatility and Stock Network Mean SD P10 P25 P50 P75 P90 Obs. Stock_Volatility Network_Effect Direct_Link Stock_Cap Panel D: Quarterly Stock Correlations and Stock Network Mean SD P10 P25 P50 P75 P90 Obs. Correlation Network_Effect Direct_Link Stock_Cap_Diff SIC SIC

31 Table 3. Correlation Matrix These tables report the correlation matrix. Panels A, B, C, and D are correlation matrices for variables in Tables 4, 5, 6 and 7, respectively. Variables are defined the same as in Table 2. Panel A: Monthly Market Return and Stock Network Market_Return (VW) Network_Effect (VW) Average_UE (VW) SMB HML MOM Market_Return (EW) Network_Effect (EW) Average_UE (EW) Market_Return (VW) 1.00 Network_Effect (VW) Average_UE (VW) SMB HML MOM Market_Return (EW) Network_Effect (EW) Average_UE (EW) Panel B: Quarterly Market Volatility and Stock Network Market_Volatility (VW) Market_Volatility (EW) Network_Effect (VW) Network_Effect (EW) SMB HML MOM MKT Market_Volatility (VW) 1.00 Market_Volatility (EW) Network_Effect (VW) Network_Effect (EW) SMB HML MOM MKT Panel C: Quarterly Stock Volatility and Stock Network Stock_Volatility Network_Effect Direct_Link Stock_Cap Stock_Volatility Network_Effect Direct_Link Stock_Cap

32 Panel D: Quarterly Stock Correlations and Stock Network Correlation Network_Effect Direct_Link Stock_Cap_Diff SIC2 SIC3 Correlation Network_Effect Direct_Link Stock_Cap_Diff SIC SIC

33 Table 4. Aggregate Market Returns and Stock Network This table reports the result of contemporaneous regressions as specified in Equation (11). The dependent variable is Market_Return, monthly value-weighted stock market return in columns (1) to (4) and equal-weighted market return in columns (5) to (8). Network_Effect represents the value of H Nη measured at the end of each month. H, the weighting scheme vector, is the weights of stocks last month market capitalization (VW) or equal weights (EW). N is the stock network matrix as defined in Equation (5). η is a vector of stocks unexpected earnings (UE) announced during each month. UE is calculated as the difference between reported earnings and analyst consensus scaled by past price. N is the stock network matrix as defined in Equation (5). H, the weighting scheme vector, equals stocks last month market capitalization in columns (1) to (4) and equals ones in columns (5) to (8). Average_UE is the average of every stock s UE in each month, valued-weighted in columns (1) to (4) and equal-weighted in columns (5) to (8). SMB, HML, MOM are Fama-French size, value, and momentum factors, respectively. t-statistics are reported in parentheses with Newey-West standard errors of 11-month lags. ***, **, and * indicate significance level of 1%, 5%, and 10%, respectively. Dependent Variable: Market_Return t (1) (2) (3) (4) (5) (6) (7) (8) Value-Weighted Equal-Weighted Network_Effect t 34.21*** 34.79*** 36.36*** 33.81* 29.45*** 25.14*** 25.11*** 25.41** (4.07) (3.17) (3.05) (1.88) (3.48) (2.82) (2.72) (2.06) SMB t ** ** ** 0.198*** 0.193*** 0.193*** (2.14) (2.21) (2.21) (8.38) (8.53) (8.51) HML t ** ** ** (-2.43) (-2.12) (-2.12) (-1.21) (-1.38) (-1.37) MOM t *** *** *** *** *** *** (-3.98) (-4.65) (-4.68) (-7.68) (-8.44) (-8.42) Market_Return t (-0.42) (-0.43) (0.97) (0.96) SMB t (0.37) (0.38) (0.81) (0.80) HML t (-1.52) (-1.46) (-1.28) (-1.28) MOM t (-1.59) (-1.59) (-0.59) (-0.55) Average_UE t (0.26) (-0.05) Constant ** *** *** *** 0.010*** *** *** *** (2.33) (3.19) (3.02) (2.89) (3.03) (4.65) (3.95) (3.76) Obs R Square