How Your Counterparty Matters: Using Transaction Networks to
Explain Returns in CCP Marketplaces
Ethan Cohen-Coley
Andrei Kirilenkoz
Eleonora Patacchinix
December 1, 2011
Abstract
We study the pro…tability of traders in two fully electronic, highly liquid markets, the Dow
and S&P 500 e-mini futures markets. We document and seek to explain the fact that traders
that transact with each other in this market have highly correlated returns. While traditional
least squares regressions explain less than 1% of the variation in trader-level returns, using the
network pattern of trades, our regressions explain more than 70% of the variation in returns.
Our approach includes a simple representation of how much a shock is ampli…ed by the network
and how widely it is transmitted. It provides a possible short-hand for understanding the
consequences of a fat-…nger trade, a withdrawing of liquidity, or other market shock. In the S&P
500 and DOW futures markets, we …nd that shocks can be ampli…ed more than 50 times their
original size and spread far across the network. We interpret the link between network patterns
and returns as re‡ecting di¤erences in trading strategies. In the absence of direct knowledge of
traders’particular strategies, the network pattern of trades captures the relationships between
behavior in the market and returns. We exploit these methods to conduct a policy experiment
on the impact of trading limits.
Keywords: Financial interconnections, contagion, spatial autoregressive models, network
centrality, trading limits.
JEL Classi…cation: G10, C21
We are very grateful to Ana Babus, Lauren Cohen, Ernst Eberlein, Rod Garratt, Thomas Gehrig, Dilip Madan,
Todd Prono, Uday Rajan, Julio Rotemberg, Jose Scheinkman, Yves Zenou as well as conference participants at Centre
for Financial Analysis and Policy (CFAP) conference on Financial Interconnections and the FRAIS conference on
Information, Liquidity and Trust in Incomplete Financial Markets for constructive comments. Nicholas Sere and
Kyoung-sun Bae provided research assistance. All errors are our own.
y
Corresponding author. Robert H Smith School of Business; 4420 Van Munching Hall, University of Maryland,
College Park, MD 20742. Email: ecohencole@rhsmith.umd.edu; tel.: +1 (301) 541-7227.
z
Commodity Futures Trading Commission; 1155 21st Street, N.W. Washington, DC 20581. Email: akirilenko@cftc.gov.
x
University of Rome La Sapienza, EIEF and CEPR. Email: eleonora.patacchini@uniroma1.it
Electronic copy available at: http://ssrn.com/abstract=1597738
1
Introduction
We study the pro…tability of traders in two fully electronic, centrally counterparty (CCP), highly
liquid markets, the Dow and S&P 500 e-mini futures markets. In these markets, as any other,
traders earn returns by buying and selling assets. Also akin to many other markets, there is a wide
variation in trader returns. We document and seek to explain the fact that traders that transact
with each other in this market have highly correlated returns. This pattern is remarkable for at
least two reason. First, in the markets that we study, because of an automatic matching algorithm,
agents cannot e¤ectively choose their partners so the correlation in returns cannot be due to social
interactions or interpersonal connections. Second, the observed correlation in returns between
traders cannot be attributed to the observed characteristics of traders. The central question of this
paper is whether and to what extent using information on the interconnection between brokers’
trades helps us to understand the trading outcomes. By doing so, we have some insight into the role
of market structure in determining liquidity provision and in understanding shock ampli…cation.
The presence of a computer match-maker linking buyers and sellers allows us to identify, in
a causal sense, the role of the network structure of contacts in shaping individual returns. In
particular, to understand the relationship between the returns of connected traders, we use the
structural properties of the entire network, i.e. all the actual (direct and indirect) connections
present in the market. This helps us both to explain the correlation between linked traders, and
also to understand the degree to which minor changes in the actions or outcomes of a single entity
can amplify into a system-wide e¤ect. Approaches for the empirical estimation of network in‡uences
are widely varied in the …nancial literature. Some use instrumental variables (Leary and Roberts,
2010), some use summary statistics of network characteristics (Ahern and Harford, 2010, Cohen et
2
Electronic copy available at: http://ssrn.com/abstract=1597738
al. 2008, 2010, Hoberg and Phillips, 2010, 2011, Hochberg et al., 2007, Lin et al. 2011, Faulkender
and Yang, 2011), others use the tools of random networks (Allen and Gale, 2000, Freixas et al,
2000, Brunnermeier and Pedersen, 2009, Amini et al. 2011, Gai et al, 2011). Unlike these, our
approach capitalizes on the network structure itself to measure the properties of the system in a
recursive manner.
Our analysis is motivated by the following simple fact. We …nd that standard least squares
regression methods using observable characteristics of the markets and traders are able to explain
less than 1% of the variation in trader-level returns. When we expand the toolset to include
the network patterns of connections, the regressions are able to explain more than 70% of the
variation in returns. Indeed, observationally equivalent traders that sit in di¤erent networks or
in di¤erent network positions earn di¤erent returns. Our incremental approach makes this clear;
while some trader characteristics are correlated with returns, considering the spatial allocation of
traders into networks as an additional source of variation greatly improves the explanatory power
of the regressions.
Following these results, we dig deeper into our modelling approach and provide an estimation
of how much a shock can be ampli…ed and how widely it can be transmitted as a function of the
network structure of traders. The link between structure and shock transmission provides some
guidance into how market structure can in‡uence risk. In e¤ect, our empirical approach provides
a short-hand for understanding the consequences of a fat-…nger trade, a withdrawing of liquidity,
or other market shock. In the S&P 500 and DOW futures markets, we …nd that shocks can be
ampli…ed as much as 50 times their original size and spread far across the network.
Why do networks matter for returns? They appear to have great signi…cance, even in these
3
two markets, in which a computer assigns trading partners by price and time priority alone. We
interpret the link between network patterns and returns as re‡ecting persistent di¤erences in trading
strategies. Without direct knowledge of traders’particular strategies, the network pattern of trades
simply captures the relationships between behavior in the market and returns. While the link
between strategies and returns is unobservable, a set of trading strategies intermediated by a
computer can lead to a pattern of trades with correlated returns for connected traders.
Our study focuses on the analysis of the distribution of returns of traders in a single asset. Wide
literatures exist that discuss the investment performance of individuals across portfolios, the price
of individual or groups of assets, etc.1 Another literature exists on the pro…tability of …nancial
intermediaries, including specialists and trading desks.2 By studying an individual asset across all
traders, we can isolate the importance of …nancial interconnections. We contribute by suggesting
that the pro…tability of trading is in‡uenced by the particular market role, as described by the
position in the network.
With the …nancial crisis and increasing concerns about …nancial integration and stability as a
leading example, a large number of theoretical papers have begun to exploit the network of mutual
exposures among institutions to explain …nancial contagion and spillovers. Allen and Babus (2009)
survey the growing literature. From an empirical point of view, there is little guidance in the
1
In studies that include portfolio management concerns, the heterogeneity in returns has been attributed to costs
di¤erences (Anand, Irvine and Puckett, Venkataraman, 2009, Perold, 1988). Often the di¤erences are found to be
explained by managerial ability in maintaining the persistence in returns over time. For mutual funds, Kacperczyk
and Seru (2007), Bollen and Busse (2005), and Busse and Irvine (2006)) show that mutual funds maintain relative
performance beyond expenses or momentum over multiple time periods.
2
Reiss and Werner (1998) suggest that interdealer trade occurs between the dealers with the most extreme inventory
imbalances. So…anos (1995) disaggregates gross trading revenues into spread and positioning revenues and argues
that, on average, about one third of spread revenues go to o¤set positioning losses. Hasbrouck and So…anos (1993) …nd
that specialists are capable of rapidly adjusting their positions toward time-varying targets, and the decomposition
of specialist trading pro…ts by trading horizon shows that the principal source of these pro…ts is short term.
4
literature on how to estimate the propagation of …nancial distress.3 We contribute to this strand
of the …nancial connections literature by providing an empirical approach able to measure carefully
the pathways of spillovers in a market with a single asset. By peroviding details on the spread of
risk and the sources of pro…tability at this level of disaggregation will help with an understanding
of systemic risk and with the development of policy.
The remainder of this paper is organized as follows. Section 2 discusses data and institutional
features of the markets that we study. Section 3 discusses the empirics of trader-level returns.
This includes a standard least-squares estimation as well as a network estimation approach. We
also show how we can improve upon our baseline network regression model and elaborate on our
estimation results to understand the di¤usion properties across the entire system following a shock.
We continue in section 4 to further highlight the role of network position to better understand
markets and trader pro…tability. Section 5 discusses the causal nature of our empirical results and
presents some additional robustness checks. Section 6 extends the work to implement a policy
experiment on the impact of trading limits. We conclude in section 7.
2
Data and Institutional Features
2.1
The CME and Futures Markets
Our data of interest are the actual trades completed on the Chicago Mercantile Exchange (CME)
for two contracts, the S&P 500 and Dow futures. The trades we observe are the result of orders
placed by traders that have been matched by a trading algorithm implemented by the CME. Using
the audit trail from the two markets, we uniquely identify two trading accounts for each transaction:
3
See for example, Boyson, Stahel and Stulz (2008) on externalities in hedge fund sector, Adrian and Brunnermeier
(2009) and Danielsson, Shin and Zigrand (2009) on the argument that risk management must be based on more than
individual institutions due to connections between them.
5
one for the broker who booked a buy and the opposite for the broker who booked a sale. For these
two markets, First In, First Out (FIFO) is used. FIFO uses price and time as the only criteria for
…lling an order: all orders at the same price level are …lled according to time priority.
Each …nancial transaction has two parties, a direction (buy or sell), a transaction ID number, a
time stamp, a quantity, and a price. We have transaction-level data for all regular transactions that
took place in August of 2008 for the September 2008 E-mini S&P 500 futures and the Dow futures
contracts. The transactions take place during August 2008 during the time when the markets for
stocks underlying the indices are open. Both markets are highly liquid, fully electronic, and have
cash-settled contracts traded on the CME GLOBEX trading platform.
Because these two markets are characterized by the use of price and time priority alone in
determining trading partners, the only phenomenon that generates networks is the pattern of
trading strategies that links traders with each other. Particular patterns of trading will lead to
di¤erent probabilities of being at the center or periphery of the network, and to distinct chances
of trading with di¤erent types of counterparties. While for each period, we do not observe the
limit order book itself, we know that transactions occurred because market orders or limit orders
were matched with existing orders in the limit order book. We can then trace the pattern of order
execution–a trading network. Figure 1 illustrates this pattern.
[insert …gure 1 here]
Empirically, we thus de…ne a trading network as a set of traders engaged in conducting …nancial
transactions within a period of time; the presence of a link is simply a re‡ection of the ex-post
realization of a cleared trade.
The choice of the period of time within which a network is de…ned is important as it contains
6
valuable information on the resulting network structure. Indeed, with more time, more transactions
are formed and more participants can form accurate beliefs about the valuation of a given asset.
Our approach is to de…ne the network as a given number of transactions among traders that
are either directly or indirectly linked. Then, throughout the remainder of the paper, we will use
a range of network densities in order to ensure that our results are robust to this choice. More
speci…cally, we designate a network as a sequence of consecutive transactions. What we will call
‘sparse’networks are de…ned as containing 250 transactions, ‘moderately dense’networks contain
500 transactions and ‘dense’networks contain 1000 transactions.
While one could imagine alternate approaches,4 our evidence supports the above choice, i.e.
a de…nition of networks as a given number of transactions. Indeed, our results on the existence
of network e¤ects are strongly robust when we vary the number of transactions. As well, the
fact that we …nd our chosen network de…nition has enormous empirical salience suggests that we
have chosen a reasonable concept for the network. In addition, there is no reason to believe that
an incorrect choice of network timing would lead to the spurious …nding of a strong relationship
between networks and returns. Indeed, the opposite is true; a randomly de…ned network will show
no evidence of network e¤ects by construction.
The networks that we de…ne are distinct from one another over time. This occurs both because
agents may not be active in each time period and because their transactions are matched by the
trading algorithm in each time period.
4
For example, an alternative would be to de…ne the network based on some period of time or number of transactions
beginning at the point of a market shock, such as a signi…cant price change.
7
2.2
Returns and Descriptive Statistics
Each broker in this market earns a return. For example, buying a contract for a price of $1 and
selling it for $1.10 yields a pro…t of $0.10 and a return of 10%. Because some positions do not
clear during a given network time period, we report realized returns when positions clear during a
network time period. When they do not clear, we report the mark-to-market returns for the trader
in question.
Our S&P 500 futures dataset consists of over 7,224,824 transactions that took place among
more than 31,585 trading accounts. The DOW futures dataset consists of 1,163,274 transactions
between approximately 7,335 trading accounts. We show in Table 1 some simple statistics of the
data for each of the two markets that we analyze. For each de…nition of networks, we compute
returns for each trader, volumes for each trader as well as the variance of returns across traders
over the course of a trading day. Returns are shown as absolute levels of holding at end of time
period based on an initial investment of $1; thus, a return of 1 indicates that the trader broke even
during the time period. Average returns vary from a loss of 4 basis points to a gain of 4 basis
points. Of course, individual level results vary more widely.
Of note is that the average return across trading accounts is below 1, suggesting that traders
with high volume, on average, earn higher returns. We report the returns unweighted by volume; the
weighted average return across traders is, by construction in futures markets, equal to 1. Also note
that the standard deviation of returns and volume is increasing in the density of the networks. As
the number of transactions increase, the variance does so as well. Notice that the mean transaction
volume declines as the density of the network (i.e. the number of transactions) increases. This
pattern re‡ects the skewness in the data. There are large numbers of transactions of low volume
8
and small negative returns, and a relatively smaller number of observations with higher volumes
and positive returns.
[insert table 1 here]
2.3
Why would individual returns be correlated?
The observed network of realized trades is a potential tool to describe the (unobserved) strategic
interactions at work in the market.
Consider a group of traders. These traders enter each day with a set of trading strategies. These
strategies can either be formal or informal, automated or manual. Indeed, the market contains
some of each of these. Among these formal strategies, for example, are algorithmic traders. These
computerized high-frequency traders composed approximately one-third of volume (Kirilenko et al,
2011) on any given day. The strategy of any given trader will depend on the anticipated strategies
of other traders as well as the observed actions during the day. As successful strategies become
known, followers emerge and copy the strategy. As long as traders either use correlated strategies
or condition their strategies on like information, their behaviors may be correlated in equilibrium
and thus as well in the observed data.
Of course, these correlated bidding patterns lead to similarity in returns. Because the matching
algorithm used by the CME is blind to identities of the traders, traders with correlated strategies
will trade with each other as well as with others. As they do so, and form links with one another,
correlation in trading strategies leads to a connection between strategies and network position.
Many traders will acknowledge that sitting between two traders with fundamental liquidity needs
can be pro…table.
Note that futures markets are zero-sum markets in aggregate. Thus, while each transaction
9
could potentially yield a pro…t for both parties, some portion of the network must absorb equal
losses for each gain. We illustrate how two traders could both pro…t from a transaction in …gure 2.
[insert …gure 2 here]
Figure 2 shows the presence of two large traders (denoted ‘A’ & ‘D’) that have fundamental
liquidity demands, one positive and one negative. Each of these participate in the futures market
by placing large one-sided orders either to buy or sell contracts.
A separate set of traders, denoted ‘B’, implements rapid o¤ers to buy and sell. The objective
of such traders is to provide the liquidity needed by the large traders with fundamentals demands.
Because the large traders may not appear on the market at precisely the same time, the liquidity
providers can extract pro…ts from the large traders by being willing to transact when needed.
The combination of the liquidity traders’actions can generate a diamond-shaped network pattern
illustrated in …gure 2, panel A. On one side, the liquidity traders buy as needed and on the other
they sell as needed. By being willing to buy and sell, the agents in the center can generate pro…ts.
Of course, knowledge that agents can achieve these pro…ts leads to a new set of trading strategies.
Figure 2 shows the emergence of additional agents, denoted ‘C’. E¤ectively, the second set of agents
hopes to intermediate between one large trader with fundamental demand and the initial set of
liquidity traders. Now, if one evaluates the correlation in returns over a given period of time of
these traders, she will observe that the pro…ts of the liquidity traders are inversely correlated with
those at the ends of the diamond. As the large traders lose money the liquidity traders earn (or
vice-versa). However, our new entrants ‘C’, over time, yield returns that are positively correlated
with the other liquidity traders.
An example is presented in Panel B. The table shows the returns of each set of traders in a
10
hypothetical case. The outcome in case 1 is that the returns of A and D are negatively correlated
with the returns of B. That is, the market will act like a shock absorber. As new shocks hit the
system, the reaction is to ameliorate the impact.
In case 2, the returns of A and D continue to be negatively correlated with B and C; however, B
and C show positively correlated returns. It is straightforward to see that as the number of traders
in the center of diamond increases, a market change that impacts one of the traders will have a
similar impact on the others.
To see how two traders could both pro…t from an interaction, consider A, B, and C. A purchases
a contract from C for $2 at time t and buys one from B for $1 at time t + 1. At time t + 2, C
purchases a contract from B for $1.25. The …nal transaction yielded C a pro…t of $0.75 and B
a pro…t of $0.25. Of course, A has lost the full dollar in the process. The trade between A and
C allows them to share the $1 gain. Repeated interactions of this type will generate a positive
correlation in returns.
These examples help understand how returns can be correlated across trading strategies, but
also importantly help illustrate how shocks can be propagated. That is, they are a representation
of the pathways of the transmission of risk in the system.
3
3.1
Empirics of Trader-Level Returns
Traditional regression models
Assume that there are N traders divided in k = 1; :::; K networks, each with nk members, i =
1; :::; nk ;
K
X
nk = N . In this section, we ignore the network structure of the connections. We
k=1
simply use the network as the relevant "market" (or trading period) for each trader. As explained
before, we consider networks as sequences of trading of 250, 500 and 1000 trades. Parsing trading
11
activity in this way allows one to avoid variations in returns that may occur solely due to the ebbs
and ‡ows of trading. We thus run least squares regressions including simple statistics of a number
of relevant characteristics to capture aggregate variations of market conditions.
For each day or half day of activity (indexed by t), we evaluate the role of a range of control
variables on returns. Our speci…cation is
ri;
;t
=
0
+
+
1 buyvolumei;k;t
4 volumet 1
f or i = 1; :::; n ;
+
+
2 sellvolumei;k;t
5 laggedreturnsi;t 1
+
+
3 variancei;t
+ :::
(1)
i; ;t
= 1; :::; K:
where buyvolume is the number of contracts purchased in the relevant trading period for each
trader i (de…ned by network k each trader belong to) and similarly sellvolume is the number of
contracts sold in the trading period. Recall that returns are calculated as mark-to-market value
of contracts at the end of the day (or half-day). As a result, buyvolume does not always equal
sellvolume during a particular trading time period. V ariance is the variance of a trader’s return
during the full (or half) trading day, volume is the net trading (buy-sell) aggregate volume in the
prior time period, and laggedreturns is the trader’s mean return in the …rst half of the trading
day. This …nal variable only appears in speci…cations estimated on data from the second half of the
day and captures potential persistence in returns. In all speci…cations, we also include a control
for trades late in the day.
Table 2 collects the estimation results. Note …rst that the vast majority of coe¢ cients have
economically insigni…cant magnitudes. With the exception of half-day returns, other variables have
essentially no impact on returns. Some variables show statistical signi…cance in places, though the
large number of observations makes this unsurprising. Second, we highlight that the adjusted R
12
squared statistics of these regressions are very small. The largest is .003. This re‡ects the fact that
individual level returns are very, very di¢ cult to explain. Indeed, there is little theory that would
point towards an ability to do so without linking the traders to particular investment strategies or
variations in access to information.
[insert table 2 here]
3.2
Network regression models
Having seen that the OLS results of the above speci…cations do a relatively poor job of characterizing
the distribution of returns, we look to augment our speci…cation to capture the information content
of the network connections.
E¤ectively, we will augment speci…cation 1 with an regressor capturing the returns of connected
agents. For example, to consider the in‡uence on i of only a single other agent (j), speci…cation 1
would read
ri; =
0+
M
X
m m
xi;
+ rj;k +
i;
;
(2)
m=1
where xm denote the set of m variables considered above and rj denotes the returns of the trading
partner. So an estimated coe¢ cient
greater than zero indicates that returns for trader j are
positively correlated with returns for trader i. Extended to a simple network of three agents (i; j; s)
i
t
t
j
t
s
the equation expands to
ri; =
0
+
M
X
m m
xi;
+
m=1
13
1 rj;k;d
+
2 rs;k;2d
+
i;
;
(3)
where the subscript d and 2d indicate agents j and s at one node and two nodes distant from i,
respectively. The coe¢ cient
whereas
2
1
captures correlation in returns between directly connected traders,
the correlation between agents further away in the network structure. These multiple
steps are important; they are similar in spirit to multiple lags in a time series regression. The set
xm now also includes additional regressors for the characteristics of every other agent. Thus, as the
number of agents increases and the network expands, we can continue to add additional regressors
to the right hand side of this speci…cation for each agent and each degree of separation from agent
i. Eventually, we will add n
1 regressors for each degree of separation, leading to a very complex
speci…cation that takes into account each type of in‡uence of every agent on every other.
To include every other agent and every degree of separation, and to simplify notation, we can
introduce a matrix that keeps track of the links between agents. This is a N square adjacency
matrix G = fgij g whose generic element gij would be 1 if i is connected to j (i.e. interacts with
j) and 0 otherwise. Here gij = 1 if trader i and j have concluded a transaction during a period of
time, and gij = 0, otherwise. This matrix represent the interaction scheme of the traders in the
market. The G matrix associated with the simple network in the picture above is:
i
i 0
G=
j 1
s 0
j
1
0
1
s
0
1
0
indicating that i trades with j; s with j; and j with i and s:
Then, we can collapse the above speci…cation with all traders at every level of interaction into
the following simpli…ed one:
ri; =
0+
M
X
m=1
m m
xi;
+
n
1 X
gi:;k
gij; rj; +
j=1
14
i;
,
for i = 1; :::; n ;
= 1; :::; K:
(4)
where ri; is the idiosyncratic return of trader i in the network k; gi:;k =
of direct links of i,
1
gi:;k
Pn
j=1 gij;
Pn
j=1 gij;
rj; is the average returns of trading partners,
is the number
i;k
is a random
error term, xm
i; is a set of M control variables at the individual and/or network level. This model is
the so-called spatial lag model or spatial autoregressive model in the spatial econometric literature
(see, e.g. Anselin 1988) and can be estimated using standard software via Maximum Likelihood.
The object of estimation is now .
Table 3 shows the estimation results of this model speci…cation vis-a-vis the previous traditional
one. We report the identical regressions as in Table 2 and the augmented regressions based on
the network approach. It is striking to see the substantive improvements in …t coming from the
additional regressor. Indeed, as we inspect the adjusted R squared coe¢ cients, we note that they
are considerably higher than the OLS speci…cation. The additional information in trading partner
returns is systematically important in predicting my own returns. The R squared coe¢ cients now
range from .05 to .37 for the S&P and .04 to .18 for the Dow. In one case, these regressions explain
more than 1/3 of the variation in trader returns.
[insert table 3 here]
In terms of the coe¢ cient of interest, , the estimates for the S&P are between .02 and .27
depending on the day and the network type; similarly, they are between .02 and .10 for the Dow.
Most of these are estimated with a very high degree of precision. This suggests that increases in
trading partner returns could be important for individual outcomes.
In order to highlight the relative importance of the network e¤ects vis-a-vis other possible
controls, such as those included in the least squares estimation table 2, we estimate model (4)
without controls. Table 4 shows the results. The coe¢ cient estimates for the network e¤ect, , as
15
well as the values of the R squared are nearly the same as in table 3, suggesting that the improved
…t of the regression is due to the network approach. The principle di¤erence that we observe is
that for some speci…cations, the t-stats are lower than they were in table 3. This reveals that while
the control variables do not explain much of the variation in returns, they absorb some estimation
noise, allowing the network e¤ects to be estimated more precisely.
[insert table 4 here]
In the remainder of this section we will improve model (4) and elaborate upon this speci…cation
to highlight the role of networks in propagating changes in returns across traders.
3.3
Weighted Networks
We extend the simple network model above to use the network equivalent of importance weights
in an OLS regression. While the baseline network model had strong results, we hypothesized that
the size of the transactions in the network would be a determinant of outcomes. Trading with large
counterparties would be di¤erent than trading with smaller ones. We will measure the importance
of traders by total trading value and replace the binary matrix G with a new matrix capturing
both the number of links and the importance of each link. Let the matrix W = GD, where G
is as de…ned above and D = fdij g is a matrix that weights the links within the network. The
scalar dij is a scaling factor, calculated as the total trading volume in the same trading period (the
network) of each i and j. Total trading volume is de…ned as the sum of all trades, both buys and
sells, made by trader i with all other traders. As a result, W = fwij g is now a weighted network
w. Figure 3 provides an illustration of the calculation of these weights. It shows a set of four
transactions amongst four traders, A,B,C and D. Each arrow is a single transaction, with the arrow
16
pointing towards the buyer of a contract. In panel A, each value along an arrow shows the number
of contracts traded. The accompanying matrix is an unweighted network representation of the
transactions, i.e. the G from above. Each cell contains a ‘1’where two brokers have transacted and
0 otherwise. Panel B shows the same set of transactions, but having along each arrow a calculation
equal to our measure of importance: (total number of contracts bought or sold by buyer + total
number of contracts bought or sold by seller)/2. These values are the ones used for the weights
D = fdij g to get a weighted accompanying matrix W .
[insert …gure 3 here]
Results for this method are presented in table 5. We replace G with W in model (4) and run
the same regression again. We again follow the format of displaying results by the density of the
network.
[insert table 5 here]
All of the results are now stronger. First, note that the estimated correlation between trader
returns is now greater than 0.9 in the S&P and greater than 0.8 in the Dow. That is, the returns
a trader earns are very similar to those of her trading partners.
Second, across densities of network structure, we …nd estimates of
that are large and always
statistically signi…cant. Across each speci…cation, the observed t-statistics increase; the estimation
is now much more precise than without the weights.
Third, these new speci…cations are able to explain a much larger fraction of the variation in the
trader-level returns. The adjusted R squared values are now uniformly above 70% in both markets.
Both the structure of the connections and their importance are important in understanding returns.
17
The last row of table 5 report values for the average multiplier, . Indeed, our network approach
(model 4) allows us to measure the aggregate amount won or lost by agents connected at any level
to a trader. Thus, if a trader wins $1, the multiplier measures how much traders in the network win
or lose. Because the coe¢ cient
measures the average correlation in returns across traders linked
by a single node in the network,
2
measures the average correlation across two links,
3
the average
across three, etc. Thus, a simple calculation allows us to measure the impact of a shock to any
given trader. Consider a shock of $1. On average, this will lead to a change in earnings of directly
connected agents of
1, agents two links away of
won or lost by a trader,
=
1
1
2
1, etc. One can see then, that for each dollar
is the aggregate amount won or lost by agents connected at any
level to the trader. For example, an estimate of
equal to 0:5 produces a multiplier of 2, suggesting
that for each dollar lost by a trader hit by an exogenous shock, the individuals connected to the
trader will lose an aggregate of 2 dollars. Because the market is a zero-sum one, if all agents are
marked-to-market at the time of the idiosyncratic loss, the two dollar losses of the trading partners
will be o¤set by two dollar of gains elsewhere in the network. Our measure is thus a calculation of
the degree of re-allocation of pro…ts.
We report the value of
below each speci…cation. We can see that the multiplier is between
16 and 66 for the S&P and 5 and 21 for the Dow. These large numbers imply that these trading
networks have very high sensitivity to shocks. Small changes to one individuals rapidly spread and
magnify. As it will be explained in greater detail in the next section, these e¤ects depends on both
the structure of the connections and on the strength of the interaction, as captured by .
The average multiplier can be interpreted as a compact statistic for a type of systemic risk of
a system. Indeed, the calculation of an average spillover following a shock de…nes the degree to
18
which idiosyncratic losses become systemic ones.
4
Networks, Centrality and Pro…tability
Now that we have established the strong empirical relationship link between structure of a network
and the returns of individual traders, we dig more deeply into the network structure to understand
to role of the individual position in the network in shaping individual returns. First, we discuss
one particular measure of network centrality, Bonacich Centrality (Bonacich 1987). This measure
is simply a characterizatio that emerges from our network model and helps to provide a simple
correspondence to returns. Second, we illustrate how this relationships between individual centrality
and returns leads to measurable distributional impacts in returns.
4.1
Centrality and returns
Di¤erent measures of individual centrality have been proposed in the network literature (Wasserman
and Faust, 1994). We consider here a particular measure, Bonacich centrality, which can be derived
from the estimation of model (4) and has a number of useful properties. First, it measures the
importance of each agent in a network taking into account not only the number of direct connections
but also their importance in terms of number of connections in a recursive manner, so that the
entire network structure is taken into account. Second, it is not parameter-free, but it depends
on the level of strategic interactions that stems form the network, as captured by our parameter
. Speci…cally, the Bonacich Centrality is a count of the number of all direct and indirect paths
starting at node i and ending at node j, where paths of length p are weighted by
p
. Recall that
this was how we compressed the large number of variables into a simple speci…cation, (4). More
paths from i to j imply a more central trader. A full description of the Bonacich measure, including
19
the connection with our model (4), is contained in Appendix A.
To illustrate the idea and its relevance to a trading network, consider the original Bonacich
(1987) example. Bonacich considers a network of individuals that communicate with each other.
The parameter
measures the probability that a communication will be transmitted by any indi-
vidual to any of his contacts. W is the expected number of these communications that are passed
on to direct contacts,
2
W 2 are the ones passed on to contacts two links away and
p
W p is the
expected number of messages that reach agents at path-length p:
In the context of a trading network of mutual exposures, the magnitude of , combined with
the network structure W each trader is embedded in, thus re‡ects the degree to which a shock
is transmitted locally or to the structure as a whole. Small values of
heavily weight the local
structure, while large values take into account the position of agents in the structure as a whole
(Bonacich, 1987). We include an illustrative …gure 4 that shows the impact of a shock to a trader
in networks with di¤erent
values. One can see that as
becomes larger, the shock transmits more
widely across the network, i.e. it impacts traders much further away in the network.
[insert …gure 4 here]
To understand the link between network topology and returns, we can use our estimated
to
calculate the Bonacich centrality for each trader in our networks (formula 7 in appendix A).5
With this distribution of positions in the network, we can look at the outcome di¤erences across
traders of di¤erent centrality levels. We do so by looking at the impact of a one unit change in
centrality on returns. Table 7 reports both the standard deviation of the centrality measure as
5
This calculation will generate a distribution of individual centralities depending on the strength of network
interactions and on the heterogeneity of network links (as captured by the estimate of and the matrix W in formula
(7), respectively).
20
well as this impact for sparse, moderately dense and dense networks.6 We report absolute changes
in returns; because the benchmark return is 1, the numbers can also be interpreted as percentage
changes. They are changes in returns over a one-day time periods. We do not normalize to an
annual basis. It appears that high returns are associated with high degrees of centrality irrespective
of network complexity.
[insert table 7 here]
We note two patterns. One, the standard deviation of the centrality measure is nearly identical
across the three network types; indeed, it’s relatively similar across markets. Two, the impact of
a one unit change (approximately 1/3 of a standard deviation), is also relatively constant across
network densities.
We highlight this …nding as it suggests, in part, that our network de…nition is e¤ective. Even
though we construct our networks based on an ad-hoc choice of transactions, the impact of the
networks that we de…ne remains consistently important throughout the measured time period.
While one could potentially improve upon the de…nition, the strength and consistency over time of
these …ndings suggests that we have captured a large portion of the network e¤ect.
Taken as a whole, this evidence indicates that the number of direct and indirection connections
in the network, as weighted by , is a relevant factor that plays a role in explaining the crosssectional variation of returns.
6
Note that we do not put centrality on the right-hand-side of our regressions. The impacts are derived from a
simple transformation of the estimated THETA from Model (4).
21
4.2
Network structure and distributional e¤ects
Our analysis so far shows to what extent network position (network centrality) of an individual
trader was important in explaining the level of individual returns. The more central a trader
emerges from the exogenous matching process, the higher his returns.
In the remainder of this section, we highlight the implication of di¤erences in network structures
in terms of the distribution of outcomes in …nancial networks. That is, is there a di¤erence in the
variance of returns for traders operating in di¤erent types of networks?
Recall …rst a few stylized facts. One, we …nd that network structure very well explains individual
level returns. Two, we …nd that the average multiplier, as measured by the ratio of an aggregate
impact to the level of an individual shock, is very high in the networks that we analyze. Three,
we …nd that an improvement in terms of centrality for an individual trader is associated with a
positive change in returns.
Given these three …ndings and the fact that futures markets are zero-sum, we can make two
claims. First, at the level of a network (250-1000 transactions), we should see that a change in
the distribution of the centrality measure should have no change on the mean return in a network.
That is, an arbitrary re-allocation of individuals around the network should change the distribution
of outcomes, but not the mean.7 Second, it thus follows that one should …nd di¤erences in the
variance of returns. We …nd evidence of these two phenomena in our data.
Figure 5 displays the results. It relates the impact of network centrality to the variance of
returns in the network, and …nds a positive relationship. It also shows that the aggregate mean
7
We discussed above that the unweighted mean of returns at the network level may not always be one, given that
some traders earn large pro…ts. Precisely, the re-allocation can impact to a small degree this unweighted return, but
cannot impact the weighted network returns–which must always be equal to one.
22
of returns remain roughly unchanged. As centrality becomes more important, the distribution of
returns widens. This is a logical implication; if being central leads to greater returns, in a zero-sum
market, this necessarily means that someone at the periphery must lose out, and the variance of
returns widens.
[insert …gure 5 here]
Technically, the relationship shows that the distribution of returns of the network with greater
sensitivity to centrality stochastically dominates (in a second order sense only) the distribution of
returns for a network with lower sensitivity to centrality.
5
Discussion and robustness checks
The validity of our analysis and its relevance for policy purposes hinges upon the correct identi…cation of the network e¤ect, .
The core problem that emerges in estimating linear-in-means models of interactions is Manski
(1993)’s re‡ection problem. This arises from the fact that if agents interact in groups the expected
mean outcome is perfectly collinear with the mean background of the group: how can we distinguish
between trader i0 s impact on j and j 0 s impact on i? E¤ectively, we need to …nd an instrument:
a variable that is correlated with the behavior of i but not of j. Cohen-Cole (2006) noted that
complex network structures can be exploited for identi…cation. Bramoullé, Djebbari and Fortin
(2009) highlighted the same phenomenon and showed that in network contexts, one observes ‘intransitivities.’ These are connections that lead from i to j then to s, but not from s to j (see
picture). Thus, we can use the partial correlation in behavior between i and j as an instrument for
the in‡uence of j on s.
23
That is, network e¤ects are identi…ed if we can …nd two agents in the economy that di¤er in the
average connectivity of their direct contacts. A formal proof is in Bramoullé, Djebbari and Fortin
(2009). As a result, the architecture of networks allows us to get an estimate of , while eluding
the “re‡ection problem.” Of course, a complex trading network such as the one we are concerned
with has a very rich structure of connections and identi…cation essentially never fails.
Another traditional concern in the assessment of network e¤ects in the social sciences is that network structure can be endogenous for both network self-selection and unobserved common (group)
correlated e¤ects. The …rst problem might originate from the possible sorting of agents. However,
given our de…nition of networks based on high-frequency data and a random matching algorithm,
we have no reason to believe that any selection e¤ects exist in this context. Agents are assigned to
trading partners as we described above, based on time and price priority alone. Even if two traders
were to attempt to time a transaction as to ensure a match, the high volume of transactions on
these markets makes this nearly impossible to complete. As such, we have a strong claim that
individuals cannot choose their network partners and thus no selection e¤ects should be present.
Network topology is exogenous here. The possible presence of unobserved correlated e¤ects instead
arises from the fact that agents in the same group tend to behave similarly because they face a
common environment or common shocks. These are typically unobserved factors. For example,
traders with similar training, that sit in similar rooms or use trading screens that show similar
types of data, may be in‡uenced in their trading patterns in ways that generate correlations in
returns. While we believe this to be very unlikely, we can control for these unobserved e¤ects by
re-estimating our model after taking deviations in returns with respect to the group-speci…c means,
i.e. from the average returns of (direct) trading partners. That is, if agents in a given empirically
24
observed network have some similarity that leads them to earn higher returns as a group, we will
average out this group-level e¤ect and look only for the presence of spillovers. Of course, our primary speci…cation already largely nets out market level returns by virtue of the fact that aggregate
market levels returns are 1. In this case, we also control for group-level unobserved heterogeneity.
That said, there is little reason to believe that in an electronically matched market one would
observe any e¤ect of this sort.
Results are in table 6 and indeed illustrate very small di¤erences from those in table 5.
[insert table 6 here]
These results are useful also for another reason. The market that we are discussing is a zerosum one; bene…ts to a given individual are necessary re‡ected in losses to another. As a result,
complementarities in returns must necessarily be re‡ected in losses elsewhere in the network. By
estimating our results in deviations from average level returns for an individual’s own ‘network,’
we handle this issue. In deviations, complementarities will no longer be re‡ected elsewhere in the
network structure and we can consequently use our results to evaluate the impact of a shock to the
system. The particular context of analysis and our approach thus enables us to uncover a causal
relationship between network structure and pro…tability.
6
A Policy Experiment
One of the advantages of this approach is that it provides a mechanism via which policy makers
and regulators can understand the impacts of their choices on the risk in the system. As a leading
example, the August, 2010 passage of the Dodd–Frank Wall Street Reform and Consumer Protection Act (Dodd-Frank) included a call for the evaluation of position limits in futures markets. The
25
impact of such limits has been …ercely debated.
In this section, we construct a counterfactual study that explores the consequences of this
policy using our framework. Our exercise runs as follows. We set an arbitrary transaction limit
for a given period of time. Given the restriction, we re-estimate our model (4) assuming that any
traders that, in the data, transact a greater number than this amount, transacted only the …xed
maximum. Speci…cally, we restrict to C the number of contracts that can be purchased in 1/10 of
a trading day, thus setting arti…cial bounds on the weights of our matrix W . This does not change
the network structure other than the weights of the links. We consider C = f2; 3; 7; 10; 20; 30; 100g
for the Dow futures market and investigate the consequences of such limits in terms of the high
returns are associated with high degrees of centrality irrespective of network complexity. estimate,
b: Figure 6 shows the results for each of the values of C.
[insert …gure 6]
The simulation has two policy interpretations. First, we see from the …gure that tighter trading
limits leads to higher average multiplier values. Above we noted that these multipliers can be
interpreted as a measure of systemic risk; however, here some additional detail is warranted. In
our context, systemic risk is a measure of the size of the passthrough that occurs following an
idiosyncratic shock. This is conceptually distinct from increases in the frequency of shocks (which
we do not address). What we observe from this exercise is that the size of shock propagation
increases as trading limits become tighter. In the case explored here, a move from no position
limits to a strict one would increase the systemic risk multiplier in the system from approximately
13 to 16.
Second, we can also infer from the exercise that tighter limits distribute the impact of the shock
26
across a wider range of market participants. That is, while a shock in the constrained world may
be widely distributed, an equivalent shock in the unconstrained world to a large trader may pass
to only a small number of counterparties. This phenomenon arises because in our experiment we
do not simulate new links between traders, the mechanism by which systemic risk increases is to
decrease the centrality of the network; that is, it downplays the importance of the traders who had
previously exceeded the limit and been quite central.
E¤ectively, this highlights that the policy comes with a distinct tradeo¤. One one hand, in
our simulation, it has the potential bene…t of dispersing adverse shocks to a wider range of market
participants. On the other hand, the limits also appear to generate aggregate larger consequences
from each shock. The …ve dollar loss may now be magni…ed to 6 or 7. The trade o¤ of the two
will determine the aggregate impact of the policy and its …nal impact will undoubtedly be market
speci…c.
7
Conclusions
Our analysis explained a conjectured but to date unproven feature of …nancial markets: returns
from trading are correlated with the position agents occupy in a trading network. Using our
network-based empirical strategy on two highly liquid …nancial markets, we are able to explain a
large portion of the individual level variation in returns. This …nding has potentially large salience.
Most importantly, one of our results is that individual level shocks are greatly ampli…ed and
spread in these markets: a one unit change in individual level returns can be ampli…ed even 50 times.
This implies very rapid propagation of shocks and little ability to avoid contagion. Because these
results are a function of the network structure, these results point policymakers in the direction of
potential interventions. Notice that the rapid spread and ampli…cation derives from the network
27
structure; adjust the structure and adjust the speed of spillovers. This points towards interventions
in the matching algorithm, potentially during times of anticipated crisis. For example, one could
alter or eliminate most of the network spillovers altogether by concentrating trading into hourly
auctions.
A long literature in sociology and economics would suggest that network patterns are important
in non-market interactions based on a variety of plausible mechanisms. These include social stigma,
information sharing, peer pressure, and more. The di¢ culty in translation of the methodologies
developed in the social science to …nancial markets, particularly electronic ones, is that there is
little basis to believe that any of the mechanisms are at work. Orders are matched at random by
a computer based on time and price priority, leaving little room for social impact even if traders
had a motivation to do so. Thus, our conclusions are statements about the empirical importance of
the networks that emerge as a result of equilibrium order strategies. We …nd that these strategies
not only lead to networks of note, but that an empirical mapping of the networks to returns shows
important e¤ects.
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8
Appendix A: Spatial autoregressive model and network centrality
For ease of interpretation, let us write model (4) in matrix notation and derive the reduced form.
The following derivations are helpful to understand why model (4) captures recursively the network
e¤ects at any degree of separation and the link with a particulat network centrality measure,
Bonacich centrality (Bonacich, 1987).
Model (4) can be written as:
r = Gr+ x + ;
where r is a N
1 vector of outcomes of N agents, x is a N
(5)
V matrix of V variables that may
in‡uence agent behavior but are not related to networks, G is the row normalized version of the G
32
matrix which is used to represent average returns and
is a N
1 vector of error terms, which are
uncorrelated with the regressors.
Given a small-enough value of
0; one can de…ne the matrix
[I
1
G]
=
+1
X
p
Gp
(6)
p=0
The p
th power of the matrix G collects the total number of paths, both direct and indirect, in
the network starting at node i and ending at node j: The parameter
is a decay factor that scales
down the relative weight of longer paths, i.e. paths of length p are weighted by
that an exact strict upper bound for the scalar
p
. It turns out
is given by the inverse of the largest eigenvalue of
G (Debreu and Herstein, 1953).
In a row-normalized matrix, such as the one used in model (4), the largest eigenvalue is 1. If j j <
1 expression (6) is well-de…ned, that is, the in…nite sum converges. The condition j j < 1 capture
the idea that connections further away are less in‡uential than direct contacts and guarantees that
the matrix [I
G]
1
is able to capture all the e¤ects that stems from a given network topology,
that is the cascades of e¤ects stemming from direct and indirect connections.
If j j > 1, the process is explosive. In a …nancial network context, it is equivalent to a complete
…nancial collapse. While interesting in its own right, we do not analyse this case here. We focus on
how, even in the absence of a complete …nancial collapse, a small shock can cascade causing large,
measurable and quanti…able damage. Therefore we consider j j < 1.
If one solves for r in model (5), the result is a reduced form relationship:
r = [I
G]
1
x + [I
G]
1
De…nition 1 (Bonacich, 1987) Consider a network g with adjacency N square matrix G and a
33
scalar
such that M(g; ) = [I
of parameter
G]
1
is well-de…ned and non-negative. The vector of centralities
in g is:
b(g; ) = [I
The centrality of node i is thus bi (g; ) =
G]
1
Pn
j=1 mij (g;
(7)
1:
), and counts the total number of paths
in g starting from i. It is the sum of all loops mii (g; ) starting from i and ending at i, and all
outer paths
P
j6=i mij (g;
) that connect i to every other player j 6= i, that is:
bi (g; ) = mii (g; ) +
X
mij (g; ):
j6=i
By de…nition, mii (g; )
1, and thus bi (g; )
1, with equality when
Therefore, once one has on hand an estimate of
= 0.
it is possible to derive the distribution of
Bonacich centralities for all the agents in the network. Observe that in the original Bonacich (1987)
paper, the centrality measure is presented for unweighted networks. However, all the techniques
apply to the weighted network case, i.e. G = W (see Newman, 2004, for a discussion).8
8
We are grateful to Jose Scheinkman for calling our attention to this.
34
Figure 1
A
B
C
D
E
F
Order Strategies
Order Submissions
Order Book
Matching Engine
D
E
A
F
B
C
Empirical Pattern (Network)
Note: Each node in the section labeled ‘order strategies’ represents a single trader’s plans for trading. The ovals beneath each trader, next to the label ‘order
submissions’ represents actual placed order. Below this, we denote with a box the complete order book. This is the aggregation at each point in time of all
the orders submitted by traders. This order book is passed through the box beneath it, which we have labeled a ‘matching engine’. This computer matches
orders based on price and time priority. Finally, beneath the matching engine, we provide a sample representation of the network patterns that could emerge
from a set of 6 completed transactions.
35
Figure 2
Panel A
Case 1
Case 2
B
C
B
C
B C
B
B
A
B
A
D
D
B C
C
B
B
B
Panel B
Case1
A
B
Action
Sell100at99
Buy100at99
Sell100at101
Case2
Returns
0.99
1.02
C
D
Buy100at101
0.99
Action
Sell100at99
Buy100at99
Sell100at100
Buy100at100
Sell100at101
Buy100at101
Returns
0.99
1.01
1.01
0.99
Note: Panel A shows two agents with fundamental liquidity needs, marked A and B, and a series of agents that have traded with them. Each edge is marked as
an arrow, pointing from the seller to the buyer. Panel B shows the same configuration with the addition of a few additional agents. The example assumes that
the market price is constant at 100.
36
Figure 3
Panel A
3
A
B
C
A B C D
4
2
Transactions
10
A B C D
1



1
1


 1
1


1 1 1 
D
Panel B
(5+7)/2=6
A
B
A
C
Bilateral Volume
(16+7)/2=11.5
(16+10)/2=13
A B C D
(16+5)/2=10.5
B
C
D
13 



6
10
.
5



6
11.5


13 10.5 11.5

D
Note: Panel A shows a set of transaction between 4 traders. Each arrow is a single transaction, with the arrow pointing towards the buyer of a contract. Each
value along an arrow shows the number of contracts traded. The accompanying matrix is an unweighted network representation of the transactions. Each cell
contains a ‘1’ where two brokers have transacted.
Panel B shows the same set of transactions. Along each arrow is a calculation equal to the (total trades of buyer + total trades of seller)/2. These values are
then used as weights in the accompanying matrix.
37
Figure 4
Impulse Response Diagram for Various Estimated Theta
Impulse Response Diagrams for Various Estimated Theta
0.12
0.1
Average Impact
0.08
θ = .7
0.06
0.04
θ = .6
0.02
θ = .5
0
0
5
10
15
20
25
30
35
40
Trader distance from shock (# network links)
45
50
Note: The figure shows the impact of a 1 unit shock to a network. Each line shows the impact of the shock for a different value of θ. The distance from shock
shows the impact as one moves away from the origin of the shock to the remainder of the network, traversing along only trading relationships. For example, 2
degrees from shock would indicate the impact of i on k, with j in between.
38
1.4
Figure 5
1.2
Mean of Returns
1
0.8
0.6
0.4
0.2
00
2.5
0.2
x 10
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.6
1.8
2
Impact of One Unit Change in Centrality
-3
Variance of Returns
2
1.5
1
0.5
00
0.2
0.4
0.6
0.8
1
1.2
1.4
Impact of One Unit Change in Centrality
Note: The top figure shows the relationship between a one-unit change in centrality and the mean of returns in the network. To calculate this, we take the
average impact of a one-unit change across all traders in a given network and plot it against the mean of returns across traders in the same network.
The bottom figure shows the relationship between a one-unit change in centrality and the variance of returns in the network. To calculate this, we take the
average impact of a one-unit change across all traders in a given network and plot it against the variance of returns across traders in the same network.
39
Figure 6
Impact of Trading Limits on Systemic Risk
17
Average Systemic Risk Multiplier
16
15
14
13
12
11
0
20
40
60
Trading Limit
80
100
Note: Figure shows the results of a simulation in which traders face trading limits. Each simulation result is an estimate of the systemic risk multiplier. The
horizontal axis, above, shows the maximum trading limit in the simulation. The vertical axis shows the average impact of the systemic risk multiplier. Limits
on the horizontal axis indicate maximum trading volume during a pre-specified time period.
40
120
Table 1: Summary Statistics
Note: As discussed in the text, sparse networks are defined as containing 250 transactions each, moderately dense networks as containing 500
transactions each and dense networks as containing 1000 transactions each. The top half of the table includes statistics from the S&P 500 e-mini
futures market. The bottom half includes statistics from the Dow futures market. The columns report the means, standard deviation, minimum and
maximum of each variable. Returns are defined as the gross return on an investment; thus a value of 1 indicates no change in value. Values greater
than one are net gains and those less than one are net losses. For each density of network in each market, we report the average daily return as well
as the total daily volume at the trader level. Thus, we report the mean return across individual level traders, where for each trader, we have
calculated their own average return over the course of the trading day. Note that these trader-level returns are unweighted by volume. Because the
futures markets are zero-sum, volume weighted returns are zero by construction. Volumes statistics are average daily volumes at the level of the
trader. Standard deviations are measured as the variance over the returns at the trader level, again unweighted. Minimums and maximums are the
smallest and largest for a trader on any day.
Mean
Standard Deviation
Min
Max
Sparse Networks
Average Returns
Volume
0.98
5.94
0.01
4.98
0.97
1.00
1.05
1215
Moderately Dense Networks
Average Returns
Volume
0.96
5.73
0.02
7.90
0.96
1.00
1.09
1,518
Dense Networks
Average Returns
Volume
0.92
5.32
0.02
12.68
0.96
1.00
1.106
2,060
Panel A
S&P 500 e-mini futures
Total number of # trading accounts
31,585
Panel B
DOW futures
Sparse Networks
Average Returns
Volume
0.99
6.39
0.03
1.42
0.99
1.00
1.02
150
Moderately Dense Networks
Average Returns
Volume
0.98
6.33
0.05
2.60
0.98
1.00
1.03
190
Dense Networks
Average Returns
Volume
0.95
5.91
0.07
4.86
0.98
1.00
1.04
341
Total number of # trading accounts
7,335
41
Table 2: Traditional estimation without network effects
Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. Each of the two panels shows three sets of results from the estimation of a
least squares specification. The dependent variable in each case is the returns of an individual trader during a window of trading. The columns distinguish between different windows of time;
250, 500 and 1000 trades for the market as a whole. For each market and each network density, we report the OLS estimation results over 21 days. T-statistics are reported below each
coefficient estimate. Below, we report the adjusted R squared value from each specification. We include control variables for sell and buy volume during trading window, variance of returns
for trader prior to trading period, volume in prior trading period, returns in the first half of the day, and an indicator for the final 2 periods of each day. We denote significance of coefficients
at the 10, 5 and 1% levels with ***, **, and *, respectively.
Panel A
250 Trades per Time Period
500 Trades per Time Period
full day
2nd half
full day
2nd half
1000 Trades per Time Period
full day
2nd half
8.63e-08***
(2.41e-08)
6.47e-08**
(3.14e-08)
2.21e-09***
(2.75e-10)
-3.99e-08*
(2.05e-08)
8.61e-08***
(2.41e-08)
6.46e-08**
(3.14e-08)
2.20e-09***
(2.74e-10)
-3.99e-08*
(2.05e-08)
0.495
(0.466)
-3.12e-09
(1.83e-08)
1.85e-08
(1.97e-08)
6.45e-09***
(1.92e-10)
-2.84e-08*
(1.48e-08)
-3.44e-09
(1.84e-08)
1.76e-08
(1.98e-08)
6.47e-09***
(1.92e-10)
-2.88e-08*
(1.48e-08)
0.174***
(0.0552)
2.03e-08
(2.87e-08)
4.01e-08
(2.71e-08)
7.10e-09***
(3.43e-10)
-3.95e-09
(2.21e-08)
2.02e-08
(2.86e-08)
4.04e-08
(2.72e-08)
7.07e-09***
(3.41e-10)
-3.76e-09
(2.21e-08)
-0.0453
(0.0318)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
0.001
0.001
0.000
0.001
0.001
0.001
S&P 500 e-mini futures
Sell Volume
Buy Volume
Variance of returns
Trailing total volume
Half-day returns
Constant
Late Day Control
R-Squared
Panel B
250 Trades per Time Period
500 Trades per Time Period
full day
2nd half
full day
2nd half
1000 Trades per Time Period
full day
2nd half
1.39e-06*
(7.58e-07)
-5.54e-07
(8.13e-07)
3.24e-07**
(1.32e-07)
-6.18e-08
(5.79e-07)
1.57e-06**
(7.79e-07)
-4.31e-07
(8.33e-07)
3.22e-07**
(1.32e-07)
-6.15e-08
(5.78e-07)
0.540
(0.644)
1.61e-06*
(9.53e-07)
-7.74e-07
(9.60e-07)
1.31e-06***
(1.25e-07)
-1.56e-07
(7.75e-07)
1.63e-06*
(9.68e-07)
-7.58e-07
(9.69e-07)
1.31e-06***
(1.25e-07)
-1.58e-07
(7.75e-07)
0.0713
(0.490)
1.30e-06
(9.92e-07)
-1.52e-06
(9.93e-07)
8.00e-07***
(7.73e-08)
-8.58e-07
(7.16e-07)
1.52e-06
(1.03e-06)
-1.40e-06
(1.01e-06)
7.99e-07***
(7.72e-08)
-8.41e-07
(7.13e-07)
0.197
(0.225)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
0.001
0.001
0.003
0.003
0.003
0.003
DOW futures
Sell Volume
Buy Volume
Variance of returns
Trailing total volume
Half-day returns
Constant
Late Day Control
R-Squared
42
Table 3: Traditional versus network estimation results
Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The columns distinguish between different levels of network structure complexity. The table reports the adjusted R2 value from table 2's OLS
specification as well as the adjusted R2 from the estimation of model (1). We highlight the adjusted R2 for the network estimation method in bold. For each type network density and each market, we report the range of adjusted R2 results across 21 trading days. We
include the same controls as in Table 1 as well as a constant.
Panel A
Sparse Networks
low
high
low
0.001
Moderately Dense Networks
high
low
0.05
0.09
0.000
0.02***
38.01
0.04***
38.02
Yes
Yes
Yes
Yes
high
low
0.001
Dense Networks
high
low
high
low
0.09
0.17
0.001
0.05***
37.13
0.08***
42.66
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
high
low
0.003
high
0.001
0.19
0.37
0.10***
38.18
0.17***
44.59
Yes
Yes
Yes
Yes
high
low
high
0.003
0.14
0.18
0.08***
9.07
0.10***
50.68
Yes
Yes
Yes
Yes
S&P 500 e-mini futures
OLS
Variance explained by specification
(adjusted R-squared)
0.001
w/ networks
Network Effect Coefficient (θ)
(t-statistic)
Constant
Control Variables
Yes
Yes
Yes
Yes
OLS
Yes
Yes
w/ networks
Yes
Yes
OLS
w/ networks
Panel B
Sparse Networks
low
high
low
0.001
Moderately Dense Networks
high
low
0.04
0.08
0.003
0.02***
7.77
0.04***
49.47
Yes
Yes
Yes
Yes
Dense Networks
high
low
0.003
0.07
0.09
0.04***
8.19
0.05***
44.81
Yes
Yes
Yes
Yes
DOW futures
OLS
Variance explained by specification
(adjusted R-squared)
0.001
w/ networks
Network Effect Coefficient (θ)
(t-statistic)
Constant
Control Variables
Yes
Yes
Yes
Yes
OLS
Yes
Yes
w/ networks
Yes
Yes
43
OLS
Yes
Yes
w/ networks
Yes
Yes
Table 4: Estimation with Network Effects
Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The columns distinguish between different levels of network structure
complexity. Each of the two panels shows three sets of results from the maximum likelihood estimation of model (4). For each type network density and each market, we report the range of
estimation results across 21 trading days. The first row shows the estimates of the parameter θ, the network effect coefficient, from the above specification. T-statistics are reported below
coefficient estimates. We denote significance of coefficients at the 10, 5 and 1% levels with ***, **, and *, respectively.
Panel A
Sparse Networks
Moderately Dense Networks
Dense Networks
250 Trades per Time Period
500 Trades per Time Period
1000 Trades per Time Period
low
high
low
high
low
high
Network Effect Coefficient (θ)
0.02
0.90
0.05***
38.24
0.05***
4.55
0.11***
46.51
0.10***
6.94
0.27***
47.60
Constant
Control Variables
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
R-Squared
0.04
0.10
0.09
0.21
0.18
0.46
S&P 500 e-mini futures
Panel B
Sparse Networks
250 Trades per Time Period
Moderately Dense Networks
500 Trades per Time Period
Dense Networks
1000 Trades per Time Period
low
high
low
high
low
high
0.02***
7.75
0.02***
50.09
0.04***
8.05
0.05***
11.38
0.08***
9.06
0.10***
50.00
Constant
Control Variables
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
R-Squared
0.04
0.04
0.07
0.08
0.14
0.17
DOW futures
Network Effect Coefficient (θ)
44
Table 5: Estimation with Weighted Network Effects
Note: This table extends the network model (4) to include weighted network to reflect the relative importance of traders in the system. Panel A shows results from the S&P 500 futures market.
Panel B shows results from the Dow futures market. Each of the two panels shows three sets of results from the estimation of model (1). The columns distinguish between different levels of
network structure complexity. This table uses a weighted matrix W defined as the element-by-element product of the adjancency matrix of realized trades and the sum of trading volume. For each
type network density and each market, we report the range of maximum likelihood estimation results across 21 trading days. The first row shows the estimates of the parameter θ, the network
effect coefficient, from the above specification. T-statistics are reported below coefficient estimates. Below, we include the adjusted R squared value from each specification and the average
systemic risk multiplier. This multiplier is total network impact of a one unit shock to an individual. Averaging across the impact for all individuals in the network produces this number, which is
equal to φ=1/(1-θ). We denote significance of coefficients at the 10, 5 and 1% levels with ***, **, and *, respectively.
Panel A
Sparse Networks
Moderately Dense Networks
Dense Networks
250 Trades per Time Period
low
high
500 Trades per Time Period
low
high
1000 Trades per Time Period
low
high
0.94***
1488.42
0.96***
2594.92
0.96***
619.46
0.98***
619.46
0.97***
516.89
0.98***
669.17
Constant
Control Variables
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
R-Squared
0.74
0.77
0.73
0.77
0.73
0.77
Average Multiplier (φ)
16.12
26.99
25.61
45.45
37.01
66.46
S&P 500 e-mini futures
Network Effect Coefficient (θ)
t - statistic
Panel B
Sparse Networks
250 Trades per Time Period
Moderately Dense Networks
500 Trades per Time Period
Dense Networks
1000 Trades per Time Period
low
high
low
high
low
high
0.82***
355.26
0.89***
475.98
0.85***
316.51
0.92***
415.36
0.90***
233.36
0.95***
309.37
Constant
Control Variables
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
R-Squared
0.71
0.80
0.71
0.79
0.71
0.78
Average Multiplier (φ)
5.52
9.43
6.71
13.15
9.80
21.26
DOW futures
Network Effect Coefficient (θ)
t - statistic
45
Table 6: Estimation with Weighted Network Effects and Network Fixed Effects
Note: This table extends the weighted network model to include network fixed effects. to include weighted network to reflect the relative importance of traders in the system. Panel A shows
results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The difference between this table and table 5 is the construction of returns. The primary results in
Table 2 used the individual level gross returns. Here we use the deviation in returns from the average return at the network level in each time period. The adjacency matrix of realized trades is a
symmetric, non-directed matrix of 1's and 0's with 1's indicating the presence of a trade and 0 the absence. For each type of network density and each market, we report the range of maximum
likelihood estimation results across 21 trading days. The first row shows the estimates of the parameter θ, the network effect coefficient, from the above specification. T-statistics are reported
below coefficient estimates. Below, we include the adjusted R squared value from each specification and the average multiplier. This multiplier is total network impact of a one unit shock to an
individual. Averaging across the impact for all individuals in the network produces this number, which is equal to 1/(1-θ). We denote significance of coefficients at the 10, 5 and 1% levels with
***, **, and *, respectively.
Panel A
Sparse Networks
Moderately Dense Networks
Dense Networks
250 Trades per Time Period
500 Trades per Time Period
1000 Trades per Time Period
low
high
low
high
low
high
0.94***
964.33
0.96***
2917.94
0.95***
544.24
0.98***
718.19
0.97***
438.62
0.99***
666.50
Constant
Control Variables
Fixed Effects
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
R-Squared
0.74
0.77
0.73
0.77
0.73
0.77
Average Multiplier (φ)
15.61
28.57
19.61
41.62
29.41
71.36
S&P 500 e-mini futures
Network Effect Coefficient (θ)
t - statistic
Panel B
Sparse Networks
250 Trades per Time Period
low
high
Moderately Dense Networks
500 Trades per Time Period
low
high
Dense Networks
1000 Trades per Time Period
low
high
0.82***
385.74
0.88***
448.79
0.84***
307.21
0.92***
411.65
0.90***
237.46
0.95***
310.21
Constant
Control Variables
Fixed Effects
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
R-Squared
0.71
0.78
0.70
0.79
0.71
0.78
Average Multiplier (φ)
5.56
8.33
6.25
12.98
10.20
19.59
DOW futures
Network Effect Coefficient (θ)
t - statistic
46
Table 7: Network topology and profitability
Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The columns distinguish between different levels of network structure
complexity. The exercise in this table is to 1) report individual level variation in centrality and 2) evaluate the difference in returns for traders with different centrality. For each type of network
density and each market, we report the range of results across 21 trading days. Individual level Bonacich centralities are calculated using the formula: b(w,θ)=[I-θW]⁻¹⋅1, where the 1 signifies a
vector of 1's. We report the standard deviation of centrality as well as the change in returns for a trader that changes his centrality by one unit.
Panel A
Sparse Networks
Moderately Dense Networks
Dense Networks
low
high
low
high
low
high
0.06
3.41
0.70
4.30
0.07
3.42
0.78
4.32
0.10
3.40
0.76
4.33
S&P 500 e-mini futures
Impact of one unit change in Bonacich Centrality
Stdeviation - Weighted Bonacich Centrality
Panel B
Sparse Networks
Moderately Dense Networks
Dense Networks
low
high
low
high
low
high
0.39
3.68
0.60
4.11
0.37
3.68
0.57
4.11
0.38
3.68
0.57
4.11
DOW futures
Impact of one unit change in Bonacich Centrality
Stdeviation - Weighted Bonacich Centrality
47