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Credit Default Swaps Drawup Networks: TooTied To Be Stable?

Rahul Kaushik, Stefano Battiston

ETH Risk Center – Working Paper Series

ETH-RC-12-013

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ETH-RC-12-013

Credit Default Swaps Drawup Networks: Too Tied To Be Stable?

Rahul Kaushik, Stefano Battiston

Abstract

Abstract

We analyse time series of CDS spreads for a set of major US and European institutions on aperiod overlapping the recent financial crisis. We extend the existing methodology of ε-drawdowns tothe one of joint ε-drawups, in order to estimate the conditional probabilities of abrupt co-movementsamong spreads. We correct for randomness and for finite size effects and we find statistically significantprobabilities of joint drawups for many pairs of CDS. We also find significant probabilities of trendreinforcement, i.e. drawups in a given CDS followed by drawups in the same CDS. Finally, we take thematrix of probability of joint drawups as an estimate of the network of financial dependencies amonginstitutions. We then carry out a network analysis that provides insights into the role of systemicallyimportant financial institutions.

Keywords: Credit Default Swaps, epsilon drawups, interdependence, trend-reinforcement, networkanalysis, bow-tie structure, time series analysis

Classifications:

URL: http://web.sg.ethz.ch/ethz risk center wps/ETH-RC-12-013

Notes and Comments:

ETH Risk Center – Working Paper Series

http://www.sg.ethz.ch

Rahul Kaushik and Stefano Battiston:Kaushik and Battiston, Credit Default Swaps Drawup Networks: Too Tied To Be Sta-ble? 2012

Credit Default Swaps Drawup Networks: Too Tied To Be Stable?

Rahul Kaushik and Stefano BattistonChair of Systems Design, ETH Zurich, [email protected], [email protected]

Abstract

We analyse time series of CDS spreads for a set of major US and European institutions on a pe-riod overlapping the recent financial crisis. We extend the existing methodology of ε-drawdowns tothe one of joint ε-drawups, in order to estimate the conditional probabilities of abrupt co-movementsamong spreads. We correct for randomness and for finite size effects and we find statistically signif-icant probabilities of joint drawups for many pairs of CDS. We also find significant probabilities oftrend reinforcement, i.e. drawups in a given CDS followed by drawups in the same CDS. Finally, wetake the matrix of probability of joint drawups as an estimate of the network of financial dependen-cies among institutions. We then carry out a network analysis that provides insights into the role ofsystemically important financial institutions.

1 Introduction

Within the field of complex networks (Caldarelli, 2007), the investigation of financial networks is cur-rently one of the emerging avenues (Schweitzer et al., 2009), also in view of the on-going global finan-cial crisis. Financial contagion on networks (Battiston et al., 2012b) differs in some important respectsfrom the well-known processes of epidemics spreading (Barrat et al., 2008; Pastor-Satorras and Vespig-nani, 2001). It also bears similarities to epidemics spreading, among which, the fact that the topologicalstructure of the network plays a crucial role in the collective dynamics and therefore in emergence ofsystemic risk. There is a body of work on networks reconstructed from correlations among equity pricesor return time series (Bonanno et al., 2003; Garas et al., 2008; Kullmann et al., 2002). The analysis ofthe minimum spanning tree provides insights into the classification of stocks and the level of correlationdepending on the market phase. Correlation analysis suffers, however, from some important limitations,the main one being that zero correlation between two series does not imply that they are independent(only the inverse is true). To overcome these limitations, here we utilise a method based on the detectionof joint ε-drawup, which allows us to estimate the probability that two series exhibit a co-movement.Moreover, in contrast to equity, CDS prices reflect the default probability of the reference entity and thusthe network constructed from CDS prices are more relevant in studying the propagation of default risk.

Our approach can be applied to construct networks of dependencies in other financial markets. In general,it applies to all domains of networks in which links are, for any reason, unobservable but the dynamics ofthe nodes reflect the dependency structure. To summarise, the contributions of the paper are the following.First, we build on the ε-drawup method (Sornette and Zhou, 2006) to estimate the probability of joint

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ε-drawups, which are essentially a particular type of co-movements across time series. Based on this,we estimate the level of the so-called interdependence and trend reinforcement in the system acrossdifferent phases of the market. In addition, we find them to be relevant indicators for the emergence ofsystemic risk, as was also suggested by previous theoretical work. Second, we construct a network ofinterdependencies among institutions and we introduce two novel centrality measures that allow for theidentification of systemically important nodes in the network. Our approach enables the disentanglementof a structure that is, only apparently, very homogenous. It also allows us to track the role of nodesevolving in time.

2 Data

Credit Default Swaps (CDS’s) are financial derivatives instruments in which the seller provides the buyerprotection against a credit event of a reference entity (see appendix C). Our aim is to analyse the timeseries data of CDS prices, or spreads, of top US and European financial institutions in the last years.The data, acquired via a subscription to Bloomberg, consists of CDS spreads of single name entitiesdenominated in US dollars and in the Euro, encompassing a total of 176 top firms in the financial sector,in the period from 2nd January 2002 until 1st December 2011. As shown in Fig. 1, the time series displaythree distinct phases. Accordingly, we divide the data into three parts: (1) January 2002 - May 2006(representative of a normal phase); (2) May 2006 - March 2009 (volatile with an upwards trend); (3)March 2009 - December 2011 (volatile with a downwards trend market scenario). The motivation to doa period-wise analysis is to extract the network structure before, during and after the crisis of 2008. Thisdata window covers a 2560 weekdays.

3 Methods

ε-drawups. For each institution’s time series we detect what we call ε-drawup’s. An ε-drawup is anextension of the notion of an ε-drawdown (Johansen, 2003). It refers to a persistent upward movement ina time series until a peak has been reached, after which the time series declines (or, has a “correction”)by more than an amplitude ε (see Fig. 2a). Since the CDS spread represents the cost of insurance, anε-drawup signifies an increase in the default probability of that institution, as perceived by the market.

We compute the ε-drawup’s in each of the time series using the following algorithm, which we describeusing the example illustrated in Fig. 2a. Suppose, we start our analysis of an ε-drawup from the firstgreen point from the left in Fig. 2a. The step are as follows: (1) We compute the local variation in thetime series for the last ten days, call it ε. (2) compute local extrema. (3) Goto first local minima, call itε-drawupcandidate (on day three, Fig. 2a). (4) Iterate to the set of local maxima and minima (occurring ondays 5 and 6 respectively, see Fig. 2a). Compute the difference between the maxima and the minima, call

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Figure 1: Time series of credit default swaps throughout the credit crisis. A plot of the CDS spreadtime series covering the financial crisis of 2008. The data ranges from January 2002 to December 2011.We can observe three market phases. Most CDS spreads peak around March 2009. The CDS prices arequoted in basis points (bp). The purpose of this plot is to highlight the market regimes, rather than theindividual CDS spread evolution. Accordingly, the CDS spreads of all the financial entities are plottedhere.

it correction amplitude (correction amplitude refers to the decline in price followed after an increase inprice). (5) Update ε by computing the local variation of the last ten days. (6) If correction amplitude ≥ ε.Then, we record the ε-drawup on the day it occurs (day 7 in Fig. 2a). And, we update ε-drawupcandidate.Otherwise, we iterate to the next minima and goto the succeeding maxima and repeat steps above.

The choice of using 10 days to compute local variations was the result of following preliminary analysis.We have computed drawups for 50 time series using various numbers of days ranging from 100 until102. At one extreme we take all the local maxima in the time series as drawups, which is not desirable(see Fig. 2a,b). At the other extreme the algorithm ignores too many drawups. ε-drawups in general canbe validated by eye only and thus we could not run an optimisation function that would maximise thenumber of “true" drawups, as a function of the number of days chosen to compute local variations. Aftera thorough visual inspection of 50 time series at various scales, we picked 10 days as the best choice of

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Figure 2: a) Illustration of the ε-drawup methodology. The * represents local extrema that were notdetected as drawups. The red-dots represent local maxima that were picked up as candidates for a ε-drawup. The green-dots represent the local minima that were picked up for a ε-drawup. Compare thedifference between the maxima and minima on days 5 and 6 respectively with ε. Since, ε > than thedifference, we iterate to the next set of local maxima and minima on days 7 and 8, keeping day 3 as theday from when we count a ε-drawup b) The plot highlights the ε-drawup methodology applied to thetime series of American International Group (AIG) and Merrill Lynch (MER)

the time window. Note that on weekends and holidays, the last traded price is carried forward; however,we have verified that this does not affect the ε - drawup algorithm.

With the above procedure, we are able to detect the ε-drawup’s in the the time series data. Once we havedetected the ε-drawup’s in i−th time series, we construct the vector vi for node i, whose length is thesame as the length of the time series, T . Also, vi(t) = 1 if there was a drawup on day t in node i, and zerootherwise.

Co-movements. When market participants buy and sell insurance on each other, their financial perfor-mances can become interdependent (see appendix C). Therefore, we are interested in detecting jointupward movements in pairs of time series. In order to detect co-movements we implement the followingalgorithm (notice, that the resulting matrices of co-movements Pτi j are square but not symmetric):

(1) Select a given node i. (2) Loop from day t = 1 till t = T and compare each vi(t) with all v j(t + τ)where j ∈ {1, ...,N} and τ ∈ {0, 1, 2, 3}. (3) If vi(t) = 1 and v j(t + τ) = 1, then countτ = countτ + 1. (4)Update a matrix of counts of joint drawup’s, i.e. Pτi j = countτ

T . (5) Repeat the steps (1)-(4) for all node i.

Correction for Randomness and Finite Size. Statistical Significance. In order to account for theco-movements that could arise by chance, for each pair (i, j) we subtract the expected number of co-movements in the case of independent events and we obtain Wτ

i j = Pτi j − Pτi Pτj . In order to correct for

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finite size effects, we carry out a permutation test. For each pair (i, j) we generate 100 permutations ofthe respective time series of ε-drawups. We compute the corresponding 100 values of Wτ,control

i j and the

value W̃i jτ,control that corresponds to a 95% confidence level. This means that if the empirical value Wτ

i j

exceeds W̃i jτ,control, it has less than 5% chance to come from the same distribution. Accordingly, we keep

the empirical value of Wτi j only if it passes this test and otherwise we set it to 0. In the following, when

we say that only statistically significant links are retained, we refer to the above filtering procedure.

Interpreting the Conditional Probability Matrix W. We now average across values of τ the filteredmatrices, i.e. Wi j = 1

4∑τ=3τ=0 Wτ

i j. Notice that the quantity represented by Wi j is not a measure of causality.However, under the assumption that the observed joint ε-drawup frequencies are an approximation ofprobabilities, each entry Wi j of the matrix W has a precise meaning: It is an estimate of the probabilityof an ε-drawup in the time series of j at a given day, conditional to an ε-drawup in time series of node iin the preceding 3 days and in the same day, averaged over the days of the time delay.

Impacting and Impacted Centrality In line with the notion of feedback centrality (e.g., PageRank, seeAppendix F), we introduce the Impacting Centrality ci,

ci =∑

j

W̃i jc(0)j + β

∑j

W̃i jc j (1)

and the Impacted Centrality bi,bi =∑

j

W̃′

i jb(0)j + β

∑j

W̃′

i jb j (2)

In the definition above: c(0)j and b(0)

j are the intrinsic centrality, which for the sake of simplicity are setto 1; the matrices are normalized so to be row-stochastic, W̃i j = Wi j/

∑l Wl j, W̃

′

i j = W′

i j/∑

l W′

l j, withW′

denoting the transpose of W; the parameter β is a dampening factor which we set to 0.85 in line withthe PageRank heuristic (Page et al., 1998). Eqn. 1 and 2 can be analytically solved to yield the solution,c = (I − W̃′)−1W̃′v and b = (I − W̃)−1W̃b. We can interpret the Impacting Centrality ci as the extentto which a firm i impacts the network via direct connections and, recursively, via indirect connections.Analogously, we can interpret the Impacted Centrality bi as the extent to which a firm i gets impacted bythe network via direct connections and, recursively, via indirect connections.

In terms of physical analogy, in the case of PageRank, it is known that the score of a node is proportionalto the expected number of visits of a random walker that is let free to navigate in the network hoppingrandomly from a node to the successor nodes. Notice that because of possible cycles in the network awalker can visit a node many times and thus the expected number of visits by random walkers can in gen-eral exceed the number of walkers. We can map the visit of the random walker into the occurrence of anε-drawup. The Impacting Centrality of a node i is then proportional to the expected number of ε-drawup’soccurring across all nodes in the network, conditional to an initial ε-drawup at node i. Conversely, theImpacted Centrality of a node i is proportional to the expected number of ε-drawup’s occurring at nodei, conditional to an initial ε-drawup at some node j , i in the network.

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Bow-tie structure A bow-tie network is a directed network consisting of four main parts, as follows. TheStrongly Connected Component (SCC): set of nodes such that each can reach any other via a directedpath; OUT: set of all nodes that can be reached, directly or indirectly, from the SCC; IN: the set of allnodes that reach the SCC directly or indirectly. The fourth and last component of the bow-tie structure,Tubes and Tendrils (TT) represent the set of all nodes that are not a part of the SCC; however, a node inthe TT can either be reached from the IN and/or OUT.

Link Pruning and Bow-tie Extraction. If a network is dense and strongly connected it is difficult tounderstand who impacts whom. We then proceed to the following link pruning. We compute the ratiobetween the Impacting and the Impacted Centrality, ri =

bici

, see Eqn. 2 & 1. If ri > 3/2 then we remove allthe incoming links of the node i. This means that all nodes that exhibit ri > 3/2 will only have outgoinglinks after the pruning. Similarly, nodes that exhibit ri < 2/3 get all their outgoing links removed. Theremaining nodes, i.e. such that 2/3 < ri < 3/2 retain both the incoming and outgoing links. With fewexceptions, this link pruning procedure extracts out of a dense strongly connected network a subnetworkwith a bow-tie structure. This is useful to highlight the role of a node. Those nodes mainly impacting theothers end up in the IN component, after the pruning. Those nodes mainly impacted by the others end upin the OUT. Those nodes being equivalently impacting and impacted end up in the SCC.

4 Interdependence and trend reinforcement

Whilst interdependence can be seen as a form of risk diversification which decreases individual risk,previous work (Battiston et al., 2012a) has demonstrated that high interdependence leads, instead, tohigher systemic risk when coupled to a so-called trend reinforcement. We thus proceed to investigatingthe presence of interdependence and trend-reinforcement in the CDS markets. In our context, trend re-inforcement refers to the tendency of an ε-drawup to be followed by another ε-drawup in the same timeseries. Interdependence refers, in contrast, to the existence of co-movements between two different timeseries (i.e. an ε-drawup followed by another one in a different time series). Here, we take the frequencyof ε-drawup’s in i as an estimate of the probability Pi that security i has a ε-drawup. Similarly, for thefrequency of joint ε-drawup’s we estimate Pi j, i.e. probability that j experiences a ε-drawup given thati experiences a ε-drawup. The expected probability of joint drawup’s in the case of two statistically in-dependent time series is Pi j = PiP j. Therefore, we take as an estimate of interdependence between twofinancial institutions, the deviation from such a case, i.e. Wi j = Pi j−PiP j. Finally, in order to account forfinite size effects, we consider only those values of Wi j that cannot be rejected based on a permutationtest and we reset to zero all the other values (see Section 3). We also account for a time lag τ = 0, 1, 2, 3days between the drawup’s and we take the average of Wi j across τ values. Analogously, we take as anestimate of the trend reinforcement for institution i the deviation: Wii = Pii−PiPi and we treat it as above.Notice that, because of the time lag τ, Wi j is not a symmetric matrix. Notice also that a positive value of

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Wi j does not imply a causality relation between the movements of i and j, but measures the dependenceof j due from i in terms of conditional probability.

The distribution of Wi j’s and Wii’s are shown in Fig. 3a, b. The histograms count only the non-zero valuesof Wi j’s and Wii’s, i.e., those that are found to be statistically significant (i.e. which pass a permutation testat 95% confidence interval, see Section 3). We find statistically significant levels of trend reinforcement,in about 50%, 72% and 80% of the nodes (respectively in period 1, 2 and 3). In Fig. 3b), the curvefor period 2 and 3 is mostly above the one for period 1. This means that the number of nodes with asignificant level of trend reinforcement increases when the market moves from the first phase to morevolatile phases. We also find statistically significant levels of interdependence in 54%, 78% and 77% ofpairs of nodes in period 1, 2, 3, respectively. The histograms of Wi j (see Fig. 3a) show that periods 2 and3 are characterised by higher frequencies. In fact, 20% of pairs in period 2 and 19% pairs of nodes inperiod 3 exhibit values of Wi j greater than the mean plus one standard deviation of period 1. Moreover,while in period 1, nearly all values of Wi j are smaller than 0.5, in period 2 and 3 there is a tail extendingup to 1.

These findings show that interdependence and trend reinforcement are indeed present in an importantmarket such as the one for CDS’s. Moreover, trend reinforcement increases from period 1 to period 2,and even more so does interdependence. Before drawing the implications of these findings in terms ofsystemic risk, we proceed to a network analysis of the structure of interdependencies among institutions.


Rahul Kaushik and Stefano Battiston:Kaushik and Battiston, CDS Networks

when the market moves from the first phase to more volatile phases. We also find a significant level ofinterdependence in 54%, 78% and 77% of pairs of nodes in period 1, 2, 3, respectively. The histogramsof Wi j (see Fig. 2b) show that periods 2 and 3 are characterised by higher frequencies. In fact, 20% ofpairs in period 2 and 19% pairs of nodes in period 3 exhibit values of Wi j greater than the mean plus onestandard deviation of period 1. Moreover, while in period 1, nearly all values of Wi j are smaller than 0.5,in period 2 and 3 there is a tail extending up to 1.

These findings show that interdependence and trend reinforcement are indeed present in an importantmarket such as the one for CDS’s. Moreover, trend reinforcement increases from period 1 to period 2,and even more so does interdependence. Before drawing the implications of these findings in terms ofsystemic risk, we proceed to a network analysis of the structure of interdependencies among institutions.

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Figure 2: a)The distribution of non-zero values of interdependence Wi j across the three periods. Thecounts in periods 2 and 3 are higher than in period 1. In addition, during periods 2 and 3 the distribu-tions of Wi j have longer tails compared to period 1. b) The distribution of non-zero values of trendreinforcement Wii across the three periods.

Network analysis and systemic risk. The CDS market can be naturally mapped into a directed andweighted network in which nodes represent institutions and edges represent interdependencies amonginstitutions. More precisely, whenever Wi j > 0, we assign a weighted edge with value Wi j from institutioni to j. Since Wi j is the probability that conditional to i experiencing a draw-up, j also experiences a draw-up with a time lag τ, it follows that the stronger the edge, the stronger the impact that i has on j. Whenlooking at the properties of connectedness of the network, we find a significant number of disconnectednodes in all three periods (81, 39, 39, respectively). Remarkably, the rest of the nodes form only onestrongly-connected component (LSCC, see Methods) encompassing, respectively, 95, 137, 137 nodes inperiod 1, 2 and 3. The density of links (i.e. the number of links over the number of possible links) in the

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Figure 3: a)The distribution of non-zero values of interdependence Wi j across the three periods. Thecounts in periods 2 and 3 are higher than in period 1. In addition, during periods 2 and 3 the distribu-tions of Wi j have longer tails compared to period 1. b) The distribution of non-zero values of trendreinforcement Wii across the three periods.

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5 Network analysis and systemic risk

The CDS market can be naturally mapped into a directed and weighted network in which nodes representinstitutions and edges represent interdependencies among institutions. More precisely, whenever Wi j > 0(recall that we have retained only the values that are statistically significant, see Section 3), we assigna weighted edge with value Wi j from institution i to j. Since Wi j is the probability that conditional toi experiencing a draw-up, j also experiences a draw-up with a time lag 0 ≥ τ ≤ 3, it follows that thestronger the edge, the stronger the impact that i has on j. When looking at the properties of connectednessof the network, we find a significant number of disconnected nodes in all three periods (81, 39, 39,respectively). Remarkably, the rest of the nodes form only one strongly-connected component (LSCC,see Methods) encompassing, respectively, 95, 137, 137 nodes in period 1, 2 and 3. The density of links(i.e. the number of links over the number of possible links) in the LSCC is high in all the three periods:0.98,0.97, 0.97. This is reflected also in the average out degree in the LSCC’s across the three periods,which is 90±8, 129±11, and 123±19. Finally, the average path length within the LSCC’s is 1.04, 1.05 and1.2, meaning that almost all the nodes in the LSCC are first neighbours to each other. In such a structure,each node has a direct impact on all the other nodes, and each of these has a further impact on all theothers. Intuitively, this finding suggests that the financial distress at one node in the SCC can quicklypropagate to all the other nodes in the LSCC and keeps reverberating through the many connections.

Indeed, recent works on systemic risk in financial networks have shown that the number of links playan ambiguous role. Few links are functional to diversify the individual risk. However, too many linksgenerate systemic risk. This holds in presence of mechanisms that either amplify the distress (such asin the case of contagion), or simply increase the persistency of the distress in time, such as trend rein-forcement (Battiston et al., 2012a). As we have seen, the CDS market exhibits a core of more than 100nodes, that is almost a fully connected graph (i.e. with maximal degree) and where in many cases linksrepresent strong interdependencies. According to the theoretical results mentioned earlier, in such a sit-uation, even small levels of amplification can make the whole system very unstable. Notice that movingfrom period 1 to period 2, the values of most CDS’s raised dramatically, in many cases by one order ofmagnitude (see Fig. 1). Looking at individual institutions it is clear that their risk of default had becomevery high. However, we see that unlike in period 2 financial institutions with the largest debt’s were inthe SCC in period 1, see Fig. 4. The top right corner of Fig. 4 represents firms that are just as vulnerableas they impact the others in the network. By superimposing the average debt levels of firms upon ri, wefind that the majority of institutions that are just as vulnerable as they impact the others in the networkfrom period 1 also have the highest debt levels. A perturbation in their debt levels would spread acrossa very large subset of the network. Since, the size of the debt of such institutions is the largest, a smallpercentage change in their debt would cause a large change the distress of the debt issued by others firmsin the network. We find that in contrast to the evolution of CDS spreads in period 1, the market dynamicsimplied ripe conditions for greater levels of systemic risk.

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In fact previous findings suggest that, in such a situation, the default of a few institutions would havetriggered a systemic default. Indeed, it is generally thought that without the massive intervention of theFederal Reserve (FED) through various emergency programs that lasted from the fall of 2008 until thesummer of 2009, (Bloomberg-News, 2011) there would have been a melt-down of the whole financialsystem.

Therefore, monitoring trend reinforcement and interdependence together with the individual riskiness ofthe participants seems to provide a valuable assessment of the level of systemic risk in a market.

6 Centrality

In order to gain insights into the systemic importance of specific nodes in the network, we proceed toinvestigate their centrality. We obviously focus only on the strongly connected component since the othernodes are isolated. The out- and in-degree of a node are the simplest measures of centrality that hold avaluable interpretation here: A high out-degree represents the ability of a node to affect many neighbourswhen it experiences a draw-up; a high in-degree corresponds to a node being affected by many nodes.Since the network is almost a complete graph, based on the out-degree, all nodes are equally systemicallyimportant and equally affected by the others. Interestingly, even the out- and in-degree strength of a node(i.e. the sum of the weights in the outgoing and incoming links), yields similar conclusions.

As an alternative approach, based on the notion of feedback centrality, for each node i we introduce anovel measure, called impacting centrality and denoted as bi, see Eqn. 2. The measure takes into account,in a recursive way, the fact that a node is more systemically important if it impacts many systemicallyimportant nodes (see Section 3). Symmetrically, we also introduce the impacted centrality of a node i,denoted as ci, see Eqn. 1. This measure captures, instead, the idea that a node is more heavily impactedif it has strong dependencies from many nodes which are in turn heavily impacted. In both cases, thevalues are normalised between 0 and 1. In analogy to the random walker for PageRank, both measureshold a physical interpretation in terms of expected numbers of ε-drawup’s (see Section 3).

Remarkably, in stark contrast to in- and out-degree, the values of these two centrality measures arebroadly spread across the range [0, 1] (see Fig. 11a, 12a, & 13a). If we focus on the ratio betweenimpacted and impacting centrality, ri =

bici

, in the scatter plot of Fig. 4, it is possible to identify threeregions (above, between and below the dotted lines), corresponding to three different roles of the nodes.Nodes are located in the top region if they have a value ri > 3/2, meaning that they impact the network1.5 times more than they are impacted by it. Symmetrically, nodes are in the bottom region if ri < 2/3.Finally, nodes that appear in the middle region are those that impact and are impacted by the network ina comparable manner. According to this classification, while in period 1 most institutions are located inthe middle region, in period 2 and 3 they progressively move to the top and the bottom region. Noticethat the impacted centrality of a node i can be viewed as the vulnerability of node i to its neighbours. We

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observe that there are many nodes in the network that not only have a high impacting centrality, but alsoa high impacted centrality. From a systemic risk perspective it is essential to study nodes that are proneto distress; however, from a policymakers perspective it is also vital to monitor nodes that are not onlyprone to distress, but that also have a high impacting centrality as distress in such nodes could lead to asystemic collapse.

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Figure 4: Scatter plot of impacting versus impacted centrality. Each institution in the CDS marketis represented by three dots depending on the period (blue, green, red refers to period 1, 2, 3, respec-tively). The size of each node is determined by the average debt of a financial institution relative to themaximum average debt of a financial institution in a given period. It can be seen that, while in period 1most institutions are located between the two dotted lines, in period 2 and 3 many of them move to thetop and bottom region. This means that ratio between the two centrality measures varies with the mar-ket phase. Few institutions of interest are labelled. For example, Bank of America (BOFA) remains inthe same region across the three periods. With reference to the subsequent bow-tie construction used inFig. 5: The scatter plot is divided into 3 regions. Nodes in the region above the line ri > 3/2 correspondto the IN. Nodes in the region 2/3 < ri < 3/2 correspond to the SCC. Nodes in the region ri < 2/3correspond to the OUT.

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7 Link Pruning and Bow-tie Extraction

In order to emphasise the 3 different roles suggested by this finding, we carry out the following linkpruning procedure. In each period, for nodes located in the top region, we remove all their incominglinks. Symmetrically, for those in the bottom region, we remove all the outgoing links. Since the initialnetwork is strongly connected, in this way. By construction, we obtain a bow-tie structure (see Section3). The position of a node in the bow-tie is related to its systemic importance. Indeed, the IN, SCC andOUT component of the bow-tie correspond to the top, middle and bottom regions of Fig. 4, respectively(e.g. the nodes in the IN are those that impact the network more than they are impacted). Note that thebow-tie structure is constructed based on the choice of impacting-impacted centrality, i.e. nodes withri > 3/2 are in the IN, nodes with 2/3 < ri < 3/2 are in the SCC, and nodes with ri > 2/3 are in theOUT. In fact, for any δ > 0, where δ ∈ (0, 1). The lines 1−δ and 1+δ would separate the nodes into threeregions. Thus, the choice of δ is based on the level of impacting-impacted centrality that is of interest,see Fig. 15 for more visualisations. In addition, if W is a directed SCC, and one truncates all incominglinks of nodes with ri < 1 − δ, and all outgoing links for nodes with ri > 1 + δ. Then, it is not always thecase that the filtered W has a non-trivial SCC (see Fig. 14a,b in the appendix for more details).

We then introduce a novel method for the visualisation of the bow-tie (Fig. 5). This enables the repre-sentation, at the same time, of a network structure, the position of the nodes in the various componentof the bow-tie, as well as their level of impacting centrality. In Fig. 5, the circle represents the SCC, thetop (bottom) section correspond to the IN (OUT). E.g. within the SCC, more central nodes are locatedtowards the centre of the circle. The colour code and the size of the dots also reflect their centrality, suchthat the red and large dots are the most central (see caption of Fig. 5). This visualisation allows to trackhow individual institutions become more or less central, or if they changed role across periods (see Fig.11b, 12b, & 13b ). In period 1, most of the nodes of the bow-tie are in the SCC (85), with 4 and 6 inthe IN and OUT respectively. Moreover, most nodes in the centre of the SCC are banks and investmentbanks, while insurance and real estate companies tend to be in the periphery of the SCC (see SI, Fig. 6b). This implies that in the normal phase most of the nodes impact the network, and are impacted by thenetwork in a comparable manner. In period 2, the bow-tie grows overall, but the SCC (97 nodes) growsproportionally less than IN (19 nodes) and OUT (22 nodes). Of the 81 nodes that were disconnected inperiod 1, twenty seven migrated to the SCC (Fig. 5). In period 3, the size of the bow-tie remains un-changed, but the SCC shrinks by about a 50% (from 97 to 47 nodes), as a result of a migration to theIN (37 nodes) and mostly to the OUT (53 nodes) (see SI, Fig. 7 b). In particular, the nodes with highimpacting centrality are now all located in the IN and not, anymore, in the SCC.

One should not forget that the original network is a strongly connected graph and the bow-tie is obtainedwith a filtering. Therefore, it is not the case that the nodes in the IN are not connected among eachother. This means that in case a few nodes would have defaulted, the others would still have been heavilyaffected. However, the observed migration of nodes implies that, compared to the normal period, there

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MS

BOFA

CINC

AIG

JPMCC

GSMERHSBC FNMADB

UBSCRDSUI

RBS

SOCGEN

BNP

LLOYDS

Figure 5: The network of the CDS reference entities from period 2. Each of the nodes represents afinancial institution. Outgoing links from nodes that are in the top, or the IN of the bow-tie structurerepresent the estimated potential impact of a financial institution to its neighbours (see Methods). Thenodes in the SCC are placed within a circle of radius one and centred at the origin. The distance ofeach node from the centre is 1−Impacting centrality. The angle increases linearly from 0 to 2π. Thus,the closer a node is to the centre the higher is impacted-impacting centrality. Similarly, nodes in theOUT and IN are placed between angles π/2- 5π/8 and 3π/2 - 13π/8 respectively. In addition, nodes inthe OUT and IN are placed with an offset of 1.1 from the origin. With the bow-tie representation weare able to visually compare the centrality of a node i with node j. Also, with this visualisation we areable to extract a network of nodes that mostly impact the others, nodes that impact just as much as theyget impacted, and nodes that only get impacted by other nodes in the network. The size and the colourof the node reflects impacted-impacting centrality of a node (nodes with larger impacted-impactingcentrality are in red). The colour assigned to links is based on where the links point to in the network.Links originating from IN to the SCC are in bright blue. Links originating in the SCC to nodes in theSCC are in green. Links that are originating in the SCC to the OUT are dull blue grey colour.

has been an increasing polarisation between nodes (IN) that predominantly impact the network, andnodes (OUT) that predominantly are impacted by the network.

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The above analysis of impacting and impacted centralities, and the bow-tie extraction allows us to movefrom an initial picture in which all nodes seemed to be equally important from the point of view ofsystemic risk, to a much more refined picture. In terms of systemic impact, we can now focus on a smallsubset of the nodes, viz. nodes that have a high impacting centrality and are located in the centre of theSCC, or in the top part of the IN. This finding is corroborated by anecdotal evidence –(see, e.g., FCIC,2011)– about the role of important actors of the credit crisis of 2008 (see appendix G). Conversely, theimpacting centrality allows also to identify nodes that suffer the most from an impact originating from theothers. Remarkably, there is no evidence of one or two nodes dominating the others in terms of systemicimportance. In contrast, we see that in each period a set of about top 19 nodes have similar values ofcentrality.

8 Conclusion

We have analysed the ε drawup’s in the CDS’s time series for the top US and EU institutions throughoutthe last 10 years. By measuring the frequency of joint drawup’s in pairs of CDS time series we haveestimated the level of interdependence and trend reinforcement in the market. According to previoustheoretical works on financial networks, the interplay of these two mechanisms is deeply linked to theemergence of systemic risk. We have found statistically significant levels of both interdependence andtrend reinforcement. Moreover, we see an increase of both in acute phases of the crisis. The result sug-gests that high interdependence and trend reinforcement together with high level of individual riskinessare a possible indicators of the level of systemic risk. Indeed, when CDS spreads were at their peak in2008, implying high risk of individual default, movements in the spread of a few institutions were verylikely to be followed by movements in another and also in the same institutions. This means that thedefault of a few players was likely to trigger a financial melt-down.

Furthermore, we have carried out what to our knowledge is the first study of the complex network ofCDS interdependencies. In order to investigate the systemic importance of individual nodes, we haveintroduced two novel measures. The impacting centrality captures, in a recursive way, how much a nodeimpacts the network. Symmetrically, the impacted centrality captures how much a node is impacted bythe network. These two measures enable the extraction of a bow-tie structure from the initial networkand to clarify the role of the nodes. We have found that in all phases of the market the top institutions bysystemic importance are not just one or two, but about 19. They all have comparable level of systemicimportance and they all are very tightly interconnected.

The specific findings of this analysis are relevant to the broad audience interested in the issue of systemicrisk and systemically important financial institutions, including policy makers. Moreover, our approachis very general and applies to any set of time series associated to units that operate in interaction. Inparticular, it is of interest for those cases where the direct interaction between units is not observable and

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the dependence has to be inferred from the dynamics. In this respect, our paper contributes to a streamof work on the observability and the reconstruction of complex networks (Clauset et al., 2008) .

Acknowledgments

The authors acknowledge the financial support from the Swiss National Science Foundation GrantCR12I1-127000) and the European Commission FET Open Project “FOC” 255987.

9 Appendix

A Motivations

Credit Default Swap’s (CDS) are important for purposes of hedging, diversification and in some instancesparticipating in markets that would be otherwise inaccessible to a subset of the market participants.

Even though CDS’s are valuable to mitigate risks, they also tend to add to the fragility of a system. Inthe absence of a proper regulatory framework, CDS’s can be used to accelerate the fragility of a firm, aswas the case with Lehman Brothers in the financial crisis of 2008.

B Background Information on CDS

A CDS is a bilateral Over the Counter (OTC) derivative. It is analogous to an insurance instrument. ACDS contract has three legs, i.e., there are three entities that formulate a CDS contract. The three partiesinvolved in a typical CDS contract are the buyer, seller and reference entity. The buyer of a CDS contractpurchases protection on the reference entity from the seller for fixed periodic payments, also calledpremiums. It is not required that a buyer of a CDS contract have a credit exposure to a reference onwhich it is buying a CDS contract. Analogously, it is equivalent to saying that one can buy an insuranceon one’s neighbours house.

In the event that the reference entity defaults on its debt to its investors, the buyer receives a one timepayment from the seller, hence the name Credit Default Swap. CDS’s can be one of the following types:single name, Index and Basket CDS. As the names might suggest, single name CDS are on a singleentity, sovereign or otherwise. Index CDS’s are issued on constituents of an index with equal weightsbeing assigned to each of the constituents. Basket CDS’s can have more than one reference entity, in facta basket of entities constitute a set of reference entities.

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In the case of basket CDS’s there are three further classifications, namely first-to-default CDS, full-basketCDS, untranched basket and a tranched basket, or Collateralised Debt Obligations (CDO). In addition,CDS contracts are negotiated privately and are bilateral in nature.

Like a swap agreement there is no initial payment that is needed to enter into a CDS contract. Unlikea corporate bond, a CDS contract enables a participant to go short in the credit of a reference entity.Also, one can enter into a CDS contract even if a corporate bond of some pre-specified maturity is notavailable.

The popularity of the CDS market can be seem from the growth of such products. International Swap andDerivatives Association (ISDA) statistics show that by the end of 2003, the notional amounts outstandingin the CDS markets stood at $3.58 trillion dollars. Compared to the year 2000, in the year 2003 the CDSmarkets gained 1.4% percent of the entire swap market. The popularity of CDS contracts declined in theyears 2009 and 2010. This was a fallout for the financial crisis of 2008. From Fig. 1, we can see that theCDS spreads were reflecting such movements in the market.

The embedded relationships that lie within a CDS contract remain opaque to the general public due to thefact that they are not only privately negotiated; but, also because information on such contracts exposesthe financial institutions to corporate attacks. In addition, even if a comprehensive dataset were available,the lack of transparency with regards to ownerships leads to different level of complexity, see Battistonet al. (2010). To further elaborate on dependence structures that can arise in the CDS markets we presenta small discussion on CDS configurations.

C CDS Configurations

In order to understand what co-dependencies can arise between CDS market participants, we have firstconstructed a representative set of configurations of connections that market participants can develop overtime in a given scenario. The motivation for such a breakdown is to account for preferences of marketparticipants that emerge given a regulatory framework, or any other market factors. For example, banks,keeping in line with their mandate, tend to be broker dealers. This implies that they are both a buyer andseller of insurance contracts. A pension fund for example, typically tends to purchase protection to hedgeits portfolio from loss due to its exposure to another entity; however, it tends to limit its transactionalactivity to being long on such instruments. Building upon the intuition we gather from the argumentsabove we can then divide dependence scenarios into three classes:

1. The set of buyers, sellers and reference entities are unique and non-overlapping sets.

2. The set of one of buyers, sellers, or reference entities overlaps with another, but not both

3. Each of the buyer can be a seller or a reference entity, or both

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We have studied the dependence in each of these classes. Since the exact and detailed information oneach of the CDS contracts is unavailable. If we restrict ourselves to the market dynamics, where each ofthe buyers, sellers and reference entities are non-separable, see C.3, we are able to explain co-movementsin CDS spreads of two reference entities; since, CDS spread data, by construction, reflects the financialhealth of reference entities.

We now elaborate on each of the configurations stated above. The CDS contracts can be bought and soldfor speculative purposes. This means that a market participant can purchase a CDS contract on anotherfirm in the absence of any credit relationship between the two market participants. In this study welimit ourselves to analysing dependence structures that do not arise out of speculative activities. In otherwords, we analyse dependence structures when market participants have pre-existing credit relationshipsand trade CDS contracts to hedge their respective risks. In addition, it is important to remember that inthe event of a default, the CDS buyer is paid approximately 60% of the notional that is embedded inthe notional of the CDS contract. This means that, if firm A is exposed to B in the amount of $100 andpurchases protection from C on B. In the event that B defaults, then C would pay $60 to A.

The kind of dependence structure that emerges within the CDS market framework would largely dependon the intersection of set of buyers, sellers and reference entities. Thus, we explore the dependencestructures that emerge when we study the permutations of possible intersecting sets of buyers, sellersand reference entities. It is important to remember here that the set of configurations that we presentherein are in no way an exhaustive set of configurations. We merely present some specific configurationsthat provide a good intuition on the dependencies that can emerge due to trading of CDS contracts. Weencourage the reader to further explore these configurations.

C.1 Buyers, Sellers and Reference entities are unique:

We start with the first set configurations that emerge when we keep the set of buyers, sellers and referenceentities as unique, i.e. the set of buyers, sellers and reference entities do not have have any commonmarket participants (see Fig. 6).

We present three pictures for three scenarios (not exhaustive) for this particular market configuration, Fig.6. Here the picture denotes some simple networks that may exist between financial entities with regardsto a single CDS transaction. The solid arrow moves from the seller of a CDS contract to the buyer. Thedotted arrow refers to the reference entity underlying the CDS contract. The red arrows represent thedirection of impact in the event the node, from which the arrow emerges, defaults.

Scenario 1: Sub picture 1In this scenario we limit each CDS transaction limited to a unique seller, buyer, and reference entity. Wefind that in the event that reference entity 1 defaults, then both seller 1, and buyer 1 would experiencean increase in their fragility. Buyer 1 would have lost 40% of its investment in reference entity 1. And,

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Figure 6: Networks that exist when the buyers, sellers and reference entities are all unique. The solidarrow moves from the seller of a CDS contract to the buyer. The dotted arrow refers to the reference entityunderlying the CDS contract. The red arrows represent the direction of impact in the event of a defaultof the node from which the arrow emerges

seller 1 would be liable for 60% of buyer 1’s exposure to reference entity 1. In this case we find that thebulk of the burden due to default lands on the shoulders of the sellers.

Scenario 2: Sub picture 2In this scenario we impose that a seller can have more than one buyer and more than one reference entityfor protection. In the event, that either of the reference entities default, the corresponding buyer andthe seller would experience an increase an fragility. Interestingly, if reference entity 1 defaults, then thefragility of buyer 2 would also get affected. Since, seller 1 is the insuring agent for reference entities 1& 2. Thus, a default in either of the reference entities would make the seller more fragile. This in turnwould lead to fragility of its other counterparties. Thus, we find that in this scenario that the fragilityof reference entity 2 can also contribute to the fragility buyer 1 as they are both connected to the sameseller.

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Scenario 3: Sub picture 3In this scenario we investigate a dependence structure whence a seller is liable to two buyers based onthe same reference entity. If reference entity 1 defaults then all the market participants experience anincrease in fragility, seller 1 being the agent that would suffer the most.

In the first part of Fig. 6 (Sub-picture 1), we see that in the event that reference entity 1 defaults, seller 1is liable to buyer 1 to amount of the contingent claim. In Fig. 6 (sub-picture 2), we see that seller 1 hassold CDS contracts to buyer 1 and 2, on reference entities 1 and 2 respectively. It is clear from this picturethat, in the unlikely event that either/both reference entities 1, 2 default; seller 1 is liable for contingentclaims against it. In Fig. 6 (sub-picture 3), seller 1 is liable for reference entity 1’s default to buyer 1 and2.

In this set of configurations we find that the burden, in the event of a default, is almost entirely upon thesellers of the CDS contracts. However, we have seen from the financial crisis of 2008, that there weremany institutions that were not necessarily sellers of CDS contracts, but also mainly buyers. A varied setof market participants were adversely affected from the fallout in the CDS market. Thus, we extend ourargument to the set of CDS configurations, where the set of buyers and sellers have a non-trivial overlap.

C.2 Buyers-Sellers non-separable

Consider sub picture 1 from Fig. 7, the arrows and lines represent the same relationships as before. FromFig. 7 (Sub picture 1) we see that (Buyer|S eller)1 has bought a CDS contract from S eller2, where thereference entity involved in the transaction is Re f erence1.

Scenario 1: Sub-picture 1The dependence structure that arises due to such a configuration is very similar to the dependence thatarises from scenario 1 whence the buyers, sellers, and reference entities are unique. Consider now thecase that Buyer2 is also a seller, (Fig. 7 Sub picture 3). In this case, the fragility of Seller 2 affects thefragility of (Buyer|S eller)1, which in turn would affects the others. Given the fragility of Seller 2, adefault of Re f erence1 would, more or less, make the entire system fragile.

Scenario 2: Sub-picture 2 In this case we find that the default of reference entity 1 would affect seller2, buyer 2, and buyer|seller 1.

Scenario 3: Sub-picture 3 In this scenario we see that a default of reference entity 1 would affect all themarket participants. However, it is worth noting here that buyer|seller 1 & 2 have sub-optimal strategiesas they are exposed to the full amount of their credit link with reference entity 1.

We find that, such a configuration could be a possible candidate for modelling dependencies in the CDSmarket; however, such a construction represents a set of sub-optimal strategies of the market participants.For instance, Buyer3, would benefit from diversifying its risk arising from the fragility of (Buyer|S eller)1.

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Figure 7: Networks that exist when the buyers and sellers are the same but different from referenceentities. The solid arrow moves from the seller of a CDS contract to the buyer. The dotted arrow refersto the reference entity underlying the CDS contract. The red arrows represent the direction of impact inthe event of a default of the node from which the arrow emerges

C.3 Buyers-Sellers-Reference non-separable

In this set of configurations we impose that the intersection of buyers, sellers and reference entities isnon-trivial.

Due to the level of complexity, we assign a label to each of the solid arrows to reflect the ref-erence entity that is involved in a particular CDS contract. Consider Fig. 8, we keep the sameconvention as before. (Buyer|S eller|Re f erence)1 buys CDS from (Buyer|S eller|Re f erence)2 due

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We see here that there is a feedback loop that emerges from the distress of (Buyer|S eller|Re f erence)4 toitself. In addition,

Scenario 3: Fig. 10 - (Buyer|S eller|Re f erence)3 defaults

Figure 10: Networks that exist when the buyers and sellers are the same but different from referenceentities. The solid arrow moves from the seller of a CDS contract to the buyer. The dotted arrow refersto the reference entity underlying the CDS contract.

In the event that (Buyer|S eller|Re f erence)3 defaults, it affects (Buyer|S eller|Re f erence)2,(Buyer|S eller|Re f erence)1, and (Buyer|S eller|Re f erence)4. Default of (Buyer|S eller|Re f erence)3

would imply that (Buyer|S eller|Re f erence)2 would be liable to (Buyer|S eller|Re f erence)1 for themajority of the loss. (Buyer|S eller|Re f erence)1 would be marginally affected by the default of(Buyer|S eller|Re f erence)3. However, (Buyer|S eller|Re f erence)1 would experience a loss due to thefact that it has sold protection to (Buyer|S eller|Re f erence)4. In the even that (Buyer|S eller|Re f erence)1

defaults due to the default of (Buyer|S eller|Re f erence)3, then (Buyer|S eller|Re f erence)4 would bedistressed due to the default of (Buyer|S eller|Re f erence)3.

We find that when we impose the intersection of the set of buyers, sellers and reference entities to be non-trivial. Then, we observe a dependence structure that would introduce interdependence in the time seriesof the market participants. With this framework, where the buyers and sellers for each CDS contract areunknown, we can still model dependencies among financial entities utilising their CDS time series data.

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D Remarks on Drawup Detection

Bloomberg has negotiated contracts with various quote providers to display data on the Bloomberg ter-minal; however, not all quote providers relax the constraint of Bloomberg being allowed to distribute thisdata to the end user. Thus, we see certain breaks in our time series, as on those days either the CDS wasnot traded (see appendix D), or is quoted by a quote provider that does not allow its data to be down-loaded via Bloomberg. To circumvent this issue, we carry forward the last traded price till a time that anew price has been reported. Thus, CDS time series data were polished beforehand to make the analysisin the ε-drawups framework possible. Not all time series exist in the same time window, thus we takethe CDS contract with the largest number of observations and use it as a reference of our time window.Thus, all time series are put in one matrix, where if a CDS time series did not exist whence the referenceexisted, then we assign a value of zero to it. This modification does not affect the dynamics of the CDS’sper se, as a spread of zero implies an absence of a swap contract anyways. Once, we have the data ina single matrix, we then iterate through all the individual time series and compute all the local extrema(see Methods). The date of each extrema is also recorded. We then proceed to computing ε-drawups. Theε parameter is a local and dynamic parameter. It is essentially the local variation of a time series in thelast ten days. We compute the ε-drawups for each security and record the date at which the ε-drawupoccurred. We then proceed to computing co-drawups pairwise. To compute common drawups we dividethe dataset into three periods (see Methods in the main paper) and compute common drawups and thedrawups experienced by each of the time series in that period. We also compute co-drawups for all thepairs with a time delay factor τ, i.e., when one security is translated by τ days w.r.t. another. We do thisexercise for all pairs in our dataset. Finally, we compute wi j’s (as before) using our count matrices.

E Control Set

After having computed a matrix of ε-drawup’s, for each security, we permute the matrix indices ofwhere the ε-drawup’s occur. This way we are re-arranging all the occurrences of ε-drawup’s in a randommanner. The reason to pursue this methodology and not generating random (or even , trend reinforcedrandom walks) is that the authors don’t wish to define the the price process as a priori, assuming randomwalks (or, trend reinforced random walks) are a good proxy for a CDS price process. In addition, we donot resample the original time series, as such a procedure introduces price movements that sometimesamplify ε-drawup’s when there in fact are none. This key point becomes even more important when wewant to develop a control set for all three periods. We perform a permutation test to filter the empiricalW′i js. We compute W′i j for each pairs of securities. To do this, we proceed with permuting the ε-drawupsin each of the securities and compute W′i j. We repeat this procedure a hundred times. With the hundredrealisations of W′i j for each pair of securities i, j, we then further filter W′i j at the 95% confidence intervalto derive a single number W∗i j for each pair of securities. We then utilise W∗i j as the control number to filter

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empirical Wi j, i.e., each empirical pair is filtered with a unique number that corresponds to the controlnumber generated from our permutation test.

F PageRank and Impacting Centrality

To compute the centrality of the nodes in the network, we take inspiration from the concept of PageRankthat was introduced in the context of the World Wide Web (WWW) to enhance users’ search experience(Page et al., 1998). The main theme of the idea revolved around determining the rank of a webpage basedupon how many sites (other than itself) point towards it. Such a rank could be used as a good proxy fordetermining a webpages’ relevance to user searches. Suppose, that the PageRank of each website, i, bedenoted as Ci. Then, Ci for websites can be defined in a network framework consisting of N vertices.Consider,

Ci = α∑j→i

C j

koj︸︷︷︸

term 1

+ (1 − α)1N︸︷︷︸

term 2

. (3)

The α in term 1 on the r.h.s. of equation 3 represents the probability of C j inherited by webpages j that arepointing to webpages i. Each webpage j contributes, proportionally C j

koj, to the webpage it points points

to. Term 2 in 3 uniformly assigns the contribution of each of the webpages j to i times the complimentaryprobability from term 1. Unlike our measure of centrality, i.e., impacted-impacting centrality, PageRankcentrality (Page et al., 1998) measure is row stochastic. Impacted-impacting centrality is analogous tothe cumulative distress in the network on account of distress in node i.

G The Bow-tie Structure & FCIC Report

A bowtie structure refers to a directed graph here, where the connected nodes are in one of the threeparts of the network: IN (nodes with outgoing links only), SCC (nodes with both incoming and outgoinglinks) and OUT (nodes with incoming links). The resulting structure resembles that of a bow-tie wherethe SCC occupies the position of a knot and the IN and OUT represent the respective wings of thetie. It is important to remember that the bow-tie structure is a construction that largely depends on thethresholds that are imposed upon the impacting-impacted centrality. We present the bow-tie structuresfrom periods 1 until 3. The nodes are in the IN, if ri > 3/2, see Fig. 11b, 12b, & 13b. The nodes inthe SCC have 2/3 < ri < 3/2, and nodes are in the OUT, if ri < 2/3. We also present the distributionsof in-degree, out-degree, impacting, and impacted centralities for all three periods. We find that eventhough in-degree and out-degree have mass of their distributions in a narrow range; the impacting andimpacted centralities across the three periods are distributed across a wider spectrum. Additionally, wepresent bow-tie structures from all three periods with varying degrees of thresholding, see Fig. 16, 17,

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18, 19, 20, 21, and 15. The existence of the bow-tie structure is not guaranteed in all graphs. Consider,Fig. 14a & b. With these counterexamples we highlight that the existence of a bow-tie structure is notassured after the separation of nodes based on their level of impacting-impacted centrality. We find thatin our network there is an SCC in all three periods after filtering the impacting-impacted centrality.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Period 1

In degreeOut degreeImpacted Centrality (b)Impacting centrality (c)

a)

MS

BOFA

CINC

AIGJPMCC

GS

MER

HSBC

FNMA

DBUBS

CRDSUI

RBSSOCGEN

BNP

LLOYDS

b)

Figure 11: a) The normalised distribution of in-degree, out-degree, impacting and impacted cen-tralities in period 1 . The bulk of the in-degree and out-degree distributions are concentrated in a narrowrange. Impacting and impacted centrality distributions are distributed across the x-axis. b) The networkof the CDS reference entities from period 1. Each of the nodes represents a financial institution. Outgo-ing links from nodes that are in the top, or the IN of the bow-tie structure represent the estimated potentialimpact of a financial institution to its neighbours (see Methods). The nodes in the SCC are placed withina circle of radius one and centred at the origin. The distance of each node from the centre is 1−Impactingcentrality. The angle increases linearly from 0 to 2π. Thus, the closer a node is to the centre the higheris impacted-impacting centrality. Similarly, nodes in the OUT and IN are placed between angles π/2-5π/8 and 3π/2 - 13π/8 respectively. In addition, nodes in the OUT and IN are placed with an offset of1.1 from the origin. With the bow-tie representation we are able to visually compare the centrality of anode i with node j. Also, with this visualisation we are able to extract a network of nodes that mostlyimpact the others, nodes that impact just as much as they get impacted, and nodes that only get impactedby other nodes in the network. The size and the colour of the node reflects impacted-impacting centralityof a node (nodes with larger impacted-impacting centrality are in red). The colour assigned to links isbased on where the links point to in the network. Links originating from IN to the SCC are in bright blue.Links originating in the SCC to nodes in the SCC are in green. Links that are originating in the SCC tothe OUT are dull blue grey colour.

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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Period 2


a)

MS

BOFA

CINC

AIG

JPMCC

GSMERHSBC FNMADB

UBSCRDSUI

RBS

SOCGEN

BNP

LLOYDS

b)

Figure 12: a) The normalised distribution of in-degree, out-degree, impacting and impacted cen-tralities in period 2 . The bulk of the in-degree and out-degree distributions are concentrated in a narrowrange. Impacting and impacted centrality distributions are distributed across the x-axis. b)The networkof the CDS reference entities from period 2. The bow-tie is constructed as described in fig. 11.

The Financial Crisis Inquiry Commission (FCIC) was established under the Fraud Enforcement andRecovery Act (Public Law 111-21) and was later passed by the Congress and officially signed and im-plemented by the President of the US in the month of May, 2009.

The goals of the FCIC was to examine the causes that led to the financial and economic crisis of 2008.During their investigation the FCIC conducted more that 700 witnesses and reviewed millions of docu-ments. In addition, it also held public hearings in New York, Washington D.C. among other regions inthe US. FCIC conducted extensive case studies on firms that it deemed as pivotal in bringing about thecrisis. These firms include: American International Group (AIG), Bear Stearns, Citigroup, CountrywideFinancial, Fannie Mae, Goldman Sachs, Lehman Brothers, Merrill Lynch, Moody’s and Wachovia.

We present brief snapshots from the FCIC report FCIC (2011) of some of the firms that the FCIC in-dicated were pivotal in financial crisis of 2008 along with the bow-tie visualisations from each of thethree periods, see Fig. 11b, 12b, & 13b. In addition we also present the degree distributions for all threeperiods, Fig. 11a, 12a, & 13a. We also present some statistics on the distributions of nodes in the variousregions of the bow-tie structure, see table 1, and the movement of some pivotal firms across the bow-tiestructure in the three periods, see table 2.

AIG:

• By 2005, AIG had written $107 billion in CDS for such regulatory capital benefits; most were with

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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Period 3


a)

MSBOFACINC

AIG

JPMCCGSMER

HSBC

FNMA

DB UBS

CRDSUI

RBS

SOCGEN

BNP LLOYDS

b)

Figure 13: a) The normalised distribution of in-degree, out-degree, impacting and impacted cen-tralities in period 3 . The bulk of the in-degree and out-degree distributions are concentrated in a narrowrange. Impacting and impacted centrality distributions are distributed across the x-axis. Unlike periods 1& 2, the in-degree and out-degree distributions are less peaked. b)The network of the CDS referenceentities from period 3. The bow-tie is constructed as described in fig. 11.

Impacting Impacted Both

Period 1 6 4 85Period 2 22 19 97Period 3 53 37 47

Table 1: Breakdown of nodes by region

European banks for a variety of asset types. That total would rise to $379 billion by 2007. Thesame advantages could be enjoyed by banks in the United States, where regulators had introducedsimilar capital standards for banks’ holdings of mortgage-backed securities and other investmentsunder the Recourse Rule in 2001. So a credit default swap with AIG could also lower Americanbanks’ capital requirements. In 2004 and 2005, AIG sold protection on super-senior CDO tranchesvalued at $54 billion, up from just $2 billion in 2003. FCIC Report, page 140.

• AIG’s business of offering credit protection on assets of many sorts, including mortgage-backedsecurities and CDOs, grew from $20 billion in 2002 to $211 billion in 2005 and $533 billion in2007, FCIC Report, page 141.

GS

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Period 1 Period 2 Period 3

AIG SCC SCC OUTGS SCC SCC SCC (periphery)

BOFA SCC SCC SCC (periphery)CINC SCC SCC SCC (periphery)

JPMCC SCC SCC IN

Table 2: Location of AIG, GS, BOFA, CINC & JPMCC across the three periods in the bow-tie structure

• Second, CDS were essential to the creation of synthetic CDOs. These synthetic CDOs were merelybets on the performance of real mortgage-related securities. They amplified the losses from thecollapse of the housing bubble by allowing multiple bets on the same securities and helped spreadthem throughout the financial system. Goldman Sachs alone packaged and sold $73 billion in syn-thetic CDOs from July 1, 2004, to May 31, 2007. Synthetic CDOs created by Goldman referencedmore than 3,400 mortgage securities, and 610 of them were referenced at least twice. This is apartfrom how many times these securities may have been referenced in synthetic CDOs created byother firms. FCIC Report, Conclusions.

• Goldman Sachs estimated that between 25% and 35% of its revenues from 2006 through 2009were generated by derivatives, including 70% to 75% of the firm’s commodities business, andhalf or more of its interest rate and currencies business. From May 2007 through November 2008,$133 billion, or 86%, of the $155 billion of trades made by Goldman’s mortgage department werederivative transactions. FCIC Report, pages 51-52.

• Goldman’s assets grew from $250 billion in 1999 to $1.1 trillion by 2007, an annual growth rateof 21%, FCIC Report, page 65.

BOFA, CINC, JPMCC

• From 1998 to 2007, the combined assets of the five largest U.S. banks: Bank of America, Citigroup,JP Morgan, Wachovia, and Wells Fargo more than tripled, from $2.2 trillion to $6.8 trillion.

• Leverage: Bank of America’s leverage rose from 18:1 in 2000 to 27:1 in 2007. Citigroup’s leverageincreased from 18:1 to 22:1, then shot up to 32:1 by the end of 2007. FCIC Report, page 65.

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1

2

3 4

5

6

a)

5 6

1 2

34

78

b)

Figure 14: a) Network W with an SCC: Dashed links are links that have been removed on the conditionthat ri < θ, where θ is some threshold. Then, we see that W no longer consists an SCC. b) Network Wwith an SCC: Dashed lines are as in a). Then we see that, W still has an SCC. In fact nodes 1,2,3, and 4remain in the SCC (as before) even after filtering links.

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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MS

MS

MS

BOFA

BOFABOFA

CINC

CINC

CINC

AIG

AIG

AIG

JPMCC

JPMCC

JPMCC

GS

GSGS

MER

MER

MER

HSBC

HSBC

HSBC

FNMA

FNMA

FNMA

DB

DBDBUBS

UBS

UBS

CRDSUI

CRDSUI

CRDSUI

RBS

RBS

RBS

SOCGEN

SOCGEN

SOCGEN

BNP

BNP

BNP

LLOYDSLLOYDS

LLOYDS

r <0.9

r >1.1

Impacted

Impa

ctin

g

r <0.8

r >1.2

r <0.7

r >1.3 r <0.6

r >1.4

r <0.5

r >1.5

r <0.4

r >1.6

r <0.3

r >1.7

r <0.2

r >1.8

r <0.1

r >1.9Period 3Period 2Period 1

Figure 15: Scatter plot of impacting versus impacted centrality. Each institution in the CDS marketis represented by three dots depending on the period (blue, green, red refers to period 1, 2, 3, respec-tively). It can be seen that, while in period 1 most institutions are located between the two dotted lines,in period 2 and 3 many of them move to the top and bottom region. This means that ratio between thetwo centrality measures varies with the market phase. Few institutions of interest are labelled. For ex-ample, Bank of America (BOFA) remains in the same region across the three periods. With referenceto the subsequent bow-tie construction used in Fig. 16, 17, 18, 19, 20, 21, and 15: The scatter plot isdivided into 3 regions for each choice of δ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} where the upperand lower regions are given by 1 +δ, and 1−δ respectively. Nodes in the region above the line ri > 1 +δ

correspond to the IN. Nodes in the region 1 − δ < ri < 1 + δ correspond to the SCC. Nodes in the regionri < 1 − δ correspond to the OUT.

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MS

BOFA

CINC

AIG

JPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNPLLOYDS

a)

MS

BOFA

CINC

AIG

JPMCC

GS

MER

HSBCFNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

b)

MS

BOFA

CINCAIG

JPMCC

GSMERHSBC

FNMA

DBUBS

CRDSUI

RBSSOCGENBNP

LLOYDS

c)

MSBOFA

CINC

AIG

JPMCC

GSMER

HSBC

FNMA

DBUBS

CRDSUI

RBSSOCGEN

BNP

LLOYDS

d)

Figure 16: Bow-tie structure from period 1 for varying values of impacting-impacted centralitiesa) The bow-tie is constructed as in Fig. 11. Nodes with 0.9 < ri < 1.1 are in the SCC, nodes withri > 1.1 are in the IN, and the nodes with ri < 0.9 are in the OUT b) The bow-tie is constructed as inFig. 11. Nodes with 0.8 < ri < 1.2 are in the SCC, nodes with ri > 1.2 are in the IN, and the nodes withri < 0.8 are in the OUT c) The bow-tie is constructed as in Fig. 11. Nodes with 0.7 < ri < 1.3 are in theSCC, nodes with ri > 1.3 are in the IN, and the nodes with ri < 0.7 are in the OUT d) The bow-tie isconstructed as in Fig. 11. Nodes with 0.7 < ri < 1.3 are in the SCC, nodes with ri > 1.3 are in the IN,and the nodes with ri < 0.7 are in the OUT

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MSBOFA

CINC

AIG

JPMCC

GS

MER

HSBC

FNMA

DBUBS

CRDSUI

RBSSOCGEN

BNP

LLOYDS

a)

MS

BOFA

CINC

AIG

JPMCC

GSMER

HSBC

FNMA

DBUBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

b)

MS

BOFA

CINCAIG

JPMCC

GS

MER

HSBC

FNMA

DBUBS

CRDSUI

RBSSOCGEN

BNP

LLOYDS

c)

MS

BOFA

CINCAIG

JPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBSSOCGEN

BNP

LLOYDS

d)


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MS

BOFA

CINC AIG

JPMCC

GS

MER

HSBC

FNMA

DB UBS

CRDSUIRBS

SOCGEN

BNP

LLOYDS

a)

MSBOFA

CINCAIG

JPMCC

GS

MER HSBCFNMA

DBUBS

CRDSUI

RBS

SOCGEN

BNPLLOYDS

b)

MSBOFA

CINCAIG

JPMCC

GS

MERHSBC FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

c)

MS

BOFA

CINCAIG

JPMCC

GS

MER

HSBC

FNMADBUBS

CRDSUI

RBS

SOCGEN

BNPLLOYDS

d)


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MS

BOFA

CINC

AIG

JPMCC

GS

MER

HSBC

FNMADB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

a)

MS

BOFA

CINC

AIG

JPMCC GS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

b)

MS

BOFA

CINC

AIG

JPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUIRBS

SOCGENBNP

LLOYDS

c)

MS

BOFA

CINC

AIG

JPMCC GS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGENBNP

LLOYDS

d)


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MS

BOFACINC

AIGJPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUIRBS

SOCGEN

BNP

LLOYDS

a)

MS

BOFA

CINC

AIG

JPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNPLLOYDS

b)

MS

BOFA

CINC

AIG

JPMCCGSMER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

c)

MS

BOFACINC

AIGJPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

d)


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MS

BOFACINC

AIGJPMCC

GS

MER

HSBC

FNMA

DB

UBS

CRDSUIRBS

SOCGEN

BNP

LLOYDS

a)

MS

BOFA

CINC

AIG

JPMCCGS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

b)

MSBOFACINC

AIG

JPMCCGS

MERHSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

c)

MS

BOFA

CINC

AIG

JPMCCGS

MER

HSBC

FNMA

DB

UBS

CRDSUI

RBS

SOCGEN

BNP

LLOYDS

d)


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References

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Battiston, S., Gatti, D., Gallegati, M., Greenwald, B., and Stiglitz, J. (2012a). Liaisons dangereuses:Increasing connectivity, risk sharing, and systemic risk. J. of Economic Dynamics and Control, forth.

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Garas, A., Argyrakis, P., and Havlin, S. (2008). The structural role of weak and strong links in a fi-nancial market network. The European Physical Journal B-Condensed Matter and Complex Systems,63(2):265–271.

Johansen, A. (2003). Characterization of large price variations in financial markets. Physica A: StatisticalMechanics and its Applications, 324(1-2):157–166.

Kullmann, L., Kertész, J., and Kaski, K. (2002). Time-dependent cross-correlations between differentstock returns: A directed network of influence. Physical Review E, 66(2):26125.

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Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic Spreading in Scale-Free Networks. PhysicalReview Letters, 86(14):3200–3203.

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Schweitzer, F., Fagiolo, G., Sornette, D., Vega-Redondo, F., Vespignani, A., and White, D. R. (2009).Economic networks: the new challenges. Science (New York, N.Y.), 325(5939):422–5.

Sornette, D. and Zhou, W. X. (2006). Predictability of large future changes in major financial indices.International Journal of Forecasting, 22(1):153–168.

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