Credit Default Swaps Drawup Networks: Too Interconnected to Be Stable?

We analyse time series of CDS spreads for a set of major US and European institutions in a period overlapping the recent financial crisis. We extend the existing methodology of -drawdowns to the one of joint -drawups, in order to estimate the conditional probabilities of spike-like co-movements among pairs of spreads. After correcting for randomness and finite size effects, we find that, depending on the period of time, 50% of the pairs or more exhibit high probabilities of joint drawups and the majority of spread series are trend-reinforced, i.e. drawups tend to be followed by drawups in the same series. We then carry out a network analysis by taking the probability of joint drawups as a proxy of financial dependencies among institutions. We introduce two novel centrality-like measures that offer insights on how both the systemic impact of each node as well as its vulnerability to other nodes' shocks evolve in time.


Background Information on CDS
A CDS is a bilateral Over the Counter (OTC) derivative. It is analogous to an insurance instrument. A CDS contract has three legs, i.e., there are three entities that formulate a CDS contract. The three parties involved in a typical CDS contract are the buyer, seller and reference entity. The buyer of a CDS contract purchases protection on the reference entity from the seller for fixed periodic payments, also called premiums. It is not required that a buyer of a CDS contract have a credit exposure to a reference on which it is buying a CDS contract. Analogously, it is equivalent to saying that one can buy an insurance on one's neighbours house.
In the event that the reference entity defaults on its debt to its investors, the buyer receives a one time payment 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 single entity, sovereign or otherwise. Index CDS's are issued on constituents of an index with equal weights being assigned to each of the constituents. Basket CDS's can have more than one reference entity, in fact a basket of entities constitute a set of reference entities.
In the case of basket CDS's there are three further classifications, namely first-to-default CDS, fullbasket CDS, 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. Unlike a 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 not available.
The popularity of the CDS market can be seem from the growth of such products. International The embedded relationships that lie within a CDS contract remain opaque to the general public due to the fact that they are not only privately negotiated; but, also because information on such contracts exposes the 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 [1].

Data set
Bloomberg has negotiated contracts with various quote providers to display data on the Bloomberg terminal; however, not all quote providers relax the constraint of Bloomberg being allowed to distribute this data to the end user. Thus, we see certain breaks in our time series, as on those days either the CDS was not traded or is quoted by a quote provider that does not allow its data to be downloaded via Bloomberg. To circumvent this issue, we carry forward the last traded price till a time that a new price has been reported. Thus, CDS time series data were polished beforehand to make the analysis in the ε-drawups framework possible. Not all time series exist in the same time window, thus we take the 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 reference existed, then we assign a value of zero to it. This modification does not affect the dynamics of the CDS's per se, as a spread of zero implies an absence of a swap contract anyways. Once, we have the data in a 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. The ε parameter is a local and dynamic parameter. It is essentially the local variation of a time series in the last ten days. We compute the ε-drawups for each security and record the date at which the ε-drawup occurred. We then proceed to computing co-drawups pairwise. To compute common drawups we divide the dataset into three periods (see Methods in the main paper) and compute common drawups and the drawups experienced by each of the time series in that period. We also compute co-drawups for all the pairs with a time delay factor τ , i.e., when one security is translated by τ days w.r.t. another. We do this exercise for all pairs in our dataset. Finally, we compute w ij 's (as before) using our count matrices.

Control Set
After having computed a matrix of ε-drawup's, for each security, we permute the matrix indices of where the ε-drawup's occur. This way we are re-arranging all the occurrences of ε-drawup's in a random manner. The reason to pursue this methodology and not generating random (or even , trend reinforced random walks) is that the authors don't wish to define the the price process as a priori, assuming random walks (or, trend reinforced random walks) are a good proxy for a CDS price process. In addition, we do not resample the original time series, as such a procedure introduces price movements that sometimes amplify ε-drawup's when there in fact are none. This key point becomes even more important when we want to develop a control set for all three periods. We perform a permutation test to filter the empirical W ij s. We compute W ij for each pairs of securities. To do this, we proceed with permuting the ε-drawups in each of the securities and compute W ij . We repeat this procedure a hundred times. With the hundred realisations of W ij for each pair of securities i, j, we then further filter W ij at the 95% confidence interval to derive a single number W * ij for each pair of securities. We then utilise W * ij as the control number to filter empirical W ij , i.e., each empirical pair is filtered with a unique number that corresponds to the control number generated from our permutation test.

PageRank and Impacting Centrality
To compute the centrality of the nodes in the network, we take inspiration from the concept of PageRank that was introduced in the context of the World Wide Web (WWW) to enhance users' search experience [2]. The main theme of the idea revolved around determining the rank of a webpage based upon how many sites (other than itself) point towards it. Such a rank could be used as a good proxy for determining a webpages' relevance to user searches. Suppose, that the PageRank of each website, i, be denoted as C i . Then, C i for websites can be defined in a network framework consisting of N vertices. Consider, The α in term 1 on the r.h.s. of equation 1 represents the probability of C j inherited by webpages j that are pointing to webpages i. Each webpage j contributes, proportionally , to the webpage it points points to. Term 2 in 1 uniformly assigns the contribution of each of the webpages j to i times the complimentary probability from term 1. Unlike our measure of centrality, i.e., vulnerability-impacting centrality, PageRank centrality [2] measure is row stochastic. vulnerability-impacting centrality is analogous to the cumulative distress in the network on account of distress in node i.

The Bow-tie Structure & FCIC Report
A bowtie structure refers to a directed graph here, where the connected nodes are in one of the three parts of the network: IN (nodes with outgoing links only), SCC (nodes with both incoming and outgoing links) and OUT (nodes with incoming links). The resulting structure resembles that of a bow-tie where the SCC occupies the position of a knot and the IN and OUT represent the respective wings of the tie. It is important to remember that the bow-tie structure is a construction that largely depends on the thresholds that are imposed upon the impacting-vulnerability centrality. We present the bow-tie structures from periods 1 until 3. The nodes are in the IN, if r i > 3/2, see Fig. S1b, S2b, & S3b. The nodes in the SCC have 2/3 < r i < 3/2, and nodes are in the OUT, if r i < 2/3. We also present the distributions of in-degree, out-degree, impacting, and vulnerability centralities for all three periods. We find that even though in-degree and out-degree have mass of their distributions in a narrow range; the impacting and vulnerability centralities across the three periods are distributed across a wider spectrum. Additionally, we present bow-tie structures from all three periods with varying degrees of thresholding, see Fig. S6, S7, S8, S9, S10, S11, and S5. The existence of the bow-tie structure is not guaranteed in all graphs. Consider, Fig. S4a & b. With these counterexamples we highlight that the existence of a bow-tie structure is not assured after the separation of nodes based on their level of impactingvulnerability centrality. We find that in our network there is an SCC in all three periods after filtering the impacting-vulnerability centrality.
The We present brief snapshots from the FCIC report [3] of some of the firms that the FCIC indicated were pivotal in financial crisis of 2008 along with the bow-tie visualisations from each of the three periods, see Fig. S1b, S2b, & S3b. In addition we also present the degree distributions for all three periods, Fig.  S1a, S2a, & S3a. We also present some statistics on the distributions of nodes in the various regions of the bow-tie structure, see table S1, and the movement of some pivotal firms across the bow-tie structure in the three periods, see   Figure S1. a) The normalised distribution of in-degree, out-degree, impacting and vulnerability centralities in period 1 . The bulk of the in-degree and out-degree distributions are concentrated in a narrow range. Impacting and vulnerability centrality distributions are distributed across the x-axis. b) The network of the CDS reference entities from period 1. Each of the nodes represents a financial institution. Outgoing links from nodes that are in the top, or the IN of the bow-tie structure represent the estimated potential impact of a financial institution to its neighbours (see Methods). The nodes in the SCC are placed within a circle of radius one and centred at the origin. The distance of each 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 vulnerability-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 of 1.1 from the origin. With the bow-tie representation we are able to visually compare the centrality of a node i with nodej. Also, with this visualisation we are able to extract a network of nodes that mostly impact the others, nodes that impact just as much as they are vulnerable, and nodes that only are vulnerable to other nodes in the network. The size and the colour of the node reflects vulnerability-impacting centrality of a node (nodes with larger vulnerability-impacting centrality 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 the SCC are in green. Links that are originating in the SCC to the OUT are dull blue grey colour. b)The network of the CDS reference entities from period 3. The bow-tie is constructed as described in fig. S1.

GS
• Second, CDS were essential to the creation of synthetic CDOs. These synthetic CDOs were merely bets on the performance of real mortgage-related securities. They amplified the losses from the collapse of the housing bubble by allowing multiple bets on the same securities and helped spread them throughout the financial system. Goldman Sachs alone packaged and sold $73 billion in synthetic CDOs from July 1, 2004, to May 31, 2007. Synthetic CDOs created by Goldman referenced more than 3,400 mortgage securities, and 610 of them were referenced at least twice. This is apart from how many times these securities may have been referenced in synthetic CDOs created by other firms. FCIC Report, Conclusions.
• Goldman Sachs estimated that between 25% and 35% of its revenues from 2006 through 2009 were generated by derivatives, including 70% to 75% of the firm's commodities business, and half or more of its interest rate and currencies business.   Figure S4. a) Network W with an SCC: Dashed links are links that have been removed on the condition that r i < θ, where θ is some threshold. Then, we see that W no longer consists an SCC. b) Network W with an SCC: Dashed lines are as in a). Then we see that, W still has an SCC. In fact nodes 1,2,3, and 4 remain in the SCC (as before) even after filtering links.  The CDS prices are quoted in basis points (bp). The purpose of this plot is to highlight the market regimes, rather than the individual CDS spread evolution. Accordingly, the CDS spreads of all the financial entities are plotted here.