The hyperbolic geometry of financial networks
TThe hyperbolic geometry of financial networks
Martin Keller-Ressel and Stephanie Nargang TU Dresden, Institute for Mathematical Stochastics, Dresden, 01062, Germany * [email protected] ABSTRACT
Based on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we demonstratefor the first time that the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e.,by hyperbolic geometry. This allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos etal., which is based on the Poincaré disc model of hyperbolic geometry. We show that the latent dimensions of ‘popularity’ and‘similarity’ in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system.In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks hasremained rather stable, while the peripheral community structure exhibits more (but still moderate) variability.
Introduction
Network models based on hyperbolic geometry have been successful in explaining the structural features of informational ,social and biological networks . Such models provide a mathematical framework to resolve the conflicting paradigms ofpreferential attachment (attraction to popular nodes) and community effects (attraction to similar nodes) in networks. Just as the geometric structure of a social network determines the diffusion of news, rumors or infective diseases betweenindividuals , the geometric structure of a financial network influences the diffusion of financial stress between financialinstitutions, such as banks. Indeed, the lack of understanding for risks originating from the systemic interaction of financialinstitutions has been identified as a major contributing factor to the global financial crisis of 2008. While many recent studieshave analysed the mechanisms of financial contagion in theoretical or simulation-based settings, less attention has been payed tothe structural and geometric characteristics of real financial networks. In particular, it has remained an open question, whetherthe paradigm of hyperbolic structure applies to financial and economic networks and what such a structure implies for financialcontagion processes.Here, we consider financial networks inferred from bank balance sheet data, as collected and made available by the EuropeanBanking Authority (EBA) within the European banking stress test and transparency exercises of 2014, 2016 and 2018.
Weshow that these networks can be embedded into low-dimensional hyperbolic space with considerably smaller distortion than intoEuclidean space, suggesting that the paradigm of latent hyperbolic geometry also applies to financial networks. Furthermore –following Papadopoulos et al. – we decompose the embedded hyperbolic coordinates into the dimensions of popularity and similarity and demonstrate that these dimensions align with systemic importance and membership in regional banking clusters respectively. Finally, the longitudinal structure of the data allows us to track changes in these dimensions over time, i.e., totrack the stability of systemic importance and of the peripheral community structure over time. Results
Inference of Financial Networks
Contagion in financial networks is a complex process, which can take place through several parallel (and potentially interacting)mechanisms and channels. These mechanisms include direct bank-to-bank liabilities , bank runs , and market-mediatedcontagion through asset sales (‘fire-sale contagion’); see also [11, p.21ff]. Here, we focus on the channel of fire-sale contagion, which has been singled out – both in simulation and in empirical studies – as a key mechanism offinancial contagion. Moreover, the propensity of fire-sale contagion can be quantified from available balance sheet data, usingliquidity-weighted portfolio overlap (LWPO) as an indicator (see Methods for details).Our inference of financial networks follows a two-stage mechanism: First, we construct a weighted bipartite networkin which banks B = ( b , . . . , b n ) are linked to a common pool of assets A = ( a , . . . , a m ) , which consist of sovereign bondsclassified by issuing country and by different levels of maturity. In the second step we perform a one-mode projection ofthis network on the node set B , using the LWPO of two banks b i , b j ∈ B to determine the weight w i j of the link between thecorresponding nodes. For any of the years y ∈ { , , } , the result is an undirected, weighted network N y of banks,in which two banks are connected if and only if they hold common assets. The link weight w i j , normalized to [ , ] , represents a r X i v : . [ q -f i n . GN ] M a y he susceptibility of two banks b i , b j to financial contagion, quantified by their LWPO. The inferred networks are very dense,i.e., almost all pairs of banks hold some common assets. However, most of the connections have very small weights, and thenetworks are dominated by a ‘sparse backbone’ of a few strong connections, which represent the dominant channels of potentialcontagion of financial distress; see Figure 1.A. Latent network geometry Figure 1.
Panel A : Edge weight distribution in the EBANetworks of years 2014, 2016 and 2018. The inter-decile rangeof the (highly skewed) distributions is indicated in red.
Panel B : Stress of network embedding into Euclidean vs.Hyperbolic geometry. Lower values of stress indicate bettergoodness-of-fit.Our first objective was to uncover the latent geometric net-work structure and to evaluate the suitability of a hyperbolicnetwork model. (See Methods for background on hyperbolicgeometry.) To this end, we calculated stress-minimizing em-beddings of the financial networks N , N and N into two-dimensional Euclidean space E and hyperbolicspace H . These methods correspond to classic multidimen-sional scaling in the Euclidean case and to the hydra+ embedding method in the hyperbolic case. The residualstress can be used as a goodness-of-fit measure betweengeometric model and true network topology. As shown inFigure 1.B the residual embedding stress from the hyper-bolic model is substantially smaller – consistently over allthree years of observation – than from the Euclidean model.This indicates that the latent geometry of the observed fi-nancial networks is much better represented by negativelycurved (hyperbolic) rather than flat (Euclidean) geometry.It is also evidence of a high degree of hierarchical organi-zation in the financial networks considered. Furthermore,as a result of the embedding we obtain for each bank node b i latent coordinates ( r i , θ i ) in the Poincaré disc model ofhyperbolic space (see Methods), which allows us to con-nect the network embedding to the popularity-vs-similaritymodel of Papadopoulos et al. The hyperbolic embedding of the full banking network of 2018 is shown in Figure 2. Theembedded network shows a clear core-periphery structure, in line with previous studies of financial networks.
A deeperanalysis of this structure is the subject of the following section.
Structural Analysis
The popularity-vs-similarity model of Papadopoulos et al. offers a direct interpretation of the latent hyperbolic networkcoordinates in the Poincaré disc in terms of their popularity dimension (the radial coordinate r ) and the similarity dimension(the angular coordinate θ ). In the context of financial networks, we hypothesized that the popularity dimension of a given bankaligns with its systemic importance, and that its similarity dimension is associated with sub-sectors of the banking system, e.g.,along geographic and regional divisions. Due to the asymmetric distribution of banks within the Poincaré disc (Figure 2), weslightly adapt the model Papadopoulos et al. and calculate the geodesic polar coordinates ( r (cid:48) i , θ (cid:48) i ) with respect to the networkcenter-of-weight, rather than the center of the Poincaré disc; see Methods for details. Rank 2014 2016 20181 Nordea * BNP Paribas * Groupe BPCE *2 Royal Bank of Scotland * UniCredit * Barclays *3 Barclays * ING Groep * Royal Bank of Scotland4 Intesa Sanpaolo Deutsche Bank * Groupe Crédit Agricole *5 UniCredit * Intesa Sanpaolo BNP Paribas *
Table 1.
For each year the five banks with the highest hyperbolic centrality(i.e. smallest r (cid:48) coordinate) are listed. Asterisks denote banks that areconsidered globally systemic relevant institutions (G-SIBs) To test the first hypothesis – the associa-tion between radial coordinate r (cid:48) and systemicimportance – we labelled a bank as system-ically important in a given year, wheneverit was included in the contemporaneous listof global systemically important banks (G-SIBs) as published by the Financial Stabil-ity Board. ; see also Table 3. Using aWilcoxon–Mann–Whitney test, we find a sig-nificant association between radial rank andsystemic importance in all years ( P < . P < . P = . r (cid:48) ) for each year. igure 2. Hyperbolic Embedding of the EBA Financial Network of 2018. Nodes are labelled by country and bank ID andcoloured according to region (see Table 3 for full names). Panel A shows the full network including the strongest links (topdecile), i.e., the connections with the largest liquidity-weighted portfolio overlap. Banks labelled as systemically important bythe Financial Stability Board (G-SIBs) are indicated by asterisks. The black cross marks the capital-weighted hyperbolic centerof the banking network. In panels B and C the Central/Eastern and the Nordic regional groups are highlighted to illustrateregional clustering. o test the second hypothesis – the association between similarity dimension θ (cid:48) and regional banking sub-sectors – weassigned banks to the following nine regional groups:Spain (ES), Germany (DE), France (FR), Italy (IT), UK and Ireland (UK/IE), Nordic Region (EE/NO/SE/DK/FI/IS),Benelux Region (BE/NE/LU), Southern/Mediterranean (GR/CY/MT/PT), Central and Eastern Europe (AT/BG/HU/LV/RO/SI).These regions are reasonably balanced in terms of the number of banks included in the EBA panel. Using ANOVA for circulardata [30, Sec. 7.4] we find a highly significant association between the angular coordinate θ (cid:48) and the regional group in allthree years considered ( P < . Network structure over time
The longitudinal structure of the data set allows us to track changes in the network structure over the whole time span ofobservations from 2014 to 2018. Note, however, that the samples of banks included by the EBA vary substantially in size and –even when restricted to the smallest sample – are not completely overlapping; see Table 2. Nevertheless, the goodness-of-fit ofthe hyperbolic model (reported in Figure 1.B) is surprisingly stable over all years. This suggests that the hyperbolic model doesindeed capture intrinsic qualities of the network, rather than relying on transitory structural artefacts.
Figure 3.
Changes in radial coordinate r (cid:48) (low values indicate highcentrality) between 2014 and 2016 (A) and 2016 and 2018 (B). Banksconsidered systemically relevant (G-SIBs) at the end of the time period aremarked in red. Nordea bank is circled in the panel A; see text forbackground.We proceed to analyze the temporal changes inthe latent radial coordinate r (cid:48) and angular coor-dinate θ (cid:48) , corresponding to changes in systemicimportance and community structure. Note thatthe small sample of banks included in the 2016stress test restricts the number of banks thatare included in this longitudinal analysis, cf.Table 2. The scatter plots in Figure 3 and thecorresponding Pearson’s correlations of . ( P < . ) and . ( P = . ) show aclear positive association between hyperboliccentrality in successive snap shots of the finan-cial networks. In panel A of Figure 3, Nordeabank can be identified as a clear outlier, mov-ing from a very central position in 2014 to a pe-ripheral position in 2016. Interestingly, Nordeawas one of just two banks (together with RoyalBank of Scotland) which were removed fromthe list of G-SIBs in the subsequent update in2018 due to decreasing systemic importance. For the angular coordinate, we account for the circular nature of the variable and compute the circular correlation of theangular coordinates between successive years. A moderate association between successive years can be observed at circularcorrelation values of 0 .
211 between 2014 and 2016 ( P = . ) and 0 .
225 between 2016 and 2018 ( P = . Discussion
Based on data from the EBA stress tests of 2014, 2016 and the transparency exercise of 2018, we have presented strong evidencethat the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolicgeometry. Calculating stress-minimizing embeddings into the Poicaré disc model of hyperbolic geometry has allowed us tovisualize this geometric structure and to connect it to the popularity-vs-similarity model of Papdopoulos et al. We find that theradial coordinate ( ‘popularity’ ) is strongly associated with systemic importance (as assessed by the Financial Stability Board)and the angular coordinate ( ‘similarity’ ) with geographic and regional subdivisions. A longitudinal analysis shows that – in theobservation period from 2014 to 2018 – systemic importance of banks within the European banking network has stayed ratherstable and has been predominated by only gradual changes. The peripheral community structure has been more variable, but as remained strongly determined by geographical divisions in all years considered.In future research we plan to study the interplay between hyperbolic network geometry and the dynamics of contagion processes.We are confident, that the empirical analysis of latent network geometry in this paper can provide the basis for new analyticmodels for the diffusion of financial stress in a banking network with hyperbolic structure.
Methods
Data Preparation and Inference of Financial Networks
The financial networks were extracted from three different publicly available data sets stemming from the stress tests (in 2014and 2016) and the EU-wide transparency exercise (in 2018) of the European Banking Authority (EBA).
The data setscontain detailed balance sheet information from all European banks (EU + Norway) included in the stress test/transparencyexercise of the EBA in the respective year. From these data sets we extracted the portfolio values of all sovereign bonds held bythe banks, split by issuing country (38 countries) and three levels of maturity (short: 0M-3M, medium: 3M-2Y, long: 2Y-10Y+),resulting in m = × =
114 different asset classes. n )
119 51 128of which included in the subseq. year 43 41
Table 2.
Sample sizes of EBA data setsFor each year, this data was stored as the weighted ad-jacency matrix P (‘portfolio matrix’) of a bipartite network.The n rows of P correspond to the banks in the sample, the m columns to the different asset classes, and the element P ik to the portfolio value (in EUR) of asset k in the balancesheet of bank i . To perform a one-mode projection of thisbipartite network, we followed Cont and Wagalath aswell as Cont and Schaanning : We computed the liquidity-weighted portfolio overlap (LWPO) of bank i and bank j as L i j = m ∑ k = P ik P jk d k , (1)where d k is the market depth for asset k . The LWPO measures the impact of a sudden liquidation of the portfolio of bank i onthe portfolio value of bank j and vice versa. Hence, it quantifies the risk of fire-sale contagion between the banks in a financialstress scenario. The market depth of asset k was estimated from P as its total volume held by all banks in the sample, i.e., as d k = ∑ ni = P ik . Writing D for the diagonal matrix of market depths, (1) can be succinctly written as matrix product L = PD − P (cid:62) .Finally, we set the link weight w i j between bank b i and b j in the one-mode projection N of the banking network equal to thenormalized LWPO between banks b i and b j , i.e., w i j : = L i j / max i , j L i j Background on hyperbolic geometry
The hyperboloid model
Hyperbolic geometry can be characterized as the geometry of a space of constant negative curvature, while the more familiarEuclidean geometry is the geometry of a flat space, i.e. a space of zero curvature. In the hyperboloid model of hyperbolicgeometry , d -dimensional hyperbolic space H d is defined as the hyperboloid H d = (cid:110) x ∈ R d + : x − x − · · · − x d = , x > (cid:111) equipped with distance d H ( x , y ) = arcosh ( x y − x y − · · · − x d y d ) . In fact, H d endowed with the Riemannian metric tensor ds = dx − dx − · · · − dx d is a Riemannian manifold and d H ( x , y ) isthe corresponding Riemannian distance. The sectional curvature of this manifold is constant and equal to −
1. Thus, H d isindeed a model of geometry of constant negative curvature. The Poincaré disc model
While the hyperboloid model is convenient for computations, a more preferable (and popular) model for visualizationsin dimension d = Poincaré disc model , which also forms the basis of the popularity-vs-similarity model ofPapadopoulos et al. . To obtain the Poincaré disc model, the hyperboloid H is mapped to the open unit disc (‘Poincaré disc’) D = (cid:8) z ∈ R : z + z < (cid:9) , parameterized by polar coordinates as z = r cos θ , z = r sin θ , using the stereographic projection (cf. [32, §4.2]) r = (cid:114) x − x + , θ = atan’ ( x , x ) , x = ( x , x , x ) ∈ H , (2) here atan’ is the quadrant-preserving arctangent. ∗ In the Poincaré disc model, the hyperbolic distance becomesd B (( r , θ ) , ( r , θ )) = arcosh (cid:18) + r + r + r r cos ( θ − θ )( − r )( − r ) (cid:19) and geodesic lines are represented by arcs of (Euclidean) circles intersected with D . Hyperbolic Embedding and Centering
Embedding into Hyperbolic Space
Network embedding methods aim to find – for each network node b i – latent coordinates x i in a geometric model space G , suchthat the geodesic distance between x i and x j in G matches – as closely as possible – a given dissimilarity measure d i j betweennodes b i and b j . Stress-minimizing embedding methods aim to minimize the stress functionalStress ( x , . . . , x n ) = (cid:115) n ( n − ) ∑ i , j (cid:16) d network i j − d geom G ( x i , x j ) (cid:17) , (3)which measures the root mean square error between given network dissimilarities and the corresponding distances in themodel space. For Euclidean geometry, this method is well-known as multidimensional scaling , or – using a weightedstress functional – as Sammon mapping . For hyperbolic space, i.e., when d geom G = d H , several optimization methods for (3)have been proposed . We use the hydra+ method implemented in the package hydra for the statistical computingenvironment R . Hyperbolic Centering
For a point cloud x , . . . , x n in H d and non-negative weights w , . . . , w n summing to one, the hyperbolic mean or hyperboliccenter of weight can be determined as follows: Calculate the weighted Euclidean mean ¯ x = ∑ w i x i , and its ‘resultant length’ ρ = (cid:112) ( ¯ x ) − ( ¯ x ) − · · · − ( ¯ x d ) , which is a measure of dispersion for the point cloud. The hyperbolic center c is thendetermined as c = ¯ x / ρ and is again an element of H d . The point cloud can be centered at c by transforming each points as ( x i ) (cid:48) = T − c x i , where T c is the hyperbolic translation matrix (‘Lorentz boost’) T c = (cid:18) c ¯ c (cid:62) ¯ c (cid:112) I d + ¯ c ¯ c (cid:62) (cid:19) with c = ( c , ¯ c ) = ( c , c , . . . , c d ) . In dimension d =
2, the stereographic projection (2) may then be applied to convert the centered coordinates ( x i ) (cid:48) to centeredpolar coordinates ( r (cid:48) i , θ (cid:48) i ) in the Poincaré disc. Application to Financial Networks
The described methods were applied to the financial networks inferred from the EBA data as follows: We converted thesimilarity weights w i j (normalized LWPO) to dissimilarities d i j = − w i j . We embedded these similarities by minimizing thestress functional (3), using the R-package hydra . For the resulting network embeddings, we calculated the capital-weightednetwork center c as the weighted hyperbolic mean with weights w i proportional to the total capital ∑ mk = P ik of bank i investedin all assets a , . . . a m . After centering at the hyperbolic center c , we calculated the coordinates ( r (cid:48) i , θ (cid:48) i ) by the stereographicprojection (2). Data Availability Statement
The data analysed during the current study are available from the website of the European Banking Authority at and https://eba.europa.eu/risk-analysis-and-data/eu-wide-transparency-exercise/2018 . References Shavitt, Y. & Tankel, T. On the curvature of the internet and its usage for overlay construction and distance estimation. In
IEEE INFOCOM 2004 , vol. 1 (IEEE, 2004). Muscoloni, A., Thomas, J. M., Ciucci, S., Bianconi, G. & Cannistraci, C. V. Machine learning meets complex networks viacoalescent embedding in the hyperbolic space.
Nat. communications , 1–19 (2017). ∗ The quadrant-preserving arctangent atan’ ( x , x ) , well-defined unless x = x =
0, returns the unique angle θ ∈ [ , π ) which solves tan θ = x / x andpoints to the same quadrant as ( x , x ) . It is commonly implemented in scientific computing environments (e.g. in MATLAB or R ) as atan2 . . Alanis-Lobato, G., Mier, P. & Andrade-Navarro, M. A. Manifold learning and maximum likelihood estimation forhyperbolic network embedding.
Appl. network science , 1–14 (2016). Papadopoulos, F., Kitsak, M., Serrano, M. Á., Boguná, M. & Krioukov, D. Popularity versus similarity in growing networks.
Nature , 537 (2012). Papadopoulos, F., Psomas, C. & Krioukov, D. Network mapping by replaying hyperbolic growth.
IEEE/ACM Transactionson Netw. (TON) , 198–211 (2015). Barabasi, A.-L. Luck or reason.
Nature , 507–509 (2012). Brockmann, D. & Helbing, D. The hidden geometry of complex, network-driven contagion phenomena.
Science ,1337–1342 (2013). Cont, R., Moussa, A. & Santos, E. B. Network structure and systemic risk in banking systems. In Jean-Pierre Fouque, J.A. L. (ed.)
Network Structure and Systemic Risk in Banking Systems (Cambridge University Press, 2010). Battiston, S., Gatti, D. D., Gallegati, M., Greenwald, B. & Stiglitz, J. E. Liaisons dangereuses: Increasing connectivity, risksharing, and systemic risk.
J. economic dynamics control , 1121–1141 (2012). Roukny, T., Bersini, H., Pirotte, H., Caldarelli, G. & Battiston, S. Default cascades in complex networks: Topology andsystemic risk.
Sci. reports , 2759 (2013). French, K. et al.
The Squam Lake report: fixing the financial system.
J. Appl. Corp. Finance , 8–21 (2010). European Banking Authority. EU-wide transparency exercise. https://eba.europa.eu/risk-analysis-and-data/eu-wide-transparency-exercise/2018.
Caccioli, F., Farmer, J. D., Foti, N. & Rockmore, D. Overlapping portfolios, contagion, and financial stability.
J. Econ.Dyn. Control. , 50–63, DOI: 10.1016/j.jedc.2014.09.041 (2015). Eisenberg, L. & Noe, T. H. Systemic risk in financial systems.
Manag. Sci. , 236–249 (2001). Brown, M., Trautmann, S. T. & Vlahu, R. Understanding bank-run contagion.
Manag. Sci. , 2272–2282 (2017). Shleifer, A. & Vishny, R. W. Liquidation Values and Debt Capacity: A Market Equilibrium Approach.
The J. Finance ,1343–1366, DOI: 10.1111/j.1540-6261.1992.tb04661.x (1992). Glasserman, P. & Young, H. P. How likely is contagion in financial networks?
J. Bank. Finance , 383–399, DOI:10.1016/j.jbankfin.2014.02.006 (2015). Cont, R. & Schaanning, E. Fire sales, indirect contagion and systemic stress testing (2017). Norges Bank Working Paper02/2017.
Cont, R. & Wagalath, L. Fire sales forensics: measuring endogenous risk.
Math. Finance , 835–866 (2016). Kruskal, J. B. & Wish, M.
Multidimensional scaling , vol. 11 (Sage, 1978).
Chowdhary, K. & Kolda, T. G. An improved hyperbolic embedding algorithm.
J. Complex Networks , 321–341 (2017). Keller-Ressel, M. & Nargang, S. Hydra: a method for strain-minimizing hyperbolic embedding of network-and distance-based data.
J. Complex Networks , cnaa002 (2020). Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. & Boguná, M. Hyperbolic geometry of complex networks.
Phys.Rev. E , 036106 (2010). Boss, M., Elsinger, H., Summer, M. & Thurner 4, S. Network topology of the interbank market.
Quant. finance , 677–684(2004). Langfield, S., Liu, Z. & Ota, T. Mapping the UK interbank system.
J. Bank. & Finance , 288–303 (2014). Mardia, K. V. & Jupp, P. E.
Directional statistics (John Wiley & Sons, 2009). Cont, R. & Wagalath, L. Running for the exit: distressed selling and endogenous correlation in financial markets.
Math.Finance: An Int. J. Math. Stat. Financial Econ. , 718–741 (2013). Ratcliffe, J. G.
Foundations of hyperbolic manifolds , vol. 3 (Springer, 1994).
Cannon, W. J., Floyd, W. J., Kenyon, R. & Parry, W. R. Hyperbolic geometry. In Silvio Levy (ed.)
Flavors of Geometry ,59–115 (MSRI Publications, 1997), 31 edn.
Borg, I. & Groenen, P. Modern multidimensional scaling: Theory and applications.
J. Educ. Meas. , 277–280 (2003). Sammon, J. W. A nonlinear mapping for data structure analysis.
IEEE Transactions on computers , 401–409 (1969).
Zhao, X., Sala, A., Zheng, H. & Zhao, B. Y. Fast and scalable analysis of massive social graphs. arXiv preprintarXiv:1107.5114 (2011).
R Core Team.
R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing, Vienna,Austria (2019).
Galperin, G. A concept of the mass center of a system of material points in the constant curvature spaces.
Commun. Math.Phys. , 63–84 (1993).
Author contributions statement
M.K.R. conceived the study, S.N. prepared the data, M.K.R. and S.N. analysed the results. All authors reviewed the manuscript.
Additional information
The author(s) declare no competing interests.
T01 Erste Group Bank AGAT08 BAWAG Group AGAT09 Raiffeisen Bank International AGAT10 Raiffeisenbankengruppe Verbund eGenAT11 Sberbank Europe AGAT12 Volksbanken VerbundBE01 Belfius Banque SABE02 Dexia NVBE04 AXA Bank Europe SABE06 KBC Group NVBE07 The Bank of New York Mellon SA/NVBE08 InvestarBG01 First Investment BankCY01 Hellenic Bank Public Company LtdCY04 Bank of Cyprus Holdings Public Limited CompanyCY05 RCB Bank LtdDE01 NRW.BankDE02 Deutsche Bank AG *DE03 Commerzbank AGDE04 Landesbank Baden-WürttembergDE05 Bayerische LandesbankDE06 Norddeutsche Landesbank-GirozentraleDE07 Landesbank Hessen-Thüringen GirozentraleDE08 DekaBank Deutsche GirozentraleDE09 Aareal Bank AGDE10 Deutsche Apotheker- und Ärztebank eGDE11 HASPA FinanzholdingDE14 Landeskreditbank Baden-Württemberg-FörderbankDE15 Landwirtschaftliche RentenbankDE16 Münchener Hypothekenbank eGDE20 DZ Bank AG Deutsche Zentral-GenossenschaftsbankDE25 Deutsche Pfandbriefbank AGDE26 Erwerbsgesellschaft der S-Finanzgruppe mbH & Co. KGDE27 HSH Beteiligungs Management GmbHDE28 State Street Europe Holdings Germany S.à.r.l. & Co. KGDE29 Volkswagen Bank GmbHDK01 Danske BankDK02 Jyske BankDK03 SydbankDK05 Nykredit RealkreditEE01 AS LHV GroupES01 Banco Santander *ES02 Banco Bilbao Vizcaya ArgentariaES03 Banco de SabadellES04 Banco Financiero y de AhorrosES07 Caja de Ahorros y M.P. de ZaragozaES08 KutxabankES09 LiberbankES11 MPCA RondaES12 Caja de Ahorros y Pensiones de BarcelonaES15 BankinterES18 Abanca Holding Financiero, S.A.ES19 Banco de Crédito Social Cooperativo, S.A.FI01 OP-Pohjola GroupFI02 Kuntarahoitus OyjFR01 La Banque PostaleFR02 BNP Paribas *FR03 Société Générale *FR06 C.R.H. - Caisse de Refinancement de l’HabitatFR08 RCI BanqueFR09 Société de Financement LocalFR12 Groupe Crédit MutuelFR13 Banque Centrale de Compensation (LCH Clearnet)FR14 Bpifrance (Banque Publique d’Investissement)FR15 Groupe BPCE *FR16 Groupe Crédit Agricole * GR01 Eurobank ErgasiasGR02 National Bank of GreeceGR03 Alpha BankGR04 Piraeus BankHU01 OTP Bank LtdIE04 AIB Group plcIE05 Bank of Ireland Group plcIE06 Citibank Holdings Ireland LimitedIE07 DEPFA BANK PlcIS01 Arion banki hfIS02 Íslandsbanki hf.IS03 LandsbankinnIT01 Intesa Sanpaolo S.p.A.IT02 UniCredit S.p.A. *IT03 Banca Monte dei Paschi di Siena S.p.A.IT04 Unione Di Banche Italiane Società Cooperativa Per AzioniIT05 Banca Carige S.P.A. - Cassa di Risparmio di Genova e ImperiaIT07 Banca Popolare Dell’Emilia Romagna - Società CooperativaIT09 Banca Popolare di SondrioIT13 Mediobanca - Banca di Credito Finanziario S.p.A.IT16 Banco BPM Gruppo BancarioIT17 Credito Emiliano Holding SpAIT18 Iccrea Banca Spa Istituto Centrale del Credito CooperativoLU01 Banque et Caisse d’Epargne de l’EtatLU02 Precision Capital S.A.LU03 J.P. Morgan Bank Luxembourg S.A.LU04 RBC Investor Services Bank S.A.LU05 State Street Bank Luxembourg S.A.MT01 Bank of Valletta plcMT02 Commbank Europe LtdMT03 MDB Group LimitedNL01 Bank Nederlandse Gemeenten N.V.NL02 Coöperatieve Centrale Raiffeisen-Boerenleenbank B.A.NL03 Nederlandse Waterschapsbank N.V.NL07 ABN AMRO Group N.V.NL08 ING Groep N.V. *NL09 Volksholding B.V.NO01 DNB Bank GroupNO02 SPAREBANK 1 SMNNO03 SR-bankPL01 PKO BANK POLSKIPL07 Bank Polska Kasa Opieki SAPT01 Caixa Geral de DepósitosPT02 Banco Comercial PortuguêsPT04 Caixa Central de Crédito Agrícola Mútuo, CRLPT05 Caixa Económica Montepio Geral, Caixa Económica Bancária SAPT06 Novo Banco, SARO01 Banca TransilvaniaSE01 Nordea Bank AB (publ) †SE02 Skandinaviska Enskilda Banken AB (publ) (SEB)SE03 Svenska Handelsbanken AB (publ)SE04 Swedbank AB (publ)SE05 Kommuninvest - groupSE06 Länsförsäkringar Bank AB - groupSE07 SBAB Bank AB - groupSI02 Nova Ljubljanska banka d. d.SI04 Abanka d.d.SI05 Biser Topco S.à.r.l.UK01 Royal Bank of Scotland Group plc †UK02 HSBC Holdings plc *UK03 Barclays plc *UK04 Lloyds Banking Group plcUK05 Nationwide Building SocietyUK06 Standard Chartered Plc *
Table 3.