Default contagion risks in Russian interbank market
aa r X i v : . [ q -f i n . R M ] J a n Default contagion risks in Russian interbankmarket
A.V. Leonidov ∗ and E.L. Rumyantsev † Theoretical Physics Department, P.N. Lebedev Physical Institute, Moscow Chair of Discrete Mathematics, Moscow Institute of Physics and Technology Center for the Study of New Media & Society, New Economic School, Moscow Laboratory of Social Analysis, Russian Endowment for Education and Science,Moscow Department of Financial Stability, Bank of Russia, Moscow
Abstract
Systemic risks of default contagion in the Russian interbank mar-ket are investigated. The analysis is based on considering the bow-tiestructure of the weighted oriented graph describing the structure ofthe interbank loans. A probabilistic model of interbank contagion ex-plicitly taking into account the empirical bow-tie structure reflectingfunctionality of the corresponding nodes (borrowers, lenders, borrow-ers and lenders simultaneously), degree distributions and disassorta-tivity of the interbank network under consideration based on empiricaldata is developed. The characteristics of contagion-related systemicrisk calculated with this model are shown to be in agreement withthose of explicit stress tests. ∗ Also at ITEP, Moscow † The material presented does not necessarily reflect the position of the Bank of Russiaon the issues under discussion. , in which the underlying interbanknetwork was considered as a weighted directed Poissonian random graphwith neither clustering nor degree-degree correlations taken into account.The focus of [15] was on systemic risk associated with the percolation phasetransition and formation of giant cluster and the related robust-yet-fragileproperty of default contagion where a small probability of a catastrophicevent goes together with its huge volume.The research following [15] was to a large extent aimed at checking theassumptions on the properties of interbank network made in [15] againstexisting data and, if necessary, incorporating the corresponding modificationsinto the theoretical formalism describing default propagation. In terms ofnetwork properties the question is thus on degree distributions, degree-degreecorrelations and clustering in realistic interbank networks, while theoreticallya central issue is that of an interplay between topological properties of anetwork and those of epidemic cascades on it.Analysis of empirical properties of the Russian interbank network wasmade in Refs. [16, 17] for the period of 01.08.2011-03.11.2011 and in Refs. [18,19] for the period of 10.2004-08.2008. It was found that this network ischaracterized by a heavy-tailed degree distribution [16, 17, 18, 19], heavy-tailed distribution of exposures [18, 19], pronounced disassortative degree-degree correlations and significant anomalous clustering [16, 17] . In [21] See also an interesting comment [10]. Similar results for degree distribution, degree-degree correlations and clustering werefound for the Brasilian interbank market in [20]. .The main objectives of this paper are the development of the above-described analytical model based on an enlarged set of empirical data on theRussian interbank market.The plan of the paper is as follows.In section 1 we discuss the structure and main characteristics of the Rus-sian money market and the data used in the analysis. In the section 2 weanalyze empirical characteristics of the deposit market from the network per-spective including its bow-tie structure, degree distributions and correlations,clustering and default propagation probabilities. In section 3 we explore thestructure of the default cluster caused by the default of a randomly cho-sen bank and after that introduce mathematical formalism for systemic riskrepresentation in chapter 4. Conclusions are presented in section 5. We are grateful to C. Borgs for pointing out this reference. Russian money market: empirics and datadescription
Russian money market consists of three main segments, the markets of de-posits, REPO and SWAP.Operations at the deposits market are uncollateralized: banks bound theirrisk in lending money to counterparties by setting limits calculated with thehelp of in-house models and taking into account expert opinions. Lendingrisks are also regulated by the special requirement of the Bank of Russiaconstraining the value of exposure to a counterparty. The uncollaterizednature of the deposit market makes it the most vulnerable with respect totrust evaporation during crises when deposit markets can freeze requiringsignificant efforts from regulators for their relaunching.Less risky is the REPO market at which collateral, usually governmentand corporate bonds and equities, is required. In operations with moneyborrowing the value of different types of collateral includes discounts thusreducing the corresponding market risk. Credit risk is more often accountedfor in the credit rate. Let us note that while at the beginning of the Russiancrisis in 2008 the REPO market in Russia did collapse, it started functioningfaster than the deposit one .The least risky is the SWAP market. Often swap operations are used asa source of short-term ruble liquidity when in exchange of rubles a lendergets foreign collateral (most often USD and EUR). SWAP operations wereattractive for Russian banks during the period of systemic liquidity deficitwhen banks are permanently in need of liquidity refinancing from centralbank.We provide comparative statistics for outstanding in different segments ofthe Russian money market in Table 1. For deposit market it includes claimson banks (resident and non-resident) in deals with Russian rubles. For SWAPmarket it includes claims on resident and non-resident banks in rubles againstUSD and euro as collateral. It is worth saying that the data includes dealsbetween banks at the OTC market as well as through Central Counterparty(CCP), the latter option was used in almost 25% of swap deals. The Repomarket data includes claims on banks in rubles. In 2013 the project of REPOdeals through Central Counterparty was launched. The total outstanding onCCP at the end of 2013 year was 1.5 bln. USD. To highlight the importanceof money market we also provide the total value of assets of the Russian The collapse of deposit and REPO market took place after the Lehman Brothersdefault. Due to efforts of Bank of Russia and Ministry of Finance the Repo market waspartly retrieved in two weeks whereas the credit risks still remained too high. \ Year 2011 2012 2013Deposit 52.6 54.1 55.0REPO 5.4 6.4 5.2SWAP 94.5 84.4 101.2Total assets 1318.9 1609.6 1742.6Table 1: Volumes of deposit, REPO and SWAP markets and total market ofbanking activities in Russia in Bln. USD [30]. Total assets of banking systemare given to compare the volume of money market with that of other bankingactivities such as corporate and retail lending, investments in securities, andothers.banking system. Some information concerning evolution of the outstandingfor different money market segments can be found in the Bank of Russia”Money market report” [30].We based our analysis on daily CBRF banking report ”Operations oncurrency and money markets” [31] containing information on all type oftransactions carried out on the OTC money market . In our analysis weconcentrate on the deposit market and take into account only uncolleter-alised deals in Russian rubles between residents. Taking into account onlythe deals between residents is due to data limitations. We also exclude dealswith Central Bank of Russia and its branches because corresponding rubleobligations are always met and therefore do not generate any risk.Analysis of other segments of the interbank market goes beyond the scopeof this paper. For an interesting analysis of the properties of the multilayernetwork including REPO, SWAP and deposit interbank markets in Italy seeRef. [32] where it was found that all segments of interbank market share suchcommon features as fat-tailed degree distributions, disassortativity and largevalues of clustering coefficients. In terms of contagion we would expect thattaking into account other money market segments may significantly influencethe corresponding exposures and amplify the volume of contagion.The data used in our analysis cover the interval from January 11 2011 tillDecember 30 2013 and contain information on interbank loans to residents of185 banks. This data corresponds to roughly 80 % of the total outstandingand can therefore be considered as representative.Having information on interbank transaction we transformed it into out- A decentralized market, without a central physical location, where market participantstrade with each other through various communication modes such as telephone, emailand proprietary electronic trading systems. An over-the-counter (OTC) market and anexchange market are the two basic ways of organizing financial markets.
For the purposes of our analysis we view the interbank credit market asa weighted oriented graph characterized by the weighted adjacency matrix W = { w ij ≥ } where link variables w ij > i towards the bank j and are computed by netting the mutualobligations of both banks on the daily basis so that a directed link i → j cor-responds to a credit to i provided by j . For a given node outgoing (out-type)links correspond therefore to its obligations towards neighboring nodes andincoming (in-type) ones to claims of the node under consideration towardsneighboring nodes. Let us note that with this choice of notations defaultcascades triggered by some initial node propagate along link’s direction. In describing systemic risks related to network topology of the interbank mar-ket it is essential to take into account the gross structure of the correspondingoriented graph represented by its bow-tie decomposition, see e.g. [33]. In theproblem under consideration, following the above-described definition of theweighted adjacency matrix W , the bow-tie structure separates, on the dailybasis, the nodes (banks) into four groups according to the type of their op-erations. The Out- and In- components include nodes having only outgoingand incoming links correspondingly, i.e. include pure borrowers and lendersrespectively. The In-Out- component includes banks having both incoming6
011 2012 2013 2014
Date O u t s t and i ng , b l n . R ub . N u m be r o f c oun t e r pa r t i e s Total outstandingOutstanding 1 − 7 daysNumber of counterparties
Figure 1: Deposit market term structure outstandingand outgoing links which are therefore both creditors and lenders. We willshow that banks belonging to this component play a crucial role in generatingsystemic risks. The last group consists of nodes without links. The bow-tiestructure includes as a particular important case the core-periphery modelwith its core belonging to the In-Out- component and periphery to the In-and Out- ones.The structural analysis of the data on interbank network shows that mostof the banks having links (60%-70%) belong to the In- component, i.e. actas pure lenders while only 10%-20% belong to the Out- component and actas pure borrowers. The number of pure borrowers and lenders displays pro-nounced seasonality so that the number of the former increases and of thelatter decreases at the beginning of the year. As to the structure of the out-standing, it is predominantly concentrated in the In-Out- component (60-70%7f total outstanding) so that the the corresponding banks have a persistenttendency of borrowing (lending) within the In-Out- component. Less im-portant from exposure point of view are the links between the In-Out andIn components (20-35% of total outstanding) and the Out and In-Out ones(2-20% of total outstanding). The links between the pure borrowers andpure lenders contain less than 5% of the total outstanding. The In-Out com-ponent contains a strongly connected one (SCC) with the size varying from10% to 15% and carrying 40-60% of the total outstanding demonstrating asignificant monopolistic power of several influential actors.
In this paragraph we discuss some most important quantitative characteris-tics of the Russian interbank network such as distributions in the number ifincoming and outgoing links, correlations in the degrees of neighboring nodesand degree of clustering.The distributions of in- and out- degrees are plotted in Figs. 2 and 3.We see that these distributions are fat-tailed . This feature is in agreementwith other results on interbank networks in the literature, see e.g. [20] andreferences therein. − − − − − − log(Number of borrowers) l og ( p ) log ( p ) =- log ( k In ) + Figure 2: Number of borrowersdistribution − − − − − log(Number of lenders) l og ( p ) log ( p ) =- log ( k Out ) + Figure 3: Number of lenders dis-tributionAn analysis shows, in agreement with the results of [20], that Russian The small number of observations in the fat-tailed part of the distributions doesn’tallow us to test a hypothesis on its power-law nature. According to [25] the sufficientnumber of observations is around 1000 while we have only 50-60. P A → B ( k, l | m, n ), where A, B refer to thecomponent of the bow-tie decomposition { I , IO , O } denoting In, In-Out andOut components respectively and the indices k, l and m, n denote the in- andout-degrees of the adjacent vertices, see Fig. 4 in which P IO → IO ( k, l | m, n ) isshown. In section 3 we will see that in order to provide a good description ofFigure 4: Bivariate distribution P IO → IO ( k, l | m, n )the empirical systemic risk characteristics one has to take into account theprobability patterns described by P IO → IO ( k, l | m, n ), etc.The interbank network is characterized by significant clustering. Thisfeature is illustrated in Fig. 5, in which time series for clustering coefficient and link probability (defined as the ratio the number of links in network tothe maximal possible number of links in network) are shown. We see that thegraph under consideration is significantly less sparse than the Erdos-Renyione fully specified by the link probability for which the clustering coefficientis simply equal to link probability. Here for simplicity we consider the graph under consideration as unoriented. . . . . . . . − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − . . . . . . . Link probabilityClustering coefficient
Figure 5: Evolution of clustering coefficient and link probability
Systemic credit risk is defined as that of cascade default of several bankstriggered by default of one or several banks on its obligations . A default isoperationally defined as an event where the Capital Adequacy Ratio (CAR)defined below in Eq. (1) falls below the minimal threshold defined by theregulator, the Central Bank of Russian Federation . The general definitionof CAR reads [34]: CAR = K − P i =1 P i P i =1 ( A i − P i ) · RW i + O , (1)where K stands for the capital, A i denotes the i -th group of assets , RW i is the corresponding risk weight and P i the corresponding provision for non- In the present study we restrict our analysis to cascades trigged by the default of asingle bank. This is also a legal reason for the Bank of Russia to revoke a license. The regulator breaks all assets into 5 types [30]. O is a collective notationfor other risk variables such as market and operational risks which also usedunder CAR calculation. As the present analysis concentrates on risks relatedto interbank loans for which the risk weight RW IC is equal to 20%, in whatfollows we shall use the following simplified version of Eq. (1): CAR = K − P IC . · ( A IC − P IC ) + ˜ O , (2)where A IC and P IC denote interbank claims and corresponding regulatorspecified provision respectively and ˜ O denotes risk variables additional tothose characterizing interbank credit obligations.After a counterparty defaults on its obligations its lenders, in accordancewith Russian regulatory document on rules for provision forming, have toform loan loss provision on deals with this counterparty. According to Rus-sian legislation banks have to establish provisions in accordance with bor-rower’s quality and quality of debt servicing. In our simulations we willassume a provision of 100% and take into account only provision for inter-bank deposit market operations considering the volume of other operationsas fixed.To assess systemic risk we calculate the size of default cluster triggeredby the bankruptcy of a particular bank. The stress-testing procedure we useis as follows: • A default of a particular bank is assumed. All its creditors form provi-sion on deals with this default counterparty and recalculate their CAR. • We check whether the new CARs meet regulatory minimum (10% fordeposit taking banks which are allowed to attract deposit from indi-viduals and 12% for other non-banking activities like depositary, set-tlement and payment). • The procedure is repeated for those creditors for which their CARs fallbelow the regulatory minimum.The procedure is repeated for each bank from Out- and In-Out- com-ponents . The resulting probability distribution over the size of defaultclusters, where size is defined as a number of banks defaulting as a conse-quence of the default of an initial default node, is shown, on the annual basis,in Table 2 and Figs. 6 , 7 (in the latter - on the log-linear scale). A first It is clear that for the type of contagion under study banks belonging to empty andIn- components do not generate any systemic risk. \ S d . . . . . . . Default cluster size P r obab ili t y Figure 6: Probability distribution fordefault cluster size − − − − − − Default cluster size l og ( P r obab ili t y ) Figure 7: Probability distribution fordefault cluster size, linear-log scaleimportant conclusion that can be drawn from the Table 2 and Figs. (6,7) isthat the distribution over the size of default clusters is approximately expo-nential. This is a natural consequence of the (approximately) exponentialdegree distribution of the nodes belonging to default clusters (not shown),see e.g. [26].Another important feature revealed by stress-testing is a higher impor-tance in terms of generating default cascades and, therefore, systemic risk,of the banks from In-Out component as compared with those from the Out-one. In Table 3 we show the percentage of cases in which a default of thenode under consideration triggers a default of another bank for the In-Out-and In- components (29 −
38% and 6 −
10% respectively). The contributionof the In-Out- component is clearly the dominant one so that in what followswe will neglect the contribution of the Out- component.From Table 3 we also see that the average stability of banks with respectto default risks as characterized by the average CAR underwent, between12omponent \ Year 2011 2012 2013In-Out 0.29 0.31 0.38Out 0.06 0.10 0.10CAR 14.7 13.7 13.5Table 3: Bow-tie breakup of percentage of cascades and CAR’s for Russianinterbank market in 2011-20132011 and 2013, a significant reduction. It is quite clear that lower values ofCAR generate larger systemic risks. Indeed, an analysis in [15] has shown adramatic dependence of contagion on capital reserves. A dependence of theaverage size of default cluster on CAR (calculated on the monthly basis) isshown in Fig. 8 . From Fig. 8 it is clearly seen that there indeed exist apronounced dependence of the magnitude of contagion on the capital reservesreflected by CAR.It is quite typical for contagion that its volume grows with increasingcentrality of the source node, see e.g. [14]. In Fig. 9 we plot a dependence ofthe average size of the default cluster on the out-degree of the bankrupt node(i.e. the number of lenders), where for each day and each bank we calculatethe number of lenders and the size of default cluster generated by this bankat this day and average over all days in a given year. We see that indeed thevolume of contagion increases with increasing out- degree and that for largeout-degrees the dependence in question is distinctly nonlinear so that thevolume of contagion shows a faster than linear growth with the out-degreeof the source node. From Fig. 9 one can also conclude that, following theabove-described reduction in CAR from 2011 to 2013, a volume of contagionhas dramatically grown within this period.The key question in developing a model for propagation of contagion isthat of topology of default clusters. In our simulation we found out thatdefault clusters combining vulnerable banks are, with very few exceptions,tree-like with the maximal length of branches equal to 4. In directed graphsthe simplest nontrivial motifs, triangles, can belong to two types, T1 and T2,differing by orientation of participating links, see Figs. 10 and 11 correspond-ingly. Let us consider for definiteness an example shown in Fig 11, wherethe starting event is a default of the bank A leading to f critical CAR lossof B, but leaving C with an admissible CAR of 10.4 % (with the regulatory The bottom and top of the box are the first and third quartiles and the band insidethe box is the median. The end of the low whisker is the lowest datum still within 1,5x interquartile range and the highest datum still within 1,5 x interquartile range of theupper quartile.” . . . . CAR A v e r age D e f au l t c l u s t e r s i z e Figure 8: Dependence of default cluster size on CARminimum set at 10 %). The consequent CAR loss of 0.6 % from the link B → C does, however lead to the default of C. Our simulations have shownthat, over all the days, the average number of triangles of the type T1 is0.06 and of the type T2 - 1.2. Both numbers are small as compared to totalnumber of default clusters per day considered in our simulations. The effec-tively tree-like structure of the default cluster that allows to neglect motifsin working out a mathematical model for default cascading. In particular,under this assumption propagation of vulnerability between the nodes canhappen only along a single link.The last key ingredient for modelling default propagation is to quantifyvulnerability of a given node with respect to default of at least one of itsneighbors. Vulnerability can most naturally be described as a probabilisticcharacteristic of link’s ability to transport contagion from one node to an-other. We call a link vulnerable if a default of the counterparty may leadto default of another counterparty through this link. The link vulnerabil-ity depends on the local geometry of a network. Generically it is defined14
20 40 60 80 100 120 140
Number of lenders A v e r age de f au l t c l u s t e r s i z e Figure 9: Dependence of default cluster size on the number of lendersas a conditional probability v ( r, s | k, l ) of default propagation from the nodewith in- and out-degrees r, s to the node with in- and out- degrees k, l . Inaddition vulnerability depends on the position of the two nodes under con-sideration within the bow-tie structure of the network so that probabilisticpattern of contagion propagation is specified by the conditional probabilities v IO → IO ( r, s | k, l ), v IO → In ( r, s | k, l ) and their more complex modifications. Thecorresponding empirical vulnerability distributions are found to differ a lot,see Refs [16, 9]. 15igure 10: Motifs of length 3 in de-fault cluster, type T1 Figure 11: Motifs of length 3 in de-fault cluster, type T2 The mathematical model we use to describe systemic risks on the Russianinterbank market is based on empirical findings described above in the Sec-tions 2, 3. Let us reiterate the main features of importance for descriptionof contagion process: • Degree distributions are fat-tailed. • The network is characterized by significant disassortative correlationsbetween adjacent nodes. • Because of significant differences in systemic risks related to the posi-tion of the infected node and its neighbors within the bow-tie structureit is natural to take this into account explicitly when building the math-ematical formalism . • Default clusters are tree-like.Let us consider a bank from the In-Out component with k + l outgoinglinks, where k of them lead to the In-Out- component and l to In- component Although the mathematical construction built in the present paper is quite naturaland is versatile enough to describe the empirical simulations of systemic risk, the questionof whether it can be simplified further is relevant and deserves further investigation. Aninteresting example of solving the problem of this kind can be found in Ref. [37]. and take a randomly chosen edge linking the chosen node toa node in the In- component which, in addition, has r − N k,l ( y ) N k,l ( y ) = ∞ X r P IO → In ( r | k, l ) (cid:0) − v IO → In ( r | k, l ) + y v IO → In ( r | k, l ) (cid:1) (3)where we have taken into account that an outgoing link under considera-tion can lead to a vulnerable bank with probability v IO → In ( r | k, l ) or safeone with probability 1 − v IO → In ( r | k, l ) and, through the conditional prob-ability P IO → In ( r | k, l ), the probabilistic interdependence of the degrees of As discussed above, nodes from the Out- component generate very small systemicrisks so that the corresponding effects will be neglected P IO → In ( r | k, l ) v IO → In ( r | k, l ).Let us now consider a randomly chosen edge linking two banks fromthe In-Out- component, see Fig. 12 b. The equation for the correspondinggenerating function M k,l ( x, y ) reads : M k,l ( x, y ) = ∞ X u,t,r P IO → IO ( u, t, r | k, l )(1 − v IO → IO ( u, t, r | k, l ))+ x ∞ X u,t,r P IO → IO ( u, t, r | k, l ) v IO → IO ( u, t, r | k, l )[ M u,t ( x, y )] u [ N u,t ( y )] t (4)The corresponding part of the default cluster can be described as a projectionof the initial network onto a graph in which a link in the original graphsurvives with the probability P IO → IO ( u, t, r | k, l ) v IO → IO ( u, t, r | k, l ).Let us define a generation function F ( x, y ) = ∞ P k,l P IO ( k, l ) x k y l for theprobability for a bank from the In-Out- component to have k and l firstneighbors from the In-Out- and In- components respectively. Then the gen-eration function for the number of vulnerable banks in the network is simply F ( { M kl ( x, y ) } , { N kl ( y ) } ) ≡ F ( M, N ) = ∞ X k,l P IO ( k, l ) [ M kl ( x, y )] k [ N kl ( y )] l (5)It is easy to see that for calculation of the mean default cluster size S onecan put y = x and compute a derivative of F ( M, N ) at point x = 1. Wehave Sdx = dF ( M, N ) y = x =1 = ∞ X k,l P IO ( k, l )( kdM k,l | x =1 + ldN k,l | x =1 ) , (6)where we have used the normalization property of generation functions M k,l | x =1 = N k,l | x =1 = 1. From Eq. (3) we get dN k,l | x =1 = ∞ X r P IO → In ( r | k, l ) v IO → In ( r | k, l ) dx, (7)so that dM k,l = ∞ X u,t α u,t,k,l dM u,t + γ k,l dx (8) The notations for the indices should be clear from Fig. 12 b. α u,t,k,l = ∞ X r uP IO → IO ( u, t, r | k, l ) v IO → IO ( u, t, r | k, l ) (9) γ k,l = ∞ X u,t,r P IO → IO ( u, t, r | k, l ) v IO → IO ( u, t, r | k, l )+ ∞ X u,t,r P IO → IO ( u, t, r | k, l ) v IO → IO ( u, t, r | k, l ) × t ∞ X r P IO → In ( r | u, t ) v IO → In ( r | u, t ) (10)It is useful to rewrite Eq. (8) in the operator form: dM = AdM + γdx, (11)where dM and γ are vectors of length k × l and A is a k × l, k × l matrixsize with the elements A ( k,l )( u,t ) = α u,t,k,l . For solution of Eq.(11) to exist itsmaximal eigenvalue λ max should satisfy λ max < . It reads: dM k,l = ∞ X u,t β k,l,u,t γ u,t dx (12)where β u,t,k,l is an element B ( u,t ) , ( k,l ) of the matrix B = ( I − A ) − . Equa-tion (12) is valid in the absence of a giant cluster and should be modified inthe percolative phase, see e.g. [35, 36].The final equation for the average default cluster size S following fromEqs. (6,7,8,11) reads S = dFdx (cid:12)(cid:12)(cid:12)(cid:12) x =1 = ∞ X k,l P IO ( k, l ) " k ∞ X u,t β k,l,u,t γ u,t + l ∞ X r ω k,l,r , (13)where ω k,l,r = P IO → In ( r | k, l ) υ IO → In ( r | k, l ) . (14)To calculate the average size of the default cluster we use empirical condi-tional probability distributions P IO → In ( r | k, l ), v IO → In ( r | k, l ), P IO → IO ( u, t, r | k, l ) In [9] it is shown that a more restrictive condition valid in the non-percolative regimereads ∞ P u,t A ( k,l )( u,t ) < k, l ). v IO → IO ( u, t, r | k, l ) that are calculated on the monthly basis. A compar-ison of the model predictions and results of stress testing for the averagedefault cluster size is shown in Fig.13. We see a very good agreement be-tween the model and experiment provided one takes into account correla-tions between the degrees of adjacent nodes captured by P IO → In ( r | k, l ) and P IO → IO ( u, t, r | k, l ) and a much poorer one when these correlations are ne-glected. The remaining deviations can be ascribed to using analytical ap-proximation appropriate to infinite graphs Another source of deviation isin neglecting triangles in default graph. . . . . . Empirical average default cluster size T heo r e t i c a l a v e r age de f au l t c l u s t e r s i z e With correlationsWithout correlations
Figure 13: Comparison of theoretical and empirical average default clustersize with and without accounting for degree-degree correlations As shown in section 3 the maximal number of branchings in default trees is 4. Conclusions
Let us formulate once again the main conclusions of the paper.In this study we have described a new analytical model of default con-tagion propagation taking into account the difference in the functional roleof the nodes (bow-tie decomposition), realistic in- and out- degree and linkweight distributions and disassortative degree-degree correlations. Its pre-dictions are shown to be in line with the results of Monte-Carlo simulations.Let us reiterate, that to build a successful model of contagion propagationin interbank networks one needs to combine empirical studies and adequatetheoretical framework. Empirical information of importance is related bothto topological characteristics of the interbank market network under con-sideration and to the process of contagion propagation from node to nodethat depends, in particular, on interplay between link and node character-istics (volume of loans and bank balance sheets respectively). It was shownthat very good description of default cascade simulation can be given in theformalism that explicitly takes into account degree distributions and degree-degree correlations and conditional probabilities of contagion propagationfrom one node to another. Let us mention that although a marked differencein conditional default probabilities for nodes belonging to different compo-nents and empirical simulation results highlighting the dominant role of thein-out component provide strong arguments for explicit account of the bow-tie structure in describing systemic contagion risks, a more detailed analysisof this issue (in particular going beyond considering the mean size of defaultclusters) is certainly desirable. We are planning to address this problemin a separate publication. The results obtained in this paper can be usedfor estimating systemic risks in interbank networks as well as for analyzingsensitivity of systemic risk with respect to changes in network topology andstability of individual banks. The results can also be used for the analy-sis of liquidity risks – with the contagion propagation through in-links andmodification of vulnerability criterion.
Acknowledgements
We are grateful to the referees for the comments that helped to clarifythe presentation of our results.
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