Assessing the risk of default propagation in interconnected sectoral financial networks
Abstract
Systemic risk of financial institutions and sectoral companies relies on their interdependencies. The interconnectivity of the financial networks has proven to be crucial to understand the propagation of default, as it plays a central role to assess the impact of single default events in the full system. Here, we take advantage of complex network theory to shed light on the mechanisms behind default propagation. Using real data from the BBVA, the second largest bank in Spain, we extract a financial network from customersupplier transactions among more than \(140\text{,}000\) companies, and their economic flows. Then, we introduce a computational model, inspired by the probabilities of default contagion, that allow us to obtain the main statistics of default diffusion given the network structure at individual and system levels. Our results show the exposure of different sectors to default cascades, therefore allowing for a quantification and ranking of sectors accordingly. This information is relevant to propose countermeasures to default propagation in specific scenarios.
Keywords
Financial networks Default analysis Financial sector analytics SIS propagation models Complex systemsAbbreviations
 BBVA
Banco Bilbao Vizcaya Argentaria, name of the second largest bank in Spain
 GLM
Generalized Linear Models
 MMCA
Microscopic Markov Chain Approach
 SIS
Susceptible–Infected–Susceptible epidemic model
 SME
Small and Mediumsized Enterprise
 VAT
ValueAdded Tax
1 Introduction
Interconnected financial networks are the fabric where economic agents from different sectors operate. One of the main challenges we face nowadays on financial networks is assessing systemic risk [1, 2, 3]. In the literature, systemic risk is defined as the probability of having large cascades of entangled economic events. Such cascades are triggered by causes that range from exogenous shocks to endogenous defaults. Besides, the succession of several defaults can jeopardize the full system because network financial interdependencies act as an economic sounding board. The interplay between the topology of the underlying interaction network and the easiness with which events propagate have proven to be essential to understand the proportion of the financial system affected by default avalanches and to assess the systemic risk [4].
Avalanches in financial systems are understood as dynamical processes that correlate individual economic states of the agents when a stress event materializes. This process resembles epidemic spreading in networks [5]. Under a simplifying assumption and to better explore the network potential for risk transmission, we model them in a similar way as epidemic spreading as was recently done in [6, 7, 8]. This is an oversimplification, the basic mechanisms of default propagation and epidemic spreading have similarities as branching process and chain reactions. Still in economic systems, especially those involving large companies such as banks, there are other mechanisms that may delay or even prevent the final default. Given the simplicity of the epidemic models and the fact that our networks are mostly formed by small and medium enterprises, we have taken this approach in the hope of getting direct information on the multilayer sectoral network interdependencies and on how the risk can pass from one to the other. The failure of one subject in the financial network generates a chain reaction through interconnections and causes shocks and therefore a default risk. This risk is understood as the incapability of one of the participants to perform their obligations, or at least to accomplish them properly, which leads to the interruption in the obligation payments of other participants.
One of the most commonly used contagion propagation models corresponds to the celebrated Susceptible–Infected–Susceptible (SIS). In a SIS model, individuals that are cured do not develop permanent immunity, but are again susceptible to the “disease”. Similarly, companies that manage to escape default by overcoming high economic stress can fall into trouble again later on. Additionally, SIS model provides valuable insights to understand how different situations may affect the outcome of the contagion process, e.g. what the most efficient technique is for isolating a limited number of companies in a given financial network to minimize the risk of observing an avalanche. Epidemic modeling is still the main application of SISlike approaches, and the main driver behind the development and refinement of this framework through time. However, the contagion analogy has been applied in different contexts and in particular in those where it is important to consider the spatial and social structure of systems. Some examples are adoption of fads and innovations [9], propagation of news and rumors [10] and information diffusion [11]. These are phenomena for which the state of the agent is affected by the interaction with its neighbors. In the financial context there is a strong causal relation between the financial and economical state of a company’s clients and how this influences its economical wellbeing [12]. This dynamics resembles a Hawkes stochastic process [13], where one event, under certain circumstances, is able to generate a new set events allowing the diffusion of a given phenomena [14]. Under this hypothesis, epidemic modeling can shed light on how systemic risk propagates through financial networks. Besides the contagion analogy, there are other similarities between the transmission of diseases and the transmission of financial distress in financial networks. For example, both are branching processes where one event produces others. But we can also observe some differences. For instance, disease transmission is usually studied as a continuous phenomena whereas financial distress is studied in a discrete time scale. Also, there are different levels of homogeneity in both cases, usually financial networks are more heterogeneous than the population networks used in disease spreading research.
In this paper we provide a mechanistic model to assess the impact of a particular diffusion process of default on financial networks. To this end, we take as basis a probabilistic computational framework named microscopic Markov chain approach (MMCA) to compute the probability of the states of individual agents in contagion processes in complex networks [15, 16, 17], and adapt its formulation to the understanding of the default propagation in financial networks. Further, we analyze the behavior of our proposed model using real data from the anonymized database of BBVA from December 2015 to December 2016, covering around 140,000 public and private Spanish firms. We set default labels to 0 or 1 based on this data, depending on whether a given company was or not in default at the beginning of the considered period. By means of this data we have access to the real network of interactions and to the initial condition for the dynamics of the default endogenous propagation.
This paper is organized as follows. First, we review previous work on default propagation in financial networks. Next, in Sect. 3, we propose a contagion model adapting the wellknown SIS Model. Section 4 provides a complete description of the data used, and its topological analysis. Section 5 includes a set of experiments to study the main characteristics of default propagation in each interconnected sector. Later, we examine the implications of using our default propagation model in Sect. 6. Finally, Sect. 7 provides some conclusions and future work.
2 Related work
The use of networks in economy and finance has a long tradition. Initial works were conceptual, like [18], where the networks were proposed as tools to represent the interactions (as links) between economic agents playing the role of nodes. When data started to become available, the popularization of complex networks brought a change of paradigm, leading to several advances in the field.
For example, the properties of the economic interchange networks between countries were studied in [19]. Also, the network formed by companies holding shares of other companies was studied for the Milan, New York and NASDAQ stock exchange markets in [20]. Interestingly, these networks show a scalefree nature, which implies that investors having a large number of connections are not uncommon. Explanations for this have been searched in the network dynamics properties mixed with a “richgetsricher” effect by means of different approaches [21, 22]. More recent models have been also proposed in [23, 24], assuming different hypothesis. A complete review of empirical economic network models can be found at [25]. Beyond the distribution of connections, other characteristics such as the level of clustering have been studied [26]. Despite all these works, there are still numerous open challenges when it comes to fully understanding the structure and dynamics of economic and financial networks [1, 27, 28].
The reason why these networks attract so much attention is that, besides economic interchanges, financial risk also propagates through them [24, 29, 30, 31, 32]. Their stability becomes thus an important question [33]. Furthermore, risk and economic distress, and even default in a second stage, can occur in cascades leading to serious systemic instabilities [32, 34]. Therefore, the resilience of the networks to contagion, as well as the circumstances under which it becomes systemic has been analyzed in many works [35, 36, 37]. Following this research line, a method called debtrank was introduced to find nodes in financial networks that can induce large cascades when perturbed [3]. This method allows to search for measures to mitigate risk propagation [38]. In the special case of networks where the nodes are banks and the links represent holding of different types of obligations, the complexity of the products traded such as derivatives [39] and the feedbackloops between solvency perception and stock and obligation values [40] can play an important role in economic distress propagation. Many of these previous works have been focused on banking [41, 42], where the risk propagation is related to the stress tests performed by central banks. These kind of models resorts on adhoc mathematical models for financial institutions. However, as it has been seen in the last crisis, the risk can spill out of the banking system to enter other economic sectors. This is why it is of high relevance to consider risk propagation in more general economic networks, including different sectors and different types of nodes, ranging from large holdings to small companies or even the final individual consumers. This is precisely the direction that we take in the present work where we use general contagion model to evaluate the default spreading in a highly heterogeneous network.
3 Default contagion model
Inspired by the Microscopic Markov Chain Approach (MMCA) designed for epidemic spreading, first we propose an adaptation of the framework for modeling the default cascades observed in the transactions between different companies in real financial networks. Then we introduce some measures to dynamically analyze the default contagion process and its functional relations with any sectoral financial network.
3.1 MMCA model for default contagion
According to the European Central Bank definition for risk classification [43], the susceptible state would correspond to a company which is in step 3 (default). In this step, the credit quality of the company is considered equivalent to a probability of default of between 0.10% and 0.40% over a oneyear horizon. Therefore, after a given period of time (12–18 months), which depends on its revenue, it can go through the step 2 (cure) and finally come back to step 1 (normal) if it proves to have a good payment behavior.
3.2 Dynamical analysis of default contagion
3.3 Dynamical properties of the network: onset slope and sensitivity
Naturally, the observed dynamical behavior is the result of an interplay between the MMCA framework and the network topology. To understand the reasons behind the different sectors’ response to default propagation, we will characterize the dynamical behavior of each sector by the onset threshold \(\mathcal{R}_{0} \) and the sensitivityS to the initial set of defaulted companies. Both metrics are descriptors of the expected behavior in the steady state regime. The onset slope is measured by estimating numerically the critical \(\beta _{c}(s)\) at which the first default cases start to appear in sector C. Practically, fixing μ, the parameter β is increased until the number of default cases in the sector in the stationary state goes over 1% of the real data defaults found in the sector, marking \(\beta _{c}(s)\). All these calculations are done in the stationary state of the system. When \(\beta _{c}(s) \) is plotted versus μ, one finds a noisy linear increase and, therefore, we define \(\mathcal{R}_{0}\) as the slope of the linear fit.
This property reveals the spreading capacity of the infectious process in each sector. Larger values imply that when the life times of defaults in the companies of the sector become shorter, one needs higher infectivity to overcome the threshold. Sectors with larger \(\mathcal{R}_{0}\) should be more resilient to general default. From the moment they start to show significant default, other sectors with lower \(\mathcal{R}_{0}\) may be in very bad shape already. Furthermore, given a certain set of parameters, an isolated default event in one of the sectors with larger \(\mathcal{R}_{0}\) can trigger an avalanche of default on weaker sectors, for which the conditions are favorable for contagion. In this sense, the \(\mathcal{R}_{0}\) value of a sector is also related to the capacity of the sector to spread default.
Regarding the sector sensitivity to default propagation, this dynamical property measures the rate of change of \(\rho (\beta ,\mu )\) at the transition point (which is normally known as β cutoff). Computing sensitivity involves fitting a linear regression to the model response and using its standardized regression coefficients as direct measures of sensitivity. Therefore this metric describes how susceptible a sector is to default, quite the opposite to \(\mathcal{R}_{0}\), which characterizes how a sector affects the system. The relationship between these two dynamical descriptors and the network structure will be explored next. As mentioned before, these are defined at the steady state, but it is also important to understand how dynamics evolve in the transient regime. This analysis will be carried out by synthetically concentrating defaulted companies (in specific proportions) in the different sectors and exploring pairwise sectoral interactions at the initial steps of the simulation.
3.4 How do sectoral properties of the nodes affect network dynamics?
4 Topological analysis of the clientsupplier network
Now we describe the sectoral financial network used in the experiments carried out in this work. To do so, we first provide all the details about network construction. Second, we report commonly used network statistical descriptors.
4.1 Network construction
Customersupplier relationships highly depend on economical sectors and the financial situation of the companies involved. To properly model this situation with real data we gathered anonymized quarterly data from the official customersupplier third party payment declarations collected by the BBVA risk management department. This declaration is used as a mechanism to avoid fraud in company VAT declarations. here, Spanish firms (our nodes) inform about their supplier payments and customer earnings. For each available company, we extracted its operating revenue and financial statement attributes: sector and default status. Collected data covers from December 2015 to December 2016. Default labels at the initial step were set to 0 or 1 depending on whether a company was in default or not at the beginning of this period. By using customersupplier relationships, and after removing selfloops, a directed and weighted network with \(142\text{,}477\) nodes and \(255\text{,}509\) edges was obtained. Direction of edges follows the path of money injection (from the customer to the supplier). All edge weights (total money transferred) were aggregated annually and normalized by its source node outstrength. Note that, both BBVA customers and noncustomers were included in a percentage of 63% and 37%, respectively. Therefore, the network contains an important percentage of missing values.
Summary of network topological measures by sector where \(\overline{k}_{\mathrm{in}}\) and \(\overline{k}_{\mathrm{out}}\) stand for the average indegree and outdegree, respectively
Sector  Size (%)  \(\overline{k}_{\mathrm{in}}\)  \(\overline{k}_{\mathrm{out}}\)  Default (%)  Rank hub  Rank auth 

Financial institutions  0.046  39.613  45.529  3.650  17  1 
Energy  0.083  12.844  8.666  1.111  14  2 
Financial services  1.165  6.300  20.265  0.786  13  3 
Utilities  1.529  5.589  5.903  1.264  11  4 
Telecoms technology & media  3.299  5.960  5.194  1.776  2  5 
Basic materials  2.745  5.789  5.350  2.782  6  6 
Transportation  4.064  5.411  4.336  1.868  1  7 
Retail  23.593  3.973  3.233  1.217  12  8 
Retailers  4.273  5.001  3.613  1.885  5  9 
Capital goods & industrial services  8.689  4.528  3.098  1.866  9  10 
Autos, components & durable goods  1.470  4.454  2.991  1.786  10  11 
Consumer & healthcare  7.055  3.259  3.770  1.539  8  12 
Construction & infrastructure  8.907  3.067  3.270  3.071  3  13 
Unknown  10.159  0.930  1.413  1.942  15  14 
Real rstate  6.843  1.517  1.844  3.603  7  15 
Leisure  12.861  2.509  2.512  1.511  4  16 
Institutions  3.219  5.547  10.764  0.535  16  17 
4.2 Statistical descriptors
5 Experimental results
We will study next three different default propagation scenarios. The first one corresponds to the classical MMCA model where all nodes share the same recovery rate. In the second one we use the heterogeneous recovery rate measure introduced in Sect. 3.2. Besides, to increase our knowledge about the role of each sector in the default propagation process, we synthetically simulate default problems in each sector to analyze the different spreading speed in the transient state. Finally, we validate our findings comparing achieved results with a null model built by rewiring the edges of our network.
5.1 Default incidences for homogeneous recovery rate
5.2 Impact of customer diversification on default incidence
In addition, we have compared the customer diversification variation model with a baseline model having constant default recovery rate μ. We simulate the latter with μ equal to the mean of \(\mu _{i}\) for the whole network, specifically with \(\mu = 0.005\). Consequently, the homogeneous μ (baseline model) used is 0.01. In our data, initial conditions for the number of companies in default at \(t_{0}\) are more concentrated in Financial Institutions, 15% (relative to the sector) and the rest varies with a default rate between 6% and 1%. However, the MMCA modeling framework does not depend on the initial conditions when the steady state is reached.
5.2.1 Sector structurefunction relationship
Summary of network measures influencing dynamic default contagion by sector
Sector  γ  \(I_{\mathrm{in}}\)  \(\mathcal{R}_{0}\)  \(S_{\mathrm{het}}^{\mathrm{rank}}\)  \(S_{\mathrm{hom}}^{\mathrm{rank}}\) 

Energy  1.33  13.44  1.74  1  1 
Financial institutions  1.27  2.39  3.81  2  2 
Basic materials  1.58  41.07  2.89  3  5 
Financial services  1.36  1.66  2.22  4  3 
Transportation  1.44  16.27  2.33  5  4 
Telecoms, technology & media  1.44  16.99  2.68  6  6 
Retailers  1.54  28.18  2.70  7  8 
Capital, goods & industrial services  1.47  9.51  2.93  8  9 
Utilities  1.44  7.15  3.43  9  7 
Autos, components & durable goods  1.52  21.24  2.82  10  10 
Retail  1.58  29.80  2.73  11  11 
Real estate  1.40  1.86  4.74  12  12 
Construction & infrastructure  1.44  17.01  4.25  13  13 
Consumer & health care  1.55  32.06  4.10  14  14 
Unknown  1.59  5.16  3.69  15  15 
Leisure  1.35  7.71  7.56  16  16 
Institutions  1.54  5.65  –  17  17 
We still lack an explanation of the observed behavior given the topological characteristics of each sector. To do so, we computed the slope coefficient of the inverse cumulative probability distribution of the companies instrength for each sector (γ). The instrength is defined as the sum of the incoming normalized weights for each company, meaning that the higher the value for a company the more probable that the default dynamics will affect it. As in most complex systems, this probability distribution is heavytailed signaling a Pareto like distribution where the slope coefficient can be computed. A smaller γ value signals that the sector is more probable to contain well connected companies (sector hubs). We observe that Energy and Financial Institutions are the most susceptible (\(S^{ \mathrm{rank}}\)) sectors, and coincide with lower values of γ (higher probability of sector hubs). The contrary happens to Institutions, and unknown sector reference. This highlights the fact that hub structures play an important role in the dynamics, and in the extent that a sector is affected by it. Clearly, leisure does not follow this explanation because it has a middlerange γ value. This could be due to the fact that it has 76% of its companies with zero instrength. However, it is naive to think that only this structural property can explain the sector response to default dynamics. In Table 2, we can observe how less susceptible sectors are more equally interconnected to other sectors (larger value of \(I_{\mathrm{in}}\)). In practice, this causes the default spreading to be less likely to find a high probability path to these sectors because their incoming weights are less concentrated. As before, economical sectors can be grouped in three blocks given its capacity to interact and affect other sectors. On one hand, energy, financial institutions, financial services, transportation, telecommunications and retailers sectors are largely affected by the others due to their large sensitivity. Besides, these sectors are highly interconnected with all the other sectors since their activity is traversal to all sectors and firms, therefore this high degree connectivity allows default to infect them easily. On the other hand, when consider leisure and unknown sectors, we observe that these are not affected by other sectors. For leisure sector, this is a consequence that most of its companies have zero instrength. Similarly, companies belonging to the unknown sector are not BBVA clients, so their information is quite limited and most of their connections are not included in the network having, both, low in and out degree. Inbetween, there are sectors, such as (public) institutions and retail, that are stable independent of the perturbed sector. The main characteristic of these sectors is their low instrength due to their customers not being companies or not having customers at all because they are public entities. In any case, default contagion does not reach these sectors and they kept healthy in all parameters setting.
Independently of the particular economic insights that may arise from these analysis, the methodology proposed in this paper has a greater advantage. It allows to perform experiments in a massive way. These large amount of data has enabled us to study how the time it takes the system to arrive to the steady state (convergence time) depends on a set of parameters such as the initial default rate or the infection rate β. This may have practical applications. For instance, if we can establish a relationship between simulations convergence time and real time, the risk departments could take advantage of this understanding to estimate the speed of default propagation among sectors.
5.3 Synthetic default assessment experiments
The methodology also allows for another kind of experiments; what we call synthetic default assessment. Previously, we have performed a sensitivity analysis using the real default distribution at the initial step. Now we are going to explore what happens when the initial default nodes are concentrated in a given sector and repeat the analysis for every sector. Previously, we have focused on the stationary state, where dynamic properties such as S and \(\mathcal{R}_{0}\) were calculated without any dependence on the initial conditions. Now, we are interested in the transient phase, when the system has not reached the stationary state yet. In this transient phase, we can gain insights on the speed of default propagation among sectors.
We observe that most of the business sectors follow the tendency that the larger the number of sectors they affect, the fewer the number of sectors that at the same time are affecting them. An example can be found in Utilities; for a 25% perturbation all other sectors except one are affected, while Utilities itself is only affected by another sector (Unknown sector). Note that this is also the case for Transportation, and seems to be a constant throughout the other sectors. This is opposite to what happens to the sectors Energy and Financial Institutions, which is affected rapidly by all the other sectors. Knowing the dependence of the default speed contagion on the business sectors may allow risk assessment models to understand the conditions to react to a sudden perturbation of a sector, or to an event that may indicate the initial stages of a sector crisis.
5.4 Validation
6 Discussion

We have analyzed the dynamics of default contagion for different values of the recovery parameter μ, dependent and nondependent on the node’s features. Our methodology allows to tune parameters individually for every company and to carry out experiments for the simulation of future scenarios. In particular, we have studied the impact of the company’s customer diversification on default propagation. A discussion on the connection between topological and dynamical properties is also included (Sect. 5.2.1);

Our methodology also allows for another kind of experiment described in Sect. 5.3, where we have focused on the dynamics of default propagation at the transient state, and its dependence on the default initial conditions.
7 Conclusions
We have proposed a computational model, based on the probabilities of default contagion, to study the default diffusion at individual and aggregated levels. We have performed massive experiments based on this model by varying several parameters such as the initial default rate, the contagion rate β and the recovery rate μ. This methodology allows us to vary the parameters at the individual level to account for a more realistic scenarios. Our results show the relationship between dynamical and topological properties for more than \(140\text{,}000\) BBVA firms aggregated at a economic sector level, and also allow us to create a ranking of sectors by sensitivity to default, which can be used in potential applications. For future work, we would like to enrich the network adding different types of payments such as national transfers or direct debits, extending in this way our computational model to a multiplex network, finally we want to enrich model parameters considering for example companies’ revenue.
Notes
Availability of data and materials
The dataset is not publicly available. It was acquired by BBVA. Any of the authors based at BBVA may be contacted for further details about the dataset.
Authors’ contributions
Conceptualization: JN, ET, PF, AA, JJR; methodology: AA; formal analysis: AB, JN, PF, ET; data curation: ET; writing (original draft preparation): PF, ET, JN; writing (review and editing): JN, PF, ET, AA, JJR; visualization: JN, AB, AM, ET; funding acquisition: JN. All authors read and approved the final manuscript.
Funding
JJR acknowledge partial funding from the Spanish Ministry of Science, Innovation and Universities, the National Agency for Research Funding AEI and FEDER (EU) under the grant PACSS (RTI2018093732BC22) and the Maria de Maeztu program for Units of Excellence in R&D (MDM20170711). AA acknowledges support by Spanish Ministry of Science, Innovation and Universities (grant PGC2018094754BC21), Generalitat de Catalunya (grant 2017SGR896), Universitat Rovira i Virgili (grant 2017PFRURVB241), ICREA Academia and the James S. McDonnell Foundation (grant 220020325). We acknowledge support of the publication fee by the CSIC Open Access Publication Support Initiative through its Unit of Information Resources for Research (URICI). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests
The authors declare that they have no competing interests.
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