Measuring systemic risk and contagion in the European financial network

11/26/2019 ∙ by Laleh Tafakori, et al. ∙ Monash University RMIT University 0

This paper introduces a novel framework to study default dependence and systemic risk in a financial network that evolves over time. We analyse several indicators of risk, and develop a new latent space model to assess the health of key European banks before, during, and after the recent financial crises. First, we adopt the measure of CoRisk to determine the impact of such crises on the financial network. Then, we use minimum spanning trees to analyse the correlation structure and the centrality of the various banks. Finally, we propose a new statistical model that permits a latent space visualisation of the financial system. This provides a clear and interpretable model-based summary of the interaction data, and it gives a new perspective on the topology structure of the network. Crucially, the methodology provides a new approach to assess and understand the systemic risk associated to a financial system, and to study how debt may spread between institutions. Our dynamic framework provides an interpretable map that illustrates the default dependencies between institutions, highlighting the possible patterns of contagion and the institutions that may pose systemic threats.



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1 Introduction

Over the last decade, the global financial system has endured several major crises, most notably the Global Financial Crisis of 2008 and the European Sovereign Debt Crisis in 2010. These events raised serious concerns regarding the resilience of the financial system against the contagion of debt during periods of economic turmoil. The collapse of Lehman Brothers on September 15, 2008, was the largest bankruptcy filing in U.S. history. This financial shock escalated into a global crisis which clearly demonstrated the frailty of the financial ecosystem, and the ineffectiveness of the current regulations. Subsequently, many repercussions were felt throughout the world by the participants of the global financial market, resulting in an excess of cross-border and cross-entity interdependencies (see De Haas and Van Horen (2012) and Acharya et al. (2014)). By the end of September 2008, the shock had rapidly spread across Europe, where the Euro area governments rescued the Belgian-French bank Dexia. Many authors agree on the fact that the increased interdependecies between the institutions has been playing a crucial role in the spread of contagion, and in forcing hasty responses to the shocks observed in the system (Aiyar (2012),Acemoglu et al. (2015)).
More recently, the research on financial networks and systemic risk has developed in many different directions (Bartram et al. (2007), Schweitzer et al. (2009), Engle et al. (2014)). A special emphasis has been dedicated to the understanding of how the topology of the system may impact the potential spread of contagion and systemic risk (Elliott et al. (2014), Acemoglu et al. (2015)). This research question is of particular importance for both regulators and financial institutions, since these should both be able to clearly identify the primary sources of systemic risk, Acharya et al. (2012), and thus how the risk may be minimised.
In practice, we are interested in extracting from the financial system a collection of summaries that can permit an assessment of the risk. In recent years, a number of different measures have been introduced or adapted for this purpose (Giudici and Parisi (2016), Battiston et al. (2012), Friel et al. (2016), Hledik and Rastelli (2018)

). In this paper, we study a dataset of default probabilities for a set of key European banks from 2005 to 2016, and we use a number of measures to assess the risk associated to each of the institutions. First, we use a collection of descriptive statistics and other known indicators that have been introduced in financial networks analyses. Then, we propose a new statistical model that aims at giving a clear and interpretable visualisation of the system by embedding it in a low dimensional space.

At the foundations of our contribution we have our constructed measures of dependencies between any given institutions. In general, these may be computed using a number of approaches. One type of approach relies on Pearson’s correlation index (see, e.g., Chi et al. (2010), Wang and Xie (2015) and Birch et al. (2016)). A second approach uses instead the partial correlation index, which measures and filters a network using partial correlation coefficients between banks. The correlation between two financial agents is frequently influenced by other financial agents, that is, two interacting financial agents may also have correlation with other financial agents (see, e.g., Mantegna and Stanley (1999)). For instance, the US stock market or the European stock markets can affect on the Hong Kong and Chinese stock markets. Therefore, one can get the pure correlation between the Chinese and Hong Kong stock markets by removing any effects of the US and European stock markets. The partial correlation coefficient quantifies the pure correlation between any two financial agents by measuring the relation between them and deducting the impact of any other financial agents. The approaches that make use of this indicator include Kenett et al. (2012), Kenett et al. (2015), Giner et al. (2018) and Wen et al. (2019). A third type of approach relies on other correlation-based network methods to construct a network using other similarity measures of correlation (Brida and Risso (2010), Matesanz and Ortega (2014)). For instance, Brida and Risso (2010) introduced the tool of symbolic time series analysis to obtain a metric distance between two different stocks; then, the authors used a minimum spanning tree for investigating the correlation structure of the 30 largest North American companies. Furthermore, Matesanz and Ortega (2014) studied the non-linear co-movements of foreign exchange markets during the Asian currency crisis by combing the minimum spanning tree and phase synchronisation coefficients. The minimum spanning tree is most frequently used since it is a robust, simple and clear tool to visualise the links.
In this paper, we transform the partial correlations to obtain the CoRisk measure (Giudici and Parisi (2016)), which quantifies the difference between the unconditional and the conditional probability of default for an institution. This measure effectively captures how much of the risk associated to an institution is due to the risk of its neighbours. We look at the pairwise CoRisk values between the European banks from 2005 to 2016, hence studying the evolution of this index over time. Then, we propose a different visualisation of the network using minimum spanning trees throughout the same time period, and highlight the importance of nodes using network centrality measures.
Furthermore, we develop a new statistical model that may be used to analyse the nodal attributes of any nonnegatively weighted network. The framework that we consider is inspired by the literature on Latent Position Models (LPMs, Hoff et al. (2002)), which has been greatly developed in the last two decades Handcock et al. (2007), Rastelli et al. (2016), Durante et al. (2017). LPMs postulate that the nodes of a graph are embedded in a latent space (generally ), and that a connection between any two nodes becomes more likely if the nodes are close to each other, and less likely if they are far apart. For example, in the context of the recent financial crises, Friel et al. (2016) introduce a type of LPM to study the boards’ compositions for companies quoted on the Irish Stock Exchange. Their approach provides an easy-to-interpret latent space representation of the Irish financial market, and it leads to the introduction of a new potential measure of financial instability. Here, our goal is to offer a new latent space perspective on our data, to ultimately assess the health of European banks during the recent crises.
The rest of this paper is structured as follows. Section 2 describes the proposed methodologies, inference and model interpretation. Section 3 shows the main empirical results for latent position model and analysing the minimum spanning tree networks by using the default probabilities and CoRisk measure. Finally, we provide conclusions with some discussions in Section 4.

2 Methodology

2.1 Default probabilities and CoRisk

CoRisk is a measure introduced by Giudici and Parisi (2016) to determine the variation in the probability of default due to contagion effects. The CoRisk measure consists of two components. Firstly, the additional risk taken on by a financial institution due to its connections with other financial institutions is known as CoRisk. Secondly, the risk caused by the financial institution to other financial institutions that are connected to is known as CoRisk.

The values for CoRisk and CoRisk (for Bank ) can be obtained using the following equations,


where represents the neighbours of bank , is the probability of default for bank and is the partial correlation value of banks and given where is a set of all other banks.

In order to determine which banks are interconnected in a financial network, a partial correlation matrix is calculated and the partial correlation values are tested for significance. Banks that have a significant partial correlation value with each other are connected by edges in the financial network. We use partial correlation instead of Pearson’s correlation as it provides a more accurate view of the correlation between two banks without the influence of external banks.

Moreover, we are interested in pairwise CoRisk values between any 2 banks to see which connections between banks are the most significant. The CoRisk value between two banks, bank and bank , is defined as follows:


where is the average probability of default of bank across a specified time period, and is the partial correlation value between bank and bank over the same time period. Intuitively, this can be interpreted as the effect that the connection of and has on the probability of default of bank .

The CoRisk measure has some useful properties. Firstly, CoRisk if and only if two banks have a negative partial correlation value. It takes a value between and , if the two banks have a positive partial correlation value instead. Secondly, the measure is not symmetric, so CoRisk CoRisk unless both banks have the same average default probability. Also, summing up the pairwise CoRisk values from bank to other banks and from other banks to bank gives us the aggregate measure of CoRisk and CoRisk values of the bank respectively.

2.2 Construction and interpretation of the adjacency matrices

The observed data consists of the values , which denote the log default probabilities, and the edges , which indicate whether two institutions are highly correlated or not. Here, represents the number of financial institutions, and determines the time period considered.

For each , the matrix can be seen as the binary adjacency matrix representing a random graph. In particular, the value is equal to one if an edge from node to node is present at time , and it is equal to zero otherwise. We obtain the matrices by threshold at the partial correlations used in the previous section, that is, we observe an edge between two financial institutions if and only if the corresponding partial correlation is greater than in absolute value. The graphs are undirected by construction, that is the edges do not have an orientation and .

2.2.1 Model

LPMs can be considered as generative models for the presence vs absence of edges in a random graph. By contrast, in this paper we are interested in modelling the log default probabilities associated to the nodes of the graph. Our modeling assumption postulates that, at each time, each institution is characterised by a vector of latent coordinates

. These values are model parameters which must be estimated from the observed data. We construct our approach following two core ideas:

  1. Institutions located close to each other will tend to exhibit similar default probabilities, that is the risk on one node will have a certain influence on the risks of the other nearby nodes.

  2. The intensity of this contagiousness is determined by the Euclidean distance between the institutions, and by whether the institutions interact with each other or not.

We introduce a new dynamic LPM to model the log default probabilities in the time periods considered. We assume that, conditionally on the latent positions, the likelihood function has the following form:


In the above equation, the summation is over all the pairs of , such that ; corresponds to one over the Euclidean distance between the nodes, and


for a model parameter .

The following proposition motivates the likelihood definition in (4).

Proposition 1

The full conditional

is a Gaussian distribution with mean

and variance

as follows:

The proof is shown in Appendix 5.1. The proposition underlines a straightforward interpretation for the model: conditionally on all parameters being fixed, the expected log default probability of an institution is equal to the weighted average of the log default probabilities of its neighbours, with weights corresponding to the similarity measure given by . This interpretation is in agreement with the first idea described in Section 2.2.1. Regarding the interpretation of , values smaller than ensure that the network neighbours will have a stronger influence than the non-neighbours. If , all the nodes contribute equally to the weighted average, and essentially the values of the adjacency matrices become irrelevant. If we obtain a degenerate framework where the non-neighbours have stronger influence than neighbours.

2.2.2 Hierarchical structure

We create a hierarchical structure and specify prior distributions on the latent positions and on the parameter

. This is in close agreement with the literature on LPMs, where Bayesian settings are most commonly used. We consider a gamma distribution on

with rate and shape both equal to one. This guarantees a certain flexibility without emphasising the values that are greater than one, since such values lead to a model with an unclear interpretation. We consider a standard Gaussian prior on the latent positions, and on the innovations of the latent positions. This means that follows a bivariate Gaussian centered in zero with the identity as covariance matrix; whereas, for , follows a bivariate Gaussian centred in with the identity as covariance matrix.

The prior distributions that we use in our setup may be regarded as informative. As we will discuss in the following sections, our framework relies on an optimisation approach: in this perspective, the prior distributions act as penalisations for the log-likelihood. As a consequence, we do not use these directly to model any prior information that we have on the model parameters, but rather we use them to penalise degenerate scenarios and to emphasise solutions that are most relevant in our context. In practice, we penalise very large values of , and we promote small innovations on the latent positions. This guarantees that the latent network snapshots remain comparable across times and it facilitates the interpretability of our results.

2.2.3 Inference and model interpretation

The likelihood function which is defined in Equation 4 specifies the density kernel only up to a proportionality constant. The associated normalising constant does not have an analytical form, and is generally difficult to approximate numerically since it involves the calculation of a -dimensional integral. Hence, in our approach we deal with a so-called intractable likelihood Møller et al. (2006), which creates a connection with a vast literature that deals with similar problems Friel and Wyse (2012).

In a Bayesian setting, an intractable likelihood leads to a so-called doubly intractable problem Murray et al. (2012)

, whereby a standard implementation of a Markov chain Monte Carlo sampler may not be used efficiently. To circumvent this limitation, we employ a pseudo-likelihood approximation:


We assume that the likelihood factorises into the product of the full conditionals of the data points. Thanks to Proposition 1, we know that each of these full conditionals is proportional to a Gaussian, and hence we are able to compute the pseudo-likelihood exactly and efficiently.

As concerns parameter estimation, we propose to approximate the maximum-a-posteriori estimator using a simulated annealing scheme. Simulated annealing can be seen as a stochastic optimisation algorithm that converges to a maximum of the objective function

Andrieu et al. (2003). In our setup, the objective function is the pseudo-posterior which can be written as follows:


where denotes a generic prior distribution.

The algorithm tries to update each of the parameters of the model in turn, by obtaining approximate samples from a tempered distribution. When an update of a latent position is attempted, the new value is sampled from a bivariate Gaussian centered in with identity covariance matrix. Let be the current value of the objective function, and be the new value of the objective after the change. Then, the proposed update is retained with probability , where is the tempering value, which decreases to zero during the procedure. The updates for are performed in an analogous way, using a log-normal proposal centered in the current value of the parameter.

The optimisation approach allows us to speed up the inferential procedure, and to bypass the likelihood unidentifiability issues that are known to arise with LPMs Shortreed et al. (2006). We run our algorithm on the dataset for iterations, and temperature values defined by , where is the iteration index. The algorithm ran in hours and minutes on a -cores machine. Figure 1 illustrates the evolution of the calculated objective function during the optimisation.

Figure 1: Pseudo-log-posterior values during the simulated annealing optimisation.

3 Empirical results

The dataset that we study consists of the probabilities of default for 31 banks across 12 countries namely Austria, Belgium, Denmark, France, Germany, Italy, Netherlands, Norway, Spain, Sweden, Switzerland and the United Kingdom based on Thomson Reuters’ structural model. Thomson Reuters evaluate the equity market’s view of credit risk via a propitiatory structural default prediction framework. Thomson Reuters produces daily updated estimates of the PD or bankruptcy within one year for more than 35,000 companies in the world, where the PDs are ranked to create 1-100 percentile scores, Pourkhanali et al. (2016). We split it into four time periods which is shown in Table 1.

Period Start Date End Date
Pre-crises 3/Jan/2005 2/Jan/2008
Financial Crisis 3/Jan/2008 2/Jun/2010
Sovereign Crisis 3/Jun/2010 2/Jan/2013
Post-crises 3/Jan/2013 17/Nov/2016
Table 1: Explanation of red four different periods in data set.

3.1 Descriptive Statistics

A summary of the values of default probabilities for each bank is provided in Table 13 in Appendix. Banks were also categorised according to their country of origin. The default probabilities for each country was then obtained by averaging the default probability values across the appropriate banks. A summary of these values are provided in Table 14 in Appendix.

Figure 2 shows the boxplots for the probabilities of default of 31 banks from 2005-2016. From these plots, it can be inferred that certain banks are more volatile and that their default probabilities change more drastically. These include Banca Monte dei Paschi di Siena (BMPS), Commerzbank AG and ING Group.

Figure 2: Boxplot of PD for 31 banks

3.2 Partial Correlation

To determine the connectedness between the banks during each period, the partial correlation values between bank and bank were calculated and tested for significance using the R statistical software.444Results are available upon request.

Our naive assumption is that during the financial and sovereign crises, the connections between banks would be closer together. Therefore, we would expect more significant partial correlations during these periods. The table of counts below demonstrates that our assumption is true during the financial crisis, but not during the sovereign crisis. However, this result alone is not sufficiently strong as it does not take into account the actual value of the partial correlations.

Precrisis Financial Crisis Sovereign Crisis Postcrisis
273 285 271 241
Table 2: Number of Significant Partial Correlations.

3.3 CoRisk Values

After the calculation of the partial networks, and using the 1-year default probability values for each bank, we are now able to calculate the CoRisk and CoRisk at each point in time. The change in CoRisk values are shown in Figure 3. These figures exhibit spikes in the CoRisk values during the financial and sovereign crises, confirming that more risk was transmitted during these periods. The CoRisk values after remain higher than they were before the crises with smaller sudden spikes. This could be due to the lingering effects of the crises and the fact that some banks are yet to stabilise.

Figure 3: Time series of CoRisk and CoRisk are shown in these plots for whole period of study.

The CoRisk matrix was also calculated based on the significant partial correlation matrix obtained earlier for each period555Results are available upon request..

3.4 Financial contagion tests

In order to see if there was a significant increase (or decrease) in the CoRisk

value between different periods, paired t-tests (the

CoRisk matrices were transformed into a vector of 961 elements) were conducted between two periods. We found that there is a significant increase in the CoRisk values during the financial crisis and also post-crises as compared to pre-crises. The results of our paired t-tests are summarised in Table 3.

CoRisk p-value
financial crisis pre-crises
pre-crises post-crises
sovereign crisis financial crisis
financial crisis post-crises
Table 3: Paired t-test.

3.5 Kendall’s tau test

We are interested in seeing if the dependency of default probabilities between banks and countries increased from the pre-crisis period to financial crisis period. We use Kendall’s tau measures the dependency between variables. In order to construct the test, sampling with replacement method was employed. Each column was re-sampled (with replacement) 200 times for the calculation of Kendall’s tau. A t-test is then performed to see if has increased. At a significant level of 5 percent, only a few pairs of countries were identified as having a significant increase in dependency. They are: Spain /Switzerland, Germany/Denmark, UK/Norway. Since our crisis period was further split into two groups as financial crisis and sovereign crisis, we then combined them to see if the pre-crisis period would have a lower compared to the combined crisis periods. However, only Switzerland/Germany and Switzerland/Norway showed a significant increase in . To confirm the robustness of this test, the re-sampling process was done a couple of times and each time the result varied. Hence we conclude that this would not be an effective way of determining the existence of contagion. In addition, among all 66 pairs of countries, the small proportion of significant pairs is another indicator that we need to explore further into the way of determining contagion effect.
The same procedure was repeated for banks. However, due to the uncertainty discussed above, we cannot draw any conclusions about the existence of contagion based on Kendall’s tau test. Therefore, we apply another approach in the next section which is called as Minimum Spanning Tree.

3.6 Minimum Spanning Tree (MST)

Network models are frequently adopted in the field of financial research due to their effectiveness at visualising large data sets. They have proven to be an effective resource in the prediction of market movements. One of the more popular methods for visualisation of financial networks and building the dependence network is the minimum spanning tree (MST) approach (Mantegna, 1999) which is designed to select or filter the information presented in the dependence (or correlation) matrix. The metric distance for creating the edges of a financial network, such as in Wang et al. (2018), uses the partial correlation values, , and is given by the following equation

In our work, we use a different distance metric, replacing the partial correlation values with CoRisk values. The formula for the distance between two edges will then be as follows,

Remark 1

Unlike the partial correlation values, the CoRisk values are not symmetric, that is . Since CoRisk takes values from to , would always be positive. Furthermore, a larger CoRisk value implies that bank has a greater impact on bank , and this is represented by a shorter distance.

A directed graph consisting of 31 nodes (31 banks) is then formed using the distances obtained (a pair of directed edges between two banks is only present if the banks are significantly correlated as determined earlier). This directed graph consists of a large number of directed edges and is difficult to visualise. Therefore, we used the R software to obtain a MST with directed edges. The MST from each of the four periods are shown in Figures 4-7.

Figure 4: Minimum Spanning Tree Pre Crisis by bank.
Figure 5: Minimum Spanning Tree Financial Crisis by bank.
Figure 6: Minimum Spanning Tree Sovereign Crisis by bank.
Figure 7: Minimum Spanning Tree Post Crisis by bank.

In this work, we consider MST approach based on Banks and countries, separately. In order to apply MST for Banks, they are colour-coded in accordance with their countries and made proportional to the value of their total assets in 2016. For example, Nordea is a Swedish bank coloured in yellow and had a total asset value of €615 billion, while British bank Barclays is coloured in green and had a total asset value of €1213 billion. The length of the edges between 2 banks is proportional to the distance metric used, and the direction shows which bank is transmitting the risk.
Figure 4 shows the MST before the global financial crisis. The Swedish bank SEB is located in the middle of the MST and the other centrally located banks are Commerzbank AG, Danske Bank and Credit Agricole. During the financial crisis, the two German banks, Deutsche Bank and Commerbank AG are the centrally located banks in the network, as seen in Figure 5. The banks also start to cluster according to their country, with the more notable ones being those from the United Kingdom and Sweden. This suggest that the banks within a country become more closely connected during the financial crisis. During the sovereign crisis, from Figure 6, the clustering effect seems to be less apparent than during the financial crisis. Commerzbank AG remains a centrally located bank, along with Credit Agricole and DNB. Figure 7 shows that the most critically important bank post-crises is BMPS, where due to its high probability of default and financial instability following the 2 crises, it has the largest impact on other European banks.
The above MSTs represent the connections between banks. To determine the relationships between countries instead, MSTs were also plotted by considering the banks located in each country. These MSTs for each of the four periods are shown below.
United Kingdom is the most centrally located country in the financial network before the financial crisis, as seen in Figure 8. It remains centrally located during the financial crisis (Figure 9) but the MST divides into two portions separated by UK, one with Belgium and the other with Norway. During the sovereign crisis (Figure 10), the major European countries, France, Germany and Italy become more centrally located. The structure of the MST remains fairly similar after the sovereign crisis (Figure 11). In the next section, we analysis MSTs by using two approaches which are called as measure of centrality and fragility.

Figure 8: Minimum Spanning Tree Pre Crisis by country.
Figure 9: Minimum Spanning Tree Financial Crisis by bank.
Figure 10: Minimum Spanning Tree Sovereign Crisis by country.
Figure 11: Minimum Spanning Tree Post Crisis by country.

3.6.1 Analysis of Minimum Spanning Tree using Fragility and Centrality measures

Fragility, as proposed by Das (2016), is the propensity for risk to spread through a network. This can lead to assess of the speed at which contagion can spread in the system. A network that has more links will transmit more risk as it spreads quicker throughout the network. Therefore, a network with a higher fragility score is more contagious.

Fragility of a network can be described by the following equation, where represents the degree of a node in the network,


Fragility values were calculated for the bank networks and country networks across the four periods. The results are summarised in the Table below.

Period Bank Network Country Network
Financial Crisis
Sovereign Crisis
Table 4: Fragility Score.
Remark 2

The fragility score gives us an overview of the transmission risk within the network, but fails to identify the sources of risk. Centrality measures are used to determine the main banks/countries that contribute to the source of the risk.

Below, we introduce some centrality measures used and the centrality scores of both banks and countries are then summarised.

The first centrality measure that we use in this work is Betweenness Centrality. Betweenness centrality for a node, , in a network is defined as the number of shortest paths between two nodes that passes through node . It is given by the following formula:


A bank with a larger value for its betweenness centrality will be located in a more central location in the network and thus be a major transmitter of risk through the network. Betweenness values were calculated for each period for both the bank and country networks.

The second centrality measure which is called as closeness centrality, measures the inverse of the sum of distances from a node to all other nodes in a network. A higher closeness value suggests that a node is close to other nodes, (for more detail see Grassi et al. (2010)). Banks with a relatively high closeness value would mean that it would require less time for it to transmit risk to other banks in the network as these banks would be ’closer’ to other banks in terms of the metric. The formula for closeness centrality is given by:


The third centrality measure is Laplacian centrality which calculates the drop in Laplacian energy (sum of squares of eigenvalues in a Laplacian matrix) when a vertex is removed, see

Qi X. (2013). It incorporates the weights of the edges in the measure, which makes it a stronger measure than betweenness centrality. The centroid value is another centrality which measures the number of interactions a node has to determine how central the node is in the network. It can be used to determine which bank is the ’co-ordinator’ within a network we use leaderRank. Banks with a relatively high centroid value will play a more central role in the financial network, Scardoni and Laudanna (2012). In order to measure the influence of a node in a directed network, by using a standard random walk procedure, we use LeaderRank. Banks with a higher LeaderRank score are more influential in the financial network.

Betweeness Closeness Laplacian Centroid LeaderRank
BMPS 5 0.0177 3276 -7 0.9913
BPM 3 0.0181 3534 -9 1.0363
BBVA 6 0.0161 2348 -15 0.8110
SAB 1 0.0181 3550 -9 1.0363
BPES 2 0.0191 4058 -9 1.1265
SAN 4 0.0172 3038 -11 0.9462
BCV 12 0.0186 3776 -9 1.0814
BARC 10 0.0177 3296 -13 0.9913
BSKP 0 0.0181 3490 -9 1.0363
BNP 11 0.0172 3026 -7 0.9462
CBK 39 0.0181 3522 0 1.0363
ACA 36 0.0181 3510 -5 1.0363
CSG 15 0.0181 3498 -11 1.0363
DANSKE 9 0.0168 2772 -9 0.9012
DBK 15 0.0172 3026 -7 0.9462
DNB 2 0.0168 2796 -11 0.9012
EBS 26 0.0196 4308 -3 1.1715
HSBC 0 0.0172 3074 -11 0.9462
ING 8 0.0186 3784 -9 1.0814
ISP 2 0.0177 3288 -11 0.9913
KBC 7 0.0191 4034 -5 1.1265
LLOY 11 0.0186 3748 -5 1.0814
LUKN 0 0.0168 2828 -15 0.9012
NDA 0 0.0161 2380 -11 0.8110
RBS 11 0.0181 3530 -7 1.0363
SEB 33 0.0186 3748 -9 1.0814
GLE 14 0.0177 3264 -9 0.9913
SGKN 8 0.0172 3046 -9 0.9462
STAN 5 0.0164 2594 -11 0.8561
SWED 9 0.0172 3022 -11 0.9462
UCG 0 0.0196 4264 -5 1.1715
Table 5: Centrality Scores for Banks: pre-crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
BMPS 10 0.0202 4510 -3 1.2382
BPM 1 0.0158 2162 -13 0.7796
BBVA 2 0.0169 2804 -13 0.9172
SAB 2 0.0165 2606 -11 0.8713
BPES 3 0.0177 3256 -13 1.0089
SAN 1 0.0182 3434 -11 1.0547
BCV 0 0.0161 2384 -15 0.8254
BARC 11 0.0169 2760 -13 0.9172
BSKP 0 0.0154 1968 -15 0.7337
BNP 7 0.0197 4216 -5 1.1923
CBK 38 0.0177 3280 -5 1.0089
ACA 15 0.0177 3236 -9 1.0089
CSG 3 0.0169 2816 -9 0.9172
DANSKE 31 0.0202 4506 -3 1.2382
DBK 16 0.0182 3494 -7 1.0547
DNB 6 0.0182 3482 -9 1.0547
EBS 11 0.0173 2986 -7 0.9630
HSBC 3 0.0173 3066 -7 0.9630
ING 20 0.0202 4506 -3 1.2382
ISP 5 0.0186 3764 -5 1.1006
KBC 9 0.0173 3078 -9 0.9630
LLOY 32 0.0182 3498 -9 1.0547
LUKN 2 0.0182 3482 -9 1.0547
NDA 2 0.0181 3494 -5 1.0547
RBS 36 0.0182 3494 -11 1.0547
SEB 7 0.0169 2808 -11 0.9172
GLE 18 0.0173 3006 -9 0.9630
SGKN 1 0.0177 3232 -7 1.0089
STAN 0 0.0177 3268 -11 1.0089
SWED 18 0.0165 2610 -7 0.8713
UCG 6 0.0173 3046 -9 0.9630
Table 6: Centrality Scores for Banks: financial crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
BMPS 27 0.0197 4284 -3 1.1681
BPM 25 0.0197 4296 -5 1.1681
BBVA 7 0.0169 2792 -15 0.8986
SAB 2 0.0186 3756 -7 1.0783
BPES 6 0.0181 3562 -9 1.0333
SAN 3 0.0161 2384 -15 0.8087
BCV 1 0.0181 3542 -9 1.0333
BARC 12 0.0173 3082 -11 0.9435
BSKP 3 0.0202 4578 -9 1.2130
BNP 21 0.0186 3808 -7 1.0783
CBK 48 0.0192 4066 0 1.1232
ACA 19 0.0182 3578 -7 1.0333
CSG 18 0.0186 3764 -7 1.0783
DANSKE 3 0.0165 2574 -13 0.8536
DBK 1 0.0173 3046 -11 0.9435
DNB 6 0.0191 4046 -3 1.1232
EBS 21 0.0186 3808 -7 1.0783
HSBC 0 0.0177 3292 -11 0.9884
ING 1 0.0181 3562 -7 1.0333
ISP 3 0.0181 3586 -15 1.0333
KBC 3 0.0165 2606 -15 0.8536
LLOY 20 0.0173 3074 -9 0.9435
LUKN 1 0.0191 4070 -9 1.1232
NDA 2 0.0165 2598 -11 0.8536
RBS 12 0.0169 2880 -11 0.8985
SEB 1 0.0161 2352 -15 0.8087
GLE 11 0.0191 4010 -9 1.1232
SGKN 2 0.0165 2594 -13 0.8536
STAN 0 0.0157 2174 -17 0.7638
SWED 3 0.0191 4058 -3 1.1232
UCG 20 0.0173 3018 -9 0.9435
Table 7: Centrality Scores for Banks: sovereign crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
BMPS 73 0.0208 4908 0 1.2056
BPM 12 0.0208 4972 -5 1.2056
BBVA 5 0.0196 4396 -11 1.1194
SAB 6 0.0168 2940 -9 0.8611
BPES 25 0.0202 4702 -9 1.1625
SAN 3 0.0186 3876 -9 1.0333
BCV 1 0.0202 4662 -3 1.1625
BARC 1 0.0165 2674 -15 0.8181
BSKP 0 0.0177 3368 -11 0.9472
BNP 1 0.0186 3852 -9 1.0333
CBK 6 0.0161 2416 -13 0.7750
ACA 3 0.0173 3206 -11 0.9042
CSG 3 0.0165 2698 -17 0.8181
DANSKE 2 0.0202 4650 -7 1.1625
DBK 25 0.0186 3868 -9 1.0333
DNB 0 0.0173 3146 -13 0.9042
EBS 4 0.0173 3134 -13 0.9042
HSBC 0 0.0186 3888 -9 1.0333
ING 5 0.0196 4384 -9 1.1194
ISP 3 0.0161 2472 -17 0.7750
KBC 6 0.0168 2888 -13 0.8611
LLOY 9 0.0177 3424 -11 0.9472
LUKN 0 0.0191 4150 -9 1.0764
NDA 1 0.0177 3364 -11 0.9472
RBS 5 0.0181 3658 -11 0.9903
SEB 0 0.0186 3876 -7 1.0333
GLE 14 0.0186 3944 -5 1.0333
SGKN 0 0.0177 3408 -17 0.9472
STAN 13 0.0202 4702 -5 1.1625
SWED 3 0.0173 3174 -7 0.9042
UCG 43 0.0197 4404 -3 1.1194
Table 8: Centrality Scores for Banks: post-crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
Italy 4 0.0544 642 -4 1.1000
Spain 1 0.0471 450 -6 0.9000
Switzerland 5 0.0544 650 -4 1.1000
United Kingdom 6 0.0589 740 0 1.2000
France 2 0.0589 744 -2 1.2000
Germany 1 0.0442 360 -4 0.8000
Denmark 1 0.0505 544 -2 1.0000
Norway 2 0.0416 274 -8 0.7000
Austria 0 0.0416 294 -6 0.7000
Netherlands 2 0.0544 638 -2 1.1000
Belgium 11 0.0544 642 -4 1.1000
Sweden 1 0.0544 646 -4 1.1000
Table 9: Centrality Scores for Countries: pre-crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
Italy 5 0.0473 434 -2 0.9000
Spain 0 0.0544 642 -4 1.1000
Switzerland 0 0.0472 450 -4 0.9000
United Kingdom 1 0.0506 548 -2 1.0000
France 3 0.0507 532 -4 1.0000
Germany 2 0.0507 552 -4 1.0000
Denmark 6 0.0545 634 -6 1.1000
Norway 3 0.0590 748 -2 1.2000
Austria 3 0.0506 524 -4 1.0000
Netherlands 2 0.0394 200 -8 0.6000
Belgium 11 0.0590 748 -2 1.2000
Sweden 0 0.0506 548 -2 1.0000
Table 10: Centrality Scores for Countries: financial crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
Italy 4 0.0590 800 0 1.0435
Spain 0 0.0589 808 -6 1.0435
Switzerland 0 0.0472 494 -6 0.7826
United Kingdom 1 0.0643 918 -2 1.1304
France 3 0.0590 804 -2 1.0435
Germany 7 0.0644 918 0 1.1304
Denmark 0 0.0544 694 -4 0.9565
Norway 0 0.0544 694 -4 0.9565
Austria 2 0.0506 592 -4 0.8696
Netherlands 0 0.0545 698 -4 0.9565
Belgium 1 0.0590 796 -6 1.0435
Sweden 0 0.0590 796 -6 1.0435
Table 11: Centrality Scores for Countries: sovereign crisis.
Betweeness Closeness Laplacian Centroid LeaderRank
Italy 6 0.0643 910 -2 1.1471
Spain 2 0.0544 682 -4 0.9706
Switzerland 1 0.0544 686 -4 0.9706
United Kingdom 3 0.0643 910 -2 1.1471
France 4 0.0590 792 -2 1.0588
Germany 4 0.0506 580 -2 0.8824
Denmark 0 0.0505 580 -6 0.8824
Norway 0 0.0589 792 -4 1.0588
Austria 0 0.0544 686 -4 0.9706
Netherlands 0 0.0544 682 -6 0.9706
Belgium 0 0.0544 686 -6 0.9706
Sweden 0 0.0544 686 -4 0.9706
Table 12: Centrality Scores for Countries: post-crisis.

The tables 5-12 show the centrality scores using the 5 aforementioned different measures. Banks/Countries that have a significant score are highlighted in bold values. The betweenness centrality measure seems to be the least correlated with the other 4 measures. This is likely due to the fact that it is the only measure that does not account the weights of edges in the financial network. For the bank network pre-crisis, Erste Group Bank and UniCredit are the more important banks within the financial network, followed by Banco Popular Espanol, Commerzbank AG and Credit Agricole. These banks are from different countries around Europe but are important financial institutions within their own country. During the financial crisis, the three critical banks are BMPS, Danske Bank and ING. This supports the fact that banks from Denmark and Netherlands were one of the worst affected banks during the financial crisis. This would have made them riskier within the financial network, resulting in higher centrality values, especially for measures that rely heavily on edge weights.
During the sovereign crisis, the banks that had the highest centrality scores were BMPS, Banca Popolare di Milano and Basler Kantonalbank. BMPS fell into severe financial trouble in 2012 due to increasing Italian government debt and lost more than a billion dollars, and was later involved in a scandal. The large risk associated with BMPS resulted in higher CoRisk values, which caused the bank to be a more central node in the financial network. BPM was also hit hard during the sovereign crisis and had large CoRisk values. These two banks continued to have difficulty after the sovereign crisis, and have the highest centrality scores for that period too. For the country network pre-crisis, the UK and France were the most important countries in the European financial network. When the financial crisis hit, Norway and Belgium became the more centrally located nodes in the network. Belgium was one of the European states most affected by the financial crisis, resulting in a higher CoRisk value between other countries. Norway, on the other hand, was not hugely affected by the financial crisis. The impact on Norway is therefore more likely to be due to its high connectedness with other European economies. Germany and UK became the countries with the highest centrality scores during the sovereign crisis. These two countries are the largest economies in Europe, and the driving force of the European economy as a whole during the sovereign crisis. In Post-crises, Italy becomes the most central country in the network due to the failure of its banking system as explained above (BMPS and BPM being contributors to the overall risk).

3.6.2 Directed Acyclic Graph: Topological Sort

A directed minimum spanning tree is also a directed acyclic graph, allowing us to do a causal analysis on the 8 graphs we had obtained earlier. The aim of this was to see which banks and countries transmitted the most risk (source nodes) to other nodes in the financial network. The key banks/countries for each period are shown below. These banks/countries are not necessarily the same as those with high centrality scores. Instead, they are financial institutions with a high importance in the European economy. For instance, Credit Agricole and Commerzbank represent the key roles played by France and Germany, which are considered as leaders of the European economy as a whole.

Key Banks transmitted the most risks for each period are as follows,

  • Pre-Crisis: Banque Cantonale Vaudoise, Barclays, Commerzbank, Credit Agricole,

  • Financial Crisis: Commerzbank, Credit Suisse, ING, RBS,

  • Sovereign Crisis: Banco Popular Espanol, Commerzbank, Credit Agricole, ING,

  • Post-crisis: BMPS, Banco Popular Espanol, Barclays, Commerzbank.

Key Countries transmitted the most risks for each period are as follows,

  • Pre-Crisis: France, Germany,

  • Financial Crisis: France, Germany,

  • Sovereign Crisis: France, Germany,

  • Post-Crisis: Italy.

3.6.3 Evolution of Country Financial Network

MSTs are not the only method of visualising the results of our analysis. Since there are only 12 countries in our study, we could analyse the complete graph to see the evolution of risk transmission during the crisis periods. Below are four such graphs, where the size of the node represents the Laplacian Centrality score and the width of the edges represent the net CoRisk (sum of CoRisk and CoRisk) between two countries. We can see that net CoRisk significantly increased from Pre-Crisis to Financial Crisis and slightly reduced during Sovereign Crisis. However, it remained at a relatively higher level in Post-Crisis period when compared with Pre-Crisis. The increased node sizes over time also revealed interesting pattern about increased Laplacian Centrality (level of influence in the network).

Figure 12: Upper figure shows Pre-Crisis and bottom plot report network in Financial Crisis.
Figure 13: Upper figure shows Sovereign Crisis, while the bottom plot shows network in Post-Crisis.

3.7 Latent position model empirical results

The estimated value of is approximately zero, meaning that only the neighbours of a financial institution can have an influence on its risk. The main information that we obtain from our fitted model are the latent positions of all financial institution in each of the periods. We show these latent spaces in Figure 14 and Figure 15, by comparing them with the corresponding country and probability of default, respectively.

Figure 14: Estimated Latent Position Model for each of the time periods, with color of the nodes representing their country.
Figure 15: Estimated Latent Position Model for each of the time periods, with color of the nodes representing the associated default probability. The arrows represent the movement of each of the nodes from the previous time frame into the next.

An interactive version of the same plots which implements more features and visualisation tools is available from the authors. In Figure 14, the countries and probabilities of default of the financial institutions are highlighted within the latent space. Before the crises, the points tend to be close to each other, signaling a dense status of the system, whereby risk may be easily spread. We note that two Swiss banks are located far from the other institutions, and that they also show a low default probability.

Starting from the second time frame (which corresponds to the 2008 financial crisis), we observe that the latent visualisation expands and creates clusters in the space. The expansion can be interpreted as a measure to counteract risk from partners, and reduce the overall correlation. Regarding the presence of clusters, our interpretation is rather straightforward: clustered institutions tend to be close to each other and will tend to be more contagious towards one another. As a consequence, they are more likely to exhibit similar default probabilities. By contrast, we expect less contagiousness between clusters, particularly if they are located far apart.

Remark 3

It is important to note that these clusters highlight some associations between banks that may play a crucial role within the financial system. These associations can help in identifying key institutions that can pose systemic risk concerns, but, also, they may help in understanding the dynamics of the spread of debt.

The Italian banks, located near the center of the space, tend to be located close to one another, signaling a high mutual exposure to others’ risks. Most Spanish banks, located in the top area, also tend to be very clustered but segregated from the rest. The Swiss banks, with the exception of CSG, are located in the outskirts of the latent space and seem to be not particularly influenced by the other institutions, throughout the study. Other associations that are exhibited in these plots are not related to countries: an example could be the association between ING and BNP after the crises in the bottom-left area, or between CBK and BSKP in the bottom-center during the 2008 financial crisis. In Figure 15, we highlight the dynamic nature of our model. Both the colors and size of the points describe the default probability associated to the bank. Also, we include two oriented segments to highlight the position changes with respect to the previous and following time frames. This plot highlights that the banks with a higher default probability tend to be located near the centre and the lower regions of the space. We note that ACA and CBK exhibit a high default probability during the crises, they are close to each other, but they are also close to a number of other banks. On the other hand, after the crises, BMPS also exhibits a high default probability while being well separated from the other institutions. This is clearly in agreement with the idiosyncratic nature of the financial distress experienced by this bank.

4 Conclusions

The recent crises have changed drastically the structure of the European financial system. While the 2008 financial crisis has hit globally, the sovereign crisis has had a more lasting impact on certain European banks. We have adopted a number of descriptive measures and model-based measures to obtain a new and different perspective on the European financial system. The measure allowed us to see which banks and countries were more systemically important within the financial network during the financial and sovereign crises. In particular, it showed how BMPS became the driving force of risk due to two factors: its connectedness and its high default probability. The combination of the visualisation tools and of the centrality measures were able to identify the main sources of the risk. Regarding our latent space model, the results show that the banks expanded in the latent space during the crisis, presumably responding to the onset of the 2008 financial crisis. After 2008, the latent point process exhibits a more sparse and clustered structure, which is known to be more resilient to targeted attacks or defaults. Our model was also able to distinguish between risk caused by the crisis and the system, and the idiosyncratic risk associated to a bank.

Regarding possible extensions of our work, we point out that the heuristic simulated annealing was used because of its speed and theoretical guarantees, however other approaches may be viable. In the Bayesian setting, one may design a sampler to obtain approximate draws from the posterior of the distribution of interest. This would also permit an assessment of the uncertainty associated to the positioning of the nodes in the latent space. In addition, our proposed solution can be implemented on a variety of financial decision-making platforms, enabling individual users to map complex financial systems and make better data-driven financial decisions.

5 Appendix

5.1 Proof of Proposition 1

Proof. We wish to study the density , up to a proportionality constant that does not depend on :





which is proportional to a Gaussian density with the following mean and variance:


5.2 Summary of Default Probabilities by Bank

Mean SD Maximum Minimum Skewness Kurtosis
Banca Monte dei Paschi di Siena (BMPS) 1.83% 2.54% 18.93% 0.04% 2.46 7.39
Banca Popolare di Milano (BPM) 0.77% 0.88% 4.6% 0.03% 1.66 2.24
Banco Bilbao Vizcaya (BBVA) 0.36% 0.43% 3.29% 0.02% 2.79 10.06
Banco de Sabadell (SAB) 0.35% 0.31% 1.51% 0.01% 1.11 1
Banco Popular Espanol S.A (BPES) 0.79% 1.02% 4.65% 0% 1.89 2.73
Banco Santander (SAN) 0.36% 0.44% 3.16% 0.02% 2.56 7.75
Banque Cantonale Vaudoise (BCV) 0.1% 0.14% 0.74% 0.01% 2.38 5.23
Barclays (BARC) 1.04% 1.53% 16.97% 0.04% 3.56 17.68
Basler Kantonalbank (BSKP) 0.08% 0.15% 0.98% 0% 2.29 4.84
BNP Paribas (BNP) 0.57% 0.76% 5.41% 0.02% 2.69 8.24
Commerzbank AG (CBK) 2.03% 3.1% 23.37% 0.09% 3.03 10.98
Credit Agricole (ACA) 1.1% 1.41% 7.78% 0.04% 1.99 3.63
Credit Suisse Group (CSG) 0.83% 0.98% 4.91% 0.05% 1.77 2.7
Danske Bank (DANSKE) 0.78% 1.71% 16.21% 0.01% 4.47 24.12
Deutsche Bank (DBK) 1% 1.52% 10.73% 0.04% 3.14 11.22
DNB ASA (DNB) 0.54% 1.13% 9.11% 0.03% 3.93 15.93
Erste Group Bank AG (EBS) 0.93% 1.39% 9.4% 0.06% 2.73 8.07
HSBC 0.2% 0.33% 2.91% 0.01% 3.82 17.08
ING group (ING) 1.12% 2.15% 21.55% 0.04% 3.99 20.3
Intesa Sanpaolo (ISP) 0.53% 0.63% 3.59% 0.02% 1.63 2.06
KBC Bancassurance Holding S.A. (KBC) 1.17% 2.09% 14.73% 0.02% 2.8 8.27
Lloyds Banking Group (LLOY) 1.26% 2.09% 16.04% 0.03% 2.7 8.95
Luzerner Kantonalbank (LUKN) 0.01% 0.01% 0.06% 0% 2.6 7.39
Nordea (NDA) 0.17% 0.21% 1.51% 0% 2.57 9.26
Royal Bank of Scotland (RBS) 1.62% 2.69% 16.11% 0.02% 2.82 8.07
SEB AB (SEB) 0.66% 1.29% 9.62% 0.03% 3.51 13.51
Societe Generale (GLE) 0.97% 1.19% 6.57% 0.03% 1.75 2.44
St. Galler Kantonalbank (SGKN) 0.07% 0.06% 0.28% 0.01% 1.52 1.3
Standard Chartered Plc (STAN) 0.41% 0.64% 3.23% 0.03% 2.45 5.21
Swedbank (SWED) 0.92% 2.4% 22.87% 0.03% 4.49 23.1
Unicredit (UCG) 1.11% 1.43% 8.36% 0.02% 1.63 1.96
Table 13: Summary of Default Probabilities by Bank.
Mean SD Maximum Minimum Skewness Kurtosis
Italy 1.06% 1.12% 5.71% 0.03% 1.48 2.06
UK 0.91% 1.37% 9.82% 0.03% 2.89 9.56
Spain 0.47% 0.48% 2.58% 0.01% 1.41 1.55
Switzerland 0.22% 0.22% 1.18% 0.03% 1.81 3.21
France 0.88% 1.08% 5.4% 0.03% 1.78 2.37
Germany 1.51% 2.21% 15.95% 0.07% 3.1 11.56
Denmark 0.78% 1.71% 16.21% 0.01% 4.47 24.12
Norway 0.54% 1.13% 9.11% 0.03% 3.93 15.93
Austria 0.93% 1.39% 9.4% 0.06% 2.73 8.07
Belgium 1.17% 2.09% 14.73% 0.02% 2.81 8.27
Sweden 0.58% 1.28% 11.07% 0.03% 4.1 18.95
Netherlands 1.12% 2.15% 21.55% 0.04% 3.99 20.3
Table 14: Summary of Default Probabilities by Country.


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