A hybrid approach to assess systemic risk in financial networks
Abstract
We propose a credit risk approach in which financial institutions, modelled as a portfolio of risky assets characterized by a probability of default and a correlation matrix, are the nodes of a network whose links are credit exposures that would be partially lost in case of neighbours’ default. The systemic risk of the network is described in terms of the loss distribution over time obtained with a multiperiod Montecarlo simulation process, during which the nodes can default, triggering a change in the probability of default in their neighbourhood as a contagion mechanism. In particular, we have considered the expected loss and introduced new measures of network stress called PDImpact and PDRank. They are expressed in monetary terms as the already known DebtRank and can be used to assess the importance of a node in the network. The model exhibits two regimes of ’weak’ and ’strong’ contagion, the latter characterized by the depletion of the loss distribution at intermediate losses in favour of fatter tails. Also, in systems with strong contagion, low average correlation between nodes corresponds to larger losses. This seems at odds with the diversification benefit obtained in standard credit risk models. Results suggest that the credit exposure network of the European global systemically important banks Â is in a weak contagion regime, but strong contagion could be approached in periods characterized by extreme volatility or in cases where the financial institutions are not adequately capitalized.
pacs:
87.15.A, 05.40.aI Introduction
One of the lessons that has been learned from the recent credit crisis is that the stability of the financial system cannot be pursued focussing exclusively on each individual bank or financial institution. A broader approach  macroprudential policy is required, as interconnections and interactions are at least as important in contributing to the overall dynamics (16)(20)(10)(14)(13). In order to limit systemic risk, defined as the risk that a considerable part of the financial system is disrupted, a number of dedicated boards and committees have been created: the Financial Policy Committee (FPC) at the Bank of England, the European Systemic Risk Board (ERSB) and the Financial Stability Oversight Council in the United States. The regulators are looking at new methodologies and ideas from different disciplines as it is felt that the existing models have shown serious limitations, being incapable of predicting the timing and the extent of the financial crisis (21). In particular techniques borrowed from network science (28) (6) have been successfully applied to the study of network resilience to external shocks (7) (29) and have proven useful in the analysis of financial systemic risk (17) (12) (22) (5) (27).
In this context, financial institutions are described as nodes in a network, connected by different kinds of edges, indicating: cross ownership (31), investments in the same set of assets (overlapping portfolios) (32) or credit exposures (8) (26). Another natural way to approach the problem is to consider the financial institutions as a portfolio of risky assets with assigned probability of default and correlations. In this context the risk is thought as the premium to insure the portfolio against the loss of a percentage of the total asset amount (18). The framework we propose in this article aims at combining the two approaches, using all the available information about the system. The original models used to study network stability (11) relied on a variant of the ’domino effect’ to propagate the stress and, if the original shocks was not big enough to start the chain reaction, no quantifiable effect could be calculated. To overcome this limitation Battistoni et al (4) introduced DebtRank, a new measure of systemic risk. The DebtRank of node , is a number measuring the fraction of the total economic value in the network that is potentially affected by the distress or the default of node . The measure presented interesting characteristics such as being expressed in monetary terms and being able to ’feel’ the stress in the network also in absence of actual defaults. However, it is not evident how, in the real world, the propagation of the stress postulated by the model would happen and how it would translate in an actual loss for the banks. In order to fill this gap Bardoscia et al (2) proposed a slightly modified model and a derivation of the dynamics for the shock propagation using basic accounting principles. To obtain their results, the authors had to make the not fully financially justified assumption that the exposures towards other banks lose their value proportionally to the loss in equity suffered by the borrowing banks, namely:
(1) 
where and are, respectively, the equity of bank and the exposure that bank has with bank at time . The above updating equation is used when bank has not defaulted in the previous time period, otherwise is set to be zero. In such approach it is also crucial to understand how the time step is defined: is it a year, a quarter or a minute? The answer is not irrelevant because one of the findings of Ref. (2) is that, no matter how small the initial shock is, if the modulus of the largest eigenvalue of the interbank leverage matrix is greater than one, at least one bank fails. This is the equivalent of the ’butterfly effect’ for banks which is clearly unrealistic in actual financial networks. The main issue relative to Eq. (1) is that it fails to consider the stochastic nature of the potential loss associated to nodes with diminished equity. We believe that the most useful objects of investigation are the probabilities of defaults of the nodes and not the assets. In introducing our model, we maintain the characteristics of DebtRank: a monetary value for the ’centrality measure’ of a node and the sensitivity to a distress of the network also in absence of actual default. In addition to this, our model uses techniques that are familiar to finance professionals (25) and its contagion mechanism is intuitive, with the default of a node increasing the probability of default of its neighbourhood. The introduction of a welldefined unit of time allows analysing the dynamic of stress propagation and modelling the financial institutions as complex agents reacting to evolving macroeconomic scenarios.
Ii The PD Model
In our model, financial institutions are described as the nodes of a network. Each node , with , is characterised, at time , by the total asset , i.e the set of anything a financial institution owns and that can be converted to cash, by a loss given default , representing the percentage of the total asset that would be lost in case of default, and by a threshold , denoting the capital of , more precisely the “tier1 capital”, which is the capital of the bank that can be used to absorb losses. Furthermore, we associate to each financial institution a probability of default per time interval . This is a crucial quantity of our model, and represents the probability that node defaults in the time interval . The links of the network are directed and weighted, and represent credit relationships, for example loans, among financial institutions. In practice, we set the weight of the link from to equal to the exposure of to the default of node .
In order to quantify systemic risk, we use a multiperiod simulation process with time iterations, during which the nodes can default, triggering an increase of the probability of default in their neighbourhood as a contagion mechanism. The time step is chosen coherently with the definition of , hence it is a well defined unit of time. In the following computations we will use year. In our model, the monetary impact on node due to the defaults of neighbouring nodes is quantified as:
(2) 
where is equal to 1 if node has defaulted at time , and is 0 otherwise. We will consider as a constant and the index in the sum includes all the nodes that had not been defaulted at the previous times . The monetary impact on node would, in turn, trigger a modification of the total assets and the thresholds, but also of the probabilities of default, of node at the next time, according to the following updating equations:
(3)  
where is in general a function of the probability of default , the impact experienced, the threshold and the total asset at the previous time. In this paper we will not take into account the dynamics of . We will set for each bank to be constant in time.
We have established a mapping between probability of default and change in equity in the context of standard financial risk management theory (3) using the Merton model (23):
(4) 
where is the cumulative gaussian distribution and, for simplicity, we have set the drift term equals to zero and considered a period of one year (see Materials and Methods). The total asset volatility and the total liability are considered constant during the simulations, with:
(5) 
Hence can be expressed as , and Eq. (4) provides an updating equation for once the right hand side is updated with the new value of obtained from Eqs. (3). We also set when the impact on the financial institution is equal to its tier1 capital , i.e to the maximum monetary loss the institution can tolerate (see Fig. 1). The procedure that we have just described, for updating the probabilities of default will be referred to as the Merton update. We have also explored a Linear update, i.e. a linear relationship between and the impact :
(6) 
with being capped to when the impact is greater or equal to (see Fig. 1).
In our simulations, at each of the time periods, the defaults are driven by a stochastic process which is assumed to follow a multivariate gaussian distribution with a correlation matrix whose elements are, for instance, calculated from the equity returns of financial institutions and , obtained from the stock market quotations (19). According to Ref. (18) we consider . In order to reflect the fact that the average correlation increases during periods of crises, it is possible to use updating equations such as:
(7) 
for the pairwise coefficient , where is the number of defaulted nodes at time . The quantity represents the average correlation increase per single default at time and needs to be calibrated (or assumed in a stress scenario). For simplicity, we have not used Eq. (7) but we have performed simulations with different values of (kept constant during the simulation).
The default of financial institution at time (more specifically in the temporal window ) happens when the corresponding drawn value of random variable in the sampling is smaller than , with being the cumulative gaussian distribution. If at least a node has defaulted at time , we update the variables of the system for the next time as in Eqs. (3). Defaulted nodes are then removed. Instead, if no node has defaulted at time , we proceed to the following time step and the new sampling with the same network and stochastic process parameters. The simulation is then continued for temporal iterations.
The loss for the entire network at time is calculated as:
(8) 
while the total loss is obtained by summing up the discounted values of the losses at the different time periods:
(9) 
where is the discount factor relative to time t (in this paper we set for each ). We have considered an ensemble of 100000 different Montecarlo realizations of the iterations of the stochastic process corresponding to a total period of 7 years, and we have calculated distributions and averages of different quantities, such as the losses, over the ensemble. We indicate with the symbol such averages. The fact that our Montecarlo simulations involve a number of time steps and in each of them the contagion is spread via a modification of the probability of default, is a crucial difference with respect to standard credit risk management techniques.
ii.1 Stress scenario: PDImpact and PDRank
In our framework, the nodes are characterized by an initial probability of default at time . Hence, even in absence of any external shock, the system can suffer losses during the simulations within the considered time frame of time periods. The loss distribution so obtained, and in particular the expected loss can be used as the baseline for comparison with the losses in presence of stress. Since a distress of the network is described as an increased probability of default of a set of nodes, , we can introduce the socalled Probability of Default Impact (PDImpact), indicated as , of the stressing perturbation onto the initial probability of default as:
(10) 
where the two terms on the right hand side are respectively the average loss of the network in the presence and absence of the additional stress .
Analogously, we can also introduce a node centrality measure, that we name the Probability of Default Rank, or PDRank, for assessing the relative importance of each financial institution. The PDRank of node is obtained multiplying the probability of default of node by the additional average loss experienced by the network due to the default of node :
(11) 
where is the initial probability vector in which the probability corresponding to node has been set to at , while is the initial probability vector where the probability corresponding to node has been set to and kept at the value for each time (the node cannot default during the simulation). Therefore, the quantities and represent respectively the average loss, during the simulation, when node defaults at time 1, and when node cannot default (the average loss that the network would suffer anyway irrespective of the node ). In practice, of node measures the expected loss “due” to node . As the already known DebtRank, it is expressed as a monetary value and can be used to rank the nodes in terms of their ’systemic risk’.
A further characterization of a network, which we name , can be obtained by quantifying the sensitivity of the system to a percentage increase of all the initial probabilities of default. Assuming an approximate linear relationship between the PDImpact obtained for an increase of the probabilities of default and the percentage of increase , we can define as follow:
(12) 
In this way represents the variation of PDImpact for a unitary percentage variation of the probabilities of default.
Results
We have applied our model to analyse the data collected by the European Banking Authority (EBA) relative to the European global systemically important banks (gsib), and we have used a new algorithm to infer the network of exposures from the incomplete data provided (see Materials and Methods). The initial values of the probabilities of default have been obtained from public information about the credit rating of the banks and from statistics available on the Fitch website (see Materials and Methods). The results in this section should be considered as an academic exercise to illustrate the characteristics of the model as we do not have the exact adjacency matrix of exposures and a number of approximations and assumptions have been made. Also it is clearly unreasonable assuming that the banks or the regulators would not react after the first defaults (replenishing their tier1 capital for example). In the following, where not specified we use an average correlation .
ii.2 Two ’regimes’: weak and strong contagion
The distributions of the total loss experienced by the network under different values of average correlations and different values of the thresholds are shown respectively in Fig. 2 and Fig. 3. We say that the system is in a ’strong contagion’ (as opposed to ’weak contagion’) regime if its ability to propagate the stress on the network affects the loss distribution, depleting the intermediate losses in favour of the extreme end of the tail. Fig. 2(b) and and Fig. 3(b), relative to the Linear update, present strong contagion characteristics. In particular Fig. 2(b) shows that lower average correlation corresponds to more pronounced losses as it is more likely to have at least one default during the first time periods and subsequently the loss is amplified by the contagion. The Merton update model in Fig. 2(a) does not show strong contagion effects with the available data. This does not mean that they cannot appear under particular circumstances. For example, the distribution corresponding to thresholds reduced by 50% or 90% in Fig. 3(a) clearly shows the depletion of the loss distribution in the middle with a marked increase in the tail. As a further example, as shown in Fig. 2(c), when the thresholds are reduced at half of their values, also the system with Merton update shows increasing losses at lower value of average .
Fig. 4 confirms an approximate linear relationship between PDImpact and a percentage increase of the initial probabilities of default, allowing the definition of the measure PDBeta (Eq. (12)) that can be used, together with the expected loss , to gauge the riskiness of a network. Defining the global asset of the network as , we have found and for the Merton update, and and for the Linear update.
ii.3 The critical nodes of the network
We can now analyse the relative contribution of the nodes to the systemic risk of the network. Table 1 reports the values of PDRank in million of EUR, obtained using respectively the Merton and the Linear update. The ranking of the most important nodes is different in the two cases and the corresponding values can vary by more than one order of magnitude for nodes with a high probability of default such as BFA with and MPS with . These nodes can act as a catalyst for a chain reaction of losses especially in a ’strong contagion’ regime: relatively small losses can have a dramatic effect on the probability of default of the impacted nodes and this explains why they are at the top of the DebtRank table in the Linear update case. The ranking implied by PDRank is different from the one that takes into consideration the total asset of the financial institutions (as in a ’too big to fail’ approach). This is evident as the PDRank definition includes the probability of default, which is not related to the total asset. We can investigate if PDRank can be explained by the probability of default multiplied by the total asset. Fig. 5 shows that this is not the case even if there is definitely a positive correlation. It is interesting to note that BFA and MPS are well above the regression line in case of the Linear update (Fig. b), which reflects once again the increased role of the probability of default in a strong contagion regime.


Iii Concluding Remarks
The model described in this paper uses different types of information regarding the financial institutions: correlation structure, probabilities of default and network of exposures. It allows following the evolution in time of the network under different stress scenarios, using multi period Montecarlo simulations. In this paper we focused our attention to average quantities to characterized the riskiness of the network, such as expected loss and the slope of the expected loss vs a percentage increase of the probabilities of default (PDBeta). However our model allows obtaining the entire loss distribution and other statistics (e.g. quantiles) could be more appropriate for particular investigations. The nodes can be ranked in a natural way with respect to their contribution to the systemic risk using the centrality measure ’PDRank’. The contagion propagation is easy to understand as it is linked to an increase of the probability of default of a node due to the default of its neighbours. We have also introduced an algorithm to recreate a network of individual exposures given the information of the total intranode assets and liabilities for each node. A ’strong contagion’ regime has been identified, characterized by larger losses at lower average correlation between nodes and by a depletion of the loss distribution at intermediate losses in favour of a fatter tail. The financial network considered does not present strong contagion effects when using the Merton update. However we have seen that such effects can appear when the banks are not sufficiently capitalized (i.e. thresholds are low). Furthermore we have kept the asset volatility constant in the Merton model. In periods with extreme volatility, would be bigger and the calculated with the Merton update (Eq. (4)) could increase significantly, moving the system toward the strong contagion regime.
Iv Materials and Methods
European global systemically important banks. As part of its mandate, the European Banking Authority collect data annually from the global systemically important banks (gsib) in the European Union and publish the results in their website ’www.eba.europa.eu’. We have chosen the data from the year 2014. The data set contains the fields ’Intrafinancial system assets’ and ’Intrafinancial system liabilities’ that we use in our model to recreate the individual exposures using the algorithm described in the next paragraph. The field ’Total exposures’ provides a proxy for the total assets (). The tier1 capital has been obtained from another study performed by EBA in cooperation with European Systemic Risk Board (ESRB): ’The EUwide stress test’, that aims at ’assessing the resilience of financial institutions to adverse market developments’.We have selected the Banks that were in both exercises and we identified 35 institutions. The initial probability of default has been obtained from the table ’Financial Institutions Average Annual Transition Matrix: 19902014’ in the document ’2015 Form NRSRO Annual Certification’ obtained from Fitch website www.fitchratings.com.
Inferring the network. We describe a new algorithm to infer the network from incomplete data. In the literature, a maximum entropy algorithm (30) has often been used but it is known that it might not represent the best choice for recreating a realistic interbank network (24) and different alternatives have been proposed (8)(15)(9). We want to capture the fact that small financial institutions are more inclined to have connections with a small number of bigger banks. The level of exposure tends to be above a certain minimum value as the creation of a credit relationship involves a ’maintenance’ cost. This was already addressed by Anand et al. (1) but here we propose an alternative algorithm that we find more intuitive and that allows controlling over the minimum exposure amount and the ’degree of attraction’ between smaller nodes and bigger ones. The main idea is to match asset with liabilities, building the adjacency matrix in steps: 1) The smaller borrower nodes choose first where to get the money from; 2) The lender (a different node) is chosen randomly with a probability that is proportional to its remaining assets to the power of alpha (alpha being the parameter for tuning the degree of attraction between heterogeneous nodes). 3) The loan amount is chosen as a percentage of the total liabilities of the borrower node and represents the minimum exposure that it is convenient to exchange, constrained by the ’residual’ assets of the lender and the ’residual’ liabilities of the borrower. 4) The adjacency matrix and the residual asset and liabilities amount are updated. 5) The process continues till all the assets are matched with all the liabilities. 6) If at the end remains one node that can borrow money only from itself, the procedure reroutes some of the previous loans so that the adjacency matrix is completed with zero values on the diagonal.
Merton Model. The Merton model can be used to evaluate the probability of default of a company described as having a single liability in terms of a bond issued and expiring at time The asset of the company is thought as following a lognormal random process with drift and volatility . A default occurs if during a simulation the asset falls below the value at time . Assuming as a constant, it is possible to calculate the probability of default as:
(13) 
is the cumulative gaussian distribution. To simplify we assume = 0 and year. Having , and it is possible to calculate and, inverting Eq. (13), . We make the assumption that doesn’t change during the simulation. It is then possible to use Eq. (13) to update the probability of default given the new values of and . The assumption of constant is not completely satisfactory as it is reasonable to expect that the volatility increases when the company approaches the default. It is possible to devise a more complex implementation of the model that includes a dynamics for .
Acknowledgements.
We are grateful for discussions with Alessandro Fiasconaro and Neofytos RodosthenousReferences
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