Balance sheets basics and the interbank network

In order to characterize the interbank market, we use the Bureau van Dijk Bankscope database that contains yearly-aggregated balance sheets information of N = 183 large European banks, from 2004 to 2013. Raw data are available from Bureau van Dijk: https://bankscope.bvdinfo.com. Refer to [43] for all the details about the handling of missing data.

For each bank i and for each year T (we omit the year label for convenience), the balance sheet composition summarizes the financial position of the bank, and consists of assets with a positive economic value (loans, derivatives, stocks, bonds, real estate, and so on) and of liabilities with a negative economic value (such as customer deposits, and debits). To disentangle the interbank market from the overall financial system, we distinguish between interbank assets A_i (corresponding to the total loans to other banks) and external (non-interbank) assets , and between interbank liabilities L_i (the total debts to other banks) and external (non-interbank) liabilities . The balance sheet identity for bank i defines its equity E_i (or tier capital) as the difference between its total assets and liabilities: (1) A bank is solvent as long as its equity is positive. In fact, negative equity means insolvency, as the bank cannot pay back its debs even by selling all of its assets. Here, following the literature on financial contagion [18, 19, 24], we take insolvency as a proxy for default. In the following, we denote the overall equity of the interbank market as E = ∑_i E_i, and its total volume as C = ∑_i A_i ≡ ∑_i L_i (see Table 1).

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Table 1. Aggregate equity and volume of the interbank market (expressed in million USD) for the various years considered.

https://doi.org/10.1371/journal.pone.0161642.t001

For each bank i, its interbank assets and liabilities are aggregates of the individual loans to and borrowings from other banks: A_i = ∑_j A_ij and L_i = ∑_j L_ij, where A_ij is the amount of the loan extended by bank i to bank j, which represents an asset for bank i and a liability for bank j: thus L_ji ≡ A_ij ∀i, j. Hence, the interbank market can be represented as a directed weighted network, in which links correspond to exposures between banks [56, 57]. However, because of privacy issues, we do not have information on the individual loans and obligations. Yet, we can reconstruct the interbank network of exposures from aggregated interbank assets and liabilities by resorting to the two-step inference procedure described in [32], which was shown to be particularly effective in assessing the presence and extent of overnight interbank loans. In a nutshell, the method estimates individual exposures as: (2) where z is a parameter that controls for the density of the network and a_ij denotes the presence of the link. Here we set a network density of 10% and generate an ensemble of 1000 interbank networks. Results presented in this paper are averaged over this ensemble (we remand the reader to [32, 33] for a detailed network analysis of reconstructed interbank markets, and to [14, 41] for a spectral analysis of their connectivity matrices). Note that the probabilistic nature of the connections is particularly fit to represent overnight positions: we assume that contracts are continuously placed and immediately resolved and rolled over, a scenario in which credit and liquidity shocks can be assumed to hit the system and spread on the same temporal scale.

Credit & funding shocks

Having built the network of interbank exposures, we now describe the credit-funding shock dynamics on the interbank market [1, 21] that will be at the basis of the Debt-Solvency Rank. Suppose a generic bank u is hit by an exogenous shock (such as the sudden devaluation of its external assets) that causes u to become insolvent. This event triggers a cascade of losses in the network, through two distinct processes.

1. Credit shock: bank u fails to meet its obligations, resulting in effective losses for its creditors. In particular, each other bank j suffers a loss equal to λA_ju, where 0 ≤ λ ≤ 1 indicates the amount of loss given default. Here we consider only uncollateralized markets (i.e., without a central counterparty that guarantees for interbank loans), and thus set λ = 1 always.
2. Funding shock: banks are unable to replace all the liquidity previously granted by u; this, in turn, triggers a fire sale of assets. In particular, each other bank j can use its cash reserves to replenish only a fraction (1 − ρ) of the lost funding from u, and its illiquid assets trade at a discount (their market value is less that their book value), so that j must sell assets worth (1 + γ)ρA_uj in book value terms—corresponding to an overall loss of γρA_uj, where the parameter γ set the change in asset prices. In the following we will focus on two cases: ρ = 1, meaning that banks actually cannot replace the lost funding from u and are thus forced to entirely replenish the corresponding liquidity by assets sales; and ρ = 0, for which there are no funding shocks as banks always have enough liquid assets to cope with these losses.

Overall, the generic bank j faces a loss of λA_ju + γρA_uj (the first term appears if j is a lender of u, and the second one if j is a borrower of u). Then, bank j also defaults if this loss exceeds E_j, and a new credit-funding shock propagates through the market, resulting in a wave of failures.

Before moving further, we discuss how to compute the coefficient γ that determines the losses due to funding shocks. Following a common approach [10, 18, 58, 59], here we assume that assets fire sales generate a linear impact on prices: given that Q = ρ∑_j A_uj is the aggregate amount of assets that need to be liquidated, we set the price impact coefficient such that when Q = 0 the relative assets price change is Δp/p = 0 (i.e., no devaluation), and when Q = C it is Δp/p = −1 (assets value vanishes). This results in a price impact Δp/p = −Q/C. The coefficient C⁻¹ in our case is of order 10⁻¹³ (see Table 1), meaning that 10bn euros of trading imbalances lead to a price change of order ten basis points—in line with [10]. Finally, to obtain the corresponding value of γ, it is sufficient to equate the relative loss γρ to the relative amount sold (1 + γ)ρ times the relative assets price change (−Δp/p), obtaining γ = [C/Q − 1]⁻¹.

The Debt-Solvency (DS) Rank

We are now equipped to apply the credit-funding shock scenario described above to the Debt Rank framework. Before starting, we recall again the basic idea outlined [40]: the probabilities of banks default can be obtained by iteratively spreading the individual banks distress levels weighted by the potential wealth affected. In its original formulation, Debt Rank considers counterparty risk in a network of long-term interbank loans (in physics terminology, the network is said to be quenched). In this situation, liabilities stay at their face value, while assets are marked to market. The microscopic dynamics of distress propagation then works as follows [41]. If a bank u is hit by a shock and its equity decreases, also if u is still solvent the market value of its obligations decreases, because u is now “closer” to default and thus less likely to meet its obligation at maturity. This results in a loss on equity for the banks that have u’s obligations in their balance sheet, which in turn decrease the market value of these banks obligations, and so forth. Now, the Debt Rank assumes that relative changes of equity translate linearly into relative changes of asset values, resulting in an impact of bank u on bank j equal to λΛ_ju = λA_ju/E_j, where Λ is the interbank leverage matrix [41, 43] (the same approach can be extended to non-linear impact functions [42]). Then, by iterating the described credit shocks dynamics, Debt Rank allows to compute the final level of financial distress in the network.

In contrast, in the interbank market the majority of contracts have overnight duration, meaning that they are placed and shortly after resolved and rolled-over: the network is said to be annealed, as links change continuously. In this situation, liabilities also change and may lead to funding shocks. In line with the original approach, we can assume that the ability of bank u to lend money to the market decreases proportionally to its equity, and that the liabilities (borrowings) of other banks towards u change in the same way. As funding shocks results in effective equity losses only in the presence of fire sales (i.e., when γ > 0), the impact of bank u on bank j in this case can be assessed as ργϒ_ju = ργA_uj/E_j. Overall, the impact of bank u on bank j due to both counterparty creditworthiness and liquidity shortage issues is λΛ_ju + ργϒ_ju. Importantly, we assume that these shocks happen on the same temporal scale. In this way, we can extend the Debt Rank formulation to a dynamics where the equity decrease of bank u causes an equity decrease both for the lenders of u because of credit shocks, and at the same time for the borrowers of u because of funding shocks.

We can formalize the above discussion as follows. The dynamics of shock propagation consists of several rounds {t}, and we are interested in quantifying the level of financial distress of each bank i and at each time step t, given by the relative change of equity: h_i(t) = 1 − E_i(t)/E_i(0). By definition, h_i = 0 when no equity losses occurred for bank i, h_i = 1 when that bank defaults, and 0 < h_i < 1 for intermediate distress levels. At step t = 0, there is no distress in the system, and thus h_i(0) = 0 ∀i. At step t = 1, an exogenous shock hits the market causing a decrease of equity for some banks; hence, the set {h_i(1)}_{i = 1, …, N} represents the initial condition of the dynamics. Subsequent values of h are obtained by spreading this shock on the system, and writing up the equation for the evolution of banks equity [41]. By defining as the set of banks that have not defaulted up to time t − 1 (and thus can still spread their financial distress at t), we assume that a generic bank j propagates shocks as long as it keeps receiving them, i.e., provided that h_j(t) > h_j(t − 1). However, such a propagation is damped with a function D[(t − t_j)/τ], where t_j denotes the time step when bank j first becomes distressed, t_j: h_j(t_j) > 0 and h_j(t_j − 1) = 0, and τ is the damping scale. Overall, we get: (3) In this expression, according to the discussion in the previous paragraphs, the individual impact of bank j on bank i at step t is λΛ_ij + ργ(t)ϒ_ij, where the fire sale devaluation factor depends on time as γ(t) = {C/[ρQ(t)] − 1}⁻¹. The aggregate amount of interbank assets potentially to be liquidated at t is in turn given by . Concerning the damping function, note that for generic bank j it plays a role only when t ≥ t_j, as otherwise h_j(t) = h_j(t − 1) = 0. Importantly, it must have the properties D(0) = 1 and : the spreading is damped more and more with the propagation step, and eventually vanishes. An exponential shape D[(t − t_j)/τ] = exp[−(t − t_j)/τ] satisfies these properties, but alternative forms can be used as well. The damping scale τ sets the mean lifetime of the spreading. In the limit τ → 0, we have exp[−(t − t_j)/τ] → δ(t − t_j): banks spread their distress only once, as in [40]. In the opposite limit τ → ∞, exp[−(t − t_j)/τ] → 1: banks always propagate received shocks until they default, as in [41].

The above described dynamics stops at t* when no more banks can propagate their distress, i.e., h_i(t*) = h_i(t* − 1) ∀i. The overall Debt-Solvency (DS) Rank of the network is then obtained as: (4) where ν_i = E_i(0)/∑_j E_j(0) = E_i(0)/E(0) is the relative importance of bank i in the system, and E(t) = ∑_i E_i(t) is the total equity of the system at time t. DS(t*) thus represents the amount of equity that is potentially at risk in the system, given an initial shock {h_i(1)}. Note that the original Debt Rank accounting only for credit shocks is recovered for λ = 1 and ρ = 0.

Group DS Rank

We first consider the simplest case of an initial shock to the system corresponding to a fixed percentage (1 − ψ) of equity loss for each bank: E_i(1) = (1 − ψ)E_i(0) ∀i. Thus, h_i(1) ≡ ψ ∀i and DS(t*) = 1 − ψ − E(t*)/E(0). Concerning the dynamics of shock propagation, we can easily study two limiting cases.

• When ψ ≃ 0, we can safely assume that no bank has failed after the first propagation step: h_i(2) < 1∀i. We thus have h_i(2) = ψ{1 + ∑_j[λΛ_ij + ργ(1)ϒ_ij]} ∀i and, after some algebra, , where γ(1) = ρψ/(1 − ρψ). For τ → 0, the dynamics stops at this stage as banks have already spread their distress once. For τ > 0, instead, the dynamics goes on and eventually banks start to default.
• When ψ ≃ 1, all banks will ultimately fail after some rounds of distress propagation. In general, if at t = t* each bank has failed, we get h_i(t*) = 1 ∀i and E(t*) = 0, so that .

Fig 1 shows the results of the group DS dynamics for two temporal snapshots of our dataset: 2008 (the year of the global financial crisis) and 2013 (the most recent year at our disposal). Panels (a) summarize the stationary profile of DS as a function of the initial shock ψ. Let us first consider the case τ = 0 of banks spreading their distress only once. As expected, for ψ ≃ 0 the system is close to the configuration of with no bank defaults. Then, as ψ increases, the overall equity loss grows, and after reaching its maximum value DS(t*) converges to the configuration with all bank defaults. The clear effect of liquidity shocks is that of increasing equity losses (especially in correspondence of the peak), and making the dynamics converge to for smaller initial shocks. Equity losses are however much heavier when subsequent rounds of shock propagation are allowed. In the extreme case τ → ∞, strikingly for year 2008 the dynamics converge to even if the initial distress is very small. By contrast, for year 2013 some banks survive, indicating a higher resilience of the system to financial distress. In any case, the dynamics needs more rounds to converge for small values of ψ, as shown in panels (b), and the fire sale devaluation factor γ eventually assumes values close to 1, as shown in panels (c): large fire sales are triggered at least once during the evolution of the system. By denoting the stationary value of DS Rank for a given value of ρ as DS^[ρ](t*), we see from panels (d) that the progressive effect of liquidity shocks, given by the difference Δ^[ρ] DS(t*) = DS^[ρ](t*) − DS^[0](t*) is almost linear in ρ, and can reach values up to 0.3 for ρ = 1. Finally, in order to gain insights on the reason why the resilience of the interbank market changed so significantly between the two considered years, we study the interbank leverage matrix, whose spectral properties determine the stability of the shock propagation dynamics [41]. By definition, the interbank leverage ratio of a bank i is given by Λ_i = ∑_j Λ_ij. Such value determines the bank resilience to credit shocks in the interbank market [41], with Λ_i < 1 representing the “safety” condition for which that bank has enough capital to cover the losses due to the simultaneous defaults of all its debtor banks. Here, in order to account also for liquidity shocks, we introduced an extended interbank leverage ratio λΛ_i + ρϒ_i, where ϒ_i = ∑_jϒ_ij and γ = 1 according to the analysis of panels (c). The shape of the probability distributions of original and extended leverage ratios in the banking system, shown in panels (e), reveals how these values were much higher in 2008 than in 2013, resulting in a more fragile system in the previous case. Indeed, since the breakdown of the interbank market at the global financial crisis, some banks started to adopt a precautionary hoarding policy (due to concerns about future liquidity needs), and other banks in turn responded to the liquidity hoarding of others by adopting the same behavior [50, 52]. While precautionary motives still affect the interbank market nowadays (its total volume being steadily decreasing since 2007, see Table 1), the resulting reduced leverage ratios resulted in a more stable system.

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Fig 1. Detailed results of group DS Rank for years 2008 (upper panels) and 2013 (lower panels).

(a) Stationary value DS(t*) as a function of the initial shock ψ, in the cases of maximal or vanishing strength of liquidity shocks (ρ = 1 or ρ = 0, respectively) and of instantaneous or vanishing damping of shocks (τ = 0 and τ = ∞, respectively). (b,c) Temporal dynamics of the system: DS and fire sale devaluation factor γ as a function of ψ for various iteration steps t, in the setting with liquidity shocks ρ = 1 and multiple shocks propagation τ = ∞. (d) Progressive effect of liquidity shocks: difference Δ^[ρ] DS(t*) between DS^[ρ](t*) (when banks must sell illiquid assets for a fraction ρ of the lost funding) and DS^[0](t*) (when no liquidity shocks occur), for ψ maximizing the overall equity loss. (e) Probability distributions of banks leverages Λ (black diamonds) and Λ + ϒ (blue circles), with relative median given by a dashed vertical line.

https://doi.org/10.1371/journal.pone.0161642.g001

While detailed results for other years can be found in S1 File, we can grasp the general systemic risk properties of the interbank market over the time span considered (2004–2013) from Fig 2, which shows color maps of the stationary group DS values for the various years and as a function of the initial shock ψ. For τ = 0, DS(t*) keeps its “inverse U” shape for all years. Yet, while in the absence of liquidity shocks systemic risk substantially decreases after 2008, these shocks do lead to an increase in equity losses that shows up especially in late years. A similar behavior is observed even more clearly for τ → ∞, as in this case the distance from the full-fail scenario after 2008 is dramatically reduced by the presence of liquidity shocks. Overall, these results point to the importance of considering liquidity shocks when assessing potential equity losses in the financial system. Notably, after 2008 liquidity risk has become more prominent for higher values of the initial shock: in complex network terminology, the system has become more robust for low ψ, yet more fragile for high ψ.

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Fig 2. Heat maps of group DS(t*) as a function of ψ and of the year.

Upper plots refer to the case of no damping (τ = ∞) and lower plots to the case of instantaneous damping (τ = 0). Then, from left to right, plots show DS(t*) with liquidity shocks and ρ = 1, no liquidity shocks ρ = 0, and the difference between these values.

https://doi.org/10.1371/journal.pone.0161642.g002

Individual DS Rank

We now move to the case in which the initial shock to the market corresponds to the failure of an individual bank u: E_u(1) = 0 ⇒ h_u(1) = 1 and E_i(1) = E_i(0)⇒h_i(1) = 0 ∀i ≠ u. To make this initial condition explicit, we use the notation X(t|u) for the value of a generic quantity X at step t an when u is the initially shocked bank. We thus have DS(t*|u) = 1 − E(t*|u)/E(0) − ν_u. Using this framework, we can compute two bank-specific risk indicators:

• Impact I, given by the relative equity loss experienced by the market after the initial default of that bank:
• Vulnerability V, namely the relative equity loss for that bank averaged over the initial defaults of all other banks:

Fig 3 shows scatter plots of impact and vulnerability of individual banks again for years 2008 and 2013 (plots corresponding to other years can be found in S2 File). First note that, as expected, vulnerability clearly increases with the leverage ratio of the bank, whereas, impact is more related to the bank size (as measured by its initial equity). According to our data, interestingly the banks with the highest leverage have relatively small size, and despite being very vulnerable they lack the potential to affect the market significantly. Concerning the different shock propagation scenarios, in general we observe that the systemic impact of banks drops significantly from 2008 to 2013, while the decrease of systemic vulnerability is less evident but for multiple shock propagation τ → ∞. The effect of liquidity shocks is in turn that of increasing both measures. In particular, in year 2013 these shocks cause an increase of 100% (for τ = 0) and of 50% (for τ → ∞) for the impact of the largest banks, again underlying the importance of fire sale spillovers for a more robust assessment of systemic risk.

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Fig 3. Scatter plot of impact and vulnerability values for each bank in years 2008 (upper panels) and 2013 (lower panels).

Here we show the cases of maximal or vanishing strength of liquidity shocks (ρ = 1 or ρ = 0, respectively) and of instantaneous or vanishing damping of shocks (τ = 0 and τ = ∞, respectively). The size of the bubble is non-linearly proportional to the initial equity of the corresponding bank: . The color of the contour is instead given by the value of the generalized leverage λΛ_i + ρϒ_i.

https://doi.org/10.1371/journal.pone.0161642.g003

We conclude by remarking that, according to our analysis, impact indeed correlates with size but is not a monotonic function of it: some banks can be more impactful on the system than others with larger equity or leverage. Thus, network features are important when determining the extent of financial contagion: with respect to banking systems regulation, relying on balance sheet constraints may be insufficient, and should be supplemented by considerations on the structure of the network [21].