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 Open Access
The joint impact of bankruptcy costs, fire sales and crossholdings on systemic risk in financial networks
 Stefan Weber^{1}Email author and
 Kerstin Weske^{1}
https://doi.org/10.1186/s4154601700209
© The Author(s) 2017
Received: 15 January 2017
Accepted: 30 May 2017
Published: 26 June 2017
Abstract
The paper presents a comprehensive model of a banking system that integrates network effects, bankruptcy costs, fire sales, and crossholdings. For the integrated financial market we prove the existence of a pricepayment equilibrium and design an algorithm for the computation of the greatest and the least equilibrium. The number of defaults corresponding to the greatest pricepayment equilibrium is analyzed in several comparative case studies. These illustrate the individual and joint impact of interbank liabilities, bankruptcy costs, fire sales and crossholdings on systemic risk. We study policy implications and regulatory instruments, including central bank guarantees and quantitative easing, the significance of last wills of financial institutions, and capital requirements.
Keywords
 Systemic risk
 Financial contagion
 Financial network
 Crossholdings
 Fire sales
 Bankruptcy costs
JEL Classification
 G01
 G21
 D85.
Introduction
“Systemic risk refers to the risk that a financial system as a whole is susceptible to failures initiated by the characteristics of the system itself.”^{1} If strong links between financial institutions are present, a shock to only a small number of entities might propagate through the system and trigger substantial financial losses. Significant dependence can thus increase the risk of a systemwide breakdown.
 (i)
We prove the existence of a clearing equilibrium that is not necessarily unique and provide an algorithm for the computation of the greatest and the least equilibrium. The equilibrium is characterized by the vector of clearing payments and the price of the commonly held illiquid asset that is exposed to price effects.
 (ii)
We study the impact of bankruptcy costs, fire sales, and crossholdings on systemic risk in numerical experiments. We demonstrate that fire sales and bankruptcy costs can trigger and amplify financial crises. Policies that mitigate their impact might significantly enhance the resilience of the financial system. Crossholdings do, in contrast, have a stabilizing effect, if they can be exchanged for liquid assets. Central banks that engage in such a market can reduce the number of defaults in the system.
 (iii)
We study policy implications and regulatory instruments, including central bank guarantees, quantitative easing, the significance of last wills of financial institutions, and the efficiency of capital requirements. We find that capital adequacy ratios based on riskweighted assets reduce systemic risk, if they are sufficiently high. However, they do not rely on any statistics that capture systemic risk in a proper way. Comparative statics show that capital adequacy ratios can be equal for varying parameters of our model that are associated with completely different levels of systemic risk. This demonstrates that classical capital adequacy ratios are a very rough instrument. A much better alternative are systemic risk measures that we analyze in the last section.
Previous papers do not allow an assessment of the robustness of their conclusions since they only focus on particular aspects of systemic risk neglecting all other driving factors. Our model, in contrast, shows to what extent causal relations that were previously discovered are preserved within a general framework; it also detects the differences that might occur. In summary, we find that many of our qualitative results are quite robust across different network structures and for a large number of driving factors. However, the relative importance of interacting contagion channels can only be characterized in the joint model. This justifies—for the first time from a general perspective—the relevance of previous approaches, but indicates at the same time that quantitative predictions and the design of regulatory policies require a more sophisticated analysis.
Literature Our approach extends the equilibrium approach of Eisenberg and Noe (2001). Their seminal paper models interbank contagion within a network of nominal liabilities and proves the existence and uniqueness of a clearing payment vector that endogenously captures losses given default. At the same time, they construct an efficient algorithm for the computation of the clearing vector. Closely related empirical studies can, e.g., be found in Cont et al. (2013), Elsinger et al. (2006), Glasserman and Young (2015), and Upper (2011). These cast doubt that empirical patterns of contagious defaults can solely be explained by networks of nominal liabilities.
In this paper, we integrate multiple interaction channels and amplifying mechanisms of contagion, including bankruptcy costs, fire sales, and crossholdings. While we investigate their joint impact, up to now the literature has only been studying these factors separately: Bankruptcy costs are, for example, considered by Elsinger (2009), Elliott et al. (2014), Rogers and Veraart (2013), and Glasserman and Young (2015); crossholdings, e.g., by Elsinger (2009), Elliott et al. (2014), Fischer (2014), Suzuki (2002), and Karl and Fischer (2014). Cifuentes et al. (2005) incorporate fire sales into the setting of Eisenberg and Noe (2001); their approach is further extended by Amini et al. (2013), Chen et al. (2016), Gai and Kapadia (2010), Nier et al. (2007), and Feinstein (2017). Most of these papers consider only one extension of the basic framework.^{2} For a detailed review of the literature see also Staum (2013).
“A fire sale is essentially a forced sale of an asset at a dislocated price. The asset sale is forced in the sense that the seller cannot pay creditors without selling assets. The price is dislocated because the highest potential bidders are typically involved in a similar activity as the seller, and are therefore themselves indebted and cannot borrow more to buy the asset. Indeed, rather than bidding for the asset, they might be selling similar assets themselves. Assets are then bought by nonspecialists who, knowing that they have less expertise with the assets in question, are only willing to buy at valuations that are much lower.”
Evidence is discussed in several papers including Brunnermeier (2009), Coval and Stafford (2007), Cont and Wagalath (2016), Jotikasthira et al. (2012), Khandani and Lo (2011), and Shleifer and Vishny (1992). In real markets, fire sales typically refer to the liquidation of portfolios. Empirical data show that this is related to increased correlations as well as price impact. A single representative illiquid asset can thus be used as a first approximation. This is the approach that we take in our model in order to keep the suggested framework simple.
Outline The paper is organized as follows. In the section “An integrated financial network model”, we present our model of the financial system and provide a preliminary analysis of net worth, price impact, and clearing payment vectors. The existence of a pricepayment equilibrium consisting of a clearing payment vector and a clearing price of the illiquid asset is demonstrated in the section “Existence of equilibria and an algorithm for their computation”. Moreover, we provide an extension of the fictitious default algorithm of Eisenberg and Noe (2001) in order to compute the greatest and least equilibrium. The section “Case studies” focuses on numerical case studies which constitute a key part of our paper. These illustrate the individual and joint impact of bankruptcy costs, fire sales, and crossholdings on systemic risk, measured as the number of defaults in the greatest pricepayment equilibrium. We describe various regulatory policies and analyze their efficiency. The main conclusions and questions for future research are discussed in the section “Conclusion”. All proofs of the results in the sections “An integrated financial network model” and “Existence of equilibria and an algorithm for their computation” are presented in the section “Proofs”.
An integrated financial network model

Direct liabilities: Banks have nominal liabilities against each other. These liabilities are promises that will only partially be fulfilled if some of the banks default.

Fire sales: If the portfolios of different banks include common assets, changes in asset prices simultaneously influence the net worths of these banks. Common holdings may give rise to substantial systemic risk, if illiquid assets are sold in large quantities and prices decrease significantly. For simplicity, our model assumes the existence of a single (representative) illiquid asset.

Crossholdings: Banks may, in addition, hold shares of each other. In this case, the net worths of banks depends on the net worths of other banks due to these crossholdings.
The single period is interpreted as a snapshot of a banking system that continues to exist afterwards. The net worth of each bank in the financial network depends on the realized payments, the price of the commonly held illiquid asset, and the net worths of the other banks. In the first step, we will describe how the value of asset holdings of an individual bank can be computed if these three key factors are exogenously fixed. In the second step, we will construct and analyze an equilibrium model that allows an endogenous computation of the net worths of all banks, a clearing payment vector, and a realized average price of the illiquid asset.
Assets and liabilities
Letting \(\mathcal {N}=\{1,\dots,n\}\) be the set of banks in the financial system, we denote by \(p\in \mathbb {R}^{n}_{+}\) the realized payments of the banks, by \(w\in \mathbb {R}_{+}^{n}\) the vector of net worths of the banks, and by \(q\in \mathbb {R}_{+}\) the price of the representative illiquid asset. In the first step, we suppose that these quantities are exogenously specified.
External Assets As suggested by Cifuentes et al. (2005), we consider banks that hold two assets which are external to the banking system: an amount of \(r\in {\mathbb {R}}^{n}_{+}\) shares of a liquid asset (e.g., cash) and \(s \in {\mathbb {R}}^{n}_{+}\) shares of an illiquid asset. Assuming that the liquid asset’s price remains constant at one monetary unit, the value of bank i’s external assets is given by r _{ i }+s _{ i } q, if the price of the illiquid asset is q.
Liabilities Each bank has nominal liabilities to the other banks for the considered time horizon. Analogous to Eisenberg and Noe (2001), we suppose that these liabilities are represented by a nominal liabilities matrix \(L\in \mathbb {R}^{n\times n}\): for all \(i,j\in \mathcal {N}\), L _{ ij }≥0 describes the nominal obligation of bank i towards bank j; no bank may hold a liability against itself, i.e., L _{ ii }=0 for all \(i\in \mathcal {N}\). In addition, banks may have further liabilities \(l\in \mathbb {R}_{+}^{n}\) to entities outside the banking system; here, the component l _{ i } is interpreted as the liability of bank i to the outside.
The vector of total liabilities \(\bar {p}\) captures all liabilities of the banks in the system; i.e., its component \(\bar p_{i}\) equals the total liabilities of bank i and is given by \(\bar {p}_{i}={\sum \nolimits }_{j\in \mathcal {N}} L_{ij}+l_{i},\) for \(i \in \mathcal {N}.\) If all banks are able to fulfill their total obligations, \(\bar {p}\) indeed equals the realized payments p of the banks. If, in contrast, some banks do not possess sufficient resources to meet their obligations, then \(p\le \bar {p},\) where the inequality is interpreted componentwise.
Following Eisenberg and Noe (2001), we assume that in case of bank i’s default, its realized payments \(p_{i}<\bar {p}_{i}\) will be distributed proportionally among its creditors according to the size of each creditor’s claim. Therefore, we define the relative liabilities matrix \(\Pi \in \mathbb {R}^{n\times n}\) by \(\Pi _{ij}=L_{ij}/\bar {p}_{i},\) if \(\bar {p}_{i}>0\), and Π _{ ij }=0, otherwise. Thus, the entry Π _{ ij } captures the size of the interbank obligations of bank i towards bank j in proportion to the size of i’s total liabilities. This implies that for a given vector of realized payments p, the value of bank i’s interbank claims is given by \({\sum \nolimits }_{j\in \mathcal {N}} \Pi _{ji}{p}_{j}.\)
CrossHoldings Each bank may hold shares of the other banks. Following Elsinger (2009), these holdings will be captured by a crossholdings matrix \(C\in \mathbb {R}^{n\times n}\): the component C _{ ij } denotes the fraction of bank i’s equity that is held by bank j. We assume that the crossholdings are nonnegative, i.e., C _{ ij }≥0 for all \(i,j\in \mathcal {N}\), and that a bank is not allowed to hold shares of itself, i.e., C _{ ii }=0 for all \(i\in \mathcal {N}\). The technical assumption \({\sum \nolimits }_{j\in \mathcal {N}}^{} C_{ij} < 1, \; i\in \mathcal {N}\), guarantees that the net worth of each bank, as introduced below, is welldefined. If both a crossholdings matrix C and a vector of positive net worths w are given, the contribution of bank i’s crossholdings to its net worth is equal to \({\sum \nolimits }_{j\in \mathcal {N}} C_{ji}w_{j}.\) We also suppose limited liability of crossholdings, i.e., if bank j’s net worth w _{ j } is negative, crossholdings of bank j do not negatively affect the net worths of other banks.
It is wellknown that crossholdings inflate the value of the financial system, see Brioschi et al. (1989), Fedenia et al. (1994), and Elliott et al. (2014). This, in particular, refers to the fact that the aggregated net worth of all banks will be larger than the value of total assets if crossholdings are present. As argued in Brioschi et al. (1989), Fedenia et al. (1994), and Elliott et al. (2014), net worths need to be adjusted by an auxiliary factor that guarantees the conservation of value in the system. The market value of bank i should thus be computed as \((1\sum _{j\in \mathcal {N}} C_{ij}) w_{i}\) for w _{ i }≥0.
Total net worth We will now describe how each bank’s net worth is calculated. In order to fulfill its obligation \(\bar {p}_{i}\), bank i will first use its liquid external assets r _{ i } and its interbank revenues \({\sum \nolimits }_{j\in \mathcal {N}} \Pi _{ji}{p}_{j}.\) If these are insufficient, the bank is left with its illiquid asset and crossholdings. We assume that bank i’s crossholdings \({\sum \nolimits }_{j\in \mathcal {N}} C_{ji}w_{j}\) can be exchanged against cash (possibly involving central banks or governments). However, we suppose that bank i can only realize a fraction of λ _{ i }∈[0,1]. An alternative way to model price impact of crossholdings liquidation via inverse demand functions is presented in “Appendix 2. Crossholdings with price impact”.
The bank is in default if it cannot cover its liabilities, i.e., if w _{ i }<0.
As mentioned before, due to crossholdings, the net worths of the banks differ from their market values, see Brioschi et al. (1989), Fedenia et al. (1994), and Elliott et al. (2014). We will, however, focus on default count statistics which can be computed in terms of the vector of net worths. Market values will only be considered explicitly in the numerical case studies.
As defined above, the net worth of bank i depends on the realized payments p, the price q of the illiquid asset, and all other banks’ net worths w. In the following sections, we provide a method to derive these three key quantities endogenously.
Net worth
Suppose first that p and q are fixed. In this situation, our aim is to define an equilibrium vector of the net worths of the banks. To simplify the notation, we write \(\boldsymbol {0}:=(0,\dots,0)^{T}, \boldsymbol {1}:=(1,\dots,1)^{T}\in \mathbb {R}^{n}\), set a∨b:=(max(a _{1},b _{1}),…, max(a _{ n },b _{ n })) for \(a,b\in \mathbb {R}^{n}\) and denote by diag (μ(p,q,w)) the diagonal matrix whose diagonal entries are given by the vector μ(p,q,w):=(μ _{1}(p,q,w),…,μ _{ n }(p,q,w))^{ T }.
Definition 2.1
The essential net worths vector is defined as a solution of the nonlinear fixedpoint problem (1). The following lemma shows that this equation always possesses a unique solution. A proof of this result is given in the section “Proofs”.
Lemma 2.2
 (a)
The essential net worths vector w ^{∗}(p,q) exists and is unique.
 (b)The essential net worths vector is increasing in payments and prices:$$\begin{array}{@{}rcl@{}} p^{1}\ge p^{2} & \quad \Longrightarrow \quad& w^{*}(p^{1},q)\ge w^{*}(p^{2},q), \\ q^{1}\ge q^{2} & \quad \Longrightarrow \quad &w^{*}(p,q^{1})\ge w^{*}(p,q^{2}). \end{array} $$
Price of the illiquid asset
We will now explain how the clearing vector p and the price q of the illiquid asset can endogenously be derived. The presence of the illiquid asset enables us to incorporate the effect of fire sales into the model. As described in the introduction, the basic economic idea is that, if a bank is unable to pay its outstanding debt in the considered period using its shares of the liquid asset and interbank payments, it can sell a proportion of its illiquid asset holdings at the current market price. This triggers an increase in the supply of the asset that can decrease its market price during times of crisis. Consequently, other banks holding the same asset are also affected by such a price decline. In particular, if they need to sell an amount of the illiquid asset themselves, the proceeds from this transaction are diminished. At the same time, the price is further pushed down.
Payment vector
In the final step, we define a pricepayment equilibrium that endogenously derives the price of the illiquid asset as well as a corresponding clearing vector. We integrate one more effect that influences the clearing process, namely bankruptcy costs. In reality, a fraction of the recovery value of the assets will be lost to the obligors in case of default due to, e.g., legal and administrative expenses. Observe that bankruptcy costs and fire sales impact systemic risk differently. Firstly, bankruptcy costs are only incurred in the case of a default, while fire sales may also occur if there are no defaults in the system. Secondly, a fire sale may affect banks that are not connected to the triggering bank via direct credit contracts. Fire sales are a global channel of contagion, while bankruptcy costs are an amplifier of credit risk.
Following Rogers and Veraart (2013), we introduce two new parameters 0≤α≤1 and 0≤β≤1, such that 1−α and 1−β determine the frictional costs due to bankruptcy: A defaulting bank will only realize a fraction α of its external asset value, i.e., the value of the liquid and illiquid asset, and a fraction β of its interbank asset value, i.e., the value of interbank claims and crossholdings. We further postulate that the clearing process satisfies the criteria of proportionality, limited liability, and absolute priority of debt, as outlined by Eisenberg and Noe (2001). Finally, we define a pricepayment equilibrium as follows.
Definition 2.3
In the combined equilibrium, the banks’ clearing payments p ^{∗} are given as the fixed point of the function χ, and a clearing price of the illiquid asset q ^{∗} is found as a fixed point of the inverse demand function f. Hence, bank i pays its total liabilities \(\bar {p}_{i}\), if its total asset value consisting of its external asset value r _{ i }+s _{ i } q ^{∗} and interbank asset value η _{ i }(p ^{∗},q ^{∗}) exceeds its liabilities. If this is not the case, bank i is in default and receives (and due to absolute priority of debt also pays out) only the given fractions α and β of its external and interbank asset value, respectively. The interbank asset value of bank i, η _{ i }(p ^{∗},q ^{∗}), depends on the proportion of crossholdings that are exchanged against liquid assets, ν _{ i }(p ^{∗},q ^{∗}), and this proportion is multiplied by λ _{ i }∈[0,1] as defined in μ _{ i }(p ^{∗},q ^{∗}). Finally, the amount of the illiquid asset that is sold in equilibrium is given by θ(p ^{∗},q ^{∗}).
The pricepayment equilibrium provides a solution to an integrated financial system which is characterized by \( (\Pi,\bar {p},r,s,\alpha,\beta,\lambda, f, C,\mathbb {I}); \) here, λ=(λ _{1},…,λ _{ n })∈[0,1]^{ n } and \(\mathbb {I}=(\mathbb {I}_{1},\dots,\mathbb {I}_{n})\in \{0,1\}^{n}\). It admits a joint analysis of a network of liabilities, bankruptcy costs, crossholdings, and fire sales as well as an analysis of models that incorporate only some of these effects. Namely, by choosing α=β=1, s=0, or C as the zero n×n matrix, we can simply exclude the corresponding extensions from our system. This shows that the models of, e.g., Eisenberg and Noe (2001), Rogers and Veraart (2013), Cifuentes et al. (2005), and Elsinger (2009) are special cases of our integrated financial system.
Existence of equilibria and an algorithm for their computation
The current section analyzes the existence and computation of equilibria. All proofs are provided in the section “Proofs”. We consider the ordered vector space \(\langle \mathbb {R}^{n+1},\le \rangle \) equipped with the partial order x≤y:⇔x _{ i }≤y _{ i } ∀i=1,…,n+1 and use the notation x<y:⇔(x≤y and ∃i:x _{ i }≠y _{ i }). We will also use this ordering on linear subspaces. The following lemma states elementary properties of the function Φ, see Definition 2.3.
Lemma 3.1
 (a)
Φ is bounded: For all \((p,q)\in [\boldsymbol {0},\bar {p}]\times [q_{\min },q_{0}]\): Φ(p,q)≥(0,q _{min}) and \(\Phi (p,q)\le (\bar {p},q_{0}),\)
 (b)
Φ is monotone: For (p ^{1},q ^{1})≤(p ^{2},q ^{2}): Φ(p ^{1},q ^{1})≤Φ(p ^{2},q ^{2}).
Lemma 3.1 enables us to apply Tarski’s fixedpoint theorem (Tarski (1955, Theorem 1)) to the function Φ proving the existence of equilibria.
Theorem 3.2
Remark 3.3
While the clearing payment vector is unique within the basic setting of Eisenberg and Noe (2001 ) under certain technical conditions, extensions allowing separately for bankruptcy costs or fire sales may lead to multiple clearing vectors, see Rogers and Veraart ( 2013 , Example 3.3) and Chen et al. ( 2016 , Example 7). This observation applies, in particular, to our integrated financial system as shown by an example in “Appendix 1. Example: pricepayment equilibria”. The example also demonstrates that the set of equilibria is not necessarily connected.
Amini et al. (2016 ) prove uniqueness of pricepayment equilibria in a model with fire sales under the additional condition on the inverse demand function f that x↦x f(x) is strictly increasing. They provide the following rational for their assumption: If there exists a subinterval I:=(x _{0},x _{1}) with I∋x↦x f(x) decreasing, rational banks would never choose to sell a suboptimal amount x∈I of the illiquid asset; they would instead liquidate less, i.e., only the quantity x _{0}.
The argument of Amini et al. (2016 ) relies on implicit assumptions. First, it requires that banks understand both the mechanism of price formation as well as their own price impact. Second, the price of the illiquid asset depends on the total quantity that is sold which would have to be known; it does not only depend on the amount that is sold by an individual bank. However, in contrast to the price, the total quantity sold is hardly observable in reality. Third, Amini et al. (2016 ) interpret the onestage model literally; the latter could also be seen as a simplified static picture of the true dynamic processes of prices sliding down the slide while banks continue to liquidate their holdings over time. For simplicity, we decided to model banks as price takers and do not impose the additional condition of Amini et al. (2016 ).
We now explain how equilibria can be determined, generalizing the fictitious default algorithm ofEisenberg and Noe (2001 ), and the procedures of Rogers and Veraart (2013 ), and Amini et al. (2013). Our method allows the computation of the greatest and least pricepayment equilibrium, see Algorithm 3.4 and Remark 3.6 below.
Algorithm 3.4
The (sloppy) notation w ^{∗}(x,y) refers to the essential net worths vector corresponding to the payment vector p ^{(k+1)}; its components are equal to x _{ i } for the defaulting banks \(i\in \mathcal {D}_{k}\) and equal to \(\bar {p}_{i}\) for the surviving banks \(i\in \mathcal {S}_{k}\).
Set k→k+1 and go to Step 1.
The algorithm works as follows. Starting with the total liabilities vector for the payments and the initial price of the illiquid asset q _{0}, it calculates the set of those banks that will default even if all other banks pay their liabilities in full. This is the set of initially defaulting banks. If there are no initially defaulting banks and, in addition, no bank has to liquidate parts of its illiquid asset holdings, we immediately arrive at the end of the clearing process and terminate. Note that leaving out the extra condition (3) in the initial step may lead to an incorrect result if the contagion cascade is solely triggered by the asset price effect. This is due to the fact that the price of the illiquid asset and the corresponding payments must be adjusted if there are banks forced to sell the illiquid asset. The adjustment is made in Step 2 by solving the fixedpoint Eqs. (4) and (5) simultaneously. Using the adjusted pricepayment pair, in Step 1 of the next round, we calculate the set of defaulting banks that corresponds to it. If this default set equals the default set from the previous round, the algorithm terminates with the current pair of payments and price of the illiquid asset. Otherwise, the procedure continues until the set of defaulting banks does not change anymore.
The following theorem extends Rogers and Veraart (2013, Theorem 3.7) to the case of crossholdings and fire sales.
Theorem 3.5
Algorithm 3.4 produces a sequence of pricepayment pairs (p ^{(k)},q ^{(k)}) that decreases to the greatest pricepayment equilibrium in at most n+1 iterations.
Remark 3.6

In the initial step: Set (p ^{(0)},q ^{(0)})=(0,q _{min}), \(\mathcal {D}_{1}=\mathcal {N}\) and terminate the algorithm if \(\mathcal {D}_{0}=\mathcal {D}_{1}\), i.e., condition (3) can be eliminated.

In Step 2: Compute the minimal solution to the system of equations.
The iterations of the procedure that computes the least pricepayment equilibrium can be viewed^{3} as a process in which financial health spreads throughout a system that is initially in default. The iterations of the algorithm that determines the greatest pricepayment equilibrium describe, in contrast, how defaults spread in a system that initially is completely healthy. As we expect the latter situation to be more likely in real world financial markets, we focus on the greatest equilibrium when conducting our numerical case studies.
Case studies
The integrated financial system is characterized by a 10tuple, \( (\Pi,\bar {p},r,s,\alpha,\beta,\lambda \), \(f, C,\mathbb {I}). \) The relative liabilities matrix Π and the crossholdings matrix C will be modeled as random quantities. In contrast to a deterministic approach, a probabilistic mechanism allows for an identification of the structure of a class of networks on the basis of appropriate statistical quantities. We choose two settings: Erdös–Rényi random networks (Erdös and Rényi (1959) ), and a twotiered (coreperiphery) random graph model adapted to German interbank market data (extracted from Craig and von Peter (2014)). We also analyze an extension to multilayer networks that capture heterogeneous business models.
Erdös–Rényi random networks
Simulation methodology
 1.
Construct an adjacency matrix \(A\in \mathbb {R}^{n\times n}\) by letting A _{ ij }, \(i\not =j\in \mathcal {N}\), be i.i.d. Bernoulli random variables, taking the value 1 with probability d _{ Π }/(n−1) and 0 with probability 1−d _{ Π }/(n−1). Set A _{ ii }=0 for all \(i\in \mathcal {N}\).
 2.
For all banks \(i\in \mathcal {N}\), set \({d_{i}^{out}={\sum \nolimits }_{j\in \mathcal {N}}^{} A_{ij}}\), and let \(\Pi _{ij}={c_{\Pi }}/{d_{i}^{out}}\) if A _{ ij }=1, otherwise Π _{ ij }=0, \(j\in \mathcal {N}\).
The parameter d _{ Π } is the average outdegree of the corresponding directed network, the parameter c _{ Π } is the row sum of the matrix A.
The crossholdings matrix C describes an interbank network as well and can be modeled according to an analogous mechanism. We denote by c∈[0,1) the corresponding level of integration, and by d∈[0,n−1] the level of diversification of the crossholdings matrix. The parameter c refers to the fraction of net worth that banks sell as crossholdings to other banks, d describes the expected number of shareholders within the interbank market. We assume that banks can liquidate crossholdings at their market value, possibly reduced to a fraction κ∈[0,1). We thus set λ _{ i }=(1−c)κ, \(i\in \mathcal {N}\). Buyers could either be market participants or a central bank that tries to stabilize the financial system.
 1.
Compute the random vector of the minimal value of assets that prevent the banks from defaulting (not considering crossholdings): \(h:=(\bar {p}  \Pi ^{T}\bar {p})\vee \boldsymbol {0}\).
 2.
Given a capital buffer δ>0, set the overall external assets to be e:=(1+δ)h.
 3.
Given a proportion ρ∈[0,1] of the illiquid asset, let r=(1−ρ)e and s=ρ e.
For simplicity, we use the parametric exponential inverse demand function f(x)= exp(−γ x); alternative inverse demand functions can also be implemented within our framework. We assume that all banks follow the same rule regarding the order of liquidation. This means that either all banks exchange their total crossholdings against cash before selling the illiquid asset (i.e., \(\mathbb {I}=\boldsymbol {1}\)), or that all banks first liquidate their total holdings of the illiquid asset before using crossholdings (i.e., \(\mathbb {I}=\boldsymbol {0}\)).
Extension parameters
Parameter  Description  Range of variation 

α  Realized fraction of external asset value in case of bankruptcy  [0.5,1] 
β  Realized fraction of interbank asset value in case of bankruptcy  [0,1] 
ρ  Proportion of the illiquid asset  [0,0.05] 
γ  Exponent of the inverse demand function  [0,1] 
c  Integration of the crossholdings matrix  [0,0.9] 
d  Diversification of the crossholdings matrix  [1,90] 
κ  Realized fraction of crossholdings’ market value  [0,1] 
\(\mathbb {I}\)  Order of liquidation  {0,1} 
Methodological Remark 4.1
The numerical case studies are conducted as follows: Π and C are randomly sampled; the derived random quantities r and s are computed from the samples. One bank \(i\in \mathcal {N}\) is uniformly sampled at random; its external asset holdings r _{ i } and s _{ i } are set to zero. This corresponds to a local shock to a single bank. For the resulting scenario, the greatest pricepayment equilibrium and the corresponding number of defaulting firms is calculated. The simulation is repeated a large number of times, and sample averages and standard deviations are computed. The exact number of the simulations that were conducted is mentioned below for each case study. We use the following notation: Parameters corresponding to a basic Eisenberg–Noe model are denoted by EN, B signifies the incorporation of bankruptcy costs, C crossholdings, and F fire sales.
Separate effects
First, we focus on the separate impact of individual model ingredients, leaving all other parameters as in the EN model. The section “Joint effects” investigates joint effects.
Bankruptcy costs and fire sales
Increasing bankruptcy costs, i.e., decreasing α and β, increases the number of defaults quite quickly, as shown by Fig. 1 a. Part (b) of Fig. 1 shows a similar phenomenon when both fire sales parameters are increased. Additionally, a clear threshold effect can be observed that separates an area of many defaults from an area of few defaults. For low parameter values of ρ and γ, the financial system exhibits the EN number of defaults (around 11). Increasing ρ and γ beyond a certain threshold boundary causes defaults of all banks in the system. The threshold curve can approximately be described by the following powerlaw function: ρ= exp(−4.3183)·γ ^{−0.4528}.
 (i)
Bankruptcy costs increase the instability of the financial system significantly. These costs are mainly incurred due to the impaired operations of financial institutions in default. Administrative and legal expenses increase significantly for such institutions. Policies that improve the efficiency of managing defaults and restructuring institutions would mitigate the consequences of financial crises. This could, for example, be achieved by limiting the complexity of financial products and the operations of financial institutions. Another promising instrument are last wills of financial institutions, approved by the regulator during their lifetime, that simplify the processes in default.
 (ii)
Illiquidity, i.e., the joint consequences of limited funding and price impact, decreases market stability. When markets dry up, the value of financial institutions that need shortterm funding might be significantly impaired. Quantitative easing would, in this case, be an appropriate instrument to stabilize the banking sector. This should include the purchase of temporarily illiquid assets.
The simulations also exhibit a sharp boundary between the regimes with few and many defaults. This indicates that regulatory policies should aim for substantial safety margins in order to create a resilient system.
CrossHoldings
Crossholdings significantly impact the number of defaults. This has implications for the policies of regulators that we will discuss below.
Crossownership in the banking sector may stabilize the financial system. However, this finding relies on the existence of a market with substantial demand for crossholdings. Our results show that regulators and central banks might use crossholdings in order to stabilize financial markets during financial crises. Regulatory policies that provide incentives to crossownership in the financial market as well as a credible promise that these shares will be purchased, e.g., by the central bank would decrease systemic risk in our model.
The benefits of crossholdings rely on the fact that they can be exchanged against cash. This effect becomes, of course, less significant if the realized fraction κ is smaller. At the same time, there might be an inverse effect on the financial institution whose shares are sold. If a large sale of its shares decreases its equity price, its solvency is not directly affected: Solvency is a function of the book value of equity—the latter being computed from a market consistent balance sheet. The book value of equity may deviate from its market value. But a decreased equity price may increase the cost of capital and thereby negatively affect the bank’s solvency indirectly. The closely related topic of the impact of equity valuation on credit supply to the real economy is, e.g., discussed inBoucinha et al. (2017).
Random fluctuations
A comparison of Figs. 4 a and 1 a leads to the following observation: For values of α and β that lead to outcomes of either a very low number of defaults (i.e., ENlevel) or the total breakdown of the system, the corresponding standard deviation is relatively low. However, in the transition area between these regimes, we observe a very high standard deviation. We observed a similar behavior when analyzing fire sales: while regimes with a very low or a very high average number of defaults exhibited low standard deviations, regimes with an intermediate average number of defaults were associated with significant fluctuations around the averages and thus with significant risk. A refined analysis^{6} shows that the empirical distribution of defaults is mainly concentrated on the extreme scenarios of few or many defaults. Medium levels of bankruptcy costs or fire sales may, on average, seem acceptable, but are in fact associated with an unstable financial system. This shows that gradual improvements of the efficiency of the operations of distressed banks or light quantitative easing do not lead to resilience. Regulatory rules and interventions must be sufficiently forceful in order to achieve the desired effect of creating a stable financial system.
In the case of crossholdings, an analysis of the average number of defaults is not sufficient. Comparing Fig. 4 b with Fig. 2 a shows that increasing integration has an overall beneficial effect on the average number of defaults, but increases the standard deviation within the considered parameter range. Features of the network thus have a quite complex impact on how financial stability and instability spread within the system. Moreover, Fig. 4 b demonstrates that increasing diversification increases the standard deviation, while diversification does almost not affect the average number of defaults for d>10. A diversification of d=10 (i.e., \(\sqrt {n}\)) seems to be an optimal level. It would be interesting to investigate if such a statement holds more generally. Regulatory incentives for crossownership in the banking sector must therefore be very carefully designed. This requires substantial future research on the exact magnitude of the impact of crossholdings in real financial networks.
Joint effects
 (i)First, consider an integrated financial system with α=0.9, β=0.9, c=0.5, d=10, and κ=0.8. The resulting average number of defaults as a function of the fire sales parameters is displayed in Fig. 5.^{7} A comparison of Figs. 1 b and 5 a reveals that the general structure of the influence of price impact is preserved if bankruptcy costs and crossholdings are added. However, the number of defaults increases and the sharp transition between the area of few defaults and the area of the breakdown of the system disappears. Similar qualitative results were all also confirmed in additional case studies based on both the ENBF and ENCF models. The results indicate that quantitative easing continues to be a suitable instrument in order to contain the number of defaults in this case.
 (ii)Second, Figs. 5 b and 6 deal with the impact of changing the order of liquidation on the number of defaults and the price of the illiquid asset as a function of the fire sales parameters ρ and γ. These figures show that, regardless of whether crossholdings or illiquid asset shares are liquidated first, the observed behavior is very similar.
 (iii)
Third, we analyze the effects of integration of crossholdings in the joint model. This leads to rather complex, but very interesting features. We investigate a network model with d=10, κ=0.8, β=0.9, γ=0.2, ρ=0.02, and varying α.
Within the considered parameter range, increasing integration decreases the average number of defaults, cf. Fig. 7 a. The nonmonotonicity observed in Fig. 2 a disappears. As expected, the higher the realized fraction α, the lower the number of defaults. However, for α≤0.8 crossholdings cannot prevent the total breakdown of the system. Observe that for the chosen value of β=0.9, the value α=0.8 corresponds roughly to the critical boundary between the regimes of a very high (α≤0.8) and a very low (α≥0.9) number of defaults in Fig. 1 a.Figure 7 b displays standard deviations. These are comparatively small for extreme regimes of the default count statistics, i.e., α∈{0.8,0.85,0.95}. For regimes with an intermediate average number of defaults, i.e., α∈{0.9,0.925}, the standard deviations are large. For fixed α≤0.9, the standard deviation increases as a function of integration c; for α≥0.925, the standard deviation is decreasing.
This different behavior can easily be understood when analyzing the empirical cumulative distribution functions (CDFs) of the number of defaults. It turns out that the distributions of the numbers of defaults are very close to Bernoulli distributions. In this case, the standard deviation is maximal for a success probability of 0.5 and monotonously decreasing if the success probability is either increased or decreased. As illustrated in Fig. 8, for α=0.9, the distribution of defaults for c=0.3 roughly corresponds to a success probability of 0.2 – 0.3; for c=0.7, to a success probability of 0.3 – 0.4. Thus, the standard deviation increases in c. In contrast, for α=0.925, the distribution of defaults for c=0.3 roughly corresponds to a success probability of 0.5 – 0.6; for c=0.7, to a success probability of 0.7 – 0.8. Thus, the standard deviation decreases in c.The approximate Bernoulli distributions are supported by the EN number of defaults and a total breakdown of the system. This shows that the system essentially randomizes between extreme states. While the probability of the negative outcome can be controlled by crossholdings for the chosen level of bankruptcy costs and fire sales, the number of defaults in this adverse scenario is not mitigated.
From a policy point of view, our numerical case studies again indicate the fundamental role played by bankruptcy costs, emphasizing the importance of efficient procedures for managing defaults and restructuring institutions. Regulators should thus implement policies that lower the costs of bankruptcy. Moreover, if bankruptcy costs are not too large, a higher integration c of crossholdings decreases both the average number of defaults and their variance. More integrated systems are thus more resilient. These results, of course, rely on the existence of a sufficiently deep and liquid market for crossholdings. If this market dries up during a crisis, central banks might buy crossholdings or provide guarantees for their purchase in order to stabilize the financial system. In our model, such a policy, however, does not seem to be efficient anymore if bankruptcy costs are too high.
Capital adequacy ratios
Capital requirements are a powerful instrument for the regulation of financial institutions. We investigate these in the section “Capital requirements” in more detail. For surveys of the literature on monetary risk measures, we refer toFöllmer and Schied (2011 ) and Föllmer and Weber (2015). A less sophisticated approach than monetary risk measures are capital adequacy ratios (CAR) based on riskweighted assets. In the current section, we show in the context of our model that these are not always welladopted for regulatory purposes.
We consider a stylized definition of CAR and ignore crossholdings. For each bank i, capital C _{ i } is computed as the sum of external asset holdings e _{ i } and promised interbank holdings \(\sum _{j\in \mathcal {N}} \Pi _{ji}\bar p_{j} \) less its liabilities \(\bar p_{i}\). We assume that riskweights^{8} are 100% and calculate riskweighted assets R A _{ i } of bank i as the sum of its illiquid assets s _{ i } and its interbank holdings \(\sum _{j\in \mathcal {N}} \Pi _{ji}\bar p_{j} \). The capital adequacy ratio of bank i is defined by C A R _{ i }=C _{ i }/R A _{ i }.
A sufficiently high CAR can indeed prevent default cascades. However, in our case study, CAR appears to be an inefficient regulatory tool that does not properly measure the driving forces behind extreme default scenarios. Apparently, the barrier between many and few defaults and the level sets of CAR are not collinear. While for small values of ρ a small CAR is sufficient, for large values of ρ a large CAR is necessary in order to stabilize the system. The results indicate that regulators should base capital regulation on more sophisticated statistics than riskweighted assets.
Coreperiphery random networks
So far, we considered homogeneous random network topologies. Recent research on financial networks, however, suggests that coreperiphery network models capture the structure of the interbank market (see, e.g.,Craig and von Peter (2014 ) for the German interbank market, and van Lelyfeld and in’t Veld (2014) for the Netherlands): These consist of a few highly connected nodes (making up the core) and a larger number of only sparsely connected nodes (referred to as the periphery). In this section, we simulate the relative liabilities matrix Π as a coreperiphery random network and observe how this affects the impact of bankruptcy costs, crossholdings, and fire sales on the number of defaults.
Simulation methodology
 1.
Adjacency matrix: We first simulate an adjacency matrix \(A\in {\mathbb {R}}^{n \times n}\), using exogenously specified connection probabilities p ^{ C C },p ^{ C P },p ^{ P C }, and p ^{ P P } for each block.
 2.
Weights: Second, we fix the value of total liabilities ℓ of all banks. A fraction c _{ Π } is allocated to the total interbank liabilities; the remaining share models external liabilities that are uniformly distributed among all banks, i.e., \(l_{i} = \frac {(1  c_{\Pi }) \cdot {\ell }}{n}\), \(i\in \mathcal {N}\). Total interbank liabilities c _{ Π }·ℓ are allocated to the four matrix blocks in fractions of x ^{ C C },x ^{ P C },x ^{ C P }, and x ^{ P P } with x ^{ C C }+x ^{ P C }+x ^{ C P }+x ^{ P P }=1. The resulting interbank liabilities per block are uniformly distributed among all existing links within the block. This is, denoting by \(l^{CP}={\sum \nolimits }_{i\in \mathcal {C}}{\sum \nolimits }_{j\in \mathcal {P}} A_{ij}\) the random number of total links in the CPblock, the corresponding entries of the nominal liabilities matrix L are \(L_{ij}=\frac {x^{CP} \cdot c_{\Pi } \cdot {\ell }}{l^{CP}}, i\in \mathcal {C}\), \(j\in \mathcal {P},\) if A _{ ij }>0, and L _{ ij }=0, otherwise. The other blocks are computed analogously. Finally, with \(\bar {p}_{i}={\sum \nolimits }_{j\in \mathcal {N}} L_{ij}+l_{i}\), the entries of the relative liabilities matrix Π are given by \(\Pi _{ij}=\frac {L_{ij}}{\bar {p}_{i}}\).
As in the other case studies, we set n=100,c _{ Π }=0.15, and ℓ=100. In addition, followingCraig and von Peter (2014) who analyzed data from the German interbank market, we choose coreperiphery parameters
\(p^{CC}=0.66, p^{CP}=0.15, p^{PC}=0.07, p^{PP}=0.001, \atop x^{CC}=0.35, x^{CP}=0.16, x^{PC}=0.47, x^{PP}=0.02.\)
We fix the number of core banks as n ^{ C }=10.^{9} The simulation methodology is analogous to the section “Simulation methodology”. Note, however, that simulation results will depend on whether a core or periphery bank is hit by an initial shock.
Results
We focus on three case studies: the separate impact of a) bankruptcy costs and b) fire sales as well as c) the joint impact of all ingredients while varying the parameters that govern the fire sales.
Comparing Figs. 10 to 1, we observe that bankruptcy costs and fire sales have a similar impact as in the Erdös–Rényi network: as expected, an increase in the number of defaults for increasing bankruptcy costs and fire sales; a clear threshold boundary separating an area of many defaults from an area of few defaults. The exact numbers are, of course, different. For example, in Fig. 10 (a) the impact of parameter β on defaults is stronger than before.
Again, our observations resemble the findings within Erdös–Rényi random networks.
The considered coreperiphery structure was calibrated to German interbank market data and presents a good description of a real world interbank market. Overall, the above comparisons indicate that the results we obtained are qualitatively similar to those within simpler Erdös–Rényi random networks. This shows that the simplified setting is already representative and relevant for the analysis of systemic risk. The policy implications discussed in the context of Erdös–Rényi random networks remain valid for the considered coreperiphery structure.
Multilayer networks
Our model can be extended to more than two layers and is capable of modeling heterogenous agents. We will illustrate this with three types of agents: banks, depositors, and borrowers. Depositors hold deposits at banks. Borrowers receive credit from the banks that act as intermediaries. For simplicity, we neglect bankruptcy costs and crossholdings and assume that the banking system consists of 20 banks that form an Erdös–Rényi random network with d _{ Π }=2 and δ=0.2. We add depositors and borrowers to the system. The total liabilities of all banks (to other banks and to the depositors) within the considered timeperiod are normalized to the total number of banks, i.e., 20. Liabilities within the banking system are set to 15% of total liabilities, i.e., c _{ Π }=0.15.
The remaining 85% are liabilities to depositors. The number of depositors is 90, and each depositor is linked to exactly two banks that are uniformly sampled at random. Each link is associated with the same liability. We assume that all liabilities are immediately due. This scenario can be interpreted as a bank run.
There are 90 borrowers outside the banking system. Each borrower is linked to two banks that are sampled uniformly at random. Borrowers’ shortterm liabilities over the considered timeperiod are 1/100 per link. Borrowers hold external assets which amount to 1+δ of their total shortterm liabilities; 40% of these are allocated to the illiquid asset.
Capital requirements
Another important regulatory tool are capital requirements for banks. In financial networks, the financial situation of a bank, of course, does not only depend on its own capital endowment, but also on the capital of other financial institutions with which it interacts directly or indirectly. Within a coreperiphery network, we discuss the role of capital on the basis of a framework suggested in Feinstein et al. (2017). We refer to this paper for further details on systemic risk measures.
In the current case study, all simulations are conducted according to the same methodology as described in the section “Simulation methodology”. However, we introduce two further parameters k ^{ C } and k ^{ P } which signify additional capital on top of the originally computed external assets e that is held by the core and periphery banks, respectively. As before, the updated amount of the external assets is then divided into liquid and illiquid assets according to the parameter ρ, and the simulations are run.
Figure (b) includes bankruptcy costs for varying β with ρ=0.5 fixed. While the capital requirements for the periphery banks are barely affected by decreasing β, the requirements for the core banks increase strongly. Since core banks are more connected than periphery banks, the impact of defaults and resulting interbank bankruptcy costs on core banks is more significant than on periphery banks. In the context of the chosen model, a regulator would be well advised if she particularly focused on strengthening the capital endowments of core banks. These can serve as a buffer that reduces systemic risk even if considerable bankruptcy costs are present.
Conclusion
 (i)
Systemic risk was studied by shocking the system and computing the average number of defaults, its variance, and distribution. Outcomes are centered on extreme scenarios. The risk of extreme adverse events is present, even if averages indicate a relatively safe system. Regulatory policies should provide substantial safety margins in order to guarantee stability.
 (ii)
Fire sales strongly increase systemic risk, while crossholdings may improve the resilience of the banking sector. Central banks might mitigate the risk of default cascades by purchasing illiquid assets and crossholdings. Quantitative easing strengthens the system.
 (iii)
Bankruptcy costs are a main driver of systemic risk. Regulators should improve the efficiency of bankruptcy procedures and limit the associated deadweight losses. Policies might include reducing the complexity of financial products as well as operational procedures and requiring last wills of financial institutions.
 (iv)
Capital requirements are a powerful instrument, but capital adequacy ratios based on riskweighted assets are an extremely rough measure of systemic risk. Instead, modern systemic risk measures that use capital efficiently could be implemented.
 (v)
We analyzed different interbank network structures and heterogeneous business models. Our qualitative results were robust. Quantitative predictions, however, require a precise specification of all driving mechanisms.
The suggested model can be used as a framework for testing the impact of regulatory policies and their robustness. It can also provide insights into the significance of the financial market architecture for systemic risk, e.g., the pros and cons of CCPs. From a statistical point of view, our comparative statics show that default cascades can be triggered by a combination of various mechanisms. In particular, bankruptcy costs and fire sales exhibit similar consequences. Their statistical estimation is a challenging issue for future research that requires further data on historical bankruptcy procedures and price impact during crises.
Proofs
Proof of Lemma 2.2.
 (a)
This follows directly from Banach’s fixedpoint theorem applied to the function Ψ.
 (b)
We prove that the essential net worths vector is increasing in the payments p. The corresponding claim for the illiquid asset’s price q can be proven analogously. To simplify the notation, we suppress the dependence on q and write w ^{∗}(p) instead of w ^{∗}(p,q) in the following. Now, for each given payment vector p, we define the following recursive sequence: Starting with \(w^{(0)}(p):=r+sq+\Pi ^{T}p\bar {p}\), we set w ^{(n)}(p)=Ψ(w ^{(n−1)}(p)), for n=1,2,…. Due to part (a), this sequence converges with \(\lim \limits _{n\rightarrow \infty }w^{(n)}(p)=w^{*}(p)\).
For two given payment vectors p ^{1}≥p ^{2}, it holds that w ^{(n)}(p ^{1})≥w ^{(n)}(p ^{2}) for all n. We prove this statement by induction. For n=0,since p ^{1}≥p ^{2}. For the induction step, n→n+1, it holds that$$w^{(0)}(p^{1})=r+sq+\Pi^{T}p^{1}\bar{p}\ge r+sq+\Pi^{T}p^{2}\bar{p}=w^{(0)}(p^{2}), $$exploiting the induction hypothesis and the fact that μ _{ i }(p,w) is by definition increasing in both components. Hence, w ^{(n)}(p ^{1})≥w ^{(n)}(p ^{2}) for all n, and this yields$$ { \begin{aligned} w^{(n+1)}(p^{1})&=r+sq+\Pi^{T}p^{1}+\text{diag}(\mu(p^{1},w^{(n)}(p^{1}))C^{T}(w^{(n)}(p^{1})\vee\boldsymbol{0})\bar{p}\\ &\ge r+sq+\Pi^{T}p^{2}+\text{diag}(\mu(p^{2},w^{(n)}(p^{2}))C^{T}(w^{(n)}(p^{2})\vee\boldsymbol{0})\bar{p}\\&= w^{(n+1)}(p^{2}), \end{aligned} } $$$$w^{*}(p^{1})=\lim\limits_{n\rightarrow\infty}w^{(n)}(p^{1})\ge\lim\limits_{n\rightarrow\infty}w^{(n)}(p^{2})=w^{*}(p^{2}). $$
Proof of Lemma 3.1.
 (a)
It holds by definition that Φ _{ i }(p,q)=χ _{ i }(p,q)≥0 and \(\Phi _{i}(p,q)\le \bar {p}_{i}\) for all i=1,…,n, since the banks will pay at most their total liabilities. For i=n+1, the boundedness follows from the definition of the inverse demand function.
 (b)Let (p ^{1},q ^{1})≤(p ^{2},q ^{2}). We have that θ(p ^{1},q ^{1})≥θ(p ^{2},q ^{2}), because the essential net worths vector is monotone in the pricepayment pairs due to Lemma 2.2 (b). Hence,since f is monotonically decreasing. For i=1,…,n, monotonicity follows from a casebycase analysis extending the arguments ofRogers and Veraart (2013 ) and Amini et al. (2013). First, assume that bank i is in default under (p ^{2},q ^{2}). This implies that it is in default under (p ^{1},q ^{1}) and that$$ \Phi_{n+1}(p^{1},q^{1})=f(\theta(p^{1},q^{1}))\le f(\theta(p^{2},q^{2}))=\Phi_{n+1}(p^{2},q^{2}), $$due to Lemma 2.2 (b). Next, assume that bank i does not default under (p ^{2},q ^{2}). Then, bank i can either survive or default under (p ^{1},q ^{1}). In the first case, monotonicity of Φ _{ i } directly follows from \(\Phi _{i}(p^{2},q^{2})=\bar {p}_{i}=\Phi _{i}(p^{1},q^{1}).\) In the second case, if bank i defaults under (p ^{1},q ^{1}) but not under (p ^{2},q ^{2}), it follows that$$ \Phi_{i}(p^{2},q^{2})=\alpha[r_{i}+s_{i} q^{2}]+\beta\eta_{i}(p^{2},q^{2})\ge\alpha[r_{i}+s_{i} q^{1}]+\beta\eta_{i}(p^{1},q^{1})=\Phi_{i}(p^{1},q^{1}), $$Here, the first inequality holds true, since bank i defaults under (p ^{1},q ^{1}). The second inequality follows from α,β≤1.$$ \Phi_{i}(p^{2},q^{2})=\bar{p}_{i}>r_{i}+s_{i} q^{1}+\eta_{i}(p^{1},q^{1})\ge\alpha[r_{i}+s_{i} q^{1}]+\beta\eta_{i}(p^{1},q^{1})=\Phi_{i}(p^{1},q^{1}). $$
Proof of Theorem 3.2
Let \(\mathcal {F}:=\{(p,q)\in [\boldsymbol {0},\bar {p}]\times [q_{\min },q_{0}] \;\; \Phi (p,q)=(p,q) \}\) denote the set of fixed points of Φ. Lemma 3.1 establishes that Φ is an increasing function on the complete lattice \(\langle [\boldsymbol {0},\bar {p}]\times [q_{\min },q_{0}],\le \rangle \) with the componentwise ≤relation. Hence, Tarski’s fixedpoint theorem Tarski (1955, Theorem 1) states that \(\mathcal {F}\) is not empty and, moreover, that \(\langle \mathcal {F},\le \rangle \) constitutes a complete lattice itself. In particular, this yields the existence of a unique greatest and least element of \(\mathcal {F}\). Since the elements of \(\mathcal {F}\) constitute by definition the pricepayment equilibria, this completes the proof.
Proof of Theorem 3.5
 (i)For the first step, we claim that$$ (p^{(k+1)},q^{(k+1)})\le \left(p^{(k)},q^{(k)}\right)\quad \forall k=0,1,\dots,n1. $$(8)We prove this statement by induction.
 (B.S.)For the base step, k=0, we need to prove that \(p^{(1)}\le p^{(0)}=\bar {p}\) and that q ^{(1)}≤q ^{(0)}=q _{0}. Regarding the payments, this clearly holds for all \(i\in \mathcal {S}_{0}\), since then \(p^{(1)}_{i}=\bar {p}_{i}=p^{(0)}_{i}\). Thus, it remains to examine the payments of the defaulting banks in \(\mathcal {D}_{0}\) and the corresponding price of the illiquid asset, which are jointly given by the maximal solution to the Eqs. (4) and (5). In order to calculate this maximal solution, we propose the following recursive procedure: Start with (x ^{(0)},y ^{(0)}), where \({x_{i}^{(0)}=p_{i}^{(0)}=\bar {p}_{i}}\) for \(i\in \mathcal {D}_{0}\) and y ^{(0)}=q ^{(0)}=q _{0}, and define the sequence (x ^{(ν)},y ^{(ν)}) recursively byfor \(i\in \mathcal {D}_{0}\) and$${\begin{aligned} x_{i}^{(\nu+1)}=\alpha\left[r_{i}+s_{i} y^{(\nu)} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{0}}^{}\Pi_{ji}x^{(\nu)}_{j}+\sum\limits_{j\in\mathcal{S}_{0}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}(x^{(\nu)},y^{(\nu)}),0\right)\right] \right] \end{aligned}} $$with \(\zeta _{i}^{(0)}(x,y)\) for all \(i\in \mathcal {S}_{0}\) defined as in Eq. (6), substituting k with 0. Next, we have to show that$$y^{(\nu+1)}=f\left(\sum\limits_{i\in\mathcal{D}_{0}}^{} s_{i}+\sum\limits_{i\in\mathcal{S}_{0}}^{}\min\left(\frac{{\zeta}_{i}^{(0)}(x^{(\nu)},y^{(\nu)})}{y^{(\nu)}},s_{i}\right)\right), $$$$ (x^{(\nu+1)},y^{(\nu+1)})\le(x^{(\nu)},y^{(\nu)}) \quad\forall \nu=0,1,\dots, $$(9)which we will prove by induction. First, for the base case, ν=0, we observe that for \(i\in \mathcal {D}_{0}\):Here, the first inequality is satisfied because 0≤α,β≤1. The second step follows from \(p_{j}^{(0)}=\bar {p}_{j}\) on \(\mathcal {S}_{0}\) and \(w^{*}_{j}(p^{(0)},q^{(0)})<0\) for \(j\in \mathcal {D}_{0}\). The last step holds according to the definition of \(\mathcal {D}_{0}\) and due to the fact that 0≤λ≤μ _{ i }(p ^{(0)})≤1.$${\begin{aligned} x_{i}^{(1)}&=\alpha\left[r_{i}+s_{i} y^{(0)} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{0}}^{}\Pi_{ji}x^{(0)}_{j}+\sum\limits_{j\in\mathcal{S}_{0}}^{}[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max(w^{*}_{j}(x^{(0)},y^{(0)}),0)] \right]\\ &\le r_{i}+s_{i} q^{(0)} +\sum\limits_{j\in\mathcal{D}_{0}}^{}\Pi_{ji}p^{(0)}_{j}+\sum\limits_{j\in\mathcal{S}_{0}}^{}\Pi_{ji}\bar{p}_{j}+\lambda \sum\limits_{j\in\mathcal{S}_{0}}^{}C_{ji}\max(w^{*}_{j}(p^{(0)},q^{(0)}),0)\\ &= r_{i}+s_{i} q^{(0)} +\sum\limits_{j=1}^{n}\Pi_{ji}p^{(0)}_{j}+\lambda\sum\limits_{j=1}^{n}C_{ji}\max(w^{*}_{j}(p^{(0)},q^{(0)}),0)<\bar{p}_{i}=p_{i}^{(0)}=x_{i}^{(0)}. \end{aligned}} $$Moreover,$$\sum\limits_{i\in\mathcal{D}_{0}}^{} s_{i}+\sum\limits_{i\in\mathcal{S}_{0}}^{}\min\left(\frac{{\zeta}_{i}^{(0)}(x^{(0)},y^{(0)})}{y^{(0)}},s_{i}\right)\ge 0, $$by definition of \({\zeta }_{i}^{(0)}(x^{(0)},y^{(0)}).\) Thus, since f is monotonically decreasing and f(0)=q _{0}, it follows that y ^{(1)}≤q _{0}=y ^{(0)}. Now, suppose that inequality (9) is satisfied up to some ν. Then, one obtains:by the induction hypothesis and Lemma 2.2 (b). This yields y ^{(ν+1)}≤y ^{(ν)}, again exploiting the induction hypothesis together with the fact that f is monotonically decreasing.$${\begin{aligned} &{\zeta}_{i}^{(0)}\left(x^{(\nu)},y^{(\nu)}\right)\\&=\max\left(\bar{p}_{i}r_{i}\sum\limits_{j\in \mathcal{D}_{0}}\Pi_{ji}x^{(\nu)}_{j}\sum\limits_{j\in\mathcal{S}_{0}} \left[\Pi_{ji}\bar{p}_{j}+\mathbb{I}_{i}\lambda C_{ji}\max\left(w^{*}_{j}(x^{(\nu)},y^{(\nu)}\right),0)\right],0\right)\\ &\ge{\zeta}_{i}^{(0)}(x^{(\nu1)},y^{(\nu1)}), \end{aligned}} $$
Analogously, it follows from the recursive definition that \(x_{i}^{(\nu +1)}\le x_{i}^{(\nu)}\) for all \(i\in \mathcal {D}_{0}\). Thus, the sequence continues to decrease for all ν. Hence, the limit \((x,y):=\lim \limits _{\nu \rightarrow \infty }(x^{(\nu)},y^{(\nu)})\) exists and solves Eqs. (4) and (5). Moreover, (x,y) is the maximal solution to these equations by construction. This completes the base step of the induction argument for the proof of (8).
 (I.S.)For the induction step, k→k+1, we first observe that by the induction hypothesis we have that \({\mathcal {D}_{k}\supseteq \mathcal {D}_{k1}}\) or, equivalently, \(\mathcal {S}_{k}\subseteq \mathcal {S}_{k1}\). Hence, for all \(i\in \mathcal {S}_{k}\): \({p_{i}^{(k+1)}=\bar {p}_{i}=p_{i}^{(k)}}\). Thus, we have to investigate the payments of the defaulting banks and the corresponding price of the illiquid asset, defined by the maximal solution to Eqs. (4) and (5). Analogous to the base step, we propose the following recursive principle to calculate this maximal solution. Start with (x ^{(0)},y ^{(0)}), where \(x_{i}^{(0)}=p_{i}^{(k)}\) for \(i\in \mathcal {D}_{k}\) and y ^{(0)}=q ^{(k)}, and define (x ^{(ν)},y ^{(ν)}) by the obvious modification of the above recursive principle:$$ {\begin{aligned} x_{i}^{(\nu+1)}=\alpha\left[r_{i}+s_{i} y^{(\nu)} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{k}}^{}\Pi_{ji}x^{(\nu)}_{j}+\sum\limits_{j\in\mathcal{S}_{k}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}\left(x^{(\nu)},y^{(\nu)}\right),0\right)\right] \right] \end{aligned}} $$(10)for \(i\in \mathcal {D}_{k}\) and$$ y^{(\nu+1)}=f\left(\sum\limits_{i\in\mathcal{D}_{k}}^{} s_{i}+\sum\limits_{i\in\mathcal{S}_{k}}^{}\min\left(\frac{{\zeta}_{i}^{(k)}(x^{(\nu)},y^{(\nu)})}{y^{(\nu)}},s_{i}\right)\right), $$(11)with \(\zeta _{i}^{(k)}(x,y)\) for all \(i\in \mathcal {S}_{k}\) defined as in Eq. (6). Again, we want to prove that$$ (x^{(\nu+1)},y^{(\nu+1)})\le(x^{(\nu)},y^{(\nu)}), \quad \nu=0,1,\dots. $$(12)First, note that for ν=0,observing \(\mathcal {D}_{k}=(\mathcal {S}_{k1}\backslash \mathcal {S}_{k})\cup \mathcal {D}_{k1}\) and \(p_{j}^{(k)}=\bar {p}_{j}\) for all \(j\in \mathcal {S}_{k1}\). Second,$$\begin{aligned} \sum\limits_{j\in\mathcal{D}_{k}}^{}\Pi_{ji}x^{(0)}_{j}=\sum\limits_{j\in\mathcal{D}_{k}}^{}\Pi_{ji}p^{(k)}_{j} &=\sum\limits_{j\in\mathcal{S}_{k1}\backslash\mathcal{S}_{k}}^{}\Pi_{ji}p^{(k)}_{j}+\sum\limits_{j\in\mathcal{D}_{k1}}^{}\Pi_{ji}p^{(k)}_{j}\\ &=\sum\limits_{j\in\mathcal{S}_{k1}\backslash\mathcal{S}_{k}}^{}\Pi_{ji}\bar{p}_{j}+\sum\limits_{j\in\mathcal{D}_{k1}}^{}\Pi_{ji}p^{(k)}_{j}, \end{aligned} $$$$\sum\limits_{j\in\mathcal{S}_{k}}^{}C_{ji}\max\left(w^{*}_{j}\left(x^{(0)},y^{(0)}\right),0\right)=\sum\limits_{j\in\mathcal{S}_{k1}}^{}C_{ji} \max\left(w^{*}_{j}\left(p^{(k)},q^{(k)}\right),0\right), $$since \(w^{*}_{j}(p^{(k)},q^{(k)})<0\) for all \(j\in \mathcal {S}_{k1}\backslash \mathcal {S}_{k}\subseteq \mathcal {D}_{k}\). We thus obtain that$$ {\begin{aligned} x_{i}^{(1)} &=\alpha\left[r_{i}+s_{i} y^{(0)} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{k}}^{}\Pi_{ji}x^{(0)}_{j}+\sum\limits_{j\in\mathcal{S}_{k}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}\left(x^{(0)},y^{(0)}\right),0\right)\right] \right]\\ &=\alpha\left[r_{i}+s_{i} q^{(k)} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{k1}}^{}\Pi_{ji}p^{(k)}_{j}+\sum\limits_{j\in\mathcal{S}_{k1}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}\left(p^{(k)},q^{(k)}\right),0\right)\right] \right]. \end{aligned}} $$(13)For \(i\in \mathcal {D}_{k1}\) this shows that \(x_{i}^{(1)}=p_{i}^{(k)}=x_{i}^{(0)}\). For the remaining case \(i\in \mathcal {D}_{k}\backslash \mathcal {D}_{k1}\), we haveHere, the first inequality holds because 0≤α,β≤1. The first stated equality follows from \(p_{j}^{(k)}=\bar {p}_{j}\) for \(j\in \mathcal {S}_{k1}\) and the fact that \(w^{*}_{j}(p^{(k)},q^{(k)})<0\) for all \(j\in \mathcal {D}_{k1}\subseteq \mathcal {D}_{k}\). The last inequality results from \(i\in \mathcal {D}_{k}\backslash \mathcal {D}_{k1}\subseteq \mathcal {S}_{k1}\).$${\begin{aligned} x_{i}^{(1)}&\le r_{i}+s_{i} q^{(k)}+\sum\limits_{j\in\mathcal{D}_{k1}}^{}\Pi_{ji}p^{(k)}_{j}+\sum\limits_{j\in\mathcal{S}_{k1}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}\left(p^{(k)},q^{(k)}\right),0\right)\right]\\ &=r_{i}+s_{i} q^{(k)}+\sum\limits_{j=1}^{n}\Pi_{ji}p^{(k)}_{j}+\lambda\sum\limits_{j=1}^{n}C_{ji}\max\left(w^{*}_{j}\left(p^{(k)},q^{(k)}\right),0\right) <\bar{p}_{i}=p_{i}^{(k)}=x_{i}^{(0)}. \end{aligned}} $$For the price of the illiquid asset, we first observe that for all \(i\in \mathcal {S}_{k}\), \(\zeta _{i}^{(k)}(x^{(0)},y^{(0)})=\zeta _{i}^{(k1)}(p^{(k)},q^{(k)})\) by using the same arguments as for Eq. (13). From this it follows thatbecause f is monotonically decreasing and the last step follows from the definition of q ^{(k)}. This proves (12) for ν=0; the arguments for ν>0 are analogous. This implies that \((p^{(k+1)},q^{(k+1)}) = {\lim }_{\nu } (x^{(\nu)}, y^{(\nu)}) \le (x^{(0)}, y^{(0)}) = (p^{(k)},q^{(k)})\). This finishes the induction step k→k+1, and thus completes the first step of the proof, i.e., the sequence of pricepayment pairs (p ^{(k)},q ^{(k)}) decreases.$$\begin{aligned} y^{(1)}&=f\left(\sum\limits_{i\in\mathcal{D}_{k1}}^{} s_{i}+\sum\limits_{i\in\mathcal{S}_{k1}\backslash\mathcal{S}_{k}}^{}s_{i}+\sum\limits_{i\in\mathcal{S}_{k}}^{}\min\left(\frac{{\zeta}_{i}^{(k1)}(p^{(k)},q^{(k)})}{q^{(k)}},s_{i}\right) \right)\\ &\le f\left(\sum\limits_{i\in\mathcal{D}_{k1}}^{} s_{i}+\sum\limits_{i\in\mathcal{S}_{k1}}^{}\min\left(\frac{{\zeta}_{i}^{(k1)}(p^{(k)},q^{(k)})}{q^{(k)}},s_{i}\right)\right)=q^{(k)}=y^{(0)}, \end{aligned} $$
 (B.S.)
 (ii)
In the second step, we need to show that (p ^{(k)},q ^{(k)})≥(p ^{+},q ^{+}) for all k=0,1,…,n. Again, this will be established by induction. For k=0 the assertion is obvious, since \((p^{(0)},q^{(0)})=(\bar {p},q_{0})\ge (p^{+},q^{+})\) by Lemma 3.1 (a).
For the induction step k→k+1, we observe that by the induction hypothesis a bank that defaults under (p ^{(k)},q ^{(k)}) does also default under (p ^{+},q ^{+}), thus \(\mathcal {D}(p^{+},q^{+})\supseteq \mathcal {D}_{k}\). Now, for \(i\in \mathcal {S}_{k}\), one has \(p_{i}^{(k+1)}=\bar {p}_{i}\ge p_{i}^{+}\). It again remains to analyze the entries of p ^{(k+1)} belonging to banks in \(\mathcal {D}_{k}\). Therefore, we reuse the recursive principle stated in Eq. (10) together with (11), and prove that for all ν=0,1,…: \(x_{i}^{(\nu)}\ge p_{i}^{+}\) (\(i\in \mathcal {D}_{k}\)), y ^{(ν)}≥q ^{+}. Now, starting again with \(x_{i}^{(0)}=p_{i}^{(k)}\) for \(i\in \mathcal {D}_{k}\) and y ^{(0)}=q ^{(k)}, we obtain for \(i\in \mathcal {D}_{k}\) thatHere, the first inequality holds because of the induction hypothesis and Lemma 2.2 (b). The second inequality follows from \(\bar {p}\ge p^{+}\) and \(w^{*}_{j}(p^{+},q^{+})<0\) for all \(j\in \mathcal {D}_{k} \subseteq \mathcal {D}(p^{+},q^{+})\). Finally, (p ^{+},q ^{+}) is a pricepayment equilibrium by assumption; thus, the last equality holds for \(i\in \mathcal {D}_{k}\subseteq \mathcal {D}(p^{+},q^{+})\) according to Definition 3.1.$${\begin{aligned} x_{i}^{(1)} &=\alpha\left[r_{i}+s_{i} q^{(k)} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{k}}^{}\Pi_{ji}p^{(k)}_{j}+\sum\limits_{j\in\mathcal{S}_{k}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}\left(p^{(k)},q^{(k)}\right),0\right)\right] \right]\\ &\ge \alpha\left[r_{i}+s_{i} q^{+} \right]+\beta\left[\sum\limits_{j\in\mathcal{D}_{k}}^{}\Pi_{ji}p^{+}_{j}+\sum\limits_{j\in\mathcal{S}_{k}}^{}\left[\Pi_{ji}\bar{p}_{j}+\lambda C_{ji}\max\left(w^{*}_{j}\left(p^{+},q^{+}\right),0\right)\right] \right]\\ &\ge \alpha\left[r_{i}+s_{i} q^{+} \right]+\beta\left[\sum\limits_{j=1}^{n}\Pi_{ji}p^{+}_{j}+\lambda \sum\limits_{j=1}^{n} C_{ji}\max\left(w^{*}_{j}\left(p^{+},q^{+}\right),0\right) \right]=p_{i}^{+}. \end{aligned}} $$Regarding the price of the illiquid asset, we first obtain by the induction hypothesis that
\({\zeta _{i}^{(k)}(p^{(k)},q^{(k)})\le \zeta _{i}^{(k)}(p^{+},q^{+})},\) for all \(i\in \mathcal {S}_{k}\). This leads tousing similar arguments as before. Hence, by the recursive definition of the sequence (x ^{(ν)},y ^{(ν)}), we see that every element of this sequence will be larger than or equal to the corresponding entries of the greatest pricepayment equilibrium. Overall, this yields (p ^{(k)},q ^{(k)})≥(p ^{+},q ^{+}) for all k=0,1,…,n, as desired.$$\begin{aligned} y^{(1)}&=f\left(\sum\limits_{i\in\mathcal{D}_{k}}^{} s_{i}+\sum\limits_{i\in\mathcal{S}_{k}}^{}\min\left(\frac{{\zeta}_{i}^{(k)}\left(p^{(k)},q^{(k)}\right)}{q^{(k)}},s_{i}\right)\right)\\ &\ge f\left(\sum\limits_{i\in\mathcal{D}(p^{+},q^{+})}^{} s_{i}+\sum\limits_{i\in\mathcal{S}(p^{+},q^{+})}^{}\min\left(\frac{{\zeta}_{i}^{(k)}\left(p^{+},q^{+}\right)}{q^{+}},s_{i}\right)\right)=q^{+}, \end{aligned} $$  (iii)
To finish this proof, we combine all previous arguments. By our first step, the payment vectors are decreasing with each iteration of the algorithm and hence, \(\mathcal {D}_{k+1}\supseteq \mathcal {D}_{k}\), which leads to two possible cases. The first is \({\mathcal {D}_{k+1}= \mathcal {D}_{k}}\). In this case, the algorithm terminates and due to the fixedpoint construction, (p ^{(k)},q ^{(k)}) is a pricepayment equilibrium as in Definition 3.1. Moreover, as (p ^{+},q ^{+}) is the unique greatest pricepayment equilibrium and by our second step (p ^{(k)},q ^{(k)})≥(p ^{+},q ^{+}), the algorithm terminates with the greatest pricepayment equilibrium. The second possibility for the sequence of default sets is that it is strictly increasing from one round to another, i.e., \(\mathcal {D}_{k+1}\supset \mathcal {D}_{k}\). This means that a new bank joined the default set, and payments and prices have to be adjusted. But since there are at most n banks that can join the default set, after at most n+1 iterations^{11} the default set does not change anymore. Thus, we eventually end up in the first possible case, finding the greatest pricepayment equilibrium.
Appendix 1. Example: pricepayment equilibria
Pricepayment equilibria are nonunique and, moreover, the set of equilibria is not necessarily connected. In the following example, we will construct a financial system in which for some \(p\in \mathbb {R}^{n}\) with p ^{−}<p<p ^{+} there does not exist a q∈[q ^{−},q ^{+}] such that (p,q) is a pricepayment equilibrium. We will also show that for given \(q\in \mathbb {R}\) with q ^{−}<q<q ^{+} it is not always possible to find p∈[p ^{−},p ^{+}] such that (p,q) is a pricepayment equilibrium.
 1.For a given clearing price q ^{∗}, the clearing payment vector p ^{∗} satisfies the relation$$ p^{*}=\chi(p^{*},q^{*}). $$(14)
For any fixed q∈[q _{min},q _{0}], we call a corresponding fixed point of Eq. (14) a clearing vector for q, denoted by p ^{∗q }. Its existence is established by Tarski’s fixedpoint theorem, analogous to the proof of Theorem 3.2.
 2.Analogously, for fixed \(p\in [\boldsymbol {0},\bar {p}]\), we define a clearing price for p by$$ q^{*p}=f(\theta(p,q^{*p})). $$(15)
Its existence follows again from Tarski’s fixedpoint theorem.
Appendix 2. Crossholdings with price impact
The net worth price vector q ^{ C } depends on the banks’ net worths vector w and vice versa. We characterize these values as a combined equilibrium:
Definition B.1
The existence of a greatest and least net worth equilibrium follows again from Tarski’s fixedpoint theorem, now applied to the function Ψ ^{ C }.
Note that the modeling framework that we consider in the numerical case studies above is a special case of the general inverse demand function. Setting λ _{ i }=(1−c)κ with 0≤κ≤1, the corresponding inverse demand function f ^{ C } is defined by its components \(f^{C}_{j}(x)=(1c)\kappa \), \(j\in \mathcal {N}\).
Declarations
Authors’ contributions
Both authors jointly conducted the research and wrote the paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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