OOptimal Network Compression
Hamed Amini ∗ Zachary Feinstein † August 21, 2020
Abstract
This paper introduces a formulation of the optimal network compression problem for financialsystems. This general formulation is presented for different levels of network compression orrerouting allowed from the initial interbank network. We prove that this problem is, generically,NP-hard. We focus on objective functions generated by systemic risk measures under systematicshocks to the financial network. We conclude by studying the optimal compression problem forspecific networks; this permits us to study the so-called robust fragility of certain networktopologies more generally as well as the potential benefits and costs of network compression.
Keywords:
Systemic risk, financial networks, portfolio compression, genetic algorithm.
The financial crisis 2007-2009 has highlighted the importance of network structure on the am-plification of the initial shock to the level of the global financial system, leading to an economicrecession. In response to market dysfunctions, the US congress enacted the largest regulations offinancial market, in the form of the "Dodd-Frank Wall Street Reform and Consumer ProtectionAct" of 2010, to ensure financial stability and reduce systemic risk. Among the regulations isthat the majority of over-the-counter (OTC) derivatives should be centrally cleared so as toreduce counterparty risk and ensure financial stability. Portfolio compression is another way tomodify the financial network structure. Several parties in the network enter into a multi-lateralnetting agreement to essentially reduce the gross exposures while keeping the net positions un-changed. The main provider of such systems is TriOptima [41], who have compressed over $1 . trillion in gross notional. ∗ Robinson College of Business, Georgia State University, Atlanta, GA 30303, USA, email: [email protected] † Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA. [email protected] a r X i v : . [ q -f i n . R M ] A ug or the purposes of this paper we consider a known initial finite network of obligationsover which we seek an “optimal” network compression. This is in contrast to the randomgraph structure considered in [2, 24, 29]. Though the initial network compression formulation ispresented without consideration of the network clearing procedure, we will primarily focus onclearing based on [23,40]. Under the DebtRank [10,11] clearing, a version of the optimal networkcompression problem as a mixed integer linear program was proposed in [20]. Other notions ofcontagion could be added to our clearing problem as well, e.g., portfolio overlap [4, 17, 25]; suchadditional avenues of contagion would influence the systemic risk and may impact the optimalcompression. We focus on the Eisenberg-Noe framework so as to remain in a, relatively, simplesetting.The optimal compression problem is related to studies in many other works in the Eisenberg-Noe clearing framework. For instance, the compression constraints can be viewed as the feasibil-ity conditions for a network reconstruction problem; [7,30,31,35,42] propose methods to sampleeither deterministically or stochastically from this feasible region. [1] considered the optimalrerouting problem of a system of identical banks under i.i.d. Bernoulli shocks. That work foundthat the completely connected system has a “robust fragility” property, i.e., it is the most stablefor small shocks but the least stable for large shocks (and vice versa for the ring network). [27]studies the sensitivity of the Eisenberg-Noe clearing payments w.r.t. the relative liability ma-trix; that work uses these sensitivities in order to find the best and worst case directions forrerouting of the liability network. [12] utilizes majorization of the financial networks in order toguarantee the relative health of two financial networks (w.r.t. the number of defaulting banks).The network compression is also related to the literature on analyzing consequences of differentnetting mechanisms in centrally cleared financial markets; see, e.g., [3, 5, 6, 8, 13, 18, 21, 22, 32].We are primarily motivated in this study by two streams of literature: that of [1] whichproposed the robust fragility of the completely connected network and those of [19, 43] whichproposed frameworks for network compression without regards for the impact on systemic risk.In this paper we seek to unify these problems into a single optimization framework, which we call optimal network compression . We further merge these problems with the systemic risk measuresof [15, 38] so as to determine the compressed network that minimizes systemic risk; as shownin [43], network compression need not improve systemic risk.The primary innovations and results of this paper are in multiple directions. First, we provethat the optimal network compression problem is generically NP-hard. This motivates us toconsider a machine learning approach to approximating the optimal financial network. Second, s in [9], we consider stress scenarios under systematic shocks. These stress scenarios allow usto analytically compute the systemic risk measures we wish to minimize; this removes the needfor Monte Carlo simulations in the computation of the objective of our optimal compressionproblem. In particular, this framework allows us to numerically generalize the results of [1]to consider the optimal rerouting problem and [43] to quantify the suboptimality of the fullcompression as utilized in [19].The organization of this paper is as follows. In Section 2, we propose the general optimalnetwork compression problem to be considered throughout this work. In so doing, we formulatefour meaningful types of “compression” problems motivated by [1,19] and prove that the optimalcompression problem is, generically, NP-hard given each of those constraint sets. We then focuson a meaningful form for the objective function, namely the systemic risk measures of [15, 38],in Section 3. Specifically, in Section 3.2, we present analytical forms for specific examples ofthese systemic risk measures under systematic shocks to the financial system. This is followedby two case studies in Section 4. First, we consider a simple three bank system that servesthe dual purposes of validating our algorithmic approach as well as testing the results of [1] inheterogeneous system with systematic shocks. Second, we study a network calibrated to the2011 European Banking Authority dataset; with this network we study the improvements insystemic risk via optimal compression and “maximal” compression (i.e., as proposed in [19]).Section 5 concludes. Throughout this work we will consider a system of n banks with obligations L ij ≥ from bank i to j ; as is typical, we assume L ii = 0 for every bank i , i.e., no bank has any obligationsto itself. Additionally, each bank i will be assumed to have liabilities external to the bankingnetwork L i ≥ . These external obligations are sometimes called societal obligations; we willinterchangeably use these terms throughout this work. The set of all such networks is denotedby L := { L ∈ R n × ( n +1)+ | L ii = 0 ∀ i } .In this section we present the primary optimization problem of interest in this work. Todo so we introduce the notion of portfolio compression which we take from [19]. In this workwe seek to find the optimal network compression problem, i.e., for an initial liability network ˜ L ∈ L , we wish to minimize some objective f : L → R . min (cid:110) f ( L ) | L ∈ C ( ˜ L ) (cid:111) . (1) he objective function f can be directly computed from the network statistics (e.g., networkentropy) or the results of systemic risk measures (see, e.g., [15, 38]). Note that in general thisfunction may also depend on the initial liability network f ( L ) = f ( L ; ˜ L ) , but we drop this fromthe notation for compactness. We will discuss the systemic risk measure based objective func-tions in the following sections. Notably, as these objective functions are in general nonconvex,this optimization problem might be hard; in fact, we will show that (generically) this problemis NP-hard given certain objectives and network compression based constraints in Theorem 2.4. Remark 2.1.
Prior works on network compression, e.g., [19], focus on removing the maximalamount of liabilities in the system subject to certain financial constraints. That is, with objective f ( L ) := (cid:80) ni =1 (cid:80) n +1 j =1 L ij . In particular, with the compression constraints highlighted within thiswork, (1) becomes a linear programming problem and can be solved in polynomial time. Other,non-optimization based, algorithms for undertaking this compression are presented in [19]. Incontrast, we are motivated, as in [43], to study partial compression to determine the optimal levelof compression; [43] focuses on conservative compression (defined below) with a strict definitionfor optimality related to the set of defaulting banks.The constraint set C ( ˜ L ) denotes the set of all possibly compressed or rerouted networksconsistent with ˜ L . Any such meaningful compression problem satisfies two properties: consistentnet liabilities and feasibility as a network. This is encoded in the following definition consistentwith prior works on network compression, e.g., [19, 43]. Definition 2.2.
Given an initial financial liability network ˜ L ∈ L , C ( ˜ L ) is a set of compressednetworks if L ∈ C ( ˜ L ) implies: • constant net liabilities : (cid:80) nj =1 [ L ij − L ji ] + L i = (cid:80) nj =1 [ ˜ L ij − ˜ L ji ] + ˜ L i for every bank i ∈ { , . . . , n } and • feasibility : L ∈ L . Compare the definition of the set of compressed networks to the General Compression Prob-lem defined in [19]. We wish to note that the set of compressed networks can often be defined asa convex polyhedron; in fact it is explicitly defined this way in [19] and every specific examplewe consider in this work follow satisfy such a structure.
Example 2.3.
In this example we consider 3 types of network compression and, fourth, thererouting problem, which we will consider throughout this work. These 3 network compressionproblems are detailed in [19] with conservative compression studied further in [43]. We describethese compression problems in varying order from most to least restrictive; though each is ank 1 Bank 2Bank 3
110 2 20330
Bank 1 Bank 2Bank 3 − µ − µ − µ − µ − µ − µ Figure 1: Example of bilateral compression with µ ∈ [0 , , µ ∈ [0 , and µ ∈ [0 , . financially meaningful, other types of compression can be implemented. To simplify notation,we will define L i := 0 for every bank i for any network L ∈ L .(i) Bilateral compression : Given an initial network ˜ L ∈ L , bilateral compression allows forthe reduction of bilateral exposures only. That is, the net obligations between banks i and j must always be kept consistent with the initial network construction; additionally, allobligations can only be reduced from their initial levels. As such, we can define bilateralcompression C B ( ˜ L ) as: C B ( ˜ L ) := (cid:110) L ∈ L | ∀ i, j : L ij − L ji = ˜ L ij − ˜ L ji , L ij ∈ [0 , ˜ L ij ] (cid:111) . Bilateral compression is special insofar as the most the network can be compressed in thisway is defined by obligations: L ij := max { , ˜ L ij − ˜ L ji } .(ii) Conservative compression : Given an initial network ˜ L ∈ L , conservative compressionallows for the reduction of cyclical exposures only. That is, the net obligations owedaround a directed cycle i → j → ... → j m → i must always be kept consistent with theinitial network construction; additionally, all obligations can only be reduced from theirinitial levels. As defined in [19], this cyclical netting rule can be encoded by the fixednet liabilities condition for every bank i . As such, we can define conservative compression C C ( ˜ L ) as: C C ( ˜ L ) := (cid:40) L ∈ L | ∀ i, j : n (cid:88) k =0 [ L ik − L ki ] = n (cid:88) k =0 [ ˜ L ik − ˜ L ki ] , L ij ∈ [0 , ˜ L ij ] (cid:41) . ank 1 Bank 2Bank 3 Bank 1 Bank 2Bank 3 − α − α − α − β − β − β Figure 2: Example of cycle compression in different models. For conservative compression: α ∈ [0 , and β ∈ [0 , . For rerouting: α ∈ ( −∞ , , β ∈ ( −∞ , and α + β = 0 . For nonconservativecompression: α ∈ ( −∞ , , β ∈ ( −∞ , and α + β ≥ . (iii) Rerouting : Given an initial network ˜ L ∈ L , rerouting allows for the rewiring of the entirenetwork. That is, all liabilities are redistributed throughout the system in such a way thatnet and gross liabilities are kept constant. As such, we can define rerouting C R ( ˜ L ) as: C R ( ˜ L ) := (cid:40) L ∈ L | ∀ i, j : n (cid:88) k =0 [ L ik − L ki ] = n (cid:88) k =0 [ ˜ L ik − ˜ L ki ] , n (cid:88) k =0 L ik = n (cid:88) k =0 ˜ L ik (cid:41) . (iv) Nonconservative compression : Given an initial network ˜ L ∈ L , nonconservative com-pression allows for the conservative compression of the rerouting problem. That is, forevery bank i , net liabilities are kept constant while gross liabilities are allowed to be re-duced from the initial setup. As such, we can define nonconservative compression C N ( ˜ L ) as: C N ( ˜ L ) := (cid:40) L ∈ L | ∀ i, j : n (cid:88) k =0 [ L ik − L ki ] = n (cid:88) k =0 [ ˜ L ik − ˜ L ki ] , n (cid:88) k =0 L ik ≤ n (cid:88) k =0 ˜ L ik (cid:41) . Sometimes one may also wish to fix the obligations L i owed to society; this is accomplished bytaking the intersection C ( ˜ L ) ∩ C ( ˜ L ) where C ( ˜ L ) := { L ∈ R n × ( n +1)+ | ∀ i : L i = ˜ L i } . We conclude this section by showing that the optimal compression problem for the constraintsets of Example 2.3 are NP-hard in general. This result motivates us to consider specific settings nd algorithms used later in this work. Theorem 2.4.
The optimal network compression problem is NP-hard for the conservative,rerouting and nonconservative compression models.
The proof of theorem is provided in Appendix A. By considering the network (relativeliabilities) entropy ( f ( L ) = − (cid:80) ni =1 (cid:80) nj =0 L ij (cid:80) nk =0 L ik log (cid:16) L ij (cid:80) nk =0 L ik (cid:17) ) as our objective function weshow that the optimal compression problem for each set of constraints is NP-hard. We prove thisby performing reduction from instances of the NP-complete subset sum problem [36]; definedby a set of positive integers S = { k , k , . . . , k n } and an integer target value θ ∈ N , we wish toknow whether there exist a subset of these integers that sums up to θ . We show that this canbe viewed as a special case for the optimal network compression for each set of constraints. Remark 2.5.
In contrast to using the maximum entropy to find the missing liabilities asin [39], we can consider the minimum entropy as our objective function for compression, seee.g. [37, 44]. Indeed, the network maximizing the entropy will be close to the complete regularnetwork. On the other hand, the network minimizing the entropy would correspond to a sparsenetwork. So it makes sense to consider it as an objective function for compression; as this isshown in [44], many of the known algorithms in pattern recognition can be characterized asefforts to minimize the entropy. Consider for example the case of similar firms all having thesame total assets and liabilities (and without obligations to society). Then it is easy to checkthat the network maximizing entropy will correspond to the complete regular network with L ij (cid:80) nk =1 L ik = n − which gives f ( L ) = n log( n − . On the other hand, the network minimizingthe entropy would correspond to the (regular) ring which corresponds to f ( L ) = 0 . Remark 2.6.
As the optimal compression problem (1) is generically nonconvex and NP-hard,we cannot rely on a gradient descent method to converge to the global optimum. As such webelieve that machine learning tools and methods would be best for solving such problems ingeneral. For this paper, as will be utilized and validated in Section 4, we will implement agenetic algorithm to solve the optimal control problem.
In this section we wish to give a specific structure to the objective function f : L → R in ouroptimal compression problem (1). Specifically, we wish to consider the network compression thatminimizes a systemic risk measure . These functions are decomposed as ρ ◦ Λ for a risk measure ρ and an aggregation function Λ . Such functions were first introduced in [15, 38] and are detailed elow; these mappings also coincide with the “insensitive systemic risk measures” of [14, 28]. Inorder to present this setting, and for the remainder of this paper, we fix some probability space (Ω , F , P ) . Let L := L (Ω , F , P ) denote those random variables that are square-integrable. In order to determine the health of a financial network, we first present a generic aggregationfunction Λ in the following definition. These aggregation functions are mappings of two argu-ments: the endowment for the banks and the liability network. The purpose of such a functionis to provide an aggregate statistic of the state of the financial system. Definition 3.1.
The mapping
Λ : R n + × L → R is a aggregation function if it is a nonde-creasing mapping in its first argument. Example 3.2.
Throughout this work we specifically consider three different aggregation func-tions that are all fundamentally associated with the clearing mechanisms of [23, 40]. That is,for recovery rates α x , α L ∈ [0 , , the clearing payments are the maximal fixed point p ( x, L ) =FIX p ∈ [0 ,L(cid:126) Ψ( p ; x, L ) for Ψ i ( p ; x, L ) = (cid:80) nj =0 L ij if x i + (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j ≥ (cid:80) nj =0 L ij α x x i + α L (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j if x i + (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j < (cid:80) nj =0 L ij . As such, the clearing procedure Ψ implies: if bank i has nonnegative wealth x i + (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j − (cid:80) nj =0 L ij ≥ then it is solvent and its wealth is equal to its total assets minus its total liabilities;if bank i has negative wealth x i + (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j − (cid:80) nj =0 L ij < then it is defaulting and itsassets are reduced by the recovery rates α x , α L . From [40], we immediately recover a greatestand least clearing solution to p = Ψ( p ; x, L ) within the lattice [0 , L(cid:126) .Let V ( x, L ) ∈ R n denote the clearing wealths from an Eisenberg-Noe style clearing procedurewith endowments x ∈ R n + and L ∈ L denote the network of obligations, i.e., V i ( x, L ) = x i + (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j ( x, L ) − (cid:80) nj =0 L ij if p i ( x, L ) = (cid:80) nj =0 L ij α x x i + α L (cid:80) nj =1 L ji (cid:80) nk =0 L jk p j ( x, L ) − (cid:80) nj =0 L ij if p i ( x, L ) < (cid:80) nj =0 L ij . (2)With this clearing procedure, we consider the following three aggregation functions: • Number of solvent banks : Λ ( x, L ) := (cid:80) ni =1 I { V i ( x,L ) ≥ } . • System-wide wealth : Λ N ( x, L ) := (cid:80) ni =1 V i ( x, L ) . External wealth : Λ ( x, L ) := (cid:80) ni =1 L i (cid:80) nj =0 L ij (cid:104)(cid:80) nj =0 L ij − V i ( x, L ) − (cid:105) .Additionally, we need to consider a risk measure ρ in order to determine the risk that thesystem is incurring. Such functions map random variables into capital requirements. Definition 3.3.
The mapping ρ : L → R is a risk measure if it satisfies the followingproperties: • normalization : ρ (0) = 0 ; • monotonicity : ρ ( X ) ≤ ρ ( Y ) if X ≥ Y a.s.; and • translative : ρ ( X + m ) = ρ ( X ) − m for m ∈ R . These risk measures may satisfy additional conditions, e.g., convexity or positive homogene-ity.
Example 3.4.
For the purposes of this work, we will focus on two standard risk measuresparameterized by γ ∈ [0 , : • Value-at-Risk : ρ VaR γ ( Z ) = − inf { z ∈ R | P ( Z ≤ z ) > − γ } =: − Z − γ . If γ = 1 then werecover the so-called worst-case risk measure : ρ VaR1 ( Z ) = − ess inf Z =: ρ WC ( Z ) . • Expected shortfall : ρ ES γ ( Z ) = − E [ Z | Z ≤ Z − γ ] . If γ = 0 then we recover the so-called expectation risk measure : ρ ES0 ( Z ) = − E [ Z ] =: ρ E ( Z ) .To measure the health of the financial system, we consider the systemic risk measures ρ ◦ Λ for risk measure ρ and aggregation function Λ . Definition 3.5.
The mapping R : ( L ) n × L → R is a systemic risk measure if it can bedecomposed into an aggregation function Λ and a risk measure ρ so that R ( X, L ) := ρ (Λ( X, L )) for every X ∈ ( L ) n and L ∈ L . Within the optimal compression problem (1), we specifically are interested in f ( L ) := ρ (Λ( X, L )) for some fixed (random) endowments X ∈ ( L ) n . That is, given a stress scenario X ∈ ( L ) n ,we seek to find the optimal financial network (subject to compression constraints) such that thesystemic risk is minimized. Remark 3.6.
We wish to highlight two special cases which relate to the notions proposed in [1].Fix X ∈ ( L ) n such that X i ≥ a.s. for every bank i . i) Consider the worst-case risk measure ρ WC ( Z ) = − ess inf Z . In the notation from [1], L ismore resilient than ˆ L if and only if ρ WC (Λ ( X, L )) ≤ ρ WC (Λ ( X, ˆ L )) .(ii) Consider the expectation risk measure ρ E ( Z ) = − E [ Z ] . In the notation from [1], L is more stable than ˆ L if and only if ρ E (Λ ( X, L )) ≤ ρ E (Λ ( X, ˆ L )) .[1] presents these notions for symmetric systems of banks with i.i.d. Bernoulli shocks X i .Notably, under such conditions, the number of solvent banks provides the full information onthe health of the system; that is, any aggregation function that depends on ( x, L ) only throughthe clearing wealths V ( x, L ) , say Λ , provides the same ordering of liability networks L, ˆ L ∈ L as Λ : ρ (Λ( X, L )) ≤ ρ (Λ( X, ˆ L )) ⇔ ρ (Λ ( X, L )) ≤ ρ (Λ ( X, L )) . We will revisit these problems, and compare our formulation with that with [1] further, inSection 4.1 below.
While the systemic risk measures provide a meaningful objective to minimize in order to optimizenetwork compression, such constructs present additional computational challenges. Namely,even a simple systemic risk measure such as ρ E ◦ Λ requires an exponential (in number ofbanks) time to compute explicitly [34]. Computationally, this can be overcome with MonteCarlo simulations though that is subject to estimation errors. Herein we will impose systematic shocks on the endowments on the banks, i.e., a comonotonic setting on the stress scenarios X ∈ ( L ) n , on an aggregate function based around the Eisenberg-Noe clearing notion. This isin contrast to [1] in which shocks were i.i.d.Throughout this section let C : R + → R n + be a nondecreasing function and q be somerandom variable such that C ( q ) ∈ ( L ) n . The stress scenario is then defined by X = C ( q ) . Forthe purposes of this section we will focus on systemic risk measures constructed from Value-at-Risk and expected shortfall (as defined in Example 3.4) and aggregate functions that dependon the endowments and liability network through Eisenberg-Noe clearing wealths only. Werefer to Example 3.2 for a brief discussion of the Eisenberg-Noe clearing problem; importantly,we define the clearing wealths V : R n + × L → R n as a mapping from the endowments andliability network. As detailed below, this setup allows for polynomial time computation of thesemeaningful systemic risk measures. Much of this section follows from the logic of [9].The systematic shock setting allows us to determine threshold market values q ∗ such that anks are on the cusp of bankruptcy; in particular, we take the view that q denotes a systematicfactor. These values are presented in Definition 3.7 below. Though presented as a mathematicalformulation, [9, Proposition 4.4] presents an iterative algorithm for finding q ∗ taking advantageof the monotonicity of C . Definition 3.7.
Define q ∗ : L → R n + so that q ∗ i ( L ) is the minimal value such that firm i issolvent under the liability network L ∈ L , i.e. q ∗ i ( L ) = inf { t ≥ | V i ( C ( t ) , L ) ≥ } . As noted above, we consider only those aggregate functions Λ whose dependence on theendowments x ∈ R n + and liability network L ∈ L come through the Eisenberg-Noe clearingwealths V ( x, L ) , i.e., Λ( x, L ) = ¯Λ( V ( x, L )) for every x ∈ R n + and L ∈ L for some monotonicfunction ¯Λ : R n → R . In this setting, the threshold values q ∗ provide a quick heuristic for thehealth of the financial system. Notably, if q ∗ i ( L ) ≥ q ∗ i ( ˆ L ) for every bank i for two financialnetworks L, ˆ L ∈ L , then Λ( C ( t ) , L ) ≤ Λ( C ( t ) , ˆ L ) for any t ∈ R + and, thus, ρ (Λ( C ( q ) , L )) ≥ ρ (Λ( C ( q ) , ˆ L )) for any nonnegative random variable q .Before proceeding to the representations for the systemic risk measures under systematicshocks, we need to introduce some notation that is provided in greater detail in [9]. Namely, wewant to consider a piecewise linear construction for the clearing wealths which follows from thefictitious default algorithm of [40]. That is, V ( x, L ) := ∆( I { V ( x,L ) < } , L ) x − δ ( I { V ( x,L ) < } , L ) for any endowment x ∈ R n + and liability network L ∈ L . In this piecewise linear construction,the mappings ∆ , δ are defined by: ∆( z, L ) := (cid:0) I − ( I − (1 − α L ) diag( z )) Π (cid:62) diag( z ) (cid:1) − ( I − (1 − α x ) diag( z )) ,δ ( z, L ) := (cid:0) I − ( I − (1 − α L ) diag( z )) Π (cid:62) diag( z ) (cid:1) − (cid:2) I − ( I − (1 − α L ) diag( z )) Π (cid:62) (cid:3) ¯ p, ¯ p := L(cid:126) , π ij := L ij ¯ p i for z ∈ { , } n denoting the set of defaulting institutions and L ∈ L is the liability network.In particular, for our comonotonic setting, we can simplify these notions as only a subset of ossible sets of defaulting institutions is possible, i.e., we define: ∆ k ( L ) := ∆( (cid:80) ni =1 I { q ∗ i ( L ) ≤ q ∗ k ( L ) } , L ) if k = 1 , , ..., nI if k = 0 ,δ k ( L ) := δ ( (cid:80) ni =1 I { q ∗ i ( L ) ≤ q ∗ k ( L ) } , L ) if k = 1 , , ..., n ( I − Π (cid:62) )¯ p if k = 0 . Finally, we will use the notation that [ k ]( L ) is the index of the k th greatest value of q ∗ ( L ) , i.e., q ∗ [1] ( L ) ≥ q ∗ [2] ( L ) ≥ ... ≥ q ∗ [ n ] ( L ) . To simplify following formulae, q ∗ [0] ( L ) ≡ + ∞ and q ∗ [ n +1] ( L ) ≡ for every liability network L ∈ L .We are now able to present the specific forms for the systemic risk measures under thesesystematic shocks. Proposition 3.8.
Consider a systematic stress scenario described by a nonnegative randomvariable q . Consider an aggregate function Λ whose dependence on the endowments x ∈ R n + and liability network L ∈ L come through the Eisenberg-Noe clearing wealths V ( x, L ) , i.e., Λ( x, L ) = ¯Λ( V ( x, L )) for every x ∈ R n + and L ∈ L for some monotonic function ¯Λ : R n → R .Let q − γ ∈ R + denote the (1 − γ ) -quantile for q , then the Value-at-Risk for level γ ∈ [0 , canbe computed as ρ VaR γ (Λ( C ( q ) , L )) = − Λ( C ( q − γ ) , L ) = − ¯Λ(∆ [ k ] ( L ) C ( q − γ ) − δ [ k ] ( L )) if q − γ ∈ [ q ∗ [ k +1] ( L ) , q ∗ [ k ] ( L )) . Additionally, the expected shortfall for level γ ∈ [0 , can becomputed as ρ ES γ (Λ( C ( q ) , L )) = − − γ n (cid:88) k =0 E (cid:104) Λ( C ( q ) , L ) I { q ∈ [ q ∗ [ k +1] ( L ) ,q ∗ [ k ] ( L )) ∩ [0 ,q − γ ] } (cid:105) = − − γ n (cid:88) k =0 E (cid:104) ¯Λ(∆ [ k ] ( L ) C ( q ) − δ [ k ] ( L )) I { q ∈ [ q ∗ [ k +1] ( L ) ∧ q − γ ,q ∗ [ k ] ( L ) ∧ q − γ ) } (cid:105) . Proof.
This follows directly from the construction of Value-at-Risk and expected shortfall andthe logic of Theorem 4.6 of [9].We conclude this section with a consideration of a special case in which the expected short-fall can be described in closed form for our example aggregation functions from Example 3.2.Specifically, we consider a case in which the systematic factor follows a lognormal distribution. orollary 3.9. Consider the setting of Proposition 3.8 in which q ∼ LogN( r − σ / , σ ) , C ( t ) := be rT + st with b ∈ R n and s ∈ R n + , and Λ takes the form of the specific aggregatefunctions provided in Example 3.2. Let Φ denote the CDF for the standard normal distributionand Φ − is the inverse CDF. For fixed level γ ∈ [0 , : ρ ES γ (Λ ( C ( q ) , L )) = − − γ n (cid:88) k =1 k × (cid:16) Φ( − d γ , [ k ] ( L )) − Φ( − d γ , [ k +1] ( L )) (cid:17) ,ρ ES γ (Λ N ( C ( q ) , L )) = − − γ(cid:126) (cid:62) n (cid:88) k =0 (cid:104) A γk ( L ) + B γk ( L ) (cid:105) ,ρ ES γ (Λ ( C ( q ) , L )) = − − γ n (cid:88) i =1 L [ i ]0 (cid:80) nj =0 L [ i ] j n (cid:88) j =0 L [ i ] j + e (cid:62) [ i ] n (cid:88) k =[ i ] (cid:104) A γk ( L ) + B γk ( L ) (cid:105) , where A γk ( L ) = (cid:0) ∆ [ k ] ( L ) be rT − δ [ k ] ( L ) (cid:1) (cid:16) Φ( − d γ , [ k ] ( L )) − Φ( − d γ , [ k +1] ( L )) (cid:17) ,B γk ( L ) = ∆ [ k ] ( L ) s (cid:16) Φ( − d γ , [ k ] ( L )) − Φ( − d γ , [ k +1] ( L )) (cid:17) ,d γ ,k ( L ) = − log( q ∗ k ( L ) ∧ q − γ ) + ( r + σ ) Tσ √ T , d γ ,k ( L ) = d γ ,k ( L ) − σ √ T ,q − γ = exp(( r − σ / T + σ √ T Φ − (1 − γ )) . Proof.
This is a direct consequence of the form for the expected shortfall as provided in Propo-sition 3.8.
In this section we will consider two case studies to demonstrate the results of this work. Asmentioned in Remark 2.6, we implement a genetic algorithm to optimize (1). First, we willpresent a small, 3 bank, financial system with heterogeneous (comonotonic) endowments. Thissystem allows us to easily present analytical results to validate the genetic algorithm we utilizeto consider optimal compression and rerouting. Additionally, this small system allows us toinvestigate the robust fragility results of [1] in a different setting to determine if those results holdfor more general settings than presented in that work. Second, we calibrate a financial systemto the 2011 European Banking Authority stress testing data. With that system we comparethe original network, the fully compressed networks, and optimally compressed networks. Asfound in [43], we find that network compression can increase systemic risk, but optimal networkcompression can find significant improvements over the original network. q x qx qx y yyL L L L L L Figure 3: Section 4.1: The generic network structure under consideration.
In [1] comparison of completely connected to ring structured networks was undertaken. Inthat work, these networks were “symmetric” in that all banks were identical in assets andliabilities, but with i.i.d. Bernoulli shocks. That work focuses on stability and resilience, i.e.,w.r.t. ρ WC ◦ Λ and ρ E ◦ Λ , respectively, as provided in Remark 3.6. Notably, [1] determines,under i.i.d. Bernoulli shocks, that the dense network is more stable if shocks are small, butthe sparse network is more stable if the shocks are large. We will explore this question furtherwith a consideration of a 3 bank system under systematic shocks that allows for: a completelyconnected network and two ring networks. The general network structure is depicted in Figure 3.For the sake of notational simplicity, we will consistently refer to bank i ∈ { , , } as the bankwith endowment equal to x i × q . Without loss of generality we assume x ≤ x ≤ x . Weadditionally assume that either x (cid:54) = x , x (cid:54) = x , or x (cid:54) = x ; if x = x = x then, due to thecomonotonic endowments inherent for a systematic shock, all rerouted networks have identicalsystemic risk. We wish to note that both the net and gross liabilities of each bank is the samein all network setups.To approach the problem of studying optimal compression and rerouting for our 3 banksystem, we first want to study the minimal solvency prices q ∗ for our three banks. In order to tudy these problems we impose the following netting conditions on the obligations: L + L = L + L , L + L = L + L , L + L = L + L . Given the network provided in Figure 3 with the aforementioned netting conditions, these valuescan be computed explicitly as: q ∗ = yx q ∗ = min { q ∗ , ( L + y )( L + L + y ) − α L L ( L + L ) α x L x +( L + L + y ) x } if 1 min { q ∗ , ¯ p (¯ p ¯ p − α L L L ) − α L ( L ( α L L L +¯ p L )+ L ( α L L L +¯ p L )) α x (¯ p L + α L L L ) x +(¯ p ¯ p − α L L L ) x + α x (¯ p L + α L L L ) x } if (cid:1) q ∗ = min { q ∗ , ¯ p (¯ p ¯ p − α L L L ) − α L ( L ( α L L L +¯ p L )+ L ( α L L L +¯ p L )) α x (¯ p L + α L L L ) x + α x (¯ p L + α L L L ) x +(¯ p ¯ p − α L L L ) x } if 1 min { q ∗ , ( L + y )( L + L + y ) − α L L ( L + L ) α x L x +( L + L + y ) x } if (cid:1) ¯ p = L + L + y , ¯ p = L + L + y , ¯ p = L + L + y and1 :([1 − α L ][ L + L ] + y )[ L x − L x ] + y ( L + L + y )[ x − x ] + α x y ( L − L ) x ≥ , (cid:1) :([1 − α L ][ L + L ] + y )[ L x − L x ] + y ( L + L + y )[ x − x ] + α x y ( L − L ) x < . These values q ∗ allow us to explicitly compute the statistics on the network as discussed inSection 3.2.To make these defaulting price levels more explicit, we wish to consider 4 simplified networkswith x i = i and ¯ p = ¯ p = ¯ p ≤ for all banks. These networks and resultant q ∗ are:(i) Completely connected:
Let L ij = for all i (cid:54) = j ∈ { , , } with y > . First wenote that, by construction, it must be that q ∗ ≥ q ∗ ≥ q ∗ in this setup for any choice ofbankruptcy costs and obligations to society. For notation to allow for easier comparisonslater on, we will denote these thresholds as q CC , q CC , q CC . q CC = y,q CC = min { q CC , y + 3 y + (1 − α L )4 y + 4 + α x } ,q CC = min { q CC , y + (2 − α L ) y + (1 − α L ))3(2 y + 2 + α x − α L } . (ii) Ring 123:
Let L = L = L = 1 and L = L = L = 0 with y > . First wenote that, by construction, it must be that q ∗ ≥ q ∗ ≥ q ∗ in this setup for any choice of ankruptcy costs and obligations to society. For notation to allow for easier comparisonslater on, we will denote these thresholds as q , q , q . q = y,q = min { q , y + 2 y + (1 − α L )2 y + 2 + α x } ,q = min { q , y + 3 y + 3 y + (1 − α L )3 y + (6 + 2 α x ) y + (3 + α x (2 + α L )) } . (iii) Ring 132:
Let L = L = L = 0 and L = L = L = 1 with y > . For notationto allow for easier comparisons later on, we will denote these thresholds as q , q , q . q = y,q = y if y ≥ (cid:16) − α x + (cid:112) (1 − α x ) + 8(1 − α L ) (cid:17) min { q , y +3 y +3 y +(1 − α L )2 y +(4+3 α x ) y +(2+ α x (3+ α L )) } if y < (cid:16) − α x + (cid:112) (1 − α x ) + 8(1 − α L ) (cid:17) ,q = min { q , y +3 y +3 y +(1 − α L )3 y +(6+ α x ) y +(3+ α x (1+2 α L )) } if y ≥ (cid:16) − α x + (cid:112) (1 − α x ) + 8(1 − α L ) (cid:17) min { q , y +2 y +(1 − α L )3 y +3+ α x } if y < (cid:16) − α x + (cid:112) (1 − α x ) + 8(1 − α L ) (cid:17) . (iv) Compressed:
Let L ij = 0 for all i (cid:54) = j ∈ { , , } with y > . For notation to allow foreasier comparison later on, we will denote these thresholds as q , q , q . q = y, q = y , q = y . As q ∗ = y for any network construction in this setup, we will compare these 4 networks forthe defaulting thresholds for banks 2 and 3 only. Figure 4 displays the default thresholds, inexcess of the fully compressed system, for the 2nd and 3rd bank ( max { q ∗ , q ∗ } and min { q ∗ , q ∗ } respectively) with α x = α L = 0 . . First, and notably, the default thresholds are lowest in thefully compressed system. This is further shown in Figure 7c in which the optimally compressednetwork with Λ is the fully compressed one. However, for comparison to [1], we also wantto investigate the rerouting problem. By investigating q ∗ we can compare the stability andresilience of financial networks to systematic Bernoulli shocks. As displayed in Figure 4, andas can be verified analytically (for any α x , α L ∈ [0 , ), q CC ≤ q for any y ≥ . That is for“small” shocks the completely connected system is always more stable and resilient than Ring123. For “large” shocks with small enough obligations to society y , we find that q CC ≤ q ; forlarger obligations to society the opposite ordering is found. That is, for “large” shocks the total y q Threshold (above fully compressed) for 2nd defaulting bank
CC1231320 (a) The excess (over the fully compressed network)in the default threshold max { q ∗ , q ∗ } − y/ for thesecond defaulting bank. y q Threshold (above fully compressed) for 3rd defaulting bank
CC1231320 (b) The excess (over the fully compressed network)in the default threshold min { q ∗ , q ∗ } − y/ for thethird defaulting bank. Figure 4: Section 4.1: Impact of obligations to society y to the default thresholds q ∗ . obligations to society can alter the stability and resilience ordering between these two networks.This, generally, coincides with the robust fragility notion from [1] in which the completelyconnected system was more robust to small shocks but more fragile to large shocks. In contrast,the opposite relations hold between the completely connected network and Ring 132; that is,we find that the ring is more stable and resilient for “small” shocks ( q CC ≥ max { q , q } )but the ordering for “large” shocks depends on the obligations to society y (for small enough y then q CC ≥ min { q , q } , for large enough y then q CC ≤ min { q , q } ). As systematicshocks and heterogeneous financial networks are vital to the consideration of financial stability,optimal compression and rerouting take on new significance since the typical heuristics in theliterature will not hold generally.We now wish to validate the performance of our genetic algorithm for finding the optimalnetworks. We will accomplish this by studying the expectation risk measure ρ E with all threeof our sample aggregation functions Λ , Λ N , Λ . The validation is accomplished by comparingthe results of the genetic algorithm with those using an interior point algorithm (initializedat L ij = for all i (cid:54) = j ). We also wish to compare these optimal networks with the 4 samplenetworks (completely connected, 2 rings, and the fully compressed system) to investigate the op-timality of these heuristic constructions. In Figure 6, we consider the optimal rerouting problemunder change of obligations to society y ; in Figure 7, we consider the optimal nonconservativecompression (with fixed obligations to society) problem under change of obligations to society y .First, and foremost, our genetic algorithm accurately matches or even outperforms the optimal q * Effects of on q * q q q (a) Effects of the network topology ( λ ) on the de-fault thresholds q ∗ for rerouting. y * E [ de f au l t s ] Effects of y on optimal network (b) Effects of obligations to society ( y ) on the moststable networks (determined by λ ∗ ) in rerouting. Figure 5: Section 4.1: Detailed visualization of the optimal rerouting problem. network using an interior point algorithm (as seen in optimal nonconservative compression with Λ ). Further, though the heuristic networks coincide with these optimal risk levels in specificcases, they do not uniformly perform as well as the optimal networks. Most interesting is theconsideration of Λ in which the optimally compressed network nearly coincides with the optimalrerouting problem for low y .Consider now the optimal compression and rerouting problems for this three bank system.As above, throughout this example, we will fix α x = α L = 0 . for simplicity of comparison.First, we will consider the optimal rerouting problem to generalize the notions from [1]. Inorder to ease the notation for the rerouting problem (with gross obligations of y for each ofthe three banks) let L = L = L = λ and L = L = L = 1 − λ . First, for the mostdirect comparison, in Figure 5a we consider how modifying λ affects the defaulting thresholds q ∗ (with y = 1 ). By inspection the most stable and resilient system is clearly for some small,but strictly positive, λ . These optimally stable networks are described in Figure 5b; such asystem is considered in which q ∼ LogN( − σ , σ ) with σ = 20% . Notably, the optimal networkdepends on the obligations to society y . In this section we demonstrate how our comonotonic approach with genetic algorithm for op-timization can be applied in a larger financial network consisting of n = 87 banks to come y -0.0200.020.040.060.080.1 E rr o r Errors in Expected Fractional Payment to Society for Rerouting
Genetic Algorithm=1/2, =1/2=1, =0=0, =1 (a) Payments to society y -0.0200.020.040.060.080.10.12 E rr o r Errors in Expected System-Wide Wealth for Rerouting
Genetic Algorithm=1/2, =1/2=1, =0=0, =1 (b) Total wealth y -0.100.10.20.30.40.50.6 E rr o r Errors in Expected
Genetic Algorithm=1/2, =1/2=1, =0=0, =1 (c) Number of defaulting (or solvent) banks
Figure 6: Section 4.1: Validation of the optimal rerouting problem.19 y -0.0200.020.040.060.080.10.120.14 E rr o r Errors in Expected Fractional Payment to Society for Nonconservative Compression
Genetic Algorithm=1/2, =1/2=1, =0=0, =1=0, =0 (a) Payments to society y -0.0500.050.10.150.20.250.30.35 E rr o r Errors in Expected System-Wide Wealth for Nonconservative Compression
Genetic Algorithm=1/2, =1/2=1, =0=0, =1=0, =0 (b) Total wealth y -0.100.10.20.30.40.50.60.7 E rr o r Errors in Expected
Genetic Algorithm=1/2, =1/2=1, =0=0, =1=0, =0 (c) Number of defaulting (or solvent) banks
Figure 7: Section 4.1: Validation of the optimal nonconservative compression problem.20 rom the 2011 European Banking Authority EU-wide stress tests. There are several previousempirical studies based on this dataset (see, e.g., [16,30]) and we calibrate this system by takingthe same approach as [26]. We consider a stylized balance sheet for each bank with only threetypes of assets and liabilities. The total interbank assets for bank i is (cid:80) nj =1 L ji while the total interbank liabilities is (cid:80) nj =1 L ij . The external risk-free assets for bank i is denoted by b i andthe external risky assets is denoted by s i . On the other hand, the external liabilities for bank i is L i and the bank i is endowed with capital C i .Note that the EBA dataset only provides the total assets A i , capital C i , and interbank lia-bilities (cid:80) nj =1 L ij for each bank i . Therefore, we will make the following simplifying assumptionssimilar to [16, 26, 33]. We assume that the external (risky) assets are the difference between thetotal assets and interbank assets. The external obligations owed to the societal node (denotedby L i ) will be assumed equal to the total liabilities less the interbank liabilities and capital.Further, we assume that the interbank assets is equal to the interbank liabilities for all banks,i.e., (cid:80) nj =1 L ij = (cid:80) nj =1 L ji for all i = 1 , . . . , n . Under these assumptions, the remainder of ourstylized balance sheet can be constructed by setting b i = 0 . × ( A i − n (cid:88) j =1 L ij ) , s i = 0 . × ( A i − n (cid:88) j =1 L ij ) , and, L i = A i − n (cid:88) j =1 L ij − C i , ¯ p i = L i + n (cid:88) j =1 L ij , which will guarantees that firm i ’s net worth is equal to its capital, i.e., C i = A i − ¯ p i .We will also need to consider the full nominal liabilities matrix L ∈ R × and not justthe total interbank assets and liabilities. To achieve this, we will use the MCMC methodologyof [30] which allows for randomized sparse structures to construct the full nominal liabilitiesmatrix consistent with the total interbank assets and liabilities. Remark that as this example isonly for illustrative purposes, we will consider only a single calibration of the interbank network.The remaining parameters of the system are calibrated as follows. We specify the systematicfactor as a lognormal distribution with parameters q ∼ LogN( r − σ , σ ) described in millionsof euros. Since during the period over which this data was collected, central banks were settinga low interest rate environment, we estimate that the risk-free interest rate is r = 0 . Finally,from comparisons to annualized historical volatility of European markets in 2011, the volatility Due to complications with the calibration methodology, we only consider 87 of the 90 institutions. DE029, LU45,and SI058 were not included in this analysis. ilateral Conservative Nonconservative-0 NonconservativeMaximally Compressed: Optimally Compressed: -0.033 -0.033 -1639.660 -5339.976Table 1: Section 4.2: Improvements in ρ ES80% ◦ Λ from the initial network L in millions of euros(i.e., negative values indicate cost savings). of the risky asset is estimated to be σ = 20% .For the purposes of this example, we consider clearing based on the pure Eisenberg-Noemechanism, i.e., with full recovery in case of default ( α x = α L = 1 ). We will consider twosystemic risk measures to optimize over: the 80% expected shortfall of the payments to society( ρ ES80% ◦ Λ ) and the number of solvent banks ( ρ ES80% ◦ Λ ). Herein we compare two types ofcompression: “maximally compressed” corresponds to compressed network that removes as muchexcess liabilities from the network as possible (as is considered in [19]) whereas “optimallycompressed” attempts to minimize the appropriate systemic risk measure. These two types ofcompression are then compared over 4 possible compression scenarios: bilateral compression( C B ( L ) ), conservative compression ( C C ( L ) ), nonconservative compression with fixed obligationsto society ( C N ( L ) ∩ C ( L ) ), and nonconservative compression ( C N ( L ) ). In particular, we wishto compare the systemic risk exhibited by the original network to those found in either themaximally compressed or optimally compressed scenarios. These results are provided in Tables 1and 2.As shown in Table 1, under maximal compression, the more relaxed the constraints the worsethe expected outcome for society in the 20% tail event; however, by using optimal compression,the systemic risk can be improved significantly under compression. Notably, the optimal bilat-eral and conservative compression only find marginal improvements in the payments to society,whereas the maximally compressed versions increase systemic risk by billions of euros. The non-conservative compressions find substantial benefits (over e e e ilateral Conservative Nonconservative-0 NonconservativeMaximally Compressed: -0.599 -2.196 -2.196 -2.196 Optimally Compressed: -0.599 -2.196 -2.196 -2.196Table 2: Section 4.2: Improvements in ρ ES80% ◦ Λ from the initial network L in In this work we presented a general formulation for the optimal network compression problemand found it to be NP-hard. We then focused on an objective function taking the form ofsystemic risk measures. In particular, we consider systematic shocks in order to find tractableanalytical forms for these systemic risk measures. Such scenarios allow us to generalize the workof, e.g., [1] to consider the robustness of various network topologies.As the optimal network compression problem is nonconvex and NP-hard in general, thechoice of optimization algorithms is of great interest. This is doubly so if idiosyncratic shocksare introduced as the computation of the systemic risk measures can be NP-hard as well in sucha setting (see, e.g., [9,34]). In this work we implemented a genetic algorithm to search for globalminima networks. We leave further research on choosing optimization procedures, especiallymachine learning methods, for future research.
References [1] Daron Acemoglu, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. Systemic risk and stabilityin financial networks.
American Economic Review , 105(2):564–608, 2015.[2] Hamed Amini, Rama Cont, and Andreea Minca. Resilience to contagion in financial net-works.
Mathematical Finance , 26(2):329–365, 2016.[3] Hamed Amini, Damir Filipović, and Andreea Minca. To fully net or not to net: Adverseeffects of partial multilateral netting.
Operations Research , 64(5):1135–1142, 2016.[4] Hamed Amini, Damir Filipović, and Andreea Minca. Uniqueness of equilibrium in a pay-ment system with liquidation costs.
Operations Research Letters , 44(1):1–5, 2016.[5] Hamed Amini, Damir Filipovic, and Andreea Minca. Systemic risk in networks with acentral node.
SIAM Journal on Financial Mathematics , 11(1):60–98, 2020.[6] Hamed Amini and Andreea Minca. Clearing financial networks: Impact on equilibriumasset prices and seniority of claims.
Available at SSRN 3632454 , 2020.
7] Kartik Anand, Ben Craig, and Goetz Von Peter. Filling in the blanks: Network structureand interbank contagion.
Quantitative Finance , 15(4):625–636, 2015.[8] Yannick Armenti and Stéphane Crépey. Central clearing valuation adjustment.
SIAMJournal on Financial Mathematics , 8(1):274–313, 2017.[9] Tathagata Banerjee and Zachary Feinstein. Pricing of debt and equity in a financial networkwith comonotonic endowments. 2019. Working paper.[10] Marco Bardoscia, Stefano Battiston, Fabio Caccioli, and Guido Caldarelli. DebtRank: Amicroscopic foundation for shock propagation.
PLOS ONE , 10(6):1–13, 2015.[11] Stefano Battiston, Michelangelo Puliga, Rahul Kaushik, Paolo Tasca, and Guido Caldarelli.DebtRank: Too central to fail? Financial networks, the FED and systemic risk.
Scientificreports , 2:541, 2012.[12] Agostino Capponi, Peng-Chu Chen, and David D. Yao. Liability concentration and systemiclosses in financial networks.
Operations Research , 64(5):1121–1134, 2016.[13] Agostino Capponi, W Allen Cheng, Sriram Rajan, et al. Systemic risk: The dynamicsunder central clearing.
Office of Financial Research, Working Paper (7 May) , 2015.[14] Çağin Ararat and Birgit Rudloff. Dual representations for systemic risk measures. 2016.Working paper.[15] Chen Chen, Garud Iyengar, and Ciamac C. Moallemi. An axiomatic approach to systemicrisk.
Management Science , 59(6):1373–1388, 2013.[16] Nan Chen, Xin Liu, and David D. Yao. An optimization view of financial systemic riskmodeling: The network effect and the market liquidity effect.
Operations Research , 64(5),2016.[17] Rodrigo Cifuentes, Hyun Song Shin, and Gianluigi Ferrucci. Liquidity risk and contagion.
Journal of the European Economic Association , 3(2-3):556–566, 2005.[18] Rama Cont and Thomas Kokholm. Central clearing of otc derivatives: bilateral vs multi-lateral netting.
Statistics & Risk Modeling , 31(1):3–22, 2014.[19] Marco D’Errico and Tarik Roukny. Compressing over-the-counter markets. 2019. Workingpaper.[20] Christian Diem, Anton Pichler, and Stefan Thurner. What is the minimal systemic risk infinancial exposure networks?
Journal of Economic Dynamics and Control , page 103900,2020.
21] Darrell Duffie, Martin Scheicher, and Guillaume Vuillemey. Central clearing and collateraldemand.
Journal of Financial Economics , 116(2):237–256, 2015.[22] Darrell Duffie and Haoxiang Zhu. Does a central clearing counterparty reduce counterpartyrisk?
Review of Asset Pricing Studies , 1:74–95, 2011.[23] Larry Eisenberg and Thomas H. Noe. Systemic risk in financial systems.
ManagementScience , 47(2):236–249, 2001.[24] Matthew Elliott, Benjamin Golub, and Matthew O. Jackson. Financial networks andcontagion.
American Economic Review , 104(10):3115–3153, 2014.[25] Zachary Feinstein. Financial contagion and asset liquidation strategies.
Operations ResearchLetters , 45(2):109–114, 2017.[26] Zachary Feinstein. Obligations with physical delivery in a multi-layered financial network.
SIAM Journal on Financial Mathematics , 10(4):877–906, 2019.[27] Zachary Feinstein, Weijie Pang, Birgit Rudloff, Eric Schaanning, Stephan Sturm, andMackenzie Wildman. Sensitivity of the Eisenberg and Noe clearing vector to individualinterbank liabilities.
SIAM Journal on Financial Mathematics , 9(4):1286–1325, 2018.[28] Zachary Feinstein, Birgit Rudloff, and Stefan Weber. Measures of systemic risk.
SIAMJournal on Financial Mathematics , 8(1):672–708, 2017.[29] Prasanna Gai and Sujit Kapadia. Contagion in financial networks. Bank of EnglandWorking Papers 383, Bank of England, 2010.[30] Axel Gandy and Luitgard A.M. Veraart. A Bayesian methodology for systemic risk assess-ment in financial networks.
Management Science , 63(12):4428–4446, 2016.[31] Axel Gandy and Luitgard Anna Maria Veraart. Adjustable network reconstruction withapplications to CDS exposures.
Journal of Multivariate Analysis , 172:193–209, 2019.[32] Paul Glasserman, Ciamac C. Moallemi, and Kai Yuan. Hidden illiquidity with multiplecentral counterparties.
Operations Research , 64(5):1143–1158, 2016.[33] Paul Glasserman and H. Peyton Young. How likely is contagion in financial networks?
Journal of Banking and Finance , 50:383–399, 2015.[34] Christian Gouriéroux, Jean-Cyprian Héam, and Alain Monfort. Bilateral exposures andsystemic solvency risk.
Canadian Journal of Economics , 45(4):1273–1309, 2012.[35] Grzegorz Hałaj and Christoffer Kok. Assessing interbank contagion using simulated net-works.
Computational Management Science , 10(2-3):157–186, 2013.
36] Richard M Karp. Reducibility among combinatorial problems. In
Complexity of computercomputations , pages 85–103. Springer, 1972.[37] Mladen Kovačević, Ivan Stanojević, and Vojin Šenk. On the hardness of entropy minimiza-tion and related problems. In , pages 512–516.IEEE, 2012.[38] Eduard Kromer, Ludger Overbeck, and Katrin Zilch. Systemic risk measures on generalprobability spaces.
Mathematical Methods of Operations Research , 84(2):323–357, 2016.[39] Paolo Emilio Mistrulli. Assessing financial contagion in the interbank market: Maximumentropy versus observed interbank lending patterns.
Journal of Banking and Finance ,35(5):1114–1127, 2011.[40] Leonard C.G. Rogers and Luitgard A.M. Veraart. Failure and rescue in an interbanknetwork.
Management Science , 59(4):882–898, 2013.[41] TriOptima. TriReduce - portfolio compression. 2016. Available .[42] Christian Upper and Andreas Worms. Estimating bilateral exposures in the German inter-bank market: Is there a danger of contagion?
European Economic Review , 48(4):827–849,2004.[43] Luitgard A.M. Veraart. When does portfolio compression reduce systemic risk? 2019.Working paper.[44] Satosi Watanabe. Pattern recognition as a quest for minimum entropy.
Pattern Recognition ,13(5):381–387, 1981.
A Proof of Theorem 2.4
We will consider the minimum relative liability entropy as the objective function: f ( L ) = − n (cid:88) i =1 n (cid:88) j =1 π ij log( π ij ) , (3)where π ij = L ij ¯ p i denotes the relative liability of firm i toward firm j . We refer to Remark 2.5for the interpretation of this objective function for network compression.Consider an instance of the NP-complete subset sum problem [36], defined by a set ofpositive integers S = { k , k , . . . , k n } and an integer target value θ ∈ N , we wish to know P P n C C αk αk αk n (1 − α ) k (1 − α ) k (1 − α ) k n Figure 8: Reduction to subset sum
NP-complete problem for optimal rerouting and nonconserva-tive compression. whether there exist a subset of these integers that sums up to θ . We will show that this can beviewed as a special case for the optimal network compression in the case of network rerouting,nonconservative and conservative compression models.Let K = k + k + · · · + k n and α = θ/K ∈ (0 , (otherwise, the subset sum decision istrivial). Given an instance of the subset sum problem, we define a corresponding instance ofthe optimal rerouting compression by considering the bipartite network of Figure 8 with twocore nodes { C , C } on one side and n periphery nodes { P , . . . , P n } on the other side. We setthe initial liabilities ˜ L as ˜ L P i ,C = αk i and ˜ L P i ,C = (1 − α ) k i for all i = 1 , . . . , n . Note that thetotal interbank receivables for C and C is respectively θ and K − θ , while the total interbankliabilities is zero for C and C . On the other hand, for all i = 1 , . . . , n , the total interbankliabilities for node P i is k i while the total interbank receivables is zero.The optimal rerouting compression model is thus equivalent to finding x i ∈ [0 , for L P i ,C = x i k i and L P i ,C = (1 − x i ) k i which satisfies (cid:80) ni =1 x i k i = θ , and minimizes f ( L ) = f ( x , . . . , x n ) = − n (cid:88) i =1 x i log ( x i ) − n (cid:88) i =1 (1 − x i ) log (1 − x i ) . Since x log( x ) + (1 − x ) log(1 − x ) ≤ for all x ∈ [0 , with equality only for x = 0 , , weinfer f ( x , . . . , x n ) ≥ , and the equality holds if and only if there exists x i ∈ { , } such that (cid:80) ni =1 x i k i = θ . Hence, if the solution to optimal rerouting compression model corresponds to f ( L ) = 0 then there exist a subset of S that sums up to θ .Further, note that in the bipartite network of Figure 8, the optimal nonconservative com-pression is equivalent to optimal rerouting. Hence, the same argument shows that the optimalnonconservative compression is NP-hard. P P n C C C k k k n k k k n k k k n K − θ θ Figure 9: Reduction to subset sum problem for optimal conservative compression model.
Hence, it only remains to prove the case of conservative compression model. Given an in-stance of the subset sum problem, we define a corresponding instance of the optimal conserva-tive compression by considering the network of Figure 9 with three core nodes { C , C , C } and n periphery nodes { P , . . . , P n } . We set the initial liabilities ˜ L as ˜ L P i ,C = k i and ˜ L P i ,C = k i for all i = 1 , . . . , n . Note that the net interbank liabilities for C and C is respectively − θ and − ( K − θ ) . On the other hand, for all i = 1 , . . . , n , the net interbank liabilities for node P i is k i .Further, for the node C the net interbank liabilities is zero.It is then easy to show that the minimum relative liability entropy can be found by setting L C ,P i = 0 for all i = 1 , . . . , n and consequently, L C ,C = 0 and L C ,C = 0 . The optimizationproblem is thus equivalent to find x i ∈ [0 , which gives L P i ,C = x i k i and L P i ,C = (1 − x i ) k i for all i = 1 , . . . , n . The x i s should satisfy (cid:80) ni =1 x i k i = θ and minimizes again f ( x , . . . , x n ) = − n (cid:88) i =1 x i log ( x i ) − n (cid:88) i =1 (1 − x i ) log (1 − x i ) ≥ . The equality holds if and only if there exists x i ∈ { , } such that (cid:80) ni =1 x i k i = θ . We concludethat if the solution to optimal conservative compression model corresponds to f ( L ) = 0 , thenthere exist a subset of S that sums up to θ , which completes the proof., which completes the proof.