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Journal ArticleDOI

A note on uniqueness of clearing prices in financial systems

AbstractThe Eisenberg and Noe (2001) model of the financial system is general- ized to the case where default is solved by means of a bankruptcy rule. For regular financial networks a unique vector of clearing prices exists if only the bankruptcy rule is strongly monotonic. This shows uniqueness of the clear- ing prices on regular financial networks for the class of equal sacrifice rules by Young (1988), and many variations of the proportional rule as in Csoka and Herings (2018). This paper disentangles the role of network topology from the way defaults are solved.

Topics: Financial networks (57%), Bankruptcy (52%), Default (50%)

Summary (2 min read)

1 Introduction

  • In the aftermath of the financial crisis in 2008 the delicate ways the players in the financial industries are intertwined is seen as the main source of the world wide spread of the shock caused by the subprime mortgage crises.
  • The major lesson for the architecture of the financial network is that it cannot only be seen as a means by which institutions and firms may diversify their risk exposures, but that instead it may also be the main cause for the amplification of risk.
  • More specifically, given such payment scheme, there are two types of nodes – the ones with a positive net value who will be able to pay all their liabilities and those with a negative net value that cannot.
  • The analysis for other monotonic bankruptcy rules is similar as the induced games also show strategic complementarities.
  • The question of uniqueness of clearing prices is also addressed by Csósak and Herings (2018), who present a discrete model that allows for decentralized clearing of the financial system.

2.1 Mathematical prerequisities

  • Let Rn denote the n-dimensional Euclidean vector space.
  • Special vector is the zero vector 0 with all zero coordinates.

2.2 Bankruptcy rules

  • 1Here I chose to use the term bankruptcy problem, but in fact the rationing problems as in Moulin (2002) or taxation problems in Young (1988) are of the same mathematical structure.
  • Well-known in the literature on taxation problems (see Young (1988), Lambert and Naughton (2009)) is the class of strictly monotonic rules which are referred to as equal sacrifice rules.
  • Basically, strong monotonicity rules out a kind of exotic rules which are strictly monotonic and do allow for zero derivatives.

2.3 The economic model

  • Where the connections or relations of agents within the network are shaped through the nominal liabilities an agent has to other agents in the system.the authors.
  • Let τ ∈ Rn+ be the vector that summarizes the total nominal obligations of the agents in the system, i.e., for i ∈ N let τi := n∑ j=1 Lij. (4) This total obligation vector τ summarizes agent-wise the payment levels required to satisfy all the contractual liabilities in the network.
  • The authors will assume that all liabilities have the same maturity date at which they become due and should be paid for.
  • On the other hand, each agent i has some justified claim Lji on pj.
  • This means that from payment pj by agent j, agent i obtains ri(Lj, pj).

2.4 Clearing payment vectors for a financial system

  • Below the authors will focus on the question whether payment vectors exist, that see to a clearing of the financial system such that two minimal requirements are satisfied.
  • First the authors will require from a payment vector that it expresses the idea of limited liability: no agent should pay more than the total of his cash inflow.
  • This holds for the partially ordered set [0, τ ].
  • For part (b) let p′ be any clearing vector.

3 Characterizing the clearing prices

  • Eisenberg and Noe (2001) characterize vectors of clearing prices using the notion of a surplus set, i.e., a set of agents S with no external obligations and a positive aggregate operation cash flow: Definition 2 Then Lemma 1 shows O(i) should contain an agent with positive equity.
  • If the set of defaulting agents is larger under pk+1 than under pk, then it must be that some agents pay their obligations in full under pk and default under pk+1, and the other agents either default or not both under pk+1 and pk.
  • And this concludes their proof by induction as also the authors have shown that {pj} is a weakly decreasing sequence.

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A note on uniqueness of clearing prices in financial systems
Koster, M.
DOI
10.2139/ssrn.3427039
Publication date
2019
Document Version
Submitted manuscript
License
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Citation for published version (APA):
Koster, M. (2019).
A note on uniqueness of clearing prices in financial systems
. SSRN.
https://doi.org/10.2139/ssrn.3427039
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Download date:09 Aug 2022

A note on uniqueness of clearing prices in
financial systems
Maurice Koster
July 26, 2019
Abstract
The Eisenberg and Noe (2001) model of the financial system is general-
ized to the case where default is solved by means of a bankruptcy rule. For
regular financial networks a unique vector of clearing prices exists if only the
bankruptcy rule is strongly monotonic. This shows uniqueness of the clear-
ing prices on regular financial networks for the class of equal sacrifice rules
by Young (1988), and many variations of the proportional rule as in Cs´oka
and Herings (2018). This paper disentangles the role of network topology
from the way defaults are solved.
Keywords: Financial networks, Systemic risk, Contagion, Clearing algorithm,
Rationing, Proportional Rule, Constrained Equal Award Rule
JEL Classification: C79, D31, D81, M41.
1 Introduction
In the aftermath of the financial crisis in 2008 the delicate ways the players in the
financial industries are intertwined is seen as the main source of the world wide
spread of the shock caused by the subprime mortgage crises. And still the intricate
way these players are connected is a main concern amongst economists and policy-
makers. Governments and central banks took extraordinary measures to bend the
impact of the crisis through monetary stimulation programmes and quantitative
easing, leaving society with costs exceeding 10 trillion dollars. Now these economic
accommodations are at the verge of being revoked, the induced outflow (or lack of
inflow) may result in liquidity disruptions which could eventually lead to similar
University of Amsterdam, Amsterdam School of Economics/CeNDEF, A: Roetersstraat 11,
1018WB Amsterdam, The Netherlands, E: mkoster@uva.nl.
1
Electronic copy available at: https://ssrn.com/abstract=3427039

detrimental effects to the financial institutions as surfaced in the years after 2008.
And it is believed that the impact of those disruptions be amplified by the fact
that worldwide debts levels hit an all-time high.
The major lesson for the architecture of the financial network is that it cannot
only be seen as a means by which institutions and firms may diversify their risk
exposures, but that instead it may also be the main cause for the amplification
of risk. The dependencies within the network may cause shocks to spread by
contagion, and lead to a cascade of defaults if not (again) prevented by public
institutions. See for instance the overview of Glasserman and Young (2016) or
Caccioli et al. (2018) which try to disentangle the problem by discussing various
ways correlations between nodes in the financial system play a role. In this paper
we will further investigate the rather simple yet seminal model of a financial system
due to Eisenberg and Noe (2001). Here a financial system is characterized by the
liability structure (who is liable to whom, and to what extent) and a description
of the aggregate external cash inflow per node, say firm or financial institution.
The authors aim at clearing this market, by determining a scheme of simultaneous
clearing prices that define the payments of each of the nodes to others. In this way
a net value for each node is defined. More specifically, given such payment scheme,
there are two types of nodes the ones with a positive net value who will be able
to pay all their liabilities and those with a negative net value that cannot. A node
is said to default in the latter case, if the total inflow of cash, i.e., the external
cashflow plus the payments to the node by others, minus the total sum of liabilities
of the node is negative. These Eisenberg and Noe clearing prices are constructed
such that (i) no node pays more than it has available, and (ii) a defaulting node
will make a maximal payment equal to its total cash inflow. Eisenberg and Noe
(2001) also propose an iterative procedure by which the clearing prices may be
calculated, and in this process defaults may occur at different stages mimicking
the indirect way financial institutions may be affected by earlier defaults. The
model allows to interpret the phase in which a financial institution defaults as a
measure of its resilience to default; the earlier a node defaults if at all the more
financial instability it can be credited. Other measures of financial instability and
assessment of systemic risk are found in Elsinger et al. (2006), Acemoglu et al.
(2015), Battiston et al. (2012).
Crucial assumption in Eisenberg and Noe (2001) is the principle of propor-
tionality; in case of a defaulting node, the corresponding clearing price is shared
proportional to the liabilities of the node to the others. Groote-Schaarsberg et al.
(2018) and Cs´oka and Herings (2018) show in a continuous and discrete setting,
respectively, that the assumption of proportionality in solving defaulting situa-
tions is not crucial at all, as the idea of clearing prices is still meaningful for other
bankruptcy rules. In accordance with Eisenberg and Noe (2001) both aforemen-
2
Electronic copy available at: https://ssrn.com/abstract=3427039

tioned works stress the fact that clearing prices may not be unique but the
resulting allocation is. This means that the net equity for an agent is the same for
each of those vectors of clearing prices. Besides that, the set of vectors of clearing
prices is well-structured as it is a completely ordered lattice with a smallest and a
largest element.
Groote-Schaarsberg et al. (2018) show that within the continuous formulation
of the model uniqueness of clearing prices is guaranteed for hierarchical structures,
i.e., problems that relate to an upper triangular matrix of liabilities. Supply chains
may have this hierarchical structure. In particular this means that uniqueness of
clearing prices is related to a network specific characteristic. In this paper, I show
that the clearing prices related to strictly monotonic bankruptcy rules are unique
for the regular financial networks discussed by Eisenberg and Noe (2001). The set
of rules that are strictly monotonic in the estate component is rich and includes
for example the equal sacrifice rules introduced by Young (1988) whereas in the
context of taxation. Regularity of the network requires for each specific node that
the aggregate operating cash flow corresponding to the set of nodes it can reach
through the liability network is positive. Importantly, regularity is a pure network
characteristic, independent from the bankruptcy rule that is used. So the contribu-
tion of this paper is also that in studying for vulnerabilities of the financial system,
network driven effects are disentangled from the way defaults are settled. Next, I
will show that for bankruptcy rules that are even strongly monotonic the iterative
procedure suggested by Eisenberg and Noe (2001) is converging in finitely many
steps so that it may be used to calculate the vector of clearing prices. A strongly
monotonic bankruptcy rule sees to it that an agent with a positive claim on a
specific agent is always credited with a minimal but positive fraction of additional
available payment under default. Basically this monotonicity property makes the
iterated mapping contracting, so that on the domain of prices there will be the
one fixed point we are looking for. The monotonicity property is a sufficient con-
dition for the results, though not necessary. As an example I discuss the financial
systems corresponding to the constrained equal award rule, which is not strictly
monotonic, and show that clearing prices may still be unique.
The uniqueness result also has some say in papers that explore other general-
izations of Eisenberg and Noe’s model. Consider for example the model including
defaulting costs by Rogers and Veraart (2013), or the model where financial insti-
tutes reinsure themselves through credit default swaps as in Schuldenzucker et al.
(2016) (see also Elliott et al. (2014)). Also it allows to generalize the characteriza-
tion of Nash equilibria in the 2 stage game proposed by Allouch and Jalloul (2018),
where the players have the choice in the first period to save or invest an amount of
capital. This game is easily generalized to general bankruptcy rules. Uniqueness
of the clearing prices assures that the players do not need to overcome a possible
3
Electronic copy available at: https://ssrn.com/abstract=3427039

coordination problem and the equilibria may be characterized in the way that is
done in Allouch and Jalloul (2018) for the proportional rule. The Nash equilibria
are characterized by the choice in the first period, to default or not. The analysis
for other monotonic bankruptcy rules is similar as the induced games also show
strategic complementarities.
The question of uniqueness of clearing prices is also addressed by Cs´osak and
Herings (2018), who present a discrete model that allows for decentralized clearing
of the financial system. This model accommodates practical situations where it
is hard to retrieve all necessary information or where defaults are not filed simul-
taneously due to timing elements. The authors concentrate on methods used in
practice, which are often a mixture of priority and proportional rules. The authors
also conclude that uniqueness of clearing prices is not guaranteed for the discrete
and decentralized model - and not for the limiting continuous framework that re-
sults from letting the smallest unit of account go to zero. A procedure is discussed
which calculates the smallest vector of clearing prices in finitely many steps for the
discrete model which may not converge in the limiting continuous model. The
result in this paper may be used to study for decentralized pricing schemes in a
continuous setup.
2 The general framework and results
2.1 Mathematical prerequisities
Let R
n
denote the n-dimensional Euclidean vector space. Special vector is the
zero vector 0 with all zero coordinates. Denote the set of all non-negative vectors
by R
n
+
:= {x R
n
: x 0}. Below we will use N = {1, 2, . . . , n} for some
integer n > 1 as notation for a set of agents. With slight abuse of notation we will
sometimes choose to denote R
N
by R
n
. For any two vectors x, y R
n
we define
vectors x y, x y R
n
such that for all i
(x y)
i
:= min{x
i
, y
i
}
(x y)
i
:= max{x
i
, y
i
}
In addition we define x
+
:= x 0 where 0 is the zero vector in R
n
such that 0
i
= 0
for all i. We will write x y iff x
i
y
i
for all i, and x < y if x
i
< y
i
for all i.
Then using this, we define R
n
+
:= {x R
n
: x 0} as the set of all non-negative
vectors, whereas R
n
++
= {x R
n
: x > 0}.
Denote by k · k the `
1
norm on R
n
so that for all x R
n
we have
kxk :=
n
X
i=1
|x
i
|.
4
Electronic copy available at: https://ssrn.com/abstract=3427039

Citations
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Posted Content
TL;DR: This work considers a situation in which agents have mutual claims on each other, summarized in a liability matrix, and analyzes decentralized clearing processes and shows the convergence of any such process in finitely many steps to the least clearing payment matrix.
Abstract: We consider a situation in which agents have mutual claims on each other, summarized in a liability matrix. Agents' assets might be insufficient to satisfy their liabilities leading to defaults. We assume the primitives to be denoted in some unit of account. In case of default, bankruptcy rules are used to specify the way agents are going to be rationed. We present a convenient representation of bankruptcy rules. A clearing payment matrix is a payment matrix consistent with the prevailing bankruptcy rules that satisfies limited liability and priority of creditors. Both clearing payment matrices and the corresponding values of equity are not uniquely determined. We provide bounds on the possible levels equity can take. We analyze decentralized clearing processes and show the convergence of any such process in finitely many steps to the least clearing payment matrix. When the unit of account is sufficiently small, all decentralized clearing processes lead essentially to the same value of equity as a centralized clearing procedure. As a policy implication, it is not necessary to collect and process all the sensitive data of all the agents simultaneously and run a centralized clearing procedure.

29 citations


References
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Journal ArticleDOI
Abstract: 1. A lattice-theoretical fixpoint theorem. In this section we formulate and prove an elementary fixpoint theorem which holds in arbitrary complete lattices. In the following sections we give various applications (and extensions) of this result in the theories of simply ordered sets, real functions, Boolean algebras, as well as in general set theory and topology. * By a lattice we understand as usual a system 21 = (A 9 <) formed by a non-empty set A and a binary relation <; it is assumed that < establishes a partial order in A and that for any two elements a f b E A there is a least upper bound (join) a u b and a greatest lower bound (meet) an b. The relations >L, <, and > are defined in the usual way in terms of <. The lattice 21 = (A, <) is called complete if every subset B of A has a least upper bound ΌB and a greatest lower bound Πβ. Such a lattice has in particular two elements 0 and 1 defined by the formulas 0 = ΓU and 1 = 11,4. Given any two elements a 9 b E A with a < b, we denote by [a 9 b] the interval with the endpoints a and b, that is, the set of all elements x E A for which a < x < b; in symbols, [ a,b] = E x [x E A and a .< x .< b ]. The system \[α,6], <) is clearly a lattice; it is a complete if 21 is complete. We shall consider functions on A to A and, more generally, on a subset B of A to another subset C of A. Such a function / is called increasing if, for any 1 For notions and facts concerning lattices, simply ordered systems, and Boolean algebras consult [l].

2,743 citations


"A note on uniqueness of clearing pr..." refers background in this paper

  • ...Theorem 1 (Tarski (1955)) Let (A,≤) be any complete lattice(3) and suppose f : A → A is monotonically increasing, i....

    [...]

  • ...Also ([0, τ ],≤) is a complete lattice, so that the implication of Tarski’s fixed point theorem (see Tarski (1955)) is that the set of fixed points of Φ is a complete lattice with respect to ≤....

    [...]

  • ...(6) Theorem 1 (Tarski (1955)) Let (A,≤) be any complete lattice3 and suppose f : A → A is monotonically increasing, i.e., for all x, y ∈ A, x ≤ y implies f(x) ≤ f(y)....

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TL;DR: By invoking a metaphor of illness, financial contagion implies an economic disorder, dislocation, or disease that spreads from one infected host to others by some mechanism.
Abstract: The phrase financial contagion draws on a concept whose root meaning lies in the field of epidemiology. Like almost all metaphors, this one has the power to illuminate and to mislead. Its referent is the spread of financial distress from one firm, market, asset class, nation, or geographical region to others. But, contagion carries with it other burdens of meaning. First, to refer to contagion, instead of merely to an epidemic, is to implicitly assert that there is a mechanism of transmission from one infected victim to other potential victims. For example, bubonic plague and malaria may give rise to epidemics, but these diseases are not contagious, being transmitted by the bite of a flea and the sting of a mosquito, rather than being spread fromone infected party to another. By contrast, some epidemics may be the result of truly contagious diseases in which the disease spreads directly from one victim to another through the direct transmittal of a pathogen, such as is the case with tuberculosis and AIDS. Second, because a contagious disease spreads from one infected host to others by some mechanism, the key to understanding such a malady is to comprehend the method of transmission. Finally, by invoking a metaphor of illness, financial contagion implies an economic disorder, dislocation, or disease.

1,558 citations


Journal ArticleDOI
TL;DR: An algorithm is developed that both clears the financial system in a computationally efficient fashion and provides information on the systemic risk faced by the individual system firms and produces qualitative comparative statics for financial systems.
Abstract: We consider default by firms that are part of a single clearing mechanism. The obligations of all firms within the system are determined simultaneously in a fashion consistent with the priority of debt claims and the limited liability of equity. We first show, via a fixed-point argument, that there always exists a "clearing payment vector" that clears the obligations of the members of the clearing system; under mild regularity conditions, this clearing vector is unique. Next, we develop an algorithm that both clears the financial system in a computationally efficient fashion and provides information on the systemic risk faced by the individual system firms. Finally, we produce qualitative comparative statics for financial systems. These comparative statics imply that, in contrast to single-firm results, even unsystematic, nondissipative shocks to the system will lower the total value of the system and may lower the value of the equity of some of the individual system firms.

1,099 citations


"A note on uniqueness of clearing pr..." refers background or methods in this paper

  • ...It is used by Eisenberg and Noe (2001) in order to define settlements after a firm defaults....

    [...]

  • ...(12) This is the sequence that Eisenberg and Noe (2001) in the model with r = r refer to as the fictitious default sequence and the machinery producing the sequence is called the fictitious default algorithm....

    [...]

  • ...In accordance with Eisenberg and Noe (2001) both aforementioned works stress the fact that clearing prices may not be unique – but the resulting allocation is....

    [...]

  • ...The following definition is the Eisenberg and Noe (2001) version, only now for general bankrutpcy rules: Definition 1 A clearing payment vector for the financial system (L, e, r) is a vector p∗ ∈ [0, τ ] that satisfies (a) Limited Liability: p∗i ≤ n∑ j=1 ri(Lj, p ∗ j) + ei (b) Absolute Priority:…...

    [...]

  • ...Eisenberg and Noe (2001) also propose an iterative procedure by which the clearing prices may be calculated, and in this process defaults may occur at different stages mimicking the indirect way financial institutions may be affected by earlier defaults....

    [...]


Journal ArticleDOI
Abstract: We provide a framework for studying the relationship between the financial network architecture and the likelihood of systemic failures due to contagion of counterparty risk. We show that financial contagion exhibits a form of phase transition as interbank connections increase: as long as the magnitude and the number of negative shocks affecting financial institutions are sufficiently small, more "complete" interbank claims enhance the stability of the system. However, beyond a certain point, such interconnections start to serve as a mechanism for propagation of shocks and lead to a more fragile financial system. We also show that, under natural contracting assumptions, financial networks that emerge in equilibrium may be socially inefficient due to the presence of a network externality: even though banks take the effects of their lending, risk-taking and failure on their immediate creditors into account, they do not internalize the consequences of their actions on the rest of the network.

1,009 citations


"A note on uniqueness of clearing pr..." refers background in this paper

  • ...Other measures of financial instability and assessment of systemic risk are found in Elsinger et al. (2006), Acemoglu et al. (2015), Battiston et al. (2012)....

    [...]

  • ...(2006), Acemoglu et al. (2015), Battiston et al. (2012). Crucial assumption in Eisenberg and Noe (2001) is the principle of proportionality; in case of a defaulting node, the corresponding clearing price is shared proportional to the liabilities of the node to the others....

    [...]

  • ...(2006), Acemoglu et al. (2015), Battiston et al. (2012). Crucial assumption in Eisenberg and Noe (2001) is the principle of proportionality; in case of a defaulting node, the corresponding clearing price is shared proportional to the liabilities of the node to the others. Groote-Schaarsberg et al. (2018) and Csóka and Herings (2018) show in a continuous and discrete setting, respectively, that the assumption of proportionality in solving defaulting situa-...

    [...]


Journal ArticleDOI
Abstract: This paper develops an analytical model of contagion in financial networks with arbitrary structure. We explore how the probability and potential impact of contagion is influenced by aggregate and idiosyncratic shocks, changes in network structure, and asset market liquidity. Our findings suggest that financial systems exhibit a robust-yet-fragile tendency: while the probability of contagion may be low, the effects can be extremely widespread when problems occur. And we suggest why the resilience of the system in withstanding fairly large shocks prior to 2007 should not have been taken as a reliable guide to its future robustness.

1,000 citations


"A note on uniqueness of clearing pr..." refers background in this paper

  • ...See for instance the overview of Glasserman and Young (2016) or Caccioli et al. (2018) which try to disentangle the problem by discussing various ways correlations between nodes in the financial system play a role....

    [...]