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A note on uniqueness of clearing prices in financial systems
Koster, M.
DOI
10.2139/ssrn.3427039
Publication date
2019
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Submitted manuscript
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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.
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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-
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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
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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