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Unformatted text preview: of i’s liabilities begins to decrease, reﬂecting the marktomarket impact of i’s deteriorating credit
quality.
In the example of Figure 2(a), suppose the central node is at its minimum capital cushion before
experiencing a shock; in other words (1 + k )¯i = 150, which implies that k = 1/14. A shock of 5 to the
p
central node’s outside assets reduces the total value of its liabilities by 5η , so each peripheral node incurs
a marktomarket loss of 5η/14. In contrast, in the original model with η = 0, the peripheral nodes do
not experience a loss unless the shock to the central node exceeds 10.
As this example illustrates, marktomarket losses from credit deterioration and bankruptcy costs
operate in qualitatively diﬀerent ways, even though both increase overall losses. Bankruptcy costs
amplify the propagation of shortfall. In contrast, accounting for credit quality eﬀectively increases the
linkages between nodes. It does so by propagating losses at higher levels of asset values, rather than
by amplifying the losses as they are propagated. Shocks can also be propagated through prices due to
common exposures or ﬁre sales that go beyond the network of direct obligations, as in Allen and Gale
(2000), Cifuentes, Ferrucci, and Shin (2005), and Caccioli et al. (2012). 7 Concluding Remarks In this paper we have argued that it is relatively diﬃcult to generate contagion solely through spillover
losses in a network of payment obligations. For contagion to occur, a shock to one node must lead to
losses at another set of nodes suﬃciently large to wipe out their initial equity (or net worth). This means
that either the initial shock must be large (and therefore improbable) relative to the net worth of the
infecting node, or that the net worth of the infected nodes must be small. More generally, the probability 27 of contagion depends on the comparative net worths of the various nodes, their level of leverage, and
the extent to which the infecting node has obligations to the rest of the ﬁnancial sector. As Theorems
1 and 2 show, one can compare the probability of contagion and direct default under a wide range of
shock distributions without knowing the topological details of the network.
The network structure matters more for the ampliﬁcation eﬀect, in which losses among defaulting
nodes multiply because of their obligations to one another. The degree of ampliﬁcation is captured by
the concept of node depth, which is the expected number of periods it takes to exit from the default set
from a given starting point. This clearly depends on the speciﬁc structure of the interbank obligations,
and is dual to the concept of eigenvector centrality in the networks literature. As Theorem 3 shows one
can place an upper bound on the ampliﬁcation eﬀect based on banks’ maximum connectedness with the
rest of the ﬁnancial system (the parameter β + ) without knowing any details of the connections within
the system.
The network structure...
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This document was uploaded on 02/20/2014 for the course ECON 101 at Pontificia Universidad Católica de Chile.
 Spring '11
 Traferri

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