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Unformatted text preview: ptcy costs expand the set of defaults (i.e., increase contagion) resulting from a given shock realization x while otherwise leaving the basic structure of the model unchanged.
In what follows, we examine the amplifying eﬀect of bankruptcy costs conditional on a default set D,
and we then compare costs with and without network eﬀects. The factor of 1 + γ already points to the
amplifying eﬀect of bankruptcy costs, but we can take the analysis further.12
Suppose, for simplicity, that the maximum shortfall of pi in (62) is not binding on any of the nodes
¯
in the default set. In other words, the shocks are large enough to generate defaults, but not so large as
12 A corresponding comparison is possible using partial recoveries at default, as in Rogers and Veraart (2012). This leads
to qualitatively similar results, provided claims on other banks are kept to a realistic fraction of a bank’s total assets. 24 to entirely wipe out asset value at any node. In this case, we have
sD = (1 + γ )[sD AD − wD + xD ].
If we further assume that ID − (1 + γ )AD is invertible, then
sD = (1 + γ )(xD − wD )[ID − (1 + γ )AD ]−1
and the systemic shortfall is
S (x) = sD · uD (γ ) = (1 + γ )(xD − wD ) · uD (γ ), (63) where the modiﬁed node depth vector uD (γ ) is given by [ID − (1 + γ )AD ]−1 · 1D . If (1 + γ )AD has
spectral radius less than 1, then
uD (γ ) = [ID + (1 + γ )AD + (1 + γ )2 A2 + · · · ] · 1D .
D
The representation in (63) reveals two eﬀects from introducing bankruptcy costs. The ﬁrst is an immediate or local impact of multiplying xi − wi by 1 + γ ; every element of sD is positive, so the outer factor
of 1 + γ increases the total shortfall. The second and more important eﬀect is through the increased
node depth. In particular, letting αD denote the cohesiveness of D as before, we can now lowerbound
the depth of each node by 1/[1 − (1 + γ )αD ].13 This makes explicit how bankruptcy costs deepen the
losses at defaulted nodes and increase total losses to the system. By the argument in Theorem 3, we get
the following comparison of losses with and without interconnections:
Corollary 4. In the setting of Theorem 3, if we introduce bankruptcy costs satisfying (1 + γ )β + < 1,
then ¯
δi ci
L
¯ ≤ 1 + [1 − (1 + γ )β + ]
Lo ci . In the EBA data in the appendix, we have β + = 0.43. If we set γ at the rather large value of 0.5 and
continue to assume δi ≤ 0.01, the corollary gives an upper bound of 1.042. In other words, even with
large bankruptcy costs, the additional expected loss attributable to the network is at most 4.2%. 5.2 Example We saw previously that in the example of Figure 2(a) we need a shock greater than 80 to the outside
assets of the central node in order to have contagion to all other nodes. Under a beta distribution
13 If the spectral radius of (1 + γ )A is less than 1, then (1 + γ )αD < 1. 25 with parameters p = 1 and q ≥ 1, this has probability (1 − 80/150)q = (7/15)q . If we...
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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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