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Unformatted text preview: o = i E [Xi ]. By the assumption of homogeneity in assets we know that E [Xi ] ∝ ci for all i. We conclude from this and (59) that
¯¯
L/Lo ≤ 1 + (1 − δi ci
+)
β ci . This completes the proof of Theorem 3.
Theorem 3 shows that the increase in losses due to network interconnections will be very small unless
β + is close to 1 or the default rates of some banks (weighted by their outside asset base) is large.
Moreover this statement holds even when the shocks are dependent, say due to common exposures, and
it holds independently of the network structure.
Some might argue that writedowns of purely ﬁnancial obligations should not be counted as part of
the systemic loss. This is tantamount to truncating node depth at 1 and thus leads to an even smaller
upper bound in Theorem 3.
To illustrate the result concretely, consider the EBA data. In this case β + = max{βi } = 0.43 where
the maximum occurs for Dexia bank. Regulatory restrictions on assets imply that any given bank is
unlikely to fail due to direct default; indeed current regulatory standards seek to make the direct default
probability of any individual bank smaller than 0.1% per year. Let us make the more conservative
assumption that the probability of default is at most 1%, that is, max{δi } < 0.01. If this standard is
met, then no matter what the interconnections might be among the banks in the data set, we conclude 22 from Theorem 3 that the additional expected loss attributable to the network is at most .01/(1 − .43),
which is about 1.7%. In other words the actual network of connections cannot increase the expected
losses by very much under the assumptions of the theorem. 5 Bankruptcy costs In this section and the next we enrich the basic framework by incorporating additional mechanisms
through which losses propagate from one node to another. We begin by adding bankruptcy costs. The
equilibrium condition (3) implicitly assumes that if node i’s assets fall short of its liabilities by 1 unit,
then the total claims on node i are simply marked down by 1 unit below the face value of pi . In practice,
¯
the insolvency of node i is likely to produce deadweight losses that have a knockon eﬀect on the shortfall
at node i and at other nodes.
Cifuentes, Ferrucci, and Shin (2005), Battiston et al. (2012), Cont, Moussa, and Santos (2010),
and Rogers and Veraart (2012) all use some form of liquidation cost or recovery rate at default in their
analyses. Cifuentes, Ferrucci, and Shin (2005) distinguish between liquid and illiquid assets and introduce
an external demand function to determine a recovery rate on illiquid assets. Elliott, Golub, and Jackson
(2013) attach a ﬁxed cost to bankruptcy. Elsinger, Lehar, and Summer (2006) present simulation results
illustrating the eﬀect of bankruptcy costs but without an explicit model. The mechanism we use appears
to be the simplest and the closest to the original EisenbergNoe setting, which facilitates the analysis of
its impact on contagion. 5.1 Shortf...
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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
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