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Unformatted text preview: hat no nodes are in default before the shock
¯
and the fact that we must have D ⊆ D − {i} for all nodes in D to default.
For the ﬁrst inequality in (10), deﬁne the shortfall at node j to be the diﬀerence sj = pj − pj . From
¯
(3) we see that the vector of shortfalls s satisﬁes
s = (sA − w + X )+ ∧ p.
¯
¯
¯
By (4) we have sj > 0 for j ∈ D and sj = 0 otherwise. We use a subscript D as in sD or AD to restrict
¯
¯
¯
a vector or matrix to the entries corresponding to nodes in the set D. Then the vector of shortfalls at
9 ¯
the nodes of D satisﬁes
sD ≤ sD AD − wD + XD ,
¯
¯
¯
¯
¯ (11) XD − wD ≥ sD (ID − AD ).
¯
¯
¯
¯
¯ (12) hence The vector sD is strictly positive in every coordinate. From the deﬁnition of βj we also know that the
¯
j th row sum of ID − AD is at least 1 − βj . Hence,
¯
¯
sj (1 − βj ) ≥ si (1 − βi ). sD (ID − AD ) · 1D ≥
¯
¯
¯
¯ (13) ¯
j ∈D From (11) it follows that the shortfall at node i is at least as large as the initial amount by which i
defaults, that is,
si ≥ Xi − wi > 0. (14) From (12)–(14) we conclude that
(Xj − wj ) = Xi − wi −
¯
j ∈D wj ≥ si (1 − βi ) ≥ (Xi − wi )(1 − βi ). (15) ¯
j ∈D −{i} This establishes (10) and the ﬁrst statement of the proposition. The second statement follows from
the ﬁrst by recalling that the shock to the outside assets cannot exceed their value, that is, Xi ≤ ci .
Therefore by (8) the probability of contagion is zero if ci < wi + (1/βi )
wi we see that this is equivalent to the condition j ∈D j ∈D wj . Dividing through by wj /wi > βi (λi − 1). The preceding proposition relates the probability of contagion from a given node i to the net worth
of the defaulting nodes in D relative to i’s net worth. The bounds are completely general and do not
depend on the distribution of shocks or on the topology of the network. The critical parameters are βi ,
the degree to which the triggering node is indebted to the ﬁnancial sector, and λi , the degree of leverage
of i’s outside assets.
The Appendix gives estimates of these parameters for large European banks, based on data from
stress tests conducted by the European Banking Authority (2011). Among the 50 largest of these banks
the average of the λi is 24.9, the average of our estimated βi is 14.9%, and the average of the products
βi (λi − 1) is 3.2. Proposition 1 implies that contagion from a “typical” bank i cannot topple a set of
banks D if the net worth of the latter is more than 3.2 times the net worth of the former, unless there
are additional channels of contagion.8
8 On average, commercial banks in the United States are leveraged only about half as much as European banks, and
their values of βi are somewhat smaller (Federal Reserve Release H.8, Assets and Liabilities of Commercial Banks in the
United States, 2012). This suggests that contagion is even less likely in the US ﬁnancial sector than in Europe. 10 3.2 Contagion with proportional shocks We can say a good deal more if we impose some structure o...
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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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