Unformatted text preview: ve to that of nodes in D has the opposite eﬀect. Importantly, the two eﬀects are mediated by βi ,
which measures the exposure of the ﬁnancial system to node i — a lower βi makes D less vulnerable
to i and makes D less sensitive to the degree of leverage at i. Thus, (20) captures the eﬀects of equity
levels, leverage ratios, and the degree of reliance on interbank lending on the risk of contagion.
By recalling that λj = cj /wj we can rewrite (20) in the equivalent form
cj λ−1 /
j
j ∈D λ−1 ≥ ci βi (1 − λ−1 ).
j
i (21) j ∈D Written this way, the condition states that contagion from i to D is weak if the average size of the nodes
in D weighted by their inverse leverage ratios (their capital ratios) is suﬃciently large relative to i; on
the right side of the inequality, ci βi measures the ﬁnancial system’s exposure to node i’s outside assets,
and the factor (1 − λ−1 ) is greater when node i is more highly leveraged. Inequality (21) is thus harder
i
to satisfy, and D more vulnerable to contagion from i, if the large nodes in D are more highly leveraged,
if node i draws more of its funding from the ﬁnancial system, or if node i is more highly leveraged.
Through (21), a key implication of Theorem 1 is that without some heterogeneity, contagion will be
weak irrespective of the structure of the interbank network:
Corollary 1. Assume that all nodes hold the same amount of outside assets ck ≡ c. Under the assumptions of Theorem 1, contagion is weak from any node to any other set of nodes.
Proof. This follows from the fact that βi (1 − λ−1 ) < 1; hence, if ck ≡ c, inequality (21) holds for all i
i
and D.
In the Appendix we apply our framework to the 50 largest banks in the stress test data from the
European Banking Authority. It turns out that contagion is weak in a wide variety of scenarios. In
particular, we analyze the scenario in which one of the ﬁve largest European banks (as measured by
assets) topples two other banks in the top 50. We ﬁnd that the probability of such an event is less than
the probability of direct default unless the two toppled banks are near the bottom of the list of 50.
In the example of Figure 2(a), with node i the central node, the left side of (20) evaluates to 20, and
the right side evaluates to 16 because βi = 2/7, λi = 15, and each of the peripheral nodes has λj = 10.
Thus, contagion is weak. 12 Proof of Theorem 1. Proposition 1 implies that contagion is weak from i to D if
P (Xi ≥ wi + (1/βi ) wj ) ≤ P (Xi > wi )
j ∈D P (Xj > wj ). (22) j ∈D On the one hand this certainly holds if wi + (1/βi ) j ∈D wj > ci , for then contagion is impossible. In this case we obtain, as in (9),
wj /wi > βi (λi − 1). (23) j ∈D Suppose on the other hand that (wi + (1/βi ) j ∈D wj ) ≤ ci . By assumption the relative shocks Xk /ck are independent and beta distributed as in (18). In the uniform case p = q = 1, (22) is equivalent to [1 − (wi /ci + (1/βi ci ) wj )] ≤ (1 − wi /ci )
j ∈D (1 − wj /cj ), (24) j ∈D We...
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 Spring '11
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