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Unformatted text preview: claim that (24) implies (22) for the full family of beta distributions in (19). To see why, ﬁrst observe
that the cumulative distribution Hp,q of hp,q satisﬁes
1 − Hp,q (y ) = Hq,p (1 − y ).
Hence (22) holds if
Hq,p (1 − wi /ci − (1/βi ci ) wj ) ≤ Hq,p (1 − wi /ci )
j ∈D Hq,p (1 − wj /cj ). (25) j ∈D But (25) follows from (24) because beta distributions with p, q ≥ 1 have the submultiplicative property
Hq,p (xy ) ≤ Hq,p (x)Hq,p (y ), x, y ∈ [0, 1]. (See Proposition 4.1.2 of Wirch 1999; the application there has q ≤ 1, but the proof remains valid for
q ≥ 1. The inequality can also be derived from Corollary 1 of Ramos Romero and Sordo Diaz 2001.) It
therefore suﬃces to establish (25), which is equivalent to
wj ≥ (1 − wi /ci )(1 − (1/βi ci )
j ∈D (1 − wj /cj )). (26) j ∈D Given any real numbers θj ∈ [0, 1] we have the inequality
(1 − θj ) ≥ 1 − θj . (27) j j Hence a suﬃcient condition for (26) to hold is that
wj ≥ (1 − wi /ci )( (1/βi ci )
j ∈D j ∈D 13 wj /cj ). (28) After rearranging terms and using the fact that λk = ck /wk for all k, we obtain (20). This concludes
the proof of Theorem 1.
From the argument following (25), it is evident that the same result holds if the shocks to each node
j are distributed with parameters pj, qj in (18) with pi ≤ minj ∈D pj and qi ≥ maxj ∈D qj .
As a further illustration of Theorem 1, suppose the nodes in D are numbered 2, . . . , m and suppose
node i = 1 receives a shock. Further suppose the outside assets are ordered c1 ≥ c2 ≥ · · · ≥ cm ; because
shocks are proportional to outside assets, the assumption that the node with the largest ck receives the
shock maximizes the chances of contagion to the other nodes.
Corollary 2. If c1 ≥ c2 ≥ · · · ≥ cm , then contagion from node 1 to nodes 2, . . . , m is weak if c2 ≥
β1 (c1 − w1 ) and cj ≥ (cj −1 − wj −1 ), j = 2, . . . , m. Contagion is impossible if, in addition, c2 − cm + wm >
β1 (c1 − w1 ).
This is a direct consequence of (24) and (26), hence we omit the details and comment on the interpretation. The lower bounds on the cj ensure that the potential spillovers from other nodes cannot
push the full set of nodes into default regardless of the network topology. Viewing the conditions in the
corollary as lower bounds on the wj suggests minimum capital requirements to ensure the resilience of
the system, based on the relative sizes of banks.
Of course these results do not say that the network structure has no eﬀect on the probability of
contagion; indeed there is a considerable literature showing that it does (see among others Haldane
and May 2011, Gai and Kapadia 2011, Georg, 2011). Rather it shows that in quite a few situations
the probability of contagion will be lower than the probability of direct default, absent some channel of
contagion beyond spillovers through payment obligations. We have already mentioned bankruptcy costs,
ﬁre sales, and mark-to-market losses as amplifying mechanisms. The models of...
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- Spring '11