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Unformatted text preview: er marking-to-market, node i’s net worth is reduced from (1) to
ci − xi + pji (x) − pi (x). (5) j =i The reduction in net worth reﬂects both the direct eﬀect of the shock component xi and the indirect
eﬀects of the full shock vector x. Note, however, that this is a statement about values ; it does not require
that the payments actually be made at the end of the period. Under this interpretation p(x) provides a
consistent re-valuation of the assets and liabilities of all the nodes when a shock x occurs.
As shown by Eisenberg and Noe, a solution to (3) can be constructed iteratively as follows. Given a
realized shock vector x deﬁne the mapping Φ : R+ → R+ as follows: pj aji + ci − xi )+ . ∀i, Φi (p) = pi ∧ (
¯ (6) j Starting with p0 = p let
p1 = Φ(p0 ), p2 = Φ(p1 ), ... (7) This iteration yields a monotone decreasing sequence p0 ≥ p1 ≥ p2 ... . Since it is bounded below it has
a limit p , and since Φ is continuous p satisﬁes (3). Hence it is a clearing vector.
We claim that p is in fact the only solution to (3). Suppose by way of contradiction that there is
another clearing vector, say p = p . As shown by Eisenberg and Noe, the equity values of all nodes
must be the same under the two vectors, that is,
p A + (c − x) − p = p A + (c − x) − p .
7 Rearranging terms it follows that
(p − p )A = p − p , where p − p = 0.
This means that the matrix A has eigenvalue 1, which is impossible because under our assumptions A
has spectral radius less than 1. 3 Estimating the probability of contagion Systemic risk can be usefully decomposed into two components: i) the probability that a given set of
nodes D will default and ii) the loss in value conditional on D being the default set. This decomposition
allows us to distinguish between two distinct phenomena: contagion and ampliﬁcation. Contagion occurs
when defaults by some nodes trigger defaults by other nodes through a domino eﬀect. Ampliﬁcation
occurs when contagion stops but the losses among defaulting nodes keep escalating because of their
indebtedness to one another. Roughly speaking the ﬁrst eﬀect corresponds to a “widening” of the crisis
whereas the second corresponds to a “deepening” of the crisis. In this section we shall examine the
probability of contagion; the next section deals with the ampliﬁcation of losses due to network eﬀects.
To estimate the probability of contagion we shall obviously need to make assumptions about the
distribution of shocks. We claim, however, that we can estimate the relative probability of contagion
versus simultaneous default with virtually no information about the network structure and relatively
weak conditions on the shock distribution.
To formulate our results we shall need the following notation. Let βi = pi /(bi + pi ) be the proportion
of i’s liabilities to other entities in the ﬁnancial system.7 We can assume that βi > 0 , since otherwise
node i would eﬀectively be outside the ﬁnancial system. Recall that wi is i’s initial net worth (before a
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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