Under this interpretation px provides a consistent re

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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 n n 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 shock hits)...
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This document was uploaded on 02/20/2014 for the course ECON 101 at Pontificia Universidad Católica de Chile.

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