The magnitude of the node depths in a default set can

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Unformatted text preview: rationale for this terminology is as follows. Consider a Markov chain on D with transition matrix AD . For each i ∈ D, ui is the expected 19 number of periods before exiting D, starting from node i.11 Expression (46) shows that the node depths measure the amplification of losses due to interconnections among nodes in the default set. We remark that the concept of node depth is dual to the notion of eigenvector centrality in the networks literature (see for example Newman 2010). To see the connection let us restart the Markov chain uniformly in D whenever it exits D. This modified chain has an ergodic distribution proportional to 1D · [ID + AD + A2 + ...], and its ergodic distribution measures the centrality of the nodes in D. It D follows that node depth with respect to AD corresponds to centrality with respect to the transpose of AD . Although they are related algebraically, the two concepts are quite different. To see why let us return to the example of Figure 2(b). Suppose that node 1 (the central node) suffers a shock x1 > 80. This causes all nodes to default, that is, the default set is D = {1, 2, 3, 4, 5}. Consider any node j > 1. In the Markov chain described above the expected waiting time to exit the set D , starting from node j , is given by the recursion uj = 1 + βj uj , which implies uj = 1/(1 − βj ) = 1 + y/55. (48) From node 1 the expected waiting time satisfies the recursion u1 = 1(100/140) + (1 + uj )(40/140). (49) u1 = 9/7 + 2y/385. (50) Hence Comparing (48) and (50) we find that node 1 is deeper than the other nodes (u1 > uj ) for 0 ≤ y < 22 and shallower than the other nodes for y > 22. In contrast, node 1 has lower eigenvector centrality than the other nodes for all y ≥ 0 because it cannot be reached directly from any other node. The magnitude of the node depths in a default set can be bounded as follows. In the social networks literature a set D is said to be α-cohesive if every node in D has at least α of its obligations to other nodes in D, that is, j ∈D aij ≥ α for every i ∈ D (Morris, 2000). The cohesiveness of D is the maximum such α, which we shall denote by αD . From (46) it follows that ∀i ∈ D, ui ≥ 1/(1 − αD ). (51) Thus the more cohesive the default set, the greater the depth of the nodes in the default set and the greater the amplification of the associated shock. Similarly we can bound the node depths from above. Recall that βi is the proportion of i’s obligations to other nodes in the financial system. Letting βD = max{βi : i ∈ D} we obtain the upper bound, 11 Liu and Staum (2012) show that the node depths can be used to characterize the gradient of the clearing vector p(x) with respect to the asset values. 20 assuming βD < 1, ∀i ∈ D, ui ≤ 1/(1 − βD ) . (52) The bounds in (51) and (52) depend on the default set D, which depends on the shock x. A uniform upper bound is given by ∀i, ui ≤ 1/(1 − β + ) where β + = max βi , (53) assuming β + < 1. We are now in a position to compare the...
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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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