OFRwp0009_GlassermanYoung_HowLikelyContagionFinancialNetworks

Part a follows from the argument in theorem 1 of

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Unformatted text preview: alls with bankruptcy costs In the absence of bankruptcy costs, when a node fails its remaining assets are simply divided among its creditors. To capture costs of bankruptcy that go beyond the immediate reduction in payments, we introduce a multiplier γ ≥ 0 and suppose that upon a node’s failure its assets are further reduced by γ pi − (ci + ¯ pj aji − xi ) , (60) j =i up to a maximum reduction at which the assets are entirely wiped out. This approach is analytically tractable and captures the fact that large shortfalls are considerably more costly than small shortfalls, where the firm nearly escapes bankruptcy. The term in square brackets is the difference between node i’s obligations pi and its remaining assets. This difference measures the severity of the failure, and the ¯ factor γ multiplies the severity to generate the knock-on effect of bankruptcy above and beyond the immediate cost to node i’s creditors. We can think of the expression in (60) as an amount of value destroyed or paid out to a fictitious bankruptcy node upon the failure of node i. 23 The resulting condition for a payment vector replaces (3) with pi = pi ∧ (1 + γ )(ci + ¯ pj aji − xi ) − γ pi . ¯ j =i (61) + Written in terms of shortfalls si = pi − pi , this becomes ¯ sj aji − wi + xi ]+ ∧ pi . ¯ si = (1 + γ )[ (62) j =i Here we see explicitly how the bankruptcy cost factor γ magnifies the shortfalls. Let Φs denote the mapping from the vector s on the right side of (62) to the vector s on the left γ side, and let Φp similarly denote the mapping of p defined by (61). We have the following: γ Proposition 3. (a) For any γ ≥ 0, the mapping Φs is monotone increasing, bounded, and continuous γ n on R+ , and the mapping Φp is monotone decreasing, bounded, and continuous. It follows that Φs has a γ γ ¯ least fixed point s and Φp has a greatest fixed point p = p − s. (b) For any s, Φs (s) is increasing in γ , γ γ and for any p, Φp (p) is decreasing in γ . Consequently, the set of default nodes under the minimal s and γ maximal p is increasing in γ . The fixed point s (and p) is unique if (1 + γ )A has spectral radius less than 1. Proof of Proposition 3. Part (a) follows from the argument in Theorem 1 of Eisenberg and Noe (2001). For part (b), write vi = ci + (pA)i − xi and observe that Φp (p)i = γ vi − γ (¯i − vi ), vi < pi ; p ¯ pi ¯ otherwise, from which the monotonicity in γ follows. The maximal fixed-point is the limit of iterations of Φp γ starting from p by the argument in Section 3 of Eisenberg and Noe (2001). If γ1 ≤ γ2 , then the iterates ¯ of Φp1 are greater than those of Φp2 , so their maximal fixed-points are ordered the same way. But then γ γ the set of nodes i for which pi < pi at the maximal fixed-point for γ1 must be contained within that for ¯ γ2 . The same argument works for Φs . Uniqueness follows as in the case without bankruptcy costs. γ This result confirms that bankru...
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