OFRwp0009_GlassermanYoung_HowLikelyContagionFinancialNetworks

OFRwp0009_GlassermanYoung_

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Unformatted text preview: Demange (2012) and Acemoglu, Ozdaglar, and Tahbaz-Salehi (2013) generate greater contagion by making debts to financial institutions subordinate to other payment obligations. With priority given to outside payments, shocks produce greater losses within the network. In practice, bank debt and bank deposits are owned by both financial institutions and non-financial firms and individuals, so characterizing seniority based on the type of lender is problematic. 3.3 Contagion with truncated shocks In this section we shall show that the preceding results are not an artifact of the beta distribution: similar bounds hold for a variety of shock distributions. Under the beta distribution the probability is zero that a node loses all of its outside assets. One could easily imagine, however, that the probability o of this event is positive. This situation can be modeled as follows. Let Xi ≥ 0 be a primary shock 14 o (potentially unbounded in size) and let Xi = ci ∧ Xi be the resulting loss to i’s outside assets. For o example Xi might represent a loss of income from an employment shock that completely wipes out i’s outside assets. Assume that the primary shocks have a joint distribution function of form F o (xo , ..., xo ) = 1 n H o (xo /ci ), i (29) 1≤i≤n where H o is a distribution function on the nonnegative real line. In other words we assume that the shocks are i.i.d. and that a given shock xo affects every dollar of outside assets ci equally. (We shall i consider a case of dependent shocks after the next result.) In general a random variable with distribution function G and density g is said to have an increasing failure rate (IFR) distribution if g (x)/(1 − G(x)) is an increasing function of x. Examples of IFR distributions include all normal, exponential, and uniform distributions and, more generally, all logconcave distributions. Observe that truncating the shock can put mass at ci and thus assign positive probability to a total loss of outside assets. Theorem 2. Assume the primary shocks are i.i.d. and IFR-distributed, and that the net worth of every node is initially nonnegative. Let D be a nonempty subset of nodes and let i ∈ D. Contagion from i to / D is impossible if wj > wi βi (λi − 1) j ∈D and it is weak if wj > wi βi j ∈D λi /λj . (30) j ∈D Corollary 3. Assume that all nodes hold the same amount of outside assets ci ≡ c. Under the assumptions of Theorem 2, contagion is weak from any node to any other set of nodes. This is immediate upon rewriting (30) as cj λ−1 / j j ∈D Proof of Theorem 2 . λ−1 ≥ βi ci . j j ∈D Through relabeling, we can assume that the source node for contagion is i = 1 and that the infected nodes are D = {2, 3, . . . , m}. By Proposition 1 we know that contagion is weak from 1 to D if wj ) ≤ P (X1 > w1 )P (X2 > w2 ) · · · P (Xm > wm ). P (X1 > w1 + (1/β1 ) 2≤j ≤m 15 (31) o Since X1 = c1 ∧ X1 , the left-hand side is zero when w1 + (1/β1 ) 2≤j ≤m w...
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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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