31 a general bound on the probability of contagion

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Unformatted text preview: , and ci is the initial value of its outside assets. The shock Xi to node i takes values in [0, ci ]. We shall assume that wi > 0, since otherwise i would already be insolvent. We shall also assume that wi ≤ ci , since otherwise i could never default directly through losses in its own outside assets. Define the ratio λi = ci /wi ≥ 1 to be the leverage of i’s outside assets. (This is not the same as i’s overall leverage, which in our terminology is the ratio of i’s total assets to i’s net worth.) 3.1 A general bound on the probability of contagion Proposition 1. Suppose that only node i receives a shock, so that Xj = 0 for all j = i. Suppose that no nodes are in default before the shock. Fix a set of nodes D not containing i. The probability that the 7 Elliott, Golub, and Jackson (2013) have a similar measure which they call the level of integration. More broadly, Shin (2012) discusses the reliance of banks on wholesale funding as a contributor to financial crises, and βi measures the degree of this reliance in our setting. 8 shock causes all nodes in D to default is at most P (Xi ≥ wi + (1/βi ) wj ). (8) j ∈D Moreover, contagion from i to D is impossible if wj /wi > βi (λi − 1). (9) j ∈D The condition in (9) states that contagion from i to D is impossible if the total net worth of the nodes in D is sufficiently large relative to the net worth of i weighted by the exposure of the financial system to node i, as measured by βi , and the vulnerability of i as measured by its leverage. A similar interpretation applies to (8). Before proceeding to the proof, we illustrate the impossibility condition in (9) through the network in Figure 2(a). The central node is node i, meaning that the shock affects its outside assets, and the remaining nodes comprise D. The relevant parameter values are βi = 2/7, λi = 15, and the net worths are as indicated in the figure. The left side of (9) evaluates to 2 and the right side to 4, so the condition is violated, and, indeed, we saw earlier that contagion is possible with a shock greater than 80. However, a modification of the network that raises the sum of the net worths of the peripheral nodes above 40 makes contagion impossible. This holds, for example, if the outside liabilities of every peripheral node are reduced by more than 5, or if the outside liabilities of a single peripheral node are reduced by more than 20. This example also illustrates that (9) is tight in the sense that if the reverse strict inequality holds, then contagion is possible in this example with a sufficiently large shock. ¯ Proof of Proposition 1. Let D(x) ≡ D be the default set resulting from the shock vector X, whose coordinates are all zero except for Xi . By assumption i causes other nodes to default, hence i itself must ¯ default, that is, i ∈ D. To prove (8) it suffices to show that βi (Xi − wi ) ≥ wj ≥ ¯ j ∈D −{i} wj . (10) j ∈D The second inequality in (10) follows from the assumption t...
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