The rst statement follows from applying 39 to both

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Unformatted text preview: al nodes and a shock greater than 10 at the central node, which has probability exp(−30µ) given i.i.d. exponential shocks. If the primary shocks have a Pareto-like tail, meaning that P (Xi > x) ∼ ax−µ (39) for some positive constants a and µ (or, more generally, a regularly varying tail), then the probability that a single shock will exceed j ∈D wj /c will be greater than the probability that the nodes in D default through multiple independent shocks, at least at large levels of the wj . However, introducing some dependence can offset this effect, as we now illustrate. To focus on the issue at hand, we take c = 1 and βi = 1. To consider a specific and relatively simple case, let Y1 , . . . , Ym be independent random variables, ˜ ˜ each distributed as tν , the Student t distribution with ν > 2 degrees of freedom. Let Y1 , . . . , Ym have ˜ a standard multivariate Student t distribution with tν marginals.10 The Yj are uncorrelated but not ˜ ˜ ˜ independent. To make the shocks positive, set Xj = Yj2 and Xj = Yj2 . Each Xj and Xj has a Pareto-like tail that decays with a power of ν/2. Proposition 2. With independent shocks Xj , m m wj ) ≥ P (Xi > j =1 P (Xj > wj ) j =1 p ˜ ˜ explicitly, (Y1 , ..., Ym ) has the distribution of (Z1 , ..., Zm )/ χ2 /ν , where the Zi are independent standard normal ν 2 has a chi-square distribution with v degrees of freedom and is independent of the Z . random variables and χv i 10 More 17 for all sufficiently large wj , j = 1, . . . , m. With dependent shocks m ˜ P (Xi > ˜ wj ) ≤ P (Xj > wj , j = 1, . . . , m) j =1 for all wj ≥ 0, j = 1, . . . , m. Proof of Proposition 2 . The first statement follows from applying (39) to both sides of the inequality. The second statement is an application of Bound II for the F distribution on p.1196 of Marshall and Olkin (1974). Thus, even with heavy-tailed shocks, we may find that default of a set of nodes through contagion from a single shock is less likely than default through direct shocks to individual nodes if the shocks are dependent. 4 Amplification of losses due to network effects The preceding analysis dealt with the impact of default by a single node (the source) on another set of nodes (the target). Here we shall examine the impact of shocks on the entire system, including multiple and simultaneous defaults. To carry out such an analysis, we need to have a measure of the total systemic impact of a shock. There appears to be no commonly accepted measure of systemic impact in the prior literature. Eisenberg and Noe (2001) suggest that it is the number of waves of default that a given shock induces in the network. Other authors have suggested that the systemic impact should be measured by the aggregate loss of bank capital; see for example Cont, Moussa, and Santos (2010). Still others have proposed the total loss in value of only those nodes external to the financial sector, i.e. firms and households. Here we shall take the s...
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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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