Let us assume that the losses at a given node i scale

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Unformatted text preview: n the distribution of shocks. The notion that contagion from i to D is weak, described informally in Section 1, can now be made precise by the condition P (Xi ≥ wi + (1/βi ) wj ) ≤ P (Xi > wi ) j ∈D P (Xj > wj ). (16) j ∈D The expression on the left bounds the probability that these nodes default solely through contagion from i, while the expression on the right is the probability that the nodes in D default through independent direct shocks. Contagion is weak if the latter probability is at least as large as the former. The assumption of independent direct shocks is somewhat unrealistic: in practice one would expect the shocks to different nodes to be positively associated. In this case, however, the probability of default from direct shocks is even larger, hence weak contagion covers this situation as well. Let us assume that the losses at a given node i scale with the size of the portfolio ci . Let us also assume that the distribution of these relative losses is the same for all nodes, and independent among nodes. Then there exists a distribution function H : [0, 1] → [0, 1] such that F (x1 , ..., xn ) = H (xi /ci ). (17) 1≤i≤n Beta distributions provide a flexible family with which to model the distribution of shocks as a fraction of outside assets. We work with beta densities of form hp,q (y ) = y p−1 (1 − y )q−1 , B (p, q ) 0 ≤ y ≤ 1, p, q ≥ 1, (18) where B (p, q ) is a normalizing constant. The subset with p = 1 and q > 1 has a decreasing density and seems the most realistic, but (18) is general enough to allow a mode anywhere in the unit interval. The case q = 1, p > 1 has an increasing density and could be considered “heavy-tailed” in the sense that it assigns greater probability to greater losses, with losses capped at 100 percent of outside assets.9 Theorem 1. Assume the shocks are i.i.d. beta distributed as in (18) 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) (19) j ∈D and it is weak if wj ≥ wi βi j ∈D (λi − 1)/λj . (20) j ∈D 9 Bank capital requirements under Basel II and III standards rely on a family of loss distributions derived from a Gaussian copula model. As noted by Tasche (2008) and others, these distributions can be closely approximated by beta distributions. 11 As noted after Proposition 1, the condition in (19) 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 and the vulnerability of i as measured by its leverage. The condition in (20) compares the total net worth of D relative to that of i with the leverage of i relative to that of the nodes in D. With other parameters held constant, increasing the relative net worth of D makes contagion weaker in the sense that it strengthens the inequality; increasing the leverage of i relati...
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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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