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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 “heavytailed” 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.
 Spring '11
 Traferri

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