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 Paretolike 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 Paretolike
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 chisquare 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 heavytailed 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.
 Spring '11
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