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Unformatted text preview: ystemic impact of a shock to be the total loss in value summed over all nodes,
including nodes corresponding to ﬁnancial entities as well as those representing ﬁrms and households.
This measure is easily stated in terms of the model variables. Given a shock realization x, the total
reduction in asset values is
i The term x = i xi + S (x) where S (x) = p
i (¯i − pi (x)). (40) xi is the direct loss in value from reductions in payments by the external sector. The term S (x) is the indirect loss in value from reductions in payments by the nodes to other nodes and
to the external sector. An overall measure of the riskiness of the system is the expected loss in value
L= (x + S (x))dF (x). (41) The question we wish to examine is what proportion of these losses can be attributed to connections
between institutions as opposed to characteristics of individual banks. To analyze this issue let x be
18 a shock and let D = D(x) be the set of nodes that defaults given x. Under our assumptions this set
is unique because the clearing vector is unique. To avoid notational clutter we shall suppress x in the
ensuing discussion.
As in the proof of Propostion 1, deﬁne the shortfall in payments at node i to be si = pi − pi , where
¯
p is the clearing vector. By deﬁnition of D,
si > 0 for all i ∈ D
si = 0 for all i ∈ D
/ (42) Also as in the proof of Proposition 1, let AD be the D × D matrix obtained by restricting the relative
liabilities matrix A to D, and let ID be the D × D identity matrix. Similarly let sD be the vector of
shortfalls si corresponding to the nodes in D, let wD be the corresponding net worth vector deﬁned in
(1), and let xD be the corresponding vector of shocks. The clearing condition (3) implies the following
shortfall equation, provided no node is entirely wiped out — that is, provided si < pi , for all i:
¯
sD AD − (wD − xD ) = sD . (43) Allowing the possibility that some si = pi , the left side is an upper bound on the right side. Recall that
¯
AD is substochastic, that is, every row sum is at most unity. Moreover, by assumption, there exists a
chain of obligations from any given node k to a node having strictly positive obligations to the external
sector. It follows that limk→∞ Ak = 0D , hence ID − AD is invertible and
D
[ID − AD ]−1 = ID + AD + A2 + ....
D (44) From (43) and (44) we conclude that
sD = (xD − wD )[ID + AD + A2 + ...].
D (45) n
Given a shock x with resulting default set D = D(x), deﬁne the vector u(x) ∈ R+ such that uD (x) = [ID + AD + A2 + ...] · 1D , ui (x) = 0 for all i ∈ D.
D (46) Combining (40), (45), and (46) shows that total losses given a shock x can be written in the form
(xi − wi )ui (x). (xi ∧ wi ) + L(x) = (47) i i The ﬁrst term represents the direct losses to equity at each node and the second term represents the
total shortfall in payments summed over all of the nodes. The right side becomes an upper bound on
L(x) if si = pi for some i ∈ D(x).
¯
We call the coeﬃcient ui = ui (x) the depth of node i in D = D(x). The...
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 Spring '11
 Traferri

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