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To illustrate the eﬀect of a shock, we consider the numerical example in Figure 2(a), which follows
the notational conventions of Figure 1. In particular, the central node has a net worth of 10 because it
has 150 in outside assets, 100 in outside liabilities, and 40 in liabilities to other nodes inside the network.
A shock of magnitude 10 to the outside assets erases the central node’s net worth, but leaves it with just
enough assets (140) to fully cover its liabilities. A shock of magnitude 80 leaves the central node with
assets of 70, half the value of its liabilities. Under a pro rata allocation, each liabilitity is cut in half,
so each peripheral node receives a payment of 5, which is just enough to balance each peripheral node’s
assets and liabilities. Thus, in this case, the central node defaults but the peripheral nodes do not. A
shock to the central node’s outside assets greater than 80 would reduce the value of every node’s assets
below the value of its liabilities.
Figure 2(b) provides a more complex version of this example in which a cycle of obligations of size
y runs through the peripheral nodes. To handle such cases, we need the notion of a clearing vector
introduced by Eisenberg and Noe (2001).
The relative liabilities matrix A = (aij ) is the n × n matrix with entries
aij = pij /pi , if pi > 0
¯¯
¯
0,
if pi = 0
¯ (2) Thus aij is the proportion of i’s obligations owed to node j . Since i may also owe entities in the external
sector, j =i aij ≤ 1 for each i, that is, A is row substochastic.6 n
Given a shock realization x = (x1 , x2 , ..., xn ) ≥ 0, a clearing vector p(x) ∈ R+ is a solution to the system
pj (x)aji + ci − xi )+ . pi (x) = pi ∧ (
¯
j
6 The row sums are all equal to 1 in Eisenberg and Noe (2001) because bi = 0 in their formulation. 6 (3) As we shall subsequently show, the clearing vector is unique if the following condition holds: from every
node i there exists a chain of positive obligations to some node k that has positive obligations to the
external sector. (This amounts to saying that A has spectral radius less than 1.) We shall assume that
this condition holds throughout the remainder of the paper. 2.2 Marktomarket values The usual way of interpreting p(x) is that it corresponds to the payments that balance the realized assets
and liabilities at each node given that: i) debts take precedence over equity; and ii) all debts at a given
node are written down pro rata when the net assets at that node (given the payments from others), is
insuﬃcient to meet its obligations. In the latter case the node is in default, and the default set is
D(p(x)) = {i : pi (x) < pi (x)}.
¯ (4) However, a second (and, in our setting, preferable) way of interpreting p(x) is to see (3) as a marktomarket valuation of all assets following a shock to the system. The nominal value pi of node i’s liabilities
¯
is marked down to pi (x) as a consequence of the shock x, including its impact on other nodes. As in our
discussion above of Figure 2(a), aft...
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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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