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Unformatted text preview: assume i.i.d.
proportional shocks to the outside assets of all nodes, then all nodes default directly with probability
[(1 − w1 /c1 ) · · · (1 − w5 /c5 )]q = [(14/15) · 0.94 ]q ≈ 0.61q > (7/15)q .
Thus contagion is weak.
Now introduce a bankruptcy cost factor of γ = 0.5 and consider a shock of 57 or larger to the
central node. The shock creates a shortfall of at least 47 at the central node that gets magniﬁed by
50% to a shortfall of at least 70.5 after bankruptcy costs. The central node’s total liabilities are 140,
so the result is that each peripheral node receives less than half of what it is due from the central
node, and this is suﬃcient to push every peripheral node into default. Thus, with bankruptcy costs, the
probability of contagion is at least (1 − (57/150))q = 0.62q , which is now greater than the probability of
direct defaults through independent shocks. A similar comparison holds with truncated exponentially
This example illustrates how bankruptcy costs increase the probability of contagion. However, it is
noteworthy that γ needs to be quite large to overcome the eﬀect of weak contagion. 6 Conﬁdence, credit quality and mark-to-market losses In the previous section, we demonstrated how bankruptcy costs can amplify losses. In fact, a borrower’s
deteriorating credit quality can create mark-to-market losses for a lender well before the point of default.
Indeed, by some estimates, these types of losses substantially exceeded losses from outright default during
2007-2009. We introduce a mechanism for adding this feature to a network model and show that it too
magniﬁes contagion. See Harris, Herz, and Nissim (2012) for a broad discussion of accounting practices
as sources of systemic risk.
The mechanism we introduce is illustrated in Figure 3, which shows how the value of node i’s total
liabilities changes with the level z of node i’s assets. The ﬁgure shows the special case of a piecewise
linear relationship. More generally, let r(z ) be the reduced value of liabilities at a node as a function of
asset level z, where r(z ) is increasing, continuous, and
0 ≤ r(z ) ≤ pi , for z < (1 + k )¯i ;
r(z ) = pi ,
¯ for z ≥ (1 + k )¯i .
p Let R(z1 , . . . , zn ) = (r(z1 ), . . . , r(zn )). Given a shock x = (x1 , . . . , xn ), the clearing vector p(z ) solves
p(x) = R(c + p(x)A + w − x).
Our conditions on r ensure the existence of such a ﬁxed point by an argument similar to the one in
26 Value of Liabilities r ( z ) pi
[1 − η (1 + k )] pi + η z −(γ + η k ) pi + (1 + γ ) z (1 + k ) pi pi Assets z Figure 3: Liability value as a function of asset level in the presence of bankruptcy costs and credit
The eﬀect of credit quality deterioration begins at a much higher asset level of z = (1 + k )¯i than
does default. Think of k as measuring a capital cushion: node i’s credit quality is impaired once its
net worth (the diﬀerence between its assets and liabilities) falls below the cushion. At this point, the
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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