2 26 value of liabilities r z pi 1 1 k pi

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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 distributed shocks. 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 ; ¯ p 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 Section 2.2. 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 quality. The effect of credit quality deterioration begins at a much higher asset level of z = (1 + k )¯i than p 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 value...
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This document was uploaded on 02/20/2014 for the course ECON 101 at Pontificia Universidad Católica de Chile.

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