Abstract
The paper proposes a simple unified model for the diffusion of defaults across financial institutions and presents some measures for evaluating the risk imposed by a bank on the system. So far the standard contagion processes might not incorporate some important features of financial contagion.
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Notes
- 1.
In the following, a \(\tilde{~}\) on variable a means that the variable is random.
- 2.
- 3.
- 4.
The process fits in the class of linear ‘threshold’ models as introduced by Granovetter [9]. More precisely, the linear model assumes a uniform distribution on the threshold, here the \(\tilde{{\mathbf{ z}}}\), and an influence of j on i, here ω ji . A node not yet ‘active’ at step t becomes active in step t if the influence of its neighbors in step t−1 is larger than its threshold, here if the sum of their liabilities, ∑ j∈D ω ji to i is larger than the threshold z i .
- 5.
For closed sets C and C′ and i not in C∩C′, surely z i ≥V i (C) or z i ≥V i (C′). Since both V i (C) and V i (C′) are larger or equal to V i (C∩C′), z i ≥V i (C∩C′): C∩C′ is closed.
- 6.
This assumes that the cost only depends on the set D and not on the precise values of z.
- 7.
Here the expression is similar to some measures of ‘power’ or ‘prestige’ developed in sociology such as the Katz prestige index [11].
- 8.
[14] also consider the loss imposed by a bank to each subsystem and derives the contribution to risk of that bank by taking an average (the Shapley value).
- 9.
Consider for example the problem of finding a subset A that maximizes Φ(A) under some constraints, say the cardinality of A less than a number m. Under sub-modularity, a fast algorithm provides an approximation for the problem. The algorithm is called ‘greedy’: it first looks for i with the largest value for Φ among the singletons, say i 1 that maximizes Φ({i}), then for j with the largest incremental change over i 1, say i 2 that maximizes Φ({i,i 1})−Φ({i 1}), and so on m times. The exact problem is known to be NP-hard in the size of n.
- 10.
Intuitively, there should be a link between the sub-modularity or super-modularity of L and the properties of the diffusion process itself, in particular the distribution of \({D}(\tilde{z})\) given A. Sub-modularity of L suggests a non-explosive dynamics of contagion. However I do not know of any result of this kind.
- 11.
What matters is the correlation that is unknown at the time of the evaluation.
- 12.
The condition is satisfied if there is a chance for the bank to default alone. This occurs for example if its initial values of the capital and liabilities are set so that a Value at Risk requirement is just binding. Specifically, given a level α, say 99 %, the level of capital, e i , and the interbank assets and liabilities are set so that the probability of default is equal to the level 1−α: F i (v i )=1−α. But VaR does not make much sense in a context with a uniform distribution.
- 13.
Most standard distributions have a log-concave density. Then the ratio \(\frac{F(x+\delta)-F(x)}{1-F(x)}\), related to the hazard rate (see for example Bergstrom and Bagnoli [3]) is increasing. Though this is compatible with the concavity of F, this suggests that concavity is far from being guaranteed.
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Acknowledgements
I would like to thank Yuan Zhang for research assistance.
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Demange, G. (2013). Diffusion of Defaults Among Financial Institutions. In: Abergel, F., Chakrabarti, B., Chakraborti, A., Ghosh, A. (eds) Econophysics of Systemic Risk and Network Dynamics. New Economic Windows. Springer, Milano. https://doi.org/10.1007/978-88-470-2553-0_1
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