Connectivity and Financial Network Shutdown

Connectivity is a measure of the number of connections in a network. It is applied here to financial network shutdown due to inter-institutional default. Since 1797 when Sir Francis Baring introduced the concept of ``lender of last resort" concepts such as "too-big-to fail", "lender-of-last-resort" and "systemic risk" have belied a need for a quantitative approach. With the increase in the market for over-the-counter inter-institutional contracts, especially the interbank market for foreign exchange, interest has increased on the consequences of an unwind in payments due to default. To date one paper, which is not purely descriptive, has been published on the effects of defaults in payments systems. That paper reports the results of three scenarios for settlement in CHIPS, one of two US dollar payments systems. By contrast, analytic results (not inferred by simulation) of the statistical properties of payments mechanisms are obtained in this paper. These results are derived given the simple clearing mechanisms of no netting and bilateral netting, and are derived using Boolean graphs. The model is applied to default propagation transition matrices. Specifically, the concepts of network architecture and connectivity of that architecture are defined. Starting from assumptions on the connection matrix. Markov transition matrices are derived where each state is the number of firms in default. Debt maturity, connectivity and initial defaults are used to calculate the transition matrices. Besides deriving the stochastic difference equations for the number of failed firms via Markov matrices, results were obtained on the effect of netting liabilities on bankruptcy propagation. The propagation of bankruptcy was compared for liabilities which offset (are netted) and those which do not. It is shown that the connectivity value which maximizes the speed of default propagation under netting is not that when liabilities do not offset. For netting liabilities this value is 1/2. For non-offsetting liabilities this value is 1. The benefit of netting given a level of connectivity, g, is the reduction in connectivity which is the connectivity squared, g'. Finally, results were obtained demonstrating that when bankruptcy propagation is fast relative to the maturity of obligations, the bankruptcy propagation over the network can be less extensive than then propagation is slower. Also, given the time scale of propagation, shortening the maturity of obligations not only may increase the extent of propagation, but definitely increases its speed.