The mechanics of direct contagion in fi nancialsystems: a fl ow network approach

This paper put forward a novel approach to the analysis of direct contagion in fi nancial networks. Financial systems are here represented as fl ow networks i.e., directed and weighted graphs endowed with source nodes and sink nodes and the propagation of losses and defaults, originated by an exogenous shock, is here represented as a ow that crosses such a network. In establishing existence and uniqueness of such a ow function, we address a know problem of indeterminacy that arise, in fi nancial networks, from the intercyclicity of payments. Sufficient and necessary conditions for uniqueness are pinned down. We embed this result in an algorithm that, while computing the propagation caused by a shock, con- trols for the emergence of possible indeterminacies. We then apply some properties of network ows to investigate the relation between the structure of a fi nancial network i.e., the size and the pattern of obligations and its exposure to default contagion. We characterise rst and nal contagion thresholds (i.e., the value of the smallest shock capable of inducing default contagion and the value of the smallest shock capable of induc- ing the default of all agents in the network, respectively) for some classes of networks, namely the complete, star-shaped, incomplete regular, and cycle-shaped networks. Finally, we show that the exposition to default contagion of a generic network both in terms of contagion thresholds and of number of defaults induced by a shock monotonically grows with the ratio between internal and external debts, where the former are the intra-network obligations and the latter are the debts that the agents in the network owe to fi nal claimants who do not belong to the network.