On Optimal Clearing Payments in Financial Networks

Modern financial networks are characterized by complex structures of mutual obligations. Such interconnections may propagate and amplify individual defaults, leading in some cases to financial disaster. For this reason, mathematical models for the study and control of systemic risk have attracted considerable research attention in recent years. One important line of research is concerned with mechanisms of clearing, that is, the mechanism by which mutual debts are repaid, in the regular regime, or in a default regime. One of the first models of a clearing mechanism was proposed by Eisenberg and Noe in [1], which introduced the concept of clearing vector of payments. In this paper, we propose a necessary and sufficient condition for the uniqueness of the clearing vector applicable to an arbitrary topology of the financial network. Further, we show that the overall system loss can be reduced if one relaxes the pro-rata rule and replaces the clearing vector by a matrix of clearing payments. This approach shifts the focus from the individual interest to the system, or social, interest, in order to control and contain the adverse effects of cascaded failures.