Interbank clearing in financial networks with multiple maturities

We consider the problem of systemic risk assessment in interbank networks in which interbank liabilities can have multiple maturities. In particular, we allow for both short-term and long-term interbank liabilities. We develop a clearing mechanism for the interbank liabilities to deal with the default of one or more market participants. Our approach generalizes the clearing approach for the single maturity setting proposed by Eisenberg and Noe [Management Sci., 47 (2001), pp. 236-249]. Our clearing mechanism focuses on the vector of each bank's liquid assets at each maturity date and develops a fixed-point formulation of this vector for a given set of defaulted banks. Our formulation is consistent with the main stylized principles of insolvency law. We show that in the context of multiple maturities, specifying a set of defaulted banks is challenging. We propose two approaches to overcome this challenge: First, we propose an algorithmic approach for defining the default set and show that this approach leads to a well-defined liquid asset vector for all financial networks with multiple maturities. Second, we propose a simpler functional approach which leads to a functional liquid asset vector which need not exist but under a regularity condition does exist and coincides with the algorithmic liquid asset vector. Our analysis permits construction of simple dynamic models and furthermore demonstrates that systemic risk can be underestimated by single maturity models.