We consider a situation in which agents have mutual claims on each other, summarized in a liability matrix. Agents’ assets might be insufficient to satisfy their liabilities, leading to defaults. In case of default, bankruptcy rules are used to specify the way agents are going to be rationed. A clearing payment matrix is a payment matrix consistent with the prevailing bankruptcy rules that satisfies limited liability and priority of creditors. Since clearing payment matrices and the corresponding values of equity are not uniquely determined, we provide bounds on the possible levels equity can take. Unlike the existing literature, which studies centralized clearing procedures, we introduce a large class of decentralized clearing processes. We show the convergence of any such process in finitely many iterations to the least clearing payment matrix. When the unit of account is sufficiently small, all decentralized clearing processes lead essentially to the same value of equity as a centralized clearing procedure. As a policy implication, it is not necessary to collect and process all the sensitive data of all the agents simultaneously and run a centralized clearing procedure. This paper was accepted by Yinyu Ye, optimization.