Financial Networks, Cross Holdings, and Limited Liability

I discuss a network of banks which are linked by financial obligations and cross holdings. Given an initial endowment the value of the obligations and the equity values of the banks are determined endogenously in a way consistent with the priority of debt and the limited liability of equity. Neither equity values nor debt values are necessarily unique. But banks with non-unique equity values are ultimately owned by banks with a zero equity value. If the value of debt claims is non-unique the obligee himself is either in default or has a non-unique equity value. I derive rather mild conditions that guarantee unique equity and debt values. Due to the cross holdings equity is not necessarily a convex and debt is not necessarily a concave function of the given endowment. An algorithm to calculate debt and equity values is developed