We investigate manipulability in the setting of financial systems by considering two weak forms of immunity: non-manipulability via merging and non-manipulability via splitting. Not surprinsingly, non-manipulability via splitting is incompatible with some basic axioms: claim boundedness, limited liability, and absolute priority. Outstandingly, we introduce a large class of financial rules that are immune to manipulations via merging. This class includes the proportional financial rule but also financial rules in accordance with parametric bankruptcy rules fulfilling non-manipulability via merging.

We analyze how interdependencies in financial networks can lead to self-fulfilling insolvencies and multiple possible equilibrium outcomes. We show that multiplicity arises if and only if there exists a certain type of dependency cycle in the network, and characterize banks' solvency in any equilibrium. We use this analysis to understand how to inject capital into banks so as to ensure solvency of all at minimum cost. We show that finding the cheapest bailout policy that prevents self-fulfilling insolvencies is computationally hard (and hard to approximate), but that the problem has intuitive solutions in specific network structures. Bailouts have an indirect value as making a bank solvent improves its creditors' balance-sheets and reduces their bailout costs, and we show how a simple algorithm that leverages these indirect benefits ensures systemic solvency at a total cost that never exceeds half of the total overall shortfall. In core-periphery networks, indirect bailouts—whereby the regulator bails out peripheral banks first as opposed to targeting core banks directly—are optimal.

We consider financial networks where agents are linked to each other via mutual liabilities. In case of bankruptcy, there are potentially many bankruptcy rules, ways to distribute the assets of a bankrupt agent over the other agents. One common approach is to first apply pairwise netting of agents that have mutual liabilities and next use the proportional rule to determine the payments on the basis of the net liabilities. We refer to this as the pairwise netting proportional rule. The pairwise netting proportional rule satisfies the basic requirements of claims boundedness, limited liability, priority of creditors, and continuity. It also satisfies the desirable properties of net impartiality, an agent that has two creditors with the same net claims pays the same amount to both creditors on top of pairwise netting, and invariance to mitosis, an agent that splits into a number of identical agents is not affecting the payments of the other agents. We demonstrate that if net impartiality and invariance to mitosis, together with the basic requirements, are regarded as imperative properties, then payments should be determined by the pairwise netting proportional rule.