[Paper Review] Sequential Defaulting in Financial Networks

Pál András Papp, Roger Wattenhofer|arXiv (Cornell University)|Nov 20, 2020

Banking stability, regulation, efficiency20 references3 citations

TL;DR

This paper studies sequential defaulting in financial networks using a reversible and a monotone model, showing that stabilization time can be exponential in the reversible model and that finding optimal default timing is NP-hard. It introduces a monotone model ensuring finite stabilization in O(n²) steps and demonstrates that early defaulting can be strategically advantageous for banks.

ABSTRACT

We consider financial networks, where banks are connected by contracts such as debts or credit default swaps. We study the clearing problem in these systems: we want to know which banks end up in a default, and what portion of their liabilities can these defaulting banks fulfill. We analyze these networks in a sequential model where banks announce their default one at a time, and the system evolves in a step-by-step manner. We first consider the reversible model of these systems, where banks may return from a default. We show that the stabilization time in this model can heavily depend on the ordering of announcements. However, we also show that there are systems where for any choice of ordering, the process lasts for an exponential number of steps before an eventual stabilization. We also show that finding the ordering with the smallest (or largest) number of banks ending up in default is an NP-hard problem. Furthermore, we prove that defaulting early can be an advantageous strategy for banks in some cases, and in general, finding the best time for a default announcement is NP-hard. Finally, we discuss how changing some properties of this setting affects the stabilization time of the process, and then use these techniques to devise a monotone model of the systems, which ensures that every network stabilizes eventually.

Motivation & Objective

• To understand the dynamics of financial networks when defaults occur sequentially rather than simultaneously.
• To investigate how the ordering of default announcements affects the final outcome, including the number of defaulting banks.
• To analyze strategic defaulting behavior from individual banks' perspectives, including optimal timing for default announcements.
• To address the instability and infinite loops in the reversible model by proposing a monotone variant that guarantees finite stabilization.
• To establish computational hardness results for finding optimal defaulting strategies and orderings.

Proposed method

• Proposes a reversible sequential model where banks update their recovery rates and can re-enter default status after exiting.
• Introduces a monotone model with a modified update rule that ensures recovery rates only decrease, preventing cycles and guaranteeing stabilization.
• Uses a branching gadget to construct systems where late defaulting avoids insolvency, demonstrating strategic benefits.
• Employs reduction techniques to prove NP-hardness of finding the optimal defaulting order and best defaulting time for a single bank.
• Constructs explicit examples with Ω(n²) stabilization time to show tightness of the upper bound in the monotone model.
• Analyzes the impact of CDSs and debt contracts on system behavior, comparing outcomes across different network configurations.

Experimental results

Research questions

• RQ1 How does the ordering of default announcements affect the final number of defaulting banks in a financial network?
• RQ2 Can a bank benefit from announcing default earlier rather than later, and under what conditions?
• RQ3 Is it computationally feasible to find the optimal defaulting order that minimizes or maximizes the number of defaults?
• RQ4 Can the sequential defaulting process in financial networks be guaranteed to stabilize, and if so, in how many steps?
• RQ5 How do the strategic incentives for defaulting differ between the reversible and monotone models?

Key findings

• The stabilization time in the reversible model can be exponentially long, depending on the default announcement ordering.
• Finding the default ordering that results in the minimum or maximum number of defaulting banks is NP-hard.
• A bank may achieve a higher recovery rate by defaulting early, as demonstrated in a constructed example with a 3/4 vs. 1/2 recovery rate outcome.
• The monotone model guarantees stabilization within O(n²) steps, and this bound is asymptotically tight, as shown by a Ω(n²) construction.
• In the monotone model, early defaulting can still be optimal, even though reversibility is removed, and this is proven via a concrete network example.
• While finding the best defaulting time for a single bank is NP-hard in the reversible model, the monotone model allows polynomial-time computation due to bounded runtime, though harder variants remain computationally challenging.