Optimal Portfolio Compression for Priority-Proportional Clearing with Defaulting Costs

Abstract

We study financial networks where banks are connected through bilateral liabilities and may default when resources are insufficient to meet obligations. We consider both the standard proportional clearing model and a priority-proportional clearing model in which banks repay creditors according to exogenously given priority classes. In such markets, portfolio compression is a process where several banks come to a netting arrangement which reduces liabilities without changing any bank's net exposure, essentially removing cycles of debt. Our goal is to understand whether portfolio compression schemes can be designed to improve clearing outcomes for a large fraction of banks. We provide a computational characterization of the benefits and limitations of compression. On the positive side, we give a polynomial-time algorithm to compute a maximal clearing outcome under priority-proportional clearing, and we show that it is possible to decide in polynomial time whether there exists a compression that limits defaults to at most one bank. On the negative side, we show that several natural optimization and decision problems are computationally intractable: deciding whether some compression can reduce the number of defaulting banks below a given threshold, or whether a specific bank can be saved from defaulting, is $\NP$-hard even in restricted settings and under proportional clearing. We further present a mixed integer linear programming (MILP) formulation that computes a compression maximizing the number of non-defaulting banks, providing a practical approach to this hard problem. Using our MILP formulation, we perform simulations on both synthetic and real-world datasets to analyze the effects of portfolio compression.

Figures (4)

• Figure 1: Illustration for Example \ref{['ex']}. Liability values are displayed alongside the corresponding arrows, shown in parenthesis for Figures (b)--(d). Initial endowments are depicted within the circles representing the banks. Figures (b), (c), and (d) show clearing vectors where the compression value along the cycle $(c_1,c_2,c_3)$ is $\varepsilon=0,1,2$, respectively, with the payment values written before the liabilities (after compression) in parenthesis.
• Figure 2: Visual representation of our results. The plots show the number of defaulting banks for various sample sizes. Results for the baseline, greedy, and MILP methods are shown in grey, red, and blue, respectively. In Figure (a), the red and grey lines overlap almost completely, showing the almost identical performance of the baseline and the greedy methods. The colored "zones" represent the $95\%$ confidence intervals.
• Figure 3: Simulation results for synthetic data generated with uniform liabilities.
• Figure 4: Simulation results for synthetic data generated with log-normal liabilities.

Theorems & Definitions (20)

• Example 1
• Theorem 1
• Lemma 1: app:prf-lemsol
• Lemma 2: app:prf-lemuniq
• Lemma 3: app:prf-lemminp
• Lemma 4: app:prf-lemzero
• Theorem 2
• Theorem 3: app:prf-thmnphfindcomp
• Theorem 4: app:prf-thmnphallbutthree
• Proposition 1