Financial institutions are interconnected by holding debt claims against each other. A default bank may cause its creditors to default, and the risk may be further propagated to up-stream institutes (risk contagion). Such interconnection is a key contributing factor to the past worldwide financial crisis. We show that a good mechanism of default liquidation may improve the total wealth of the financial system and therefore may curb the risk contagion. We formulate this problem as a nonlinear optimization problem with constraints and propose an optimal liquidation policy to minimize the system’s loss. We show that the problem resembles a Markov decision problem (MDP) and therefore we can apply the direct-comparison based optimization approach to solve this problem. Higher order directional derivatives and some optimality properties are obtained. Furthermore, we derive an iterative algorithm which combines both the policy iteration and the gradient based approach to find a local optimal policy, and under some conditions, a global optimal policy. Our work provides a new direction in curbing the risk contagion in financial networks; and it illustrates the advantages of the direct-comparison based approach, originated in the field of discrete event dynamic system, in nonlinear optimization problems.

Introduction

This paper is motivated by two recent developments in financial engineering and performance optimization. First, financial institutions are interconnected by borrowing-and-lending activities among themselves or holding marketable securities against each other (Chen, Liu, & Yao, 2014). Such interconnection is a critical influencing factor to the past worldwide financial crisis and the European sovereignty debt crisis, and could potentially threaten the stability of financial networks. For example, a default bank may cause its creditors to default, and the risk may be further propagated to up-stream institutes. During default liquidation, it is extremely significant to curb such risk contagion among financial networks. The goal of this paper is to propose a new liquidation scheme based on performance optimization and to provide computation algorithms that solve the problem. The problem formulated above has a large dimension and many highly nonlinear constraints, and we need to develop an efficient algorithm for an optimal solution. On this side, a direct-comparison based approach has been developed in the past years to the optimization of nonlinear problems and has been successfully applied to many problems, such as optimization of singular controlled diffusion processes (Ni & Fang, 2013), MDP with long-run average criterion (Cao, 2007, 2015), and variance criterion (Xia, 2016), and nonlinear performance with probability distortion (Cao & Wan, 2017). In this paper, we show that the special features of the financial risk contagion problem make it possible to be solved by the direct-comparison based approach, leading to some new insights to the problem. While network connections have a positive effect by diversifying risk (Haldane, 2009; Summer, 2013), it can have a negative effect by adding spreading channels for risks. When a global shortage for liquidity happens, the systemic risk will be transmitted through the risk-sharing mechanism (Allen & Gale, 2000; Leitner, 2005; Rochet & Tirole, 1996). Such risk contagions may result in consecutive consequences for the financial system, such as declines in asset prices, higher price volatility, more bank defaults, and market inefficiency (Allen & Gale, 2004; Brunnermeier & Pedersen, 2009; Holmstrom & Tirole, 2000).

The law and economics of default resolution in bankruptcy typically presume a single debtor that owes multiple creditors. But multiple debtors owe multiple creditors within interbank payment networks like Fedwire and CHIPS, as well as in decentralized networks created by complex interbank loan obligations. Nash Bargaining theory has previously been used to justify two popular rules for resolving default of a single debtor. I describe how to extend these rules to financial networks, but find that Nash Bargaining theory does not justify applying those rules to resolve defaults in those networks.

“Systemic risk refers to the risk that a financial system as a whole is susceptible to failures initiated by the characteristics of the system itself.”1 If strong links between financial institutions are present, a shock to only a small number of entities might propagate through the system and trigger substantial financial losses. Significant dependence can thus increase the risk of a system-wide breakdown. Financial institutions influence each other via direct or indirect channels such as credit contracts, similar asset portfolios that are jointly exposed to price impact in market crises, and cross-shareholdings. Frictions like, e.g., bankruptcy costs may amplify the impact of the effect of the firms’ interaction. The aim of the current paper consists in constructing and analyzing a comprehensive model that integrates all effects mentioned above. This multi-factor setting allows to assess regulatory policies in a robust manner. To the best of our knowledge, such a contribution is still missing in the literature.

(i) We prove the existence of a clearing equilibrium that is not necessarily unique and provide an algorithm for the computation of the greatest and the least equilibrium. The equilibrium is characterized by the vector of clearing payments and the price of the commonly held illiquid asset that is exposed to price effects.

(ii) We study the impact of bankruptcy costs, fire sales, and cross-holdings on systemic risk in numerical experiments. We demonstrate that fire sales and bankruptcy costs can trigger and amplify financial crises. Policies that mitigate their impact might significantly enhance the resilience of the financial system. Cross-holdings do, in contrast, have a stabilizing effect, if they can be exchanged for liquid assets. Central banks that engage in such a market can reduce the number of defaults in the system.

(iii) We study policy implications and regulatory instruments, including central bank guarantees, quantitative easing, the significance of last wills of financial institutions, and the efficiency of capital requirements. We find that capital adequacy ratios based on risk-weighted assets reduce systemic risk, if they are sufficiently high. However, they do not rely on any statistics that capture systemic risk in a proper way. Comparative statics show that capital adequacy ratios can be equal for varying parameters of our model that are associated with completely different levels of systemic risk. This demonstrates that classical capital adequacy ratios are a very rough instrument. A much better alternative are systemic risk measures that we analyze in the last section.