Contagion! Systemic Risk in Financial Networks

Attempts to define systemic risk are summarized and found to be deficient in various respects. This introductory chapter, after considering some of the salient features of financial crises in the past, focusses on the key characteristics of banks, their balance sheets and how they are regulated.

Network effects such as default contagion and liquidity hoarding are transmitted between banks by direct contact through their interbank exposures. During asset fire sales, shocks are transmitted indirectly from a bank selling assets to other banks via the impact on the price of their common assets. Banks maintain safety buffers in normal times, but these may be weakened or fail during a crisis. Asset prices that are relatively stable in normal times may collapse during a crisis. Banks react to such stresses by making large adjustments to their balance sheets. Such adjustments send further shocks to their counterparties both directly through their exposures and indirectly via asset price impact, creating a cascade. All these cascade mechanisms can be modelled mathematically starting from a common framework. In such models, the eventual extent of a crisis is a fixed point or equilibrium of a cascade mapping. Towards the end of the chapter, a proposal is made that the properties of cascade mappings can be most clearly understood when implemented on very large random financial networks.

The right kind of connectivity turns out to be both necessary and sufficient for large scale cascades to propagate in a network. After outlining percolation theory on random graphs, we develop an idea known as “bootstrap percolation” that proves to be the precise concept needed for unravelling and understanding the growth of simple network cascades. These principles are illustrated by the famous Watts model of information cascades.

This chapter realizes the central aim of the book, which is to understand a simple class of cascades on financial networks as a generalization of percolation theory. The main results apply to random financial networks with locally tree-like independence and characterize zero-recovery default cascade equilibria as fixed points of certain cascade mappings. The proofs of the main results follow a new and distinctive template presented here for the first time, that has the important virtue that its logic extends to LTI financial networks of arbitrary complexity. Numerical computations, both large network analytics and finite Monte Carlo simulations, verify that essential characteristics such as cascade extent and cascade frequency can be derived from the properties of the cascade fixed points.

The prospects are considered for extending the mathematical framework of cascade mechanisms on locally tree-like random financial networks to address problems of real financial importance.