Sensitivity and computational complexity in financial networks

Abstract

Abstract. Determining the causes of instability and contagion in financial networks is necessary to inform policy and avoid future financial collapse.In the American Economic Review, Elliott, Golub and Jackson proposed a simple model for capturing the dynamics of complex financial networks. In Elliott, Golub and Jackson's model, the institutions in the network are connected by linear dependencies (cross-holdings) and if any institution's value drops below a critical threshold, its value suffers an additional failure cost. This work shows that even in this simple model there are fundamental barriers to understanding the risks that are inherent in a network.First, if institutions are not required to maintain a minimum amount of self-holdings, any change in investments by a single institution can have an arbitrarily magnified influence on the net worth of the institutions in the system. This implies that if institutions have small self-holdings, then estimating the market value of an institution requires almost perfect information about every cross-holding in the system. Second, even if a regulator has complete information about all cross-holdings in the system, it may be computationally intractable to estimate the number of failures that could be caused by a small shock to the system.

DStress

“…The algorithm is also not guaranteed to converge after n steps, as each step can cause a valuation drop even beyond the discontinuous drop. However, as shown in [39], it converges to its final value monotonically, and thus a limited number of iterations provides a good approximation result. Figure 2(b) shows an implementation in DStress.…”

“…It is well known that the answer to many questions about graph-shaped data can change radically when even a single edge is added or removed, and thus such questions cannot be answered with differential privacy. However, the TDS is an exception: adding or removing edges does not disproportionally affect the TDS [39]. The privacy guarantee that results when we add noise to the TDS is called dollar-differential privacy; it was first introduced in [30].…”

Forecasting financial crashes with quantum computing

“…Following Ref. [10], we further define the equity value V i of institution i as V i = k D ik p k + j C ij V j , i.e., the value of institution i due to ownership of assets and cross-holdings. In matrix notation, we can then write V = D p+C V , such that V = (I−C) −1 D p. As explained in Ref.…”

“…In matrix notation, we can then write V = D p+C V , such that V = (I−C) −1 D p. As explained in Ref. [10], matrix I−C is guaranteed to be invertible. Additionally, the market value v i of institution i is its equity value rescaled with its self-ownership, i.e., v i = C ii V i .…”

“…As explained in Refs. [10,21], the model further introduces the notion of failure. This means that if the market value of an institution drops below a certain critical threshold, then the institution suffers an extra discontinuous loss in equity value.…”