The impact of network inhomogeneities on contagion and system stability

Abstract

This work extends the contagion model introduced by Nier et al. (J Econ Dyn Control 31(6):2033–2060, 2007. http://www.bankofengland.co.uk/publications/workingpapers/wp346.pdf) to inhomogeneous networks. We preserve the convenient description of a financial system using a sparsely parameterized random graph, but add several relevant inhomogeneities. These include well-connected banks, financial institutions with disproportionately large interbank assets, and big banks that focus on wholesale and retail customers. These extensions significantly enhance the model’s generality as they reflect realistic inhomogeneities that have a potentially decisive impact on a system’s stability. We find that large and well-connected banks have a surprisingly modest impact. However, institutions with disproportionately large interbank assets significantly increase the risk of contagion in networks. Moreover, these effects can be partly compensated by a redistribution of equity capital, even without increasing the total amount. However, the level of regulatory capital should be defined according to the interbank market position of a bank, and not the size of the bank.

Appendix

1.1 Determination of the adjusted connection probability $p^{*}$

While riskiness and the inhomogeneous distribution of the external assets can be implemented in a deterministic way, the introduction of the node property, connectivity, depends on a technical parameter, namely the adjusted connectivity probability, $p^{*}$. In order to ensure a stable model evaluation, this parameter (which is only defined via an expected value) must be determined in a reliable way. An iterative numerical approach is implemented to determine $p^{*}$: We start with an estimated value. The number of network links, Z, is calculated for up to 10.000 configurations generated for $p^{*}$. Then, the average of the observed Z-values is interpreted as the expected value of the left-hand side of Eq. (10). These two steps are performed within a secant-like method to adjust $p^{*}$ so that, finally, Eq. (10) approximately holds.

Table 4 Accuracy of the determination of the adjusted connection probabilities, $p^{*}$, for different values of the variance, $\sigma _{p}^{2}$, of the inner connectivity

Table 4 shows our approximated values for the adjusted connection probability, $p^{*}$, for different levels of $\sigma _{p}^{2}$. Note that for the chosen model parameters, the targeted value for the expected number of links is 495. The small variations of $p^{*}$ and Z indicate that $p^{*}$ is determined with high accuracy, even if the variance $\sigma _{p}^{2}$ exceeds the width of the reasonable interval, [0; 1].