Assessing interbank contagion using simulated networks

Abstract

This paper presents a new approach to randomly generate interbank networks while overcoming shortcomings in the availability of bank-by-bank bilateral exposures. Our model can be used to simulate and assess interbank contagion effects on banking sector soundness and resilience. We find a strongly non-linear pattern across the distribution of simulated networks, whereby only for a small percentage of networks the impact of interbank contagion will substantially reducoe average solvency of the system. In the vast majority of the simulated networks the system-wide contagion effects are largely negligible. The approach furthermore enables to form a view about the most systemic banks in the system in terms of the banks whose failure would have the most detrimental contagion effects on the system as a whole. Finally, as the simulation of the network structures is computationally very costly, we also propose a simplified measure—a so-called Systemic Probability Index—that also captures the likelihood of contagion from the failure of a given bank to honour its interbank payment obligations but at the same time is less costly to compute. We find that the SPI is broadly consistent with the results from the simulated network structures.

Appendix

Proof of theorem 3.1

We focus on the triggering bank $I$. Let us define a mapping $\Psi _I:[0,1]^N\rightarrow [0,1]^N$ as

$$\begin{aligned} \Psi _{Ij}(z)=\left\{ \begin{array}{ll} \mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j),&{}\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)>\gamma \\ \min \left\{ \gamma ,B(z)+\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)\right\} ,&{}\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)\le \gamma \\ \end{array}\right. \end{aligned}$$

where $A(z)$ is an isotone, positive mapping and $B(z)$ a given mapping (in $\mathbb R$).

Suppose that $z_1\in [0,1]^N$ is such that $z_{1i}\ge \gamma$ and $z_1\succ z_2$. Then, $\Psi _I(z_1)\succ \Psi _I(z_2)$. It follows from the fact that $A(\cdot )$ is isotone and positive. In fact, $A(\cdot )$ and $B(\cdot )$ both depend on $j$ but we drop the index for brevity. Let us notice, that for $e$ being a unit vector (e.g. $e^{(k)}:=[\underbrace{0\dots 0}_{k}\ 1\ \underbrace{0\dots \ 0}_{N-k-1}]$), $e^{(k)}\preceq \Psi _I(e^{(k)})$, since by definition $\Phi _I$ is bounded by 0 and 1. If $\Psi _Ij(e^{(k)})\ge \gamma$, then the sequence $\Psi _Ij(e^{(k)}),\ \Psi _Ij\circ \Psi _Ij(e^{(k)}),\dots ,\ \Psi _Ij\circ \dots \circ \Psi _Ij(e^{(k)}),\dots$ is non-decreasing and, since is bounded by 1, it converges. It is, then, sufficient to prove the theorem by showing that $\Psi _I$ is isotone if $A(z)$ is replaced by $[z_1L_1l_1,\dots ,z_NL_Nl_N]^{\top }$. But trivially, $A_j(z)$ is increasing in every $z_i$. This completes the proof.

Remark 6.1

Why $(P_{Ij}^{(k)})$ may not be globally convergent? Set $b:\!=\![z_1L_1l_1\,\dots \,z_N$ $L_Nl_N]^{\top }$. Let $B(z)$ be replaced by

$$\begin{aligned} \left[ \begin{array}{cc} \omega (P^{geo}_{\cdot 1},d_{\cdot 1},b)\frac{K_1}{1+\left| \frac{C_1-b\cdot \mathbf E [\pi ^G_{\cdot 1}]}{\sqrt{b^2\cdot D^2[\pi ^G_{\cdot 1}]}}\right| ^3}\\ \vdots \\ \omega (P^{geo}_{\cdot N},d_{\cdot N},b)\frac{K_1}{1+\left| \frac{C_N-b\cdot \mathbf E [\pi ^G_{\cdot N}]}{\sqrt{(b)^2\cdot D^2[\pi ^G_{\cdot N}]}}\right| ^3}\\ \end{array}\right] . \end{aligned}$$

Let us represent $B$ in the following way (we slightly abuse the notation introducing $z$ to power $n^{\text{ th }}$, i.e. $z^n:=[z_1^n,\dots ,x_N^n]^{\top }$):

$$\begin{aligned} B(z)=B^1(z)B^2(z) \end{aligned}$$

where

$$\begin{aligned} B^1(z)&= \frac{Q^{(21)}\cdot z^3}{\sqrt{Q^{(22)}\cdot z^2}^3}\\ B^2(z)&= \frac{1}{1+\frac{C_j-Q^{(23)}\cdot z}{\sqrt{Q^{(24)}\cdot z^2}}} \end{aligned}$$

for positive vectors $Q^{(21)}, Q^{(22)}, Q^{(23)}$ and $Q^{(24)}$. We determine a region where $B$ is increasing. Namely, differentiating $B^1$ with respect to $z_i$ (in the set $\{z|\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)<\gamma \}$), one observes that it is increasing if

$$\begin{aligned} 3Q^{(21)}_iz_i^2\sqrt{Q^{(22)}\cdot z^2}^3-3Q^{(21)}\cdot z^3\sqrt{Q^{(22)}\cdot z^2}Q^{(22)}_iz_i>0. \end{aligned}$$

It happens for $z$ bounded from $0^N$, i.e. for all $i\in \{1,\dots ,N\}$ satisfying

$$\begin{aligned} z_i>\frac{Q^{(22)}_i\sum _{m\ne i}Q^{(21)}_mz_m^3}{Q^{(21)}_i\sum _{m\ne i}Q^{(22)}_mz_m^2}. \end{aligned}$$

In case of $B^2$ the differentiation with respect to $z_i$ brings us to the following inequality

$$\begin{aligned} Q^{(23)}_iz_i\sqrt{Q^{(24)}\cdot z^2}+(C_j-Q^{(23)}\cdot z)\frac{Q^{(24)}_iz_i}{\sqrt{Q^{(24)}\cdot z^2}}>0 \end{aligned}$$

that translates into increasing $B^2$. The sufficient condition for the inequality to hold is $C_j-Q^{(23)}\cdot z>0$.