Measuring systemic risk and contagion in the European financial network

Abstract

This paper introduces a novel framework to study default dependence and systemic risk in a financial network that evolves over time. We analyse several indicators of risk, and develop a new latent space model to assess the health of key European banks before, during and after the recent financial crises. We propose a new statistical model that permits a latent space visualisation of the financial system. This provides a clear and interpretable model-based summary of the interaction data, and it gives a new perspective on the topology structure of the network. Crucially, the methodology provides a new approach to assess and understand the systemic risk associated with a financial system, and to study how debt may spread between institutions. Our dynamic framework provides an interpretable map that illustrates the default dependencies between institutions, highlighting the possible patterns of contagion and the institutions that may pose systemic threats.

Appendices

Appendices

Appendix A: Proof of Proposition 1

Proof

We wish to study the density $\pi \left( y_i^{(t)} \big | \mathbf{y }_{-(i,t)}, \mathbf{Z }, \mathbf{X } \right)$, up to a proportionality constant that does not depend on $y_i^{(t)}$:

$$\begin{aligned} \begin{aligned} \pi \left( y_i^{(t)} \big | \mathbf{y }_{-(i,t)}, \mathbf{Z }, \mathbf{X } \right)&= \frac{ \pi \left( \mathbf{y } \big | \mathbf{Z }, \mathbf{X } \right) }{ \pi \left( \mathbf{y }_{-(i,t)} \big | \mathbf{Z }, \mathbf{X } \right) } \propto f\left( \mathbf{y } \big | \mathbf{Z }, \mathbf{X } \right) \\&\propto \exp \left\{ - \frac{1}{2}\sum _{t=1}^{T} \sum _{i=1}^N \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^N x_{ij}^{(t)} \eta _{ij}^{(t)} \left( y_i^{(t)} - y_j^{(t)} \right) ^2\right\} \\&\propto \exp \left\{ - \frac{1}{2} \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^N x_{ij}^{(t)} \eta _{ij}^{(t)} \left( y_i^{(t)} - y_j^{(t)} \right) ^2\right\} \\&\propto \exp \left\{ - \frac{1}{2}\left[ {\mathcal {A}}_{i}^{(t)} \left( y_i^{(t)}\right) ^2 - 2{\mathcal {B}}_{i}^{(t)} y_i^{(t)} \right] \right\} \end{aligned} \end{aligned}$$

(16)

where

$$\begin{aligned} {\mathcal {A}}_{i}^{(t)}= & {} \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^N x_{ij}^{(t)} \eta _{ij}^{(t)}, \\ {\mathcal {B}}_{i}^{(t)}= & {} \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^N x_{ij}^{(t)} \eta _{ij}^{(t)} y_j^{(t)}. \end{aligned}$$

Then:

$$\begin{aligned} \begin{aligned} \pi \left( y_i^{(t)} \big | \mathbf{y }_{-(i,t)}, \mathbf{Z }, \mathbf{X } \right)&\propto \exp \left\{ - \frac{1}{2}\left[ {\mathcal {A}}_{i}^{(t)} \left( y_i^{(t)}\right) ^2 - 2{\mathcal {B}}_{i}^{(t)} y_i^{(t)} \right] \right\} \\&\propto \exp \left\{ - \frac{{\mathcal {A}}_{i}^{(t)}}{2}\left[ \left( y_i^{(t)}\right) ^2 - 2\frac{{\mathcal {B}}_{i}^{(t)}}{{\mathcal {A}}_{i}^{(t)}} y_i^{(t)} + \left( \frac{{\mathcal {B}}_{i}^{(t)}}{{\mathcal {A}}_{i}^{(t)}}\right) ^2 \right] \right\} \\&\propto \exp \left\{ - \frac{{\mathcal {A}}_{i}^{(t)}}{2}\left[ y_i^{(t)} - \frac{{\mathcal {B}}_{i}^{(t)}}{{\mathcal {A}}_{i}^{(t)}} \right] ^2 \right\} \\ \end{aligned} \end{aligned}$$

(17)

which is proportional to a Gaussian density with the following mean and variance:

$$\begin{aligned} \mu _i^{(t)} = \frac{{\mathcal {B}}_{i}^{(t)}}{{\mathcal {A}}_{i}^{(t)}}, \quad \quad \quad \nu _i^{(t)} = \frac{1}{{\mathcal {A}}_{i}^{(t)}}. \end{aligned}$$

$\square$

Appendix B: Summary of default probabilities by bank

See Tables 16 and 17.

Table 16 Summary of default probabilities by bank

Table 17 Summary of Default Probabilities by country