How to measure interconnectedness between banks, insurers and financial conglomerates
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Gaël Hauton
Abstract
Financial institutions’ interconnectedness is a key component of systemic risk. However there is still no consensus on its measurement. Using a unique database of network of exposures of French financial institutions, we compare three strategies to measure interconnectedness: closeness of exposure distributions, identification of core-periphery structure and contagion models. The closeness of exposure distributions is adequate to identify outlier institutions. The “core-periphery” structure, usually applied to banking network, is still valid with insurance companies. However this approach is not immune to size effect. This result contrasts with previous analyses where size was not accounted for. Contagion-based stress-tests are the best suited to capture institutions’ systemic fragility, emphasizing their importance as a supervisory tool.
A Structure identification
Craig and von Peter [9] propose the methodology in the light core-periphery structure identification. Our contribution is to introduce a censoring threshold to define the observed adjacency matrix. We transpose the methodology in the case of the complete core-periphery structure.
A.1 Identification of the complete core-periphery structure
“Complete core-periphery structure” refers to the structure proposed in [20]. Let us denote by h the size of the core (h for hubs). If the core institutions are firstly indexed, the theoretical adjacency matrix
where all the off-diagonal coefficients of
To compare this structure with our observation, the first step is to define a distance measure. Let us for instance consider the
where
To build a distance measure, we aggregate the number of pairwise errors analyzing the
The distance is the sum of the coefficients of the error matrix over the number of links:
Determining which institutions are in the core and which ones are in the periphery is seeking the partition which minimizes the distance
A.2 Identification of the light core-periphery structure
“Light core-periphery network” refers to the structure proposed in [9]. Let us denote by h the size of the core (h for hubs). If the core institutions are firstly indexed, the theoretical adjacency matrix
where all the off-diagonal coefficients of
In the case of a light core-periphery structure, the aggregate error matrix for an observed adjacency matrix
The distance is the sum of the coefficients of the error matrix over the number of links:
Determining which institutions are in the core and which ones are in the periphery is seeking the partition which minimizes the distance
B Robustness checks for core-periphery identification
Since the sample size is small, we use a grid to search for the best threshold. The original algorithm proposed by Craig and von Peter [9] relies on simulated annealing in order to deal with a large sample size. Tables 6–8 present robustness checks for the complete core-periphery structure identification. Tables 9–11 present robustness checks for the light core-periphery structure identification.
Complete core-periphery structure identification based on volume exposures – robustness check. Thresholds areexpressed in mn €. The last row indicates the distance of fit. Source: ACPR data, authors’ computations.
| Threshold (mn €) | 0 | 0.5 | 0.75 | 1 | 1.25 | 1.5 | 1.75 | 2 | 3 | 4 | 5 | 6 |
| Number of conglomerates in the core | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 4 | 3 | 2 | 2 |
| Number of pure banks in the core | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Number of pure insurers in the core | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Distance (%) | 14.6 | 17.9 | 12.5 | 11.3 | 9.5 | 10.0 | 16.7 | 18.8 | 19.2 | 17.7 | 45.5 | 50.0 |
Complete core-periphery structure identification based on credit risk exposures – robustness check. The last rowindicates the distance of fit. Source: ACPR data, authors’ computations.
| Threshold (%) | 0 | 1 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
| Number of conglomerates in the core | 2 | 3 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 |
| Number of pure banks in the core | 3 | 2 | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| Number of pure insurers in the core | 7 | 2 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| Distance (%) | 13.1 | 27.2 | 48.1 | 53.6 | 61.1 | 70.0 | 66.7 | 60.0 | 50.0 | 50.0 | 66.7 |
Complete core-periphery structure identification based on funding risk exposures – robustness check. The last row indicates the distance of fit. Source: ACPR data, authors’ computations.
| Threshold (%) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Number of conglomerates in the core | 5 | 5 | 5 | 4 | 3 | 2 | 2 | 2 | 1 | 1 | 1 |
| Number of pure banks in the core | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Number of pure insurers in the core | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Distance (%) | 13.1 | 19.4 | 18.9 | 26.5 | 32.8 | 34.7 | 33.3 | 37.0 | 39.0 | 37.5 | 37.8 |
Light core-periphery structure identification based on volume exposures – robustness check. Thresholds are expressed in mn €. The last row indicates the distance of fit. Source: ACPR data, authors’ computations.
| Threshold (mn €) | 0 | 0.5 | 0.75 | 1 | 1.25 | 1.5 | 1.75 | 2 | 3 | 4 | 5 | 6 |
| Number of conglomerates in the core | 6 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 3 | 2 | 2 |
| Number of pure banks in the core | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Number of pure insurers in the core | 5 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Distance (%) | 13.1 | 12.6 | 9.7 | 7.6 | 4.8 | 5.0 | 8.3 | 12.5 | 11.5 | 5.9 | 9.1 | 12.5 |
Light core-periphery structure identification based on credit risk exposures – robustness check. The last row indicates the distance of fit. Source: ACPR data, authors’ computations.
| Threshold (%) | 0 | 1 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
| Number of conglomerates in the core | 2 | 1 | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| Number of pure banks in the core | 3 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Number of pure insurers in the core | 7 | 4 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| Distance (%) | 13.1 | 25.2 | 44.2 | 64.3 | 72.2 | 80.0 | 100.0 | 100 | 100 | 100 | 100 |
Light core-periphery structure identification based on funding risk exposures – robustness check. The last rowindicates the distance of fit. Source: ACPR data, authors’ computations.
| Threshold (%) | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 | 0.1 |
| Number of conglomerates in the core | 5 | 5 | 5 | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 |
| Number of pure banks in the core | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Number of pure insurers in the core | 5 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Distance (%) | 12.3 | 21.0 | 22.4 | 30.9 | 36.2 | 40.8 | 39.6 | 45.7 | 51.2 | 55.0 | 64.9 |
Systemic importance and systemic fragility – robustness check 1/2. Note: On the top-left panel, one conglomerate leads 18 others institutions to lose more than 1% of their equity. This same conglomerate suffers losses higher than 1% of its equity when seven other institutions individually default. Source: ACPR data, authors’ computations.
Systemic importance and systemic fragility – robustness check 2/2. Note: On the top-left panel, one conglomerate leads 12 others institutions to lose more than 10% of their equity. This same conglomerate suffers losses higher than 10% of its equity when three other institutions individually default. Source: ACPR data, authors’ computations.
Acknowledgements
We are very grateful to the anonymous referees and Michael Gordy (Editor) for their helpful suggestions and remarks. We thank I. Alves, M. Billio, S. Darolles, C. Gouriéroux, C. Labonne and the participants of the ESRB Workshop (Frankfurt, 2014), the 7th Risk Forum (Paris, 2014), the 14th CREDIT Conference (Venice, 2014), the 3rd EBA Workshop (London, 2014) and the 6th IMF Forum (London, 2014) for their comments. The opinions expressed in the paper are only those of the authors and do not necessarily reflect those of the Autorité de Contrôle Prudentiel et de Résolution (ACPR).
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