# CoRisk: Credit Risk Contagion with Correlation Network Models

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Proposed Methodology

## 4. Empirical Findings

#### 4.1. Data

#### 4.2. Correlation Networks

#### 4.3. VAR Model Estimation

#### 4.4. CoRisk as a New Centrality Measure

#### 4.5. Robustness

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Partial correlation network referred to 10 countries: five core economies (France, Germany, Japan, the United Kingdom and the United State) and five peripheral economies (Greece, Ireland, Italy, Portugal and Spain). Green lines stand for positive partial correlations, red lines for negative correlations; the thicker the line, the stronger the connection; the larger a node, the higher the corresponding idiosyncratic probability of default.

**Figure 2.**Partial correlation networks referred to the 10 considered countries in three different time periods (financial-crisis, sovereign-crisis, post-crisis). Green lines stand for positive partial correlations, red lines for negative correlations; the thicker the line, the stronger the connection; the larger a node, the higher the corresponding idiosyncratic probability of default.

**Figure 3.**$CoRisk$ and Total Risk for peripheral countries. For each considered peripheral country, the graphs report the observed CDS spreads (black line), the estimated autoregressive CDS spread components (green line) and the estimated contemporary CDS spread component ($CoRisk$, red line).

**Figure 4.**$CoRisk$ and Total Risk for core countries. For each considered core country, the graphs report the observed CDS spreads (black line), the estimated autoregressive CDS spread components (green line) and the estimated contemporary CDS spread component ($CoRisk$, red line).

**Figure 5.**August 2011, predictive performance for peripheral countries. For each considered peripheral country, the graphs report the observed CDS spreads at time t (black line), the CDS spreads predicted at time $t-1$ using the proposed model (red line) and the CDS spreads predicted at time $t-1$ using only the autoregressive component (blue line).

**Figure 6.**August 2011, predictive performance for core countries. For each considered core country, the graphs report the observed CDS spreads at time t (black line), the CDS spreads predicted at time $t-1$ using the proposed model (red line) and the CDS spreads predicted at time $t-1$ using only the autoregressive component (blue line).

**Figure 7.**August 2012, predictive performance for peripheral countries. For each considered peripheral country, the graphs report the observed CDS spreads at time t (black line), the CDS spreads predicted at time $t-1$ using the proposed model (red line) and the CDS spreads predicted at time $t-1$ using only the autoregressive component (blue line).

**Figure 8.**August 2012, predictive performance for core countries. For each considered core country, the graphs report the observed CDS spreads at time t (black line), the CDS spreads predicted at time $t-1$ using the proposed model (red line) and the CDS spreads predicted at time $t-1$ using only the autoregressive component (blue line).

**Table 1.**Summary of statistics for the CDS spreads from 1 July 2006 to 31 December 2016. Means, standard deviations, minimum and maximum values are all expressed in basis points (bps).

Country | Mean | St. Dev. | Min | Max |
---|---|---|---|---|

France | 40.17 | 34.12 | 1.47 | 198.68 |

Germany | 20.30 | 17.48 | 1.29 | 91.37 |

Greece | 3708.36 | 7534.43 | 4.71 | 23,571.95 |

Ireland | 179.22 | 218.78 | 1.66 | 1193.98 |

Italy | 133.18 | 105.94 | 5.29 | 501.52 |

Japan | 51.68 | 32.99 | 2.45 | 160.43 |

Portugal | 276.86 | 308.39 | 3.86 | 1554.02 |

Spain | 124.26 | 105.85 | 2.34 | 504.15 |

United Kingdom | 41.45 | 31.32 | 1.19 | 164.63 |

United States | 26.96 | 17.36 | 1.07 | 100.25 |

**Table 2.**Estimated partial correlations between contemporaneous country spread effects, and the corresponding significance t-test values (in parentheses). Bold values indicate not significant values at the 1% level.

Country | France | Germany | Greece | Ireland | Italy | Japan | Portugal | Spain | UK | USA |
---|---|---|---|---|---|---|---|---|---|---|

France | 1.00 | 0.67 | −0.34 | 0.05 | 0.35 | 0.21 | 0.17 | 0.21 | −0.22 | −0.19 |

(47.90) | (−19.48) | (2.86) | (19.80) | (11.57) | (9.34) | (11.59) | (−12.00) | (−10.36) | ||

Germany | 0.67 | 1.00 | 0.15 | 0.27 | −0.05 | −0.05 | −0.24 | −0.27 | 0.61 | −0.04 |

(47.95) | (8.26) | (14.80) | (−2.69) | (−2.91) | (−13.32) | (−15.13) | (41.04) | (2.21) | ||

Greece | -0.34 | 0.15 | 1.00 | −0.24 | 0.10 | −0.01 | 0.14 | 0.50 | −0.10 | 0.01 |

(−19.48) | (8.26) | (−13.35) | (5.71) | (−0.91) | (7.75) | (30.73) | (−5.56) | (0.84) | ||

Ireland | 0.05 | 0.27 | −0.24 | 1.00 | −0.61 | −0.24 | 0.73 | 0.54 | −0.10 | 0.28 |

(2.86) | (14.80) | (−13.80) | (−40.99) | (−13.33) | (55.88) | (33.71) | (−5.36) | (15.24) | ||

Italy | 0.35 | −0.05 | 0.10 | −0.61 | 1.00 | −0.18 | 0.55 | 0.55 | 0.12 | 0.15 |

(19.80) | (−2.69) | (5.71) | (−40.99) | (−9.58) | (34.81) | (34.66) | (6.57) | (8.17) | ||

Japan | 0.21 | −0.05 | −0.01 | −0.24 | −0.18 | 1.00 | 0.36 | 0.21 | 0.13 | 0.28 (15.31) |

(11.57) | (−2.91) | (−0.91) | (−13.33) | (−9.58) | (20.23) | (11.33) | (7.06) | (15.31) | ||

Portugal | 0.17 | −0.24 | 0.14 | 0.73 | 0.55 | 0.36 | 1.00 | −0.33 | −0.07 | −0.21 (−11.52) |

(9.34) | (−13.32) | (7.75) | (55.88) | (34.81) | (20.23) | (−18.59) | (−3.90) | (−11.52) | ||

Spain | 0.21 | −0.27 | 0.50 | 0.54 | 0.55 | 0.21 | −0.33 | 1.00 | 0.02 | 0.06 |

(11.59) | (−15.13) | (30.73) | (33.71) | (34.66) | (11.33) | (−18.59) | (1.12) | (3.43) | ||

UK | −0.22 | 0.61 | −0.10 | −0.10 | 0.12 | 0.13 | −0.07 | 0.02 | 1.00 | 0.52 |

(−12.00) | (41.04) | (−5.56) | (−5.36) | (6.57) | (7.06) | (−3.90) | (1.12) | (32.06) | ||

USA | −0.19 | 0.04 | 0.01 | 0.28 | 0.15 | 0.28 | −0.21 | 0.06 | 0.52 | 1.00 |

(−10.36) | (2.21) | (0.84) | (15.24) | (8.17) | (15.31) | (−11.52) | (3.43) | (32.06) |

**Table 3.**Comparison between the two components of risk obtained with our proposed structural VAR model ($CoRisk$ and autoregressive parts) and the eigenvector centrality measures.

Country | Corisk | Autoreg | Eigenvector Centrality |
---|---|---|---|

France | 18.49 | 23.43 | 0.98 |

Germany | 17.43 | 12.12 | 0.82 |

Greece | 400.80 | 2454.28 | 0.16 |

Ireland | 58.77 | 118.89 | 0.60 |

Italy | −5.98 | 86.46 | 0.84 |

Japan | −2.62 | 33.44 | 0.51 |

Portugal | −9.14 | 183.05 | 0.66 |

Spain | −45.28 | 81.70 | 0.81 |

United Kingdom | 0.23 | 27.59 | 0.63 |

United States | 3.94 | 17.59 | 0.53 |

**Table 4.**August 2011: comparison between the root mean square errors (RMSE) obtained with a model composed by the solely autoregressive component (RMSE only autoregressive), and with our full structural VAR model (RMSE full).

Country | RMSE Only Autoregressive | RMSE Full |
---|---|---|

France | 3.86 | 3.42 |

Germany | 0.36 | 0.29 |

Greece | 5.65 | 4.33 |

Ireland | 2.86 | 2.82 |

Italy | 1.07 | 0.42 |

Japan | 0.22 | 0.23 |

Portugal | 1.63 | 2.38 |

Spain | 0.13 | 0.16 |

United Kingdom | 0.36 | 0.25 |

United States | 0.07 | 0.08 |

**Table 5.**August 2012: comparison between the root mean square errors (RMSE) obtained with our full structural VAR model (RMSE full) and with a model composed by the solely autoregressive component (RMSE only autoregressive).

Country | RMSE Only | RMSE Full |
---|---|---|

France | 0.83 | 0.57 |

Germany | 0.25 | 0.18 |

Greece | 104.31 | 71.04 |

Ireland | 0.82 | 0.18 |

Italy | 0.56 | 1.32 |

Japan | 0.55 | 0.53 |

Portugal | 1.43 | 1.19 |

Spain | 2.15 | 1.89 |

United Kingdom | 0.08 | 0.04 |

United States | 0.21 | 0.19 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Giudici, P.; Parisi, L. CoRisk: Credit Risk Contagion with Correlation Network Models. *Risks* **2018**, *6*, 95.
https://doi.org/10.3390/risks6030095

**AMA Style**

Giudici P, Parisi L. CoRisk: Credit Risk Contagion with Correlation Network Models. *Risks*. 2018; 6(3):95.
https://doi.org/10.3390/risks6030095

**Chicago/Turabian Style**

Giudici, Paolo, and Laura Parisi. 2018. "CoRisk: Credit Risk Contagion with Correlation Network Models" *Risks* 6, no. 3: 95.
https://doi.org/10.3390/risks6030095