# Interconnected Risk Contributions: A Heavy-Tail Approach to Analyze U.S. Financial Sectors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Risk Measurement Framework

#### 3.1. Multiple Risk Measures

**Definition 1.**

**Definition 2.**

#### 3.2. Shapley Value Methodology

## 4. Empirical Analysis

#### 4.1. The Data

**Table 1.**Summary statistics of the U.S. sector indexes from 2 January 1992 till 28 June 2013. The eighth column, denoted by “1% Str. Lev.”, is the 1% empirical quantile of the returns distribution, while the last column, denoted by “JB”, is the value of the Jarque–Berá test-statistics.

Name | Min | Max | Mean × ${10}^{3}$ | SD | Skewness | Kurtosis | 1% Str.Lev. | JB |
---|---|---|---|---|---|---|---|---|

Banks | −0.318 | 0.377 | 0.930 | 0.042 | −0.050 | 19.786 | −0.106 | 13,325.840 |

Fin. srvs | −0.243 | 0.242 | 1.588 | 0.037 | −0.008 | 8.326 | −0.096 | 1,341.471 |

Life insur. | −0.379 | 0.336 | 1.566 | 0.043 | −0.657 | 24.121 | −0.124 | 21,178.483 |

Non-life Insur. | −0.272 | 0.157 | 1.076 | 0.030 | −0.661 | 12.363 | −0.085 | 4,228.914 |

**Figure 1.**Cumulative returns of the different sectors: banks (blue line), financial services (red line), life insurance (dark line) and non-life insurance (green line). Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description.

#### 4.2. Estimation Results

**Table 2.**Log-likelihood values, Akaike (AIC) and Schwarz (BIC) information criteria of the proposed Student-t MS model with different number of components. Bold face indicates the selected model.

L | Log-Likelihood | AIC | BIC |
---|---|---|---|

2 | 11,889.544 | −23,713.088 | −23,410.697 |

3 | 11,713.089 | −23,320.177 | −23,053.355 |

4 | 11,969.138 | −23,788.276 | −23,546.954 |

5 | 11,946.912 | −23,695.824 | −23,197.420 |

6 | 11,973.013 | −23,696.025 | −23,066.727 |

**Table 3.**Maximum likelihood parameter estimates of the selected Student-t MS model with four components.

**µ**

_{l}, for $l=1,2,\cdots ,L$, denote the location parameters, while the diagonal matrices

**Λ**

_{l}and the full matrices

**Ω**

_{l}for $l=1,2,\cdots ,L$ are such that ${\sum}_{l}={\mathsf{\Lambda}}_{l}{\mathsf{\Omega}}_{l}{\mathsf{\Lambda}}_{l}$, where ${\sum}_{l}$ is the scale matrix of the Student-t distribution.

µ $\phantom{\rule{3.33333pt}{0ex}}\times {\mathbf{10}}^{\mathbf{3}}$ | Banks | Fin. Srvs | Life Insur. | Non-life Insur. |
---|---|---|---|---|

State 1 | −6.8905 | −0.2288 | −8.1268 | −2.7188 |

State 2 | −4.5542 | −4.2744 | −1.9875 | −3.4971 |

State 3 | 1.0459 | 1.8574 | 1.2758 | 1.9089 |

State 4 | 3.5818 | 4.4010 | 4.3724 | 2.9295 |

$\Lambda \times {\mathbf{10}}^{\mathbf{3}}$ | Banks | Fin. Srvs | Life Insur. | Non-life Insur. |

State 1 | 13.8891 | 7.0203 | 15.6855 | 4.3998 |

State 1 | 1.4302 | 1.6766 | 1.0333 | 1.0232 |

State 1 | 0.9998 | 0.6871 | 1.2204 | 0.3585 |

State 1 | 0.3427 | 0.5197 | 0.2827 | 0.3137 |

${\Omega}_{1}$ | Banks | Fin. Srvs | Life Insur. | Non-life Insur. |

Banks | 1.0000 | |||

Fin. srvs | 0.8399 | 1.0000 | ||

Life insur. | 0.8511 | 0.8248 | 1.0000 | |

Non-life insur. | 0.6966 | 0.8013 | 0.8015 | 1.0000 |

${\Omega}_{2}$ | Banks | Fin. Srvs | Life Insur. | Non-life Insur. |

Banks | 1.0000 | |||

Fin. srvs | 0.8573 | 1.0000 | ||

Life insur. | 0.7654 | 0.7771 | 1.0000 | |

Non-life insur. | 0.7683 | 0.7881 | 0.8205 | 1.0000 |

${\Omega}_{3}$ | Banks | Fin. Srvs | Life Insur. | Non-life Insur. |

Banks | 1.0000 | |||

Fin. srvs | 0.8807 | 1.0000 | ||

Life insur. | 0.8537 | 0.8939 | 1.0000 | |

Non-life insur. | 0.7827 | 0.8472 | 0.8555 | 1.0000 |

${\Omega}_{4}$ | Banks | Fin. Srvs | Life Insur. | Non-life Insur. |

Banks | 1.0000 | |||

Fin. srvs | 0.8620 | 1.0000 | ||

Life insur. | 0.7337 | 0.7678 | 1.0000 | |

Non-life insur. | 0.7250 | 0.7622 | 0.7792 | 1.0000 |

**Table 4.**Maximum likelihood estimates of the degrees of freedom

**ν**, the initial probability

**δ**and the transition probability matrix $\mathbf{\text{Q}}$ of the Markov chain for the selected Student-t MS model with four components.

ν | State 1 | State 2 | State 3 | State 4 |

15.6839 | 10.2542 | 11.0300 | 9.9473 | |

δ | State 1 | State 2 | State 3 | State 4 |

0.0000 | 1.0000 | 0.0000 | 0.0000 | |

$\mathbf{\text{Q}}$ | State 1 | State 2 | State 3 | State 4 |

State 1 | 0.8934 | 0.1066 | 0.0000 | 0.0000 |

State 2 | 0.0244 | 0.9608 | 0.0071 | 0.0077 |

State 3 | 0.0000 | 0.0000 | 0.9919 | 0.0081 |

State 4 | 0.0000 | 0.0052 | 0.0022 | 0.9926 |

**Λ**), as opposed to negative returns, where standard deviations are substantially lower. States 2 and 3 identify periods of low volatility associated with moderately negative and positive mean returns, respectively. This essentially implies that States 1 and 4 can be identified as periods of financial turbulence and stability, while States 2 and 3 are regimes where the financial system transits just before or immediately after a crisis period. This latter observation can be evinced also by inspecting Figure 2 displaying the Markovian predicted and smoothed probabilities of being in a given state at each time period, denoted by $\mathbb{P}\left(\right)open="("\; close=")">{S}_{t}\mid {\mathcal{I}}_{T}$. During the 2007–2008 GFC, for example, we observe that the predictive probability (red line) of being in State 1 (turbulence) is larger than 99%. During the period immediately before the 2007 crisis, covering most of the 2006 and 2007 years, till the collapse of the Bear Stearns hedge fund in August 2007, the system visits the transitory State 2, which corresponds to the pre-crisis regime. The same empirical findings are confirmed by inspecting the smoothed probabilities; the grey area in Figure 3. All of those results document the importance of choosing the right model specification in order to understand the global dynamics of the economic system.

**Ω**). As expected, correlations are higher during crisis periods, while during more stable phases, variances are relatively low and the contagion effect is less marked.

**Figure 2.**Total risk evaluated by multiple-conditional value-at-risk (CoVaR) (top) and multiple-conditional expected shortfall (CoES) (bottom) for the different sectors: banks (blue line), financial services (red line), life insurance (dark line) and non-life insurance (green line). Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description.

**Figure 3.**Smoothed probabilities of visiting the states of the Markovian chain $\mathbb{P}\left(\right)open="("\; close=")">{S}_{t}=j\mid {\mathcal{I}}_{T}$ for $j=1,\cdots ,4$ and $t=1,2,\cdots ,T$ (from top to bottom) implied by the Student-t model with four components. The superimposed red lines represent the predictive probabilities $\mathbb{P}\left(\right)open="("\; close=")">{S}_{t}=j\mid {\mathcal{I}}_{t-1}$ for $j=1,\cdots ,4$ and $t=1,2,\cdots ,T$. States 1 and 4 are identified as periods of financial turbulence and stability, respectively, while States 2 and 3 are regimes where the financial system transits just before or immediately after a crisis period. Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description.

**ν**and the transition probabilities

**Q**of the hidden Markov chain. Looking at the

**ν**parameters’ estimate, fat-tails have been detected, and this is in line with empirically-observed stylized facts. Moreover, the introduction of conditional Student-t distributions increases the state persistence significantly, resulting in longer and more stable volatility periods. This is confirmed by the transition matrix estimate. The large off-diagonal transition probabilities in all states, except the first one, confirm the large persistence of the transitory states (2 and 3), as well as State 4 of financial stability. On the contrary, State 1 of financial crisis is characterized by a smaller probability value on the main diagonal, denoting a small level of persistence of the crisis periods. The crisis state is also characterized by a moderately large probability to move to State 2, suggesting that after coming out from a crisis, the system enters a period of “moderate” financial turbulence.

#### 4.3. Risk Contributions

**Figure 4.**Shalpey value ${\Delta}^{\mathsf{M}}$-CoVaR of the different sectors against banks (

**a**), financial services (

**b**), life insurance (

**c**) and non-life insurance (

**d**). Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description. (a) Financial services (light gray), life (gray) and non-life (dark) insurance against banks; (b) banks (light gray), life (gray) and non-life (dark) insurance against financial services; (c) banks (light gray), financial services (gray) and non-life insurance (dark) against the life insurance index; (d) banks (light gray), financial services (gray) and life insurance (dark) against the non-life insurance index.

**Figure 5.**Shalpey value ${\Delta}^{\mathsf{M}}$-CoES of the different sectors against banks (

**a**), financial services (

**b**), life insurance (

**c**) and non-life insurance (

**d**). Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description. (a) Financial services (light gray), life (gray) and non-life (dark) insurance against banks; (b) banks (light gray), life (gray) and non-life (dark) insurance against financial services; (c) banks (light gray), financial services (gray) and non-life insurance (dark) against the life insurance index; (d) banks (light gray), financial services (gray) and life insurance (dark) against the non-life insurance index.

**Figure 6.**Comparisons of Shapley values ${\Delta}^{\mathsf{M}}$CoVaR for the different sectors. Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description. (

**a**) Distress of banks on financial services (dark) and vice versa (light gray); (

**b**) distress of banks on life insurance (dark) and vice versa (light gray); (

**c**) distress of banks on non-life insurance (dark) and vice versa (light gray); (

**d**) distress of financial srvs on life insurance (dark) and vice versa (light gray); (

**e**) distress of financial servis on non-life insurance (dark) and vice versa (light gray); (

**f**) distress of life insurance on non-life insurance (dark) and vice versa (light gray).

**Figure 7.**Comparisons of Shapley values ${\Delta}^{\mathsf{M}}$CoES for the different sectors. Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description. (

**a**) Distress of banks on financial services (dark) and vice versa (light gray); (

**b**) distress of banks on life insurance (dark) and vice versa (light gray); (

**c**) distress of banks on non-life insurance (dark) and vice versa (light gray); (

**d**) distress of financial services on life insurance (dark) and vice versa (light gray); (

**e**) distress of financial services on non-life insurance (dark) and vice versa (light gray); (

**f**) distress of life insurance on non-life insurance (dark) and vice versa (light gray).

**Table 5.**Comparison of the Shapley value ${\Delta}^{\mathsf{M}}$CoVaR (light gray) and the Adrian and Brunnermeier’s standard ΔCoVaR approach (light gray) for all of the sectors against each other. Vertical dotted lines represent major financial downturns: see Figure 1 for a detailed description.

Banks | Fin. Srvs | Life Ins. | non-life Ins. | |

Banks | ||||

Fin. Srvs | ||||

Life Ins. | ||||

non-life Ins. |

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## A. Index Descriptions

- Banks. Index name: Dow Jones U.S. Banks Total Stock Market Index (DWCBNK). Index description: banks providing a broad range of financial services, including retail banking, loans and money transmissions.
- Financial services. Index name: Dow Jones U.S. Financial Services Total Stock Market Index (DWCGFN). Index description:
- -
- Asset managers: companies that provide custodial, trustee and other related fiduciary services; includes mutual fund management companies.
- -
- Consumer finance: credit card companies and providers of personal finance services, such as personal loans and check cashing companies.
- -
- Specialty finance: companies engaged in financial activities not specified elsewhere. Includes companies not classified under equity investment instruments or non-equity investment instruments engaged primarily in owning stakes in a diversified range of companies.
- -
- Investment services: companies providing a range of specialized financial services, including securities brokers and dealers, online brokers and security or commodity exchanges.
- -
- Mortgage finance: companies that provide mortgages, mortgage insurance and other related services.

- Life insurance. Index name: Dow Jones U.S. Life Insurance Total Stock Market Index (DWCINL). Index description: companies engaged principally in life and health insurance.
- Non-life insurance. Index name: Dow Jones U.S. Life Insurance Total Stock Market Index (DWCNLI). Index description:
- -
- Full line insurance: insurance companies with life, health, property and casualty and reinsurance interests, no one of which predominates.
- -
- Insurance brokers: insurance brokers and agencies.
- -
- Property and casualty insurance: companies engaged principally in accident, fire, automotive, marine, malpractice and other classes of non-life insurance.
- -
- Reinsurance: companies engaged principally in reinsurance.

## Conflicts of Interest

## References

- J.D. Cummins, and M.A. Weiss. “Systemic Risk and the Insurance Industry.” J. Risk Insur. 81 (2014): 489–527. [Google Scholar] [CrossRef]
- J.D. Cummins, and M.A. Weiss. “Systemic Risk and the insurance Industry.” In Handbook of Insurance, 2nd ed. Edited by Forthcoming in Georges Dionne. NY, USA: Springer, 2013, pp. 745–794. [Google Scholar]
- R. Pozsar, T. Adrian, A. Ashcraft, and H. Boesky. “Shadow Banking.” In Federal Reserve Bank of New York Staff Report No. 458. NY, USA, 2010. [Google Scholar]
- S. Harrington. “The financial crisis, systemic risk, and the future of insurance regulation.” J. Risk Insur. 76 (2009): 785–819. [Google Scholar] [CrossRef]
- S. Markose, S. Giansante, and R.A. Shaghaghi. ““Too interconnected to fail”: Financial network of U.S. CDS Market: topological fragility and systemic risk.” J. Econ. Behav. Organ. 83 (2012): 627–646. [Google Scholar] [CrossRef] [Green Version]
- M. Billio, M. Getmansky, A.W. Lo, and L. Pellizon. “Econometric measures of connectedness and systemic risk in the finance and insurance sectors.” J. Financ. Econ. 104 (2012): 535–559. [Google Scholar] [CrossRef]
- Z. Adams, R Füss, and R. Gropp. “Modeling spillover effects among financial institutions: A State-dependent Sensitivity Value-at-Risk Approach.” J. Financ. Quant. Anal., 2015. forthcoming. [Google Scholar] [CrossRef]
- V.V. Acharya, L.H. Pedersen, T. Philippon, and M. Richardson. “Measuring systemic Risk.” In Working Paper, Federal Reserve Bank of Cleveland. OH, USA, 2010. [Google Scholar]
- N. Hautsch, J. Schaumburg, and M. Schienle. “Financial network systemic risk contributions.” Rev. Financ. 18 (2014): 1–54. [Google Scholar] [CrossRef]
- O. Bernal, J.-Y. Gnabo, and G. Guilmin. “Assessing the contribution of banks, insurance and other financial services to systemic risk.” J. Bank. Financ. 47 (2014): 270–287. [Google Scholar] [CrossRef]
- E.C. Brechmann, K. Hendrich, and C. Czado. “Conditional copula simulations for systemic risk stress testing.” Insur.: Math. Econ. 53 (2013): 722–732. [Google Scholar] [CrossRef]
- H. Chen, J.D. Cummins, K.S. Viswanathan, and M.A. Weiss. “Systemic risk and the interconnectedness between banks and insurers: An econometric analysis.” J. Risk Insur., 2013. [Google Scholar] [CrossRef]
- N. Podlich, and M. Wedow. “Are insurers SIFIs? A MGARCH model to measure interconnectedness.” Appl. Econ. Lett. 20 (2013): 677–681. [Google Scholar] [CrossRef]
- T. Adrian, and M.K. Brunnermeier. “CoVaR.” In Working Paper Federal Reserve Bank of New York. 2014. [Google Scholar]
- M. Bernardi, G. Gayraud, and L. Petrella. “Bayesian tail risk interdependence using quantile regression.” Bayesian Anal., 2015. forthcoming. [Google Scholar] [CrossRef]
- C. Castro, and S. Ferrari. “Measuring and testing for the systemically important financial institutions.” J. Emp. Financ. 25 (2013): 1–14. [Google Scholar] [CrossRef]
- G. Girardi, and A.T. Ergün. “Systemic risk measurement: multivariate GARCH estimation of CoVaR.” J. Bank. Financ. 37 (2013): 3169–3180. [Google Scholar] [CrossRef]
- M. Jäger-Ambrożewicz. “Closed form solutions of measures of systemic risk.” Ann. Univ. Sci. Budapestinensis Rolando Eötvös Nomin. Sect. Comput. 39 (2013): 215–225. [Google Scholar]
- M.A. Sordo, A. Suárez–Llorens, and A.J. Bello. “Comparison of conditional distributions in portfolios of dependent risks.” Insur.: Math. Econ., 2014. fortcoming. [Google Scholar] [CrossRef]
- D. Bisias, M. Flood, A.W. Lo, and S. Valavanis. “A survey of systemic risk analytics.” Annu. Rev. Financ. Econ. 4 (2012): 255–296. [Google Scholar] [CrossRef]
- M. Bernardi, A. Maruotti, and L. Petrella. “Multivariate Markov-Switching models and tail risk interdependence.” Preprint arXiv:1312.6407 [stat.ME]. 2013. Available online: http://arxiv.org/abs/1312.6407 accessed on 2 April 2015).
- Z. Cao. “Multi–CoVaR and Shapley value: A systemic risk measure.” Banq. France Work. Pap., 2013. [Google Scholar]
- J. Bulla. “Hidden Markov Models with t Components. Increased Persistence and Other Aspects.” Quant. Financ. 11 (2011): 459–475. [Google Scholar] [CrossRef] [Green Version]
- C.W.J. Granger, and Z. Ding. “Some properties of absolute return: An alternative measure of risk.” Ann. Econ. Stat. 40 (1995): 67–91. [Google Scholar]
- C.W.J. Granger, and Z. Ding. Stylised Facts on the Temporal and Distributional Properties of Daily Data from Speculative Markets. Unpublished Paper; San Diego, CA, USA: Department of Economics, University of California, 1995. [Google Scholar]
- T. Rydén, T. Teräsvirta, and S. Asbrink. “Stylized facts of daily return series and the hidden Markov model.” J. Appl. Econ. 13 (1998): 217–244. [Google Scholar] [CrossRef]
- L. Shapley. “A value for n–person Games.” Ann. Math. Stud. 28 (1953): 307–317. [Google Scholar]
- O. Cappé, E. Moulines, and T. Rydén. “Inference in Hidden Markov Models.” Berlin, Germany: Springer Series in Statistics, Springer–Verlag, 2005. [Google Scholar]
- W. Zucchini, and I. MacDonald. Hidden Markov Models for Time Series: An Introduction Using R, 2009.
- P. Dymarski. Hidden Markov Models, Theory and Applications. Rijeka, Croatia: Intech, 2011, pp. 207–222. [Google Scholar]
- G. Amisano, and J. Geweke. “Hierarchical Markov Normal Mixture models with applications to financial asset returns.” J. Appl. Econom. 26 (2011): 1–29. [Google Scholar] [CrossRef]
- J. Geweke, and G. Amisano. “Comparing and evaluating Bayesian predictive distributions of asset returns.” Int. J. Forecast. 26 (2010): 216–230. [Google Scholar] [CrossRef]
- A.P. Dempster, N.M. Laird, and D.B. Rubin. “Maximum likelihood from incomplete data using the EM algorithm (with discussion).” J. R. Stat. Soc. Ser. B 39 (1977): 1–39. [Google Scholar]
- N. Tarashev, C. Borio, and K. Tsatsaronis. “Attributing systemic risk to individual institutions: Methodology and policy applications.” BIS Work. Pap. No. 308, 2010. [Google Scholar] [CrossRef]
- T.R. Berry–Stölzle, G.P. Nini, and S. Wende. “External financing in the life insurance industry: evidence from the financial crisis.” J. Risk Insur. 81 (2011): 529–562. [Google Scholar] [CrossRef]
- T. Rydén. “EM versus Markov chain Monte Carlo for estimation of Hidden Markov models: A computational perspective.” Bayesian Anal. 3 (2008): 659–688. [Google Scholar] [CrossRef]
- G.B. Gorton, and A. Metrick. “Securitised Banking and the Run on Repo. NBER Working Papers No. 15223, National Bureau of Economic Research, Inc.” J. Financ. Econ., 2009. forthcoming. [Google Scholar]
- M. Bell, and B. Keller. Insurance and Stability: The Reform of Insurance Regulation. Zurich, Switzerland: Zurich Financial Services Group, 2009. [Google Scholar]

© 2015 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 license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bernardi, M.; Petrella, L.
Interconnected Risk Contributions: A Heavy-Tail Approach to Analyze U.S. Financial Sectors. *J. Risk Financial Manag.* **2015**, *8*, 198-226.
https://doi.org/10.3390/jrfm8020198

**AMA Style**

Bernardi M, Petrella L.
Interconnected Risk Contributions: A Heavy-Tail Approach to Analyze U.S. Financial Sectors. *Journal of Risk and Financial Management*. 2015; 8(2):198-226.
https://doi.org/10.3390/jrfm8020198

**Chicago/Turabian Style**

Bernardi, Mauro, and Lea Petrella.
2015. "Interconnected Risk Contributions: A Heavy-Tail Approach to Analyze U.S. Financial Sectors" *Journal of Risk and Financial Management* 8, no. 2: 198-226.
https://doi.org/10.3390/jrfm8020198