# The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio

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## Abstract

**:**

## 1. Introduction

## 2. Insurance Framework

#### 2.1. The Technical Risk Premium

#### 2.1.1. One policy case

#### 2.1.2. The case of a portfolio of N policies

#### 2.2. Cost of Capital and Risk Loading

#### 2.3. Risk Measures

## 3. Stochastic Modeling

#### 3.1. The First Model, under the iid Assumption

#### 3.1.1. Numerical Application

Number of Losses | Policy Loss | Probability Mass | cdf |
---|---|---|---|

k | $l\phantom{\rule{3.33333pt}{0ex}}X\left(\omega \right)$ | $\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{P}}[{S}_{1n}=k]$ | $\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{P}}[{S}_{1n}\le k]$ |

0 | 0 | 33.490% | 33.490% |

1 | 10 | 40.188% | 73.678% |

2 | 20 | 20.094% | 93.771% |

3 | 30 | 5.358% | 99.130% |

4 | 40 | 0.804% | 99.934% |

5 | 50 | 0.064% | 99.998% |

6 | 60 | 0.002% | 100.000% |

**Table 2.**The risk loading per policy as a function of the number N of policies in the portfolio (with $n=6$).

Risk Measure | Number N of Policies | Risk Loading R per Policy with Probability | ||
---|---|---|---|---|

ρ | p = 1/6 | p = 1/4 | p = 1/2 | |

VaR | ||||

1 | 3.000 | 3.750 | 4.500 | |

5 | 1.500 | 1.650 | 1.800 | |

10 | 1.050 | 1.200 | 1.350 | |

50 | 0.450 | 0.540 | 0.600 | |

100 | 0.330 | 0.375 | 0.420 | |

1,000 | 0.102 | 0.117 | 0.135 | |

10,000 | 0.032 | 0.037 | 0.043 | |

TVaR | ||||

1 | 3.226 | 3.945 | 4.500 | |

5 | 1.644 | 1.817 | 1.963 | |

10 | 1.164 | 1.330 | 1.482 | |

50 | 0.510 | 0.707 | 0.675 | |

100 | 0.372 | 0.425 | 0.476 | |

1,000 | 0.116 | 0.134 | 0.154 | |

10,000 | 0.037 | 0.042 | 0.049 | |

$\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{E}}\left[L\right]/N$ | 10.00 | 15.00 | 30.00 |

#### 3.2. Introducing a Structure of Dependence to Reveal a Systemic Risk

#### 3.2.1. A Dependent Model, but Conditionally Independent

#### 3.2.2. Numerical Application

**Table 3.**For Model Equation (11), the risk loading per policy as a function of the probability of occurrence of a systemic risk in the portfolio using VaR and TVaR measures with $\alpha =99\%$. The probability of giving a loss in a state of systemic risk is chosen to be $q=50\%$.

Risk Measure | Number N of Policies | In a Normal State | Risk Loading R with Occurrence of a Crisis State | |||
---|---|---|---|---|---|---|

ρ | $\tilde{p}=0$ | $\tilde{p}=0.1\%$ | $\tilde{p}=1.0\%$ | $\tilde{p}=5.0\%$ | $\tilde{p}=10.0\%$ | |

VaR | ||||||

1 | 3.000 | 2.997 | 4.469 | 4.346 | 5.693 | |

5 | 1.500 | 1.497 | 2.070 | 3.450 | 3.900 | |

10 | 1.050 | 1.047 | 1.770 | 3.300 | 3.450 | |

50 | 0.450 | 0.477 | 1.410 | 3.060 | 3.030 | |

100 | 0.330 | 0.327 | 1.605 | 3.000 | 2.940 | |

1,000 | 0.102 | 0.101 | 2.549 | 2.900 | 2.775 | |

10,000 | 0.032 | 0.029 | 2.837 | 2.866 | 2.724 | |

TVaR | ||||||

1 | 3.226 | 3.232 | 4.711 | 4.755 | 5.899 | |

5 | 1.644 | 1.707 | 2.956 | 3.823 | 4.146 | |

10 | 1.164 | 1.266 | 2.973 | 3.578 | 3.665 | |

50 | 0.510 | 0.760 | 2.970 | 3.196 | 3.141 | |

100 | 0.372 | 0.596 | 2.970 | 3.098 | 3.020 | |

1,000 | 0.116 | 0.396 | 2.970 | 2.931 | 2.802 | |

10,000 | 0.037 | 0.323 | 2.970 | 2.876 | 2.732 | |

$\phantom{\rule{1.em}{0ex}}\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{E}}\left[L\right]/N$ | 10.00 | 10.02 | 10.20 | 11.00 | 12.00 |

#### 3.2.3. A More Realistic Setting to Introduce a Systemic Risk

#### 3.2.4. Numerical Application

**Table 4.**For Model Equation (14), the risk loading per policy as a function of the probability of occurrence of a systemic risk in the portfolio using VaR and TVaR measures with $\alpha =99\%$. The probability of giving a loss in a state of systemic risk is chosen to be $q=50\%$.

Risk Measure | Number N of Policies | In a Normal State | Risk Loading R with Occurrence of a Crisis State | |||
---|---|---|---|---|---|---|

ρ | $\tilde{p}=0$ | $\tilde{p}=0.1\%$ | $\tilde{p}=1.0\%$ | $\tilde{p}=5.0\%$ | $\tilde{p}=10.0\%$ | |

VaR | ||||||

1 | 3.000 | 2.997 | 2.969 | 4.350 | 4.200 | |

5 | 1.500 | 1.497 | 1.470 | 1.650 | 1.800 | |

10 | 1.050 | 1.047 | 1.170 | 1.350 | 1.500 | |

50 | 0.450 | 0.477 | 0.690 | 0.990 | 1.200 | |

100 | 0.330 | 0.357 | 0.615 | 0.945 | 1.170 | |

1,000 | 0.102 | 0.112 | 0.517 | 0.882 | 1.186 | |

10,000 | 0.032 | 0.033 | 0.485 | 0.860 | 1.196 | |

100,000 | 0.010 | 0.008 | 0.475 | 0.853 | 1.199 | |

TVaR | ||||||

1 | 3.226 | 3.232 | 4.485 | 4.515 | 4.448 | |

5 | 1.644 | 1.792 | 1.870 | 2.056 | 2.226 | |

10 | 1.164 | 1.252 | 1.342 | 1.604 | 1.804 | |

50 | 0.510 | 0.588 | 0.824 | 1.183 | 1.408 | |

100 | 0.375 | 0.473 | 0.740 | 1.118 | 1.358 | |

1,000 | 0.116 | 0.348 | 0.605 | 1.013 | 1.295 | |

10,000 | 0.037 | 0.313 | 0.563 | 0.981 | 1.276 | |

100,000 | 0.012 | 0.301 | 0.550 | 0.970 | 1.269 | |

$\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{E}}\left[L\right]/N$ | 10.00 | 10.02 | 10.20 | 11.00 | 12.00 |

Risk Measure | Number N of Policies | In a Normal State | Risk Loading R with Occurrence of a Crisis State | |||
---|---|---|---|---|---|---|

ρ | $\tilde{p}=0$ | $\tilde{p}=0.1\%$ | $\tilde{p}=1.0\%$ | $\tilde{p}=5.0\%$ | $\tilde{p}=10.0\%$ | |

VaR | ||||||

1 million | 0.330 | 0.357 | 0.615 | 0.945 | 1.170 | |

10 million | 0.330 | 0.357 | 0.615 | 0.945 | 1.155 | |

20 million | 0.330 | 0.357 | 0.615 | 0.945 | 1.170 | |

TVaR | ||||||

1 million | 0.375 | 0.476 | 0.738 | 1.115 | 1.358 | |

10 million | 0.374 | 0.472 | 0.739 | 1.117 | 1.357 | |

20 million | 0.375 | 0.473 | 0.740 | 1.118 | 1.358 | |

$\phantom{\rule{1.em}{0ex}}\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{E}}\left[L\right]/N$ | 10.00 | 10.02 | 10.20 | 11.00 | 12.00 |

## 4. Comparison and Discussion

Model | $\mathit{\text{I}}\phantom{\rule{-2.0pt}{0ex}}\mathit{\text{E}}\left[\mathit{L}\right]/\mathit{N}$ | $\mathit{v}\mathit{a}\mathit{r}\left(\mathit{L}\right)/{\mathit{N}}^{\mathbf{2}}$ |
---|---|---|

Equation (7) | $ln\phantom{\rule{3.33333pt}{0ex}}p$ | $\frac{{l}^{2}n}{N}\phantom{\rule{3.33333pt}{0ex}}p(1-p)$ |

Equation (11) | $ln\phantom{\rule{3.33333pt}{0ex}}\left(\tilde{p}\phantom{\rule{3.33333pt}{0ex}}q\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}(1-\tilde{p})\phantom{\rule{3.33333pt}{0ex}}p\right)$ | $\frac{{l}^{2}n}{N}\left(q(1-q)\tilde{p}+p(1-p)(1-\tilde{p})\right)+{\mathbf{l}}^{\mathbf{2}}{\mathbf{n}}^{\mathbf{2}}{(\mathbf{q}-\mathbf{p})}^{\mathbf{2}}\tilde{\mathbf{p}}(\mathbf{1}-\tilde{\mathbf{p}})$ |

Equation (14) | $ln\phantom{\rule{3.33333pt}{0ex}}\left(\tilde{p}\phantom{\rule{3.33333pt}{0ex}}q\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}(1-\tilde{p})\phantom{\rule{3.33333pt}{0ex}}p\right)$ | $\frac{{l}^{2}n}{N}\left(q(1-q)\tilde{p}+p(1-p)(1-\tilde{p})\right)+{\mathbf{l}}^{\mathbf{2}}\mathbf{n}\phantom{\rule{3.33333pt}{0ex}}{(\mathbf{q}-\mathbf{p})}^{\mathbf{2}}\tilde{\mathbf{p}}(\mathbf{1}-\tilde{\mathbf{p}})$ |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- R.J. Caballero, and P. Kurlat. “The ‘Surprising’ Origin and Nature of Financial Crises: A Macroeconomic Policy Proposal.” Financial Stability and Macroeconomic Policy. Federal Reserve Bank of Kansas City 2009. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1473918 (accessed on 17 November 2009).
- R. Cont, A. Moussa, and E. Santos. “Network structure and systemic risk in banking systems.” In Handbook of Systemic Risk. Edited by L. Fouque. Cambridge, UK: Cambridge University Press, 2013. [Google Scholar]
- V.V. Acharya, L.H. Pedersen, T. Philippon, and M. Richardson. “Measuring Systemic Risk.” Preprint. 2010. Available online: http://pages.stern.nyu.edu/sternfin/vacharya/public_html/MeasuringSystemicRisk_final.pdf (accessed on 24 March 2014).
- M.K. Brunnermeier, and P. Cheridito. “Measuring and allocating systemic risk.” Working Paper. 2013. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2372472 (accessed on 24 March 2014).
- M.K. Brunnermeier, and Y. Sannikov. “A Macroeconomic Model with a Financial Sector.” Working Paper. 2013. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2160894 (accessed on 24 March 2014).
- Z. He, and A. Krishnamurthy. “A Macroeconomic Framework for Quantifying Systemic Risk.” Chicago Booth Paper No. 12-37. 2014. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2133412 (accessed on 24 March 2014).
- D. Martinez-Miera, and J. Suarez. “A Macroeconomic Model of Endogenous Systemic Risk Taking.” Working Paper. 2012. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2153575 (accessed on 24 March 2014).
- M. Billio, M. Getmansky, A.W. Lo, and L. Pelizzon. “Economic Measures of the Systemic Risk in the Finance and Insurance Sectors.” NBER Working Paper Series no. 16223. 2010. Available online: http://www.nber.org/papers/w16223 (accessed on 24 March 2014).
- D. Bisias, M. Flood, A.W. Lo, and S. Valavanis. “A survey of systemic risk analytics.” Office of Financial Research Working Paper. 2012. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1983602 (accessed on 24 March 2014).
- C.T. Brownlees, and R. Engle. “Volatility, Correlation and Tails for Systemic Risk Measurement.” Working Paper. 2012. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1611229 (accessed on 24 March 2014).
- D. Duffie. “Systemic Risk Exposures: A 10 by 10 by 10 Approach.” NBER Working Paper No. 17281. 2011. Available online: http://www.nber.org/papers/w17281 (accessed on 24 March 2014).
- S. Giglio, B. Kelly, and S. Pruitt. “Systemic Risk and the Macroeconomy: An Empirical Evaluation.” Chicago Booth Paper No. 12-49. 2013. Available online: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2158347 (accessed on 24 March 2014).
- R. Bürgi, M. Dacorogna, and R. Iles. “Risk Aggregation, dependence structure and diversification benefit.” In Stress testing for financial institutions. Edited by D. Rösch and H. Scheule. London, UK: Incisive Media, 2008. [Google Scholar]
- S. Emmer, M. Kratz, and D. Tasche. “What Is the Best Risk Measure in Practice? A Comparison of Standard Measures.” ArXiV pre-print arXiv:1312.1645. 2013. Available online: http://arxiv.org/abs/1312.1645 (accessed on 24 March 2014).
- S Emmer, and D. Tasche. “Calculating Credit Risk Capital Charges with the One-factor Model.” J. Risk 7 (2004): 85–103. [Google Scholar]
- M.M. Dacorogna, and C. Hummel. “Alea jacta est, an illustrative example of pricing risk.” In SCOR Technical Newsletter. Zurich, Switzerland: SCOR Global P&C, 2008, pp. 1–4. [Google Scholar]
- A. McNeil, R Frey, and P. Embrechts. Quantitative Risk Management. Princeton, NJ, USA: Princeton University Press, 2005. [Google Scholar]
- P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. “Coherent measures of risks.” Math. Financ. 9 (1999): 203–228. [Google Scholar] [CrossRef]
- M. Busse, M. Dacorogna, and M. Kratz. “Does risk diversification always work? The answer through simple modeling.” SCOR Pap. 24 (2013): 1–19. [Google Scholar]
- B.S. Everitt, and D.J. Hand. Finite Mixture Distributions. London, UK: Chapman & Hall, 1981. [Google Scholar]

^{1}We use here the word “risk” instead of “loss”. In fact, these two words are used for one another in an insurance context.

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## Share and Cite

**MDPI and ACS Style**

Busse, M.; Dacorogna, M.; Kratz, M.
The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio. *Risks* **2014**, *2*, 260-276.
https://doi.org/10.3390/risks2030260

**AMA Style**

Busse M, Dacorogna M, Kratz M.
The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio. *Risks*. 2014; 2(3):260-276.
https://doi.org/10.3390/risks2030260

**Chicago/Turabian Style**

Busse, Marc, Michel Dacorogna, and Marie Kratz.
2014. "The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio" *Risks* 2, no. 3: 260-276.
https://doi.org/10.3390/risks2030260