# Generating VaR Scenarios under Solvency II with Product Beta Distributions

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## Abstract

**:**

## 1. Introduction

## 2. The Monte Carlo Algorithm

- Choose an index I randomly according to a uniform distribution over $\{1,\dots ,n\}$.
- Generate independently d random variables ${Z}_{1}$, …, ${Z}_{d}$ where ${Z}_{k}$ follows a Beta distribution with parameters $(m+1){\widehat{F}}_{k}({x}_{kI})$ and $(m+1)\left(1-{\widehat{F}}_{k}({x}_{kI})\right)$ (product beta distribution).
- Set ${Y}_{k}:={\widehat{F}}_{k}^{-1}({Z}_{k})$.

## 3. Case Study

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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No. | Risk ${\mathit{X}}_{1}$ | Risk ${\mathit{X}}_{2}$ |
---|---|---|

1 | 0.468 | 0.966 |

2 | 9.951 | 2.679 |

3 | 0.866 | 0.897 |

4 | 6.731 | 2.249 |

5 | 1.421 | 0.956 |

6 | 2.040 | 1.141 |

7 | 2.967 | 1.707 |

8 | 1.200 | 1.008 |

9 | 0.426 | 1.065 |

10 | 1.946 | 1.162 |

11 | 0.676 | 0.918 |

12 | 1.184 | 1.336 |

13 | 0.960 | 0.933 |

14 | 1.972 | 1.077 |

15 | 1.549 | 1.041 |

16 | 0.819 | 0.899 |

17 | 0.063 | 0.710 |

18 | 1.280 | 1.118 |

19 | 0.824 | 0.894 |

20 | 0.227 | 0.837 |

$\mathit{\mu}$ | $\mathit{\sigma}$ | |
---|---|---|

$ln({X}_{1})$ | 0.0954 | 1.1909 |

$ln({X}_{2})$ | –0.0437 | 0.2857 |

m = 15 | m = 20 | m = 25 | m = 30 | m = 50 | m = 100 | Kernel Density | |
---|---|---|---|---|---|---|---|

${\widehat{\mathrm{VaR}}}_{0.05}$ | 13.987 | 12.978 | 12.347 | 12.016 | 11.341 | 10.908 | 11.754 |

${\widehat{\mathrm{VaR}}}_{0.01}$ | 40.637 | 31.235 | 26.989 | 23.966 | 19.498 | 16.580 | 17.272 |

${\widehat{\mathrm{VaR}}}_{0.005}$ | 60.752 | 44.270 | 36.410 | 30.846 | 23.390 | 18.864 | 19.087 |

Bernstein | NB Rook, a = 7 | NB UF, a = 7 | NB Rook, a = 15 | NB UF, a = 15 | |
---|---|---|---|---|---|

${\widehat{\mathrm{VaR}}}_{0.05}$ | 7.166 | 6.885 | 7.016 | 6.974 | 7.155 |

${\widehat{\mathrm{VaR}}}_{0.01}$ | 15.634 | 15.973 | 15.744 | 15.877 | 16.059 |

${\widehat{\mathrm{VaR}}}_{0.005}$ | 21.105 | 20.801 | 21.311 | 20.256 | 21.733 |

Gamma Rook, a = 7 | Gamma UF, a = 7 | Gamma Rook, a = 15 | Gamma UF, a = 15 | |
---|---|---|---|---|

${\widehat{\mathrm{VaR}}}_{0.05}$ | 9.330 | 10.072 | 9.522 | 10.191 |

${\widehat{\mathrm{VaR}}}_{0.01}$ | 18.113 | 21.224 | 18.550 | 21.428 |

${\widehat{\mathrm{VaR}}}_{0.005}$ | 22.933 | 28.123 | 23.079 | 28.588 |

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**MDPI and ACS Style**

Pfeifer, D.; Ragulina, O.
Generating VaR Scenarios under Solvency II with Product Beta Distributions. *Risks* **2018**, *6*, 122.
https://doi.org/10.3390/risks6040122

**AMA Style**

Pfeifer D, Ragulina O.
Generating VaR Scenarios under Solvency II with Product Beta Distributions. *Risks*. 2018; 6(4):122.
https://doi.org/10.3390/risks6040122

**Chicago/Turabian Style**

Pfeifer, Dietmar, and Olena Ragulina.
2018. "Generating VaR Scenarios under Solvency II with Product Beta Distributions" *Risks* 6, no. 4: 122.
https://doi.org/10.3390/risks6040122