# The Effect of Non-Proportional Reinsurance: A Revision of Solvency II Standard Formula

## Abstract

**:**

## 1. Introduction

## 2. Some Notations and Preliminaries

- $\tilde{K}$ is a r.v. denoting the claim count. As is usually provided in literature (see Daykin et al. (1994) and Savelli and Clemente (2009)), the number of claims’ distribution follows the Poisson law ($\tilde{K}\sim Poi(n\xb7\tilde{q})$), with an expected number of claims n disturbed by a structure variable $\tilde{q}$.
- $\tilde{q}$ is a mixing variable (or contagion parameter) (see Gisler (2009), Meyers and Shenker (1982), and Savelli and Clemente (2009)) that describes the parameter uncertainty on the number of claims. It is assumed that $E\left(\tilde{q}\right)=1$ and that the r.v. is defined only for positive values.
- ${\tilde{Z}}_{h}$ is the amount of the claim h (severity). As usual, ${\tilde{Z}}_{1},{\tilde{Z}}_{2},\dots $ are mutually independent and identically distributed r.v.’s, each independent of the number of claims $\tilde{K}$. We denote with ${a}_{j,\tilde{Z}}$ raw moments of order j.

## 3. Non-Proportional Reinsurance in QIS5

- The standard deviation of the ratio of gross losses to gross premiums $\frac{{\tilde{X}}_{pre,i}^{gross}}{{V}_{pre,i}^{gross}}$ is modified to take into account XL through the net-to-gross ratio of coefficients of variation: $\frac{C{V}_{{\tilde{X}}_{pre,i}^{net}}^{CPP}}{C{V}_{{\tilde{X}}_{pre,i}^{gross}}^{CPP}}$. QIS5 implicitly assumes that expected losses decrease in the same proportion as the volume measure after reinsurance: $E\left[\frac{{\tilde{X}}_{pre,i}^{net}}{{\tilde{X}}_{pre,i}^{gross}}\right]=\frac{{V}_{pre,i}^{net}}{{V}_{pre,i}^{gross}}$. In other words, Equation (7) assumes that loading coefficients of the reinsurer are equal to loading coefficients of the cedent.
- The size of insurers’ portfolio is not considered. Risk mitigation usually has a different effect because of a different weight of pooling and non-pooling risk (see, e.g., International Actuarial Association (2004)).
- Both the gross and net reinsurance distributions are assumed as lognormal. The skewness of this distribution being strictly related to the coefficient of variation5, QIS5 assumes that skewness ${\gamma}_{\tilde{X}}$ reduces in a similar way to CV (${\gamma}_{{\tilde{X}}_{pre,i}^{net}}\approx \frac{C{V}_{{\tilde{X}}_{pre,i}^{net}}}{C{V}_{{\tilde{X}}_{pre,i}^{gross}}}{\gamma}_{{\tilde{X}}_{pre,i}^{gross}}$) for usual volatilities. The XL acting on the tail of the severity distribution, it should be evaluated if the mitigation of skewness is greater than the mitigation of volatility.

## 4. Non-Proportional Reinsurance in Commission Delegated Regulation

## 5. A Proposal of an Alternative Non-Proportional Factor

## 6. Some Examples of Numerical Impact

## 7. Conclusions

## Conflicts of Interest

## References

- Beard, R. E., T. Pentikäinen, and E. Pesonen. 1984. Risk Theory, 3rd ed. London: Chapman & Hall. [Google Scholar]
- Bühlmann, Hans. 1970. Mathematical Methods in Risk Theory. New York: Springer. [Google Scholar]
- Clark, David R. 1996. Basics of Reinsurance Pricing. CAS Study Notes 41–43: 1–52. [Google Scholar]
- Clemente, Gian Paolo, Nino Savelli, and Diego Zappa. 2015. The Impact of Reinsurance Strategies on Capital Requirements for Premium Risk in Insurance. Risks 3: 164–82. [Google Scholar] [CrossRef]
- Clemente, Gian Paolo, and Nino Savelli. 2017. Actuarial Improvements of Standard Formula for Non-Life Underwriting Risk. In Insurance Regulation in the European Union Solvency II and Beyond. Basingstoke: Palgrave Macmillan. [Google Scholar]
- Coutts, Stewart M., and Timothy R. H. Thomas. 1997a. Capital and risk and their relationship to reinsurance programmes. Paper presented at 5th International Conference on Insurance Solvency and Finance, London, UK, June 17. [Google Scholar]
- Coutts, Stewart M., and Timothy R. H. Thomas. 1997b. Modelling the impact of reinsurance on financial strength. British Actuarial Journal 3: 583–653. [Google Scholar] [CrossRef]
- Daykin, Chris D., Teivo Pentikäinen, and Martti Pesonen. 1994. Practical Risk Theory for Actuaries, Monographs on Statistics and Applied Probability 53. London: Chapman & Hall. [Google Scholar]
- De Lourdes Centeno, Maria. 1995. The effect of the retention limit on risk reserve. ASTIN Bulletin 25: 67–74. [Google Scholar] [CrossRef]
- EIOPA. 2017. Consultation Paper on EIOPA’s First Set of Advice to the European Commission on Specific Items in the Solvency II Delegated Regulation. Technical Report. Frankfurt: European Insurance and Occupational Pensions Authority (EIOPA). [Google Scholar]
- EIOPA. 2018. Consultation Paper on EIOPA’s Second Set of Advice to the European Commission on Specific Items in the Solvency II Delegated Regulation. Technical Report. Frankfurt: European Insurance and Occupational Pensions Authority (EIOPA). [Google Scholar]
- European Commission. 2015. Commission Delegated Regulation (EU) 2015/35 supplementing Directive 2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II), 10 of October 2014. Official Journal of the EU 58: 1–797. [Google Scholar]
- European Commission. 2010. Quantitative Impact Study 5—Technical Specifications. Brussels: European Commission. [Google Scholar]
- Gisler, Alois. 2009. The Insurance Risk in the SST and in Solvency II: Modelling and Parameter Estimation. Helsinki: Astin Colloquium. [Google Scholar]
- Hurlimann, Werner. 2005. Excess of loss reinsurance with reinstatements revisited. Astin Bulletin 35: 211–38. [Google Scholar] [CrossRef]
- Klugman, Stuart A., Harry H. Panjer, and Gordon E. Wilmot. 2008. Loss Models: From Data to Decisions, 3rd ed. Wiley Series in Probability and Statistics; Hoboken: John Wiley & Sons. [Google Scholar]
- International Actuarial Association. 2004. A Global Framework for Insurer Solvency Assessment. Report of Insurer Solvency Assessment Working Party. Ottawa: International Actuarial Association. [Google Scholar]
- Meyers, Glenn, and Nathaniel Shenker. 1982. Parameter Uncertainty in the Collective Risk Model, Casualty Actuarial Society, Discussion Paper Program. pp. 253–300. Available online: https://www.casact.org/pubs/proceed/proceed83/83111.pdf (accessed on 1 May 2018).
- Savelli, Nino, and Gian Paolo Clemente. 2011. Hierarchical Structures in the aggregation of Premium Risk for Insurance Underwriting. Scandinavian Actuarial Journal 3: 193–213. [Google Scholar] [CrossRef]
- Savelli, Nino, and Gian Paolo Clemente. 2009. Modelling Aggregate Non-Life Underwriting Risk: Standard Formula vs Internal Model. Giornale dell’Istituto Italiano degli Attuari 72: 301–38. [Google Scholar]
- Wills Re. 2017. Global Reinsurance and Risk Appetite Report. Technical Report. Available online: http://www.willisre.com/documents/Media_Room/Publication/Willis%20Re%20Global%20Reinsurance%20and%20Risk%20Appetite%20Survey%202017.pdf (accessed on 1 May 2018).

1. | For comments about main drawbacks in the quantification of the capital requirement in non-life insurance, see also Clemente and Savelli (2017). |

2. | In the following, a tilde indicates random variables. |

3. | Three alternative methodologies are provided in QIS5 for both premium and reserve risk. |

4. | |

5. | Given a random variable $\tilde{X}$ distributed as lognormal, we have that skewness is ${\gamma}_{\tilde{X}}=C{V}_{\tilde{X}}*(3+C{V}_{\tilde{X}}^{2})$. |

6. | |

7. | Similar approaches for the calibration of parameters have been used in Clemente et al. (2015) and Savelli and Clemente (2011). |

**Figure 4.**$N{P}_{LoB}$ according to several values of standard deviation of structure variable (${\sigma}_{q}$).

**Figure 6.**$N{P}_{LoB}$ according to different retention limits (${L}_{LoB}=E({Z}_{pre,LoB}^{gross})+S\xb7\sigma ({Z}_{pre,LoB}^{gross})$).

**Figure 7.**Skewness of (gross or net) total losses under either lognormal distribution or collective risk model.

LoB | Next-Year Premiums | Best Estimate | Total | Weight on Total | ||
---|---|---|---|---|---|---|

MTPL | 544.76 | 725.93 | 1270.68 | 26.2% | 35.0% | 61.2% |

GTPL | 133.17 | 570.42 | 703.60 | 6.4% | 27.5% | 33.9% |

MOD | 79.15 | 22.60 | 101.76 | 3.8% | 1.1% | 4.9% |

Total | 757.08 | 1318.95 | 2076.04 | 36.5% | 63.5% | 100.0% |

LoB | n | ${\mathit{\sigma}}_{\tilde{\mathit{q}}}$ | g | m | ${\mathit{CV}}_{\tilde{\mathit{Z}}}$ | i | $\mathit{\lambda}$ | c |
---|---|---|---|---|---|---|---|---|

MTPL | 100,000 | 7% | 2% | 4000 | 6 | 3% | 5.0% | 14.0% |

GTPL | 10,000 | 8% | 2% | 10,000 | 10 | 3% | −10.0% | 29.0% |

MOD | 20,000 | 14% | 2% | 2500 | 2 | 3% | 10.0% | 27.0% |

LoB | ${\mathit{NP}}_{\mathit{LoB}}^{\mathit{QIS}5}$ | ${\mathit{NP}}_{\mathit{LoB}}^{\mathit{DA}}$ | ${\mathit{NP}}_{\mathit{LoB}}^{\mathit{C}}$ | ${\mathit{NP}}_{\mathit{LoB}}^{\mathit{C}2}$ |
---|---|---|---|---|

MTPL | 65.83% | 64.51% | 98.08% | 97.52% |

GTPL | 53.37% | 51.09% | 75.49% | 74.10% |

MOD | 92.21% | 93.24% | 99.93% | 99.02% |

© 2018 by the author. 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**

Clemente, G.P.
The Effect of Non-Proportional Reinsurance: A Revision of Solvency II Standard Formula. *Risks* **2018**, *6*, 50.
https://doi.org/10.3390/risks6020050

**AMA Style**

Clemente GP.
The Effect of Non-Proportional Reinsurance: A Revision of Solvency II Standard Formula. *Risks*. 2018; 6(2):50.
https://doi.org/10.3390/risks6020050

**Chicago/Turabian Style**

Clemente, Gian Paolo.
2018. "The Effect of Non-Proportional Reinsurance: A Revision of Solvency II Standard Formula" *Risks* 6, no. 2: 50.
https://doi.org/10.3390/risks6020050