# Premium Risk Net of Reinsurance: From Short-Term to Medium-Term Assessment

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## Abstract

**:**

## 1. Introduction

## 2. A Model in Order to Assess Risk Profile of Non-Life Insurers in Presence of Reinsurance

#### 2.1. Quota Share

#### 2.2. Excess of Loss

## 3. Solvency II Regulation for Non-Life Insurance

**s**= $[{\sigma}_{1},\dots ,{\sigma}_{h},\dots ,{\sigma}_{L}]$ using a fixed12 correlation matrix

**R**:

- 4) Motor Vehicle Liability insurance (MVL): obligations which cover all liabilities arising out of the use of motor vehicles operating on land;
- 5) Other Motor insurance (OM): obligations which cover all damage to or loss of land vehicles;
- 8) General Liability insurance (GL): obligations which cover all liabilities other than those in the LoB 4.

**s**of Equation (29) according to Solvency II is given by:

## 4. Numerical Analysis for Risk Assessment Gross and Net of Reinsurance Mitigation

## 5. Numerical Analysis in Case of Different Portfolio Mix

**R**.

## 6. Medium-Term Assessment and ORSA

## 7. A Medium-Term Model for the Net Risk Reserve

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CRM | Collective Risk Model |

GWP | Gross Written Premiums |

SF | Standard Formula |

IM | Internal Model |

SCR | Solvency Capital Requirement |

LoB | Lines of Business |

OM | Other Motor insurance |

MVL | Motor Vehicle Liabilities insurance |

GL | General Liabilities insurance |

QS(H) | Quota Share Reinsurance (High Pricing) |

XL(H) | eXcess of Loss Reinsurance (High Pricing) |

ORSA | Own Risk and Solvency Assessment |

SR | Solvency Ratio |

## References

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1 | When a tilde is taken on a character, then it will mean that it is a random variable. |

2 | Claims cost of the year incorporate both payments for claims incurred during the year and the provisions for new claims ($\tilde{{X}_{t}}={\tilde{X}}_{t,h}^{paid,CY}+{\tilde{V}}_{t,h}^{S}$). Regarding premium risk, both payments and reserves for claims incurred in previous year are necessarily covered by initial claims reserve. Indeed, their volatility attains to reserve risk. |

3 | Earned premiums are the difference between written premium of the year and the one-year change in premium reserve for unearned premiums and unexpired risk $({B}_{t,h}={B}_{t,h}^{written}-{\tilde{V}}_{t,h}^{P}+{V}_{t-1,h}^{P})$. |

4 | Reinsurance commissions can also be stochastic: ${\tilde{C}}_{t,h}^{RE}={\tilde{c}}_{t,h}^{RE}\xb7{B}_{t,h}^{RE}$. |

5 | See (Daykin et al. 1994) and (Klugman et al. 2008) for an extensive treatment of CRM. |

6 | See (Clemente et al. 2015). |

7 | We avoid to refer to h-th LoB in some formulas of this subsection for practical purpose. |

8 | See (Clemente et al. 2014) and EIOPA Calibration Paper on Premium and Reserve Risk. |

9 | It measures the potential loss in value of a risky asset or portfolio over a defined time horizon for a given confidence level. |

10 | Standard formula has been calibrated by EIOPA through the four Quantitative Impact Studies (QIS), carried on the whole EU insurance market on a voluntary basis by solo/group insurers. |

11 | It is worth mentioning that, under the assumption of a Log-Normal distribution for the aggregate claims cost, this multiplier holds only when ${\sigma}_{Pre}$ is roughly 14.47%. It turns to overestimate capital requirement for big insurers they usually have smaller volatility coefficients. |

12 | Correlation parameters for two different segments are given by the correlation matrix set out in Annex IV of delegated acts. |

**Figure 1.**Expected single claim size ${m}_{M}$ for different ${c}_{\tilde{Z}}$ and retention limit (by multiplicative factor k).

**Figure 2.**Gross-to-net ${a}_{2,\tilde{Z}}$ (

**Left**) and ${r}_{2,\tilde{Z}}$ (

**Right**) for different ${c}_{\tilde{Z}}$ and retention limit (by multiplicative factor k).

**Figure 3.**Gross-to-net coefficient of variation (CoV) for different ${c}_{\tilde{Z}}$ and retention limit (by the multiplicative factor k).

**Figure 4.**Simulated distributions of Aggregate Claims Cost for Tau (for each Line of Business (LoBs)).

**Figure 6.**Simulated Net technical result distributions for Tau (green lines on x-value 0), according to different reinsurance strategies.

**Figure 10.**Simulated Gross and Net technical result distributions for Tau (green lines on value 0), during the following 4 years—Baseline Portfolio mix (the k-th row represents $\tilde{{Y}_{k}}$ distribution).

**Figure 11.**${\tilde{u}}_{t}$, for t = 0, 1, 2, and 3 Gross of Reinsurance-Baseline portfolio mix (${u}_{0}=25\%$).

**Figure 12.**${\tilde{u}}_{t}$, for t = 0, 1, 2, and 3 Net of Quota Share (QS) Reinsurance-Baseline portfolio mix (${u}_{0}=25\%$).

**Figure 13.**${\tilde{u}}_{t}$, for t = 0, 1, 2, and 3 net of Excess of Loss (XL) Reinsurance-Baseline portfolio mix (${u}_{0}=25\%$).

LoB | OM | MVL | GL |
---|---|---|---|

Standard deviation | 8% | 10% | 14% |

LoBs | Omega | Tau | Epsilon | All Insurers | |||
---|---|---|---|---|---|---|---|

${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t},\mathit{h}}/{\sum}_{\mathit{h}}{\mathit{B}}_{\mathit{t},\mathit{h}}$ | |

MVL | 600 | 630 | 300 | 315 | 60 | 63 | 60% |

OM | 200 | 210 | 100 | 105 | 20 | 21 | 20% |

GL | 200 | 210 | 100 | 105 | 20 | 21 | 20% |

TOTAL | 1000 | 1050 | 500 | 525 | 100 | 105 | 100% |

Insurer | LoBs | ${\mathit{n}}_{\mathit{t}}$ | ${\mathit{\sigma}}_{\mathit{q}}$ | g | ${\mathit{m}}_{\mathit{t}}$ | ${\mathit{c}}_{\mathit{Z}}$ | i | $\mathit{\lambda}$ | c |
---|---|---|---|---|---|---|---|---|---|

Omega | MVL | 114,846.03 | 7.9% | 1.95% | 4000 | 7 | 3% | 2.8% | 21.3% |

OM | 51,594.74 | 12.1% | 1.95% | 2500 | 2 | 3% | 8.9% | 29.8% | |

GL | 14,260.81 | 14.7% | 1.95% | 10,000 | 12 | 3% | $-4.4\%$ | 31.8% | |

Tau | MVL | 57,423.74 | 7.9% | 1.95% | 4000 | 7 | 3% | 2.8% | 21.3% |

OM | 25,797.01 | 12.1% | 1.95% | 2500 | 2 | 3% | 8.9% | 29.8% | |

GL | 7130.40 | 14.7% | 1.95% | 10,000 | 12 | 3% | $-4.4\%$ | 31.8% | |

Epsilon | MVL | 11,484.60 | 7.9% | 1.95% | 4000 | 7 | 3% | 2.8% | 21.3% |

OM | 5159.47 | 12.1% | 1.95% | 2500 | 2 | 3% | 8.9% | 29.8% | |

GL | 1426.08 | 14.7% | 1.95% | 10,000 | 12 | 3% | $-4.4\%$ | 31.8% |

LoBs | Quota Share | eXcess of Loss | |||||
---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathit{h}}$ | $\mathbf{\Delta}{\mathit{c}}_{\mathit{h}}^{\mathit{RE},1}$ | $\mathbf{\Delta}{\mathit{c}}_{\mathit{h}}^{\mathit{RE},2}$ | ${\mathit{M}}_{\mathit{t},\mathit{h}}$ | ${\mathit{m}}_{\mathit{M}}$ | ${\mathit{\lambda}}_{\mathit{h}}^{\mathit{R}\mathit{E},1}$ | ${\mathit{\lambda}}_{\mathit{h}}^{\mathit{RE},2}$ | |

MVL | 95% | 0 | $20\%\xb7{c}_{\mathrm{MVL}}$ | 424,000 | 3831 | 2.8% | 5% |

OM | 90% | 0 | $20\%\xb7{c}_{\mathrm{OM}}$ | 27,500 | 2398 | 8.9% | 1% |

GL | 85% | 0 | $20\%\xb7{c}_{\mathrm{GL}}$ | 1,810,000 | 9393 | $-4.4\%$ | 10% |

**Table 5.**Coefficient of variation (CoV) and skewness of simulated distribution for each LoB (Baseline Portfolio Mix).

Insurers | LoBs | ${\tilde{\mathit{X}}}_{\mathit{t},\mathit{h}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{t},\mathit{h}}^{\mathbf{QS}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{t},\mathit{h}}^{\mathbf{XL}}$ | |||
---|---|---|---|---|---|---|---|

CoV | $\mathit{\gamma}$ | CoV | $\mathit{\gamma}$ | CoV | $\mathit{\gamma}$ | ||

Omega | MVL | 8.1% | 0.17 | 8.1% | 0.17 | 7.9% | 0.14 |

OM | 12.0% | 0.23 | 12.0% | 0.23 | 11.9% | 0.22 | |

GL | 16.3% | 1.15 | 16.3% | 1.15 | 13.9% | 0.28 | |

TOTAL | 8.4% | 0.32 | 8.4% | 0.30 | 8.0% | 0.15 | |

Tau | MVL | 8.4% | 0.19 | 8.4% | 0.19 | 8.1% | 0.14 |

OM | 12.0% | 0.23 | 12.0% | 0.23 | 12.0% | 0.23 | |

GL | 19.7% | 3.17 | 19.7% | 3.17 | 14.8% | 0.28 | |

TOTAL | 8.8% | 0.37 | 8.7% | 0.34 | 8.1% | 0.15 | |

Epsilon | MVL | 10.2% | 0.82 | 10.2% | 0.82 | 8.9% | 0.16 |

OM | 12.4% | 0.25 | 12.4% | 0.25 | 12.3% | 0.25 | |

GL | 32.4% | 6.02 | 32.4% | 6.02 | 20.4% | 0.46 | |

TOTAL | 11.9% | 1.93 | 11.6% | 1.74 | 9.3% | 0.25 |

LoBs | Omega | Tau | Epsilon | All Insurers | |||
---|---|---|---|---|---|---|---|

${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t},\mathit{h}}/{\sum}_{\mathit{h}}{\mathit{B}}_{\mathit{t},\mathit{h}}$ | |

MVL | 500 | 525 | 250 | 263 | 50 | 52 | 50% |

OM | 400 | 420 | 200 | 210 | 40 | 42 | 40% |

GL | 100 | 105 | 50 | 52 | 10 | 11 | 10% |

TOTAL | 1000 | 1050 | 500 | 525 | 100 | 105 | 100% |

Insurers | LoBs | ${\tilde{\mathit{X}}}_{\mathit{t},\mathit{h}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{t},\mathit{h}}^{\mathbf{QS}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{t},\mathit{h}}^{\mathbf{XL}}$ | |||
---|---|---|---|---|---|---|---|

CoV | $\mathit{\gamma}$ | CoV | $\mathit{\gamma}$ | CoV | $\mathit{\gamma}$ | ||

Omega | MVL | 8.2% | 0.19 | 8.2% | 0.19 | 8.0% | 0.17 |

OM | 12.0% | 0.24 | 12.0% | 0.24 | 11.9% | 0.24 | |

GL | 21.3% | 3.62 | 21.3% | 3.62 | 14.8% | 0.27 | |

TOTAL | 8.6% | 0.24 | 8.5% | 0.23 | 8.2% | 0.18 | |

Tau | MVL | 8.5% | 0.38 | 8.5% | 0.38 | 8.1% | 0.15 |

OM | 12.0% | 0.25 | 12.0% | 0.25 | 12.0% | 0.24 | |

GL | 24.9% | 5.91 | 24.4% | 5.91 | 16.4% | 0.33 | |

TOTAL | 8.9% | 0.30 | 8.9% | 0.29 | 8.4% | 0.18 | |

Epsilon | MVL | 10.6% | 1.00 | 10.6% | 1.00 | 9.0% | 0.17 |

OM | 12.2% | 0.24 | 12.2% | 0.24 | 12.1% | 0.23 | |

GL | 42.3% | 10.5 | 43.0% | 10.5 | 25.6% | 0.64 | |

TOTAL | 11.1% | 0.99 | 11.0% | 0.92 | 9.5% | 0.23 |

**Table 8.**Liabilities oriented Portfolio Mix in terms of Gross premium volumes (amounts in mln of Euro).

LoBs | Omega | Tau | Epsilon | All Insurers | |||
---|---|---|---|---|---|---|---|

${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t}}$ | ${\mathit{B}}_{\mathit{t}+1}$ | ${\mathit{B}}_{\mathit{t},\mathit{h}}/{\sum}_{\mathit{h}}{\mathit{B}}_{\mathit{t},\mathit{h}}$ | |

MVL | 500 | 525 | 250 | 263 | 50 | 52 | 50% |

OM | 100 | 105 | 50 | 52 | 10 | 11 | 10% |

GL | 400 | 420 | 200 | 210 | 40 | 42 | 40% |

TOTAL | 1000 | 1050 | 500 | 525 | 100 | 105 | 100% |

**Table 9.**CoV and skewness of simulated distribution for each LoB (Liabilities oriented Portfolio Mix).

Insurers | LoBs | ${\tilde{\mathit{X}}}_{\mathit{t},\mathit{h}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{t},\mathit{h}}^{\mathbf{QS}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{t},\mathit{h}}^{\mathbf{XL}}$ | |||
---|---|---|---|---|---|---|---|

CoV | $\mathit{\gamma}$ | CoV | $\mathit{\gamma}$ | CoV | $\mathit{\gamma}$ | ||

Omega | MVL | 8.2% | 0.17 | 8.2% | 0.17 | 8.0% | 0.15 |

OM | 12.1% | 0.25 | 12.1% | 0.25 | 12.1% | 0.24 | |

GL | 14.7% | 0.54 | 14.7% | 0.54 | 13.4% | 0.25 | |

TOTAL | 9.1% | 0.27 | 8.9% | 0.26 | 8.6% | 0.16 | |

Tau | MVL | 8.5% | 0.22 | 8.5% | 0.22 | 8.1% | 0.15 |

OM | 12.2% | 0.24 | 12.2% | 0.24 | 12.1% | 0.24 | |

GL | 16.2% | 1.04 | 16.2% | 1.04 | 13.9% | 0.28 | |

TOTAL | 10.0% | 0.93 | 9.9% | 0.86 | 8.9% | 0.21 | |

Epsilon | MVL | 10.8% | 1.14 | 10.8% | 1.14 | 9.1% | 0.18 |

OM | 12.8% | 0.25 | 12.8% | 0.25 | 12.7% | 0.25 | |

GL | 28.3% | 4.79 | 28.3% | 4.79 | 17.1% | 0.34 | |

TOTAL | 13.9% | 2.19 | 13.6% | 1.37 | 10.5% | 0.27 |

**Table 10.**Solvency Ratio (SR) for Tau during the following 3 years (amounts in mln of Euro) (Baseline Ptf—20% GL).

t | 0 | 1 | 2 | 3 | |
---|---|---|---|---|---|

Gross | OF | 125.0 | 134.5 | 144.5 | 154.9 |

SCR | 92.8 | 96.1 | 101.4 | 105.7 | |

SR | 135% | 140% | 142% | 147% | |

Net QS High | OF | 125.0 | 131.8 | 141.7 | 152.1 |

SCR | 85.9 | 89.5 | 93.4 | 98.2 | |

SR | 145% | 147% | 152% | 155% | |

Net XL High | OF | 125.0 | 133.5 | 143.4 | 153.8 |

SCR | 75.8 | 79.2 | 82.9 | 87.5 | |

SR | 165% | 169% | 173% | 176% |

t | 0 | 1 | 2 | 3 | |
---|---|---|---|---|---|

Gross | OF | 125.0 | 141.4 | 158.3 | 175.9 |

SCR | 77.6 | 81.0 | 84.6 | 88.5 | |

SR | 161% | 174% | 187% | 199% | |

Net QS High | OF | 125.0 | 137.8 | 151.2 | 165.1 |

SCR | 73.0 | 76.4 | 78.3 | 82.6 | |

SR | 171% | 180% | 193% | 200% | |

Net XL High | OF | 125.0 | 140.6 | 156.9 | 173.7 |

SCR | 66.4 | 68.9 | 71.6 | 74.3 | |

SR | 188% | 204% | 219% | 234% |

**Table 12.**SR for Tau during the following 3 years (amounts in mln of Euro) (Liabilities Ptf—40% GL).

t | 0 | 1 | 2 | 3 | |
---|---|---|---|---|---|

Gross | OF | 125.0 | 126.9 | 129.0 | 131.3 |

SCR | 114.6 | 118.6 | 120.8 | 124.9 | |

SR | 109% | 107% | 107% | 105% | |

Net QS High | OF | 125.0 | 124.6 | 124.3 | 124.1 |

SCR | 103.2 | 107.0 | 107.8 | 113.7 | |

SR | 121% | 116% | 115% | 109% | |

Net XL High | OF | 125.0 | 125.5 | 126.2 | 127.0 |

SCR | 92.9 | 95.8 | 98.5 | 101.8 | |

SR | 135% | 131% | 128% | 125% |

Mix | $1-\mathit{\u03f5}$ | Gross | Net QSH | Net XLH | |||
---|---|---|---|---|---|---|---|

$0.5\%$ | $5\%$ | $0.5\%$ | $5\%$ | $0.5\%$ | $5\%$ | ||

Baseline | $T=1$ | 92.8 | 48.6 | 85.9 | 46.1 | 75.8 | 42.3 |

$T=2$ | 127.9 | 64.6 | 119.8 | 62.6 | 105.5 | 57.1 | |

$T=3$ | 147.5 | 74.7 | 140.1 | 73.6 | 118.3 | 65.0 | |

Motor | $T=1$ | 77.6 | 40.8 | 73.0 | 39.5 | 66.4 | 36.1 |

$T=2$ | 102.5 | 49.4 | 98.5 | 49.6 | 89.0 | 45.6 | |

$T=3$ | 115.5 | 49.8 | 111.7 | 52.0 | 101.3 | 46.0 | |

Liabilities | $T=1$ | 114.5 | 63.4 | 103.2 | 58.8 | 92.9 | 55.7 |

$T=2$ | 148.7 | 81.5 | 136.9 | 74.4 | 114.4 | 69.2 | |

$T=3$ | 186.7 | 108.7 | 172.7 | 103.3 | 151.3 | 94.6 |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Pallaria, A.; Savelli, N.
Premium Risk Net of Reinsurance: From Short-Term to Medium-Term Assessment. *Risks* **2019**, *7*, 72.
https://doi.org/10.3390/risks7030072

**AMA Style**

Pallaria A, Savelli N.
Premium Risk Net of Reinsurance: From Short-Term to Medium-Term Assessment. *Risks*. 2019; 7(3):72.
https://doi.org/10.3390/risks7030072

**Chicago/Turabian Style**

Pallaria, Antonio, and Nino Savelli.
2019. "Premium Risk Net of Reinsurance: From Short-Term to Medium-Term Assessment" *Risks* 7, no. 3: 72.
https://doi.org/10.3390/risks7030072