# Effect of Stop-Loss Reinsurance on Primary Insurer Solvency

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## Abstract

**:**

## 1. Introduction

## 2. The Risk Process Model

## 3. Stop-Loss Reinsurance and (In)Finite-Time Ruin Probability

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Remark**

**2.**

## 4. Finite-Difference Approximations of Finite-Time Ruin Probability

#### 4.1. Description of the Numerical Scheme

**Lemma**

**1.**

**Proof.**

#### 4.2. Numerical Experiments Verifying Convergence

#### 4.3. Results for Finite-TIME Ruin Probability $\mathsf{\Psi}(u,T)$

## 5. Finite-Time Ruin Probability with and without Stop-Loss Reinsurance

## 6. Finite-Time Ruin Probability, Stop-Loss Retention Level and Initial Capital

## 7. Stop-Loss Reinsurance and The Primary Insurer Solvency

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | Note the difference between Figure 5 and Figure 7. In here, we present how time horizon affects ruin probability, which is not the same concept as how big is the retention level of stop-loss reinsurance. Referring to the formula linking the critical time ${t}_{0}$ and the retention level B, we can see that the retention level is a function of time and initial capital; since we are changing both of these parameters here, each point on this surface represents a different retention level, which results in a different story of how stop-loss affects ruin probability. |

## References

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**Figure 2.**Convergence using different S for approximating the convolution term. We note that for mid-point and right–point schemes, the convergence is clearly linear. For the left-point scheme, it appears to be sub–linear.

**Figure 4.**Convergence as $K\to \infty $. Here, the convergence is exponential. It appears that K of 10 already produces errors of order ${10}^{-4}$ in the ruin probability, and the choice of $\tau $ and h are the determining factors for producing the required accuracy.

**Figure 5.**Plot of finite-time ruin probability: this is a numerical approximation showing how the initial capital and the time horizon affect the finite-time ruin probability. Initial capital varies from 10 to 20, time horizon ranges from 0 to 3, $h=0.005$, $\tau =0.01$, $K=12$, and other parameters are as in Table 1.

**Figure 7.**Plot of finite-time ruin probability with stop-loss reinsurance: this is a numerical approximation showing how initial capital and stop-loss retention level influence finite-time ruin probability. Initial capital varies from 3 to 10, the retention level from 5 to 20, $h=0.005$, $\tau =0.01$, $K=12$, and all other parameters are as in Table 1.

**Table 1.**Constants used in the numerical experiments: $T>0$ is the (finite) time horizon, c is the premium rate, $\lambda $ is the claim intensity, $1/\beta $ is the mean claim size and ${u}_{0}$ is the largest initial capital for which we wish to approximate the probability of ruin in finite time.

Parameter | ${u}_{0}$ | T | c | $\lambda $ | $\beta $ |

Value | 5 | 5 | 20 | 5 | 0.5 |

**Table 2.**Finite and infinite-time ruin probability under different parameter settings. For the parameters, u is the initial capital, c is the premium rate, $\lambda $ is the claim intensity, and $1/\beta $ is the mean claim size.

Parameters | Time Horizon, T (Years) | ||||||
---|---|---|---|---|---|---|---|

$(\mathit{u},\mathit{c},\mathit{\lambda},\mathit{\beta})$ | 0.1 | 0.5 | 1 | 2 | 4 | 8 | Infinity |

(5, 20, 5, 0.5) | 0.035165 | 0.099974 | 0.125627 | 0.139186 | 0.142946 | 0.143313 | 0.143252 |

(5, 15, 5, 0.5) | 0.039238 | 0.137707 | 0.195929 | 0.244954 | 0.275293 | 0.287726 | 0.289732 |

(5, 20, 7, 0.5) | 0.053630 | 0.174861 | 0.238538 | 0.288632 | 0.318013 | 0.329485 | 0.330657 |

(5, 20, 5, 0.3) | 0.095225 | 0.287057 | 0.391502 | 0.484876 | 0.557218 | 0.606357 | 0.649001 |

(10, 20, 5, 0.5) | 0.004292 | 0.020611 | 0.031224 | 0.038415 | 0.040771 | 0.041025 | 0.041042 |

(10, 15, 5, 0.5) | 0.004855 | 0.031306 | 0.058143 | 0.088889 | 0.112544 | 0.123639 | 0.125917 |

(10, 20, 7, 0.5) | 0.007359 | 0.046122 | 0.081174 | 0.117582 | 0.143278 | 0.154477 | 0.156191 |

(10, 20, 5, 0.3) | 0.027800 | 0.128481 | 0.211560 | 0.303877 | 0.386723 | 0.448424 | 0.505442 |

**Table 3.**Initial capital corresponding to a 0.5% ruin probability for different parameter values of the risk process. For the parameters, c is the insurer premium rate, $\lambda $ is the claim intensity, $1/\beta $ is the mean claim size, and $\alpha $ is the reinsurer premium rate.

Panel A: Without Reinsurance | ||||||
---|---|---|---|---|---|---|

Parameters | Time Horizon, $\mathit{T}$ (Years) | |||||

$(\mathit{c},\mathit{\lambda},\mathit{\beta})$ | 0.1 | 0.5 | 1 | 2 | 4 | 8 |

(20, 5, 0.5) | 9.593921 | 14.188087 | 16.232585 | 17.713893 | 18.320151 | 18.405424 |

(15, 5, 0.5) | 9.893007 | 15.643934 | 19.054614 | 22.792824 | 26.261153 | 28.565091 |

(20, 7, 0.5) | 10.857631 | 17.401518 | 21.315918 | 25.605376 | 29.565199 | 32.094999 |

(20, 5, 0.3) | 16.802917 | 27.698801 | 34.573297 | 40.518959 | 46.216685 | 52.393767 |

Panel B: With Reinsurance and $\mathit{T}=\mathbf{1}$ | ||||||

Retention Level, ${\mathit{B}}_{\mathit{\alpha}}$ | ||||||

$\mathit{\alpha}=\mathbf{0.3}$ | 1 | 2 | 4 | 6 | 8 | 10 |

(20, 5, 0.5) | 3.532088 | 3.521673 | 3.500712 | 6.379115 | 4.432340 | 11.204400 |

(15, 5, 0.5) | 3.699426 | 3.689711 | 3.670160 | 6.463373 | 3.168102 | 10.553199 |

(20, 7, 0.5) | 4.997532 | 4.994197 | 4.987510 | 5.584031 | 3.922885 | 8.700725 |

(20, 5, 0.3) | 7.370037 | 7.367306 | 7.361851 | 7.357099 | 7.292685 | 6.793047 |

$\mathit{\alpha}=\mathbf{0.9}$ | ||||||

(20, 5, 0.5) | 3.712315 | 3.683212 | 3.623897 | 6.436518 | 4.343473 | 11.204400 |

(15, 5, 0.5) | 3.867475 | 3.840345 | 3.785043 | 6.474379 | 2.375872 | 10.553199 |

(20, 7, 0.5) | 5.082380 | 5.072757 | 5.053388 | 5.607053 | 3.322549 | 6.186034 |

(20, 5, 0.3) | 7.456775 | 7.448370 | 7.431625 | 7.415705 | 7.327712 | 6.533600 |

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**MDPI and ACS Style**

Constantinescu, C.; Dias, A.; Li, B.; Šiška, D.; Wang, S.
Effect of Stop-Loss Reinsurance on Primary Insurer Solvency. *Risks* **2022**, *10*, 193.
https://doi.org/10.3390/risks10100193

**AMA Style**

Constantinescu C, Dias A, Li B, Šiška D, Wang S.
Effect of Stop-Loss Reinsurance on Primary Insurer Solvency. *Risks*. 2022; 10(10):193.
https://doi.org/10.3390/risks10100193

**Chicago/Turabian Style**

Constantinescu, Corina, Alexandra Dias, Bo Li, David Šiška, and Simon Wang.
2022. "Effect of Stop-Loss Reinsurance on Primary Insurer Solvency" *Risks* 10, no. 10: 193.
https://doi.org/10.3390/risks10100193