# Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach

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## Abstract

**:**

## 1. Introduction

## 2. Ruin Probability: A Probabilistic Approach

#### 2.1. Problem Reduction

- (i)
- ${u}_{1}/a\ge {u}_{2}/(1-a)$ and
- (ii)
- ${u}_{1}/a<{u}_{2}/(1-a)$.

#### 2.2. Ruin Probability for Exponential Claims

**Theorem**

**1.**

**Proof.**

#### 2.3. Comparison with Other Results

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A comparison of the ruin probability values obtained using Theorem 1 (Method 1) and Formula (26) (Method 2) in correct and incorrect versions and using Monte Carlo simulations (’Exact’ result, 10,000 repetitions), upper panels; relative errors compared with Method 1, bottom panels. The left panels correspond to exponential claims with $\beta =2$ and the right panels to $\beta =0.5$. Common parameters: ${\theta}_{1}=10\%$, ${\theta}_{2}=3\%$, $a=80\%$, and $\lambda =10$.

**Figure 2.**A comparison of the joint ruin probability values obtained with the use of Theorem 1 (Method 1) for various values of the $\beta $ parameter of the exponential claim amount random variable, $\beta \in (0.5,2)$, and for various values of the a parameter of the quota-share contract proportion, $a\in (55\%,95\%)$. Common parameters are ${u}_{1}=10$, ${\theta}_{1}=10\%$, ${\theta}_{2}=3\%$, and $\lambda =10$. Note that in the left panel, the initial capital of the reinsurer is ${u}_{2}={u}_{1}/3$, while in the right panel, it is ${u}_{2}={u}_{1}/6$.

**Figure 3.**Phase diagram of admissible relative safety loading parameters (${\theta}_{1}$,${\theta}_{2}$). Note that the assumptions of Theorem 1 correspond to the green area given by ${\theta}_{1}>{(1+{\theta}_{2})}^{2}-1$. Note that in the region of ${\theta}_{1}\le {(1+{\theta}_{2})}^{2}-1$, which is depicted in grey, our approach is not applicable; then, the first case of Theorem 2 in Avram et al. (2008a) can be applied.

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**MDPI and ACS Style**

Burnecki, K.; Teuerle, M.A.; Wilkowska, A.
Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach. *Risks* **2021**, *9*, 86.
https://doi.org/10.3390/risks9050086

**AMA Style**

Burnecki K, Teuerle MA, Wilkowska A.
Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach. *Risks*. 2021; 9(5):86.
https://doi.org/10.3390/risks9050086

**Chicago/Turabian Style**

Burnecki, Krzysztof, Marek A. Teuerle, and Aleksandra Wilkowska.
2021. "Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach" *Risks* 9, no. 5: 86.
https://doi.org/10.3390/risks9050086