# A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Insurance Risk Model

#### Cumulative Parisian Ruin

## 3. A VaR-type Risk Measure Derived from Cumulative Parisian Ruin

#### 3.1. Stochastic Dominance

**Theorem**

**1.**

- (i)
- Let $\left\{{X}_{1},{X}_{2},\dots ,{X}_{m}\right\}$ and $\left\{{Y}_{1},{Y}_{2},\dots ,{Y}_{m}\right\}$ be two finite sets of independent random variables such that ${X}_{i}{\u2aaf}_{st}{Y}_{i}$, for each $i=1,\dots ,m$. Then, for any increasing function $g:{\mathbb{R}}^{m}\to \mathbb{R}$, we have$$g\left({X}_{1},{X}_{2},\dots ,{X}_{m}\right){\u2aaf}_{st}g\left({Y}_{1},{Y}_{2},\dots ,{Y}_{m}\right).$$
- (ii)
- Consider two sequences of random variables $\left\{{X}_{1},{X}_{2},\dots \right\}$ and $\left\{{Y}_{1},{Y}_{2},\dots \right\}$ and two random variables X and Y such that$${X}_{n}\stackrel{d}{\to}X\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{Y}_{n}\stackrel{d}{\to}Y,$$
- (iii)
- Let the positive integer-valued random variable N be independent of the family of random variables $\left\{{C}_{1},{C}_{2},\cdots \right\}$ and define $S={\displaystyle \sum _{i=1}^{N}}{C}_{i}$. Define similarly $\tilde{S}={\displaystyle \sum _{i=1}^{\tilde{N}}}{\tilde{C}}_{i}$.If $N{\u2aaf}_{st}\tilde{N}$ and ${C}_{i}{\u2aaf}_{st}{\tilde{C}}_{i}$ for each i, then$$S{\u2aaf}_{st}\tilde{S}.$$

#### 3.2. Properties of the Risk Measure ${\rho}_{\u03f5}^{(r,t)}$

**Theorem**

**2.**

- (i)
**Invariance by translation:**For $a>0$,$${\rho}_{\u03f5}^{(r,t)}\left[L+a\right]={\rho}_{\u03f5}^{(r,t)}\left[L\right]-a.$$- (ii)
**Positive homogeneity:**For $b>0$,$${\rho}_{\u03f5}^{(r,t)}\left[bL\right]=b{\rho}_{\u03f5}^{(r,t)}\left[L\right].$$- (iii)
**Monotonicity:**If $L{\u2aaf}_{st}\tilde{L}$, then$${\rho}_{\u03f5}^{(r,t)}\left[L\right]\le {\rho}_{\u03f5}^{(r,t)}\left[\tilde{L}\right].$$

**Proof.**

#### 3.3. Relationship with Other Risk Measures

**Proposition**

**1.**

**Remark**

**1.**

## 4. Example: Cramér–Lundberg Model with Exponential Claims

**Theorem**

**3**

**Corollary**

**1.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Beekman, John A. 1985. A series for infinite time ruin probabilities. Insurance: Mathematics and Economics 4: 129–34. [Google Scholar] [CrossRef]
- Cramér, Harald. 1930. On the Mathematical Theory of Risk. Alingsås: Centraltryckeriet. [Google Scholar]
- Guérin, Hélène, and Jean-François Renaud. 2017. On the distribution of cumulative Parisian ruin. Insurance: Mathematics and Economics 73: 116–23. [Google Scholar] [CrossRef]
- Goovaerts, Marc, and F. Etienne De Vylder. 1984. Dangerous Distributions and Ruin Probabilities in the Classical Risk Model. Paper presented at the 22nd International Congress of Actuaries, Sydney, Australia, October 21–27; pp. 111–20. [Google Scholar]
- Denuit, Michel, Jan Dhaene, Marc Goovaerts, and Rob Kaas. 2005. Actuarial Theory for Dependent Risks: Measures, Orders and Models. New York: Wiley. [Google Scholar]
- Shaked, Moshe, and George Shanthikumar. 2007. Stochastic Orders. New York: Springer. [Google Scholar]
- Kaas, Rob, Marc Goovaerts, Jan Dhaene, and Michel Denuit. 2008. Modern Actuarial Risk Theory Using R. Berlin/Heidelberg: Springer. [Google Scholar]
- Loisel, Stéphane. 2005. Differentiation of some functionals of risk processes, and optimal reserve allocation. Journal of Applied Probability 42: 379–92. [Google Scholar] [CrossRef]
- Loisel, Stéphane, and Julien Trufin. 2014. Properties of a risk measure derived from the expected area in red. Insurance: Mathematics and Economics 55: 191–9. [Google Scholar] [CrossRef]
- Lefèvre, Claude, Julien Trufin, and Pierre Zuyderhoff. 2017. Some comparison results for finite-time ruin probabilities in the classical risk model. Insurance: Mathematics and Economics 77: 143–49. [Google Scholar] [CrossRef]
- Lundberg, Filip. 1903. I. Approximerad framställning af sannolikhetsfunktionen. II. Återförsäkring af kollektivrisker. Akademisk afhandling, etc. Uppsala: Almqvist & Wiksell. [Google Scholar]
- Renaud, Jean-François. 2014. On the time spent in the red by a refracted Lévy risk process. Journal of Applied Probability 51: 1171–88. [Google Scholar] [CrossRef]
- Mitric, Ilie-Radu, and Julien Trufin. 2016. On a risk measure inspired from the ruin probability and the expected deficit at ruin. Scandinavian Actuarial Journal 2016: 932–51. [Google Scholar] [CrossRef]
- Trufin, Julien, Hansjoerg Albrecher, and Michel Denuit. 2011. Properties of a risk measure derived from ruin theory. The Geneva Risk and Insurance Review 36: 174–88. [Google Scholar] [CrossRef]

**Figure 1.**A sample path of a Cramér–Lundberg process ${X}_{t}$. The time of ruin ${\tau}_{0}^{-}$ is in red and the time of cumulative Parisian ruin ${\kappa}_{r}$ is in blue.

**Figure 2.**The probability of cumulative Parisian ruin for the Cramér–Lundberg process with $\alpha =1/8$, $\lambda =2$, $c=17$, $r=1$, $x=10$ and $t=10$.

**Figure 3.**Risk measures ${\rho}_{\u03f5}^{(r,t)}$ and ${\zeta}_{\u03f5}^{\left(t\right)}$ for the Cramér–Lundberg process with $\alpha =1/8$, $\lambda =2$, $c=17$, $t=10$, $r=2$, and $\u03f5=0.3$.

© 2018 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**

Lkabous, M.A.; Renaud, J.-F.
A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model. *Risks* **2018**, *6*, 85.
https://doi.org/10.3390/risks6030085

**AMA Style**

Lkabous MA, Renaud J-F.
A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model. *Risks*. 2018; 6(3):85.
https://doi.org/10.3390/risks6030085

**Chicago/Turabian Style**

Lkabous, Mohamed Amine, and Jean-François Renaud.
2018. "A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model" *Risks* 6, no. 3: 85.
https://doi.org/10.3390/risks6030085