# Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- Generalizing the set of ceded loss functions. Cai, et al. [12] and Cheung [14] considered the set of all the increasing and convex functions as their feasible ceded loss function class. While, Lu, et al. [16] took the set of all the increasing and concave functions as their feasible ceded loss function class. Chi and Weng [15] extended their feasible ceded loss function class to$$\left\{f\right(x\left)\right|0\le f\left(x\right)\le x,\phantom{\rule{0.166667em}{0ex}}\mathrm{both}x-f\left(x\right)\mathrm{and}f\left(x\right)/x\text{}\mathrm{are}\text{}\mathrm{increasing}\text{}\mathrm{function}\text{}\mathrm{of}\text{}x\}.$$Chi and Tan [17] further extended their feasible ceded loss function class to$$\left\{f\right(x\left)\right|0\le f\left(x\right)\le x,\phantom{\rule{0.166667em}{0ex}}\mathrm{both}x-f\left(x\right)\mathrm{and}f\left(x\right)\mathrm{are}\text{}\mathrm{increasing}\text{}\mathrm{function}\text{}\mathrm{of}\text{}x\}.$$Other more general feasible ceded loss function class$$\left\{f\right(x\left)\right|0\le f\left(x\right)\le x,\phantom{\rule{0.166667em}{0ex}}x-f\left(x\right)\phantom{\rule{0.166667em}{0ex}}(\mathrm{or}f\left(x\right)\left)\mathrm{is}\text{}\mathrm{increasing}\text{}\left(\mathrm{left-continuous}\right)\text{}\mathrm{function}\text{}\mathrm{of}\text{}x\right\}$$
- (b)
- Generalizing the premium principles. To our knowledge, the most widely used premium principle in the existing works turns out to be the expected premium principle, see Cheung, et al. [19], Lu, et al. [16], Cai, et al. [5], Chi and Tan [17], etc.. Assa [20], Zheng and Cui [21], Cui, et al. [22] extended their premium principle to the distortion premium principle. Zhu, et al. [23] further extended their premium principle to very general one that satisfies three mild conditions: distribution invariance, risk loading and preserving the convex order, see also Chi and Tan [24].
- (c)
- Generalizing the risk measures. Using the VaR, CTE, AVaR, respectively, Hu, et al. [25], Cai and Tan [26], Cai, et al. [12], Cheung [14] and Chi and Tan [24] found the optimal reinsurance contract. In Asimit, et al. [27], a quantile-based risk measure was adopted in accordance with the insurer’s appetite. Assa [20], Zheng and Cui [21] and Cui, et al. [22] generalized their risk measures to the distortion risk measures. Cheung, et al. [19] further extended the problem by using a general law-invariant convex risk measure.
- (d)
- Constraints involved. Borch [1] (and also Arrow [4]) showed that, subject to a budget constraint, the stop-loss policy is an optimal reinsurance contract for the ceding company when the risk is measured by variance (or by a utility function). Reinsurance optimization problems involving premium constraint were also considered in Gajek and Zagrodny [10], Zhou, et al. [28], Zheng and Cui [21], Cui, et al. [22] and Cheung and Lo [18]. Cheung, et al. [29] introduced a reinsurer’s probabilistic benchmark constraint of his potential loss. In Tan and Weng [30], a profitability constraint was proposed.

## 2. The Mathematical Presentation of the Reinsurance Optimization Problem

## 3. Characterizing the Optimal Reinsurance Strategy

**Lemma**

**1.**

**Proof.**

- (a)
- If ${S}_{X}^{-1}\left(\alpha \right)\ge {\tau}_{f}\ge {S}_{X}^{-1}\left(\beta \right)$, then ${f}_{c,d}\left(x\right)\ge f\left(x\right)$ for any $x\in [{S}_{X}^{-1}\left(\alpha \right),supX]$. Hence (9) implies that$$\begin{array}{c}\hfill {V}_{\alpha ,\beta}\left(f\right)-{V}_{\alpha ,\beta}\left({f}_{c,d}\right)\ge 0.\end{array}$$
- (b)
- If ${\tau}_{f}\ge {S}_{X}^{-1}\left(\alpha \right)\ge {S}_{X}^{-1}\left(\beta \right)$, then it follows from (8) and (9) that$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}{V}_{\alpha ,\beta}\left(f\right)-{V}_{\alpha ,\beta}\left({f}_{c,d}\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\frac{1}{\alpha}\left({\int}_{{S}_{X}^{-1}\left(\beta \right)}^{supX}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)-{\int}_{{S}_{X}^{-1}\left(\beta \right)}^{{S}_{X}^{-1}\left(\alpha \right)}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}-\frac{1}{\alpha}{\int}_{{S}_{X}^{-1}\left(\beta \right)}^{{S}_{X}^{-1}\left(\alpha \right)}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& \ge & \phantom{\rule{-8.5359pt}{0ex}}0,\hfill \end{array}$$
- (c)
- If ${S}_{X}^{-1}\left(\alpha \right)<{S}_{X}^{-1}\left(\beta \right)$, then (8) and (9) imply that$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}{V}_{\alpha ,\beta}\left(f\right)-{V}_{\alpha ,\beta}\left({f}_{c,d}\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\frac{1}{\alpha}\left({\int}_{{S}_{X}^{-1}\left(\alpha \right)}^{{S}_{X}^{-1}\left(\beta \right)}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)+{\int}_{{S}_{X}^{-1}\left(\beta \right)}^{supX}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\frac{1}{\alpha}{\int}_{{S}_{X}^{-1}\left(\alpha \right)}^{{S}_{X}^{-1}\left(\beta \right)}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& \ge & \phantom{\rule{-8.5359pt}{0ex}}0,\hfill \end{array}$$

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 4. Sensitivity Analysis

- (i)
- According to Lemma 1, Problem (3) is equivalent to$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}\underset{{f}_{c,d}\in {\mathcal{H}}_{0}}{inf}\left(\frac{1}{\alpha}{\int}_{0}^{\alpha}[{S}_{X}^{-1}\left(\lambda \right)-{f}_{c,d}\left({S}_{X}^{-1}\left(\lambda \right)\right)]d\lambda +\frac{1+\theta}{\beta}{\int}_{0}^{\beta}{f}_{c,d}\left({S}_{X}^{-1}\left(x\right)\right)dx\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\underset{(c,d)\in D}{inf}(\frac{1}{\alpha}{\int}_{0}^{\alpha}[{S}_{X}^{-1}\left(\lambda \right)-[({S}_{X}^{-1}\left(\lambda \right)\wedge c)+{({S}_{X}^{-1}\left(\lambda \right)-d)}_{+}]]d\lambda \hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}+\frac{1+\theta}{\beta}{\int}_{0}^{\beta}[({S}_{X}^{-1}\left(x\right)\wedge c)+{({S}_{X}^{-1}\left(x\right)-d)}_{+}]dx).\hfill \end{array}$$We take the infimum given by (13) as the bivariate function of $(\alpha ,\beta )$, and numerically investigate how sensitively the value of $(\alpha ,\beta )$ will affect our optimal value. The corresponding numerical results are given by Figure 1 below. It seems that significant differences in the size of optimal value can be obtained depending on the value of $(\alpha ,\beta )$. It also seems that the optimal value increases as α and β approaches 0.
- (ii)
- Consider the following optimization problem,$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}\underset{f\in \mathcal{H}}{inf}\left({\int}_{0}^{1}\frac{1}{\alpha}{\int}_{0}^{\alpha}[{S}_{X}^{-1}\left(\lambda \right)-f\left({S}_{X}^{-1}\left(\lambda \right)\right)]d\lambda d\mathsf{\Phi}\left(\alpha \right)+\frac{1+\theta}{\beta}{\int}_{0}^{\beta}f\left({S}_{X}^{-1}\left(x\right)\right)dx\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\underset{(c,d)\in D}{inf}({\int}_{0}^{1}\frac{1}{\alpha}{\int}_{0}^{\alpha}[{S}_{X}^{-1}\left(\lambda \right)-[({S}_{X}^{-1}\left(\lambda \right)\wedge c)+{({S}_{X}^{-1}\left(\lambda \right)-d)}_{+}]]d\lambda d\mathsf{\Phi}\left(\alpha \right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}+\frac{1+\theta}{\beta}{\int}_{0}^{\beta}[({S}_{X}^{-1}\left(x\right)\wedge c)+{({S}_{X}^{-1}\left(x\right)-d)}_{+}]dx),\hfill \end{array}$$

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- K. Borch. “An attempt to determine the optimum amount of stop loss reinsurance.” In Proceedings of the Transactions of the 16th International Congress of Actuaries, Brussels, Belgium, 15–22 June 1960; pp. 597–610.
- M. Kaluszka. “Optimal reinsurance under mean-variance premium principles.” Insur. Math. Econ. 28 (2001): 61–67. [Google Scholar] [CrossRef]
- R. Kaas, M. Goovaerts, J. Dhaene, and M. Denuit. Modern Actuarial Risk Theory. Boston, MA, USA: Kluwer Academic Publishers, 2001. [Google Scholar]
- J. Cai, Y. Fang, Z. Li, and G. Willmot. “Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability.” J. Risk Insur. 80 (2013): 145–168. [Google Scholar] [CrossRef]
- Y. Chi. “Reinsurance arrangements minimizing the risk-adjusted value of an insurer’s liability.” Insur. Math. Econ. 42 (2012): 529–557. [Google Scholar] [CrossRef]
- K.C. Cheung, W.F. Chong, and S.C.P. Yam. “The optimal insurance under disappointment theories.” Insur. Math. Econ. 64 (2015): 77–90. [Google Scholar] [CrossRef]
- M. Kaluszka. “An extension of Arrow’s result on optimality of a stop-loss contract.” Insur. Math. Econ. 35 (2004): 527–536. [Google Scholar] [CrossRef]
- M. Kaluszka. “Optimal reinsurance under convex principles of premium calculation.” Insur. Math. Econ. 36 (2005): 375–398. [Google Scholar] [CrossRef]
- L. Gajek, and D. Zagrodny. “Optimal reinsurance under general risk measures.” Insur. Math. Econ. 34 (2004): 227–240. [Google Scholar] [CrossRef]
- S.D. Promislow, and V.R. Young. “Unifying framework for optimal insurance.” Insur. Math. Econ. 36 (2005): 347–364. [Google Scholar] [CrossRef]
- J. Cai, K.S. Tan, C.G. Weng, and Y. Zhang. “Optimal reinsurance under VaR and CTE risk measures.” Insur. Math. Econ. 43 (2008): 185–196. [Google Scholar] [CrossRef]
- A. Balbás, B. Balbás, and A. Heras. “Optimal reinsurance with general risk measures.” Insur. Math. Econ. 44 (2009): 374–384. [Google Scholar] [CrossRef]
- K.C. Cheung. “Optimal reinsurance revisited-a geometric approach.” ASTIN Bull. 40 (2010): 221–239. [Google Scholar] [CrossRef]
- Y. Chi, and C. Weng. “Optimal reinsurance subject to Vajda condition.” Insur. Math. Econ. 53 (2013): 179–189. [Google Scholar] [CrossRef]
- Y. Chi, and K.S. Tan. “Optimal reinsurance under VaR and CVaR risk measures: A simplified approach.” ASTIN Bull. 41 (2011): 487–509. [Google Scholar]
- K.C. Cheung, and A. Lo. “Characterizations of optimal reinsurance treaties: A cost-benefit approach.” Scand. Actuar. J., 2015. [Google Scholar] [CrossRef]
- K.C. Cheung, K.C.J. Sung, S.C.P. Yam, and S.P. Yung. “Optimal reinsurance under general law-invariant risk measures.” Scand. Actuar. J. 2014 (2014): 72–91. [Google Scholar] [CrossRef]
- Z. Lu, L. Liu, Q. Shen, and L. Li. “Optimal reinsurance under VaR and CTE risk measures when ceded loss function is concave.” Commun. Stat. Theory Methods 43 (2014): 3223–3247. [Google Scholar] [CrossRef]
- H. Assa. “On optimal reinsurance policy with distortion risk measures and premiums.” Insur. Math. Econ. 61 (2015): 70–75. [Google Scholar] [CrossRef]
- Y.T. Zheng, and W. Cui. “Optimal reinsurance with premium constraint under distortion risk measures.” Insur. Math. Econ. 59 (2014): 109–120. [Google Scholar] [CrossRef]
- W. Cui, J.P. Yang, and L. Wu. “Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles.” Insur. Math. Econ. 53 (2013): 74–85. [Google Scholar] [CrossRef]
- Y. Zhu, Y. Chi, and C. Weng. “Multivariate reinsurance designs for minimizing an insurer’s capital requirement.” Insur. Math. Econ. 59 (2014): 144–155. [Google Scholar] [CrossRef]
- Y. Chi, and K.S. Tan. “Optimal reinsurance with general premium principles.” Insur. Math. Econ. 52 (2013): 180–189. [Google Scholar] [CrossRef]
- X. Hu, H. Yang, and L. Zhang. “Optimal retention for a stop-loss reinsurance with incomplete information.” Insur. Math. Econ. 65 (2015): 15–21. [Google Scholar] [CrossRef]
- J. Cai, and K.S. Tan. “Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures.” ASTIN Bull. 37 (2007): 93–112. [Google Scholar] [CrossRef]
- A. Asimit, A. Badescu, and T. Verdonck. “Optimal risk transfer under quantile-based risk measurers.” Insur. Math. Econ. 53 (2013): 252–265. [Google Scholar] [CrossRef]
- K.J. Arrow. “Optimal insurance and generalized deductibles.” Scand. Actuar. J. 1 (1974): 1–42. [Google Scholar] [CrossRef]
- C.Y. Zhou, W.F. Wu, and C.F. Wu. “Optimal insurance in the presence of insurer’s loss limit.” Insur. Math. Econ. 46 (2010): 300–307. [Google Scholar] [CrossRef]
- K.C. Cheung, F. Liu, and S.C.P. Yam. “Average Value-at-Risk minimizing reinsurance under Wang’s premium principle with constraints.” ASTIN Bull. 42 (2012): 575–600. [Google Scholar]
- K. Tan, and C. Weng. “Enhancing insurer value using reinsurance and Value-at-Risk criterion.” Geneva Risk Insur. Rev. 37 (2012): 109–140. [Google Scholar] [CrossRef]
- V.R. Young. “Premium principles.” Encycl. Actuar. Sci. 3 (2004): 132–1331. [Google Scholar]
- H. Föllmer, and A. Schied. StochastiC Finance: An Introduction in Discrete Time, 4th ed. Berlin, Germany: Walter de Gruyter, 2016. [Google Scholar]
- J. Belles-Sampera, M. Guillén, and M. Santolino. “Beyond Value-at-Risk: GlueVaR Distortion Risk Measures.” Risk Anal. 34 (2014): 121–134. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The horizontal axis represents β, the vertical axis represents α, the lateral axis represents the optimal value given by (13). The Greek θ is set to be 0.1. The distribution of X is exponential: ${S}_{X}\left(x\right)={e}^{-\frac{1}{100}x}$.

**Figure 2.**The horizontal axis represents β, the vertical axis represents the optimal value given by (14). The Greek θ is set to be 0.1. The distribution of X is exponential: ${S}_{X}\left(x\right)={e}^{-\frac{1}{100}x}$.

© 2016 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**

Chen, M.; Wang, W.; Ming, R. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and *TVaR* Premium Principle. *Risks* **2016**, *4*, 50.
https://doi.org/10.3390/risks4040050

**AMA Style**

Chen M, Wang W, Ming R. Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and *TVaR* Premium Principle. *Risks*. 2016; 4(4):50.
https://doi.org/10.3390/risks4040050

**Chicago/Turabian Style**

Chen, Mi, Wenyuan Wang, and Ruixing Ming. 2016. "Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and *TVaR* Premium Principle" *Risks* 4, no. 4: 50.
https://doi.org/10.3390/risks4040050