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

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## Abstract

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## 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$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}{V}_{\alpha ,\beta}\left(f\right)-{V}_{\alpha ,\beta}\left({f}_{c,d}\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\frac{1}{\alpha}\left(\right)open="("\; close=")">{\int}_{{S}_{X}^{-1}\left(\beta \right)}^{supX}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)-{\int}_{{S}_{X}^{-1}\left(\beta \right)}^{{S}_{X}^{-1}\left(\alpha \right)}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)\hfill \end{array}$$
- (c)
- If ${S}_{X}^{-1}\left(\alpha \right)<{S}_{X}^{-1}\left(\beta \right)$, then (8) and (9) imply that$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}{V}_{\alpha ,\beta}\left(f\right)-{V}_{\alpha ,\beta}\left({f}_{c,d}\right)\hfill \\ \hfill \phantom{\rule{-8.5359pt}{0ex}}& =& \phantom{\rule{-8.5359pt}{0ex}}\frac{1}{\alpha}\left(\right)open="("\; close=")">{\int}_{{S}_{X}^{-1}\left(\alpha \right)}^{{S}_{X}^{-1}\left(\beta \right)}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)+{\int}_{{S}_{X}^{-1}\left(\beta \right)}^{supX}[{f}_{c,d}\left(x\right)-f\left(x\right)]d{F}_{X}\left(x\right)\hfill \end{array}$$

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 4. Sensitivity Analysis

- (i)
- According to Lemma 1, Problem (3) is equivalent to$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}\underset{{f}_{c,d}\in {\mathcal{H}}_{0}}{inf}\left(\right)open="("\; close=")">\frac{1}{\alpha}{\int}_{0}^{\alpha}[{S}_{X}^{-1}\left(\lambda \right)-{f}_{c,d}\left({S}_{X}^{-1}\left(\lambda \right)\right)]d\lambda +\frac{1+\theta}{\beta}{\int}_{0}^{\beta}{f}_{c,d}\left({S}_{X}^{-1}\left(x\right)\right)dx\hfill \end{array}$$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,$$\begin{array}{ccc}\hfill \phantom{\rule{-8.5359pt}{0ex}}& & \phantom{\rule{-8.5359pt}{0ex}}\underset{f\in \mathcal{H}}{inf}\left(\right)open="("\; close=")">{\int}_{0}^{1}\frac{1}{\alpha}{\int}_{0}^{\alpha}[{S}_{X}^{-1}\left(\lambda \right)-f\left({S}_{X}^{-1}\left(\lambda \right)\right)]d\lambda d\mathsf{\Phi}\left(\alpha \right)+\frac{1+\theta}{\beta}{\int}_{0}^{\beta}f\left({S}_{X}^{-1}\left(x\right)\right)dx\hfill \end{array}$$

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**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: ${S}_{X}\left(x\right)={e}^{-\frac{1}{100}x}$.

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**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