# Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Proposition**

**1.**

**Proof.**

- If $1-{\alpha}_{c}\ge {S}_{X}\left(0\right)$ and $1-{\alpha}_{r}\ge {S}_{X}\left(0\right)$, then ${a}_{c}={a}_{r}=0$. Thus,
- when $\beta <1/2$, the solution is ${f}^{*}\left(x\right)=x$;
- when $\beta =1/2$, the objective function is always zero.

- If $1-{\alpha}_{c}<{S}_{X}\left(0\right)$ and $1-{\alpha}_{r}\ge {S}_{X}\left(0\right)$, then ${a}_{c}>0$ and ${a}_{r}=0$. Thus,
- when $\beta <1/2$, the optimal ceded function is ${f}^{*}\left(x\right)=x$;
- when $\beta >1/2$, the form of the optimal ceded function is similar to the case when $\beta =1$, with only the risk and the profit of the cedent considered (the solution for the latter case can be found in Case 2 of Section 3.2 and Section 4.2 below);
- when $\beta =1/2$, the optimal ceded function is ${f}^{*}\left(x\right)=x$.

- If $1-{\alpha}_{c}\ge {S}_{X}\left(0\right)$ and $1-{\alpha}_{r}<{S}_{X}\left(0\right)$, then ${a}_{c}=0$ and ${a}_{r}>0$. Thus,
- when $\beta <1/2$, the form of the optimal ceded function is similar to the case when $\beta =0$, with only the risk and the profit of the reinsurer being considered (the solution for the latter case can be found in Case 3 of Section 3.2 and Section 4.2 below).
- when $\beta =1/2$, the optimal ceded function is ${f}^{*}\left(x\right)=0$ for all x.

## 3. Optimal Reinsurance Policies When $\mathit{f}\in {\mathcal{C}}^{\mathbf{1}}$

#### 3.1. Functional Form of the Ceded Function

#### 3.1.1. Case 1: $\beta >1/2$

**Remark**

**1.**

#### 3.1.2. Case 2: $\beta <1/2$

#### 3.1.3. Case 3: $\beta =1/2$

#### 3.2. Parameter Values of the Optimal Ceded Function

#### 3.2.1. Case 1: $\beta >1/2$ and ${\alpha}_{c}<{\alpha}_{r}$

**Theorem**

**1.**

- 1.
- $c=1$ and $d={d}^{*}$ when ${\theta}^{*}<{S}_{X}\left(0\right)$ and $U\left({\theta}^{*}\right)<Q(\beta ,{a}_{c},{a}_{r})$;
- 2.
- $c\in [0,1]$ is any constant and $d={d}^{*}$ when ${\theta}^{*}<{S}_{X}\left(0\right)$ and $U\left({\theta}^{*}\right)=Q(\beta ,{a}_{c},{a}_{r})$;
- 3.
- $c=1$ and $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$ and $(1+\theta )\mathbf{E}\left[X\right]<Q(\beta ,{a}_{c},{a}_{r})$;
- 4.
- $c\in [0,1]$ is any constant and $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$ and $(1+\theta )\mathbf{E}\left[X\right]=Q(\beta ,{a}_{c},{a}_{r})$.

**Proof.**

**Remark**

**2.**

- When $\beta =1$, then only the cedent is considered. In this case, $Q(\beta ,{a}_{c},{a}_{r})={a}_{c}$ and ${f}^{*}\left(x\right)={(x-{d}^{*})}_{+}$ when $U\left({\theta}^{*}\right)<{a}_{c}$. Therefore, when $U\left({\theta}^{*}\right)>{a}_{c}$, then ${f}^{*}\left(x\right)=0$ for all x, and the primary insurance company will not purchase any reinsurance policy. This result agrees with those derived by [11,12].
- When $\beta \searrow 1/2$, then $Q(\beta ,{a}_{c},{a}_{r})\sim ({a}_{c}-{a}_{r})/(4\beta -2)<0$, and the optimal value of c is zero. Therefore, ${f}^{*}\left(x\right)=0$ for all x.
- The value of ${d}^{*}$ in the excess-of-loss reinsurance policy does not depend on the choice of β whenever $U\left({\theta}^{*}\right)\le Q(\beta ,{a}_{c},{a}_{r})$.

#### 3.2.2. Case 2: $\beta >1/2$ and ${\alpha}_{c}>{\alpha}_{r}$

**Theorem**

**2.**

- $c=1$ and $d={d}^{*}$ when $1-{\alpha}_{r}<{\theta}^{*}<{S}_{X}\left(0\right)$ and $Q(\beta ,{a}_{c},{a}_{r})>U\left({\theta}^{*}\right)$;
- $c\in [0,1]$ is any constant and $d={d}^{*}$ when $1-{\alpha}_{r}<{\theta}^{*}<{S}_{X}\left(0\right)$ and $Q(\beta ,{a}_{c},{a}_{r})=U\left({\theta}^{*}\right)$;
- $c=1$ and $d={a}_{r}$ when ${\theta}^{*}<1-{\alpha}_{r}<{\theta}_{\beta}^{*}$, and ${a}_{c}>{U}_{\beta}(1-{\alpha}_{r})$;
- $c\in [0,1]$ is any constant and $d={a}_{r}$ when ${\theta}^{*}<1-{\alpha}_{r}<{\theta}_{\beta}^{*}$, and ${a}_{c}={U}_{\beta}(1-{\alpha}_{r})$;
- $c=1$ and $d={d}_{\beta}^{*}$ when $1-{\alpha}_{c}<{\theta}_{\beta}^{*}<1-{\alpha}_{r}$ and ${a}_{c}>{U}_{\beta}\left({\theta}_{\beta}^{*}\right)$;
- $c\in [0,1]$ is any constant and $d={d}_{\beta}^{*}$ when $1-{\alpha}_{c}<{\theta}_{\beta}^{*}<1-{\alpha}_{r}$ and ${a}_{c}={U}_{\beta}\left({\theta}_{\beta}^{*}\right)$;
- $c=1$ and $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$ and $Q(\beta ,{a}_{c},{a}_{r})>(1+\theta )\mathbf{E}\left[X\right]$;
- $c\in [0,1]$ is any constant and $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$ and $Q(\beta ,{a}_{c},{a}_{r})=(1+\theta )\mathbf{E}\left[X\right]$.

**Proof.**

**Remark**

**3.**

- When $\beta =1$, then ${\theta}_{\beta}^{*}={\theta}^{*}$ and ${U}_{\beta}\left(x\right)=U\left(x\right)$. Thus, the result is exactly the same as in the first bullet at the end of Case 1 above. The value of ${\alpha}_{r}$ makes no difference here because only the cedent’s risk is considered when $\beta =1$.
- When $\beta \searrow 1/2$, then $Q(\beta ,{a}_{c},{a}_{r})\sim ({a}_{c}-{a}_{r})/(4\beta -2)\nearrow \infty $, ${\theta}_{\beta}^{*}\nearrow \infty $ and ${U}_{\beta}({\theta}_{\beta}^{*})=0$. Therefore, Parts (1) and (3) of Theorem 2 apply. We have:$${f}^{*}\left(x\right)=\left\{\begin{array}{cc}{(x-{d}^{*})}_{+}\hfill & \mathrm{when}\phantom{\rule{1.em}{0ex}}{\theta}^{*}>1-{\alpha}_{r},\hfill \\ {(x-{a}_{r})}_{+}\hfill & \mathrm{when}\phantom{\rule{1.em}{0ex}}{\theta}^{*}<1-{\alpha}_{r}.\hfill \end{array}\right.$$

#### 3.2.3. Case 3: $\beta <1/2$ and ${\alpha}_{c}<{\alpha}_{r}$

**Theorem**

**3.**

**Remark**

**4.**

- When $\beta \nearrow 1/2$, then ${g}_{3}^{\prime}\left(\eta \right)\to ({a}_{r}-{a}_{c})/2>0$. In this case, ${\eta}^{*}=0$ and the optimal reinsurance policy is ${f}^{*}\left(x\right)={(x-{a}_{r})}_{+}.$
- When $\beta =0$ and only the reinsurer’s risk is considered, Theorem 3 holds with $Q(\beta ,{a}_{c},{a}_{r})={a}_{r}$.

#### 3.2.4. Case 4: $\beta <1/2$ and ${\alpha}_{c}>{\alpha}_{r}$

**Theorem**

**4.**

#### 3.3. An Illustrative Example

#### 3.3.1. Scenario A: ${\alpha}_{c}=0.95$ and ${\alpha}_{r}=0.99$

- For $\beta \in (0.654,1]$, the insurer is “more important”. As a result, it retains the “good” risk in the layer of losses $(0,{S}_{X}^{-1}\left({\theta}^{*}\right))$ and cedes the rest. For $\beta \in [0,0.5)$, the reinsurer is “more important”, and it assumes the risk above ${a}_{r}$. As a result, the chance of a payment is so small that its VaR does not increase; it actually reduces to $-10$ because of the collected premium. For $\beta \in (0.5,0.654)$, no agreement is reached between the two parties.
- From Table 1, we see that when β gets larger and the cedent becomes increasingly important, then ${\mathrm{VaR}}_{{\alpha}_{c}}\left({C}_{{f}_{1A}^{*}}\right)$ decreases, whereas ${\mathrm{VaR}}_{{\alpha}_{r}}\left({R}_{{f}_{1A}^{*}}\right)$ increases.
- When $\beta =0.5$ and $\beta =0.654$, the optimal ceded functions are only partially specified, and the risk of the two parties varies in some range. For example, when $\beta =0.5$, then ${\mathrm{VaR}}_{{\alpha}_{c}}\left({C}_{{f}_{1A}^{*}}\right)$ is maximized by choosing ${f}_{1A}^{*}\left(x\right)={(x-4605.2)}_{+}$ because the cedent is choosing a maximal ceded function and paying a maximal reinsurance premium (within the partially-specified optimal ceded functions). However, its $\mathrm{VaR}$ does not reduce with such a high deductible value. On the other hand, ${\mathrm{VaR}}_{{\alpha}_{c}}\left({C}_{{f}_{1A}^{*}}\right)$ is minimized with ${f}_{1A}^{*}\left(x\right)=0$, within the partially-specified optimal ceded functions.

#### 3.3.2. Scenario B: ${\alpha}_{c}=0.99$ and ${\alpha}_{r}=0.95$

## 4. Optimal Reinsurance Policy When $\mathit{f}\in {\mathcal{C}}^{\mathbf{2}}$

#### 4.1. Functional Form of the Ceded Function

#### 4.1.1. Case 1: $\beta >1/2$

#### 4.1.2. Case 2: $\beta <1/2$

#### 4.1.3. Case 3: $\beta =1/2$

#### 4.2. Parameter Values of the Optimal Ceded Function

#### 4.2.1. Case 1: $\beta >1/2$ and ${\alpha}_{c}<{\alpha}_{r}$

**Theorem**

**5.**

- 1.
- $d={d}^{*}$ when $1-{\alpha}_{c}<{\theta}^{*}<{S}_{X}\left(0\right)$;
- 2.
- $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$.

**Proof.**

#### 4.2.2. Case 2: $\beta >1/2$ and ${\alpha}_{c}>{\alpha}_{r}$

**Theorem**

**6.**

- 1.
- $d={d}^{*}$ when $1-{\alpha}_{r}<{\theta}^{*}<{S}_{X}\left(0\right)$;
- 2.
- $d={a}_{r}$ when ${\theta}^{*}<1-{\alpha}_{r}<{\theta}_{\beta}^{*}$;
- 3.
- $d={d}_{\beta}^{*}$ when ${\theta}^{*}<1-{\alpha}_{r}$ and $1-{\alpha}_{c}<{\theta}_{\beta}^{*}<1-{\alpha}_{r}$;
- 4.
- $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$.

**Proof.**

#### 4.2.3. Case 3: $\beta <1/2$ and ${\alpha}_{c}<{\alpha}_{r}$

**Theorem**

**7.**

- 1.
- $d={d}^{*}$ when $1-{\alpha}_{c}<{\theta}^{*}<{S}_{X}\left(0\right)$;
- 2.
- $d={a}_{c}$ when ${\theta}^{*}<1-{\alpha}_{c}<{\theta}_{\beta}^{*}$;
- 3.
- $d={d}_{\beta}^{*}$ when ${\theta}^{*}<1-{\alpha}_{c}$ and $1-{\alpha}_{r}<{\theta}_{\beta}^{*}<1-{\alpha}_{c}$;
- 4.
- $d={a}_{r}$ when ${\theta}^{*}<1-{\alpha}_{c}$ and ${\theta}_{\beta}^{*}<1-{\alpha}_{r}$;
- 5.
- $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$.

#### 4.2.4. Case 4: $\beta <1/2$ and ${\alpha}_{c}>{\alpha}_{r}$

**Theorem**

**8.**

- 1.
- $d={d}^{*}$ when $1-{\alpha}_{r}<{\theta}^{*}<{S}_{X}\left(0\right)$;
- 2.
- $d={a}_{r}$ when ${\theta}^{*}<1-{\alpha}_{r}<{S}_{X}\left(0\right)$;
- 3.
- $d=0$ when ${\theta}^{*}\ge {S}_{X}\left(0\right)$.

#### 4.3. The Illustrative Example Continued

#### 4.3.1. Scenario A: ${\alpha}_{c}=0.95$ and ${\alpha}_{r}=0.99$

- Since the cedent and the reinsurer have more choices when $f\in {\mathcal{C}}^{2}$, their VaRs under the optimal reinsurance policy ${f}_{2A}^{*}$ are lower than the corresponding ones under ${f}_{1A}^{*}$. In particular, the reinsurer’s risk is reduced significantly even when $\beta =1$.
- For $\beta \in [0,0.5)$, the reinsurer assumes the “good” risk in the layer $(0,{S}_{X}^{-1}\left({\theta}^{*}\right))$, as well as losses greater than $4422.9$. The former layer creates profit, and the latter layer does not contribute to its VaR because the chance of penetration is too small compared with the probability level ${\alpha}_{r}$ used in its VaR.
- For $\beta \in (0.5,1)$, the insurer retains the “good” risk in the layer $(0,{S}_{X}^{-1}\left({\theta}^{*}\right))$, as well as the losses greater than $2813.4$. The former layer creates profit, and the latter layer does not contribute to its VaR because the chance of penetration is too small compared with the probability level ${\alpha}_{c}$ used in its VaR.

#### 4.3.2. Scenario B: ${\alpha}_{c}=0.99$ and ${\alpha}_{r}=0.95$

## 5. A Numerical Comparison of the Optimal Reinsurance Policies in ${\mathcal{C}}^{\mathbf{1}}$ and ${\mathcal{C}}^{\mathbf{2}}$

- The efficient frontier for the $\mathrm{VaR}$s of the two parties with $f\in {\mathcal{C}}^{1}$ is represented by the path from $A=(1182.32,3422.85)$ to $B=(2995.72,0)$ and then to $C=(3005.73,-10)$. Note that the points between A and B represent the VaRs of the two parties resulting from the optimal policies obtained with $\beta =0.5$. The points between B and C represent the VaRs of the two parties resulting from the optimal policies obtained with $\beta =0.654$.
- The efficient frontier for the VaRs of the two parties when $f\in {\mathcal{C}}^{2}$ is represented by the path from $D=(1122.32,1873.41)$ to $E=(3025.41,-29.68)$.
- For the quota-share reinsurance with ${f}_{1}\left(x\right)=ax$ where a ranges from zero to one, the VaRs of the two parties go from B to $F=(1200,3405.2)$. When $f\in {\mathcal{C}}^{1}$, the quota-share reinsurance policy is quite close to the efficient frontier.
- For the excess-of-loss reinsurance ${f}_{2}\left(x\right)={(x-d)}_{+}$ with d ranging from zero to ${a}_{r}=4605.2$, the VaRs of the two parties go along the path $F\to A\to G\to C$ with $G=(3055.47,1545.43)$.

- The efficient frontier for the VaRs of the two parties with $f\in {\mathcal{C}}^{1}$ is represented by the path from $A=(1182.32,1813.41)$ to $B=(3055.73,-60)$.
- The efficient frontier for the VaRs of the two parties when $f\in {\mathcal{C}}^{2}$ is represented by the path from $C=(1170.33,1825.38)$ to $D=(3073.43,-77.67)$. In fact, it can algebraically be shown that the path from B to A is actually a part of the path from D to C. That is, by allowing $f\in {\mathcal{C}}^{2}$, the efficient frontier is extended from the path $B\to A$ to the path $D\to C$.
- For the quota-share reinsurance with the parameter a ranging from zero to one, the VaRs of the two parties are represented by the path from $E=(4605.7,0)$ to $F=(1200,1795.7)$. We see that when ${\alpha}_{c}>{\alpha}_{r}$, the quota-share reinsurance policies are not efficient.
- For the excess-of-loss reinsurance with the parameter d ranging from zero to ${a}_{c}=4605.2$, the VaRs of the two parties change along the path $F\to A\to B\to E$. We see that setting $d\in (0,{a}_{r})$ is quite efficient, whereas setting $d\in ({a}_{r},{a}_{c})$ is not.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{c}}}\left({\mathit{C}}_{{\mathit{f}}_{1\mathit{A}}^{*}}\right)$ | ${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{r}}}\left({\mathit{R}}_{{\mathit{f}}_{1\mathit{A}}^{*}}\right)$ | |
---|---|---|

$\beta \in [0,0.5)$ | 3005.73 | $-10$ |

$\beta =0.5$ | between $2995.73$ and $3005.73$ | between $-10$ and 0 |

$\beta \in (0.5,0.654)$ | 2995.73 | 0 |

$\beta =0.654$ | between $1182.32$ and $2995.73$ | between 0 and $3422.85$ |

$\beta \in (0.654,1]$ | 1182.32 | 3422.85 |

${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{c}}}\left({\mathit{C}}_{{\mathit{f}}_{1\mathit{B}}^{*}}\right)$ | ${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{r}}}\left({\mathit{R}}_{{\mathit{f}}_{1\mathit{B}}^{*}}\right)$ | |
---|---|---|

$\beta \in [0,0.5)$ | 3055.73 | $-60$ |

$\beta =0.5$ | between $1182.32$ and $3055.73$ | between $-60$ and $1813.41$ |

$\beta \in (0.5,1]$ | 1182.32 | 1813.41 |

${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{c}}}\left({\mathit{C}}_{{\mathit{f}}_{2\mathit{A}}^{*}}\right)$ | ${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{r}}}\left({\mathit{R}}_{{\mathit{f}}_{2\mathit{A}}^{*}}\right)$ | |
---|---|---|

$\beta \in [0,0.5)$ | 3025.41 | $-29.68$ |

$\beta =0.5$ | between $1122.32$ and $3025.41$ | between $-29.68$ and $1873.41$ |

$\beta \in (0.5,1]$ | 1122.32 | 1873.41 |

${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{c}}}\left({\mathit{C}}_{{\mathit{f}}_{2\mathit{B}}^{*}}\right)$ | ${\mathbf{VaR}}_{{\mathit{\alpha}}_{\mathit{r}}}\left({\mathit{R}}_{{\mathit{f}}_{2\mathit{B}}^{*}}\right)$ | |
---|---|---|

$\beta \in [0,0.5)$ | 3073.43 | $-77.67$ |

$\beta =0.5$ | between $1170.33$ and $3073.43$ | between $-77.67$ and $1825.38$ |

$\beta \in (0.5,1]$ | 1170.33 | 1825.38 |

© 2017 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/).

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**MDPI and ACS Style**

Jiang, W.; Ren, J.; Zitikis, R. Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account. *Risks* **2017**, *5*, 11.
https://doi.org/10.3390/risks5010011

**AMA Style**

Jiang W, Ren J, Zitikis R. Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account. *Risks*. 2017; 5(1):11.
https://doi.org/10.3390/risks5010011

**Chicago/Turabian Style**

Jiang, Wenjun, Jiandong Ren, and Ričardas Zitikis. 2017. "Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account" *Risks* 5, no. 1: 11.
https://doi.org/10.3390/risks5010011