# Pareto-Optimal Reinsurance Revisited: A Two-Stage Optimisation Procedure Approach

Authors: Ying Fang; Lu Wang; Zhongfeng Qu
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this paper, based on the Tail-Value-at-Risk (TVaR) measure, we revisit the Pareto-optimal reinsurance policies for the insurer and the reinsurer via a two-stage optimisation procedure. To reduce ex-post moral hazard, we assume that reinsurance contracts satisfy the principle of indemnity and the incentive compatible constraint which have been advocated by Huberman et al. (1983). We show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form. The proposed class of premium principles encompasses the net premium principle, expected value premium principle, TVaR premium principle, generalized percentile premium principle, and so on. We further use the TVaR premium principle and the expected value premium principle as examples to illustrate the two-stage optimisation procedure by deriving explicitly the Pareto-optimal reinsurance policies. We extend the results by Cai et al. (2017) when the expected value premium principle is replaced by the TVaR premium principle.


---

## Body

## 1. Introduction

The study of optimal reinsurance design has drawn great interest from both academics and practitioners since the seminal work by Borch [1] and Arrow [2]. There have been many important literatures and conclusions about this problem in the past few decades. For example, by minimizing the variance of the insurer’s retained loss, Borch [1] showed that the stop-loss reinsurance is optimal under the expected value premium principle. Arrow [2] obtained that the optimal reinsurance policy is a stop-loss reinsurance strategy when the optimisation criterion is to maximize the expected utility function of the insurer. Both these results have been extended in a number of important directions. For example, Young [3] generalized Arrow’s result under Wang’s premium principle. Kaluszka [4] generalized Borch’s result under mean-variance premium principles. Kaluszka and Okolewski [5] showed that the limited stop-loss and the truncated stop-loss are the optimal contracts under a number of criteria including the maximization of the expected utility, the stability, and the survival probability of the insurer for a fixed reinsurance premium calculated according to the maximal possible claims principle. Cai and Tan [6] developed two new optimisation criteria for deriving the optimal retentions by minimizing the Value-at-Risk (VaR) and the conditional tail expectation (CTE) of the total risk of an insurer. In recent years, VaR and CTE have been used as optimisation criteria to study the optimal reinsurance strategy. For example, by minimizing VaR and CTE of an insurer’s total cost, Cai et al. [7] derived the optimal reinsurance strategies in the set of increasing and convex ceded loss functions. The optimal reinsurance strategy depends on the confidence level of risk measurement, and it can be a stop-loss reinsurance strategy, a quota-share reinsurance strategy, or a change-loss reinsurance strategy. Bernard and Tian [8] provided alternative risk transfer mechanisms on the capital market when the optimal reinsurance is arranged under tail risk measures. Cheung [9] gave a geometric approach to revisit the optimal reinsurance problem and generalized the results in [7] by studying the VaR-minimization problem with Wang’s premium principle. Chi and Tan [10] analyzed the VaR- and CVaR-based optimal reinsurance models over different classes of ceded loss functions with increasing generality. However, the above statements only consider the insurer, and from the point of view of the reinsurer, the optimal policy may not be optimal. For example, Vajda [11] showed that the optimal reinsurance strategy is a quota-share reinsurance instead of a stop-loss reinsurance when the optimisation criterion is to minimize the variance of the loss of the reinsurer. Thus, an optimal reinsurance contract for the insurer may not be optimal for the reinsurer and it might be unacceptable for the reinsurer. Then, an interesting question about optimal reinsurance is to design a reinsurance contract so that it considers the interests of both the insurer and the reinsurer.Borch [1] first studied the optimal reinsurance strategy that consider the interests of both the insurer and the reinsurer. He discussed the optimal quota-share retention and stop-loss retention that maximize the product of the expected utility functions of the two parties’ wealth. Kaishev [12] analyzed the optimal reinsurance contracts under which the finite horizon joint survival probability of the two parties is maximized. Under the general reinsurance principles, Cai et al. [13] took maximization of the joint survival probability and the joint profitable probability of both the insurer and the reinsurer as the optimisation criterion to give a sufficient condition of the optimal reinsurance existence. To maximize the joint survival probability of the insurer and the reinsurer, Fang and Qu [14] studied the optimal policy of combination of quota-share reinsurance and stop-loss reinsurance. Fang et al. [15] studied the optimal reinsurance models from the perspective of both the insurer and the reinsurer by minimizing their total costs under the criteria of the loss function which is defined by the joint Value-at-Risk. Cai et al. [16] studied the optimal reinsurance strategy, which was based on the minimum convex combination of the VaR of the insurer and the reinsurer under two types of constraints. Lo [17] discussed the generalized problems in [16] by using the Neyman–Pearson approach. Based on the optimal reinsurance strategy in [16], Jiang et al. [18] proved that the optimal reinsurance strategy is a Pareto-optimal reinsurance policy and gave optimal reinsurance strategies using the geometric method. Cai et al. [19] studied the Pareto optimality of reinsurance arrangements under general model settings and obtained the explicit forms of the Pareto-optimal reinsurance contracts under the TVaR measure and the expected value premium principle. Jiang et al. [20] studied the optimal reinsurance with constraints under the distortion risk measure. By the geometric approach, Fang et al. [21] studied Pareto-optimal reinsurance policies under general premium principles and gave the explicit parameters of the optimal ceded loss functions under the Dutch premium principle and Wang’s premium principle. Lo and Tang [22] characterized the set of Pareto-optimal reinsurance policies analytically and visualized the insurer-reinsurer trade-off structure geometrically. Huang and Yin [23] studied two classes of optimal reinsurance models from perspectives of both insurers and reinsurers by minimizing their convex combination where the risk is measured by a distortion risk measure and the premium is given by a distortion premium principle.In this paper, based on the TVaR measure, we revisit the Pareto-optimal reinsurance policies for the insurer and the reinsurer via a two-stage optimisation procedure which was proposed by Asimit et al. [24]. To reduce ex-post moral hazard, we assume that reinsurance contracts satisfy the principle of indemnity and the incentive compatible constraint which have been advocated by Huberman et al. [25]. We first show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form such as (10). We emphasize that there are many premium principles which satisfy this property such as the net premium principle, expected value premium principle, TVaR premium principle, and generalized percentile premium principle. Then, we take the TVaR premium principle and the expected value premium principle as examples to illustrate the two-stage optimisation procedure by deriving explicitly the parameters of the Pareto-optimal reinsurance policies.It is worth noting that the Pareto-optimal reinsurance contracts under the TVaR measure and the expected value premium principle has been obtained in [19]. We reexamine this problem for two reasons. First, it should be emphasized that the results can be achieved by using a different approach based on a two-stage optimisation procedure. Second, and more importantly, the two-stage optimisation procedure is more intuitive and can analyze Pareto-optimal reinsurance policies with other reinsurance premium principles.The remaining arrangements of this paper are as follows. In Section2, we introduce some definitions and model formulation and then we show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form such as (10). Based on the TVaR measure, we obtain the Pareto-optimal policies under the TVaR premium principle and the expected value premium principle in Section 3. In Section 4, we give illustrative numerical examples. Section 5 concludes the paper. Finally, all the proofs are given in the Appendix.


## 2. Model Formulation

LetX be the amount of loss faced by the insurer in a given time period. Suppose that X is a nonnegative random variable with a distribution function FXx=PX≤x and survival function SXx=1−FXx. In addition, the value of the right endpoint XF of the distribution function FXx can be either finite or infinite, where XF≔infz:Fz=1. Under a reinsurance arrangement, RX and IRX represent the ceded loss and the retained loss of the insurer, respectively, where IRX=X−RX. Functions Rx and IRx are called the ceded loss function and the retained loss function. The principle of indemnity, which is widely used in insurance and reinsurance, requires the indemnity to be nonnegative and less than the initial loss. Mathematically, we should have 0≤Rx≤x. Let πRX be the reinsurance premium. Under such a setting, the total losses of the insurer and the reinsurer are MR≔X−RX+πRX and NR≔RX−πRX, respectively.In this paper, besides the principle of indemnity, we also assume that reinsurance contracts satisfy the incentive compatible constraint which has been advocated by Huberman et al. [25] to reduce ex-post moral hazard. This means that the more the realized loss, the more paid by both the insurer and the reinsurer. Mathematically, this implies that both the ceded loss function and the retained loss function should be increasing. Therefore, throughout the paper, we assume that the admissible set of ceded loss functions is given by(1)ℱ=Rx:0≤Rx≤x,bothRxandIRxare increasing functions.It was shown by Chi and Tan [10] that all functions Rx∈ℱ are Lipschitz continuous and differentiable almost everywhere.In insurance and finance, risk measures such as VaR and TVaR have been widely used for quantifying risks. Now, we give a brief description of VaR and TVaR measures:Definition 1.
VaR. For a random variable X, VaR is defined as(2)VaRpX≔infx∈R:PX≤x≥p,where 0<p<1 represents a confidence level of the loss variable X.Definition 2.
TVaR. For a random variable X, TVaR is defined as(3)TVaRpX≔11−p∫p1VaRsXds=VaRpX+11−pEX−VaRpX+,where 0<p<1 represents a confidence level of the loss variable X.Remark 1.
(1)
By the definitions of the VaR and the TVaR, distinctly,TVaRp evaluates the expected loss amount incurred among the worst 1−p% scenarios under a confidence level p. Therefore, the TVaR represents a more precise risk measurement than the VaR.(2)
When1−p≥SX0, we have VaRpX=0. Therefore, in order to avoid a trivial case, we assume that 1−p∈0,SX0.
In this paper, we assume that the confidence levels of the insurer and the reinsurer are possibly different. Letαc and αr denote the confidence levels of the insurer and the reinsurer, respectively. Therefore, the total loss of the insurer and the reinsurer under the TVaR measure is(4)TVaRαcMR=VaRαcMR+11−αcEMR−VaRαcMR+,TVaRαrNR=VaRαrNR+11−αrENR−VaRαrNR+.
Next, we study Pareto-optimal reinsurance policies whereby the risk is measured by the TVaR. For our model, a reinsurance policy with the ceded loss functionR∗x is called Pareto optimal if there is no other admissible ceded loss function R∈ℱ such that TVaRαcMR≤TVaRαcMR∗ and TVaRαrNR≤TVaRαrNR∗ and at least one of the inequalities is strict. A general approach to identify Pareto-optimal reinsurance policies is to minimize a convex combination of the TVaRs of the two parties. The result can be found in [18–20].Proposition 1.
All Pareto-optimal reinsurance policiesR in ℱ can be determined by solving the problem:(5)minR∈ℱβTVaRαcMR+1−βTVaRαrNR,where β∈0,1.In view of Proposition1, throughout the rest of this paper, we only need to determine optimal reinsurance policies by solving the optimisation problem (5). Define(6)VR=βTVaRαcMR+1−βTVaRαrNR.Then, by translation invariance and comonotonic additivity of TVaR, we have(7)VR=βTVaRαcX−βTVaRαcRX+1−βTVaRαrRX+2β−1πRX.Therefore, the optimisation problem (5) becomes(8)minR∈ℱHR,where(9)HR=−βTVaRαcRX+1−βTVaRαrRX+2β−1πRX.In this paper, we determine the Pareto-optimal reinsurance policies via a two-stage optimisation procedure which was developed in [24]. The first stage is solving an infinite-dimensional problem, while the second stage becomes a classical constrained optimisation problem. The first stage can be solved as shown in Proposition 1 in [24], and we now present it as a lemma.Lemma 1.
Letf⋅ be a real-valued function defined on s1,s2 with 0≤s1≤s2≤1. Then,(10)minR∈ℱ∫s1s2fsRVaRsxds,subjecttoRVaRs1X=ξ1,RVaRs2X=ξ2,is uniquely solved by(11)R∗X;ξ1,ξ2=X−VaRs1X+ξ1∧ξ2,iffs<0for alls1≤s≤s2,ξ1+X−VaRs2X+ξ2−ξ1+,iffs>0for alls1≤s≤s2,where ξ1,ξ2 are some constants such that 0≤ξ2−ξ1≤VaRs2X−VaRs1X.Note that(12)HR=∫αc1−β1−αcRVaRsXds+∫αr11−β1−αrRVaRsXds+2β−1πRX.By Lemma1, we know that the Pareto-optimal reinsurance policy exists if the reinsurance premiums πRX can be expressed as an integral form such as (10). Next, we give several premium principles which satisfy this property:(1)
Net premium principle:πX=EX. Since Rx is a nondecreasing continuous function, then πRX=∫01RVaRsXds.(2)
Expected value premium principle:πX=1+θEX, where θ∈0,1 is a safety loading coefficient. Therefore, πRX=1+θ∫01RVaRsXds.(3)
TVaR premium principle:πX=1+θ/1−α∫α1VaRsXds, where α∈0,1 is a confidence level and θ∈0,1 is a safety loading coefficient. Since Rx is a nondecreasing continuous function, then πRX=1+θ/1−α∫α1RVaRsXds.(4)
Generalized percentile premium principle:πX=EX+βFX−11−p−EX with 0<β,p<1. Since FX−11−p=VaR1−pX, then πRX=1−βERX+βRVaR1−pX.In the following sections, we take the TVaR premium principle and the expected value premium principle as examples to illustrate the two-stage optimisation procedure.


## 3. Pareto-Optimal Reinsurance Policy

In this section, we determine the Pareto-optimal reinsurance policies under the TVaR premium principle and the expected value premium principle. The TVaR premium principle was first proposed by Young [26]. It can be viewed as an extended version of the expected value premium principle, that is, letting α=0 gives the expected value premium principle.


### 3.1. Pareto-Optimal Reinsurance Policies under TVaR Principle

Under the TVaR premium principle, the optimisation problem (8) becomes(13)minR∈ℱHR,where HR=−βTVaRαcRX+1−βTVaRαcRX+2β−11+θTVaRαRX. From the mathematical point of view, the confidence level α can be larger than confidence levels αc and αr. However, α is usually smaller while αc and αr are usually larger in practice. So, we assume further α<minαc,αr to avoid complex and lengthy discussions in this section.For simplicity, we define the following notations:(14)a=VaRαX,ac=VaRαcX,ar=VaRαrX,ξ=RVaRαX,ξc=RVaRαcX,ξr=RVaRαrX,m=2β−11+θ1−α−β1−αc,n=2β−11+θ1−α+1−β1−αr,sβ=1−β−1m,tβ=1−βn.Next, we divide our discussion into three cases: ①ac<ar; ② ar<ac; ③ ac=ar. Then, we obtain the following three theorems.Theorem 1.
Under the conditionac<ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2 and β−1/1−αr<m, then(15)R∗x=x∧VaRsβX,when1+θ1−αc>1−α,x∧VaRθ+α/1+θX,when1+θ1−αc≤1−α.(2)
If0≤β<1/2 and β−1/1−αr>m, then R∗x=x.(3)
If0≤β<1/2 and β−1/1−αr=m, then R∗x=xIa≤x≤ar+RxIx>ar, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(4)
Ifβ=1/2, then R∗x=RxIa≤x≤ac+u1Ix>ac, where u1 is an arbitrary constant in u,ac−a+u, u is an arbitrary constant in 0,a, and Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(5)
If1/2<β≤1 and m>0, then R∗x=0.(6)
If1/2<β<1 and β−1/1−αr<m<0, then(16)R∗x=0,when1+θ1−αc≥1−α,x−VaRθ+α/1+θX+∧VaRsβX−VaRθ+α/1+θX,when1+θ1−αc<1−α.(7)
If1/2<β≤1 and m<β−1/1−αr, then R∗x=x−VaRθ+α/1+θX+.(8)
If1/2<β<1 and m=β−1/1−αr, then R∗x=x−VaRθ+α/1+θX+Ia≤x≤ar+RxIx>ar, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(9)
If1/2<β<1 and m=0, then R∗x=0.(10)
Ifβ=1 and 1+θ1−αc=1−α, then R∗x=RxIx>ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 2.
Under the conditionar<ac, the Pareto-optimal reinsurance policies are given as follows:(1)
Ifβ=0 and 1+θ1−αr=1−α, then R∗x=xIa≤x≤ar+RxIx>ar, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
If0≤β<1/2 and n>β/1−αc, then R∗x=x∧VaRθ+α/1+θX.(3)
If0<β<1/2 and 0<n<β/1−αc, then(17)R∗x=x∧VaRθ+α/1+θX+x−VaRtβX+,when1+θ1−αr<1−α,x,when1+θ1−αr≥1−α.(4)
If0<β<1/2 and n=β/1−αc, then R∗x=x∧VaRθ+α/1+θXIa≤x≤ac+RxIx>ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(5)
If0≤β<1/2 and n<0, then R∗x=x.(6)
If0<β<1/2 and n=0, then R∗x=xIa≤x≤arorx>ac+RxIar<x≤ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(7)
Ifβ=1/2, then R∗x=RxIa≤x≤ar+x−ar+u2Ix>ar, where u2 is an arbitrary constant in u,ar−a+u, u is an arbitrary constant in 0,a, and Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(8)
If1/2<β≤1 and n>β/1−αc, then R∗x=0.(9)
If1/2<β≤1 and 0<n<β/1−αc, then(18)R∗x=x−VaRθ+α/1+θX+,when1+θ1−αr≤1−α,x−VaRtβX+,when1+θ1−αr>1−α.(10)
If1/2<β≤1 and n=β/1−αc, then R∗x=RxIx>ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 3.
Under the conditionac=ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2, then(19)R∗x=x,when1+θ1−αc>1−α,xIa≤x≤ac+RxIx>ac,when1+θ1−αc=1−α,x∧VaRθ+α/1+θX,when1+θ1−αc<1−α,whereRx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
Ifβ=1/2, the objective function is identical to 0 and the problem is trivial.(3)
If1/2<β≤1, then(20)R∗x=0,when1+θ1−αc>1−α,RxIx>ac,when1+θ1−αc=1−α,x−VaRθ+α/1+θX+,when1+θ1−αc<1−α,whereRx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.


### 3.2. Pareto-Optimal Reinsurance Policies under Expected Value Principle

In this section, we reexamine an optimal reinsurance problem studied in [19], in which the objective is to find the optimal reinsurance contracts that minimize the TVaR of the total risk exposure under the expected value premium principle. We provide a more intuitive approach to solve the problem by using a two-stage optimisation method. Under the expected value principle, the optimisation problem (5) becomes(21)minR∈ℱ−βTVaRαcRX+1−βTVaRαrRX+2β−11+θERX.For simplicity, we define the following notations:(22)m0=2β−11+θ−β1−αc,n0=2β−11+θ+1−β1−αr,s0β=1−β−1m0,t0β=1−βn0.Theorem 4.
Under the conditionac<ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2 and β−1/1−αr<m0<0, then(23)R∗x=x∧VaRs0βX,when1+θ1−αc>1,0,whenSX0≤θ∗and1+θ1−αc≤1,x∧VaRθ/1+θX,whenSX0>θ∗and1+θ1−αc≤1.(2)
If0≤β<1/2 and m0<β−1/1−αr, then R∗x=x.(3)
If0≤β<1/2 and β−1/1−αr=m0, then R∗x=xIx≤ar+RxIx>ar, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(4)
Ifβ=1/2, then R∗x=RxIx≤ac+u3Ix>ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ and u3∈0,ac.(5)
If1/2<β≤1 and m0>0, then R∗x=0.(6)
If1/2<β<1 and β−1/1−αr<m0<0, then(24)R∗x=0,when1+θ1−αc≥1,x∧VaRs0βX,whenSX0≤θ∗and1+θ1−αc<1,x−VaRθ/1+θX+∧VaRs0βX−VaRθ/1+θX,whenSX0>θ∗and1+θ1−αc<1.(7)
If1/2<β≤1 and m0<β−1/1−αr, then(25)R∗x=x,whenSX0≤θ∗,x−VaRθ/1+θX+,whenSX0>θ∗.(8)
If1/2<β<1 and m0=β−1/1−αr, then(26)R∗x=xIx≤ar+RxIx>ar,whenSX0≤θ∗,x−VaRθ/1+θX+Ix≤ar+RxIx>ar,whenSX0>θ∗,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(9)
If1/2<β<1 and m0=0, then R∗x=0.(10)
Ifβ=1 and 1+θ1−αc=1, then R∗x=RxIx>ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 5.
Under the conditionar<ac, the Pareto-optimal reinsurance policies are given as follows:(1)
Ifβ=0 and 1+θ1−αr=1, then R∗x=xIx≤ar+RxIx>ar, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
If0≤β<1/2 and n0>β/1−αc, then(27)R∗x=0,whenSX0≤θ∗,x∧VaRθ/1+θX,whenSX0>θ∗.(3)
If0<β<1/2 and 0<n0<β/1−αc, then(28)R∗x=x,when1+θ1−αr≥1,x−VaRt0βX+,whenSX0≤θ∗and1+θ1−αr<1,x∧VaRθ/1+θX+x−VaRt0βX+,whenSX0>θ∗and1+θ1−αr<1.(4)
If0<β<1/2 and n0=β/1−αc, then(29)R∗x=RxIx>ac,whenSX0≤θ∗,x∧VaRθ/1+θXIx≤ac+RxIx>ac,whenSX0>θ∗,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(5)
If0≤β<1/2 and n0<0, then R∗x=x.(6)
If0<β<1/2 and n0=0, then R∗x=xIx≤arorx>ac+RxIar<x≤ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(7)
Ifβ=1/2, then R∗x=RxIx≤ar+x−ar+u4Ix>ar, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ and u4∈0,ar.(8)
If1/2<β≤1 and n0>β/1−αc, then R∗x=0.(9)
If1/2<β≤1 and 0<n0<β/1−αc, then(30)R∗x=x−VaRt0βX+,when1+θ1−αr≥1,x,whenSX0≤θ∗and1+θ1−αr<1,x−VaRθ/1+θX+,whenSX0>θ∗and1+θ1−αr<1.(10)
If1/2<β≤1 and n0=β/1−αc, then R∗x=RxIx>ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 6.
Under the conditionac=ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2, then(31)R∗x=x,when1+θ1−αc>1,xIx≤ac+RxIx>ac,when1+θ1−αc=1,0,whenSX0≤θ∗and1+θ1−αc<1,x∧VaRθ/1+θX,whenSX0>θ∗and1+θ1−αc<1,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
Ifβ=1/2, the objective function is zero and the problem is trivial.(3)
If1/2<β≤1, then(32)R∗x=0,when1+θ1−αc>1,RxIx>ac,when1+θ1−αc=1,x,whenSX0≤θ∗and1+θ1−αc<1,x−VaRθ/1+θX+,whenSX0>θ∗and1+θ1−αc<1,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Remark 1.
By comparing our results with those in [19], we would like to point out the relationship between the two articles. First, Cai et al. [19] give the explicit forms of the Pareto-optimal reinsurance contracts under the expected value premium principle by the construction method. In our paper, we use the two-stage optimisation procedure. This technique is intuitive and applicable when the expected value premium principle is replaced by other premium principles. Using this technique, we extend the results in [19] under the TVaR premium principle. Second, under the expected value premium principle, Cai et al. [19] derived the optimal ceded loss functions without considering the relationship between SX0 and 1 in their Theorems 1 and 2. However, we discuss the relationship between them and derive different optimal ceded functions from theirs in the case SX0<1. By comparison, we find that our result is more reasonable.


## 3.1. Pareto-Optimal Reinsurance Policies under TVaR Principle

Under the TVaR premium principle, the optimisation problem (8) becomes(13)minR∈ℱHR,where HR=−βTVaRαcRX+1−βTVaRαcRX+2β−11+θTVaRαRX. From the mathematical point of view, the confidence level α can be larger than confidence levels αc and αr. However, α is usually smaller while αc and αr are usually larger in practice. So, we assume further α<minαc,αr to avoid complex and lengthy discussions in this section.For simplicity, we define the following notations:(14)a=VaRαX,ac=VaRαcX,ar=VaRαrX,ξ=RVaRαX,ξc=RVaRαcX,ξr=RVaRαrX,m=2β−11+θ1−α−β1−αc,n=2β−11+θ1−α+1−β1−αr,sβ=1−β−1m,tβ=1−βn.Next, we divide our discussion into three cases: ①ac<ar; ② ar<ac; ③ ac=ar. Then, we obtain the following three theorems.Theorem 1.
Under the conditionac<ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2 and β−1/1−αr<m, then(15)R∗x=x∧VaRsβX,when1+θ1−αc>1−α,x∧VaRθ+α/1+θX,when1+θ1−αc≤1−α.(2)
If0≤β<1/2 and β−1/1−αr>m, then R∗x=x.(3)
If0≤β<1/2 and β−1/1−αr=m, then R∗x=xIa≤x≤ar+RxIx>ar, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(4)
Ifβ=1/2, then R∗x=RxIa≤x≤ac+u1Ix>ac, where u1 is an arbitrary constant in u,ac−a+u, u is an arbitrary constant in 0,a, and Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(5)
If1/2<β≤1 and m>0, then R∗x=0.(6)
If1/2<β<1 and β−1/1−αr<m<0, then(16)R∗x=0,when1+θ1−αc≥1−α,x−VaRθ+α/1+θX+∧VaRsβX−VaRθ+α/1+θX,when1+θ1−αc<1−α.(7)
If1/2<β≤1 and m<β−1/1−αr, then R∗x=x−VaRθ+α/1+θX+.(8)
If1/2<β<1 and m=β−1/1−αr, then R∗x=x−VaRθ+α/1+θX+Ia≤x≤ar+RxIx>ar, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(9)
If1/2<β<1 and m=0, then R∗x=0.(10)
Ifβ=1 and 1+θ1−αc=1−α, then R∗x=RxIx>ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 2.
Under the conditionar<ac, the Pareto-optimal reinsurance policies are given as follows:(1)
Ifβ=0 and 1+θ1−αr=1−α, then R∗x=xIa≤x≤ar+RxIx>ar, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
If0≤β<1/2 and n>β/1−αc, then R∗x=x∧VaRθ+α/1+θX.(3)
If0<β<1/2 and 0<n<β/1−αc, then(17)R∗x=x∧VaRθ+α/1+θX+x−VaRtβX+,when1+θ1−αr<1−α,x,when1+θ1−αr≥1−α.(4)
If0<β<1/2 and n=β/1−αc, then R∗x=x∧VaRθ+α/1+θXIa≤x≤ac+RxIx>ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(5)
If0≤β<1/2 and n<0, then R∗x=x.(6)
If0<β<1/2 and n=0, then R∗x=xIa≤x≤arorx>ac+RxIar<x≤ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(7)
Ifβ=1/2, then R∗x=RxIa≤x≤ar+x−ar+u2Ix>ar, where u2 is an arbitrary constant in u,ar−a+u, u is an arbitrary constant in 0,a, and Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(8)
If1/2<β≤1 and n>β/1−αc, then R∗x=0.(9)
If1/2<β≤1 and 0<n<β/1−αc, then(18)R∗x=x−VaRθ+α/1+θX+,when1+θ1−αr≤1−α,x−VaRtβX+,when1+θ1−αr>1−α.(10)
If1/2<β≤1 and n=β/1−αc, then R∗x=RxIx>ac, where Rx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 3.
Under the conditionac=ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2, then(19)R∗x=x,when1+θ1−αc>1−α,xIa≤x≤ac+RxIx>ac,when1+θ1−αc=1−α,x∧VaRθ+α/1+θX,when1+θ1−αc<1−α,whereRx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
Ifβ=1/2, the objective function is identical to 0 and the problem is trivial.(3)
If1/2<β≤1, then(20)R∗x=0,when1+θ1−αc>1−α,RxIx>ac,when1+θ1−αc=1−α,x−VaRθ+α/1+θX+,when1+θ1−αc<1−α,whereRx is any increasing 1-Lipschitz continuous function such that R∗x∈ℱ.


## 3.2. Pareto-Optimal Reinsurance Policies under Expected Value Principle

In this section, we reexamine an optimal reinsurance problem studied in [19], in which the objective is to find the optimal reinsurance contracts that minimize the TVaR of the total risk exposure under the expected value premium principle. We provide a more intuitive approach to solve the problem by using a two-stage optimisation method. Under the expected value principle, the optimisation problem (5) becomes(21)minR∈ℱ−βTVaRαcRX+1−βTVaRαrRX+2β−11+θERX.For simplicity, we define the following notations:(22)m0=2β−11+θ−β1−αc,n0=2β−11+θ+1−β1−αr,s0β=1−β−1m0,t0β=1−βn0.Theorem 4.
Under the conditionac<ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2 and β−1/1−αr<m0<0, then(23)R∗x=x∧VaRs0βX,when1+θ1−αc>1,0,whenSX0≤θ∗and1+θ1−αc≤1,x∧VaRθ/1+θX,whenSX0>θ∗and1+θ1−αc≤1.(2)
If0≤β<1/2 and m0<β−1/1−αr, then R∗x=x.(3)
If0≤β<1/2 and β−1/1−αr=m0, then R∗x=xIx≤ar+RxIx>ar, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(4)
Ifβ=1/2, then R∗x=RxIx≤ac+u3Ix>ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ and u3∈0,ac.(5)
If1/2<β≤1 and m0>0, then R∗x=0.(6)
If1/2<β<1 and β−1/1−αr<m0<0, then(24)R∗x=0,when1+θ1−αc≥1,x∧VaRs0βX,whenSX0≤θ∗and1+θ1−αc<1,x−VaRθ/1+θX+∧VaRs0βX−VaRθ/1+θX,whenSX0>θ∗and1+θ1−αc<1.(7)
If1/2<β≤1 and m0<β−1/1−αr, then(25)R∗x=x,whenSX0≤θ∗,x−VaRθ/1+θX+,whenSX0>θ∗.(8)
If1/2<β<1 and m0=β−1/1−αr, then(26)R∗x=xIx≤ar+RxIx>ar,whenSX0≤θ∗,x−VaRθ/1+θX+Ix≤ar+RxIx>ar,whenSX0>θ∗,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(9)
If1/2<β<1 and m0=0, then R∗x=0.(10)
Ifβ=1 and 1+θ1−αc=1, then R∗x=RxIx>ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 5.
Under the conditionar<ac, the Pareto-optimal reinsurance policies are given as follows:(1)
Ifβ=0 and 1+θ1−αr=1, then R∗x=xIx≤ar+RxIx>ar, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
If0≤β<1/2 and n0>β/1−αc, then(27)R∗x=0,whenSX0≤θ∗,x∧VaRθ/1+θX,whenSX0>θ∗.(3)
If0<β<1/2 and 0<n0<β/1−αc, then(28)R∗x=x,when1+θ1−αr≥1,x−VaRt0βX+,whenSX0≤θ∗and1+θ1−αr<1,x∧VaRθ/1+θX+x−VaRt0βX+,whenSX0>θ∗and1+θ1−αr<1.(4)
If0<β<1/2 and n0=β/1−αc, then(29)R∗x=RxIx>ac,whenSX0≤θ∗,x∧VaRθ/1+θXIx≤ac+RxIx>ac,whenSX0>θ∗,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(5)
If0≤β<1/2 and n0<0, then R∗x=x.(6)
If0<β<1/2 and n0=0, then R∗x=xIx≤arorx>ac+RxIar<x≤ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(7)
Ifβ=1/2, then R∗x=RxIx≤ar+x−ar+u4Ix>ar, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ and u4∈0,ar.(8)
If1/2<β≤1 and n0>β/1−αc, then R∗x=0.(9)
If1/2<β≤1 and 0<n0<β/1−αc, then(30)R∗x=x−VaRt0βX+,when1+θ1−αr≥1,x,whenSX0≤θ∗and1+θ1−αr<1,x−VaRθ/1+θX+,whenSX0>θ∗and1+θ1−αr<1.(10)
If1/2<β≤1 and n0=β/1−αc, then R∗x=RxIx>ac, where Rx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Theorem 6.
Under the conditionac=ar, the Pareto-optimal reinsurance policies are given as follows:(1)
If0≤β<1/2, then(31)R∗x=x,when1+θ1−αc>1,xIx≤ac+RxIx>ac,when1+θ1−αc=1,0,whenSX0≤θ∗and1+θ1−αc<1,x∧VaRθ/1+θX,whenSX0>θ∗and1+θ1−αc<1,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.(2)
Ifβ=1/2, the objective function is zero and the problem is trivial.(3)
If1/2<β≤1, then(32)R∗x=0,when1+θ1−αc>1,RxIx>ac,when1+θ1−αc=1,x,whenSX0≤θ∗and1+θ1−αc<1,x−VaRθ/1+θX+,whenSX0>θ∗and1+θ1−αc<1,whereRx is an increasing 1-Lipschitz continuous function such that R∗x∈ℱ.Remark 1.
By comparing our results with those in [19], we would like to point out the relationship between the two articles. First, Cai et al. [19] give the explicit forms of the Pareto-optimal reinsurance contracts under the expected value premium principle by the construction method. In our paper, we use the two-stage optimisation procedure. This technique is intuitive and applicable when the expected value premium principle is replaced by other premium principles. Using this technique, we extend the results in [19] under the TVaR premium principle. Second, under the expected value premium principle, Cai et al. [19] derived the optimal ceded loss functions without considering the relationship between SX0 and 1 in their Theorems 1 and 2. However, we discuss the relationship between them and derive different optimal ceded functions from theirs in the case SX0<1. By comparison, we find that our result is more reasonable.


## 4. Numerical Examples

In this section, we give two numerical examples to illustrate the applications of the results obtained in previous sections.Example 1.
(TVaR principle). Assume that the loss variableX is exponentially distributed with the survival function SXx=e−0.001x. In this section, we assume θ=0.2 and α=0.2; then a=223.1. Using the results in Theorems 1, 3, and 3, we have the following cases.Case 1.
αc=0.95 and αr=0.99. In this case, a=223.1, ac=2995.7, ar=4605.2, TVaRαcX=3995.7, and TVaRαrX=5605.2. The optimal ceded loss function R∗x is shown in Table 1, and the various key values of R∗x are shown in Table 2.
From Table1, we know that the optimal reinsurance policy depends on the combining coefficient β. From Table 2, obviously, with the increase in the weight coefficient β, the loss of the insurer TVaRαcMR∗ is decreasing while the loss of the reinsurer TVaRαrNR∗ and the mean premium EπR∗ are increasing, especially more intuitive when β∈0.5,0.8419. Note that we ignore the key values at the endpoints 0.5 and 0.8419 because the Pareto-optimal reinsurance policy at endpoints 0.5 and 0.8419 is uncertain.Table 1
R∗x with αc<αr under exponential distribution.
β∈0,0.5R∗x=x∧405.5β=0.5R∗xis unspecified,223.1≤x≤2995.7R∗x=u1,x>2995.7,∀u1∈223.1,2995.7β∈0.5,0.8419R∗x=x−405.5+∧VaRsβX−405.5,∀VaRsβX∈2995.7,4605.2β=0.8419R∗x=x−405.5+,223.1≤x≤4605.2, R∗xis unspecified,x>4605.2β∈0.8419,1R∗x=x−405.5+Table 2
Various key values ofR∗x with αc<αr under exponential distribution.
TVaRαcMR∗TVaRαrNR∗EπR∗β∈0,0.54058−62.3467.8β∈0.5,0.84192330.5↓1590.51665.2↑33214.7925↑985β∈0.8419,11405.54199.71000Case 2.
αc=0.99 and αr=0.95.
In this case,a=223.1, ac=4605.2, ar=2995.7, TVaRαcX=5605.2, and TVaRαrX=3995.7. The optimal ceded loss function R∗x is shown in Table 3, and the various key values of R∗x are shown in Table 4.Table 3
R∗x with αc>αr under exponential distribution.
β∈0,0.1581R∗x=x∧405.5β=0.1581R∗x=x∧405.5,223.1≤x≤4605.2, R∗xis unspecified,x>4605.2β∈0.1581,0.5R∗x=x∧405.5+x−VaRtβX+,∀VaRtβX∈2995.7,4605.2β=0.5R∗xis unspecified,223.1≤x≤4605.2R∗x=x−2995.7+u2,x>4605.2,∀u2∈223.1,2995.7β∈0.5,1R∗x=x−405.5+Table 4
Various key values ofR∗x with αc>αr under exponential distribution.
TVaRαcMR∗TVaRαrNR∗EπR∗β∈0,0.15815667.5−62.3467.8β∈0.1581,0.54682.5↓3133122.7↑862.7482.8↑542.8β∈0.5,11405.52590.21000Case 3.
αc=αr=0.95.
In this case,a=223.1, ac=ar=2995.7, and TVaRαcX=TVaRαrX=3995.7. The optimal ceded loss function R∗x is shown in Table 5, and the various key values of R∗x are shown in Table 6.Table 5
R∗x with αc=αr under exponential distribution.
β∈0,0.5R∗x=x∧405.5β=0.5R∗x is unspecifiedβ∈0.5,1R∗x=x−405.5+Table 6
Various key values ofR∗x with αc=αr under exponential distribution.
TVaRαcMR∗TVaRαrNR∗EπR∗β∈0,0.54058−62.3467.8β=0.54058↓1405.5−62.3↑2590.2467.8↑1000β∈0.5,11405.52590.21000Remark 2.
Under the expected value premium, assume that the loss variableX is exponentially distributed with the survival function SXx=e−0.001x and θ=0.2. Using the results in Theorems 4, 5, and 6 we get the same results as in [19].Example 2.
(expected value premium principle). Assumeθ=0.2 and the loss variable X with the survival function:(33)SXx=1,x<0,0.25,x=0,0.75e−0.001x,x>0.
Using the results in Theorems4, 5, and 6, we have the following cases.Case 4.
αc=0.95 and αr=0.99. In this case, ac=2708.1, ar=4317.5, TVaRαcX=3708.1, and TVaRαrX=5317.5. The optimal ceded loss function R∗x is shown in Table 7, and the various key values of R∗x are shown in Table 8.Table 7
R∗x with αc<αr.
β∈0,0.5R∗x=0β=0.5R∗xis unspecified,x≤2708.1R∗x=u3,x>2708.1,∀u3∈0,2708.1β∈0.5,0.84R∗x=minx,VaRs0βX,∀VaRs0βX∈2708.1,4317.5β=0.84R∗x=x,x≤4317.5, R∗xis unspecified,x>4317.5β∈0.84,1R∗x=xTable 8
Various key values ofR∗x with αc<αr.
TVaRαcMR∗TVaRαrNR∗EπR∗β∈0,0.53708.100β∈0.5,0.841840↓10881868.1↑3429.5840↑888β∈0.84,19004417.5900Case 5.
αc=0.99 and αr=0.95.
In this case,ac=4317.5, ar=2708.1, TVaRαcX=5317.5, and TVaRαrX=3708.1. The optimal ceded loss function R∗x is shown in Table 9, and the various key values of R∗x are shown in Table 10.Table 9
R∗x with αc>αr.
β∈0,0.1599R∗x=0β=0.1599R∗x=0,x≤4317.5R∗xis unspecified,x>4317.5β∈0.1599,0.5R∗x=x−VaRt0βX+,∀VaRt0βX∈2708.1,4317.5β=0.5R∗xis unspecified,x≤2708.1R∗x=x−2708.1+u4,x>2708.1,∀u4∈0,2708.1β∈0.5,1R∗x=xTable 10
Various key values ofR∗x with αc>αr.
TVaRαcMR∗TVaRαrNR∗EπR∗β∈0,0.15995317.500β∈0.1599,0.54329.5↓2768.1188↑94012↑60β∈0.5,19002808.1900Case 6.
αc=αr=0.95.
In this case,ac=ar=2708.1 and TVaRαcX=TVaRαrX=3708.1. The optimal ceded loss function R∗x is shown in Table 11, and the various key values of R∗x are shown in Table 12.
It is worth mentioning that the distribution in Example 2 is not applicable in [19], and it violates the meaning of the ceded loss function. In addition, note that the parameter β and the confidence levels of TVaRs have significant influences on the Pareto-optimal contracts. If β is small, the weight of the reinsurer is larger than the insurer, and then the reinsurer bears less losses. Conversely, if β is large, the weight of the insurer is larger than the reinsurer, and then the reinsurer bears more losses. If αc<αr, which means that the TVaR standard of the reinsurer is higher than the insurer, then the reinsurer bears less losses. If αc>αr, which means that the TVaR standard of the insurer is higher than the reinsurer, then the reinsurer bears more losses.Table 11
R∗x with αc=αr.
β∈0,0.5R∗x=0β=0.5R∗x is unspecifiedβ∈0.5,1R∗x=xTable 12
Various key values ofR∗x with αc=αr.
TVaRαcMR∗TVaRαrNR∗EπR∗β∈0,0.53708.100β=0.53708.1↓9000↑2808.10↑900β∈0.5,19002808.1900


## 5. Conclusion

In this paper, based on the TVaR measure, we show that the Pareto-optimal reinsurance policies must exist for the insurer and the reinsurer under a class of premium principle, such as the net principle, expected value premium principle, TVaR principle, and generalized percentile. Using a two-stage optimisation procedure, we derive explicitly the Pareto-optimal reinsurance policies under the TVaR principle. Since the expected value premium principle can be viewed as a special case of the TVaR principle, then lettingα=0 in the TVaR principle gives Pareto-optimal reinsurance policies for the expected value premium principle. We extend the results in [19]. Compared with the method used in [19], using the two-stage optimisation method to derive the Pareto-optimal strategy is simpler and more intuitive. Furthermore, by comparing the results in [19] with ours, Cai et al. [19] derived the optimal ceded loss functions without considering the relationship between SX0 and 1, while we discuss the relationship between SX0 and 1 and derive different optimal ceded functions from theirs in the case SX0<1.We also wish to point out that further research on this topic is needed. First, the risk measure TVaR can be generalized to coherent risk measures. Although some papers have been devoted to deriving optimal reinsurance under coherent risk measures, the optimal reinsurance study still lacks of available analyze tools. Since nonlinear expectation is an essential feature of coherent risk measures, maybe we can draw support from nonlinear expectation; research literatures in this regard are [27–31], etc. Second, we can analyze risk with the strategies of dividend and reinsurance. For more references on the dividend, refer to [32–34], etc. Third, in most of the optimal reinsurance problems, it is assumed that the distributions of the insurer’s risks are known. However, in practice, only incomplete information on the distributions is available. How to obtain optimal reinsurance contracts with incomplete information is also an interesting topic. An attempt to such a problem is to use the statistical methods. For more references on statistical methods, see, e.g., [35–37]. We hope that these important open problems can be addressed in the future research. We also believe that this article will foster further research in this direction.



---

*Source: 3061298-2020-09-29.xml*