Distributionally Robust Reinsurance with Glue Value-at-Risk and Expected Value Premium

: In this paper, we explore a distributionally robust reinsurance problem that incorporates the concepts of Glue Value-at-Risk and the expected value premium principle. The problem focuses on stop-loss reinsurance contracts with known mean and variance of the loss. The optimization problem can be formulated as a minimax problem, where the inner problem involves maximizing over all distributions with the same mean and variance. It is demonstrated that the inner problem can be represented as maximizing either over three-point distributions under some mild condition or over four-point distributions otherwise. Additionally, analytical solutions are provided for determining the optimal deductible and optimal values.


Introduction
Reinsurance has played a crucial role in facilitating the transfer of risks from insurers to reinsurers.The optimization of reinsurance strategies has been the subject of extensive research, with significant contributions made by [1,2].Since then, numerous extensions and advancements have been proposed in this field.
Some researchers have devoted their attention to identifying the most favorable reinsurance contract through the examination of multiple criteria.These criteria encompass minimizing regulatory capital and mitigating the insurer's risk exposure with respect to specific risk measures.The first proposal of the optimal reinsurance problem for VaR and CVaR risk measures was put forward by [3].They analyzed the class of stop-loss policies based on the expected value premium principle.Ref. [4] expanded the scope of optimal reinsurance selection to a broader set.Additionally, Refs.[5][6][7] have investigated various approaches and models to determine the optimal reinsurance contract in accordance with the selected risk measure.For a comprehensive overview of optimal reinsurance design with a focus on risk measures, we highly recommend consulting the survey paper [8].
Given the practical difficulties in obtaining the precise distribution of losses, a widely adopted approach to tackle model uncertainty is the utilization of a moment-based uncertainty set.This approach entails considering distributions that adhere to particular moment constraints.An example of this is seen in [9], where optimal reinsurance incorporating stop-loss contracts was examined while accounting for incomplete information regarding the loss distribution.In a similar vein, Ref. [10] examined model uncertainty in the insurance context by maximizing over a finite set of probability measures.In the research conducted in [11], the investigation of VaR-based and CVaR-based reinsurance took place, while taking into account model uncertainty related to stop-loss reinsurance contracts.The authors successfully provided an analytical solution to the reinsurance problem.Moreover, in the study conducted in [12], a distributionally robust reinsurance problem incorporating expectiles was examined, and the optimization problem was numerically solved.
In this paper, we investigate a reinsurance problem incorporating the risk measure Glue Value-at-Risk (GlueVaR) under the framework of distributional robustness.GlueVaR, initially introduced in [13] as a function with four parameters, enables the customization of risk measures to specific contexts by adjusting these parameters.The practical aspects of tailsubadditivity were analyzed by the authors of [14], who also examined the subadditivity and tail-subadditivity of risk aggregation.In a recent study, Ref. [15] examined the optimal reinsurance problem using a novel combination of risk measures and derived closedform solutions for optimal reinsurance policies.They also explored the application of VaR and GlueVaR as specific instances of risk management.However, it is important to note that their analysis assumed knowledge of the risk distribution.In our research, we possess partial knowledge about the distribution of losses, specifically, regarding the mean and variance.The primary contribution of this paper is to establish that the worst-case distribution falls within the category of three-point or four-point distributions, thereby transforming the infinite-dimensional inner problem into a finite-dimensional one.As a result, we obtain closed-form solutions for the optimal deductible and optimal value.Our findings build upon the conclusions presented in [11], thus extending their implications.
The remaining sections of this paper are structured as follows.Section 2 presents the definition and properties of GlueVaR and formulates our distributionally robust reinsurance problem as a minmax problem.In Section 3, we address the distributionally robust reinsurance problem by identifying the optimal deductibles and optimal values.Section 4 provides an in-depth discussion on the optimal solution, and concludes the paper with closing remarks.Some detailed proofs can be found in the Appendixes A and B.

Risk Measures
A risk measure is a mathematical function that assigns a non-negative real number to a specific level of risk.Within the financial and insurance sectors, two commonly utilized risk measures are Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).For a deeper understanding of CVaR, please consult the discussion provided in [16].
For a random variable X, VaR is defined in two versions: left-continuous VaR (VaR − ) and right-continuous VaR (VaR).They are defined as Both of these measures serve as a scientific foundation for determining the initial capital required by a company to withstand risks.However, it should be noted that VaR − and VaR do not satisfy subadditivity, and they cannot accurately calculate extreme risks.
RVaR includes right-continuous VaR and CVaR as its limiting cases, and the two versions of VaR do not impact the values of RVaR and CVaR.
Moving on to another perspective, Ref. [18] defines a family of risk measures using the concept of a distortion function.A distortion function is an non-decreasing function g : [0, 1] → [0, 1] such that g(0) = 0 and g(1) = 1.The corresponding distortion risk measure, denoted as ρ g [•], is defined as follows: where F(t) is the survival function of X.For a more comprehensive understanding of distortion risk measures, see [19,20].

Distributionally Robust Reinsurance with GlueVaR
Suppose we have a non-negative ground-up loss, denoted as X, defined on an atomless probability space (Ω, F , P).The insurer has the option to transfer a portion of the loss, denoted as I(X), to a reinsurer in exchange for a reinsurance premium, denoted as π(I(X)).A stop-loss reinsurance contract is defined as I(X) = (X − d) + , where d ∈ [0, ∞] is referred to as the deductible.When d = 0, the identity function I(X) is equal to X, which signifies that stop-loss reinsurance defaults to full reinsurance.Conversely, if d = +∞, I(X) equals 0, indicating that no reinsurance is purchased.In a stop-loss reinsurance arrangement, the total retained loss for the insurer can be expressed as Here, θ denotes a loading factor.
Given a pair of non-negative mean and standard deviation, denoted as (µ, σ), we define the uncertainty set F (µ, σ) as follows: The distributionally robust reinsurance problem based on the risk measure ρ(•) is defined as min where X F is the loss with distribution function F. An optimal solution to (2) is called an optimal deductible, denoted by d * .A distribution that solves is referred to as a worst-case distribution, and ρ ↑ (d) is called the worst-case value corresponding to a given d.In other words, we have ρ ↑ (d * ) = min d 0 ρ ↑ (d).
The distributionally robust reinsurance with ρ = VaR and ρ = CVaR has been studied by [11], and when ρ is an expectile-based risk measure, it was studied by [12].Since Glue-VaR can incorporate more information about agents' attitudes towards risk, we consider the distributionally robust reinsurance problem with GlueVaR.By utilizing the translation invariance of GlueVaR, model (2) can be reduced to min where GlueVaR c := GlueVaR r 1 ,r 2 α,β with 0 α β 1 and 0 r 1 r 2 1.
From the expression in Equation ( 1), α and β represent the levels of riskiness in GlueVaR.The larger the values of α and β, the more risk the insurer will transfer to the reinsurer.Conversely, if α and β are small enough, the insurer will assume all the risk, aligning with the following proposition.Proposition 1.For d 0, if the following conditions hold then the optimal deductible for problem is decreasing in d 0, we can use Equations ( 1) and ( 3) to obtain the following expression: Next, let us consider the following three cases: Taking the derivative with respect to d, we have From this, we can observe that h c (X F , d) 0. This implies that h c (X F , d) is decreasing for d F −1 θ 1+θ , and Thus, the derivative with respect to d is We will demonstrate that h c (X F , d) is decreasing in d by considering the following three subcases.Subcase 1: By Condition (4), we have (1 In this subcase, we have From this, we can observe that h c (X Combining the above three cases, we conclude that h c (X F , d) is decreasing for d 0.

Main Results
As indicated by Proposition 1, the optimal deductible under certain moderate conditions is d * = ∞ for the case where (1 1, implying a scenario with no reinsurance purchase.However, practical instances often present α values near 1 and relatively minor θ values.This makes the condition (1 − α)(1 + θ) > 1 rarely satisfied.Consequently, our primary focus in this section is on cases where (1 Moving forward, we employ a methodological approach in examining the issue.In Section 3.1, we transform problem (3) into a finite-dimensional and more computationally amenable problem.Subsequently, in Section 3.2, we deduce the worst-case scenario for problem (5).Lastly, within Section 3.2, we present the optimal deductible for the optimization problem represented by (3).

The Problem Formulation
We first consider the inner problem of (3), which is defined as follows: We demonstrate that under mild conditions, if (1 − α)r 1 (1 − β)r 2 , the worst-case distribution of the problem (5) can be restricted to the class of three-point distributions F 3 (µ, σ).Conversely, when (1 − α)r 1 (1 − β)r 2 , the worst-case distribution of problem ( 5) can be confined to the class of four-point distributions F 4 (µ, σ).
We first present the following two lemmas, which will be utilized in the proof of the subsequent propositions.
Similar to the proof of Lemma 1, we can obtain the following Lemma.
The next proposition states that if (1 − α)r 1 (1 − β)r 2 , then the worst-case distribution of optimization problem (5) can be limited to the set of three-point distributions F 3 (µ, σ).
Proof.It suffices to show that for each F ∈ F (µ, σ), there exists a distribution To construct F, we consider the following three cases.
We state that E obviously holds.Using Hölder's inequality, we obtain that Hence, Var( X) Var(X) = σ 2 .By the definition of X as in (7), we obtain and Consequently, J 1 0. In a similar manner, we can confirm that J 2 0 and The second equality holds because . Define a discrete random variable X(ω) as in Equation (7).Similar to the proof of (i), it is easy to show E[ X] = µ, Var( X) σ 2 , and where the inequality follows from If Var( X) < σ 2 , by Lemma 1, there exists a three-point distribution Similar to case (i), we have E( where p ) is continuous and increasing in ε 0. Next, we consider the following two subcases.(iii.a)If there exists an ε ∈ [0, x 1 /(1 − p)] such that Var(X ε ) = σ 2 , then the distribution of X ε is the desired two-point distribution.(iii.b)Otherwise,we have Var(X ε 0 ) < σ 2 with ε 0 := x 1 /(1 − p).In this case, for q ∈ [0, 1 − p], we define X q = (0, p; d, q; y, with y := (µ − dq)/(1 − p − q) > d.Then X 0 and X ε 0 have the same distribution and EX q = µ, h c (d, X q ) h c (d, X ε 0 ).Noting that the variance Var(X q ) is continuous in q ∈ [0, 1 − p] and lim q→1−p Var(X q ) = ∞, there exists an q * ∈ [0, 1 − p] such that Var(X q * ) = σ 2 .Then the distribution of X q * is the desired distribution F * .
Corollary 2. For d 0, we have From the proof of Proposition 2, we can obtain the explicit form of the worst-case distributions.Specifically, we have the following result as a corollary of Proposition 2.
, the worst-case distribution F = (x 1 , p 1 ; x 2 , p 2 ; x 3 , p 3 ) ∈ F 3 (µ, σ) of problem (5) must be in the following two sets: Proof.In the proof of Proposition 2 case (iii.b), for the X q * founded in the proof, we consider the following three cases.
(i) If q * > β − p, then VaR β (X q * ) = d.By applying case (i) of the proof of Proposition 2, we obtain a three-point distribution in In this case, by applying case (ii) of the proof of Proposition 2, we obtain a three-point distribution in In this case, we can apply case (iii) of the proof of Proposition 2. Specifically, we obtain a random variable X q 1 defined by ( 9) with q 1 + p 1 > q * + p.By iterating this process, we can obtain a distribution in case (i), (ii), or case (iii.a) of the proof of Proposition 2, such that the objective function h c (X, d) is not less than that of the original one.
The proposition below asserts that if (1 − α)r 1 (1 − β)r 2 , then the worst-case distribution of optimization problem (5) can be limited to the set of four-point distributions F 4 (µ, σ).Proposition 3.For d 0, we assume that (1 − α)r 1 (1 − β)r 2 .The optimization problem (5) can be equivalently represented as follows: Proof.To prove this, it is sufficient to demonstrate that for every F ∈ F (µ, σ), there exists a distribution F ∈ F 4 (µ, σ) such that h c (d, X F ) h c (d, X F ).We can derive the following expression: To construct F, we consider the following three cases.
(i) If VaR β (X) d, let U ∼ U[0, 1] be a uniform random variable that is comonotonic with X. Define the events and We define a discrete random variable as follows: It can be stated that E[ X] = µ, Var( X) σ 2 , and h c ( X F, d) h c (X F , d).By applying the Hölder inequality, we obtain that Thus, Var( X) Var(X) = σ 2 .From the definition of X, it follows that it follows that h c ( X F, d) h c (X F , d).
If Var( X) = σ 2 , then F ∈ F 4 (µ, σ), and h c ( X F, d) h c (X F , d).If Var F ( X) < σ 2 , by Lemma 2, there exists a four-point distribution . Define a discrete random variable X(ω) as in (7), and we can determine that h c ( X F, d) h c (X F , d).If Var( X) = σ 2 , then F ∈ S 3 (µ, σ), and h c ( X F, d) h c (X F , d).If Var( X F ) < σ 2 , by Corollary 2, there exists a three-point distribution , the proof is similar to the proof of (iii) in Proposition 2, so we omit it here.
Corollary 3 implies that when (1 , and we can express the supremum as follows: Next, we demonstrate that sup F∈F ∧ 3 (µ,σ) h c (X F , d) is increasing with respect to σ.This implies that the uncertainty set F ∧ 3 (µ, σ) can be substituted with the following set: Specifically, we can state the following result.
For ξ ∈ [0, 1], let us define the following functions: Next, we define the constants: It can be shown that d 2 d 2 and d 3 d 3 .Additionally, According to Corollary 4, we have VaR ↑ α (d) = CVaR ↑ α (d).Therefore, we rewrite the above equation as Theorem 2. For d 0, under the conditions (1

Optimal Deductible
In this subsection, we provide the explicit solutions for the optimization problems (3) and (5).From Proposition 1, we know that the worst-case value GlueVaR ↑ c (d) is decreasing in d, and the optimal deductible is d * = ∞.Based on this, we can state the following theorem.
then an optimal deductible of the optimization problem (3) is d * = ∞, and the optimal value is given by Proof.By Proposition 1, the optimal deductible of the optimization problem is d * = ∞.Therefore, based on Proposition 2, we obtain the optimal value given by (19).
1, regardless of whether (1 − α)r 1 (1 − β)r 2 or not, the optimal deductibles are the same, and the optimal values are equal.Specifically, we have the following theorem.
and the optimal value is given by according to Corollary 4, the worst-case value of the problem for GlueVaR r 1 ,r 2 α,β is exactly the same as that of CVaR α and VaR α .Therefore, the optimal deductible remains the same in both cases.Referring to Theorem 1 of [11], the optimal deductible is given by (20), and the optimal value is given by (21).

Conclusions
In this paper, we examine a distributionally robust reinsurance problem with the risk measure, GlueVaR (Generalized Value-at-Risk), under model uncertainty.As demonstrated in the main results, the parameters θ and (α, β, r 1 , r 2 ) significantly impact the optimal reinsurance design.If θ is sufficiently large such that (1 r 1 , then the insurer will not transfer any risk to the reinsurer.A smaller value of θ leads to a lower optimal deductible.The parameters α and β represent the levels of GlueVaR used to measure riskiness.Higher values of α and β indicate that the insurer transfers more risk to the reinsurer.This observation aligns with the results stated in Proposition 1 and Theorem 4. The values of r 1 and r 2 determine the weights of CVaR β , CVaR α , and VaR α .When (1 − α)(1 + θ) > 1 and r 1 < (1 + θ)(1 − β), the optimal deductible is infinite (d * = ∞).This means that the insurer will not purchase reinsurance.When (1 − α)(1 + θ) 1, regardless of whether r 1 < (1 + θ)(1 − β) holds or not, the optimal deductible is zero (d * = 0) when θ σ 2 /µ 2 .
By demonstrating that the worst-case distribution must fall within the set of four-point distributions or three-point distributions, we have effectively transformed the infinitedimensional minimax problem into a finite-dimensional optimization problem.This reduction allows us to solve the problem in a tractable manner.We have provided closed-form optimal deductibles and optimal values for the distributionally robust reinsurance problem with GlueVaR.Our result generalizes the result in [11], although our proof differs from that in [11].

d d 2 .
Then, in the interval (d 1 , d 2 ], there exists a unique d such that ∂h ↑ c (d) ∂d | d= d = 0. Furthermore, we have that ∂h ↑ Consequently, h ↑ c (d) initially decreases for d ∈ (0, d] and then increases for d ∈ [ d, ∞).Therefore, the minimum value of h ↑ c (d) is attained at d. Solving the equation ∂h ↑ c (d) ∂d = 0, we can determine the value of d, and