Bowley reinsurance with asymmetric information under rein-surer’s default risk

The method, theorem and simulation study for Bowley reinsurance under default risk.


Introduction
Reinsurance is an effective risk management tool in today's complex business environment and has been an important research field. Since the seminal paper of Borch [1] , optimal (re)insurance design problems have been widely studied in the literature. Borch [1] showed that stop-loss reinsurance is optimal by minimizing the variance of an insurer's total risk exposure under the expected value premium principle. Borch's result has constantly been extended in many directions. One direction is to consider the expected utility maximization [2−4] . Another direction is to consider the same optimal problem under the risk management framework. For example, Cai et al. [5] and Cheung [6] minimized the value-at-risk (VaR) and tail value-at-risk (TVaR) of the insurer's total risk exposure. Cui et al. [7] and Cheung et al. [8] considered the problem under general convex risk measures including VaR, TVaR and distortion risk measures as special cases.
As noted by Borch [1] , an insurer and a reinsurer may have conflicting interests under a reinsurance contract. If they cooperate, in the same work, Borch gained the optimal retention of the quota-share and stop loss reinsurance contracts to maximize the product of individual expected utility at the end. Following this way, this research line leads to Paretooptimal [9−11] . Later, Borch [12] considered that a reinsurance contract could be attractive to one party, but may not be acceptable to another. One of the best ways to solve this non-cooperative conflict between reinsurer and insurer is the theory of Nash equilibrium [13,14] . Under different assumptions, Aase [13] and Boonen et al. [14] adopted the Nash bargaining framework to price reinsurance contracts. Except for Nash equilibrium, some other important equilibrium concepts such as the Bowley solution, the Stackelberg game and the principalagent problem are also used in optimal reinsurance theory. For instance, in a Stackelberg game, one player (the 'leader') chooses first, and all other players (the 'followers') move after the leader. In other words, there is an order of the game; once the buyer chooses her indemnity, then the seller cannot modify its premium [15−17] . And under the principal-agent problem, the principal (monopoly) has the right to determine the optimal insurance contracts and the corresponding premiums charged to each type of agents. Meanwhile, the principal can only rely on the prior knowledge of the proportion of each type, not the hidden characteristics of any single agent [18,19] .
Recently, there has been increasing interest in studying Bowley reinsurance since the paper of Cheung et al. [16] . Cheung et al. focused on preferences given by distortion risk measures. As far as we know, Chan and Gerber [20] first used the nature of the reinsurer's monopolistic and built a Bowley solution (Stackelberg equilibria) of equilibrium reinsurance arrangements to maximize the expected utility of both the insurer and reinsurer in order, and then back. Motivated by Boonen et al. [21] and Boonen and Zhang [22] , we consider Bowley reinsurance solutions with asymmetric information under the reinsurer's default risk with a general pricing principle. Asymmetric information refers to that the insurer is given a distortion risk measure but the reinsurer does not have any idea about the preferences of the insurer. The reinsurer sets the premium principle first to the insurer, and the insurer decides his own optimal reinsurance indemnity based on the premium principle. For the proceedings of Bowley solutions, we refer to Boonen and Zhang [22] .
However, in the above mentioned papers involving Bowley solutions, it is assumed that the reinsurer will be able to compensate for the losses they commit. In reality, the reinsurer could fail to pay the promised part of loss if the loss is huge, which is denoted as default risk. There is an extensive literature on default risk [24−27] . Therefore, the aim of the paper is to fill the gap between default risk and the Bowley solutions. We follow the framework of Asimit et al. [24] and Lo [27] . Note that there is a difference between this paper and Asimit et al. [24] in calculating the premium principle. In this paper, the premium principle bases on the promised part of loss whereas Asimit et al. [24] considered the default risk. The reason for this is that even if there exists default in a (re)insurance contract, the insurer does not know in advance. Our results indicate that the optimal reinsurance indemnity depends on the default rate, i.e., with the increase in the default rate, the insurer cedes less risk to the reinsurer and retains more risk.
Finally, examples are also given to illustrate the main results, where the explicit expressions for the optimal reinsurance treaties are provided. This paper is organized as follows. Section 2 states the asymmetric information and default risk problem studied in this paper. Section 3 solves this problem. In section 4 we give two examples to illustrate the main result when the two distortion functions of the insurer are ordered. Section 5 concludes the paper.

Preliminaries
(Ω, F , P) Throughout, let be a probability space. We assume that the total loss faced by the insurer is denoted by a bounded, non-negative random loss variable , and its cumulative distribution function and survival function are denoted by and , respectively. We assume that both the insurer and the reinsurer are known about and . The quantile of at level is denoted by . Denote as the indicator function such that for and for . We also denote the following two sets G g is a non − decreasing and left − continuous}, and G = {g : [0, 1] → R + |g is continuous and concave, g(0) = 0}.

Indemnities, distortion risk measures and premium principles
To manage risk explosion, the insurer cedes the risk to the reinsurer, where is called a ceding function. Avoiding moral hazard or insurance swindles [28] , we impose an incentive compatibility constraint on , i.e., we assume that with F ={ f : R Under the incentive compatibility constraint, it is easy to see that is almost everywhere differentiable on . Moreover, there exists a Lebesgue integrable function such that h f h where is the slope of the ceded loss function . Zhuang et al. [29] termed as marginal indemnification function. Next, we discuss the preferences of the insurer. Following most of the literature, we assume that the insurer wants to minimize its distortion risk measure. The formal definition is given below, For or , a functional of a nonnegative random variable is given by ρ g ∈ G d ρ g whenever the integral exists. When , we call as distortion risk measure.
Note that, the class of distortion risk measure is a big umbrella under which many common risk measures fall, for example the most notably VaR, whose definition is recalled below.
The value-at-risk (VaR) of a random variable at the probability level of is the left-continuous inverse of its distribution function at : The VaR is a distortion risk measure with distortion function .
When the insurer cedes loss to a reinsurer, the reinsurer charges premium in turn. In this paper, we assume that the premium principle is comonotonically additive and law invariant, but not necessarily monotone. In particular, for any ceded loss function and , we define the reinsurance premium principle where satisfies (1) and the second equality follows from Lemma 1 below. Note that we do not require to be monotone because need not be monotone. The above distortion-deviation premium principle was recently introduced by Liang et al. [23] who derived the form by analogy with the mean-standard deviation premium principle. The distortiondeviation premium principle encompasses a large class of premium principles typically used in the literature, such as the expected premium principle, Wang's premium principle [30] , Gini premium principle, the absolute deviation premium principle [31] and so on. Liang et al. [23] termed (3) as canonical representation. They showed that there exist two functions such that where is non-decreasing and is a deviation distortion, i.e., and . Finally, we end this subsection with a lemma, which is a useful tool for our following analysis.
Let be a non-decreasing, absolutely continuous function with and be a non-negative random variable. Then, for or , we have ρ

Bowley reinsurance solutions with asymmetric information under default risk
The main goal in this paper is to incorporate default risk under the setting of Bowley reinsurance with asymmetric information. We follow the framework of Asimit et al. [24] , who was motivated by the recent implementation of the Solvency II regulatory framework in the countries of the European Union. Assume that the reinsurer operates in a VaR-regulated environment and thus prescribes its regulatory capital in accordance with the -level VaR of the reinsured loss for some large probability level (e.g., in Solvency II). If represents the reinsurance indemnity function and is the recovery rate of the loss given default, then the insurer is in effect compensated for . Obviously, the no-default scenario is recovered if we set . It is easy to see that the greater is, the less likely the default is. For more on default risk, we refer to Cai et al. [26] and Lo [27] .
Under the setting of Bowley reinsurance, we assume that the reinsurer does not have any idea about the distortion risk measures used by the insurer, while the reinsurer only knows that the insurer may have finitely many possible distortion risk measures. For brevity of our result, we consider the case in which there are only two possible distortion risk measures of the insurer. To be precise, the reinsurer holds the opinion that the insurer minimizes either or with probability and , respectively, where and are the two possible distortion functions adopted by the insurer.
Recall that there exists default risk on the reinsurer. Denote the reinsurance indemnity function . Then the total retained loss for the insurer is equal to , rather than , where is the distortion-deviation premium principle given by (3). Note, we assume that even if default risk exists, the premium principle calculated based on rather than . This is different from Asimit et al. [24] . Because we believe that even if there is a default risk in the future, the reinsurer charges the most desirable premium at first. In conclusion, the two-step game played by the insurer and the reinsurer is formalized as follows: • (Decision problem faced by the insurer) For any given provided by the reinsurer, the insurer chooses the optimal ceded loss function by solving where or , depending on the type of the insurer. The reinsurer is uncertain about the type of the insurer, but knows the distortion functions and and probability . Thus for the reinsurer, the goal is to select the optimal reinsurance premium generating function by maximizing the expected net profit. Then, the optimization problem of interest is where denotes the aggregate administrative cost paid by the reinsurer if the insurer purchases the policy , for .
In the absence of default, i.e., , Boonen and Zhang [22] also considered the above two-step game. When the insurer chooses the indemnity function that is optimal for him/her, the reinsurer will know the type of the insurer that is revealed by indemnity selection. Thus, the problem of this paper is summarized as follows: where is the expected net profit of the reinsurer in (5)  For (4), similar problem has been solved in the literature [16,24,27,29] . We state it in the following proposition and provide a self-contained proof.
g r ∈ G f gr ,gi j Proposition 1. For any , the optimal ceded loss function that solves problem (4) is given by where the function is defined as and is any measurable function with , for . Proof. Due to translation invariance, we can rewrite (4) as ).

X
Being non-decreasing, 1-Lipschitz functions of the groundup loss , the three random variables ρ gi j are all comonotonic. By virtue of the comonotonic additivity and positive homogeneity of , we further have Apply Lemma 1 to the two absolutely continuous functions (noting that is merely a constant) whose derivatives are almost everywhere equal to respectively. Then, To minimize the objective function in the form of the above equation over all , we use the method that minimizes the integrand. To this end, we can easily see that Thus, the solution is given by (6), and we complete the proof.
Remark 1. We consider only two parties, namely an insurer and a default-prone reinsurer. It is interesting to consider the insurance-reinsurance model in which three agents, namely a policyholder, an insurer and a default-prone reinsurer, coexist [27] . This will be our future work to consider the insurance-reinsurance model under Bowley question.
In Boonen et al. [21] and Boonen and Zhang [22] , they considered Bowley reinsurance without reinsure's default risk. The former paper considered the administrative cost of offering the compensation is proportional to the expectation of the ceded loss, i.e., for any , where is a fixed constant representing the cost coefficient, and the latter paper considered the administrative cost of offering the compensation to be proportional to a distortion risk measure. To make our model generality, we follow the assumption of the latter paper, i.e., let for any , where is a fixed constant and . Then, for , we have . Thus, we rewrite the objective in (5) as Under such indifference circumstances, it is common in the literature to achieve definiteness assuming that , which means that the insurer is 'willing to' act in favor of the reinsurer [16,21,22,32] . In this way, we shall set in the sequel. Under this setup, problem (5), by Lemma 1, boils down to solving where is the Radon measure on such that for .

Main result
In this section, we provide our main result for problem (7). We assume . It will be helpful to define, for , These sets allow us to state Theorem 1, which provides the Bowley solutions under asymmetric information and reinsurer's default risk. Note that Theorem 1 is provided with respect to since there exists a one-to-one correspondence between and . In Remark 2, we express the solution with respect to . G * r ∈ G Theorem 1. The solution set to problem (7) contains those such that

Moreover, we also define
if t ∈ C, and Moreover, for any of these , we have, for , The proof is similar to that of Theorem 3.1 in Boonen and Zhang [22] .
We use the technique of path-wise optimization. Equation (7) is written as Now, we solve the maximization problem path wise, and therefore, we first fix . Next, we construct solutions of . We consider three different cases.
For all , it holds that is strictly increasing on and on , and on . Thus, the maximum value of is either located at the possible discontinuities, and , or it is zero. Hence, For all , it holds that is strictly increasing on , and on . Thus, the maximum value of is either located at, , or it is zero. Hence, If , then is solved by . If , then , and it is thus solved by any .
For all , it holds that strictly increases on and on , and on . Thus, the maximum value of is either located at the possible discontinuities, and , or it is zero. Hence, If and , then is solved by . Likewise, if and , then is solved by . If and , then is solved by either or .
Finally, if , then , and it is thus solved by any .
Now we constructed the solutions of for all . Let such that it solves for all . Note that is a solution to . Thus, Thus, the inequalities can be replaced by equalities, and we conclude the proof of the premium generating functions Bowley solutions under the reinsurer's default risk.
For a fixed premium generating function , the optimal indemnity functions and follow from Proposition 1. This concludes the proof.

Remark 2.
From Theorem 1 and , we obtain , i.e., if t ∈ C, and Zou et al.
When setting , we recover Theorem 3.1 [22] under the same premium principle.
While the function is merely used as an ancillary function to construct the Bowley solutions under default risk, it has an interpretation as follows [22] . At a given value is the marginal profit that the reinsurer makes by choosing instead of . Therefore, if is positive (negative), then it is profitable for the reinsurer to select the premium generating function that makes the type 2 (1) insurer indifferent between buying or not buying marginal reinsurance. While, for the marginal profit, reinsurance prices often make one type of insurer 'indifferent', this does not imply that the insurer will be indifferent between insuring or not insuring. In fact, since it may hold that for some , the insurer can strictly profit from buying reinsurance.

Two examples with ordered distortion functions of the insurer
In this section, two examples are provided for illustrating Theorem 1 in Section 3. These two examples have order distortion functions i.e., for all . Then . For ease of implementing the calculation, we first set so that and . Under these circumstances, the function simplifies as and are defined in Section 3.
From Theorem 1, we obtain that, for any optimal and , When , i.e., there is no reinsurer's default risk, then we recover the results obtained by Boonen and Zhang [22] . Additionally, we remark that there exists an error in Section 4 of Boonen and Zhang [22] for and . The correct forms are given in Eq. (8) and (9). p For the first example below, as we will observe, the value of probability plays a key role in determining the optimal premium generating function and the corresponding ceded loss functions.
Example 1. Suppose that the risk has an exponential distribution with mean 1, the recovery rate and the reinsurer sets its regulatory capital at the level of . Then . Assume that the distortion functions of the insurer are given by and , for , where and . Clearly, , for all , which implies . For , the solutions of the equations and on t ∈ (0, 1) t 1 = 0.8877 p are and , respectively. According to the definitions of sets and , we first need to determine the signs of the function for , and then obtain the explicit expressions of these three sets. Consider the following three values of the probability : . In this case, we calculate that Figure 1(a) plots the function for . From figure 1(a) x ∈ R + Furthermore, the optimal ceded loss functions for are given by x + := max{x, 0} where , and This means that the type of our optimal ceded loss function is a traditional stop-loss policy. Moreover the expected net profit can be calculated as .
. In this case, we have Figure 1( are given by Hence, the stop-loss contract is signed between the reinsurer and the type 1 insurer, while a two layer stop-loss policy is provided for the type 2 insurer. Moreover the net gain can be calculated as . . In this case, we have Figure 1(c) plots the function for . From the figure, we know that and . Premium generating functions in Bowley solutions with default risk are then given by Furthermore, the optimal ceded loss functions for are given by Hence, the stop-loss contract is signed between the reinsurer and the type 1 insurer. When the probability of type 1 insurer is sufficiently high, for example , the type 2 insurer will not cede any function. Furthermore, we do not know exactly the critical point to the insurers that cede no loss. This will be our future study. Moreover the profit acquired by the reinsurer can be computed as .
δ δ δ The following example illustrates how the default rate affects the optimal contract form. We will see that has the greatest impact on the type 1 insurer, and the optimal contract forms depend on the value of . X β ∈ (0, 1) δ ∈ (0, 1) Example 2. Suppose that the risk is a non-negative continuous random variable possibly with a jump point at 0. The reinsurer sets its regulatory capital at and the recovery rate . The distortion functions of the insurer are given by and for , respectively. We assume that . Clearly, under this setting, we have , that is, and . Let with . If , then , and We have for and for . Thus the following relationship between and holds: .
The right hand side can be written as δ f * To analysis how influences , we consider the following cases: (i) Suppose that , . If , let