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This paper investigates optimal reinsurance strategies for an insurer which cedes the insured risk to multiple reinsurers. Assume that the insurer and every reinsurer apply the coherent risk measures. Then, we find out the necessary and sufficient conditions for the reinsurance market to achieve Pareto optimum; that is, every ceded-loss function and the retention function are in the form of “multiple layers reinsurance.”

The insurance and reinsurance industry has an irreplaceable role in global economy risk compensation. With global warming and the earth being ever-active, various types of extreme disasters are happening more frequently; therefore, research on loss allocation by reinsurance is of more practical significance. In October 2012, hurricane Sandy landed in the United States and more than 650,000 houses were destroyed, resulting in economic losses of up to $70 billion. Insurance paid 18.75 billion US dollars, of which 60% were paid by primary insurers while the other 40% were by reinsurance ones. In 2001, the 911 terrorist attacks suffered by the United States caused huge losses and insurance indemnity reached 26.8 billion US dollars. Of these, 35% were paid by primary insurance companies and 65% by reinsurance ones (the data comes from Global Reinsurance forum). The examples above show the ceded-loss strategy of reinsurance companies plays a crucial role in reducing losses encountered in catastrophes. The superiority of reinsurance mechanism is mainly reflected in the following two aspects. First, the reinsurance can reduce the exposure risk and thus stabilize underwriting income. Catastrophe risk is characteristic of low occurrence and huge loss. Consequently, a single claim may lead to financial crisis of the entire insurance company or even bankruptcy. Therefore, the reinsurance strategy allows insurance companies to avoid or reduce the impact of catastrophe risks. Undoubtedly, for some newly established insurance companies, the reinsurance can significantly improve their risk tolerance abilities. Second, by undertaking reinsurance, the primary insurance companies not only increase income the primary insurance companies not only increase income.

With the development of economy, the value of the subject matter insured has been increasing. The solvency faced by separate original insurers is facing challenges; thus, seeking multiple insurers in international reinsurance market to share the risk and revenue is becoming popular. The total worldwide premium income from reinsurance has risen from 20% in the 1980s of last century to 40% today, showing that the value of reinsurance has improved significantly. Appropriately designed reinsurance programs make it possible to fully realize the advantages of the reinsurance mechanism. However, an improper reinsurance program probably leads to high risk aggregation and loss of quality services, and it will further affect underwriting profits and even endanger the solvency. Thus, the studying on the optimal reinsurance design has practical significance.

For individuals and the whole society, risks exist objectively. The final risk takers often include policy holders, insurers, reinsurers, financial market investors, and also local and central government. Among them, the policy holders, insurers, reinsurers, and financial market investors are market-based sharers, whereas local and central governments provide non-market-based sharing. Non-market-based sharing may lead to distortions of the market; therefore, market-based sharing is relatively more efficient than it. Currently, the insurance risk sharing through capital market is still in its infant stage. The core of market-based loss sharing lies in sharing by both the insurers and the international reinsurance markets. Therefore, from the perspective of theoretical research, optimal reinsurance design has always been one of the most fascinating problems in the actuarial research fields.

Great attention has been paid from both academics and practitioners to the study of optimal reinsurance problems since the seminal work of Borch [

The proposed problem can be classified as a multiobjective optimization problem. This paper just uses the idea of multiobjective optimization but does not involve the algorithm. In 1896, the earliest idea of multiobjective optimization originates from the French economist Pareto, who proposed it in the study of economic equilibrium. Subsequently, this problem has gradually developed in three different areas, namely, operations research, economics, and psychology. Currently, multiobjective programming has been applied to fields like engineering [

All the optimal reinsurance models in the aforementioned literature are formulated under the assumption that the insurer cedes risks to a single reinsurer. In practice, with the development of the insurance industry, the insurance coverage is also expanding. Therefore, the insurer prefers finding multiple reinsurers to share the risks and the underwriting revenue. Nowadays, the scale of reinsurance is becoming larger and larger; thus, a perfect market of reinsurance has been created. There exist various types of reinsurance contracts in such a market; most of them are in the case that the insurer cedes the risk to several reinsurers. Thus, we will focus on the problem of “ceding the risk to multiple reinsurers.”

In this paper, we apply the principle in microeconomics that “Pareto optimum achieves the optimal Nash equilibrium” as a starting point and then analyze the situation in which the original insurers allocate risks to several reinsurers. Assume that the insurer and all reinsurers use coherent distortion risk measures; we demonstrate the necessary and sufficient conditions for the reinsurance market to achieve Pareto optimum. That is, every reinsurance function should be “multiple-layer reinsurance.” The novelty and contribution of this paper are summarized in the following aspects.

First, we study the optimal reinsurance on a global perspective. Different from the literatures mentioned above, we focus on the optimization problem of the insurer with multireinsurers. This assumption is closer to the insurance practice since the insurer’s optimal reinsurance contract does not always mean the best choice for the reinsurer. Furthermore, if the insurer only seeks a single reinsurer in the reinsurance contracts, that may lead to serious problems such as inadequate solvency and credit risks. Thus, the primary insurers often seek multiple reinsurers to share the risks.

Second, our reinsurance model design does not have any assumptions on the premium principles and is thus broadly applicable. In previous literatures, the optimal designs of multireinsurers have all been based on certain specific premium principle assumption. Many of them indicate that the future research is needed for their optimum reinsurance designs to be applicable under other premium principles. This paper has addressed such a problem. By reducing the assumptions on the premium principles, our research makes it more generally applicable.

Third, the reinsurance model has outstanding economic significance. To achieve the optimal reinsurance design with multireinsurers, the concept “Pareto optimum” from microeconomics is adopted in this paper. This makes our proposed reinsurance model different from those in existing actuarial literatures and possess more economic value.

This paper proceeds as follows. Our optimal reinsurance model is formulated in Section

Consider an insurer suffers a potential loss

While the insurer can reduce its risk exposure by ceding part of the loss, it incurs an additional cost in the name of reinsurance premium. Assume that the reinsurance premium for the

Assume that there exists an increasing function

Now let us present the concept of “Pareto optimum” before the reinsurance model is established. Popularly, assume that there exist a group of people and resources which can be allocated. Changing from one dispensing state to another, no one can be made better off by making someone worse off. Such a state or distribution of economic resources is called the Pareto optimum. Later researchers have conducted a series of studies on such a state; the most famous one is American economist John. F. Nash. Combining game theories, he proved that Pareto optimum is a specific form of “Nash equilibrium” which means the economic activities are the most efficient and ideal.

Given a random return variable

In reinsurance market, all insurers and reinsurers pay more attention to the risks. As an alternative, we rewrite Definition

Since now the problem involves multiple reinsurers, we need to redefine the risks undertaken by the insurers and reinsurers after reinsurance. After the insurer cedes the risk

On the other hand, the

Next, assume that the insurers and all reinsurers adopt coherent distortion risk measures, which can be denoted as

Consequently, the solution to problem (

Before analyzing the models above, the definitions and notations of some set are given, which will help us prove the relevant issues and provide follow-up solutions. Denote sets

Each

Firstly, for any rational number satisfying

According to the continuity of

Additionally, for any

Now let us analyze the set

Similarly, (

Now we construct the ceded-loss functions which can achieve the Pareto optimum.

Assume that all generation functions

First we explain that the numbers

On the other hand, according to the definition of

Corresponding to each function above, we need to find out the nonnegative constants

The definition of

Next, combining (

In addition, (

We get

Next, the risk measures will be transformed in the following form which can be handled more easily. Recalling that

Since the coherent risk measures have the property of “additivity for the comonotonic random vector” (for the definition and related proposition of coherent risk measures, see Section 5.1 in Dhaene et al. [

Intuitively, when the generating functions are small, the corresponding distortion risk measures are also small; that is, the risk levels are relatively lower. The theorem above shows that various participants in the reinsurance market only bare the loss when their risk measures are on the “smallest intervals,” and the reinsurance market achieves the Pareto optimum in such a situation. More specifically, (

At last, we apply two numerical examples to show how to find the optimal reinsurance functions under the Pareto optimum, which may help readers have a better understanding of our work.

Assume that the loss

First, we derive the form of the reinsurance function under Pareto optimum. Based on the form of the distortion risk measure above,

Next, the exact form of

In step two, we get that the unique root of

Based upon Theorem

In this example, we only replace the distributions of loss based on the example above. Assume that

This study has an important effect on the risk management and strategic planning of insurance and reinsurance companies and provides guidance for the regulatory authorities to optimize the reinsurance market. In practice, reinsurance should consider some restrictions, such as solvency and default risk. In theory, this paper considers only one kind of Nash equilibrium; thus, more optimal reinsurance strategies can be studied under generalized Nash equilibrium in the future.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper was supported by the Fundamental Research Funds for the Central Universities (JBK160101) and the Youth Foundation for Humanities and Social Sciences of Education of China (15YJC630105).