This paper aims to provide a practical optimal reinsurance scheme under particular conditions, with the goal of minimizing total insurer risk. Excess of loss reinsurance is an essential part of the reinsurance market, but the concept of stoploss reinsurance tends to be unpopular. We study the purchase arrangement of optimal reinsurance, under which the liability of reinsurers is limited by the excess of loss ratio, in order to generate a reinsurance scheme that is closer to reality. We explore the optimization of limited stoploss reinsurance under three risk measures: value at risk (VaR), tail value at risk (TVaR), and conditional tail expectation (CTE). We analyze the topic from the following aspects: (1) finding the optimal franchise point with limited stoploss coverage, (2) finding the optimal limited stoploss coverage within a certain franchise point, and (3) finding the optimal franchise point with limited stoploss coverage. We provide several numerical examples. Our results show the existence of optimal values and locations under the various constraint conditions.
Reinsurance, an agreement between insurers and reinsurers that allows insurers to transfer and diversify away a certain amount of risk, is the primary risk management tool used by insurance companies. The amount that an insurer pays to transfer risk to the reinsurer is known as the reinsurance premium. The losses caused by accidents that meet the requirements in the reinsurance contract (and that are borne by the reinsurer) are known as reinsurance recoverable. The insurer aims to reduce its compensation expenses to the greatest extent possible.
However, the situation is everchanging, and optimal reinsurance has become a popular topic for both researchers and practitioners. This has resulted in a plethora of important insights. The earliest study on optimal reinsurance focused on safety loading, Borch [
Thus, limited stoploss reinsurance, which is the focus of this study, is the most practical realworld solution. In this study, we regard stoploss reinsurance as a special case. We aim to solve the problem and provide optimal reinsurance advice to achieve optimal risk transfer under different risk measurement models and limiting conditions. The remainder of this paper is organized as follows. Section
To illustrate the concept behind limited stoploss reinsurance, consider an example involving coverage for the total amount of claim
In this paper, subscripts
Figure
Relation between
Let
In reinsurance studies, the risk measure is used to ensure optimal decision making (see, e.g., Cai et al. [
According to Hang [
Plane (
From (
From (
According to the transformation among risk measure models (Hang [
We define risk measures in this paper as per Hang [
Under risk measure
We let
At any given point in time on the reinsurance market, some reinsurance policies are not active. For example, the coverage of reinsurance in the market is generally given as
As we noted previously,
Next, we establish that
Note that (
Let
If (
As mentioned in the previous sections, districts i and iii are open sets. However,
When
If
When
The proof indicates the existence of
Note that insurance companies may consider a reinsurance position where the franchise point is the profit equilibrium point, because they can obtain a certain profit before the threshold. Beyond this threshold, profits will be reduced or even reach a deficit. Thus, the best option is appropriate reinsurance coverage
Next, we let
If (
According to (
When
After analyzing the optimal decision under given franchise point
Under the condition of (
The optimal value of
Given a company’s risk preference
In this subsection,
Under the condition of (
Figure
In this subsection,
If
From (
In district ii, the partial derivatives in
For a given
Under the condition of (
According to (
No global optimum
Assume that the optimal solution of
Assuming that the optimal solution is in the interior of district ii, it must be a saddle point according to the extreme value theory of multivariate functions. Moreover,
We assume that the claim ratio of some insurance is gamma (4.1405, 0.1796) and
Note in Figure
Then, as shown in the contour map in Figure
Figure
Equations (
We assume that the claim ratio of some insurance is gamma (4, 0.125). We have
Optimal
For the risk distribution discussed above, note that (
The model definition indicates that the CTE risk measures are more accurate than those of TVaR. However, the analysis and examples above show that discontinuous CTE can hinder optimal decision making to some extent.
We mentioned the optimization problem regarding reinsurance premium budget constraints briefly during the introduction. Porth et al. [
The CTE in Porth et al. [
Similarly, we assume that the claim ratio of some insurance is gamma (4.1405, 0.1796) and
Under different financial constraints of
We have mentioned the numerical solution of the equation several times throughout this paper. But it is ultimately unnecessary. For example, for a given
Compared with stoploss reinsurance (
Table
Existence condition of optimal solution.
Risk 
Certain 
Certain 
Comprehensive 

VaR 


No 
TVaR 

—  — 
CTE  Lemma 
—  — 
In this study, note that safety loading
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was funded by the Youth Project of the National Natural Science Foundation of China (71102125) and the MOE Project of the Key Research Institute of Humanities and Social Sciences at Universities (13JJD790041). The authors are grateful for the support provided by the Beijing Education Committee through the Young Talents Plan Project.