Quota-share and stop-loss reinsurance combination based on Value-at-Risk (VaR) optimization

Every insurance companies have a capacity limit related to the maximum claim that can be borne. Therefore, insurance companies need to reinsure risks to reinsurance companies. Besides of quota-share, types of reinsurance contracts that commonly used is stop-loss. The quota-share reinsurance premium is proportional based on the amount claim that is covered, but not safe against a large claim. While for stop-loss, the reinsurance premium is relatively large but safe for a large claim. So, this paper combines both types of reinsurance to cover the shortcomings with their respective strengths. After being combined, it is necessary to determine the optimal quota-share proportion and stop-loss retention. One criterion of determines optimal proportion and retention is based on Value-at-Risk (VaR) optimization. With the reinsurance premium as a constraint, this optimization problem is solved for each type of reinsurance combination, be it quota-share before stop-loss or stop-loss before quota-share. From each of these types combinations, the result is optimal quota-share proportion and stop-loss retention, so as produce a minimum VaR value from the borne risk by insurance companies. by comparing the results of VaR optimization of these combinations, stop-loss before quota-share is obtained resulting in a more minimum VaR value.


Introduction
Insurance companies provide risk management services to policyholders. When a policyholder suffered a loss, a claim will be submitted to the insurance company. In some cases, not all claims can be borne fully by insurance companies, especially for large claims that exceed the retention limit. Retention is a large risk that can be borne by the insurance company itself. Therefore, in covering the risks of the policyholders, the insurance company protects itself by using reinsurance services. Stop-loss scheme is one of the commonly used type of insurance, after the quota-share scheme.
Both for quota-share and stop-loss have pros and cons from the perspective of the insurance company. Gavranovic et al. [1] modeled the quota-share reinsurance where the reinsurance premiums paid by insurance companies is proportional to the claims they bear. While for stop-loss reinsurance as discussed by Cai et al. [2], reinsurance premiums are relatively large or expensive, depending on the retention set by the insurance company itself. In addition, quota share reinsurance is not safe against large total claims, because it can produce a loss ratio that is greater than what the insurance company can bear. However, in stop-loss reinsurance, the insurer determines the limit of claim coverage, so it is safe against large total claims. By combining the two types of reinsurance, it is expected that the After the insurance company shares the risks it has borne with the reinsurance company, as collateral for the transferred risk, the insurance company must pay the reinsurance premium. Reinsurance premiums are calculated using certain principles, among them is the expectation value principle.
Because risk limitation means a reduction in profits, reinsurance premiums must be within reasonable limits. That is, the smaller the retention determined by the insurance company, the lower the insurance company's ability to bear the risk and reinsurance premiums to be paid will be even greater, and vice versa. Therefore, it is important for insurance companies to determine the optimal retention value. One way is to measure losses that can be borne by insurance companies. Losses incurred by insurance companies are measured using certain methods, one of them is Value-at-Risk (VaR) method [3]. The aim of this study is to analyze the possible order of combination between quota-share and stop-loss that minimize VaR of the insurer.

Quota-share reinsurance policy under VaR
Quota-share reinsurance is a reinsurance agreement where the insurance company surrenders a part of the insurance value to the reinsurance company in a fixed percentage. Suppose that ܺ is a total claim submitted by policyholders. The insurance company bears the claim in the amount of proportion ‫‬ and the reinsurance company bears proportion ͳ െ ‫,‬ represented by the formulation as the following: where ܺ ூ is a total claim that the insurer borne and ܺ ோ denotes a total claim that the reinsurer borne. The VaR of total cost model at a confidence level ͳ െ ߙǡ Ͳ ൏ ߙ ൏ ͳǡ is defined as where ߜ denotes the reinsurance premium and ܺ ் means total cost model i.e. total claims to be borne and reinsurance premium to be paid. From the overall possible value of ‫‬ ‫א‬ ሾͲǡͳሿǡ an optimal value of ‫,‬ i.e. ‫‬ ‫כ‬ ‫א‬ ሾͲǡͳሿǡ will be sought in order that the VaR value of total cost is minimum., that is Assume that the reinsurance premium ߜሺήሻ satisfies ߜሺͲሻ ൌ Ͳ and positive homogeneity ߜሺ‫ܺ‬ሻ ൌ ‫ߜ‬ሺܺሻ for a constant ‫‬ ͲǤ Then the optimal quota-share coefficient, ‫‬ ‫כ‬ ǡ depends on ߜሺܺሻ and ܸܴܽ ఈ ሺܺሻ as in the following [4], This formulation states that if the reinsurance premium is smaller than ܸܴܽ ఈ ሺܺሻ, to minimize ܸܴܽ ఈ ሺܺ ் Ǣ ‫‬ ‫כ‬ ሻ, all claims should be reinsured to the reinsurer. If the reinsurance premium is equal to ܸܴܽ ఈ ሺܺሻ, for any value of ‫‬ ‫כ‬ ‫א‬ ሺͲǡͳሻ, ܸܴܽ ఈ ሺܺ ் Ǣ ‫‬ ‫כ‬ ሻ will be minimum. If the reinsurance premium is greater than ܸܴܽ ఈ ሺܺሻ, to minimize ܸܴܽ ఈ ሺܺ ் Ǣ ‫‬ ‫כ‬ ሻǡ the insurer must borne all claims.

Stop-Loss reinsurance policy under VaR
Stop-loss reinsurance is a reinsurance contract where the insurance company will get a compensation from the reinsurance company if the amount of the claim submitted by the policyholder exceeds the retention of a certain nominal that can be borne by the insurance company. The formulation of this policy can be written as: where ‫ݍ‬ represents retention of the insurer and ܺ is a total claim submitted by policyholders. The VaR of total cost model at a confidence level ͳ െ ߙǡ Ͳ ൏ ߙ ൏ ͳǡ is defined as where ܺ ் means total cost model i.e. total claims to be borne and reinsurance premium to be paid.
Among possible values of ‫ݍ‬ ‫א‬ ሾͲǡ λሻǡ an optimal value of ‫ݍ‬ i.e. ‫ݍ‬ ‫כ‬ ‫א‬ ሾͲǡ λሻ will be sought in order to minimize the VaR of total cost, that is, The optimal retention ‫ݍ‬ ‫כ‬ in (9) exists if and only if and hold [4]. The optimal retention ‫ݍ‬ ‫כ‬ is given by and the minimum VaR of ܺ ் is Hence, if equation 10 and equation 11 are satisfied, the optimal retention is ‫ݍ‬ ‫כ‬ ൌ ܸܴܽ ఈ ቀ ଵ ଵାఘ ቁ and VaR of total cost model reach a minimum.

Mean-VaR optimization for quota-share before stop-loss
This combination of quota-share and stop-loss discusses about the optimization of VaR for the borne risk of insurer when applying quota-share at the first claim and then stop-loss to optimize the VaR, involving reinsurance premium as a constraint. The borne risk of insurer is ܺ ூ ൌ ‫‪ǡ‬ݍܺ‬ where an optimal ‫‬ is ‫‬ ‫כ‬ ‫א‬ ሾͲǡͳሿ and an optimal ‫ݍ‬ is ‫ݍ‬ ‫כ‬ ‫א‬ ሾͲǡ λሻ such that subject to where ܲ is the reinsurance premium, ߩ Ͳ is safety loading factor depends on the company's operating costs, ߤ is expectation of ܺǡ and ‫ݍ‬ ത is a value of ‫ݍ‬ when ‫‬ ൌ ͳ which satisfies To identify optimum ‫‬ and ‫ݍ‬ǡ a constraint graph will be made by considering ‫‬ as the ‫ݔ‬ െaxis and the point where the value of ‫‪ሻ‬ݍܺ‪ሺ‬ܧ‬ changes i.e.
as the ‫ݕ‬ െaxis, where ‫‬ ‫א‬ ሾͲǡͳሿ and ‫ݍ‬ ‫א‬ ሾͲǡ λሻǤ For a certain value of ‫ݍ‬ǡ the graphic between ‫‬ and is illustrated in figure 1: The next step to solving the optimization problem is searching for points that produce a minimum VaR. This is given by Lemma 1.
From both cases, the minimum value of ܸܴܽ ఈ ሺ‫‬ ‫כ‬ ‫ݍܺ‬ ‫כ‬ ሻ is attained when ‫‬ and ‫ݍ‬ are minimum. Since ‫‬ and ‫ݍ‬ are minimum, so does . Therefore, the minimum value of ܸܴܽ ఈ ሺ‫‬ ‫כ‬ ‫ݍܺ‬ ‫כ‬ ሻ is attained at the boundary line of ‫‪ሻ‬ݍܺ‪ሺ‬ܧ‬ ൌ ߤ െ ܲ Ǥ After finding that the VaR value is minimum at the boundary line of ‫‪ሻ‬ݍܺ‪ሺ‬ܧ‬ ൌ ߤ െ ܲ ǡ the next step is to find the optimal proportion ‫‬ and retention ‫ݍ‬Ǥ This is explained in the following Theorem 3.1.

Theorem 1
If an insurer takes quota-share before stop-loss to minimize the VaR at a confident level ͳ െ ߙ, then •

Mean-VaR optimization for stop-loss before quota-share
Similar with the first combination of quota-share and stop-loss, in this section we will discuss the optimization of VaR for the borne risk of insurer when applying quota-share first for the combination and then stop-loss, involving reinsurance premium as a constraint. The retained risk of the insurer is ܺ ூ ൌ ‫ܺ‬ ሺͳ െ ‫‬ሻሺܺ‫ݍ‬ሻǡ where an optimal ‫‬ is ‫‬ ‫כ‬ ‫א‬ ሾͲǡͳሿ and an optimal ‫ݍ‬ is ‫ݍ‬ ‫כ‬ ‫א‬ ሾͲǡ λሻ such that

Figure 2. Graph of ሺͳ െ ‫‬ሻ‫ܧ‬ሺሺܺ െ ‫ݍ‬ሻ ା ሻ ܲ
To identify an optimum ‫‬ and ‫ݍ‬ǡ a constraint graph will be made by considering ‫‬ as the ‫ݔ‬ െaxis and the point where the value of ‫ܧ‬൫ܽܺ ሺͳ െ ܽሻሺܾܺሻ൯ changes i.e. ‫ݍ‬ as the ‫ݕ‬ െaxis, where ‫‬ ‫א‬ ሾͲǡͳሿ and ‫ݍ‬ ‫א‬ ሾͲǡ λሻǤ The graphic between ‫‬ and ‫ݍ‬ is illustrated in figure 2.
The next step to solving the optimization problem is searching for points that minimize the VaR . This is given by Lemma 2.

Lemma 2
The

Case study
Assume that the insurer's total loss ܺ in one period follows a lognormal distribution with mean ߤ of 5, and the variance ߪ ଶ of 4Ǥ The expectation of total loss ܺ is ͳͲͻǡ͵ and the VaR of ܺ at different confidence level are presented in table 1.
Given several different levels of ܲ less than ߤ, or ‫ܧ‬ሺܺሻ as reinsurance premium that must be paid by an insurer to the reinsurance company, then the retention ‫ݍ‬ ത can be found such that satisfy ‫ܧ‬ሺሺܺ െ ‫ݍ‬ ത ሻ ା ሻ ൌ ܲ Ǥ In will decrease. For each pair of reinsurance premium ܲ and confidence level ߙǡ we calculated an optimal reinsurance be it quota-share (QS), stop-loss (SL), quota-share before stop-loss (QSbSL), or stop-loss before quota-share (SLbQS); and the minimal VaR value of the retained risk of the insurer. The result is presented in Table 4 for the combination of stop-loss after quota-share, and Table 5 presents the result for the combination of quota-share after stop-loss.
According to table 4. for the confidence level of ߙ ൌ ͷ Ψ and constraint reinsurance premium ܲ ͺͲͲǡ the optimal reinsurance is a pure stop-loss policy, with the retention ‫ݍ‬ ͺͲǤͺʹ and the minimal VaR of retained risk is ͺͲǤͺʹǤ Keeping ܲ the same and increasing ߙ to ͳͲ Ψǡ the optimal reinsurance turned to quota-share with the proportion ‫‬ ͲǤʹ and the minimal VaR reduces to ͷʹͲǤͻʹǤ As for stop-loss before quota-share combination, if the insurer set the confidence level ߙ ͳ Ψ and ܲ ൌ ͲͲǡ the optimal reinsurance is pure stop-loss with retention ‫ݍ‬ ൌ ʹͶʹͷǤͷͺ and the minimal VaR of retained risk is ʹͶʹͷǤͷͺ (table 5). Increasing ߙ to ͳͲ Ψ while keeping ܲ the same, the optimal reinsurance is quota-share after stop-loss with the proportion ‫‬ ͲǤͶͳͳ and retention ‫ݍ‬ ͳͲͻǤͶͷǡ and the minimal VaR reduces to ͺͷͷǤ͵ͷǤ

Conclusion
The minimal VaR of the insurer depends on the proportion of quota-share and retention of stop-loss that varies according to the condition of ܸܴܽ ఈ ሺܺሻǤ For ܸܴܽ ఈ ሺܺሻ which is greater or smaller than function of the expected total loss, reinsurance premium, and stop-loss retention when quota-share is not applied, the minimal VaR are the same in both quota-share before stop-loss and stop-loss before quota-share. However, if ܸܴܽ ఈ ሺܺሻ does not meet these conditions, the minimal VaR of retained risk can yield a smaller value in stop-loss after quota-share policy. Therefore, it is recommended for an insurer to apply quota-share after stop-loss to minimize the VaR of the insurer.