Abstract
In this paper, we study the optimal retentions for an insurer with a compound fractional Poisson surplus and a layer reinsurance treaty. Under the criterion of maximizing the adjustment coefficient, the closed form expressions of the optimal results are obtained. It is demonstrated that the optimal retention vector and the maximal adjustment coefficient are not only closely related to the parameter of the fractional Poisson process, but also dependent on the time and the claim intensity, which is different from the case in the classical compound Poisson process. Numerical examples are presented to show the impacts of the three parameters on the optimal results.
1. Introduction
In various geophysical applications, it is observed that the interarrival times between extreme events are power-law distributed, and the exponentially distributed interarrivals cannot be applied [1]. Musson et al. [2] studied the earthquake interarrival times for several regions in Japan and Greece and found that a lognormal distribution provided a good fit. Salim and Pawitan [3] investigated the hourly rainfall data in the southwest of Ireland by a generalized Bartlett–Lewis model with Pareto storm interarrival time. Stoynov et al. [4] proposed an approach for modeling the flood arrivals on Chinese rivers Yangtze and Huanghe by switch-time distributions, which can be considered as distributions of sums of random number exponentially distributed random variables.
Considering the importance of quantifying the stochastic behavior of extreme events in actuarial sciences, Beghin and Macci [5] deal with a fractional Poisson model for insurance, in which the interarrival times between claims are assumed to have Mittag-Leffler distribution instead of the exponential distribution as in the classical Poisson model. Inspired by this work and motivated by the use of the fractional Poisson process in modeling extreme events, such as earthquakes and storms, Biard and Saussereau [6] initiatively described surplus processes of insurance companies by compound fractional Poisson processes, and some results for ruin probabilities are also presented under various assumptions on the distribution of the claim sizes. Different from the case in the classical compound Poisson process (CPP), the compound fractional Poisson process (CFPP) becomes nonstationary [6] and is no longer Markovian [7]. The long-range dependence and the short-range dependence of the CFPP are studied in [6, 8], the estimation of parameters is given by [9], and the convergence of quadratic variation is investigated by [10]. To complete the review of the existing literature on the CFPP, we refer the reader to [11–18].
In this paper, we model the surplus process of an insurance company by the abovementioned CFPP proposed by [6], which can be expressed aswhere is the initial capital, is the constant premium rate, and , represents the size of the th claim and the claim sizes are assumed to be independent and identically distributed nonnegative variables with a common distribution function . The counting process is the fractional Poisson process that was first defined in [11, 19] as a renewal process with Mittag-Leffler waiting time. Specifically, it has independent and identically distributed interarrival times between two claims with distribution given byfor and , whereis the Mittag-Leffler function ( denotes the Euler gamma function) defined for any complex number . With , the time of the th jump, the process defined by is the so-called fractional Poisson process of parameter . It includes the usual Poisson process when .
This paper supposes the insurer reinsures his or her risk by a layer reinsurance treaty. As in [20, 21], we assume that the common distribution function of is such a continuous function that for and for , here , and ; that the moment generating function of , , exists for for some ; and that . Let be the expected value of . Denote the decision variables representing the layer retention by and . The ceded loss function is the layer reinsurance in the form of where , and . Thus, the insurer will retain from the ith claimThen are i.i.d. strictly positive random variables and independent of the claim counting process .
Assume that the reinsurance premium is charged by the expected value principle, and denote the expected value of by . Then the premium income rate becomeswhere denotes the security loading of the insurer, and is the security loading of the reinsurer. As usual, we assume that . Note that the following inequality should be held,Otherwise, the insurance company faces ruin with probability one.
Thus, the reserve process of the insurer with the layer reinsurance policy can be represented by
Now define the ruin time byand define the ruin probability by
2. Optimal Results
In this section, we devote to get the explicit expressions for the optimal retentions in the layer reinsurance treaty. It is difficult to derive the explicit expression of the ruin probability in the CPP and even more difficult in CFPP. We consider the optimal retentions to maximize the adjustment coefficient, i.e., to maximize the coefficient which satisfies the following inequality
Lemma 1. If satisfies the equationwhich is an implicit equation with respect to ; then the inequality (12) follows.
Proof. Assume (13) holds; we prove the inequality (12) by mathematical induction (see [22] for the CPP case). Let be the probability that ruin occurs on the nth claim or before with an initial surplus u. Clearly, andFurthermore, from we haveTo complete the nontrivial part of the mathematical induction, we apply the total probability formula with respect to the arrival time and the size of the first claim. Then, we obtainwhere the last equation is obtained from equation (4.15) in [23]. Thus, the inequality (12) follows immediately from (13).
Since , (13) is equivalent toSubstituting (7) into (19) yields Our goal is to maximize , i.e., to find the optimal retention , such thatNote that the left-hand side of (19) is a concave function and the right-hand side is a convex function, with respect to r. Therefore, there are at most two solutions to (19), and the left-hand side of (20) is nonpositive at , i.e., is the solution to or, equivalently, where
Next we adopt the method used by [21] to determine the optimal retention level .
Lemma 2. Denote the maximizer of with and being and , respectively. Then, is the solution to the following equation with respect to ,and .
Proof. By differentiating with respect to , we havewhich means that for any fixed .
Then, differentiating with respect to and combining with (27), we obtainNote thatholds for any , and we have and thusBy replacing back into (27), we can derivewhich completes the proof of Lemma 2.
Since and , by (8), we haveDenote .
According to Lemma 2, we know that to solve the optimization problem (23) is equivalent to solving the equation: or, alternatively,where is a univariate function of determined by (32). In fact, we have Lemmas 3–5.
Lemma 3. Equation (32) has a unique positive root for any given .
Proof. For any given , define the left-hand side of (32) by , i.e.,It is not difficult to see thatMoreover, note that ; we know that is a strictly decreasing function in . Thus, it completes the proof of Lemma 3.
Lemma 4. The function is strictly decreasing in and .
Proof. Rewrite (32) as By differentiating both sides of (39) with respect to , we haveThen, we find that
Lemma 5. The equation has a unique positive root .
Proof. Differentiating with respect to , by (39), we have Hence, from Lemma 4 we know that . Moreover, we have and , which can be seen from (39). Thus, for any , it is held that andIf , it is easy to see that and . Therefore, the proof of Lemma 5 is completed.
Now, we can conclude the main result of this paper.
Theorem 6. Let be the unique positive root of the equation . Then the optimal layer reinsurance retention level of the compound fractional Poisson surplus (1) to maximize the adjustment coefficient is , and the maximal adjustment coefficient is the unique positive root of (32).
Since the CFPP degenerates into the classical CPP when , we immediately obtain the following corollary from Theorem 6.
Corollary 7. Let be the unique positive root of the equationThen the optimal layer reinsurance retention level of the classical compound Poisson surplus to maximize the adjustment coefficient is , and the maximal adjustment coefficient is the unique positive root of (32), i.e.,
Remark. It is not difficult to see that Corollary 7 is in fact Theorem 4.3 of [21]. By comparing the obtained Theorem 6 in this paper for CFPP with those in [21] for CPP, we find that the optimal retention level and the maximal adjustment coefficient here not only depend on the parameter of the fractional Poisson process, but also depend on the claim intensity and are both relevant to time t, which should be more realistic. In fact, the claim intensity is a very important parameter for estimating the ruin probability and by the dynamic reinsurance strategy the change of the insurer’s best risk position is reflected with respect to time.
To illustrate the impact of replacing the exponential distributed interarrivals by the general Mittag-Leffler distributed interarrivals, as well as the claim intensity and the time , on the optimal results, we give some numerical examples and compare the optimal retention levels and the maximal adjustment coefficient with different parameters , , and .
3. Examples
Assume that the insurer has an initial capital , that the claim size has a uniform distribution on the interval , and that , . We compute the values of , and the upper bound of ruin probability with different parameter values of , , and . To this end, we need to solve the following equations:for different given with and .
By applying numerical method, the results for different cases are given in Table 1.
From Table 1, it is not difficult to see that the impacts of the parameter on the optimal retention level and the upper bound of ruin probability are significant, and the impacts of the parameters and are also obvious. Specifically, if the risk process of the insurance company obeys the compound fractional Poisson model and the compound Poisson model is used, then the insurer may take more risk and the ruin probability is overestimated or underestimated. Even with the CFPP, the optimal strategy should vary timely and according to the change of the claim intensity.
4. Conclusion
To characterize and to disperse the extreme event risk that the insurer may face in practice, this paper models the underwriting risk as a compound fractional Poisson process and studies the optimal retentions with a layer reinsurance treaty. At first, the equation that the adjustment coefficient of the compound fractional Poisson process should satisfy is given and proved. Secondly, to overcome the difficulties caused by the newly adopted model, some lemmas are given, and the closed form expressions of the optimal retention levels are obtained. It is found that the optimal retention level and the maximal adjustment coefficient here relate to the parameter of the fractional Poisson process, the time , and the claim intensity , which are all absent in the optimal results for the classical compound Poisson process. Finally, numerical examples demonstrate the impacts of the three parameters on the optimal results, respectively. The obtained results in this paper may help the insurers, especially the ones who underwrite extreme risk, to make more appropriate decisions in reinsurance contracts.
Data Availability
There is a numerical example in this article; the parameter data used to support the findings of this study are included within the article. For the software (Matlab) code data to obtain the numerical results in the example, it will be available upon request by contact with the corresponding author.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by Anhui Provincial Natural Science Foundation (No. 1608085QG169) and Humanity and Social Science Youth Foundation of Ministry of Education (No. 17YJC630212).