# Optimal Layer Reinsurance for Compound Fractional Poisson Model

Authors: Jiesong Zhang
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## 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.


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## Body

## 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 as(1)u+ct-∑i=1NhtXi,t≥0,where u is the initial capital, c is the constant premium rate, and Xi, i=1,2,3,⋯ represents the size of the ith claim and the claim sizes are assumed to be independent and identically distributed nonnegative variables with a common distribution function F. The counting process Nht 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 τi between two claims with distribution given by(2)Prτ>t=Eh-λthfor λ>0 and 0<h≤1, where(3)Ehz=∑k=0∞zkΓ1+hkis the Mittag-Leffler function (Γ denotes the Euler gamma function) defined for any complex number z. With Tn=τ1+τ2+⋯τn, the time of the nth jump, the process Nhtt≥0 defined by (4)Nht=max⁡n≥0:Tn≤t=∑k≥11Tk≤tis the so-called fractional Poisson process of parameter h. It includes the usual Poisson process when h=1.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 Fx of Xi is such a continuous function that F0=0,0<Fx<1 for 0<x<M and Fx=1 for x≥M, here M=inf⁡{x:Fx=1, and 0<M≤+∞; that the moment generating function of Fx, MXr, exists for r∈-∞,r∞ for some 0<r∞≤∞; and that limr→r∞⁡MXr=+∞. Let μ be the expected value of Xi. Denote the decision variables representing the layer retention by d1 and d2. The ceded loss function is the layer reinsurance in the form of (5)gx=min⁡x-d1+,d2-d1=x-d1+-x-d2+,where {x}+=max⁡{x,0}, and 0<d1≤d2≤M. Thus, the insurer will retain from theith claim(6)Xid1,d2=Xi∧d1+Xi-d2+,i=1,2,⋯,Nht.Then {Xid1,d2} are i.i.d. strictly positive random variables and independent of the claim counting process Nht.Assume that the reinsurance premium is charged by the expected value principle, and denote the expected value ofXid1,d2 by μd1,d2. Then the premium income rate becomes(7)cd1,d2=λhth-1Γ1+hμ1+θ-1+ηλhth-1Γ1+hEXi-Xid1,d2=θ-ηλhth-1Γ1+hμ+1+ηλhth-1Γ1+hμd1,d2=λhth-1Γ1+hθ-ημ+1+ημd1,d2where θ=c/μΓ1+h/λhth-1-1 denotes the security loading of the insurer, and η is the security loading of the reinsurer. As usual, we assume that η>θ. Note that the following inequality should be held,(8)θ-ημ+1+ημd1,d2>0.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(9)Utd1,d2=u+cd1,d2t-∑i=1NhtXid1,d2.Now define the ruin time by(10)τd1,d2=inf⁡t≥0:Utd1,d2<0,and define the ruin probability by(11)ψu=ψd1,d2u=Pr⁡τd1,d2<∞∣U0d1,d2=u.


## 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 coefficientR which satisfies the following inequality(12)ψu≤e-Ru.Lemma 1.
IfR=Rcd1,d2≥0 satisfies the equation(13)λ∫0∞erxdPXid1,d2x=λ+r∙cd1,d2h,which is an implicit equation with respect to r; then the inequality (12) follows.Proof.
Assume (13) holds; we prove the inequality (12) by mathematical induction (see [22] for the CPP case). Let ψnu be the probability that ruin occurs on thenth claim or before with an initial surplusu. Clearly, (14)0≤ψ0u≤ψ1u≤⋯≤⋯ψnu≤⋯and(15)limn→∞ψnu=ψu.Furthermore, from (16)ψ0u=1,u<00,u≥0we have(17)ψ0u≤e-Ru.To 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 obtain(18)ψnu=∫0∞∑k=1∞-1k-1khλkthk-1Γ1+hk∫0∞ψn-1u+cd1,d2t-xdPXid1,d2xdt≤∫0∞∑k=1∞-1k-1khλkthk-1Γ1+hke-Ru-R∙cd1,d2tdt∫0∞eRxdPXid1,d2x=e-Ru∙Ee-τ∙R∙cd1,d2∙∫0∞eRxdPXid1,d2x=e-Ru∙λλ+R∙cd1,d2h∙∫0∞eRxdPXid1,d2x,where the last equation is obtained from equation (4.15) in [23]. Thus, the inequality (12) follows immediately from (13).SinceMXid1,d2r=∫0∞erxdPXid1,d2(x), (13) is equivalent to(19)r∙cd1,d2h=λMXid1,d2r-1.Substituting (7) into (19) yields (20)λhth-1Γ1+hθ-ημ+1+ημd1,d2hrh-λMXid1,d2r-1=0.Our goal is to maximize Rcd1,d2, i.e., to find the optimal retention d1∗,d2∗, such that(21)RC≔RCd1∗,d2∗=supd1,d2Rcd1,d2.Note that the left-hand side of (19) is a concave function and the right-hand side is a convex function, with respect tor. Therefore, there are at most two solutions to (19), and the left-hand side of (20) is nonpositive at r=RC, i.e., RC is the solution to (22)supd1,d2λhth-1Γ1+hθ-ημ+1+ημd1,d2hrh-λMXid1,d2r-1=0,or, equivalently, (23)supd1,d2gd1,d2=0,where (24)gd1,d2=λhth-1Γ1+hθ-ημ+1+ημd1,d2hrh-rλ∫0d11-Fxerxdx+∫d2M1-Fxex+d1-d2rdx.Next we adopt the method used by [21] to determine the optimal retention level d1∗,d2∗.Lemma 2.
Denote the maximizer ofgd1,d2 with d1 and d2 being d¯1 and d-2, respectively. Then, d¯1 is the solution to the following equation with respect to d1,(25)h1+ηhth-1Γ1+hhλrθ-ημ+1+η∫0d11-Fxdxh-1=erd1,and d-2=M.Proof.
By differentiatinggd1,d2 with respect to d1, we have(26)∂gd1,d2∂d1=hrh1+ηλhth-1Γ1+hhθ-ημ+1+ημd1,d2h-11-Fd1-rλ1-Fd1erd1+r∫d2M1-Fxex+d1-d2rdx,which means that (27)h1+ηhth-1Γ1+hhλrθ-ημ+1+ημd-1,d2h-1-erd-11-Fd-1=r∫d2M1-Fxex+d-1-d2rdx,for any fixed d2.
Then, differentiatinggd¯1,d2 with respect to d2 and combining with (27), we obtain(28)∂gd-1,d2∂d2=hrh1+ηλhth-1Γ1+hhθ-ημ+1+ημd-1,d2h-1Fd2-1-rλerd-1Fd2-1-r∫d2M1-Fxex+d-1-d2rdx=rλh1+ηhth-1Γ1+hhrλθ-ημ+1+ημd-1,d2h-1-erd-1Fd2-1+r2λ∫d2M1-Fxex+d-1-d2rdx=rλr∫d2M1-Fxex+d-1-d2rdx1-Fd1Fd2-1+r2λ∫d2M1-Fxex+d-1-d2rdx=r2λFd2-11-Fd-1+1∫d2M1-Fxex+d-1-d2rdx.Note that(29)Fd2-11-Fd1+1≥0holds for any d2≥d1, and we have (30)r2λFd2-11-Fd-1+1∫d2M1-Fyey+d-1-d2rdy≥0,and thus(31)d-2=M.
By replacingd2=M back into (27), we can derive(32)h1+ηhth-1Γ1+hhλrθ-ημ+1+η∫0d-11-Fxdxh-1-erd-1=0,which completes the proof of Lemma 2.Since(33)μd1,d2=∫0d11-Fxdx+∫d2M1-Fxdxand d-2=M, by (8), we have(34)μd¯1,M=∫0d¯11-Fxdx>η-θ1+ημ>0.Denote d_1=inf⁡d¯1d¯1satisfies(34).According to Lemma2, we know that to solve the optimization problem (23) is equivalent to solving the equation: (35)λhth-1Γ1+hθ-ημ+1+η∫0d-11-Fxdxhrh-rλ∫0d-11-Fxerxdx=0,or, alternatively,(36)Gd-1≔λhth-1Γ1+hθ-ημ+1+η∫0d-11-Fxdxh-λ∫0d-11-Fxerxdxrh-1=0,where r=rd-1 is a univariate function of d-1 determined by (32). In fact, we have Lemmas 3–5.Lemma 3.
Equation (32) has a unique positive root r=RC for any given d¯1∈d_1,M.Proof.
For any givend¯1∈d_1,M, define the left-hand side of (32) by Hr, i.e.,(37)Hr=h1+ηhth-1Γ1+hhλrθ-ημ+1+η∫0d-11-Fxdxh-1-erd-1.It is not difficult to see that(38)limr↓0Hr=+∞andlimr↑∞Hr=-∞.Moreover, note that 0<h≤1; we know that Hr is a strictly decreasing function in r. Thus, it completes the proof of Lemma 3.Lemma 4.
The functionr=rd-1 is strictly decreasing in d1 and ∂r/∂d-1<0.Proof.
Rewrite (32) as (39)h1+ηhth-1Γ1+hhλθ-ημ+1+η∫0d-11-Fxdxh-1=erd-1rh-1.By differentiating both sides of (39) with respect to d¯1, we have(40)hh-11+η2λh-1hth-1Γ1+hhθ-ημ+1+η∫0d-11-Fxdxh-21-Fd-1-erd-1rh-2=d1erd-1-erd-1h-1r-1rh-1∂r∂d-1.Then, we find that(41)∂r∂d-1=-h1-h1+η2λh-1hth-1Γ1+hhθ-ημ+1+η∫0d-11-Fxdxh-2∙1-Fd1e-rd-1+1rh-2∙rhrd-1+1-h<0.Lemma 5.
The equationGd¯1=0 has a unique positive root d¯1C∈d_1,M.Proof.
DifferentiatingGd¯1 with respect to d¯1, by (39), we have (42)G′d-1=hλhth-1Γ1+hhθ-ημ+1+η∫0d-11-Fxdxh-11+η1-Fd-1-λ∂r∂d-1∙∫0d-1x1-Fxerxdx+1-Fd1erd-1rh-1-h-1∫0d-11-Fxerxdxrh-2r2h-2=hλhth-1Γ1+hhθ-ημ+1+η∫0d-11-Fxdxh-11+η1-Fd-1-λ∫0d-1x1-Fxerxdx∂r/∂d-1+1-Fd-1erd-1rh-1+λh-1∫0d-11-Fxerxdxr-1rh-1∂r∂d-1=hλhth-1Γ1+hhθ-ημ+1+η∫0d-11-Fxdxh-11+η1-Fd-1-λ∫0d-1x1-Fxerxdxrh-1-h-1∫0d-11-Fxerxdxr-1rh-1∂r∂d-1-λ1-Fd-1erd-1rh-1=λ1-Fd-1erd-1rh-1-λ1-Fd-1erd-1rh-1-λ∫0d-1x1-Fxerxdxrh-1-h-1∫0d-11-Fxerxdxr-1rh-1∂r∂d-1=-λ∫0d-1x1-Fxerxdxrh-1+1-h∫0d-11-Fxerxdxr-1rh-1∂r∂d-1.Hence, from Lemma 4 we know that G′d¯1>0. Moreover, we have limd¯1↓d_1⁡rd¯1=+∞ and limd¯1↑M⁡rd¯1=0, which can be seen from (39). Thus, for any h∈0,1, it is held that (43)limd¯1↓d_1Gd¯1≤-limd1↓d_1λr1-h∫0d¯11-Fxdx=-∞,and(44)limd¯1↑MGd¯1=limd-1↑M⁡λhth-1Γ1+hθ-ημ+1+η∫0d-11-Fxdxh-λ∫0d-11-Fxerxdxrh-1=λhth-1Γ1+hθ-ημ+1+ηEX1∧Mh-limd1↑M⁡λr1-h∫0M1-Fxerxdx=λhth-1Γ1+hθ-ημ+1+ημh>0.If h=1, it is easy to see that limd¯1↓d_1⁡Gd¯1≤-limd1↓d_1⁡λ∫0d¯11-Fxdx<0 and limd¯1↑M⁡Gd¯1=λθμ>0. Therefore, the proof of Lemma 5 is completed.Now, we can conclude the main result of this paper.Theorem 6.
Letd¯1C be the unique positive root of the equation Gd1=0. Then the optimal layer reinsurance retention level of the compound fractional Poisson surplus (1) to maximize the adjustment coefficient is d¯1C,M, and the maximal adjustment coefficient RC is the unique positive root of (32).
Since the CFPP degenerates into the classical CPP whenh=1, we immediately obtain the following corollary from Theorem 6.Corollary 7.
Letd¯1C be the unique positive root of the equation(45)θ-ημ+1+η∫0d-11-Fxdx-∫0d-11-Fxerxdx=0.Then the optimal layer reinsurance retention level of the classical compound Poisson surplus to maximize the adjustment coefficient is d¯1C,M, and the maximal adjustment coefficient is the unique positive root of (32), i.e., (46)RC=ln⁡1+ηd¯1C.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 d¯1C,M and the maximal adjustment coefficient RC here not only depend on the parameter h of the fractional Poisson process, but also depend on the claim intensity λ and are both relevant to timet, 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 intensityλ and the time t, on the optimal results, we give some numerical examples and compare the optimal retention levels and the maximal adjustment coefficient with different parameters h, λ, and t.


## 3. Examples

Assume that the insurer has an initial capitalu=5, that the claim size Xi has a uniform distribution on the interval 0,4, and that θ=0.4, η=0.5. We compute the values of d¯1C, RC and the upper bound of ruin probability with different parameter values of h, λ, and t. To this end, we need to solve the following equations:(47)1.5hhth-1Γ1+hhλRC-0.2+1.5d-1C-d-1C28h-1-eRC∙d-1C=0,λhth-1Γ1+h-0.2+1.5d-1C-d-1C28h-λ-1-4RC+eRC∙d-1C4RC-RC∙d-1C+14RC1+h=0,for different given h,λ,t with d¯1C∈d_1,M=0.1356,4.0000 and RC>0.By applying numerical method, the results for different cases are given in Table1.Table 1
Optimal retention levels and the upper bounds with different parameters.
t=1  λ=2 d - 1 C  RC e - u R C h=0.5  λ=2 d - 1 C  RC e - u R C h=0.5  t=1 d - 1 C  RC e - u R C h t λ 0.1 0.1489 0.0001 0.6 0.2508 0.0162 1.6 0.2514 0.0166 1.8234 0.8241 0.8021  0.2 0.1659 0.0014 0.7 0.2521 0.0210 1.7 0.2523 0.0220 1.3195 0.7728 0.7629  0.3 0.1878 0.0063 0.8 0.2531 0.0259 1.8 0.2532 0.0263 1.0144 0.7308 0.7273  0.4 0.2163 0.0174 0.9 0.2540 0.0309 1.9 0.2540 0.0310 0.8099 0.6955 0.6946  0.5 0.2548 0.0360 1.0 0.2548 0.0360 2.0 0.2548 0.0360 0.6650 0.6649 0.6649  0.6 0.3084 0.0606 1.1 0.2555 0.0411 2.1 0.2555 0.0412 0.5606 0.6383 0.6377  0.7 0.3861 0.0876 1.2 0.2561 0.0462 2.2 0.2562 0.0467 0.4870 0.6150 0.6128  0.8 0.5002 0.1088 1.3 0.2567 0.0513 2.3 0.2569 0.0525 0.4436 0.5940 0.5895  0.9 0.6557 0.1109 1.4 0.2572 0.0564 2.4 0.2574 0.0584 0.4399 0.5752 0.5682  1 0.8053 0.0807 1.5 0.2576 0.0613 2.5 0.2579 0.0644 0.5035 0.5583 0.5486From Table1, it is not difficult to see that the impacts of the parameter h on the optimal retention level and the upper bound of ruin probability are significant, and the impacts of the parameters t and λ 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 parameterh of the fractional Poisson process, the time t, and the claim intensity λ, 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.



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