Expected Present Value of Total Dividends in the Compound Binomial Model with Delayed Claims and Random Income

Acta Mathematica Scientia 2013,33B(6):1639–1651 http://actams.wipm.ac.cn

EXPECTED PRESENT VALUE OF TOTAL DIVIDENDS IN THE COMPOUND BINOMIAL MODEL WITH DELAYED CLAIMS AND RANDOM INCOME∗

ª¡¢)1 Xiaoyun MO (£§©)1,2 ¤  Xiangqun YANG (¨¦¥)1†

Jieming ZHOU ( Hui OU ( )1

1. College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing Ministry of Education of China, Hunan Normal University, Changsha 410081, China 2. Department of Mathematics, Hunan University of Finance and Economics, Changsha 410205, China E-mail: [email protected]; [email protected]; [email protected]; [email protected]

Abstract In this paper, a compound binomial model with a constant dividend barrier and random income is considered. Two types of individual claims, main claims and byclaims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. The premium income is assumed to another binomial process to capture the uncertainty of the customer’s arrivals and payments. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are Kn distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends. Key words compound binomial model; main claim; by-claim; dividend; random income 2010 MR Subject Classification

1

60J27; 60J28; 62P05; 60K39

Introduction

In recent years, insurance claims may be delayed due to various reasons. Since the work by Waters and Papatriandafylou [1], risk models with this special feature have been discussed by many authors in the literature. Indeed, a frame work of delayed claims is built by introducing two kinds of individual claims, namely main claims and by-claims, and allowing possible delays ∗ Received

May 18, 2012; revised January 6, 2013. This is supported by the NSFC (11171101), Doctoral Fund of Education Ministry of China (20104306110001), the Graduate Research and Innovation Fund of Hunan Province (CX2011B197). † Corresponding author: Xiangqun YANG.

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of the occurrences of by-claims. A significant amount of work has been done in this area. See, for example, Yuen and Guo [2], Yuen et al. [3], Xiao and Guo [4], Xie and Zou [5], etc. Dividend strategy for insurance risk models were first proposed by De Finetti [6] to reflect more realistically the surplus cash flows in an insurance portfolio, and he found that the optimal strategy must be a barrier strategy. From then on, barrier strategies have been studied in a number of papers and books. Claramunt et al. [7] calculated the expected present value of dividends in a discrete time risk model with a barrier dividend strategy. Other risk model involving dividend payments were studied by Gerber and Shiu [8], Zhou and Guo [9], Fang and Wu [10], Wu and Li [11] and the references therein. All risk models described in the paragraph above relied in the assumption that the premium is collected with a positive deterministic constant rate. Thus, if no other investment income is taken into account, the insurer’s income will be proportional to the time interval. However, it is evident that the deterministic premium income fails to capture the uncertainty of the customers’ arrivals. To reflect the cash flows of the insurance company more realistically, in recent years, risk models with random income have been one of the major interests in the risk theory literature. Bao and Liu [12] studied a compound binomial risk model with delayed claims and random income. Recursive equations for both the probability of ultimate ruin and the joint distribution of the surplus one period prior to ruin and the deficit at ruin were obtained. Dong et al. [13] calculated the Gerber-Shiu function in a risk process with random income and a constant barrier. Related works can be found in Bao [14], Bao and Ye [15], Yang and Zhang [16] and the references therein. The model proposed in this paper is a generalization of compound binomial risk model with paying dividends and delayed claims. It seems to be the first risk model with delayed claims, a constant dividend barrier and random income. We show that, the explicit expression for the expected present value of total dividends in this risk model can be obtained. The work of this paper can be seen as a complement to the work of Bao and Liu [12] that calculated the expected present value of total dividends in the compound binomial model with delayed claims and random income. The model considered in this paper is also related to the one considered by Wu and Li [11]. Although both models employ a discrete time risk model with dividends and delayed claims, our model differs from the one by Wu and Li [11] as follows. Our model is more general than that of Wu and Li [11] in that we assume the premium income process is a binomial process with a certain parameter, while the premium is a positive deterministic constant in Wu and Li [11]. The rest of this paper is organized as follows. Section 2 defines the model of this paper, describes various payments, including the premiums, claims and dividends, and lists the notation. In Section 3, difference equations with certain boundary conditions are developed for the expected present value of total dividend payments prior to ruin. Then an explicit expression is derived, using the technique of generating functions. In Section 4, closed-form solutions for the expected present value of dividends are obtained for two classes of claim size distributions. Numerical examples are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends in Section 4.

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The Model

Here, we consider a discrete time compound binomial risk model with random income. Throughout, denote by N the set of nature numbers and N+ = N/{0}. We assume that the premium income process is a binomial process {Mk , k ∈ N} with parameter p0 (0 < p0 ≤ 1). More precisely, in any time period, we denote by p0 the probability that a premium of size 1 is received and q0 = 1 − p0 the probability that no premium is received.

It is assumed that there are two types of insurance claims, namely the main claims and by-claims. For a detailed description and the intuition for this model, see Yuen and Guo [2]. In any time period, the probability of having a main claim is p (0 < p < 1) and the probability of no main claim is q = 1 − p. Thus the number of main claims up to time k is also a binomial process {Nk , k ∈ N} with parameter p. Independence is assumed between {Mk , k ∈ N} and {Nk , k ∈ N}. Let Sk be the total amount of claims up to the end of the kth time period, k ∈ N+ , and S0 = 0. We define Sk = SkX + SkY , where SkX and SkY are the total main claims and by-claims, respectively, in the first k time periods. The random variables {Xi , i ∈ N+ } denote the sizes of the main claims, independent of {Nk , k ∈ N}, are independent and identically distributed (i.i.d.) positive and integer valued random variables with the same distribution as the generic random variable X and the common probability function (c.p.f.) of X is given by fm = P (X = m) for m = 1, 2, · · ·. The correspond∞ P ing probability generating function (p.g.f.) and mean of X are defined as fe(z) = fm z m and µX =

∞ P

m=1

mfm , respectively.

m=1

The amounts of by-claims, denoted by {Yj , j ∈ N+ }, independent of {Nk , k ∈ N}, are also i.i.d. random variables distributed as a positive and integer valued random variable Y . Denote ∞ P their c.p.f. by gn = P (Y = n) for n = 1, 2, · · ·, and write the p.g.f. and mean as e g(z) = gn z n and µY =

∞ P

n=1

ngn , respectively. Xi and Yj are independent of each other for all i and j. One

n=1

main claim induces one by-claim, which occurs simultaneously with probability θ (0 ≤ θ ≤ 1), that is to say, the by-claim may be delayed with probability 1 − θ. We only consider a delay of one time period in this paper. Similar to the model of Wu and Li [11], we assume that premiums are received at the beginning of each time period, and all claim payments are made only at the end of each time period. We introduce a dividend policy to the company that certain amount of dividends will be paid to the policyholder instantly, as long as the surplus of the company at time k is higher than a constant dividend barrier b (b > 0). It implies that the dividend payments will only possibly occur at the beginning of each period, right after receiving the premium payment. The surplus at the end of the kth time period, Ub (k), is then defined to be, for k = 1, 2, · · ·, Ub (k) = u + Mk − Sk − D(k),

Ub (0) = u,

(2.1)

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where D(k) is the sum of the total dividend payments in the first k periods, with the definition D(k) = D1 + D2 + · · · + Dk ,

D(0) = 0,

denote by Dk the amount of dividend paid out in period k, for k = 1, 2, · · ·, with definition Dk = max{Ub (k − 1) + ηk − b, 0}, for k ∈ N+ and D0 = 0, where b is a fixed non-negative integer, {ηk , k ≥ 1} is a series of randomized decision functions that are mutually independent, identically distributed and independent of Sk . In detail, we denote by ηk = 1 the event where a premium of size 1 is received at time k and denote by ηk = 0 the event where no premium is received at time k. By the assume that the premium income process is a binomial process {Mk , k ∈ N} with parameter p0 (0 < p0 ≤ 1), we have P (ηk = 1) = p0 , P (ηk = 0) = q0 , where q0 = 1 − p0 . Here, the initial surplus is u, u = 0, 1, · · · , b. It is easy to see that EMk = kp0 and ESk+1 = kp(µX + µY ) + pµX + pθµY . Therefore, we further assume p0 > p(µX + µY ). This assumption ensures that E(Mk − Sk ) > 0 for k ∈ N+ and hence the safety loading is positive. Define Tu,b := inf{k ≥ 1 : Ub (k) < 0} to be the time of ruin. Let υ be a constant annual discount rate for each period. Then the expected present value of the dividend payments due until ruin is  TX  u,b k−1 V (u; b) := E υ Dk Ub (0) = u . k=1

3

The Expected Present Value of Dividends

To study the expected present value of dividends payments, V (u; b), we need to study the claim occurrences in two scenarios (see Yuen and Guo [2]). In the first scenario, if a main claim occurs in a certain time period, its associated by-claim also occurs in the same period. Thus the surplus process is renewed at the beginning of the next time period. The second scenario is simply the complement of the first one, i.e., given a main claim, its associated by-claim will occur one period later. Conditional on the second scenario, we define a complementary surplus process as follows: Ub∗ (k) = u + Mk − Sk − D∗ (k) − Y,

k = 1, 2, · · · ,

(3.1)

with Ub∗ (0) = u, where D∗ (k) is the sum of dividend payments in the first k periods, and Y is a random variable following the probability function gn , n = 1, 2, · · ·, and is independent of all other claim amounts random variables Xi and Yj for all i and j. The corresponding expected present value of the dividend payments is denoted by V ∗ (u; b). Then conditioning on the occurrences of claims at the end of the first time period, we can set up the following difference equations for V (u; b) and V ∗ (u; b):  X V (u; b) = υ q0 qV (u; b) + p0 qV (u + 1; b) + q0 pθ V (u − m − n; b)fm gn m+n≤u

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+p0 pθ

X

V (u + 1 − m − n; b)fm gn + q0 p(1 − θ)

m+n≤u+1

+p0 p(1 − θ)

u+1 X

m=1

u X

m=1

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V ∗ (u − m; b)fm

 V ∗ (u + 1 − m; b)fm , u = 0, 1, 2, · · · , b − 1, V ∗ (0; b) = υp0 qV (0; b)g1 ,

(3.2) (3.3)

and for u = 1, 2, · · · , b − 1,

 u u+1 X X V (u; b) = υ q0 q V (u − n; b)gn + p0 q V (u + 1 − n; b)gn ∗

n=1

X

+q0 pθ

n=1

m+n+l≤u

V (u − m − n − l; b)fm gn gl

X

+p0 pθ

m+n+l≤u+1

+q0 p(1 − θ) +p0 p(1 − θ)

V (u + 1 − m − n − l; b)fm gn gl

X

m+l≤u

V ∗ (u − m − l; b)fm gl

X



∗

m+l≤u+1

V (u + 1 − m − l; b)fm gl ,

  V ∗ (b; b) = q0 V ∗ (b; b) + p0 1 + q0 V ∗ (b; b) + p0 V ∗ (b − 1; b) .

(3.4) (3.5)

Rewrite the equation (3.2) as  X V (u; b) = υ q0 qV (u; b) + p0 qV (u + 1; b) + pθ V (u − m − n; b)hX m gn m+n≤u

+p(1 − θ)

u X

m=0



V ∗ (u − m; b)hX m , u = 0, 1, 2, · · · , b − 1,

(3.6)

where hX m = q0 fm + p0 fm+1 , f0 = 0, m = 0, 1, 2, · · ·. Then the equation (3.4) can be represented as  X u V ∗ (u; b) = υ q V (u − n; b)hYn + pθ n=0

+p(1 − θ)

X

m+l≤u

X

m+n+l≤u

V (u − m − n − l; b)hX m gn gl

 V ∗ (u − m − l; b)hX g m l , u = 0, 1, 2, · · · , b − 1,

(3.7)

where hYn = q0 gn + p0 gn+1 , g0 = 0, n = 0, 1, 2, · · ·. From (3.2) and (3.4) one can rewrite V ∗ (u; b) as ∗

V (u; b) =

u X

n=1

V (u − n; b)gn .

This result can also be obtained from model (3.1) as ∗

V (u; b) = E[V (u − Y ; b] =

u X

n=1

V (u − n; b)gn .

(3.8)

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Substituting (3.8) into (3.6) gives V (0; b) = υq0 qV (0; b) + υp0 qV (1; b) + υ 2 p20 pq(1 − θ)V (0; b)f1 g1 , and for u = 1, 2, · · · , b − 1,

X

V (u; b) = υq0 qV (u; b) + υp0 qV (u + 1; b) + υp

m+n≤u

V (u − m − n; b)hX m gn ,

(3.9)

(3.10)

with a new boundary condition:   V (b; b) = q0 V (b; b) + p0 1 + q0 V (b; b) + p0 V (b − 1; b) .

(3.11)

Remark 3.1 When p0 = 1, that is to say, the premium received in each time period is one. In this case, our model is the same as in Wu and Li [11], the equation (3.9) becomes V (0; b) = υqV (1; b) + υ 2 pq(1 − θ)V (0; b)f1 g1 , and for u = 1, 2, · · · , b − 1, (3.10) becomes V (u; b) = υqV (u + 1; b) + υp

X

m+n≤u+1

V (u + 1 − m − n; b)fm gn ,

with a new boundary condition: V (b; b) = 1 + V (b − 1; b). The above three equations are the same as in Wu and Li [11]. To obtain an explicit expression for V (u; b) from (3.9) and (3.10), we define a new function W (u) that satisfies the following difference equation, W (0) = υq0 qW (0) + υp0 qW (1) + υ 2 p20 pq(1 − θ)W (0)f1 g1 , and for u = 1, 2, · · ·, W (u) = υq0 qW (u) + υp0 qW (u + 1) + υp

X

m+n≤u

W (u − m − n)hX m gn .

(3.12)

(3.13)

The solution of (3.13) is uniquely determined by the initial condition W (0). Moreover, apart from a multiplicative constant, the solution of (3.12) and (3.13) is unique (see Wu and Li [11]). Therefore, we can set W (0) = 1. It follows from the theory of difference equations that the solution to (3.9) and (3.10) with boundary condition (3.11) is of the form V (u; b) = C(b)W (u),

(3.14)

 −1 where C(b) = p0 · (1 − q0 − p0 q0 )W (b) − p20 W (b − 1) . ∞ P f (z) := Let the generating function of W (u) be W W (u)z u , −1 < R(z) < 1. Similarly, u=0

∞ P m ∞ X (z) := hf hX is probability generating function (p.g.f.) of {hX mz m }m=0 . Furthermore, we m=0

construct two new generating functions e h(z, 1) := q + pfe(z)e g(z) and e h(z, k) := [e h(z, 1)]k . We e denote the probability function of h(z, k) by h(i, k). Yuen and Guo [2] have commented that h(i, k) is the probability function of the total claims in the first k time periods in the compound

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binomial model with individual claim amount X1 + Y1 . In the following theorem, we show that V (u; b) can be expresses explicitly in terms of h(i, k). Theorem 1 For the expected present value of dividend payments, V (u; b), of model (2.1), we have, for u = 1, 2, · · · , b, V (u; b) = C(b)

∞ X ∞ X i=0 k=i

  υ k+1 p0 qCki q0k−i pi0 υp0 p(1 − θ)f1 g1 h(i + u, k) − h(i + u + 1, k) , (3.15)

where C(b) = p0 ·

(

∞ X ∞ X i=0 k=i

−p20 h(i

  υ k+1 p0 qCki q0k−i pi0 υp0 p(1 − θ)f1 g1 (1 − q0 − p0 q0 )h(i + b, k)

)−1    2 + b − 1, k) − (1 − q0 − p0 q0 )h(i + b + 1, k) − p0 h(i + b, k) ,

and V (0; b) = C(b). Proof Multiplying both sides of (3.13) by z u and summing over from 1 to ∞, we get ∞ X

∞ X

W (u)z u = υq0 q

u=1

W (u)z u + υp0 q

u=1

+υp

∞ X

W (u + 1)z u

u=1

∞ X X

u=1 m+n≤u

u W (u − m − n)hX m gn z .

We can immediately get X (z)e f (z) − W (0)] = υp0 qz −1 [W f (z) − W (0) − W (1)z] + υpW f (z)hf (1 − υq0 q)[W g(z).

From (3.12) and the fact that W (0) = 1, the above equation simplifies to

X (z)e f (z) = (1 − υq0 q)W (0) − υp0 qz −1 (W (0) + W (1)z) [1 − υq0 q − υp0 qz −1 − υphf g (z)]W

= υp0 q[υp0 p(1 − θ)f1 g1 − z −1 ],

(3.16)

X (z) = q fe(z) + p z −1 fe(z). where hf 0 0 Then the above equation can be rewriting as

f (z) = W =

υp0 q[υp0 p(1 − θ)f1 g1 − z −1 ]     1 − υq0 q + pfe(z)e g(z) − υp0 q + pfe(z)e g(z) z −1 υp0 q[υp0 p(1 − θ)f1 g1 − z −1 ] . 1 − υq0 e h(z, 1) − υp0 e h(z, 1)z −1

(3.17)

Rewriting [1 − υq0 e h(z, 1) − υp0 e h(z, 1)z −1 ]−1 in terms of a power series in z, we have f (z) = υ 2 p2 qp(1 − θ)f1 g1 W 0 −υp0 q

∞ X

k=0

∞ X

k=0

υk e h(z, k)(q0 + p0 z −1 )k

υk e h(z, k)(q0 + p0 z −1 )k z −1 .

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Comparing the coefficients of z u in both sides gives, for u = 1, 2, · · · , b, W (u) = υ 2 p20 qp(1 − θ)f1 g1 −υp0 q =

∞ X

∞ X ∞ X

υ k h(i, k)Cki−u q0k−i+u pi−u 0

i=u k=i−u

∞ X

υ k h(i, k)Cki−u−1 q0k−i+u+1 pi−u−1 0

i=u+1 k=i−u−1 ∞ ∞ XX  υ k+1 p0 qCki q0k−i pi0 υp0 p(1 i=0 k=i

 − θ)f1 g1 h(i + u, k) − h(i + u + 1, k) .

The above result together with (3.14) gives us the explicit expression for V (u; b) as in (3.15). 2

4

Two Classes of Claim Size Distributions

In this section, we consider two special cases for the distribution of X1 + Y1 such that W (u) has a rational generating function which can be easily inverted. One case is that the probability function of X1 + Y1 has finite support such that its p.g.f. is a polynomial, and the other case is that X1 + Y1 has a discrete Kn distribution, i.e., the p.g.f. of X1 + Y1 is a ratio of two polynomials with certain conditions. 4.1 Claim Amount Distributions with Finite Support Now assume that the distribution of X1 + Y1 has finite support, e.g., for N = 3, 4, · · ·, (f ∗ g)x = P (X1 + Y1 = x) = πx ,

x = 2, 3, · · · , N − 1,

where ∗ denotes convolution. Then e h(z, 1) = q + p

N −1 X x=2

z x πx ,

−1 < R(z) < 1

(4.1)

f (z) in (3.17) simplifies to is a polynomial of degree N − 1. Hence W f (z) = W

υp0 q[υp0 p(1 − θ)π2 − z −1 ] h i h i NP −1 NP −1 1 − υq0 q + p z x πx − υp0 q + p z x πx z −1 x=2

x=2

p0 q[1 − υp0 p(1 − θ)π2 z] = . q0 pπN −1 (z − R1 )(z − R2 ) · · · (z − RN ) Let

    N −1 N −1 X X x x DN (z) := z − υq0 z q + p z πx − υp0 q + p z πx , x=2

x=2

−1 < R(z) < 1

(4.2)

is a polynomial of degree N . R1 , R2 , · · · , RN are the N roots of the equation of DN (z) = 0 in the whole complex plane. Further, if R1 , R2 , · · · , RN are distinct, then by partial fractions, we have N p 0 q X ai f W (z) = , q0 pπN −1 i=1 Ri − z

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where ai =

1647

υp0 p(1 − θ)π2 Ri − 1 , i = 1, 2, · · · , N. N Q (Ri − Rj ) j=1,j6=i

f (z) yields Inverting the p.g.f. W W (u) =

N X i=1

p0 qai R−u−1 , q0 pπN −1 i

u = 0, 1, 2, · · · .

(4.3)

 Now V (u; b) = C(b)W (u), for u = 0, 1, 2, · · · , b − 1 and as V (b; b) = q0 V (b; b) + p0 1 +  q0 V (b; b) + p0 V (b − 1; b) , then for u = 0, 1, 2, · · · , b, V (u; b) =

=

p0 W (u) (1 − q0 − p0 q0 )W (b) − p20 W (b − 1) N P p0 ai Ri−u−1 i=1

N P

i=1



 ai (1 − q0 − p0 q0 )Ri−1 − p20 Ri−b

.

(4.4)

Example 1 In this example, we assume f1 = g1 = 1. Then the p.g.f. of W (u) in (3.17), has a simplified expression vp0 q[vp0 p(1 − θ)z − 1] z − vq0 z(q + pz 2 ) − vp0 (q + pz 2 ) vp0 q[1 − vp0 p(1 − θ)z] = . vq0 pz 3 + vp0 pz 2 + (vq0 q − 1)z + vp0 q

f (z) = W

(4.5)

Let R1 , R2 , R3 be the solutions of the equation vq0 pz 3 + vp0 pz 2 + (vq0 q − 1)z + vp0 q = 0. Then by partial fraction, (4.5) can be rewritten as   a1 a2 a3 f (z) = p0 q W + + , q0 p R1 − z R2 − z R3 − z where

a1 =

vp0 p(1 − θ)R1 − 1 vp0 p(1 − θ)R2 − 1 vp0 p(1 − θ)R3 − 1 , a2 = , a3 = . (R1 − R2 )(R1 − R3 ) (R2 − R1 )(R2 − R3 ) (R3 − R1 )(R3 − R2 )

Substituting them into (4.4) gives, for u = 0, 1, 2, · · · b, n o. V (u; b) = p0 a1 R1−u−1 + p0 a2 R2−u−1 + p0 a3 R3−u−1 n a1 [(1 − q0 − p0 q0 )R1−1 − p20 ]R1−b + a2 [(1 − q0 − p0 q0 )R2−1 − p20 ]R2−b o +a3 [(1 − q0 − p0 q0 )R3−1 − p20 ]R3−b .

(4.6)

In Example 1, we assume p0 = 0.45, p = 0.2, υ = 0.75, b = 10, then have R1 = −3.4484353306, R2 = 2.1985922990 and R3 = 0.4316612134. Table 1 summaries the results for V (u; 10) for θ = 0, 0.25, 0.5, 0.75, 1 and u = 0, 1, · · · , 10. The numbers show that the higher the initial surplus of the insurance company, the higher the expected present value of the dividend payments prior to the time of ruin. They also show that V (u; 10) is increasing as the probability of the delay of by-claims is increasing, i.e., θ is decreasing.

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Table 1 V (u; 10)

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Values of V (u; 10) when f1 = g1 = 1

θ=0

0.25

0.5

0.75

1

u=0

0.0008179898

0.0008118982

0.0008058967

0.0007999832

0.0007941559

1

0.0019746123

0.0019736081

0.0019726188

0.0019716439

0.0019706833

2

0.0046954665

0.0046944975

0.0046935428

0.0046926021

0.0046916752

3

0.0109081188

0.0109078266

0.0109075387

0.0109072550

0.0109069755

4

0.0252910744

0.0252908984

0.0252907250

0.0252905542

0.0252903858

5

0.0585975772

0.0585975097

0.0585974431

0.0585973774

0.0585973128

6

0.1357529980

0.1357529635

0.1357529296

0.1357528961

0.1357528632

7

0.3144913279

0.3144913131

0.3144912985

0.3144912842

0.3144912700

8

0.7285613416

0.7285613342

0.7285613268

0.7285613196

0.7285613124

9

1.6878085628

1.6878085584

1.6878085542

1.6878085500

1.6878085459

10

3.9100307850

3.9100307807

3.9100307764

3.9100307722

3.9100307681

Kn Claim Amount Distributions Li [17] studies a class of discrete Sparre Andersen risk models in which the claims interarrival times are Kn distributed. This class of distributions includes geometric, negative binomial, discrete phase-type, as well as linear combinations (including mixtures) of these. For the two independent claim amount random variables X1 and Y1 , if they have Kn distributions, so does their sum. Therefore, in this subsection, we assume that (f ∗ g)x = P (X1 + Y1 = x) is Kn distributed for x = 2, 3, · · ·, and n = 1, 2, · · ·, i.e., the p.g.f. of f ∗ g is given by z 2 En−1 (z) 1 fe(z)e g(z) = Q , R(z) < min{ : 1 ≤ i ≤ n}, (4.7) n qi (1 − zqi ) 4.2

i=1

where 0 < qi < 1, for i = 1, 2, · · · , n and En−1 (z) = with En−1 (1) =

n Q

i=1

f (z) = W

n−1 P k=0

z k ek is a polynomial of degree n − 1

f (z) can be transformed to the following rational function (1 − qi ). Then W [v 2 p20 pq(1 − θ)f1 g1 z − vp0 q]

(1 − vq0 q)z

n Q

(1 − zqi ) − vp0 q

i=1

n Q

i=1

(1 − zqi ) −

n Q

i=1

(1 − zqi )

.

vq0 pz 3 En−1 (z)

−

vp0 pz 2 En−1 (z) (4.8)

Since the denominator of the above equation is a polynomial of degree n + 2, it can be facn+2 Q tored as −vq0 pen−1 (z − Ri ), where R1 , R2 , · · · , Rn+2 are the n + 2 zeros of the denominator. i=1

We remark that −vq0 pen−1 = (−1)n+1 vp0 q/

f (z) = W

 n+2 Y i=1

Ri

n+2 Q i=1

 [1 − vp0 p(1 − θ)f1 g1 z] n+2 Q i=1

f (z) simplifies to Ri . Then W n Q

(zqi − 1)

i=1

(z − Ri )

=

 n+2 Y i=1

Ri

 n+2 X i=1

ri , Ri − z

(4.9)

No.6

1649

J.M. Zhou et al: EXPECTED PRESENT VALUE OF TOTAL DIVIDENDS

where ri =

[vp0 p(1 − θ)f1 g1 Ri − 1] n+2 Q

n Q

j=1

j=1,j6=i

(Ri − Rj )

 n+2 Y

 n+2 X

f (z) gives Inverting W W (u) =

Ri

i=1

(Ri qj − 1)

ri Ri−u−1 ,

i=1

i = 1, 2, · · · , n + 2.

,

u = 0, 1, 2, · · · .

  Now that V (b; b) = q0 V (b; b) + p0 1 + q0 V (b; b) + p0 V (b − 1; b) , then finally we have V (u; b) =

n+2 P i=1

n+2 P i=1

p0 ri Ri−u−1

  ri (1 − q0 − p0 q0 )Ri−1 − p20 Ri−b

,

u = 0, 1, 2, · · · , b.

(4.10)

Example 2 In this example, we assume that the main claim X1 follows a geometric distribution with fx = β(1 − β)x−1 , 0 < β < 1, x = 1, 2, · · ·, and the by-claim Y1 follows a geometric distribution with gx = γ(1 − γ)x−1 , 0 < γ < 1, x = 1, 2, · · ·, so that fe(z)e g(z) =

βγz 2 . [1 − (1 − β)z][1 − (1 − γ)z]

Here n = 2, q1 = 1 − β, q2 = 1 − γ, and En−1 (z) = βγ. Let R1 , R2 , R3 are the three roots of the equation (1 − vq0 q)z[1 − (1 − β)z][1 − (1 − γ)z] − vp0 q[1 − (1 − β)z][1 − (1 − γ)z] − vq0 pz 3 βγ − vp0 pz 2 βγ = 0. Similar to derive (4.9), we have f (z) = R1 R2 R3 [vp0 p(1 − θ)βγz − 1] W

3 X i=1

ri , (z − Ri )

where [(1 − β)R1 − 1][(1 − γ)R1 − 1] [(1 − β)R2 − 1][(1 − γ)R2 − 1] , r2 = , (R1 − R2 )(R1 − R3 ) (R2 − R1 )(R2 − R3 ) [(1 − β)R3 − 1][(1 − γ)R3 − 1] r3 = . (R3 − R1 )(R3 − R2 ) r1 =

f (z) gives Inverting W

W (0) = R1 R2 R3

3 X

ri Ri−1 ,

i=1

and W (u) = R1 R2 R3

3 X i=1

ri [1 − vp0 p(1 − θ)βγRi ]Ri−u−1 , u = 1, 2, · · · .

Then by V (u; b) =

p0 W (u) , (1 − q0 − p0 q0 )W (b) − p20 W (b − 1)

(4.11)

1650

ACTA MATHEMATICA SCIENTIA

Vol.33 Ser.B

we have p0

V (0; b) = R1 R2 R3

3 P

i=1

p0 V (u; b) =

3 P

i=1 3 P

i=1

,

(4.12)

ri [1 − vp0 p(1 − θ)βγRi ][(1 − q0 − p0 q0 ) − p20 Ri ]Ri−b−1 ri [1 − vp0 p(1 − θ)βγRi ]Ri−u−1

ri [1 − vp0 p(1 − θ)βγRi ][(1 − q0 − p0 q0 ) − p20 Ri ]Ri−b−1

, u = 1, 2, · · · , b. (4.13)

As an example, let p0 = 0.55, p = 0.2, v = 0.95, β = γ = 0.8, b = 10, we get R1 = −14.3131552685, R2 = 1.3269447164, R3 = 0.7749429464 and r1 = 0.0632255226, r2 = 0.0625079024, r3 = −0.0857334250. The values of V (u; 10) for θ = 0, 0.25, 0.5, 0.75, 1 and u = 0, 1, · · · , 10 are listed in Table 2. We observe the same features as in Example 1, that V (u; 10) is an increasing function with respect to u, and a decreasing function over θ. Table 2 V (u; 10)

Values of V (u; 10) for geometric distributed claims

θ=0

0.25

0.5

0.75

1

u=0

0.4075929798

0.4020865972

0.3967270086

0.3915084212

0.3864253428

1

0.6143578382

0.6127810510

0.6112462992

0.6097519239

0.6082963525

2

0.9018843041

0.9002832057

0.8987247908

0.8972073750

0.8957293615

3

1.2419717815

1.2407769222

1.2396139165

1.2384815074

1.2373785031

4

1.6618513753

1.6609280350

1.6600293099

1.6591542287

1.6583018704

5

2.1890680068

2.1883452987

2.1876418572

2.1869569220

2.1862897722

6

2.8584135496

2.8578340512

2.8572700016

2.8567207912

2.8561858419

7

3.7138696109

3.7133879266

3.7129190835

3.7124625749

3.7120179202

8

4.8115259631

4.8111049323

4.8106951256

4.8102961002

4.8099074362

9

6.2232590885

6.2228669143

6.2224851950

6.2221135181

6.2217514924

10

8.0414409067

8.0410487325

8.0406670132

8.0402953363

8.0399333105

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