Dividend problems in the dual risk model

Transcription

1 Dividend problems in the dal risk model Lordes B. Afonso, Ri M.R. Cardoso & Alfredo D. gídio dos Reis CMA and FCT, New University of Lisbon & CMAPR and ISG, Technical University of Lisbon, Portgal Abstract We consider the compond Poisson dal risk model, dal to the well known classical risk model for insrance applications, where premims are regarded as costs and claims are viewed as profits. The srpls can be interpreted as a ventre capital like the capital of an economic activity involved in research and development. Like most athors, we consider an pper dividend barrier so that we model the gains of the capital and its retrn to the capital holders. By establishing a proper and crcial connection between the two models we show and explain clearly the dividends process dynamics for the former model, properties for different random qantities involved as well as their relations. Using or innovative approach we derive some already known reslts and go frther by finding several new ones. Varios athors have addressed the model, we start by making a short overview and then go frther. We stdy different rin and dividend probabilities, sch as the calclation of the probability of a dividend, nmber of dividends, expected and amont of dividends as well as the time of getting a dividend. We obtain some integro-differential eqations for the above reslts and also Laplace transforms, then we can get either nmerical or analytical reslts for cases where soltions and/or inversions are possible. Keywords: Dal risk model; classical risk model; rin probabilities; dividend probabilities; disconted dividends; dividend amonts; nmber of dividends. The athors grateflly acknowledge financial spport from FCT-Fndação para a Ciência e a Tecnologia Project reference PTDC/G-CO/18481/28. This work was partially spported by the Fndação para a Ciência e a Tecnologia Portgese Fondation for Science and Technology throgh Pst-O/MAT/UI297/211 CMA. 1

2 1 Introdction We consider in this manscript the dal risk model, as explained, for instance, by Avanzi et al. 27. Using their langage, the srpls or eqity of a company is explained by the eqation, Ut = ct + St, t, 1.1 where is the initial srpls, c is a constant meaning the rate of expenses, {St, t } is a compond Poisson process with parameter λ and density fnction px, x >, of the positive gains, with mean p 1 we therefore assme that it exists. Its distribtion fnction is denoted as P x. The expected increase per nit time, given by µ = [S1] c = λp 1 c, is positive, that is c < λp 1. All these qantities have a corresponding meaning in the well known classical continos time risk model, also known as the Cramér-Lndberg risk model, for insrance applications. For the remainder of or text we will refer this latter model as simply the standard risk model shortly SM. For those sed to work with it we note that the income condition, c < λp 1, is reversed. A few athors have addressed the dal model simply DM, we can go back to Gerber 1979 who called it the negative claims model, see pp , also see Bühlmann 197. We can go even frther back to athors like Cramér 1955, Takács 1967 and Seal Avanzi et al. 27, Section 1, explains well where applications of the dal model are said to be appropriate. We jst retain a simple bt illstrative interpretation, the srpls can be considered as the capital of an economic activity like research and development where gains are random, at random instants, and costs are certain. More precisely, the company pays expenses which occr continosly along time for the research activity and gets occasional profits according to a Poisson process. This model has been recently sed by Bayraktar and gami 28 to model capital investments. Indeed, recently the model has been targeted with several developments, involving the present vale of dividend payments and/or dividend strategies. We nderline the cited work by Avanzi et al. 27 and Avanzi 29, an excellent review paper. Other works are of importance, of which we coment appropriately some lines below. Important financial applications of the model rled by 1.1 are the modeling of ftre dividends of the investments. So, we add an pper barrier, the dividend barrier, noted as b. We refer to the pper graph in Figre 1 [see also Figre 1 of Avanzi et al. 27]. On the instant the srpls pcrosses de barrier a dividend is immediately paid and the process re-starts from level b. We can also consider the case b <, however an immediate dividend is paid and the process starts from b, see Avanzi et al. 27. This makes the sitation less interesting from or point of view. In this manscript we are not interested on strategies of dividend payments bt jst the random amonts, once defined their level b. We will consider the payments either disconted or not. Several papers have been pblished recently sing this model with an pper dividend barrier, where the calclation of expected amonts of the disconted paid dividends is targeted. Higher moments have also been considered. See Avanzi et al. 27, Avanzi and Gerber 28, Cheng and Drekic 28, Gerber and Smith 28, Ng 29 and Ng 21. Yang and Zh 28 compte bonds for the rin probability. Song et al. 28 consider Laplace transforms for the calclation of the expected dration of negative srpls. Cheng 211 also deals with negative srpls excrsions related problems. 2

3 For those works as well as in ors where the dividend barrier b is the key point, it is important to emphasize two aspects: we are going to consider two barriers, one reflecting and another absorbing, the dividend barrier b and the rin level, respectively. In the case of the pper barrier b the process restarts at level b if this is overtaken by a gain. As mentioned above, this is becase an immediate amont of srpls in excess of b is paid in the form of a dividend, it s a pay-back capital. It is not the case with the rin level which makes the process to die down. Indeed, this happens with probability 1 we ll come back to this isse later in the text. To achieve a payable dividend the process mst not be rined previosly. Frthermore, nder the conditions stated the process, sooner or later, will reach one of the two barriers, we mean, with probability one the process reaches a barrier. In this paper we focs on the connection between the SM and the DM, and based on this we work on nknown problems, however having present some known reslts from a different viewpoint, which in some cases have interesting interpretations. We will nderline these points apropriately. We base or research on the insights and ideas known from the classical risk model. This is a key point for or research. We first do a brief srvey of the known reslts from the literatre, then we make important connections between the classical and the dal model featres. Afterwards, we make or own developments. We consider important that known reslts can be looked from or point of view so that or frther developments are better taken and nderstood. Let s now consider some of the basic definitions and notation for the dal risk model, those which we address throghot this paper. Some specific qantities we will define and denote on the appropriate section only. First, consider the process as driven by qation 1.1, free of the dividend barrier. Let τ x = inf {t > : Ut = U = x}, be the time to rin, this is the sal definition for the model free of the dividend barrier τ x = if Ut t. Let [ ] ψx, δ = e δτ x Iτ x < U = x, where δ is a non negative constant. ψ, δ is the Laplace transform of time to rin τ x. If δ = it redces to the probability of ltimate rin of the process free of the dividend barrier, when δ > we can see ψ, δ as the present vale of a contingent claim of 1 payable at τ x, evalated nder a given valation force of interest δ [see Ng 21]. Let s now consider an pper level b in the model, see the pper graph of Figre 1, we don t call it yet a dividend barrier. Let T x = inf {t > : Ut > b U = x} be the time to reach an pper level b x for the process which we allow to contine even if it crosses the rin level. De to the income condition T x is a proper random variable since the probability of crossing b is one. Dividend will only be de if T x < τ x and rin will only occr prior to that pcross otherwise. Whenever we refer to conditional random variables, or distribtions, we will denote them by adding a tilde, like T x for T x T x < τ x. Let s now introdce into the model the barriers b, as a dividend barrier, and the rin barrier, respectively reflecting and absorbing, sch that if the process isn t rined it will 3

4 reach the level b. Here, an immediate dividend is paid by an amont in excess of b, the srpls is restored to level b and the process resmes. We will be mostly working the case < b. Let denote the probability of reaching b before rin occrring, for a process with initial srpls, and ξ, b = 1 χ, b. We have χ, b = Pr T < τ. Becase of the existence of the barrier b ltimate rin has probability 1. The rin level can be attained before or after the process is reflected on b. Then the probability of ltimate rin is χ, b + ξ, b = 1. Let D = {UT b and T < τ } be the dividend amont and its distribtion fnction be denoted as G, b; x = PrT < τ and UT b + x, b with density g, b; x = d dxg, b; x. G, b; x is a defective distribtion fnction, clearly G, b; = χ, b. We refer now to the pper graph in Figre 1. If the process crosses b for the first time before rin at a random instant, say T 1, then a random amont, denoted as D 1 is paid. The process repeats, now from level b. The random variables D i and T i, i = 1, 2,..., respectively dividend amont i and waiting time ntil that dividend, make a bivariate seqence of independent random variables { } T i, D i i=1. We mean, D i and T i are dependent in general bt D i and T j, i j, are independent. This follows from the Poisson process properties. This is well known from the classical Cramér-Ldberg risk model. Frthermore, if we take the sbset { } T i, D i we have now a seqence of independent i=2 and jointly identically distribted random variables and independent of the T 1, D 1, the bivariate random variables only have the same joint distribtion if = b. To simplify notations we set that T i, D i is distribted as Tb, D b, i = 2, 3,..., and T 1, D 1 is distribted as T, D. Let M denote the nmber of evental dividends of the process. Total amont of disconted dividends at a force of interest δ > is denoted as D, b, δ and D, b = D, b, + is the ndisconted total amont. Their n-th moments are denoted as V n ; b, δ and V n ; b, respectively. For simplicity denote as V ; b, δ = V 1 ; b, δ. The prpose of this work is to find new reslts for the different qantities of interest arond the dividend problem in the dal risk model as well as to give a new insight for already known reslts. A key in or work is the interface we establish between the SM and the DM. Despite the reversed income condition, many qantities can be characterized throgh featres well known from the literatre regarding the SM. For instance, the single dividend amont random variable can be viewed as the severity of rin in the SM, althogh adapted to allow a second barrier. Or contribtion in this work can be split into three different levels. The first contribtion is the new insight we give by connecting the DM and the SM. Althogh this is basically a tool for later developments it is of interest. This is done in Section 3. On a second stage, we develop a new insight and new formlae for an already known problem: expectations for the disconted ftre dividends. The way we do allow s to find separate formlae for the different stages of dividends, nlike the known formlae from the literatre where they appear as an aggregate. This is done in Section 3. On a 3rd level we develop expressions for new qantities sch as probability of getting a dividend, nmber of dividends, amont of a single dividend. The first two qantities are stdied in Section 5 whereas Section 6 deals 4

5 with the single dividend amont. All of this is done by connecting both the above models. Before, in Section 2 we make an overview of the literatre and reslts for the model. The last Section is devoted to work illstrative examples. 2 Paper review and reslts We present here known reslts from the literatre particlarly those related to or developments. We are interested on working on the different random variables defined in the previos section and expectations on dividends. It is not or concern dividend strategies, so we omit findings related. Using a martingale argment Gerber 1979 showed that the rin probability is given by ψ := ψ, = e R, 2.1 where R is the niqe positive root of the eqation λ e Rx pxdx 1 + cr =. 2.2 We can se a standard probabilistic argment instead, we show it here as the method is going to be sed later in the text for other prposes. If there are no gains ntil t = /c rin level is crossed. By considering whether or not a gain occrs before time t, we have ψ = e λt + t making s = ct and rearranging we get ce λ c ψ = c + λ e λ s c Differentiating with respect to we get λ e λ t pxψ ct + x dxdt, ce λ λ c ψ c + ceλ d c d ψ = λeλ c pxψs + x dxds. from which we get the following integro-differential eqation λψ + c d d ψ = λ pxψ + x dx, pxψ + x dx, 2.3 with the bondary conditions ψ = 1 and ψ =. Now, it s easy to set that ψ + x = ψψx since rin to occr from initial level + x mst first cross level and from there get rined de to the independent increments property of the Poisson process. Hence, c d ψ d = λψ pxψxdx 1 d d log ψ = λ A 1 c ψ = e λ c A 1, 5

6 where A = pxψxdx A is not dependent on. If we set R = λ c A 1, then we get 2.1. Ng 29 generalized the above probability for a positive δ, ψ, δ, which is given by ψ, δ = e R δ, where R δ is the niqe positive root sch that λ e Rδx pxdx 1 + cr δ = δ. Reslts below concern expections, moments, for disconted dividends by integro-differential eqations which in some cases shown can be solved analytically. As far as expectations of total dividends is concerned, sing a direct and standard approach, by considering possible single gains/jmps events over a small time interval, Avanzi et al. 27 fond that V ; b, δ = b + V b; b, δ if > b, and = cv ; b, δ + λ + δv ; b, δ λ λ b b V + y; b, δpydy 2.4 b + ypydy λv b; b, δ [1 P b ], if < < b, noting that V ; b, δ =, since rin is immediate if =. Besides, Avanzi et al. 27 fond soltions for eqation 2.4 when exponential or mixtres of exponential gains size distribtions are considered. Ng 21 shows soltions when individal gains are phasetype distribted. For higher moments of disconted dividends, Cheng and Drekic 28 with a similar procedre show integro-differential eqations similar to 2.4 V n ; b, δ = n j= n b n j V j b; b, δ if > b, and j = cv n; b, δ + λ + nδv n ; b, δ λ λ n j= n V j b; b, δ j + b b V n + y; b, δpydy 2.5 y b + n j pydy, if < < b. as well as soltions for combinations of exponentials distribted gains size and for jmp size distribtions with rational Laplace transforms. Also, they work an approximation method. We also can get the above reslts sing the approach sed for getting Connecting the classical and the dal model In or frther developments it s crcial the connection between the classical and the dal model as we want to transport methods and reslts from the first to the second, which has had extensive treatment. So, we pt aside together both models, at a first stage we consider 6

7 the models free of barriers. As widely known, the standard Cramér-Lndberg risk model is rled by the eqation U t = + c t S t, t, 3.1 where the qantities involved have similar characteristics althogh different interpretations to those corresponding to the dal model. To emphazise that we denote the corresponding qantities with the same letter bt coming with a. Apart form their application and interpretation, an important difference between the two models comes with the income condition as noted in the first paragraph of Section 1. We recall that in the SM model that condition comes expressed as c > λ p 1, which is reversed in or DM model. In the DM this conditions assres that the srpls ltimately tends to infinity with probability one if no barriers are added. The reversed condition in the DM is intended to achieve the same target, if it wasn t so the DM wold be of diffi clt application, investers woldn t get dividends as they cold wish. To compare and relate both models we have to set the DM income condition on both now. If we first consider first the model withot barriers we can relate the two models in the following way U t = + ct St = b + ct St, t, b >. 3.2 Now, let s get back to or dividend problem and pt the barriers back in order to establish the wanted relation between these two models. We refer to Figre 1. In the DM we consider it with an pper dividend barrier and a rin barrier. The first is reflecting and the second is absorbing. In the corresponding positive claims model SM, the corresponding dividend barrier can be seen as the rin barrier of a srpls starting from initial srpls b. The other mentioned barrier sally is not considered in the SM, and it may jst correspond to an pper line at level b. See again Figre 1. As far as the DM is considered, we note that if the rin level wasn t absorbing, i.e., the process wold contine if the rin level was achieved, then the pper level b wold be reached with probability 1, de to the income condition. However, we follow the model defined by Avanzi et al. 27 that we shold only pay dividends if the process isn t rined. Perhaps we cold work with negative capital, bt that is ot of scope in this work [that kind problem is addressed by Cheng 211]. We are only interested working over the set of the sample paths of the srpls process that do not lead to rin. We need to calclate the probability of the srpls process reaches the barrier b before crossing the level zero. This probability does not correspond to the srvival probability, from initial level. Look at Figre 1, pper graph again. If we trn it pside down rotate 18 o and look at it from right to left we get the classical model shape, where level b is the rin { level, } is the initial srpls, becoming b, and the level is a pper barrier. Di i=2 is viewed as a seqence of i.i.d. severity of rin random variables from initial srpls zero and D 1 the independent, bt not identically distribted, severity of rin random variable from intitial srpls b. Similarly, we have that { } T i can be viewed as a seqence i=2 of i.i.d. random variables meaning time of rin from initial srpls, independent of T 1 which in trn represents the time of rin from initial srpls b. The connection between the two models is briefly mentioned by Avanzi 29 Section 3.1, however not clearly. It is implicit here that in the case of the classical model whenever rin occrs, the srpls is replaced at level. 7

8 Ut b τ t D 1 D 2 D3 T 1 T 2 T 3 U t t b b D 1 D 2 D 3 Figre 1: Classical vs dal model 8

9 We need to consider some reslts on the severity of rin expectations on the disconted severity of rin from the classical risk model adapted to allow an absorbing pper barrier b >. The following reasoning follows from Dickson and Waters 24, Section 4, we adapted to consider the barrier b we refer to Figre 1, U t graph. Now, we present some new definitions, valid for this section only. Let Y denote the deficit at rin and T the time of rin given an initial srpls. We denote the defective distribtion of the deficit Y as G ; x with density g ; x [G ; = ψ, no barrier b considered]. Now, define φ n, b, δ = [e δt Y n ], n =, 1, 2,..., as an expectation of a disconted power of the severity of rin φ 1, b, δ is the expected disconted severity of rin. Let t denote the time that the srpls takes to reach b if there are no claims, so that + ct = b. Using the same approach to set Formla 2.1, by conditioning on the time and the amont of the first claim, φ n, b, δ = + t t and by sbstitting s = + ct, φ n, b, δ = 1 c + 1 c λe λt +ct e δt φ n + ct y, b, δpydydt 3.3 λe λt b +ct λe λ+δ b λe λ+δ e δt y ct n pydydt s c s c s s φ n s y, b, δpydyds y s n pydyds. Differentiating and rearranging, we get the following integro-differential eqation, with bondary condition φ n b, b, δ =, d d φ n, b, δ = λ + δ φ c n, b, δ λ φ c n y, b, δpydy λ y n pydy. c 3.4 If we denote by φ n s; b, δ the Laplace transform of φ n, b, δ with respect to after extending the domain of from b to, i.e., we have φ n s; b, δ = e s φ n ; b, δ d c [ s φ n s; b, δ φ n ; b, δ ] = 3.5 λ + δ φ n s; b, δ λ φ n s; b, δ ps λ e s y n py dy d, where ps is the Laplace transform of the density px.the doble integral can be reexpressed by sing reslts in Section 1 of Willmot et al. 25, which are also sed in Cheng and Drekic 28, involving the link between stop-loss moments and eqilibrim distribtions. So, let P n y be the n-th eqilibrim fnction of P y if it exists sch that y P n y = [1 P n 1t] dt, for y, n = 1, 2,... [1 P n 1 t] dt 9

10 where P y = P y. The corresponding density and Laplace transform are given by, denoted as p n y and p n s respectively, p n y = 1 P n 1 y [1 P n 1t] dt, p ns = Using qation 1.5 of Willmot et al. 25, we obtain y n py dy = p n 1 P n = p n p n+1 if the n-th moment p n = [X n ] exists. Since e s y n py dy d = p n p n+1 s then, from 3.5 we have, 1 p n 1 s s [1 P n 1t] dt. [1 P n t] dt, 1 p n s [1 P n 1 t ] dt = p n, s φ n s, b, δ = cφ 1 p n, b, δ λp ns n s. 3.6 cs λ + δ + λ ps φ n, b, δ can be fond sing the bondary condition above. These reslts are going to be sed for comptation of expectations of times and dividends in the dal model, it will become clear in the next section. Finally, we recall the probability that the srpls attains the level b withot first falling below zero, χ, b = δ /δ b, where δ. is the infinite non-rin probability. Its complementary comes ξ, b = 1 χ, b [please see Dickson 25, Section 8.2, for details]. We will be sing similar probabilities in the dal model. 4 A new approach for the expected disconted dividends For b, looking at Figre 1 it s easy to set the present vale of the total dividends in infinite time, or the total disconted dividends amonts, D, b, δ. Its expected vale comes, [ ] V ; b, δ = [D, b, δ] = e δ i ] j=1 T j Di = [e δ i j=1 T j Di = e δt 1 D 1 + e δt 1 i=1 + e δt 1 e δt 2 i=1 e δt 2 D 2 e δt 3 D becase the pairs of random variables are independent T i, D i, i = 1, 2,..., two by two. Note that T i and D i are dependent in general, they have similar properties as time to rin and its severity in the classical case. Besides, T i, D i, i = 2, 3,..., are also identically distribted, i.e. T b, D b d = T i, D i, i = 2, 3,..., also Ti d = Tb, i = 2, 3,... Set T, D d = T1, D 1, we can write V ; b, δ = e δt D + e δt + e δt e δt b = e δt D + e δt e δt b D b 2 e δt b D b +... e δt b D b 1 + e δt e δt b e δt b D b i= i e b δt 4.1

11 Hence e δt b D V ; b, δ = e δt D + e δt b 1 e δt. 4.2 b A similar expression can be fond in Dickson and Waters 24 relating disconted time and severity of rin in the classical model with a dividend strategy. We only need to evalate e δt D, e δt b D b, e δt and e δt b. To compte the above qantitives, and therefore V ; b, δ, we can make se of xpression 3.3 and qation 3.4 at the end of Section 3, φ n b, b, δ = e δt bd n for n =, 1 and = or b. Alternatively, we can invert 3.6. In the simpler case = b, we have d= d T 1, D 1 Tb, D b and T 1 = Tb, and the above formla simplifies to V b; b, δ = e δt bd b 1 e δt. 4.3 b Then we have V ; b, δ = b + e δt bd b 1 e δt b if b, becase V b; b, δ is 4.3. Note that Formla 4.2 can be written as V ; b, δ = e δt D + e δt V b; b, δ, which means that the expected vale of disconted ftre dividends eqals the expected vale of the disconted first dividend pls the disconted dividends from the process restarting from b. Unlike Avanzi et al. 27, V ; b, δ comes as a fnction of the expected disconted first dividend amont, of the Laplace transform of the time to get a first dividend and similar qantities for other ftre dividends and respective times. Then we have to find soltions for all these. For instance, with these formlae we can see the contribtion of the first dividend amont disconted to the global ftre dividends. In the last section we can see some nmerics. Using the same method we can compte higher moments. For instance, if we want to compte the variance of the accmlated disconted dividends we need to compte V 2 ; b, δ. Let Z i be the disconted dividend i so that Z i = e δ i j=1 T j Di. Then, V 2 ; b, δ = [D, b, δ 2 ] = [Zi 2 ] + 2 [Z i Z j ]. i=1 i=1 j=i+1 Using the above properties on the distribtions T b, D b d = T i, D i, i = 2, 3,..., and T, D d = T 1, D 1 as well as independence in the seqence, we can write [Zi 2 ] = e 2δT [Z 2 1] = e 2δT D 2 e 2δT b. i 2 e 2δT b Db 2, i = 2, 3, 4,... 11

12 Hence, with Now, i=1 [Zi 2 ] = e 2δT D 2 + e 2δT e 2δT b Db 2 = e 2δT D 2 i=1 j=i+1 [Z 1 Z j ] = e 2δT D [Z i Z j ] = e 2δT [Z i Z j ] = + e 2δT e 2δT bdb 2 1 e 2δT. b [Z 1 Z j ] + j=2 e δt b D b e 2δT b D b i=2 j=i+1 e δt b D b j 2 e b δt, j 2 e 2δT b for i < j, i = 2, 3,... Then [Z 1 Z j ] = e 2δT D e δt b D b 1 e δt, b j=i+1 j=2 j i+1 e b δt = i=2 j=i+1 k= k e δt 1 b = 1 e δt, b i=2 [Z i Z j ], [Z i Z j ] = e 2δT e 2δT bd b e δt b D b 1 e δt b = e 2δT e 2δT bd b e δt b D b [1 e δt b ] [1 e 2δT b]. i 2 e 2δT b i 2 e δt b j i 1, i=2 i 2 e 2δT b Fnction V 2 ; b, δ can be expressed in terms of φ n, b, δ in the following way: V 2 ; b, δ = φ 2 b, b, 2δ + φ b, b, 2δφ 2, b, 2δ 1 φ, b, 2δ [ φ1 b, b, 2δφ + 2 1, b, δ 1 φ, b, δ + φ b, b, 2δφ 1, b, 2δφ 1, b, δ 1 φ, b, δ1 φ, b, 2δ ]. 4.4 We see that we need to evalate the following six different qantities e 2δT, e 2δT b, e 2δT D, e 2δT b D b, e 2δT b Db 2, e 2δT D 2, apart from those for needed for the first moment e δt D, e δt b D b, e δt and e δt b. If we want to consider jst the total of a finite nmber of expected dividends, denoting as V, b, δ, n, we can compte easily sing 4.1 [ n ] V ; b, δ, n = e δ i j=1 T j Di = i=1 e δt D + e δt n 1 e δt b D b i=1 i e δt b = e δt D + e δt e δt b 1 e δt b n D b 1 e δt, n = 2, 3, b 12

13 and V ; b, δ, 1 = e δt D. The above expression enhances the tility of or approach with new formlae, becase it allows to compte the contribtion of each dividend to the aggregate expectation as given by Avanzi et al On the nmber of dividends As we said at the end of Section 1 to get a dividend is necessary that the process reaches/crosses the level b before rin, which occrs with probability χ, b. The complementary probability is ξ, b. Finding closed forms for ξ, b, or χ, b, isn t as straightfoward as the similar qantities in the classical model referred at the end of Section 3. Using the sal approach, to reach rin level prior to dividend level is possible with or withot a gain at time t : ct =. Then, for < < b: from which we find or t ξ, b = e λt + λξ, b + c d ξ, b = λ d λξ, b + c d ξ, b = λ d b ct λe λt pxξ ct + x, b dxdt, b b pxξ + x, b dx py ξy, b dy 5.1 where the bondary conditions is ξ, b = 1. Setting ξ + x, b = ξ, b xξx, b = ξx, b ξ, b we get Likewise, we can get or λξ, b + c d ξ, b = λξ, b d + c d ξ, b = ξ, b λ d d d log ξ, b = λ c λχ, b + c d χ, b = λ d λχ, b + c d χ, b = λ d b b b b b pxξx, b dx pxξx, b dx 1 pxξx, b dx 1. pxχ + x, b dx + λ [1 P b ] py χy, b dy + λ [1 P b ] where the bondary conditions is χ, b =. We can compte Laplace transforms on qation 5.1 as an alternative method to find ξ, b. We can a se a method of change of variable already sed by Avanzi et al. 27, Section 6, retrieved by Cheng and Drekic 28 and mentioned in the review paper by Avanzi 29. In that eqation replace by z = b and define z, b = ξb z, b = ξ, b. 13

14 This change of variable analytically is like setting the relation between the two models, classical and dal. Note that b, b = ξ, b = 1. The corresponding integro-differential eqation for z, b is λz, b c z z z, b λ pz yy, b dy =. In fnction z, b extend the range of z from z b to z and denote the reslting fnction by ɛz, then compte its Laplace transform, denoted as ɛs, so that Hence, λ ɛs c [s ɛs ɛ] λ ɛs ps =. ɛs = cɛ cs λ + λ ps, 5.2 where ɛ = ξb, b note that ɛb = b, b = ξ, b = 1. When ps is a rational fnction we can invert ɛs to find a soltion for ɛz. Finally ξ, b = ɛb for b. Now let s consider the mltiple dividend sitations and let M, the nmber of dividends to be claimed. Then, Pr [M = ] = ξ, b and Pr[M 1] = 1 ξ, b = χ, b. Besides, Pr[M = 1] = χ, bξb, b, first the process crosses b a dividend is paid, then restarts from b and after that is rined. Pr[M = 2] = χ, bχb, bξb, b, and so on. In smmary, we get Pr[M = ] = ξ, b Pr[M = k] = χ, bχb, b k 1 ξb, b, k = 1, 2,... M follows a zero-modified geometric distribtion if = b we get a geometric distribtion with Pr[M = k] = χb, b k ξb, b, k =, 1, 2,... The total amont of dividend gains not disconted, D, b follows a compond zero-delayed geometric distribtion. 6 On the dividend amont distribtion Now we are going to work on the distribtion of the random variable D, distribtion and density fnctions denoted as G, b; x and g, b; x, respectively see Section 1. First we will set the probability ξ, b and g, b; x. Consider the process free of the barrier b, jst consider b as a fixed level. We can write, considering that rin can occr before or after crossing b, ψ = ξ, b + g, b; xψb + xdx = ξ, b + ψb g, b; xψxdx ξ, b = e R e Rb ḡ, b; R 6.1 setting ψx = e Rx and rearranging, where ḡ, b; R is the Laplace transform of the density g, b; x evalated at R. Back to the sal model, we can compte an integro-differential eqation for G, b; x sing the standard procedre. Conditioning on the first claim we get, where t is sch that ct =, [ t b ct ] b ct+x G, b; x = λe λt pyg ct + y, b; x dy + pydy dt. 14 b ct

15 Rearranging and differentiating with respect to, we obtain the following integro-differential eqation λg, b; x+c G, b; x = λ b with bondary condition G, b; x =. We get easily λg, b; x + c g, b; x = λ py Gy, b; x dy +λ [P b + x P b ], 6.2 b pyg + y, b; x dy + λpb + x. 6.3 We can compte Laplace transforms for G, b; x by methods similar to those sed for 5.2. Let Gz, b; x = Gb z, b; x. Then Gb, b; x = G, b; x =. Then, from 6.2 we get λgz, b; x c z z Gz, b; x λ pz ygy, b; x dy λ [P z + x P z] = Let ρz, x be the correspondent fnction arising from extending the range of z. Laplace transforms we get easily, [ ] ps λ ρs, x c [s ρs, x ρ, x] λ ρs, x ps + λ ˆps, x =, s where Hence, ps s = ρs, x = e sz P z dz e sz ρz, x dz Taking ˆps, x = e sx e sy P y dy. 6.4 ρs, x = x cρ, x + λ [ ps/s ˆps, x] cs λ + λ ps Likewise, for the density g, b; x from 6.3, setting γz, x = gb z, b; x, λγz; x c z z γz; x λ pz yγy; x dy λpz + x = from which we get the Laplace transform for γz, x 6.5 γs, x = cγ, x λ ps, x cs λ + λ ps with Note that ps, x = e sx e sy py dy. x x = γs, x. x ρs, 15

16 Similarly to the classical model with a probability argment we can write G, b; x = g, b; x = b b g, ; yg + y, b; x dy + b +x b g, ; ydy g, ; yg + y, b; x dy + g, ; b + x Let s now consider the process contining even if rin occrs. The process can cross for the first time the pper dividend level before or after having rined. Then we can write the proper distribtion of the amont by which the process first pcrosses b, denoted as H, b; x = Pr [UT b + x] H, b; x = H, b; x T < τ χ, b + H, b; x T > τ ξ, b = G, b; x + ξ, bh, b; x. The second eqation above simply means that the probability of the amont by which the process first pcrosses b is less or eqal than x, eqals the probability of a dividend claim less or eqal than x pls the probability of a similar amont bt in that case it cannot be a dividend. This second probability can be compted throgh the probability of first reaching the level, ξ, b, times the probability of an pcrossing of level b by the same amont x bt restarting from, H, b; x. We can compte H, b; x throgh expressions known for the distribtion of the severity of rin obtained from the positive claims model recall that the income condition is reversed, making it a proper distribition fnction. Then we get the refers to the classical case eqivalent to For = b we have G b ; x = G, b; x + ξ, bg b; x G, b; x = G b ; x ξ, bg b; x. 6.6 Hb, b; x = Gb, b; x + ξb, bh, b; x Gb, b; x = G ; x ξb, bg b; x. From here differentiate and take Laplace transforms and evalate at R we get then se 6.1 to get gb, b; x = g ; x ξb, bg b; x ḡb, b; R = g ; R ξb, bg b; R, ξb, b = [ 1 g ; R ] e Rb 1 g b; Re Rb. g ; x = p 1 1 [1 P x] is the severity density in the positive claims model whose Laplace transform is g ; R = 1 1 e Rx pxdx = c Rp 1 λp 1 sing 2.2. We still need to compte g b; R, clearly, it is not a trivial calclation since it s a Laplace transform of the severity of rin with a positive initial srpls in the positive claims model. If px is exponential then g ; R = g ; R, bt that is the trivial example. 16

17 7 Illstrations 7.1 xponential jmps We consider the case when gain amonts are exponentially distribted, that is py = αe αy, y >, with α >. Using qation 3.4 and then 4.2 we get e δt D n V ; b, δ = λ α = φ n b, b, δ = n! α n λ c e r 2 e r 1 r 1 + αe r 2b r 2 + αe r 1b e r 2 e r 1 cr 1 + α λe r 2b cr 2 + α λe r 1b, where r 1 < e r 2 > are soltions of the eqation s 2 + α λ + δ s αδ c c =. xpression above for V ; b, δ is eqivalent to the one by Avanzi et al. V 2 ; b, δ can be evalated sing 4.4, after some simplification we get 27. Fnction V 2, b, δ = 2 cλ α 2 r 1 + αe r1b r 2 + αe r2b e s2 e s1 cr 1 + α λe r1b cr 2 + α λe r2b cs 1 + α λe s2b cs 2 + α λe s1b where s 1 and s 2 are the roots of the eqation s 2 + α λ + 2δ s 2αδ c c For the comptation of χ, b and ξ, b, from 5.1 we get, where R = λ/c α, χ, b = λ λe R λ αce Rb = ξ, b = λe R αce Rb λ αce Rb. It is mch easier to find ξ, b by sing 5.2. We get ɛ = α + ReR α α + Re Rb α. Ths ξ, b = ɛb. To find the soltion for the distribtion of a single amont of dividend, G, b; x, we can se the Laplace transforms dealt in Section 6. After some algebra, we get ρ, x = 1 e αx [1 + α α + ReR α + Re Rb α ]. Finally we get, G, b; x = ρb, x = 1 e αx λ λe R λ αce Rb. 17

18 Note that we have, for the conditional d.f., denoted as G, b; x, G, b; x = G, b; x χ, b = 1 e αx. We cold get the same sing 6.6. There is a correspondence to the classical model. De to the memoryless property of the exponential distribtion, the conditional distribtion of the single dividend amont has the same distribtion of the single gains amont, conditional on the pcross of level b prior to rin. Consider now the conditional distribtions, given that T < τ. Note that in view of the above we have that where e δt D T n < τ e δt T < τ = λ c = φ nb, b, δ χ, b = D T n < τ λ e r2 e r 1 c r 1 + αe r2b r 2 + αe r 1b = D T n < τ e δt T < τ χ, b e r 2 e r 1 r 1 + αe r 2b r 2 + αe r 1b χ, b xpression 7.1 shows that the conditional variables, given that T < τ, of time to dividend and dividend amont are independent, respectively T and D. This case is the analoge to the classical risk model, between the conditional severity of rin and time to rin, given that rin occrs [see Gerber 1979, for instance]. Hence, xpression 7.2 gives the Laplace transform of the time to dividend. To get some nderstanding abot the figres we can expect for some of the above qantities we show some nmbers We set a sitation where on average we have one jmp per nit time with an average amont of one again. That is λ = α = 1 with c =.75 and δ =.2. We show in Table 7.1 figres for V ; b,.2 for diferent vales of and b. For instance, an investment of three capital nits, three times greater than the average jmp with a dividend barrier of six the doble of the investment is expected to give back a present vale of capital retrn of more than seven nits. It is more than the doble of the invested capital. Bt with higher dividend levels the retrned capital descreases. For more details please see Avanzi et al. 27 as they find optimal vales for the expectation. Table 7.2 shows disconted expected vale of a ftre dividend payment. Note that for = 1 and b = 1 for instance, we get a disconted retrn of abot 45% of the invested capital on the first dividend. Table 7.3 shows figres for the probability of getting a dividend payment, in all cases for the same parameter and argment vales. In the of the last table we add a colmn for the probability of non-rin χ, = 1 ψ. 7.2 Combination of exponentials We se an illstration example also worked by Avanzi et al. 27, where px = 3e 3x/2 3e 3x, x >. 18

19 We consider the same parameter vales like in the previos example. Also, the individal gain sizes have mean one, however this example has qite different featres. So, in addition to the qantities V ; b,.2, e δt D and χ, b like are shown for the previos example, we thoght of some interest to show the qantities e δt and D. We frther show a graph with different plots for the probability density fnction of the first dividend amonts g, 1, x, for some given vales of the initial srpls see Figre 2. Table show vales for V ; b,.2, e δt D, e δt, D and χ, b, respectively. In particlar, e δt means the average disconted amont of a first dividend of a one nit amont, and it gives the idea of how mch a dividend of one nit is disconted, on average. D is the ndisconted average dividend. For this example, when compared to the previos one, respective vales for the probability of a first dividend χ, b are higher. References Avanzi, B. 29. Strategies for dividend distribtion: A review, North American Actarial Jornal 132, Avanzi, B. and Gerber, H.U. 28. Optimal dividends in the dal model with diffsion, ASTIN Blletin, 38 2, Avanzi, B., Gerber, H.U. and Shi,.S.W. 27. Optimal dividends in the dal model, Insrance: Mathematics and conomics, 41 1, Bayraktar,. and gami, M. 28. Optimizing ventre capital investment in a jmp diffsion model, Mathematical Methods of Operations Research 67 1, Bühlmann, H Mathematical Methods in Risk Theory, Springer Verlag, New York. Cramér, H Collective Risk Theory: A Srvey of the Theory from the Point of View of the Theory of Stochastic Process, Ab Nordiska Bokhandeln, Stockholm. Cheng,.C.K A nifying approach to the analysis of bsiness with random gains, Scandinavian Actarial Jornal, forthcoming, available from Cheng,.C.K. and Drekic, S. 28. Dividend moments in the dal risk model: exact and approximate approaches, ASTIN Blletin 38 2, Dickson, D.C.M. 25. Insrance Risk and Rin, International series on Actarial Science, Cambridge University Press. Dickson, D.C.M. and Waters, H.W. 24. Some optimal dividends problems, ASTIN Blletin 34 1, Gerber, H.U An Introdction to Mathematical Risk Theory, S.S. Hebner Fondation for Insrance dcation, University of Pennsylvania, Philadelphia, Pa. 1914, USA. Gerber, H.U. and Smith, N. 28. Optimal dividends with incomplete information in the dal model, Insrance: Mathematics and conomics 43 2,

20 Ng, A.C.Y. 29. On a dal model with a dividend threshold, Insrance: Mathematics and conomics 44 2, Ng, A.C.Y. 21. On the pcrossing and downcrossing probabilities of a dal risk model with phase-type gains, ASTIN Blletin 4 1, Seal, H.L Stochastic Theory of a Risk Bsiness, Wiley, New York Song, M., W, R. and Zhang, X. 28. Total dration of negative srpls for the dal model, Applied Stochastic Models in Bsiness and Indstry 24 6, Takács, L Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York. Willmot, G.., Drekic, S. and Cai, J. 25. qilibrim compond distribtions and stoploss moments, Scandinavian Actarial Jornal, Yang, H. and Zh, J. 28. Rin probabilities of a dal Markov-modlated risk model, Commnications in Statistics, Theory and Methods 37, Lordes B. Afonso Ri M.R. Cardoso Alfredo D. gídio dos Reis Depart. de Matemática Depart. de Matemática Department of Management and CMA and CMA ISG and CMAPR Facldade Ciências e Tecnologia Facldade Ciências e Tecnologia Technical University of Lisbon Universidade Nova de Lisboa Universidade Nova de Lisboa Ra do Qelhas Caparica Caparica Lisboa Portgal Portgal Portgal lbafonso@fct.nl.pt rrc@fct.nl.pt alfredo@iseg.tl.pt 2

21 Table 7.1: Vales for V ; b,.2 for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b Table 7.2: Vales for e δt D for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b

22 Table 7.3: Vales for χ, b for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b Table 7.4: Vales for V ; b,.2 for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b Table 7.5: Vales for e δt D for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b

23 Table 7.6: Vales for e δt for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b Table 7.7: Vales for D for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b Table 7.8: Vales for χ, b for =1, 3, 5, 1, 15, 2; b =2, 3, 6, 1, 3, 4 b

24 Figre 2: g, 1; x, p.d.f.of the first dividend. 24