Model Uncertainty within the Tweedie Exponential Dispersion Family

Transcription

1 Model Uncertanty wthn the Tweede Exponental Dsperson Famly Danel H. Ala 1 Maro V. Wüthrch 2 Department of Mathematcs, ETH Zurch, 8092 Zurch, Swtzerland Conference Paper January 13, 2009 Abstract The use of generalzed lnear models GLM to estmate clams reserves has become a standard method n nsurance. Most frequently, the exponental dsperson famly EDF s used; see England, Verrall [2]. We study the so-called Tweede EDF and test the senstvty of the clams reserves and ther mean square error of predctons MSEP over ths famly. Furthermore, we develop approxmatons for the clams reserves and the MSEPs for members of the Tweede famly that are dffcult to obtan n practce, but are close enough to models for whch clams reserves and MSEP estmatons are easy to determne. As a result of multple case studes, we fnd that clams reserves estmaton s relatvely nsenstve to whch dstrbuton s chosen amongst the Tweede famly, n contrast to the MSEP, whch vares wdely. Keywords: Clams Reservng, Exponental Dsperson Famly, Model Uncertanty, Power Varance Functon, Tweede s Exponental Dsperson Models, Predcton Error. 1 danel.ala@math.ethz.ch 2 maro.wuethrch@math.ethz.ch 1

2 1 Introducton The use of generalzed lnear models GLM n actuaral scence s well developed and broadly accepted. Not only does the framework of GLM allow for flexblty n parameter and model selecton, n some cases, such as wth the chan ladder method, GLM recovers tradtonal methods for clams reserves estmaton. For a comprehensve reference of GLM, see McCullagh, Nelder [10]. In ths paper, we study the exponental dsperson famly EDF and ts role n modellng clams reserves; see Jørgensen [5], [6] for more on the EDF and Renshaw [15], Haberman, Renshaw [4], England, Verrall [2] and Wüthrch, Merz [20] for applcatons to nsurance. We focus on a specal member of the EDF, the so-called Tweede exponental dsperson model. Besdes contanng many standard models, such as the Gaussan, Posson and Gamma, we are partcularly nterested n the compound Posson models; Mllenhall [11] provdes an excellent revew of these models. For specfc applcatons of the Tweede compound Posson model, see for nstance Jørgensen, De Souza [7], Smyth, Jørgensen [16], and Wüthrch [19]. The famly of Tweede exponental dsperson models s controlled by a model parameter p. For example, p = 1 corresponds to the overdspersed Posson model. In ths paper, we calculate the senstvty of the clams reserves wth respect to ths model parameter. Peters et al. [14] and Ggante, Sgalott [3] have also tackled ths ssue of model uncertanty, the former optng for a Bayesan Markov Chan Monte Carlo smulaton approach averagng over p, and the latter addressng the ssue wthn a GLM framework, usng an teratve procedure to solve for p usng quas-lkelhood functons as ntroduced by Wedderburn [18] and Nelder, Pregbon [12]. In our approach, we drectly work wth the lkelhood functon and rather than solve for p, we fnd the clams reserves n terms of p,.e. for a fxed dstrbutonal model. Besdes clams reserves senstvty, we also nvestgate the senstvty of the mean square error of predcton MSEP wth respect to p. We conclude that, based on multple case studes, the clams reserves are rather nsenstve to the choce of p and hence fnd that there s only moderate model uncertanty 2

3 when modellng wthn the Tweede exponental dsperson famly. In contrast however, we fnd that the MSEP s hghly senstve to the model parameter p. Ths has mportant consequences for solvency consderatons and the requred rsk bearng captal. Organsaton of the paper. In Secton 2 we descrbe the data and the model assumptons. Maxmum lkelhood estmaton MLE of the underlyng model parameters s dscussed n Secton 3. In Sectons 4 and 5 we study the senstvty of the clams reserves and the MSEP, respectvely, wth respect to the model parameter p. In Secton 6, we provde one of the earler mentoned case studes to hghlght the performance of the approxmatons derved prevously. 2 Data and Model 2.1 Setup Let X, denote the ncremental payments of accdent year and development year, {0, 1,..., I} and {0, 1,..., J}. We assume that the data s gven by a clams development trangle,.e. I = J. Ths means that our data are gven by an upper trangle, D I = {X,, + I}, and that we are nterested n predctng the ncremental payments for the correspondng lower trangle {X,, + > I} at tme I. See Fgure 1 for a graphc representaton of the data. The outstandng loss labtles are gven by R = X,. These are the future cashflows at tme I. We are gong to predct these outstandng loss labltes, R, wth a predctor ˆR, the so-called clams reserves, that s based on the nformaton D I avalable at tme I. 2.2 The Exponental Dsperson Famly Nelder, Wedderburn [13] establshed the framework of GLM and the so-called analyss of devance. These concepts were orgnally developed for exponental famles of 3

4 accdent development year year J 0 realzatons of. r.v. X,, + I. predcted r.v. X,, I + > I Fgure 1: Clams development trangle. dstrbutons, yet extended to a wder class of dstrbutons, termed dsperson models. Here, we work wthn the framework of ths broader famly of dstrbutons but focus on an mportant sub-class called the Tweede exponental dsperson models, ntroduced n Tweede [17]. Model Assumptons 2.1 Exponental Dsperson Model A random varable X, follows an exponental dsperson model f t has generalzed densty f x; θ,, φ {, w, = exp x θ, bθ, } c x; φ,, w, φ, w, where w, > 0 denotes a known weght, θ, s the canoncal parameter, φ, > 0 the dsperson parameter. The functon b s a twce dfferentable general functon that determnes the more specfc famly the dstrbuton falls nto and c s a sutable normalzng constant. Ths generalzed densty s ether defned wth respect to Lesbegue measure or the countng measure. Moreover, the doman of x depends on the choce of b. As noted above, the functon b determnes to whch specfc famly the exponental dsperson dstrbuton belongs. Lkewse, one can specfy the structure of the underlyng unt varance functon, V, defned as, V µ = b b 1 µ; see Jørgensen [6], Theorem We focus on unt varance functons of the power varety, namely V µ = µ p for p, 0] [1, ; see Jørgensen [6], Propos- 4

5 ton 4.2, regardng the possble values of p. The famly so defned s known as the Tweede EDF. Specfc values of p correspond to specfc dstrbutons, for example when p = 0, 1, 2, 3, we recover the Gaussan, Posson, Gamma, and nverse Gaussan dstrbutons, respectvely. As such, the parameter p plays a central role of model uncertanty for dstrbutons wthn the Tweede class. Model Assumptons 2.2 Tweede Exponental Dsperson Model A random varable X, follows a Tweede exponental dsperson model f t s an exponental dsperson model wth parameter θ, Θ p and functon b defned as b p θ, = 1 2 p 1 pθ, 2 p 1 p, p / 0, 1] [2], b 1 θ, = exp θ,, p = 1, b 2 θ, = log θ,, p = 2, where R, for p = 0, 1, [0,, for p < 0, Θ p =, 0, for 1 < p 2,, 0], for 2 < p <. The specfcaton on b s made so that the unt varance functon, V, has a power structure wth power p, that s, b p mples that V µ = µ p. Under these assumptons, X, has expectaton and varance gven by E[X, ] = m, = b pθ,, VarX, = φ, w, b pθ, = φ, w, V m, ; see, for example, Bühlmann, Gsler [1], Theorem 2.2. In the remander of ths paper we assume that φ, w, s constant and defne φ = φ, w,. Ths assumpton mples that φ cancels n the MLE of θ,, whch substantally smplfes the analyss. 5

6 3 Maxmum Lkelhood Estmators 3.1 Clams Reserves Under Model Assumptons 2.2 wth φ = φ, w,, we estmate the model parameters usng MLE. The log-lkelhood functon for D I s gven by lθ,, φ = l DI θ,, φ = 1 φ X,θ, b p θ, + log cx, ; φ. + I At the moment we have I unknown parameters θ, and only I + 1I + 2/2 observatons. Therefore we ntroduce addtonal model structure to obtan a more parsmonous model. As s standard n clams reservng modellng, we choose a multplcatve model m, = µ γ, where µ > 0 s the exposure of accdent year and γ > 0 descrbes the clams development pattern. Ths mples that the canoncal parameter, θ, = b p 1 m,, s gven by θ, = log µ γ, p = 1, θ, = µ γ 1 p, p 1. 1 p The MLEs of the parameters µ and γ can now be obtaned by settng the followng system of equatons equal to zero: µ lθ,, φ = 1 φ γ lθ,, φ = 1 φ I =0 I =0 X, θ, µ b pθ, µ X, θ, γ b pθ, γ, {0,..., I},, {0,..., J}. We further need to ntroduce a constrant to obtan a unque soluton to these equatons. Usng µ 0 = 1, we obtan the MLEs see, for example, Wüthrch, Merz [20], formulas 5.49 and 5.50, ˆµ I =0 I ˆγ =0 ˆγ 2 p = ˆµ 2 p = I x,ˆγ 1 p, {1,..., I}, =0 I =0 ˆµ 0 = 1. x, ˆµ 1 p, {0,..., J}, 1 6

7 Note that for p = 1, the above system corresponds to the overdspersed Posson model, whch yelds the chan ladder clams reserves see Mack [9] or Lemma 2.16 n Wüthrch, Merz [20]. From the equatons gven n 1, t s clear that ˆµ and ˆγ are functons of p and consequently, model uncertanty wthn the Tweede famly may be expressed n terms of ther dervatves wth respect to p. Note that n 1, φ cancels and consequently has no nfluence on the parameter estmaton of µ and γ. However, an estmate of φ s requred to estmate the predcton uncertanty. We could estmate φ wth MLE but ths nvolves an nfnte summaton that s often dffcult to evaluate; see Peters et al. [14]. Therefore, we prefer usng Pearson resduals for the estmaton of φ. Furthermore, Pearson resduals are standard outputs n all GLM software tools and are wdely accepted n practce. We obtan the followng estmate: ˆφ = 1 d + I X, ˆm, 2, V ˆm, where d = I+1I+2 2 2I 1 s the degrees of freedom of the model and ˆm, = ˆµ ˆγ. Wth these parameter estmates, we can predct the outstandng loss labltes, R, wth the clams reserves, ˆR, gven by ˆR = ˆµ ˆγ. The clams reserves ˆR = ˆRp depend on p, our am s a senstvty analyss n p. 3.2 Asymptotc Propertes of the MLE The proposton below yelds the asymptotc behavour of MLEs; see, for example, Lehmann [8], Theorem Proposton 3.1 Assume X 1,..., X n are..d. wth densty f ζ from the exponental dsperson famly, wth parameters ζ = ζ 1,..., ζ m T. Furthermore, ˆζ = ˆζ 1,..., ˆζ m T s the MLE of ζ, 7

8 then, d n ˆζ ζ N 0, Hζ 1, as n, where we defne the Fsher nformaton matrx by Hζ = Hζ r,s=1,...,m wth, [ ] 2 Hζ r,s = E ζ logf ζ X. ζ r ζ s In our case we use of the notaton ζ = ζ 1,..., ζ 2I+1 T = µ 1,..., µ I, γ 0,..., γ I T. Before dervng the Fsher nformaton matrx, H, we provde the followng necessary partal dervates: θ, = µ p µ θ, γ γ 1 p, = µ 1 p γ p, b p θ, = µ 1 p µ b p θ, γ γ 2 p, = µ 2 p γ 1 p. From these, and the dervatves of the log-lkelhood functon wth respect to the underlyng parameters, we obtan Hζ r,r = ζ p r φ Hζ s,s = ζ p s φ I r s=0 2I+1 s r=1 ζ 2 p I+1+s, r {1,..., I}, ζ 2 p r, s {I + 1,..., 2I + 1}, 2 Hζ r,s = 1 φ ζ1 p r ζs 1 p, r {1,..., 2I + 1}, s {1,..., 2I + 1 r}, r, s / {1,..., I} {1,..., I}. The remanng entres of the 2I + 1 2I + 1 matrx H are zero. Note that we omt ˆµ 0 n our constructon of H because ts ncluson would mply H to be sngular. We estmate the MSEP usng the above result. Note that H depends on φ and ζ. By replacng these parameters by ther estmates we obtan the estmated Fsher nformaton matrx Ĥ = Ĥˆζ, whch s a functon of p. Of specfc mportance are the estmates of the covarances of the MLEs, ˆζ. Proposton 3.1 provdes the followng estmator, Ĉovˆζ r, ˆζ s = Ĥˆζ 1 r,s, r, s {1,..., 2I + 1}. 3 Before studyng the effect of the model parameter p on the MSEP, we frst study the senstvty of the clams reserves wth respect to the choce of p. 8

9 4 Senstvty of the Clams Reserves wth respect to p As stated n prevous sectons, assumng a dstrbuton from the Tweede EDF, we can estmate the parameters µ and γ, from whch we can predct the outstandng loss labltes R wth the clams reserves ˆR = ˆRp. We analyze the senstvty of ˆR wth respect to p by a Taylor expanson. The second order Taylor expanson around p s gven by wth, ˆR p + ε = ˆRp + ˆR p ε + ˆR p ε 2, 4 2 ˆR p = ˆR p = ˆµ ˆγ + ˆµ ˆγ, ˆµ ˆγ + 2ˆµ ˆγ + ˆµ ˆγ. The frst and second dervatves are provded n Lemmas 4.1 and 4.2 below. 4.1 Reserves Approxmaton usng Frst Order Taylor Expanson We begn by studyng the frst order Taylor expanson, gven by omttng the last term n 4. To approxmate the clams reserves usng a frst order Taylor expanson, we need to calculate the frst dervatves wth respect to p of the MLEs ˆζ. Dfferentatng the equatons gven n 1 wth respect to p provdes: I I ˆµ ˆγ 2 p ˆγ + ˆγ p =0 =0 I I ˆγ ˆµ 2 p ˆµ + ˆµ p =0 =0 ˆµ 0 = 0. [2 pˆµ ˆγ 1 px, ] = I =0 ] logˆγ ˆγ [ˆµ ˆγ p ˆγ x,, {1,..., I}, 5 [2 pˆγ ˆµ 1 px, ] = I =0 ] logˆµ ˆµ [ˆγ ˆµ p ˆµ x,, {0,..., J}, 9

10 To solve the above system of equatons we defne a 2I +2 2I +2 matrx A, whose components are the followng: a, = I =0 a,i++1 = 1ˆγ p a I++1, = 1ˆµ p a I++1,I++1 = I =0 a 0,0 = 1. ˆγ 2 p, {1,..., I}, 2 pˆµ ˆγ 1 px,, {0,..., I}, {0,..., I }, 2 pˆµ ˆγ 1 px,, {0,..., I}, {0,..., I }, ˆµ 2 p, {0,..., I}, The remanng entres of the matrx are zero. In addton to the matrx A, we defne column vectors ˆζ = ˆµ 0,..., ˆµ I, ˆγ 0,..., ˆγ I T and α = 0, α 1,..., α I, β 0,..., β I T, where α = β = I =0 I =0 logˆγ ˆγ ˆγ p ˆµ ˆγ x,, logˆµ ˆµ ˆµ p ˆµ ˆγ x,, Usng matrx notaton, we rewrte equatons gven n 5 as Aˆζ = α. {1,..., I}, {0,..., I}. Lemma 4.1 The frst dervate of the MLE ˆζ s gven by ˆζ = A 1 α, where A and α are defned as gven above. In the followng subsecton, we study the second order approxmaton. 4.2 Reserves Approxmaton usng Second Order Taylor Expanson What remans to be provded n order to use the second order Taylor approxmaton s the second dervatve of the MLE ˆζ wth respect to p. Rather than dfferentat- 10

11 ng equatons gven n 5, we frst rewrte them usng smplfyng notaton already ntroduced above for matrx A. We have I ˆµ a, + ˆγ a,i++1 = α, {1,..., I}, =0 I ˆγ a I++1,I++1 + ˆµ a I++1, = β, {0,..., I}, Takng dervates we obtan: where ˆγ ˆµ =0 I a, + =0 I a I++1,I++1 + =0 I κ = α a,ˆµ ˆγ a,i++1, =0 ˆµ 0 = 0. ˆγ a,i++1 = κ, {1,..., I}, ˆµ a I++1, = λ, {0,..., I}, 6 ˆµ 0 = 0, I {1,..., I}, λ = β a I++1,I++1ˆγ ˆµ a I++1,, {0,..., I}. =0 Hence, we need to fnd the dervates of a,, a,i++1, a I++1, a I++1,I++1, α, and β wth respect to p. They are gven n Appendx A. We defne column vectors, ˆζ = ˆµ 0,..., ˆµ I, ˆγ 0,..., ˆγ I T formulate the equatons gven n 6 as where A s as prevously defned. and κ = 0, κ 1..., κ I, λ 0,..., λ I T, so that we can Aˆζ = κ, Lemma 4.2 The second dervate of the MLE ˆζ s gven by ˆζ = A 1 κ, where A and κ are defned as gven above. 11

12 Remark 4.3 ˆµ k Of course ths can nductvely be expanded to any other dervatves and ˆγ k, k 3, where the rght-hand sdes n equatons gven n 6 become approprate functons dependng on ˆµ l and ˆγ l, l < k. 5 Senstvty of the MSEP wth respect to p Before studyng the dervate of the MSEP wth respect to p, we need to estmate the MSEP. The condtonal MSEP for predctor ˆR of the outstandng loss labltes R s defned as follows, msep R DI ˆR = E [ ˆR ] [ R 2 DI = E ˆµ ˆγ X, 2 DI ]. Snce the predctor ˆµ ˆγ s D I measurable, we decompose t nto terms referred to as condtonal process varance and condtonal estmaton error; see Wüthrch, Merz [20], Secton 3.1. We get the followng decomposton: msep R DI ˆR = Var DI X, + ˆµ ˆγ E[X, D I ] 2. Furthermore, due to the ndependence assumptons on X,, we obtan msepp = msep R DI ˆR = VarX, + ˆµ ˆγ E[X, ] 2. Stayng wthn the framework of Tweede s EDF, we see that as wth the clams reserves, the estmate of the MSEP depends on the model parameter p. Most often n presentng results, the square root of the MSEP s gven, we also follow ths conventon. Denotng the estmators for the process varance by pvp and for the estmaton error as êep, we decompose the estmated condtonal MSEP as follows: where msep 1 2 p 1 = pvp + êep, 2 msep 1 2 p msep 1 2 p = pvp + êep The frst order Taylor expanson of msep 1 2 p around p s gven by 1 2 msep 1 2 p + ε = msep 1 2 p + msep 1 2 p ε

13 The dervates of the estmators of the process varance and the estmaton error are provded below n Lemma 5.1 and 5.2. Note that we can also fnd the second order Taylor expanson, msep 1 2 p + ε = msep 1 2 p + msep 1 2 p ε + msep 1 2 p ε 2 2 ; 8 ths requres the calculaton of pvp and êep, whch s rather nvolved. Some formulas requred for the second order approxmaton are gven n Appendx C. We hghlght the performance of the second order approxmaton n the case study of Secton Process Varance Determnng an estmate of the process varance s relatvely easy. Indeed, Var X, = VarX, = φµ γ p. To estmate ths quantty we replace the parameters by ther estmates, whch gves pvp = Var X, = ˆφˆµ ˆγ p. Note that ˆµ, ˆγ, as well as ˆφ are functons of p. To obtan the dervatve of pvp, we start by obtanng the dervate of ˆφ wth respect to p: ˆφp = 1 d + I y,x, ˆµ ˆγ 2 2y, x, ˆµ ˆγ ˆµ ˆγ + ˆµ ˆγ, p+1 where y, = ˆµ ˆγ p, and y, = y,logˆµ ˆγ p y p ˆµ ˆγ + ˆµ ˆγ. Usng the above we obtan the followng lemma., Lemma 5.1 The dervate of pvp wth respect to p s gven by pvp = Var where y, are defned above. X, = ˆφ y, ˆφ y, y, 2, 13

14 5.2 Estmaton Error It s standard to estmate the estmaton error by ts expected value; see England, Verrall [2]. Hence, we estmate by E [ ] 2 ˆµ ˆγ E[X, ] = 2 ˆµ ˆγ E[X, ] k+l>i E[ ˆµ ˆγ E[X, ] ˆµ kˆγ l E[X k,l ] ]. Note that the predctor ˆµ ˆγ s not necessarly unbased for E[X, ] = µ γ. Ths bas s for typcal clams reservng data of neglgble order. One uses the approxmaton 2 ˆµ ˆγ E[X, ] Covˆµ ˆγ, ˆµ kˆγ l ; k+l>i see, for example, Wüthrch, Merz [20], Secton As an estmator of the above covarances we use Ĉovˆµ ˆγ, ˆµ kˆγ l = ˆγ ˆγ l Ĉovˆµ, ˆµ k + ˆµ kˆγ Ĉovˆµ, ˆγ l + ˆµ ˆγ l Ĉovˆµ k, ˆγ + ˆµ ˆµ k Ĉovˆγ, ˆγ l ; see Appendx B for detals. The estmated covarance terms on the rght-hand sde of the above equalty are provded n 3. We hence obtan [ ] êep = Ê 2 ˆµ ˆγ E[X, ] = Ĉovˆµ ˆγ, ˆµ kˆγ l, whch has as ts dervatve êep = k+l>i k+l>i ˆγ ˆγ l + ˆγ ˆγ l Ĉovˆµ, ˆµ k + ˆγ ˆγ l Ĉovˆµ, ˆµ k + ˆµ kˆγ + ˆµ kˆγ Ĉovˆµ, ˆγ l + ˆµ kˆγ Ĉovˆµ, ˆγ l + ˆµ ˆγ l + ˆµ ˆγ l Ĉovˆµ k, ˆγ + ˆµ ˆγ l Ĉovˆµ k, ˆγ + ˆµ ˆµ k + ˆµ ˆµ k Ĉovˆγ, ˆγ l + ˆµ ˆµ k Ĉovˆγ, ˆγ l. 14

15 To obtan the above we need only provde the dervatves wth respect to p of the covarances of the MLEs. Ths nvolves the estmated Fsher nformaton matrx Ĥ = Ĥˆζ. Denote wth Ĥ the matrx contanng the dervatves of the estmated covarances: Ĥ = Ĥ 1 = Ĥ 1 Ĥ Ĥ 1. The dervatves of the entres of Ĥ wth respect to p are gven as follows, see 2, Ĥ = ˆφ! 2I+1 r X ˆζ p r,r ˆφ 2 r ˆζ s 2 p + 1ˆφ 2I+1 r X p 2 p ˆζ r 2 p p ˆζ s ˆζ s + ˆζ r, r {1,..., I}, s=i+1 s=i+1 Ĥ = ˆφ! 2I+1 s X ˆζ 2 p s,s ˆφ 2 r ˆζ s p + 1ˆφ 2I+1 s X 2 p p ˆζ r p 2 p ˆζ s ˆζ s + ˆζ r, s {I + 1,..., 2I + 1}, r=1 r=1 Ĥ = ˆφ! r,s ˆφ ˆζ1 p 2 r ˆζ s 1 p + 1ˆφ 1 p 1 p ˆζ r 1 p 1 p ˆζ s ˆζ s + ˆζ r, r {1,..., 2I + 1}, where, s {1,..., 2I + 1 r} and r, s / {1,..., I} {1,..., I}, p ˆζ r = logˆζ r ˆζ p r + pˆζ r 1 p ˆζ r, for r {1,..., 2I + 1}. Usng the above, we obtan the followng lemma. Lemma 5.2 The dervate of êep wth respect to p s gven by, êep = k+l>i ˆγ ˆγ l + ˆγ ˆγ lĥ 1,k + ˆγ ˆγ l Ĥ,k + ˆµ kˆγ + ˆµ kˆγ Ĥ 1,I+l + ˆµ kˆγ Ĥ,I+l +ˆµ ˆγ l + ˆµ ˆγ lĥ 1 I+,k + ˆµ ˆγ l Ĥ I+,k + ˆµ ˆµ k + ˆµ ˆµ kĥ 1 I+,I+l + ˆµ ˆµ k Ĥ I+,I+l, where Ĥ and Ĥ are defned above,, k {1,..., I} and, l {0,..., I}. Note: The second order Taylor expanson 8 s provded n Appendx C. 6 Case Study We analyze the standard dataset from Wüthrch, Merz [20], see Table 1 below. We center our examples around p = 1 and p = 2, correspondng to the overdspersed Posson and the Gamma dstrbutons. The clams reserves and the MSEP for these 15

16 models are attanable wth relatve ease usng most standard statstcal software packages. / ,946,975 3,721, , , ,704 62,124 65,813 14,850 11,130 15, ,346,756 3,246, , ,797 67,824 36,603 52,752 11,186 11, ,269,090 2,976, , , ,703 65,444 53,545 8, ,863,015 2,683, , , ,976 88,340 4, ,778,885 2,745, , , , , ,184,793 2,828, , , , ,600,184 2,893, , , ,288,066 2,440, , ,290,793 2,357, ,675,568 Table 1: Observed ncremental payments X,. 6.1 Estmatng wth p = 1, the Overdspersed Posson Dstrbuton Clams Reserves Usng the statstcal software R we obtan the MLEs of the underlyng parameters µ and γ. They, as well as the frst and second dervatves are found n Table 2. Note that for p = 1, the estmates of the underlyng parameters could also have been obtaned usng the classcal chan ladder method; see Corollary 2.18 n Wüthrch, Merz [20]. Recall the admssble values of p, p, 0] [1,, mplyng that we study postve ε when usng p = 1 as the focal pont of our approxmaton. / ˆµ ˆµ ˆµ ˆγ ˆγ ˆγ ,572, , , ,237, , , ,835 41,156 24, ,836 14,891 6, ,501 9, ,540 4, ,870 2,931 3, , , , , Table 2: Estmates of parameters µ and γ and ther dervatves for the case p = 1. The clams reserves under the assumpton that the Tweede exponental dsperson 16

17 model has p = 1,.e. that t s overdspersed Posson dstrbuted, s 6, 047, 059. In Table 3 we hghlght the performance of the approxmaton n relaton to the true values under the varous levels p, notce the accuracy of the second order approxmaton. Fgure 2 presents these results graphcally. Moreover, the clams reserves ˆRp are rather nsenstve to the choce of p. p Exact 1st Order 2nd Order Exact 1st Order 2nd Order Reserve Approx. Approx. msep 1/2 Approx. Approx ,047,059 6,047,059 6,047, , , , ,043,385 6,043,459 6,043, , , , ,039,560 6,039,859 6,039, , , , ,035,577 6,036,259 6,035, , , , ,031,429 6,032,660 6,031, , , , ,027,113 6,029,060 6,027, , , , ,022,621 6,025,460 6,022, , , , ,017,951 6,021,860 6,018, , , , ,013,100 6,018,260 6,013, , , , ,008,070 6,014,660 6,008, , , , ,002,865 6,011,060 6,003, , , , ,997,497 6,007,460 5,998, , , , ,991,983 6,003,860 5,993, , , , ,986,347 6,000,260 5,987, , , , ,980,624 5,996,660 5,982, , , , ,974,856 5,993,060 5,976, , , , ,969,088 5,989,461 5,970, , , , ,963,380 5,985,861 5,964, , , , ,957,772 5,982,261 5,958, , ,230 1,019, ,952,316 5,978,661 5,952,325 1,037, ,527 1,087,804 Table 3: True and approxmated clams reserves and MSEP Predcton Uncertanty Table 3 hghlghts the results of the MSEP 1 2 approxmaton usng p = 1. Fgure 3 presents these results graphcally. It s evdent that the MSEP 1 2 s not stable n p, one cannot devate too far from p = 1.e. ε cannot be too far from 0 when usng the frst order approxmaton. Furthermore, t seems that the MSEP 1 2 s near a local mnmum at p = 1. Under the assumpton of the overdspersed Posson dstrbuton, the estmates of the dsperson parameter φp and ts dervates were 17

18 Reserve Approxmaton Exact Reserve and Approxmate Reserve exact frst order approx. second order approx p Fgure 2: True and approxmated clams reserves. found to be, ˆφ1 = 14, 714, ˆφ 1 = 197, 314 and ˆφ 1 = 2, 678, 513. The second order approxmaton far outperforms the frst, but as prevously stated s also more strenous to calculate. MSEP^1/2 Approxmaton Exact MSEP^1/2 and Approxmate MSEP^1/ p exact frst order approx. second order approx. Fgure 3: True and approxmated MSEP

19 6.2 Estmatng wth p = 2, the Gamma Dstrbuton Clams Reserves The MLEs of the underlyng parameters and ther frst and second order dervates are presented n Table 4. The clams reserves under the assumpton that the Tweede exponental dsperson model has p = 2 s 5, 947, 049. In Table 5 we hghlght the performance of the approxmaton for the clams reserves n relaton to the true values under the varous levels p. Fgure 4 presents these results graphcally. / ˆµ ˆµ ˆµ ˆγ ˆγ ˆγ ,999, ,394-1,316, ,426,601 33, , , , ,086-7,201-47, ,788-15,077-29, ,394-7,532-16, ,418 3,227-13, ,159 1,090-3, ,226 1,409-5, , Table 4: Estmates of parameters µ and γ and ther dervatves for the case p = 2. Reserve Approxmaton Exact Reserve and Approxmate Reserve exact frst order approx. second order approx p Fgure 4: True and approxmated clams reserves. 19

20 p Exact 1st Order 2nd Order Exact 1st Order 2nd Order Reserve Approx. Approx. msep 1/2 Approx. Approx ,997,497 5,993,507 6,002, , , , ,991,983 5,988,345 5,995, , , , ,986,347 5,983,183 5,988, , , , ,980,624 5,978,021 5,981, , , , ,974,856 5,972,859 5,975, , , , ,969,088 5,967,697 5,969, , , , ,963,380 5,962,535 5,963, , , , ,957,772 5,957,373 5,957, , , , ,952,316 5,952,211 5,952,319 1,037,959 1,033,788 1,038, ,947,049 5,947,049 5,947,049 1,117,386 1,117,386 1,117, ,941,997 5,941,887 5,941,995 1,205,544 1,200,984 1,205, ,937,171 5,936,724 5,937,158 1,303,693 1,284,581 1,302, ,932,570 5,931,562 5,932,537 1,413,275 1,368,179 1,407, ,928,178 5,926,400 5,928,133 1,535,917 1,451,777 1,522, ,923,961 5,921,238 5,923,946 1,673,439 1,535,375 1,645, ,919,883 5,916,076 5,919,976 1,827,850 1,618,972 1,777, ,915,901 5,910,914 5,916,222 2,001,354 1,702,570 1,917, ,911,966 5,905,752 5,912,684 2,196,368 1,786,168 2,067, ,908,033 5,900,590 5,909,364 2,415,529 1,869,766 2,225, ,904,057 5,895,428 5,906,260 2,661,728 1,953,364 2,392,460 Table 5: True and approxmated clams reserves and MSEP Predcton Uncertanty The approxmatons of the MSEP usng the Gamma dstrbuton are presented n Table 5. Fgure 5 presents these results graphcally. Under the assumpton of the Gamma dstrbuton, the estmates of the dsperson parameter, φp, and ts dervate were found to be, ˆφ2 = , ˆφ 2 = and ˆφ 2 = Remark 6.1 Notce that n addton to the fact that due to our boundary condton of ˆµ 0 = 1, all dervates of ˆµ 0 equal zero, also all dervatves of ˆγ J equal zero. Ths s the case snce ˆγ J = X 0,J, whch s D I measurable,.e. ˆγ J s constant. Ths fact shows up clearly n Tables 2 and 4. 20

21 MSEP^1/2 Approxmaton Exact MSEP^1/2 and Approxmate MSEP^1/ exact frst order approx. second order approx p Fgure 5: True and approxmated MSEP 1 2. Acknowledgements Both authors are grateful to Prof. Dr. Paul Embrechts for provdng valuable nsghts related to ths paper. Furthermore, the frst author s greatly ndebted to ACE Lmted for provdng the fnancal support necessary for the completon of ths work. References [1] Bühlmann, H. and Gsler, A A Course n Credblty Theory and ts Applcatons. Sprnger, Berln. [2] England, P.D. and Verrall, R.J Stochastc clams reservng n general nsurance. Brtsh Act. J., 8, [3] Ggante, P. and Sgalott, L Model Rsk n clams reservng wth generalzed lnear models. Gornale dell Insttuto Italano degl Attuar, 68, [4] Haberman, S. and Renshaw, A.E Generalzed lnear models and actuaral scence. The Statstcan, 45,

22 [5] Jørgensen, B Exponental dsperson models. J. Roy. Statst. Soc., 49, [6] Jørgensen, B The Theory of Dsperson Models. Chapman & Hall, London. [7] Jørgensen, B. and De Souza, M.C.P Fttng Tweede s compound Posson model to nsurance clams data. Scand. Actuar. J., [8] Lehmann, E.L Theory of Pont Estmaton. Wley & Sons, New York. [9] Mack, T A smple parametrc model for ratng automoble nsurance or estmatng IBNR clams reserves. ASTIN Bulletn, 21, [10] McCullagh, P. and Nelder, J.A Generalzed Lnear Models, 2nd Edton. Chapman & Hall, London. [11] Mllenhall, S.J A systematc relatonshp between mnmum bas and generalzed lnear models Proceedngs of the Casualty Actuaral Socety, 86, [12] Nelder, J.A. and Pregbon, D An extended quas-lkelhood functon. Bometrka, 74, [13] Nelder, J.A. and Wedderburn, R. W. M Generalzed lnear models. J. Roy. Statst. Soc., 135, [14] Peters, G.W., Shevchenko, P.V., and Wüthrch, M.V Model rsk n clams reservng wthn Tweede s compound Posson models. ASTIN Bulletn, to appear. [15] Renshaw, A.E Modellng the clams process n the presence of covarates. ASTIN Bulletn, 24, [16] Smyth, G.K, Jørgensen, B Fttng Tweede s compound Posson model to nsurance clams data: dsperson modellng. ASTIN Bulletn, 32,

23 [17] Tweede, M.C.K An ndex whch dstngushes some mportant exponental famles. In Statstcs: Applcatons and New Drectons Eds. J.K. Ghosh and J. Roy, Indan Statstcal Insttute, Calcutta. [18] Wedderburn, R.W.M Quas-lkelhood functons, generalzed lnear models and the Gauss-Newton method. Bometrka, 61, [19] Wüthrch, M.V Clams reservng usng Tweede s compound Posson model. ASTIN Bulletn, 33, [20] Wüthrch, M.V. and Merz, M Stochastc Clams Reservng Methods n Insurance. Wley & Sons, West Sussex, England. A Dervatves of A, α and β In ths appendx, we provde the dervatves of the entres of the matrx A and the dervates of the α and β, requred n Secton 4.2. a, = a I++1,I++1 = I =0 I =0 a,i++1 = 2 pˆµ ˆγ 1 p ˆγ 2 p logˆγ + 2 pˆγ, {1,..., I}, ˆγ ˆµ 2 p logˆµ + 2 pˆµ, {0,..., I}, ˆµ logˆγ + 1 pˆγ ˆγ 1 px,ˆγ p logˆγ + pˆγ + x,ˆγ p, ˆγ a I++1, = 2 pˆγ ˆµ 1 p logˆµ + 1 pˆµ 1 px, ˆµ p logˆµ + pˆµ ˆµ + ˆµ + 2 pˆµ ˆγ 1 p {0,..., I}, {0,..., I }, + ˆγ + 2 pˆγ ˆµ ˆµ 1 p + x, ˆµ p, {0,..., I}, {0,..., I }, 23

24 and, α = β = I [logˆγ ˆµ =0 logˆγ x,ˆγ 1 p =0 ˆγ 2 p logˆγ + 2 pˆγ ˆγ logˆγ + 1 pˆγ ˆγ I [logˆµ ˆγ ˆµ 2 p logˆµ + 2 pˆµ logµ x, ˆµ 1 p logˆµ + 1 pˆµ ˆµ ˆγ + ˆµ + logˆγ ˆµ ˆγ 2 p ˆγ ˆγ ] x, ˆγ 1 p, {1,..., I}, ˆγ ˆµ ˆµ + ˆγ + logˆµ ˆγ ˆµ 2 p ˆµ ] ˆµ x, ˆµ 1 p ˆµ, {0,..., I}. B Covarance Approxmaton In ths appendx, we am to show that, We begn as follows, Covˆµ ˆγ, ˆµ kˆγ l ˆγ ˆγ l Ĉovˆµ, ˆµ k + ˆµ kˆγ Ĉovˆµ, ˆγ l + ˆµ ˆγ l Ĉovˆµ k, ˆγ + ˆµ ˆµ k Ĉovˆγ, ˆγ l. Covˆµ ˆγ, ˆµ kˆγ l = Covexp{logˆµ ˆγ }, exp{logˆµ kˆγ l } = µ γ µ k γ l Covexp{logˆµ ˆγ logµ γ }, exp{logˆµ kˆγ l logµ k γ l } µ γ µ k γ l Cov1 + logˆµ ˆγ logµ γ, 1 + logˆµ kˆγ l logµ k γ l = µ γ µ k γ l Covlogˆµ ˆγ, logˆµ kˆγ l = µ γ µ k γ l Covlogˆµ + logˆγ, logˆµ k + logˆγ l = µ γ µ k γ l Covlogˆµ, logˆµ k + Covlogˆµ, logˆγ l + Covlogˆγ, logˆµ k + Covlogˆγ, logˆγ l, where we have used the lnearzaton expz 1 + z for z 0. We proceed wth the covarance terms remanng on the rght-hand sde. The calculatons are analogous, hence we only provde the detals for approxmatng Covlogˆµ, logˆµ k, Covlogˆµ, logˆµ k = Cov1 + logˆµ logµ, 1 + logˆµ k logµ k Covexp{logˆµ logµ }, exp{logˆµ k logµ k } 1 = Covˆµ, ˆµ k, µ µ k 24

25 where we agan have used the lnearzaton expz 1 + z for z 0. As a fnal step of the approxmaton, we replace unknown varables and unknown covarances wth estmates thereof. C MSEP Approxmaton usng Second Order Taylor Expanson In ths secton we provde the man formulas for the second order Taylor expanson 8. msep 1 2 p + ε = msep 1 2 p + msep 1 2 p ε + msep 1 2 p ε 2 2. msep 1 2 p = msep 2 p pvp + êep msep 2 p pvp + êep. ˆφ = 1 d pvp = 1 y ˆφ 2 y, ˆφ, y, y, 2 y, 3 y, y, 2 y,x, ˆµ ˆγ 2 4y,x, ˆµ ˆγ ˆµ ˆγ + ˆµ ˆγ + I +2y, ˆµ ˆγ + ˆµ ˆγ 2 x, ˆµ ˆγ ˆµ ˆγ + 2ˆµ ˆγ + ˆµ ˆγ. ˆφ. y, = y, y p+1 p, 1 p logˆµ ˆγ + p + 1 y, ˆµ ˆγ + ˆµ ˆγ 2 ˆµ ˆγ + ˆµ ˆγ + p ˆµ ˆγ + 2ˆµ ˆγ + ˆµ ˆγ. êep = k+l>i ˆγ ˆγ l + 2ˆγ ˆγ l + ˆγ ˆγ l Ĥ 1,k + 2ˆγ ˆγ l + ˆγ ˆγ l Ĥ,k + ˆγ ˆγ l Ĝ,k +ˆµ kˆγ + 2ˆµ kˆγ + ˆµ kˆγ Ĥ 1,I+l + 2ˆµ kˆγ + ˆµ kˆγ Ĥ,I+l + ˆµ kˆγ Ĝ,I+l +ˆµ ˆγ l + 2ˆµ ˆγ l + ˆµ ˆγ l Ĥ 1 I+,k + 2ˆµ ˆγ l + ˆµ ˆγ l ĤI+,k + ˆµ ˆγ l Ĝ I+,k +ˆµ ˆµ k + 2ˆµ ˆµ k + ˆµ ˆµ k Ĥ 1 I+,I+l + 2ˆµ ˆµ k + ˆµ ˆµ k ĤI+,I+l + ˆµ ˆµ k Ĝ I+,I+l. 25

26 Ĝ = 2 Ĥ Ĥ Ĥ Ĥ 1 Ĥ 1 Ĥ 1 Ĥ 1 2 Ĥ 1. Ĥ 2 r,r Ĥ 2 s,s Ĥ 2 r,s = = = ˆφ ˆφ 2 + 1ˆφ ˆφ ˆφ 2 + 1ˆφ ˆφ ˆφ 2 + 1ˆφ 2I+1 r X s=i+1 2I+1 r X s=i+1 2I+1 s X r=1 2I+1 s X r=1 2I+1 r ˆζ r p ˆζ s 2 p + 2 ˆφ ˆφ X ˆφ 3 s=i+1 p ˆζ r ˆζ 2 p 2 s p ˆζ r + 2 ˆζ 2 p s 2I+1 s ˆζ r 2 p ˆζ s p + 2 ˆφ ˆφ X ˆφ 3 r=1 2 p ˆζ r ˆζ p 2 s ˆζ1 p r ˆζ s 1 p + 2 ˆφ ˆφ 1 p ˆζ r ˆζ 1 p 2 s 2 p ˆζ r + 2 ˆφ ˆζ1 p 3 r 1 p ˆζ r + 2 ˆζ p s ˆζ s 1 p 1ˆφ2 ˆζ 1 p s + ˆζ p r ˆζ s 2 p 1ˆφ2 2I+1 r X s=i+1! + ˆζ 2 p r p ˆζ r ˆζ 2 p s 2 ˆζ s p 1ˆφ2 2I+1 s X r=1! 2 p + ˆζ r ˆζ 1 p r 1 p 1 p ˆζ s ˆζ r 2 ˆζ p s 2 ˆζ s 1 p +! ˆζ p r ˆζ s 2 p +, r {1,..., I}, ˆζ 2 p r ˆζ s p + p ˆζ r 2 p ˆζ r, s {I + 1,..., 2I + 1}, 1 p ˆζ r ˆζ 1 p s!!, r {1,..., 2I + 1}, s {1,..., 2I + 1 r} and r, s / {1,..., I} {1,..., I}. ˆζ 2 p s ˆζ p s!!!! p ˆζ r 2 = 2ˆζ r 1 p ˆζ r p ˆζ r logˆζ r + p ˆζ r ˆζ 1 p r + ˆζ 1 p r ˆζ r. 26