Prediction of Disability Frequencies in Life Insurance


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1 Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng Fran Weber Maro V. Wüthrch October 28, 2011 Abstract For the predcton of dsablty frequences, not only the observed, but also the ncurred but not yet reported (IBNyR) clams have to be taen nto account. In the present paper we dscuss an IBNyR model whch allows to ncorporate nformaton of an external economc factor. The am s to fnd such an economc factor whch precedes the dsablty frequency n the sense that ts tme shfted process s strongly correlated to the occurrence of dsablty clams. Ths economc factor wll then serve as a prendcator to mprove the qualty of the dsablty frequency predcton. 1 Introducton We consder the predcton of dsablty clams and frequences n lfe nsurance. A dsablty clam occurs, f an nsured becomes partally or completely unable to wor for an extended tme perod, and f he does not reactvate or de wthn a gven watng perod (watng perod s sometmes also called deferred perod or qualfcaton perod). Such a dsablty clam may be caused by scness or an accdent. As a consequence, the nsured receves dsablty benefts accordng to the nsurance contract and hs degree of dsablty. The clams frequency s an essental trgger of the dsablty rs of a gven portfolo. predcton of ths clams frequency, however, s a complcated matter because of late reportng and recordng of clams. Ths problem arses from the delay between the occurrence of a clam and ts recordng n the nsurer s admnstratve system,.e. untl t s avalable for statstcal analyss. Such delays, whch can be months or sometmes even years, may have several causes: One essental cause for late reportng s the uncertanty whether the wor ncapacty due to ether scness or an accdent leads to a long term dsablty. Another reason s that usually clams are not reported before the end of the watng perod. Once the nsurer s aware of a (potental) clam, t needs to be verfed whether the nsured s actually enttled to any dsablty benefts. comprehensve medcal nvestgaton whch s often tmeconsumng. AXA Wnterthur, GeneralGusanStr. 40, Postfach 357, 8401 Wnterthur, Swtzerland ETH Zurch, RsLab, Department of Mathematcs, 8092 Zurch, Swtzerland The Frequently ths process ncludes a 1
2 Dsregardng ths late reportng and recordng problem would lead to a systematc underestmaton of dsablty frequences, and therefore, to an underestmaton of the dsablty rs. For ths reason t s mportant to predct the number of those dsablty cases, whch have already occurred, but are stll not recorded n the nsurer s admnstratve system. Ths corresponds to the wellnown tas of estmatng the number of ncurred but not yet reported (IBNyR) clams n nonlfe nsurance. The analyss of ths predcton problem s usually based on a socalled runoff scheme. consder the matrx {N,j ; 0, j I}, where N,j denotes the number of the dsablty cases whch have occurred n tme perod and whch are recorded n the nsurer s admnstratve system durng the accountng perod = + j,.e. whch have a recordng delay j. Thus, the rows of ths matrx correspond to the occurrence perods of the clams, whereas the columns j agree wth ther reportng and recordng delays. The last row = I of the scheme refers to the latest complete tme perod before the statstcs are establshed. Hence, the upper runoff trangle D I = {N,j ; + j I, 0, j I}, (1.1) contans the numbers of the observed dsablty cases. In the followng, the number I s always assumed to be suffcently large, such that all clams have been reported after I tme perods. For our numercal analyss we wll consder 2 dfferent tme perods (ntervals): (1) years and (2) quarters, respectvely, see Secton 3, below. A numercal example s gven n Table 1. Apart from the recorded dsablty cases N,j D I, ths scheme also contans the correspondng numbers s of nsured lves n occurrence perods (as an exposure and volume measure). The tme perods n Table 1 are calendar years. Note that we occurrence reportng delay j underlyng perod exposure s Table 1: Clams development trangle (yearly tme perods). We 2
3 have shfted the occurrence perod labelng from {0,..., I} (wth I = 11) to {1997,..., 1997+I} because n the sequel ths wll allow us to relate these occurrence perods to economc factors. The predcton of the number of cases, whch have already occurred but are not yet recorded n the nsurer s admnstratve system, corresponds to the tas of predctng the outcome of the random varables n the lower runoff trangle DI c = {N,j ; + j > I, 0, j I}. (1.2) To ths end we rely on three dfferent models: 1. Posson model. Ths method, that s frequently used n nonlfe nsurance, allows for occurrence perod and recordng delay j modelng. Dependences n the accountng perods = + j are not consdered. Note, however, that these accountng perod dependences are crucal for the predcton of dsablty frequences. For ths reason, the Posson model s not approprate for our purposes. 2. Verbee model. Ths s an alternatve method for the predcton of dsablty frequences. It allows for recordng delay j and accountng perod = + j modelng, but t neglects dependences on occurrence perods. 3. Full model. Ths s a generalzaton of both the Posson model and the Verbee model. It allows for the modelng of all three drectons, j and = + j. The crucal pont n our analyss s that the accountng perod dependences are strongly correlated wth precedng economc factors. The ncluson of such economc factors nto our model wll mprove the qualty of the dsablty frequency predctons. Therefore, we concentrate on the Full Model n the analyss, whle the Posson and Verbee models only serve as benchmars. Organzaton of the paper. In Secton 2 we frst present the three dfferent models mentoned above. Addtonally, the basc propertes of these methods are dscussed. In Secton 3 the models are appled to the data presented n Table 1. Here we consder tme perods of a year and a quarter. In Secton 4 t s shown how the qualty of dsablty frequency predctons can be mproved by the consderaton of addtonal external nformaton, such as precedng economc factors. In our concludng Secton 5 we gve a summary of the results and propose further possble steps. 2 The models In ths secton we defne the three models mentoned above. The frst two models are used as benchmar models. Ther parameters wll be estmated wth maxmum lelhood estmaton (MLE) methods, whereas the thrd model (Full Model) wll be put nto a Bayesan framewor. 3
4 2.1 Posson model Model 2.1 (Posson Model) Assume that there are fxed exposures s > 0, and parameters p, γ j > 0 (, j = 0,..., I) wth I j=0 γ j = 1 (normalzaton) such that N,j are ndependent Posson dstrbuted random varables wth mean s p γ j for, j = 0,..., I. The Posson Model s wellestablshed n nonlfe nsurance clams reservng, see for example EnglandVerrall [2]. It has the followng propertes: The total number of clams n occurrence perod up to reportng delay j satsfes C,j = j =0 N, Po ( s p ) j γ. (2.1) Henceforth, the expected total number of dsablty clams n occurrence perod s gven by =0 E [C,I ] = s p. (2.2) Ths shows that the p s are the dsablty frequences and the γ j s gve the reportng pattern. Parameter estmaton s done wth MLE methods, see WüthrchMerz [11], Secton 2.3. The MLE s for p and γ j, gven the observatons D I, can be found analytcally, see Mac [6] and WüthrchMerz [11], Secton 2.4: We ntalze (usng the normalzaton) p P 0 = I j=0 N 0,j and γ I P = N 0,I s 0 s 0 p P, (2.3) 0 where the superscrpt P s used for estmators from the Posson Model. Then we obtan the MLE s teratvely for n = 1,..., I by p P n = I n j=0 N n,j ( s n 1 I j=i n+1 γp j ) and γ P I n = n =0 N,I n n =0 s p P. (2.4) The MLE predctors for DI c are then n the Posson Model 2.1 gven by the forecast (the notaton Ê[ ] ndcates that we estmate the mean E[ ]) N,j P = Ê [N,j D I ] = s p P γ j P for + j > I, (2.5) and the dsablty frequences p are estmated by the MLE s p P. The choce of the Posson dstrbuton s justfed by the fact that t may serve as approxmaton to the bnomal dstrbuton for large portfolo sze s and small dsablty frequency p. 2.2 Verbee model Model 2.2 (Verbee [9] Model) Assume that there are parameters λ > 0 ( = 0,..., 2I) and γ j > 0 (j = 0,..., I) wth I j=0 γ j = 1 (normalzaton) such that N,j are ndependent Posson dstrbuted random varables wth mean λ +j γ j for, j = 0,..., I. 4
5 In the Verbee Model the expected total number of dsablty clams E [C,I ] = s p from the Posson Model, see (2.2), s replaced by the expresson E [C,I ] = I λ +j γ j. (2.6) j=0 If there are no accountng perod effects,.e. λ = λ for all, then we are n the Posson Model 2.1 wth s p = λ. Henceforth, the Verbee Model 2.2 allows for accountng perod effects λ modelng. The parameters are agan estmated wth MLE methods. They can be calculated analytcally gven D I, see Verbee [9]: We ntalze (usng the normalzaton) λ V I = I =0 N I, and γ I V = N 0,I, (2.7) λ V I where the superscrpt V s used for estmators from the Verbee Model. Then we terate for n = I 1,..., 0 n λ V =0 n = N n, 1 I j=n+1 γv j and γ V n = I n =0 N,n I λ. (2.8) =n V Note that the nformaton n the Verbee Model s not suffcent to predct the IBNyR clams D c I. For the predcton of N,j, + j > I, we need to have an estmate for λ +j whch unfortunately s not gven by (2.7)(2.8) for + j > I. Often n practce λ, for > I, s estmated by lnear regresson from λ V 0,..., λ V I. We denote the resultng lnear regresson estmates by λ V and hence obtan the Verbee predctor of N,j by N V,j = Ê [N,j D I ] = λ V +j γ V j for + j > I, (2.9) and the dsablty frequences p are estmated by (see also (2.6)) for gven fxed exposures s > 0. p V = I j=0 λ V +j γv j s, (2.10) 2.3 Full model For the Full Model we tae a Bayesan pont of vew. The Bayesan pont of vew has several advantages, e.g. rather smple numercal algorthms lead to the Bayesan predctors and we do not only get pont predctors for the varables of nterest, but full posteror dstrbutons that allow for Bayesan nference on parameters. Model 2.3 (Full Model) Assume there exst fxed exposures s > 0, = 0,..., I. For gven parameters Θ = (π 0,..., π I, γ 0,..., γ I, λ 0,..., λ 2I ) the random varables N,j are ndependent Posson dstrbuted wth mean s π λ +j γ j for, j = 0,..., I. All the components of Θ are ndependent and postve P a.s. Moreover, λ has a pror normalzaton,.e. E[λ ] = 1 for = 0,..., 2I. 5
6 Remars. Condtonally gven the parameters Θ, we have an ndependent Posson cells model that allows for modelng of all three drectons: occurrence perod π, reportng delay γ j and accountng perod λ wth = + j. Due to the fact that we do not now the true parameters Θ, we choose a pror dstrbuton for Θ hghlghtng our nowledge and parameter uncertanty. Then we can do Bayesan nference on Θ, gven the observatons D I. If we have no accountng perod effects, we set λ 1 and then we are n the Bayesan verson of the Posson Model 2.1, see also England et al. [3]. Our am s to see whether λ dffers from 1,.e. whether we have accountng perod effects. Note that γ j s not normalzed and therefore π s not a dsablty frequency as n the Posson Model 2.1. The Bayesan predctor n the Full Model 2.3 for + j > I s gven by N F,j = E [N,j D I ] = s E [π γ j D I ], (2.11) λ beng ndependent of D I for > I. The estmator for the dsablty frequency p s gven by I p F j=0 = N,j + I N j=i +1,j F = C I,I + E [π γ j D I ]. (2.12) s s j=i +1 Note that the last term on the rghthand sde cannot be further calculated analytcally because there s an mpled posteror dependence between π and γ j, gven D I. Moreover, a Bayesan nference analyss for the accountng perod parameter λ, I, allows to compare: s E [λ D I ] approxmately equal to 1? (2.13) Applcaton of the Full Model n practce. In order to apply the Full Model 2.3 we need to specfy the pror dstrbutons for Θ. Often there s no canoncal choce for the pror dstrbutons. Therefore, often one maes a choce that allows for an easy nference analyss n (2.11) and (2.13). We choose ndependent gamma dstrbutons for all components of Θ wth pror means E[π ], E[γ j ] = γ j P and E[λ ] = 1, (2.14) wth γ j P gven by the MLE s of the Posson Model 2.1 and wth coeffcents of varaton stll to be determned. The posteror dstrbuton Θ D I can then be calculated numercally. The two most popular methods are the Gbbs samplng method and the Marov chan Monte Carlo (MCMC) method. Because these methods are wellestablshed n the lterature, see AsmussenGlynn [1], Gls et al. [4] and Scolln [8] we are not gong to further dscuss these smulaton methods here. In our analyss we have used the MetropolsHastngs [7, 5] algorthm for the MCMC smulaton method n a smlar fashon as n Wüthrch [10]. Therefore, for the explct mplementaton we refer to the latter reference. 6
7 3 Dsablty frequency estmaton: an example In our ntal analyss we study the data gven n Table 1. These data comprse the observed (reported) number of dsablty clams over the observaton perod from 1997 untl 2008 together wth the exposure s that counts the number of persons nsured. Due to confdentalty reasons the data were scaled wth a factor. In a frst step we have used the three models, the Posson Model 2.1, the Verbee Model 2.2 and the Full Model 2.3. We have appled these three models to yearly (y) and quarterly (q) data (tme perods), resultng n the dsablty frequency estmates p P y, p P q, p V y, p V q, and p F y, p F q, (3.1) where the frst upper ndex denotes the method and the second the perod length. 3.1 Posson and Verbee models p Posson (yearly) p Posson (quarterly) Fgure 1: Dsablty frequency estmates p P y, p P q from the Posson model. Fgure 1 gves the dsablty frequency estmates for the Posson Model 2.1. We observe that the estmates become more volatle for shorter tme perods. Ths already ndcates that the Posson model on short tme perods s not approprate because the parameter uncertantes are too large and because the Posson model reacts too senstvely to small numbers of observatons. Moreover, f we would consder monthly data, we would observe strong seasonal effects whch are smoothed n the quarterly and yearly vew. The same fgure for the Verbee Model 2.2, see Fgure 2, gves a much more stable pcture. We see that the accountng perod parameter λ s able to smooth the estmaton also for smaller tme perods. 7
8 p Verbee (yearly) p Verbee (quarterly) Fgure 2: Dsablty frequency estmates p V y, p V q from the Verbee model. If we compare the quarterly dsablty frequency estmates p P q and p V q we see that the estmaton levels are very smlar, see Fgure 3. The dfference n the late occurrence perods ( ) comes from the fact that the lnear regresson n the Verbee method s more conservatve compared to the Posson method. Overall we see a decrease n the dsablty frequences between 2003 and 2008 whch has manly to do wth the fact that the economy has recovered after the fnancal crss n Ths gves already a frst hnt that the dsablty frequences follow economc factors wth some delay. 3.2 Full model In order to apply the Full Model 2.3 we need to specfy the parameters of the pror gamma dstrbutons of Θ. Ths was already partly done n (2.14). Furthermore, expert judgment says E[π ] = 1.5%, std(π ) = 0.1% and Var(γ j ) = ( γ P j ) 2. (3.2) Henceforth, the coeffcent of varaton of λ remans to be specfed. We choose fve dfferent values: Vco(λ ) = std(λ ) E[λ ] = std(λ ) { 1, 0.5, 0.2, 0.1, 0.01 }. (3.3) A coeffcent close to 0 mples that λ fluctuates very lttle around 1. Then we calculate numercally p F y and p F q for these fve values of Vco(λ ). The results for the estmated quarterly dsablty frequences p F q are provded n Fgure 4. We see that the dfferent choces of the coeffcent of varaton only have an nfluence on the recent occurrence perods where only lttle nformaton s avalable. Moreover, we can compare the dfferent methods (Posson, Verbee and Full Model). The results for the quarterly frequences p P q, p V q p F q wth std(λ )=0.2 are presented n Fgure 5. 8
9 dsablty rate wthout IBNR estmates p Posson p Verbee Fgure 3: Quarterly dsablty frequency estmates p P q, p V q model. from the Posson and Verbee! "#! "# $ %& ' ( ) ' * ( +, %. + /0 1 %# $ %# 2! ""! "! &%! &$! &#! &"! &! %! $! #! "! &''( &''% &''' " " & " " " ) " # " * " $ " ( " % " ' +,./01&2 +,./01!*2 +,./01!"2 +,./01!&2 +,./01! &2 from the Full Model wth dfferent stan Fgure 4: Quarterly dsablty frequency estmates p F q dard devatons for λ, see (3.3). 9
10 One mght queston ths comparson because n the Posson and the Verbee models we use MLE for the parameter estmaton and n the Full Model we use Bayesan nference methods. However, we would le to emphasze that these two estmaton methods often gve very smlar results, see also the analyss n England et al. [3].! "#! "# $ %& ' ( ) ' * ( +, %. + /0 1 %# $ %# 2! ""! "! &%! &$! &#! &"! &! %! $! #! "! &''( &''% &''' " " & " " " ) " # " * " $ " ( " % " ' +,.//,0123/ +, ,839:33; Fgure 5: Comparson of the quarterly dsablty frequency estmates p F q, p P q, p V q for the three models. For the Full Model we chose std(λ ) = 0.2. In addton, the MCMC smulaton method gves the whole posteror dstrbuton of the number of IBNyR clams at tme I R = N,j, (3.4) +j>i condtonally gven D I. The emprcal results for the quarterly data wth std(λ ) = 0.2 are provded n Fgure 6. That s, n the full Bayesan model we obtan the whole emprcal posteror dstrbuton for the number of IBNyR clams. Ths dstrbuton now allows for the calculaton of any rs measure, e.g. the ValueatRs for the number of dsablty clams on a 95% level. 4 Improvement of dsablty frequency predctons Both n the Full Model 2.3 and n the Verbee Model 2.2 we analyze the accountng perod parameters λ, whch are estmated n the Full Model by Bayesan nference as λ F = E [λ D I ], (4.1) see (2.13), and n the Verbee Model by the MLE s as λ V. We provde the Bayesan estmators λ F y F q F q V q and λ for std(λ ) = 0.2 n Fgure 7. If we normalze both λ and λ to emprcal mean zero and emprcal varance 1 we obtan the results presented n Fgure 8. 10
11 #)#"! "# $ %& ' ( ) ' * ( +, %. + /0 1 %# $ %# 2 #)#* #)#! #)#( #)#' #!"# $"# %!"# %$"# $!"# $$"# &!"# &$"# '#!"# '#$"# ''!"# ''$"# '(!"# '($"# Fgure 6: Posteror dstrbuton of R D I for std(λ ) = Lambda (yearly) Lambda (quarterly) Fgure 7: Comparson of the estmators F y F q λ and λ for std(λ ) =
12 Lambda Full Model Lambda Verbee Fgure 8: Comparson between V q F q λ and λ (both normalzed). Frst conclusons. In Fgure 8 we see a very smlar behavor for λ, I, n both models. Therefore, the man queston s whether ths accountng perod pattern s related to economc factors? If t s related to economc factors, the nowledge of these economc factors may help to mprove the estmatons and predctons of the accountng perod parameters λ, > I. In our analyss we have explored the followng economc factors: unemployment rate, credt spread, SMI stoc maret ndex and scness daly allowance ndex. The most convncng one n our analyss was the credt spread that we have obtaned between corporate bonds and federal government bonds. For ths credt spread we have observed hgh correlatons wth the accountng perod parameter λ and also an approprate tme lag (ths s dscussed below). Surprsngly, the unemployment rate dd not have any predctve power for mprovng dsablty rate forecasts. The (quarterly) credt spread graph s provded n Fgure 9. We especally see the hgh spreads n the fnancal dstress perods after 2000 and n We denote the quarterly tme seres of these credt spreads by (S t ) t T, where T denotes the set of tme ponts where the credt spread s avalable. Questons. Is ths credt spread (S t ) t T correlated wth ( λ F q ) =0,...,I? If yes, what s the optmal tme shft (tme lag) between t and? What can we learn from the credt spread (S t ) t T for ( λ F q ) =I+1,...,2I? 12
13 ! "# $ %& ' ( ) ' * ( +, %. + /0 1 %# $ %# 2 $ # "!!%%#!%%&!%%'!%%% "! " # " & " ' " % Fgure 9: Quarterly credt spreads n %. Analyss 1. In a frst analyss we calculate the emprcal correlatons between ) q (S ) =,...,I and ( λf, (4.2) =0,...,I where {0,..., 16} denotes the tme shft n quarters between these two tme seres. The emprcal correlatons as a functon of {0,..., 16} are provded n Fgure 10.!(! "# $ %& ' ( ) ' * ( +, %. + /0 1 %# $ %# 2!'!&!%!$!#!"! " # $ % & ' ( ) * " "" "# "$ "% "& "' Fgure 10: Emprcal correlaton between the credt spread S and tme shft (n quarters). λ F q as a functon of the We see that a tme shft of 5 quarters gves an emprcal correlaton of almost 70%! Ths suggests that the credt spread (S ) runs about 5 quarters ahead of the accountng perod 13
14 effects ( λ F q ), and henceforth, the nowledge of the credt spread should mprove the dsablty frequency estmatons for the next accountng perods > I. Analyss 2. In a second analyss we mae a lnear regresson Ansatz: Assume that there exst parameters α, β such that λ F q = α + β S + ε, (4.3) where ε denotes the error term and s agan the tme shft. Usng the mnmal least squares (MLS) method we can determne the optmal α and β. We observe a maxmal slope β and a mnmal pvalue (for the null hypothess β = 0 under Gaussan error terms ε ) agan for a shft of 5 quarters. The correspondng estmates are α = and β = (4.4) and the pvalue s less than 0.1%. Henceforth, from the observaton (S ) I and wth the shft of = 5 we obtan the lnear regresson estmators (see also Fgure 11) λ reg I+t = α + β S I+t for t = 1,...,. (4.5) &!"! "# $ %& ' ( ) ' * ( +, %. + /0 1 %# $ %# 2 &!* &!) &!( &!' &!& &!!%!$!#!" & ' ( Fgure 11: Regresson lne for λ F q, see (4.3)(4.5). In vew of (2.11) and (2.12), ths provdes the (economc factors) mproved predctors N reg,j = reg s E [π γ j D I ] λ +j, for I < + j I +, (4.6) N reg,j = s E [π γ j D I ], for + j > I +. (4.7) The estmator for the dsablty frequency p s then gven by p reg = I j=0 N,j + I j=i +1 N reg,j s. (4.8) 14
15 ! "#! "# $ % & ' ( )"* +,."/ 0 )"/ 1 1! ""! "! &%! &$! &#! &"! &! %! $! #! "! &''( &''% &''' " " & " " " ) " # " * " $ " ( " % " ' +,.//,0123/ +,.//,0123/,4567,839:3;;51< Fgure 12: Quarterly dsablty frequency estmates n the Full Model compared to the mproved estmates whch are based on the lnear regresson (4.3)(4.6). The numercal results are presented n Fgure 12. Interpretaton of Fgure 12. We see n Fgure 12 that the ncluson of the credt spread nformaton gves much more conservatve dsablty frequency predctons. Ths comes from the fact that we now nclude nformaton about the fnancal crss nto the dsablty frequency predcton. Ths fnancal crss wll ncrease the frequences, and f we would neglect ths nformaton we would clearly underestmate the number of dsablty clams. 5 Conclusons and outloo In a frst study we analyze a dsablty development model that models all three tme drectons: occurrence perod, reportng perod and accountng perod. Bayesan nference methods allow to predct the number of dsablty clams and we fnd that the accountng perod parameter s a relevant parameter. In a second analyss we then study how ths accountng perod parameter s related to economc tme seres. We see that the credt spread runs fve quarters ahead of the accountng perod parameter. Therefore, credt spread nformaton allows to mprove dsablty frequency predctons (note that the credt spread s often used as an ndcator for future economc developments). In a next step one should merge the economc tme seres model and the dsablty frequency model nto a full stochastc framewor that also allows to model economc tme seres stochastcally. Ths would allow for the analyss of predcton uncertanty and also relax the..d. as 15
16 sumpton of the accountng perod parameters (whch s too restrctve). Fnally, one should not only model the number of dsablty clams but also ther clam szes (dsablty benefts). References [1] Asmussen, S., Glynn, P.W. (2007). Stochastc Smulaton. Sprnger. [2] England, P.D., Verrall, R.J. (2002). Stochastc clams reservng n general nsurance. Brtsh Act. J. 8/3, [3] England, P.D., Verrall, R.J., Wüthrch, M.V. (2011). Bayesan overdspersed Posson model and the BornhuetterFerguson clams reservng method. Preprnt. [4] Gls, W.R., Rchardson, S., Spegelhalter, D.J. (1996). Marov Chan Monte Carlo n Practce. Chapman & Hall. [5] Hastngs, W.K. (1970). Monte Carlo samplng methods usng Marov chans and ther applcatons. Bometra 57, [6] Mac, T. (1991). A smple parametrc model for ratng automoble nsurance or estmatng IBNR clams reserves. ASTIN Bulletn 21/1, [7] Metropols, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E. (1953). Equaton of state calculatons by fast computng machnes. J. Chem. Phys. 21/6, [8] Scolln, D.P.M. (2001). Actuaral modelng wth MCMC and BUGS. North Amercan Actuaral J. 5/2, [9] Verbee, H.G. (1972). An approach to the analyss of clams experence n motor lablty excess of loss rensurance. ASTIN Bulletn 6/3, [10] Wüthrch, M.V. (2010). Accountng year effects modellng n the stochastc chan ladder reservng method. North Amercan Actuaral J. 14/2, [11] Wüthrch, M.V., Merz, M. (2008). Stochastc Clams Reservng Methods n Insurance. Wley. 16
Prediction of Disability Frequencies in Life Insurance
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