Stochastc reservng case study usng a Bayesan approach Prepared by Bartosz Pwcewcz Presented to the Insttute of Actuares of Australa 16 th General Insurance Semnar 9-12 November 2008 Coolum, Australa Ths paper has been prepared for the Insttute of Actuares of Australa s (Insttute) XVIth General Insurance Semnar 2008. The Insttute Councl wshes t to be understood that opnons put forward heren are not necessarly those of the Insttute and the Councl s not responsble for those opnons. Bartosz Pwcewcz The Insttute wll ensure that all reproductons of the paper acknowledge the Author/s as the author/s, and nclude the above copyrght statement: The Insttute of Actuares of Australa Level 7 Challs House 4 Martn Place Sydney NSW Australa 2000 Telephone: +61 2 9233 3466 Facsmle: +61 2 9233 3446 Emal: actuares@actuares.asn.au Webste: www.actuares.asn.au
Stochastc reservng case study usng a Bayesan approach Abstract There are a number of quanttatve approaches used for stochastc reservng, uncertanty assessment and rsk margn modellng n general nsurance. Australan actuares are qute famlar wth technques such as bootstrappng, stochastc chan ladder or the Mack method. However, the Bayesan approach to stochastc reservng s not well as understood n the actuaral communty. Ths approach s relatvely new and has been extensvely researched overseas. Ths paper ams to ntroduce the Bayesan approach to those unfamlar wth t n the context of stochastc reservng. The dscusson s supported by a case study and suggestons on how the theory can be deployed n practce. Keywords: Bayesan approach, stochastc reservng; rsk margns; Page 2 of 26
Stochastc reservng case study usng a Bayesan approach Table of Contents 1. Introducton...4 2. Overvew of the theory...5 2.1. General Bayesan modellng process...5 2.2. Mathematcs of the Bayesan approach...7 2.3. A non-reservng example...8 2.4. MCMC methods and the Gbbs sampler...9 2.5. Advantages and dsadvantages...11 3. Case study...13 3.1. Overvew of mplemented models and the modellng strategy...13 3.2. Data...16 3.3. Stochastc modellng results...16 4. Summary and conclusons...19 5. Acknowledgements...20 6. Appendx...21 7. Bblography...25 Page 3 of 26
Stochastc reservng case study usng a Bayesan approach 1. Introducton Actuares recognsed the dffculty nvolved n the estmaton of general nsurance labltes a long tme ago. The frst actuaral research papers consderng the uncertanty nherent n the general nsurance reservng process were wrtten n the early 1980s. In essence, stochastc reservng s an attempt to quantfy ths uncertanty and estmate the dstrbuton underlyng the reserves for a partcular nsurance portfolo. The scope of stochastc reservng s broader than for tradtonal reservng methods, whch are generally concerned wth the estmaton of a central estmate (e. the mean of the underlyng dstrbuton of outcomes). Whle dscussng stochastc reservng, t s mportant to dfferentate between a model and the approach whch s used to mplement ths model. In ths context, the model s a statstcal model descrbng the underlyng nsurance clam process (e.g. the over-dspersed Posson) and the approach s a method/technque used to deploy such a model n a partcular reservng stuaton (e.g. assessng uncertantes nherent n the past data). Arguably there s no sngle model that suts all reservng problems. A robust approach should be flexble enough to accommodate any model and perform well n all reservng stuatons. In Australa, stochastc reservng has come under ncreased scrutny followng the 2002 APRA reforms and the ntroducton of a requrement to nclude a rsk margn n the provson for nsurance labltes for regulatory purposes. Most recently, APRA has also released gudelnes regardng nternal economc captal models. Ths more recent development may encourage further research nto and use of stochastc reservng technques. The purpose of ths paper s to present a stochastc reservng framework that uses Bayesan technques (Bayesan stochastc reservng). Bayesan stochastc reservng has been extensvely researched overseas but not n Australa. In partcular, the paper wll address the followng man topcs n the context of a reservng case study: an overvew of the mechancs and theory underlyng Bayesan stochastc reservng; and an llustraton of how two reservng actuaral methods, the chan ladder and Bornhuetter-Ferguson methods, can be mplemented usng ths approach. Although the case study presents an mplementaton of the Bayesan approach n the context of outstandng clam labltes, ths approach can be equally used for premum (or unexpred rsk) labltes. The paper s essentally structured nto two man parts. The frst part gves an overvew of key theoretcal concepts underpnnng Bayesan stochastc reservng. The second part presents an applcaton of ths approach to a reservng case study usng the statstcal software package WnBUGS. The paper s concluded wth an Appendx showng the WnBUGS code used for the case study. Page 4 of 26
Stochastc reservng case study usng a Bayesan approach 2. Overvew of the theory In the past, the mplementaton of Bayesan approaches has been lmted by the lack of effcent computatonal/samplng methods. Developments n computng power and some sgnfcant advances n the understandng of samplng algorthms n the early 1990s led to an ncrease n the applcaton of Bayesan approaches to a wde range of practcal problems, ncludng stochastc reservng. Snce then a number of papers on Bayesan stochastc reservng have been wrtten outlnng Bayes theory and proposng varous Bayesan models. I have ncluded some of these papers n the Bblography secton. I encourage readers nterested n obtanng a thorough understandng of Bayesan stochastc reservng to read the followng papers: de Alba [6], England and Verrall [10], Ntzoufras and Dellaportas [17], Scollnk[22], Verrall [23] and Verrall and England [24]. I consder these papers to provde a good descrpton of the theory and so I have lmted my dscusson n ths secton of the paper to a hgh-level outlne of the key concepts and mechansms and concluded wth a dscusson of the advantages and dsadvantages of Bayesan stochastc reservng. 2.1. General Bayesan modellng process Before consderng the techncal detals underlyng Bayesan stochastc reservng, t s mportant to dscuss the steps nvolved n a typcal Bayesan modellng process. Conceptually, the process comprses sx steps 1, presented n Fgure 2.1 and dscussed below: Fgure 2.1: General Bayesan modellng process (schematcs) Step 1 Specfy a probablty dstrbuton for the data, gven some unknown parameters (data dstrbuton) Step 2 Specfy pror probablty dstrbutons for the parameters of the data dstrbuton (pror dstrbutons) Step 3 Derve the lkelhood functon of the parameters, gven the data (lkelhood functon) Step 4 Combne the lkelhood functon wth the pror dstrbutons to derve the posteror jont dstrbuton of the parameters, gven the data (posteror dstrbuton) Step 5 Obtan parameters from the posteror dstrbuton Step 6 Obtan forecasts usng the predctve dstrbuton derved by combnng the posteror dstrbuton wth the pror dstrbutons 1 Ths process s effectvely a consequence of the Bayes theorem and emprcal Bayes methods, for more detal refer to Carln and Lous [5] (frst two chapters n partcular). Note that England [8] also outlned a smlar, but more detaled process. Page 5 of 26
The frst two steps are concerned wth the specfcaton of a statstcal model. An example of such a model could be the over-dspersed Posson model (ODP model) mplemented n the second part of ths paper or any other statstcal model defned n terms of steps 1 and 2. A detaled dscusson of how to select an approprate statstcal model gven a partcular reservng stuaton s outsde the scope of ths paper. However ths process would nvolve a degree of actuaral judgement and some goodness of ft testng. Whle the data dstrbuton s lkely to be completely defned by a partcular statstcal model (e.g. ODP dstrbuton n the ODP model), the specfcaton of pror dstrbutons for the model parameters can be qute flexble. For example, one can use non-nformatve (or vague) pror dstrbutons, where parameters have large varances and do not contrbute any nformaton to the posteror dstrbuton. Alternatvely, nformatve (or strong) pror dstrbutons could be used for whch parameters have small varances and so nfluence the shape of the posteror dstrbuton. For some statstcal models, t s also possble to assume dfferent statstcal dstrbutons for pror dstrbutons (e.g. Gamma, Normal, Lognormal etc.), further nfluencng the shape of the posteror dstrbuton. Step 4 essentally leads to the estmaton of the posteror dstrbuton and arses from the applcaton of Bayes theory. The key s that the posteror dstrbuton s proportonal to the product of the lkelhood functon and the pror dstrbutons 2. The complexty of step 5 depends on two man factors: the number of parameters underlyng a partcular statstcal model and the shape of the posteror dstrbuton. If there s only one parameter and the posteror dstrbuton can be easly recognsed as a standard statstcal dstrbuton the estmaton of the parameter s farly straghtforward as shown n secton 2.3. However, f there are multple parameters and the posteror dstrbuton s not recognsable, as s the case for the majorty of actuaral statstcal models, the dervaton of the parameters requres a specal samplng algorthm. Markov chan Monte Carlo (MCMC) methods have proven very useful n ths context. Secton 2.4 gves an overvew of one such samplng technque, the Gbbs sampler. In the last step, the forecasts of new observatons not ncluded n the exstng dataset are obtaned, ncludng parameter and process error. As for step 5, the complexty assocated wth forecastng vares consderably dependng on whether the predctve dstrbuton s recognsable. If the predctve dstrbuton s recognsable, as n the example presented n secton 2.3, the requred parameters of the predctve dstrbuton can be estmated drectly and the new observatons are straghtforward to obtan. If however ths dstrbuton s not a standard statstcal dstrbuton t s necessary to smulate the new observatons from the data dstrbuton, condtonal on the smulated (n step 5) parameters. Ths process requres a generc samplng algorthm such as Adaptve Rejecton Samplng 3 (ARS), whch generates samples from non-standard statstcal dstrbutons. As an extenson of step 6, t s also possble to assume a partcular standard statstcal dstrbuton as the predctve dstrbuton and use the output from a generc sampler to parameterse t. For example, England and Verrall [10] used the Gamma dstrbuton as the predctve dstrbuton n ther mplementaton of the ODP model. 2 For more detal see http://en.wkpeda.org/wk/posteror_probablty, and the expresson of the Bayes' theorem n terms of lkelhood shown n http://en.wkpeda.org/wk/bayes%27_theorem 3 For more detal see Glks and Wld [13] Page 6 of 26
2.2. Mathematcs of the Bayesan approach Bayes theory wll be descrbed here very brefly and my man focus s on ts applcaton to stochastc reservng. For a comprehensve dscusson on Bayes theory and methods, the reader s referred to statstcs textbooks (e.g. Berger [2], Bernardo and Smth [3] or Carln and Lous [5]). In probablty theory, Bayes' theorem consders the condtonal and margnal probabltes of two random events and t can be stated n a number of dfferent ways. For stochastc reservng, the most convenent presentaton s n terms of probablty dstrbuton functons and the lkelhood functon. Let s consder a smple reservng project shown n Table 2.1, where {C j : = 1,, n; j= 1,, n} are random varables representng clam payments or any other data commonly used n actuaral trangulaton analyss. Orgn Perod Table 2.1: Example of a reservng project Development perod 1 2 3 n 1 C 11 C 12 C 13 C 1n 2 C 21 C 22 C 23 C 2n 3 C 31 C 32 C 33 C 3n n C n1 C n2 C n3 C nn If we further note that c={c j : + j n + 1} s the upper-left trangle ncorporatng the observed payment (or other) data, the reservng problem s to estmate the unobserved values n the lower-rght trangle. Usng the notaton of Bayes theory 4 and the general step-by-step process ntroduced n secton 2.1, ths reservng problem can be approached as follows: Steps 1 to 2: Assume that each C j (whether n the upper-left or lower-rght trangle) follows a probablty dstrbuton f(c j /θ), where θ denotes a vector of parameters descrbng a partcular clam process generatng C j, and all parameters are dstrbuted accordng to a pror dstrbuton functon π(θ). Steps 3: Calculate the lkelhood functon L(θ/c) for the parameters gven the observed data: L ( θ / c) = f ( C / θ) + j n + 1 j Steps 4: Gven the data dstrbuton and the pror dstrbuton, the posteror dstrbuton f(θ/c) s proportonal to the product of the lkelhood functon and the pror dstrbuton: 4 For example see http://en.wkpeda.org/wk/bayes_theorem and the bblography referred to on ths webste Page 7 of 26
( θ/ c) L( θ/ c) π( θ) f Steps 5: Parameters θ are obtaned from f(θ/c) and used n the next step. Step 6: As noted by de Alba [6], f we were nterested n nference about the parameters θ we could end our modellng process at step 5 and look at the propertes of f(θ/c). However f our am s predcton, as n the case of stochastc reservng, then the known data C j (for +j n+1) s used to predct unobserved values n the lower-rght trangle C j (for +j > n+1) by means of the predctve dstrbuton: f ( Cj c) f ( Cj / θ) f ( θ / c) / = dθ, for, j = 1,, n and +j > n+1 In some smple cases such as the ones presented n the next secton, t may be possble to obtan a closed and recognsable form for the posteror and predctve dstrbutons analytcally. However ths s mpossble for many actuaral problems and approxmaton procedures such as MCMC methods and generc samplng algorthms are requred nstead. 2.3. A non-reservng example Ths secton shows how the general Bayesan modellng process and the mathematcal formulae shown n secton 2.2 can be appled to a partcular Bayesan model, the Posson- Gamma model. Ths model s often used for llustraton purposes n texts on Bayesan approaches (e.g. Carln and Lous [5]). Other smple Bayesan models also nclude the Beta- Bnomal model and the Gaussan-Gaussan model. Posson-Gamma model Let s assume that ndvdual data ponts x ( = 1,, n) n a partcular dataset have Posson dstrbutons wth parameter θ and the parameter θ follows a Gamma dstrbuton wth some known parameters α and β. In mathematcal terms ths could be wrtten as: x / θ ~ Independent Posson θ / α, β ~ Gamma ( α, β ) ( θ ) Followng steps 3 to 5 of the Bayesan modellng process, the calculatons proceed as follows: The lkelhood functon s gven by n x θ θ e L θ x, where x denotes a vector of all data ponts x. ( / ) = = 1 x! Page 8 of 26
The posteror dstrbuton s proportonal to the product of ths lkelhood functon and the pror dstrbuton: f n x θ α α 1 βθ / x θ e, = 1 x! Γ( α ) ( θ, α, β ) θ e and ths can be smplfed as β f n α + x 1 x. = 1 ( β + n) θ ( θ /, α, β ) θ e The smplfed expresson on the rght hand sde can be recognsed as a Gamma dstrbuton: n θ / x, α, β ~ Gamma α + x, β + n = 1 In the fnal step we obtan the predctve dstrbuton for forecastng x ~, new data ponts not ncluded n the exstng data set. Ths s acheved by ntegratng the product of the Posson dstrbuton for the data and the posteror Gamma dstrbuton. The resultng predctve dstrbuton s f ( ~ x / ) ( ~ α ~ 1 x Γ α1 + x ) β 1 1 x = ( 1) ( ~ x 1) 1 1 1, Γ α Γ + + β + β1 n x = 1 where α = + and = β + n 1 α β 1 Ths dstrbuton can be recognsed as a Negatve Bnomal dstrbuton wth parameters y and q ( y q) ~ x ~ NegatveBnomal,, where y = α 1 and 1 q = 1+ β 1 2.4. MCMC methods and the Gbbs sampler The dscusson n ths secton largely follows Walsh [25], who gves a good and farly detaled descrpton of MCMC methods, ncludng the Gbbs sampler 5. My ntenton s to present here a hgh-level overvew of the Gbbs samplng approach and llustrate t wth a general example. As noted n secton 2.1, Gbbs samplng could be used n step 5 of the general Bayesan modellng process. The key dffculty wth the applcaton of Bayesan methods s the ablty to sample from a jont posteror dstrbuton of parameters. Ths task becomes partcularly complcated f there 5 The Gbbs sampler was ntally developed n the context of mage processng (Geman and Geman [11]). Gelfand and Smth [12] then showed how the method could be appled to a wder range of Bayesan problems. Page 9 of 26
are multple parameters and the dstrbuton tself s non-standard. MCMC methods have been developed to tackle such practcal mplementaton problems effcently. In order to llustrate how the Gbbs sampler tackles ths problem let us consder a bvarate jont dstrbuton f(x,y), for whch we wsh to derve one or both margnal dstrbutons, f(x) and f(y). The key to the Gbbs algorthm s that t only consders unvarate condtonal dstrbutons (e. f(x/y) and f(y/x)), whch are far easer to compute than the margnal dstrbutons va ntegraton of the jont densty (e.g. f(x) = f(x,y)dy). In other words, ths bvarate problem s broken up nto a sequence of unvarate problems. The sampler starts wth some ntal arbtrary value y 0 for y and obtans x 0 by generatng a random varable from the condtonal dstrbuton f(x/y = y 0 ). The sampler then uses x 0 to generate a new value of y 1, drawng from the condtonal dstrbuton based on the value x 0, f(y/x = x 0 ). If f(x/y) and f(y/x)) are standard statstcal dstrbutons (e.g. Gamma) then these draws are farly straghtforward to obtan. However, f these dstrbutons are nonstandard then Gbbs samplng s combned wth a generc samplng algorthm such as ARS or more effcent Random Walk Metropols algorthms 6 to draw randomly from the condtonal dstrbutons. If ths process runs for k teratons a k x 2 grd wth values s populated, where the rows of the grd relate to teratons of the Gbbs sampler, and the columns relate to varables x and y. The samplng process s as follows: x f ( x / y = y 1) ( y / x ) ~ y ~ f = x It s worth hghlghtng that at each teraton only the most recent nformaton to date for the other varable s used, whch s the same as n case of a Markov chan. Once a suffcent number of draws (so called burn-n sample) s smulated from the jont dstrbuton, the Gbbs sequence converges to a statonary (equlbrum or target) dstrbuton that s ndependent of the startng/ntal values. In practce, t s common to run the sampler for an addtonal m smulatons after the burn-n thus ensurng a random sample from the jont dstrbuton. An alternatve approach would be to generate several samples of length m each startng from a dfferent ntal value. There are a number of convergence dagnostcs used to assess whether a Gbbs sample has converged. These nclude some formal tests such as the Geweke test or the Raftery- Lews test 7. However, one should always vsually nspect parameter values generated by the Gbbs sampler plotted aganst the number of teratons and/or check the autocorrelaton n the smulated sample of parameter values. It s worth notng that these and other tests are readly avalable wthn WnBUGS. The Gbbs sampler has been mplemented n a statstcal software WnBUGS developed by the MRC Bostatstcs Unt at the Unversty of Cambrdge. The BUGS (Bayesan nference Usng Gbbs Samplng) project started n 1989 and ts key purpose has been to desgn a 6 E.g. Adaptve Rejecton Metropols Samplng (Glks, Best and Tan [14]) 7 See Walsh [25] for a descrpton of these tests Page 10 of 26
flexble software for the Bayesan analyss of complex statstcal models usng MCMC methods. The BUGS Project webste s found at www.mrc-bsu.cam.ac.uk/bugs/. The WnBUGS software s supplemented wth an extensve manual ncludng examples and tps on how a Bayesan model can be mplemented wthn WnBUGS. In addton, there have been a number of papers dscussng the use of WnBUGS n the context of varous actuaral applcatons (e.g. Scollnk [21]). Although WnBUGS s a robust and farly straghtforward software to use, t s not free from some practcal mplementaton ssues e.g.: Standard statstcal dstrbutons and models are easy to mplement wthn WnBUGS. For non-standard dstrbutons some workaround s requred usng the so-called zeros or ones trcks. There can be consderable numercal overflow/underflow ssues, slowng down the smulaton process or n some cases makng t mpossble to run. It s often a good dea to scale down or up all numbers, so there are no very large or very small values handled by WnBUGS. The error messages are sometmes qute unhelpful. 2.5. Advantages and dsadvantages Concludng the theoretcal part of the paper, t s mportant to consder some key propertes of Bayesan stochastc reservng (both advantages and possble dsadvantages). These are summarsed n Table 2.2 below. Table 2.2: Advantages and dsadvantages of Bayesan stochastc reservng Advantages Requres a completely specfed statstcal model, ensurng clarty of underlyng assumptons Flexble n applcaton and not lmted to any partcular model Allows explct modellng of varous sources of uncertanty Allows ncorporaton of adjustment for uncertantes not ncluded n the past data Automatcally produces full dstrbuton of outcomes Does not generate pseudo data and so s not mpacted by ssues sometmes affectng bootstrappng (e.g. a lmted set of combnatons of resduals, the possblty of negatve pseudo-data at the start of a trangle) Farly easy to mplement usng ether WnBUGS or a varety of programmng languages, once the underlyng theory s understood Dsadvantages Theoretcally more sophstcated approach The mechansm underlyng the Bayesan approach s less open to manpulaton than bootstrappng methods 8 Selecton of pror dstrbutons may be problematc May be seen as a black box approach 8 Note that ths dsadvantage could be also seen as an advantage, snce t lmts mplementaton errors. Page 11 of 26
The above table shows that Bayesan stochastc reservng offers some benefts compared to other stochastc reservng methods. In partcular, ths approach s more flexble than for example the bootstrappng approach. England and Verrall [10] showed that the Bayesan and bootstrappng approaches essentally produce the same results, when non-nformatve pror dstrbutons are used wthn the Bayesan approach. However f one would lke to ncorporate judgement regardng parameters/parameter dstrbutons underlyng a partcular statstcal model or combne together several statstcal models, the Bayesan approach s the preferred (or n most cases the only) opton. Any stochastc reservng technque only ncorporates uncertantes nherent n the past data and an allowance for any other uncertantes s often made separately. The key beneft of the Bayesan approach s that t provdes a flexble and sound mechansm to allow for these other uncertantes. Actuaral judgement and external nformaton regardng uncertantes not reflected n the past data can be allowed for wthn Bayesan stochastc reservng n a number of ways and there are papers that have shown some examples of such mplementatons e.g.: Verrall and England [24] showed how external nformaton and actuaral judgement could be ncorporated n the development factors for a partcular Bayesan model (the Negatve Bnomal model) underlyng the chan ladder method. Verrall [23] showed how external nformaton and actuaral judgement about accdent years could be ncorporated nto two Bayesan models (the Negatve Bnomal and Over-dspersed Posson models) usng the mechansm of the Bornhuetter-Ferguson method. Scollnk [22] also showed how the mechansm of the Bornhuetter-Ferguson method could be used to ncorporate external nformaton and actuaral judgement about accdent years nto the dstrbuton of outstandng losses. Note that ths method has been appled n the case study n the second part of ths paper. Although the above papers are qute comprehensve, I acknowledge that there s stll more research to be conducted n ths area. The key dsadvantage of Bayesan stochastc reservng, whch may dscourage actuares from choosng ths approach, s ts apparent complexty compared to other stochastc reservng methods. The mathematcs looks qute complcated and the mplementaton may requre some sophstcated samplng algorthms (e.g. MCMC methods or ARS). However, I beleve that once some basc concepts from Bayes theory are understood, the mplementaton becomes farly straghtforward, especally when one uses WnBUGS or other software e.g. Igloo wth ExtrEMB [16]. Page 12 of 26
Stochastc reservng case study usng a Bayesan approach 3. Case study Ths part of the paper presents an applcaton of the Bayesan approach n a stochastc reservng context. All modellng for the purpose of ths case study has been conducted usng WnBUGS. The WnBUGS code used s shown n the Appendx. The case study uses a trangulaton of clam payments relatng to Automatc Facultatve General Lablty (excludng Asbestos and Envronmental) from the Hstorcal Loss Development Study [15], prevously used by other authors ncludng England and Verrall [9] and Verrall [23]. In addton, I have created dummy earned premums for each accdent year. The trangle and earned premums are shown n Table 3.1. 3.1. Overvew of mplemented models and the modellng strategy Ths secton essentally follows the general Bayesan modellng process dscussed n secton 2.1, unless noted otherwse. In order to present Bayesan stochastc reservng n the most accessble way, I have chosen a farly straghtforward statstcal model underlyng the chan ladder method. Ths model has then been extended usng the mechancs of the Bornhuetter-Ferguson method ncorporated wthn a Bayesan approach. The statstcal chan ladder model s specfed as an ODP model, defned by Renshaw and Verrall [19]. The Bayesan Bornhuetter-Ferguson extenson (BF model) s based on an approach prevously consdered by Scollnk [22]. Over-Dspersed Posson Chan Ladder Model The key to the ODP model s the dervaton of the over-dspersed Posson dstrbuton, whch follows from an observaton that f X ~ Posson (µ), then Y=φX has the over-dspersed Posson dstrbuton, wth mean φµ and varance φ 2 µ. φ s called the over-dsperson parameter and s generally greater than 1. The ODP model for the chan ladder method can be specfed as follows: Cj / x, y, ϕ ~ ndependent over-dspersed Posson, wth mean x y j, and j= x y ~ ndependent non-nformatve Gamma dstrbutons 9, j n 1 y j = 1 x and y are parameter vectors relatng to the rows (orgn years 1,, n) and columns (development years 1,, n), respectvely, of the data trangle. The row parameters x can be nterpreted as expected ultmate clams cost for the -th accdent year and the column parameters y j as the proporton of ultmate clams emergng n the j-th development year. The above specfcaton means that past and future ncremental payments follow ndependent over-dspersed Posson and the row and column parameters have nonnformatve pror dstrbutons. For smplcty, the over-dsperson parameter φ s constant across all development perods and estmated outsde the model usng maxmum lkelhood estmaton. As an alternatve, one could assume φ vares by development year (as shown 9 Refer to the WnBUGS code for detals Page 13 of 26
by England and Verrall [10]) and/or follows some pror dstrbuton 10 as s the case for the row and column parameters. Usng the general Bayesan notaton ntroduced n secton 2.2 and consstent wth steps 5 and 6 of the general Bayesan modellng process: The posteror dstrbuton for the row and column parameters satsfes the followng proportonalty: f ( x, y / c, ϕ ) f ( C / x, y, ϕ ) π ( x) π ( y) + j n + 1 j For the unobserved values n the lower-rght trangle C CL j (for +j > n+1), the predctve dstrbutons are: f CL ( Cj / c, ϕ ) f ( Cj / x, y, ϕ ) f ( x, y / c, ϕ ) = dxdy Snce the posteror dstrbuton f(x,y/c,φ) cannot be recognsed as a standard statstcal dstrbuton and there are multple parameters, t s necessary to mplement ths model n WnBUGS, where the Gbbs sampler along wth ARS are used to generate random draws of row and column parameters from ths dstrbuton. The generated parameters are CL ncorporated nto the ODP dstrbutons for unobserved values C j to derve the dstrbuton of undscounted outstandng clams 11. Bornhuetter-Ferguson Model As already noted, the BF model s an extenson of the chan ladder model. The basc mechansm to derve the ultmate clams cost for each -th accdent year s consstent wth the Bornhuetter-Ferguson method (Bornhuetter and Ferguson [4]) and I assume that the reader s already famlar wth ths approach. The stochastc element s added to the Bornhuetter-Ferguson method va assumed ultmate loss ratos for each accdent year. In addton, the proportons of the BF-based ultmate clams cost emergng n each j-th development year are derved from the chan ladder factors smulated as part of the ODP model. BF The modellng process to obtan the unobserved values C j (for +j > n+1) s slghtly dfferent from the general Bayesan modellng process descrbed n secton 2.1. For ths reason, I nclude here a more detaled explanaton of the varous steps of ths process. 10 Such an approach was proposed by Scollnk (see Antono, Berlant and Hoedemakers [1]), where φ was treated as the thrd parameter n the ODP model and followed a Gamma dstrbuton. 11 For completeness, t s worth notng that the mplementaton n WnBUGS uses a quas-lkelhood approach so that the observed and future payments are not restrcted to postve ntegers. See Verrall [23] and England and Verrall [10] for more dscusson. Page 14 of 26
In The pror parameters of the model are the ntal ultmate loss ratos LR based on some external knowledge. The observed data are the ultmate loss ratos obtaned usng the ODP model LR. The statstcal model s specfed as follows: ODP LR LR In ODP ~ Independent Normal / LR In ~ Independent ( µ, σ (1) ) In Normal( LR, σ ) (2) 12 ODP In Although the data dstrbuton (of LR / LR ) assumes Normalty, the In uncondtonal on the LR are derved from the ODP chan ladder model. In ODP The posteror dstrbuton ( LR LR ) f / s obtaned as follows: In ODP In ODP In ( LR / LR ) L( LR LR ) f ( LR ) f / ODP LR The specfcaton of the model and the non-normal dstrbuton of In ODP LR mean that ths posteror dstrbuton ( LR LR ) In ODP LR uncondtonal on f / s complex and the Gbbs sampler and ARS wthn WnBUGS are requred to generate samples of the assumed ultmate loss ratos, e. the ntal ultmate loss ratos condtonal on the ODP-based ultmate loss ratos. The smulated assumed ultmate loss ratos are appled to earned premums to derve BFbased ultmate clam costs across dfferent accdent years. The unobserved values C BF j (for +j > n+1) are then obtaned by applyng the chan ladder factors smulated as part of the ODP model. It s worth notng that these factors are consstent wth the LR used to In ODP obtan the posteror dstrbuton ( LR LR ) f /. It s also mportant to hghlght a partcular credblty mechansm mplemented wthn the BF model. The relatvty between standard devatons of the two normal dstrbutons controls how much weght s gven to the ntal ultmate loss ratos and to the ODP-based ultmate loss ratos. 13 ODP In If standard devatons σ (2) selected for the normal dstrbuton of LR / LR are relatvely lower than σ (1) the resultng assumed ultmate loss ratos are closer to the smulated ODP-based ultmate loss ratos. ODP 12 For the purpose of the case study, I have arbtrarly assumed that standard devatons for both normal dstrbutons are equal to 7.1% and the mean of the dstrbuton of ntal ultmate loss ratos s 71% across all accdent years. Ths mean has been based on the total ultmate loss rato mpled by the determnstc chan ladder (see Table 3.1). 13 Note that ths relatvty s expressed by the weght parameter n the WnBUGS code. Page 15 of 26
3.2. Data Table 3.1 shows a trangle of ncremental payments wth reserves estmated usng the determnstc chan ladder. Earned premums, ultmate loss ratos and chan ladder factors are also shown. Table 3.1: Case study ncremental clam payments and determnstc chan ladder results Accdent Development Year Outstandng Earned Ultmate Year 1 2 3 4 5 6 7 8 9 10 clams premum loss rato 1 5,012 3,257 2,638 898 1,734 2,642 1,828 599 54 172 0 28,975 65% 2 106 4,179 1,111 5,270 3,116 1,817-103 673 535 154 20,478 82% 3 3,410 5,582 4,881 2,268 2,594 3,479 649 603 617 28,984 83% 4 5,655 5,900 4,211 5,500 2,159 2,658 984 1,636 38,432 75% 5 1,092 8,473 6,271 6,333 3,786 225 2,747 47,290 61% 6 1,513 4,932 5,257 1,233 2,917 3,649 24,308 80% 7 557 3,463 6,926 1,368 5,435 23,228 76% 8 1,351 5,596 6,165 10,907 30,721 78% 9 3,133 2,262 10,650 29,611 54% 10 2,063 16,339 29,407 63% Chan ladder factors 2.9994 1.6235 1.2709 1.1717 1.1134 1.0419 1.0333 1.0169 1.0092 52,135 301,434 71% The chan ladder factors n the above table were derved usng the standard chan ladder approach ncorporatng all years of data. No judgement was appled to adjust any of these factors. 3.3. Stochastc modellng results Table 3.2 presents the results from the two Bayesan models dscussed n secton 3.1, ncludng the mean, standard devaton, coeffcent of varaton and 75 th percentle. In addton, the mean ntal, ODP and BF ultmate loss rato are ncluded for comparson. Accdent Table 3.2: Results for the Bayesan Over-Dspersed Posson and Bornhuetter-Ferguson models Standard Devaton ODP model Coeffcent of Varaton 75 th Percentle Standard Devaton BF model Coeffcent of Varaton 75 th Percentle Mean Ultmate Loss Rato ODP Intal model BF model Year Mean Mean 1 0 NA NA NA 0 NA NA NA 71% 65% 65% 2 164 619 378% 0 144 356 247% 107 71% 82% 82% 3 641 1,201 187% 1,087 585 713 122% 799 71% 83% 83% 4 1,688 1,892 112% 2,174 1,609 1,104 69% 2,124 71% 75% 75% 5 2,815 2,343 83% 4,347 2,984 1,375 46% 3,711 71% 61% 62% 6 3,707 2,553 69% 5,434 3,447 962 28% 4,009 71% 80% 79% 7 5,521 3,233 59% 7,607 5,258 1,110 21% 5,916 71% 77% 76% 8 11,070 5,266 48% 14,130 10,420 1,891 18% 11,540 71% 79% 77% 9 10,800 6,293 58% 14,130 12,370 2,577 21% 13,780 71% 55% 60% 10 17,200 14,320 83% 23,910 17,720 6,413 36% 20,850 71% 66% 67% Total 53,606 19,660 37% 64,120 54,538 9,626 18% 59,980 71% 71% 72% There are several mportant observatons that can be made regardng the results of ths analyss: The calculated total mean for the ODP model s hgher than that obtaned from the determnstc chan ladder presented n Table 3.1. There are several reasons for ths dfference ncludng smulaton error, choce of pror dstrbutons for the row/column parameters and the model lnkng the parameters to the mean (Verrall [23]). The results from the ODP model are smlar to the results obtaned by England and Verrall [9] usng the bootstrappng method for the same ODP model and the same Page 16 of 26
dataset. Ths s a consequence of the choce of non-nformatve pror dstrbutons for the row and column parameters. In such a stuaton, the forecast future payments are drven by the data ncluded n the trangle (through the lkelhood functon) and are not nfluenced by the pror dstrbutons of parameters. The total mean from the BF model s hgher than for the ODP model. Ths s drven by the last two accdent years where the ODP-based ultmate loss ratos are lower than the correspondng ntal ultmate loss ratos. The coeffcent of varaton for the BF model s lower than the coeffcent of varaton for the ODP model. Ths s due to the choce of farly low standard devatons for the ntal ultmate loss ratos. The last three columns show the mean ntal, ODP-based and BF-based (or assumed) ultmate loss ratos. The BF-based ultmate loss rato s always between the mean ntal and ODP-based ultmate loss ratos. Ths s due to the mechansm of the Bayesan BF model mplemented n the case study. The assumed ultmate loss ratos are ntally based on the smulated ntal ultmate loss ratos and then updated to reflect the smulated ODP-based ultmate loss ratos. The extent to whch the mean BF-based ultmate loss rato s closer to ether the mean ntal or the mean ODPbased ultmate loss rato s controlled by the relatvty between assumed standard devatons σ (1) and σ (2), ntroduced n secton 3.1. Fgure 3.1 below compares the probablty dstrbuton functons of the total undscounted outstandng clams produced by both Bayesan models. These dstrbutons are automatcally produced by the Bayesan approach. Fgure 3.1: Probablty dstrbutons for the total undscounted outstandng clams 3,000 2,500 2,000 Frequency 1,500 1,000 500 0 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000 180,000 200,000 220,000 240,000 Outstandng Clams ODP model BF model In lne wth the results presented n Table 3.2, both dstrbutons are postvely skewed. However, the dstrbuton produced by the BF model exhbts a lower level of skewness. Ths s due to the Normal dstrbuton used for the ntal ultmate loss ratos, whch effectvely reduces the skewness nherent n the probablty dstrbuton functon obtaned from the ODP model. All of the results presented were produced usng 60,000 MCMC teratons. The burn-n sample to remove the mpact of ntal values for the parameters and ensure convergence was 25,000 teratons. I have also conducted varous checks of convergence of the Markov chan, ncludng a vsual nspecton of plots of generated values and autocorrelatons. It s Page 17 of 26
worth notng that these and other tests are readly avalable through WnBUGS as part of the standard modellng output. The presented results were based on a partcular specfcaton of the two Bayesan models. It s however possble to make some further modfcatons to these models, ncludng the ntroducton of nformatve pror dstrbutons for the row and column parameters n the ODP model or changng standard devatons σ (1) and σ (2) n the BF model. More sgnfcant changes could also nclude: Introducton of a dstrbuton for the over-dsperson parameter n the ODP model Choce of a non-normal dstrbuton for the ntal ultmate loss rato n the BF model Choce of other than Gamma dstrbutons for the row and column parameters n the ODP model Page 18 of 26
Stochastc reservng case study usng a Bayesan approach 4. Summary and conclusons The man purpose of ths paper s to provde a hgh-level overvew of the theory underlyng the Bayesan approaches and to show how they could be appled n the context of stochastc reservng. In partcular, the theoretcal dscusson s supported by a case study where two statstcal models (underlyng the chan ladder and Bornhuetter-Fergusson methods) have been mplemented usng Bayesan approaches. I acknowledge that these models have a number of lmtatons as hghlghted by other researchers. In partcular, the ODP model s not generally approprate for data wth negatve ncremental amounts (e.g. ncurred cost data), whle the Normal dstrbuton assumed for the ntal ultmate loss ratos n the BF model may not be sutable, gven the postve skewness nherent n loss dstrbutons observed n general nsurance. It s however mportant to note that the presented models are farly smple to apply and n my opnon qute adequate for llustraton purposes n ths paper. In practce, t may be more approprate to ncorporate modfcatons mentoned n sectons 3.1 and 3.3 or to use a completely dfferent Bayesan model. My experence s that Bayesan methods are not commonly used n stochastc reservng n Australa. I hope that ths paper has hghlghted some of the key benefts of these approaches and wll encourage Australan actuares to nclude these methods n ther stochastc reservng toolkt. As noted n prevous sectons, I have ncluded the WnBUGS code used for the case study n the Appendx. I would encourage practtoners to download a free copy of WnBUGS and experment wth ths code. I have also ncluded a comprehensve bblography on Bayesan stochastc reservng for readers nterested n further research, ncludng some papers on the applcatons of WnBUGS n actuaral scence. One of the key benefts of Bayesan stochastc reservng s ts flexblty and, n partcular, capablty to ncorporate actuaral judgement and external nformaton nto the stochastc reservng process. The case study also llustrated how a Bayesan BF model could be used to ncorporate actuaral judgement/external nformaton. In partcular, my specfcaton of ths model led to a dramatc decrease n the volatlty of smulated undscounted outstandng clams compared to the results from the ODP model. Havng sad that, I beleve that there s stll more research requred n the context of the mplementaton of actuaral judgement and external nformaton nto Bayesan stochastc reservng process. Page 19 of 26
Stochastc reservng case study usng a Bayesan approach 5. Acknowledgements I would lke to express my grattude and apprecaton to the peer-revewers of ths paper Karl Marshall and Peter England. They have both provded valuable peer-revew feedback and general support n wrtng ths paper. I also would lke to thank my employer, Quantum, for the support provded n wrtng ths paper. Whlst, as the author, I have made every effort to provde accurate and current nformaton I do not warrant that the nformaton contaned heren s n every respect accurate and complete. I expressly dsclam any responsblty for any errors or omssons or for any relance placed on ths paper. If you have any comments on or queres about the paper, please feel free to emal me at: bartoszp@quantum.com.au Page 20 of 26
Stochastc reservng case study usng a Bayesan approach 6. Appendx Ths secton presents WnBUGS code used n the case study. Ths code s based on the materal prevously presented by Verrall[23] and Scollnk[22]. There are several aspects of the exstng code where further model extensons could be ncorporated ncludng: Replacng the ODP dstrbuton wth a Gamma dstrbuton for future observatons, so that forecast ncremental payments look more realstc Incorporaton of a full Bayesan approach for the over-dsperson parameter, as descrbed by Sollnk n Antono, Berlant and Hoedemakers [1] Changes to the weght parameter and the dstrbuton for ntal and assumed ultmate loss ratos Choce of dfferent pror dstrbutons for the row and column parameters n the ODP model It s mportant to note that f a dfferent data set s used wth the code below the followng nputs wll need to be changed or adjusted: Parameters a[1] and ph (England and Verrall [10] show formulas how ph can be estmated from the data) Scale and shape parameters of the Gamma pror dstrbutons for parameters a and p1 Intal values for a and p1 model { # ODP model for data: # x are row parameters # y are column parameters # ph s the over-dsperson or scale parameter and t s a plug-n estmate n the code below for( n 1 : 55 ) { Z[] <- C[]/1000 log(mu[]) <- x[row[]] + y[col[]] # Zeros trck to cope wth non-postve nteger data: zeros[] <- 0 zeros[] ~ dpos(possmean[]) PossMean[] <- (mu[] - Z[]*log(mu[]) + loggam(z[] + 1))/ph # MINUS log lkelhood } # ODP model for future observatons: for( n 56 : 100 ) { mu2[] <- mu[]/ph C[] ~ dpos(mu2[]); log(mu[]) <- x[row[]] + y[col[]] Z[] <- ph*c[] } for( n 1 : 100 ) { ft[] <- Z[]*1000 } ph <- 1.08676 # as per Verrall [23] a[1] <- 18.834 # set equal to the ncremental payment n the latest development year of the frst accdent year x[1] <- log(a[1]) Page 21 of 26
# Pror dstrbutons for row parameters: for (k n 2:10) { a[k] ~ dgamma(0.000001,0.0000001) x[k] <- log(a[k]) } # Pror dstrbutons for column parameters: # p1 are ntermedate parameters to derve ncremental pad developments # p are ncremental payout ratos for (k n 1:10) { s <- sum(p1[1:10]) for (k n 1:10) { p1[k] ~ dgamma(0.00001,0.0001) } p[k] <- p1[k]/s y[k] <- log(p[k]) } #Chan ladder factors requred for the BF method # pc are cumulatve payout ratos # CLfact are chan ladder factors # CumCLfact are cumulatve chan ladder factors pc[1] <- p[1] for (k n 2:10) {pc[k] <- pc[k-1] + p[k]} for (k n 1:9) {CLFact[k] <- pc[k+1]/pc[k]} for (j n 1:9) { for( n 1 : 10 ) { for ( n 1:10) {CLFactByAccYr[,j] <- CLFact[j]} } for( j n 1 : 9 ) {CumCLfact[, j ] <- prod(clfactbyaccyr[, j : 9 ] )} } #Chan ladder ODP outstandng clams by accdent year and n total: CLOS[1] <- 0 CLOS[2] <- ft[56] CLOS[3] <- sum(ft[57:58]) CLOS[4] <- sum(ft[59:61]) CLOS[5] <- sum(ft[62:65]) CLOS[6] <- sum(ft[66:70]) CLOS[7] <- sum(ft[71:76]) CLOS[8] <- sum(ft[77:83]) CLOS[9] <- sum(ft[84:91]) CLOS[10] <- sum(ft[92:100]) CLTotalOS <- sum(clos[2:10]) #BF model for future observatons: # lossrato[, 1 ] are ntal ultmate loss ratos # lossrato[, 2 ] are ODP ultmate loss ratos # lossrato[, 3 ] are ntermedate step to acheve assumed loss ratos; # note that the am s actually to obtan # lossrato[, 3 ] ~ dnorm( lossrato[, 1 ], weght*taulr [ ] ); however ths s not possble and # the zeros trck needs to be appled nstead # weght s a parameter that specfes how much weght s gven to the ntal ultmate loss rato, # the lower t s the less weght s gven to the ODP ultmate loss rato weght <- 1 for( n 1 : 10 ) { lossrato[, 1 ] ~ dnorm( meanlr[ ], taulr [ ] ) taulr [ ] <- 1 / pow(stdlr [ ], 2) lossrato[, 2 ] <- (sum( CLOS[ ] )+padtodate [ ]) / premum[ ] Page 22 of 26
lossrato[, 3 ] <- cut( lossrato[, 2 ] ) # Use the zeros trck code below n place of lne above n order to avod defnng # the lossrato[, 3 ] node twce. zero[ ] <- 0 zero[ ] ~ dpos( ph2[ ] ) ph2[ ] <- weght * taulr[ ] * ( lossrato[, 3 ] - lossrato[, 1 ] ) * ( lossrato[, 3 ] - lossrato[, 1 ] ) * 0.5 ultmate[ ] <- premum[ ] * lossrato[, 1 ] } for( n 1 : 10 ) { BFY[, 1 ] <- ultmate[ ] * 1/CumCLfact[, 1 ] for( j n 2 : 9 ) {BFY[, j ] <- ultmate[ ] * ( 1/CumCLfact[, j ] - 1/CumCLfact[, j - 1 ] )} BFY[, 10 ] <- ultmate[ ] * ( 1-1/CumCLfact[, 9 ] ) } #BF outstandng clams by accdent year and n total: BFOS[1] <- 0 for( n 2 : 10 ) {BFOS[ ] <- sum( BFY[, 10 + 2 - : 10 ] )} BFTotalOS <- sum(bfos[1:10]) } # DATA lst( row = c(1,1,1,1,1,1,1,1,1,1, 2,2,2,2,2,2,2,2,2, 3,3,3,3,3,3,3,3, 4,4,4,4,4,4,4, 5,5,5,5,5,5, 6,6,6,6,6, 7,7,7,7, 8,8,8, 9,9, 10, 2, 3,3, 4,4,4, 5,5,5,5, 6,6,6,6,6, 7,7,7,7,7,7, 8,8,8,8,8,8,8, 9,9,9,9,9,9,9,9, 10,10,10,10,10,10,10,10,10), col = c(1,2,3,4,5,6,7,8,9,10, 1,2,3,4,5,6,7,8,9, 1,2,3,4,5,6,7,8, 1,2,3,4,5,6,7, 1,2,3,4,5,6, 1,2,3,4,5, 1,2,3,4, 1,2,3, 1,2, 1, 10, 9,10, 8,9,10, 7,8,9,10, 6,7,8,9,10, 5,6,7,8,9,10, 4,5,6,7,8,9,10, 3,4,5,6,7,8,9,10, 2,3,4,5,6,7,8,9,10), C = c(5012,3257,2638,898,1734,2642,1828,599,54,172, 106,4179,1111,5270,3116,1817,-103,673,535, 3410,5582,4881,2268,2594,3479,649,603, 5655,5900,4211,5500,2159,2658,984, 1092,8473,6271,6333,3786,225, 1513,4932,5257,1233,2917, Page 23 of 26
557,3463,6926,1368, 1351,5596,6165, 3133,2262, 2063, NA, NA,NA, NA,NA,NA, NA,NA,NA,NA, NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA,NA,NA,NA), padtodate=c(18834,16704,23466,27067,26180,15852,12314,13112,5395,2063), premum=c(28975,20478,28984,38432,47290,24308,23228,30721,29611,29407), meanlr=c(0.71,0.71,0.71,0.71,0.71,0.71,0.71,0.71,0.71,0.71), stdlr=c(0.071,0.071,0.071,0.071,0.071,0.071,0.071,0.071,0.071,0.071)) # INITIAL VALUES # Note that for the loss ratos n the BF model, the ntal values need to be generated wthn WnBUGS. # Alternatvely they could be add here as another vector lst( a = c(na,20,20,20,20,20,20,20,20,20), p1 = c(0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1 ), C = c(na,na,na,na,na,na,na,na,na,na, NA,NA,NA,NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA,NA, NA,NA,NA,NA,NA, NA,NA,NA,NA, NA,NA,NA, NA,NA, NA, 0, 0,0, 0,0,0, 0,0,0,0, 0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0)) Page 24 of 26
Stochastc reservng case study usng a Bayesan approach 7. Bblography 1. Antono, K., J. Berlant and T. Hoedemakers (2005): Dscusson of papers already publshed: A Bayesan Generalzed Lnear Model for the Bornhuetter-Ferguson Method of Clams Reservng, R. J. Verrall, July 2004. North Amercan Actuaral Journal. Avalable from: http://www.soa.org/lbrary/journals/north-amercan-actuaraljournal/2005/july/naaj0503-8.pdf 2. Berger, J. O. (1985): Statstcal Decson Theory and Bayesan Analyss. 2 nd edton New York, Sprnger-Verlag 3. Bernardo, J. M. and A. F. M. Smth (1994): Bayesan Theory. New York, John Wley & Sons 4. Bornhuetter, R. L. and R. E. Ferguson (1972): The Actuary and IBNR. Proceedngs of the Casualty Actuaral Socety 59: pages 181 95 5. Carln B. P. and T. A. Lous (2000): Bayes and Emprcal Bayes Methods for Data Analyss: texts n statstcal scence. 2 nd edton, Chapman and Hall/CRC 6. de Alba, E. (2002): Bayesan Estmaton of Outstandng Clam Reserves, North Amercan Actuaral Journal 6 (4): pages 1-20 7. Dellaportas, P. and A. F. M. Smth (1993): Bayesan nference for generalzed lnear and proportonal hazards models va Gbbs samplng. Appled Statstcs, 42(3): pages 443-459 8. England, P. D. (2002): Bayesan Stochastc Reservng: Background Statstcs. EMB presentaton 9. England, P. D. and R. J. Verrall (2002): Stochastc clams reservng n general nsurance. Insttute of Actuares and Faculty of Actuares. Avalable from http://www.actuares.org.uk/ data/assets/pdf_fle/0014/31721/sm0201.pdf 10. England, P. D. and R. J. Verrall (2006): Predctve dstrbutons of outstandng clams n general nsurance. Annals of Actuaral Scence, 1, II: pages 221-270 11. Geman, S. and D. Geman (1984): Stochastc relaxaton, Gbbs dstrbuton and Bayesan restoraton of mages. IEE Transactons on Pattern Analyss and Machne Intellgence 6: pages 721 741 12. Gelfand, A. E. and A. F. M. Smth (1990): Samplng-based approaches to calculatng margnal denstes. J. Am. Stat. Asso. 85: pages 398 409 13. Glks, W. R. and P. Wld (1992): Adaptve rejecton samplng for Gbbs samplng. Appled Statstcs, 41(2): pages 337-348 14. Glks, W. R., Best N. G. and K. K. C. Tan (1994): Adaptve Rejecton Metropols Samplng wthn Gbbs Samplng. Appled Statstcs 44: pages 455 472 15. Hstorcal Loss Development Study (1991). Rensurance Assocaton of Amerca. Washngton D.C. 16. Igloo Professonal wth ExtrEMB (2007). Igloo Professonal wth ExtrEMB v3.1.7. EMB Software Ltd, Epsom, U.K. 17. Ntzoufras, I. and P. Dellaportas (2002): Bayesan modellng of outstandng labltes ncorporatng clam count uncertanty. North Amercan Actuaral Journal, 6(1): pages 113-136 18. O Dowd C., Smth A. and P. Hardy (2005): A Framework for Estmatng Uncertanty n Insurance Clams Cost. Proceedngs of the 15 th General Insurance Semnar 19. Renshaw, A. E. and R. J. Verrall (1998): A Stochastc Model Underlyng the Chan Ladder Technque, Brtsh Actuaral Journal 4(4): pages 903 23 Page 25 of 26
20. Rsk Margns Taskforce (2008): A Framework for Assessng Rsk Margns. Insttute of Actuares Australa, 16 th General Insurance Semnar 21. Scollnk, D. P. M. (2001): Actuaral modellng wth MCMC and BUGS. North Amercan Actuaral Journal, 5(2): pages 96-125 22. Scollnk, D.P.M. (2004): Bayesan Reservng Models Inspred by Chan Ladder Methods and Implemented Usng WnBUGS. Avalable from: http://www.soa.org/lbrary/proceedngs/arch/2004/arch04v38n2_3.pdf 23. Verrall, R. J. (2004): A Bayesan generalzed lnear model for the Bornhuetter- Ferguson method of clams reservng. North Amercan Actuaral Journal, 8(3) 24. Verrall, R. J. and P. D. England (2005): Incorporatng expert opnon nto a stochastc model for the chan-ladder technque. Insurance: Mathematcs and Economcs 37: pages 355 370 25. Walsh, B. (2004): Markov Chan Monte Carlo and Gbbs Samplng. Lecture Notes, Department of Statstcs at Columba Unversty. Avalable from: http://www.stat.columba.edu/~lam/teachng/neurostat-spr07/papers/mcmc/mcmcgbbs-ntro.pdf Page 26 of 26