Chain Ladder Method: Bayesian Bootstrap versus Classical Bootstrap

Transcription

1 Chan Ladder Method: Bayesan Bootstrap versus Classcal Bootstrap Gareth W. Peters 1,2 Maro V. Wüthrch 3 Pavel V. Shevchenko 2 1 UNSW Mathematcs and Statstcs Department, Sydney, 2052, Australa; emal: peterga@maths.unsw.edu.au 2 CSIRO Sydney, Locked Bag 17, North Ryde, NSW, 1670, Australa 3 ETH Zurch, Department of Mathematcs, CH-8092 Zurch, Swtzerland Preprnt submtted to Elsever February 28, 2009

2 Chan Ladder Method: Bayesan Bootstrap versus Classcal Bootstrap Gareth W. Peters 1,2 Maro V. Wüthrch 3 Pavel V. Shevchenko 2 Abstract The ntenton of ths paper s to analyse the mean square error of predcton MSEP) under the dstrbuton-free chan ladder DFCL) clams reservng method. We compare the estmaton obtaned from the classcal bootstrap method wth the one obtaned from a Bayesan bootstrap. To acheve ths n the DFCL model we develop a novel approxmate Bayesan computaton ABC) samplng algorthm to obtan the emprcal posteror dstrbuton. We need an ABC samplng algorthm because we work n a dstrbuton-free settng. The use of ths ABC methodology combned wth bootstrap allows us to obtan samples from the ntractable posteror dstrbuton wthout the requrement of any dstrbutonal assumptons. Ths then enables us to calculate the MSEP and other rsk measures lke Value-at-Rsk. Key words: Clams reservng, dstrbuton-free chan ladder, mean square error of predcton, Bayesan chan ladder, approxmate Bayesan computaton, Markov chan Monte Carlo, adapton, annealng, bootstrap Preprnt submtted to Elsever February 28, 2009

3 1. Motvaton The dstrbuton-free chan ladder model DFCL) of Mack [13] s probably the most popular model for stochastc clams reservng. In ths paper we use a tme seres formulaton of the DFCL model whch allows for bootstrappng the clams reserves. An mportant aspect of ths model s that t can provde a ustfcaton for the classcal determnstc chan ladder CL) algorthm whch s not founded on an underlyng stochastc model. Moreover, t allows for the study of predcton uncertantes. Note that there are dfferent stochastc models that lead to the CL reserves see for example Wüthrch-Merz [22], Secton 3.2). In the present paper we use the DFCL formulaton to reproduce the CL reserves. Ths paper analyses the parameter estmates n the DFCL model, the assocated clams reserves and the mean square errors of predcton MSEP) from both the frequentst perspectve and a contrastng Bayesan vew. In dong so we analyse CL pont estmators for parameters of the DFCL model, the resultng estmated reserves and the assocated MSEP from the classcal perspectve. These nclude bootstrap estmated predcton errors obtaned va a process of condtonal back propagaton. These classcal frequentst estmators are compared to Bayesan pont estmators. The Bayesan estmates consdered are the maxmum a posteror MAP) and the mnmum mean square error MMSE) estmators and the assocated Bayesan estmated reserves condtonal on such pont estmators. In addton, snce n the Bayesan settng we obtan samples from the posteror on the parameters we use these along wth the MSEP obtaned by the pont estmators to obtan assocated posteror predctve ntervals to be compared wth the classcal bootstrap procedures. We then robustfy the predcton of reserves and analyse the resultant MSEP when we ntegrate out the nfluence of the predcton of unknown varance parameters n the DFCL model, agan achevable snce n the Bayesan settng we obtan samples from the ont posteror for the CL factors and the varances. Ths requres that we develop a Bayesan CL model for the DFCL model whch makes no parametrc assumptons on the form of the lkelhood functon, see also Gsler-Wüthrch [11]. Ths s unlke the works of Yao [23] and Peters-Shevchenko-Wüthrch [19] that assume explct dstrbutons n order to construct the posteror dstrbutons n the Bayesan context. Instead we demonstrate how to work drectly wth the ntractable lkelhood functons and the resultng ntractable posteror dstrbuton, usng approprate dstance functons. In ths regard we demonstrate that we do not need to make any parametrc assumptons to perform posteror nference, avodng the often poor model assumptons made, as for example n the paper of Yao 3

4 [23]. To acheve ths we develop a novel approxmate Bayesan computaton ABC) samplng algorthm to obtan samples from the posteror. We develop the frst use of ABC methodology combned wth bootstrap whch allows us to obtan samples from the ntractable posteror dstrbuton wthout the requrement of any dstrbutonal assumptons or even the requrement on generatng of observatons from the ntractable lkelhood model. Instead, we smply requre that certan model assumptons are not volated n our condtonal back propagaton bootstrap procedure. In ths context we analyse several aspects of the ABC methodology, ncludng senstvty studes and convergence dagnostcs, before we make use of the algorthm to perform a detaled comparson wth classcal models. Outlne of ths paper. The paper begns wth a presentaton of the clams reservng problem and then presents the model we shall consder. Ths s followed by the descrpton of the classcal chan ladder algorthm and the constructon of a Bayesan model that can be used to estmate the parameters of the model. The Bayesan model s constructed n a dstrbuton-free settng. Followng ths s a dscusson on classcal versus Bayesan parameter estmators along wth a bootstrap based procedure for estmaton of parameter uncertanty n the classcal settng. The next secton presents the methodology of ABC coupled wth a novel bootstrap based samplng procedure whch wll allow us to work drectly wth the dstrbuton-free Bayesan model. We then llustrate the developed algorthm on a synthetc data set and the real data set, comparng performance to the classcal results and those obtaned va credblty theory. 2. Clams development trangle and DFCL model We brefly outlne the clams development trangle structure we utlse n formulaton of our models. Assume there s a run-off trangle contanng clams development data wth the structure gven n Table 1. Assume that C, are cumulatve clams wth ndces {0,...,I} and {0,...,J}, where denotes the accdent year and denotes the development year cumulatve clams can refer to payments, clams ncurred, etc). We make the smplfyng assumpton that the number of accdent years s equal to the number of observed development perods.e. I = J. At tme I, we have observatons D I = {C, ; + I}, 2.1) 4

5 accdent development years year I 0 1 observed random varables C, D I.. to be predcted C, D c I I 1 I Table 1: Clams development trangles. and for clams reservng at tme I we need to predct the future clams DI c = {C, ; + > I, I, J}. 2.2) Moreover, we defne for {0,...,I} the set B = {C,k ; + k I,0 k }, e.g. B 0 s the frst column n Table Classcal chan ladder algorthm In the classcal determnstc) chan ladder algorthm there s no underlyng stochastc model. It s rather a recursve algorthm that was used to estmate the clams reserves and whch has proved to gve good practcal results. It smply nvolves the followng recursve steps to predct unobserved cumulatve clams n D c I. Set Ĉ,I = C,I and for > I Ĉ, = Ĉ, 1 f CL) 1 wth CL factor estmates fcl) 1 = I =0 C, I =0 C. 2.3), 1 Snce ths s a determnstc algorthm t does not allow for quantfcaton of the uncertanty assocated wth the predcted reserves. To analyse the assocated uncertanty there are several stochastc models that reproduce the CL reserves: for example Mack s dstrbuton-free chan ladder model [13], the over-dspersed Posson model see England-Verrall [5]) or the Bayesan chan ladder model see Gsler-Wüthrch [11]). We use a tme seres formulaton of the Bayesan chan ladder model n order to use bootstrap methods and Bayesan nference Bayesan DFCL model We use an addtve tme seres verson of the Bayes chan ladder model Model Assumptons 3.1, n Gsler-Wüthrch [11]). 5

6 Model Assumptons We defne the CL factors by F = F 0,...,F ) and the standard devaton parameters by Ξ = Ξ 0,...,Ξ ). We assume ndependence between all these parameters,.e., the pror densty of F,Ξ) s gven by πf, σ) = =0 πf ) πσ ), 2.4) where πf ) denotes the densty of F and πσ ) denotes the densty of Ξ. 2. Condtonally, gven F = f = f 0,...,f ) and Ξ = σ = σ 0,...,σ ), we have: Cumulatve clams C, n dfferent accdent years are ndependent. Cumulatve clams satsfy the followng tme seres representaton C,+1 = f C, + σ C, ε,+1, 2.5) where condtonally, gven B 0, we have that the resduals ε, are..d. satsfyng E [ε, B 0, F,Ξ] = 0 and Var [ε, B 0, F,Ξ] = 1, 2.6) and P [C, > 0 B 0, F,Ξ] = 1 for all,. Remark. Note that the assumptons on the resduals are slghtly nvolved n order to guarantee that cumulatve clams C, are postve P-a.s. Corollary 2.2. Under Model Assumptons 2.1 we have that condtonally, gven D I, the random varables F 0,...,F, Ξ 0,...,Ξ are ndependent. Proof of Corollary 2.2. The proof s completely analogous to the proof of Theorem 3.2 n Gsler-Wüthrch [11] and follows from pror ndependence of the parameters and the fact that C,+1 only depends on F, Ξ and C, Markov property). In partcular, Corollary 2.2 says that we obtan the followng posteror dstrbuton for F, Ξ), gven D I, π f, σ D I ) = =0 π f D I )π σ D I ). 2.7) Ths has mportant mplcatons for the ABC samplng algorthm developed below. In order to perform the Bayesan analyss we make explct assumptons on the pror dstrbutons of F,Ξ). 6

7 Model Assumptons 2.3. In addton to Model Assumptons 2.1 we assume that the pror model for all parameters {0,...,J 1} s gven by: F Γ α, β ), where Γ α, β ) s a gamma dstrbuton wth mean E [F ] = α β = see 2.3)) and large varance to have dffuse prors. f CL) The varances Ξ 2 IGa, b ), where IGa, b ) s an nverse gamma dstrbuton wth [ ] mean E Ξ 2 = b /a 1) = σ 2CL) see 2.8), below) and large varance. Remarks. 1) The lkelhood model s ntractable, meanng that no densty can be wrtten down analytcally n the DFCL model. We have only made dstrbutonal assumptons on the parameters F,Ξ) but not on the observable cumulatve clams C,. Therefore, a full Bayesan analyss usng analytc posteror dstrbutons cannot be performed. One way out of ths dlemma would be to formulate a full Bayesan model by makng dstrbutonal assumptons ths s, e.g., done n Yao [23]) but then the model s no longer dstrbuton-free. Another approach would be to use credblty methods see Gsler-Wüthrch [11]) but ths only gves statements for second moments. In the present set up we develop ABC methods that allow for a full dstrbutonal answer for the posteror wthout makng explct dstrbutonal assumptons for the cumulatve clams C,. 2) We select the prors to ensure that we mantan several relevant aspects of the DFCL model. Frstly, t s mportant to utlse prors that enforce the strct postvty of the parameters f, σ > 0. We note here that the parametrc Bayesan model developed n Yao [23] faled n ths aspect and therefore we develop an alternatve pror structure that satsfes these requred propertes of the DFCL model. Secondly, our prors are chosen such that they do not contan nformaton,.e. we assume dffuse prors wth large varances Classcal and Bayesan parameter estmators In the classcal CL method, the CL factors are estmated by parameters are estmated by, see e.g. 3.4) Wüthrch-Merz [22], f CL) gven n 2.3). The varance σ 2CL) = 1 I 1 I 1 =0 C, C,+1 C, ) 2 CL) f. 2.8) Note that ths estmator s only well-defned for < I 1. There s a vast lterature and dscusson on the estmaton of tal parameters. We do not enter ths dscusson here but we 7

8 smply choose the estmator gven n Mack [13] for the last varance parameter whch s defned by σ 2CL) = mn { σ 4CL) J 2 σ 2CL) J 3, σ 2CL) J 3, σ2cl) J 2 }. 2.9) In a Bayesan nference context one calculates the posteror dstrbuton of the parameters, gven D I. As n 2.7) we denote ths posteror by π f, σ D I ). Then, there are two commonly used pont estmators n Bayesan analyss that correspond to the posteror mode MAP) and the posteror mean MMSE), respectvely. Gven the posteror ndependence see Corollary 2.2) they are gven by: f MAP) = arg max f π f D I ), 2.10) σ MAP) = arg max σ π σ D I ), 2.11) and f MMSE) = σ MMSE) = f π f D I ) df = E [F D I ], 2.12) σ π σ D I )dσ = E [Ξ D I ]. 2.13) Note that for dffuse pror we fnd see Corollary 5.1 n Gsler-Wüthrch [11]) f MMSE) f CL). 2.14) Hence, usng Corollary 2.2, we obtan the approxmaton E [C,J D I ] = E [E [C,J D I, F,Ξ] D I ] = C,I E = C,I =I C,I =I E [F D I ] = C,I f CL) = Ĉ,J, =I =I F D I f MMSE) 2.15) where on the last lne we have an asymptotc equalty f the dffusvty of the prors πf ) tends to nfnty. Ths s exactly the argument why the Bayesan CL model can be used to ustfy the CL predctors, see Gsler-Wüthrch [11]. 3. Bootstrap and mean square error of predcton Assume that we have calculated the Bayesan predctor or the CL predctor gven n 2.15). Then we would lke to determne the predcton uncertanty,.e., we would lke to study the 8

9 devaton of C,J around ts predctor. If one s only nterested n second moments, the socalled condtonal mean square error of predcton MSEP), one can often estmate the error terms analytcally. However, other uncertanty measures lke Value-at-Rsk VaR) can only be determned numercally. A popular numercal method s the bootstrap method. The bootstrap technque was developed by Efron [3] and extended by Efron-Tbshran [4] and Davson-Hnkley [1]. Ths procedure allows one to obtan nformaton regardng an aggregated dstrbuton gven a sngle realsaton of the data. To apply the bootstrap procedure one ntroduces a mnmal amount of model structure such that resamplng observatons can be acheved usng observed samples of the data. In ths secton we present a bootstrap algorthm n the classcal frequentsts approach,.e., we assume that the CL factors F = f and the standard devaton parameters Ξ = σ gven n Model Assumptons 2.1 are unknown constants. The bootstrap then generates synthetc data denoted by DI that allow for the study of the fluctuatons of f CL) and σ 2CL) for detals see Wüthrch-Merz [22], Secton 7.4). In the presented text we restrct ourselves to the condtonal resamplng approach presented n Secton of Wüthrch-Merz [22]. Non-parametrc classcal bootstrap condtonal verson). 1. Calculate estmated resduals ε, for + I, > 0, condtonal on the estmators and σ 2CL) 0: and the observed data D I: CL) ε, = ε, f 1, σcl) 1 ) = C, CL) f 1 C, 1 σ CL) 1 C, These resduals ε, ) + I gve the emprcal bootstrap dstrbuton F DI. 3. Sample..d. resduals ε, F DI for + I, > Generate bootstrap observatons condtonal resamplng) f CL) 0: C, = f CL) 1 C, 1 + σ CL) 1 C, 1 ε,, whch defnes DI = DI f CL), σ CL) ). Note that for the uncondtonal verson of bootstrap we should generate C, CL) = f 1 C, 1 + σcl) 1 C, 1 ε,. For a dscusson on ths approach see Secton of [22]. 5. Calculate bootstrapped CL parameters f f = σ 2 = I 1 =0 C,+1 I 1, =0 C, 1 I 1 =0 and σ2 I 1 9 by C, C,+1 C, f ) 2.

10 6. Repeat steps 3-5 and obtan emprcal dstrbutons from the bootstrap samples Ĉ,J, f and σ 2. These are then used to quantfy the parameter estmaton uncertanty. Ths non-parametrc classcal bootstrap method can be seen as a frequentst approach. Ths means that we do not express our parameter uncertanty by the choce of an approprate pror dstrbuton. We rather use a pont estmator for the unknown parameters and then study the possble fluctuatons of ths pont estmator. The man dffculty now s that the non-parametrc bootstrap method, as descrbed above, underestmates the true uncertanty. Ths comes from the fact that the estmated resduals ε,, n general, have varance smaller than 1 see formula 7.23) n Wüthrch-Merz [22]). Ths means that our estmated resduals are not approprately scaled. Therefore, frequentsts use several dfferent scalngs to correct ths fact see formula 7.24) n Wüthrch-Merz [22] or England- Verrall [5]). Here, we use a dfferent approach by ntroducng the Bayesan bootstrap method, see Secton 4 below. Frequentst bootstrap estmates. Let us for the tme-beng concentrate on the condtonal MSEP gven by msep C,J D I Ĉ,J ) [ ) ] 2 = E C,J Ĉ,J D I Ĉ,J) 2 = Var C,J D I ) + E [C,J D I ]. 3.1) The frst term s known as the condtonal process varance and the second term as the parameter estmaton uncertanty. In the frequentsts approach,.e. for gven determnstc F = f and Ξ = σ, ths can be calculated, see Wüthrch-Merz [22], Secton 3.2. Namely, the terms are gven by and Var C,J D I ) = E [C,J C,I ] ) 2 ) 2 E [C,J D I ] Ĉ,J = C 2,I =I =I f σ 2/f2 def. = C,I Γ I, 3.2) E [C, C,I ] f CL) =I 2 def. = C 2,I I. 3.3) The process varance 3.2) s estmated by replacng the parameters by ts estmators, VarC,J D I ) = Ĉ,J ) 2 =I σ 2CL) CL) / f ) 2 Ĉ, def. = C,I Γfreq I. 3.4) 10

11 The parameter estmaton error s more nvolved and there we need the bootstrap algorthm. Assume that the bootstrap method gves T bootstrap samples 1) f,..., T) f. Then the parameter estmaton error 3.3) s estmated by the sample varance of the product of the bootstrap observaton chan ladder parameter estmates 1) f,..., T) f, whch gves the estmator C,I 2 freq I. Bayesan estmates. In the Bayesan setup,.e. choosng pror dstrbutons for the unknown parameters F and Ξ, we obtan a natural decomposton of the condtonal MSEP. msep C,J D I E [C,J D I ]) = Var C,J D I ) 3.5) = E [Var C,J D I, F,Ξ) D I ] + Var E [C,J D I, F,Ξ] D I ). The average process varance s gven by see Wüthrch-Merz [22], Lemma 3.6) 1 E [Var C,J D I, F,Ξ) D I ] = C,I E F m Ξ 2 Fn 2 D I 3.6) = C,I 1 =I m=i =I m=i E [F m D I ] E [ Ξ 2 ] D I n=+1 n=+1 E [ Fn 2 ] def. D I = C,I ΓBayes, where we have used Corollary 2.2. The parameter estmaton error s gven by Var E [C,J D I, F,Ξ] D I ) = C,I 2 Var F D def. I = C,I 2 Bayes I, 3.7) =I where we have used 2.15). Usng posteror ndependence, see Corollary 2.2, we obtan for the last term C,I 2 Bayes I = C,I 2 =I E [ F 2 ] D I =I E [F D I ] ) In order to calculate these two terms gven n 3.6) and 3.8) we need to calculate the posteror dstrbuton of F,Ξ), gven D I. Snce we do not have a full dstrbutonal model, we cannot wrte down the lkelhood functon, whch would allow for analytcal solutons or Markov chan Monte Carlo MCMC) smulatons. Therefore we ntroduce the ABC framework whch allows for dstrbuton-free smulatons usng approprate bootstrap samples and a dstance metrc. Ths we are gong to dscuss n the next secton. Credblty Estmates. As mentoned prevously, we can also consder the credblty estmates gven n Gsler-Wüthrch 11

12 [11]. As long as we are only nterested n the second moments,.e. condtonal MSEP, we can also use credblty estmators, whch are mnmum varance estmators that are lnear n the observatons. For dffuse prors we obtan the approxmaton gven n Corollary 7.2 of Gsler-Wüthrch [11] where Γ cred I = cred I = msep C,J D I E [C,J D I ]) = C,I Γcred + C,I 2 cred I, 3.9) =I =I 1 m=i f CL) m σ 2CL) n=+1 CL) f ) 2 σ 2CL) + I 1 =0 C, ) ) CL) f n ) 2 σ n 2CL) + I n 1 =0 C,n, 3.10) =I CL) f ) ) In the results secton we compare the frequentst bootstrap approach, the credblty approach and the ABC bootstrap approach that s descrbed below see Table 7 below). Note that the ABC bootstrap approach can also be appled to other rsk measures, not ust MSEP. Ths s unlke the credblty approach whch only apples to MSEP estmates. For example, f we fx a securty level 95% we can ask for the VaR on that level, whch s defned by ) { [ ] } VaR 0.95 C,J E [C,J D I ] D I = mn x; P C,J E [C,J D I ] > x D I ) 4. ABC for ntractable lkelhoods Obtanng samples { f t), σ 2t)} whch are realsatons of a random vector dstrbuted wth t=1:t a posteror dstrbuton π f, σ D I ) n the DFCL model s dffcult snce the lkelhood s ntractable. Hence, standard approaches such as Markov chan Monte Carlo MCMC) algorthms see, e.g., Glks et al. [10]) cannot be drectly used. Other authors have avoded ths dffculty by makng dstrbutonal assumptons for the form of the lkelhood. Ths then volates a dstrbuton-free CL model assumpton but allows for relatvely standard samplng procedures to be appled. Note that a specfc Gaussan assumpton does not allow for skewness. Here, we do not make any such assumptons and nstead we work n a truly dstrbuton-free model usng approxmate Bayesan computaton ABC) to facltate samplng from an ntractable posteror dstrbuton. To sample from the posteror n our DFCL model we develop a novel formulaton of the ABC methodology based on the bootstrap and condtonal back transformaton procedure, smlar to that dscussed n Secton 3. ABC methods am to sample from posteror dstrbutons n the presence of computatonally ntractable lkelhood functons. For an applcaton n rsk modellng of ABC technques see 12

13 Peters-Ssson [15]. In ths artcle we present a novel ABC-MCMC algorthm. Alternatvely, sequental Monte Carlo SMC) based algorthms whch can mprove smulaton effcency can be found n Del Moral et al. [2], Ssson et al. [21], Peters et al. [16],[17] and Maroram et al. [14]. These algorthms are beyond the scope of the present paper. We provde a basc argument for how the ABC methodology works n Appendx A. For a gven observaton y we want to sample from πθ y) wth an ntractable lkelhood functon. We assume that Sy) s a suffcent statstc for the model from whch we assume data y s a realsaton. We defne a hard decson functon gx, y) = I{ρSx), Sy)) < ǫ}x) for a gven tolerance level ǫ > 0 and a dstance metrc ρ, ), where I{ } s the ndcator functon whch equals 1 f the event s true and 0 otherwse. As demonstrated n Appendx A we use the approxmaton, see A.3)-A.4), πθ y) gx, y) πθ) πx θ) dx gx, y) πθ) πx θ) dxdθ = πθ)e [gx, y) θ], 4.1) E [gx, y)] where X πx θ) and θ πθ). In the next step the numerator of 4.1) s approxmated usng the emprcal dstrbuton,.e. πθ)e [gx, y) θ] πθ) 1 L L l=1 ) g x l) θ), y, 4.2) where x l) θ)..d. πx θ). Fnally, we need to consder the denomnator E [gx, y)]. In general ths has a non-trval form that cannot be calculated analytcally. However, snce we use an MCMC based method the denomnators cancel n the accept-reect stage of the algorthm. Therefore the ntractablty of the denomnator does not mpede samplng from the posteror. Therefore, we use πθ y) gx, y) πθ) πx θ) dx gx, y) πθ) πx θ) dxdθ πθ)e [gx, y) θ] πθ) 1 L L l=1 ) g x l) θ), y 4.3) n order to obtan samples from πθ y). Almost unversally, L = 1 s adopted to reduce computaton but on the other hand ths wll slow down the rate of convergence to the statonary dstrbuton. Note that sometmes one also uses softer decson functons for gx, y). The role of the dstance measure ρ s evaluated by Fan et al. [6], we further extend ths analyss to the class of models consdered n ths paper. We analyse several choces for the dstance measure ρ such as Mahlanobs dstance, scaled Eucldean dstance and the Manhattan Cty Block dstance. Fan et al. [6] demonstrate that t s not effcent to utlse the standard Eucldean dstance, especally when summary statstcs consdered are on dfferent scales. 13

14 Addtonally, usng an ABC-MCMC algorthm, t s mportant to assess convergence dagnostcs. Partcularly when usng ABC-MCMC where seral correlaton n the Markov chan samples can be sgnfcant f the sampler s not desgned carefully. We assess autocorrelaton of the smulated Markov chan, the Geweke [9] tme seres statstc and the Gelman-Rubn [7] R- statstc convergence dagnostc n an ABC settng. Concludng: We apply three dfferent technques n order to treat the ntractable lkelhood: 1) we use ABC to get a handle on the lkelhood, 2) therefore we need synthetc samples from the DFCL model gven realsatons of the parameters, these come from the bootstrap algorthm and 3) by usng a well understood MCMC based samplng algorthm we obtan cancellaton of the non-analytc normalzng constants. Next we dscuss the ABC-MCMC algorthm n detal ABC algorthmc choces for the tme seres DFCL model We start wth the choces of the ABC components. Data set: The observatons are gven by the cumulatve clams denoted by D I. Generaton of a synthetc data set: Note that n ths settng not only s the lkelhood ntractable but addtonally the generaton of a synthetc data set D I parameter values F,Ξ s not straghtforward. The synthetc data set D I gven the current s generated usng the bootstrap procedure descrbed n Secton 3. Note that both the bootstrap resdual ε, and the bootstrap samples DI are functons of the parameter choces, see page 9. Therefore we generate for gven F = f and Ξ = σ the bootstrap resduals ε, = ε, f 1, σ 1 ) and the bootstrap samples DI = D I f, σ) accordng to the non-parametrc bootstrap see page 9) where we replace the CL parameter estmates f CL), σ CL) ) by the parameters θ = F,Ξ). Summary statstcs: In order to defne the decson functon g we ntroduce summary statstcs, see Appendx A for detals of the role these summary statstcs play n the ABC approxmaton. For the observed data D I we defne the vector S D I ; 0, 1) = S 1,...,S n+2 ) = C 0,1,...,C 0,J, C 1,1,...,C 0,,...,C I 2,1, C I 2,2, C I 1,1 ; 0, 1), where n denotes the number of resduals ε,. For gven θ = F,Ξ) we generate the bootstrap sample D I = D I F,Ξ) as descrbed above. The correspondng resduals ε, = 14

15 ε, F 1, Ξ 1 ) should also be close to the standardzed observatons. Therefore we defne ts emprcal mean and standard devaton by µ = µ F,Ξ) = 1 n s = s F,Ξ) = ε, F 1, Ξ 1 ), 4.4), 1 n 1 ε, F 1, Ξ 1 ) µ F,Ξ)) 2 Hence, the summary statstcs for the synthetc data s gven by, 1/2. 4.5) S D I; µ, s ) = C 0,1,...,C 0,J, C 1,1,...,C 0,,...,C I 2,1, C I 2,2, C I 1,1; µ, s ). Dstance metrcs: Mahlanobs dstance and scaled Eucldean dstance Here we draw on the analyss of Ssson et al. [6] who propose the use of the Mahlanobs dstance metrc gven by, ρ S D I ; 0, 1),S D I; µ, s )) = [S D I ; 0, 1) S D I; µ, s )] Σ 1 D I [S D I ; 0, 1) S D I; µ, s )], where the covarance matrx Σ DI s an approprate scalng descrbed n Appendx B. The scaled Eucldean dstance s obtaned when we only consder the dagonal elements of the covarance matrx Σ DI. Note, the covarance matrx Σ DI provdes a weghtng on each element of the vector of summary statstcs to ensure they are scaled approprately accordng to ther nfluence on the ABC approxmaton. There are many other such weghtng schemes one could conceve. Manhattan Cty Block dstance We consder the L 1 -dstance gven by n+2 ρ S D I ; 0, 1),S DI; µ, s )) = S D I ; 0, 1) S DI; µ, s ). Decson functon: We work wth a hard decson functon gven by =1 g D I, D I) = I {ρ S D I ; 0, 1),S D I; µ, s )) < ǫ}. 15

16 Tolerance schedule: We use the sequence ǫ t = max { 20, t, ǫ mn}. Note, the use of an ABC-MCMC algorthm can result n stckng of the chan for extended perods. Therefore, one should carefully montor convergence dagnostcs of the resultng Markov chan for a gven tolerance schedule. There s a trade-off between the length of the Markov chan requred for samples approxmately from the statonary dstrbuton and the bas ntroduced by non zero tolerance. In ths paper we set ǫ mn va prelmnary analyss of the Markov chan sampler mxng rates for a transton kernel wth coeffcent of varaton set to one. We note that n general practtoners wll have a requred precson n posteror estmates, ths precson can be drectly used to determne, for a gven computatonal budget, a sutable tolerance ǫ mn. Convergence dagnostcs: We stress that when usng an ABC-MCMC algorthm, t s crucal to carefully montor the convergence dagnostcs of the Markov chan. Ths s more mportant n the ABC context than n the general MCMC context due to the possblty of extended reectons where the Markov chan can stck n a gven state for long perods. Ths can be combatted n several ways whch wll be dscussed once the algorthm s presented. The convergence dagnostcs we consder are evaluated only on samples post annealng of the tolerance threshold and after an ntal burnn perod once tolerance of ǫ mn s reached. If the total chan has length T, the ntal burnn stage wll correspond to the frst T b samples and we defne T = T T b. We denote by {θ t) } t=1: T the Markov chan of the -th parameter after burnn. The dagnostcs we consder are gven by, Autocorrelaton. Ths convergence dagnostc wll montor seral correlaton n the Markov chan. For gven Markov chan samples for the -th parameter {θ t) } t=1: T we defne the based autocorrelaton estmate at lag τ by ÂCFθ, τ) = 1 T τ)ˆσ θ ) T τ t=1 [θ t) µθ )][θ t+τ) µθ )], 4.6) where µθ ) and ˆσ θ ) are the estmated mean and standard devaton of θ. Geweke [9] tme seres dagnostc. For parameter θ we calculate: 16

17 1. Splt the Markov chan samples nto two sequences, {θ t) } t=1:t1 and {θ t) } t=t : T, such that T = T T 2 + 1, and wth ratos T 1 / T and T 2 / T fxed such that T 1 + T 2 )/ T < 1 for all T. ) 2. Evaluate µ and µ sequence. θ T 1 θ T 2 ) correspondng to the sample means on each sub 3. Evaluate consstent spectral densty estmates for each sub sequence, at frequency 0, denoted ŜD0; T 1, θ ) and ŜD0; T 2, θ ). The spectral densty estmator consdered n ths paper s the classcal non-parametrc perodogram or power spectral densty estmator. We use Welch s method wth a Hannng wndow, for detals see Appendx C. 4. Evaluate convergence dagnostc gven by Z T = µ ) µ θ T 2 θ T 1 ŜD0;T 1,θ )+T 1 ). For T one has accordng to the central T ŜD0;T 2,θ ) lmt theorem that Z T N0, 1) f the sequence {θ t) } t=1: T s statonary. Gelman-Rubn [7] R-statstc dagnostc. Ths approach to convergence analyss requres that one runs multple parallel ndependent Markov chans each startng at randomly selected ntal startng ponts, we run 5 chans. For comparson purposes we splt the total computatonal budget of T nto T1 = T 2 =... = T 5 = T 5. We compute the convergence dagnostc for parameter θ wth the followng steps: 1. Generate 5 ndependent Markov chan sequences, producng the chans for parameter θ, denoted {θ t),k } t=1:t k for k {1,...,5}. ) 2. Calculate the sample means µ θ T k for each sequence, and the overall mean ) µ θ T. 3. Calculate the varance of the sequence means, ) )) µ µ θ T 2 def. = B /T k k=1 θ T k ) 4. Calculate the wthn-sequence varances ŝ θ 2 T k for each sequence. 5. Calculate the average wthn-sequence varance, k=1 ŝ2 θ T k ) def. = W. 6. Estmate the target posteror varance for parameter θ by the weghted lnear ) combnaton, σ 2 θ T = T k 1 T k W + 1 T k B. Ths estmate s unbased for samples whch are from the statonary dstrbuton. In the case n whch not all sub chans have reached statonarty, ths overestmates the posteror varance for a fnte T but t asymptotcally, T, converges to the posteror varance. 17

18 7. Improve on the Gaussan estmate of the target posteror gven by ) N µ θ T, σ 2 θ T )) by accountng for samplng varablty n the estmates of the posteror mean and varance. Ths can be acheved by makng a Student-t ) approxmaton wth locaton µ θ T, scale V and degrees of freedom df, each gven respectvely by; ) V = σ 2 θ T + B T and df = 2 V 2 ) Var V = 1 5 Var V ) ) T1 1 2 Var T T 1 1) Ĉov 25T 1 24T 1 1) µ 25T 1, where the varance s estmated as, ŝ 2 θ T k ŝ 2 θ T k ) θ T Ĉov ), µ )) ) B 2 T 2 ŝ 2 θ T k θ T )) ) )), µ θ T. 4.7) Note, the covarance terms are estmated emprcally usng the wthn sequence estmates of the mean and varance obtaned for each sequence. 8. Calculate the convergence dagnostc, R = V df W df 2), where as T one can prove that R 1. Ths convergence dagnostc montors the scale factor by whch the current dstrbuton for θ may be reduced f smulatons are contnued for T. We begn by presentng the ABC-MCMC based algorthm we propose for samplng from the posteror dstrbuton presented n Secton ABC-MCMC to sample from π f, σ D I ) We develop an ABC-MCMC algorthm whch has an adaptve proposal mechansm and annealng of the tolerance durng burnn of the Markov chan. Havng reached the fnal tolerance post annealng, denoted ǫ mn, we utlze the remanng burnn samples to tune the proposal dstrbuton to ensure an acceptance probablty between the range of 0.3 and 0.5 s acheved. The optmal acceptance probablty when posteror parameters are..d. Gaussan was proven to be at 0.234, see Gelman et al. [8]. Though our problem does not match the requred condtons for ths proof, t provdes a practcal gude. To acheve ths, we tune the coeffcent of varaton of the proposal, n our case the shape parameter of the gamma proposal dstrbuton. We mpose an addtonal constrant that the mnmum shape parameter value s set at γ mn for {1,...,2J}. ABC-MCMC algorthm usng bootstrap samples. 18

19 ) 1. For t = 0 ntalze the parameter vector randomly, ths gves θ 0) 1:2J = f 0) 0:, σ0) 0:. Intalze the proposal shape parameters, γ γ mn 2. For t = 1,...,T ) a) Set θ t) 1:2J = b) For = 1,...,2J θ t 1) 1:2J ) for all {1,...,2J}.. Sample proposal θ from a Γγ, θ t) /γ )-dstrbuton. We denote the gamma proposal densty by K θ ; γ, θ t) /γ ). Ths gves proposed parameter vector ) θ = θ t) 1: 1, θ, θt). +1:2J. Condtonal on θ = θ t) 1: 1, θ, θt) +1:2J ), generate synthetc bootstrap data set D I = D I θ ) usng the bootstrap procedure detaled n Secton 3 where we replace the CL parameter estmates f CL), σ CL) ) by the parameters θ.. Evaluate summary statstcs S D I ; 0, 1) and S D I ; µ ; s ) and correspondng decson functon gd I, DI ) as descrbed n Secton 4.1. v. Accept proposal wth ABC acceptance probablty ) ) π θ A θ t) 1:2J, θ ) = mn 1, K θ t) ; γ, θ /γ ) ) gd π θ t) K θ ; γ, θ t) I, DI) /γ. That s, smulate U U0, 1) and set θ t) = θ f U < A θ t) 1:2J, θ ). v. If 100 t T b and ǫ t = ǫ mn then check to see f requre tunng of the proposal. Defne the average acceptance probablty over the last 100 teratons of updates for parameter by ā t 100:t) 0.9γ f ā t 100:t) γ = and consder the adapton, 1.1γ f ā t 100:t) > 0.5, γ otherwse. Then set the proposal shape parameter as, γ = max{γ, γmn }. < 0.3 and γ > γ mn, The ABC-MCMC algorthm presented can be enhanced by utlzng an dea of Gramacy et al. [12] n an ABC settng. Ths nvolves a combnaton of temperng the tolerance {ǫ t } t=1:t and mportance samplng correctons. 19

20 5. Example 1: Synthetc data The tunng of the proposal dstrbuton n ths study s done for the smplest base dstance metrc, the weghted Eucldean dstance. To study the effect of the dstance metrc n a comparatve fashon we shall keep the proposal dstrbuton unchanged. The frst example we present has a clams trangle of sze I = J = 9. In ths example we fx the true model parameters, denoted by f = f 0,...,f ) and σ 2 = σ0 2,.. ).,σ2 and gven n Table 2, used to generate the synthetc data set Generaton of synthetc data To generate the synthetc observatons for D I we generate randomly the frst column, denoted B 0. Then condtonal on ths realzaton of B 0 we make use of the model gven n 2.1) to generate the remanng columns of D I, ensurng the model assumptons are satsfed. Ths requres settng C,0 suffcently large, for approprate choces of f and σ 2, and then sample..d. realzatons of ε, U [ 3, 3 ] whch are used to obtan D I, see the observatons n Table Senstvty analyss and convergence assessment We perform a senstvty analyss, studyng the mpact of the dstance metrc on the mxng of the Markov chan n the case of ont estmaton of the chan ladder factors and the varance parameters. The pre-tuned coeffcent of varaton of the Gamma proposal dstrbuton for each parameter of the posteror was performed usng the followng settngs; T b = 50, 000, T = 200, 000, ǫ mn = 0.1 and ntal values γ = 1 for all {1,...,2J}. Addtonally, the pror parameters for the chan ladder factors F were set as α, β) = 2, 1.2/2) and the parameters for the varance parameters Ξ 2 were set as a, b) = 2, 1/2). After tunng the proposal dstrbutons durng burnn and roundng the shape parameters we found that γ = 10 for all {1,...,2J} produced average acceptance probabltes for each parameter between 0.3 and 0.5. Then keepng the proposal dstrbuton constant and usng a common data set D I we ran 3 versons of the ABC-MCMC algorthm for 200,000 samples correspondng to; 1. Scaled Eucldean dstance and ont estmaton of posteror for F,Ξ 2 2. Mahlanobs dstance modfed) and ont estmaton of posteror for F,Ξ 2 3. Manhattan Cty Block dstance and ont estmaton of posteror for F,Ξ 2. 20

21 We estmate the three convergence dagnostcs we presented n Secton 4.1. The results of ths analyss are presented as a functon of Markov chan teraton t post burn of 50,000 samples. Autocorrelaton Functon: In Fgure 1 we demonstrate the estmated autocorrelaton functons for the Markov chans of the random varables F 0 and Ξ 2 0. Snce the posteror parameters n ths model are ndependent t s sutable to analyze ust the margnal parameters to get a reasonable estmate of the mxng behavor of the MCMC-ABC algorthm on all the posteror parameters. The results demonstrate the degree of seral correlaton n the Markov chans generated for these parameters as a functon of lag tme τ. The hgher the decay rate n the tal of the estmated ACF as a functon of τ, the better the mxng of the MCMC algorthm. Due to the ndependence propertes of ths model there s lttle dfference between results obtaned for Scaled Eucldean and Mahlanobs dstances. As shown n Appendx A the estmate of the covarance matrx s dagonal on all but the rght lower 2 2 block. Hence, we recommend usng the smple Scaled Eucldean dstance metrc as t provded the best trade-off between smplcty and mxng performance. Geweke Tme Seres Dagnostc: Fgure 2 presents results for the Geweke tme seres dagnostc. Agan, we present the results for the random varables F 0 and Ξ 2 0. Note, we used the posteror mean as the sample functon and a set of ncreasng values for T from T b + 5, 000 ncreasng n steps of 5,000 samples to T. In each case we splt the chan n each wndow gven by {θ t) } t=1:t1 and {θ t) } t=t : T accordng to recommendatons from Geweke et al. [9]. We then calculate the convergence dagnostc Z T whch s the dfference between these 2 means dvded by the asymptotc standard error of ther dfference. As the chan length ncreases T, then the samplng dstrbuton of Z N0, 1) f the chan has converged. Hence values of Z T n the tals of a standard normal dstrbuton suggest that the chan was not fully converged early on.e. durng the 1st wndow). Hence, we plot Z T scores versus ncreasng T and montor f they le wthn a 95% confdence nterval Z T [ 1.96, 1.96]. The results n Fgure 2 demonstrate that clearly the convergence propertes of the dstance functons dffer. Agan ths s more materal n the Markov chan for the varance parameter when compared to the Markov chan results for the chan ladder factor. The man pont we note s that agan one would advse aganst use of the Cty block dstance metrc. Gelman and Rubn R statstc: In Fgure 3 we present the Gelman and Rubn convergence dagnostc. To calculate ths we ran 20 chans n parallel, each of length 10,000 samples and for each chan we dscarded 250 samples as burnn. We then estmated the R statstc as a functon of smulaton tme post burnn. In Fgure 3 we demonstrate the convergence rate of 21

22 the R statstc to 1 for each dstance metrc on ncreasng blocks of 200 samples. Usng ths summary statstc all three dstance metrcs are very smlar n terms of convergence rate of the R statstc to 1. Overall, these three convergence dagnostcs demonstrate the smple scaled Eucldean dstance metrc s the superor choce. Secondly, we see approprate convergence of the Markov chans under three convergence dagnostcs whch test dfferent aspects of the mxng of the Markov chans, gvng confdence n the performance of the MCMC-ABC algorthm n ths model Parameter estmaton Bayesan estmates In ths secton we present results for the Scaled Eucldean dstance metrc, wth a Markov chan of length 200,000 samples dscardng the frst 50,000 samples as burnn. In Tables 4 and?? we present the Chan Ladder parameter estmates for the DFCL model and the assocated parameter estmaton error. We defne the followng quanttes: MAP) f σ 0:, MMSE) f σ 0:, σ f σ 0: and [ˆq 0.05, ˆq 0.95 ] σ 0: denote respectvely the Maxmum a-posteror, Mnmum Mean Square Error, posteror standard devaton of the condtonal dstrbuton of chan ladder factor F and the posteror coverage probablty estmates at 5% of the condtonal dstrbuton of chan ladder factor F, each of these estmates s condtonal on knowledge of the true σ 0:. MAP) f, fmmse), σ f and [ˆq 0.05, ˆq 0.95 ] denote the same quanttes for the uncondtonal dstrbuton after ont estmaton of F 0:, Ξ 0:. Ave.[A θ 1:2J, f )] and Ave.[A θ 1:2J, σ )] denotes the average acceptance probablty of the Markov chan. σ 2MAP), σ 2MMSE), σ σ 2 and [ˆq 0.05, ˆq 0.95 ] denotes the same quanttes for the chan ladder varances as those defned above for chan ladder factors. For the frequentst approach we obtan the standard error n the estmates by usng 1,000 { } { } bootstrap realzatons of D s) I to obtan fccl) s=1:1,000 s), σ 2CCL) s). We use these s=1:1,000 bootstrap samples to calculate the standard devaton n the estmates of the parameters n the classcal frequentst CL approach, present n brackets.) next to ther correspondng estmators. The standard errors n the Bayesan parameter estmates are obtaned by blockng the Markov chan nto 100 blocks of length 1,500 samples and estmatng the posteror quanttes on each block. 22

23 6. Example 2: Real Clams Reservng data In ths example we consder estmaton usng real clams reservng data from Wüthrch-Merz [22], see Table 4. Ths yearly loss data s turned nto annual cumulatve clams and dvded by 10,000 for the analyss n ths example. We use the analyss from the prevous study to ustfy use of the ont MCMC-ABC smulaton algorthm wth a Scaled Eucldean dstance metrc. We pre-tuned the coeffcent of varaton of the Gamma proposal dstrbuton for each parameter of the posteror. Ths was performed usng the followng settngs; T b = 50, 000, T = 200, 000, ǫ mn = 10 5 and ntal values γ = 1 for all {1,...,2J}. Here we make a strct requrement of the tolerance level to ensure we have accurate results from our ABC approxmaton. Addtonally, ) CL) the pror parameters for the chan ladder factors F were set as α, β ) = 1, f and the ) parameters for the varance Ξ 2 prors were set as a, b ) = 1, σ CL). The code for ths problem was wrtten n Matlab and t took approxmately 10 mn to smulate 200,000 samples from the MCMC-ABC algorthm, on a Intel Xeon 3.4GHz processor wth 2Gb RAM. After tunng the proposal dstrbutons durng burnn we obtaned rounded shape parameters γ 1:9 = [50, 100, 500, 500, 5, 000, 20, 000, 100, 000, 2, 000, 000, 3, 000, 000] provded average acceptance probabltes between 0.3 and 0.5. In Fgures 4 we present box-whsker plots of estmates of the dstrbutons of the parameters F 0:, Ξ 0: obtaned from the MCMC-ABC algorthm, post burnn. Fgure 5 presents the Bayesan MCMC-ABC emprcal dstrbutons of the ultmate clams, C,J for = 1,...,I. In Table 5 we present the predcted cumulatve clams for each year along wth the estmates for the chan ladder factors and chan ladder varances under both the classcal approach and the Bayesan model. We see that wth ths farly vague pror specfed, we do ndeed obtan convergence of the MCMC-ABC based Bayesan estmates f MMSE), σ MMSE) to the classcal estmates f CL), σ CL). In Fgure 6 we present a study of the hstogram estmate of the margnal posteror dstrbuton for chan ladder factor π f 0 D I, ǫ mn). The plot was obtaned by samplng from the full posteror π f, σ D I, ǫ mn) for each specfed tolerance value, ǫ mn. Then the samples for the partcular chan ladder parameter n each plot are turned nto a smoothed hstogram estmate for each epslon and plotted. We observe a sgnfcant dfference n the parameter uncertanty as reflected by the change n the posteror precson as the tolerance decreases. Hence, the tolerance wll mpact the msep. Mnmzng ths mpact, nvolves managng the computatonal budget to acheve mxng at a gven tolerance level whlst mnmzng the tolerance level. 23

24 Ultmately, we would lke an algorthm whch could work well for any ǫ mn, the smaller the better. However, we note that wth a decreasng ǫ mn n the sampler we present n ths paper, one must take addtonal care to ensure the Markov chan s stll mxng and not stuck n a partcular state, as s observed to be the case n all MCMC-ABC algorthms. To avod ths acknowledged dffculty wth MCMC-ABC requres that one ether runs much longer MCMC chans or t requres the use of more sophstcated samplng algorthms such as SMC Samplers PRC-ABC based algorthms, see Ssson, Peters, Fan and Brers 2008). Ths s well beyond the scope of ths paper and wll be the subect of future papers and nvestgaton n ths context. In Table 6 we present the predctve VaR at 95% and 99% levels for the ultmate predcted clams, obtaned from the MCMC-ABC algorthm. These are easly obtaned under the Bayesan settng, we smply used the MCMC-ABC posteror samples to explctly obtan samples from the full predctve dstrbuton of the cumulatve clams after ntegratng out the parameter uncertanty numercally. In addton to ths we present the analyss of the MSEP under the bootstrap frequentst procedure and the Bayesan MCMC-ABC and credblty estmates for the total predcted cumulatve clams for each accdent year. We also present results for the sum of the total cumulatve clams for each accdent year. We can make the followng conclusons from these results: 1. The process varance for each C,J demonstrate that the unscaled condtonal frequentst bootstrap and the credblty estmates are very close for all accdent years. The Bayesan results compare favorably. 2. The results for the parameter estmaton error for the predcted cumulatve clams C,J demonstrate for small the Bayesan approach outperforms the frequentst approach. However, the Bayesan approach produces performance whch worsens as ncreases, relatve to the credblty approach. Ths could be a result of the tolerance level settng n the ABC algorthm. In future work t wll be nterestng to study the effect of the tolerance level on the parameter estmaton error. Fgure 6 demonstrates numercally that the tolerance ǫ should affect the parameter estmaton error n the Bayesan approach. Ths study requres sophstcated algorthms such as those found n Ssson, Peters, Fan and Brers 2008). 3. The total results for the process varance for C = C,J demonstrate that the frequentst and credblty results are very close. Addtonally, Bayesan total results are largest followed by credblty and then frequentst estmates whch s n agreement wth theoretcal bounds. 24

25 4. The total results for the parameter estmaton error for C = C,J demonstrate that frequentst uncondtonal bootstrap procedure results n the lowest total error. The Bayesan approach and credblty total parameter errors are close. Addtonally, we note that the results n Wüthrch-Merz, 2008) Table 7.1 for the total parameter estmaton error under an uncondtonal frequentst bootstrap wth unscaled resduals s also very close to the total obtaned for the frequentst approach. 7. Dscusson Ths paper has presented a dstrbuton-free clams reservng model under a Bayesan paradgm. We have then present a novel advanced MCMC-ABC algorthm to obtan estmates from the resultng ntractable posteror dstrbuton of the chan ladder factors and chan ladder varances. We assessed several aspects of ths algorthm, ncludng the propertes of the convergence of the MCMC algorthm as a functon of the dstance metrc approxmaton n the ABC component. We studed the performance on a synthetc data set generated from known parameters, n order to demonstrate the accuracy of ths methodology. Next we assessed a real clams reservng data set and compared the results we obtaned for predcted cumulatve ultmate clams to those obtaned va classcal chan ladder methods and va credblty theory. We note that ths clearly demonstrates our algorthm s workng accurately and provdes us not only wth the ablty to obtan pont estmates for the frst and second moments of the ultmate cumulatve clams, but an accurate emprcal approxmaton of the entre dstrbuton of the ultmate clams. Ths s valuable for many reasons, ncludng predcton of reserves whch are not based on centralty measures, such as the tal based VaR results we present. We have demonstrated a unque and accurate way n whch one can bypass the need to make dstrbutonal approxmatons. Acknowledgements The fst author thanks ETH FIM and ETH Rsk Lab for ther generous fnancal assstance whlst completng aspects of ths work at ETH. The frst author also thanks the Department of Mathematcs and Statstcs at the Unversty of NSW for support through an Australan Postgraduate Award and to CSIRO for support through a postgraduate research top up scholarshp. Fnally, ths materal was based upon work partally supported by the Natonal Scence Foundaton under Grant DMS to the Statstcal and Appled Mathematcal Scences Insttute, North Carolna, USA. Any opnons, fndngs, and conclusons or recommendatons expressed n ths materal are those of the authors) and do not necessarly reflect the vews of the Natonal 25

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