2. The Munich chain ladder method

Transcription

1 ntoduction ootstapping has become vey popula in stochastic claims eseving because of the simplicity and flexibility of the appoach One of the main easons fo this is the ease with which it can be implemented in a speadsheet in ode to obtain an appoximation to the estimation eo of a fitted model in a statistical context Futhemoe, it is also staightfowad to extend it to obtain the appoximation to the pediction eo and the pedictive distibution of a statistical pocess by including simulations fom undelying distibutions Theefoe, bootstapping is a poweful tool fo the most popula subect fo eseving puposes in geneal insuance, the pediction eo of the eseve estimates t should be emphasised that to obtain the pedictive distibution, athe than ust the estimation eo, it is necessay to extend the bootstap pocedue by simulating the pocess eo t is also impotant to ealise that bootstapping is not a model, and theefoe it is impotant to ensue that the undelying eseving models ae coectly calibated to the obseved data n this pape, we do not addess the issue of model checking, but simply show how a bootstapping pocedue can be applied to the Munich chain ladde model n the aea of non-life insuance eseving, thee ae pimaily two types of data used n addition to the paid claims tiangle, thee is fequently a tiangle of incued data also available The taditional appoach is to fit a model to eithe paid o incued claims data, sepaately One of the most popula methods used fo eseving is the chain ladde technique While we do not believe that this is the most appopiate appoach fo all data sets, it has etained its populaity fo a numbe of easons Fo example, the paametes ae undestood in a pactical context, it is flexible and it is easy to apply This pape concentates on methods which have a chain ladde stuctue, and in this context, two types of appoaches exist: deteministic methods such as chain ladde, and the ecently developed stochastic chain ladde eseving models When the chain ladde technique is used (eithe as a deteministic appoach o using a stochastic model, one set of data will be omitted - eithe the paid o the incued data can be used, but not both at the same time Obviously, this does not make full use of all the data available and esults in the loss of some infomation contained in those data This leads us to conside whethe it is possible to constuct a model fo both data sets, and to a consideation of the dependency between the two un-off tiangles, which is not staightfowad This issue also aises when taditional methods ae applied sepaately to each tiangle, which poduces inconsistent pedicted ultimate losses n esponse to this issue, uag and Mack (2004 poposed a diffeent appoach within a egession famewok, consideing the likely coelations between paid and incued data uag and Mack (2004 called this new method as the Munich chain ladde (ML model t is this model that is the subect of this pape, and we show how the pedictive distibution may be estimated using bootstapping Thus, in this pape an adapted bootstap appoach is descibed, combined with simulation fo two dependent data sets The speadsheets used in this pape can be used in pactice fo any data sets, and ae available on equest fom the authos The pape is set out as follows Section 2 biefly descibes the ML model using a notation appopiate fo this pape n section 3, the basic algoithm and methodology of bootstapping is explained Section 4 shows how to obtain the estimates of the pediction eos and the empiical pedictive distibution using the adapted bootstapping and simulation methods n Section 5, two numeical examples ae povided including the data fom Mack and also some eal London maket data Finally, section 6 contains a discussion and conclusion

2 2 The Munich chain ladde method The ML model aims to poduce a moe consistent ultimate loss pediction when modelling both paid and incued claim data t is specially designed to deal with the coelation between paid and incued claims as the taditional models, such as chain ladde model, sometimes poduce unsatisfactoy esults by ignoing this dependence t should be emphasized that the paid and incued claims fom the same calenda yeas ae not coelated t is that the paid claims (incued claims ae coelated to the incued claims (paid claims fom the next (pevious calenda yea The fundamental stuctue of the ML model is the same as Mack s distibution-fee chain ladde model n the othe wod, the chain ladde development factos in the ML model ae obtained by Mack s distibution-fee appoach Moe details of Mack s model ae contained in Mack (993 Moeove, the ML model adusts the chain ladde development factos using the coelations between the obseved paid and incued claims The adusted chain ladde development factos theefoe become individual not only fo diffeent development yeas but also fo diffeent accident yeas The coelation adustment is caied out within a linea modelling famewok This is explained in moe detail in the sections 2 and 22 2 Notation and Assumptions Fo ease of notation, we assume that we have a tiangle of data Although the data could be classified in diffeent ways, we efe to the ows as accident yeas and the columns as development yeas Denote as cumulative paid claims and as cumulative incued claims occued in accident yea i, development yea, whee i n a nd i+ fo the obseved data The aim of the chain ladde technique and of ML is to estimate the data up to development yea n This poduces estimates fo i n a nd i+ 2 n, and we theefoe efe to the complete ectangle of data in the assumptions: i, n Mack s distibution-fee chain ladde method models the patten of the development factos in a i, + egession famewok, which ae defined as F, fo paid claims and F, fo incued claims Also the atios of paid divided by incued claims and the invese ae intoduced as and, espectively Futhemoe, define the obseved data up to calenda yea k as k { i + k} { : i + k} and { : i + k} k these, espectively Assumptions A (Expectations k i, i :,, fo paid, incued claims and both of (A Fo n thee exists a constant f such that (fo i,, n - 2 -

3 E F f This assumption is fo paid claims t is necessay to make anothe analogous assumption fo incued claims since both data sets ae taken into account (A2 Fo n, thee exists a constant f such that (fo i,, n E F f n ode to analyse the two un-off tiangles dependently, the following assumptions ae also equied (A3 Fo n, thee exists a constant q such that (fo i,, n E q (A4 Fo n, thee exists a constant q such that (fo i,, n E q Assumptions (Vaiances ( Fo n, thee exists a constant σ such that (fo i,, n Va F ( σ 2 Again, the analogous assumption fo the incued claims is made as follows (2 Fo n, thee exists a constant σ such that (fo i,, n Va F ( σ 2 Also, fo the atios of incued to paid and vice vesa, the following vaiance assumptions ae made (3 Fo n, thee exists a constant τ such that (fo i,, n Va ( τ 2 (4 Fo n, thee exists a constant τ such that fo ( i,, n - 3 -

4 Va ( τ 2 Assumptions (ndependence ( The andom vaiables petaining to diffeent accident yeas fo paid claims, ie,2,, n n,2,, n, ae stochastically independent { },,{ } (2 The andom vaiables petaining to diffeent accident yeas fo incued claims, ie,2,, n n,2,, n, ae stochastically independent { },,{ } Using assumptions A to, the eason esiduals used in ML model can be defined as shown in equations (4 to (44 These esiduals ae a cucial pat of the bootstapping pocedues descibed in section 4 F E F, Va F (2 E, Va (22 and F E F, (23 Va F E (24 Va Assumptions D (oelations (D Thee exists a constant ρ such that (fo i, n E, ρ (25 The following equation states that the constant ρ is in fact the coelation coefficient between the esiduals Note that since the esiduals have vaiance, the coelation is equal to the covaiance The poof can be found in uag and Mack (2004 ov o Va o F (26,,,,,, ρ Va F - 4 -

5 uag and Mack (2004 uses equation (45 to deive expected ML paid development factos adusted by the coelation as shown in equation (46 ( E F, E F o F,, E + (27 (D2 Analogously to assumption (D, fo the incued claims it is assumed that thee exists a constant ρ such that (fo i, n E, ρ (28 Similaly, the constant esiduals ie ρ measues the coelation coefficient o the covaiance between the Va ov o o F, i,, i,,,,, ρ Va F (29 Hence, the expected ML incued development factos adusted by the coelation can be deived fom equation (47 as follows, ( E F,, i i, E F + ov F,, E (20 22 Unbiased Estimates of the aametes n this section, we summaise the unbiased estimates of the paametes deived by uag and Mack (2004 Fo the paid data, estimates ae equied fo the paametes of the development factos, the vaiance constants and also the coelation coefficient The estimates of the paid development facto paametes can be intepeted as weighted aveages of the obseved development factos o, using as the weights: F and i, + i fˆ F (2-5 -

6 n i q ˆ (22 The unbiased estimates of the vaiance constants coefficient ae as follows: and n 2 ( ˆ ( F fˆ n 2 σ (23 2 ( ˆ ( qˆ + 2 τ n (24 Hence the eason esiduals ae and ˆ F f (25 ˆ σ qˆ (26 τ Finally, the estimate of the coelation coefficient is given in equation (45 ˆ ρ i, i, ( 2 (27 Similaly, fo incued data, the estimates of the development facto paametes can be intepeted as weighted aveages of the development factos F o, using as the weights: and i, + i fˆ F (28 qˆ (29 Also the unbiased estimates fo the vaiance paametes ae defined as follows: - 6 -

7 and n 2 2 ( ˆ ( F fˆ n σ (220 2 ( ˆ ( qˆ + 2 τ n (22 Hence the eason esiduals ae and ˆ F f (222 ˆ σ ˆ q (223 τ And finally, the estimate of the coelation coefficient is given in equation (224 ˆ ρ i, i, ( 2 (224 Assumptions A in section 2 have defined the expectations of the development factos, given ust the data in the espective tiangles n ode to poduce a single estimate based on the data fom both paid and incued data, uag and Mack (2004 also consides the expectations of the development factos given both tiangles and define E F λ and E F λ Using plug-in estimates fom equations (49 to (45, the estimates of the paid ML development factos ae calculated using equation fom (46: ˆ ˆ ˆ σ λ ˆ ( ˆ f + ρ q (225 ˆ τ Similaly, plug-in estimates fom equations (46 to (422 ae used in equation (48 so that the estimates of the incued development factos ae ˆ ˆ ˆ σ λ ˆ ( ˆ f + ρ q (226 ˆ τ 3 ootstapping and laims Reseving ootstapping is a simulation-based appoach to statistical infeence t is a method fo poducing sampling distibutions fo statistical quantities of inteest by geneating pseudo samples, which ae - 7 -

8 obtained by andomly dawing, with eplacement, fom obseved data n simple tems, bootstapping is a e-sampling pocedue and all the pseudo samples geneated by bootstapping ae subsets of the obseved sample o identical to the obseved sample t should be emphasized that bootstapping is a method athe than a model ootstapping is useful only when the undelying model is coectly fitted to the data, and bootstapping is applied to data which ae equied to be independent and identically distibuted The bootstapping method was fist intoduced by Efon (979 and a good intoduction to the algoithm can be found in Efon and Tibshiani (993 Fo the pupose of claity we begin by giving a geneal bootstapping algoithm and biefly eviewing pevious applications of bootstapping to claims eseving n section 4, we show how an algoithm of this type can be applied to the ML Suppose we have a sample X and we equie the distibution of a statistic θˆ The following thee steps compise the simplest bootstapping pocess: Daw a bootstap sample X { X, X 2,, X n } fom the obseved data X { X, X 2,, X n } 2 alculate the statistic inteest ˆ θ fo the fist bootstap sample X { X, X 2,, X n } 3 Repeat steps and 2 N times y epeating steps and 2 N times, we obtain a sample of the unknown statistic θˆ, calculated fom N pseudo samples, ie θ { ˆ θ, ˆ θ,, ˆ 2 θ N } When N 000, the empiical distibution constucted fom θ { ˆ θ, ˆ θ,, ˆ 2 θ N } can be taken as the appoximation to the distibution fo the statistic inteest θˆ Hence all the quantities of the statistic inteest θˆ can be obtained since such a distibution contains all the infomation elated to θˆ The above bootstapping algoithm can be applied to the pediction distibutions fo the best estimates in stochastic claims eseving subect England and Veall (2007 contains an excellent eview on the application n addition, Lowe (994, England and Veall (999 and inheio (2003 ae also good esouces fo moe details England and Veall (2007 showed how bootstapping can be used fo ecusive models, following on fom the ealie papes (England and Veall, 999 and England 2002 which applied bootstapping to the ove-dispesed oisson model t should be noted hee that the eason esiduals ae commonly used athe than the oiginal data in the Genealized Linea Model (GLM famewok The eason esiduals ae equied in ode to scale the esponse vaiables in the GLM so that they ae identically distibuted This is necessay because the bootstap algoithm equies that the esponse vaiables ae independent and identically distibuted To ou knowledge, thee has not been any consideation of bootstapping fo dependent data in the actuaial liteatue t should be noted hee that a model taking account of all infomation available could be potentially vey valuable, even when the data is dependent in pactice The dependence makes it even difficult to calculate the pediction eo theoetically Fo these easons, we believe that adopting bootstap method fo these models is wothy of investigation, paticulaly in ode to obtain the pedictive distibution of the estimates of outstanding liabilities - 8 -

9 4 ootstapping the Munich chain ladde model This section consides bootstapping the ML model n section 4 we descibe the methodology and in section 42 we give the algoithm that is used 4 Methodology The method of bootstapping stochastic chain ladde models can be seen in a numbe of diffeent contexts England and Veall (2007 categoize the models as ecusive and non-ecusive and show how bootstapping methods can be applied in eithe case Since we ae dealing with ecusive models hee, we follow England and Veall and conside the obseved development link atios athe than the claims data themselves n othe wods, fo Mack s distibution-fee chain ladde model the link i, + atios, F, ae andomly dawn against, noting that E [ F ] E f Fo this eason, the bootstap estimates of the development factos which ae obtained by taking weighted aveages of the bootstapped obseved link atios, F, use athe than as the weights Howeve, this method cannot be simply extended to the ML model, since this model is designed fo dealing with two sets of coelated data, the paid and incued claims This means that it is not possible to use the nomal bootstap appoach: the independence assumption cannot be met any moe n ode to addess the poblem of how to adapt the existing bootstap appoach in ode to cope with the ML model fo dependent data sets, the consideation of the coelation is cucial t should be noted that the coelation which is obseved in the data epesents eal dependence between the paid and incued data, and the model is specifically designed because of this dependence Theefoe, it should emain unchanged within any e-sampling pocedue The staightfowad solution is to daw samples paiwise so that the coelation between the two dependent oiginal data sets will not be boken when geneating a sampling distibution fo a statistic inteest Obviously, when bootstapping the ecusive ML model, the esiduals of the paid and incued link atios ae equied instead of the aw data The question aises of how to deal with these esiduals in ode to meet the equiement of not beaking the obseved dependence between paid and incued claims, The answe is to goup all the fou sets of esiduals calculated in the ML model, ie the paid and incued development link atios, the atios of incued ove paid claims fom the pevious yeas and its evese, individually This is because that the paid claims (incued claims ae coelated to the incued claims (paid claims that ae fom only the next (pevious yea and doing this will peseve the equied dependence And also the coelation coefficient of paid and incued claims is equal to the coelation coefficient of those esiduals, as stated in equations (26 and (29 n fact, in the case of the paid claims data, the tiangle containing the esiduals of the obseved paid link atios and the tiangle containing the esiduals of the atios of incued ove paid (except the fist column, ae paied togethe, individually, with the same dimensions And it is the same pocedue fo the incued claims data Howeve, note that the atios of the paid ove incued claims and the evese, indicate the same infomation Theefoe, the atios should emain unchanged when paiing them with paid and incued claims with the same dimensions The consequence of this is that all the fou sets of esiduals fo paid, incued link atios and the atios of incued ove paid claims and the evese should all be gouped togethe f - 9 -

10 Note hee that an altenative appoach would be to goup thee sets of esiduals: the esiduals of the paid and incued link atios and eithe the esiduals of the paid ove incued atios o the evese This would poduce the same esults as gouping fou sets of esiduals as the esiduals of paid ove incued atios and the eseve can always be calculated fom each othe Howeve, it is simple to goup all the fou sets as the calculation of the fouth set of esiduals is natually skipped in this case Obviously, this combines the fou esiduals tiangles into one new tiangle that consists of these gouped esiduals and we name it as the gouped esidual tiangle n each unit fom this gouped esidual tiangle, the esiduals ae fom the same accident and development yea and coespond to paid and incued claims Theefoe, the new gouped esidual tiangle contains all the infomation available and meanwhile maintains the obseved dependence When applying bootstapping, this gouped tiangle is consideed as the obseved sample And the new geneated pseudo samples ae obtained by andom dawing, with eplacement, fom this gouped tiangle The e-sampled incued and paid tiangles can be obtained by sepaating the pais in the pseudo sample geneated as above and backing out the esidual definition The ML appoach can then be applied to calculate all the statistics of inteest fo the e-sampled paid and incued tiangles ie the coelation coefficient fo paid and incued, the paid and incued development factos, the atios of paid ove incued o the invese, and the vaiances Finally, adusting the paid and incued development factos by the coelation coefficient using the ML appoach, the bootstapped ML eseve estimates ae obtained This completes a single bootstap iteation Again, the bootstap method povides only the estimation vaiance of the ML model n ode to include the pediction eo and estimate the pedictive distibution fo the ML estimates of outstanding liabilities, an additional step is added at the end of each of the bootstap iteation, which is to add the pocess vaiance to the estimation vaiance n the context of bootstapping, the way to do this is to simulate new obsevations fom an assumed pocess distibution with the mean and vaiance obtained in the same bootstap iteation n this pape, an undelying nomal distibution is assumed fo both the cumulative paid o incued claims and moe details ae given in the section 42 n ode to obtain a easonable appoximation to the pedictive distibution, at least 000 pseudo samples ae equied Fo each of the pseudo samples, the ow totals and oveall total of outstanding liabilities ae stoed so that the sample means, sample vaiances and the empiical distibutions can be calculated and plotted They ae taken as the appoximations to the best estimates of outstanding liabilities, the pediction eos and the pedictive distibutions of the outstanding liabilities Also, an estimate of any equied pecentile and confidence inteval can be calculated fom the pedictive distibution n ode to satisfy the assumption that the sample is identically distibuted in the bootstapping pocedue, the eason esiduals ae calculated and used As in England and Veall (2007, we use the eason esiduals of the obseved development factos athe than those fo the actual claims, since we ae using ecusive models Note that a bootstap bias coection is also needed, and the simplest way to do this is to multiply the esiduals by n, whee p is the numbe of ( n p paametes estimated in the model and n is the numbe of esiduals n addition to dawing the gouped sample fo bootstapping coelated data sets, thee ae also two othe pactical points that should be mentioned The fist is to note that the fitted values ae obtained by stating fom the final diagonal in each tiangle and woking backwads, by dividing by the development factos The second is that the zeo esiduals which appea in both tiangles ae also left out - 0 -

11 42 Algoithm This section povides the algoithm, step by step, which is needed in ode to implement the bootstap pocess intoduced in section 4, - Apply the ML model to both the cumulative paid and incued claims data to obtain the esiduals fo all the fou sets atios, the paid, incued link atios, the paid ove incued atios and the evese They can be obtained fom following equations: ˆ F f, ˆ σ qˆ, τ ˆ F f and ˆ σ ˆ q τ - Adust the eason esidual estimates by multiplying ( n p n to coect the bootstap bias, whee p is the numbe of paametes estimated in the model and n is the numbe of esiduals - Goup all the fou esiduals, ie, {(, ( ( ( },, U, and togethe We wite this as - Stat the iteative loop to be epeated N times ( N 000 This consists of the following steps: Randomly sample fom the gouped esiduals with eplacement, denoted as {(,( ( ( },, U the gouped esiduals is ceated, fom the gouped tiangle so that a pseudo sample of 2 alculate the ML eseves fo the pseudo samples of paid and incued claims by fitting the ML model, as shown below alculate the coesponding coelation coefficient fo the e-sampled data using the ( pseudo esiduals, (, ( and ( as follows, ( ˆ ρ i, ( ( i, ( 2 and ( ˆ ρ ( ( i, 2 ( ( i, ( ( ( ( whee,, and denote the e-sampled esiduals, which ae obtained by simply sepaating U (,(,(,( { } alculate the pseudo samples of the fou tiangles fo the paid, incued link atios, the atios of paid ove incued and the evese by inveting the eason esiduals definition as follows: - -

12 and ( F ( F ( ˆ σ ˆ + f, ( ( ˆ σ ˆ + f, ( ( ˆ τ q +, ( ˆ τ + qˆ alculate the weighted and weighted aveage of the bootstap paid and incued development factos as follows: and ˆ, ( qˆ ( i, + ( f ( F ˆ, ( q ( i i, + ( f ( F ˆ Note that the weights used hee ae fom the oiginal data sets and not fom the pseudo samples alculate the bootstap development factos adusted by the coelation coefficient between the pseudo samples as follows: and ˆ ˆ τ ( ( ˆ ( ˆ σ λ ( ˆ ( ˆ f + ρ q ( ˆ ( ˆ σ λ ( ˆ ( ˆ f ρ ( ˆ + q, ˆ τ fo the e-sampled bootstap paid and incued un-off tiangles, espectively alculate the futue cumulative payment using the following equations ( ˆ i n i ( fˆ, + 2 i+ * i, i+ and ( ˆ ( ( i fˆ * ˆ, + and ( ( ˆ i n i fˆ, + 2 i+ * i, i+ and ( ˆ ( ( i fˆ * ˆ, whee > n i + +, - 2 -

13 3 Simulate a futue payment fo each cell in the lowe tiangle fo both paid and incued claims, fom the pocess distibution with the mean and vaiance calculated fom pevious step To do this, the following steps ae equied: Fo the one-step-ahead pedictions fom the leading diagonal, a nomal distibution is assumed ie fo, 2 i n ( 2 ( ˆ λ ( ˆ σ ~ Nomal, and ( ˆ ~ Nomal λ,(( ˆ σ 2 i, + i 2 i, + i i, + i + i i, + i i, + i 2 i, + i i, + i + i i, + i fo paid claims fo incued claims Fo the two-step-ahead pedictions up to the n-step-ahead pedictions, nomal distibutions ae still assumed, but with the mean and vaiance calculated fom pevious pediction instead of the obseved data ie fo 3 k n and n k + 3 n, ( 2 ( λ ( ˆ σ ~ Nomal ˆ ˆ, ˆ and ( ˆ ˆ ~ Nomal λ,(( ˆ σ 2 ˆ kl kl, kl, l kl, kl kl, kl, l kl, fo paid claims, fo incued claims - Sum the simulated payments in the futue tiangle by oigin yea and oveall to give the oigin yea and total eseve estimates espectively - Stoe the esults, and etun to the stat of the iteative loop 5 Examples This section illustates the bootstapping appoach to the ML and uses two numeical examples to assess the esults The fist example uses the data fom uag and Mack (2004 Example 2 uses maket data fom Lloyd s which have been scaled fo confidentiality easons These data elate to aggegated paid and incued claims fo two Lloyd s syndicates, categoized at isk level Example is included in ode to illustate the esults fo the oiginal set of data used by uag and Mack (2004 The pupose of example 2 is to illustate that the ML model does not necessaily povide bette esults in all situations The indications fom ou esults that it pefoms bette when the data have less inheent vaiability and ae less umpy 5 Example n this section, we use the data fom uag and Mack (2004 0,000 bootstap simulations wee caied out, and the data and esults ae shown in Tables to 5-3 -

14 Table aid laim Data fom uag and Mack ( ,804,970 2,024 2,074 2,02 2,3 866,948 2,62 2,232 2,284 2,348,42 3,758 4,252 4,46 4,494 2,286 5,292 5,724 5,850,868 3,778 4,648,442 4,00 2,044 Table 2 ncued laim Data fom uag and Mack ( ,04 2,34 2,44 2,74 2,82 2,74,844 2,552 2,466 2,480 2,508 2,454 2,904 4,354 4,698 4,600 4,644 3,502 5,958 6,070 6,42 2,82 4,882 4,852 2,642 4,406 5,022 Tables and 2 show the data, and in ode to obseve the un-off natue of the data in a staightfowad way, figues and 2 show plots of the data fom table and 2, espectively Fom figues and 2, it can be seen that the data ae stable and not too much spead out Figue aid laims aid laims laims Development Yea - 4 -

15 Figue 2 ncued laims ncued laims laims Development Yea The bootstap methodology descibed in this pape has been applied, and the esults ae shown in Tables 3 and 4 Table 3 simply shows that the theoetical ML eseves (fom uag and Mack and the mean of the bootstap distibutions ae close to each othe in both cases of paid and incued claims Table 4 displays the bootstap pediction eo of the ML eseves poected by both paid and incued claims These ae displayed both in absolute tems, and as a pecentage of the Reseves Table 3 ootstap Reseves and ML Reseves ootstap ML Reseves Reseves aid ncued aid ncued Yea Yea Yea Yea Yea Yea Yea 7 5,52 5,655 5,505 5,606 Total 6,893 7,75 6,846 7,63-5 -

16 Table 4 ootstap ediction Eo ediction Eo ediction Eo % aid ncued aid ncued Yea 0 0 0% Yea % 5% Yea % 52% Yea % 26% Yea % 35% Yea % 3% Yea % 2% Total % % t is inteesting to compae the esults with those fom the chain ladde model Fo this, we used the bootstap esults based on the ove-dispesed oisson o ove-dispesed negative binomial models descibed in England and Veall (2007 Since the pupose of the ML model is to use moe data to impove the estimation of the eseves, it is expected that the pediction eos should be lowe than the staightfowad L model This is confimed fo these data by Table 5, which shows that the pediction eo of the ML eseves is lowe than the pediction eo of L eseves As has been said above, this should not be supising since the ML eseves use moe infomation than the L eseves Table 5 ootstap edictions of L and ML Models ootstap L ediction Eo % ootstap ML ediction Eo % aid ncued aid ncued Yea - 0% - 0% Yea 2 45% 9% 4% 5% Yea 3 33% 96% 42% 52% Yea 4 2% 38% 2% 26% Yea 5 8% 62% 24% 35% Yea 6 3% 47% 3% 3% Yea 7 22% 4% 3% 2% Total 6% 3% % % n figue 3, the distibutions of the ML and L eseve poections fo paid and incued, ae plot espectively in ode to compae the esults moe staightfowadly Figue 3 shows that the paid and incued best eseve estimates ae vey close when using ML appoach And on the contast, the paid and incued best eseve estimates, poected by L method, ae much futhe compaed with the ML case Moe impotantly, the L method povides a much moe spead out DF gaph than the ML appoach, both in paid and incued cases This means that the ML is a bette method as it not only bidge the gap between paid and incued best eseves estimates, but also poduces a smalle uncetainty aound the best eseve estimates - 6 -

17 Figue 3 edictive Distibutions of Oveall Reseves A compaison between L and ML poections ML aid ML ncued L aid L ncued 52 Example 2 n this section, a set of aggegate data fom Lloyd s syndicates ae consideed n this case, the data ae not as stable o well-behaved and the esults ae quite diffeent Tables 6 and 7 show the data, which ae plotted in figues 4 and 5 t can be seen fom these figues that the data ae much moe unstable and moe spead out compaed with the pevious two examples Table 6 Scaled Aggegate aid laims at Risk Level fom Lloyd s 84,845 3,748 5,400 6,23 9,006 9,699 0,008 0,035 0,068 55,483 3,768 7,899 8,858 3,795 5,360 5,895 9, ,287 0,635 6,02 22,77 28,825 29,828 30, ,038 2,879 6,329 4,366 6,20 922,693 3,523 4,64 6,43 8, ,424 5,649 7, ,980 5,062 24,782 3,

18 Table 7 Scaled Aggegate ncued laims at Risk Level fom Lloyd s,530 8,238 0,564 2,332 2,73 0,576 0,630 0,36 0,325 0,280,505 6,247 8,728 0,500 5,24 6,720 6,845 6,829 9,675 2,505 6,50 7,937 22,43 29,5 33,336 32,62 3, ,748 9,984 3,67 6,523 7,807 8,959 2,285 4,36 6,432 8,834 2,092 5, ,549 7,24 2,422 3,58,27 2,657 6,87, ,533 6,423 2,023 5,45,779 Figue 4 Scaled aid laims fom Lloyd s Maket aid laims laims Development Yea Figue 5 Scaled ncued laims fom Lloyd s Maket ncued laims laims Development Yea - 8 -

19 The ML model still poduces consistent ultimate loss pedictions fo this data set, as shown in table 8 Howeve, the pediction eo contained in table 9, estimated by the bootstap ML appoach, appeas to be much highe than fo the pevious examples Table 8 ootstap Reseves and ML Reseves ootstap ML Reseves Reseves aid ncued aid ncued Yea Yea Yea 3 4,77 3,945 3,97 3,974 Yea 4 7,39 4,548 6,692 4,306 Yea 5 8,366 9,007 7,223 8,200 Yea 6 0,708 8,492 0,456 8,34 Yea 7 4,29,553 4,430,29 Yea 8 9,670 8,845 9,004 9,05 Yea 9 23,980 9,987 23,584 9,85 Yea 0 27,90 24,542 28,90 24,633 Total 6,459 9,386 2,822 89,35 Table 9 ootstap ediction Eo ediction Eo ediction Eo % aid ncued aid ncued Yea 0 0-0% Yea % 92% Yea 3 3,489 3,693 84% 94% Yea 4 5,630 3,059 77% 67% Yea 5 8,02 6,905 99% 77% Yea 6 6,346 5,83 53% 69% Yea 7 8,7 7,320 3% 63% Yea 8 7,507 9,29 8% 05% Yea 9 3,947 4,92 33% 7% Yea 0 5,698 36,989 85% 5% Total 86,509 47,804 74% 52% This becomes even cleae when compaing with the chain ladde technique Table 0 shows that the ML model poduces a highe pediction eo than the L model The conclusion fom this is that although the ML model uses moe data, and should be expected to poduce lowe pediction eos, this is not always the case in pactice We believe that the eason fo this is that the assumptions made by the ML model the specific dependencies assumed ae not as stong as expected in this case A conclusion fom this is that the data have to be examined caefully befoe the ML model is applied - 9 -

20 Table 0 ootstap edictions of L and ML Models ootstap L ediction Eo % ootstap ML ediction Eo % aid ncued aid ncued Yea - 0% - 0% Yea 2 48% 64% 3% 92% Yea 3 02% 88% 84% 94% Yea 4 99% 57% 77% 67% Yea 5 92% 28% 99% 77% Yea 6 7% 37% 53% 69% Yea 7 72% 36% 3% 63% Yea 8 62% 49% 8% 05% Yea 9 97% 55% 33% 7% Yea 0 85% 03% 85% 5% Total 66% 36% 74% 52% Again, in figue 6, a moe staightfowad compaison is povided t can be seen that the L model poduces less vaiable foecasts than the ML model fo this set of data: the uncetainty o the pediction vaiance ae elatively smalle than the ML appoach Figue 6 edictive Distibutions of Oveall Reseves A compaison between L and ML poections ML aid ML ncued L aid L ncued

21 6 onclusion This pape has shown how a bootstapping appoach can be used to estimate the pedictive distibution of outstanding claims fo the ML model The model deals with two dependent data sets, ie the paid and incued claims tiangles, fo geneal insuance eseving puposes We believe that bootstapping is well-suited fo these puposes fom a pactical point of view, since it avoids complicated theoetical calculations and is easily implemented in a simple speadsheet This pape adapts the method by taking account of the dependence obseved in the data and maintaining it by esampling paiwise A numbe of examples have been given, which show that the ML model does not always poduce supeio esults to the staightfowad chain ladde model As a consequence, we believe that it is impotant fo the data to be caefully checked to test whethe the dependency assumptions of the ML model ae valid fo each data set befoe it is applied - 2 -

22 Refeences Efon, (979 ootstap methods: anothe look at the ackknife Annals of Statistics 7, -26 Efon, and Tibshiani RJ (993 An intoduction to the bootstap New Yok: hapman and Hall England, D and Veall, RJ (999 Analytic and bootstap estimates of pediction eos in claims eseving nsuance: Mathematics and Economics 25, England, D and Veall, RJ (2002 Stochastic laims Reseving in Geneal nsuance (with discussion itish Actuaial Jounal 8, pp England, D (2002 Addendum to Analytic and bootstap estimates of pediction eos in claims eseving nsuance: Mathematics and Economics 3, England D and Veall, RJ (2007 edictive Distibutions of Outstanding Liabilities in Geneal nsuance Annals of Actuaial Science nstitute of Actuaies (2, Lowe, J (994 A pactical guide to measuing eseve vaiability using: ootstapping, Opeational Time and a distibution fee appoach oceedings of the 994 Geneal nsuance onvention, nstitute of Actuaies and Faculty of Actuaies Mack, T (993 Distibution-fee calculation of the standad eo of chain ladde eseve estimates ASTN ulletin 23 (2, Muphy, DM (994 Unbiased Loss Development Factos oc AS 8, inheio, JR, Andade e Silva, JM, enteno, ML (2003 ootstap methodology in claim eseving Jounal of Risk and nsuance, 70 (4, uag, G and Mack, T (2004 Munich hain Ladde latte DGVFM 26,