Afonso = art. 20. Lo u rd e s B. Afo n s o, Alfredo D. Eg í d i o d o s Reis

Transcription

1 Afonso art. 0 Numercal evaluaton of contnuous tme run probabltes for a portfolo th credblty updated premums by Lo u rd e s B. Afo n s o, Alfredo D. Eg í d o d o s Res a n d Ho a r d R. Waters Abstract The probablty of run n contnuous and fnte tme s numercally evaluated n a classcal rs process here the premum can be updated accordng to credblty models and therefore change from year to year. A major consderaton n the development of ths approach s that t should be easly applcable to large portfolos. Our method uses as a frst tool the model developed by Afonso et al. (009), hch s qute flexble and allos premums to change annually. We extend that model by ntroducng a credblty approach to experence ratng. We consder a portfolo of rss hch satsfy the assumptons of the Bühl mann (967, 969) or Bühlmann and Straub (970) credblty models. We compute fnte tme run probabltes for dfferent scenaros and compare th those hen a fxed premum s consdered. Ke y o rd s Probablty of run; fnte tme run probablty; credblty premums; Bühlmann s model; Bühlmann-Straub s model; large portfolos; numercal evaluaton.. In t ro d u c t o n We compute fnte tme run probabltes for a contnuous tme compound Posson rs model for a portfolo of rss here the premums can be updated accordng to the Bühlmann (967, 969) or Bühlmann and Straub (970) credblty models. We compare these results th those hen the total net premum s fxed. Our approach s based on the frameor developed by Afonso et al. (009), hch uses a combnaton of smulaton and approxmaton to calculate the fnte and contnuous tme run probablty for a rs model here the premum s updated from year to year but s ept constant durng the year. Astn Bulletn 40(), do: 0.43AST by Astn Bulletn. All rghts reserved.

2 l.b. afonso, a.d. egdo dos res and h.r. aters The problem of calculatng the probablty of run for a rs process here the premum s updated accordng to a credblty model has been consdered by varous authors. Dubey (977) assumes that the premum s adjusted accordng to a functon of the clams frequency and contans some nterestng theoretcal results. We go further by consderng the past aggregate amounts and extendng the procedure to the Bühlmann-Straub model. Asmussen (999) compares the behavour of run probabltes n the classcal and a modfed model th adapted premum rules based, for example, on past clams statstcs. He consders the Cramér-Lundberg asymptotc formula and n hs study he separates lght-taled and heavy-taled clams behavour. Tsa and Parer (004) has a target smlar to ours, based hoever on the dscrete tme surplus model, focusng totally on Monte Carlo smulaton, and here the premum s updated accordng to Bühlhlmann s credblty model only. Trufn and Losel (009) has a frameor smlar to Tsa and Parer (004) and addresses problems smlar to those consdered by Asmussen (999). In the next secton e set out our basc methodology and ntroduce assumptons and defntons. In Secton 3 e descrbe a portfolo hch satsfes the assumptons of Bühlmann s credblty model. We sho ho to calculate the probablty of run n contnuous and fnte tme usng to approaches to the calculaton of the annual premum: a classcal approach, here mportant characterstcs of the portfolo are assumed to be non th certanty and the premum does not change from year to year, and a credblty approach here the net premum s updated usng Bühlmann s credblty model. Numercal results are presented for the portfolo. Secton 4 follos a smlar pattern to Secton 3, but the portfolo s no set up to satsfy the assumptons of the Bühllmann-Straub credblty model. The major queston of nterest s, Ho s the probablty of run affected f e update premums accordng to a credblty model? We can pose ths queston n terms of the portfolo as a hole, or n terms of each ndvdual rs thn the portfolo. It s not possble to gve a defntve anser to ether queston, but our examples provde some nsght and the methodology set out n ths paper provdes a ay of nvestgatng ths queston further.. Basc frameor In ths secton e summarze brefly our basc frameor for the calculaton of the probablty of run n fnte and contnuous tme. We do ths n the context of a sngle rs, but n our major applcatons n Sectons 3 and 4 e ll extend t to a portfolo of rss. Full detals of the basc methodology for a sngle rs can be found n Afonso et al. (009). Consder the surplus process for a sngle rs over an n-year perod. We denote by S( t ) the aggregate clams up to tme t, so that S( 0 ) 0, and by Y the aggregate clams n year,,, n, so that Y S( ) S( ).

3 numercal evaluaton of contnuous tme run probabltes 3 We assume that {Y } n s a sequence of..d. random varables, each th a compound Posson dstrbuton hose frst three moments exst. Let P denote the premum charged n year and let U( t ) denote the nsurer s surplus at tme t, 0 t n. We assume premums are receved contnuously at a constant rate throughout each year. The ntal surplus, u ( U( 0 )), and the ntal premum, P, are non. For,, n, e assume that P s a - functon of { U ( j)} j, the surplus at the end of each of the precedng years. For any tme t, 0 t n, U( t ) s calculated as follos: - U( t) u + P + ( t - + ) P -S( t) (.) j j here s the nteger such that t! [, ), and here 0 j Pj 0. For, the surplus level U( ), and hence the premum P, are random varables. Where e sh to refer to a partcular realzaton of these varables, e ll use the loer case letters u( ) and p, respectvely. We denote by c( u, n ) the probablty of run n contnuous tme thn n years. We also need the probablty of run thn year, gven the surplus u( ) at the start of the year, the surplus u( ) at the end of the year and the rate of premum ncome p durng the year, here u( ), u( ) 0; ths s denoted c( u( ),, u( )). We can calculate, more precsely, estmate, c( u, n ), as follos: ( a ) We smulate the annual aggregate clams, Y, Y,, Y n, N tmes, approxmatng the dstrbuton of each Y by a translated gamma dstrbuton th matchng moments. ( b ) Gven the values of the annual aggregate clams for smulaton j, e can calculate successvely u( ), p, u( ), p 3,, u( n ), p n, u( n ). For that smulaton e then compute the probablty of run, denoted as c j ( u, n ). ( c ) If u( ) < 0 for any, e set c j ( u, n ) for that smulaton and move to the next smulaton. ( ) If u( ) 0 for each, e calculate the probablty of run c( u( ),, u( ) ) thn each year [, ] condtonal on the process startng at u( ) and endng at u( ). Afonso et al. (009) sho ho to approxmate t usng a translated gamma process approxmaton to the contnuous tme surplus process thn the year. For ths smulaton, e set n % cj ( u, n) - ( - c( u( -),, u( ))) ( ) Our estmate of c( u, n ) s then the mean of the estmates from each smulaton, {c j ( u, n )} N j, and e can also calculate the standard error of ths estmate.

4 4 l.b. afonso, a.d. egdo dos res and h.r. aters 3. Run probablty th Bühlmann s credblty model 3.. Prelmnares In ths secton e dscuss the effect on the probablty of run for a portfolo of rss of updatng premums accordng to the Bühlmann credblty model. We do ths through a numercal example based on a portfolo of fve rss. The portfolo s specfed n Secton 3.. In Secton 3.3 e sho ho to calculate the probablty of run n the case here the premums, collectve and ndvdual, are not updated annually. Ths s a classcal rs theory approach. In Secton 3.4 e sho ho to calculate the probablty of run n the case here the ndvdual premums, and hence the collectve premum, are updated annually usng Bühlmann s credblty model. The numercal results are dscussed n Secton The portfolo Our portfolo s specfed as follos: The annual aggregate clams from rs,,,,, n year, denoted Y, have a compound Posson dstrbuton. The Posson parameter s l and ndvdual clam amounts have a lognormal dstrbuton th parameters q and ( so that, for example, the mean of a sngle clam s exp( q ). For each of the fve rss, the parameter q has the value ndcated n Table 3.. Table 3. Values o f t h e r s parameter, q. Rs 3 4 q The annual aggregate clams from dfferent rss are ndependent. The annual aggregate clams from the same rs n dfferent years are ndependent. We suppose that e already have fve years data for ths portfolo and that e measure tme n years from hen the portfolo as frst nsured, so that no s tme, hch s the start of the sxth year. We are nterested n the probablty of run n contnuous tme over the next 0 years, that s, beteen tmes and. There s an ntal surplus, u, currently avalable for the hole portfolo. When consderng sngle rss, e assume an ntal surplus u s assgned to each rs.

5 numercal evaluaton of contnuous tme run probabltes We consder to models for the Posson parameters, l : N l s constant and equal to 000 each year for each rs. In ths case e denote the common value l; N l s a random varable and {{l } } n s a set of..d. random varables, each th a U( 800, 00 ) dstrbuton. Note that ths portfolo has been constructed to satsfy the assumptons of the Bühlmann credblty model. For scenaro N: E[Y q ] E[E[Y q ] l ] E[l ] exp( q ) 000 exp( q ) V[ q ] V[E[( Y q ) l ]] + E[V[( Y q ) l ]] V[l exp( q )] + E[l exp( q )] exp( q ) ( V[l ] + E[l ]) exp( q ) so that both E[Y q ] and V[Y q ] are some functons of the rs parameter q, as requred. Note that scenaro N s a specal case of scenaro N th V[l ] 0. We can smulate the aggregate clams for each of the fve rss and for each of the years, 0 n the future and fve n the past. It s convenent to do ths by assumng each Y has a translated gamma dstrbuton, t s smple and a good ft [see Afonso (008)]. The steps n ths smulaton process are frst to smulate a value for l ( f necessary ), to calculate the parameters of the translated gamma dstrbuton hch matches the frst three moments of Y ( gven the value of l ) and fnally smulatng the value of ( the translated gamma approxmaton to ) Y The classcal approach We no put ourselves n the place of the actuary settng premums for each of the fve rss for each future year for ths portfolo. We assume n ths subsecton that the actuary nos ( precsely ) the expected value of the aggregate clams for the portfolo each year, e[ Y ]. The premum charged each year for ths portfolo s P, here: P ( +h(u)) e Y G here the premum loadng factor, h( u ), s a functon of the ntal surplus, u, and s taen from Table 3.. Note that snce under scenaro N the expected

6 6 l.b. afonso, a.d. egdo dos res and h.r. aters value of the Posson parameter s the same as the constant value for N, the value of e[ Y ] s the same under N and N. Hence the premum s the same n both cases. The loadng factors, h( u ), have been chosen so that the probablty of ultmate run for the portfolo under N s approxmately 0.0. Ths eases the comparson of results for dfferent ntal surpluses. See Afonso et al. (009). The premum for the portfolo does not change from year to year and s receved contnuously at constant rate. Where e requre a premum for each ndvdual rs n the year, e assume ths s P. Table 3.. Premum loadng factors for Secton 3.3. u h( u ) To calculateestmate the probablty of run thn 0 years for ths portfolo, e smulate the future aggregate clams a large number of tmes, 0000 n our example, and use the methodology outlned n Secton appled to the total aggregate clams for the portfolo each year. We do ths for the to scenaros N ( fxed l ) and N ( varable l ). By applyng ths methodology to the smulated aggregate clams for each rs, and assumng an ntal surplus u and annual premum P, e can estmate the probablty of run thn 0 years for each rs. The numercal results based on ths approach are set out n Table 3.3 n the columns headed P N and P N. Note that for the portfolo the values of c( u, 0 ) for P are close to 0.0 as ntended. We refer to the approach n ths subsecton as the classcal approach snce t has many elements of classcal rsrun theory: some propertes of the aggregate clams dstrbuton are non th certanty ( the mean n our case ), the premums are constant and any data s gnored n terms of decson mang The credblty approach In ths subsecton e assume the actuary taes a dfferent, and more realstc, approach. The actuary has no specfc nformaton about any parameter values. The aggregate annual clams for each of the fve rss for each of the past fve years are non no and, th each successve year, the aggregate clams for that year for each rs are non at the end of that year. The actuary assumes that these fve rss satsfy the assumptons of the Bühlmann credblty model, as set out, for example, n Norberg (979), Secton 3C, and updates the annual net premum for each of these rss n accordance th ths model. Let P denote the premum calculated for rs,,,,, at the start of year, 6, 7,,. We assume that ths premum has a constant loadng factor, h( u ), hch depends on the ntal surplus and s the same for

7 numercal evaluaton of contnuous tme run probabltes 7 each of the rss n each year. We denote by P the correspondng net premum, so that: P P ( + h( u ) ) Then the annual net premum s calculated as follos: P Z Y, + ( Z ) E, here: Y, - E Z s t - j Y Y j -, - - ( -) ( - )( - + s t ),, ( - ) - j j, - ( Y - Y ), s maxe ( Y - Y ) -,0. 4 o Ths s the Bühlmann credblty premum th the usual estmators for the structural parameters. See, for example, Norberg (979), Secton 3D. Note that the total net premum for year, P, s equal to - j Y j ( -), hch s the natural estmate of the mean annual aggregate clams for the portfolo based on the data observed so far. For each scenaro N and N e calculateestmate the probablty of run thn 0 years for ths portfolo and also for each rs separately by smulatng the past and future annual aggregate clams 0000 tmes. Here e use the methodology outlned n Secton th the annual premums updated as descrbed above. The numercal results based on ths approach are set out n Table 3.3 n the columns headed P N and P N. 3.. Results Table 3.3 shos numercal results for both the classcal and credblty approaches under scenaros N and N. These results are estmates of the probablty of run, c( u,0 ), and the standard devaton of each estmate, SD[c( u, 0 )], for each ndvdual rs and for the portfolo, for dfferent values of the ntal surplus, u. The same set of 0000 smulatons of {Y } for N and N ere used to calculate the probablty of run for cases P and P. We mae the follong comments on the results n Table 3.3:

8 8 l.b. afonso, a.d. egdo dos res and h.r. aters Table 3.3. Secton 3: estmates and standard devatons of c( u,0 ). P N P N P N P N u c( u,0 ) SD[c( u,0 )] c( u,0 ) SD[c( u,0 )] c( u,0 ) SD[c( u,0 )] c( u,0 ) SD[c( u,0 )] E E E E E E E E E E E E E E E E E E E E-06 Port E E E E E E E E E E E E E E E E E E E E E E E E-06 Port E E E E E E E E E E E E E E E E E E E E E E E E-06 Port E E E E E E E E E E E E E E E E E E E E E E E E-06 Port E E E E E E E E E E E E E E E E E E E E E E E E-06 Port E E E E-06 ( ) The pattern of results n the table, for the portfolo and for the ndvdual rss, s the same for all values of u. ( ) Comparng portfolo values for scenaros P ( classcal ) and P ( credblty ), e see that the probablty of run s hgher for P relatvely much hgher for N ( fxed Posson parameter ) than N ( varable Posson parameter ). ( ) The portfolo values for scenaro P N are very much hgher than for P N. We ould expect ths. The extra varablty resultng from the varable Posson parameter has not been offset by any ncrease n the premum.

9 numercal evaluaton of contnuous tme run probabltes 9 The model for a varable Posson parameter, based on the unform dstrbuton, may not be reasonable n practce. Hoever, the ncrease n the values from P to P s a remnder that the classcal assumpton of a fxed ( and non ) Posson parameter may be very msleadng. See Dayn et al. (996), page 39. ( v ) For scenaros P, the results for ndvdual rss vary dely. Ths s because the rss are dfferent dfferent expected clam amounts but have been assgned, somehat arbtrarly, the same premum, P, and ntal surplus, u. It s notceable that under P, credblty adjusted premums, the values of c( u, 0 ) are all much closer to each other. In other ords, the credblty adjustment s orng qucly to assgn an approprate premum to each rs. ( v ) The standard devatons of c( u, 0 ) are all very small. A more detaled analyss of those smulatons leadng to end of year, rather than thn year, run shos that: ( a ) The credblty premum n the year before run s alays less than the fxed premum, by about %, and that the aggregate clams n the year of run are on average 0% hgher than expected. Ths sheds lght on pont ( ) above. ( b ) The proporton of c( u, 0 ) due to end of year, rather than thn year, run s smlar for P and P but decreases th the ntal surplus, rangng from 0.96 ( u 0 ) to 0.67 ( u 40 ) for N and from 0.9 ( u 0 ) to 0.3 ( u 40 ) for N. ( c ) The average tme to run ncreases a lttle th u but s smlar for dfferent combnatons of P, P, N and N, rangng from to. years. See Afonso (008) for full detals. 4. Run Probablty th the Bühlmann-Straub credblty model 4.. The portfolo In ths secton e consder the calculaton of the probablty of run, n contnuous and fnte tme, for a portfolo of rss hch satsfy the assumptons of the Bühlmann-Straub credblty model. Our approach s smlar to that used n Secton 3 a numercal study based on a specfed portfolo except that e ll present numercal results only for the portfolo, and not for the ndvdual rss. We consder the portfolo to be more nterestng than the ndvdual rss. We start by specfyng our portfolo. We have a portfolo of fve rss, for each of hch e have fve years past clams data. Tme s ( agan ) measured n years from hen the data ere collected so no s tme. For rs,,,,, clams data for year,,

10 0 l.b. afonso, a.d. egdo dos res and h.r. aters,, conssts of the total aggregate clams, Y, and an assocated rs volume, or eght,. The scaled aggregate clams, Y, s denoted. We assume that the rs volumes for future years, 6, 7,, are non-random and non at the start of the relevant year. We assume: The annual aggregate clams from rs,,,,, n year, Y, have a compound Posson dstrbuton. The Posson parameter s l and ndvdual clam amounts have a lognormal dstrbuton th parameters q and The parameters l are constant and equal to 0 for all rss and all years. For each of the fve rss, the parameter q has the value ndcated n Table 3.. The annual aggregate clams from dfferent rss are ndependent. The annual aggregate clams from the same rs n dfferent years are ndependent. There s an ntal surplus, u, currently avalable for the hole portfolo. We are nterested n the probablty of run n contnuous tme over the next 0 years, c( u,0 ). We consder three cases for the rs volume, see Table 4. for values. W, 30,, 0, 3, 60, 4, 0,, 40. The rs volumes vary among the rss but are constant for each year. The domnant rs, rs, has a small rs parameter, q. W, 30,, 40, 3, 60, 4, 0,, 0. The rs volumes vary among the rss but are constant for each year. The domnant rs, rs, has a large rs parameter, q. W3 For ths case, the rs volumes have been generated by assumng, U( 0.,,., ) here, s the rs volume of case W. Note that ths portfolo has been constructed to satsfy the assumptons of the Bühlmann--Straub credblty model, snce: e[ q ] e[ Y q ] l exp( q ) 0 exp( q ) v[ q ] l exp( q ) 0 exp( q )

11 numercal evaluaton of contnuous tme run probabltes Table 4.. Secton 4: Rs volumes,, by c a s e, r s a n d ye a r. Rs Rs Volume Year 3 4 Total W,, W,, W so that both E[ q ] and V[ q ] are some functons of the rs parameter q, as requred. Throughout Secton 4 e ll assume that the premum loadng factor appled to annual net premums calculated for the portfolo s alays 0%. As n Secton 3, e ll calculate c( u, 0 ) usng dfferent approaches to the calculaton of the annual premum for the portfolo. In ths secton e ll use a classcal approach, here the actuary nos the expected value of the aggregate annual clams for each of the fve rss, an ntermedate approach, here the actuary nos the structure of the portfolo, and a credblty approach, here the net premum s updated at the start of each year accordng to the Bühlmann-Straub credblty model. These approaches are descrbed n Sectons 4., 4.3 and 4.4, respectvely. An mportant dstncton beteen these approaches s that for the classcal and ntermedate approaches the actuary has some pror nformaton about the portfolo and taes no account of the data. For the credblty approach, the actuary s only nformaton about the portfolo comes from the data tself. The numercal results are presented and dscussed n Secton 4.. The calculaton of c( u, 0 ) proceeds as n Secton 3. For each year, 6, 7,,, and rs,,,, e smulate Y by smulatng from a translated

12 l.b. afonso, a.d. egdo dos res and h.r. aters gamma dstrbuton th the same frst three moments. The surplus at the start of the year, u( ), the values of the correspondng rs volumes,, and the total gross premum to be charged n the year, p, are all non at the start of the year. The surplus at the end of the year s u( ), here: u() u( - ) + p - Y. If u( ) s negatve, run has occurred. If u( ) s non-negatve, e can calculate the probablty of run thn the year, c( u( ),, u( ) ), by approxmatng the dstrbuton of the total aggregate clams n year, Y,, by a translated gamma dstrbuton th the same frst three moments. We can then calculate c( u, 0 ) as n Secton. 4.. The classcal approach For the classcal approach e assume our actuary nos precsely the values of the expected unscaled annual aggregate clams for each of the fve rss, E[. ]. Ths s a slghtly stronger assumpton than n Secton 3.3, here e assumed that only the expected annual aggregate clams for the hole portfolo as non. The gross annual premum for the hole portfolo n year, 6,7,,, s gven by: 4.3. The ntermedate approach ( + 0.) e[ ]. For the ntermedate approach e assume the actuary nos the structure of the portfolo n the sense that ( s )he nos that The underlyng Posson parameter for all rss n all years, before scalng by the rs volume, s 0. 40% of the rss have expected clam amount exp ( ). 40% of the rss have expected clam amount exp ( ). 0% of the rss have expected clam amount exp ( ). Hoever, the actuary does not no hch rs has hch expected clam amount ( and does not learn by loong at the data ). Hence, n year, the total premum charged s p, here: 0 p ( + 0.) e o(0.4[ exp( ) + exp( )] + 0. exp( )).

13 numercal evaluaton of contnuous tme run probabltes The credblty approach For the credblty approach, e assume the actuary nos only the past aggregate clams and correspondng rs volumes for each of the fve rss ( as ell as the rs volumes at the start of each future year ). In partcular, the actuary does not no the rs parameter, q, for rs. The actuary assumes the rss satsfy all the condtons for the Bühlmann-Straub credblty model, and updates each year the net premum for each rs accordngly. The gross annual premum for the hole portfolo n year, 6, 7,,, s gven by: ( 0.) P C + here P C s the net credblty premum for rs n year. Ths premum s calculated as follos: t l + l l s m z z z ( P z.. C j j l l l + -. z j m ) e o c t s s ( (, max z z c r j j j ) ) _ f e p o ) * 3 4

14 4 l.b. afonso, a.d. egdo dos res and h.r. aters It can be seen that e are usng the usual estmators thn the Bühlmann- Straub model. See, for example, Bühlmann and Gsler (00), Theorem 4. and Secton Numercal results Table 4. shos ( estmated ) values of c( u, 0 ), together th the correspondng standard errors of the estmates, for selected values of the ntal surplus, u, and for: Three cases for the rs volumes, W, W and W3. Three approaches to the calculaton of the net premums: the classcal approach, labelled P3, the ntermedate approach, labelled P4, and the credblty approach, labelled P. All the values are based on the same set of 0000 smulatons of the scaled aggregate annual clams. It can be seen from Table 4. that the rs volumes for scenaro W3 are broadly smlar to, but more varable than, those for W. From Table 4. e can see that ths varablty of the rs volumes has a neglgble effect on the probablty of run. The results n Table 4. have very small standard errors, as as the case n Secton 3. Table 4.. Secton 4: Estmates and standard devatons of c( u,0 ). Rs Volume u P3 P4 P c( u,0 ) SD[c( u,0 )] c( u,0 ) SD[c( u,0 )] c( u,0 ) SD[c( u,0 )] W W W E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-09

15 numercal evaluaton of contnuous tme run probabltes One dfference beteen the results n Table 3.3 and those n Table 4. s that n Secton 3 e chose the premum loadng factors so that, for a gven ntal surplus, the probablty of run n the classcal case th fxed Posson parameter as approxmately 0.0. Ths s not the case n Secton 4, here the premum loadng factor s alays 0%. Consequently, n Table 4. c( u, 0 ) s alays a decreasng functon of u. The mportant features of the results n Table 4. are: ( ) The results for the ntermedate case, P4, are good for W and W3, but poor for W. Ths because n ths case the actuary s lucy that both W and W gve hgher eghts to the rss th loer expected clam amounts recall that our actuary does not no the expected clam amount for the ndvdual rss and does not learn from the data. ( ) The results for the classcal and credblty cases are very smlar, th the credblty case alays gvng a slghtly hgher run probablty. For the classcal case, our actuary has precse noledge of the expected clam amounts for each of the rss. For the credblty case, the fve years of data are suffcent to allo our actuary to calculate approprate premums. More detaled analyss of the smulatons hch lead to end of year run sho that for all scenaros: ( a ) End of year run occurs almost alays thn the frst to years. ( b ) The average aggregate clams n the year of run s hgher than the overall expected aggregate clams. ( c ) For the credblty case, P, the average net premum n the year of run s loer than the overall expected aggregate clams. Ths mples that n these cases run occurs hen a bad year ( hgher than expected clams ) follos one or more good years. See Afonso (008) for more detals relatng to cases P4 and P. Concludng remar One of our objectves has been to devse a methodology hch can be used to calculate the probablty of run for large portfolos. See Afonso et al. (009) To ease the presentaton e have llustrated our methodology usng portfolos th just fve rss. Increasng the number of rss ould ncrease the tme needed to produce results, but only lnearly. On the other hand, ncreasng the sze of the portfolos by ncreasng the Posson parameter for the expected number of clams ould have no effect on calculaton tme. Ac n o l e d g e m e n t s The authors gratefully acnoledge fnancal support from Fundação para a Cênca e a Tecnologa ( programme FEDERPOCI 00 ).

16 6 l.b. afonso, a.d. egdo dos res and h.r. aters References Afo n s o, L.B. ( 008 ) Evaluaton of run probabltes for surplus processes th credblty and surplus dependent premums. PhD thess, ISEG, Lsbon. Afo n s o, L.B., Eg í d o d o s Res, A.D. and Waters, H.R. ( 009 ) Calculatng contnuous tme run probabltes for a large portfolo th varyng premums. ASTIN Bulletn, 39( ), Asmussen, S. ( 999 ) On the run problem for some adapted premum rules, MaPhySto Research Report No.. Unversty of Aarhus, Denmar. Avalable at MPS-RR999.pdf. Bü h l m a n n, H. ( 967 ) Experence ratng and credblty. ASTIN Bulletn, 4( 3 ), Bü h l m a n n, H. ( 969 ) Experence ratng and credblty. ASTIN Bulletn, ( ), 7-6. Büh l m a n n, H. and Gsler, A. ( 00 ) A Course n Credblty Theory and ts Applcatons, Sprnger, Berln. Büh l m a n n, H. and Str a u b, E. ( 970 ) Glaubürdget für Schadensätze, Mttelungen der Verengung Schezerscher Verscherungsmathemater, 970, -33. Translated to Englsh by C.E. Broos Credblty for Loss Ratos ARCH, 97. Day n, C.D., Pe n t ä n e n, T. and Pesonen, M. ( 996 ) Practcal Rs Theory for Actuares. Chapman and Hall, London. Dubey, A. ( 977 ) Probablté de rune lorsque le paramètre de Posson est ajusté a posteror, Mttelungen der Verengung Schezerscher Verscherungsmathemater, 77, 3-4. No r b e rg, R. ( 979 ) The credblty approach to experence ratng, Scandnavan Actuaral Journal, 979, 8-. Tr u f n, J. and Lo s e l, S. ( 009 ) Ultmate run probablty n dscrete tme th Bühlmann credblty premum adjustments, Worng paper. Avalable at hal en. Ts a, C. and Pa r e r, G. ( 004 ) Run probabltes: classcal versus credblty, 004 NTU Internatonal Conference on Fnance. Lourdes B. Afonso Depart. de Matemátca and CMA Faculdade Cêncas e Tecnologa Unversdade Nova de Lsboa 89-6 Caparca, Portugal E-Mal: lbafonso@fct.unl.pt Alfredo D. Egído dos Res Depart. of Mathematcs ISEG and CEMAPRE Techncal Unversty of Lsbon Rua do Quelhas Lsboa, Portugal E-Mal: alfredo@seg.utl.pt Hoard R. Waters Depart. of Actuaral Mathematcs and Statstcs and The Maxell Insttute for Mathematcal Scences Herot-Watt Unversty Rccarton Ednburgh EH4 4AS, Scotland E-Mal: H.R.Waters@ma.h.ac.u