Tail Factor Convergence in Sherman s Inverse Power Curve Loss Development Factor Model

Transcription

1 Tal Fator Covergee Sherma s Iverse Power Curve Loss Developmet Fator Model Jo Evas ABSTRACT The fte produt of the age-to-age developmet fators Sherma s verse power urve model s prove to overge to a fte umer whe the power parameter s less tha, ad alteratvely to dverge to fty whe the power parameter s or greater For the overget parameter values, a smple formula s derved, terms of ay fte produt of age-to-age fators, for the edpots of a terval otag the lmt of the fte produt These edpots overge to the lmt as the fte tme utoff pot reases For ay fte tme utoff, the produt of age-to-age fators les elow the terval, ad thus the lower edpot of the terval s always a etter estmate of the lmt tha the fte produt tself Several umeral examples are luded for llustrato The overgee odto ad the terval formula are applale to the seleto of a fte utoff age, revew of the reasoalty of the overgee rate, ad atual umeral alulatos of the tal fator KEYWORDS Tal fator, verse power urve Copyrght 204 Natoal Coul o Compesato Isurae, I All Rghts Reserved VOLUME 9/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 227

2 Varae Advag the See of Rs Bagroud ad troduto Sherma 984) foud that a verse power urve of the form + a t + ) ft empral age-to-age loss developmet fators etter tha several other as futoal forms he tested Lowe ad Mohrma 985) expressed oer aout the overgee of the produt of the age-to-age fators Boor 2006, p 373), ad the CAS Tal Fator Worg Party 203, p 52) oted that there has ee o ow losed-form expresso that approxmates the tal geerated y the verse power urve I prate, the age-to-age developmet fators produed y the urve are multpled together out to some fte age utoff, suh as t = 80, to produe a umulatve developmet fator The mpat of fators eyod that age to ultmate, or the tal fator eyod the utoff, ths ase t = 8, s assumed to e eglgle Alteratvely, f the mpat of the tal fator s ot eglgle, the some other modelg osderato must form the seleto of the utoff tme The potetal dager the assumpto of eglgle tal fator mpat s llustrated Tale ad Fgure Two dfferet sets of parameters share the same tal age-to-age fator of 0 at t = ad the same umulatve fator of 30 from t = to 00 However, whle the umulatve fator for Example, usg power parameter = 40, grows oly a lttle past t = 00, Tale Examples of apparetly overget ad dverget tal fators for the verse power urve model Parameter Values Parameters Example Example 2 a Cumulatve Developmet Fators From to Example Example , , , E+04,000, E+3 0,000, E+4 Example 2, usg = 05, appears to zoom toward fty the tal Ths paper uses as real aalyss Rud 976 eg a stadard textoo referee) to prove that the fte produt of the age-to-age fators overges to a fte umer whe the power parameter s less tha, ad dverges to + whe Note, ths paper we refer to a sequee that reases wthout ay upper oud as dvergg to +, or havg a Fgure Examples ad 2 from Tale 00, C umulatve Developmet Fator From to 0,000000, Example Example ,000 0,000 00, CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE 2

3 Tal Fator Covergee Sherma s Iverse Power Curve Loss Developmet Fator Model lmt of + Furthermore, whe <, for ay fte produt of the age-to-age fators up to a spef age, there s a smple formula for a terval otag the lmt of the fte produt As reases, the terval eomes tghter ad the edpots eah overge to the lmt of the fte produt The lower edpot of ths terval s always a etter estmate of the fte produt tha the fte produt of the ageto-age fators, whh s always less tha the lower edpot It s worth otg aga that tal dvergee does ot eessarly mea the model s vald, ut smply that ay spef fte utoff pot should e otherwse justfed For a overget tal, ether a utoff pot must stll e justfed y some other osderato or are must e tae that the tal fator past the utoff s reasoaly lose to The terval estmate derved ths paper a help aswer the latter questo The proof of overgee/dvergee s lad out Seto 2, wth the proof of several useful lemmas Appedx A The terval estmate s derved Seto 22 Numeral examples of the progressve overgee/dvergee of the fte produt ad the terval estmate of the fte produt for several sets of parameters are show Seto 23 2 Covergee theorem ad lmt estmato Followg the otatoal ovetos of the reet CAS Tal Fator Worg Party 203), the remader of ths paper, d, stead of t, s used for age or tme 2 Statemet ad proof of prmary theorem Frst we wll set up a defto for the fte produt of the age-to-age fators the verse power urve model ) Defto: F a,, ) = + ad + ) where a > d= 0,, ad 0 are real umers ad s a postve teger Note, ths defto ludes ases where d egs at a hgher value tha, as the parameter a e reased to hadle suh ases It s also worth otg that ad + ) > 0, a ey fat that wll e used susequet dervatos Theorem ) If the lm F a,, ) = + ) If < the lm F a,, ) = Fa,, ) < + exsts ) For ay sequee of umers x > 0 where =,, ad 2 the equalty + x ) > + x holds aordg to Lemma A3 Applyg ths we have F a,, ) = + ad + ) d= d= + + lm d d= + + ) d= + ad + ) = + a d If the > = + aordg to Lemma A, ad osequetly lm F a,, ) = + ) By Lemma A2, log + x) < x for ay x > 0, so log + ad + ) ) < ad + ) Summg over log F a,, ) = log + ad + ) d gves ) + d= d= + = + lm d = + d= < ad + ) = a d If < the L a d exsts ad s less tha + aordg to Lemma A Now ote that log F a,, ) s a reasg sequee, eause + ad + ) > mples that log + ad + ) ) > 0, ad s ouded y L Cosequetly, lm log F a,, ) exsts ad s less tha + So lm F a,, ) = Fa,, ) exsts ad s less tha + 22 A terval estmate for the fte produt lmt For the overget ase of <, t s possle to ostrut a useful terval estmate for the fte VOLUME 9/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 229

4 Varae Advag the See of Rs produt The followg deftos are oveet for spefyg terval estmates Defto: The tal upper oud fator s U a,, ) = exp a ) Defto: The tal lower oud fator s L a,, ) = a ) Theorem 2 Let Fa,, ) = lm F a,, ) If < the: ) ) lm U a,, ) = lm L a,, ) = ) Fa,, ) L a,, )F a,, ), U a,, ) F a,, )) ) + < 0 mples that lm a + ) + ad osequetly lm exp a + ) + ) + < 0 mples lm a + + ) + lm a ) = ) Fa,, ) = F a,, ) + ad + ) d= ) = 0 = Tag the logarthm of the tal fator ad applyg oudg tehques desred Lemmas A ad A2, log + ad + ) ) < d= ) ad a d a + ) = < + d= + d= + + Expoetatg produes + ad + ) d= + = ) exp a ) Cosequetly, Fa,, ) < U a,, )F a,, ) Smlarly, usg tehques from Lemmas A ad ) A3 produes + ad + ) d= + < > + ad + ) d= + = + a d a ) > Cosequetly, d= Fa,, ) > L a,, )F a,, ) Ths ompletes the proof of Theorem 2 The lower edpot of the estmato terval s always a etter estmate of the fte produt Fa,, ) tha smply usg the fte produt F a, ), se L a,, ) > ad osequetly F a,, ) < L a,, )F a,, ) < Fa,, ) The tal oud fators are omputatoally smple eve for large values of ad gve a measure of the relatve wdth of the estmato terval pror to dog the omputatoally tese alulato of the fte produt For example, to aheve a erta target U for the upper + ) log U) + oud requres + a ) A more relevat measure of relatve error, ut wthout ay smple formula for that the author s aware of, s the rato of the tal upper oud fator to the tal lower oud fator U a,, )/L a,, ) = + ) exp a ) a Example : A upper oud fator target set at U = 0 for the parameter values a = , = 40, ad = requres 78 However, y = 29 the rato of the tal upper oud fator to the tal lower oud fator s aout 0 23 More umeral examples Tale 2 shows sx dfferet sets of parameters, eah of whh produes a age-to-age fator at d = of 0 ad a umulatve fator from d = to 00 of 30 The parameter sets are dexed y a set of values {20, 5,, 0, 09, 06} for the power parameter For = the dvergee happes very slowly, ut for = the overgee happes remaraly slowly To aheve U a,, ) 0 for the = parameter set would requre , although y U a,, )/ L a,, ) 0, stll a astroomally slow rate of overgee 230 CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE 2

5 Tal Fator Covergee Sherma s Iverse Power Curve Loss Developmet Fator Model Tale 2 Examples of fte developmet fator produts ad terval estmates for fte developmet fator produts Parameters Parameter Values Example 3 Example 4 Example 5 a Cumulatve Developmet Fator Produt Ifte Produt Lower Boud, Ifte Produt Upper Boud) Example 3 Example 4 Example , 43) , 589) , 3877) , 428) , 585) , 3868) , 423) , 58) , 3858), , 423) , 580) , 3856) 0, , 423) , 580) , 3856) 00, , 423) , 580) , 3856),000, , 423) , 580) , 3856) 0,000, , 423) , 580) , 3856) 00,000, , 423) , 580) , 3856),000,000, , 423) , 580) , 3856) Parameters Parameter Values Example 6 Example 7 Example 8 a Cumulatve Developmet Fator Produt Example 6 Example 7 Example , , , ,000, E+05 0,000, E+4 00,000, E+37,000,000, E+94 VOLUME 9/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 23

6 Varae Advag the See of Rs Aowledgmet The author s greatly thaful to Joh Roertso, Da Corro, ad Le Her for revew ad ommets o ths paper Appedx A Lemmas Lemma A Let e a postve teger ad l > 0 = l ) If the lm =+ = l ) If < the lm <+ exsts It suffes to show overgee or dvergee for lm =+ l se l s a fte umer For > ad 0,, ad therefore lm =+ l = + For > ad < 0, s a strtly dereasg futo of, ad therefore there s a sadwh + equalty t dt < < t dt ad osequetly + t dt < < t dt =+ l l Solvg the tegrals whe l ) + l ) l l + < < ) =+ l + l ) + + For <, tag lmts produes + l < < + lm =+ l + I ths ase, the upper oud of the equalty s a fte umer ad mples overgee to a fte umer se the sequee of partal sums the mddle s o-dereasg For < < 0, tag lmts results lm =+ l = +, se ths ase the lower oud of the ) l + ) earler equalty dverges lm + = + For the ase =, tegrato of the earler equalty leads to log + ) < < ) l + Oe aga tag lmts leads to =+ l lm =+ l log l = + from the lower oud of the equalty dvergg + lm log l + ) = + Lemma A2 If x > 0, the log + x) < x If t > the /t < 0, ad osequetly + x t ) dt < 0 + x + x So, dt t dt < 0 ad solvg the tegrals produes log + x) x < 0 Lemma A3 + x ) > + x > 0, =,,, ad teger 2 x for ay sequee of umers We proeed y duto For = 2, se x x 2 > 0, t follows that + x + x 2 + x x 2 > + x + x 2 Assume the oluso of the lemma s true for where 2 We wll show that the lemma s the true for + + I geeral + x ) = + x ) + x ) = + x ) + x + x ) + But x + + x ) > x + so + x ) + x x ) > + x + x = + x, whh estalshes the lemma CASUALTY ACTUARIAL SOCIETY VOLUME 9/ISSUE 2

7 Tal Fator Covergee Sherma s Iverse Power Curve Loss Developmet Fator Model Referees Boor, J, Estmatg Tal Developmet Fators: What to Do Whe the Tragle Rus Out, Casualty Atuaral Soety Forum, Wter 2006 CAS Tal Fator Worg Party, The Estmato of Loss Developmet Tal Fators: A Summary Report, Casualty Atuaral Soety Forum, Fall 203 Lowe, S P, ad D F Mohrma, Dsusso of Extrapolatg, Smoothg ad Iterpolatg Developmet Fators, Proeedgs of the Casualty Atuaral Soety 72, 985, pp Rud, W, Prples of Mathematal Aalyss 3rd ed), New Yor: MGraw-Hll, 976 Sherma, R E, Extrapolatg, Smoothg ad Iterpolatg Developmet Fators, Proeedgs of the Casualty Atuaral Soety 7, pp 22 55, 984 VOLUME 9/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 233