JAB Chain. Long-tail claims development. ASTIN - September 2005 B.Verdier A. Klinger
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1 JAB Chan Long-tal clams development ASTIN - September 2005 B.Verder A. Klnger
2 Outlne Chan Ladder : comments A frst soluton: Munch Chan Ladder JAB Chan
3 Chan Ladder: Comments Black lne: average pad to ncurred (/I) rato CL keeps a below or above-average /I rato
4 Outlne Chan Ladder: comments A frst soluton: Munch Chan Ladder JAB Chan
5 A frst soluton : Munch Chan Ladder Idea: apply a correcton to the Chan Ladder development factors (CL DF) dependng on the /I rato. Ths ntends to ncrease the DF (for UW year and DY ) f the prevous /I (for UW year and DY -) s lower than the average and vce-versa: Low /I rato Relat. hgh ncurred Hgher future payments Hgh DF
6 A frst soluton : Munch Chan Ladder Y-axs: f f f Where s the DF average e CL. X-axs: / / I I Where / I s the /I average rato.
7 A frst soluton : Munch Chan Ladder DF resdual = DF Average yelds DF = Average DF Resdual and gven the /I rato resdual Corrected DF = CL Correcton
8 A frst soluton : Munch Chan Ladder To be accurate MCL does the same for ncurred and pad but uses I/ ratos for the pad process and the /I rato for the ncurred process. The formula s: Expected pad development factors assumng that both pad and ncurred processes (B (s)) are known up to DY s CL DF Regresson lne slope Varance factor (Scale parameter) I/ resdual
9 Munch Chan Ladder: Results Ultmate /I ratos convergng towards 00% consstent development of pad and ncurred.
10 Outlne Chan Ladder: comments A frst soluton: Munch Chan Ladder JAB Chan
11 JAB Chan We further developped the MCL method to the JAB method by takng nto account: tme-varyng slopes ntegraton nto one model ont estmaton of all factors
12 JAB Chan Tme-varyng slopes MCL uses only one slope n the correcton term: ths mght be a lttle bt too rough. We expect the ncurred process more nformatve durng the frst DYs when the pad are stll low. Thus havng a sngle coeffcent s averagng the correcton over the whole development perod. On the other hand estmatng a dfferent coeffcent for each DY may be dffcult for the last DYs (few data) smoothng.
13 JAB Chan Integraton nto one model the ncurred process s thought to be nformatve to the pad process but the latter does not necessary add some relevant nformaton to the ncurred process. the ncurred process s modeled separately and dfferently e.g. by ncorporatng also calendar year effects.
14 JAB Chan Jont estmaton of all factors MCL les on regresson resdual analyss and thus mples estmatng some parameters calculatng the resduals then deducng the slopes to fnally predct the future pad and ncurred amounts. We adopted a sngle model: smultaneous estmaton of all parameter takng nto account possble dependences between them. drect calculatons for model valdaton and standard error
15 JAB Chan We propose a model n accordance wth the three prevous remarks: the correcton appled to the DF depends on the DY the ncurred process s modeled separately we use a one-step model ( ) ( ) q Q ε β α = f ε = Chan ladder I I f I I ε = JAB Chan ladder Ths last equaton can be rewrtten to: ( ) { } [ ] = = = n n q Q 2 0 ~ where σ ε ε β α K K heteroscedastcty: the varance s proportonal to the payments
16 JAB Chan The model allows us to assume smoothngly varyng α and β coeffcents. The sequence of states s defned by two lnear transton equatons: α β = α εα where εα ~ = β ε β where ε β ~ 2 [ o σ α ] 2 [ 0 σ ] α and β are defned as random walks whle the noses are assumed to be whte nose processes such that ther expected value s zero and ther varance s constant. β
17 JAB Chan arameters estmaton We try to mnmze the followng quantty: ( ) ( ) ( ) ( ) = q Q M β β σ α α σ α β ω β α ˆ measures the ft between the model predctons and realty penalzes volatle estmatons of and Ths can be acheved n a smple way by rewrtng ths nto a matrx form.
18 JAB Chan: results arameters estmaton: not constant growng (n absolute value) convergng towards 0
19 JAB Chan: results Ultmate pad and ultmate /I ratos JAB predctons close to the MCL one greater than CL ultmate /I ratos around 00% consstent development of pad and ncurred processes
20 JAB Chan: results atterns
21 Outlook Ths dea frst wth Munch Chan Ladder and now wth our method s relatvely recent and the feld s stll to be explored. Here are some potental developments among others: ncurred DF could be ncorporated and estmated smultaneously n another model. extenson to non-gaussan pad and ncurred processes nstead of assumng the coeffcents to vary over the DY the coeffcent β may also be assumed to vary a dfferent tme scales or even other metrc varables.
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