Application of skewness to non-life reserving. ASTIN colloquium 2012

Transcription

1 Applcaton o ewne to non-le reervng ASTN colloquum 0 SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0

2 Reerve r dtrbuton Why an nteret on ewne? dtrbuton Same mean Same ov Sewne gve an ndcator o the aymmetry o the r dtrbuton..00% 6.00%.00% 6.00% LogN Gamma Pareto.00% 6.00%.00% 0'000'000 '000'000 4'000'000 6'000'000 8'000'000 0'000'000 '000'000 4'000'000 6'000'000 8'000'000 0'000' % SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0

3 SOR Swtzerland Ltd Audt ommttee Meetng Augut UWY Dvpt Ultmate Reerve r dtrbuton What do we now n a chan-ladder ramewor? - - or σ 4 5 Bet Etmate R Reerve From Mac (99): n n n n R me wth R me R me () σ σ

4 Reerve r dtrbuton What do we now n other ramewor? Bornhuetter Ferguon Bet etmate nown Etmate o the tandard devaton nown (ee Mac 008) Mx Bornhuetter Ferguon / han-ladder Bet etmate nown Hybrd chan-ladder method provde an etmate o the tandard devaton (ee Arbenz 00) GLM baed on ncremental trangle Bet etmate nown Derent etmate o the tandard devaton gven (See Merz-Wüthrch 008) ape-od Bet etmate nown No etmate o the tandard devaton SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 4

5 Reerve r dtrbuton How do we get the dtrbuton today? Aume a lognormal dtrbuton wth mean gven by Bet Etmate and tandard devaton gven by Mac tandard equaton (ee equaton () on lde ). Boottrappng technque baed on Pearon redual (ee England and Verall 006) Generalzed Lnear Model baed on ncremental trangle (ee Merz and Wüthrch 008) Uual aumpton: The dtrbuton o the random element o the ncremental clam X belong to the Exponental Dperon Famly (e.g. Poon Gamma ) Model o Salzman Wüthrch Merz on hgher moment o the lam Development Reult n General nurance (ASTN Bulletn 0) Two model aumed or the dtrbuton o the ndvdual clam development actor F : Gamma and Lognormal model All o the above model ue ome dtrbutonal aumpton. However n Madrd lat year a preentaton ndcated that the attempt to bactet the hypothe o a lognormal dtrbuton or the reervng r wa not ucceul (See Meyer 0). SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 5

6 Reerve r dtrbuton Why t mportant to now more about t? The reervng r dtrbuton ued n many ntance: alculaton o the Solvency aptal n nternal model Ue n Solvency tandard model Example Bet Etmate Standard devaton Lognormal Gamma µ σ 0.0 θ 0' VaR 99% 5'089'7 4'840'064 % derence on the VaR 99% dependng only on the choce o the dtrbuton SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 6

7 Reervng r dtrbuton Sewne: A propoal or a new approach UWY Dvpt Aumpton on ewne: where: SK SK(... ) [ Var(... )] Ultmate 4 5 (... ) E ( E(... )) depend on but not on [ ]... For one development year wthn a trangle the ewne the ame or any UWY. ontext : Reervng portolo whch r are mlar or every UWY. SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 7

8 8 SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 Sewne : Ue o the new approach Wth the above aumpton under a Mac model or the volatlty we have: [ ] [ ] [ ] uch that S SK SK Var SK σ γ σ γ γ Then t poble to how that the etmator below unbaed.: or - S omment: The ormula or ha a mlar hape a the ormula or The ormula or ue the uual weghted average (power.5) o cubc derence. A or the ormula o outler can play a maor role n the etmaton o 4 The homogenety ormula n term o power o ept n the above ormula. S σ S σ S

9 9 SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 Sewne : Hnt to the proo o unbaedne We want to how that We have: S S E or ) ( - B E B S E Eq [ ] B E B E B E B E Eq ) ( For each element o (Eq) t poble to nd a ormula. Overall mplcaton are poble. { } - B wth

10 Sewne : How to ue the new parameter S? Law o total ewne: ( A) E[ SK( A B) ] SK[ E( A B) ] cov[ E( A B) var( A B) ] SK n the current context we have the ollowng cheme (wthout any mathematcal ormalm): SK SK SK ( ) E[ SK(... )] SK[ E(... )] cov[ E(... ) var(... )] ( ) E S SK[ ] cov[ σ ] ( ) S E SK[ ] σ var[ ] No eay ormula or uch an equaton Smulaton neceary SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 0

11 SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 Sewne : Smulaton to the ultmate UWY Dvpt Ultmate - - or σ 4 5 or - S Supportng dtrbuton to etmate the dtrbutonal properte o. Recurve calculaton (ee Mac 999) Aumpton: The choen upportng dtrbuton hould not be nluencng the overall mulated ewne.

12 SOR Swtzerland Ltd Audt ommttee Meetng Augut ξ ξ ξ 4 ξ 5 ξ 6 ξ 7 ξ 8 ξ 9 ξ 0 Aymmetry Sewne : Smulaton to the ultmate Generalzed Pareto Dtrbuton ) ; ; ( or ) ; ; ( GEV GPD µ µ Parameter o all dtrbuton at tep are nown Etmaton o parameter o all dtrbuton at tep ) ; ; ( GPD U µ µ S σ ξ Varance σ µ

13 SOR Swtzerland Ltd Audt ommttee Meetng Augut ξ ξ ξ 4 ξ 5 ξ 6 ξ 7 ξ 8 ξ 9 ξ 0 Aymmetry Sewne : Smulaton to the ultmate Generalzed Extreme Value Dtrbuton ) ; ; ( or ) ; ; ( GEV GPD µ µ Parameter o all dtrbuton at tep are nown Etmaton o parameter o all dtrbuton at tep ) ; ; ( ln GEV U µ µ Γ Γ Γ Γ Γ Γ S σ ξ - - Γ Γ Varance σ Γ µ

14 Sewne : Applcaton on one trangle han-ladder ramewor '098 '475 '489 '489 '489 '489 '489 '489 '786 '95 '70 '09 ' '89 '5 ' σ SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 4

15 Sewne : Applcaton on one trangle mulaton NA ξ κ NA GPD κ GEV κ ξ S σ omment: Sewne ξ o development year and are the mot gncant. Negatve ewne alo preent on certan development year or whch the tudy o underlyng data could be neceary. SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 5

16 Sewne : Applcaton on one trangle - Reult Actual data GPD model GEV model Mean Mac me Mean Derence to actual Me Derence to actual Mean Derence to actual Me Derence to actual % 0 0.0% 0-0.% 0-0.% % 0.5% 0 -.0% 0-0.% % 0 -.0% 0 0.5% 0-0.4% % 6-0.6% 0.0% 6-0.9% % 4 0.4% 9-0.8% 4-0.% % 8 0.8% 5 -.5% 8 0.% % % 0.5% 60-0.% % 0 0.% 75-0.% 0 -.% % 8 0.8% 69-0.% 9 0.5% Total % 70.8% % 7.% omment: mulaton gve content reult wth Mac ormulac etmate. SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 6

17 Sewne : Applcaton on one trangle - Reult Actual data GPD model GEV model Mean Mac me Sewne Sewne Total ξ 9 NA omment: Overall we have a potve ewne a expected. ontency o overall ewne between GPD and GEV model: The underlyng ewne upport dtrbuton doe not eem to play a gncant role. SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 7

18 Sewne : Applcaton on one trangle - Reult Lognormal Gamma µ σ 0.59 θ 5.68 Smulaton GPD Smulaton GEV Sewne.840 Sewne VaR 99% '44 '4 '5 '8 Lognormal and Gamma dtrbuton parameter are choen on the ba o the mean (494) and o the me (74) correpondng to the overall reerve r dtrbuton. omment: VaR 99% etmated wth a tted Lognormal or Gamma dtrbuton hgher than the VaR 99% comng rom the mulaton. SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 8

19 Reerve r dtrbuton oncluon and next tep Th only a rt tep to try and ncreae the nowledge o the reerve r dtrbuton charactertc. Bad new: The LogNormal aumpton ued to model the reerve r may not be approprate Good New: The LogNormal aumpton ued to model the reerve r conervatve Next tep: Study o Kurto Ue o co-ewne to etmate accdent year ewne ntead o mulatng reult? Revew o general ewne/urto properte o aggregated accdent year reerve See you n Den Haag Holland next year SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 9

20 Reerence MAK Thoma 99 : Dtrbuton-ree calculaton o the tandard error o chan-ladder reerve etmate ASTN Bulletn Vol No 99 MAK Thoma 999: The tandard error o chan-ladder reerve etmate Recurve calculaton and ncluon o a tal actor ASTN Bulletn Vol 9 No 999 MAK Thoma 008 : The Predcton Error o Bornhuetter/Ferguon ASTN Bulletn Vol 8 No 008 WÜTHRH M. V. MERZ M. 008: Stochatc lam Reervng Method n nurance Wley hcheter SALZMANN R. WÜTHRH M. V. MERZ M. 0: Hgher Moment o the lam Development Reult n General nurance ASTN Bulletn Vol 4 No 0 MEYERS G. SH P. 0: The Retropectve Tetng o Stochatc Lo Reerve Model ASTN olloquum 0 Madrd ARBENZ P. SALZMANN R. 00: A robut dtrbuton-ree lo reervng method wth weghted data- and expert-relance SOR Swtzerland Ltd Audt ommttee Meetng Augut 8 0 0