Insurance Risk EC for XL contracts with an inflation stability clause

Transcription

1 suce Rs E fo L cotcts wth flto stlt cluse 40 th t. AT oll. Mdd Jue 9-0 opght 008 FR Belgum V "FRGlol".

2 Aged upemposed flto / stlt cluse suce Rs olutos o-lfe Rs / tdd ppoch o-lfe Rs / tochstc ppoch goss of Re o-lfe Resuce Rs Resuce Rs fo lmted L cotcts Exteded d vte logoml ppoxmtos opght 008 FR Belgum V "FRGlol".

3 upemposed flto / stlt cluse Hghe losses e suect to hghe flto th clsscl flto the soclled supemposed flto Mesuemet: Msteso lm ost dex Les of Busess Resuce mpct Assume fxed deductle of L cotct ove clms developmet peod f clm exceeds the deductle the the esue must cove ll futue ceses due to supemposed flto: flto mol hd opght 008 FR Belgum V "FRGlol". Potecto gst flto mol hd: flto stlt cluse Futue flto s shed etwee the cedg comp d the esue Exmple: dte of pmet stlt cluse dex the deductle d lmt of L cotct usg the to of the sum of ctul pmets to the sum of flto dusted pmets

4 suce Rs olutos Resuce outeptes suce & Fcl evets lfe: deth suvvl dslt o-lfe: ccdets popet & llt clms fcl: e.g. teest-te chges Resuce Mets Resuce otcts Behvo lpse execsg optos e e e 3 e come E@R Vlue Rs Tme estvt V@R 3 A suce s lss d epotg s comto of: Vlue come hge Vlue estvt Rs sstemtc devtos fom ometc lfe tles pmete s model s etc. o-lfe pemum & esevg s pocess s esuce s ctstophe s etc. Egs t s. 3 Vlue t s. opght 008 FR Belgum V "FRGlol". 4

5 o-lfe Rs / tdd ppoch tdd ppoch ol cuet R fo comed pemum & eseve s V σ : Vlue of comed volume mesue fo cuet e : Vlue of comed stdd devto fo cuet e V@R mesue R σ V ρ α σ ρ α exp : cuet e olvec ptl Requemet R vlue-t-s of uexpected cese log-omll dstuted comed loss to t the cofdece level α99.5% Detls of specfcto { } Φ α l σ σ : Q500 dstöm0. Hdoo of olvec Futhe edgs : The olvec Hdoo Rs Boos hp. & 3 opght 008 FR Belgum V "FRGlol".

6 o-lfe Rs / tochstc ppoch goss of Re put: Le of Busess LoB chctestcs : dom ume of clms e.g. Posso dstuto : dom dvdul clms d... detcl dstuto : dom sum of ultmte ggegte pd clms t... : clms pmet dtes t : fl settlemet dte c... : clms pmet ptte f 0... : flto dex umultve Pd d cued lms f c... f 0 opght 008 FR Belgum V "FRGlol".... R... d... : futue dvdul pd clms : futue dvdul cumultve pd clms : futue dvdul clms eseves : futue dvdul eseve devto ptte

7 : futue dvdul cued clms mplfg fomuls: cumultve pd outstdg wth the flto dusted fctos Vlue of Aggegte umultve Pd d cued lms o-lfe Rs / tochstc ppoch goss of Re R d d f f c... 0 opght 008 FR Belgum V "FRGlol". : futue ggegte cumultve pd clms : futue ggegte cued clms come: cemetl Vlues of Agg. umultve Pd d cued lms

8 o-lfe Rs / tochstc ppoch goss of Re 3 Aggegte Loss Reseves d cemetl Aggegte Loss Reseves R... R R... R d RM goss of Resuce Model ssumpto: detemstc ptte of futue eed pemums The the futue uexpected cese of losses e gve [ L ] E[ ] R E[ R ] E[ ]... L E olvec ptl Requemets R s goss of Resuce V@R esp. E@R: R VR 0 VRα [ ] E[ ] Rs mg RM goss of Resuce estcted to pemum & eseve s: VR VR RM v R : cost-of-cptl te; v : s-fee dscout te o f o R VR [ ] E[ ]... VR α f opght 008 FR Belgum V "FRGlol".

9 o-lfe Resuce Rs put: splttg of LoB chctestcs etwee edet d Resue c : splttg of dvdul clms c : splttg of ultmte ggegte pd clms c plttg of cemetl Aggegte cued lms c c c c... R d RM of Resuce olvec ptl Requemets R s of Resuce V@R esp. E@R: R [ ] E[ ] 0... VR VRα Rs mg RM of Resuce estcted to pemum & eseve s: RM VR o v R f VR opght 008 FR Belgum V "FRGlol".

10 o-lfe Resuce Rs Exmples: splttg dvdul/ggegte clms etwee edet d Resue Popotol Resuce Exmple : quot-she c q q Exmple : suplus R h o-popotol Resuce Exmple 3: excess-of-loss R c m h R le mxml mout sue s wllg to p fo ech polc sum sued of the polc ht the -th dvdul clm Exmple 4: stop-loss c m d mx splttg of ultmte ggegte pd clms h c m d mx d0 d0 opght 008 FR Belgum V "FRGlol".

11 Lmted L cotct wth flto stlt cluse Gve e the followg put chctestcs: : clms pmet dtes : clms pmet ptte : flto dex : supemposed flto dex Resuce Rs fo lmted L cotct l m m t... c... f... 0 s... 0 opght 008 FR Belgum V "FRGlol". The the cumultve pd d cued loss sequeces e gve wth m m... m m l l 0 0 f f s s c d s s c... 0

12 Resuce Rs fo lmted L cotct Exmple: compoud Posso Peto esuce model d... ~ Posso λ [ ] : Peto clms wth dstuto : ume of clms > OP Recll tht VR depeds upo the cemetl cued losses α... eed to ow the vte dstutos of the dom loss vectos... Pgmtc ppoch: vte logoml ppoxmtos γ x F x OP x OP > 0 γ > Bsed o fomuls fo the me coeffcet of vto of d coelto coeffcets etwee d detls ppe opght 008 FR Belgum V "FRGlol".

13 Resuce Rs fo lmted L cotct 3 Exteded olvec stdd ppoch mple uvte logoml ppoxmto sed o the me d stdd devto of the sped Dffcult : egtve loss speds ovestted loss eseves Appl logoml ppoxmto to the poft sped: Poposto A. Let Z e loss wth o-eo fte me d vce [ ] 0 Z V[ Z] / { } { } [ ] exp Φ α l [ ] Φ Φ α l VR VR R Z R Z se : E Z > α α ε [ ] 0 V[ Z] / se : E Z > R VR α { } { } [ ] exp Φ ε l [ ] Φ Φ ε l VR Z R Z α ε Dffcult : udeestmto of the VR d VR s mesues opght 008 FR Belgum V "FRGlol".

14 Bvte logoml ppoxmto close to olvec stdd model Assume follows vte logoml dstuto Dstuto d stop-loss tsfom of the vte logoml sped Assume vte logoml wth pmetes.e. stdd vte oml wth coelto Let deote the suvvl fucto Resuce Rs fo lmted L cotct 4 ρ l l Z Z ρ [ ] E P F π... opght 008 FR Belgum V "FRGlol". Let deote the suvvl fucto d the stop-loss tsfom of the sped Poposto B. tegl epesettos wth exp exp ' Φ Φ Φ d A d A F d A F ϕ ϕ ρ ρ π ϕ π exp l ρ ρ A

15 pecl cse: Mge s fomul Altcl ppoxmtos Deg L d Zhou008. de: qudtc expso of the uxl fucto oud wth Poposto B. d ode closed-fom ppox. detls ppe ooll B.3 st ode closed-fom ppox. geeled Mge fomul Resuce Rs fo lmted L cotct 5 Φ Φ exp exp 0 ρ ρ ρ ρ π c A c opght 008 FR Belgum V "FRGlol". ooll B.3 st ode closed-fom ppox. geeled Mge fomul Popet : s Mge s fomul Popet : s ecess VR/VR codtos. exp exp Φ Φ Φ Φ F ρ ρ π π F π