Estimation Des Lois Maintien Incapacité - NoteTechnique BCAC

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Estimation Des Lois Maintien Incapacité - NoteTechnique BCAC

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bpsis_cp_lancement_avenir_naoassur_e Mediapart Du 5 emp_etude_2015_2016Decembre 2016

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BCAC��B/��

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BCAC��B/��

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Laun lmontant oi EVIN (lmioin89-1009 du 31 décembre 1989) et ses décrets d' a ppl i c ati o n, en exi g i m um pour l e s provi s i o ns techni q ues d' i n capaci t é-i n val i d i t é, ont, d part, imposé la technique dite de capitalisation pour la gestion de ce risque et, d'a 

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derni e rs seront l i s sés par l a méthode de l i s sage non paramétri q ue de Whi t ta Henderson à deux dimensions. Dans l e but de prendre compte de l ’ h étérogénéi t é de notre popul a ti o n, on s’ i n téres dans un second temps, au modèle de Cox et au celui d’Aalen. Le premier mo Sign up to vote on this title

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BCAC��B/��

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Lad’uconstructi o n d’ u ne tabl e de mai n ti e n repose sur des données i s sues de l ’ o bserva n échanti l o n. Dans cette parti e , ces données seront détai l é es dans un premi e r Ensuite, un travail préliminaire de traitement des données aberrantes sera effe 

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BCAC��B/��

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Les17/04/2012. fichiers reçusEn revanche, contiennentILlen’s asinpasistresétésurvenant à parti r du 01/01/2005 j u sq tenu compte des personnes entrée incapacité (temporaire et permanente) après le 31/12/2011 dans le but d’évite 

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Les contrats traités sont de diverses franchises. Celles-ci varient entre 0 et 195 jours Toutefois, la franchise n’est pas toujours précisée (valeur manquante) ce qui pos 

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BCAC��B/��

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Dans nos fi c hi e rs de données, certai n s arrêts étai e nt déj à présents au début d péri o de d’ o bservati o n. Les données sont di t es tronquées pui s que nous n’ e xpl o i t e l2005. ’information relative à ces arrêts que partiel ement, c’est-à-dire à partir du 1er jan Sign up to vote on this title

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BCAC��B/��

(1) L’arrêt

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de travai l a débuté et s’ e st termi n é avant l a péri o de d’ o bservat L’information quant au sinistre ne sera pas utilisée lors de l’analyse ; (2) L’arrêt de travail a débuté avant le début de la période d’observation, mais s terminé durant cette dernière. L’information, bien que tronquée, sera utilisée à com Sign up to vote on this title

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BCAC��B/��

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parla réparti leur suppressi o n bi a i s erai t l e refl e t que pourrai t avoi r notre base. Pour t i o n des si n i s tres suppri m és sel o n l e s di f férentes caractéri s ti q ues comparée à cel l e de l a réparti t i o n du portefeui l e . Par exempl e , si tous siavoinisrtresle sexesuppride ml’aésssurés’avérai e nt être des si n i s tres sur l e s femmes, l ’ e ffet que p sur le maintien serait faussé. Sign up to vote on this title

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BCAC��B/��

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Nous vérifions que : DT ATDT début IJDTin IJDT passage InvalDT sortie Inval Pour le cas d’incapacité, nous prenons comme date d’entrée la date d’arrêt en tra Sign up to vote on this title

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BCAC��B/��

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Nous présenterons, dans cette parti e , une séri e de stati s ti q ues descri p ti v es sur l a obtenue grâce aux retrai t ements. Cel a nous permettra d’ a voi r une mei l connaissance de notre portefeuil e de sinistres. 

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BCAC��B/��

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200 � 180 � � � � 160 � � �140 �


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BCAC��B/��

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Après avoir étudié l ’ i n f l u e ce de l ’ â ge d’ e ntrée en arrêt de travai l sur l a duré mai n tien, i l est l e temps d e vérif i er à quel point l e sexe de l ’ a ss ré peut i m pa l’ancienneté en incapacité. 

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 10000 � 8000 �

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BCAC��B/��

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Sign up to vote on this title 450000 400000 350000

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BCAC��B/��

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Lesmaincaractéri s ti q ues de l ’ a ssuré (âge, sexe, CSP) i n fl u ence non seul e ment sur l a duré ti e n mai s aussi el l e s peuvent être pri s e en compte pour fi x er certai n s paramè liés à la garantie souscrite à savoir la franchise. Sign up to vote on this title

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BCAC��B/��

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900 � 800

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BCAC��B/��

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Effecti v ement, nous avons un vol u me très i m portant mai s si on prend, par exemp casest équi des cadres pour l e ri s que d’ i n val i d i t é, ces derni e rs ne représentent que 5, 9 % ce v al e nt à 1526 arrêts. Ce vol u me ne peut être suffi s ant pour avoi r des bon estimations. Sign up to vote on this title

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BCAC��B/��

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16.04

��  Sign up to vote on this title

Lad’uneduréehypothèse de l’incapaci t é est très courte, el l e est l i m i t ée à 3 ans. Ce qui rend l e ri s de l o i i n adaptée très él e vé si on opte pour un modèl e paramétri Dans ce sens, nous avons choisi de commencer notre étude par l’un des estimateurs 

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BCAC��B/��

16.04

ip, 1iq,  SS(ut) (t) Unquitconcept cl é est l a foncti o n de hasard, qui expri m e l a probabi l i t é i n sta stantané ntané ter un état en , conditionnellement à la survie jusqu’à la date t. Sign up to vote on this title

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BCAC��B/��

16.04

3.

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Le principe de l’estimation de Kaplan-Meier repose sur une idée simple :

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BCAC��B/��

16.04

Nous pouvons expri m er l e s condi t i o ns de premi e r ordre de ce probl è me maximisation comme suit : ∂lnL d ,  n, d, 0  ∂q, q, 1q , Sign up to vote on this title

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BCAC��B/��

16.04

 S(t)   1 n, d,dc, , tr,

 On notera que la méthode de Kaplan-Meier présentée ci-avant, a été développée dan Sign up to vote on this title Useful

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BCAC��B/��

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Onasymptoti peut montrer que l ’ e sti m ateur de Kapl a n-Mei e r véri f i e un théorème de norm q ue gl o bal e . Ce théorème consti t ue une extensi o n du théorème de Don dans le cas où une censure aléatoire droite est présente. 

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BCAC��B/��

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Lesles taux tauxdedesorti sortieebruts ont étécalcestiuléms parés âgel’estiparmateur âge avecde Kaplun pasan-Meijournal i e r. On représent er. 

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BCAC��B/��

16.04

robustesse de l ’ e sti m ati o n. Pl u s parti c ul i è rement, el l e nous permet de cerner l e s zon soumises à un risque d’échantil onnage. Une représentati o n graphi q ue peut nous ai d er à anal y ser l a pl a ge stabl e que utilisera par la suite. Pour ceci, les variances grandissant exponentiel ement, on appl Sign up to vote on this title

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BCAC��B/��

16.04

OnWhicherche donc à appl i q uer un l i s sage à deux di m ensi o ns. L’ a daptati o n de l a méth t taker-Henderson dans l e cas bi d i m ensi o nnel cause pl u s de probl è mes prati q que théoriques. Pour plus de précisions, on se reportera à F. Planchet 2004. Sign up to vote on this title

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BCAC��B/��

16.04

  S      Δq, 

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On cherche à minimiser la fonction suivante :

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16.04


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BCAC��B/��

16.04

Onstatiretistiqeuent riaignoureuse si cette derni è re esti m ati o n. Cependant, on attendra de mener une é avant de se prononcer définitivement. Sign up to vote on this title

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BCAC��B/��

16.04

0,9 0,8 0,7 � � 0,6


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BCAC��B/��

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L’enobjterme ectif ded’alarrêtsegmentati o n est d’ o bteni r des popul a ti o ns l e s pl u s homogènes possi de travai l tout en tenant compte des facteurs consi d érés com discriminants à priori et de permettre une modélisation adéquate des lois. 

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BCAC��B/��

16.04

Notons :  ∑∑,  Le nombre total de sorties observées  ∑  ( ,)  Le nombre total de sorties théoriques  


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BCAC��B/��

16.04

D’maiaprès l e graphe sui v ant, i l sembl e que l e sexe de l ’ a ssuré soi t i n dépendant de ntien dans l’arrêt de travail : Sign up to vote on this title

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BCAC��B/��

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Pour l a catégori e soci o -professi o nnel l e (CSP), nous al l o ns nous i n téresser aux catégo suil’imvpact antesde: lleasCSPcadres, l e s non cadres et l ’ e nsembl e du personnel . Avant d’ e xami n e sur le maintien est significatif ou non significatif selon le test de 

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BCAC��B/��

16.04

Mai n tenant, nous dési r ons obteni r pour chacune de ces segmentati o ns une tabl e mai n ti e n à deux di m ensi o ns (anci e nneté et âge). Or, pour avoi r des résul t ats fi a bl e robustes, il faut faire en sorte d’avoir pour chaque âge un volume de données suffisan Sign up to vote on this title

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BCAC��B/��

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Leparamétri modèleqdeue régressi o n de Cox ou modèl e à hasard proporti o nnel est un modèl e s qui i n trodui t un effet mul t i p l i c ati f des covari a bl e s par rapport à fonction de hasard de base (fonction pour laquelle l’ensemble des valeurs 

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BCAC��B/��

16.04

  Lt,…,t/β,…,βh z (t) exp β     

  S(t) ( ) 

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La vraisemblance dépend de la fonction de hasard de base qui n’est pas spécifiée. P

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BCAC��B/��

16.04

3.

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Lesou decoeffi c i e nts de régressi o n sont obtenus en maxi m i s ant cette vrai s embl a nce part façon équivalente la log-vraisemblance partiel e. 

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BCAC��B/��

16.04

 Onmaxinotemumβde l’evraistimsateur du maxi m um de vrai s embl a nce contrai n t et β l ’ e sti m ateu emblance non contraint. Rapport des maxi m a de vrai s embl a nce : L’ i d ée i c i est de comparer vraisemblances contraintes et non contraintes et d’accepter l’hypothèse nul Sign up to vote on this title

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BCAC��B/��

16.04

valsortieurse audetemps cettet()covari a bl e sur l ’ e nsembl e des i n di v i d us exposés au ri s qu ( temps de l’observation de l’individu i). s  δ z z(())  Sign up to vote on this title

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16.04


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BCAC��B/��

16.04

Une fois les coefficients estimés, on obtient facilement notre loi de maintien. Toutefoi s , avant d’ a l e r pl u s l o i n dans l ’ i n terprétati o n du modèl e , i l est uti l e de vo lgl’hoypothèse d’ u n modèl e de HP est véri f i é e. Cette véri f i c ati o n est à fai r e sur l e mo bal, avant même d’interpréter les tests (qui ne sont pas valables lorsque l’hypoth Sign up to vote on this title

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16.04


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BCAC��B/��

��. ��

16.04


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Leéchelmodèl e à hasard addi t i f d’ A al e n permet de mesurer l ’ e ffet des covari a bl e s sur l e absol u e et non pl u s de mani è re rel a ti v e comme pour l e modèl e de Cox. De p les coefficients de régression estimés varient en fonction du temps. 

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BCAC��B/��

16.04

�)

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Soient : T : La durée passée en incapacité de l’individu .


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BCAC��B/��

16.04

() avec les éléments ()  ()  1… peut alors  Leestivecteur col o nne m é. Pour cal c ul e r ces esti m ati o ns, nous cal c ul o ns d’ a bord l a (1) matri c e qui est définie comme suit :  ()  ()(1,  ()) Pour la è ligne de () Sign up to vote on this title

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BCAC��B/��

16.04

2.

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Dans cette parti e , nous al l o ns présenter l e s tests permettant l a val i d ati o n du mo additif d’Aalen. 

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BCAC��B/��

�� 2.

16.04

�� )


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Les fichiers réalisés lors du modèle de Cox sont utilisés pour le modèle d’Aalen

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BCAC��B/��

16.04

12 000 10 000 � 8 000 � � �


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BCAC��B/��

16.04

tabl e des coef f i c i e nts corre spondante. Les ci n q tabl e s des coeffi c i e nts permettan segmentation ainsi que la logénérale de maintien sont présentées en nnexe. Icipuinous représenterons l e s oi s de mai n ti e n de di f férentes val e urs de ovari a bl e (l e s la CSP) afin de les com arer à la loi de maintien globale. On pren comme âge, Sign up to vote on this title

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BCAC��B/��

16.04

12 000

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BCAC��B/��

��.

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16.04


Pour chaque i n di v i d u en état d’ i n capaci t é, l e s entrepri s es d’ a ssurance doi v ent consti deux types de provi s i o ns techni q ues : une provi s i o n d’ i n capaci t é en cours et provision d’invalidité en attente (couvrant le risque que l’incapable puisse dev 

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BCAC��B/��

16.04

Enfflux,inl,’ulen BCAC préconi s e d ne consi d érer non pas des fl u x, mai s l a moyenne de étant versé en déb ut de péri o de, l’ a utre en fi n de période. C eci a pour obj de diminuer l’écart dû à la d scrétisation de l’intégral. Sign up to vote on this title

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BCAC��B/��

16.04

16,00 14,00 12,00


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BCAC��B/��

16.04

Sign up to vote on this title 18,00 16,00

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BCAC��B/��

16.04

Toutefoi s , el l e ne permet pas de bi e n comparer l e s coeffi c i e nts en termes d’ a nci e nn Par conséquent, nous procédons à une comparaison par âge. Sign up to vote on this title

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BCAC��B/��

16.04

14,00 12,00 10,00


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16.04

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Après avoir calculé les coefficients PM pour les hommes et les femmes, nous av

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BCAC��B/��

16.04

Ondespeut dédui r e que pour un portefeui l e contenant pl u s d’ h ommes âgés (60 ) femmes âgées, l ’ a ssureur qui prend en consi d érati o n l e sexe de l ’ a ssuré provi s i o plprudent us quedecelprendre ui qui calencconsi ule sesdératiprovion sleiosexe ns àdepartil’assuré r les sicoeffi c i e nts gl o baux. C’ e st l e nombre des i n capabl e s est considérable. Sign up to vote on this title

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BCAC��B/��

16.04

18,00 16,00 14,00 12,00


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BCAC��B/��

16.04

Ce(68%) dernieter se810compose de 2241 assurés en état d’ i n capaci t é dont 1731 sont des fem des hommes (32%). Les cadres et l e s non-cadres représen respecti v ement 26, 6 % et 51, 2 % du total des i n capabl e s. On i g nore l a CSP des incapables restant (soit 22,2%). Sign up to vote on this title

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BCAC��B/��

16.04

Ondu sexe. remarque que l ’ i m pact de l a CSP sur l e provi s i o nnement est pl u s i m portant que c Ce derni e r peut ne pas être pri s en consi d érati o n pui s que son i n fl u ence su provisions à constituer est négligeable. Pour conclure, les résultats obtenus sont cohérents avec les tests statistiques Sign up to vote on this title

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BCAC��B/��

16.04

��


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Estimation Des Lois Maintien Incapacité - NoteTechnique BCAC

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