NON-HOMOGENEOUS SEMI-MARKOV REWARD PROCESS FOR THE MANAGEMENT OF HEALTH INSURANCE MODELS.

Transcription

1 NON-HOOGENEOU EI-AKO EWA POCE FO THE ANAGEENT OF HEATH INUANCE OE. Jacque Janen CEIAF ld Paul Janon 84 e 9 6 Charlero EGIU Fax: E-mal: ceaf@elgacom.ne and amondo anca Unverà a apenza parmeno d aemaca per le econ Economche Fnanzare ed Acurave va del Caro aurenzano 9 6 oma ITAY Telephone: Fax: E-mal: rmanca@cec.eco.unroma. Arac How mple and naural o apply NHP o acuaral cence howed n he paper. Two model ueful o olve Permanen Healh Inurance PHI prolem are propoed. The econd one a generalzaon of he fr and perm o ake n accoun no only he nured age a uually done n he leraure u alo o follow he me evoluon. The reward rucure perm o conder mulaneouely he fnancal developmen and he llne evoluon of he healh nurance conrac. Th gve he poly o conrol he dynamc fnancal equlrum.

2 Keyword ochac procee healh nurance em-arkov reward procee permanen healh nurance. Inroducon The fr applcaon of em-arkov Proce P n acuaral feld wa gven y J. Janen []. any auhor uccevely ued hee procee and her generalzaon for acuaral applcaon ee Hoem [6] Carravea e omnc anca [] ahn alcer [4]. In ome ook alo hown how pole o ue hee procee n acuaral cence ee Pacco Olver [3] CI2 [5]. Thee procee can e generaled nroducng a reward rucure ee for example Howard [7]. n h way are defned he Homogeneou em-arkov eward Procee HP. The cree Tme Non-Homogeneou em-arkov eward Procee TNHP were nroduced n e omnc anca [4]. A he auhor knowng hee procee n acuaral feld were nroduced only for he conrucon of heorecal model ha were no ye appled ee e omnc anca Granaa [3] Janen anca [9]. The applcaon propoed n hoe paper were n penon feld. How mple and naural o apply NHP o acuaral cence wll e hown. The paper wll propoe wo model ueful n Permanen Healh Inurance PHI prolem. The econd one a generalzaon of he fr and perm o ake n accoun no only he nured age a uually done n he leraure u alo o follow he me evoluon. The reward rucure perm o have drecly he fnancal evoluon of he healh nurance conrac. Th gve he poly o conrol he dynamc fnancal equlrum. 2. A general acuaral model I poed ha n he llne developmen are expeced m ae.. The fr m- ae are characerzed y dfferen llne level.. The ae correpond o healh ae no llne he oher one correpond o dfferen llne age ha gve dfferen nurance clam. The m-h ae an aorng ae repreen he dead ae. I uppoed ha n he ae he nured people pay conruon n he oher m-2 ae he ge money. The money pad n

3 form of an annuy ha can e gven daly weekly or monhly dependng on he conrac rucure. The model hown n he followng graph. Fg.. m ae model for healh nurance. I o prece ha n h cae he arc are weghed and her wegh are he change ae proale hee proale change n funcon of he evoluon of he emporal varale ha condered h varale can e he age or he me. Furhermore alo he node ha repreen he model ae are weghed and her wegh repreen he reward pad or receved n each ae. Thee reward can e fxed for each ae or can change n he me evoluon of he model. 3. The cree Tme non-homogeneou em-arkov Proce TNHP In h par wll e decred he TNHP. Thee procee were nroduced n e omnc Janen [2] prevouely he connuou one cae wa defned n Iofecu anu [8]. Fr he ochac proce defned. X n n N a random varale r.v. wh he e of ae E{ 2 m} repreenng he ae a he n-h ranon. T n n N an oher r.v. wh e of ae equal o N where T n repreen he me of he n-h ranon

4 X n : Ω E T : Ω Ν. n The proce X n T n a non-homogeneou markovan renewal proce. The kernel Q [Q j ] aocaed o he proce defned n he followng way: Q j P[ X n j T n X n T n ] and reul [8]: p j lm Q j ; j E N where P [p j ] he ranon marx a me of he emedded non-homogeneou arkov chan. Furhermore neceary o nroduce he proaly ha proce wll leave he ae n a me : P[ T n X n T n ]. Ovouly reul ha: m 3. j Q j. Furhermore he followng proale are condered: j P[ X n j T n X n T n ] Thee proale can e gven n funcon of he Q j :

5 3.2 j Qj f Q Q f 2... j j Now pole o defne he proaly druon of he wang me n each ae gven ha he ae uccevely occuped known: G j P[ T n X n X n j T n ]. Ovouly he relaed proale can e oaned y mean of he followng formula: 3.3 G j Q / pj f pj. f p j j Now he TNHP Z Z N can e defned. I repreen for each wang me he ae occuped y he proce. The ranon proale are defned n he followng way: p j P[ Z j Z ]. They are oaned olvng he followng evoluon equaon: 3.4 pj δ j p E j where δ j repreen he Kronecker ymol. I o prece ha n our prolem p j repreen he proaly o go from he ae o he ae j gven ha a me he yem wa n he ae and a me n he ae j where he ae repreen a llne degree n correpondence of whch change he reward. 4. The reward nroducon. Now a reward rucure wll e condered ee ne Oak [] h rucure conneced wh he Z proce. In h way a TNHP wll e condered.

6 Th proce conder each me ha he yem pae for a gven ae he reward ha receved or pad n he ae. repreen he reward receved or pad n he -h ae. The followng formula repreen he evoluon equaon of he TNHP: 4. E E. In our cae he meanng of he equaon 4. he followng one. The lef memer repreen he um of he paymen ae or lale ha were done from he me up he me gven ha a me he proce wa n he ae. In he fr elemen of he rgh de of 4. he erm - repreen he proaly o reman n he ae. In h cae a reward wa pad for - perod. The econd elemen repreen he reward value receved or pad n he ae up o he fr change of ae. A la he hrd elemen repreen he um he paymen receved or pad n he ae ved afer leavng. The followng equaon repreen he dcouned cae of he TNHP. In h way 4.2 nroduce he poly o dcoun he reward: 4.2 E a a ] r ] r E r The only dfference wh he 4. ha he reward are dcouned a me. The nere rae r uppoed n a fr approach o e conan. Each me ha he yem n he ae gven a reward ha hould e dcouned for each epoch a me. Th can e done y mean of a ]r preen value of an unary annuy. Clearly n he acuaral applcaon he non dcounng cae ha no relevance. Ovouly pole o conder reward ha change n he me. In h cae 4. and 4.2 can e repecvely wren n he followng way:

7 4.3 E E 4.4 E r r E r. 4.4 can e furherly generalzed uppong o have he followng erm rucure of mpled forward rae r r 2 r. enong y: < h f r h f h h he dcouned facor relaed o he mpled forward rae rucure he evoluon equaon of he TNHP can e wren: 4.5 E E. Th equaon n marx form can e wren: 4.6

8 or equvalenly: O O * * I I I I 4.7 O O

9 O O where: m m m m m O and:. 3 2 m

10 Furhermore he repreen he uual row column marx produc and he elemen for elemen produc. 5. A non-homogeneou em-arkov ochac nere rae approach. A ochac erm rucure of mpled forward rae nroduced n h par. The rucure wll e conruced y mean of TNHP. In h cae he ae of he proce wll e: { σ σ 2 } F K σ k where he σ repreen all he pole mpled ochac nere rae and k gve he numer of he mpled nere rae. The evoluon equaon of he THP wll e he followng one: φ δ j j E φ j where φ j repreen he proaly ha a me he mpled nere rae wll e σ j gven ha he mpled nere rae wa σ a me. The relaed mean dcoun rucure a me h wll e conruced n he followng way: 5. h h k φj σ j j Where h repreen he mean dcounng facor for a me h gven ha a me he nere rae wa σ and he um nde he parenhe n 5. gve he mean nere rae a epoch gven ha a epoch he nere rae wa σ. A la he evoluon equaon 3.7 ecome:

11 5.2 E E uppoed ha σ wll e he mpled nere rae a me. A la conderng he 4.3 he followng reul are oaned: 5.3 k η φ where η σ wa he known rae of nere a me. I o oulne ha o oan he ochac erm rucure of mpled forward rae neceary o olve he evoluon equaon of TNHP gven n A healh nurance TNHP. The concep of TNHP and TNHP were nroduced n he paragraph 3 and 4. How o apply hee model n a healh nurance prolem wll e gven n he nex wo par. I wa howed ha a P follow he evoluon of he r.v. couple X n T n. ore precely X n repreen he ae of he yem a he n-h ranon and T n he me of he n-h ranon. To apply he model n he healh nurance envronmen wll poed ha T n repreen he age of he nured peron a he n-h ranon. In a fr approach he me developmen wll e gnored. For h reaon all he reward and he nere rae wll e uppoed conan n he me. In h lgh he equaon 4.2 can e ued o decre he evoluon of our model. In ha equaon we have ha:

12 repreen he preen value a me of all he um pad and/or receved n - perod y a peron ha a me had age and wa n he ae he age meaured n me perod. a ]r repreen he preen value of he reward pad n he cae n whch here wan change ae from he age up o he age. E a ] r repreen he preen value of he reward pad n he ae o a peron ha wa n h ae from he age up o he age and a age wen n he ae. E r repreen he preen value of he reward pad afer he fr change of he ae o a peron ha wa from he age up o he age n he ae and a age wen n he ae. To apply he model neceary o know he Q j n fac j and a reul from he 3. and 3.2 can e oaned y mean of and from 3.3 reul: Q j Qj Gj pj. o neceary o evaluae he ncreang d.f. G j * and he non-homogeneou arkov chan P. Th can e done y mean of raw daa n whch repored for each peron of he condered populaon he age of enrance n each ae durng h/her lfe. The model ha are uually ued o manage hee prolem are connuou me model nead he propoed model a dcree me model. The evoluon of a llne a connuou phenomenon u n PHI real applcaon he phenomenon condered dcree n me he mo frequen dcree me ep he week. For h reaon he model doen mple

13 mplfcaon for he conderaon of dcree me. Furhermore he conderaon of connuou me model nvolve he compuaon of ranon nene. Thee proale are generally evaluaed y mean of negro-dfferenal equaon ha nvolve numercal dffcule o ge he oluon. The only dffculy ha he applcaon of a TNHP model nvolve he grea ze of daa ha are neceary o evaluae he Q j and he dffcule o ge he daa n he rgh way. In h cae he wo dffcule are le relevan ecaue n he heal nurance model he numer of ae mall furhermore uually he evoluon of he llne for each peron known. Th fac nvolve ha he daa are naurally aved n he way ha are ueful for he applcaon of em-arkov proce. 7. Generalzed healh nurance TNHP model. In he prevou paragraph wa preened a TNHP model n whch wa no condered he me and T n repreened he age durng he llne evoluon. Th mplfcaon uual n he conrucon of healh nurance model. In h par wll e preened a model ha wll conder oh he evoluon of me and of he age of he nured peron. In h way wll e pole o ake n accoun me dependen reward and erm rucure of mpled forward rae. To oan h knd of model neceary o generalze he one gven n he prevou paragraph. A mlar generalzaon wa gven n [6] for he applcaon of TNHP n penon feld. Takng n accoun h generalzaon he formula 4.5 can e wren n he followng way: 7. E E In h cae and repreen he me and repreen he age of he nured peron a me. o a me > he wll have age. In h cae oo here wll no e dffcule o ge rgh daa ecaue he dae of he v are known wh he evoluon of he llne and he age of he populaon peron.

14 In he cae of ochac nere rae he 7. ecome: E E uppoed ha σ wa he nere rae a me. A la a for 5.2 knowng ha σ η wa he rae of nere a me oaned: 7.2 φ. k η 8. Concluon Applyng he TNHP model for each year a he propoed acuaral prolem we are ale n he fr approach o compue he elemen of he or N. They repreen he preen value of he paymen ae or lale ha were done from he age o he age for a peron ha wa n he ae a age. In h way pole o ge he expoon of he nurer company repec each gven nured poon. In h approach no pole o conder he me evoluon of he yem. The formula ueful o ake n accoun he me evoluon were gven n he prevou paragraph. The 7. conder he me evoluon of nere rae and of he reward. In 7.2 an mpled ochac nere rae rucure condered. In h way he me evoluon conder oh he demographc and he fnancal rk n a naural way. To ge he oluon neceary o olve he evoluon equaon of he TNHP ha follow he ochac nere rae developmen and afer a generalzed TNHP evoluon equaon ha olve n a complee way he prolem ha we are facng. I o oulne ha alo when he auhor olved he prolem of he me developmen of a penon fund y mean of a TNHP hey go mlar reul [6]. In her opnon he

15 NHP a very mporan ool for he conrucon of acuaral model. Now hey are workng o gve a general model ha could e appled o meaure he any knd of acuaral rk. An oher drecon n whch he auhor are workng he conrucon of a he algorhm and he relaed compuer program ueful for he reoluon of THP and TNHP n he way o apply o real prolem her em-arkov model. EFEENCE []. Carravea. e omnc. anca emmarkov proce n ocal ecury prolem n Caher du C.E..O. 98. [2]. e omnc Janen Jacque An algorhmc approach o non-homogeneou em- arkov procee. Inurance: ahemac and Economc vol [3]. e omnc anca.. Granaa The ynamc of Penon Fund n a ochac Envronmen candnavan Acuaral Journal 992. [4]. e omnc anca. A Compuaonal Procedure for he Aynoc Analy of Homogeneou em-arkov Procee ac & Proaly eer Norh Holland 984. [5]. e omnc anca. A Compuaonal Procedure for he Aynoc Analy of Homogeneou em-arkov Procee ac & Proaly eer Norh Holland 984. [6] J.. Hoem Inhomogeneou em-arkov procee elec acuaral ale and duraon-dependence n demography n T.N.E. Grevlle Populaon ynamc Academc Pre [7]. Howard ynamc proalc yem vol II Wley 972. [8] A. Iofecu anu Non homogeneou em-arkov procee ud. ere. a. vol [9] J. Janen e. anca A ealc Non-Homogeneou ochac penon Fund odel on cenaro a candnavan Acuaral Journal [] J. Janen Applcaon de proceu em-markoven à un proléme d nvaldé ullen de l Aocaon oyale de Acuare elge [] J. Janen. anca e G. e edc Fnancal operaon evaluaon: a em-arkov approach Proc. AFI ympoum ruel 995.

16 [2] H. ne and. Oak arkovan decon procee Elever 97. [3] E. Pacco e A. Olver Inroduzone alla eora auarale delle acurazon d perone Quadern dell UI 42 Pagora Edrce ologna 997. [4] I. ahn and Y. alcer ochac model for a penonale ervce Oper. e [5] CI2 Connuou oraly Invegaon epor numer The analy of permanen healh nurance daa. The Inue of Acuare and he Faculy of Acuare 99