Statistical Analysis and Empirical Study for Life Insurance

Transcription

1 Vo 3, No Iraioa Joura of Busiss ad Maagm Saisica Aaysis ad Emirica Sudy for Lif Isurac Xiag Jiag, Jia Ni Guagi Mdica ad Pharmacuica Cog, Naig 532, Chia T: E-mai: Wii Wu Guagi Idusria Vocaioa Tchica Cog, Naig 53, Chia T: E-mai: Absrac Accordig o h ucraiy of isurac accid, i his aric, w iroduc h imorac of isurac ad h aicaio of robabiiy saisics i h isurac, brify dscrib svra aramr mods abou if disribuio (i if mods), discuss h sabishm ad us of h if ab, ad saisicay aayz h rofi ad oss ad isurd amou i if isurac Th rsu coud ossss crai dircioa maigs ad rfrc fucio for racica wors Kywords: Lif isurac, Probabiiy saisics, Saisica aaysis, Lif mod, Emirica Sudy Iroducio Th robabiiy saisics is aid abroad o isurac, icudig accrua ad discou, surviva mod ad if ab, saisica aaysis of surviva ad dah isurac ad saisica aaysis of rmium ad comsaio Isurac is a bargai which rad o righ ad obigaio bw isurr ad isurac aica This bargai rguas i gra ha h isurac aica mus ay h isurr crai amou rmium, ad h isurr romis o ay h isurd amou wh h covioa isurac accid occurs Th isurac cagoris ar ui abudac, bcaus of h diffr sadards, isurac ca divids io rory isurac, iabiiy isurac ad if isurac Th firs ad scod isurac ar cad o-if isurac, h hird is cad if isurac Th cacuaio of if isurac rmium is a comicad usio irsy, h comosiios of rmium ar drmid by may facors Th isurac amou usd o ay for h rmium simy is cad aura rmium Bu h isurr ados from isurac aica is o oy h aura rmium, bu aso various fs ha ach oraio roducs ad various ossib riss Morovr, as h mai body of coomic, h vur isurac ag shoud cosidr crai rofis A hos comoss gross rmium Th gross rmium ca b cacuad basd o aura rmium Tha is, cacuas h aura rmium accordig o h uivac uaio firsy, ad h adds h cssary riss, fs, rofis ad ohr facors Ad h cacuaio of gross rmium ca aso b cacuad by usig h uivac uaio basd o h cosrvaiv fs irs ra ad wighd coffici of dah ra Th disic characrisic of isurac is h ucraiy haig of h isurac accid, such as o s if, raffic accid ad corms This characrisic aso drmis h imora ro of robabiiy saisics i isurac cagoris sig, isurac oraio adig ad ohr ascs So, if you wa o b a viab acuary, you mus b o o of saisics 2 Surviva mod Amog h cagoris of isurac, if isurac is a radiioa ad aso imora isurac Lif isurac ras o o s if characrisic, hrfor, i is cssary o rsarch o s characrisic ad surviva mod 2 Rmaiig Lifim Suosig T () is h isurr s rmaiig ifim, aary, i is a o-gaiv coiuous radom variab Marig h f () is h disribuio dsiy fucio of T (), ad is disribuio fucio is ( ) P{ T ( ) } f ( ) d By a aaracs, wh, T () dos h rmaiig ifim of oaa ifa, ad ( ) dos h disribuio fucio of oaa ifa s rmaiig ifim 22 Commo mars A rs, iraioa acuaria scic uss h radiioa mar (Zhag, 23, P39-83), whr dos h 76

2 Iraioa Joura of Busiss ad Maagm Ocobr, 28 robabiiy of yas od if s dah ra i yars, dos h robabiiy of yas od if s surviva ui yars ar, s dos h robabiiy of h yas od if s surviva i + s yars ad dah ra i yars afr ha, + dos h yas od if s codiio robabiiy of h dah ra o h codiio of is + s ag s Ad h, hr ar () (), ( ), (2) s P{sT s + } s+ s, (3) - ( s + ) P{T > s + T ( ) > s}, + s ( s) (4) ( s + ) ( s) P{T s + T ( ) > s}, + s ( s) If, aways is omid, shorig as, ad, c s 23 Ecaio of Rmaiig ifim If suosig is h caio of isura s rmaiig ifim wh h is yas od, h i ca b asiy rovd, f ( ) d ( ( ) d d 24 Dah isiy Dfi h yas od if s dah isiy (Zhag, 23, P39-83) as f ( ) d ( ()) W ca + ( ) d s ha, h dah robabiiy of yas od if i h scio of (, + d ) ca b rssd as foows, P{T + d} i + d + d Thrfor, h caio of yas od if s rmaiig ifim ca aso cacuas Th dah isiy ca aso b dfid as d +, imm h igra of his formua, w ca g d ha { } + sds I is obvious ha h dah isiy of yas od if s rmaiig im ad h disribuio fucio of i ar dfid cusivy by ach ohr 3 Svra aramr mods abou if disribuio 3 Lif mod of D Moivr (724) Lif mod of D Moivr cosidrs ha hr is a maima agω, ad h rmaiig ifim T () of yas od if i h scio of (, ω ) obys h uaiy disribuio, i f ( ), < < ω, so w ca g h disribuio ω fucio ( ), < < ω, ad h dah isiy is +, < < ω ω ω 32 Lif mod of Gomrz (825) (Zou, 25) This mod cosidrs ha h dah isiy icrasig is oia icras, ha is, + + BC > B > C > By comarig wih Lif mod of D Moivr, if mod of Gomrz rfcs h cours of if rfraby, ad hrows off h maimum ag assumio Th cacuaio bcoms asy Th disribuio fucio of if mod of Gomrz is B ( ) ( C ) I 86, Maham dd h mod of C + Gomrz H suosd ha h dah isiy was + A + BC > A > B >, C >, ad h h disribuio fucio was B ( ) A ( C ) If c, h dah isiy of Gomrz s ad C Maham s if mod ar cosa Th h disribuio fucio of T () is oia disribuio fucio Ahough i is sim wih mah, i ca o rfc h if-sa of o 33 Lif mod of Wibu (Zou, 25) I 939, Wibu u forward ha h dah isiy of o was o oia icras, bu icras wih owr, 77

3 Vo 3, No Iraioa Joura of Busiss ad Maagm ha is + + ) K > > Ad h disribuio fucio of i is K + ( ) [( + ) ] Th + abov-miod if mod is vry hfu 34 Aicaio isac of if mod Eam : Suosd ha h dah of o aggrga obys h mod of Maham, w g h daa, 3, 2 4, 3 6,h cacua o g 4 Souio, Accordig o h mod of dah ad h ow daa, sabish h foowig sysm of uaios, 2 A + BC 3 A + BC 4 A + BC , 4, 6, Sovig h abov sysm of uaios, w ca g A 2, B 25, C 2, ad h h dah isiy + is ,h 25 ( ) 2 (2 ),wha foows 2 is (2 ) Igra rmaiig ifim Dfiiio K ( ) [ T ( )] is igra yars which yas od o wi iv hrough, somims cad rmaiig ifim Th robabiiy disribuio of igra radom variab is { ( ) } P{ T ( ) + } +, (,,2, Λ, ) Th caio of ) is cad h yas od o s c igra rmaiig ifim, sig as Th P{ ) }, or + P{ ) } Th advaag of usig h igra rmaiig ifim of yas od o is ha i rdigss h rmaiig ifim of yas od o, ad i is covi o h cacuaio of caio of rmaiig ifim Maig S( ) T( ) ), S() is h fracio ar of h yars which yas od o has ivd I is asy o s ha S() is a coiuous radom variab o h igra of (,) Assumig ha ) ad S() ar idd, h h codiioa disribuio of S() do o dd o ) wh h vau of ) is giv, ad u + P{ S() u ) } + Prsuoss abov formua ua o H (u), H (u) is a crai fucio which has ohig o do wih Thr wih, u + H( ), +,2, Λ Suosig H ( u) u, ha is, S() is a homogous disribuio i h igra of (,), h + V ar ( T ( )) Var( )) Bcaus ha h disribuio of ) a dds o, hrfor, ca srucur a ids of ab by mas of Th commo if ab of isurac is ab of 4 Th comosiio ad aicaio of if ab Through h rsarch o h disribuio fucio of if, ha is a radom variab, h robabiiy dsiy fucio ad caio, h robabiiy of crai o s dah i crai im or i isaaous ca b rssd as survivorshi fucio ad som scia symbos Tha is o say, w ca sima h if sa of isura i oioa ag I racic, w of us h if o sad for his Th survivorshi fucio ad if ab ar cocd wih ach ohr, ad h survivorshi fucio ca b cacuad hrough if ab Lif ab ar summary ab which b sabishd accordig o h saisic daa of ach ag s dah i crai im 4 Aufbau rici of if ab O h basis of rici of argr umbr, h surviva robabiiy of ach ag o ca b cacuad by obsrvd daa (simaig frucy by frucy) Th commo symbos icud, w-bor if umbr is, is h ag, ad is h umos ag 78

4 Iraioa Joura of Busiss ad Maagm Ocobr, Characrisics of if ab Th Aufbau rici of if ab is sim, ad is o drmid by h oa disribuio (o-aramr mhod) Th surviva caio umbr of w-bor if wi iv o ag X is s(), h dah caio umbr of w-bor if wi di bw h ag of ad + is d s Esciay, wh, mar as d - +,d - + Th sum of yars which w-bor if iv hrough o h igra of ad + is L dy, ad h y oa umbr of h idividua s rmaiig if, which ca iv o ag, is T ydy, h ca g T 43 Aicaio isac of if ab Eam 2: Th ow vau is ),as cacua h foowig vaus, () d 3, 2 3,3 3, 3, (2) h dah robabiiy of 2 yars od o i h igra of 5 o 55, (3) h avrag if of h o Souio: () d 3 3-3, /7, / /7, 3 (2) /6, (3) T 5 ( 5 ) d Aufbau rici of scio-uima if ab (as s i Tab ) Th hah of w moys which g hrough h frsh hysica amiaio cs h hah of od moys which go hrough h hysica amiaio a a og im ago Th forc of scio wi disaar wih im 45 Th aicaio of scio-uima if ab Th aicaio of if ab ad h rsu of cacuaio rsss as h ab 2 46 Th assumio abou fracio ag 46 Surviva siuaio i if ab Usig bacgroud, if ab rovid h disribuio of igra ag, bu somim w d o aaysis h surviva siuaio, so w sc h disribuio of crai ag basd o h wo daa bordr uo, ad sima h surviva siuaio of fracio ag 462 Basic ricis ad commo mhod (iroaio mhod) Basic ricis ad commo mhod (iroaio mhod) icud () Assumio of h uaiy disribuio (iar iroaio), s(+) (-)s()+s(+), <<, (2) Assumio of h dah forc (gomrica iroaio), s(+) s() (-) s(+), <<, (3) Assumio of Baducc (harmoious iroaio), +, << s( + ) s( ) s( + ) Eam 3: Th ow vau is ), as cacua h foowig vaus udr hr diffr fracio ag, 5 3, 525 5, 35 Souio, udr h hr assumios, () Bcaus 3 3 3, so, , UDD 5 3, 4 5 3C , 7 79

5 Vo 3, No Iraioa Joura of Busiss ad Maagm 5 3 Baducci (2) , Bcaus 5 5, 5 5 9, 55, 45 so, UDD , C +9 ( ) 5422, i Baducc (3) 35 UDD 3, C -( 3 )- 69, 7 35 Baducci Th isurac rofi ad oss aaysis basd o cra imi horm 5 Th cra imi horm ad is maig Th cra imi horm ois ou ha if o radom variab is iducd by umrous radom facors, ad vry succssiv chag has fw fucios, so i coud dduc ha h radom variab which dscribs h radom homa obys h orma schoo So o ruir h sum of radom variabs i crai ara, w shoud oy sadardiz i ad aroimay comus i by h orma schoo Th cra imi horm has dircioa maig for h isurac idusry, ad hrough i, w ca sima ad rdic h rofi ad oss of o isurac comay, ad h aw of argr umbrs is h basd o sabish h modr isurac idusry Th foowig am aias o h imora fucio ad cocr aicaio of h aw of argr umbrs ad h cra imi horm i h isurac idusry 52 Eam aaysis ad souio Eam 4: Suos ha hr ar o a ar i h isurac i o isurac comay, ad vry o ays rmium of 2 Yua o h comay vry yar, ad vryo s dah robabiiy i o yar is 6, ad his famiy umbrs coud draw Yua from h isurac comay wh h dis, h aswr h usios, () how big h oss robabiiy of h isurac comay is?, ad (2) how big h robabiiy ha h rofi of h isurac comay i o yar is o ss ha 4 Yua? 4 Souio, Suosig ha h umbr of h dah i o i o yar is X, so X ~ B(, ),, 6,, ad h yar icom of h isurac comay is 2 X () If h isurac comay oss moy i busiss, so 2 X <, ad X > 2 rom h cra imi horm, X { 2} P > P > Φ( ) Φ(77693) So h isurac comay woud o oss moy i busiss (2) If h yar rofi of h isurac comay is o ss ha 4 Yua, so 2 X 4 ad X 8 X P{ 8} P Φ( ) Φ(259) So h robabiiy ha h yar rofi of h isurac comay is o ss ha 4 Yua is Rsu aaysis ad discussio Through aaysis ad souio of h am, w coud cary udrsad why so may isurac comais com io isc, bcaus h oss robabiiy of isurac comay amos is zro, ad why isurac y is br for h isurac comay i so may isurac ys ushd by h isurac comais a rs, ad w coud f ha h robabiiy saisics is surroudig us, ad is aicaio is vry siv

6 Iraioa Joura of Busiss ad Maagm Ocobr, 28 6 Saisica aaysis of isurd amou i if isurac Th dah isurac is h isurac y aig huma dah as h isurac sadard, ad if h isura dis i h im imi of corac, h isurr shoud ay isurd amou o h isura, ad if h isura iv afr h im imi of corac, h isurr ds o ay h isurd amou Thrfor, h cofirmaio of isurd amou dds o h im imi of corac ad h robabiiy characr of isura s rmidig if, ad i aso shoud cosidr may facors such as h rofi of isurac comay, h gross icom ad ayou Accordig o diffr aym ims of isurd amou, h aym mhod of isurd amou ca b dividd io wo sors, i ayig i h a of h yar wh h isura dis, ad ayig wh h isura dis irs, w iroduc h firs sor Suosig ha h corac rguas ha afr h isura isurs his if wh h is yars od, if h dis i h fuur yars, h isurr shoud ay h isurd amou of o moy ui o h isura i h yar h dis, ad if h isura coud iv afr + yars, h isurr woud ay ohig Suos b () dos h isurd amou aid i h h yar, v () dos h discou facor i h h yar So,, b() < K ( ),, ad h rs vau of h isurd amou is od as Z ( (, ) >, )), K, so ( ) ( ) v, < ), Z, ( )) b( ) v, ) > Th caio ad h variac of h rs isurd amou vau rscivy ar A ( ) E[ Z ad, ω ( ))] v P{ ) < + } v ( + ) 2, ( )] EZ, ( ) A ( ) + v A ( ) Var [ Z No, If i h abov formua is ua o h imi ag, h isurac is h dah isurac for if, ad h rs ω caio vau of h isurd amou is ( ω ) EZ, ω ( )) v + A X + I h dah isurac for if, h aym of h isurd amou is h a of h dah yar, so udr h hyohsis ha h isurd amou is o moy ui ad h irs is h cosa, h foowig raioshi coud com io isc A ( ) v + ν A ( ω) ω + 7 Cocusios Emirica sudy idicad ha hrough amos 2 yars dvom, h acuaria mahmaics had graduay dvod o b a rofssioa subjc from a sor of scia comuaio mhod, ad is aicaio rag was graduay adig, ad i had crai dircioa maigs ad br rfrcd fucio for h racica wor of isurac Bcaus of h gh, i h aric, w oy iroduc fw mos basic ad sim cos abou isurac, ad h sudy ad aicaio abou h igraio of isurac, robabiiy saisics ad acuaria mahmaics shoud b furhr discussd by us Rfrcs HUGrbr (999) Th Mahmaics of Comoud Irs Shaghai, Shaghai Word Boo Pubishig Comay Oc, 999 P62-89 Li, Wira (26) Th Paramr Esimaio of Dah orc ucio Basd o Scio-Uima Lif Tab Joura of Hifi Uivrsiy of Tchoogy (Naura Scic) No6 P Mi, Xiaoi (24) Saisics i Lif Isurac Thory Bijig Saisics No Wagya (28) Acuaria Scic of Lif Isurac Bijig, Chia Rmi Uivrsiy Prss May, 28 P2-7 Wi, Zogshu (24) Tuoria of Probabiiy Thory ad Mahmaica Saisics Bijig, Highr Educaio Prss Aug, 24 Zhag, Miyu (23) Sudy of Rsidua Lif ucio i Riabiiy Thory Joura of Lazhou Uivrsiy of Tchoogy No5 P39-83 Zhou, Jiagiog (26) Th Cosrucig Thory of Lif Tab (2d Ediio) Tiaji, Naai Uivrsiy Prss P9-3 Zou, Gogmig (25) Surviva Mods ad Thir Esimaio Shaghai, Shaghai Uivrsiy of iac & 8

7 Vo 3, No Iraioa Joura of Busiss ad Maagm Ecoomics Prss Aug, 25 Tab Scio-uima if ab Ag ara Dah roorio Surviva Numbr a h bgiig of h rm Dah umbr i h rm Surviva yars i h ag ara Sum of rmaiig if Avrag rmaiig if a h bgiig of h rm -+ r ds T Tab 2 Us ad comuaio of scio-uima if ab [] [+] [+2] [+3] [+4] [+5] Tab 3 ucio comariso aaysis of if ab udr hr hyohss fucio v disribuio cosa dah forc Bucci -- ( ) - - ( ) y+ y -- y ( y ) + ( ) fr() --u 2 [ ( ) ] 82