Unit-Linked Life Insurances as a Form of Protection for the Future

Transcription

1 IN paper verson IN elecronc verson Economc and Envronmenal udes Vol. 17, No. 1 (41/217), 35-46, March 217 Un-Lnked Lfe Insurances as a Form of Proecon for he Fuure Magdalena HOMA Unversy of Wroclaw, Poland Absrac: Insurances wh capal funds (un-lnked, UFK),.e. lfe and annuy savng and coverng nsurances combned wh shares n seleced nvesmen funds of fnancal nsuons are consdered as addonal form of penson savng. However, hey are frequenly seen as uneconomcal producs, whch s caused manly by he fac ha nsurance polces compensaons does no lve up o expecaons of he nsured. I became an mpulse o mplemen he analyss of cash flow characersc of such nsurance ypes, carry ou an accurae valuaon of he un-lnked nsurance porfolo, as well as o nvesgae fnancal surplus n relaon o a radonal nsurance wh guaraneed sum. Keywords: un-lnked nsurance, European opon, valuaon, value of nsurance, Mone-Carlo mehod JEL codes: C58, G22, G17 1. Inroducon Insurances relaed o he fnancal marke can be consdered an exenson of radonal lfe nsurances. uch conracs, so called un-lnked nsurances (UFK), are complex nsurance producs combned wh nvesmen funds. In oher words, hey are lfe and annuy savng nsurances negraed wh shares n seleced nvesmen funds of fnancal nsuons. Neverheless, hey are frequenly hough of as uneconomcal producs, whch s caused manly by he fac ha he polcy compensaon does no fulfl expecaons of he nsured. I should be poned ou ha n un-lnked nsurances, n conras o radonal lfe nsurances, he sum of nsured s no usually guaraneed by he nsurer and depends on he markeable secures value n whch funds are deposed, herefore s a random varable. Nex, s mporan o realze ha nsurances wh capal fund Correspondence Address: Magdalena Homa, Insue of Economc cences/ Faculy of Law, Admnsraon and Economcs, Unversy of Wroclaw, Unwersyecka 22/26, Wrocław, Poland. E-mal: magdalena.homa@uwr.edu.pl 217 Opole Unversy

2 MAGDALENA HOMA brng many nvesmen possbles, bu n each case here s a fnancal rsk deermnng he unlnked polces value. Ths ssue has been addressed n leraure, akng no consderaon formal, legal, acuaral and fnancal aspecs, manly from he perspecve of he nsurer. In Poland nsurances wh capal fund are nsurances n whch he rsk s aken by he nsured, who s consequenly responsble for any negave oucomes of her decsons. Thus has become a premse o analyses hese nsurance ypes from he pon of vew of he nsured and her benefs. For hs reason valuaon of cash flow of he seleced un-lnked producs was mplemened, and on hs bass he value of reference porfolo was deermned, accoun o be aken of boh nsurance and fnancal rsks. Moreover, dsbursemens from enerng a lfe nsurance wh capal fund agreemen was compared o hose provded by radonal lfe nsurances wh fxed ne premums. 2. Lfe nsurance wh capal fund Fnancal nsuons havng un-lnked on offer are nsurance companes. However, hs nsurances are no separae producs; hey supplemen basc producs,.e. coverng lfe nsurances. The dea of un-lnked nsurances s deermned by her complex savngs-nvesmen naure. In shor, hey allow he nsured o nves a fracon of he premum no funds of varous rsk levels offered by nsurance companes. In hs way hey flexbly combne he aspecs of nsurng and savng, a he same me creang many opporunes of managng he saved capal. In Poland such agreemens enable he nsured o collec savngs n an ndvdual nvesmen porfolo, comprsng varous nvesmen funds managed by exernal fund nsurance companes ndependen from he nsurer. Wha s more, he srucure of un-lnked polces s open and ransparen, whch gves he nsured he possbly of adjusng he conen of he porfolo accordng o changng marke condons. Consequenly, profs ganed by hese funds can vary. Un-lnked nsurance are avalable predomnanly whn he scope of ndvdual lfe nsurances n he followng forms (Moller T.,effensen M. 27): o Unlmed lfe nsurance. o Lmed lfe and annuy nsurance. o avng nsurance. Regardless of he form, he nsurance wh capal fund s an agreemen concernng lfe and annuy concluded beween he nsured and he nsurer, accordng o whch he nsured pays premums and 36

3 UNIT-LINKED LIFE INURANCE A A FORM OF PROTECTION FOR THE FUTURE n urn he nsurance company provdes he nsured wh ceran paymens. In conras o radonal lfe nsurances, n whch he nsurance cos (expressed n he pad premum) s consan durng he enre nsurance perod and s no a consequence of he rsk level, bu of he average rsk durng he enre nsurance perod, un-lnked nsurances ake no accoun he age of he nsured, so he cos s subjec o paymens, neres rae, admnsrave coss, and he rsk of deah, accdens, ec. As far as paymens are concerned, radonal nsurances spulae her amoun n he agreemen as he nsurance sum, whereas un-lnked nsurances deermne he amoun of provsons basng on he value of he nvesmen porfolo, as well as on rends concernng he prce of a gven fund, whch are dependen on nsurance polces. Due o he nvesmen sraegy, he nsurance capal funds dffer n erms of rsk, and hus one can dsngush balanced, deb, cash and shares funds. 3. Invesmen porfolo and s value n UFK nsurances In un-lnked nsurances lke n radonal nsurances, he nsurance company oblges self o he dsbursemen of paymens accordng o he ype of conrac: o Due o he nsurance ermnaon (whle he nsured s sll alve) (UD). o In case of deah durng he nsurance perod (UZ). Conrary o radonal lfe nsurances, n un-lnked nsurances he momen of dsbursemen, as well as he amoun of he paymen, are random. A he momen of an even covered by he nsurance he marke value of he nsurance porfolo s pad ou by he nsurer. Therefore he paymen amoun depends eher on an ndex developmen or on a specfc porfolo value. In relaon o such consrucon of un-lnked he paymen s a funcon of a cumulaed nvesmen,.e. he value of nsurance porfolo (Bacnello 23): B f ( FV ) where: FV value of he nsurance (reference) porfolo a he momen of. In pracce mos frequenly he funcon f akes one of he followng forms (Bacnello 23): f ( lub f FV ) maxg, FV ) FV ( FV, where: G guaraneed sum dependen on he amoun of he nsurance premum. In he former case he dsbursemen amoun depends only on he value of he reference porfolo, whereas n he laer he mnmal nsurance sum s guaraneed by he nsurer. The second nsance refers o a suaon n whch a he momen of an even covered by he nsurance he hgher value 37

4 MAGDALENA HOMA s pad ou,.e. he mnmal guaraneed nsurance sum wh added marked value of he nsurance porfolo. The majory of Polsh nsurance companes end o mnmze he coverng aspec of nsurance, a he same me focusng on he savng feaure. Consequenly, agreemens of hs ype do no mply any assurance, whch leads o mplemenng he frs presened formula whle payng ou he nsurance amoun. The value of he nsurance porfolo n such nsurances ( ) s random and depends on: o The fund prce a he momen of. o The pas uns prce (covered by he premums). o The nvesmen sze. Thus he nsurance porfolo value s acheved by nvesng a fracon of he pad nsurance premum. The fracon of he premum nvesed a he momen s marked as Π and referred o as he nvesmen premum. Accordngly, he nsured bulds her reference porfolo value by nvesng n chosen asses (eher nvesmen funds or shares), wh prce deermned as he process ( ). The value balance a he momen (.e. a he end of he year) s descrbed by he followng formula (Homa 213): FV FV 1 porfolo value n he momen 1 1 FV 1 1 gans from he nvesmen n he year changes on he accoun n he year Hence he porfolo value s he sum of porfolo values n he perod before achevng he ncreased value due o he profs from nvesmens and changes on he accoun. A each momen nsured purchases respecvely Π ( ) asse uns for a prce Π. o a he momen he porfolo comprses Π uns, where each of hem possesses he value. Therefore he reference porfolo value a he momen of he polcy ermnaon equals: he FV T n T 1 1 whch means ha he dscouned porfolo value (n relaon o ) equals he dscouned value of all nvesmens. Thus he reference porfolo value a any me can be descrbed by he followng formula: FV mn{ } 1 ( ) 38

5 UNIT-LINKED LIFE INURANCE A A FORM OF PROTECTION FOR THE FUTURE Takng no accoun he money value changng over me, he marke value of he dsbursed paymen a he momen equals: V ( T ) ( ) n 1 T BT, Makng he valuaon one should ake no consderaon no only he marke value, bu also he money value changng over me and he rsk relaed o he subjec of nsurance. As far as lfe and annuy nsurances are concerned, would be also he moraly rsk. If hese wo ypes of rsk are consdered o be ndependen, he valuaon a he end of he nsurance perod would be mplemened by spulang he acuaral value n he followng way: ( T ) ( T BT, E K T E V n1 ( T ) E ( x T x ) ) p n1 T In case of dsbursemen valuaon a he momen of deah of he nsured a any me durng he nsurance perod,.e. n he perod,, he probably of deah s aken no accoun. Bearng n mnd all of he above, can be concluded ha a person who purchases a un-lnked polcy can expec a surplus o he guaraneed nsurance sum n he radonal lfe nsurance only n he suaon when he nsurance porfolo value exceeds he guaraneed value. Due o he fac ha he un-lnked nsurances srucure mples fnancal rsk, he nsured should conrol he rsk ncluded n he fracon of he premum whch consues he nvesmen porfolo value. The hgher he premum, he greaer he nvesmen porfolo value, and he nsured receves a compensaon resulng from he surplus comng from he nvesmen porfolo value. 4. Fnancal and moraly marke model As an example boh radonal and un-lnked ermnaed agreemens concernng lfe and annuy nsurance were examned, wh he premse ha n he case of deah a he momen Θ/{ } he nsurer wll dsburse a sum resulng from he porfolo value: ( 1) k ( ) j ( ( 1 j ) ) 39

6 MAGDALENA HOMA Analogcal paymen would be ssued also n he suaon of he nsured beng alve a he momen of he conrac ermnaon. Makng he valuaon of boh radonal and un-lnked nsurances, one has o ake no accoun no only he money value change over me, bu also an accurae descrpon of moraly dynamcs. I has been herefore acceped ha on he marke here s an accoun wh fxed neres rae and connuous capalzaon, whch means ha he moneary un value process equals (Weron 1998): B e e ln( 1r ) Furhermore, n order o creae an accurae descrpon of he moraly rae n he populaon sample aged from 3 o 8, he Gomperz-Makeham law was mplemened, whch allowed o oban he probably of deah. The rae of deceases can be descrbed by he formula (Bowers 1997): x x A Bc Basng on Lfe Table for men, a funcon approxmaon was carred ou, whch led o he followng esmaors of he bgges plausbly of he funcon s parameers: A,4; B,34674; c 1 Therefore was possble o oban he probably of boh beng alve and dead used n he example. Addonally, n un-lnked nsurances a fracon of he pad nsurance premum (so called savng premum) s nvesed by he nsurance company no he fnancal marke. Thus he effecveness of nsurance companes s ncreased, and her marke offer becomes more compeve. Neverheless, snce nsurance companes do no know n advance consequences of such an acvy, he rsk seems o be aken by he nsured. In hs case he lfe nsurance cash value s nfluenced by boh posve and negave effecs of hs acvy. Also, was assumed ha he nsured can dfferenae profs resulng from he varable porfolo value by mplemenng varous nvesmen sraeges. In he analyss he bes UFK funds n he followng four porfolo groups were aken no accoun: o AK_UN general shares, o MI_T mxed of balanced growh, o MI_ZR mxed balanced, o PD_UN general of deb The analyss of funds rsk was conveyed n hese groups wh regard o basc measures: sandard devaon, harp s ndex, and Informaon Rao (IR). The fund wh he hghes noe was chosen from each group; oucomes are presened n he able 1. 4,6

7 UNIT-LINKED LIFE INURANCE A A FORM OF PROTECTION FOR THE FUTURE Table 1. Indces of he bes funds n groups (1) (4) Group The bes fund n a group AK_UN AXA Quercus Aggressve UFK harpe s Index IR % WIG MI_ZR Nordea Balanced Porfolo MI_T Nordea Balanced Growh Porfolo PD_UN HDI General Novo Deb ecures ource: Own elaboraon based on he daa from Declared Benchmark 5% WIG + 5% Polsh Treasury Bonds Marke Index 2% WIG + 8% Polsh Treasury Bonds Marke Index 1% EFFA Bond Indces Poland Lqud All > 1Yr, Wha follows, n order o carry ou he valuaon of he un-lnked flow, as well as o calculae he amoun of paymen, a smulaon of he prces process has o be mplemened (wh aenon pad o he presumed fnancal marke model). The Black-choles (Black-Meron-choles) model wh horzon T has been adoped as he fnancal marke model. I has been assumed ha an deal marke s beng deal wh, where here are uns of he fund chosen by he nsured wh he prce esmaed by he formula (Mlevsky, alsbury 26): d ( ) d ( ) dw where s he Wener process (he geomercal Brownan moon), he presumed prce of he dervave conrac, he acual prce of he dervave conrac, - he process average, sandard devaon of he process. The only soluon of he above dfferenal equaon s followng process whch has lognormal dsrbuon: 2 exp W o he smples model of prce evoluon, he geomercal Brownan moon, has been accouned for. Upon applcaon of hs mahemacal model he followng formula MC smulang (Glasserman 24) he fuure value of he dervave conrac can be creaed: k k 1 2 exp r where means ndependen values generaed from he normal dsrbuon, s he rsk-free neres rae. k k 1 k k 1 k 41

8 MAGDALENA HOMA 5. mulaon of porfolo and paymen value for an exemplary un-lnked nsurance n relaon o nvesmen sraegy Regardless of he form,.e. radonal or un-lnked lfe nsurance should be a long-erm soluon. Therefore n he valuaon of he capal ncreasng effecveness whn he un-lnked sysem a long me horzon has been appled, accordng o whch he un-lnked paymen amoun (.e. he reference porfolo value) and a radonal lfe and annuy nsurance sum were compared. I was acceped ha he nsured s a man aged x who has purchased an n-year lfe and annuy nsurance polcy wh he nsurance sum G = 1 moneary un. Nex, he un-lnked acuaral value of reference porfolo was calculaed by applyng he Mone Carlo mehod, wh varous nvesmen sraeges mplemened. The capalsaon compled o he agreed scenaro: 1 8 eraons, for 2-, 3-, 4-, and 5-year long un-lnked conracs (Table 2). Table 2. Acuaral value of reference porfolo wh dfferen sraeges a he momen Porfolo n=2 n=3 n=4 n=5 AK_UN 151, ,51 156, ,63 MI_ZR 1498,28 152,22 15,8 1497,55 MI_T 15, , , ,5 PD_UN 1476, 1393,17 147, ,56 ource: Own elaboraon. Takng no consderaon he above daa concernng he acuaral value of he un-lnked lfe nsurance porfolo a me, can be presumed ha boh he duraon of he nsurance, as well as s nvesmen sraegy appled by he nsured, do no sgnfcanly deermne he polcy value. More mporanly, he expeced value n each of he presened cases exceeds he radonal nsurance sum, whch means ha regardless of he chosen nvesmen sraegy, he nsured can ancpae a surplus n comparson o radonal lfe nsurances. Conrary o radonal nsurances s no a suffcen evdence, because he porfolo value s a random varable, whch can ake dfferen value durng he nsurance perod. Therefore momen of paymen s also mporan and we examned he acuaral value of he paymen durng he perod of nsurance for n=2 and n=5 (Table 3 and 4). 42

9 UNIT-LINKED LIFE INURANCE A A FORM OF PROTECTION FOR THE FUTURE Table 3. Acuaral value of reference porfolo wh dfferen sraeges and = n=2 AK_UN MI_ZR MI_T PD_UN =5 79,1 78,86 79,2 697,99 =1 912,4 91,14 912,12 898,87 =15 117, ,67 117, ,6 =2 151, ,28 15, ource: Own elaboraon. Table 4. Acuaral value of reference porfolo wh dfferen sraeges and = n=5 AK_UN MI_ZR MI_T PD_UN =5 259,3 23,41 23,65 23,19 =1 336,71 39,4 334,99 26,71 =15 432,62 473,3 413,41 334,46 =2 556,73 549,69 551,93 475,17 =25 78,29 64,48 76,2 79,51 =3 955,22 913,84 98,35 98,3 = , ,35 117,79 111,68 =4 1289, , , ,3 =45 141, ,2 1353, ,27 =5 1513, , ,5 1489,56 ource: Own elaboraon. Table 5. Acuaral value of surplus wh dfferen sraeges and = nvesmen porfolo AK_UN MI_ZR MI_T PD_UN acuaral paymen -29,9-291,14-29,98-32,1 =5 error,41,35,24,9 acuaral paymen -87,6-89,86-87,88-11,13 =1 error,79,67,45,17 acuaral paymen 17,75 164,67 17,27 168,6 =15 error 1,31 1,13,74,28 acuaral paymen 51,49 498,28 5, =2 error 2,9 1,8 1,11,42 ource: Own elaboraon. Comparng un-lnked nsurance and classc lfe nsurance analyzed also surplus values wh s probably durng he nsurance perod. Oucomes of he smulaon, mplemened wh he MC mehod for a 2-year long UFK agreemen wh capalsaon compable wh he agreed scenaro for dfferen sraeges along wh he esmaon error, are presened n Table 5. Analogcal smulaons were carred ou for un-lnked wh he nsurance perod = 3, 4 and 5, respecvely, and he oucomes are presened n he followng fgures. 43

10 MAGDALENA HOMA Fgure 1. urplus un-lnked nsurance vs classc lfe nsurance durng he nsurance perod wh =,,, =5 =1 =15 =2 =5 =1 =15 =2 =25 =3 AK_UN MI_ZR MI_T PD_UN AK_UN MI_ZR MI_T PD_UN =5 =1 =15 =2 =25 =3 =35 =4 =5 =1 =15 =2 =25 =3 =35 =4 =45 =5 AK_UN MI_ZR MI_T PD_UN AK_UN MI_ZR MI_T PD_UN ource: own elaboraon. The obaned oucomes ndcae explcly ha he un-lnked nsurance reference porfolo value exceeds he nsurance sum only afer ceran me whn he agreemen duraon, whch means ha only hen he nsured can expec a fnancal surplus resulng from un-lnked. Also, he long-erm feaure has been confrmed. Therefore an nsured person resgnng from he un-lnked polcy oo early, regardless of he agreed nvesmen sraegy, s no only dsadvanaged n comparson o radonal nsurances, bu also does no ge he pad premums back. 44

11 UNIT-LINKED LIFE INURANCE A A FORM OF PROTECTION FOR THE FUTURE 6. Concluson Havng consdered he obaned oucomes can be saed ha regardless of he agreed way of capalsng, he nsured can conrol he surplus resulng from he UFK nsurance porfolo value compared o radonal nsurance. The amoun of he UFK nsurance dsbursemen a he momen of polcy ermnaon exceeded he nsurance sum guaraneed by radonal nsurances ndependenly of he rsk-averson of he nsured and he agreed nvesmen sraegy. Neverheless, should be poned ou ha he surplus over he guaraneed sum s presen n he mddle of he nsurance perod, and s dynamcs s nfluenced by he nvesmen sraegy. Therefore, akng no accoun he long-erm characer of UFK, he combnaon of coverng and savng s a favourable soluon. Fnally, whle choosng UFK one has o bear n mnd ha hese are long-erm producs, and n hs framework hey can be consdered an effecve way of collecng penson savngs. Leraure Balloa, L.; Habermann,. (26). The far valuaon problem of guaraneed annuy opons: he sochasc moraly envronmen case. Insurance Mahemacs & Economcs 38. Bacnello, A. R., (23). Prcng guaraneed lfe nsurance parcpang polces wh annual premums and surrender opon. Norh Amercan Acuaral Journal 7(3). Bacnello, A.R.; C. Perna; M. bllo (28). Chaper: A full Mone Carlo approach o he valuaon of he surrender opon embedded n lfe nsurance conracs n Mahemacal and ascal Mehods for Insurance and Fnance. prnger. Bowers, N.L.; Gerber, H.U.; Hckman, J.C; Jones, D.A.; Nesb, C. (1997). Acuaral mahemacs. chaumburg: The ocey of Acuares. Glasserman, P. (24). Mone Carlo mehods n fnancal engneerng. ochasc Modellng and Appled Probably 53. Homa, M. (213). Rozkład wypłay w ubezpeczenu na życe z funduszem kapałowym a ryzyko fnansowe. PN UE Zagadnena akuaralne - eora prakyka 312. Homa, M Propozycja modyfkacj składk neo w ubezpeczenach na życe z funduszem kapałowym uwzględnająca dodakowe ryzyko fnansowe. Quanve Mehods n Economcs 15(3). Mlevsky, M.; alsbury T.. (26). Fnancal valuaon of guaraneed mnmum whdrawal benefs. Insurance: Mahemacs and Economc 38(1). Moller, T. (23). Indfference prcng of nsurance conracs In a producs pace model: applcaons. Insurance Mahemacs & Economc 32. Moller, T.; effensen, M. (27). Marke valuaon mehods n lfe and penson nsurance. Cambrdge: Cambrdge Unversy Press. Weron, A.; Weron, R. (1998). Inżynera fnansowa. Wycena nsrumenów pochodnych, symulacje kompuerowe, saysyka rynku. Warszawa: Wydawncwo Naukowo-Technczne. 45

12 MAGDALENA HOMA Ubezpeczena na życe z funduszem kapałowym jako forma zabezpeczena na przyszłość reszczene Ubezpeczena z funduszem kapałowym UFK czyl oszczędnoścowo-ochronne ubezpeczene na życe dożyce połączone z udzałam w wyselekcjonowanych funduszach nwesycyjnych nsyucj fnansowych wskazywane są jako dodakowa forma oszczędzana na emeryurę. Coraz częścej jednak pojawają sę opne wskazujące na neopłacalność ego ypu produków, a wynkają one przede wszyskm z faku, że wypłaa z yułu zawarej polsy ne spełna oczekwań ubezpeczonych. anowło o przesłankę do przeprowadzena analzy przepływów penężnych charakerysycznych dla ego ypu ubezpeczeń oraz dokonane prawdłowej wyceny warośc porfela ubezpeczenowego UFK oraz zbadane nadwyżk fnansowej w sosunku do klasycznego ubezpeczena z gwaranowaną sumą. łowa kluczowe: ubezpeczene un-lnked, opcja europejska, wycena, warość ubezpeczena, meoda Mone-Carlo. 46