A New Method to Evaluate Equity-Linked Life Insurance
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1 Coneporary Manageen Research Pages -, Vol. 0, No., March 04 do:0.790/cr.097 A New Mehod o Evaluae Equy-Lnked Lfe Insurance Mng-Shann Tsa Naonal Unversy of Kaohsung E-Mal: ssa@nuk.edu.w Shh-Cheng Lee Yuan-Ze Unversy Unversy of Adelade E-Mal: sclee@saurn.yzu.edu.w Jun-Ln Chen Unversy of Adelade E-Mal: alex.chen@adelade.edu.au Shu-Lng Wu Nan Shan Lfe Insurance Co., Ld E-Mal: bbkoala07@nanshanlfe.co.w ABSTRACT Ths paper provdes a new ehod o apprase equy-lnked lfe nsurance producs. Usng he xure oraly odel o deerne he deah and survval probably, we calculae he preu for a deah cla and a survval cla for equy-lnked lfe nsurance polces. The nuercal resul shows ha he preu s hgher han ha calculaed usng he Brennan and Schwarz odel. As a xed survval funcon can solve he probles of he rgd applcaon n Lfe Tables, he preu fro our ehod s ore accurae for lfe nsurance. Thus, our ehod can help lfe nsurance copanes avod poenal losses fro sprced producs. Keywords: Equy-Lnked Lfe Insurance, Mxure Moraly Model, Moraly Rae
2 Coneporary Manageen Research 4 INTRODUCTION Lfe nsurance has prevaled for ore han 00 years. Tradonal nsurance provdes only consan benefs when he nsured person des. However, because of he reendous developen of fnancal arkes n recen decades, lfe nsurance s desgned no only o provde lfe proecon, bu also o represen an poran nvesen conrac o sasfy he needs of nvesors. For exaple, he os faous nnovave lfe nsurance produc s called equy-lnked lfe nsurance. To apprase such nsurance, wo sources of rsk us be consdered: he payoff uncerany and he e uncerany. As he payoff of an equy-lnked conrac wh a nu guaranee s jus lke an opon, researchers always use opon prcng heory o solve he proble of payoff uncerany (see Brennan and Schwarz, 976 & 979; Bacnello and Oru, 99; Grosen and Jorgensen, 997; Persson and Aase, 997; Nonnenacher and Rub, 998; Bacnello, 00). Meanwhle, scholars always use Lfe Tables o solve he proble of e uncerany and deerne he preu for nsurance. Because of edcal echnology advances and he decreasng brh rae, he dependency rao has gradually grown hroughou he world. Ths rao s also expeced o rse progressvely for he foreseeable fuure. Therefore, s ncorrec o evaluae lfe nsurance producs usng he radonal prcng ehod. Furherore, he lfe nsurance copany experences losses due o naccurae prcng resuls. Two probles arse fro obanng he oraly rae fro Lfe Tables. Frs, he daa n Lfe Tables only reflec hsorcal oraly nforaon. One canno oban a forecased oraly rae for evaluang he nsurance. Second, we canno oban oraly nforaon beween wo ages because Lfe Tables only presen he oraly rae for a dscree perod. Thus, we argue ha he preu for lfe nsurance gh be underesaed usng Lfe Tables. To ore accuraely esae he preu, we need o carefully consder how o forecas he oraly rae accuraely. A presen, several well-known oraly funcons are suppored for esang oraly raes, such as Webull-ype for chldren, nverse-webull-ype for eenagers, and Goperz-ype for aduls (see Brllnger, 96; Tenenben and Vanderhoof, 980; Weersrand, 98; Lee and Carer, 99; Carrere, 99, 994; and Jasulewcz, 997). To he bes of our knowledge, no research has cobned hese oraly odels o evaluae he equy-lnked lfe nsurance. Ths paper uses he xure oraly funcon repored by Carrere (99, 994) o oban he oraly rae. Then, under he assupon of a sochasc neres rae, we oban he preu for equy-lnked lfe nsurance wh a nu guaranee. The advanages of hs odel are ha one can ore precsely prce an nsurance conrac usng forecased oraly rae and ore clearly analyze he nfluence of he change of paraeers of oraly rae on nsurance.
3 Coneporary Manageen Research 5 Therefore, researchers can forecas he change of he paraeer of he oraly rae and oban ore accurae preus for nsurance producs. To he bes of our knowledge, hs s he frs paper o use he forecased oraly rae n he valuaon of equy-lnked lfe nsurance producs. As we know, he lfe nsurance syse provdes poran socal welfare for a counry. Our odel s sgnfcan because he dependency rao s gradually growng n he world. If he valuaon odel for lfe producs gnores hs rend, he lfe nsurance copany gh suffer losses fro sellng hese lfe producs, whch can even cause bankrupcy for he copany and furher nfluence he socal welfare as well as hur he econoc developen of a counry. The paper s organzed as follows. In Secon, we descrbe he odels for equy-lnked lfe nsurance and he xure oraly funcon. In Secon, we use he open daa fro Lfe Tables o esae he paraeers of he xure oraly odel and deerne he value of lfe nsurance usng a nuercal ehod. Secon 4 gves he concluson. MODEL We defne he saes on he probably space ( Ω, Φ, P) as Φ = Ψ F, where Ω represens sae space; Ψ represens he σ -feld of he deah a e ; F represens he σ -feld for nflraon of oher nforaon on he bond a e ; P represens he physc easure; and Q represens he arngale easure. Jarrow, Lando and Yu (005) ephaszed ha he equvalence beween he physcal and arngale easures holds under he assupon of a well-dversfed porfolo. As he oraly rsk can be dversfed by sellng he sae knd of conrac o enough people, we can oban he arke prces of nsurance under he assupon of a well-dversfed porfolo. Therefore, he physcal easure s dencal o he arngale easure. The oraly rsks can be calculaed fro arke nforaon. Le G ( be he survval funcon a e ; F ( s he deah probably a e, where F( = ; and f ( s he F( probably densy funcon for deah, where f ( =. We assue ha he lfee s ndependen of he neres rae and porfolo. The payoff for equy-lnked lfe nsurance s dependen on he perforance of he porfolo. We denoe he nal porfolo value as y ( a e. Under he arngale easure, he change n porfolo value y can be descrbed as follows: dy = r( d + σ y dz y (, () y
4 Coneporary Manageen Research 6 where σ y s he nsananeous varance of porfolo and dz y ( s he sandard Brownan oon of porfolo. The nsurer provdes a nu guaranee for equy-lnked lfe nsurance. To splfy, we assue he nu guaranee s equal o a poron (θ ) of he nal porfolo value. The basc dea of prcng a conngen payoff usng he rsk-neural probably easure s ha one can perfecly replcae he payoff of he conngen cla wh a self-fnancng sraegy. When a payoff s conngen on he oraly of he holder, perfec hedgng usng a self-fnancng radng sraegy becoes possble because he oraly rsk canno be raded on he fnancal arke. One soluon for overcong hs proble s o consder he rsk-nzng hedgng sraegy (see Ln and Tan, 00). Hence, he deah cla a e s s descrbed as an opon. Usng he R pv, s, s: opon-prcng heory, he expeced presen value of he deah cla, ( ) where R pv (, = y N( d ( ) B(, θ y( N( d ( ) + B(, θy( ), () ( lnθ lnb(, + 0.5σ x ( s d ( = σ s ; x σ x = σ y + σ r ( ρσ r ( σ y s a coponen varance of he porfolo and he neres rae, ρ s he correlaon beween he porfolo and he neres rae, N ( ) s he cuulave noral dsrbuon funcon, and B, s represens he value of a zero coupon bond wh aury dae s n e. ( ) r( u) du B, I should be noed ha he zero coupon bond s defned as (, = E( e ) where r (u) s he sochasc spo neres rae. We can use neres rae odels, such as Vascek s (977) odel and he Cox, Ingersoll, and Ross (985) odel, o descrbe he value of a zero coupon bond. The preu for equy-lnked lfe nsurance a nal e usng he rsk-nzng hedgng sraegy s descrbed as follows: where T D = Rpv (, T) T + Rpv (, f ( s ds, () G ( T s he survval probably a e T condonal on e, and s
5 Coneporary Manageen Research 7 f ( s s he deah probably densy funcon a e s condonal on e. The xure survval odel s a xure of hree dsnc dsrbuons: a Webull dsrbuon for chldhood oraly, an nverse Webull dsrbuon for eenage oraly, and a Goperz dsrbuon for adulhood oraly. The xure survval odel can be specfed as follows: where = ψ G (, (4) = ψ s he probably of dfferen perods, and ψ + ψ + ψ = ; and G ( s a specfc survval funcon, =,,, whch represens he funcons of Webull, nverse-webull, and Goperz, respecvely. The survval probably densy funcon of Webull, nverse-webull, and Goperz are descrbed as follows: G σ = exp( ( ) ), (5) ( G σ = exp( ( ) ), and ( = exp(exp( ) exp( )), (6) σ σ where and σ, =,,, are he locaon paraeers and dsperson paraeers. Therefore, he deah probably densy funcon s descrbed as follows: ( ) f ( = = ψ f (, (7) where = σ σ f( = ( ) exp( ( ) ), (8) σ f σ σ = ( ) exp( ( ) ), and σ (
6 Coneporary Manageen Research 8 f( = exp( + exp( ) exp( )). σ σ σ σ Consderng ha he nsured s alve a e, he condonal survval probably densy funcon and condonal deah probably densy funcon are descrbed as follows: G ( s = (9) and f ( f ( s =. (0) NUMERICAL ANALYSIS We used he Lfe Tables of he USA and Tawan n he year 005 o esae he paraeers of he survval funcon. The daa were obaned fro he Huan Lfe-Table Daabase (hp:// We copared he preus n our odel wh he odel used by Brennan and Schwarz (976). The esaed paraeers for ales are presened n Table. Table Paraeer Esaes for he Mxure Moraly Model Counry σ σ σ ψ ψ ψ USA Tawan We ade he followng assupons: The survval cla and deah cla are boh $00,000; he age of he nsured s 5 years old; he varance of he porfolo s 0%; and he rao of parcpaon s equal o. In addon, for splcy, we le ρ = 0 and obaned he zero coupon bond under a consan spo neres rae (.e.,.5%). The calculaed values of nsurance preus are presened n Table, whch shows ha he preu for a deah cla s hgher n our odel. Moreover, he oal preu calculaed fro our odel s hgher han ha calculaed fro he Brennan and Schwarz odel. Therefore, he preu for lfe nsurance gh be underesaed usng he radonal ehod wh Lfe Tables.
7 Coneporary Manageen Research 9 Table Coparson of Proposed Model wh Brennan and Schwarz s (976) Model for USA and Tawan Daa Age Toal preu Our odel Brennan and Schwarz (976) Survval preu Dead preu Toal preu Survval preu Dead preu USA 5,8 07,90 4,48,90 07,80 4,70 5,046 0,700 0,46,9 0,40 9, ,685 9,7,48,97 9, 0,86 Tawan 5,56 06,080 6,456,9 05,90 6,00 5,48 99,705,7,9 00,070, ,00 87,78 8,,9 88,4,49 CONCLUSION A far preu wll encourage people o anan lfe nsurance. Ths paper consrucs a odel o evaluae he far prce for equy-lnked lfe nsurance. Copared wh radonal research ehods, we adoped a xure odel of he Webull funcon, nverse-webull funcon, and Goperz survval funcon o f he paern of oraly raes n Lfe Tables. We hen used he paraeers esaed o calculae he survval probably and deah probably. Fnally, we deerned he preus for endowen nsurance and equy-lnked lfe nsurance. The oal preu calculaed usng our odel s hgher han ha calculaed usng he Brennan and Schwarz odel. We beleve ha usng a xed survval funcon can solve he probles of he rgd applcaon n Lfe Tables. Thus, hs ehod gves a ore accurae preu for lfe nsurance. Fro he anageral vewpon, our odel can no only help he lfe nsurance copany avod losses fro naccurae valuaons of an equy-lnked lfe nsurance produc, bu also asss he rsk anager of he lfe nsurance copany o effcenly conrol he oraly rsk relaed wh hs produc. Because we found he oal preu calculaed fro our odel o be hgher han ha calculaed fro he Brennan and Schwarz odel, we also sugges ha sakeholders can rase he curren preus o avod losses seng fro he evaluaon of he preus of relaed lfe producs usng radonal odels.
8 Coneporary Manageen Research 0 REFERENCES Bacnello, A.R., and Oru, F. (99a). Prcng equy-lnked lfe nsurance wh endogenous nu guaranees. Insurance: Maheacs and Econocs,, hp://dx.do.org/0.06/ (9)904-h Bacnello, A.R. (00). Far valuaon of a guaraneed lfe nsurance parcpang conrac ebeddng a surrender opon. The Journal of Rsk and Insurance, 70, hp://dx.do.org/0./ Brennan, M.J., and Schwarz, E.S. (976). The prcng of equy-lnked lfe nsurance polcy wh an asse value guaranee. Journal of Fnancal Econocs,, 95-. hp://dx.do.org/0.06/ x(76) Brennan, M.J., and Schwarz, E.S. (979). Alernave nvesen sraeges for he ssues of equy-lnked lfe nsurance polces wh an asse value guaranee. Journal of Busness, 5, 6-9. Brllnger, D.R. (96). A jusfcaon of soe coon laws of oraly. Transacons of he Socey of Acuares,, 6-6. Carrere, J.F. (99). Paraerc odels for lfe ables. Transacons of he Socey of Acuares, 44, Carrere, J.F. (994). A selec and ulae paraerc odel. Transacons of he Socey of Acuares, 46, Cox, J.C., Ingersoll, J.E., and Ross, S.A. (985). A heory of he er srucure of neres raes. Econoerca, 5, hp://dx.do.org/0.07/94 Grosen, A., and Jorgensen, P.L. (997). Valuaon of early exercsable neres rae guaranees. The Journal of Rsk and Insurance, 64, hp://dx.do.org/0.07/576 Jarrow, R.A., Lando, D., and Yu, F. (005). Defaul rsk and dversfcaon: heory and eprcal plcaons. Maheacal Fnance, 5, -6. hp://dx.do.org/0./j x Jasulewcz, H. (997). Applcaon of xure odels o approxaon of age-a-deah dsrbuon. Insurance: Maheacs and Econocs, 9, 7-4. hp://dx.do.org/0.06/s (96) Lee, R.D., and Carer, L.R. (99). Modelng and forecasng U.S. oraly. Journal of he Aercan Sascal Assocaon, 87, hp://dx.do.org/0.07/900 Ln, X.S., and Tan, K.S. (00). Valuaon of equy-ndexed annues under sochasc neres raes. Norh Aercan Acuaral Journal, 7, 7-9. Nonnenacher, D.J.F., and Rub, J. (998). Rsk odels n he conex of equy-lnked lfe nsurance conracs. The Geneva Papers on Rsk and Insurance Theory,,
9 Coneporary Manageen Research Persson, S.A., and Aase, K.K. (997). Valuaon of he nu guaraneed reurn ebedded n lfe nsurance producs. Journal of Rsk and Insurance, 64, hp://dx.do.org/0.07/5888 Tenenben, A., and Vanderhoof, I.T. (980). New aheacal laws of selec and ulae oraly. Transacons of he Socey of Acuares,, 9-8. Vascek, O.A. (977). An equlbru characerzaon of he er srucure. Journal of Fnancal Econocs, 5, hp://dx.do.org/0.06/ x(77)9006- Weersrand, W.H. (98). Paraerc odels for lfe nsurance oraly daa: Goperz s law over e. Transacons of he Socey of Acuares,,
10 Coneporary Manageen Research
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