Installment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate

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1 Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi Egieerig Uiversity, Harbi, 5, Chia a *Correspodig author Keywords: Stochastic iterest rate, Istallmet paymet, Combie life isurace, Net level premium, Reserve. bstract. ctuarial theory i a stochastic iterest rate eviromet is a active research area i life isurace. Istallmet joit life isurace actuarial theories are oe of the ey cotets i actuarial theory. I this study, a iterest force accumulatio fuctio model with a Wieer process ad a Poisso process is proposed as the basis for the istallmet joit life isurace actuarial models. The icreasig life isurace actuarial models with the cosumer price idex are approximated. With the proposed model, the et sigle premium, et level premium, the reserves ad the ris of loss model are provided. The actuarial models i the paper provide a feasible method to calculate the life isurace premium. Itroductio The actuarial origiates from the calculatio of life isurace premium, but i Chia, life isurace idustry develops slowly. Util the late 98s, Chia itroduces the life isurace actuarial sciece, ad the scholars begi to study o the life isurace actuarial sciece. I foreig coutries, Pajer [] (98) built the life isurace actuarial models uder the stochastic iterest rate ad sudde mortality rate, ad he studied the statistical properties of the iterest rate ad the auity. Beema [] (99) advaced a ew model. The model raised stochastic pheomea of iterest rate ad the future life, ad Beema draw the expressio of auity ad et sigle premium. I Chia, Qigrog Jiag [3](997) for the first time regarded the iterest rate as the idepedet ad idetical distributio, the he calculated the correspodig premium with this coditio. Wejiog He [4] (998) ad Qigrog Jiag used the Gauss process to simulate chages i the iterest rate. They calculated the et sigle premium ad the variace. The literature simulates stochastic iterest rate i may ways, such as Time Series alysis, auto-regressive itegrated movig average, Browia movemet ad so o. I this paper, we use the stadard Wieer process ad the Poisso process to simulate the fluctuatio of the iterest rate. The ti-iflatio Life Isurace ctuarial Model Symbol Descriptio is the force of iterest; i is the aual iterest rate; x is the idividual who is at the age of x ; ft x t is the probability desity fuctio of T x; ft xy t is the probability desity fuctio of T xy ; x is the ability of the death whe a idividual is at the age of x ; T x is the future livig time of the idividual who is at the age of x ; t p x is probability that a idividual aged x will live aother t years, the q t x p ; t x 4. The authors - Published by tlatis Press 3

2 T xy is the joit life of the couple, x is the husbad who is at the age of x, y is the wife who is at the age of y. The ti-iflatio Model The policy-holder is at the age of x, his future livig time is T. Let the policy-holder buys years term life isurace, the holder pays the fee m times per year for the same iterval, the holder pays the equal fee at the begiig of each iterval. The paymet deadlie is years, if the holder dies durig the years, he stops payig. Let Ct be the paymet fuctio, the stochastic variable of the paymet s preset value is C T Z T e T T () Cosumer price idex (CPI) is the macroecoomic idicator of the geeral household cosumptio of goods ad services. I order to deal with iflatio, isurace compaies lauch a variable life isurace; the isured amout will chage with the time. Table. The real purchasig power of two id s life isurace fee (Uit: Te Thousad yua) C t C t e The loss of The loss of.6t C t C t..6. I Tab., C t is the equal isurace ad C t e.6t is the variable isurace. s ca be see from Tab., the loger the equal life isurace duratio is, the more moey the policy-holder loses. While the variable life isurace has a better ability to resist the ris, it ca effectively resist the.6t damage of iflatio. From Tab., let Ct e be the paymet of variable isurace i this paper. The Force of Iterest ccumulatio Fuctio I the paper, we use the Wieer process ad Poisso process to simulate the fluctuatio of the iterest rate. Let the force of iterest accumulatio fuctio be t, t is give by t tw t Pt t () where Wt follows stadardized Wieer process, Pt follows Poisso process ad the parameter is, W t ad Pt are mutual idepedet., ad are all real umber, which are equal or greater tha. Lemmas Lemma. ssumig that W t follows stadardized Wieer process, Pt follows Poisso process ad the parameter is, W t ad P t are idepedet., ad are real umber, which are equal or greater tha. If t tw t Pt t, the 3

3 b t b E e exp bt b t te Fb, t. [5] (3) Lemma. Let i ad j satisfy i j, the exp,,.[5] (4) E b i b j F b j i F b i Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Net Sigle Premium Let xy : be joit life isurace et sigle premium, xy : is give by t Txy E Z : t E E E C t e E E W Z t f t dt xy W P T P t E E W P C t e t pxt py x t y t dt CtEe p p dt t x t y t x t y t (5) ccordig to Lemma ad Lemma, we ca get exp t x t y xt yt C xy: t t t t e p p dt (6) ccordig to Formula (3), xy : is expressio as xy: C t F t p p dt, t x t y xt yt (7) Survivorship uity ual Survivorship uity. x ad y are i the state of joit life, they pay the fee at the begiig of the year, the amout is a uit of discrete joit life auity. We mar the amout as xy :, xy : is give by xy: px pye e exp e px py px pyf, (8) Mothly Survivorship uity. x ad y are i the state of joit life, they pay the fee at the begiig of the moth, the amout is a uit of discrete joit life auity. We mar the amout as & is expressio as &, a xy : a xy : xy: p p E e x y p exp x py e 33

4 p p F, x y (9) Net Level Premium Net Level Premium of ual Paymet of Life Isurace. x ad y are i the state of joit life. Let P xy : P xy be the aual et level premium, : is give by t t t e p p t x t y xtyt exp : C t xy dt P xy : exp xy : p p t t te x y C t F t p p dt t x t y, xtyt p p F, x y () Net Level Premium of Mothly Paymet of Life Isurace. x ad y are i the state of joit life, let be the mothly et level premium, is give by P xy :, t t x t y t P xy : xy: C t F t p x p y dt P xy: xy: p p F, x y Reserve of the Isurace Compay Reserve of ual Paymet of Life Isurace. I the state of joit life, let clxy be the future loss of isurace compay at the time c, clxy is give by () J clxy CcJe P xy : xy: J () where J is the future livig time of xy at the time c. Let c V xy be the reserve of the isurace compay at the time c, we ca get V E( L ) c xy c xy c c ( ct) ( ) ( ) ( ) ( xy: CctEe ) tpxc tpyc xctyctdtp E IJ Ke c, t xc t yyc xct yc+ t CtF t p p dt, t x t y y xt y+ t C t F t p p dt c t xc t yc px pyf, p p F, (3) 34

5 Reserve of Mothly Paymet of Life Isurace. I the state of joit life, let be the future loss of isurace compay at the time c, is give by clxy clxy J clxy C c J e P xy: xy J : (4) Let V be the reserve of the isurace compay at the time c, we ca get c xy c c xy, t xc t yc xctyct V C c t F t p p dt, c t x t y x t yt px pyf CtF t p p dt pxc pyc F,, (5) Coclusios This paper studies the istallmet joit life isurace actuarial models with the stochastic iterest rate. Firstly, accordig to the aalysis of Chia's cosumer price idex data, the paper poits out the impact of price icreases to policy-holders, the put the cosumer price idex ito the life isurace actuarial models ad costruct life isurace actuarial models. The model has the capacity of ati-iflatioary. Secodly, the paper builds joit life isurace actuarial models. It discusses the aual ad mothly paymet with the stochastic iterest rate. Thirdly, this paper preset the model about the isurace compay reserves. The model we proposed provides a method to determie the loss of life isurace compay i future. cowledgemet This research was fiacially supported by the Graduate Traiig Fud of Harbi Egieerig Uiversity. Refereces [] Pajer H H, Bellhouse D R. Stochastic modelig of iterest rate with applicatios to life cotigecies [J]. Joural of Ris ad Isurace, pp. 9-, 98. [] Beema J, Fuellig C P. Iterest ad mortality radomess i some auities [J]. Isurace: Mathematics ad Ecoomics, 9(), 85-96P, 99. [3] Qigrog Jiag. Whole Life Isurace with the Stochastic Iterest Rate [J]. Joural of Hagzhou Uiversity, 4(), pp.95-99, 997. [4] Wejiog He, Qigrog Jiag. Icreasig Life Isurace with the Stochastic Iterest Rate [J]. Joural of pplied Mathematics, 3(), pp.45-5, 998. [5] Jig Wei, Yogmao Wag. The Reserve of Cotiuous Icreasig Life Isurace with the Stochastic Iterest Rate [J]. Joural of Yasha Uiversity, 3(), pp.-4, 6. 35

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