OPTIONS ON PENSION ANNUITY

Transcription

1 06 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 OPION ON PNION NNUIY hulamih. Goss *, Rami Yosef **, Ui Benzion *** bsac We inoduce a uoean (eoic) call oion on a ension annuiy. he oion gives is owne he igh o buy, fo a secified lum sum, an odinay annuiy (ension) ha sas a a secified fuue dae ( eiemen age ). hus, insead of conibuing monhly o a ension fund, one could buy his oion and insue he ems of hei eiemen. We ice he oions unde sochasic inees aes in boh discee and coninuous ime egimes. In he discee ime case, we use an ode auoegessive ocess (R()). In he coninuous ime case, we use simulaions of he sho inees ae accoding o vaious models, such as CIR, and simulaions of GRCH ocess esimaed fom eal daa. Key wods: uoegessive Pocess R(); GRCH uoegessive Pocess; uoean call oion; ochasic Inees Raes. JL Classificaion: G.. Inoducion We oose (eoic) call oions on ension annuiies, which ae, in fac, ension insuance. We ice hem unde sochasic inees aes discee and coninuous ime egimes, using eal make daa. Oions on ension annuiies ovide ension insuance, and have ecenly been discussed by Milevsky and Pomislow (Milevsky and Pomislow, 00) and Yosef, Benzion, and Goss (Yosef, Benzion, and Goss, 003). Ohes, including Lee (Lee, 00), Cains (Cains, 00), Balloa and Habeman (Balloa, and Habeman, 003), Pelsse (Pelsse, 003), Wilkie, Waes, and Yang (Wilkie, Waes, and Yang, 003), and Boyle and Hady (Boyle and Hady, 003) invesigaed simila oions, called guaaneed annuiy oions (GO s). he GO oion gives he oion holde he igh o eceive a eiemen he geae of a cash aymen equal o he cuen value of he invesmen in he equiy fund and he eeced esen value of he life annuiy obained by conveing his invesmen a he guaaneed ae. he GO oion diffes fom he oion ha we oose in ha i fuhe gives he olicyholde he igh o choose eihe he annuiy based on cuen make aes, o an annual aymen using he guaaneed annuiy inees ae. call oion on a ension annuiy elaces in effec he adiional ension insuance model. Unde he adiional model a eson ays monhly emia ino his ension fund houghou his woking life, and eceives a monhly benefi of B saing a a edefined age of eiemen. Unde he lan ha we oose, a eson aged,, buys a uoean call oion on his ension annuiy. his call oion allows he oion holde o buy his ension annuiy a a secified sike ice, a o io o he age of eiemen. he oion holde is eniled o eceive he ension benefis only if he insued is sill alive a eiemen age, and hen houghou he insued s life in eiemen. hus insead of conibuing monhly aymens o hei ension fund houghou hei woking lives, individuals could buy ou oosed oion as insuance, elacing he monhly emia by a wo-insallmen sysem: one a age when he oion is bough, and one a eecise ime a o io o eiemen age, when he oion is eecised a he esecified sike ice. Yosef, Benzion, and Goss, (Yosef, Benzion, and Goss, 003) used a consan inees ae o simlify he icing of oions on adiional ension conacs in discee ime. In his a- hulamih. Goss, Rami Yosef, Ui Benzion, 007. * Bauch College of he Ciy Univesiy of New Yok, U. ** Ben-Guion Univesiy of he Negev, Isael. *** Ben-Guion Univesiy of he Negev, Isael.

2 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, e we oose o eend hei wok in wo diecions: we elace he consan inees ae by sochasic inees aes and we do his, boh in discee and coninuous ime egimes. Milevsky and Pomislow, (Milevsky and Pomislow, 003) consideed boh sochasic discee and he coninuous inees aes, bu eaed imaily oions on ue endowmens. ising eamens, including hose of GO s, evaluae ension annuiy oions using a heoeical Maingale measue ha is had o esimae fom obseved daa. hei fomulae deend diecly on he secial oeies of he Maingale measue and do no hold fo he emiical measue associaed wih eal wold inees aes. We ovide a valuaion of annuiy oions ha is comuable fom he emiically esimable sho ae ocess and is associaed em sucue of inees aes. Ou fomulae ae geneal and do no deend on any aicula model used o fi emiically obseved inees aes o he em sucues of inees aes. hey ovide a modelindeenden comuaional vehicle ahe han an elici fomula. he convegence of infinie sums involved in he discee ime case, and he inegals in he coninuous ime case, deends only on he finie lifeime of he insued, and equies no ea assumions. We fis esen discee (ecion ) and coninuous ime (ecion 3) fomulae fo comuing he value of a adiional ension insuance in which he insued ays a single emium a age o eceive a monhly ension benefi B fom he age of eiemen hough deah, unde any sochasic sucue fo he sho and em sucue of inees aes. We hen use he fomulae o ice oions on ension annuiy lans. We aly hese fomulae o simulaed daa fom discee and coninuous ime models. he daa on which we base ou simulaions is he monhly Bank of Isael nominal bank ae seies fom /985 o /00 fo he discee ime simulaions. Fo he coninuous ime we use he monhly uodolla annualized deosi inees aes fo 4/953 hough 5/003 ublished by he Fedeal Reseve in is H.5 daabase. he choice of daa is quie abiay, ece ha i seves o fi a sochasic model fo he sho aes fom wo diffeen souces, oviding esimaed aamee values fo he simulaions ha follow. We hen esen ables fo he ension annuiy oion mean values and sandad deviaions as a funcion of insued age, sike ice, sike ime, mean sho inees ae, and inees ae ocess volailiy. consan volailiy discee auoegessive ocess [ee (Panje and Bellhouse, 980) and (Pake, 994)] of ode wo adequaely fis he Isaeli daa. Fo he uodolla daa we ied vaious em sucue models based on sochasic volailiy sho ae models found elsewhee in he lieaue [e.g., CIR model, )Co, Ingesoll, and Ross, 985)], bu found ha none of hese models fi ou emiical daa vey well. Chan, Kaolyi, Longsa and andes (Chan, Kaolyi, Longsa and andes, 99) have obained simila esuls fo diffeen emiical daa. We heefoe esoed o he sochasic volailiy RCH-GRCH discee aoimaions o he coninuous models. he aamees we esimaed fo he RCH-GRCH models wee hen used fo geneaing daa in he coninuous ime simulaion. We assumed ha eal wold invesos have obseved he sho ae sucue fo a long ime and have lean he laws of moion and he aamees of he dynamics of he sho ae. hese laws induce eecaions abou fuue sho aes ha in un induce a em sucue wih imlied fowad aes, which we use in icing o discouning cash flows. In ohe wods, we use emiical daa o esimae he eal-wold sho ae ocess, and use hese eecaions o deemine he em sucue and is associaed fowad aes. Because he uodolla deosi ae (o Bank of Isael inees ae) ocess embodies he same isk levels as he cash flows we iced, we can now use he imlied fowad aes,,... o ice ou cash flows. We emhasize ha emia and annuiy oion ices deend on wo sochasic henomena: he inees ae ocess and he insued lifeime, which iself may be modeled as a sochasic ocess because lifeime disibuions may vay in ime. In ou simulaions we simulaed sochasic ae ocesses using esimaed aamees fom emiical daa. Howeve, ahe han model he suvival disibuion, we used ublished Isaeli and U life-ables fo comuing condiional suvival obabiliies in he ime-indeenden, non-dynamic suvival case. Life ables used by he Isaeli insuance indusy unil aoimaely en yeas ago wee in fac Biish ables. he mos commonly used able has been he ( ) able, which we use in ou simulaions in he discee

3 08 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 ime case. In he coninuous ime simulaions we used boh meican life ables, and meican inees ae daa. meican life daa wee obained fom he U Decennial life ables fo , able fo he oal oulaion, fo comuing condiional suvival obabiliies in he imeindeenden, sochasic suvival case. he choice of life-ables ove secific obabiliy disibuions, o secific dynamic suvival ocesses, as was done by Milevsky and Pomislow (Milevsky and Pomislow, 00), was indicaed by he fac ha insuance comanies adiionally use life-ables ahe han fied suvival disibuions. lso, he esenaion of ou fomulae in ems of life-able, ime-invaian, and condiional obabiliies, emis a simle comuaional adjusmen o incooae ime deenden life-able condiional obabiliies. We shall indicae he changes equied in he esenaion ha follows. Finally, we noe ha ohe cones in which wo o moe sochasic henomena goven he life of a call oion have been sudied in he lieaue. wo imoan ones ae oions on defaulable bonds [see e.g., Duy and ingleon (Duy and ingleon, 997)], and oions on sian echange aes in a wo-cuency economy [see e.g., (Nielsen and andmann, 00)]. In he fis, he defaul hazad is used o adjus he insananeous inees ae. In he second, boh he foeign and domesic cuency zeo-couon bond ices ae used o model he echange aes. In boh cases he wo sochasic henomena ae assumed o be indeenden, even hough he assumion is quesionable in boh cases. I is fa moe easonable in ou case. Ou esenaion is divided ino discee and coninuous ime modeling of fowad aes, wih coesonding fomulae and simulaions. he models acually emloyed in he discee case ae fied volailiy auoegessive ocesses, wheeas he discee aoimaions used in he coninuous ime fowad ae ocesses ae sochasic volailiy models. Noneheless, a quick comaison of esuls esened in ables and 7 fo he wo cases, fo idenical ages a annuiy oion uchase, idenical sike ices and aveage fowad aes, ae ahe simila. his is ue desie he fac ha he souces of emiical daa fom which volailiy aamees wee esimaed ae diffeen, and he fis is based on a consan volailiy auoegessive ocess of ode wo, and he second is based on a sochasic volailiy, auoegessive model of ode wo. he life ables used in he wo cases belong of couse o he diffeen counies of oigin of he daa. Ou simulaion esuls dislay mean and sandad deviaions fo annuiy oion ices as funcions of age a annuiy oion uchase, sike ice, and aveage fowad ae. We emhasize ha he sandad deviaions eoed ae he sandad deviaions of he (sochasic) oion ice as esimaed fom he simulaion. We found i o be vey sable as he numbe of simulaions inceased. I is an inheen oey of he oion ice ha deends on he wo sochasic henomena of age a deah and fowad inees aes. his is he case because we used aamee values ha wee esimaed fom eal daa in wo vey diffeen counies and economies. No amoun of ea simulaions will educe hese sandad deviaions. hese elaively lage sandad deviaions ae an inheen oey of oions on annuiies in a sochasic inees ae economy.. Picing adiional Pension nnuiy Plans: he ochasic Discee Foce of Inees Case Conside a adiional ension insuance in which an insued eson, age, buys ension insuance wih a single emium, guaaneeing a B annuiy fom he age of eiemen,, hough deah. ssume ha inees aes change a discee ime oins, and le he seies...,, 0,,,..., denoe he sochasic seies of fowad inees aes, whee 0 a age of he insued. he isk embodied in hese aes is he isk of cash flows we ice. I is convenien a his oin, foeseeing he need o model he sochasic fowad inees ae, o inoduce insead he foce of inees seies log, () and he cumulaive foce of inees

4 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, , i i () which ae no necessaily osiive quaniies bounded beween zeo and one, and hus easie o model. ee Panje and Bellhouse (Panje and Bellhouse, 980) and Pake (Pake, 994)... Picing Pension Plans Unde Discee ochasic Inees Raes Given he seies of cumulaive foce of inees, he single emium aid by an insued aged, o eie a age, >, and eceive B monhly heeafe unil he age a deah, denoed by, is given by B, e (3) whee,...,,...,,, 0 denoe he sochasic seies of foces of fowad inees aes, which we shall now denoe simly by. Noice ha is andom and is eeced value is, by he chain ule of condiional eecaions, and he indeendence assumed beween and, )], ( [ ] [ (4) whee efes o he eecaion wih esec o he sochasic foce of inees, and efes o he eecaion wih esec o he age a deah whose condiional obabiliy, given he eson had aleady suvived ill age, ha he will suvive beyond is denoed, as usual, by. We emhasize ha is a comlicaed (no mulilicaively seaable) funcion of and of. heefoe, even hough we assume ha and ae indeenden, we canno use he oduc ule fo eecaions, and we eso insead o he use of he chain ule. We hen obain, ] [ 0 0 M B e B (5) whee ) (u M denoes he momen geneaing funcion of a u. Fuhemoe,, VR VR VR (6) see Ross (00). Fom (3) we comue ), ( 0 0 M B (7) ), ( ) ( 0 0 M M B (8) and VR, ) ( 0 0 e e B (9) so ha VR is given by he sum in (8) and (0) below

5 0 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 M ( ) 0 VR B. (0) ( ) 0 0 M o fa we have no assumed any secific obabilisic sucue fo he inees ae, and no fom fo he suvival disibuion which we shall comue using life ables. We now secialize ou esuls o he case of auoegessive seies of ode, R() fo he foce of inees seies. s we discoveed boh in he cases of Isaeli sho-ae daa, and U sho ae daa, R() ocesses fi o he foce of inees daa led o fied ocess aamees ha uon simulaion yielded a lage facion of negaive inees aes. Howeve, R() ocesses led o vey few negaive aes ha could be uned ino zeo wihou subsanially aleing he ocess. ecifically, we fi he ocess, () whee he innovaions ae indeenden nomal 0, vaiables, is he fied mean of he ocess, and and ae he auo egession aamees of odes and esecively ha saisfy he equiemens,,, insuing ha he seies is saionay. Fo his ocess, he momen geneaing funcion is, a any fied ime, is given by uu G( u) M ( u) e, () wih G( ) G ( ) ( ) G( ), G i ) i i i i( i, whee and ae eciocals of he oos of he auoegessive ocess chaaceisic equaion ( ) ( ) 0, (3) (ee Panje and Bellhouse (Panje and Bellhouse, 980) and Pake (Pake, 994) and ( ), (4) whee and ae he auo egession aamees. ince we can also wie M ( u) e s u u G( s) G( ) s we have an elici fomula fo ] [ u ( G( s )), (5) and VR ] in he nomal auoegessive [ R() case as well. We emhasize again ha in hese models volailiy is assumed o be fied... Oions on adiional Pensions he ochasic Discee-ime Case he ice of a uoean call oion on a discouned ension annuiy fo a aicula ealizaion, o samle ah of is C Ke, (6) whee denoes he andom ime of deah, which is known o eceed,, K is he sike ice of he oion, and a ma( 0, a). Recall ha a he ime of uchase of

6 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 he call oion, and a eiemen, he eecise dae of he oion. lso when. Using he usual condiional eecaion k C, P Be K, (7) k 0 and k C P Be Ke. (8) k 0 We do no fuhe secialize his fomula o he R() Gaussian case because he heavyside (lus) funcion evens us fom wiing his eession fo he condiional eecaion of C, in ems of known quaniies like Gaussian momen geneaing funcions. his fomula is howeve easily amenable o simulaion..3. imulaion udies: he Discee Consan Vaiance Case Conside he case of insued individuals whose ages ae as secified in he able below and who ae ineesed in buying a uoean call oion on hei ension annuiy of $000 e monh which will be aid fom hei eiemen o hei deah. ssume ha age of eiemen is 65, and he foce of inees follows a consan vaiance R() auoegessive model wih aamees and and e yea. hese aamees wee obained uon fiing an R() model o he foce of inees o he monhly Isaeli nominal bank-ae inees seies fom /985 o /00. his ae incooaes he aoiae make isk emium elevan o he cash flows we ice. he obseved oigin of his aameeizaion heled insue ha few negaive inees aes wee oduced in he simulaion, and few inees aes lage han wee obained in he simulaion. he aamee fom () was chosen o coesond o 3%, 5%, and 0% annual ae esecively. he single mean emium fo he ension lan, fo he hee aveage aes, and he hee ages a uchase ime, ae given in able. he esuls eoed in able may be comaed wih he coesonding values unde fied inees ae of 3%, 5%, and 0% annual ae, esecively. Fo eamle, he mean ice of he ension lan, a 0% inees ae, fo an insued aged 40, is $,4($3,33), wheeas he ice of he same lan fo ha individual a a fied ae of 0% is $3,; imilaly, a age 50, wih he same mean ae, he mean ice was $3,5($8,7), and he fied ae ice was $33,880. he values in aenheses efe o he simulaion sandad deviaions, modeaing some of he disceancy obseved in he mean sochasic ice and he fied ae ice. In his simulaion sudy we acually simulaed he foce of inees seies only, as we used acual life ables ha gave us he condiional eecaion of he single emium given he foce of inees seies elicily. he simulaion sandad deviaions we eo in aenheses unde he mean emia in able, ae in fac esimaes of he condiional sandad deviaion of he emium given he foce of inees seies. ll simulaion esuls eoed in his ae eesen he oucome of 0,000 simulaion uns fom he samle ah of he inees aes seies. In he same simulaion we also comued he mean ice of he uoean oion as a funcion of he mean foce of inees, he age of he insued a he ime of uchase of he oion and he sike ice, when he oion is eecised a eiemen ime. he esuls ae hen comaed o he coesonding values unde non-sochasic inees aes. gain, he eeced ices unde he ime-invaian inees ae and he sochasic R() foce of inees seies wih fied vaiance ae sikingly simila. he simulaion sandad deviaions eoed ae again esimaes of he condiional sandad deviaions of he oion ice given he inees ae seies.

7 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 able he mean single emium by age a uchase, and by annual inees ae wih an R() consan volailiy model fo he inees ae* ge \ Mean 0% 5% 3% 30 $4,3 ($,45) $34,048 ($,644) $79,8 ($7,833) 40 $,4 ($3,33) $54,870.9 ($6,8) $05,390 ($33,05) 50 $3,5 ($8,7) $9,09 ($5,94) $44,889 ($40,795) * he numbe in aenheses eesens he simulaion sandad deviaion fo he coesonding aamee. he auoegession aamees ae , and Mean cos and sandad deviaion of a uoean call oion by sike ices, age a uchase, and mean inees aes** able ge K (C) (C) (C) mean ae 0% 5% 3% 30 $30,000 $3,506 ($,08) $3,58 ($966) $3,03 ($906) $9,4 ($8,754) $8,04 ($8,404) $6,643 ($8,06) $70,564 ($,5) $67,70 ($0,47) $64,90 ($9,74) 40 $30,000 $9,35 ($,9) $8,668 ($,6) $8,00 ($,034) $47,69 ($,9) $45,4 ($,477) $43,60 ($,043) $93,636 ($3,67) $89,88 ($,89) $86,7 ($,83) 50 $30,000 $5,44 ($4,967) $3,630 ($4,683) $,84 ($4,409) $79,044 ($5,74) $75,8 ($5,58) $7,439 ($4,590) $7,066 ($5,588) $,903 ($4,80) $6,80 ($4,054) ** he numbe in aenheses eesens he simulaion sandad deviaion fo he coesonding aamee. he auoegession aamees ae , and he esuls eoed in able may again be comaed o he coesonding values unde fied inees ae of 3%, 5%, and 0% annual ae, esecively. Fo eamle, he mean ice of he ension lan, a 5% inees ae, fo an insued age 30, wih a sike ice of $30,000 and ae $9,4($8,754) and $8,04($8,404) esecively. he coesonding values fo fied inees ae wee $9,995 and $8,4. Wheeas he ice of he same lan fo ha individual a a fied ae of 0% is $3,. imilaly, a age 50, wih he same mean ae, he mean ice was $3,5($8,7), and he fied ae ice was $33,880. gain we noe ha hese esuls may be aially elained by ice volailiy in he sochasic case, and he uwad end due o inees ae volailiy is again noiceable. We emak ha he fomulae we esened fo he eeced value of a single emium, and he eeced cos of a single oion on he ension benefi, can be eended o he dynamic suvival case whee he suvival disibuion deends on ime. o visualize he siuaion, conside he discee-ime sochasic ocess of suvival ha sas a ime 0 a sae 0 (alive), and ansiions ino sae (dead), a some ime. ae is absobing fo his suvival ocess. We assume he ocess o be Makovian. he n-se ansiion obabiliy fo his ocess P [ X ( n) X ( ) 0] may deend on calenda ime, making he ocess a nonhomogeneous Makov ocess. Wien diffeenly,

8 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, P [ n] P [ X ( n) X ( ) 0]. hus in fomulae (5) and (8) we need o elace P [ ] ( by ) P [ P [ ] and ] by P [ ]. P [ ].4. imulaion udies: he Discee Consan Vaiance Case -R() In ode o sudy he effec of lage volailiy on he single emium of a ension annuiy lan, and he ice of a uoean oion on i, we simulaed homoscedasic seies wih he same auoegessive aamees as befoe, bu wih diffeen sandad deviaions. We wee seiously limied in vaying because he simulaed seies ended o elode when he sandad deviaion was subsanially lage han he emiical one fied o eal daa. Because he sandad deviaions hus emained in a faily sho ineval, we eo only he esuls fo he smalles and lages we used and able 3 he mean single emium by age a uchase, and mean annual inees ae fo an R() consan volailiy ae model fo wo values of ge \ Mean ae 0% 5% 3% $4,096 ($65) 30 $4,40 ($,364) $3,34 ($,74) $3,698 ($,4) $75,435 ($5,86) $76,663 ($6,896) $,99 ($659) 40 $,88 ($3,48) $53,65 ($3,3) $54,4 ($7,38) $0,795 ($6,496) $04,90 ($34,095) $3,08 ($,59) 50 $3,468 ($7,73) $90,900 ($4,686) $9,9 ($3,893) $4,78 ($7,548) $44,585 ($38,575) he mean emium was only slighly aised due o an inceased sandad deviaion. Noe he almos linea effec of he ae ocess sandad deviaion on he emium simulaion sandad deviaion. When he fome was mulilied by 5, he lae was also aoimaely mulilied by 5. his linea elaionshi is no eac, bu held in ou simulaions fo he inemediae sandad deviaions as well. Using he same aamees, and he same simulaion uns, we also esimaed he mean ice of a uoean oion on he ension conacs iced in able 3. gain we may comae he esuls wih consan inees ae, and unde wo volailiy values, unde he same se of condiions on age a uchase, mean inees ae, eecise ime, and sike ice. Remakably, he mean ice of he oion inceased slighly wih a highe ae ocess sandad deviaion, and is sandad deviaion was aoimaely ooional o he ocess sandad deviaion.

9 4 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 able 4 Mean ice and sandad deviaion of a uoean call oion by sike ice and mean ge (C) (C) (C) K,, 0% 5% 3% $3,304 ($94) 30 $30,000 $3,3 ($984) $7,675 ($,650) $7,858 ($8,350) $66,07 ($3,97) $66,70 ($0,) $3,067 ($8) $3,08 ($9) $6,3 ($,580) $6,498 ($7,997) $63,485 ($3,83) $63,966 ($9,48) $,833 ($70) $,847 ($860) $4,984 ($,5) $5,54 ($7,65) $60,795 ($3,695) $6,64 ($8,77) $9,034 ($468) 40 $30,000 $9,050($,376) $46,04 ($,408) $46,3 ($,86) $90,94 ($4,74) $90,430 ($3,987) $8,385 ($439) $8,400 ($,8) $43,77 ($,309) $43,876 ($,680) $86,485 ($4,58) $86,76 ($3,6) $7,745 ($4) $7,759 ($,083) $4,548 ($,) $4,649 ($,86) $8,80 ($4,43) $83,046 ($,35) $5,68 ($,049) 50 $30,000 $5,36 ($5,343) $77,99 ($3,39) $78,96 ($6,949) $5,04 ($5,407) $5,53 ($7,54) $3,359 ($996) $3,45 ($5,077) $74,73 ($3,0) $74,373 ($6,40) $0,05 ($5,6) $0,37 ($6,794) $,575 ($945) $,639 ($4,8) $70,40 ($3,4) $70,598 ($5,869) $4,96 ($5,0) $5,75 ($6,083) * he numbe in aenheses eesens he simulaion sandad deviaion fo he coesonding aamee. Dae.5. Pension Oion Pices as Funcion of ike Pice and biay ecise We now allow uchases of annuiy ension oions o eecise hei oions conac befoe eiemen. hus eecise ime fo ension oions may be anywhee beween uchase ime and, he age of eiemen. he mean oion ice fo sike ime s, s, is hen k P[ ] Be Ke s k0 ( C ) (9)

10 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, gain, his fomula is comleely geneal, and does no deend on any aicula model fo inees aes. I is sicly a comuaional fomula, and is no aiculaly useful fo heoeical evaluaions. We esen ou simulaion esuls fo vaious eecise imes in able 5. quick eusal hough he able eveals ha as eeced, as he eecise age deceases fom he age of eiemen o he age a uchase, he mean ice and is sandad deviaion also decease. Ohe able esuls equie few commens. Obviously, when eecise age is smalle han age a uchase, he emium and he oion ice ae null. s eeced boh he mean emium and he mean ice and he coesonding sandad deviaions ae highly affeced by he eecise dae, in he edicable diecion. able 5 Mean ice and sandad deviaion of he ice, mean inees ae and eecise dae fo an R() inees ae* ge K, ecise ge (C) (C) (C) 0% 5% 3% 45 $30 ($37) 30 $30,000 $,90 ($89) 45 $ ($0) $,703 ($789) 45 $.6 ($6) $,7 ($688) 45 $676 ($,63) 40 $30,000 $6,65 ($,394) 45 $98 ($46) $4,579 ($,74) 45 $.5 ($07) $3,47 ($,935) 45 $0 ($0) 50 $30,000 $6,664 ($6,76) 45 $0 ($0) $,99 ($5,84) 45 $0 ($0) $8,38 ($5,390) $,4 ($8,696) $6,588 ($8,687) $9,05 ($8,5) $4,368 ($8,6) $5,834 ($7,786) $,0 ($7,854) $36,778 ($4,63) $43,68 ($3,) $3,84 ($3,850) $40,0 ($3,046) $5,940 ($3,33) $36,4 ($,5) $0 ($0) $7,394 ($9,930) $0 ($0) $66,43 ($9,599) $0 ($0) $60,5 ($9,53) $63,30 ($,804) $67,09 ($,398) $58,46 ($,96) $63,378 ($0,670) $53,739 ($0,594) $59,730 ($9,965) $85,0 ($8,38) $90,35 ($7,063) $78,64 ($7,95) $85,337 ($6,36) $7,59 ($7,0) $80,4 ($5,677) $0 ($0) $,866 ($3,45) $0 ($0) $5,958 ($3,885) $0 ($0) $09,76 ($3,58) * he numbe in aenheses eesens simulaion sandad deviaion, e.g. means, eecise dae.

11 6 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, ochasic Inees Raes: he Case of Coninuous ime 3.. Coninuous Models: ome Hisoy Financial conacs ha deend on levels of inees aes, wih zeo-couon discoun bonds being he ime eamle, bu also including oions on ension annuiy conacs, have aaced much aenion in he financial lieaue in ecen yeas. Picing hese conacs equied he fomulaion of adequae sochasic models fo sho-em inees aes in coninuous ime. Recen models fo he sho ae ocess, o he fowad-ae ocess, may be divided ino wo gous ha ae disinguished imaily by he way hei volailiy is modeled. hei basic sucue is ha of a coninuous auoegession, usually of ode, [see, e.g., Koedijk, Nissen, choman, and Wol (Koedijk, Nissen, choman, and Wol, 997), and he many efeences heein]. Olde models, such as Meon s (Meon, 973) and Vasicek s (Vasicek, 977), deived via an agumen of a simle economic equilibium, feaue a fied volailiy. Co, Ingesoll, and Ross (CIR, 985), deived he fis sochasic volailiy model fom fis economic inciles. hei is a coninuous ime auoegessive model wih condiional volailiy, given he as of he ocess u o ime, jus io o ime, ha is he squae oo of he sho ae a ime. Chan, Kaolyie e al. (o. ci.) have genealized he CIR model ino a family of coninuous auoegessive models wih sochasic volailiy, ha include many of he imoan models in use u o ha ime. he family is secified by he sochasic diffeenial equaion d ) d db, (0) ( whee,, and 0, and B is a sandad Bownian Moion. he sochasic volailiy is adaed o he B ocess (essenially deemined a ime by is hisoy u o ime ), whee he condiional volailiy isv *. he aamee in his model has been 0. dubbed he elasiciy aamee because i conols he disance he ocess eaches in he osiive half-sace, and he fequency of is waves. he CIR ocess feaues Chan, Kaolyie e al. (o. ci.) have found he value. 5 o bes fi hei daa of emiical monhly inees-aes of U -bills. ubsequen sudies have disued his finding, suggesing ha he high value of he elasiciy aamee gamma was in fac a esul of model missecificaion ha did no ake ino accoun he new egime imosed by he Fedeal Reseve on inees aes [see Koumos (Koumos, 998) and efeences heein], and in fac he CIR model, feauing 5 fis he daa locally fa bee. In fowad ae simulaion of hese coninuous models, he ule aoimaing discee model ( ) Z fo...,,,0,,,..., () whee Z eesens an indeenden, idenically disibued sequence of Nomal (0, ) deviaes, is used. Moe ecen emiical sudies of sho-em inees aes (ee Koumos (Koumos, 998, 000) and Bali (Bali, 003)) sugges ha discee ime aoimaions o such coninuous ocesses by R() o R() models wih RCH o GRCH ems fo he condiional volailiy given he as of he inees ae ocess, fi much emiical daa fa bee. he lae ocesses ae eoed o accoun fo olonged walks obseved in emiical inees ae seies in he osiive half-sace, wihou causing fequen negaive aes. he simles insance of a linea RCH-GRCH (discee ime) model is an auoegessive model R(k) k 0 j ( j ), j ()

12 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, whee he eos V 0. 5 Z and he Z ae indeenden idenically disibued sandad nomal vaiaes and he volailiy ocess V, in he GRCH (, q) secificaion, is given by he auoegession V V, (3) 0 j j j j j j q wih fied aamee vecos and. Moe ecen volailiy models ha ae linea combinaions of elasiciy ems, and RCH-GRCH ems, have also been oosed by Bali (o. ci.). He also oosed wo-faco models which descibe he inees ae ocess using a coninuous auoegession wih eos given by one Bownian Moion, and a condiional volailiy model, also an auoegession, wih anohe Bownian Moion, indeenden of he fis. lhough hese wo-faco models seem omising, we do no usue hem in his wok. 3.. Picing adiional Pension nnuiy Plans and uoean Oions on nnuiy Pension Plans In ou modeling of he dynamic fowad inees ae, o inees foce, in coninuous ime, and fiing i o emiical daa, we have, of couse, used he discee ule aoimaion descibed in equaion (). he emiical daa we used fo fiing a coninuous model, via is discee aoimaion, ae he monhly uodolla annualized deosi inees ae seies, fom 6/953 o 6/003, ublished by Fedeal Reseve in is H.5 daabase. We hen followed he fowad inees ae model fiing by a simulaion using he aamees we have esimaed, and evenually comued he single emium fo a ension lan, and he ice of a uoean oion on his ension lan. Because we used discee aoimaions o he coninuous ocess, he main diffeence in he coninuous ime case fom he discee case is ha in he lae we used sochasic volailiy models, wheeas in he fome we used fied volailiy models. s we saw ealie he fied volailiy models emied he sudy of he deendence of oion ices on imoan aamees such as vaiance size, sike ime, and age a uchase of oion. he sochasic volailiy models oduce chaoic volailiy ha ohibis he sudy of hese aamees, because small changes in aamee values lead o he elosion of he seies of aes. We have ied o fi combinaion models o all of hese ossible models, and found, using maimum likelihood, he RCH-GRCH volailiy models o fi ou emiical daa bes, and hen lead o moe successful simulaions ha oduce easonable seies of foce of inees, and hen of inees, ha do no hi below zeo, o above one oo ofen. he fomulae we obained in he discee sochasic case can easily be modified o fi he coninuous sochasic fowad inees ae case. We use he same noaion fo he coninuous foce of inees, and cumulaive foce of inees, and allow hem o follow an unsecified sochasic model. We assume ha he coninuous sochasic suvival (ime of deah) vaiable, which follows an unsecified coninuous disibuion, is indeenden of he sochasic foce of inees ocess. he single emium aid by an insued aged o uchase an annuiy insuance ha would ay B fom age 65, say, ill his deah, ovided he is alive a age, is andom and is given by he inegal: B e[ ( )] d, (4) whee () denoes he cumulaive sochasic foce of inees a ime, and. he comlee coninuous ime seies will again be denoed by. Hee ( ) log( ( )), (5)

13 8 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 whee () is he sochasic fowad inees ae, and he cumulaive foce of inees is given by ( ) ( u) du. (6) 0 B is he fied benefi aid e uni ime fom eiemen ime onwad, ill he deah ime of he insued. By he indeendence beween and () we can wie he eecaion of as [ ] [ ( )] u B F / F e 0 u0 duf ( ) d, whee F denoes he cumulaive disibuion funcion of. his eecaion may also be eessed in ems of he momen geneaion funcion M ( ) ( u) of () a u. Fo Gaussian ocesses he inne eecaions, being he momen geneaing funcion of a Gaussian vaiable, ake on a simle eonenial fom, and deending on he densiy f () he inegal may be comued elicily. In any case Mone Calo simulaion is saigh fowad. Fuhemoe, we can wie again VR VR VR and eess i in ems of join and maginal momen geneaing funcions, as in he discee case. he icing fomula fo he coninuous ime case emains fomula (6) bu is eecaion becomes C f ( ) P 0 Be u du Ke f Noe ha ( ) fo is no a hazad ae, in conas o Milevsky and Pomislow s (o. ci.) fomula, and is also non-sochasic. Howeve, if we assume ha he disibuion P[ ] of is sochasic, ha is suvival changes dynamically wih ime, he only change we would have o make in he las fomula is o add an eecaion wih esec o he disibuion f of ( ). We also noe ha unlike Milevsky and Pomislow (o. ci.), we have no caied P[ ] ou ou comuaions unde he Maingale measue. he comuaion unde he Maingale measue simlifies he fomula subsanially, by moving he eecaion unde he inne inegal sign. he fomula as i sands seves howeve vey well fo simulaion uoses. Once we choose a model fo he foce of inees () via a sochasic diffeenial equaion, we elace he lae by is discee aoimaion, which we hen use wih idenical ime inevals (e.g., monhs) in boh he life ables and he foce of inees discee ocess imulaions in he Coninuous ime Case We fis fi seveal models o he Fedeal Reseve monhly inees daa. he bes fi was obained by an RCH-GRCH R() model of ode ( =, q = ), wih RCH aamee , GRCH aamee , , mean ,. 46, 0.470, and hese esimaed aamees wee hen used in a simulaion o comue he mean single emium of a lan (able d. (7) (8) (9)

14 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, ) and he ice of he lan (able 7) as a funcion of age a uchase, and mean ae. o educe he size of hese ables, we assume ha he oion is eecised a eiemen ime, and included only hee sike amouns. lhough he esuls eoed in he wo ables below canno be diecly comaed o he coesonding ables fo ime-invaian volailiy, one can noneheless noe he emakably small sandad deviaion elaive o he mean amoun, fo boh he emium and he oion ice in he sochasic volailiy case. he mean single emium by age a uchase and mean annual inees ae fo RCH GRCH (,), R() inees ae model* able 6 ge \ Rae 0% 5% 3% 30 $4,0 ($433) $37,30 ($3,73) $89,390 ($9,48) 40 $,50 ($,097) $6,383 ($5,808) $,49 ($,78) 50 $35,00 ($,564) $06,50 ($8,339) $70,83 ($3,83) * he numbe in aenheses eesens he simulaion sandad deviaion fo he coesonding aamee. able 7 Mean cos and sandad deviaion of a uoean call oion by sike ices and mean inees aes fo RCH-GRCH (,), R() inees ae model* ge K (C) (C) (C) Mean ae 0% 5% 3% 30 $30,000 $3,73 ($9) $3,478 ($73) $3,36 ($56) $3,484 (,59) $3,09 ($,40) $9,7 ($,3) $79,50 ($6,4) $76,70 ($5,944) $73,97 ($5,750) 40 $30,000 $0,58 ($73) $9,587 ($68) $8,93 ($64) $54,469 ($3,87) $5,4 ($3,73) $49,835 ($3,593) $09,3 ($7,793) $05,389 ($7,565) $0,589 ($7,34) 50 $30,000 $8,645 ($,773) $6,765 ($,689) $4,905 ($,607) $9,543 ($5,793) $88,578 ($5,60) $84,648 ($5,45) $5,04 ($9,583) $46,69 ($9,353) $4,389 ($9,7) * he numbe in aenheses eesens he simulaion sandad deviaion fo he coesonding aamee. 4. Conclusions In his ae we analyzed he behavio of a ecenly inoduced ension insuance insumen, a uoean call oion defined on ension annuiy. Unde his lan, insued aies can buy an oion on hei ension annuiy benefi, ganing hem he oouniy o buy hei discouned annual ension annuiy benefi io o o a he age of eiemen fom he oions wies, a a defined sike ice. he analysis was caied ou unde sochasic inees aes. We consideed a vaiey of diffeen sochasic models fo inees ae, and in all models we found ha as he sandad. he use of his uoean call oion fo ensions is a new mehod which enables individuals o subscibe o a ension annuiy a a lae age, fiing he ems of aymen in advance, while he cuen value aid by he individual is somewha highe han sandad ension annuiy.

15 0 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007 Because Milevsky & Pomislow s (Milevsky & Pomislow, 00) wok eaed he elaed oblem of valuing uoean-syle oions on he moaliy-coningen claim ha ays one lum sum uon suviving a e-secified eiod, we eview ou conibuion elaive o hei sudy. he insuance oduc hey eaed is also known as an endowmen olicy, in conas o adiional life annuiy, and is quie simila o a zeo-couon bond. Milevsky and Pomislow go on o egad, unde ceain odeing condiions, an oion on a adiional ension lan as a baske of such endowmen lans of diffeen mauiies. In he discee ime case only a one yea hoizon is acually woked ou. hei agumen fo elacing he ension annuiy by a baske of oions aeas o sugges ha he sae sace of inees aes mus also be discee. In he coninuous ime case, Milevsky and Pomislow ado Due and ingleon s (Due and ingleon, 997) aoach o he valuing of uoean oions on defaulable bonds. Hee Milevsky and Pomislow elace he defaul hazad by he insued moaliy hazad. Fo he single endowmen lan hey osi a Co, Ingesoll, and Ross (o. ci.) model fo he sochasic inees ae, and an indeenden Bownian moion wih a linea end, fo he logaihm of he sochasic hazad ae. No eension is acually given fo an oion on a ension lan in coninuous ime. small simulaion sudy comues he ice of a baske of oions on endowmen olicies as a funcion of hei fied volailiy of hei inees ae ocess. lhough a fomula is develoed fo he value of an endowmen oion in ems of he Maingale measue ha uns he inees ae ocess ino a maingale, he simulaion aeas o simly use he CIR model unde he usual measue, wihou making use of he fomula. he main diffeence beween ou aoach and ha of Milevsky and Pomislow (o. ci.) is in he fac ha we give geneal fomulae fo he valuaion of uoean oions on ension lans diecly, wihou assuming any aicula fom fo eihe he suvival disibuion o he inees ae ocess. Boh aoaches assume indeendence beween suvival and inees ae. In he discee case we also ovide vaiance fomulae fo he (sochasic) ice acually aid by he insued who buys an oion a some ime befoe eiemen. imila fomulae ae also ossible in he coninuous ime case. Refeences..G. Bali. Modeling he ochasic Behavio of ho-em Inees aes: Picing Imlicaions fo Discoun Bonds. // he Jounal of Banking and Finance, 003. Volume 7, Balloa, L.,. Habeman. Valuaion of Guaaneed nnuiy Convesion Oions // Insuance, Mahemaics and conomics, Boyle, P., M. Hady. Guaaneed nnuiy Oions // Woking ae, Univesiy of Waeloo, Canada. 4. Cains. Family of Models fo he em-ucue of Inees Raes: n licaion o Guaaneed nnuiy Oions // Pesenaion a he IM 00 Confeence, Lisbon. 5. Chan, K.C., Kaolyi, G.., Longsa, F.., and.b. andes. n miical Comaison of lenaive Models of he ho-em Inees Rae // he Jounal of Finance, 99. Volume 47, Co, C.J., Ingesoll, J..,.. Ross. heoy of he em ucue of Inees Raes // conomeica, 985. Volume 53, Due, D., K. ingleon. n conomeic Model of he em ucue of Inees Rae wa Yields // he Jounal of Finance, 997. Volume 5, Koedijk, K.G., Nissen, F.G.J.., choman, P.C., C.C.P., Wolff. (997). he dynamics of ho-em Inees Rae Volailiy econsideed // uoean Finance Review, G. Koumos. he Volailiy of Inees Raes coss Mauiies and Fequencies // he Jounal of Fied Income, 998. Decembe G. Koumos. Modeling ho-em Inees Rae Volailiy: Infomaion hocks Vesus Inees Rae Levels // he Jounal of Fied Income, 000. Mach

16 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 007. P. Lee. Picing and Reseving fo guaaneed nnuiy Oion // Pesenaion fo Den Danske kuafoening, 00. h:// R.C. Meon. heoy of Raional Oion Picing // Bell Jounal of conomics and Managemen cience, , Milevsky, M.., D.. Pomislow. Moaliy Deivaives and Oion o nnuiies // Insuance: Mahemaics and conomics, 00. Volume Nielsen,.J., K. andmann. Picing of sian change Rae Oions Unde ochasic Inees Raes as a um of Oions // Finance and ochasics, 00. Volume Panje, H.H., D.R. Bellhouse. ochasic Modelling of Inees Raes wih licaions o Life Coningencies // Jounal of Risk and Insuance, 980. Volume G. Pake. wo ochasic oaches fo Discouning cuaial Funcions // sin Bullein, 994. Volume 4, Pelsse. Picing and Hedging Guaaneed nnuiy Oions via aic oion Relicaion. Insuance: Mahemaics and conomics, 003. Volume Ross. Inoducion o Pobabiliy Models. New Yok: cademic Pess, Inc., O. Vasicek. n quilibium Chaaceizaion of he em ucue // Jounal of Financial conomics, 977. Volume Wilkie, D., Waes, H.,. Yang. Reseving Picing and Hedging fo Policies wih Guaaneed nnuiy Oions // Biish cuaial Jounal, 003. o aea.. Yosef, R., Benzion, U., and.. Goss. Picing of a uoean Call Oion on Pension nnuiy Insuance // he Jounal of Insuance Issue, 003. Fohcoming, ing 004.