Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies * Daniel Bauer Research raining Group Ul Universi Helholzsraße 8 8969 Ul Geran Phone: 49 (73 5388 Fax: 49 (73 53239 Daniel.Bauer@uni-ul.de lexander Kling Insiu für Finanz- und uarwissenschafen Helholzsraße 22 898 Ul Geran Phone: 49 (73 53242 Fax: 49 (73 53239.Kling@ifa-ul.de Jochen Russ Insiu für Finanz- und uarwissenschafen Helholzsraße 22 898 Ul Geran Phone: 49 (73 53233 Fax: 49 (73 53239 J.Russ@ifa-ul.de bsrac Variable nnuiies wih ebedded guaranees are ver popular in he US-are. here exiss a grea varie of producs wih boh guaraneed iniu deah benefis (GMDB and guaraneed iniu living benefis (GMLB. lhough several approaches for pricing soe of he corresponding guaranees have been proposed in he acadeic lieraure here is no general fraewor in which he exising varie of such guaranees can be priced consisenl. he presen paper fills his gap b inroducing a odel which peris a consisen and exensive analsis of all pes of guaranees currenl offered wihin Variable nnui conracs. Besides a valuaion assuing ha he policholder follows a given sraeg wih respec o surrender and wihdrawals we are able o price he conrac under opial policholder behavior. Using boh Mone-Carlo ehods and a generalizaion of a finie esh discreizaion approach proposed b ansanen and Luarinen (24 we find ha soe guaranees are overpriced whereas ohers e.g. guaraneed annuiies wihin guaraneed iniu incoe benefis (GMIB are offered significanl below heir risneural value. Kewords: Ris-Neural Valuaion Guaraneed Miniu Benefis Ebedded Opions Variable nnui Mone-Carlo-Siulaion Discreizaion Mehods * he auhors han Hans-Joachi Zwiesler for useful insighs and coens. Corresponding and presening auhor
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies Inroducion Variable nnuiies i.e. deferred annuiies ha are fund-lined during he deferen period were inroduced in he 97s in he Unied Saes (see Sloane (97. Saring in he 99s insurers included cerain guaranees in such policies nael guaraneed iniu deah benefis (GMDB as well as guaraneed iniu living benefis (GMLB. he GMLB opions can be caegorized in hree ain groups: Guaraneed iniu accuulaion benefis (GMB provide a guaraneed iniu survival benefi a soe specified poin in he fuure o proec policholders agains decreasing soc ares. Producs wih guaraneed iniu incoe benefis (GMIB coe wih a siilar guaraneed value G a soe poin in ie. However he guaranee onl applies if his guaraneed value is convered ino an annui using given annuiizaion raes. hus besides he sandard possibiliies o ae he are value of he fund unis (wihou guaranee or conver he are value of he fund unis ino a lifelong annui using he curren annui conversion raes a ie he GMIB opion gives he policholder a hird choice nael convering soe guaraneed aoun G ino an annui using annuiizaion raes ha are fixed a incepion of he conrac (. he hird ind of guaraneed iniu living benefis are so-called guaraneed iniu wihdrawal benefis (GMB. Here a specified aoun is guaraneed for wihdrawals during he life of he conrac as long as boh he aoun ha is wihdrawn wihin each polic ear and he oal aoun ha is wihdrawn over he er of he polic sa wihin cerain liis. Coonl guaraneed annual wihdrawals of up o 7% of he (single up-fron preiu are guaraneed under he condiion ha he su of he wihdrawals does no exceed he single preiu. hus i a happen ha he insured can wihdraw one fro he polic even if he value of he accoun is zero. Such guaranees are raher coplex since he insured has a broad varie of choices. Mos of he earlier lieraure on Variable nnuiies e.g. Renz Jr. (972 or Greene (973 is epirical wor dealing wih produc coparisons raher han pricing issues. I was no unil recenl ha he special pes of guaranees were discussed b praciioners (cf. JPMorgan (24 Lehan Brohers (25 or analzed in he acadeic lieraure. Milevs und Posner (2 price various pes of guaraneed iniu deah benefis. he presen closed for soluions for his ianic Opion 3 in case of an exponenial orali law and nuerical resuls for he ore realisic Goperz-Maeha law. he find ha in general hese guaranees are overpriced in he are. In Milevs und Salisbur (22 a odel for he valuaion of cerain GMLB and GMDB opions is presened in a fraewor where he insured has he possibili o pariall surrender he polic. he auhors call his a Real Opion o Lapse 4. he presen closed 3 he auhors denoe his opion as ianic Opion since he paen srucure falls beween European and erican Opions and he paen is riggered b he decease of he insured. 4 heir Real Opion is a financial raher han a real opion in he classical sense (cf. Mers (977. - 2 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies for soluions in he case of an exponenial orali law consan surrender fees and no auri benefis. I is shown ha boh he value and he opial surrender sraeg are highl dependen on he aoun of he guaranee and of he surrender fee. In Milevs und Salisbur (26 he sae auhors price GMB opions. Besides a saic approach where deerinisic wihdrawal sraegies are assued he calculae he value of he opion in a dnaic approach. Here he opion is valuaed under opial policholder behavior. he show ha under realisic paraeer assupions opiall a leas he annuall guaraneed wihdrawal aoun should be wihdrawn. Furherore he find ha such opions are usuall underpriced in he are. In spie of hese approaches for he pricing of several opions offered in Variable nnuiies here is no general fraewor in which he exising varie of such opions can be priced consisenl and siulaneousl. he presen paper fills his gap. In paricular we presen a general fraewor in which an design of opions and guaranees currenl offered wihin Variable nnuiies can be odeled. Besides he valuaion of a conrac assuing ha he policholder follows a given sraeg wih respec o surrender and wihdrawals we are able o deerine an opial wihdrawal and surrender sraeg and price conracs under his raional sraeg. he res of he paper is organized as follows. In Secion 2 we give a brief overview over he exising fors of guaranees in Variable nnuiies. Secion 3 inroduces he general pricing fraewor for such guaranees. e show how an paricular conrac can be odeled wihin his fraewor. Furherore we explain how a given conrac can be priced assuing boh deerinisic wihdrawal sraegies and opial sraegies. he laer is referred o as he case of raional policholders. Due o he coplexi of he producs in general here are no closed for soluions for he valuaion proble. herefore we have o rel on nuerical ehods. In Secion 4 we presen a Mone Carlo algorih as well as a discreizaion approach based on generalizaions of he ideas of ansanen und Luarinen (24. he laer enables us o price he conracs under he assupion of raional policholders. Our resuls are presened in Secion 5. e presen he values for a varie of conracs analze he influence of several paraeers and give econoic inerpreaions. Secion 6 closes wih a suar of he ain resuls and an ouloo for fuure research. 2 Guaraneed Miniu Benefis his Secion inroduces and caegorizes predoinan guaranees offered wihin Variable nnui conracs. fer a brief inroducion of Variable nnuiies in general in Secion 2. we dwell on he offered Guaraneed Miniu Deah Benefis (Secion 2.2 and Guaraneed Miniu Living Benefis (Secion 2.3. e explain he guaranees fro he cusoer s poin of view and give an overview over fees ha are usuall charged. - 3 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies 2. Variable nnuiies Variable nnuiies are deferred fund-lined annui conracs usuall wih a single preiu paen up-fron. herefore in wha follows we resric ourselves o single preiu policies. hen concluding he conrac he insured are frequenl offered opional guaranees which are paid for b addiional fees. he single preiu P is invesed in one or several uual funds. e call he value of he insured s individual porfolio he insured s accoun value. Cusoers can usuall influence he ris-reurn profile of heir invesen b choosing fro a selecion of differen uual funds. ll fees are aen ou of he accoun b cancellaion of fund unis. Furherore he insured has he possibili o surrender he conrac o wihdraw a porion of he accoun value (parial surrender or o annuiize he accoun value afer a iniu er. he following echnical ers are needed o describe he considered guaranees: he rache benefi base a a cerain poin in ie is he axiu of he insured s accoun value a cerain previous poins in ie. Usuall i denoes he axiu value of he accoun on all pas polic anniversar daes. his special case is also referred o as annual rache benefi base. In order o siplif noaion in wha follows we onl consider producs wih annual rache guaranees. Furherore he roll-up benefi base is he heoreical value ha resuls fro copounding he single preiu P wih a consan ineres rae of i % p.a. e call his ineres rae he roll-up rae. 2.2 Guaraneed Miniu Deah Benefis If he insured dies during he deferen period he dependans obain a deah benefi. hen Variable nnuiies were inroduced a ver siple for of deah benefi was predoinan in he are. However since he id 99s insurers sared o offer a broad varie of deah benefi designs (cf. Lehann Brohers (25. he basic for of a deah benefi is he so-called Reurn of Preiu Deah Benefi. Here he axiu of he curren accoun value a ie of deah and he single preiu is paid. he price for his ind of benefi usuall is alread included in he charges of he conrac i.e. his opion is available wihou addiional charges. noher varian is he nnual Roll-Up Deah Benefi. Here he deah benefi is he axiu of he roll-up benefi base (ofen wih a roll-up rae of 5% or 6% and he accoun value. pical fee for ha deah benefi wih a roll-up rae of 6% is approxiael.25% p.a. of he accoun value (see e.g. JPMorgan (24. If he conrac conains an nnual Rache Deah Benefi he deah benefi consiss of he greaer of he annual rache benefi base and he curren accoun value. he charges for his pe of deah benefi are siilar. - 4 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies Furherore he varian Greaer of nnual Rache or nnual Roll-Up Deah Benefi is offered. ih his ind of opion he greaer of he roll-up benefi base and he annual rache benefi base bu a leas he curren accoun value is paid ou as he deah benefi. ih a roll-up rae of i6% insurers picall charge abou.6% p.a. for his guaranee (see e.g. JPMorgan (24. 2.3 Guaraneed Miniu Living Benefis I was no unil he lae 99s ha Guaraneed Miniu Living Benefis have been offered in he are. oda GMLB are ver popular. he wo earlies fors Guaraneed Miniu ccuulaion Benefis (GMB and Guaraneed Miniu Incoe Benefis (GMIB originaed alos a he sae ie. Boh guaranees offer he insured a guaraneed auri benefi i.e. a iniu benefi a he auri of he conrac. However wih he GMIB his guaranee onl applies if he accoun value is annuiized. Since 22 a new for of GMLB is offered he so-called Guaraneed Miniu ihdrawal Benefi (GMB. Here he insured is eniled o wihdraw a pre-specified aoun annuall even if he accoun value has fallen below his aoun. hese guaranees are exreel popular. In 24 69% of all Variable nnui conracs sold included a GMB opion. Each of he 5 larges Variable nnui providers offered his ind of guaranee a his ie (cf. Lehann Brohers (25. 2.3. Guaraneed Miniu ccuulaion Benefis (GMB Guaraneed Miniu ccuulaion Benefis are he siples for of guaraneed living benefis. Here he cusoer is eniled o a inial accoun value G a auri of he conrac. Usuall G is he single preiu P soeies a roll-up benefi base. he corresponding fees var beween.25% and.75% p.a. of he accoun value (cf. Mueller (26. 2.3.2 Guaraneed Miniu Incoe Benefis (GMIB auri of a Variable nnui wih a GMIB he policholder can as usual choose o obain he accoun value (wihou guaranee or annuiize he accoun value a curren are condiions (also wihou an guaranee. However he GMIB opion offers an addiional I choice: he policholder a annuiize soe guaraneed aoun G a annuiizaion raes ha have been specified up-fron. herefore his opion can also be inerpreed as a guaraneed annui saring a where he annui paens have alread been specified a. Noe ha if he accoun value a auri is below he guaraneed value I G he cusoer I G canno ae ou he guaraneed capial as a lup su bu onl in he for of an annui a he pre-specified annuiizaion raes. hus he opion is in he one a ie if he - 5 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies resuling annui paens exceed he annui paens resuling fro convering he acual accoun value a curren annui raes. I he guaraneed aoun G usuall is a roll-up benefi base wih e.g. i 5% or 6% or a rache benefi base. Soeies here is no one specified auri bu he policholder can annuiize wihin a cerain (ofen raher long ie period. he offered roll-up raes frequenl exceed he ris-free rae of ineres whereas he pre-specified annuiizaion facors are usuall raher conservaive. hus a auri he opion igh no be in he one even if he guaraneed aoun exceeds he accoun value. Furherore he pricing of hese guaranees is ofen based on cerain assupions abou he cusoers behavior raher han assuing ha everbod exercises he opion when i is in he one. Such assupions of course reduce he opion value. 5 Depending on he specific for of he guaranee he curren fees for GMIB conracs picall var beween.5% and.75% p.a. of he accoun value. 2.3.3 Guaraneed Miniu ihdrawal Benefis (GMB Producs wih a GMB opion give he policholder he possibili o wihdraw a specified aoun (usuall he single preiu in sall porions. picall he insured is eniled G o annuall wihdraw a cerain proporion x of his aoun even if he accoun value has fallen o zero. auri he policholder can ae ou or annuiize an reaining funds if he accoun value did no vanish due o such wihdrawals. Recenl several fors of so-called Sep-up GMB opions have been inroduced: ih one popular version he oal guaraneed aoun which can be wihdrawn is increased b a predefined raio a cerain poins in ie if no wihdrawals have been ade so far. In wha follows we will onl analze his for of Sep-up GMB. lernaivel here are producs in he are where a cerain poins in ie he reaining oal guaraneed aoun which can be wihdrawn is increased o he axiu of he old reaining guaraneed aoun and he curren accoun value. he laes developen in his area are so-called GMB for life opions where onl soe axiu aoun o be wihdrawn each ear is specified bu no oal wihdrawal aoun. his feaure can be analzed wihin our odel b leing and. Fro a financial poin of view GMB opions are highl coplex since he insured can decide a an poin in ie wheher and if so how uch o wihdraw. he are currenl offered for beween.4% and.65% p.a. of he accoun value. However Milevs and Salisbur (26 find ha hese guaranees are subsaniall underpriced. he conclude G G 5 Cf. Milevs and Salisbur (26. - 6 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies ha insurers eiher assue a subopial cusoer behavior or use charges fro oher (overpriced guaranees o cross-subsidize hese guaranees. hile his suar of GMDB and GMLB opions covers all he basic designs a coplee descripion of all possible varians would be beond he scope of his paper. hus soe producs offered in he are a have feaures ha differ fro he descripions above. 6 Our odel and noaion presened in he following Secion is designed o cover all he guaranees described in his Secion as special cases. Of course he underling general fraewor allows for an specific variaions of he guaranees ha igh deviae fro he producs described above. 3 General Valuaion Fraewor for Guaraneed Miniu Benefis 3. he Financial Mare s usual in his conex we assue ha here exiss a probabili space (ΩFQ equipped ( [ wih a filraion F I ] where Q is a ris-neural easure under which paen sreas can be valued as expeced discouned values. 7 iplies ha he financial are is arbirage free. e use a ban accoun nuéraire process which evolves according o Exisence of his easure also ( [ B ] as he db B r d B >. ( Here r denoes he shor rae of ineres a ie. e furher assue ha he underling uual fund S of he Variable nnui is odeled as a righ-coninuous F adaped sochasic process wih finie lef hand liis (RCLL. 8 In paricular he discouned asse process assue S B. S B [ ] is a Q-aringale. For convenience we 6 For curren inforaion regarding Variable nnui producs pes of guaranees and curren fees we refer e.g. o www.annuifi.co. 7 his is a consequence of he ris-neural valuaion forula cf. Bingha and Kiesel (24. 8 For our nuerical calculaions we assue ha S evolves according o a geoeric Brownian oion wih consan coefficiens. - 7 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies 3.2 Model for he Insurance Conrac ihin he following odel an ind of Variable nnui conrac which conains one or ore of he guaranees inroduced in Secion 2 can be represened. In our nuerical analsis we resric ourselves o conracs wih a os one GMDB and one GMLB opion. e consider a Variable nnui conrac wih a finie ineger auri which is aen ou a ie for a single preiu P. lhough he odel generall allows for flexible expiraion opions in order o siplif he noaion we onl consider a fixed auri. e denoe he accoun value b and ignore an up-fron charges. herefore we have P. During he er of he conrac we onl consider he charges which are relevan for he guaranees i.e. coninuousl deduced charges for he guaranees and a surrender fee. he surrender fee is charged for an wihdrawal of funds fro he conrac excep for guaraneed wihdrawals wihin a GMB opion. he coninuousl deduced guaranee fee ϕ is proporional o he accoun value and he surrender fee s is proporional o he respecive aoun wihdrawn. In order o valuae he benefis of he conrac we sar b defining wo virual accouns: denoes he value of he cuulaive wihdrawals up o ie. e will refer o i as he wihdrawal accoun. Ever wihdrawal is credied o his accoun and copounded wih he ris-free rae of ineres up o auri. ie zero we have. Siilarl b D we denoe he value of he deah benefis paid up o ie. nalogousl o he wihdrawals we credi deah benefi paens o his deah benefi accoun and copound he value of his accoun wih he ris-free rae unil ie. Since we assue he insured o be alive a ie zero we obviousl have D. In order o describe he evoluion of he conrac and he ebedded guaranees we also need he following processes: he guaraneed iniu deah benefi a ie is denoed b a ie is given b D { ; } G D G. hus he deah benefi ax. e le G D if he conrac conains one of he described GMDB opions (cf. Secion 2.2 oherwise we le G. he evoluion of over ie depends on he pe of he GMDB opion included in he conrac. I will be described in deail in Secion 3.3. he guaraneed auri benefi of he GMB opion is denoed b for possible changes of he guaranee over he er of he conrac we le D G D G. In order o accoun ( G [ ] represen he evoluion of his guaranee (see Secion 2.3. for deails. e have G for conracs wih one of he described GMB opions and G for conracs wihou a GMB opion. - 8 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies nalogousl we le I G denoe he guaraneed auri benefi ha can be annuiized in he case of a GMIB opion and odel is developen b I ( G [ ] G I I and G for conracs wih and wihou a GMIB opion respecivel.. lso we have G Finall o be able o represen GMB opions we inroduce he processes ( [ ] and E ( [ ]. denoes he reaining oal aoun ha can be wihdrawn afer ie and G E G G is he axiu aoun ha can be wihdrawn annuall due o he GMB opion. If he conrac conains a GMB we le G and G x where x is he porion of he preiu ha can be wihdrawn annuall. For conracs wihou GMB we le G E G Secion 3.3. E. he evoluion over ie of hese processes is also explained in deail in Due o he Marov-proper 9 of he underling processes all inforaion available a ie is copleel conained in he so-called sae variables E G and. o siplif noaion we inroduce he following sae vecor I D E ( D G G G G G. 3.3 Evoluion of he Insurance Conrac During he er of he conrac here are four possible pes of evens: he insured can wihdraw funds as a guaraneed wihdrawal of a GMB opion perfor a parial surrender i.e. wihdraw ore han he guaraneed wihdrawal aoun copleel surrender he conrac or pass awa. For he sae of siplici we assue ha all hese evens can onl occur a a polic anniversar dae. herefore a ineger ie poins 2... for all sae variables we disinguish beween ( and ( i.e. he value iediael before and afer he occurrence of such evens respecivel. he saring values a of all accouns and processes describing he conrac were given in Secion 3.2. Now we will describe heir evoluion in wo seps: Firs for 2... he developen wihin a polic ear i.e. fro o ( - is specified. Subsequenl we will D G I G D G G 9 See Secion 5.3.2 in Bingha and Kiesel (24. - 9 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies describe he ransiion fro ( - o ( which depends on he pe of guaranees included in he conrac and he occurrence of he described evens. Finall we describe he auri benefis of he conrac. 3.3. Developen beween and ( - s indicaed in Secion 3. he price of he underling uual fund evolves sochasicall over ie. hus aing ino accoun coninuous guaranee fees ϕ for he accoun value we have S ϕ e. (2 S he accouns and D are copounded wih he ris-free rae of ineres i.e. r s ds r s ds e and D D e. he developen of he processes and depends on he specificaion of he corresponding GMDB GMB and GMIB opion: if he corresponding guaraneed benefi is he D / / I single preiu or if he opion is no included we le G G D / / I benefi is a roll-up base wih roll-up rae i we se G G guaranees we have D / / I G G D G D / / I G. If he guaraneed ( i. For rache since he rache base is adjused afer possible wihdrawals and herefore considered in he ransiion fro ( - o ( (cf. Secion 3.3.2. G E G I G D / / I D / / I / E / E he processes and do no change during he ear i.e. G G. 3.3.2 ransiion fro ( - o ( he polic anniversar dae we disinguish four cases: a he insured dies wihin he period (] Since our odel onl allows for deah a he end of he ear ding wihin he period (] is equivalen o a deah a ie. he deah benefi is credied o he deah benefi accoun and will hen be copounded wih he ris-free rae unil auri : D D D ax{ G ; }. Since afer deah no fuure benefis are possible we le / I / / D / E as well as. he wihdrawal accoun where possible prior G wihdrawals have been colleced will no be changed i.e. copounded unil auri.. his accoun will be - -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies b he insured survives he ear (] and does no ae an acion (wihdrawal surrender a ie Here neiher he accoun D nor is changed. hus we have D D and / I / D have G G. For he GMB GMIB and GMDB wihou a rache pe guaranee we also / I / D. If however one or ore of hese guaranees are of rache pe { } / I / D / I / D we adjus he corresponding guaranee accoun b G ax G. ; If he conrac includes a GMB opion wih sep-up and is a sep-up poin he GMB processes are adjused according o he sep-up feaure bu onl if here were no pas wihdrawals: If denoes he facor b which he oal aoun o be wihdrawn is i increased (cf. Secion 2.3.3 we ge / E / E an oher case we have G G. ( E G G Ι { } i and G. In x G c he insured survives he ear (] and wihdraws an aoun wihin he liis of he GMB opion wihdrawal wihin he liis of he GMB is a wihdrawal of an aoun E in G G since he wihdrawn aoun a neiher exceed he axial annual { } E ; E G G wihdrawal aoun nor he reaining oal wihdrawal aoun. he accoun value is reduced b he wihdrawn aoun. In case he wihdrawn aoun exceeds he accoun value he accoun value is reduced o. hus we have ax { ; E }. lso he reaining oal wihdrawal aoun is reduced b he wihdrawn aoun i.e. G G E. Furherore he wihdrawn aoun is credied o he wihdrawal accoun: E. he axial annual wihdrawal aoun as well E E as he deah benefi accoun reain unchanged: G and D D. G Usuall living benefi guaranees (GMB and GMIB and in order o avoid adverse selecion effecs also he guaraneed deah benefis are reduced in case of a wihdrawal. e will resric our consideraions o a so-called pro raa adjusen. Here guaranees which are no of rache pe are reduced a he sae rae as he accoun value i.e. - -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies / I / D / I / D G G. If one or ore of he guaranees are of rache pe for he respecive guaranees we le G / I / D ax G. / I / D ; d he insured survives he ear (] and wihdraws an aoun exceeding he liis of he GMB opion firs noe ha his case includes he following cases as special cases: d he conrac does no coprise a GMB opion and an aoun E is wihdrawn. < < d2 GMB opion is included in he conrac bu he insured wihdraws an aoun E E wih in { G G }. < < E > ; d3 he insured surrenders b wihdrawing he aoun E. e le 2 E E E E where in { G G }. Consequenl is he porion of E ; he wihdrawal wihin he liis of he GMB opion. If he conrac does no include a E GMB opion we obviousl have. E s in case c he accoun value is reduced b he aoun wihdrawn i.e. E and he wihdrawn aoun is credied o he wihdrawal accoun. However he insured has o pa a surrender fee for he second coponen which leads o 2 E E ( s. he deah benefi accoun reains unchanged i.e. D D. Besides pro raa adjusens here are also reducions b he so-called dollar ehod. Here all he respecive / I / D / I / D processes are reduced b he wihdrawn aoun i.e. G ax[ G E ]. In order o odel and evaluae producs where he dollar ehod or an oher reducion schee applies he respecive forulas can be adjused. E If he conrac coprises a GMB opion and if { G G } ; < in as well as G hen a wihdrawal of E is wihin he liis of he GMB and does no lead o a surrender of he conrac. However his case is covered b case c. - 2 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies gain he fuure guaranees are odified b he wihdrawal: For he guaranees which are no of rache pe we have / I / D / I / D G G whereas for he rache pe guaranees we le G / I / D ax G. / I / D ; E For conracs wih a GMB wihdrawing an aoun { G G } E > ; fuure guaraneed wihdrawals. e consider a coon ind of GMB 2 guaraneed fuure wihdrawals are reduced according o in also changes opion where he G in G E ; G i.e. he wihdrawal aoun is reduced b he higher of a pro raa reducion and a reducion according o he dollar ehod. For fuure annual guaraneed aouns we use E E G G. 3 3.3.3 Mauri Benefis a If he conrac neiher coprises a GMIB nor a GMB opion he auri benefi L is sipl he accoun value i.e. L. In conracs wih a GMB opion he survival benefi { } a auri is a leas he GMB hus L ax ; G. Insured holding a GMIB opion can decide wheher he wan a lup su paen of he accoun value or annuiize his aoun a curren annuiizaion raes. lernaivel he can annuiize he guaraneed annuiizaion aoun a pre-specified condiions. If we denoe 4 b ä and ä he annui facors when annuiizing a he curren and he curren guar guaraneed pre-specified condiions respecivel he value of he guaraneed benefi a I ä curren auri is given b G. hus a financiall raional acing cusoer will chose he ä guar annui whenever we have G I ä ä curren guar I I ä curren is given b L ax ; G. ä guar >. herefore he value of he benefi a ie 2 Cf. Pioneer (25 pp. 36. 3 Cf. Pioneer (25 page 36f. lso a reducion of he for G E E G G is frequenl offered. G 4 Here an annui facor is he price of an annui paing one dollar each ear. - 3 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies If he conrac conains boh a GMB and a GMIB opion he auri value of he conrac I L ax L ; L. is { } 3.4 Conrac Valuaion e ae he coon assupion ha financial ares and bioeric evens are independen. Furherore we assue ris-neurali of he insurer wih respec o bioeric riss (cf. ase and Persson (994. hus he ris-neural easure for he cobined are (insurance and financial are is he produc easure of Q and he usual easure for bioeric riss. In order o eep he noaion siple in wha follows we will also denoe his produc easure b Q. Even if ris-neurali of he insurer wih respec o bioeric ris is no assued here are sill reasons o eplo his easure for valuaion purposes (see Møller (2. Le x be he insured s age a he sar of he conrac and p x denoe he probabili for a x -ear old o survive ears. B q x we denoe he probabili for a ( x -ear old o die wihin he nex ear. he probabili ha he insured passes awa in he ear (] is hus given b p x q x. he liiing age is denoed b ω i.e. survival beond age ω is no possible. 3.4. Valuaion under Deerinisic Policholder Behavior firs we assue ha he policholder s decisions (wihdrawal/surrender are deerinisic i.e. we assue here exiss a deerinisic sraeg which can be described b ( ( a wihdrawal vecor IR 5 ξ ξ;...; ξ. Here ξ denoes he aoun o be wihdrawn a he end of ear if he insured is sill alive and if his aoun is adissible. If he aoun ξ is no adissible he larges adissible aoun E < ξ is wihdrawn. In paricular if he conrac does no conain a GMB opion he larges adissible aoun is E { } in ξ ;. full surrender a ie is represened b ξ. B ( Ψ Ψ Ψ IR... we denoe he se of all possible deerinisic sraegies. In paricular ever deerinisic sraeg is F -easurable. If a paricular conrac and a deerinisic sraeg are given hen under he assupion ha he insured dies in ear { 2... ω x } he auri-values L ;ξ ;ξ and D ;ξ are specified for each pah of he soc price S. hus he ie zero value including all opions is given b: 5 Here denoes he non negaive real nubers (including zero; furherore we le IR IR IR { }. - 4 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies. ; ; ; ; ; ; ; ; ; ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ω D L e E p D L e E q p D L e E q p V ds r Q x ds r Q x x x ds r Q x x s s s (3 3.4.2 Valuaion under Probabilisic Policholder Behavior B probabilisic policholder behavior we denoe he case when he policholders follow cerain deerinisic sraegies wih cerain probabiliies. If hese deerinisic sraegies and he respecive probabiliies are nown ( he value of he conrac under probabilisic policholder behavior is given b ( ( j j j IR ( ( ( ;...;ξ ξ ξ n j... 2 ( j p ξ n j j p ( ξ ( ( j n j j V p V ξ ξ. (4 his value also adis anoher inerpreaion: if he insurer has derived cerain forecass for he policholders fuure behavior wih respec o wihdrawals and surrenders and assigns he respecive relaive frequencies as probabiliies o each conrac hen he su of he probabilisic conrac values consiues exacl he value of he insurer s whole porfolio given ha he forecas is correc. hus his cuulaive value equals he coss for a perfec hedge of all liabiliies if policholders behave as forecased. However in his case he ris ha he acual clien behavior deviaes fro he forecas is no hedged. 3.4.3 Valuaion under Sochasic Policholder Behavior ssuing a deerinisic or probabilisic cusoer behavior iplies ha he wihdrawal and surrender behavior of he policholders does no depend on he evoluion of he capial are or equivalenl on he evoluion of he conrac over ie. sochasic sraeg on he oher hand is a sraeg where he decision wheher and how uch one should be wihdrawn is based upon he inforaion available a ie. hus an adissible sochasic sraeg is a discree F easurable process (X which deerines he aoun o be wihdrawn depending on he sae vecor. hus we ge: ( Ε X.... 2 For each sochasic sraeg (X and under he hpohesis ha he insured deceases in ear { x } 2...ω he values ( ;(X L ( ;(X and ( ;(X D are specified for an given pah of he process S. herefore he value of he conrac is given b: - 5 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies e le V ω x r ds x x Q s ( (X p q E e ( L ( (X ( (X D ( (X. (5 Ξ denoe he se of all possible sochasic sraegies. hen he value V of a conrac assuing a raional policholder is given b V supv ((X. (6 (X Ξ 4 Nuerical Valuaion of Guaraneed Miniu Benefis For our nuerical evaluaions we assue ha he underling uual fund evolves according o a geoeric Brownian oion wih consan coefficiens under Q i.e. ds S rd σ dz S (7 where r denoes he (consan shor rae of ineres. hus for he ban accoun we have r B e. Since he considered guaranees are pah-dependen and raher coplex i is no possible o find closed for soluions for heir ris-neural value. herefore we have o rel on nuerical ehods. e presen wo differen valuaion approaches: in Secion 4. we presen a siple Mone Carlo algorih. his algorih quicl produces accurae resuls for a deerinisic probabilisic or a given F easurable sraeg. However Mone Carlo ehods are no preferable o deerine he price for a raional policholder. hus in Secion 4.2 we inroduce a discreizaion approach which addiionall enables us o deerine prices under opial policholder behavior. 4. Mone-Carlo Siulaion 8 Le (X : IR IR IR a F easurable wihdrawal sraeg. B Iô s forula (see e.g. Bingha and Kiesel (24 we obain he ieraion 2 S ϕ σ e exp r z ; z ~ S ϕ 2 σ N ( iid which can be convenienl used o produce realizaions of saple pahs of he 6 underling uual fund using Mone Carlo Siulaion. For an conrac conaining Guaraneed Miniu Benefis for an saple pah and for an ie of deah we obain ( j a 6 For an inroducion o Mone Carlo ehods see e.g. Glasseran (23. - 6 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies he evoluion of all accouns and processes eploing he rules of Secion 3. Hence ( j ( j ( j realizaions of he benefis l ( (X w ( (X d ( (X a ie given ha he insured dies a ie are uniquel defined in his saple pah. hus he ie zero value of hese benefis in his saple pah is given b v [ (X ] ω ( x ( j r ( j ( j ( j (X p x q ( ( ( x l (X w (X d J ( i Hence V ( v ( e. (X (X is a Mone-Carlo esiae for he value of he conrac J j where J denoes he nuber of siulaions. However for he evaluaion of a conrac under he assupion of raional policholders following an opial wihdrawal sraeg Mone-Carlo siulaions are no preferable. 4.2 Mulidiensional Discreizaion pproach ansanen and Luarinen (24 presen a valuaion approach for paricipaing life insurance conracs including a surrender opion which is based on discreizaion via a finie esh. e exend and generalize heir approach in several regards: we have a ulidiensional sae space and hus need a ulidiensional inerpolaion schee. In addiion heir odel does no include fees. herefore we odif he odel such ha he guaranee fee ϕ and he surrender fee s can be included. Finall wihin our approach a sraeg does no onl consis of he decision wheher or no o surrender. e raher have an infinie aoun of possible wihdrawal aouns in ever period. Even hough we are no able o include all possible sraegies in a finie algorih we sill need o consider nuerous possible wihdrawal sraegies. e sar his Secion b presening a quasi-analic inegral soluion o he valuaion proble of Variable nnuiies conaining Guaraneed Miniu Benefis. Subsequenl we show how in each sep he inegrals can be approxiaed b a discreizaion schee which leads o an algorih for he nuerical evaluaion of he conrac value. e resric he presenaion o he case of a raional policholder i.e. we assue an opial wihdrawal sraeg. However for deerinisic probabilisic or sochasic wihdrawal sraegies he approach wors analogousl afer a sligh odificaion of he funcion F ~ in Secion 4.2.3. 4.2. quasi-analic soluion he ie value V of a conrac depends solel on he sae variables a ie I D E ( D G G G G G. Since besides he sae variables change deerinisicall beween wo polic anniversaries he value process V is a funcion of and he sae vecor a he las polic anniversar ( i.e. V V ;. - 7 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies he discree poins in ie 2... we disinguish he value righ before deah ( benefi paens and wihdrawals V V ; and he value righ afer hese evens ( V V. If he insured does no die in he period ( ] E o he nowledge of he wihdrawal aoun and he accoun value deerine he developen of he sae variables fro (. e denoe he corresponding ransiion funcion b Siilarl b f ( ( ( ] wihin. f ( ( E we denoe he ransiion funcion in case of deah B siple arbirage arguens (cf. ansanen and Luarinen (24 we can conclude ha V is a coninuous process. Furherore wih Iô s forula (see e.g. Bingha and Kiesel (24 one can show ha he value funcion τ V for all [ τ saisfies a Blac-Scholes parial differenial equaion (PDE which is slighl odified due o he exisence of he fees ϕ. Hence here exiss a funcion v : IR IR IR wih V ( τ a v ( τ a τ [ a IR and v saisfies he PDE. 2 2 2 d v dv 2σ a 2 dv rv dτ d a da ( r ϕ a (8 wih he boundar condiion v ( a ( q V ( f ( a q V ( f ( a x E x which in paricular is dependen on he insured s survival. a IR hus we can deerine he ie-zero value of he conrac V b he following bacward ieraion: : V L D. auri we have ( -: Le ( V a ie (- be nown for all possible values of he sae vecor. hen he ie (- value of he conrac is given b he soluion v ( a of he PDE (8 wih boundar condiion - 8 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies v ( a ( q sup V ( f ( a q V ( f ( a. x E x E IR r ϕ 2 2 soluion of he PDE (8 can be obained b defining υ : ρ : σ υ r and σ 2 2 2 g ( x x e σ υ ρτ τ v σx ( τ e. hen diensional hea equaion li g ( τ ( σx v ( e xυ ρ τ x e σ and g saisfies a one- 2 d g dg (9 2 2 dx d a soluion of which is given b 7 2 ( x u (( τ g ( τ x exp g ( u du. ( 2π (( τ 2 hus we have 2 ( logλ (( τ ρ (( τ υ v ( a e exp λ v 2 2 2π (( τ σ 2 σ ( λa dλ. ( B subsiuing 2 λ( u exp σ u r ϕ σ we obain 2 V ( ( ( sup ( ( ( q x V f E u r E ir e u Φ q V ( f ( λ( u x λ (2 du where Φ denoes he cuulaive disribuion funcion of he sandard noral disribuion. 4.2.2 Discreizaion via a Finie Mesh In general he inegral (2 canno be evaluaed analicall. herefore we have o rel on nuerical ehods o find an approxiaion of he value funcion on a finie esh. Here a finie esh is defined as follows: Le ( 8 Y IR be he se of al possible sae vecor values. e denoe a finie se of possible values for an of he eigh sae variables as a se of esh basis values. Le a se of esh basis values for each of he eigh sae variables be given. Provided ha he Caresian produc of hese eigh ses is a subse of Y we denoe i b 7 Cf. heore 3.6 of chaper 4 Karazas and Shreve (99. - 9 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies Y Grid and call i a Y -esh or sipl a esh or a grid. n eleen of is called a grid poin. For a given grid we ierae he evaluaion bacwards saring a. auri he value funcion is given b: Grid Grid ( Grid D L V. e repea he ieraion sep described above ies and hereb obain he value of he conrac a ever ineger ie poin for ever grid poin. In paricular we obain he ie zero value of he conrac V. ihin each ie period we have o approxiae he inegral ( wih he help of nuerical ehods. his will be described in he following Secion. 4.2.3 pproxiaion of he Inegral Following ansanen and Luarinen (24 for IR a and a given sae vecor we define he funcion ( ( ( ( ( (. sup ~ x E IR E x a f V q a f V q a F du u F u e (3 hus (2 is equivalen o ( ( r Φ ( ~ ( λ Grid for V where 2 2 exp ( σ ϕ σ λ r u u as above. In order o evaluae he inegral we evaluae he funcion ( a F ~ for each and for a selecion of possible values of he variables a. In beween we inerpolae linearl. Grid hus le and Grid ax > a axial value for a be given. e spli he inerval in M subinervals via [ ax ] { } M M... 2 : ax α. Le. hen for an can be approxiaed b ( F ~ α γ IR a ( a F ~ ( ( [ ( [ [ ] [ [ ] [ ( ( ( ( ~ ax ax a b a b a b a b a a a a a F M M M M M M M M M M Ι Ι Ι Ι α α α α γ γ α α α γ γ γ α α α γ - 2 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies where b γ ( γ γ... M ; M b M... M ; b M b M and I denoes he indicaor funcion. hus we have V b and b ( γ γ M ax M ϕ r ( a [ ( Φ( Φ( ( Φ( a e b u σ u σ b e u Φ( u ] ax r ϕ σ where u u log and u M. σ M a σ σ 2 Defining b b we obain V ( M ϕ r e ( b b ( Φ( u σ e ( b b Φ( u [ ( ]. Hence i suffices o deerine he values γ F ( α { 2 M } ~.... hen deerining he γ heoreicall he funcion f E has o be evaluaed for an possible wihdrawal aoun E. For our ipleenaion we resric he evaluaion o a finie ~ aoun of relevan values E-. Furherore due o he definiion of (see (3 i is necessar o evaluae V afer he ransiion of he sae vecor fro ( o (. Since he sae vecor and hus he arguens of he funcion are no necessaril eleens of Grid V ( has o be deerined b inerpolaion fro he surrounding esh poins. F e inerpolae linearl in ever diension. Due o he high diensionali of he proble he copuaion ie highl depends on he inerpolaion schee. In order o reduce calculaion ie and he required eor capaci we reduced he diensionali b onl considering he relevan accouns for he considered conracs. In paricular when he deah benefi accoun D is sricl posiive i.e. if he insured has died before ie he accoun value will be zero. Conversel as long as is greaer han zero D reains zero i.e. he insured is sill alive a ie. hus he diensionali can alwas be reduced b one. Furherore in our nuerical analses we onl consider conracs wih a os one GMDBopion and a os one GMLB-opion. herefore b onl considering he relevan sae variables we can furher reduce he diensionali o a axiu of 4. However for a conrac wih er o auri of 25 ears depending on he specific ind of conrac he calculaion of one conrac value under opial policholder sraeg on a single CPU sill aes beween 5 and 4 hours. - 2 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies 5 Resuls e use he nuerical ehods presened in Secion 4 o calculae he ris-neural value of Variable nnuiies including Guaraneed Miniu Benefis for a given guaranee fee ϕ. e call a conrac and also he corresponding guaranee fee fair if he conrac s ris-neural value equals he single preiu paid i.e. if he equilibriu condiion P ( ϕ V V holds. Unless saed oherwise we fix he ris-free rae of ineres r 4% he volaili σ 5% he conrac er 25 ears he single preiu aoun P he age of he insured x 4 he sex of he insured ale he surrender fee s 5% and use bes esiae orali ables of he Geran socie of acuaries (DV 24 R. For conracs wihou GMB we analze wo possible policholder sraegies: Sraeg assues ha cliens neiher surrender nor wihdraw one fro heir accoun. Sraeg 2 assues deerinisic surrender probabiliies which are given b 5% in he firs polic ear 3% in he second and hird polic ear and % hereafer. In addiion we calculae he risneural value of soe policies assuing raional policholders. For conracs wih GMB we assue differen sraegies which are described in Secion 5.2.4. 5. Deerining he fair Guaranee Fee In a firs sep we analze he influence of he annual guaranee fee on he value of conracs including hree differen inds of GMB opions. For conrac he guaraneed auri value is he single preiu (one-bac guaranee conrac 2 guaranees an annual rache base whereas a roll-up base a a roll-up rae of i 6% is considered for conrac 3. Figure shows he corresponding conrac values as a funcion of he annual guaranee fee assuing neiher surrenders nor wihdrawals. - 22 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies 8 6 4 2 8 6 4.%.5%.%.5% 2.% 2.5% 3.% 3.5% 4.% 4.5% 5.% 6% Roll-Up annual rache one-bac guaranee preiu Figure : Conrac value as a funcion of he annual guaranee fee For conrac a guaranee fee of ϕ.7% leads o a fair conrac. he fair guaranee fee increases o.76% in he rache case. he ris-neural value of conrac 3 exceeds for all values of ϕ. hus under he given assupions here exiss no fair guaranee fee for a conrac including a 6% roll-up GMB. s a consequence such guaranees can onl be offered if he guaranee coss are subsidized b oher charges or if irraional policholder behavior is assued in he pricing of he conrac. 5.2 Fair Guaranee Fees for Differen Conracs 5.2. Conracs wih a GMDB Opion e analze hree differen conracs wih a iniu deah benefi guaranee. Conrac provides a one-bac guaranee in case of deah conrac 2 an annual rache deah benefi and conrac 3 a 6% roll-up benefi. - 23 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies able shows fair guaranee fees for hese conracs under he wo policholder sraegies described above. conrac sraeg : no wihdrawals or surrenders 2: deerinisic surrender probabili Mone-bac guaranee Rache benefi base 6% roll-up benefi base.%.4%.4% < % < %.5% able : Fair guaranee fee for conracs wih GMDB under differen consuer behavior ssuing ha cusoers neiher surrender heir conracs nor wihdraw an one before auri he fair guaranee fee for all hese conracs is raher low. However he guaraneed deah benefi included in conrac 3 is significanl ore expensive han he oher guaranees. If policholders surrender heir conracs a he surrender raes assued in sraeg 2 he fair guaranee fee srongl decreases for wo reasons: Policholders pa fees before surrendering bu will no receive an benefis fro he corresponding opions. Secondl surrender fees can be used o subsidize he guaranees of he cliens who do no surrender. For conracs and 2 surrender fees exceed he value of he reaining cliens opions. hus he ris-neural value of he conrac exceeds he single preiu even if no fee is charged for he opion. hus our resuls are consisen wih Milevs and Posner (2 who find ha GMDB opions are generall overpriced in he are. Overall he guaranee fees are raher low since a benefi paen is onl riggered in he even of deah. here is no possibili for raional consuer behavior in ers of exercising he opion when i is in he one. he onl wa of raional policholder behavior is surrendering a conrac when he opion is far ou of he one: I is opial o surrender he conrac if he expeced presen value of fuure guaranee fees exceeds he value of he opion plus he surrender fee. However for he considered surrender charge of 5% surrendering a conrac is alos never opial. hus he conrac value for a raional policholder hardl differs fro he value under sraeg. However for lower surrender charges policholder behavior would be ore iporan. 5.2.2 Conracs wih a GMB Opion e analze hree differen conracs wih a iniu accuulaion benefi guaranee. gain conrac provides a one-bac guaranee a he end of he accuulaion phase conrac 2 an annual rache guaranee and conrac 3 a 6% roll-up benefi base. he value of hese conracs under policholder sraeg has been displaed as a funcion of ϕ in Figure above. - 24 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies able 2 shows he fair guaranee fee for hese hree conracs under he wo given policholder sraegies. In addiion we show he fair guaranee fee if an addiional 6% rollup deah benefi is included (coluns wih DB. conrac Mone-bac guaranee Rache benefi base 6% roll-up benefi base sraeg w/o DB wih DB w/o DB wih DB w/o DB wih DB : no wihdrawals or.7%.23%.76%.94 --- --- surrenders 2: deerinisic surrender probabili < %.2%.57%.74% --- --- able 2: Fair guaranee fee for conracs wih GMB under differen consuer behavior he fair guaranee fees for he conracs differ significanl. For he one-bac guaranee he fair guaranee fee is below.25% even if he GMDB opion is included. he fee for he rache guaranee is significanl higher. Even under sraeg 2 and wihou addiional deah benefi i exceeds.5%. In an case he fair guaranee fee of he rache guaranee is a leas four ies as high as he corresponding fair guaranee fee of he one-bac guaranee. 6% roll-up GMB canno be offered even under he assued surrender paern. he addiional fee for deah benefi (difference beween coluns wih DB and w/o DB alwas exceeds he fair guaranee fee of he pure deah benefi guaranee shown in able and is hardl reduced b he assued surrenders. Furher analses showed ha raional policholder behavior hardl influences he ris-neural value of he conracs: he values under opial policholder behavior are ver close o he values under sraeg (no surrender or wihdrawal. his is no surprising since for he one-bac guaranee surrender is rarel opial due o he raher high surrender charges. In he case of a rache guaranee he acual guaranee level is annuall adjused o a poeniall increasing fund value. hus he guaranee is alwas a or in he one a a polic anniversar dae. However as explained above surrendering is usuall onl opial if he opion is ou of he one. 5.2.3 Conracs wih a GMIB Opion GMIB opion gives he policholder he possibili o annuiize he iniu benefi base a an annui facor ha is fixed a. heher or no he opion is in he one depends on boh he fund value and he raio of he guaraneed annui facor and he curren annui facor a annuiizaion. Usuall he guaraneed annui facor is calculaed based on ä curren conservaive assupions which are supposed o lead o a raio ä : <. However ä guar increasing longevi and decreasing ineres raes a change his raio during he er of he conrac and ae he guaranee exreel valuable a annuiizaion. - 25 -
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies e analze hree differen GMIB-conracs for differen values of ä. gain he iniu benefi base for conrac is he single preiu conrac 2 includes an annual rache guaranee whereas conrac 3 coes wih a 6% roll-up benefi base. he hree conracs are analzed wih and wihou he addiional GMDB opion fro he previous Secion. he respecive fair guaranee fees are shown in able 3. conrac Mone-bac guaranee Rache benefi base 6% roll-up benefi base sraeg w/o DB wih DB w/o DB w/o DB wih DB w/o DB : no ä.2.4%.3%.55%.83% --- --- wihdrawals ä..7%.23%.76%.94% --- --- or surrenders ä.8.3%.8%.25%.4% --- --- ä.6.%.6%.5%.9% 2.32% 3.76% 2: ä.2.4%.8%.24%.4% --- --- deerinisic ä. < %.2%.57%.74% --- --- surrender ä.8 < %.%.5%.29% > 4% > 4% probabili ä.6 < %.8% < %.%.45%.88% able 3: Fair guaranee fee for conracs wih GMIB under differen consuer behavior Obviousl for ä he fair guaranee fees are he sae as for he corresponding GMB opions. he value of he guaranee highl depends on he value of ä. Since bes esiaes abou fuure orali raes are subjec o high uncerain his assupion bears a significan ris for he insurer ha canno be hedged wih exising financial insruens. he difference beween he fair guaranee fee wih or wihou surrender is huge. hus basing he produc calculaion on esiaes abou fuure policholder behavior bears a significan non-diversifiable ris for he insurer. For an ä he values of he hree conrac pes differ considerable. Under sraeg here is no fair guaranee fee for a conrac wih 6% roll-up guaranee for ä.8 i.e. he expeced presen value of he guaraneed annuiies exceeds he single preiu. For ä.6 he fair guaranee fee equals 2.32% and is uch higher han pical charges for hese opions in he are. Even under sraeg 2 he fair guaranee fee is abou wice as high as he opion price observed in he are. hus here is evidence ha insurers base heir calculaions no onl on he assupion of irraional surrender behavior. he a also assue oher irraionaliies e.g. ha policholders ae he lup su paen (i.e. he accoun value wihou guaranee even if he annuiizaion opion is in he one. For he reason described in Secion 5.2.2 here is alos no difference beween raional policholder behavior and sraeg for conracs wih a one-bac or a rache guaranee. However in he case of a 6% roll-up benefi base raional policholder behavior increases he fair guaranee fee fro 2.32% o over 4%. hus here have o be an scenarios where i is opial o surrender he conrac i.e. he expeced presen value of fuure guaranee fees exceeds he value of he opion plus he surrender fee. - 26 -