A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *

Transcription

1 A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone: ( , Fax: ( DBauer@gsu.edu Alexander Kling Insiu für Finanz- und Akuarwissenschafen Helmholzsraße 22, 898 Ulm, Germany Phone: 49 ( , Fax: 49 ( A.Kling@ifa-ulm.de Jochen Russ Insiu für Finanz- und Akuarwissenschafen Helmholzsraße 22, 898 Ulm, Germany Phone: 49 ( , Fax: 49 ( J.Russ@ifa-ulm.de Absrac Variable Annuiies wih embedded guaranees are very popular in he US marke. here exiss a grea variey of producs wih boh, guaraneed minimum deah benefis (GMDB and guaraneed minimum living benefis (GMLB. Alhough several approaches for pricing some of he corresponding guaranees have been proposed in he academic lieraure, here is no general framework in which he exising variey of such guaranees can be priced consisenly. he presen paper fills his gap by inroducing a model, which permis a consisen and exensive analysis of all ypes of guaranees currenly offered wihin Variable Annuiy conracs. Besides a valuaion assuming ha he policyholder follows a given sraegy wih respec o surrender and wihdrawals, we are able o price he conrac under opimal policyholder behavior. Using boh, Mone-Carlo mehods and a generalizaion of a finie mesh discreizaion approach, we find ha some guaranees are overpriced, whereas ohers, e.g. guaraneed annuiies wihin guaraneed minimum income benefis (GMIB, are offered significanly below heir risk-neural value. * he auhors hank Hans-Joachim Zwiesler for useful insighs and commens. Corresponding auhor - -

2 Inroducion Variable Annuiies, i.e. deferred annuiies ha are fund-linked during he defermen period were inroduced in he 97s in he Unied Saes (see Sloane (97. Saring in he 99s, insurers included cerain guaranees in such policies, namely guaraneed minimum deah benefis (GMDB as well as guaraneed minimum living benefis (GMLB. he GMLB opions can be caegorized in hree main groups: Guaraneed minimum accumulaion benefis (GMAB provide a guaraneed minimum survival benefi a some specified poin in he fuure o proec policyholders agains decreasing sock markes. Producs wih guaraneed minimum income benefis (GMIB come wih a similar guaraneed value G a some poin in ime. However, he guaranee only applies if his guaraneed value is convered ino an annuiy using given annuiizaion raes. hus, besides he sandard possibiliies o ake he marke value of he fund unis (wihou guaranee or conver he marke value of he fund unis ino a lifelong annuiy using he curren annuiy conversion raes a ime, he GMIB opion gives he policyholder a hird choice, namely convering some guaraneed amoun G ino an annuiy using annuiizaion raes ha are fixed a incepion of he conrac (. he hird kind of guaraneed minimum living benefis are so-called guaraneed minimum wihdrawal benefis (GMWB. Here, a specified amoun is guaraneed for wihdrawals during he life of he conrac as long as boh he amoun ha is wihdrawn wihin each policy year and he oal amoun ha is wihdrawn over he erm of he policy say wihin cerain limis. Commonly, guaraneed annual wihdrawals of up o 7% of he (single up-fron premium are guaraneed under he condiion ha he sum of he wihdrawals does no exceed he single premium. hus, i may happen ha he insured can wihdraw money from he policy, even if he value of he accoun is zero. Such guaranees are raher complex since he insured has a broad variey of choices. Mos of he earlier lieraure on Variable Annuiies, e.g., Renz Jr. (972 or Greene (973 is empirical work dealing wih produc comparisons raher han pricing issues. I was no unil recenly, ha he special ypes of guaranees were discussed by praciioners (cf. JPMorgan (24, Lehman Brohers (25, or analyzed in he academic lieraure. Milevsky und Posner (2 price various ypes of guaraneed minimum deah benefis. hey presen closed form soluions for his ianic Opion 3 in case of an exponenial moraliy law and numerical resuls for he more realisic Gomperz-Makeham law. hey find ha in general hese guaranees are overpriced in he marke. In Milevsky und Salisbury (22, a model for he valuaion of cerain GMLB and GMDB opions is presened in a framework where he insured has he possibiliy o parially surrender he policy. he auhors call his a Real Opion o Lapse 4. hey presen closed 3 he auhors denoe his opion as ianic Opion since he paymen srucure falls beween European and American Opions and he paymen is riggered by he decease of he insured. 4 heir Real Opion is a financial raher han a real opion in he classical sense (cf. Myers (

3 form soluions in he case of an exponenial moraliy law, consan surrender fees and no mauriy benefis. I is shown ha boh, he value and he opimal surrender sraegy, are highly dependen on he amoun of he guaranee and of he surrender fee. Ulm (26 addiionally considers he real opion o ransfer funds beween fixed and variable accouns, and analyzes he impac of his opion on he GMDB rider and conrac as a whole, respecively. In Milevsky und Salisbury (26, he same auhors price GMWB opions. Besides a saic approach, where deerminisic wihdrawal sraegies are assumed, hey calculae he value of he opion in a dynamic approach. Here, he opion is valuaed under opimal policyholder behavior. hey show ha under realisic parameer assumpions opimally a leas he annually guaraneed wihdrawal amoun should be wihdrawn. Furhermore, hey find ha such opions are usually underpriced in he marke. In spie of hese approaches for he pricing of several opions offered in Variable Annuiies, here is no general framework in which he exising variey of such opions can be priced consisenly and simulaneously. he presen paper fills his gap. In paricular, we presen a general framework in which any design of opions and guaranees currenly offered wihin Variable Annuiies can be modeled. Asides from he valuaion of a conrac assuming ha he policyholder follows a given sraegy wih respec o surrender and wihdrawals, we are also able o deermine an opimal wihdrawal and surrender sraegy, and price conracs under his raional sraegy. he res of he paper is organized as follows. In Secion 2, we give a brief overview over he exising forms of guaranees in Variable Annuiies. Secion 3 inroduces he general pricing framework for such guaranees. We show how any paricular conrac can be modeled wihin his framework. Furhermore, we explain how a given conrac can be priced assuming boh, deerminisic wihdrawal sraegies and opimal sraegies. he laer is referred o as he case of raional policyholders. Due o he complexiy of he producs, in general here are no closed form soluions for he valuaion problem. herefore, we have o rely on numerical mehods. In Secion 4, we presen a Mone Carlo algorihm as well as a discreizaion approach based on generalizaions of he ideas of anskanen und Lukkarinen (24. he laer enables us o price he conracs under he assumpion of raional policyholders. Our resuls are presened in Secion 5. We presen he values for a variey of conracs, analyze he influence of several parameers and give economic inerpreaions. Secion 6 closes wih a summary of he main resuls and an oulook for fuure research. 2 Guaraneed Minimum Benefis his Secion inroduces and caegorizes predominan guaranees offered wihin Variable Annuiy conracs. Afer a brief inroducion of Variable Annuiies in general in Secion 2., we dwell on he offered Guaraneed Minimum Deah Benefis (Secion 2.2 and Guaraneed Minimum Living Benefis (Secion 2.3. We explain he guaranees from he cusomer s poin of view and give an overview over fees ha are usually charged

4 2. Variable Annuiies Variable Annuiies are deferred, fund-linked annuiy conracs, usually wih a single premium paymen up-fron. herefore, in wha follows we resric ourselves o single premium policies. When concluding he conrac, he insured are frequenly offered opional guaranees, which are paid for by addiional fees. he single premium P is invesed in one or several muual funds. We call he value A of he insured s individual porfolio he insured s accoun value. Cusomers can usually influence he risk-reurn profile of heir invesmen by choosing from a selecion of differen muual funds. All fees are aken ou of he accoun by cancellaion of fund unis. Furhermore, he insured has he possibiliy o surrender he conrac, o wihdraw a porion of he accoun value (parial surrender, or o annuiize he accoun value afer a minimum erm. he following echnical erms are needed o describe he considered guaranees: he rache benefi base a a cerain poin in ime is he maximum of he insured s accoun value a cerain previous poins in ime. Usually, i denoes he maximum value of he accoun on all pas policy anniversary daes. his special case is also referred o as annual rache benefi base. In order o simplify noaion, in wha follows, we only consider producs wih annual rache guaranees. Furhermore, he roll-up benefi base is he heoreical value ha resuls from compounding he single premium P wih a consan ineres rae of i % p.a. We call his ineres rae he roll-up rae. 2.2 Guaraneed Minimum Deah Benefis If he insured dies during he defermen period, he dependans obain a deah benefi. When Variable Annuiies were inroduced, a very simple form of deah benefi was predominan in he marke. However, since he mid 99s, insurers sared o offer a broad variey of deah benefi designs (cf. Lehmann Brohers (25. he basic form of a deah benefi is he so-called Reurn of Premium Deah Benefi. Here, he maximum of he curren accoun value a ime of deah and he single premium is paid. he price for his kind of benefi usually is already included in he charges of he conrac, i.e. his opion is available wihou addiional charges. Anoher varian is he Annual Roll-Up Deah Benefi. Here, he deah benefi is he maximum of he roll-up benefi base (ofen wih a roll-up rae of 5% or 6% and he accoun value. A ypical fee for ha deah benefi wih a roll-up rae of 6% is approximaely.25% p.a. of he accoun value (see, e.g., JPMorgan (24. If he conrac conains an Annual 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 ype of deah benefi are similar

5 Furhermore, he varian Greaer of Annual Rache or Annual Roll-Up Deah Benefi is offered. Wih his kind 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. Wih a roll-up rae of i6%, insurers ypically charge abou.6% p.a. for his guaranee (see, e.g., JPMorgan ( Guaraneed Minimum Living Benefis I was no unil he lae 99s ha Guaraneed Minimum Living Benefis have been offered in he marke. oday, GMLB are very popular. he wo earlies forms, Guaraneed Minimum Accumulaion Benefis (GMAB and Guaraneed Minimum Income Benefis (GMIB originaed almos a he same ime. Boh guaranees offer he insured a guaraneed mauriy benefi, i.e. a minimum benefi a he mauriy of he conrac. However, wih he GMIB, his guaranee only applies if he accoun value is annuiized. Since 22, a new form of GMLB is offered, he so-called Guaraneed Minimum Wihdrawal Benefi (GMWB. Here, he insured is eniled o wihdraw a pre-specified amoun annually, even if he accoun value has fallen below his amoun. hese guaranees are exremely popular. In 24, 69% of all Variable Annuiy conracs sold included a GMWB opion. Each of he 5 larges Variable Annuiy providers offered his kind of guaranee a his ime (cf. Lehmann Brohers ( Guaraneed Minimum Accumulaion Benefis (GMAB Guaraneed Minimum Accumulaion Benefis are he simples form of guaraneed living A benefis. Here, he cusomer is eniled o a minimal accoun value G a mauriy of he conrac. Usually, G A is he single premium P, someimes a roll-up benefi base. he corresponding fees vary beween.25% and.75% p.a. of he accoun value (cf. Mueller ( Guaraneed Minimum Income Benefis (GMIB A mauriy of a Variable Annuiy wih a GMIB, he policyholder can as usual choose o obain he accoun value (wihou guaranee or annuiize he accoun value a curren marke condiions (also wihou any guaranee. However, he GMIB opion offers an addiional I choice: he policyholder may annuiize some guaraneed amoun G a annuiizaion raes ha have been specified up-fron. herefore, his opion can also be inerpreed as a guaraneed annuiy, saring a, where he annuiy paymens have already been specified a. Noe ha if he accoun value a mauriy is below he guaraneed value G, he cusomer I canno ake ou he guaraneed capial G as a lump sum bu only in he form of an annuiy a he pre-specified annuiizaion raes. hus, he opion is in he money a ime if he I - 5 -

6 resuling annuiy paymens exceed he annuiy paymens resuling from convering he acual accoun value a curren annuiy raes. I he guaraneed amoun G usually is a roll-up benefi base wih, e.g., i 5% or 6%, or a rache benefi base. Someimes here is no one specified mauriy, bu he policyholder can annuiize wihin a cerain (ofen raher long ime period. he offered roll-up raes frequenly exceed he risk-free rae of ineres, whereas he pre-specified annuiizaion facors are usually raher conservaive. hus, a mauriy he opion migh no be in he money, even if he guaraneed amoun exceeds he accoun value. Furhermore, he pricing of hese guaranees is ofen based on cerain assumpions abou he cusomers behavior raher han assuming ha everybody exercises he opion when i is in he money. Such assumpions reduce he opion value. 5 Depending on he specific form of he guaranee, he curren fees for GMIB conracs ypically vary beween.5% and.75% p.a. of he accoun value Guaraneed Minimum Wihdrawal Benefis (GMWB Producs wih a GMWB opion give he policyholder he possibiliy o wihdraw a specified W amoun G (usually he single premium in small porions. ypically, he insured is eniled W o annually wihdraw a cerain proporion x W of his amoun G, even if he accoun value has fallen o zero. A mauriy, he policyholder can ake ou or annuiize any remaining funds if he accoun value did no vanish due o such wihdrawals. Recenly, several forms of so-called Sep-up GMWB opions have been inroduced: Wih one popular version, he oal guaraneed amoun which can be wihdrawn is increased by a predefined raio a cerain poins in ime, if no wihdrawals have been made so far. In wha follows, we will only analyze his form of Sep-up GMWB. Alernaively, here are producs in he marke, where a cerain poins in ime, he remaining oal guaraneed amoun which can be wihdrawn is increased o he maximum of he old remaining guaraneed amoun and he curren accoun value. he laes developmen in his area are so-called GMWB for life opions, where only some maximum amoun o be wihdrawn each year is specified bu no oal wihdrawal amoun. W his feaure can be analyzed wihin our model by leing G and. From a financial poin of view, GMWB opions are highly complex, since he insured can decide a any poin in ime wheher and, if so, how much o wihdraw. hey are currenly offered for beween.4% and.65% p.a. of he accoun value. However, Milevsky and Salisbury (26 find ha hese guaranees are subsanially underpriced. hey conclude ha insurers eiher assume a subopimal cusomer behavior or use charges from oher (overpriced guaranees o cross-subsidize hese guaranees. 5 Cf. Milevsky and Salisbury (

7 While his summary of GMDB and GMLB opions covers all he basic designs, a complee descripion of all possible varians would be beyond he scope of his paper. hus, some producs offered in he marke may have feaures ha differ from he descripions above. For curren informaion regarding Variable Annuiy producs, ypes of guaranees, and curren fees, we refer, e.g., o Our model and noaion presened in he following Secion is designed o cover all he guaranees described in his Secion as special cases. Of course, he underlying general framework allows for any specific variaions of he guaranees ha migh deviae from he producs described above. 3 A General Valuaion Framework for Guaraneed Minimum Benefis 3. he Financial Marke As usual in his conex, we assume ha here exiss a probabiliy space (Ω,F,Q equipped wih a filraion F ( [, ] I, where Q is a risk-neural measure under which, according o he Risk-Neural valuaion formula (cf. Bingham and Kiesel (24, paymen sreams can be valued as expeced discouned values. Exisence of his measure also implies ha he financial marke is arbirage free and ha here exiss some self-financing invesmen B, sraegy which allows he insurer o hedge his liabiliies. We use a bank accoun ( [ ] he numéraire process, which evolves according o as db B r d, B >. ( Here, r denoes he shor rae of ineres a ime. We furher assume ha he underlying muual fund S of he Variable Annuiy is modeled as a righ-coninuous F adaped sochasic process wih finie lef limis (RCLL. 6 In paricular, he discouned asse process S B. S B [ ], is a Q-maringale. For convenience, we assume 3.2 A Model for he Insurance Conrac In wha follows, we presen a model suiable for he descripion and valuaion of variable annuiy conracs. Wihin his framework, any combinaion of guaranees inroduced in 6 For our numerical calculaions, we assume ha S evolves according o a geomeric Brownian moion wih consan coefficiens

8 Secion 2 can be represened. In our numerical analysis however, we resric ourselves o conracs wih a mos one GMDB and one GMLB opion. We consider a Variable Annuiy conrac wih a finie ineger mauriy, which is aken ou a ime for a single premium P. Alhough he model generally allows for flexible expiraion opions, in order o simplify he noaion, we only consider a fixed mauriy. We denoe he accoun value by A and ignore any up-fron charges. herefore, we have A P. During he erm of he conrac, we only consider he charges which are relevan for he guaranees, i.e. coninuously deduced charges for he guaranees and a surrender fee. he surrender fee is charged for any wihdrawal of funds from he conrac excep for guaraneed wihdrawals wihin a GMWB opion. he coninuously deduced guaranee fee ϕ is proporional o he accoun value and he surrender fee s is proporional o he respecive amoun wihdrawn. In order o valuae he benefis of he conrac, we sar by defining wo virual accouns: W denoes he value of he cumulaive wihdrawals up o ime. We will refer o i as he wihdrawal accoun. Every wihdrawal is credied o his accoun and compounded wih he risk-free rae of ineres up o mauriy. A ime zero, we have W. Similarly, by D we denoe he value of he deah benefis paid up o ime. Analogously o he wihdrawals, we credi deah benefi paymens o his deah benefi accoun and compound he value of his accoun wih he risk-free rae unil ime. Since we assume he insured o be alive a ime zero, we obviously have D. In order o describe he evoluion of he conrac and he embedded guaranees, we also need he following processes: he guaraneed minimum deah benefi a ime is denoed by G D a ime is given by { A ; } D. hus, he deah benefi max G. We le G D A if he conrac conains one of he D D described GMDB opions (cf. Secion 2.2, oherwise we le G. he evoluion of G over ime depends on he ype of he GMDB opion included in he conrac. I will be described in deail in Secion 3.3. he guaraneed mauriy benefi of he GMAB opion is denoed by G. In order o accoun for possible changes of he guaranee over he erm of he conrac, we le ( [ ] A A G, represen he evoluion of his guaranee (see Secion 2.3. for deails. We have G A A A for conracs wih one of he described GMAB opions and G for conracs wihou a GMAB opion

9 I Analogously, we le G denoe he guaraneed mauriy benefi ha can be annuiized in G, I he case of a GMIB opion and model is developmen by ( [ ] G I I and G for conracs wih and wihou a GMIB opion, respecively. A. Also, we have Finally, o be able o represen GMWB opions, we inroduce he processes ( G [, ] E ( G [, ]. W and W G denoes he remaining oal amoun ha can be wihdrawn afer ime, and E G is he maximum amoun ha can be wihdrawn annually due o he GMWB opion. If he conrac conains a GMWB, we le G W A and G E xw A, where x W is he porion of he premium ha can be wihdrawn annually. For conracs wihou GMWB, we le W E G G. he evoluion over ime of hese processes is also explained in deail in Secion 3.3. Due o he Markov-propery 7 of he underlying processes, all informaion available a ime is A I D W compleely conained in he so-called sae variables A, W, D, G, G, G, G and E G. o simplify noaion, we inroduce he following sae vecor A I D W E ( A, W, D, G, G, G, G G y,. 3.3 Evoluion of he Insurance Conrac During he erm of he conrac here are four possible ypes of evens: he insured can wihdraw funds as a guaraneed wihdrawal of a GMWB opion, perform a parial surrender, i.e. wihdraw more han he guaraneed wihdrawal amoun, compleely surrender he conrac, or pass away. For he sake of simpliciy, we assume ha all hese evens can only occur a a policy anniversary dae. herefore, a ineger ime poins,2,...,, for all sae variables we disinguish beween ( and (, i.e. he value immediaely before and afer he occurrence of such evens, respecively. 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 developmen wihin a policy year, i.e. from o ( - is specified. Subsequenly, we will 7 See Secion in Bingham and Kiesel (

10 describe he ransiion from ( - o (, which depends on he ype of guaranees included in he conrac and he occurrence of he described evens. Finally, we describe he mauriy benefis of he conrac Developmen beween and ( - As indicaed in Secion 3., he price of he underlying muual fund evolves sochasically over ime. hus, aking ino accoun coninuous guaranee fees ϕ, for he accoun value we have S ϕ A e. (2 A S he accouns W and D are compounded wih he risk-free rae of ineres, i.e. r s ds r s ds W W e and D D e. D A I he developmen of he processes G, G and G depends on he specificaion of he corresponding GMDB, GMAB and GMIB opion: if he corresponding guaraneed benefi is he D / A / I D / A / I single premium or if he opion is no included, we le G G. If he guaraneed D / A / I D / A / I benefi is a roll-up base wih roll-up rae i, we se G G ( i D / A / I D / A / I. For rache guaranees, we have G G, since he rache base is adjused afer possible wihdrawals, and herefore considered in he ransiion from ( - o ( (cf. Secion W E he processes G and G do no change during he year, i.e ransiion from ( - o ( A he policy anniversary dae, we disinguish four cases: a he insured dies wihin he period (,] W / E W / E. G G Since our model only allows for deah a he end of he year, dying wihin he period (,] is equivalen o a deah a ime. he deah benefi is credied o he deah benefi accoun and will hen be compounded wih he risk-free rae unil mauriy : D D D max{ G ; A }. Since afer deah, no fuure benefis are possible, we le A / I / W / D / E A as well as G. he wihdrawal accoun, where possible prior wihdrawals have been colleced, will no be changed, i.e. W compounded unil mauriy. W. his accoun will be - -

11 b he insured survives he year (,] and does no ake any acion (wihdrawal, surrender a ime Here, neiher he accoun D nor W is changed. hus, we have A A, W W A / I / D D D and. For he GMAB, GMIB, and GMDB, wihou a rache ype guaranee, we also A / I / D have G G. If, however, one or more of hese guaranees are of rache ype, A / I / D A / I / D we adjus he corresponding guaranee accoun by max{ G A } G. ; If he conrac includes a GMWB opion wih sep-up and is a sep-up poin, he GMWB processes are adjused according o he sep-up feaure, bu only if here were no pas wihdrawals: If iw denoes he facor, by which he oal amoun o be wihdrawn is ( W W increased (cf. Secion 2.3.3, we ge G G Ι { } i W W and W / E W / E any oher case, we have G G. E W G x W G. In c he insured survives he year (,] and wihdraws an amoun wihin he limis of he GMWB opion A wihdrawal wihin he limis of he GMWB is a wihdrawal of E W { G G } E min ; wihdrawal amoun G an amoun, since he wihdrawn amoun may neiher exceed he maximal annual E W nor he remaining oal wihdrawal amoun G. he accoun value is reduced by he wihdrawn amoun. In case he wihdrawn amoun exceeds he accoun value, he accoun value is reduced o. hus, we have A { A E } max ;. Also, he remaining oal wihdrawal amoun is reduced by he W W wihdrawn amoun, i.e. G G E. Furhermore, he wihdrawn amoun is credied o he wihdrawal accoun: W W E. he maximal annual wihdrawal amoun as well E E as he deah benefi accoun remain unchanged: G G and D D. Usually, living benefi guaranees (GMAB and GMIB and, in order o avoid adverse selecion effecs, also he guaraneed deah benefis are reduced in case of a wihdrawal. We will resric our consideraions o a so-called pro raa adjusmen. Here, guaranees which are no of rache ype are reduced a he same rae as he accoun value, i.e. - -

12 A G A / I / D A / I / D G. If one or more of he guaranees are of rache ype, for he A respecive guaranees, we le G A A / I / D max A G. 8 A A / I / D ; d he insured survives he year (,] and wihdraws an amoun exceeding he limis of he GMWB opion A firs, noe ha his case includes he following cases as special cases: d he conrac does no comprise a GMWB opion and an amoun wihdrawn. < < A E is d2 A GMWB opion is included in he conrac, bu he insured wihdraws an amoun < < A E W E wih { G G } E > ; min. d3 he insured surrenders by wihdrawing he amoun E A 9. We le E W E, where min { G G } 2 E E E ;. Consequenly, E is he porion of he wihdrawal wihin he limis of he GMWB opion. If he conrac does no include a GMWB opion, we obviously have E. As in case c, he accoun value is reduced by he amoun wihdrawn, i.e. A A E, and he wihdrawn amoun is credied o he wihdrawal accoun. However, he insured has o pay a surrender fee for he second componen which leads o ( s 2 W W E E. he deah benefi accoun remains unchanged, i.e. D D. 8 Besides pro raa adjusmens, here are also reducions by he so-called dollar mehod. Here, all he respecive A / I / D A / I / D processes are reduced by he wihdrawn amoun, i.e. G max[ G E, ]. In order o model and evaluae producs where he dollar mehod or any oher reducion scheme applies, he respecive formulas can be adjused. 9 E W If he conrac comprises a GMWB opion and if { G G } A < W G A min ; as well as, hen a wihdrawal of E A is wihin he limis of he GMWB and does no lead o a surrender of he conrac. However, his case is covered by case c

13 Again, he fuure guaranees are modified by he wihdrawal: For he guaranees which are no of rache ype, we have A / I / D A A / I / D G G, whereas for he rache ype A guaranees, we le G A A / I / D max A G. A A / I / D ; E W For conracs wih a GMWB, wihdrawing an amoun { G G } E > min ; also changes fuure guaraneed wihdrawals. We consider a common kind of GMWB opion, where he W W W A guaraneed fuure wihdrawals are reduced according o G min G E ; G, A i.e. he wihdrawal amoun is reduced by he higher of a pro raa reducion and a reducion according o he dollar mehod. For fuure annual guaraneed amouns, we use E E A G G. A Mauriy Benefis a If he conrac neiher comprises a GMIB nor a GMAB opion, he mauriy benefi L is simply he accoun value, i.e. L A. In conracs wih a GMAB opion, he survival benefi A a mauriy is a leas he GMAB, hus { } A L A ; G guar max. Insured holding a GMIB opion can decide wheher hey wan a lump sum paymen of he accoun value A or annuiize his amoun a curren annuiizaion raes. Alernaively, hey can annuiize he guaraneed annuiizaion amoun a pre-specified condiions. If we denoe by ä curren and ä guar he annuiy facors 2 when annuiizing a he curren and he guaraneed, pre-specified condiions, respecively, he value of he guaraneed benefi a I ä curren mauriy is given by G. hus, a financially raional acing cusomer will chose he ä annuiy, whenever we have G I ä ä curren guar I I ä curren is given by L max A ; G. ä guar > A. herefore, he value of he benefi a ime Cf. Pioneer (25, pp. 36. Cf. Pioneer (25, page 36f. Also, a reducion of he form G E E G W W G is frequenly offered. G 2 Here, an annuiy facor is he price of an annuiy paying one dollar each year

14 If he conrac conains boh, a GMAB and a GMIB opion, he mauriy value of he conrac A I is L max{ L ; L }. 3.4 Conrac Valuaion We make he common assumpion ha financial markes and biomeric evens are independen. Furhermore, we assume risk-neuraliy of he insurer wih respec o biomeric risks (cf. Aase and Persson (994. hus, he risk-neural measure for he combined marke (insurance and financial marke is he produc measure of Q and he usual measure for biomeric risks. In order o keep he noaion simple, in wha follows, we will also denoe his produc measure by Q. Even if risk-neuraliy of he insurer wih respec o biomeric risk is no assumed, here are sill reasons o employ his measure for valuaion purposes as i is he so-called variance opimal maringale measure (see Møller (2 for he case wihou sysemaic moraliy risk and Dahl and Møller (26 in he presence of sysemaic moraliy risk. Le x be he insured s age a he sar of he conrac and a x -year old o survive years. By q x p x denoe he probabiliy for, we denoe he probabiliy for a ( x -year old o die wihin he nex year. he probabiliy ha he insured passes away in he year (,] is hus given by p x q x. he limiing age is denoed by ω, i.e. survival beyond age ω is no possible Valuaion under Deerminisic Policyholder Behavior A firs, we assume ha he policyholder s decisions (wihdrawal/surrender are deerminisic, i.e. we assume here exiss a deerminisic sraegy which can be described by IR. 3 Here, ξ denoes he amoun o be wihdrawn a he end of year, if he insured is sill alive and if his amoun is admissible. If he amoun a wihdrawal vecor ξ ( ξ ;...; ξ ( ξ is no admissible, he larges admissible amoun E < ξ is wihdrawn. In paricular, if he conrac does no conain a GMWB opion, he larges admissible amoun is E { ; A } A full surrender a ime is represened by ξ. By ( min ξ. Ψ Ψ Ψ IR... we denoe he se of all possible deerminisic sraegies. In paricular, every deerminisic sraegy is F -measurable. If a paricular conrac and a deerminisic sraegy are given, hen, under he assumpion ha he insured dies in year {,2,..., x } ω, he mauriy-values L ;ξ, W ;ξ and 3 Here, IR denoes he non negaive real numbers (including zero; furhermore we le IR IR { }

15 - 5 - ;ξ D are specified for each pah of he sock price S. hus, he ime zero value including all opions is given by:. ; ; ; ; ; ; ; ; ; ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ω D W L e E p D W L e E q p D W L e E q p V ds r Q x ds r Q x x x ds r Q x x s s s ( Valuaion under Probabilisic Policyholder Behavior By probabilisic policyholder behavior, we denoe he case when he policyholders follow cerain deerminisic sraegies wih cerain probabiliies. If hese deerminisic sraegies ( ( j j j IR ( ( ( ;...;ξ ξ ξ, n j,...,,2 and he respecive probabiliies ( j p ξ are known ( n j j p ( ξ, he value of he conrac under probabilisic policyholder behavior is given by ( ( j n j j V p V ξ ξ. (4 his value also admis anoher inerpreaion: if he insurer has derived cerain forecass for he policyholders fuure behavior wih respec o wihdrawals and surrenders, and assigns he respecive relaive frequencies as probabiliies o each conrac, hen he sum of he probabilisic conrac values consiues exacly he value of he insurer s whole porfolio given ha he forecas is correc. hus, his cumulaive value equals he coss for a perfec hedge of all liabiliies, if policyholders behave as forecased. However, in his case he risk ha he acual clien behavior deviaes from he forecas is no hedged Valuaion under Sochasic Policyholder Behavior Assuming a deerminisic or probabilisic cusomer behavior implies ha he wihdrawal and surrender behavior of he policyholders does no depend on he evoluion of he capial marke or, equivalenly, on he evoluion of he conrac over ime. A sochasic sraegy on he oher hand, is a sraegy where he decision wheher and how much money should be wihdrawn is based upon he informaion available a ime. hus, an admissible sochasic sraegy is a discree F measurable process (X, which deermines he amoun o be wihdrawn depending on he sae vecor y. hus, we ge: ( y Ε, X,,...,,2.

16 For each sochasic sraegy (X and under he hypohesis, ha he insured deceases in year {,2,..., ω x }, he values L ( ;(X, W ( ;(X and D ( ;(X are specified for any given pah of he process S. herefore, he value of he conrac is given by: V ω x r ds x x Q, s ( (X p q E e ( L (,(X W (,(X D ( (X. (5 We le Ξ denoe he se of all possible sochasic sraegies. hen he value V of a conrac assuming a raional policyholder is given by (X Ξ ((X V supv. (6 4 Numerical Valuaion of Guaraneed Minimum Benefis For our numerical evaluaions, we assume ha he underlying muual fund evolves according o a geomeric Brownian moion 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 bank accoun we have r B e. Since he considered guaranees are pah-dependen and raher complex, i is no possible o find closed form soluions for heir risk-neural value. herefore, we have o rely on numerical mehods. We presen wo differen valuaion approaches: in Secion 4., we presen a simple Mone Carlo algorihm. his algorihm quickly produces accurae resuls for a deerminisic, probabilisic or a given F measurable sraegy. However, Mone Carlo mehods are no preferable o deermine he price for a raional policyholder. hus, in Secion 4.2, we inroduce a discreizaion approach, which addiionally enables us o deermine prices under opimal policyholder behavior. 4. Mone-Carlo Simulaion 8 Le (X : IR IR IR a F measurable wihdrawal sraegy. By Iô s formula (see, e.g. Bingham and Kiesel (24, we obain he ieraion A 2 S ϕ σ A e A exp r z ; z ~ S ϕ 2 σ N (, iid, - 6 -

17 which can be convenienly used o produce realizaions of sample pahs ( j a of he underlying muual fund using Mone Carlo Simulaion. 4 For any conrac conaining Guaraneed Minimum Benefis, for any sample pah, and for any ime of deah, we obain he evoluion of all accouns and processes, employing he rules of Secion 3. Hence, ( j ( j ( j realizaions of he benefis l (,(X w (,(X d (,(X a ime, given ha he insured dies a ime are uniquely defined in his sample pah. hus, he ime zero value of hese benefis in his sample pah is given by v [ ] ω x ( j x x, r ( j ( j ( j ((X e p q l (,(X w (,(X d ( (X J ( i Hence, V ((X v ((X is a Mone-Carlo esimae for he value of he conrac, J j where J denoes he number of simulaions. However, for he evaluaion of a conrac under he assumpion of raional policyholders following an opimal wihdrawal sraegy, Mone-Carlo simulaions are no preferable. 4.2 A Mulidimensional Discreizaion Approach anskanen and Lukkarinen (24 presen a valuaion approach for paricipaing life insurance conracs including a surrender opion, which is based on discreizaion via a finie mesh. We exend and generalize heir approach in several regards: we have a mulidimensional sae space, and, hus, need a mulidimensional inerpolaion scheme. In addiion, heir model does no include fees. herefore, we modify he model, such ha he guaranee fee ϕ and he surrender fee s can be included. Finally, wihin our approach a sraegy does no only consis of he decision wheher or no o surrender. We raher have an infinie number of possible wihdrawal amouns in every period. Even hough we are no able o include all possible sraegies in a finie algorihm, we sill need o consider numerous possible wihdrawal sraegies. We sar his Secion by presening a quasi-analyic inegral soluion o he valuaion problem of Variable Annuiies conaining Guaraneed Minimum Benefis. Subsequenly, we show how in each sep he inegrals can be approximaed by a discreizaion scheme which leads o an algorihm for he numerical evaluaion of he conrac value. We resric he presenaion o he case of a raional policyholder, i.e. we assume an opimal wihdrawal sraegy. However, for deerminisic, probabilisic or sochasic wihdrawal sraegies he approach works analogously afer a sligh modificaion of he funcion F ~ in Secion For an inroducion o Mone Carlo mehods see, e.g., Glasserman (

18 4.2. A quasi-analyic soluion he ime value V of a conrac depends solely on he sae variables a ime A I D W E ( A, W, D, G, G, G, G G y,. Since besides A, he sae variables change deerminisically beween wo policy anniversaries, he value process V is a funcion of, A and he sae vecor a he las policy anniversary, i.e. V V (, A ; y. A he discree poins in ime benefi paymens and wihdrawals V V (, A ; y V V (, A, y. If he insured does no die in he period (, ],2,...,, we disinguish he value righ before deah and he value righ afer hese evens, he knowledge of he wihdrawal amoun E and he accoun value o (. We denoe he corresponding ransiion funcion by f E ( A (,, y A y Similarly, by f ( A, y ( A, y wihin (, ]. A deermine he developmen of he sae variables from we denoe he ransiion funcion in case of deah By simple arbirage argumens (cf. anskanen and Lukkarinen (24, we can conclude ha V is a coninuous process. Furhermore, wih Iô s formula (see, e.g. Bingham and Kiesel (24 one can show ha he value funcion V τ for all τ [, saisfies a Black-Scholes parial differenial equaion (PDE, which is slighly modified due o he exisence of he fees ϕ. Hence, here exiss a funcion v : IR IR IR wih V ( τ, a, y v ( τ, a τ [,, a IR and v saisfies he PDE. dv d v dv rv dτ da da σ a 2 ( r ϕ a (8 wih he boundary condiion v (, a ( q V (, f ( a, y q V (, f ( a, y x E x, a IR, which, in paricular, is dependen on he insured s survival. For a derivaion and inerpreaion of he PDE (8 and he boundary condiion, see Ulm (26. hus, we can deermine he ime-zero value of he conrac V by he following backward ieraion: : A mauriy, we have V ( A, y L W D,

19 -k: V, k k a ime (-k be known for all possible values of he sae vecor. hen, he ime (-k value of he conrac is given by he soluion v k, a of he PDE (8 wih boundary condiion Le ( k, A y ( v ( k, a ( q sup V ( k, f ( a, y q V ( k, f ( a, y. x k E k k x k E k IR r ϕ 2 2 A soluion of he PDE (8 can be obained by defining υ :, ρ : σ υ r and σ x x g x e σ υ ρτ σ x x ( τ, v τ, e. hen, g x e σ υ ρ σ lim ( τ, v, e and g saisfies a one- ( dimensional hea equaion, τ ( ( 2 d g dg, (9 2 2 dx d a soluion of which is given by 5 2 ( x u ( τ g ( τ, x exp g (, u du. ( 2π (( τ 2 ( hus, we have v 2π (( τ exp 2 σ 2 ( 2 ( logλ ( τ ρ (( τ (, a e υ λ v 2 σ ( λa dλ k. ( By subsiuing 2 λ( u exp σ u r ϕ σ, we obain 2 V ( k, A k, y ( ( k sup (, ( (, q x V k f u A y k E k k k r E k ir e u Φ q V ( k, f ( λ( u A, y x k k λ (2 du, k where Φ denoes he cumulaive disribuion funcion of he sandard normal disribuion. 5 Cf. heorem 3.6 of chaper 4, Karazas and Shreve (

20 4.2.2 Discreizaion via a Finie Mesh In general, he inegral (2 canno be evaluaed analyically. herefore, we have o rely on numerical mehods o find an approximaion of he value funcion on a finie mesh. Here, a finie mesh is defined as follows: Le ( 8 Y IR be he se of al possible sae vecor values. We denoe a finie se of possible values for any of he eigh sae variables as a se of mesh basis values. Le a se of mesh 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 by Grid Y and call i a Y -mesh or simply a mesh or a grid. An elemen of Grid is called a grid poin. For a given grid Grid, we ierae he evaluaion backwards saring a. A mauriy, he value funcion is given by: V ( A, y L W D, y Grid,. We repea he ieraion sep described above imes and hereby obain he value of he conrac a every ineger ime poin for every grid poin. In paricular, we obain he ime zero value of he conrac V. Wihin each ime period, we have o approximae he inegral ( wih he help of numerical mehods. his will be described in he following Secion Approximaion of he Inegral Following anskanen and Lukkarinen (24, for a IR and a given sae vecor define he funcion y k, we ~ F k ( a, y k ( q sup V ( k, f ( a, y q V k, f ( a, y x k E k E k IR k x k (. k (3 hus, (2 is equivalen o V r ~ ( k A, y e Φ( u F ( ( u A, y du k k, λ for y k Grid k, k 2 where λ( u exp σ u r ϕ σ as above. In order o evaluae he inegral, we evaluae he funcion F 2 ~ k ( a, y k for each y k Grid k and for a selecion of possible values of he variables a. In beween, we inerpolae linearly. hus, le y k k Grid k and A max >, a maximal value for a, be given. We spli he inerval Amax ~ [, A ] in M subinervals via α : m, m {,,2 M }. Le F (, y max hen, for any IR ~ a, F ( a, y k m,..., M k k can be approximaed by m k α m k γ

21 ~ F k M a α m ( a, y k γ m ( γ m γ m Ι [ αm, αm m γ ( γ M γ M Ι [ A, ( a ( a M [ bm, a bm, ] Ι [ α, ( a [ b, a b, ] [, ( a, m α m M M Ι Amax m M where b γ m ( γ γ m a α α α M α α M m M m, m m m,,..., M m ; M, b M, m,..., M ; b M, b M, and I denoes he indicaor funcion. hus, we have V max b and b ( γ γ M m, m m, Amax M ϕ r ( k, a, y [ ( Φ( Φ( ( Φ( k a e bm, u m σ u m σ bm,e u m Φ( u m ] m Amax m r ϕ σ where u, u m log and u M. σ M a σ σ 2 Defining b b, we obain V,, ( k, A k, y k M ϕ r A k e ( bm, bm, ( Φ( u m σ e ( bm, bm, Φ( u m m [ ( ]. Hence, i suffices o deermine he values γ F ( α, y, m {,,2 M } ~ m k m k,...,,. When deermining he γ m, heoreically he funcion f E has o be evaluaed for any possible k wihdrawal amoun E k. For our implemenaion, we resric he evaluaion o a finie ~ amoun of relevan values E -k. Furhermore, due o he definiion of F k (see (3, i is necessary o evaluae V afer he ransiion of he sae vecor from ( k o ( k. Since he sae vecor and, hus, he argumens of he funcion are no necessarily elemens of Grid k, V ( k, A k, y k has o be deermined by inerpolaion from he surrounding mesh poins. We inerpolae linearly in every dimension. Due o he high dimensionaliy of he problem, he compuaion ime highly depends on he inerpolaion scheme. In order o reduce calculaion ime and he required memory capaciy, we reduced he dimensionaliy by only considering he relevan accouns for he considered conracs. In paricular, when he deah benefi accoun D is sricly posiive, i.e. if he insured has died before ime, he accoun value A will be zero. Conversely, as long as A is greaer han zero, D remains zero, i.e. he insured is sill alive a ime. hus, he dimensionaliy can always be reduced by one

22 Furhermore, in our numerical analyses, we only consider conracs wih a mos one GMDBopion and a mos one GMLB-opion. herefore, by only considering he relevan sae variables, we can furher reduce he dimensionaliy o a maximum of 4. However, for a conrac wih erm o mauriy of 25 years, using abou 4, o 65, laice poins, 6 seps for he numerical calculaion of he inegral, and a discreizaion of he opimal sraegy o 52 poins, he calculaion of one conrac value under opimal policyholder sraegy on a single CPU (Inel Penium IV 2.8 GHz,. GB RAM sill akes beween 5 and 4 hours. 5 Resuls We use he numerical mehods presened in Secion 4 o calculae he risk-neural value of Variable Annuiies including Guaraneed Minimum Benefis for a given guaranee fee ϕ. We call a conrac, and also he corresponding guaranee fee, fair if he conrac s risk-neural value equals he single premium paid, i.e. if he equilibrium condiion P V V ( ϕ holds. Unless saed oherwise, we fix he risk-free rae of ineres r 4%, he volailiy σ 5%, he conrac erm 25 years, he single premium amoun P,, he age of he insured x 4, he sex of he insured male, he surrender fee s 5%, and use bes esimae moraliy ables of he German sociey of acuaries (DAV 24 R. For conracs wihou GMWB, we analyze wo possible policyholder sraegies: Sraegy assumes ha cliens neiher surrender nor wihdraw money from heir accoun. Sraegy 2 assumes deerminisic surrender probabiliies which are given by 5% in he firs policy year, 3% in he second and hird policy year, and % hereafer. In addiion, we calculae he riskneural value of some policies assuming raional policyholders. For conracs wih GMWB, we assume differen sraegies which are described in Secion Deermining he fair Guaranee Fee In a firs sep, we analyze he influence of he annual guaranee fee on he value of conracs including hree differen kinds of GMAB opions. For conrac, he guaraneed mauriy value is he single premium (money-back 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 assuming neiher surrenders nor wihdrawals

23 V 8, 6, 4, 2,, 8, 6, 4,.%.5%.%.5% 2.% 2.5% 3.% 3.5% 4.% 4.5% 5.% ϕ 6% Roll-Up annual rache money-back guaranee premium 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 risk-neural value of conrac 3 exceeds, for all values of ϕ. hus, under he given assumpions here exiss no fair guaranee fee for a conrac including a 6% roll-up GMAB. As a consequence, such guaranees can only be offered if he guaranee coss are subsidized by oher charges or if irraional policyholder behavior is assumed in he pricing of he conrac. 5.2 Fair Guaranee Fees for Differen Conracs 5.2. Conracs wih a GMDB Opion We analyze hree differen conracs wih a minimum deah benefi guaranee. Conrac provides a money-back guaranee in case of deah, conrac 2 an annual rache deah benefi and conrac 3 a 6% roll-up benefi. able shows fair guaranee fees for hese conracs under he wo policyholder sraegies described above. conrac sraegy : no wihdrawals or surrenders 2: deerminisic surrender probabiliy Money-back guaranee Rache benefi base 6% roll-up benefi base.%.4%.4% < % < %.5% able : Fair guaranee fee for conracs wih GMDB under differen consumer behavior

24 Assuming ha cusomers neiher surrender heir conracs nor wihdraw any money before mauriy, he fair guaranee fee for all hese conracs is raher low. However, he guaraneed deah benefi included in conrac 3 is significanly more expensive han he oher guaranees. If policyholders surrender heir conracs a he surrender raes assumed in sraegy 2, he fair guaranee fee srongly decreases for wo reasons: Policyholders pay fees before surrendering bu will no receive any benefis from he corresponding opions. Secondly, 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 remaining cliens opions. hus, he risk-neural value of he conrac exceeds he single premium even if no fee is charged for he opion. hus, our resuls are consisen wih Milevsky and Posner (2, who find ha GMDB opions are generally overpriced in he marke. Overall, he guaranee fees are raher low, since a benefi paymen is only riggered in he even of deah. here is no possibiliy for raional consumer behavior in erms of exercising he opion when i is in he money. he only way of raional policyholder behavior is surrendering a conrac when he opion is far ou of he money: I is opimal 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 almos never opimal. hus, he conrac value for a raional policyholder hardly differs from he value under sraegy. However, for lower surrender charges, policyholder behavior would be more imporan Conracs wih a GMAB Opion We analyze hree differen conracs wih a minimum accumulaion benefi guaranee. Again, conrac provides a money-back guaranee a he end of he accumulaion phase, conrac 2 an annual rache guaranee and conrac 3 a 6% roll-up benefi base. he value of hese conracs under policyholder sraegy has been displayed as a funcion of ϕ in Figure above. able 2 shows he fair guaranee fee for hese hree conracs under he wo given policyholder sraegies. In addiion, we show he fair guaranee fee if an addiional 6% rollup deah benefi is included (columns wih DB. conrac Money-back guaranee Rache benefi base 6% roll-up benefi base sraegy w/o DB wih DB w/o DB wih DB w/o DB wih DB : no wihdrawals or.7%.23%.76% surrenders 2: deerminisic surrender probabiliy < %.2%.57%.74% able 2: Fair guaranee fee for conracs wih GMAB under differen consumer behavior

25 he fair guaranee fees for he conracs differ significanly. For he money-back guaranee, he fair guaranee fee is below.25%, even if he GMDB opion is included. he fee for he rache guaranee is significanly higher. Even under sraegy 2 and wihou addiional deah benefi i exceeds.5%. In any case, he fair guaranee fee of he rache guaranee is a leas four imes as high as he corresponding fair guaranee fee of he money-back guaranee. For a roll-up rae of 6%, he value of he pure mauriy guaranee wihou fund paricipaion (i.e. ϕ % exceeds, under boh surrender scenarios. hus, even under he assumed surrender paern, a 6% roll-up GMAB canno be offered a a fair price. he addiional fee for deah benefi (difference beween columns wih DB and w/o DB always exceeds he fair guaranee fee of he pure deah benefi guaranee shown in able, and is hardly reduced by he assumed surrenders. Furher analyses showed ha raional policyholder behavior hardly influences he risk-neural value of he conracs: he values under opimal policyholder behavior are very close o he values under sraegy (no surrender or wihdrawal. his is no surprising since for he money-back guaranee, surrender is rarely opimal due o he raher high surrender charges. In he case of a rache guaranee, he acual guaranee level is annually adjused o a poenially increasing fund value. hus, he guaranee is always a or in he money a a policy anniversary dae. However, as explained above, surrendering is usually only opimal if he opion is ou of he money Conracs wih a GMIB Opion A GMIB opion gives he policyholder he possibiliy o annuiize he minimum benefi base a an annuiy facor ha is fixed a. Wheher or no he opion is in he money depends on boh, he fund value and he raio of he guaraneed annuiy facor and he curren annuiy facor a annuiizaion. Usually, he guaraneed annuiy facor is calculaed based on conservaive assumpions which are supposed o lead o a raio ä ä ä curren : <. However, increasing longeviy and decreasing ineres raes may change his raio during he erm of he conrac and make he guaranee exremely valuable a annuiizaion. We analyze hree differen GMIB-conracs for differen values of ä. Again, he minimum benefi base for conrac is he single premium, conrac 2 includes an annual rache guaranee whereas conrac 3 comes wih a 6% roll-up benefi base. he hree conracs are analyzed wih and wihou he addiional GMDB opion from he previous Secion. he respecive fair guaranee fees are shown in able 3. Obviously, for ä, he fair guaranee fees are he same as for he corresponding GMAB opions. he value of he guaranee highly depends on he value of ä. Since bes esimaes abou fuure moraliy raes are subjec o high uncerainy, his assumpion bears a significan risk for he insurer ha canno be hedged wih exising financial insrumens. guar