GMWB For Life An Analysis of Lifelong Withdrawal Guarantees

Transcription

1 GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees Daniela Holz Ulm Universiy, Germany Alexander Kling *) Insiu für Finanz- und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany phone: , fax: Jochen Ruß Managing Direcor Insiu für Finanz- und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany phone: , fax: Key words: variable annuiies, guaraneed minimum living benefis, risk-neural valuaion, embedded opions Absrac We analyze he laes guaranee feaure in he variable annuiies marke: guaraneed minimum wihdrawal benefis for life or guaraneed lifelong wihdrawal benefis. This opion gives he clien he righ o deduc a cerain amoun annually from he policy s accoun value unil deah even if a uni-linked accoun value drops o zero. We show how such producs can be analyzed wihin a general framework presened in Bauer e al. (26). We price he embedded guaranee for differen produc designs and parameers under deerminisic and opimal clien behaviour. *) Corresponding Auhor

2 1 Inroducion Variable annuiies, i.e. deferred annuiies ha are fund-linked during he accumulaion period were inroduced in he 197s in he Unied Saes (see Sloane (197)). Since he 199s, cerain opional guaranees are usually offered in such policies: guaraneed minimum deah benefis (GMDB) as well as guaraneed minimum living benefis (GMLB). The GLWB opions can be caegorized in hree main groups: Guaraneed minimum accumulaion benefis (GMAB) provide a guaraneed minimum survival benefi a some or several specified poins of ime, guaraneed minimum income benefis (GMIB) offer a guaraneed lifelong fixed annuiy saring a he end of he defermen period. However, GMIB producs offer no guaraneed lump sum benefi. Finally, in guaraneed minimum wihdrawal benefis (GMWB) some 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. A deailed produc descripion and exensive lieraure overview is given in Bauer e al. (26) and is herefore omied in his paper. In he presen paper, we look a he laes varian of GMWB-guaranees, so called GMWB for life. Such producs have recenly been inroduced in he US, Asia and Europe. As he name suggess, GMWB for life or Guaraneed Lifelong Wihdrawal Benefis (GLWB) offer a lifelong wihdrawal guaranee. Therefore, here is no limi for he oal amoun ha is wihdrawn over he erm of he policy. Usually, wih an immediae GLWB, annually a cerain percenage of he single premium (he guaraneed amoun) can be wihdrawn from he policy. This percenage rae may depend on he age of he insured. If a he ime of deah here is a remaining accoun value, hen his value is paid o he beneficiary as deah benefi. If, however, due o declining sock markes and/or such wihdrawals he accoun value of he policy becomes zero while he insured is sill alive, hen he insured can coninue o wihdraw he guaraneed amoun annually unil deah. The insurer charges a fee for his guaranee which is usually a fix percenage of he policyholder s accoun value per annum. In deferred versions of he produc, he produc is fund linked wih or wihou guaranee during he defermen period. The accoun value a he end of he defermen period (or some guaraneed amoun if guaranees are included in he defermen period as well) is hen reaed like a single premium o an immediae GLWB conrac. The res of he paper is organized as follows. In Secion 2, we give a deailed descripion of GLWB producs. Secion 3 inroduces our model. Here, we build on a general model inroduced in Bauer e al. (26) and show, how GLWB producs can be included in heir model. Due o he complexiy of he producs, in general here are no closed form soluions for he valuaion problem. Therefore, we have o rely on numerical mehods and presen our valuaion approach in Secion 4. We show how a given conrac can be priced wih Mone Carlo mehods assuming deerminisic as well as opimal policyholder behavior. For he laer, we inroduce numerical mehods for he deerminaion of such opimal sraegies. Finally, in Secion 5 we presen he resuls of our analyses ha are derived in a Mone Carlo framework. We give he fair value of he guaranee 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.

3 2 Guaraneed Minimum Wihdrawal Benefis for Life In his Secion, we explain he concep of GLWB wihin variable annuiies. We sar wih a very brief descripion of variable annuiies in general and oher ypes of guaranees offered wihin such producs and refer he reader o Bauer e al. (26) for more deails. Variable Annuiies are deferred, fund-linked annuiy conracs, usually wih a single premium paymen up-fron. Therefore, 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. The 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 from his 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. If he insured dies during he defermen period, he beneficiary obains a deah benefi ha depends on he accoun value. If a guaraneed minimum deah benefi is chosen, hen he deah benefi paid is he higher of he accoun value and some specified guaraneed value. Since he lae 199s, guaraneed minimum living benefis are offered in he marke. The wo earlies forms, guaraneed minimum accumulaion benefis (GMAB) and guaraneed minimum income benefis (GMIB) offer he insured a guaraneed mauriy benefi, i.e. a minimum benefi a he mauriy T of he conrac (or wihin a cerain period of ime). However, wih he GMIB, his guaranee only applies if he accoun value is annuiized. Since 22, a new form of GLWB is offered: so-called guaraneed minimum wihdrawal benefis (GMWB). Producs wih a GMWB opion give he policyholder he W possibiliy o wihdraw a specified amoun G (ha usually coincides wih he single premium) in small porions. Typically, he insured is eniled o annually wihdraw a W cerain porion x W of his amoun G, even if he accoun value has fallen o zero, W unil he oal wihdrawals reach G. A mauriy, he policyholder can ake ou or annuiize any remaining funds if he accoun value did no vanish due o wihdrawals. These guaranees are exremely popular. In he firs half of 25, more han 3 ou of 4 variable annuiy conracs sold included a GMWB opion. Each of he 15 larges variable annuiy providers offered his kind of guaranee a ha ime (cf. Lehmann Brohers (25)). Since GMWB producs are so popular, insurers developed several versions of his produc o be more compeiive. The laes version ha is now already offered in he USA, Asia and Europe includes a W lifelong wihdrawal guaranee: The oal amoun o be wihdrawn G is no limied. Insurers ha inroduced his ype of guaranee (GLWB) were immediaely able o significanly increase heir new business volume. Usually, he annual amoun o be wihdrawn in a GLWB produc is a cerain percenage x WL of he single premium P. Any remaining accoun value a he ime of deah is paid o he beneficiary as deah benefi. If, however, he accoun value of he policy drops o zero while he insured is sill alive, he insured can sill coninue o wihdraw he guaraneed amoun annually. The

4 insurer charges a fee for his guaranee which is usually a pre-specified annual percenage of he accoun value. Deferred versions of he produc also exis in he marke. Here, he produc is fund linked during he defermen period. The accoun value a he end of he defermen period is hen reaed like a single premium o an immediae GLWB conrac. Someimes, cerain guaranees are also included in he defermen period: if he accoun value a he end of he defermen period is lower han some guaraneed amoun, han his guaraneed amoun is ransferred o he GLWB payou phase. Furhermore, differen addiional feaures like sep-ups and roll-ups are offered wih GLWBs: If a sep-up is included, he guaraneed annual wihdrawal amoun can be increased a pre-specified poins in ime. A hese sep-up daes, he guaraneed annual wihdrawal amoun is increased if he porion x WL of he curren accoun value exceeds he previous guaraneed annual wihdrawal amoun. 1 Therefore, sep-ups only occur if he policyholder s funds yield high performance and he accoun value has no been decreased heavily due o previous wihdrawals. Common sep-up feaures are, e.g., annual rache guaranees. If a roll-up is included wihin GLWB, he annual guaraneed wihdrawal amoun is increased by a fixed percenage every year during a cerain ime period bu only if he policyholder has no sared wihdrawing money. Therefore, roll-ups are commonly used as an incenive o he policyholder no o wihdraw money from he accoun in he firs years. 3 The Model 3.1 The Financial Marke We assume ha here exiss a probabiliy space (Ω,F,Q) wih a filraion F = ( I ) [, T ] and a risk neural probabiliy measure Q. Under his risk-neural measure, paymen sreams can be valued as expeced discouned values using he risk-neural valuaion formula (cf. e.g. Bingham and Kiesel (24)). Exisence of his measure also implies an arbirage free financial marke. Thus, for every regular derivaive, here exiss some self-financing invesmen sraegy which replicaes he payoff of he derivaive. This allows he insurer o hedge he liabiliies. We assume a numéraire process ( B ) [, T ] which evolves according o db = r d, B B >, (1) where r denoes he shor rae of ineres a ime. For all our numerical calculaions, r we assume r = r. Thus, for he bank accoun we have B = e. We furher assume he exisence of some risky asse S ha serves as underlying muual fund for he considered variable annuiy conracs. S evolves according o a Geomeric Brownian moion wih consan coefficiens under Q, i.e. 1 Typically, hese sep-up daes are annually or every hree or five years on he policy anniversary dae.

5 ds = rd + σ dz, S = 1, (2) S where σ denoes he (consan) volailiy of he risky asse. Under he risk-neural S probabiliy measure, he discouned asse process B is a maringale. A =, [, T ] we assume S = B 1. = 3.2 A Model for he Insurance Conrac In Bauer e al. (26), a general model for he descripion and valuaion of variable annuiy conracs was inroduced. Wihin his framework, any conrac wih one or several living benefi guaranees and/or a guaraneed minimum deah benefi can be represened. In he numerical analysis however, only conracs wih a raher shor finie ime horizon were considered. In paricular, GLWB were no analyzed. In wha follows, we herefore describe how GLWB-producs can be included in his model. We refer o Bauer e al. (26) for he explanaion of oher living benefi guaranees and more deails on he model. Le o x be he insured s age a he sar of he conrac, o p x be he probabiliy for a x -year old o survive he nex years, o q x + be he probabiliy for a ( x + ) -year old o die wihin he nex year, and o ω be he limiing age of he moraliy able, i.e. he age beyond which survival is impossible. The probabiliy ha an insured aged x a incepion passes away in he year (,+1] is hus given by p x q x +. The limiing age ω allows for a finie ime horizon T = ω x. We denoe he single premium invesed ino he conrac a ime = by P. A denoes he accoun value a ime. Throughou he paper, we assume ha all fees and charges are deduced coninuously as a percenage ϕ of he accoun value and no upfron charges exis. This leads o A = P. Besides, we allow for a surrender fee s which is charged as a percenage of any wihdrawal of funds exceeding guaraneed wihdrawals wihin a GMWB or GLWB opion. Following he noaion in Bauer e al. (26), we denoe he value of some wihdrawal accoun a ime by W. Wihdrawals up o ime are credied o his accoun and compounded wih he risk-free rae of ineres. A incepion, we have W =. Similarly, deah benefis paid up o ime are accumulaed in he deah benefi accoun D which is also compounded by he risk-free rae unil ime T. A ime =, we have D =. Throughou his paper, we focus on guaraneed minimum wihdrawal benefis and, in addiion, allow for guaraneed minimum deah benefis. We use he following noaions again following Bauer e al. (26): If he conrac includes a GMDB opion, he deah benefi a ime is given by he greaer of he curren accoun value and he guaraneed minimum deah benefi base D D G, i.e. max { A ; } G. If a GMDB is included,

6 he iniial amoun of he guaraneed minimum deah benefi base is given by G D = A if no saed oherwise. Conracs wihou a GMDB rider simply pay ou he curren fund ne asse value in he case of deah. Such conracs are modeled by leing G = for all. D W For he modeling of GMWB or GLWB opions, we inroduce wo processes G and E W G. We call G he oal amoun guaraneed for fuure wihdrawals. For sandard GMWB opions, we usually have G W = A a incepion. For GLWB, he oal amoun W of wihdrawals is unlimied and hus G = for all. The maximum amoun ha may E be wihdrawn annually due o he GMWB or GLWB-opion is called G. A = i is given by some percenage of he single premium P, i.e. G E = xw A. Please noe ha we use he same sae variables for he modeling of GMWB and GLWB opions since we do no consider conracs wih boh, a GMWB and a GLWB-opion. Such conracs could also be modeled in our framework by inroducing separae processes W E G and G for each opion. 2 Finally, we call y (,,, D, W, E AW DG G G ) = he sae vecor a ime conaining all informaion abou he conrac a ha poin in ime. Since we resric our analyses o single premium conracs and do no allow for addiional premium paymens, policyholder acions during he life of he conrac are limied o wihdrawals. Depending on he amoun of money wihdrawn, he policyholder can wihdraw funds as a guaraneed wihdrawal of a GMWB/GLWB opion, perform a parial surrender, i.e. wihdraw more han he guaraneed wihdrawal amoun, or compleely surrender he conrac. For he sake of simpliciy, we allow for wihdrawals a policy anniversaries only. Also, we assume ha deah benefis are paid ou a policy anniversaries if he insured person has died during he previous year. Thus, he value of he sae variables described above may have disconinuiies a imes = 1,2,..., T. Thus, a each policy anniversary, we have o disinguish beween he value of he respecive variable ( ) immediaely before and he value ( ) + afer wihdrawals and deah benefi paymens. During he year, all processes are subjec o capial marke movemens and may herefore also change beween wo policy anniversaries. In wha follows, we describe he developmen beween wo policy anniversaries and he ransiion a policy anniversaries for differen conrac designs. From hese, we are finally able o deermine all benefis for any given policy holder sraegy and any capial marke pah. This allows for he valuaion of such conracs in a Mone-Carlo framework Developmen beween wo Policy Anniversaries We assume ha he annual guaranee fee ϕ is deduced from he policyholder s accoun value on a coninuous basis. Thus, he developmen of he accoun value be- 2 Since G W is no needed for GLWB opions, wo processes G E and one process G W are sufficien.

7 ween wo policy anniversaries is given by he developmen of he underlying fund afer deducion of he guaranee fee, i.e. + S + 1 ϕ A + 1 = A e. (3) S As described earlier, he wihdrawals and deah benefis are compounded on he accouns W and D a he risk-free rae of ineres r. This leads o r s ds + r + s ds W = W e and D = D e Throughou his paper, we use reurn of premium deah benefis if a GMDB opion is included. Thus, he value of he guaraneed minimum deah benefi base is no D/ E/ W D/ E/ W+ changed beween wo policy anniversaries which leads o G+ 1 = G. Since wihdrawals only occur a ineger imes, he processes G W and G E only change during he year if some roll-up feaure is included. The ime horizon wihin which roll-ups occur is usually limied o a cerain number of years WL from he beginning of he wihdrawal period. Roll-ups only are applied if no wihdrawals have W W+ been made so far. We denoe he annual roll-up rae by i w and le G+ 1 = G ( 1+ iw ) E E+ W W+ and G+ 1 = G ( 1+ iw ) if W = and WL. If no roll-up is included, G+ 1 = G. Besides roll-ups, wihdrawal guaranees are ofen equipped wih so called sep-up feaures. If a sep-up is included, wihdrawal guaranees can only be increased a he policy anniversaries. Thus, sep-ups are described in he following secion Transiion a a Policy Anniversary A he policy anniversaries, we have o disinguish he following four cases: a) The insured has died wihin he previous year (-1,] If he insured person has died wihin he previous policy year, he deah benefi is + credied o he deah benefi accoun D : max{ D D = D + G ; A }. Wih he paymen of he deah benefi, he insurance conrac maures. Thus, he policyholder s accoun value and all riders aached o i are erminaed, i.e. A + D =, G + W =, G + =, and E G + =. Wihdrawals ha have been made earlier remain on he wihdrawal accoun + W : W = W. b) The insured has survived he previous policy year and does no wihdraw any money from his accoun a ime If no deah benefi is paid ou o he policyholder and no wihdrawals are made from he conrac, he accoun value as well as he wihdrawal and he deah benefi accoun remain unchanged, i.e. A = A, D = D and W = W. Also he guaraneed minimum deah benefi doesn change in his case which leads o D + D G = G. If he conrac includes a wihdrawal guaranee wih sep-up and is a sep-up poin, he oal amoun available for fuure wihdrawals is increased if he accoun value exceeds he curren value of he guaranee accoun, i.e. G = max { G ; A }. Noe ha W+ W + in he case of GLWB, his value remains unchanged (a infiniy). The annual guaran-

8 eed wihdrawal amoun E { } E+ + G = max G ; x A. W E G may also be increased. I is given by c) The insured has survived he previous policy year and a he policy anniversary wihdraws an amoun wihin he limis of he wihdrawal guaranee If he insured has survived he pas year, no deah benefis are paid and herefore + E D = D. Any wihdrawal E below he guaraneed annual wihdrawal amoun G and W lower han he oal wihdrawal amoun G reduces he accoun value by he wihdrawn amoun. Of course, we do no allow for negaive policyholder accoun values + and hus ge A = max{ ; A E}. The wihdrawal accoun is increased by he amoun wihdrawn, i.e. W + = W + E. In he case of a GMWB opion, he remaining oal wihdrawal amoun is reduced by W+ W he amoun wihdrawn, i.e. G = G E. For GLWB-guaranees, he oal amoun of W+ W wihdrawals is unlimied and hus remains unchanged G = G =. For boh riders, E+ E he maximal annual wihdrawal amoun G = G remains unchanged. If, however, a sep-up feaure is included and is a sep-up poin, he oal amoun available for fuure wihdrawals can be increased as described in b), i.e. G = max { G ; A } (for W+ W + GMWB only) and he new annual guaraneed wihdrawal amoun G E is given by E+ E + G = max G ; x A. { } W Wih any wihdrawals, he guaraneed deah benefis are reduced a he same rae as + D+ A D he accoun value, i.e. G = G. A d) The insured has survived he previous policy year and a he policy anniversary wihdraws an amoun exceeding he limis of he wihdrawal guaranee + In his case again, no deah benefis are paid and herefore D = D. Wihdrawals exceeding he limis of he wihdrawal guaranee always lead o a parial or full surrender of he annuiy conrac, depending on he amoun of money wihdrawn and on he amoun remaining wihin he policyholder s accoun. Any wihdrawal E exceeding he limis of he wihdrawal guaranee can be separaed 1 E W ino wo pars, he guaraneed amoun E { } + 1 = min G + 1 ; G + 1 and he exceeding par 2 1 E = E 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 afer deducion of surrender charges for he exceeding par. 3 Thus, we ge W = W + E + E 1 s. ( ) 3 Noe ha surrender charges only apply o he exceeding par, Therefore surrender charges were no considered in case c) above.

9 In he case of a GMWB opion, usually he remaining oal wihdrawal amoun is reduced by he wihdrawn amoun, bu a leas by he same percenage, by which he + accoun value is reduced. 4 W+ W A W From his we ge G = min G E; G. For GLWB, A he oal amoun of wihdrawals is unlimied and hus again remains unchanged W+ W E G = G =. For boh riders, he maximum annual wihdrawal amoun G + is reduced by he same percenage, by which he accoun value is reduced, i.e. + E+ A E G = G. If a sep-up feaure is included and is a sep-up poin, he oal A + E+ A E + amoun available for fuure wihdrawals may be increased: G = max G ; xw A. A As in case c), wih any wihdrawals, he guaraneed deah benefis are reduced a he + D+ A D same rae as he accoun value, i.e. G = G. A Mauriy Benefis a T A mauriy of he conrac a = T = ω - x, he insured has eiher died or surrendered he conrac. Thus, all insurance benefis have already been credied o D or W and no addiional final paymen is given o he policyholder. We herefore call W T and D T he mauriy benefi of he conrac. 4 Valuaion of he Guaranees Assuming independence beween financial markes and moraliy and risk-neuraliy of he insurer wih respec o moraliy risk, we are able o use he produc measure of he risk-neural measure of he financial marke and he moraliy measure. In wha follows, we denoe his produc measure by Q. 4.1 Deerminisic Policyholder Behavior If we assume deerminisic policyholder behavior, any wihdrawal sraegy can easily be described by using a wihdrawal vecor ξ = ( ξ ;...; ξ ) ( ) T T IR where ξ denoes he deerminisic wihdrawal amoun a he end of year, if he insured is sill alive. Of course, if any such wihdrawal exceeds he guaraneed annual wihdrawal amoun, he wihdrawal leads o a parial or even full surrender. By allowing for ξ =, a full surrender a ime can also be represened wihin such a sraegy. Since deerminisic sraegies are already specified a ime =, every deerminisic sraegy is F - measurable. We denoe he se of all possible F -measurable sraegies by Ψ = Ψ Ψ ( ) T T IR For any given sraegy and under he assumpion ha he _ 4 Whenever a non-guaraneed wihdrawal occurs, fuure guaranees may be reduced. We here describe a so-called pro-raa reducion which is he predominan form in he marke. 5 Here, IR + denoes he non negaive real numbers (including zero); furhermore we le IR + = IR + { }.

10 insured dies in year { 1,2,..., ω x }, he mauriy-values W _ ;ξ and _ T D T ;ξ are specified for each capial marke pah. Thus, he ime zero value assuming he given policyholder sraegy including all opions is given by he risk-neural discouned expeced value of hese mauriy values: T _ ω x rds s V ξ 1px q x 1E + Q e WT ; ξ DT ; ξ = +. = 1 (4) 4.2 Probabilisic Policyholder Behavior If policyholders follow cerain deerminisic sraegies wih cerain probabiliies, we call his behavior probabilisic behavior. Policyholder sraegies are sill F - measurable bu several such sraegies are now weighed by cerain probabiliies. We ( j ) ( j ) denoe he corresponding deerminisic sraegies by = ( ξ ;...;ξ ) ( ) T j = 1,2,...,n and he respecive probabiliies by ( _ j ) ξ, n ( j ) p ξ where of course j = 1 1 T p ( j ) ξ IR + = 1. For any probabilisic sraegy, he value of he conrac under probabilisic policyholder behavior is given by n _ ( j ) ( ) V = p j ξ V ξ. (5) j = Sochasic Policyholder Behavior We call a policyholder sraegy sochasic if he decision wheher and how much money should be wihdrawn depends on he accoun value or oher informaion available a ime. Thus, sochasic policyholder sraegies are no necessarily F - measurable. However, we sill assume some F -measurable process (X), which deermines he amoun o be wihdrawn depending on he sae vecor y a ime. Thus, we ge: X (, y ) = Ε, = 1,2,..., T. Assuming ha he insured dies in year { 1,2,..., ω x }, for each sochasic sraegy (X) he values W T ( ;(X) ) and D T ( ;(X) ) are specified for any capial marke pah. Therefore, he value of he conrac following some sochasic sraegy (X) is given by: T ω x rds s V ((X)) = 1px q ( ( ) ( )) x 1 E + Q e WT,(X) DT,(X) +. (6) = We le Ξ denoe he se of all admissible sochasic sraegies. Then he value V of a conrac assuming a raional policyholder is given by V = supv ((X) ). (7) (X) Ξ Even hough he value of he conrac under raional policyholder behavior can easily be defined, he respecive raional sraegy is no obvious and canno be easily deermined. In he following secion, we describe how Mone-Carlo simulaion can be used o approximae opimal policyholder sraegies. The ideas are based on Anderson (1999).

11 4.4 Deermining he Conrac Value using Mone Carlo Mehods By Iô s formula (see, e.g. Bingham and Kiesel (24)), he ieraion 2 S ϕ + σ A + 1 = A e = A exp r + z + 1 ; z + 1 ~ N (,1) S ϕ 2 σ iid, ( j ) can be convenienly used o produce realizaions of sample pahs a of he policyholder s accoun using Mone Carlo Simulaion. 6 For any capial marke developmen and for any ime of deah, he evoluion of all accouns and processes is deermined by he rules given in Secion 3.2. Thus, realizaions of he benefis ( j) ( j) wt (,(X) ) + dt (,(X) ) a ime T are uniquely defined for any capial marke scenario and he ime zero value of hese benefis in his sample scenario is given by ω x ( j) rt ( j) ( j) v ((X)) = e 1px q ( ) ( ) x+ 1 wt,(x) + dt,(x). = 1 J 1 ( i ) Hence, V ((X) ) = v ((X) ) is a Mone-Carlo esimae for he value of he conrac J j = 1 wih J denoing he number of simulaions. A he end of each year he policyholder can decide wha amoun o wihdraw from he accoun. In Milevsky and Salisbury (26) he auhors prove ha wihin heir model an opimal sraegy for a GMWB conrac can only be achieved by wihdrawing eiher nohing or he guaraneed annual wihdrawal amoun or he oal accoun value. In conras o a GMWB, for a GLWB wihdrawing nohing can never be opimal unless roll-ups or sep-ups are included. Therefore in an opimal sraegy, he policyholder can only wihdraw he guaraneed amoun or surrender he conrac. The propery ha opimal sraegies can only be achieved by wihdrawing he guaraneed annual wihdrawal amoun or he oal accoun value also holds wihin our model: Firs, a wihdrawal below he guaraneed annual wihdrawal amoun can never be opimal since no adjusmens are made for fuure wihdrawal guaranees in his case. Hence, when wihdrawing less han he guaraneed annual wihdrawal he fuure guaranees are he same han when wihdrawing he full guaraneed amoun. However, in he laer case, he accoun value is lower and hus he value of fuure wihdrawal guaranees is higher. Furhermore, if a GMDB is included, he so-called addiional deah benefi, i.e. he par of he deah benefi ha exceeds he fund value, is reduced. Due o he maringale propery of he underlying asse process and he guaranee fee ha is deduced from he accoun value, he value of he addiional deah benefi is never greaer han he wihdrawal amoun iself. Second, if i is opimal for he policyholder o wihdraw more han he guaraneed annual wihdrawal amoun, han i has o be opimal o compleely surrender he conrac since all changes of he sae variables occur on a pro-raa basis only. In he case of surrender, he policyholder receives he accoun value afer deducion of surrender charges; in reurn he waives all claims arising from he GLWB-opion, i.e. he lifelong guaraneed annual wihdrawals and fuure deah or surrender benefis. 6 For an inroducion o Mone Carlo mehods see, e.g., Glasserman (23).

12 Thus, under opimal behaviour, he policyholder would wihdraw exacly he annual guaraneed amoun each year unil he value of he underlying fund less surrender fee exceeds he expeced value of fuure benefis. Then, he would surrender he conrac. We now describe how opimal policyholder sraegies can be found using Mone- Carlo simulaions. The ask is o maximize a conrac s value by surrendering a an opimal poin of ime allowing for F -measurable sraegies only. x 2 We call K K1... K x 1 R ω = ω an exercise sraegy. Following his sraegy, he conrac is surrendered a ime if and only if he conrac has no been surrendered before and A K, i.e. 1 K ( A ) = 1 and 1 ( ) = K A u u u { 1,..., 1}. The value of he conrac under some given sraegy K is hen given by V ( K ) as defined above. For his given sraegy, he conrac is surrendered a he sopping ime τ( K) : = inf{ {1,..., ω x 1} A K} or no surrendered a all, i.e. τ( K): = ω x, if A K } { 1,..., ω x 1. This value can be easily calculaed by Mone Carlo mehods. In wha follows we explain how an opimal sraegy can be approximaed. Obviously, ω x 2 an opimal sraegy has o be of he form K = [ k1, ) x... x[ kω x 1, ) R : if i is opimal o surrender a some fund value, i has o be opimal o surrender a any higher fund value, as well, since a higher fund value leads o a higher surrender value if he conrac is surrendered, and a lower value of fuure guaranees as well as a higher fuure guaranee fee if he conrac is no surrendered. Analogously, if i is opimal no o surrender he conrac a a given value, i is also opimal no o surrender a any lower fund value. For he deerminaion of he opimal surrender boundaries k1,..., kω x 1 we use he following backward inducion algorihm: Firs, we deermine he opimal sraegy a ime = ω - x - 1 by maximizing he conrac value following a given sraegy [ k ω x, ) for k 1 ω x. Second, we deermine 1 some opimal sraegy a = ω - x - 2 by maximizing he conrac value following a given sraegy [ kω x 2, ) [ kω x 1, ) for k ω x 2, ec. Thus, by repeaedly maximizing he conrac value for he opimal surrender boundaries, we are able o deermine opimal sraefies. 5 Resuls We use he numerical mehods presened in Secion 4 o calculae he risk-neural value of variable annuiies including GMWB or GLWB guaranees for a given guaranee fee ϕ. We call a conrac and he corresponding guaranee fee fair if he conrac s risk-neural value equals he single premium paid, i.e. if P = = ( ϕ). V V Unless saed oherwise, we use a risk-free rae of ineres r of 4%, a volailiy σ of 15%, and a single premium P = 1,. Furhermore, we le he age of he insured x = 65, and he surrender fee s =1%. Moreover, we use bes esimae moraliy ables of he German sociey of acuaries (DAV 24 R) for a male insured.

13 5.1 Deerminaion of he Fair Guaranee Fee To illusrae how he fair guaranee fee can be derived wihin our framework, in a firs sep, we analyze he influence of he guaranee fee on he value of conracs for hree differen kinds of GLWB guaranees. Conrac 1 conains a plain vanilla guaranee, some guaraneed annual wihdrawal amoun of 5% of he single premium, conrac 2 conains a roll-up a a rae of i = 6% for a maximum of 5 years, whereas an annual sep-up is considered in conrac 3. We assume deerminisic cusomer behavior: For conrac 1 and conrac 3, he insured wihdraws he guaraneed annual amoun of 5 - beginning immediaely in he firs year of he conrac; for conrac 2, we assume ha he insured person does no access his guaraneed amoun for he firs five years and hen sars o wihdraw he guaraneed amoun of 669 per annum (which has been increased due o he roll-up feaure). For all hree versions, we assume ha he conrac is no surrendered. Figure 1 shows he corresponding conrac values as a funcion of he annual guaranee fee. Figure 1 Conrac values as a funcion of he guaranee fee for differen versions of GLWB For conrac 1, he conrac value equals 1, a a guaranee fee of 43, hus his is he fair guaranee fee. For conracs 2 and 3, he fair guaranee fee amouns o 48 bps and 8 bps, respecively. 5.2 Fair Guaranee Fees for Differen Conracs In his secion we analyze conracs wih differen GLWB-versions. We sar wih conracs conaining a roll-up feaure and hen move on o conracs wih differen sepup / rache feaures.

14 5.2.1 GLWB wih Roll-Up In his secion, we analyze and compare hree differen conracs. Conrac 1 guaranees lifelong wihdrawals of 5% of he iniial premium (no roll-up). Conrac 2 provides an annual increase of he guaraneed wihdrawal amoun of 6% for a maximal period of 5 years, as long as no wihdrawals are made. The hird conrac comes wih a 5% roll-up rae and a roll-up period of 1 years. We consider all hree conracs wih and wihou a money-back GMDB opion. We assume he following policyholder behavior: In conrac 1 he policyholder wihdraws 5% of he iniial premium up o his deah saring immediaely. For conrac 2 and 3 he insured sars he wihdrawals of 669 and 814, respecively a he end of he roll-up period (5 and 1 years, respecively) and hen wihdraws annually and lifelong he corresponding guaraneed annual wihdrawal amoun. Table 1 shows he fair guaranee fee for hese hree conracs wih and wihou an addiional GMDB opion. conrac no roll-up 6% roll-up benefi for 5 years 5% roll-up benefi for 1 years sraegy w/o DB wih DB w/o DB wih DB w/o DB wih DB Wihdrawals of guaraneed amoun saring a he end of roll-up period 43 bps 48 bps 47 bps 52 bps 28 bps 35 bps Table 1: Fair guaranee fee for GLWB conracs wih roll-up The resuls show ha i is no always opimal o pospone wihdrawals unil he end of he roll-up period even if guaraneed annual wihdrawal amouns are increased in his case. The value of conrac 3 under he assumed sraegy is significanly lower han he value of conrac 1. On he one hand, a roll-up rae exceeding he risk free rae leads o an increase in he value of he guaranee if wihdrawals are posponed. On he oher hand however, he insured becomes older during he waiing period. Wih increasing age, he expeced number of wihdrawals unil deah decreases and so does he value of he guaranee. Apparenly, for our hird conrac, he laer effec is dominan. The addiional fee for deah benefi (difference beween columns wih DB and w/o DB ) increases as we move from conrac 1 o 2 and 3. This is due o he fac ha every wihdrawal leads o a reducion of he GMDB value. Since we assume fewer and laer wihdrawals in conracs 2 and 3, he corresponding GMDB opion is more valuable GLWB wih rache guaranee For he analysis of conracs wih sep-up feaure, we again compare hree differen GLWB conracs. Conrac 1 has no sep-up feaure and hus coincides wih conrac 1 from he previous subsecion. In conrac 2 a sep-up is possible every 5 h anniversary of he policy following he rules described in secion 3. Finally, conrac 3 comes wih an annual sep-up. Each of he conracs is analyzed wih and wihou an addiional money-back GMDB.

15 We assume he policyholder o annually wihdraw he guaraneed amoun unil deah (no surrender). If he guaraneed amoun is increased by a sep-up, we assume ha he policyholder immediaely increases he annual wihdrawal amoun o he new guaraneed amoun. The fair guaranee fees of he conracs are shown in Table 2. conrac sraegy Wihdrawals of guaraneed amoun beginning in year 1 no sep-up 5 year sep-up annual sep-up w/o DB wih DB w/o DB wih DB w/o DB wih DB 43 bps 48 bps 55 bps 62 bps 8 bps 88 bps Table 2: Fair guaranee fee for GLWB conracs wih differen sep-up feaures Obviously he sep-up feaure provides addiional value for he policyholders. Our analysis shows ha he more frequenly sep-ups are provided, he higher he value of he feaure and hus he fair guaranee fees are. The addiional fee for he deah benefi (difference beween columns wih DB and w/o DB ) is roughly equal for all hree conracs (abou 1% of he fair fee wihou GMDB). The effec observed in secion for conracs wih roll-up can no be found here, since in scenarios where he GMDB opion is in he money (declining fund values), he hree conracs develop similarly. 5.3 Sensiiviy Analysis wih respec o he Guaraneed Annual Wihdrawal Amoun In his secion, we look a he influence of he annual maximum guaraneed wihdrawal amoun on he fair guaranee fee for he plain vanilla GLWB conrac (i.e. neiher a roll-up nor a sep-up is included). We consider annual wihdrawal amouns of x WL =3%, x WL =4%, x WL =5%, and x WL =6%. The fair guaranee fees are displayed in Table 3. conrac sraegy Wihdrawals of guaraneed amoun annually beginning in year 1 x WL = 3% x WL = 4% x WL = 5% x WL = 6% 3 bps 12 bps 43 bps 117 bps Table 3: Influence of he maximum annual wihdrawal amoun on he fair guaranee fee for a GLWB conrac The maximum annual wihdrawal amoun noably influences he fair guaranee fee. Raher low annual wihdrawal raes of 3% lead o a low fair guaranee fee of 3 bps, while a fee of 117 bps is necessary o back a GLWB opion wih 6% annual wihdrawals. Unlike in a GMWB conrac, he oal wihdrawal amoun is no resriced. As a consequence, he influence of he annual maximum wihdrawal amoun in a GLWB conrac is considerably higher han in a GMWB conrac. For more deails on he comparison of GMWB and GLWB see Secion 5.7.

16 5.4 Sensiiviy Analysis wih respec o he Insured s Age In he previous secions we assumed he insured o be 65 years old. We now deermine he influence of he age on he conrac value. In paricular wih respec o lifelong guaranees, he age is of significan influence on he conrac value, since moraliy raes roughly increase exponenially in age. We calculae conrac values for insured persons aged 55, 65, 75, 85 and 95 years, respecively. The policyholder s sraegy is assumed o be he same in all conracs, namely lifelong wihdrawals of he guaraneed annual amoun from he beginning of he conrac and no surrender. Table 4 shows he fair guaranee fees for hese conracs. Age sraegy Wihdrawals of guaraneed amoun annually (5) beginning in year bps 43 bps 11 bps ~.5 bps ~.5 bps Table 4: Fair guaranee fee for conracs wih GLWB under differen ages of he insuran Since wihdrawals are guaraneed lifelong, he fair guaranee fee is decreasing in he insured s age wihin he considered age inerval. While he fair guaranee fee for a 55 year old amouns o 15 bps i decreases o less han 1 bp for an 85 or 95 year old person. These resuls explain why mos of he curren GLWB producs require a minimum age of 6 years when wihdrawals sar. Some alernaive o requiring for a minimum age would be o link he guaraneed annual wihdrawal amoun o he age a wih wihdrawals are acually sared. Thus, we now fix a guaranee fee of 5 bps and deermine he fair wihdrawal rae for differen ages, i.e. he wihdrawal rae for which he conrac s risk-neural value coincides wih he premium paid. The fair wihdrawal raes are displayed in Table 5. age sraegy Wihdrawals of guaraneed amoun annually beginning in year % 5.2% 6.6% 9.2% 13.4% Table 5: Fair wihdrawal rae for GLWB conracs for a guaranee fee of 5 bps and differen ages Obviously, for he same fee, higher wihdrawal raes can be guaraneed o older people. Whils for a 55 year old only an annual wihdrawal amoun of 44 can be guaraneed, his amoun increases o 134 for a 95 year old. 5.5 Analysis of he Longeviy Risk As menioned above, moraliy raes are an imporan facor for he calculaion of such conracs. So far, we used he moraliy able DAV 24R for our calculaions. An increase of longeviy ha exceeds he rend embedded in his able would have a negaive impac on he insurer s profiabiliy. To analyze his risk we calculae he fair guaranee fee assuming differen moraliy probabiliies. Table 6 gives he fair guaranee fee for he plain vanilla GLWB conrac for differen ages assuming he moral-

17 iy able DAV 24R as above and addiionally under he assumpion ha moraliy raes drop o 7% of he raes given in his able. age moraliy able DAV 24R 15 bps 43 bps 11 bps ~.5 bps ~.5 bps 7% of DAV 24R 138 bps 58 bps 2 bps ~1.5 bps ~.5 bps Table 6: Fair guaranee fee of a GLWB conrac for differen ages and differen moraliy scenarios The resuls show ha he value of lifelong wihdrawal guaranees is raher sensiive wih respec o longeviy. For a 55-year old, he fair guaranee fee increases by almos one hird o 138 bps, and for an 75-year old, he fair guaranee fee doubles if moraliy raes drop by 3%. 5.6 Sensiiviy Analysis wih respec o Capial Marke Parameers In his secion we analyze he influence of he capial marke parameers r and σ on he conrac value. Up o now we based our calculaions on he assumpion of r=4% and σ=15%. We consider he plain vanilla GLWB conrac assuming a cusomer who wihdraws he guaraneed annual amoun and does no surrender. We vary he riskfree rae of ineres r as well as he volailiy σ. Table 7 shows he fair guaranee fee for differen combinaions of he capial marke parameer values. risk-free rae r=3% r =4% r =5% volailiy σ = 1% 48 bps 19 bps 7 bps σ = 15% 85 bps 43 bps 2 bps σ = 2% 124 bps 69 bps 41 bps Table 7: Influence of he capial marke parameers r and σ on he fair guaranee fee for a GLWB conrac As expeced, he fair guaranee fee is decreasing in he risk-free rae of ineres and increasing in he volailiy since, on he one hand, he risk-neural value of a guaranee decreases wih increasing ineres raes; and, on he oher hand, opions are more expensive when volailiy increases. Changes in volailiy have a remendous impac on he opion values and, hus, on he fair guaranee fee. A incepion of he conrac and wih some producs also during he erm of he conrac, he insured has he possibiliy o influence he volailiy by choosing he underlying fund from a predefined selecion of muual funds. Since for some producs offered in he marke he fees do no depend on he fund choice, his possibiliy presens anoher valuable opion for he policyholder. For any risk-free rae r, he fair guaranee fee for σ = 2% is more han wice as high as he fee for σ =1%. Thus, one imporan risk managemen ool for insurers offering variable annuiy guaranees is he sric limiaion and conrol of he ypes of underlying funds offered wihin hese producs.

18 An alernaive would be o link he guaraneed annual wihdrawal amoun o he fund s expeced volailiy. One insurer gives a guaraneed annual wihdrawal amoun of 5% of he single premium paid for hree differen funds wih differen sock raios beween. However, he fee increases in he fund s sock raio. Besides, a fourh fund wih an even higher sock raio is offered. However, he guaraneed annual wihdrawal amoun is reduced o 4.5% if his fund is chosen. Therefore, in a nex sep we deermine he fair wihdrawal rae for differen capial marke parameers for a guaranee fee of 5 bps. The resuls are shown in Table 8. risk-free rae r=3% r =4% r =5% volailiy σ = 1% 4.9% 5.7% 6.6% σ = 15% 4.6% 5.2% 5.8% σ = 2% 4.1% 4.7% 5.1% Table 8: Influence of he capial marke parameers r and σ on he fair wihdrawal rae for a GLWB conrac (guaranee fee 5 bps) Obviously, for a given guaranee fee he fair wihdrawal rae is increasing in he riskfree rae of ineres and decreasing in he volailiy. 5.7 Comparison of GLWB and GMWB As menioned above, GLWB-opions are a recen variaion of GMWB-producs. In wha follows, we compare he wo produc ypes. In a firs sep, we analyze he value of a conrac wih GLWB under differen wihdrawal raes. The values for conracs under he wihdrawal raes 3%, 4% and 5% as a funcion of he annual guaranee fee are shown in Figure 2. Figure 2 Value of GLWB conracs as a funcion of he annual guaranee fee The hree curves are nearly parallel. The resuls show, as expeced, ha he conrac is significanly more valuable for a higher annual wihdrawal rae. In a second sep, we analyze he influence of he annual wihdrawal rae on a GMWB conrac. Figure 3 displays he conrac values for wihdrawal raes of 5%, 6% and 7%.

19 Figure 3 Value of GMWB conracs as a funcion of he annual guaranee fee For small guaranee fees, he conrac values are very close; only for higher guaranee fees, he gap beween he differen conracs increases and he conrac wih a wihdrawal rae of 7% is significanly more valuable han he ones wih lower raes. The obvious difference beween Figure 2 and Figure 3 is due o he differen srucure of he GMWB and GLWB opions. In a GMWB he oal wihdrawal amoun is limied (for example by he iniial premium); a higher annual wihdrawal rae in a conrac wih GMWB resuls in a lower number of wihdrawals. By conras, high wihdrawals ha are guaraneed wihin a GLWB are guaraneed unil deah and herefore increase he value of he opion significanly. Finally, we compare he fair wihdrawal rae wihin he GLWB and he GMWB opion for given guaranee fees. As a policyholder s sraegy we assume ha he annual guaraneed amoun is wihdrawn unil deah wihin a GLWB and unil he oal wihdrawal amoun has been reached wihin a GMWB. For he analysis we choose a guaranee fee of 12 bps, which is he fair guaranee fee for a common produc in he US-marke, a 7% GMWB; furher, we examine he fair guaranee fee for he GLWB wih a wihdrawal rae of 5%, which is 43 bps. The resuls are shown in Table 9. Benefi Guaraneed Lifeime Wihdrawal Benefi Guaranee fee 12 bps 4.% 7.% 43 bps 5.% 12.5% Guaraneed Minimum Wihdrawal Benefi Table 9: Fair wihdrawal rae for GLWB and GMWB under differen guaranee fees For he guaranee fee of 12 bps, he fair wihdrawal rae of a GLWB-opion is 4%. This means, for he same guaranee fee he policyholder can wihdraw 4 annually wihin a GLWB and 7 in a GMWB. Furhermore, a conrac wih 5% GLWB corresponds o a GMWB conrac wih 12.5% guaraneed annual wihdrawal. This valuaion is of relevance, as some insurers offer heir cliens he choice beween a GMWB and a GLWB wih differen wihdrawal raes.

20 5.8 Resuls under Opimal Cusomer Behavior In his las subsecion, we show he resuls for opimal cusomer behavior ha have been derived using he mehods described in Secion 4. Firs we calculae he bounds k for he opimal sraegy in a conrac ha conains a GLWB for a guaranee fee of 5 bps. Figure 4 displays he bounds k, which sar a abou 18,7 in he firs year and decrease slighly in he beginning and sronger as he insured s age increases. Figure 4 Opimal Surrender Sraegy for a GLWB conrac and a guaranee of 5 bps The fair guaranee fee under opimal policyholder behavior amouns o 49 bps. This is slighly lower han he curren fees of mos insurance companies for his benefi. I is noable ha he fair guaranee fee under opimal cusomer behavior differs only slighly from he fair fee assuming deerminisic behavior (annual wihdrawal, no surrender). Bauer e al (26) found much larger differences beween deerminisic and opimal clien behavior for GMAB, GMIB and GMWB producs. The reasons for his is, as we have seen above, ha in GLWB producs, he clien has fewer choices han in oher producs since opimal behavior is always characerized by eiher wihdrawing he guaraneed amoun or surrendering he conrac. 6 Summary and Oulook Guaraneed lifelong wihdrawal benefis are he laes innovaion in he variable annuiy marke. The producs are very popular since hey cover longeviy risk like a regular annuiy bu combine his feaure wih a permanen availabiliy of he remaining accoun value (if posiive). However, our analyses have shown ha such producs are raher risky for he insurer: Whils hese producs are much less sensiive wih respec o clien behavior han oher guaranees ypically embedded in variable annuiies (cf. Bauer e al. (26)), he sensiiviy wih respec o changes in ineres raes and fund volailiies as

21 well as moraliy raes is significan. This is paricularly dangerous considering he long ime horizon of he producs and he fac ha according o marke survey (cf. Lehman Brohers (25)) ofen only dela risk is hedged. Since our asse model is raher simple, a worhwhile exension migh be an analysis of such producs in a Lévy-ype framework wih sochasic ineres raes. Also, in paricular for GLWB producs, i would be ineresing o see how our resuls change in a model wih sochasic moraliy raes (cf. e.g. Cairns e al. (25) or Bauer e al. (27)).

22 References Aase, K.K. und Persson, S.A. (1994): Pricing of Uni-Linked Insurance Policies. Scandinavian Acuarial Journal, 1, Anderson, L. (1999): A Simple Approach o he Pricing of Bermudan Swapions in he Muli-facor Labor Marke Model. Journal of Compuaional Finance, 3, Bauer, D.; Kling, A. and Russ, J. (26): A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies. Working Paper, Ulm Universiy. Bauer D.; Börger M.; Ruß, J. and Zwiesler, H-J. (27): The Volailiy of Moraliy. Working Paper, Ulm Universiy. Bingham, N.H. und Kiesel, R. (24): Risk-Neural Valuaion Pricing and Hedging of Finanicial Derivaives. Springer Verlag, Berlin. Cairns, A.J., Blake, D., Dowd, K., 25b. Pricing Deah: Frameworks for he Valuaion and Securiizaion of Moraliy Risk. ASTIN Bullein, 36: Dahl, M., Møller, T., (26): Valuaion and hedging life insurance liabiliies wih sysemaic moraliy risk. Insurance: Mahemaics and Economics, 39, Glasserman, P. (23): Mone Carlo Mehods in Financial Engineering. Series: Sochasic Modelling and Applied Probabiliy, Vol. 53. Springer Verlag, Berlin. Lehman Brohers (25): Variable Annuiy Living Benefi Guaranees: Over Complex, Over Popular and Over Here? European Insurance, 22. April 25. Milevsky, M. und Salisbury, T.S. (26): Financial valuaion of guaraneed minimum wihdrawal benefis. Insurance: Mahemaics and Economics: 38, Møller, T. (21): Risk-minimizing hedging sraegies for insurance paymen processes. Finance and Sochasics, 5, Sloane, W. R. (197): Life Insurers, Variable Annuiies and Muual Funds: A Criical Sudy. Journal of Risk and Insurance, 37, S. 99.