The bequest motive and single people s demand for life annuities

Transcription

1 The beques moive and single people s demand for life annuiies Carlos Vidal-Meliá and Ana Lejárraga-García 1 Absrac. The main objecive of his pap is o go deep ino he annuiy puzzle by inroducing he alruisic and sraegic beques moive and demining wheh his really is a relevan issue affecing he heoreical decision o purchase life annuiies. Wih his end in view we develop an opimizaion model based on ha firs pu forward by Lejárraga e al. (22), hen add o i elemens from oh models such as Friedman and Warshawsky s (199) and Jousen s (1998 and 21) which include he beques moive. We also analyse welfare by calculaing he equivalen wealh in diffen conexs: he possibiliy of having access o an acuarially fair life annuiy or programmed wihdrawal marke, he incorporaion of so-called marke impfecions, and he inclusion of he assumpion ha he individuals already have par of heir wealh in pre-exising life annuiies. Keywords:. Capializaion, Pension Funds, Phased Wihdrawal, Reiremen, Uiliy. 1 Inroducion I is well known ha he are various reasons for individuals low level of demand for annuiies. Fundamenally hese are: 1) The already exiss a cain level of proecion agains he risk of longeviy in he annuiies of he public pension sysem which provides a minimal level of resources in reiremen. 2) The annuiy marke is no "acuarially fair". This is a consequence of he applicaion of adminisraive and managemen coss by insurance firms. 3) The problem of advse selecion. This obliges insurs o use moraliy and survival ables ha are diffen from hose applicable o he genal populaion. 4) Individuals use family self-insurance. This allows hem o ake advanage of he possibiliies of join consumpion, sharing ou financial resources among he membs of he family, heby reducing he araciveness of annuiies. 5) The pcepion of he accumulaed reiremen fund in he form of lump sum allows he individual o face he appearance of unforeseen fuure expendiures such as exraordinary medical coss no coved by he usual medical insurance, he cancellaion of an ousanding morgage, or simply he possibiliy of going on a longdesired journey. 6) The diffen ways in which annuiies are axed. In some counries ax regulaions penalize annuiies; in Belgium, for insance, his is he main reason why hey are unpopular. 7) In addiion, according o Impavido e al. (23), he low level of demand for annuiies may be relaed o radiional producs ha seem o provide consums wih only a binomial allocaion of risk. Eih self insurance or complee insurance agains he mos imporan ypes of risks are available in many counries. All hese aspecs have been deal in he work of Beniez-Silva (23), Brown (1999, 21, 23), Brown and Poba (2), Friedman and Warshawsky (199), Kolikoff and Spivak (1981), Lejárraga e al. (22), Michell e al. (1999), Vidal e al. (23), and Yaari (1965). Anoh facor ha could be decisive in he choice of he mode of wihdrawing he funds available is he exisence of moivaions o leave a beques o he heirs, alhough he are quie conradicory resuls in he liaure concning his aspec. The "annuiy puzzle" is ha he empirical evidence shows he exreme rariy of volunary privae individual annuiy conracs even hough, according o Yaari (1965), individuals would be be off holding only annuiized asses in he absence of a beques moive, or a porfolio of annuiized and radiional asses in he presence of a beques moive. I mus be emphasized ha in Yaari s model he purchase of life annuiies is opimal und he following assumpions: 1) Consums maximize he expeced (Neumann-Morgensn) uiliy wih separabiliy and addiiviy. 2) The only risk faced by consums is longeviy risk. 3) The are no oh asses, such as housing, ha off anoh source of annuiy ha is characisically uncorrelaed. 4) The individual is single and has no descendens. 5) The individual has no access o pensions in he form of life annuiies from he firs pillar of he social securiy sysem. 6) The annuiy marke is acuarially fair. This simulaneously implies ha: a) The moraliy ables ha insurance firms apply coincide wih he consum s probabiliies. b) The echnical ines rae ne of adminisraive coss coincides wih he risk free rae. 7) The is only one convenional asse which pays a given ines rae. 8) Consums can borrow and lend a his same rae. Davidoff e al. (23) show ha he condiions und which he purchase of annuiies is opimal are no as demanding as hose se ou by Yaari (1965). If financial markes are complee, one only requires ha no beques moive exiss and ha he expeced rae of reurn on annuiies is grea han ha on a refence financial asse. Parial annuiizaion is opimal when he condiion of complee insurance marke is relaxed. The is a curious siuaion in Swizland, Bul (23), which conradics he annuiy puzzle : mos individuals choose o receive annuiies, even hough hey can already coun on a firs pillar of suppor wih a high degree of proecion agains longeviy in he form of large annuiies, and hough benefis received in he form of capial sums receive far be ax reamen. 1 Faculad de Economía. Deparameno de Economía Financia. Univsidad de Valencia. Avenida de los Naranjos, s/n Valencia (Spain). [email protected] (Corresponding auhor). [email protected] BELGIAN ACTUARIAL BULLETIN, Vol. 4, No. 1, 24

2 The aim of he presen sudy is o go deep ino clarifying he "annuiy puzzle" hrough he inroducion of he beques moive 1, boh alruisic (he pension simply wans o leave a beques o his family wihou expecing anyhing in reurn) and sraegic (he pension wans o give his family an incenive o look af him in his old age by promising o leave hem a beques), and demine wheh his is really a major facor influencing he heoreical decision o acquire annuiies. To bring he model clos o realiy, he possibiliy is also consided ha he individual can already coun on a pre-exising life annuiy, and ha he annuiy marke is no acuarially fair. The work is srucured as follows. Af his brief Inroducion, Secion 2 presens an opimizaion model based on ha firs pu forward by Lejárraga e al. (22), supplemened by elemens from oh models ha include he beques moive, in paricular, hose of Friedman and Warshawsky (199) and Jousen (1998 and 21). The following hree secions analyse an individual s welfare by calculaing he equivalen wealh in diffen conexs: wih he possibiliy of access o an acuarially fair annuiy or programmed wihdrawal marke (Secion 3), incorporaing so-called marke impfecions (Secion 4), and assuming ha individuals already hold par of heir wealh in pre-exising annuiies (Secion 5). Finally, Secion 6 presens he conclusions, policy recommendaions and fuure research. 2 The model One of he reasons why i has no been cusomary in he analysis of welfare o include he exisence of he moivaion o leave a beques could well be for he sake of mainaining he analyical simpliciy of he model. Also, howev, Beniez-Silva (23), i could be due o he lack of consensus abou he relevance of he beques moive in an individual s decision wih respec o conracing annuiies, and abou how o model his beques facor. Brown (1999) quesions he imporance of he beques moive in influencing marginal annuiy purchasing decisions neih he presence of children nor of he beques moive are deminans in a life annuiy purchasing decision. The auhor saes ha a simple life cycle model wihou bequess gives predicions ha are consisen wih marginal annuiy purchasing behaviour, and is hefore a useful firs approximaion o behaviour. These resuls coincide wih hose of Hurd (1987), who concludes ha beques moives have no significan effec on he marginal financial behaviour of eldly individuals. This is furh suppored in a la work, Hurd (1989), which finds ha many bequess are apparenly accidenal, resuling from he uncainy in he ime of deah, wihou he being any evidence for real beques moives. These findings are in conradicion wih he conribuions of Bnheim (1991) and Lain and Jus (1996), who claim ha beques moives do indeed influence he decision o purchase annuiies. Also, Friedman and Warshawsky (199) conclude ha he presence of a beques moive will reduce or eliminae he 1 Acorrding o Impavido e al. (23), he are hree main reasons for bequess: uncain lifeimes, alruism, and sraegic behavior owards heirs. Uncain lifeimes and incomplee insurance markes resul in involunary bequess as individuals need o save for precauionary moives. If insurance and capial markes are impfec, uninsured risks relaing o healh and longeviy may give rise o precauionary moives for presving wealh in old age. We do no consid uncain lifeimes in his pap. demand for annuiies, even when heir reurn surpasses he real marke ines rae. In a previous work, Friedman and Warshawsky (1988), hese auhors had indicaed ha he inacion of a delibae beques moive and he acuarial unfairness of he annuiies offed by insurance firms could lead individuals o rejec purchasing hem. In a more recen work, Brown (21) inroduces he beques moive ino he single pson model, wihou disinguishing beween alruisic and sraegic moives. The resuls indicae ha in vy few cases does he obligaory purchase of annuiies lead o low welfare han when no annuiies are purchased, alhough he welfare gains are less han hose obained in he no-inhiance life cycle model. According o Bnheim (1991), if he reiree can genae savings ha are exnal o he life annuiies and/or purchase addiional, annually renewable, life insurance, no major diffences resul in welfare gains by incorporaing a beques moive. The consumpion opimizaion model wih a beques moive is based on ha developed by Lejárraga e al. (22), i being necessary o inroduce a funcion ha represens he uiliy of he wealh exising a each poin in ime ha will allow he individual o bequeah an amoun o he heirs on he dae of deah. Le U(C ) be he welfare funcion of he piod corresponding o age, defined ov he consumpion (C ); V(W ) he welfare funcion for he piod corresponding o age, defined ov he wealh (W ); δ he pure rae of ime prefence, i.e., he classical exponenial discoun facor of fuure uiliy; ω an individual s maximum lifespan; and e r he reiremen age. The expeced uiliy (EU) is given by: = whe (ages in years): ω 1 = U ( C ) (1 + δ ) V ( W h (1 + δ ) + 1 ) Pe + r Pe q r EU (1) W +1 : Wealh corresponding o age P : Probabiliy ha an individual of age e r survives +1- e r years more. q : Probabiliy ha an individual dies a age. h : Relaive weigh of he uiliy of he beques consided by he individual a age wih respec o he expeced uiliy corresponding o he flow of consumpion a age : C. -P q : Probabiliy ha an individual of age e r survives o age bu dies before reaching age +1. I has been assumed ha he discoun rae of he fuure uiliy of he wealh o bequeah, δ, coincides wih he rae of prefence of he uiliy of consumpion, since he exiss no jusificaion for any possible alnaive values. This is he assumpion used by mos researchs, including Brown (21), Friedman and Warshawsky (199), Hurd (1989), and Fisch (1973). Jousen (1998 and 21), howev, consids ha δ, coincides wih he real marke ines rae. One of he demining elemens in he consumpion opimizaion model when he beques moive is included is he parame h, which indicaes he individual s assessmen of he amoun of 6

3 wealh ha could be bequeahed a any given ime o his or h heirs. Friedman and Warshawsky (199) consid ha his parame does no depend on age, and ha i is relaed o he beques-consumpion raio corresponding o he final piod: W ω / C ω-1. Likewise Brown (21) and Jousen (1998) assume i o be consan ov he life cycle. Oh researchs, Hurd (1989 and 1999) for insance, do no weigh he uiliy of bequess relaive o ha of consumpion, and apply o boh he same valuaion on he par of he individual. Fisch (1973) and Yaari (1965) consid he parame ha reflecs he value given o he possibiliy of leaving a beques, h, o be a hump-shaped funcion, due o he grea imporance ha individuals give he beques in he mid-years of life when family membs have a grea dependence on hem, decreasing in reiremen. This is valid for he case of he individual s moivaions for leaving a beques being alruisic. If, howev, he reiree s ends are sraegic in he sense of seeking o encourage family membs o care for him or h in old age in exchange for he promise of a beques, i would be more appropriae o consid ha he parame h increases wih age. 2.1 The individual does no have access o he annuiy marke Wih he assumpions ha individuals have no access o he annuiy marke and ha hey value he exisence of a beques o leave heir heirs, he consumpion opimizaion model is, following Friedman and Warshawsky (199): max C ω 1 = U ( C ) r 1 e P e r (1 + δ ) V ( W + 1) h + 1 e P r e q r (1 + δ ) W 1+ r) W (2) s.. C = ( + 1 (3) W, (4) whe: r: Nominal expeced risk free rae (assumed consan ov he reiree s lifeime). Only he uiliy diving from he possible bequess once deah occurs is consided. No accoun is aken of he fac ha in cain cases he individual may wan o make gifs of wealh while sill alive, insead of waiing for he beques o be made once dead. Making a gif of money before deah could come abou because of he diffen ax reamen applied o he wo siuaions, or simply because of he individual s prefence for his or h relaives o enjoy as soon as possible he goods (financial wealh) ha he or she is able o give hem in life. 2.2 The individual can conrac a single-premium life annuiy Life annuiies are a ype of pension markeed by insurance firms in which, in exchange for an iniial premium, hey commi hemselves o paying cain piodic amouns unil he deah of he policyhold, heby aking on he annuian s longeviy risk. In he absence of annuiies, reirees could reduce heir annual consumpion in response o he uncainy in he dae of heir deah. They hen, howev, would risk dying before consuming all heir available wealh, which represens a cos of he missed opporuniy for consumpion. This cos will be less if he individuals value he wealh ha remains a he ime of heir deah as a beques. Wih he assumpions ha he individual has access o he annuiy marke and has a beques moive, he consumpion opimizaion model is based on maximizing he objecive funcion given in expression (2), bu subjec o he following consrains: s.. C = W ) ( 1+ r) W+ 1 + Ae (1 + α (5) r W, (6) Expressed in his form, consrain (5) is similar o ha in Brown e al. (1999, 2), wih W being he iniial wealh af he deducion of he wealh allocaed o annuiizaion, and A he iniial amoun of he life annuiies purchased in exchange for a single premium W ANNUITIES, given, when he exiss no revsion on any beneficiary, by he following formula: A W = ANNUITIES ω + α (1 ) θ = [(1 + i) ] ( -1) + 1 P * (7) A = A ( 1+ α ), (8) This akes ino accoun boh he adminisraive coss (he m θ) and he survival and moraliy probabiliies ha he insur applies P * 1 in pricing he annuiy ( + ) which may diff from he consum s subjecive probabiliies as well as he variaions ha may exis beween he real echnical ines rae guaraneed by he insur (i) and he nominal marke ines rae. The parame α represens he annual accumulaed growh of he annuiy. Two cases are consided in he calculaions ha follow: α equal o zo, wih he annuiy consan in nominal ms; and α equal o he expeced Consum Price Index (CPI), wih he annuiy consan in real ms. Alhough he CPI is he aken as consan, he resuls would be similar if i we allowed o vary, assuming ha he annuiy is pfecly indexed o he CPI. This is possible, Michell and McCarhy (22a and 2b) and James and Song (21), in counries such as he Unied Saes, he Unied Kingdom, Chile, Israel, and more recenly France, in which he govnmen issues bonds fully indexed o he CPI. This allows insurs o off annuiy policies in which he variaion of he annual conribuion or m depends on he variaion in he CPI for each fiscal year. 2.3 The individual can purchase a phased wihdrawal conrac In a programmed wihdrawal conrac, Devesa and Vidal (21), he reiree mainains an individual accoun of annuiizaion capial, and can wihdraw each year an amoun equal o he quoien of he 7

4 accumulaed fund in ha accoun and he sum needed o pay ou one uni of pension, as a funcion of he real reurn on he accumulaed fund, of he life expecancy in he corresponding year, and of he echnical ines rae used which will vary each year according o he evoluion of he markes. Programmed wihdrawal, Valdés and Edwards (1998) and Valdés- Prieo (22), allows a beques o be lef ha will be equal o he unused balance of he individual accoun, bu he dae of paymen and he amoun of ha beques are random. An alnaive in his sense is he (fixed or variable) life insurance policy, which has he advanage ha he reiree can se he amoun and he paymen dae. Und his assumpion, he maximizaion problem including he beques moive becomes wih max C W ω 1 = U ( C ) + 1 (1 + δ ) V ( W W h (1 + δ ) s.. PW + 1 PW P ) + Pe q r C + A = W 1+ r) W +1 W (9) ( (1) W, (11) = W W (12) = W 1+ i) A PW PW 1 ( 1, (13) whe W PW is he accumulaed fund in he annuiizaion accoun a he beginning of he year corresponding o age, and i is he real expeced reurn on he annuiizaion accoun. A each age, he maximum annuiy ha can be wihdrawn, A, will be given by: A = ω W (1 + α) PW s= [(1 + i) ] (1 + i) θ, s s+ 1 s+ 1 P * (14) The individual is assumed always o wihdraw he maximum pmied annuiy (A ), and o hold an iniial unannuiized wealh of W. I is also assumed ha if he consumpion in he year corresponding o age is less han he amoun of he programmed wihdrawal, A, he surplus will be added o he unannuiized wealh. This assumpion is easily relaxed by adding a new variable o he model he wealh wihdrawn from he annuiizaion accoun und he consrain ha is amoun is less han he maximum pmied annuiy. has a beques moive, and has no access o he annuiy marke, he model has he following consrains: whe and wih s.. C + R = W 1+ r) W +1 ( (15) W, (16) W = W UW, (17) W = W UW + W PA, (18) W UW : Level of iniial unannuiized wealh. W PA : Level of iniial wealh in pre-exising annuiies. R is a pos-payable life annuiy, assumed o dive from a preexising public pension scheme, given by he expression: R W = ω (1 + π ) = [ + ] (1 r) R = R PA p g (19) ( 1+ π ), (2) in which he wealh providing his pre-exising annuiy is evaluaed according o he genal moraliy probabiliies ( +1- P g ) and he real marke ines rae, and he annuiy is aken o be CPI-indexed (π ). Similarly, in he case ha he individual can volunarily conrac a life annuiy, he model involves maximizing he same objecive funcion (2), bu subjec o he following consrains: C + R + A = W 1+ r) W +1 whe W is he iniial unannuiized wealh: ( (21) W, (22) W = W UW (23) and W = W UW + W PA + W ANNUITIES (24) and A is calculaed from expressions (7) and (8). 2.5 The uiliy funcion and he soluion of he model Und he assumpion of consan relaive risk avsion (which implies decreasing absolue risk avsion), he uiliy funcion corresponding o he individual s consumpion is given by he expression: 2.4 The individual has a pre-exising life annuiy Considing he annuiy purchasing decision of an individual who already holds par of his or h wealh in a pre-exising life annuiy, 8

5 whe: C + 1 (1 + π ) 1 β U ( C ) = C Ln + 1 (1 + π ) 1 β, 1, if β 1 if β = 1 (25) β >, he Pra-Arrow coefficien, represening boh risk avsion and he invse of he elasiciy of inemporal consumpion subsiuion. In pracice, he avsion coefficien and he consumpion subsiuion elasiciy do no have o be invsely relaed, nor even necessarily linked. While subsiuion elasiciy reflecs consum prefences for diffen piods, he risk avsion coefficien indicaes how individuals wish o move heir consumpion beween alnaive saes of naure. Despie hese limiaions, he poenial uiliy funcion wih decreasing absolue risk avsion and consan relaive risk avsion has become he mos widely used assumpion in he financial (and even he macroeconomic) liaure in he inemporal conex. The is currenly a rend owards he revision of his concep. Davidoff e al. (23) explore he possibiliy ha he uiliy depends on a sandard of living, i.e., ha he piod s uiliy is a funcion of presen and fuure consumpion. Ponzeo (23), using he recursive uiliy funcion of Epsein and Zin, aemps o separae risk avsion from subsiuion elasiciy. The uiliy diving from he beques ha remains a each ime is given by he same isoelasic funcion, in which deah has been assumed o occur half way hrough he corresponding year (uniform disribuion of deahs), so ha he accumulaed wealh on he dae of deah will be equal o ha a he beginning of he piod +1, discouned for one half year: W +1 [(1+r)] -1/2. Hence, he funcion ha gives he value of he uiliy of he beques is given by he expression: V (W + 1 [(1 + r) ] 1/ W (1 + π ) 1 β ) = W + 1 Ln + 1 (1 + π ) 2 [(1 + r) ] 1 β 1/ 2, 1, if β 1 if β = 1 (26) The level of risk avsion ha is used in he beques uiliy funcion is he same as ha applied o consumpion. This is he criion adoped by mos researchs, Brown, Fisch (1973), and Friedman and Warshawsky (199) for insance, alhough for Fisch (1973), he level of risk avsion migh be expeced o be high in he beques han in he consumpion uiliy funcion, since individuals may be less willing o accep risk wih respec o he welfare of heir heirs han wih respec o heir own welfare. Jousen (1998) akes he compleely conrary sandpoin, considing ha individuals are less avse o risk wih respec o gifs and bequess han in heir psonal consumpion, jusifying he use of zovalued risk avsion parame refleced in he choice of a quasilinear beques uiliy funcion. Hurd (1989), oo, consids ha he level of risk avsion associaed wih he beques moive is equal o zo. The mahemaical model was implemened in he language of he LINGO suie of opimizaion programs. The resuls of he compuaions are presened in he various figures and ables. 2.6 Opimal consumpion pah wih a beques, for each ype of pension The soluion of he model yields he opimal consumpion pah ha maximizes he individual s expeced uiliy including a possible beques. The following assumpions and parame values we used in he calculaions: 1. The GRMF-95 survival and moraliy ables. 2. The insurance firm applies no ype of coss in he singlepremium life annuiy conrac θ = 1). 3. The survival probabiliies ha he insur uses in seing he premium coincide wih he consum s probabiliies ( +1- P * = +1- P ). 4. The nominal marke ines rae, r, coincides wih he annuiy s echnical ines rae, and is equal o 4.545%. 5. The level of risk avsion β akes he hree values (.7, 2.9, and 4.4). While he is no consensus in he liaure on which values should be used for he degree of avsion o risk, he presen sudy uses values close o hose employed in Brown (1999). 6. Reiremen age for boh men and women is Expeced inflaion rae equal o 1.5%. 8. Rae of prefence according o he individual s level of δ = λ ( 1+ r) 1, impaience given by he expression [ ] whe he values of he parame λ (2, 1, and.25) classify he individuals as, respecively: (A) vy impaien, (C) indiffen o impaience, and (E) vy paien. In mos of he paps cied he level of impaience is no usually emphasized. Seldom are impaien or vy impaien individuals consided which could be due o he fac ha hey presen grea difficulies o calculae. Two alnaives we also aken ino accoun for he funcion h. On he one hand, when he exis alruisic moives for he beques, he funcion was aken o decrease af he age of 65, since his is he individual s inacive phase. On he oh hand, if he exiss a sraegic ines in bequeahing wealh in exchange for possible assisance from he family in old age, he funcion h was aken as increasing wih age. In paricular, he values of h are he following: a. Alruisic beques moives: h = h 1 /1.2 ; h ω = 2, < ω (27) b. Sraegic beques moives: h h 1.2 ; h 2, > e r (28) = 1 e r = 9

6 As was noed above, he is no consensus in he liaure on how o model beques moives, and he values consided for he parame weighing he uiliy of beques relaive o he uiliy of consumpion are quie disparae. Thus, Brown (21) uses wo diffen hypoheses.5 and 1. Fisch (1973) consids values in a range of approximaely or wih he rae of consumpion prefence hypohesis 2 saring from he age of 65. The beques parame 3 applied in he model of Jousen (1998) is equal o 5.5*1-5. Finally, Friedman and Warshawsky (199) demine he opimal pcenage of wealh o annuiize assuming ha he beques parame can vary beween and 1. Figure 1 shows he opimal consumpion pah ha maximizes he expeced uiliy for each of he ypes of pension sudied, for a man who wans o leave a beques o his heirs, considing he moives o be alruisic in he firs case 4, and sraegic in he second During he firs years, he level of consumpion is, in all he assumpions, less han he corresponding level in he individual case, since in his model he amoun of wealh remaining in each piod conribues posiively o he expeced uiliy ha is o be maximized, so ha he individual prefs o give up a par of his consumpion in exchange for more wealh o bequeah. Wha phaps aracs he aenion mos in Figure 1 is he opimal consumpion profile in he case whe an individual does no have access o he annuiies marke (1), changing from concave o convex because i deals wih exreme cases. The convex profile represens an individual who is vy impaien o consume, who places much more value on presen rah han fuure consumpion and, in addiion, does no have much avsion owards ouliving his resources. Because of his and his probabiliies of survival, which are much high in he years immediaely following reiremen, his consumpion is concenraed in he iniial years. An individual wih lile impaience for consumpion, on he oh hand, places more value on fuure rah han presen consumpion. Wih a high level of avsion o he risk of longeviy, his individual ends o hold back on consumpion o cov he numb of years he may possibly live. Table 1 shows he diffences beween opimal consumpion for each ype of pension, wih special emphasis on he beques moive. Alhough Figure 1 gives an idea of he pah of opimal consumpion, due o he being only small diffences i is difficul in some cases o disinguish which individual consumes more, he one wih an alruisic moive or he one wih a sraegic moive. As Table 1 shows, i can be said ha in genal ms he individual wih an alruisic moive will always consume more han he individual wih a sraegic moive, alhough some excepions can be found which will depend on he ype of pension and he combinaion of impaience for consumpion and he degree of risk avsion. In hose cases whe consumpion is high for an individual wih sraegic moives, i always begins a a more 2 To demine he range of values used by Fisch (1973), accoun was aken of he effec of he uiliy funcion discoun rae for each of he wo hypoheses consided (approximaely 4.17% and 8.7%, respecively) which is incorporaed ino he weigh of he beques uiliy funcion. 3 Beques moive is in he form of a linear beques uiliy m. The parame on he linear beques uiliy m is 5.5* Throughou he sudy, individual case is used o ref o he siuaion of a single individual wih no beques moive. advanced age - from ages 78, 92, 15 and 81 years respecively for No annuiies, Unindexed annuiy, Indexed annuiy and Programmed wihdrawal. This consumpion behavior is pfecly undsandable since he pension wih a sraegic beques moive wans o build up savings during he iniial years and heby give his family an incenive for looking af him. 3 Equivalen wealh In his pap he gain in welfare is compued by means of he equivalen wealh, a measure of wha would be he level of wealh required o be on he same expeced uiliy curve in whichev of he cases analysed. This measure is aimed a demining how much he individual avse o risk and wih a beques moive would value he possibiliy of purchasing eih a life annuiy or a programmed wihdrawal conrac and hus being able o proec him or hself agains he risk of excessive longeviy, including boh financial and psychological parames. The equivalen wealh is given by µw, whe he coefficien µ is defined as: W + W W µ = = 1+ (29) W W and W is he amoun of addiional wealh he consum would need o have access o so ha, following his or h opimal consumpion rae in any one of he cases consided, he same level of uiliy would be aained as when he consumpion rae is opimized in any oh of he cases. The gain in welfare, expressed as a pcenage of he iniial wealh W, is given by he raio W/W. More precisely, he equivalen wealh will be demined by he pcenage µ ij which saisfies: whe: UE ( ij W ) = UE j ( W ), i, j = 1,2,3,4 i µ (3) UE 1 (W): Expeced uiliy dived from he opimal consumpion rae ha he individual would choose in Case 1 (no access o he annuiy marke), for an iniial wealh a he age of reiremen equal o W. UE 2 (W): Expeced uiliy dived from he opimal consumpion rae ha he individual would choose in Case 2 (purchasing an unindexed life annuiy wih a single premium equal o he oal wealh a reiremen W). UE 3 (W): Expeced uiliy dived from he opimal consumpion rae ha he individual would choose in Case 3 (purchasing an indexed life annuiy wih a single premium equal o he oal wealh a reiremen W). UE 4 (W): Expeced uiliy dived from he opimal consumpion rae ha he individual would choose in Case 4 (purchasing a programmed wihdrawal conrac wih he wealh a reiremen). The parame µ ij can ake hree values: if i is grea han one, hen alnaive j is prefred ov i; if i is less han one, he conrary is he case; and if i is equal o one, he is indiffence o which of he wo alnaives is chosen. 1

7 3.1 Equivalen wealh wih he beques moive and life annuiies In his subsecion, he same parame values and assumpions ha we applied in Secion 2.6 are used o compue he value of he equivalen wealh for men and women wih alruisic and wih sraegic beques moives. Tables 2-3 lis he resuls comparing he purchase of an indexed or unindexed life annuiy wih he siuaion in which he individual has no access o he annuiy marke. The equivalen wealh is also compared wih he same siuaions when he model does no include he beques moive (lised und 'individual'). The beques moive always noiceably reduces he equivalen wealh wih respec o he individual case model, i.e., i makes annuiies less aracive, increasing he numb of individual profiles who will pref no o purchase an annuiy. This exends even o hose who, showing lile risk avsion, feel lile impaience for consumpion as in he case of men who purchase fully CPI-indexed annuiy conracs. As also is he case for individuals wih no beques moive, he welfare aained wih annuiies increases wih increasing risk avsion and decreasing impaience for consumpion. Also, unlike he case in which he individual has no beques moive, he resuls indicae ha i is prefable o purchase an unindexed han an indexed annuiy in pracically all he profiles. The reason is ha, in he iniial years, he indexed annuiy provides a small income flow, so ha he individual does no have he desired level of wealh in ord o leave a beques o he heirs. Only in he case of sraegic beques moives can grea welfare be aained wih he purchase of an indexed annuiy, as long as he level of impaience for consumpion is vy low. This is because, in he case of sraegic beques moives, fuure bequeahable wealh is valued more han curren wealh, and he indexed annuiy allows consumpion o be raionalized in he iniial years in ord o provide grea income in he fuure, heby increasing he amoun of bequeahable wealh. When he level of risk avsion is low and he beques moive is alruisic, women achieve grea welfare han men by purchasing life annuiies. People wih alruisic beques moives aain less welfare wih he purchase of an annuiy han hose wih sraegic moives. This is logical since he alruisic individual values he beques more in he iniial years, and hus consumes less in his piod. The consequence is a decrease in he uiliy diving from ha consumpion, which is ha wih greaes weigh in he expeced uiliy As was noed above, an increase in he parame h reduces he welfare obained by purchasing life annuiies when he beques moive is included. Grea values of his parame lead o noably small values of he equivalen wealh. As an example, Table 2, for a man wih a level of risk avsion of.7, who is vy impaien o consume (case A), and who has an alruisic beques moive, he equivalen wealh decreases by 11% from.81 o.711 when he value of he weigh of he beques in he expeced uiliy is aken o be consan and equal o 2. If he level of risk avsion is equal o 2.9 and he aiude owards consumpion is neural (case C), he decrease is somewha less 8% from an equivalen wealh of wih he value of he parame used in he presen pap o wih a beques parame equal o 2. In all he ables he values for he equivalen wealh of he nonindexed (indexed) annuiies ha are grea han hose for indexed (non-indexed) annuiies have been undlined. Values less han 1, which mean ha wealh wihou annuiies is prefable, are shown in bold. Anoh way o measure he prefence ha individuals show in accessing he annuiy marke is o calculae he maximum pcenage of wealh accumulaed a reiremen ha hey are willing o annuiize. To demine his maximum pcenage i has o be esablished ha he wealh allocaed o annuiies is a conrol variable of he problem se. I should no be consided as aking on a fixed value equal o he iniial wealh he individual has available a he ime of reiremen; his assumpion is made o obain he value of equivalen wealh. As was noed before, he mahemaical models in his pap have been ranslaed ino LINGO sofware programming language, and his program was used o obain he numical resuls shown in he various ables. More deails can be found in Lejárraga (23). As can be seen in Table 4, he grea heir impaience o consume, he small he annuiizaion pcenage. Reirees, boh men and women, wih a vy low level of risk avsion and a high level of impaience o consume (β =.7 and case A) would pref o consume heir accumulaed wealh direcly and would allocae no amoun o a life annuiy. As indicaed by Brown (21), in he presence of a beques moive, grea welfare gains resul when only a cain pcenage of he wealh available a reiremen is annuiized insead of he whole of ha wealh. Indeed, he only case in which he opimal choice is 1% annuiizaion is ha of an individual wih a sraegic beques moive, a vy low level of risk avsion, and who is vy paien wih respec o consumpion. Grea risk avsion implies a grea annuiizaion pcenage, excep in he case ha he reiree has a sraegic beques moive and is vy paien wih respec o consumpion, i.e., in he case whe he level of risk avsion has grea weigh in he beques uiliy funcion han in he consumpion uiliy funcion. Also, individuals wih an alruisic beques moive will always annuiize a small pcenage han hose wih a sraegic moive. 3.2 Equivalen wealh wih he beques moive and programmed wihdrawal In a programmed wihdrawal conrac, when he real reurn obained on he annuiizaion accoun is equal o he real marke ines rae, one obsves in Tables 5-6 ha from a cain level of risk avsion and impaience for consumpion he individual obains he same uiliy by purchasing his ype of pension as when no annuiy is purchased. This is because he opimal consumpion rae in hese cases is less han he amoun of he available annuiy, so ha he wo siuaions are valued he same. When, howev, he individual shows lile risk avsion and is also vy impaien o consume (case A), he uiliy of no purchasing annuiies is grea han ha wih a programmed wihdrawal conrac. Unlike he life annuiies case, programmed wihdrawal is no always more aracive when he beques moives are sraegic. Insead, one obsves ha individuals who are vy impaien for consumpion and, in genal, wih lile risk avsion obain grea welfare wih his ype of pension when heir beques moive is alruisic han when i is sraegic. 11

8 Likewise, only individuals wih lile risk avsion and grea impaience for consumpion obain grea welfare hrough an unindexed programmed wihdrawal conrac han hrough an indexed conrac. This is diffen from he life annuiies case, in which an unindexed annuiy was prefable for pracically all he individual profiles. The level of risk avsion is he demining facor in he level of welfare ha is aained, since, when i is high, he same equivalen wealh is obained independenly of he individual s profile of impaience for consumpion. In pracically all cases for boh men and women, when he level of risk avsion is low, he equivalen wealh is grea for an indexed programmed wihdrawal conrac when a beques moive is included. This is because in he individual case no valuaion is made of he funds ha remain in he individual annuiizaion accoun, and hence hey do no conribue o he expeced uiliy. When he level of risk avsion is high, he welfare aained is independen of he consumpion impaience profile, in all hree beques moive conexs alruisic, sraegic, and no moive o leave a beques. In he individual case, grea welfare always resuls from purchasing a life annuiy han a programmed wihdrawal conrac. In he beques moive conexs, howev, he conrary is he case for reirees showing a low level of risk avsion and grea impaience o consume. This prefence is more marked in he alruisic case when hey value he beques more in he firs years of reiremen han in he sraegic case. This is so even when he expeced reurn on he individual annuiizaion accoun coincides wih he real marke ines rae. 4 Equivalen wealh and marke impfecions 5 Annuiy markes, Blake (1999), are insufficienly developed even in many of he financially mos advanced counries. The is enough empirical evidence in he economics liaure o show ha people who volunarily decide o purchase annuiies live long han he avage genal populaion. In heir innaional sudy, Michell and McCarhy (22b) show ha he advse selecion associaed wih volunary purchass of life annuiies reduces he expeced moraliy raes by a leas 25%. Furh evidence relaing o advse selecion, can be found in paps by Mung (22), Poba (21) and Villeneuve (23). Friedman and Warshawsky (199) and more recenly Michell e al. (1999) have shown ha annuiies in he Unied Saes end o be more expensive han would be expeced from he life expecancy of he genal populaion. In paricular, Michell e al. (1999) calculae ha, for a 65-year-old male, he value of he annuiy is 15-25% less han would correspond o he genal moraliy ables. They conclude ha abou half of his reducion is because he annuians live long han he US mean, and ha he res is impuable o adminisraive coss. Finkelsein and Poba (2) find ha he value of annuiies conraced volunarily by 65-year-old men in he Unied Kingdom is 1-15% less han ha diving from he use of he moraliy ables of he ovall populaion. More han 6% of his reducion is 5 Some auhors show heir complee disagreemen wih he m impfecion, and simply aribue he decrease in he amoun of he annuiies on he marke ov he acuarially fair amoun o he price ha has o be paid o he company for assuming he financial risk and he risk of longeviy. due o he long life expecancy of he annuians compared wih he populaion mean. The exiss innaional evidence, Devesa e al. (22) and (23), ha he ransformaion of he accumulaed savings of an individual annuiizaion accoun ino a life annuiy may involve high coss for he paricipan. Togeh wih he moraliy 6 and he inmediaion coss, anoh major facor in demining life annuiies is he echnical ines rae ha is applied. While individuals migh no undsand how he use of a moraliy rae or he adminisraive coss applied by he insurance company influence he price ha hey pay for he annuiies, hey may be vy sensiive wih respec o guaraneed ines raes. One final aspec ha could be consided an impfecion of he annuiies marke is he fac ha in some counries - Belgium, for insance, as already menioned in he inroducion - annuiies are axed a a high rae han oh ypes of pension. The inclusion in he model of he exisence of so-called marke impfecions, Tables 7-8, leads o low values of he equivalen wealh han in he conex of acuarially fair markes. In paricular, i was consided ha he combined effec of he diffen moraliy and survival probabiliies used by he insurance firm in markeing life annuiies, he applicaion of adminisraive coss, and he possible diffences beween he ines rae guaraneed in he annuiy and he marke rae (he m θ in expression (7)), reduces he annuiies by 15% wih respec o he acuarially fair value. In a similar way as occurred in he case of an individual who has no beques moive, he reducion in he value of he equivalen wealh caused by inroducing marke impfecions coincides wih he pcenage decrease consided for he annuiy wih respec o he "acuarially fair" value, i.e., 15% for all he individual profiles ha we analyzed. I is imporan o noe ha, in almos all cases, individuals wih lile risk avsion would pref no o purchase a life annuiy. The rend of he prefences beween indexed and unindexed annuiies is he same for he diffen individual profiles as in he "acuarially fair" case. As also in his la case, when marke impfecions are included, women wih a low level of risk avsion aain grea welfare wih life annuiies, wheh indexed or no, han men do. In no case, howev, do hey pref he purchase of annuiies ov he consumpion of curren wealh. 5 Equivalen wealh and pre-exising life annuiies In some counries he is a cain freedom of choice, and he can be pre-exising annuiies from he firs pillar of he public pension sysem. This is he case in Argenina, Cosa Rica, and he Unied Kingdom. One mus also include he case of counries, 6 As Pire Devold has poined ou, he use of diffen moraliy ables is no really he impfecion. If he ables used by he insur give a fair esimae of he moraliy rae of people choosing annuiies (which is low), he is no impfecion even hough i means ha ables oh han hose coving he genal populaion are used. The impfecion arises from he exra safey margin he insur can include, which is no jusified judging by he expience of annuians. 12

9 Spain for example, which have a volunary sysem ha complemens he public pension sysem, in which he benefis are exclusively in he form of join annuiy wih coningen survivor benefi, wih he paricipan having he possibiliy of ransforming he accumulaed savings a reiremen ino an addiional life annuiy. 5.1 Wihou marke impfecion The resuls lised in Tables 9-1 indicae ha in he presence of pre-exising annuiies he is a reducion in he welfare aained wih he purchase of an acuarially fair life annuiy. The reducion is up o 22% and 15% for men and women, respecively, wih respec o he value aained when no beques moive is consided. In genal, consums wih lile risk avsion independenly of heir aiude owards consumpion, or who are vy impaien wih respec o consumpion independenly of heir aiude owards risk, will lose welfare by acquiring addiional life annuiies. I is noable ha, considing indexed annuiies, when an individual is already receiving par of his or h wealh in he form of life annuiies, he welfare aained wih he purchase of an addiional annuiy decreases af a cain level of risk avsion excep for women wih a sraegic beques moive, i.e., in hese cases he level of risk avsion has more weigh in he beques uiliy funcion han in he consumpion uiliy funcion. 5.2 Pre-exising life annuiies and marke impfecions I was consided ha he reiree already receives a par of his or h wealh in paricular 5% in he form of a life annuiy linked o he expeced CPI, and also purchases a life annuiy ha includes marke impfecions. The resuls Tables show a reducion in he welfare aained wih respec o he same siuaion excep ha he life annuiy he reiree can purchase is "acuarially fair". As was found when he values of he equivalen wealh wih no beques moive we compued, he was an even more specacular increase in individuals who heoreically would pref no o purchase a life annuiy since ha would produce a sharp decline in he resuling welfare. The individuals wih greaes risk avsion and wih vy lile impaience for consumpion, for whom he purchase of a life annuiy would sill be prefable, would achieve negligible welfare gains, even hough hese would always be grea in he sraegic beques moive case han in he alruisic case. The foregoing resuls grealy clarify he annuiy puzzle for single individuals. The successive incorporaion of marke elemens ogeh wih he considaion of he beques moive pracically resolve he puzzle, since und hese condiions vy few individuals would be willing o purchase life annuiies, as is indeed he case in realiy when people are allowed freedom of choice. 6 Conclusions, policy recommendaions, and fuure research The presen pap has conribued o clarifying he "annuiy puzzle" by including he beques moive. To his end, he evaluaion was presened of an ample se of assumpions ha have received lile aenion in he economics liaure. Also, some ligh was shed on wheh he beques moive in iself is really a major facor influencing he heoreical decision o purchase annuiies. In all he conexs and individual profiles consided, people wih an alruisic moive o bequeah par of heir wealh obain less welfare by purchasing a life annuiy han hose who have a sraegic moive, alhough he exis excepions o his rule in he programmed wihdrawal ype of conrac. The basic model, which does no include he exra characisics of marke impfecions and pre-exising annuiies, gives he bes measure of he rue impac of he beques moive on he decision o purchase annuiies. In he absence of infence from hese oh characisics, i seems o indicae ha he beques moive in isolaion is no really a major facor, alhough i has o be noed ha: 1) I reduces he araciveness of annuiies, increasing he numb of individual profiles ha will pref no o purchase an annuiy, exending o hose who, showing lile risk avsion, even feel lile impaience for consumpion. 2) Unlike he case in which he individual has no ines in leaving a beques, he resuls indicae ha for pracically all he profiles he purchase of an unindexed annuiy has more uiliy han ha of an indexed annuiy. Only in he case when he moives for he beques are sraegic, can grea welfare be aained wih he purchase of an indexed annuiy, and hen only as long as he is vy lile impaience for consumpion. 3) Wih respec o he programmed wihdrawal ype of pension, only individuals wih lile risk avsion and who are vy impaien for consumpion would pref an indexed programmed wihdrawal conrac ov an unindexed conrac. The level of risk avsion is he demining facor, since for high levels of his parame he resuling equivalen wealh is he same, independenly of he individual s profile of impaience for consumpion. 4) Reirees who show a low level of risk avsion and a grea impaience for consumpion would pref he programmed wihdrawal conrac ov life annuiies. This is so even when he reurn on he individual annuiizaion accoun coincides wih he real marke ines rae. The genalized model, wih all he characisics included marke impfecions and pre-exising annuiies pracically resolves he so-called "annuiy puzzle", since und hese condiions vy few individuals would be willing o purchase life annuiies, as is usually he case in realiy when hey have freedom of choice. The resuls lend suppor o he conclusions drawn from previous models in he sense ha: (i) a cain flexibiliy seems advisable in regulaing he ypes of pension schemes so as o accommodae individual circumsances and public policy objecives; and (ii) in defined conribuion schemes ha complemen defined benefi pension sysems, in which individuals already hold a major porion of heir wealh in he form of life annuiies, freedom should be oal, wih no obligaion o allocae any amoun whasoev o purchasing addiional life annuiies. 13

10 Finally, furh research could focus on: a) Developing a model o include he beques moive from he sandpoin of couples. The difficuly in his case would be o demine he level of welfare provided by life annuiies, as his would require more assumpions o be made regarding he srucure of boh he uiliy funcion and he beneficiaries' pensions. b) Modeling he fac ha life annuiies are virually he only produc ha can guaranee an ines rae in he long-m. Howev, in he model followed in his pap all he oh ypes of pension have been assigned a fixed reurn, which invalidaes he supposed supioriy of life annuiies whe his aspec is concned. This could be vy imporan when financial markes are undgoing piods of insabiliy (which is almos always) and for hose pensions who are more avse o risk. ACKNOWLEDGEMENTS The auhors wish o express heir graiude for he invaluable commens and suggesions made by José Enrique Devesa-Carpio, Pire Devold, Inmaculada Domínguez-Fabián, Pe Hall, Salvador Valdés-Prieo and one anonymous refee. Ana Lejárraga-García acknowledges financial suppor received from FUNCAS (Fundación de cajas de ahorro confedadas in Spain). One preliminary vsion of his pap was presened a he 7 h Spanish-Ialian confence on Financial Mahemaics in Cuenca, Spain. Any rors are enirely hose of he auhors. REFERENCES [1] Beniez-Silva, H. (23): The Annuiy Puzzle Revisied. Michigan Reiremen Research Cenre. WP [2] Bnheim, D. (1991): How srong are beques moives on esimaes of he demand for life insurance?. Journal of Poliical Economy 93 (6), [3] Blake, D. (1999): Annuiy markes: Problems and soluions. The Geneva Paps on Risk and Insurance (24) 3, [4] Brown, J. R. (1999): Privae Pensions, Moraliy Risk, and he Decision o Annuiize. Naional Bureau of Economic Research. WP [5] Brown, J. R. (21): Redisribuion and insurance: Mandaory annuiizaion wih moraliy heogeneiy. Cen for Reiremen Research on Pensions a Boson College. WP [6] Brown, J. R. (23): Redisribuion and insurance: Mandaory annuiizaion wih moraliy heogeneiy. Journal of Risk and Insurance, 7 (1), [7] Brown, J. R., O.S. Michell and J.M. Poba (1999): The Role of Real Annuiies and Indexed Bonds in an Individual Accouns Reiremen Program. Naional Bureau of Economic Research. WP- 75. [8] Brown, J. R., O.S. Michell and J.M. Poba (2): Moraliy Risk, Inflaion Risk, and Annuiy Producs. Naional Bureau of Economic Research. WP [9] Brown, J. R and J.M. Poba (2): Join life annuiies and annuiy demand by married couples. The Journal of Risk and Insurance 67 (4), [1] Bul, M. (23): Mandaed Annuiies in Swizland. Mímeo. Univsié de Lausanne. [11] Davidoff, T, J. Brown and P. Diamond (23): Annuiies and Individual Welfare. Cen for Reiremen Research on Pensions a Boson College. WP [12] Devesa, J.E. and C. 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11 Triese. Diparameno di Maemaica Applicaa alle Scienze Economiche Saisiche e Auariali, (8), [39] Villeneuve, B. (23): Mandaory Pensions and he Inensiy of Advse Selecion in Life Insurance Markes Journal of Risk and Insurance, 7 (3), [4] Yaari, M. E. (1965): Uncain lifeime, life insurance and he Theory of he Consum. The Review of Economic Sudies 32 (9), APPENDIX Consumpion Consumpion Consumpion Vy impaien, bea =.7, Alruisic Age [1] [2] [3] [4] Indiffen, bea = 2.9, Alruisic Age [1] [2] [3] [4] Vy paien, bea = 4.4, Alruisic Age [1] [2] [3] [4] Consumpion Consumpion Consumpion Vy impaien, bea =.7, Sraegic Age [1] [2] [3] [4] Indiffen, bea = 2.9, Sraegic Age [1] [2] [3] [4] Vy paien, bea = 4.4, Sraegic Age [1] [2] [3] [4] Figure 7 1. Opimal consumpion pah for men wih beques moive. (1)No annuiies, (2)Unindexed annuiy, (3)Indexed annuiy, (4)Programmed wihdrawal. 7 Due o he large numb of graphs ha would be needed o represen he consumpion flow associaed wih each of he resuls shown in he ables, we decided o selec hose we consided he mos represenaive o reflec he variaions in his consumpion flow, when he main parames ha are used o demine hem are modified. 15

12 Table 1: Comparison beween opimal consumpions wih beques moives Type of pension Vy impaien Indiffen Vy paien bea =.7 bea = 2.9 bea = 4.4 No annuiies Al > S; years Al >S; years Al > S; years Al < S; years Al < S; years Al < S; 11 years Unindexed annuiy Al > S; years Al > S; years Al > S; years Al < S; years Al < S; years Indexed annuiy Al > S; years Al > S; years Al > S; years Al < S; years Programmed wihdrawal Al = S; 65-8 years Al >S; years Al < S; years Al >S; years Al < S; years Al > S; years Al < S; 11 years δ Table 2: Equivalen wealh for men. β Alruisic Individual Sraegic A) Unindexed annuiy A C E B) Indexed annuiy A C E δ Table 3: Equivalen wealh for women. β Alruisic Individual Sraegic A) Unindexed annuiy A C E B) Indexed annuiy A C E δ β Table 4: Opimal pcenage of wealh allocaed o purchasing an indexed annuiy. Men (%) Women (%) Alruisic Sraegic Alruisic Sraegic A C E

13 Table 5: Equivalen wealh for men. Alruisic Individual Sraegic δ β i=3% i=4% i=3% i=4% i=3% i=4% A) Unindexed programmed wihdrawal A C 1, E 1, B) Indexed programmed wihdrawal A C E δ β Table 6: Equivalen wealh for women. Alruisic Individual Sraegic i=4.55% i=5.56% i=4.55% i=5.56% i=4.55% i=5.56%,7 2,9,7 2,9,7 2,9,7 2,9,7 2,9,7 2.9 A) Unindexed programmed wihdrawal A.973 1, 1,73 1,11,913 1, 1,19 1,11,954 1, 1,59 1,11 C 1. 1, 1,12 1,11,997 1, 1,18 1,11 1, 1, 1,16 1,11 E 1. 1, 1,14 1,11 1, 1, 1,11 1,11 1, 1, 1,17 1,11 B) Indexed programmed wihdrawal A C E δ β Table 7: Equivalen wealh for men wih 15% reducion facor. Alruisic Individual Sraegic A) Unindexed annuiy A C E B) Indexed annuiy A C E δ Table 8: Equivalen wealh for women wih 15% reducion facor. β Alruisic Individual Sraegic A) Unindexed annuiy A C E B) Indexed annuiy A C E

14 δ Table 9: Equivalen wealh wih 5% of he wealh in pre-exising annuiies for men. Alruisic Individual Sraegic β A) Unindexed annuiy A C E B) Indexed annuiy A C E δ Table 1: Equivalen wealh wih 5% of he wealh in pre-exising annuiies for women. Alruisic Individual Sraegic β A) Unindexed annuiy A C E B) Indexed annuiy A C E δ Table 11: Equivalen wealh wih 5% of he wealh in pre-exising annuiies for men wih 15% reducion facor. Alruisic Individual Sraegic β A) Unindexed annuiy A C E B) Indexed annuiy A C E, δ Table 12: Equivalen wealh wih 5% of he wealh in pre-exising annuiies for women wih 15% reducion facor. Alruisic Individual Sraegic β A) Unindexed annuiy A C E B) Indexed annuiy A C E