Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

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1 Life-Cycle Asse Allocaion wih Annuiy Markes: Is Longeviy Insurance a Good Deal? by WOLFRAM HORNEFF, RAIMOND MAURER, and MICHAEL STAMOS November 25 Compeiive Paper Wolfram Horneff Goehe Universiy of Frankfur Deparmen of Finance Keenhofweg 39 (Uni-PF 58), 654 Frankfur Germany T: F: Raimond Maurer (corresponding auhor) Goehe Universiy of Frankfur Deparmen of Finance Keenhofweg 39 (Uni-PF 58), 654 Frankfur Germany T: F: Michael Samos Goehe Universiy of Frankfur Deparmen of Finance Keenhofweg 39 (Uni-PF 58), 654 Frankfur Germany T: F:

2 Life-Cycle Asse Allocaion wih Annuiy Markes: Is Longeviy Insurance a Good Deal? November 25 Absrac We show how an individual wih uninsurable labor income, borrowing consrains, and Epsein-Zin uiliy funcion opimally spreads her financial wealh across socks, bonds and life-annuiies over he life-cycle when he finie invesmen horizon is sochasic. In spie of asymmeric moraliy beliefs, public pensions, and beques moives, we find ha he individual sars ou wih annuiizing 8.59 percen of her accumulaed financial wealh five years prior o reiremen and coninues gradually annuiizing hereafer. Our welfare analysis shows ha he presence of an annuiy marke increases he individual s welfare by 4.4 percen a age 8 and by 3.7 percen a age 9. JEL Codes: D9 G G22 H55 J26. Keywords: Porfolio Choice, Dynamic Asse Allocaion, Life-Cycle, Insurance, Annuiies, Pensions, Reiremen, Reiremen Policies. 2

3 I Inroducion Nowadays, he burden of old-age provision has o be increasingly borne by individuals as naions worldwide experience a shif from roubled public pay-as-you-go o privaely funded pension sysems. In addiion, employers are moving from professionally managed defined benefi o defined conribuion plans for heir employees. Thus, personal financial planning over he enire life cycle is more and more becoming he sole responsibiliy of he individual. The reasonable accumulaion and decumulaion of savings has become a life-ask. Faced wih his onerous ask, he individual has o deermine how o consume and how o spread lifesavings across differen asse classes. Throughou her work life, he individual has o build up a sufficien capial sock for reiremen. During reiremen, she has o spread her accumulaed wealh over he remaining years of life. She has o assume he role of a risk manager for hree main risks in order o ensure sable consumpion over her lifeime and o guaranee beques if desired: labor income risk, capial marke risk and longeviy risk. The sharp rise in life expecancies in he pas few decades makes he risk of exhausing he reiremen capial sock more criical han ever before. Insurance producs like life-annuiies can hedge his longeviy risk away. A life-annuiy is a financial conrac beween an insured person and an insurer ha pays ou a periodic amoun for as long as he annuian is alive, in exchange for an iniial premium (Brown e al., 2). The insurance providers hemselves can hedge he guaraneed annuiy paymens by pooling he longeviy risks. From he individual s perspecive, he premium paid iniially canno be recovered. This inflexibiliy is said o be he main disadvanage of annuiy purchases compared o flexible wihdrawal plans. Many sudies compare life-annuiy purchases wih radiional asse classes. In an early sudy, Yaari (965) finds ha all asses should be annuiized if he individual is a raional invesor wihou a beques moive. In his model, he invesor is only exposed o longeviy risk 3

4 and all annuiies are fairly priced from an acuarial sandpoin. Yaari s (965) resul has been subjec o exensive research in he public economics and insurance lieraure. Brown e al. (25) show ha he condiions under which he purchase of an annuiy is opimal are no as demanding as he ones se ou in Yaari (965). If here is no beques moive and he reurn on he annuiy is greaer han ha of he reference asse, an individual will fully annuiize her wealh in he presence of a complee marke. Parial annuiizaion may become opimal, if he condiion of complee insurance markes is relaxed. If here is a beques moive for invesors, complee annuiizaion will be subopimal. Richard (975) was he firs o include he uncerainy of he ime of deah in a coninuous life-cycle framework and o exend Meron s (97) model o include insananeous life insurance. However, his framework lacks he realism of an acual insurance marke because Richard (975) models insananeous life insurance and annuiy demand symmerically. Mos recenly, Blake e al. (23), Milevsky and Young (22/23), Kapur and Orszag (999), Dushi and Webb (24), Kingson and Thorp (25) and Sabile (23) invesigaed annuiizaion sraegies in a uiliy framework. However, mos of he above menioned sudies solely focus on he reiremen phase. Our main conribuion is he inroducion of a life-annuiy marke o a realisic life-cycle framework. We provide insigh ino how a pruden invesor should opimally spread her wealh across bonds, socks, and consan real life-annuiies. Our model incorporaes uninsurable income during work life, borrowing consrains, and sochasic ime of deah. The individual s asse allocaion and savings decisions are driven by hree moives. The firs is he precauionary savings moive due o he uninsurable risky income. The invesor we consider wans o save and inves her asses in such a way ha she can hedge adverse developmens in her income sream. This moive was firs described by Deaon (99) and Carroll (997). The second is he reiremen savings moive because pension income afer reiremen is lower han he labor income in he preceding accumulaion phase. In order o 4

5 smooh consumpion over ime and o cushion he drop of income due o reiremen, he individual consumes less when she is young in order o consume her savings when she is old. These wo moives can be summarized by he consumpion smoohing moive. The hird moive is he beques moive because he individual migh gain uiliy from leaving esae o her heirs. The life-cycle asse allocaion and consumpion model we use is of he discree ime ype and is excep for he annuiy marke similar o he models used in he recen lifecycle lieraure, e.g. Berau and Haliassos (997), Campbell e al. (999), Davis and Willen (2), Gomes and Michaelides (23, 25), Cocco, Gomes, and Maenhou (25), Dammon, Spa, and Zhang (24), Cocco (24), Yao and Zhang (25). We assume ha he individual has Epsein-Zin uiliy (Epsein and Zin, 989) and can realize uiliy from bequeahing her heirs. An imporan implicaion of our life-cycle model is ha he invesor is in conrol and has he flexibiliy o spend annuiy payous from previously purchased annuiies. She can use hem eiher o consume, o purchase bonds and socks or even o purchase addiional annuiies. This flexibiliy was negleced in recen sudies such as Blake e al. (23), Milevsky and Young (22), Kingson and Thorp (25) and Sabile (23) which assumed ha annuiy payous can be used for consumpion purposes only. The focus of hose sudies is on he opimal sochasic and deerminisic swiching ime o an annuiy. In our model he invesor can gradually purchase annuiies a any age and is no resriced o he decision wheher o swich compleely o an annuiy a a cerain poin in reiremen. Previous sudies looking a gradual invesmen in life-payou annuiies include Kapur and Orszag (999) and Milevsky and Young (23). However, hese auhors se heir opimizaion problem up as a coninuous ime model wih ime-addiive CRRA preferences, focusing merely on he reiremen period. In conras, we can show how an individual chooses 5

6 her asse allocaion and annuiy purchases during her life in a realisically calibraed life-cycle framework. Our model demonsraes ha decisions in he reiremen phase canno be separaed from decisions in he accumulaion phase. We carried ou a sensiiviy analysis for () risk aversions, (2) he srengh of beques moives, (3) moraliy asymmery and (4) public pensions. Our findings indicae a very imporan role of he risk aversion parameer for deermining he annuiy demand. Even individuals wih a low risk aversion purchase annuiies a very high ages bu o a lesser exen. Our analysis shows a srong relaionship beween he srengh of he beques moive and asse allocaion. We sill find a subsanial demand for annuiies in he presence of beques moives. We also allow he individual o value expeced uiliy via a subjecive force of moraliy, while he annuiy is priced by using an annuian moraliy able. We show ha hese asymmeries of moraliy beliefs can conribue o explaining why individuals who believe hemselves o be less healhy han average are less likely o buy annuiies. However, he effec of moraliy asymmery has a small impac on annuiy demand. Alhough preexising public pensions have he same payou srucure as annuiies, we sill find individuals purchasing annuiies. In a final welfare analysis, we verify ha a subsanial demand for annuiies goes hand in hand wih a considerable equivalen increase in financial wealh due o he presence of annuiy markes. Especially for senior individuals annuiy markes imply a considerable equivalen increase in financial wealh ranging from 4.4 percen a age 8 up o 3.7 percen a age 9. The remainder of his aricle is organized as follows. In secion II, we describe he invesor s opimizaion problem, he model calibraion, and he numerical opimizaion mehod. In secion III, we show he resuls for our base-line case wih and wihou annuiy markes. Secion IV coninues wih a robusness analysis and secion V wih a welfare analysis. Secion VI concludes he aricle. 6

7 7 II. The Model A. Preferences The model is ime discree wih =,,T, where is he adul age of he individual and can be calculaed as acual age minus 9. The individual lives up o T years. The individual has a subjecive probabiliy s p ha she survives unil given ha she is alive a. Furhermore, he individual has Epsein-Zin uiliy defined over a single non-durable consumpion good. Le C be he consumpion level and B he beques a ime. Then Epsein-Zin preferences as in Epsein and Zin (989) are described by ( ) ( ) ( ) ψ ρ ψ ρ ρ ψ ρ β β / / / / = k B k p V p E C p V s s s, () where ρ is he level of relaive risk aversion, ψ is he elasiciy of ineremporal subsiuion, β is he discoun facor and k he srengh of he beques moive. Since = s T p () reduces in T o ( ) ψ ρ ψ ρ ψ ρ β / / / / = k B k E C V T T T T, which gives us he erminal condiion for V T. B. Labor Income Process During he accumulaion phase ( K) he individual earns uninsurable, real labor income Y. Being consisen wih Cocco e al. (25) he process of labor income follows ( ) ( )., exp N P P PU f Y = = f() is a deerminisic funcion of age o recover he hump shape of income sream. P is a permanen componen wih innovaion N and U is a ransiory shock. The logarihms of N

8 and U are normal disribued wih means zero and wih volailiies σ N, σ U respecively. The shocks are assumed o be uncorrelaed. In reiremen ( > K) we assume for he sake of simpliciy ha he individual receives consan pension paymens Y ς exp( f ( K )) P K = afer reiremen, where ζ is he consan replacemen raio. Clearly, i migh be worhwhile deermining he reiremen age K and labor supply endogenously. This quesion is beyond he scope of his analysis since we focus on he asse allocaion decision. C. Incomplee Annuiy Marke The individual can inves in an incomplee insurance marke by purchasing consan real payou life-annuiies. A life-annuiy is a financial conrac beween an individual and an insurer ha pays ou a periodic amoun for as long as he annuian is alive, in exchange for an iniial premium (Brown e al., 2). The insurance providers hemselves can hedge he guaraneed annuiy paymens by pooling he longeviy risks of many annuians. Conrary o liquid invesmens, he iniial premium canno be recovered by he individual laer on. The acuarial premium PR of a life-annuiy wih paymens L is given by: PR = L, a where a is he annuiy facor for an individual wih adul age is a s a s = ( δ ) pu R f, s= u= where a p u are he survival probabiliies used by he life-annuiy provider and δ is he expense facor. Thus, he annuiy facor is he expense facor imes he sum of he discouned expeced payous. Annuiies define an asse class wih cerain reurn characerisics because paymens are condiional on survival. The reurn on capial of hose who die is allocaed o he living members of a cohor. The survivors reurn from he one-period annuiy is R p > R. f f 8

9 6,5% 6,% 5,5% 5,% 4,5% 4,% 3,5% 3,% 2,5% 2,% Figure. Implied longeviy yield. The solid line depics he implied longeviy yield of a female annuian for ages over 65. The dashed line shows he implied longeviy yield of a male annuian who bough he annuiy a age 65. The resuling incremenal reurn he annuian receives above he ineres rae is called moraliy credi. The older he individual, he lower he survival probabiliy p, he higher is he incremenal reurn. The moraliy credi neglecs he iniial loss of he whole lump sum paid o he insurance company. The implied longeviy yield (figure ), an inernal rae of reurn, accouns for he loss of he iniial premium. One way o compue he implied longeviy yield of an annuiy purchase beween wo poins in ime is o solve he following equaion as derived in Milevsky (25): ξδ a Δ a e =. ξ ξ The longeviy yield ξ is he consan rae ha has o be earned on a porfolio in order o be as well off, afer a period of Δ, as if he individual purchased an annuiy iniially. The equaion assumes ha he invesor confines herself o a self-annuiizaion sraegy in order o accumulae sufficien wealh o purchase an annuiy afer a period of Δ, where he erm self- 9

10 annuiizaion refers o he drawdown scheme ha replicaes he payou srucure of a fixed annuiy. Figure () shows he longeviy yield for boh male and female annuian who purchased a life-annuiy a age 65. The implied longeviy yield increases every year he annuian survives and oulives her peers. Clearly, he male annuian has a higher implied longeviy yield because of his lower survival probabiliies. In urn, his means ha life annuiies are a compleely differen asse depending on he invesor s sex and individual survival probabiliies. D. Capial Marke The individual can inves in he wo radiional financial asses: riskless bonds and risky socks. The real bond gross reurn denoes R f, and he real risky sock reurn in is R. The risky reurn is lognormal disribued wih expeced reurn μ and volailiy σ. Le φ n (φ u ) denoe he correlaion beween he sock reurns and he permanen (ransiory) income shocks. E. Moraliy To give moraliy a funcional form we use he Gomperz law for he sake of convenience and because of is widespread use in he insurance and finance lieraure. Using Gomperz law allows us o model he asymmery beween he insurer s view on moraliy and he annuian s beliefs abou her own healh in a simple and consisen way. The funcional form of he s subjecive force of moraliy λ and he force of moraliy for compuing annuiy premiums a λ are hen specified by i i m λ = i b exp, i = a,s. b

11 Parameers m i and b i deermine he shape of he force of moraliy funcion. The survival probabiliies can now be expressed as follows: i i = i m p exp λ sds = exp exp exp i i. b b Addiionally, we model he subjecive force of moraliy as linear ransformaions of he force of moraliy used for annuiy pricing o incorporae asymmeric moraliy beliefs. Then we ge for he subjecive force of moraliy and he subjecive probabiliies: s a λ = νλ, p = exp ( ν ) p. s a F. Wealh Accumulaion A each poin in ime he invesor has o make a decision how o spread cash on hand W across bonds, socks, annuiies, and consumpion. Therefore, he budge consrain is W = M S PR C, (2) where M S denoe he value of financial wealh, M is he absolue wealh amoun invesed in bonds and S he amoun invesed in socks. PR is he amoun ha he invesor pays for an annuiy and C is consumpion. The individual s cash on hand in is given by W M R S R L Y, = f where M R S R denoe he nex period value of financial wealh, L is he sum of f annuiy paymens which he invesor ges from previously purchased annuiies and Y is her labor income. The sum of annuiy paymens follows he process L = L PR a, where L is he sum of all annuiy paymens from annuiies purchased before and PR a is he annuiy paymen purchased in. In he invesor has o make a new decision how o spread her cash on hand W across bonds, socks, annuiies, and consumpion. A his poin

12 we wan o highligh our assumpion ha he invesor is no resriced o use annuiy payous for consumpion purposes only, as in Blake e al. (23), Milevsky and Young (22), Kingson and Thorp (25) and Sabile (23). The invesor has full flexibiliy in spending he annuiy payous. They can be used o consume, o purchase bonds or socks or even o purchase addiional annuiies. Addiionally, we impose borrowing consrains: M, S, PR, (3) since we do no allow he invesor o borrow agains fuure labor income and o sell lifeannuiies. Hence, from he individual s perspecive, he premium paid iniially canno be recovered. If she dies, her beques B will be given by he remaining financial wealh B M R f S = R. G. The Numerical Soluion of he Opimizaion Problem The problem of he individual is o choose in each year how much she consumes, saves in socks and bonds, and how much she invess in life-annuiies. Thereby she maximizes () under consideraion of he budge and shor-selling resricions (2) and (3). The opimal policy depends on four sae variables: he permanen income P, cash on hand W and annuiy payous from previously purchased annuiies L and age. As an analyic soluion o his problem does no exis, we use dynamic programming echniques o maximize he value funcion by backward inducion. Firs of all, he curse of dimensionaliy (Bellman, 96) can be parly miigaed by reducing he sae space by one sae variable. We exploi he scale independence of he opimal policy if we rewrie all variables using lower-case leers as raios of he permanen income componen P. 2

13 3 The goal funcion () can hen be rewrien as ( ) ( ) ( ) ψ ρ ψ ρ ρ ψ ρ β β / / / / ),, ( ),, ( = k b k p l w v p E c p l w v s s s, (4) where he only sae variables are normalized cash on hand and normalized annuiy payous. The evoluion of he sae variables and resricions are hen given by [ ]( ) ( ) ( ) [ ]( ) ( ) ( ). / exp / exp,, K a PR l l K K f l R s m R w K N a PR l l K U f l N R s m R w pr s m c pr s m w f f = = < = < = = ς We solve he problem in a hree-dimensional sae space by backward inducion. The coninuous sae variables normalized wealh w and normalized annuiy payous l have o be discreized and he only discree sae variable is age. For each grid poin we calculae he opimal policy and he value of he value funcion. Thereby he expecaion operaor in (4) is compued by resoring o gaussian quadraure inegraion and he opimizaion is done by numerical consrained minimizaion. We derive he policy funcions ),, ( w l s, ),, ( w l m, ),, ( w l pr, ),, ( w l c and he value funcion ),, ( w l v by cubic-splines inerpolaion. H. Calibraion of Parameers in he base-line case The following specificaions of parameers define he analyzed base-line case. The individual life-span ranges from age 2 o age (T = 8) a mos. Reiremen sars a age 65 (K = 46). Hence, work life is 45 years long while he maximum lengh of he reiremen phase is 36 years. The preference parameers are se o sandard values found in he life-cycle lieraure (e.g. Gomes and Michaelides, 25): coefficien of relaive risk aversion ρ = 5, elasiciy of ineremporal subsiuion ψ =.2, discoun facor β =.96, and beques weigh

14 k = 2. Since he empirical evidence on beques moives is somewha ambiguous (e.g., Bernheim e al., 985, and Hurd, 987), we display resuls for various degrees of beques srengh. The deerminisic age-dependen labor income funcion f() is aken from Cocco e al. (25). The funcional dependence reproduces a hump shaped income profile. Like Gomes and Michaelides (25) we selec volailiy parameers for individuals wih high school educaion bu wihou college educaion and se hem o σ u =.5 and σ n =. which is in line wih he esimaes found by Gourinchas and Parker (22). The replacemen raio including accumulaed pensions from Social Securiy bu excluding volunary annuiizaion is se o 68.2 percen as currenly esimaed by Cocco e al. (25). Since we expec a very srong relaionship beween he replacemen raio and he opimal life annuiy allocaion we do some sensiiviy analysis below he curren figure of 68.2 percen. Thus, we are able o analyze scenarios of sinking public pensions o various degrees. We se he real ineres rae R f o 2 percen, he equiy premium μ - R f o 4 percen and sock volailiy σ o 8 percen which is in line wih he recen life-cycle lieraure. The correlaion beween he sock reurns and he ransiory (permanen) income shocks φ n (φ u ) is zero. The expense facor δ is se o 7.3 percen for female annuians. This facor is aken from he 995 annuiy value per premium dollar compued on an afer ax basis by Michell e al. (999). We refer he ineresed reader o his aricle for a greaer discussion of he explici and implici coss relaed o annuiies. Applying nonlinear leas squares we fi he Gomperz force of moraliy o wo discree moraliy ables: he 996 US Annuiy 2 Aggregae Basic and he 2 Populaion Basic moraliy able. The leas square mehod gives us he following parameers for he 996 US Annuiy 2 Basic (female) able a m f = 9.5, a b f = 8.73 respecively. For he 2 Populaion Basic moraliy able we compue parameers for 4

15 females a m f =86.85, a b f = 9.98 respecively. While he firs discree moraliy able is used for annuiy pricing, he second moraliy able describes he individual s subjecive moraliy beliefs in he base-line case. In he laer analysis of asymmeric moraliy beliefs we also consider a case wih lower han populaion survival probabiliies. To double he force of moraliy we se he parameer v o wo. III. Resuls wih Annuiy Marke and wihou Annuiy Marke for he Base-line Case A. Presence of Annuiy Markes. A.. Policy Funcions. For our base-line case of an average US female wih a high school degree, he policy funcions show how he individual is influenced in her decision making by each sae variable and age as well. Figure (2) depics he opimal consumpion level c ( w, l, ), annuiy purchases pr ( w, l, ), bond holdings m ( w, l, ) and sock invesmens s ( w, l, ) in four separae graphs. Policies are condiional on surviving o a specific age. The policy sars a age 2 and ends a age. We se he sae variable annuiy paymens l o zero when ploing figure (2). This means ha no annuiies have been purchased before. The opology of he consumpion policy in graph (A) of figure (2) is almos fla for mos of he age-wealh saes excep for sligh increases wih higher wealh levels and surges for a very old individual unil age. The individual consumes only a small par of wealh on hand during mos of her life. This is because he individual seeks o cushion shor run adverse developmens in he income sream and especially he drop in reiremen income relaive o labor income by saving financial wealh. Furhermore, she wans o leave a cerain amoun of financial wealh o bequeah her heirs. If he individual urns very old and has a lo of cash on hand, he reiree will sar consuming more of her financial wealh since her moraliy becomes very high and longeviy risk less criical. Due o her beques moive she never consumes her whole financial wealh and always reserves a cerain amoun for beques. 5

16 4 (A) (B) 4 c 2 pr w age 8 4 w age 8 (C) (D) 4 4 m 2 s w age 8 w age 8 Figure 2. Opimal policy space. Opimal policy funcions for (A) consumpion level, (B) annuiy purchases, (C) bond invesmen, and (D) sock invesmen. The x-axis represens he individual s age, he y-axis he level of normalized cash on hand. Thus, she aims a achieving a precision landing in erms of consumpion and financial wealh according o her beques moive. Sock invesmens in graph (D) swing up as he level of financial wealh rises for any given age while in general sock invesmens decrease wih age. Wih sock holdings decreasing boh bonds and annuiies become more imporan over he remaining life-cycle. This resul is in line wih recen life-cycle lieraure and wih recommendaions made by praciioners as well as policy makers. The reason is ha he young individual is over-invesed in her human capial which is he presen value of labor and pension income. Even hough labor income is risky and uninsurable, bonds are considered as a closer subsiue for human capial han socks during work life because he implici discouning of fuure income is more 6

17 considered han is volailiy. During he reiremen phase human capial represens he presen value of he riskless pension income. Then, human capial is an implici annuiy holding because i perfecly resembles is payou srucure. Her human capial decreases wih age and hence he implici holdings in bonds and annuiies as well. In urn, wealh is increasingly composed of explici holdings of he laer wo asses. Considering graph (B) of figure (2), he reader can infer he policy funcions for new annuiy purchases. Even hough he recen reiremen lieraure regarding annuiies and common wisdom suggess reaing payou life-annuiies as a vehicle o realize consumpion afer he individual reires, we find ha she acually wans o subsanially purchase annuiies from age 6 on for mos saes of financial wealh. The lower he financial wealh, he laer he individual sars o buy annuiies. If financial wealh is sufficienly high annuiy purchases will rise unil he female reiree becomes 8 years old and will sar decreasing hereafer. Ye, he individual never buys annuiies if financial wealh remains very low. The reason is ha bonds and socks are preferred o annuiies because he individual has a beques moive and her pension income crowds ou he annuiy demand. Ineresingly, if he level of financial wealh rises, he demand for annuiies will surge relaive o bonds. In he las period, a age, he individual does no purchase annuiies anymore because she canno survive anoher period in our model and annuiy payous canno be ransferred o her heirs. Comparing graph (B) and graph (C) he reader can clearly see ha annuiy purchases are realized a he expense of bond savings. From age 6 on, he increasing moraliy credi and he need o hedge longeviy risk make annuiies more aracive relaive o bond savings unil age 8. A he end of he life-cycle, he invesor reduces her life annuiy purchases and shifs back o bonds in order o be able o leave beques for her heirs. Bond invesmens again become more aracive relaive o annuiies a he very end of he life-cycle because he beques moive becomes sronger and longeviy risk is less criical. Wih moraliy being especially high a 7

18 he end of he life-cycle, he moraliy credi iself is sill no high enough o avoid a decrease in life-annuiy purchases and an increase in bond invesmens. For he case wih previously purchased annuiies (l > ) he shape of he annuiy policy is similar excep ha he amoun of new annuiy purchases would decrease. A2. Life-Cycle Profiles and Asse Allocaion. In order o compue he expeced life-cycle profile, we resored o Mone Carlo mehods. We simulaed, life cycles for he baseline case scenario o compare he expeced consumpion, wealh, income, annuiy purchase, and annuiy payou pah Figure 3. Expeced life-cycle profile wih annuiies. The dashed line depics he expeced financial wealh pah. The doed line is he expeced consumpion pah. The solid line reflecs he expeced labor and pension income. The solid line wih aserisks represens annuiy purchases, and he line wih crosses annuiy payous. Clearly, he income profile is hump shaped. Firs her income increases hen i slighly backslides. A he beginning of reiremen, her las income is replaced by a pension paymen ha is exacly 68.2 percen of her previous labor income. While here is a sharp drop in 8

19 income, he consumpion pah remains smooh. We find ha he female invesor saves from her labor income unil she urns 5 years old in expecaion. Thereafer, she already sars divesing in order o realize consumpion before he acual reiremen begins (please see consumpion-income-raio in able ()). Even so, her financial wealh increases unil she reaches age 6 and i peaks a.73 imes he labor income (compare able ()). Up o his poin she wihdraws jus from capial gains. Financial wealh and beques poenial remain a a subsanial level unil age. The firs ime he invesor is expeced o purchase annuiies is age 6. She uses 8.59 percen of her cash on hand in order o buy annuiies (compare able ()). Throughou reiremen, she coninues annuiizing par of her wealh in expecaion Figure 4. One rajecory of he individual s life-cycle. The dashed line depics he financial wealh pah. The doed line is he consumpion pah. The solid line reflecs he labor and pension income. The solid line wih aserisks represens annuiy purchases and he line wih crosses annuiy payous. Sample pahs also show ha annuiizaion is pursued sep by sep over ime (compare he example rajecory of figure (4)). Swiching sraegies (Blake e al. (23), Milevsky and Young (22), Kingson and Thorp (25) and Sabile (23)) are herefore generally subopimal in our model. 9

20 Figure 5. Expeced asse allocaion. Lef graph: he upper righ cu area depics he purchases of new annuiies relaive o he sum of sock holdings, bond holdings and annuiy purchases. The middle area shows he fracion of bonds, he boom area reflecs he fracion of socks. Righ graph: he upper righ cu area depics he presen value of annuiies relaive o all invesmen holdings (socks, bonds and annuiy presen value). The middle area shows he fracion of bonds, he boom area is he fracion of socks. The savings behavior in our model also suggess ha he division of he life-cycle ino work life a.k.a. accumulaion phase and reiremen a.k.a. decumulaion phase is no fully adequae since disinvesing and annuiizaion can occur prior o reiremen. Expeced asse allocaions are given in figure (5). The lef hand graph highlighs new annuiy purchases relaive o he sum of sock holdings, bond holdings and annuiy purchases. For ages over sixy he individual buys annuiies wih iniially high and hen coninuously decreasing purchases. The righ hand graph displays he annuiy fracion no as newly purchased annuiies, bu as he presen value of all annuiies bough. For ages over sixy he asse allocaion shifs from bonds o annuiies in expecaion. This subsiuion effec is in line wih he policy funcions given in figure (2). From age 6 on, annuiies become more aracive relaive o bonds because he moraliy credi is now high enough o compensae he individual for he inflexibiliy drawbacks of annuiies. 2

21 Table Base case simulaion resuls Wealh Conribuion and Wihdrawals Annuiy Purchases Asse Allocaion Consumpion Cash on Hand- Income-Raio Conribuion- Cash on Hand Raio Conribuion- Income-Raio Conribuion over Income and Annuiy Payous Annuiy Purchases- Cash on Hand Raio Annuiized Wihdrawal Fracion Fracion of Sock Invesmens Fracion of Bond Invesmens Fracion of he Presen Value of Annuiies Consumpion- Cash on Hand Raio Consumpion- Income-Raio Consumpion over Income and Annuiy Payous % % 97.98% 2.2% % 46.99% % % 99.97%.3% % 32.9% % % 99.78%.22% % 23.4% % % 97.3% 2.69% % 7.67% % % 9.45% 9.55% % 4.2% % % 8.43% 8.57% % 2.4% % % 72.69% 27.3% %.9% % 92.95% 64.74% 7.43% 7.83%.9% % 52.39% 57.23% 8.99% 33.78% 3.84% % 53.7% 52.% 3.8% 44.9% 6.8% % 49.72% 45.97% 4.6% 49.97% 9.77% % 45.9% 4.9% 7.86% 52.5% 22.44% % 38.98% 35.46% 3.78% 5.76% 24.68% % 3.37% 32.44% 2.26% 46.3% 26.6% % 29.8% 3.5% 3.3% 37.72% 28.9% This able repors he summary resuls of he Mone-Carlo simulaion for he base-line case. The parameers for he base-line case are given in secion III A. All repored figures in his able show averages which are based on, simulaed life-cycles. The able repors he cash on hand-income-raio; conribuion-cash on hand-raio; conribuion-income-raio; conribuion over income and annuiy payous; annuiy purchases-cash on hand-raio; annuiized wihdrawal fracion; fracion of sock invesmens; fracion of bond invesmens; fracion of presen value of annuiies; consumpion-cash on hand-raio; consumpion-income-raio; consumpion over income and annuiy payous. 2

22 Surprisingly, as he individual ges older, he weigh of he annuiy presen value reduces while he fracion of bond holdings increases, even hough he individual coninues purchasing new annuiies. The higher he age, he lower becomes he annuian s survival probabiliy and hence he value of already purchased annuiies. This decrease in he value of annuiy holdings ouweighs he new annuiy purchases. A a very high age bonds are more aracive relaive o annuiies and socks because longeviy risk becomes smaller and he impac of he beques moive sronger. Unil age 4 he fracion of socks is around percen a young ages and monoonically decreases o percen. This resul is again in line wih he previous lifecycle lieraure and recommendaions of policy makers promoing decreasing sock exposures. B. Absence of Annuiy Markes. B.. Policy Funcions. The firs graph of figure (6) shows he opimal consumpion policy ha can be easily compared o he case wih annuiy markes because of he similariy beween he opimal consumpion rules. Bu he policy recommends consuming less a he end of he lifecycle han in a siuaion wih annuiy markes. This resul does no come as a surprise since he key insigh from invesing in annuiies is o realize life-long sreams of consumpion. Wihou annuiy markes longeviy risk prevails while beques and exhausion of financial wealh becomes criical. The policy for sock invesmens appears similar o he case wih annuiy markes. Again, shrinking human capial is responsible for he decreasing sock exposure over ime. The individual canno fall back on annuiy payous and has o rely on bonds o miigae longeviy risk. The risk of running ou of funds and he possibiliy of no meeing he beques moive can be more effecively reduced by purchasing bonds insead of socks. 22

23 (A) w age 8 (B) (C) w age 8 w age 8 Figure 6. Opimal policy space. Opimal policy funcions for (A) consumpion level, (B) bond invesmen, and (C) sock invesmen. The x-axis represens he individual s age. The y-axis-represens he level of normalized cash on hand. B.2. Life-Cycle Profiles and Asse Allocaion. The lef hand graph of figure (7) highlighs he much higher financial wealh when he invesor is abou o reire compared o he case wih annuiy markes. Acually, he financial wealh peaks a age 64 when i is 28.7 imes he iniial income compared o he peak (25.65 imes he iniial income) a age 6 wih annuiy markes. The individual needs higher financial wealh o miigae par of he longeviy risk and he relaed risk of leaving no beques for he heirs. Unlike he annuiy case he absolue level of consumpion decreases a he end of he life-cycle. 23

24 Figure 7. Expeced life-cycle profile and asse allocaion wihou annuiy marke. Lef graph: he dashed line depics he expeced financial wealh pah. The doed line is he expeced consumpion pah. The solid line reflecs he expeced labor and pension income. Righ graph: he uppermos cu area depics he invesmens in bonds. The boom area is he invesmen holdings in socks. The righ graph of figure (7) displays he uncondiional mean asse allocaion in equiies and bonds. As in he case wih annuiy markes he expeced fracion invesed in socks is around percen for individuals unil age 4, and hen decreases coninuously down o 27 percen. IV. Opimal Expeced Life-cycle Annuiy Invesmens for Alernaive Cases A. Risk Aversion Varying he level of risk aversion dramaically changes he opimal asse allocaion of he individual. We find ha he less risk-averse individual hardly buys annuiies in expecaion. Only a age 88 is she willing o buy a small amoun of annuiies. Expeced financial wealh of less risk-averse individuals is also relaively high compared o he base-line case. This resul does no sem from high conribuions bu raher from a high fracion of socks in financial wealh. A higher degree of risk aversion han in he base-line case goes hand in hand wih a very high demand for annuiies which e.g. a age 6 amouns o 67 percen of financial wealh. 24

25 Low risk aversion Moderae risk aversion High risk aversion (ρ = 2) (ρ = 5) (ρ = ) Figure 8. Life-cycle profiles and annuiy purchases for parameers of relaive risk aversion. The lef hand graph displays a low risk aversion of ρ = 2 while he middle graph reflecs a moderae level (ρ = 5). The high level of risk aversion (ρ = ) is displayed in he righ hand graph. Dashed lines are financial wealh. Doed lines reflec consumpion, solid lines income, and he solid lines wih aserisks reflec he annuiy purchases. The lines wih crosses show he annuiy payous. Also he desire for precauionary savings is much higher han in he base-line case. Srikingly, he high risk-averse invesor uses cash flows from annuiies and public pensions o reinves hem ino financial wealh mainly consising of bonds o ensure leaving sufficien beques o her heirs. B. Implicaions of Beques Moives Empirical sudies such as Kolikoff and Summers (98) found ha almos 8 percen of he oal accumulaed wealh in he Unied Saes is due o inergeneraional ransfers. This sylized, empirical fac raises he quesion as o wheher bequess are accidenal or inenional. The lieraure on inenional bequess disinguishes beween alruisic and sraegic beques moives as opposie ends of he specrum. 25

26 k = k = 2 k = Figure 9. Life-cycle profiles and annuiy purchases for differen beques moives. The lef hand graph shows he case in which he individual does no have any beques moive a all. The middle graph displays he base-line case, while he case of he srong beques moive is shown in he righ graph. Dashed lines are financial wealh. Doed lines reflec consumpion, solid lines income, and he solid lines wih aserisks reflec he annuiy purchases. The lines wih crosses show he annuiy payous. For insance, Abel and Warshawsky (988) sudy he alruisic beques moive in a reduced form and find a joy of giving parameer ha is of a subsanial magniude. Bernheim e al (985) analyze he sraegic beques moive and discover empirical evidence. By conras, Hurd (987) does no find any evidence of beques moives because he paern of asse decumulaion is similar among differen household sizes. In addiion, Hurd (989) can suppor his prior findings by showing ha he naure of mos bequess is accidenal because he dae of deah is uncerain o an individual. Since he resuls of hese sudies seem somewha ambiguous, we presen cases wih varying beques moives in our model. The lef graph of figure (9) illusraes he case in which he individual has no beques (k = ). We find ha expeced financial wealh is he lowes in his case compared o he cases wih k = 2 and k = 4. A age 83 she has exhaused her financial wealh compleely. This also means ha she won have anyhing o bequeah hereafer. From ha age on she sops purchasing annuiies and exclusively uses public pension income as well as annuiy income 26

27 from previously purchased annuiies for consumpion purposes only. If she survives unil age 97 she sars purchasing annuiies again because he moraliy credi of annuiies is exraordinarily high due o he small survival probabiliies. In his way she booss consumpion possibiliies condiional on her survival. In cases wih beques, she never exhauss her financial wealh compleely. She always keeps a cerain liquid capial sock in bonds and socks on hand in order o guaranee beques in case she dies. The higher he beques moive, he more he individual saves in socks and bonds (dashed line in figure (9)). She prefers liquid financial wealh o annuiy paymens ha las a life-ime and canno be ransferred o her heirs. Our resuls suppor common wisdom ha annuiy purchases are especially preferable o oher asse classes if he individual has no beques moive. Wih decreasing beques moive, he individual increases he weigh of annuiies purchases relaive o he size of he asse porfolio and purchases more annuiies over ime. However, even wih moderae or high beques moives here is a remarkable demand for annuiies. Surprisingly, he absolue demand for annuiies a age 6 is higher in he case in which he individual has a moderae beques moive compared o he case wih no beques moive. The reason is ha he individual wihou a beques moive already sars o purchase annuiies a age 59. C. Asymmery in Moraliy Beliefs We use he discree moraliy 996 US Annuiy 2 able for pricing annuiies. Survival probabiliies enering he uiliy funcion as well as he compuaion of he annuiy premium are idenical for he base-line case. Applying nonlinear leas square we fi he Gomperz force of moraliy o wo discree moraliy ables: he 996 US Annuiy 2 Aggregae Basic and he 2 Populaion Basic moraliy able. Figure () shows he fied condiional survival probabiliies for females from he ime hey are born o he age of. 27

28 Survival Probabiliies Figure. Fied survival probabiliies. The doed line shows he condiional survival probabiliies according o he fied US Annuian 2 Gomperz law. The dashed line depics he survival probabiliies, if he Gomperz law is fied o he 2 populaion basic moraliy able. The solid line is a linear ransformaion (ν = 2) of he Gomperz law for he US Annuian 2 moraliy able. Survival probabiliies for he ransformaion (ν = 2) range for mos ages below he 2 populaion basic moraliy able. US Annuian 2 survival probabiliies are by far higher han he 2 populaion basic survival probabiliies and in paricular higher han he ransformaion (ν = 2). Insurance companies calculae annuiy premiums from higher survival probabiliies as a resul of he adverse selecion process since individuals who believe hemselves o be healhier han average are more likely o buy more annuiies (e.g. Brugiavini (993)). The magniude of asymmery in moraliy beliefs is refleced by he implici coss he annuian has o bear when she purchases an annuiy. The higher he asymmery in moraliy beliefs, he higher are he implici coss of annuiies from he view of he individual. 28

29 996 US Annuiy 2 Aggregae Basic 2 Populaion Basic Doubled force of moraliy Figure. Life-cycle profiles and annuiy purchases for differen subjecive survival probabiliies. The lef graph displays he case in which survival probabiliies are idenical o he underlying moraliy 996 US Annuiy 2 Aggregae Basic able used for annuiy pricing. The middle graph reflecs he base-line case wih 2 populaion basic probabiliies. The righ hand graph shows he case when he force of moraliy is wice as high as in he Annuian 2 able. Dashed lines are financial wealh. Doed lines reflec consumpion, solid lines income, and he solid lines wih aserisks reflec he annuiy purchases. The lines wih crosses show he annuiy payous. These higher implici coss make he purchase of annuiies more unaracive because of reduced moraliy credis resuling in lower annuiy demand. However, even a female wih a doubled force of moraliy buys considerable amouns of annuiies since she is sill willing o accep high premiums o hedge longeviy risk. D. Differen Levels of Public Pensions As benefis from public pensions are idenical o payou srucures of life-annuiies, he laer produc is undoubedly a perfec subsiue for public pensions. The obvious difference is in he way of funding he fuure pension paymens. 29

30 ξ =.682 ξ =.6 ξ = Figure 2. Life-cycle profiles and annuiy purchases for differen replacemen raios. The righ and he middle graph display he case in which public pensions are cu o a replacemen raio of 5 and 6 percen, respecively. The lef hand graph displays he base-line case. Dashed lines are financial wealh. Doed lines reflec consumpion, solid lines income, and he solid lines wih aserisks reflec he annuiy purchases. The lines wih crosses show he annuiy payous. Mos public pension sysems are based on an iner-generaional conrac whereby he generaion of Social Securiy conribuors finances he generaion of public pension beneficiaries. The pay-as-you-go public pension sysems are running ino rouble since longer life-expecancies and lower birh raes lead o decreasing raio of conribuors o beneficiaries. Realizing he circumsances, we analyze wo cases in which he replacemen raio ζ of our model is cu from 68.2 percen firs o 6 percen and hen o 5 percen. We assume ha he labor income process remains he same o reflec consan Social Securiy axes and decreasing public pension paymens. On he conrary, life-annuiy payous are funded by he beneficiary herself. Once she pays he annuiy premium o he insurance company she receives annuiy payous unil she passes away. Figure (2) shows he crowding-in effec ino he annuiy markes. As anicipaed here is a subsanial increase in expeced annuiy purchases when public pensions are cu. This means ha he individual wans a subsiue for public pension cus. Direc invesmens in bonds and socks increase o a moderae exen. For 3

31 example in he case ξ =.5 he individual a age 6 is expeced o purchase 56.8 percen more annuiies han in he base-line case while invesmens in bonds and socks rise moderaely by 2.7 percen. V. Welfare Analysis The subsanial demand for annuiies suggess ha considerable uiliy gains can be generaed hrough he presence of annuiy markes. We do a welfare analysis similar o Michell e al. (999). In our analysis we firs compue he expeced uiliy of individuals living in a world wih and wihou access o annuiy markes separaely. Of course, he expeced uiliy is always higher for individuals who can volunarily purchase annuiies. Then we compue he equivalen increase in financial wealh for every age in order o measure he expeced uiliy gains in moneary unis. The equivalen increase in financial wealh refers o he compensaion an individual requires o achieve he same uiliy level in a world wihou annuiy markes as in he presence of hem. Therefore, we equae he expeced uiliy values of individuals wih and wihou access o annuiy markes by raising he financial wealh of individuals in he no-annuiy case. For he base-line case, boh graphs in figure (3) depic he equivalen increase in financial wealh due o he presence of annuiy markes from he very beginning on. Figure (3) shows ha individuals who can buy annuiies realize equivalen increases in financial wealh every year of heir lifeime. Even a young ages individuals gain from annuiy markes because hey anicipae he indirec uiliy gains emerging a he end of heir life-cycle. Especially for old individuals annuiy markes imply a considerable wealh increase from 4.4 percen (or 3.36 imes he firs income) a age 8 up o 3.7 percen (or 3.22 imes he iniial income) a age 9. The presence of annuiy markes allows he individual o finance consumpion and beques more effecively han in he case wihou annuiies. 3

32 Figure 3. Equivalen relaive and absolue increase in financial wealh. Lef graph: he solid line displays he percenage increase in financial wealh which he individual who canno buy annuiies needs in order o achieve he same uiliy as he individual who can buy annuiies. Righ graph: he solid line displays he increase in financial wealh as muliple of iniial income which an individual wihou access o annuiy markes requires in order o achieve he same level of uiliy as he individual who can buy annuiies. Figure 4. Consumpion Perceniles wih and wihou Annuiy Markes. Solid lines show he, 5 and 99 percenile of he consumpion disribuion in he case wih annuiy markes and doed lines he perceniles for he case wihou annuiy markes. 32

33 Table 2 Equivalen increase in financial wealh (percenage poins) Cases: Base-line case Low risk aversion (ρ = 2) High risk aversion (ρ = ) No beques moive (k = ) High beques moive (k = 4) Bad survival probabiliies (ν = 2) Good survival probabiliies (p s = p a ) Lowes pension income (ζ =.5) Lower pension income (ζ =.6) This able repors welfare gains in he presence of annuiy markes for he differen ypes of individuals we considered. Welfare gains are compued as he equivalen percenage increase in financial wealh an individual wihou access o annuiy markes would need in order o aain he same expeced uiliy as in he case wih annuiy markes. The compuaion is done for age 6, 7, 8 and 9. We assume ha individuals have aced opimally unil he specific year. This sems from he fac ha she can profi from consumpion of life long annuiy paymens and hedge he longeviy risk away. Anoher way of looking a i is o undersand he effec of he longeviy yield ha increases wih he holding period of life-annuiies because he individual oulives her peers. The equivalen increase in financial wealh can be aribued o advanages in consumpion possibiliies gained from he presence of he annuiy markes. Figure (4) demonsraes ha wihou annuiy markes he individual s disribuion of consumpion is decreasing when she ges very old because he individual s financial wealh shrinks whereas he beques moive is more significan. However, in he case wih annuiy markes he individual purchases annuiies in a way ha he consumpion disribuion does no decrease. We also calculaed he equivalen increases in financial wealh for all cases given in secion IV by comparing he uiliy wih ha of individuals who do no have access o annuiy 33

34 markes. Table (2) shows ha for all cases he paern of equivalen increases in financial wealh over he life-cycle is similar o he one in he base-line case. Even in he cases wih high public pensions, high beques moives and high degree of asymmeric moraliy beliefs annuiy markes deliver subsanial equivalen increases in financial wealh. VI. Conclusion This aricle inroduces incomplee annuiy markes o he life-cycle lieraure and in urn lifecycles o he insurance lieraure. Life-cycle modeling in his conex becomes necessary as separae analyses of he accumulaion and decumulaion period can be misleading for he invesor because decisions in boh phases are inrinsically ied o each oher. Our analysis offers insighs ino he individual s demand for life annuiies and he way she gains uiliy from annuiy markes over he life-cycle. We find ha he individual has demand for boh flexible wihdrawal possibiliies from financial wealh and inflexible annuiies. Srikingly, individuals sar wih high annuiy purchases before reiring and coninue annuiizing gradually and slowly over he remaining life-ime. Hence, i is shown ha swiching sraegies canno be opimal. Compuaions of equivalen wealh increases show ha life-annuiies are indeed a good deal for he whole specrum of individuals considered excep for hose wih low risk aversion. This is somewha surprising, since we considered individuals wih already high pension income, srong beques moives, and asymmeric moraliy beliefs. The individual can realize significan increases in equivalen financial wealh if she oulives mos of her peers and benefis from he relaed moraliy credi. Annuiy payous enable her o hedge longeviy risk as much as hey conribue o enjoying a sable consumpion sream during he whole reiremen period. Fuure work suggess iself. Firs, he subopimaliy of annuiy swiching sraegies is a research projec we are currenly working on, since hese sraegies have been considered in 34

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