Research. Michigan. Center. Retirement

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1 Michigan Universiy of Reiremen Research Cener Working Paper WP Opimizing he Reiremen Porfolio: Asse Allocaion, Annuiizaion, and Risk Aversion Wolfram J. Horneff, Raimond Maurer, Olivia S. Michell, and Ivica Dus MR RC Projec #: UM06-11

2 Opimizing he Reiremen Porfolio: Asse Allocaion, Annuiizaion, and Risk Aversion Wolfram J. Horneff Johann Wolfgang Goehe Universiy of Frankfor Raimond Maurer Johann Wolfgang Goehe Universiy of Frankfor Olivia S. Michell Wharon School, Universiy of Pennsylvania Ivica Dus Johann Wolfgang Goehe Universiy of Frankfor July 2006 Michigan Reiremen Research Cener Universiy of Michigan P.O. Box 1248 Ann Arbor, MI hp:// (734) Acknowledgemens This work was suppored by a gran from he Social Securiy Adminisraion hrough he Michigan Reiremen Research Cener (Gran # 10-P ). The findings and conclusions expressed are solely hose of he auhor and do no represen he views of he Social Securiy Adminisraion, any agency of he Federal governmen, or he Michigan Reiremen Research Cener. Regens of he Universiy of Michigan David A. Brandon, Ann Arbor; Laurence B. Deich, Bingham Farms; Olivia P. Maynard, Goodrich; Rebecca McGowan, Ann Arbor; Andrea Fischer Newman, Ann Arbor; Andrew C. Richner, Grosse Poine Park; S. Marin Taylor, Gross Poine Farms; Kaherine E. Whie, Ann Arbor; Mary Sue Coleman, ex officio

3 Opimizing he Reiremen Porfolio: Asse Allocaion, Annuiizaion, and Risk Aversion Wolfram J. Horneff, Raimond Maurer, Olivia S. Michell, Ivica Dus Absrac Reirees mus draw down heir accumulaed asses in an orderly fashion so as no o exhaus heir funds oo soon. We derive he opimal reiremen porfolio from a menu ha includes payou annuiies as well as an invesmen allocaion and a wihdrawal sraegy, assuming risk aversion, sochasic capial markes, and uncerain lifeimes. The resuling porfolio allocaion, when fixed as of reiremen, is hen compared o phased wihdrawal sraegies such a self-annuiizaion plan or he 401(k) defaul paern encouraged under US ax law. Surprisingly, he fixed percenage approach proves appealing for reirees across a wide range of risk preferences, supporing financial planning advisors who ofen recommend his rule. We hen permi he reiree o swich o an annuiy laer, which gives her he chance o inves in he capial marke and be on deah. As risk aversion rises, annuiies firs crowd ou bonds in reiree porfolios; a higher risk aversion sill, annuiies replace equiies in he porfolio. Making annuiizaion compulsory can also lead o subsanial uiliy losses for less risk-averse invesors. Auhors Acknowledgemens This research was conduced wih suppor from he Social Securiy Adminisraion via he Michigan Reiremen Research Cener a he Universiy of Michigan under subconrac o he Johann Wolfgang Goehe-Universiy of Frankfur and a TIAA-CREF Insiue gran o he Naional Bureau of Economic Research. Addiional suppor was provided by he Pension Research Council a The Wharon School of he Universiy of Pennsylvania, and he Friz-Thyssen Foundaion. Opinions and errors are solely hose of he auhors and no of he insiuions wih whom he auhors are affiliaed. This is par of he NBER Program on he Economics of Aging Horneff, Maurer, Michell, and Dus. All Righs Reserved.

4 Opimizing he Reiremen Porfolio: Asse Allocaion, Annuiizaion, and Risk Aversion Baby Boomers nearing reiremen are now argeed by compeing financial service providers seeking o help hem manage heir money in heir golden years. Employer-based pensions are also swiching from defined benefi o defined conribuion plans, furher underscoring reirees need for insighs regarding how hey migh conver heir accumulaed asses ino a sream of reiremen income wihou exhausing heir funds oo soon. On he one hand, insurers offer life annuiies as he preferred disribuion mechanism. On he oher, muual fund providers propose phased wihdrawal plans as he beer alernaive. This paper compares differen reiremen payou approaches o show how people can opimize heir reiremen porfolios by simulaneously using invesmen-linked reiremen rules along wih life annuiies. To explore his issue, we firs evaluae payou producs using he defaul paern adoped under US ax law for defined conribuion or 401(k)-ype pension porfolios. This permis us o deermine wheher hese wihdrawal rules sui a broad range of invesors, and we illusrae he drawback of sandardizing wihdrawal rules. Nex, we show ha reiremen planning would no involve a simple choice beween annuiizing all one s money versus selecing a phased wihdrawal plan, bu raher i requires a combined porfolio consising of boh annuiies and muual fund invesmens. Using a lifeime uiliy framework, we compare he value of purchasing a sand-alone life annuiy versus a phased wihdrawal sraegy backed by a properly diversified invesmen porfolio, as well as combinaions of hese wo producs. This framework also enables us o demonsrae he welfare implicaions of making annuiizaion compulsory a a specific age, as is currenly he case in Germany and he UK. Prior Sudies The simples form of life annuiy is a bond-like invesmen wih longeviy insurance proecing he reiree from ouliving her resources, guaraneeing lifeime level paymens o he annuian. 1 Insurers hedge hese conracs by pooling he longeviy risks across a group of annuiy purchasers. Sandard economic heory eaches us ha life annuiies will be valued by risk-averse reirees, inasmuch as hese conracs provide a seady income for life and hence hey 1 Accordingly, life annuiies are similar o public defined benefi pensions wih respec o heir payou srucure.

5 2 proec he reiree agains he risk of exhausing her asses. 2 Thus Yaari (1965) showed ha he reiree maximizing a ime separable uiliy funcion wihou a beques moive would buy annuiies wih all her wealh, given a single risk-free asse and facing acuarially fair annuiies; he approach has been exended by Davidoff e al. (2005) who again predics full annuiizaion. Ye available evidence from mos counries indicaes ha very few reirees acually purchase annuiies wih heir disposable wealh. Effors o explain his so-called annuiy puzzle have noed some disadvanages of annuiizaion; for example, buyers lose liquidiy because he asses usually canno be recovered even o mee special needs (e.g. in he case of poor healh; c.f. Brugiavini 1993). The presence of a beques moive also reduces reiree desires o annuiize wealh, and in he US, more han half of he elderly anicipae leaving a beques worh more han $10,000 (Bernheim, 2001; Hurd and Smih, 1999). Oher explanaions for why people may be relucan o buy annuiies include high insurance company loadings; he abiliy o pool longeviy risk wihin families; asymmeric moraliy expecaions beween annuiy buyers and sellers; and he exisence of oher annuiized resources (e.g. Social Securiy or employer-sponsored pensions; c.f. Brown and Poerba, 2000; Michell e al., 1999). In addiion, annuiies appear relaively expensive in a low ineres rae environmen, as compared o equiy-based muual fund invesmens. And i also mus be noed ha, in he US a leas, many payou annuiies sold by commercial insurers are fixed in nominal erms, so he annuiy purchaser does no paricipae in sock marke performance (c.f. Davidoff e al., 2005). Anoher reason people may no annuiize is ha hey believe hey will do beer by coninuing o inves heir reiremen asses, making wihdrawals periodically over heir remaining lifeimes. Doing his is no so simple, however, as he reiree mus selec boh an invesmen sraegy how much o inves in socks and bonds and a wihdrawal rae, spelling ou how much of her balance o spend per year. Financial advisors ofen recommend rules of humb, for insance dividing he porfolio roughly 60% socks/40 % bonds and a spending rule of 4-5% of he balance per year (Polyak, 2005; Whiaker, 2005). Compared o buying a fixed life annuiy, such an invesmen-linked phased wihdrawal sraegy has several advanages: i provides greaer liquidiy, paricipaion in capial marke reurns, possibly higher consumpion while alive, and he chance of bequeahing asses in he even of early deah. Ye a phased 2 See he sudies reviewed in Michell e al. (1999).

6 3 wihdrawal acic also exposes he reiree o invesmen risk and i offers no longeviy pooling, so he reiree could possibly oulive her asses before her uncerain dae of deah. Thus any wihdrawal plan which includes some risky invesmens and also requires he reiree o draw a fixed amoun from her accoun each period involves a sricly posiive probabiliy of hiing zero before he reiree dies. The risk of running ou of money can be parially miigaed by linking he drawdown o he fund balance each period, hough of course his will produce benefi flucuaions which migh fall subsanially below wha he life annuiy paymen would have been. Prior sudies have compared he pros and cons of specific phased wihdrawal plans wih life annuiies ha pay fixed benefis (see Table 1). For insance, some auhors calculae he probabiliy of running ou of money before he reiree s uncerain dae of deah, using assumpions abou age, sex, capial marke performance, and iniial consumpion-o-wealh raios. 3 These analyses also show how an opimal asse mix can be se o minimize he probabiliy of zero income. Follow-on work by Dus e al. (2005) exended his research by quanifying risk and reurn profiles of fixed versus variable wihdrawal sraegies using a shorfall framework. On he reurn side, ha sudy quanified he expeced presen value of he beques poenial and he expeced presen value of benefi paymens; conversely, i measured he risk as he iming, probabiliy, and magniude of a loss when i occurs, compared o a fixed annuiy benchmark. Table 1 here A naural nex quesion o address is wheher reirees migh benefi from following a mixed sraegy, where he porfolio migh involve boh a life annuiy and a wihdrawal plan. A mixed sraegy seems inuiively appealing as i reduces he risk of paymens falling below an annuiy benchmark and i also enhances payous early on. 4 I is also ineresing ha some governmens have mandaed ha ax-qualified reiremen saving plans include a mandaory annuiy ha sars afer an iniial phased wihdrawal phase. For example, in he UK, accumulaed pension asses had o be mandaorily annuiized by age 75 (his rule expired in 2006). 3 See for insance Albrech and Maurer (2002); Ameriks e al. (2001); Bengen (1994, 1997); Chen and Milevsky (2003); Ho e al. (1994); Hughen e al. (2002); Milevsky (1998, 2001); Milevsky and Robinson (2000); Milevsky e al. (1997); and Pye (2000, 2001). 4 See Blake e al. (2003); Milevsky and Young (2002); Kingson and Thorp (2005); Milevsky e al. (2006); and Dus e al. (2005). An alernaive acic would be o annuiize gradually (c.f. Kapur and Orszag, 1999); Milevsky and Young (2003) show ha purchasing consan life annuiies is a barrier conrol problem.

7 4 Germany s Rieser plans provide a ax inducemen if life annuiy paymens begin o pay ou a age 85 (wihdrawn amouns mus eiher be consan or rising, prior o annuiizaion.) In he US, of course, annuiizaion is no compulsory for 401(k) plans; as a resul, mos reirees roll hem over o an Individual Reiremen Accoun and manage he funds hemselves, subjec o he ax laws requiring minimum disribuions o begin a age 70 ½. Despie he growing ineres in he reiremen payou problem, prior sudies have no ye fully evaluaed he pros and cons of purchasing a sand-alone life annuiy versus a phased wihdrawal sraegy backed by a properly diversified invesmen porfolio, as well as combinaions of hese wo producs. In wha follows, we show ha he appropriae mix depends on he reiree s aiude oward risk as well as key assumpions regarding he capial marke and acuarial ables. Comparing Alernaive Payou Rules Our model assumes ha he reiree is endowed wih an iniial level wealh V 0. This can be eiher used o purchase a cos PR 0 a single-premium life annuiy-due paying a consan nominal annual benefi, or o finance a phased-wihdrawal schedule of paymens unil he funds are exhaused (Dus e al., 2005). In wha follows, we focus on he case of he female reiree, inasmuch as longeviy risk is more imporan for women han for men. The Consan Life Annuiy. When he consumer purchases a life annuiy, i pays her a consan amoun A condiional on her survival: A & 1 = A = PR ax. Using he acuarial principle of equivalence, we can deermine he gross single premium of he annuiy by calculaing he presen value of expeced benefis paid o he annuian (including expense loadings). The annuiy facor a& x & for he reiree of age x is given by: 5 w x 1 a& = ( 1+ δ ) & x (1 + ) = 0 px r, (1) where w is he assumed las age (radix) of he moraliy able; p x = p x p x+-1 is he probabiliy ha a reiree of age x will survive o age x +, where p x are he year-o-year survival probabiliies for an individual aged x; δ is he expense facor; and r is he yield on a zero 5 Here we resric our analysis o consan nominal annuiies during he payou phase; furher research will consider variable annuiies.

8 5 coupon bond mauring a ime aken from he curren ineres rae erm srucure. 6 Survival probabiliies used o price he annuiy are aken he female US Annuian 2000 moraliy able provided by he Sociey of Acuaries. Given hese assumpions, and an expense facor of 7.3 percen (Michell e al., 1999), we compue he yearly fixed nominal payou a he beginning of each year for life as $7.2 per $100 premium. 7 This consan payou life annuiy consiues an asse class wih a unique reurn profile, as paymens are condiional on he annuian s survival. The capial of hose who die is allocaed across surviving members of he cohor. Accordingly, a survivor s one-period oal reurn from an annuiy is a funcion of her capial reurn on he asses plus a moraliy credi. Oher hings equal, he older he individual, he higher is he moraliy credi. Alernaive Phased Wihdrawal Plans. If he reiree insead pursued a phased wihdrawal plan, she can selec eiher a fixed or a variable wihdrawal paern. If she elecs he fixed benefi approach, she will pay herself a consan benefi B = min( B, V ) unil she dies or exhauss her reiremen asses (here V is he value of he reiremen wealh a he beginning of year = 0, 1, jus before ha year s paymen). In wha follows, B is se o equal he iniial payou of a life annuiy available for he same iniial value V. The idea of he fixed benefi rule is o replicae he payou from a life annuiy as long as he funds permi (someimes ermed a self-annuiizaion sraegy), while a he same ime reaining liquidiy and some beques poenial in he even of an early deah. Of course he risk of such a self-annuiizaion sraegy is ha poor invesmen reurns could drive V o zero while he reiree is sill alive. If she elecs a variable phased wihdrawal plan, several opions are available. The hree we explore in deail here are he fixed percenage rule, he 1/T rule, and he 1/E(T) rule. Under he firs, a consan fracion is wihdrawn each period from he remaining fund wealh; ha is, he benefi-wealh raio is fixed over ime so ha: B V ω = ω. = (2) This wihdrawal rule has he advanage of simpliciy, requiring no informaion regarding he maximum possible duraion of he payou phase or he reiree s personal characerisics. For example, ω can be se a he fracion which equals he life annuiy payou divided by iniial 6 To model he erm srucure of risk free ineres raes we assume a Vasicek model and use he corresponding spo raes o specify he discoun facors. Deails on parameerizaion are given in Appendix A. 7 This is consisen wih curren quoes; see hp://

9 6 wealh. 8 Alernaively, he 1/T rule deermines he wihdrawal fracion according o he maximum possible duraion of he plan, or for example, o he oldes age in a moraliy able. Therefore he wihdrawal fracion under he 1/T framework is no consan bu raher rises wih age. Formally, he benefi-wealh raio a he beginning of year ( = 0, 1, T-1) of his reiremen plan is given according o: B V 1 = ω =. (3) T Finally, he 1/E(T) wihdrawal rule akes ino accoun he reiree s remaining life expecancy in a dynamic way. Then, for a reiree of age x, her benefi-o-wealh raio in period condiional on survival is given as: 9 B V 1 = ω =. (4) E[ T ( x + )] The shorer is her expeced remaining lifeime E[T(x+)], he higher he fracion ha she will wihdraw from her accoun. The 1/E(T) wihdrawal rule is akin o he 401(k) rule, requiring reirees o begin consuming asses from age 70½ o ensure ha hey will consume heir axqualified pension accouns insead of leaving hem as bequess for heir heirs. The female US 2000 Annuian Table is used for expeced remaining lifeimes. Figure 1 displays he reiree s wihdrawal rae for he hree variable wihdrawal rules. The fla line for he fixed percenage rule conrass wih he rising fracion wih age for boh he 1/T and 1/E(T) rules. The 1/T rule sars ou wih a small wihdrawal fracion and remains moderae for many years before rapidly increasing o reach a benefi-o-wealh raio of one a age T = 100, i.e. he maximum age assumed in our uiliy analysis. By conras, he 1/E(T) rule sars wih a moderae wihdrawal percenage and is less convex han he 1/T rule; consequenly he 1/E(T) pah involves an earlier porfolio drawdown as compared o he 1/T rule. Figure 1 here Expeced Benefis and Value a Risk under Alernaive Payou Paerns. A reiree who pursues a phased wihdrawal plan mus allocae her remaining asses across a porfolio of socks and bonds. To model he payou implicaions of alernaive invesmen choices, we assume ha he 8 The firs rae (ω-rule) is hen equal o he 1/ä x+ rule used in Blake e al and in Milevsky and Young This assumes p x is he condiional probabiliy ha an x-year old woman will aain age x +, so he complee w x expecaion of life is calculaed as [ ] T ( x + ) = = E. p x 0

10 7 sochasic dynamics of he marke reurns of boh asse classes follow a muli-dimensional geomeric random walk wih drif. We calibrae he model for US daa, using ime series for large cap equiies and long erm bonds ranging from 1974 o 2004 (deails appear in Appendix A). We assume ha he reiremen asses are rebalanced coninuously o mainain an equiy/bond asse spli of 60/40%, as his is commonly recommended by financial advisers for reiremen porfolios. Figure 2 compares expeced benefi pahs for he various disribuion programs o he life annuiy profile, condiional on survival. Focusing firs on he fixed benefi rule, in he firs year of her reiremen, he reiree s mean benefis equal her life annuiy payou; his is sensible as his rule was designed o mimic he fixed life annuiy unil funds are exhaused. A some poin, however, expeced paymens mus decrease, reflecing he risk of running ou of money. The fixed percenage rule also sars in he firs year wih a benefi equal o he life annuiy payou, by consrucion. Thereafer, mean benefis rise as he reiree ages, because he pension accoun s expeced gross rae of reurn exceeds he consan benefi-o-wealh-raio. Figure 2 here The oher wo payou paerns behave somewha differenly. Compared o he oher payou plans, he 1/T rule offers lower expeced benefis unil age 74, bu expeced benefis rise exremely quickly afer ha, and o very high levels. This occurs because he 1/T rule pays he reiree only a small amoun of money during he firs par of he reiremen period, in fac, less han her porfolio s annual expeced reurn. Accordingly, he reiree coninues o build up saving in earlier years which can boos her expeced benefis laer. The 1/E(T) rule begins wih a lower annual payou, which hen rises above he fixed annuiy paymen when he reiree is sill raher young (age 69). Thereafer, he 1/E(T) benefis peak (a age 88) and decline; as less wealh remains in he accoun, a some poin expeced benefis mus fall, alhough he wihdrawal fracion increases. I is also insrucive o repor a wors-case risk measure for he phased wihdrawal plans. Figure 3 depics he probable minimum benefi (o a confidence level of α = 1%) compared o he life annuiy profile. This PMB meric is defined as follows: P(B < PMB, 1-α ) = α = 1% (5) The PMB, 99% of a disribuion program represens he firs percenile of he payou disribuion in each period, condiional on survival. In oher word, if he reiree is sill alive years afer

11 8 reiremen, she would receive a payou from he payou program equal o or higher han he PMB, 99% wih a probabiliy of 99%. This meric looks a he lower ail of he payou disribuion, so i can be inerpreed as a wors-case risk measure. Figure 3 here I is imporan o noe ha, iniially, he probable minimum benefi for he fixed benefi rule is he same as he annuiy paymen, bu i quickly falls over ime and becomes zero a age 80, i.e. he reiree runs ou of money. By conras, he benefis in wors-case siuaions for all he variable wihdrawal plans are well below he annuiy paymens during he firs 20 years of reiremen. The probable minimum benefis of he 1/T as well as of he 1/E(T) rules are much lower han he annuiy paymen early on, and hey increase hereafer. A age 85, he probable minimum benefi for he 1/T even exceeds he annuiy paymen. On he conrary, he 1/E(T) rule never exceeds he annuiy paymen. The probable minimum benefi of he fixed percenage rule remains a a very low level and never recovers. In summary, all he disribuion programs examined incorporae wors-case risk profiles ha are remarkably high for reirees. A Uiliy Approach o Disribuion Rules The expeced benefi and he probable minimum benefi merics described above are useful in exploring risk/reurn radeoffs of differen payou sraegies. Nex we urn o a uiliybased approach which permis us o assess how a reiree migh evaluae hese disribuion programs while aking ino accoun risk aversion and ime preference. Impac of Risk Aversion on he Choice of Disribuion Rule. To undersand how he various disribuion paerns would be assessed by people wih differen levels of risk aversion, we adop an addiively ime-separable uiliy funcion of he Consan Relaive Risk Aversion (CRRA) class. 10 As above, B denoes he nominal level of benefis from a phased wihdrawal plan, while A represens he benefis from a life annuiy a ime. Here V represens he value of he remaining asses in he reiremen accoun, which also represens he beques should he reiree die. We assume he reiree s objecive funcion U is defined over oal benefis received and beques lef a deah, and i akes he form: This value funcion is also consisen wih oher sudies which invesigae payou sraegies including annuiies; c.f. Table 1 as well as Dushi and Webb(2004) and Milevsky and Young (2003). 11 In our model seup, he reiree uses all payous only for consumpion purposes.

12 9 U E K = = 0 β p ( B + A ) 1 γ 1 γ s V + ( ) k β 1 p x+, (6) 1 γ γ 1 s s px+ i x+ i= 0 1 where β reflecs he ime preference of he invesor (se o 0.96, in line wih Blake e al. 2003). The srengh of he beques moive is represened by k (which can range from 0 o 1). The uiliy of benefis in period is weighed by he condiional probabiliy p x ha a woman of age x a he beginning of he reiremen phase is sill alive a. The parameer γ reflecs he individual s coefficien of relaive risk aversion (RRA) and also her willingness o engage in ineremporal subsiuion in consumpion. The parameer plays an imporan role in evaluaing he various disribuion programs when he payous are uncerain because of sochasic asse reurns. In wha follows, we repor resuls using a range of risk aversion coefficiens from 1 o 10. We classify as leas risk averse hose wih γ below 1; he moderaely risk averse have γ from 1 o 5; and very risk averse individuals have γ above Implemening his approach requires ha we firs selec he opimal saic (bu coninuously rebalanced) asse allocaion of socks and bonds for each wihdrawal rule, for each level of risk aversion, and holding oher parameers fixed. Nex, we compue analyically he expeced lifeime uiliy given his asse allocaion paern for each phased wihdrawal rule (excep for he fixed benefi rule). We hen ransform his uiliy level ino an equivalen nominal annuiy income sream for life. 13 The resuling cerainy equivalens can hen be direcly compared o he nominal life annuiy benchmark. Finally, as a benchmark for he convenional payou paern, we also compue an opimal (variable) wihdrawal plan for an individual wihou access o an annuiy marke for every level of risk aversion using sochasic dynamic programming. The sochasic componen of he problem arises from uncerainy regarding dae of deah as well as uncerain asse reurns. This means ha we selec boh he opimal wihdrawal paern for ω and is associaed asse allocaion pah o maximize he expeced lifeime uiliy funcion given in (6) (see Appendix B for deail). The annuiy-equivalen income sream can be inerpreed as he lifelong nominal annuiy sream ha would provide he same level of lifeime uiliy o he reiree, if she lacked access o an annuiy marke. 12 To price he annuiy, we use female annuian moraliy ables from he Sociey of Acuaries; he female 2000 Populaion moraliy able is used o weigh uiliy (see 13 See Cocco e al. (2005) for a similar use of he equivalen consan consumpion sream (equivalen annuiy sream).

13 10 Figure 4 displays he resuls. 14 The graph confirms ha risk aversion plays an imporan role in influencing he preferred payou (as in Brown e al. 2001). Furhermore, he fixed benefi rule is consisen only wih very low levels of risk aversion (<1), as i exposes he reiree o he risk of ouliving her asses. Also a hose risk aversion levels, he fixed benefi approach is dominaed by all he oher payou rules and he annuiy pah as well. Surprisingly, he fixed percenage rule is preferred across a wide specrum of risk preferences. I dominaes he 1/T rule for all levels of risk aversion considered, and i is more appealing han he annuiy for low/moderae levels of risk aversion. In his sense, our findings are supporive of hose in he financial planning indusry who propose such a fixed benefi rule. Figure 4 here We also find ha he 1/T rule is clearly he leas preferred of all he variable payou rules. By conras, he 1/E(T) rule does appeal o low and moderaely risk averse reirees, bu i is unfavorable for he very risk averse. For exremely high levels of risk aversion, he 1/E(T) rule is he leas aracive of all variable payou rules. The opimal wihdrawal plan provides higher uiliy for low/moderae reires han he oher payou rules and also han he annuiy. Only he very risk-averse will find he fixed annuiy appealing, given hese parameers. To illusrae he relaive magniudes, consider a 65-year old reiree wih moderae risk aversion (γ =3). Relaive o buying an annuiy, she would be 16.8% beer off if she seleced he fixed percenage rule; 34.7% worse if she adoped he 1/T rule; 9.7% beer if she adoped he 1/E(T) rule; and 30.4% beer if she seleced he opimal wihdrawal plan. For he exremely riskaverse reiree (γ=9), relaive o buying an annuiy, she would be 17.6% worse off if she seleced he fixed percenage rule; 52.4% worse if she adoped he 1/T rule; 63.4% worse off if she adoped he 1/E(T) rule; and 3.1% worse if she seleced he opimal wihdrawal plan. Figure 5 shows he opimal asse allocaion associaed wih each payou program. Firs, we show ha he asse allocaion for he fixed benefi rule (relevan only o he exremely risk preferring) has equiy exposure of abou 47%. Second, he asse allocaion paern is idenical for all of he variable phased wihdrawal plans, bu he paern varies wih γ. For values of risk aversion up o 2, he reiree holds 100% equiies, and as risk aversion rises, her preferred equiy 14 Similar o he case wihou a beques moive, we compued cerainy equivalens for k = 1. The level of beques moive is insensiive o he order of choice regarding a specific reiremen rule.

14 11 exposure falls. 15 I is ineresing ha he 60/40% sock/bond porfolio commonly recommended by financial advisers is appropriae only for hose wih risk aversion of around 4, bu he curve slopes slowly so even very risk averse consumers will sill hold 40% of heir asses in equiies. Figure 5 here Blending Porfolios of Annuiies and Wihdrawal Rules. To deermine wheher a blended porfolio migh provide greaer uiliy han sand-alone sraegies, we evaluae approaches ha combine boh a life annuiy produc and a phased wihdrawal rule. The 1/E(T) is a naural payou rule o focus on, in view of he fac i mimics he defaul paern under US ax law. 16 We consider firs he case where he reiree a age 65 elecs how much o annuiize and how much o mainain in her wihdrawal accoun. 17 Nex, we allow he reiree o defaul ino he 1/E(T) plan a reiremen, and hen she is permied o swich he remainder of her wealh ino an annuiy a some laer poin. 18 Figure 6 compares he resuls in he case where he blending rule mus be se a reiremen. Those wih low risk aversion do no annuiize, bu as γ rises o 1.5, he demand for annuiies rises srongly. The reiree wih γ = 4 invess 62.6% of her wealh in annuiies and 37.4% is held in he phased wihdrawal plan and held in equiies. In his sense, he annuiy crowds ou bonds and he wihdrawal plan, as risk aversion rises. Figure 6 here Table 2 summarizes resuls when he reiree says in he phased wihdrawal plan a age 65 ye she is permied o annuiize compleely a some laer dae; his sraegy is now compared o he iniial blending sraegy. The reiree may defer annuiizaion if she wans o coninue o paricipae in he equiy marke, or if she seeks o ensure ha she can bequeah some asses o her heirs. 19 We ake he invesmen weighs and he wihdrawal fracions from he previous wihdrawal analysis and ry o deermine when she would swich fully ino annuiies. We consruced ineremporal porfolios of annuiies and he 1/E(T) rule and compued he welfare 15 The equiy porion never falls o zero, because bonds are also somewha risky. 16 Milevsky and Young (2002) find ha he opimal wihdrawal fracion for a log invesor in a deerminisic swiching blending is idenical o he IRA defaul case if he risk free rae is se o zero. Blake e al. (2003) s wihdrawal rule collapses o he 1/E(T)-rule if he acuarial rae of reurn is se o zero. Dus e al. (2005) find appealing characerisics of he 1/E(T)-rule in a shorfall framework. 17 The program spans he sae space by drawing 10,000 random numbers for each of he 36 reiremen years. Then we opimize he fracion invesed in bonds and equiies as well as he amoun used for purchasing a life annuiy. 18 The program spans he sae space by drawing 10,000 random numbers for each of he 36 reiremen years. Then we compare he uiliy oucomes for all possible swiching ages and selec he swiching year wih he highes uiliy. 19 Fuure work could consider how he purchase of life insurance would change his problem.

15 12 gains compared o annuiizing immediaely. The welfare loss from annuiizing immediaely is equivalen o he real opion value of delaying annuiizaion (chrisened by Milevsky and Young 2002 he Real Opion Value o Delay Annuiizaion or RODA). The reiree immediaely annuiizes if her γ exceeds 4.5. For insance, he reiree wih moderae risk aversion (γ = 3) pospones annuiizing her wealh unil she urns 80. Compared o a mandaory annuiizaion a age 65, she gains almos 11% from deferring annuiizaion for 15 years. As risk aversion decreases, he uiliy gain over he benchmark rises subsanially as well as he relaed swiching age. Table 2 here The magniude of he welfare gains for differen levels of risk aversion ha we esimae are in line wih hose from Milevsky (2006) and Milevsky and Young (2002), hough our capial marke model allows for more realism and our wihdrawal rule follows he defaul paern mandaed by he US ax law. Making swiching mandaory a a cerain age, say a age 75 as was rue in he UK unil recenly, penalizes everyone ha has a lower risk aversion han γ = 3.5. A swiching age of 85 as in Germany, however, does no harm any reiree who considers a complee swiching sraegy. Consrucing porfolios over ime works well for risk preferring reirees, bu moderae o exremely risk-averse reirees will benefi from blending wihdrawal sraegies and life annuiies iniially. Even a reiree wih a risk aversion of 10 would sill hold a very small fracion of her asses in a phased wihdrawal plan, in order o be beer off by almos 4% compared o immediae and complee annuiizaion. The preceding analysis has assumed ha he reiree iniially deermines her opimal swiching age and does no accoun for fuure circumsances. Accordingly, nex we le he reiree reac o changes in he yield curve using dynamic programming echniques. 20 Figure 7 shows he resuling swiching froniers for wo differen reirees. The lower lef region of he fronier indicaes he area in which he reiree will wan o coninue o wihdraw asses from her reiremen accoun. The upper righ area shows he region in which he reiree has already annuiized her asses because he shor rae realizaion has been sufficienly high. No surprisingly, he higher he ineres rae, he sooner he reiree annuiizes her enire asse base. 20 The expecaion operaor in he Bellman equaion is compued by resoring o Gaussian quadraure inegraion and by cubic-splines inerpolaion. We hen derive he opimal annuiizaion age by comparing he value of coninuing wih he 1/E(T) wihdrawal plan o he uiliy derived from swiching o life-annuiies compleely. We used 40 saes for he shor rae and a binary indicaor variable I o deermine he swiching ino life annuiies sraegy.

16 13 However, a more risk-loving reiree will also demand a higher shor rae han her risk-averse counerpar. The swiching fronier iself is concave because he moraliy credi increases over ime and replaces cos advanages formerly generaed by he relaed shor rae. Accordingly, Figure 7 shows he combined effec of ineres rae level and moraliy credi. The lower is risk aversion, he higher he shor rae mus be o induce he reiree o annuiize her asses. Figure 7 here Furher Resuls. This secion describes oher resuls of policy ineres (no repored here in deail bu available on reques). Firs, disregarding adminisraive coss included in he insurance premium only modesly alers our resuls. Second, we have also compued cerainy equivalence values for he various payou sraegies assuming a posiive beques moive. Of course, all wihdrawal plans seem more appealing once a beques moive is aken ino accoun. Furher, he lower he level of risk aversion, he larger he gains resuling from applying a paricular wihdrawal rule. For insance, he fixed benefi approach becomes 57% more aracive wih a beques moive, compared o he equivalen annuiy sream. Full annuiizaion a he beginning of he reiremen period is inapplicable for γ 1. Neverheless, he level of he beques moive is insensiive o he order of choice regarding specific reiremen rules. Las, we have also examined he sensiiviy of resuls o asymmery of moraliy beliefs. Assuming ha he 1/E(T) rule is adjused for higher survival probabiliies wih age, he female reiree who was exacly as healhy as he insurance company assumed when i se he annuiy premium would annuiize her wealh earlier. Conversely a reiree in worse-han-average healh would end o pospone annuiizaion by up o wo years, depending on her level of risk aversion. Bu if, due o regulaory reasons, he 1/E(T) rule is no adjused for higher survival probabiliies, reirees would prefer o annuiize laer. Accordingly, his example shows ha asymmery regarding moraliy beliefs can conribue o explaining why individuals who believe hemselves o be less healhy han average are more likely o pospone annuiizaion. Conclusions and Discussion Global disinermediaion rends imply ha workers are increasingly reaching reiremen age wih subsanial reiremen accumulaions which hey will be called on o manage hemselves. As a resul, hey need advice how o opimally conver accumulaed asses ino a sream of reiremen income so as no o exhaus heir funds oo soon. Our uiliy framework

17 14 enables us o compare he value of purchasing a sand-alone life annuiy, versus a phased wihdrawal sraegy backed by a properly diversified invesmen porfolio, as well as combinaions of hese wo acics. We show ha he appropriae mix depends on reiree aiudes oward risk (as well as he underlying economic and demographic assumpions). Specifically, comparing wihdrawal rules ofen cied by financial planners and policymakers, such as he 1/E(T), he fixed percenage, he 1/T, and he fixed benefi rule, we find ha: Somewha surprisingly, he fixed percenage rule is appealing for reirees across a wide range of risk preferences. In his sense, his rule is supporive of some in he financial advice indusry who propose such a rule. The 1/E(T) rule appeals o low/moderaely risk-averse reirees, bu i is unfavorable for he very risk-averse. The fixed benefi rule is no appealing for mos reirees as i exposes hem o he risk of ouliving heir asses, while he 1/T performs worse han any oher variable wihdrawal rule for a broad range of invesors. For he hree variable disribuion plans, he opimal asse allocaion is idenical given a risk aversion level. Specifically, for low risk aversion, equiies dominae and bonds play a greaer role as risk aversion rises. By conras, in he fixed benefi case, even risk-averse reirees are led o hold high fracions in bonds. Nex, we compare sand-alone wihdrawal rules versus immediae annuiizaion of he enire porfolio. Consisen wih previous sudies, we show ha annuiies are aracive as a sand-alone produc when he reiree has sufficienly high risk aversion and lacks a beques moive. Wihdrawal plans dominae annuiies for low/moderae risk preferences, because he reiree can gain by invesing in he capial marke and from being on deah. Finally, we examine combinaion/mixed sraegies where reirees may boh inves some of heir asses and also buy a payou annuiy. In he case where he annuiizaion decision occurs a he poin of reiremen, we find ha: Annuiies become appealing for hose wih moderae risk aversion, when reirees can hold boh annuiies and phased wihdrawal plans as a mixed sraegy. Wihdrawal plans are now aracive for highly risk-averse reirees. From an asse allocaion perspecive, annuiies firs crowd ou bonds when risk aversion rises. As risk aversion increases furher, annuiies replace equiies in he overall porfolio.

18 15 When a reiree is permied o swich ino an annuiy a some poin afer he reiremen dae, we find ha he opimal annuiizaion age is sensiive o he degree of risk aversion and ineres raes in he following manner: Less risk-averse reirees will wai longer unil hey swich o an annuiy. Very risk-averse individuals will be willing o annuiize in a low ineres rae environmen, bu higher ineres raes are required o induce annuiizaion among risk preferrers. Our resuls are relevan o a wide range of financial service providers and regulaors in he reiremen markeplace. Money managers and insurers should noe ha many reirees hold subopimal asse allocaions, as we show ha annuiies firs crowd ou bonds as risk aversion rises, and a higher levels of risk aversion, hey replace equiies in he reiree s porfolio. Making annuiizaion compulsory can also lead o subsanial uiliy losses for less risk-averse invesors, if annuiizaion is forced oo early. Bu i would appear ha annuiizaion a age 85 could be a sensible annuiizaion poin if beques moives are disregarded (and given our model parameerizaions). Mandaory annuiizaion has recenly been implemened in Germany a exacly his age for ax-shelered Rieser Personal Pension accouns; and in he U.S. compulsory annuiizaion was recommended by he recen Commission o Srenghen Social Securiy. Thus far our model indicaes ha reirees find equiy-linked phased wihdrawal plans aracive because invesors are assumed o access he equiy marke only by using a phased wihdrawal plan. This assumpion is realisic, insofar as in he US mos payou annuiies are nominal (Brown e al. 2001). Fuure research will explore he role of equiy-linked variable payou annuiies in reiree porfolios, o he exen ha hey complee he underlying asse srucure by including an equiy premium and a moraliy credi.

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