CEBR/CESIFO CONFERENCE ON PENSION REFORM

Transcription

1 CEBR/CESIFO CONFERENCE ON PENSION REFORM Copenhagen, June 2005 Being on Deah and Capial Markes in Reiremen: A Shorfall Risk Analysis of Life Annuiies versus Phased Wihdrawal Plans Ivica Dus, Raimond Maurer and Olivia S. Michell CESifo Poschingersr. 5, Munich, Germany Phone: +49 (89) Fax: +49 (89) office@cesifo.de Inerne: hp://

2 Being on Deah and Capial Markes in Reiremen: A Shorfall Risk Analysis of Life Annuiies versus Phased Wihdrawal Plans Ivica Dus, Raimond Maurer, and Olivia S. Michell Ivica Dus Johann Wolfgang Goehe-Universiy of Frankfur Deparmen of Finance Keenhofweg 139 (Uni-PF 58), Frankfur Germany T: F: dus@wiwi.uni-frankfur.de Raimond Maurer (corresponding auhor) Johann Wolfgang Goehe-Universiy of Frankfur Deparmen of Finance Keenhofweg 139 (Uni-PF 58), Frankfur Germany T: F: Rmaurer@wiwi.uni-frankfur.de Olivia S. Michell The Wharon School, Universiy of Pennsylvania 3641 Locus Walk, 307 CPC Philadelphia PA T: 215/ F: 215/ michelo@wharon.upenn.edu Acknowledgmens 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 Universiy of Pennsylvania. Addiional suppor was provided by he Cener for Financial Sudies of he Universiy of Frankfur and he Pension Research Council of he Wharon School a he Universiy of Pennsylvania. Daa collecion was faciliaed by he German Invesmen and Asse Managemen Associaion (BVI). Research for he paper was underaken while he second auhor was a Mezler Visiing Professor a he Deparmen of Insurance and Risk Managemen a he Wharon School. We are graeful for commens provided by Neil Dohery, Alex Muermann, Chris Robinson, Sephen Shore, and Ken Smeers. 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.

3 2 Being on Deah and Capial Markes in Reiremen: A Shorfall Risk Analysis of Life Annuiies versus Phased Wihdrawal Plans Absrac Reirees mus draw down heir accumulaed asses in an orderly fashion, so as no o exhaus heir funds oo soon. We compare alernaive phased wihdrawal sraegies o a life annuiy benchmark using German daa; one paricular phased wihdrawal rule seems aracive, as i offers relaively low expeced shorfall risk, good expeced payous for he reiree during his life, and some beques poenial; resuls are similar for he US case. Delayed annuiizaion may also appeal, as i offers higher expeced benefis wih lower expeced shorfalls. JEL Codes: G22 Insurance; G23 Pensions; J26 Reiremen and Reiremen Policies; J32 Pensions; H55 Social Securiy and Public Pensions

4 1. Inroducion Economiss ofen advise reirees seeking o spread heir asses over heir reiremen period o purchase a life annuiy, which 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., 2001). Annuiy paymens may be fixed in nominal erms (fixed annuiy); hey can rise a a pre-specified fixed nominal escalaion rae (graded annuiy); or hey can be indexed o inflaion (real annuiy). Alernaively, hey may reflec he reurn of a specific asse porfolio which backs he (variable) annuiy, or hey can depend on he insurance company s experience wih moraliy, invesmen reurns, and expenses (paricipaing annuiy). As Michell e al. (1999) noe, he essenial aracion of a life annuiy is ha he individual is proeced agains he risk of ouliving his own asses, given uncerainy abou his remaining lifeime, by pooling longeviy risk across a group of annuiy purchasers. Yaari (1965) shows ha risk-averse reirees wihou a beques moive facing annuiy markes ha charge acuarially fair premiums, should annuiize 100 percen of heir wealh. Though life annuiies provide invaluable longeviy insurance, hey also have some disadvanages. Mos obviously, he purchaser faces loss of liquidiy and conrol over his asses, because he lump sum premium canno be recovered afer purchase of he annuiy, irrespecive of special needs (e.g. o cover unexpeced and uninsured medical coss; c.f. Brugiavini 1993). Also, if he annuiy paymens are coningen on he individual s survival, here is no chance of leaving a beques for one s heirs. Oher explanaions for why people may be relucan o buy annuiies are he high adminisraive coss levied by insurance companies (Michell e al., 1999), he abiliy o pool longeviy risk wihin families (Brown and Poerba, 2000; Kolikoff and Spivak, 1981), and he presence of oher annuiized resources from Social Securiy or employer-sponsored defined benefis plans (Munnell e al., 2002).

5 2 As an alernaive o buying a life annuiy, one migh self-annuiize following a phased wihdrawal approach. Here he reiree allocaes his wealh endowmen across various asse caegories (e.g. equiy, bonds, cash) ypically included in a family of muual funds where he asses will earn uncerain raes of reurn. A cerain amoun of he invesed funds can hen be wihdrawn periodically for consumpion purposes. An advanage of he phased wihdrawal sraegy, as compared o a life annuiy, is ha i offers greaer liquidiy, he possibiliy of greaer consumpion while alive, and he possibiliy of bequeahing some of he asses in he even of early deah. On he oher hand, relying on a seady asse drawdown wihou any insurance provides no pooling of longeviy risk, so he reiree could oulive his asses before his uncerain dae of deah. Anoher wihdrawal rule, for example, consuming a specified fracion of he remaining fund wealh each period, avoids he risk of ouliving one s oal asses, as long as he benefi-o-wealh raio is lower han one. Ye sochasic invesmen reurns will mean ha pension asses change over ime, and he periodically wihdrawn amoun could be subsanially lower or higher han he benefi payable under a life annuiy. Such phased wihdrawal approaches are becoming popular in many counries, promped by a round of pension reforms in Europe and America. For example, in Germany, recenly-inroduced Rieser plans offer a ax inducemen for volunary saving in individual reiremen accouns (IRA) during he worklife, underscoring he governmen s ineres in boosing asse accumulaion for he aging populaion (Börsch-Supan e al., 2003). A reiremen, hiry percen of he accumulaed asses in he IRA may be wihdrawn as a lump-sum disribuion; he remainder mus be aken as a life annuiy (offered by a commercial insurance company) or paid ou according o a phased wihdrawal plan (ypically offered by muual fund and/or bank providers) of which par revers o an annuiy as of age 85. In he UK, personal pensions are now he norm, and here oo, a porion of he accumulaed asse mus be annuiized by age 75. In Canada, he reiree a age 69 mus eiher buy an annuiy wih his ax-shelered saving or creae a discreionary managed wihdrawal plan (Milevsky and Rob-

6 3 inson, 2000). In he US, no compulsory annuiizaion is required for 401(k) plans; raher, mos reirees roll over heir pension asses ino an Individual Reiremen Accoun and manage he funds hemselves in old age. Togeher, hese rends signal a growing ineres in helping reirees manage he asse decumulaion process. To compare alernaive reiremen asse decumulaion sraegies quaniaively, we require a formal risk/reurn framework for decision-making under uncerainy. One approach aken by financial economiss is o maximize he expeced discouned value of a (ime separable) uiliy funcion for uncerain fuure benefis and (if necessary) for a beques. For example, Blake e al. (2003) evaluae differen wihdrawals plans assuming ha mandaory annuiizaion is required a age 75, using a consan relaive risk aversion uiliy framework. Milevsky and Young (2003) use a similar objecive funcion o deermine he opion value of deferring annuiizaion. A shorcoming of such an approach, however, paricularly in he world of financial reiremen planning, is ha decisionmakers ofen lack an explici measure of reirees risk preferences (Pye 2000). For his reason, an alernaive approach using risk-value (or risk-reurn) models are appealing, in ha hey use explici measures of risk and value along wih a funcion reflecing he radeoff beween hese wo. To he exen ha individuals prefer more reurn o less, and less risk o more, we can derive a parial ordering of opporuniies wihin a risk-reurn dominance conex, even if he exac uiliy weighs for risk and reurn are unknown. Depending on which risk meric is seleced and how we formulae he radeoff beween risk and reurn, a risk-value model can be consisen wih he expeced uiliy approach of choice. 1 1 Perhaps he mos widely used risk-reurn model in he area of finance is he classic mean-variance porfolio analysis which is, iner alia, consisen wih a quadraic uiliy funcion. For a discussion of he compaibiliy of expeced uiliy models and muliparameer rade-off choice models, see Schneeweiß (1967) and Sarin and Weber (1993). A lexicographic ordering of risks would generally be inconsisen wih expeced uiliy: e.g. if he decisionmaker will only accep a maximum level of risk, independen of reurn.

7 4 In wha follows, we herefore ake a risk-value approach. Here he reurn is he expeced level of benefis as well as he expeced possibiliy of beques, and he risk is he possibiliy of no reaching a benchmark or desired level of consumpion. Previous sudies aking his ack focus on he probabiliy of consumpion shorfall as he operaive risk measure (Ho e.al 1994, Bengen 1994, 1996, Milevsky e al. 1997, Milevsky and Robinson 2000, Milevsky 1998, 2001, Ameriks e al. 2001, Pye 2001, Hughen e al. 2002, or Albrech and Maurer 2002). Assuming ha he reiree consumes a fixed amoun a specific poins in ime from a self-managed pension accoun, hose sudies calculae he probabiliy of running ou of money before an uncerain dae of deah using alernaive assumpion abou he asse allocaion, he iniial consumpion-o-wealh raio, and he opimal waiing ime before swiching he reiremen wealh ino an annuiy. Our work exends his lieraure in several direcions. Firs, we examine he risk and reurn profiles of several variable self-annuiizaion sraegies ha provide paymens according o a predeermined benefi-o-wealh raio. Second, we address a major shorcoming of he shorfall-probabiliy risk measure, namely ha i ignores he size of he possible loss ha may be experienced. In pracice, of course, boh heoreical and empirical argumens sugges ha invesors ake boh he probabiliy and he amoun of a possible shorfall ino consideraion. Our conribuion exends prior work by looking no only a he probabiliy of a consumpion shorfall, bu also by considering he size of he shorfall when i occurs. Third, we examine how he resuls change if a mandaory annuiizaion rule were imposed akin o hose in he recen German and UK pension reforms. Fourh, we evaluae he impac of allowing he annuiizaion dae o be endogenous, along wih he asse allocaion decision. We illusrae how he risk of a consumpion shorfall and reurn profiles of fixed and variable phased wihdrawal sraegies compare o he life annuiy, and indicae wha dominan sraegies migh be. In he remainder of his paper, we describe several wihdrawal sraegies and illusrae heir implicaions assuming capial and insurance marke condiions relevan o he German markeplace.

8 5 We adop hese so as o be informaive abou alernaive payou opions ha migh be conemplaed under he German Rieser plans when hey reach mauriy. Mos resuls focus on an age-65 male reiree, bu we also provide findings for oher ages and for women. Resuls are firs given using a fixed asse allocaion paern, and subsequenly, asses are permied o be allocaed opimally. A final secion summarizes and concludes. 2. The Case of Phased Wihdrawal We assume ha he reiree is endowed wih an iniial level wealh V 0 ha he can use o buy a single-premium immediae life annuiy paying a consan annual real benefi B a he beginning of each year for life, wih no beques. We denoe his as he benchmark annuiy, and refer o Appendix A regarding he pricing of such an insurance produc using assumpions abou moraliy, loadings, and ineres raes. If he reiree does no annuiize his wealh, he mus allocae his reiremen money across various financial asses such as equiies and bonds (represened here as muual funds); hereafer, he can wihdraw a cerain amoun a he beginning of each year for consumpion purposes. Throughou, we assume ha payous are axed as ordinary income; herefore axes will no change he desirabiliy of volunary annuiizaion or sysemaic wihdrawal from a self-managed reiremen accoun Wihdrawal Plans wih Fixed Benefis Under a fixed benefi rule, a he beginning of each year he reiree will sell as many fund unis as required o reach he same yearly benefis paid by he life annuiy, eiher unil he dies or he reiremen asses are exhaused. Formally, he benefis B a he beginning of each year are given by: B = min( B, V ), (1) 2 This is accurae for he German conex; for more on annuiy ax reamen in he US see Brown e al. (1999). Hugen e al. (2002) sudy he cash-flows of various wihdrawal raes (as a percenage of iniial porfolio value) wihin an ex pos conex using hisorical reurns daa on common socks and bonds before and afer axes.

9 6 where V is he value of he reiremen accouns asses wealh a he beginning of year ( = 0, 1, ) jus before he wihdrawal B for ha year is made. The reiree faces an ineremporal budge consrain such ha wealh nex period V +1 equals wealh oday V, less wha is subraced for benefi paymens B, imes he (inflaion adjused) porfolio reurn R +1 over he period, or zero if he fund is exhaused: ( V B)(1 + R+ 1) V > B V + 1 = ( V B ) (1 + R+ 1) =. (2) 0 V B. Noe ha he benefi paid B depends on he value of he reiremen asses used o finance wihdrawals, V. If hese asses are risky, benefi payous are exposed o uncerain capial marke reurns. The idea of he fixed benefi rule is o replicae he payou from a life annuiy (self-annuiizaion) as long as he funds permi, while a he same ime offering liquidiy and some beques poenial in he even of an early deah. Neverheless, he risk of such a self annuiizaion sraegy is ha adverse capial markes linked o longeviy oucomes migh produce a siuaion where V his zero and herefore B = B +1 = = 0, while he reiree is sill alive Phased Wihdrawal Rules wih Variable Benefis Under a variable phased wihdrawal plan, he reiree receives an ex ane fixed fracion of he reiremen asses remaining each period (as in Meron 1971). Due o he sochasic naure of capial markes, he value of he reiree s fund is exposed o posiive as well as negaive flucuaions. Consequenly, he level of benefi paymens under a variable wihdrawal plan also flucuaes in andem wih he accoun value. The pah of benefis payable under a variable phased wihdrawal rule can be formalized as follows. Le V be he value of he reiremen asses a he beginning of period ( = 0, 1, ) before he wihdrawal B for ha year is made. A he beginning of period, an ex ane specified fracion ω (0 < ω 1) is wihdrawn from curren wealh; hence he reiree receives a paymen according o:

10 7 B = ω V (3) Furher le R +1 denoe he reurn of he funds over he period. Then, he ineremporal budge consrain of he reiremen accoun is given by: V V B ) (1 + R ) = (1 ω ) V (1 R ). (4) + 1 = ( Noe ha if he asses of he reiremen accoun are invesed in risky asses, boh he benefis B as well as he beques poenial V are random variables. In wha follows, we focus aenion on hree specific wihdrawal rules ha generae variable benefis: he fixed percenage rule, he 1/T rule, and he 1/E(T) rule. Each is discussed in urn. Fixed Percenage Wihdrawal Rule: Here a consan fracion is wihdrawn each period from he remaining fund wealh; ha is, he benefi-wealh raio is fixed over ime: B V ω = ω. = (5) This wihdrawal rule has he advanage of simpliciy, requiring no informaion regarding he maximum possible duraion of he payou phase or he reiree s demographic characerisics. "1/T Rule" Wihdrawal Rule: The idea behind his rule is o se he wihdrawal fracion according o he maximum possible duraion of he plan, denoed by T. One way is o se T equal o he oldes age assumed in a moraliy able; anoher is o fix i a he reiree s life expecancy as of his reiremen dae (Brown e al., 1999). In he firs case, he maximum number of paymens T is given by he limiing age l of he moraliy able minus he curren age of he reiree x plus one (T l x +1). The reiree ges a fracion of 1/T of his iniial pension accoun as he firs paymen, he second paymen is worh 1/(T 1) of he remaining asses, and so forh unil he reiree eiher passes away or reaches he plan s limiing age l. Formally, he benefi-wealh raio a he beginning of year ( = 0, 1, T-1) of his reiremen plan is given according o:

11 8 B V 1 = ω =. (6) T In conras o he fixed percenage rule discussed above, he wihdrawal fracion is no consan, bu raher rises wih age. 1/E[T(x)]" Wihdrawal Rule: This rule, which we will call he 1/E(T) rule for shor, akes ino accoun he reiree s remaining life expecancy in a dynamic way. Now, he wihdrawal fracion is no longer deermined by he maximum lengh of he plan, bu insead i is a funcion of he reiree s remaining life expecancy. Le p x represens he condiional probabiliy ha an x-year old man will E T ( x + ) aain age x +, he complee expecaion of life is calculaed as [ ] = = l x 0 p x where l is he maximum age according o a moraliy able. Then, for an a reiremen x-year old man, he benefio-wealh raio in period afer reiremen, condiional on he fac ha he is sill alive, is given as: B V 1 = ω =. (7) E[ T ( x + )] The shorer his expeced remaining lifeime, he higher he fracion he will wihdrawal from his pension accoun. The 1/E(T) wihdrawal rule is used in he US during he decumulaion phase of 401(k) plans, where he ax auhoriy seeks o ensure ha reirees consume heir ax-qualified pension accouns insead of leaving hem as bequess for heir heirs (see Munnell e al., 2002). 3. Risk and Reward Analysis of Phased Wihdrawal Plans Condiional on Survival 3.1 Research Design To compare he risk and value characerisics of he four phased wihdrawal rules of ineres, i is useful o begin wih an assessmen of expeced payous condiional on reiree survival. For he momen, herefore, we focus only on he risk resuling from capial markes and suppress moraliy. To do so, we assume a 65-year old male reiree who seeks o compare benefis under he four

12 9 phased wihdrawal plans given an iniial asse balance. His reiremen asses are rebalanced annually o mainain an asse pool spli evenly beween socks and bonds, consisen wih recommendaions by financial advisors. 3 The analysis o follow uses assumpions drawn from he German capial and annuiy marke environmen; laer, we offer some comparisons wih US assumpions. The annuian moraliy able is provided by he German Sociey of Acuaries and used o calculae survival probabiliies and expeced lifeime (in he 1/E(T) case). Since his able ends a age 110, we se l = 110 for he 1/T rule. For he fixed percenage wihdrawal rule, we selec ω = 5.82%, since his benefi-o-wealh raio produces an iniial payou equal o he life annuiy in he firs year of he plan. In he case of he fixed benefi rule, we assume ha he iniial wihdrawal coninues unil he reiree dies or he accoun is exhaused. We compare he risk and reurn paerns ha emerge under hese alernaive phased wihdrawal paerns o hose from a fixed real annuiy providing lifelong consan payous. When focusing on risks and benefis, he compuaions eiher assume ha he reiree is alive, or conversely, we evaluae he beques poenial if he reiree is assumed o pass away a a specific age. To do so, we specify an exogenous srucure on he ex-ane probabiliy disribuion governing he financial uncerainy of fuure reurns and esimae he parameers of such a model from independen (e.g. yearly) hisorical observaions of real reurns. Wih such a model in place, i is possible o look ino he fuure and compue he expeced benefi paymens and differen shorfall-risk measures of he four wihdrawal plans in which we are ineresed. Implemening i relies on he assumpion ha he sochasic specificaion of he asse values in he reiremen accoun follows a geomeric random walk 3 Feldsein e al. (2001) and Ibboson (2003) assume ha reirees hold heir non-annuiized asses in a 60% sock, 40% bond porfolio. Here, for illusraive purposes, we use a more conservaive spli, consisen wih he posiion recommended by he Presiden s Commission o Srenghen Social Securiy (see Cogan and Michell 2003). Some financial advisers propose ha invesors hold equiies equal o 100 minus heir age; see Canner e al. (1997) or Vora and McGinnes (2000). The number 100 can be jusified as i is he maximum age used in mos populaion moraliy ables, bu annuian moraliy ables ofen have a maximum age years higher.

13 10 wih drif, a sandard assumpion in financial economics. This implies ha he yearly log-reurns are serially independen and idenically normally disribued wih given mean and covariance. We also use German hisorical ime series over he period for he German Equiy Index (DAX) and he German Bond Index (REXP) as proxies for sock and bond invesmens. The DAX represens an index porfolio of German blue-chip socks, and he REXP represens a porfolio of German governmen bonds. Each of hese indices is adjused for capial gains as well as dividends and coupon paymens (on a pre-ax basis). To accoun for poenial adminisraive coss, we subrac he equivalen of 0.5% p.a. from he yearly porfolio reurn. Subsequenly, asse reurns are adjused for inflaion by using he German Consumer Price Index. These yearly daa produce esimaes for he real log average rae of reurn for socks of 6.18 percen and 3.96 percen for bonds, respecively. The corresponding volailiies are percen for socks and 5.07 percen for bonds, and he correlaion-coefficien is Since we assume normally disribued log reurns, i.e. I = ln(1 + R ) ~ N(µ, σ ), hese parameers imply a real log mean rae of reurn on he fify-fify sock-bond porfolio of µ = 5.81 percen wih a sandard deviaion of σ = percen. Noe ha his produces an expeced gross rae of reurn of E(1 + R ) = E[exp(I )] = exp[ *0.1328²] = Assuming 4 ha he normaliy propery also holds for he log porfolio reurns, i is sraighforward o develop an analyical closed form soluion for he probabiliy disribuion of fuure benefis of he differen variable phased wihdrawal rules (see Appendix B for deails). However, because he value of he reiremen accouns value migh hi zero, he ineremporal budge consrain in equaion (2) for he fixed benefi rule is no (log)linear, and fuure benefis are pah-dependen. Hence, for he fixed benefis wihdrawal plan, he probabiliy disribuion of fuure benefis is unknown. As a resul, esimaes for he differen risk and reurn measure 4 This assumpion is widely used in he sraegic asse allocaion lieraure (e.g. Feldsein e al or Campbell and Viceira 2002) and i can be jusified by a Taylor approximaion of he nonlinear funcion relaing log-individual-asse reurns o log porfolio reurns. For deails see Campbell and Viceira (2002), p and Campbell e al. (2001).

14 11 use Mone-Carlo simulaion o generae a large number (i.e. 100,000) of pahs for he evoluion of he wihdrawal plan. 3.2 Analysis of Expeced Benefis Figure 1 depics he Expeced Benefis profiles condiional on survival under all four phased wihdrawal rules; in each case payous are compared o he annuiy profile. Focusing on he fixed benefi rule, we see ha in he firs year, mean benefis are (by consrucion) equal o he annuiy payou. Thereafer, however, expeced paymens from he plan are decreasing, reflecing he risk of running ou of money. The fixed fracion rule also sars wih a benefi equal o he life annuiy payou, and afer ha, mean benefis slighly rise as he reiree ages. This is due o he fac ha he pension accoun s expeced gross rae of reurn is 6.92% p.a., which exceeds he consan benefi-owealh-raio of 5.82% p.a. (i.e *( ) = > 1). Figure 1 here By conras, he 1/T rule pays a much lower expeced benefi up o he age of 80, bu hereafer, he expeced benefi rises exremely quickly and o very high levels. This can be explained by he low wihdrawal fracions under his rule, during he firs par of he reiremen plan. Up o age 95, he benefi-o-wealh raio is lower han he expeced rae of reurn; consequenly, he expeced value of he pension asses grows over ime. Reserves buil up in earlier ages can be used o increase he expeced benefis in laer years. The 1/E(T) rule sars a a level of abou 85% of he annuiy paymen and increases o 100% when he reiree aains age 70. This payou approach reaches is maximum expeced paymen of abou 150% a age of 83. Afer ha poin, expeced paymens monoonically decrease, reaching he life annuiy benefi level a age 91. For ages older han 100, he 1/E(T) rule would expose he reiree o very low benefis, asympoically approaching zero. Only for he firs six years of he reiremen plan will he benefi-o-wealh raio be lower han he expeced reurn earned on pension asses. If he reiree survives unil age 71, his expeced lifeime is

15 12 abou 15 years, resuling in a wihdrawal fracion of 6.66% which is abou he same as he expeced rae of reurn. Beyond ha age, he wihdrawal fracion grows ever larger han he expeced asse reurns backing benefi paymens. For some ime (i.e. up o age 83), he increasing wihdrawal fracions produce increasing expeced benefis. Bu because less and less wealh is lef in he fund, a some poin (here age 83) he expeced benefi amouns decrease alhough he wihdrawal fracion increases Shorfall Risk Analysis In general, shorfall risk is associaed wih he possibiliy of somehing bad happening, in oher words, falling below a required arge reurn. Reurns below he arge (losses) are considered o be undesirable or risky, while reurns above he arge (gains) are desirable or non-risky. In his sense, shorfall-risk-measures are called relaive or pure measures of risk. 5 To analyze his risk in he case of our phased wihdrawal sraegies, we employ several differen shorfall risk measures. Shorfall Probabiliy: We begin wih he shorfall probabiliy, defined as: SP(B ) = P(B < z). (8) This measures he probabiliy ha he periodic wihdrawal B is smaller han a chosen benchmark z, which is here he paymen provided by he life annuiy. Figure 2 depics he SP for he fixed benefi rule, he fixed fracion rule, he 1/T approach, and he 1/E(T) rule, compared o he annuiy benefi. In he firs year, all he sraegies excep he fixed benefi program face a high probabiliy of shorfall; he only reason he fixed benefi approach does no is ha i is se, by consrucion, o pay he iniial annuiy value as long as he funds are no exhaused. Accordingly, he fixed benefi program offers a shorfall probabiliy close o zero a he 5 The concep of shorfall risk was inroduced in he area of finance by Roy (1952) and Kaaoka (1963), and i was expanded and heoreically jusified by Bawa (1978) and Fishburn (1977, 1982, 1984). I was widely applied o invesmen asse allocaion by Leibowiz e al. (1996) and used by Leibowiz and Krasker (1988) and Maurer and Schlag (2003) among ohers o judge he long erm risk of socks and bonds. In addiion Libby and Fishburn (1977); Kahneman and Tversky (1979); Laughhunn e al. (1980) and March and Shapira (1987) show ha in empirical business decisionmaking, many individuals judge he risk of an alernaive relaive o a reference poin.

16 13 beginning of he reiremen period, bu his risk meric begins o rise over ime, reaching abou 20% around age 85. By conras, boh he 1/T and 1/E(T) rules have high shorfall probabiliies early in he reiremen period. This is because a reiree invesing his asses in a muual fund hoping o generae he same paymen offered by he life annuiy mus wihdraw abou 6.50% of he fund annually. Bu he wihdrawal fracions under he 1/T and he 1/E(T) rules are smaller early in reiremen, meaning ha he wealh remaining grows quickly. Consequenly he SP declines over ime, hough he wihdrawal fracion is growing. The reiree ha wihdraws a fixed fracion each year faces a risk profile ha is remarkably high for all ages. In early years, he probabiliy of receiving a benefi below he benchmark life annuiy is abou 50%, gradually increasing o abou 54% a he end of he period. Figure 2 here Anoher ineresing finding has o do wih he gradien of he SP under he 1/E(T) rule. Early in he reiremen period here is a fas decline in his risk, bu if he reiree is sill alive a age 83, he SP begins o rise quickly due o he special consrucion of his spending rule. In conras o he 1/T rule, expeced paymens a he beginning of he plan are already higher, meaning ha few reserves are buil up in he beginning of he plan. Also, he 65-year-old reiree has an expeced remaining lifeime of 19 years, and his expeced remaining lifeime decreases over ime, especially afer he age of 80. The shorer is he remaining expeced lifeime, he more wealh will be wihdrawn in he 1/E(T) case. As he wihdrawal fracions increase, less and less wealh is lef in he fund; a some poin, wealh remaining is insufficien o provide high enough paymens, so he shorfall probabiliy again rises. Shorfall Measures Tha Incorporae Severiy: As Bodie (2001: 308) noes, a major shorcoming of he popular SP risk meric is ha i compleely ignores how large he poenial shorfall migh be. Tha is, he shorfall probabiliy answers he quesion how ofen consumpion falls shor, bu

17 14 no how bad he loss is if i occurs, under each of he differen wihdrawal rules. A shorfall risk meric ha considers boh he probabiliy and he average size of he shorfall when i occurs is he Shorfall Expecaion (SE): SE(B ) = E[max(z - B,0)] = MEL(B ) SP(B ). (9) The SE is he sum of losses weighed by heir probabiliies, and hence i is a measure of he uncondiional average loss. As equaion (9) shows, he SE is he produc of he shorfall probabiliy and he condiional expeced shorfall given he occurrence of a shorfall. This measure is also known as he Mean Excess Loss and i is defined as MEL(B ) = E[ z B B < z ], i.e. he MEL answers he quesion of how badly on average he sraegy performs (see Arzner e al. 1999). In Figure 3 we plo he shorfall expecaion resuls for he various wihdrawal sraegies of ineres, namely he fixed benefi rule, he fixed fracion rule, he 1/T approach, and he 1/E(T) rule. The SEs can be compared for each acic o he annuiy benefi, all condiional on survival. Here we see ha he fixed benefi rule has a very low shorfall expecaion hrough abou age 83, whereas he 1/T rule is iniially he riskies wih a 60% SE. I akes a very long ime unil he SE of he 1/T rule declines o a negligible level, older han age 90 for he case under sudy. The fixed fracion and he 1/E(T) rules boh have SEs below 20% hrough a leas age 80, bu he 1/E(T) rule again races ou wha is perhaps unexpeced behavior afer falling o low levels hrough abou age 84, he risk begins o rise subsanially 20 years afer reiremen, and i has he highes expeced shorfall for he long-lived individual. Figure 3 here 3.4. Analysis of Expeced Bequess The oher aspec of hese rules, of course, is ha he reiree mus in effec compare his own consumpion wih he poenial value of any beques going o his heirs should he die. Figure 5 illusraes he expeced beques under he various formulaions, condiional on deah. The paern exhib-

18 15 iing mos sabiliy is he fixed fracion rule, bu he oher hree are highly divergen. For example, he 1/T expeced beques follows an ineresing pah, rising during he early reiremen period when wihdrawals are small. Abou 35 years afer reiremen, however, he expeced beques begins o decline very quickly a fac ha is direcly aribuable o he consrucion of his plan. The older a reiree ges, he more he or she wihdraws from his accoun: hus five years before he plan ends, he reiree wihdraws 1/5 (or 20%) of he remaining wealh. If he reiree should, by chance, live beyond age 110, his approach offers no coninued paymen or beques poenial. The 1/E(T) rule also offers only a very low beques poenial afer reaching a limiing age. In conras wih he 1/T acic, however, he 1/E(T) plan offers lower expeced inheriance a every age. Paricularly if he reiree does no die unil 20 years ino reiremen, he inheriance will decline dramaically. Figure 4 here 4. Risk-Minimizing Phased Wihdrawal Sraegies Thus far, our analysis has assumed ha he reiree holds his pension asses in a fixed-weigh porfolio comprised of 50% socks and 50% bonds; accordingly he payous in reiremen ake ino accoun only capial marke uncerainy, wihou permiing invesmen opimizaion around risk/reward radeoffs. In his subsecion, we exend he analysis by considering moraliy risk and wo addiional phased wihdrawal rules ha permi he reiree o opimize he design of he wihdrawal paerns. In he nex subsecion, we furher vary he porfolio s invesmen weighs o aain a risk-minimizing saic asse allocaion. The porfolio weighs are herefore deermined endogenously (excluding shor-selling), following Albrech and Maurer (2002). Finally, he following subsecion examines he impac of mandaory shifing o annuiizaion a a specific age. This is currenly required in ax qualified German Rieser plans a he age of 85 and for UK income drawdown plans a he age 75.

19 Opimized Wihdrawal Rules in a Risk-Reurn Conex To evaluae how he relaive ranking of he alernaive wihdrawal rules migh change wih an endogenous asse mix in he reiree s invesmen fund and oher plan design parameers, i is useful o define he expeced presen value of he shorfall, called here EPVShorfall : l = x p xse(b ) EPVShorfall = (1 + R ) 0 f (10) Here, SE(B ) = E[max(z B, 0)] denoes he expeced shorfall wih respec o he arge z, which is he benefi flow of he benchmark life annuiy. Possible expeced shorfalls are weighed by he condiional probabiliy p x ha a man aged x a he beginning of he reiremen phase is sill alive, if a shorfall occurs. All possible expeced shorfalls are discouned back o he beginning of he reiremen period using he risk-free ineres rae R f (i.e. assuming a fla erm srucure of real ineres raes) and summed over he maximum lengh of he moraliy able used. This useful summary measure of he risk associaed wih a phased wihdrawal sraegy may be inerpreed as he lump sum premium ha would be required for he reiree o ransfer his shorfall risk o an insurer, assuming acuarially fair pricing and no addiional loading. Given his funcion, we minimize i wih regard o asse allocaion and oher plan design parameers, o derive he paerns mos amenable o alernaive wihdrawal rules. Previous sudies, mos noably Milevsky (1998), Milevsky and Robinson (2000) and Albrech and Maurer (2002), approach he issue of opimal fixed benefi wihdrawal rules by adoping he crierion of conrolling he probabiliy of a consumpion shorfall in reiremen. On he oher hand, as we have argued, his perspecive does no accoun he iming and magniude of he loss when i happens, which our risk measure does. To exend he approach, we propose wo addiional reward measures associaed wih each opimized phased wihdrawal sraegy, namely, he expeced presen

20 17 value of benefis received during life (EPVBenefis) and he expeced presen value of bequess a deah (EPVBeques). These are defined, respecively, as: l = x p x E(B ) EPVBenefis, and (11) = (1 + R ) 0 f EPVBeques = l x = 1 1 p x q x+ (1 + R E(V ) f ) (12) Here, he EPVBenefis is similar o he money s worh concep used by Michell e al. (1999); i reflecs he expeced presen value of benefi paymens condiional on survival. Finally, EBVBeques measures he expeced presen value of he inheriance ha he reiree would pass on o heirs in he even of his deah. 6 Using hese, we develop wo opimized rules, namely he Fixed Percen Opimized rule, and he 1/T Opimized rule. The firs minimizes he expeced presen value of he shorfall by joinly selecing he opimal consan wihdrawal fracion and an asse allocaion. This relaxes he consan wihdrawal rule menioned earlier by endogenizing he wihdrawal fracion. Compared o he non-opimized Fixed Percen rule, we expec ha having wo addiional parameers, he fracion consumed as well as he asse allocaion, will be more successful in conrolling boh moraliy and capial marke risk. The second rule, denoed as 1/T Opimized, minimizes he EPVShorfall by joinly selecing he maximum duraion of he plan condiional on survival, and he asse allocaion. We expec ha he 1/T Opimized rule will permi more consumpion when he probabiliy is high 6 These merics are useful as compared o a specific uiliy funcion for several reasons. Firs, he risk measures are consisen wih expeced uiliy analysis, since hey are he primiives ha ener ino uiliy maximizers objecive funcions (Brachinger and Weber 1997). For example, one could imagine ha a reiree rades off expeced benefi paymens versus he expeced shorfall vis a vis he benchmark annuiy; his risk value model is consisen wih a uiliy funcion suggesed by Fishburn (1977). Furher any paricular funcional form mus embody specific radeoffs beween risk and reurn componens, whereas our approach can remain agnosic abou he specific weighs aached o each (Sarin and Weber 1993). We choose risk minimizaion as our objecive funcion because i is consisen wih many prior sudies (c.f. Albrech and Maurer, 2002; Chen and Milevsky, 2003; Milevsky and Robinson, 1994), and i is also consisen wih convenional wisdom offered by financial planners when providing advice regarding reiremen income payous (c.f. Ameriks, 2004; Ameriks e al., 2001; Ibboson Associaes, 2003).

21 18 ha he reiree remains alive, as compared o he non-opimized 1/T rule, bu i will also offer lower expeced bequess Comparaive Resuls: Annuiy versus Phased Wihdrawal Plans Table 1 repor resuls for he various wihdrawal rules of ineres, allowing opimized asse allocaion. These may be compared o he benchmark case of a life annuiy benefi given in Row 1; here, we find ha a 65-year old male who paid 100 for an immediae real annuiy will receive annual benefis of 5.82 for life. By consrucion, boh he EPVShorfall and EPVBeques are zero for he annuiy purchase; he EPVBenefis measure is slighly below 100 due o he annuiy load assumed. Row 2 repors resuls for a phased wihdrawal program where he Fixed Benefi is se equal o he annuiy a 5.82 as before; of course, he reiree may run shor of funds. The opimized asse allocaion associaed wih minimizing he EPVShorfall for his Fixed Benefi wihdrawal plan consiss of 25% socks and 75% bonds, and associaed wih his plan is an expeced shorfall worh 3.24 per 100 of iniial asses. As long as he reiree lives, he can expec benefis oaling (in presen value). The presen value of he beques ha his heirs can expec is quie large, a (or more han half he iniial invesmen). Clearly, unless he reiree has an enormous ase for bequess, annuiizaion would be judged far superior o aking a fixed benefi a 5.82 per annum unil he fund is likely exhaused. Table 1 here Rows 3 and 4 of Table 1 display resuls for wo Fixed Percenage sraegies. The firs is deermined by selecing a fixed percenage rule ha pays ou a firs-year benefi equivalen o he 5.82 real lifelong annuiy payable o a 65-year old male paying 100. Given his consan benefiwealh-raio (i.e. ω = 5.82%), we solve for he opimal asse mix minimizing he EPVShorfall. The second sraegy selecs a fixed fracion ha is now also opimized wih regard o EPVShorfall. Wha is differen here is ha boh he asse allocaion and he wihdrawal fracion are simulane-

22 19 ously opimized a he beginning of he reiremen phase. These wo rows indicae ha, in boh cases, he risk measured by he EPVShorfall is almos four imes as large as under he Fixed Benefi approach. Offseing his could be he higher benefi sream condiional on survival and higher beques value o he heirs. Boh fixed percenage sraegies have slighly higher equiy exposures (abou 35%) han he fixed benefi approach (25%). This conrass wih he high equiy exposures recommended by Albrech and Maurer (2002) and Vora and McGinnes (2000) who use a fixed benefi wihdrawal approach. Of course, an opimized sraegy ha permis a fixed percenage payou of 7% of he accoun annually has a lower expeced shorfall and higher expeced benefis han he non-opimized sraegy. Nex we urn o he wo 1/T rules, where again he firs simply ses T o he maximum plan duraion (he oldes age in he moraliy able), and opimizes asse allocaion so as o minimize he EPVshorfall. The second rule endogenously evaluaes boh he asse allocaion and he plan duraion ha minimizes EPVShorfall. I is ineresing ha he simple 1/T rule (Row 5) resuls in he highes equiy exposure, and i is also unlikely o be preferred by many: his is because he size of he expeced shorfall is he larges of hose considered ( 34 of he iniial 100 asse), and he expeced benefis are he lowes of hose examined. The only clear gainers are likely o be he heirs. We conras his wih he paern ha would resul from opimizing he maximum plan duraion, which he reiree could do if he had Social Securiy or welfare o live on in he even ha his asse were exinguished bu he were sill alive. This would occur around age 87, according o he program compued. Row 6 indicaes using he 1/T rule opimized for asse allocaion and he dae of running ou of asses offers lower risk, higher expeced han he annuiy, a reasonable beques, and he asse allocaion is no oo risky (16% equiy and 84% bonds). Finally we urn o Row 7 for he 1/E(T) rule, which is consisen wih he phased wihdrawal scheme for 401(k) pension plans allowed by he US ax auhoriy. This is an ineresing sraegy,

23 20 because i offers quie low expeced shorfalls and 8% higher expeced benefis han he life annuiy, while sill affording a decen beques poenial. The asse allocaion implied is raher conservaive, wih 22% in equiy and 78% in bonds. Overall, looking across he phased wihdrawal plans, here is no clearly dominan sraegy, since all involve radeoffs beween risk, benefi, and beques measures, and individual preferences may vary. Neverheless, he 1/E(T) rule seems relaively appealing as compared o he ohers, as long as he reiree has only a moderae appeie for bequess. The second panel of Table 1 repors resuls for a female age-65 reiree considering he same phased wihdrawal paerns. To summarize resuls, we find ha women generally confron lower expeced shorfall risks and anicipae higher EPVBenefis. This is because lower female moraliy ranslaes ino a lower iniial annuiy paymen; i.e. her acuarially fair benefi is 5.02 per year for a 100 purchase (versus he male payou of 5.82). Consequenly, variable wihdrawal plans have he woman wihdraw less early in life, leaving more asses in he fund o earn fuure capial marke reurns. Since he woman also is expeced o live longer, she will more likely be alive o reap he fruis of he invesmen. We would herefore predic, and he resuls confirm, ha he 1/E(T) rule is more aracive o women han men, since i offers raher low expeced shorfalls, and 20% higher expeced benefis as compared o he annuiy, while sill affording a decen beques poenial. I is also ineresing ha he asse allocaion sraegies for women are similar o hose for men. Thus far, he analysis has assumed he reiree begins he payou phase a age 65, bu i may be of ineres o explore how phased wihdrawal paerns migh change for alernaive reiremen ages. Table 2 displays he findings for a male reiring a age 60 or age 70, which can be direcly compared wih he op panel of Table 1. The resuls show ha he phased wihdrawal paerns are unambiguously more aracive for an age-60 reiree, as compared o he 65-year old. In oher words, all expeced shorfall risk measures are lower, expeced benefi payous o he living reirees are higher, and expeced bequess are similar; furhermore, he porfolios are slighly ligher in equi-

24 21 ies. This is because he moraliy drag for he life annuiy purchased by a younger person, and herefore he benchmark, is subsanially lower. By conras, higher moraliy faced by a 70-year old reiree produces a higher benchmark annuiy which ranslaes ino greaer EPVShorfalls, lower expeced benefis, and also lower expeced bequess. This is despie having 10-15% higher equiy exposure. This leads us o conclude ha annuiizaion would be relaively more appealing o older reirees, as compared o phased wihdrawal paerns. Table 2 here Thus far he annuiy benchmark has been compued using he sex-specific moraliy able relevan o he individual making he purchase. Bu in some conexs, insurers are required o use a unisex moraliy able when pricing annuiies: for example, his is rue in he US if an annuiy is purchased wih company-based pension accumulaions (McGill e al., 2004). Likewise in he UK, unisex ables are used o price annuiies in he Personal Pension arrangemens. A unisex moraliy able is generaed by averaging moraliy probabiliies for men and women a each age. Naurally, such a able booss he annuiy paid o a female reiree and reduces he male s benefi, as compared o using sex-specific ables. In our conex, if German moraliy ables were used o value unisex payous (as per Appendix A), a 100 annuiy purchased by a female would have benefi payous ha are 7% higher han oherwise, whereas paymens o he male would be 7.7% lower. Ye he surprise is ha women are no necessarily gainers, depending on he phased wihdrawal paern seleced. This is because adoping a unisex able for annuiizaion changes he annuiy payou benchmark, while he phased wihdrawal plan sill embodies he purchaser s sex-specific moraliy able. Table 3 illusraes his case, where he annuiy benefi is now (by consrucion) equal for men and women, a 5.37 annually for a 100 purchase. For men, expeced shorfalls under all wihdrawal paerns are lower, expeced benefis are lower, and bequess are higher. The paern is he opposie for women: expeced shorfalls are higher, expeced benefis are higher, and

25 22 bequess lower. In oher words, if a governmen mandaed a unisex able for annuiy pricing ye sill permied phased wihdrawal paerns, women who coninue o selec a phased wihdrawal paerns are exposed o greaer risk. This resul migh be surprising o hose who advocae unisex ables in reiremen accouns, hough i flows from he fac ha unisex ables subsidize women annuians, on average. Table 3 here 4.3 Phased Wihdrawal Plans wih Mandaory Deferred Annuiies. The resuls above sugges ha some reirees migh prefer o engage in a mixed sraegy ha is, o underake phased wihdrawals during he early porion of he reiremen period and hen o swich o an annuiy hereafer. Furhermore, some researchers have suggesed ha such a mixed sraegy would be aracive: i enhances he payou early on, in exchange for relaively low risk, and i also adds he insurance feaure laer in life (Blake e al., 2003; Milevsky, 1998). In addiion, as noed earlier, some governmens have recenly required ha he elderly annuiize afer a phased income drawdown period. To examine he risks and rewards associaed wih phased wihdrawal followed by mandaory annuiizaion a some laer age, we now revisi our calculaions under each wihdrawal rule bu assume ha annuiy purchase is required if he individual is sill alive a eiher age 75 or 85. Two approaches are considered. In he firs case, which we call he swiching sraegy, a reiree would follow he relevan phased wihdrawal rule unil reaching he mandaory swiching age. Again as he benchmark, we use he real annuiy ha he reiree could have purchased a age 65, o compare our new resuls wih prior findings. If, a he swiching poin, he fund is inadequae o purchase his real annuiy, he gap represens a shorfall; conversely, if he accoun holds more han is needed o buy he benchmark annuiy, his excess can be allocaed o increase he beques or used for higher consumpion. In he following, we assume ha an excess (if any) is used o increase he level of he

26 23 annuiy saring a age 75 or 85, enhancing he EPVBenefis raher han EPVBeques measure. For he second case, we examine an immediae purchase deferral sraegy. In his case, he reiree purchases an annuiy on reiremen, wih deferred payous beginning a age 75 (or 85). The deferred annuiy benefi is se equal o he benchmark ha he reiree could have received if he iniiaed annuiy paymens a age 65. I is worh noing ha i is unclear wha one migh expec from hese swiching sraegies, in erms of risks and rewards. Some analyss sugges ha swiching may be a preferred sraegy, relying on he fac ha he moraliy drag rises wih age; annuiies pay ou more for a given premium, he older one is when purchasing hem (Milevsky, 2001). On he oher hand, ha analysis focuses only on he probabiliy of a shorfall bu does no weigh he size of he loss, condiional on he shorfall occurring. By delaying annuiizaion, he reiree can benefi from capial marke reurns if hey are favorable, so benefi paymens can be higher while he lives, or bequess higher if he dies. Ye delaying annuiizaion also exposes him o shorfall risk. Table 4 repors findings for he male reiring a age 65, making he decision o swich from a phased wihdrawal o an annuiy a eiher age 75 (or 85). Comparing resuls in Panel A of Tables 1 and 4, we see ha if delayed annuiizaion is available, his generally increases he value of he EP- VBenefis amoun and shrinks he EPVShorfall, boh of which are beneficial. The EPVBeques falls, indicaing ha he deferred annuiizaion sraegy is likely o be mos aracive o hose seeking o secure consumpion while alive, wihou compleely sripping heir heirs of some unexpended funds. In oher words, he risk/reurn profile of he phased wihdrawal plan ha includes a delayed annuiy is enhanced, as compared o no annuiy, a he cos of a smaller beques poenial. Also ineresing is he fac ha swiching o an annuiy laer in life (i.e. a age 85; compare panels A and B in Table 4) raises he equiy share of he porfolio slighly, bu grealy enhances he bond exposure. Also, buying he annuiy laer obviously promise more beques poenial, a he cos of higher shorfall.

27 24 Table 4 here Table 5 displays resuls for a 65-year old male purchasing a deferred annuiy a he beginning of he reiremen period, wih annuiy payous commencing a age 75 (or 85) assuming he is alive. In conras o he mandaory annuiizaion sraegy, we see ha he risk and reurn profile depends heavily on he chosen wihdrawal rule. In he case of he 1/T rule combined wih a deferred annuiy payable from age 75, he logical sraegy is o consume all remaining wealh using he phased wihdrawal acic by age 74, secure in he knowledge ha one is proeced agains longeviy risk hereafer. This paern provides a benefi sream worh slighly more han he real annuiy, and i offers low shorfall risk and low expeced bequess. This is an imporan resul since i indicaes he advanage of allowing flexibiliy unil age 75, paired wih proeced consumpion afer ha age. Similar resuls hold if he deferred annuiy were o begin a age 85, wih slighly higher benefi and beques levels a he expense of somewha higher shorfalls. By conras, he 1/E(T) rule combined wih a deferred annuiy a age 75 provides he reiree wih relaively low payous up o age 75, producing a high EPVShorfall, bu afer ha age, benefis flow from boh he annuiy and he phased wihdrawal plan which raises EPVBenefis (and higher poenial bequess). Delaying he annuiy payou dae o age 85 insead of 75 exposes he reiree o much higher shorfall risk, along wih higher possible wealh for he heirs. Table 5 here 4.4. Comparaive Resuls In addiional analyses no repored here (bu available on reques), we have also explored he sensiiviy of our resuls o a range of alernaive capial and annuiy marke scenarios. An ineresing experimen develops he environmen ha migh be relevan o US reirees: here, life expecancy is longer han in Germany, loadings are lower, and he capial marke presens differen risk/reurn characerisics. In his simulaion, we use he US Annuian 2000 Basic Male moraliy able along