No. 2004/01. Betting on Death and Capital Markets in Retirement: A Shortfall Risk Analysis of Life Annuities versus Phased Withdrawal Plans

Transcription

1 No. 2004/01 Being on Deah and Capial Markes in Reiremen: A Shorfall Risk Analysis of Life Annuiies versus Phased Wihdrawal Plans Ivica Dus, Raimond Maurer, Olivia S. Michell

2 Cener for Financial Sudies The Cener for Financial Sudies is a nonprofi research organizaion, suppored by an associaion of more han 120 banks, insurance companies, indusrial corporaions and public insiuions. Esablished in 1968 and closely affiliaed wih he Universiy of Frankfur, i provides a srong link beween he financial communiy and academia. The CFS Working Paper Series presens he resul of scienific research on seleced opics in he field of money, banking and finance. The auhors were eiher paricipans in he Cener s Research Fellow Program or members of one of he Cener s Research Projecs. If you would like o know more abou he Cener for Financial Sudies, please le us know of your ineres. Prof. Dr. Jan Pieer Krahnen Prof. Volker Wieland, Ph.D.

3 CFS Working Paper No. 2004/01 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 January 2004 Absrac: How migh reirees consider deploying he reiremen asses accumulaed in a defined conribuion pension plan? One possibiliy would be o purchase an immediae annuiy. Anoher approach, called he phased wihdrawal sraegy in he lieraure, would have he reiree inves his funds and hen wihdraw some porion of he accoun annually. Using his second acic, he wihdrawal rae migh be deermined according o a fixed benefi level payable unil he reiree dies or he funds run ou, or i could be se using a variable formula, where he reiree wihdraws funds according o a rule linked o life expecancy. Using a range of daa consisen wih he German experience, we evaluae several alernaive designs for phased wihdrawal sraegies, allowing for endogenous asse allocaion paerns, and also allowing he worker o make decisions boh abou when o reire and when o swich o an annuiy. We show ha one paricular phased wihdrawal rule is appealing since i offers relaively low expeced shorfall risk, good expeced payous for he reiree during his life, and some beques poenial for he heirs. We also find ha unisex moraliy ables if used for annuiy pricing can make women s expeced shorfalls higher, expeced benefis higher, and bequess lower under a phased wihdrawal program. Finally, we show ha delayed annuiizaion can be appealing since i provides higher expeced benefis wih lower expeced shorfalls, a he cos of somewha lower anicipaed bequess. JEL Classificaion: G22, G23, J26, J32, H55 Keywords: Insurance, Pensions, Reiremen and Reiremen Policies, Social Securiy and Public Pensions 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 , U.S.A., T: 215/ F: 215/ , mailo:michelo@wharon.upenn.edu

4 3 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 January 2004 Policy Summary Reirees ofen face he quesion of how o draw down asses ha hey have accumulaed over heir worklives. Many recommend ha people should purchase a life annuiy o proec hem agains longeviy risk, bu here we explore an alernaive sraegy called a self-annuiizaion or phased wihdrawal approach. Here he reiree allocaes his funds across various asse caegories (e.g. equiy, bonds, cash) and periodically wihdraws a porion of he invesed funds for consumpion purposes. The advanage of such a phased wihdrawal sraegy, as compared o a life annuiy, is ha i offers greaer liquidiy, he possibiliy of greaer consumpion while alive as well as he possibiliy of bequeahing some of he asses in he even of early deah. Ye relying on income from asses wihou any insurance provides no pooling of longeviy risk. Our paper explores several alernaive wihdrawal rules ha rely no on some fixed amoun per period, bu raher on consuming a specified fracion of he remaining fund wealh each period. This alernaive approach avoids he risk of ouliving one s oal asses, as long as he benefi-o-wealh raio is lower han one. Neverheless, due o sochasic invesmen reurns, he value of he pension accouns asses change over ime implying ha he periodically wihdrawn amoun mus vary in andem and i could be subsanially lower or higher han he benefi payable under a life annuiy. To evaluae differen decumulaion opions on a quaniaive basis, we adop a risk-value (or riskreurn) model which uses an explici measure of risk, an explici measure of value, and a funcion reflecing he rade-offs beween value and risk. Mos relevan o policymakers is our finding ha mandaing annuiizaion afer a phased wihdrawal period can be quie appealing in erms of risk. This is of paricular ineres since his approach has recenly been implemened in boh he UK and Germany; some annuiizaion has also been recommended by he US Commission o Srenghen Social Securiy. We also find ha requiring unisex ables for annuiy pricing exposes women o poenially greaer risk. Finally, our resuls speak o he asse mix reirees will opimally wan o hold: laer annuiizaion (say, a age 85) would imply a larger fracion of he financial asses would be held in bonds.

5 4 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 January 2004 Execuive Summary Reirees ofen face he issue of how o draw down asses ha hey have accumulaed over heir worklives. While economiss ofen recommend ha hey purchase a life annuiy, which covers hem agains longeviy risk, hese financial insrumens have some disadvanages. A buyer faces loss of liquidiy and conrol over his asses, and in many cases annuiies do no leave money for bequess. By conras, in some European counries, policymakers have permied alernaive income wihdrawal paerns for asse pools dedicaed o old-age consumpion. This paper focuses on rules similar o hose adoped in Germany under he so-called Rieser plans, where some porion of he funds can be aken as a lump-sum and some oher porion mus be annuiized. Similar rules are in place in he UK and in Canada. A key aspec of he reiree s decision during he payou phase is how o inves his or her reiremen plan asses, and also how payous should be srucured so as o balance consumpion flows versus beques inenions wihou running ou of money. We explore an alernaive sraegy o buying a life annuiy called a self-annuiizaion or phased wihdrawal approach. Here he reiree allocaes his funds across various asse caegories (e.g. equiy, bonds, cash) and periodically wihdraws a porion of he invesed funds for consumpion purposes. The advanage of such a phased wihdrawal sraegy, as compared o a life annuiy, is ha i offers greaer liquidiy, he possibiliy of greaer consumpion while alive as well as he possibiliy of bequeahing some of he asses in he even of early deah. Ye relying on income from asses wihou any insurance provides no pooling of longeviy risk. Consequenly, if he reiree consanly consumes an equal amoun from his accoun, he could oulive his asses before his uncerain dae of deah, paricularly in he even of long-run low invesmen reurns. We develop several alernaive wihdrawal rules ha rely no on some fixed amoun per period, bu raher on consuming a specified fracion of he remaining fund wealh each period. This alernaive approach avoids he risk of ouliving one s oal asses, as long as he benefi-o-wealh raio is lower han one. Neverheless, due o sochasic invesmen reurns, he value of he pension accouns asses change over ime implying ha he periodically wihdrawn amoun mus vary in andem and i could be subsanially lower or higher han he benefi payable under a life annuiy. So as o evaluae he differen decumulaion opions on a quaniaive basis, we inroduce a formal risk/reurn framework for decision making under uncerainy. Here we adop a risk-value (or riskreurn) model which uses an explici measure of risk, an explici measure of value, and a funcion reflecing he rade-offs beween value and risk. Since reirees will end o prefer more reurn o less, and less risk o more, oher hings equal, we can derive a parial-ordering of opporuniies wihin a risk-reurn dominance conex. Whereas previous sudies have focused on he probabiliy of consumpion shorfall as he operaive risk measure, we exend he lieraure in several direcions. Firs, we examine he risk and reurn profiles of several variable self-annuiizaion sraegies ha provide paymens according o predeermined benefi-o-wealh raio. Second, we address a major

6 5 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 is o go beyond prior work by looking no only a he probabiliy of a consumpion shorfall, bu also consider 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 regulaion. 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. Our analysis shows ha a phased wihdrawal sraegy paying he same benefi as he annuiy exposes reirees o he risk of ouliving heir asses while sill alive. A phased wihdrawal plan using a fixed benefi-o-wealh raio avoids he risk of running ou of money, since benefis flucuae in andem wih he pension fund s value. Bu a fixed benefi wihdrawal rule affords lower risk han variable wihdrawal rules, if one uses a moraliy-weighed shorfall-risk measure (which includes boh shorfall probabiliy and magniude of loss). We also show ha mandaory deferred annuiizaion wih a fixed wihdrawal rule can enhance expeced payous and cu expeced shorfall risk bu a he cos of reduced expeced bequess, as compared o no annuiy. For a variable wihdrawal plan, a simple deferred annuiizaion may no reduce risk: raher, i requires opimizaion of he benefi o wealh raio. We furher explore using an 1/E(T) phased wihdrawal rule, which offers relaively low expeced shorfall risk, good expeced payous for he reiree during his life, and some beques poenial for his heirs. Bu if mandaory annuiies are combined wih his phased wihdrawal plan, we find he 1/E(T) rule o be less aracive. We also find ha he opimized 1/T rule and he fixed benefi rule boh have appealing risk characerisics, paricularly when combined wih a mandaory deferred annuiy. Relevan o policymakers is our finding ha mandaing annuiizaion afer a phased wihdrawal period can be quie appealing in erms of risk. This is of paricular ineres since his approach has recenly been implemened in boh he UK and Germany. A degree of mandaed annuiizaion has also proposed for he US by he recen Commission o Srenghen Social Securiy in he US conex. The presen paper also implies ha a governmen mandae requiring ha unisex ables be adoped for annuiy pricing (as in he UK) exposes women who eleced a phased wihdrawal plan o greaer risk. Finally, our resuls have implicaions for he asse mix reirees will opimally wan o hold: laer annuiizaion (say, a age 85) would imply a larger fracion of he financial asses would be held in bonds.

7 6 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 1. Inroducion In reiremen, many people face he quesion of how o draw down asses ha hey have accumulaed over heir worklives. Economiss ofen sugges ha a sensible approach is o purchase a life annuiy. An annuiy is a financial conrac beween an insured person and an insurance company ha pays ou a periodic amoun for as long as he annuian is alive, in exchange for an iniial premium (Brown e al., 2001: p. 1). The paymens may be fixed in nominal erms (fixed annuiy), or hey migh rise a a pre-specified fixed nominal escalaion rae (graded annuiy), or hey could be indexed o inflaion (real annuiy) keeping he reiree s sandard of living consan. Alernaively, hey migh 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 ha canno be replicaed by pure invesmen vehicles, hey also have some disadvanages. Firs, 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 expendiures for uninsured medical coss). 1 Second, in is simples form, where income paymens are coningen on he individual s survival, here is no chance of leaving money for heirs, even in he case of he annuian s early deah. Oher explanaions for why individuals will 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). Recen developmens in European pension sysems have focused aenion on alernaive income wihdrawal paerns for asse pools dedicaed o old-age consumpion. In Germany, so-called Rieser plans offer ax inducemens for volunary saving in individual pension accouns (IPA) during he worklife, underscoring he governmen s ineres in boosing asse accumulaion in an aging populaion (Börsch-Supan e al., 2003a, b). When he age of reiremen is reached, weny percen of he accumulaed asses in he IPA can be aken as a lump-sum disribuion. The res mus be drawn down in he form of a lifelong annuiy (offered by a commercial insurance company) or a phased wihdrawal plan (ypically offered by muual fund and/or bank providers) which mus parly rever ino an annuiy a he age of 85. In he UK, personal pensions have also grown in populariy 1 See Brugiavini (1993) for a heoreical model in which he healh saus of he reiree is sochasic.

8 7 (Blake e al., 2003). As in Germany, a porion of he accumulaed asse can be aken as a lump sum, while wih he res, one is legally obliged o buy an annuiy by he age of 75. In Canada, a age 69 reirees mus eiher buy an annuiy wih heir ax-shelered savings or creae a discreionary managed wihdrawal plan (Milevsky and Robinson, 2000). In he US, no compulsory annuiizaion is required for 401(k) plans a reiremen; insead, many workers roll over heir funds as a lump sum ino an Individual Reiremen Accoun which manage hemselves in old age. Though some researchers have explored aspecs of he accumulaion phase in hese accouns (e.g. Maurer and Schlag, 2003; Blake e al., 2001), hus far, relaively lile aenion has been devoed o he payou phase. A key aspec of he reiree s decumulaion process is he decision of how o inves hese reiremen plan asses and how o srucure payous during he reiremen period, so as o bes balance consumpion flows versus beques inenions wihou running ou of money. An alernaive sraegy o buying a life annuiy is associaed wih wha has been called self-annuiizaion or phased wihdrawal approach (c.f. Milevsky and Robinson, 2000). A reiremen, he wealh endowmen is allocaed 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. The paricular advanage of such a phased wihdrawal sraegy, as compared o he life annuiy, is ha i offers greaer liquidiy, he possibiliy of greaer consumpion while alive as well as he possibiliy of bequeahing some of he asses in he even of early deah. On he oher hand, relying on income flows wihdrawn direcly from an IRA wihou any insurance provides no pooling of longeviy risk. Consequenly, if he reiree consanly consumes an equal amoun from his accoun, he could oulive his asses before his uncerain dae of deah, paricularly in he even of long-run low invesmen reurns. An alernaive wihdrawal rule is o no ake ou some fixed amoun per period, bu raher o consume a specified fracion of he remaining fund wealh each period. This second sraegy, in conras o he fixed wihdrawal echnique, avoids he risk of ouliving one s oal asses, as long as he benefi-o-wealh raio is lower han one. Neverheless, due o sochasic invesmen reurns, he value of he pension accouns asses change over ime implying ha he periodically wihdrawn amoun mus vary in andem and i could be subsanially lower or higher han he benefi payable under a life annuiy. To be able o evaluae he differen decumulaion opions on a quaniaive basis, i is necessary o inroduce a formal risk/reurn framework for decision-making under uncerainy. The sandard approach in financial economics 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) use a uiliy funcion of he consan relaive risk class (CRRA), o evaluae differen wihdrawals plans assuming mandaory annuiizaion is required a age 75. Milevsky and Young (2003) use a similar objecive funcion o deermine he value of he opion o defer annuiizaion. A shorcoming of such an approach, especially in he pracical world, is ha he decision-maker rarely has explici measures of risk preferences wihou knowing he shape of his uiliy funcion. As Pye 2000 poined ou, neiher endowmen fund managers nor financial planners are using hese models o help make decisions. As a resul, risk-value (or risk-reurn) models of choice have he advanage of developing an explici measure of risk, an explici measure of value, and a funcion reflecing he rade-offs beween value and risk. Clearly, individuals prefer more reurn o less and less risk o more, oher hings equal. This propery allows a parial-ordering of opporuniies wihin a riskreurn dominance conex, even if he exac preference weighs for he risk and reurn radeoff are unknown. Depending on which risk meric is seleced and how he rade-off beween risk and reurn

9 8 is formulaed, a risk-value model can bu need no be consisen wih he expeced uiliy approach of choice (Sarin and Weber 1993). 2 In his paper, we ake a risk-value approach, whereby 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. 3 Assuming ha he reiree consumes a fixed real amoun a specific poins in ime from a self-managed pension accoun, hese auhors calculae he probabiliy of running ou of money before he 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 selfannuiizaion sraegies ha provide paymens according o 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 is o go beyond prior work by looking no only a he probabiliy of a consumpion shorfall, bu also consider 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 regulaion. 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. The remainder of his paper is divided ino four secions. The nex secion describes several differen wihdrawal sraegies. To illusrae heir implicaions, we assume condiions wih respec o capial and insurance markes producs and pricing found in he German annuiy and capial markeplace. 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. Secion hree repors resuls using a fixed asse allocaion paern, and Secion four permis asses o be allocaed opimally. A final secion summarizes and concludes. 2. The Case of Phased Wihdrawal 2.1 Wihdrawal Plans wih Fixed Benefis We assume ha he reiree is endowed wih an iniial wealh of V 0 ha he can use o buy a singlepremium immediae life annuiy paying consan annual real benefis B a he beginning of each year, for life wih no beques. We denoe his as he benchmark annuiy and refer he reader o Appendix A regarding he pricing of such an insurance produc using assumpion abou moraliy, 2 Perhaps he mos widely used risk-reurn model in he area of finance is he classic mean-variance porfolio analysis elaboraed by Markowiz (1952), which is, iner alia, consisen wih a quadraic uiliy funcion (see Campbell and Viceira, 2002, p. 24). A general analysis of condiions regarding he compaibiliy of muliparameer rade-off models of choice wih he expeced uiliy model is given in Schneeweiß (1967). 3 See for insance Milevsky e al. (1997), Milevsky and Robinson (2000), Milevsky (1998, 2001), Ameriks e al. (2001), Pye (2000, 2001) and Albrech and Maurer (2002).

10 9 loadings and ineres raes o discoun fuure annuiy paymens. If he reiree does no annuiize his wealh, he invess he reiremen asses in various financial asses (e.g. equiies, bonds, cash) ypically represened by a family of muual funds, and hen he wihdraws a cerain amoun a he beginning of he year for consumpion purposes. Throughou he paper, we assume ha benefis are axed as ordinary income; herefore axes will no change he desirabiliy of volunary annuiizaion or sysemaic wihdrawal from a self-managed reiremen accoun. 4 Under he fixed benefi rule, he reiree will sell a he beginning of each year as many fund unis as required o reach he same yearly benefis paid by he life annuiy, unil eiher 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) 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 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 V of he reiremen asses used o finance wihdrawals. If hese asses are risky, he benefi payous are exposed o uncerain capial marke reurns. The idea of he fixed benefi rule is o replicae he income from a life annuiy as long as he funds permi, while a he same ime offering some beques poenial in he even of an early deah. Neverheless, he risk of he fixed benefi rule 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 no a fixed benefi amoun per period, bu raher an ex ane fixed fracion of he reiremen asses remaining each period. This benefiwealh raio can be consan, increasing, or decreasing over ime. 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 accouns value. Depending on he wihdrawal fracion and he realized reurns of he reiremen accouns asses, benefi paymens could be subsanially lower or higher han paymens from a life annuiy a some poin during he pos reiremen phase. A variable phased wihdrawal plan and a variable annuiy have in common he fac ha hey pay pension benefis ha vary wih uncerain invesmen reurns. Neverheless, he former offers he possibiliy of bequeahing he remaining value of he reiremen accoun in he case of he reiree s deah, while he laer does no. 4 This is accurae for he German conex; for more on annuiy ax reamen in he US see Brown e al. (2001).

11 10 The pah of benefis payable using 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 every period, an ex ane specified fracion ω (0 < ω 1) is wihdrawn from curren wealh; hence he reiree receives a paymen according o: 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 = ( If he reiree dies a he beginning of period +1, V +1 represens he beques poenial for his heirs. Noe ha if he asses of he pension accoun are invesed in risky asses (e.g. socks and/or bonds), he reurns are also uncerain, and herefore boh he pension 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, i.e. he benefi-wealh raio is fixed over ime: B V ω = ω. = This wihdrawal rule has he advanage of simpliciy, requiring no informaion regarding he maximum possible duraion of he payou phase or he reiree s characerisics (i.e. age, sex). "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. (6) 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: B V 1 = ω =. (7) T In conras o he fixed percenage rule discussed above, he wihdrawal fracion is no consan bu raher increases wih age. Wha his means is ha he longer he reiree survives, he higher he wihdrawal fracion will be. For example, if l = 110 and x = 65 he firs wihdrawal fracion a age 65 is ω 0 = 1/46 = 2.17%, he second a age 66 is ω 1 = 1/45 = 2.22%, and a age 101 he benefi o wealh raio is ω 101 = 10%. The rule pay ou all of he remaining wealh of he reiremen accoun (i.e. ω 110 = 100%) by he age of 110 no beques poenial is lef in conras o he fixed percenage rule. (5)

12 11 1/E[T(x)]" Wihdrawal Rule: This rule, which we will call he 1/E(T) rule for shor, akes ino consideraion 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 by he reiree s life expecancy remaining. Le p x represens he condiional probabiliy ha an x-year old man will aain age x +, he complee expecaion of life is calculaed as: [ ] T ( x + ) = = l x E (8) p x 0 where l is he maximum age according o a moraliy able. Then, for an a reiremen x-year old man, he benefi-o-wealh raio in period afer reiremen, condiional on he fac ha he is sill alive, is given as: B V 1 = ω =. (9) E[ T( x + )] The shorer his expeced remaining lifeime, he higher he fracion he will wihdrawal from his pension accoun. Therefore, he wihdrawal fracion rises wih he age of he reiree. Since he reiree s life expecancy is less han he maximum age of he moraliy ables, he benefi-o-wealh raio of he 1/E(T) rule exceeds ha of he 1/T rule, in general. 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, i is useful o begin wih an assessmen of expeced payous condiional on reiree survival (Secion 4 generalizes resuls wih moraliy-weighed risk and reward compuaions). 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 phased wihdrawal plans given an iniial asse balance. The plan asses are rebalanced annually o mainain an asse pool spli evenly beween socks and bonds, consisen wih recommendaions by financial advisors (asse allocaion is opimized in he nex secion). 5 We employ an annuian moraliy able provided by he German Sociey of Acuaries 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. 5 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 could (probably) be jusified wih he maximum age used in mos populaion moraliy ables bu we noe ha annuian moraliy ables generally have a maximum age of years higher.

13 12 We nex assess he risk and reurn paerns ha emerge under hese alernaive phased wihdrawal paerns (before axes), compared o 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 Geomeric Brownian moion, a sandard assumpion in financial economics (which can be raced back o Bachellier, 1900). This implies ha he yearly log-reurns are i.i.d. and normally disribued. 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. 6 Subsequenly, asse reurns are adjused for inflaion by using he German Consumer Price Index. These yearly daa produce esimaes (before axes) for he real log average rae of reurn, he volailiy and he correlaion-coefficiens of socks and bonds as repored in Appendix C: Since we assume normally disribued log reurns, i.e. I = ln(1 + R ) ~ N(µ, σ ), hese parameers imply a real log average rae of reurn on he fify-fify sock-bond porfolio of µ = 5.52 percen wih a sandard deviaion of σ = %. Noe ha his produces an expeced gross rae of reurn of E(1 + R ) = E[exp(I )] = exp[ *0.1378²] = Assuming ha he normaliy propery also holds for he log porfolio reurns, 7 i is sraighforward o develop an analyical closed form soluion for he probabiliy disribuion of fuure benefis of he differen variable phased wihdrawal rules since he ineremporal budge consrain given in equaion (4) is (log)linear (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, even under he assumpion of independen and idenically disribued log porfolio reurns, for he fixed benefis wihdrawal plan he probabiliy disribuion of fuure benefis is unknown. Therefore, o obain esimaes for he differen risk and reurn measure we use Mone-Carlo simulaion o generae a large number (i.e. 100,000) of pahs for he evoluion of he wihdrawal plan. 8 6 Feldsein e al. (2001) p. 60 use a similar procedure o accoun for adminisraion coss. 7 This assumpion is widely used in sraegic asse allocaion (e.g. Feldsein e al. (2001) or Campbell and Viceira (2002)) and can be jusified by a Taylor approximaion of he nonlinear funcion relaing log-individual-asse reurns o log porfolio reurns. For full deails see Campbell and Viceira (2002), p and Campbell e al. (2001). 8 Milevsky and Robinson (2000) have developed an analyical approximaion mehod based on momen-maching echniques and he reciprocal gamma disribuion and herefore can avoid Mone Carlo-simulaion.

14 Analysis of Expeced Benefis Figure 1 depics he Expeced Benefis profiles condiional on survival, for he four phased wihdrawal rules, as compared o he annuiy profile. Focusing on he fixed benefi rule shows ha in he firs year, mean benefis are (by consrucion) equal o he annuiy benefi. However, in he following years, he 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 6.66% p.a., which exceeds he consan benefi-o-wealh-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, reaching almos 700% of he annuiy paymen lae in life. This can be explained by he low wihdrawal fracions of 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 (i.e. 6.66%); 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% if he reiree reaches age 70. This payou approach reaches is maximum expeced paymen of abou 150% a age of 83. Afer his age, he expeced paymens are monoonously decreasing, reaching he level of he life annuiy a age 91. A ages older han 100, following he 1/E(T) rule would leave he reiree very exposed o quie low benefis, asympoically approaching 0. Noe ha he wihdrawal fracion under he 1/E(T) rule is higher han under he 1/T rule. 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 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 Shorfall Probabiliy In accordance wih oher fields of research, as well as wih convenional wisdom, 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-riskmeasures are called relaive or pure measures of risk. 9 To analyze his risk in he case of our 9 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 (1975) 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); Laughhuun 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.

15 14 phased wihdrawal sraegies, we employ several differen shorfall risk measures. We begin wih he shorfall probabiliy, defined as: SP(B ) = P(B < z). (10) 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 Shorfall Probabiliy for he fixed benefi rule, he fixed fracion rule (5.82%), he 1/T approach and he 1/E(T) rule, as compared o he annuiy benefi. In he firs year, all he sraegies excep he fixed benefi program face a high probabiliy of shorfall, and 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 have no been exhaused. The fixed benefi program offers a Shorfall Probabiliy close o zero a he 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 very 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 shorfall probabiliy 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 60% a he end of he period. 10 Figure 2 here Anoher ineresing finding has o do wih he gradien of he Shorfall Probabiliy under he 1/E(T) rule. Early in he reiremen period here is a very fas decline in his risk, bu if he reiree is sill alive a age 83, he SP begins o rise very 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 increases. 10 This resuls from he lognormal disribuion of fuure benefis which become increasingly skewed o he righ, he longer he reiree remains alive. Noe ha he expeced level reurn (i.e. exp( *0.1378²) - 1 = 6.66%) of he reiremen accoun asses is greaer han he wihdrawal fracion (i.e. 5.82%), bu he median level reurn (i.e. exp(0.0552) - 1 = 5.68%) is slighly below he wihdrawal fracion, so he shorfall probabiliy rises wih age.

16 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. The shorfall probabiliy answers he quesion how ofen consumpion falls shor, bu no how bad he loss is if i occurs, under each of he differen wihdrawal rules. To provide informaion abou he poenial exen of a shorfall, we nex calculae he Mean Excess Loss (MEL) as an addiional risk measure. Formally his risk meric is given by: MEL(B )=E[ z B B < z ]. (11) I indicaes he expeced loss wih respec o he benchmark, under he condiion ha a shorfall occurs. Therefore, given a loss, he MEL answers he quesion how badly on average does he sraegy perform; i MEL can be characerized as a wors case risk measure, which is highly sensiive wih respec o realisaions a he ail of he disribuion (i.e. large-scale shorfalls). 11 An addiional shorfall risk measure ha links boh he probabiliy and he exen of he condiional shorfall in an inuiive way is he Shorfall Expecaion (SE): SE(B ) = E[max(z - B,0)] = MEL(B ) SP(B ). (12) The shorfall expecaion is he sum of losses weighed by heir probabiliies, and hence i is a measure of he uncondiional average loss. As equaion (12) shows, he SE is simply he produc of he shorfall probabiliy and he mean level of shorfall given he occurrence of a shorfall. 12 In Figure 3 we plo he Mean Excess Loss resuls for he various wihdrawal sraegies of ineres, namely he fixed benefi rule, he fixed fracion rule (5.82%), he 1/T approach and he 1/E(T) rule. Here we compare he MEL for each acic versus he annuiy benefi. Resuls are similar in form: ha is, in he firs year, all sraegies bu he fixed benefi program have a posiive MEL, since he fixed benefi approach pays for as long as possible an amoun equal o he iniial annuiy. The 1/T rule has a paricularly high iniial MEL, a 60% of he value of he annuiy paymen, and his falls only o 30% some 15 years ino he reiremen period. Boh he 1/E(T) and he fixed fracion rules have 30% MEL profiles hrough abou age 90, bu hen he 1/E(T) rule confrons he reiree wih a rapidly rising mean excess loss aaining close o 100% lae in life. By conras, he 1/T plan faces he reiree wih a gradually declining expeced loss afer age 90, falling o abou 30%. The resuls make clear ha from a wors-case risk perspecive, he fixed fracion rule, and he 1/E(T) rule, are no proper financial insrumens for insurance agains longeviy. Figure 3 here 11 The MEL is closely conneced wih he Tail Condiional Expecaion (TCE), which is given by TCE = E(R R < z) = z MEL. The TCE has some favourable feaures, e.g. i is (in conras o he shorfall probabiliy) a coheren risk measure wih respec o he axioms developed by Arzner e al. (1999). 12 In addiion, he SE is relaed o he price of a (derivaive) financial conrac which allows he annuian o ransfer he downside risk of a phased wihdrawal plan ino he capial marke. For example, if he reiree buys a pu opion paying P = max(z B, 0) a ime, hen he is compleely hedged agains he risk ha he benefi from he wihdrawal plan is lower han he paymen from he benchmark annuiy z. Noe ha he fuure benefis are direcly relaed o he marke value of he reiremen accouns asses V. Using sandard argumens from opion pricing heory, he price of such a (European) pu opion is given by p 0 = E Q [max(z B, 0)]/exp(R f ), where E Q denoes he expecaion operaor wih respec o he risk adjused ( maringale ) probabiliies and R f is he risk free ineres rae.

17 16 The profiles for he Shorfall Expecaion appear in Figure 4, and i will be recalled ha hese combine he Shorfall Probabiliy and he Mean Excess Loss, all condiional on survival. This graph underscores he paerns revealed by he wo previously analyzed risk measures. Now 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 hypoheical individual under sudy. The fixed fracion and he 1/E(T) rules have a SE of less han 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 4 here 3.4. Analysis of Expeced Bequess The oher side of he sory behind hese rules is ha he reiree mus in effec compare his own consumpion wih he poenial value of he beques ha would go o he heirs if he should die. Figure 5 illusraes he expeced beques under he various formulaions, condiional on deah. The paern exhibiing mos sabiliy is he fixed fracion rule, bu he oher hree are highly divergen. For example, he 1/T rule shows an ineresing pah, firs rising in 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=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 rule, 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 5 here

18 17 4. Risk-Minimizing Phased Wihdrawal Sraegies 4.1. Opimized Wihdrawal Rules in a Risk-Reurn Conex Thus far, our analysis has assumed ha he reiree holds his pension plan asses in a fixed-weigh porfolio comprised of 50% socks and 50% bonds over a fixed invesmen horizon. Thus he payous during reiremen ake ino accoun only capial marke uncerainy, and here was no possibiliy of opimizaion around risk/reward radeoffs. In his secion, we exend he analysis by including a consideraion of moraliy risk, and furher we discuss wo addiional phased wihdrawal rules ha permi he reiree o opimize he design of he wihdrawal paerns. In he nex subsecion, our analysis varies he invesmen weighs of he associaed wih sock, bonds, and cash invesmens, o aain a risk-minimizing saic asse allocaion. The porfolio weighs are herefore deermined endogenously (excluding shor-selling), following Albrech and Maurer (2002). In he following subsecion, we go on o examine 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. In boh counries, he resricion of mandaory swiching has already considerable criicism in he public debae (c.f. Blake e al. 2003; Börsch-Supan and Wilke, 2003b). 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 Here, SE(B ) = E[max(z B, 0)] denoes he expeced shorfall wih respec o he arge z, which is equal o he benefi provided by he benchmark life annuiy. The possible expeced shorfall in year are weighed by he condiional probabiliy p x ha a man aged x a he beginning of he reiremen phase is sill alive, in he case when 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. 13 Given his funcion, we minimize i wih regard o asse allocaion and oher plan design parameers, o derive he asse allocaion 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 benefis 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 for he size of he loss when i happens, which our risk measure does. f (13) 13 Noe, if SE(B )/(1+R f ) is calculaed wih respec o he corresponding risk adjused ( maringale ) probabiliies (consisen wih an arbirage free capial marke) of he underlying asse process, i is consisen wih he price of an European pu opion which pays he difference if a shorfall happens in year afer reiremen. Then he EPVShorfall is he value of a porfolio consising of (European) pu opions weighed by survival probabiliies p x.