No. 2007/21 Money in Motion: Dynamic Portfolio Choice in Retirement. Wolfram J. Horneff, Raimond H. Maurer, Olivia S. Mitchell, and Michael Z.

Transcription

1 No. 2007/2 Money in Moion: Dynmic Porfolio Choice in Reiremen Wolfrm J. Horneff, Rimond H. Murer, Olivi S. Michell, nd Michel Z. Smo

2 Cener for Finncil Sudie The Cener for Finncil Sudie i nonprofi reerch orgnizion, uppored by n ociion of more hn 20 bnk, inurnce compnie, induril corporion nd public iniuion. Eblihed in 968 nd cloely ffilied wih he Univeriy of Frnkfur, i provide rong link beween he finncil communiy nd cdemi. The CFS Working Pper Serie preen he reul of cienific reerch on eleced opic in he field of money, bnking nd finnce. The uhor were eiher pricipn in he Cener Reerch Fellow Progrm or member of one of he Cener Reerch Projec. If you would like o know more bou he Cener for Finncil Sudie, plee le u know of your inere. Prof. Dr. Jn Pieer Krhnen Prof. Volker Wielnd, Ph.D.

3 CFS Working Pper No. 2007/2 Money in Moion: Dynmic Porfolio Choice in Reiremen* Wolfrm J. Horneff, Rimond H. Murer 2, Olivi S. Michell 2, nd Michel Z. Smo 4 My 2007 Abrc: Reiree confron he difficul problem of how o mnge heir money in reiremen o o no oulive heir fund while coninuing o inve in cpil mrke. We poi dynmic uiliy mximizer who mke boh e locion nd llocion deciion when mnging her reiremen finncil welh nd nnuiie, nd we prove h he cn benefi from boh he equiy premium nd longeviy inurnce in her reiremen porfolio. Even wihou beque, he will no fully nnuiize; rher, her opiml ock llocion moun iniilly o more hn hlf of her finncil welh nd decline wih ge. Welfre gin from hi regy cn moun o 40 percen of finncil welh (depending on rik prmeer nd oher reource). In prcice, i urn ou h mny reiree will do lmo well by purching vrible nnuiy inveed 60/40 in ock/bond. JEL Clificion: G, G23, G22, D4, J26, H55 Keyword: Porfolio Choice, Prive Finncil Service, Inurnce, Peronl Finnce, Reiremen Policie, Socil Securiy * Thi reerch w conduced wih uppor from he US Socil Securiy Adminirion vi he Michign Reiremen Reerch Cener he Univeriy of Michign. Addiionl reerch uppor w provided by he Germn Invemen nd Ae Mngemen Aociion (BVI), he Germn Reerch Foundion (DFG), he Friz-Thyen Foundion, he Obervoire de l Eprgne Européenne (OEE), nd he Penion Reerch Council (PRC) The Whron School of he Univeriy of Pennylvni. We re greful for ueful commen from John Amerik, Dvid Bbbel, Jeffrey Brown, Neil Dohery, Chriin Gollier, nd Alexnder Muermnn. Thi i pr of he NBER Progrm on he Economic of Aging. Opinion nd error re olely hoe of he uhor nd no of he iniuion wih whom he uhor re ffilied Horneff, Murer, Michell, Smo. All righ reerved. Finnce Deprmen, Goehe Univeriy, Keenhofweg 39 (Uni-PF. 58), Frnkfur m Min, Germny, e-mil: horneff@finnce.uni-frnkfur.de 2 Finnce Deprmen, Goehe Univeriy, Keenhofweg 39 (Uni-PF. 58), Frnkfur m Min, Germny, e-mil: rmurer@wiwi.uni-frnkfur.de 3 Dep of Inurnce nd Rik Mngemen, The Whron School, Univeriy of Pennylvni, 3620 Locu Wlk, 3000 SH-DH, Phildelphi, PA 904, e-mil: michelo@whron.upenn.edu. 4 Finnce Deprmen, Goehe Univeriy, Keenhofweg 39 (Uni-PF. 58), Frnkfur m Min, Germny, e-mil: mo@finnce.uni-frnkfur.de

4 Money in Moion: Dynmic Porfolio Choice in Reiremen A Bby Boomer move ino heir 60, increing enion i being devoed o he propec of $5 rillion in e h reiree will be required o mnge over wh promie o be very long period (McKiney, 2006). A finncil produc h offer ome promie in hi regrd i he lifeime pyou nnuiy, which repreen promie by n inurer o py benefi over he reiree remining lifeime, no mer how long he live. Lifeime nnuiie cn ugmen reiremen ecuriy, hey llow he reiree o proec gin ouliving her e while enhncing expeced reurn hrough he morliy credi. Mo nnuiie currenly old in he Unied Se re of he vrible ype where purcher cn inve in cpil mrke e, ye hee hve been lile udied by reercher. Ined, he bulk of previou udie hve focued on fixed pyou nnuiie, or nnuiie h py fl benefi. 2 Alo, lile prior nlyi h explored opiml dynmic porfolio choice in reiremen involving vrible pyou nnuiie. 3 Our pper fill hi reerch gp by exploring how vrible pyou nnuiie cn help reiree cce ock mrke during he decumulion phe of her lifecycle, while ill permiing her o ke dvnge of he ddiionl expeced reurn nd hedge morliy rik uing nnuiy produc. Specificlly, we derive n opiml endogenou grdul nnuiizion regy nd e llocion pern for rik-vere reiree fcing ochic lifeime nd endowed wih penion The morliy credi refer o he exr income received by urviving nnuiy buyer reuling from hring in he commied e of deceden; c.f. Brown e l. (200), Feldein nd Rnguelov (200), nd Michell e l. (999). 2 For innce, Du, Murer, nd Michell (2005); Gerrrd, Hbermn, nd Vign (2004); Horneff, Murer, Michell, nd Du (2006); Kpln (2006) Kpur nd Orzg (999); Kingon nd Thorp (2005; Milevky (998); Milevky nd Young (2002); Milevky, Moore, nd Young (2006); Michell, Poerb, Wrhwky, nd Brown (999); nd Sbile (2003). 3 Severl pper hve explored vrible nnuiie bu dop rericive nnuiizion nd e llocion pern; c.f. Blke, Cirn nd Dowd (2003); Browne, Milevky nd Slibury (200); Brown, Michell nd Poerb (200); Chrup nd Milevky (2002); Feldein nd Rnguelov (200); Koijen, Nijmn, nd Werker (2006); Milevky (2002);nd Milevky nd Young (2006 nd b).

5 2 income. She mu mke boh n e locion choice, regrding how much of her fund o hold in liquid finncil welh veru vrible pyou nnuiie, nd lo n e llocion deciion, where he elec how much of her porfolio o hold in riky ock veru rikle bond. Adding vrible nnuiy o he reiremen porfolio cn be preferred o direc invemen, ince he nnuiy py uperior expeced reurn due o he fc h urviving nnuin receive he diribued fund of hoe who die premurely. On he oher hnd, once he reiree buy n nnuiy, he nnuiy principl cn no longer be recovered. In conr o previou udie, we how h he reiree will no fully nnuiize immediely on reiremen, even if he lck beque moive. Rher, he will iniilly, prilly nnuiize nd hold percen of her remining finncil e in ock. A he ge, he will grdully buy more nnuiie nd opimlly replce ock wih bond in he porfolio. Nex, we how how hee choice repond o cpil mrke hock: in priculr, he reiree ock frcion vrie counercycliclly, finding h diverge from oher udie which fvor procyclicl buy-nd-hold regy. Our welfre nlyi indice h rericing reiree o imple phed wihdrwl pln i diincly ubopiml, uch h hey would hve o be given up o 40 percen more iniil welh, o leve hem well off wih vrible pyou nnuiie. The fc h vrible nnuiie cn ubnilly enhnce reiremen ecuriy i priculrly relevn o policymker, including hoe in he US who eek o implemen funded Peronl Reiremen Accoun (PRA) in revmped Socil Securiy yem. For innce Feldein nd Rnguelov (200) noe h reiree who purche immedie nnuiie inveed 60/40 in ock/bond could be niciped o fre beer hn under he curren Socil Securiy yem rule. Our nlyi ke hi ep furher by evluing he poenil for welfre gin

6 3 from endogenou opiml grdul nnuiizion nd dynmic e mngemen over he enire reiremen period. In wh follow, we fir offer brief lierure overview nd dicuion of he mechnic of vrible pyou nnuiie. Nex we derive nd dicu he opiml nnuiizion nd e llocion regy for be ce. 4 Subequenly, we exmine he welfre gin from expnding he e pce uing vrible nnuiie. A finl ecion conclude. Prior Sudie Yri (965) imporn udy on how o proec gin ouliving one money uggeed h rionl reiree lcking beque moive would nnuiize ll her e. In hi frmework, he inveor i expoed only o morliy rik nd ll nnuiie re firly priced from n curil ndpoin (oher ource of rik due o inere re, ock, nd inflion re omied). In n imporn recen exenion of h work, Dvidoff, Brown, nd Dimond (2005) conclude h reiree will ill fully nnuiize finncil welh in he preence of complee mrke if here i no beque moive, when he ne reurn on he nnuiy i greer hn h of he reference e. Pril nnuiizion could be opiml if he umpion bou complee mrke i relxed, or if he inveor h beque moive. The complee mrke umpion require h here will be erm life nnuiie vilble for purche h py ou ech fuure de. Bu ince rel-world nnuiy provider end o offer fixed pyou, everl reercher hve formuled more reliic dynmic porfolio choice model h incorpore minly fixed pyou nnuiie. Iniilly, hi lierure umed rericed nnuiizion regy, where ll welh mu be eiher fully nnuiized immediely on 4 There i no nlyic oluion for hi problem in eiher dicree or coninuou ime, o we employ numericl dynmic opimizion echnique in wh follow.

7 4 reiremen or ler; more recen udie hve endogenized he nnuiizion deciion, llowing reiree o grdully nnuiize ny ge nd repeedly. 5 Le well udied i he propoiion h nnuiy purcher my do beer if hey urn o vrible pyou nnuiie, ined of merely fixed pyou produc. Some reerch compre vrible pyou nnuiie o oher e cle on n eiher-or bi: for innce Blke, Cirn, nd Dowd (2003) how h equiy-linked vrible nnuiie hould ppel o mny reiree (cro rnge of rik preference), compred o eiher phed wihdrwl pln or fixed pyou nnuiy. Brown, Michell, nd Poerb (200) how h vrible pyou nnuiie cn hedge inflion rik beer hn equiy-linked nnuiie, by inveing he underlying fund in inflion-indexed bond. A he me ime, hough, hey find h pure equiy-linked nnuiie cn genere greer uiliy hn rel nnuiie for brod rnge of rik verion prmeer. Uing Mone Crlo imulion, Milevky (2002) compre rik/reurn chrceriic of vrible pyou nnuiy pyou rem wih fixed nd ecling nnuiie, nd he rgue h vrible pyou nnuiie my hedge inflion beer hn ecling nnuiie. While hee udie nlyze porfolio choice problem which include vrible pyou nnuiie, hey re ill rericive in imporn wy. For innce, wih repec o he e locion deciion, hey ll require h finncil welh mu be fully nnuiized, ye hi i unlikely in prcice. Regrding he nnuiizion iming deciion, mo prior udie require h nnuiie be bough immediely on reiremen; for innce Milevky nd Young (2002, 2006b) endogenize he iming of he nnuiy purche bu hey do no llow for pril nd grdul nnuiizion. Alo, few udie hve endogenized he opiml e llocion of he nnuiy porfolio, nd he one h do, reric he reiree o o-clled money mix pln, which i 5 For udie nlyzing porfolio choice problem wih grdul nnuiizion we refer he inereed reder o Milevky nd Young (2003), nd Horneff, Murer, nd Smo (2006, 2006b).

8 5 one-ime irreverible buy nd hold porfolio of fixed nd fully equiy-linked nnuiie. 6 In wh follow, we exend prior lierure in everl wy. Fir, we endogenize he reiree nnuiizion (or e locion) deciion, permiing her o dynmiclly nd grdully buy vrible pyou nnuiie by drwing down her finncil welh during reiremen. Accordingly, we do no force full nnuiizion, bu rher we llow boh pril nd grdul nnuiizion nd model vrible pyou nnuiie on n equl fooing wih ock nd bond. Second, we opimize he e llocion deciion e endogenouly nd dynmiclly, in boh he non-nnuiized finncil porfolio nd he vrible pyou nnuiy. Conequenly, he reiree my reblnce her e mix wihin he vrible pyou nnuiy opimlly ccording o e vrible, where prior work opimized only he iniil buy nd hold mix beween equiy-linked nd conn nnuiie. 7 Third, we conribue o he lierure by incorporing beque moive explicily, o chrcerize he rdeoff beween he nnuiy morliy credi nd he beque poenil of non-nnuiized finncil welh. Finlly, we exmine he eniiviy of key deciion o he reiree rik verion nd ring level of welh. The Mechnic of Vrible Pyou Annuiie Our work focue on vrible pyou nnuiie which re finncil conrc beween reiree nd n inurer. In reurn for he iniil nnuiy premium PR 0, he nnuiy buyer receive pre-pecified number of fund uni n in ech period { 2,,..., }. Here T i umed o be he reiree mximum poible urvivl ge, nd {,..., T } i her ochic de of deh, 6 See Chrup nd Milevky (2002), Koijen, Nijmn, nd Werker (2006), Browne, Milevky, nd Slibury (2003), nd Milevky nd Young (2006). None of hee endogenize he e locion nd llocion deciion dynmiclly, we do below. Browne e l. (2003) e welfre loe from ylized ce where only fixed/equiy linked nnuiie cn be exchnged for ech oher. 7 By opimizing he porfolio wihin he underlying nnuiy fund we byp he ylized umpion of Browne e l. (2003) in which only fixed nd equiy-linked nnuiie re exchnged.

9 6 diribued ccording o p which refer o he cumulive urvivl probbiliy h n x x individul of ge x urvive o ge x. In hi eup, he pymen PO pid o he nnuin in yer, condiionl on her being live, i PO Z n, where Z i he price of one nnuiy fund uni ime which evolve ccording o Z Z R, nd R i he gro reurn on invemen of he nnuiy fund in period. The gro reurn cn be ochic when he underlying fund inide he nnuiy wrpper re held in riky e. Thu, while he number of fund uni received i deerminiic, he pyou in monery erm i ochic when he nnuiy i inveed in riky e. If he inurnce pool i ufficienly lrge, he premium for he pyou proce () cn be clculed ccording o he curil principle of equivlence: PR () 0 N 0Z 0 where N T 0 x px n i he number of fund uni needed o bck he expeced fuure pyou rem n, for { 2,,..., }. Then he premium pid for vrible pyou nnuiy only depend on he curren price of he nnuiy fund, he fund uni rem, nd urvivl probbiliie. Simple lgebr yield PR 0 T x px PO R 0,, where, Z Z 0 i he cumulive R 0 reurn of he nnuiy fund. In oher word, he premium cn be inerpreed he um of dicouned nd urvivl probbiliy-weighed pyou. Rerrnging he premium funcion yield PR 0 T p R0, x PO x. 8 Thi expreion how h he dicoun re i higher hn he pure 8 If he nnuiy fund were inveed only in rikle bond, we would obin he clicl reul for conn pyou nnuiie: T x px PO PR0, where R f i he rikle growh re. R f

10 7 mrke reurn ince x p x ; hi morliy credi i ribuble o he e hred mong urviving member of he inurnce pool when member die. Accordingly, vrible pyou nnuiie cnno be repliced hrough direc invemen in ock nd bond. The Aumed Inere Re (AIR) nd Pyou. Pricing vrible pyou nnuiie doe no require ny umpion bou he funcionl form in generl, inofr he proce for he fund uni n i concerned. Bu in prcice, i i convenien o inroduce funcionl form for n which relie on he Aumed Inere Re (AIR), erm convenionlly found in inurnce pricing. I i lo ueful o derive how he AIR influence he diribuion of pyou over ime. 9 In inurnce erminology, he umed inere re refer o he hrinkge re of he number of fund uni over ime: AIR n n. Then he recurive evoluion of pyou i given by: R PO PO. (2) AIR I i nurl o refer he AIR he umed inere re becue he condiionl expeced PO E R pyou growh i E. To produce pyou rem conn in expecion, he PO AIR AIR mu be e excly equl o he expeced reurn on he e of he nnuiy fund E R. When hi i he ce, he reducion in he number of fund uni due o pyou excly offe heir expeced price incree due o invemen reurn. Hving fixed AIR implie h he number of fund uni evolve exponenilly (increing or decreing) ccording 9 A 4 percen AIR i common in he US indury (c.f. he Vngurd nd TIAA-CREF vrible pyou nnuiy webie); he US Nionl Aociion of Inurnce Commiioner require h he AIR no exceed nominl 5 percen.

11 8 o he rule n n ( AIR). The premium PR i hen: PR 0 Z n T 0 x p x AIR, (3) where T x p x AIR i imilr o he uul fixed nnuiy fcor, bu he rikle dicoun fcor i now replced by he AIR in he ce of he vrible pyou nnuiy. Following he ndrd pproch, he probbiliy diribuion for nnuiy pyou over ime my be pecified in uming h price of he nnuiy fund uni evolve ccording o muliplicive rndom wlk wih log-norml reurn; in oher word, Z Z R Z e, wih Z, where 0 i he nnuiy fund men log-reurn, i he ndrd deviion of he fund log-reurn, nd i n i.i.d. ndrd norml diribued hock. Thee umpion llow u o derive he probbiliy diribuion for nnuiy pyou over ime. Specificlly, from n (2), he pyou i equl o PO R ( AIR) wih he men nd ndrd deviion: E PO n ( AIR). Hence, PO i log-normlly diribued 2 ( ) 2 2 ( ) e, nd STDPO EPO e The -percenile of he pyou diribuion in i hen: P ( PO ) n ( AIR). (4) N ( ) e, (5) where N ( ) i he invere of he norml diribuion funcion. To illure how vrible nnuiy pymen cn vry given lernive e llocion, we refer o Figure. Here Pnel () rce he 0 h /90 h percenile of pymen oucome for

12 9 $00,000 immedie vrible nnuiy, given hree lernive ock llocion of 0, 60, nd 00 percen, repecively. We ume h he expeced rel porfolio reurn wih 00 (60) percen ock i 6 (4.4) percen wih ndrd deviion of 8 (0.8) percen; 0 he AIR i 4 percen nd he rikle re 2 percen, repecively. (The vrible nnuiy wih 0 percen ock nd he AIR e o 4 percen i equivlen o rikle nnuiy wih hrinking pyou.) The nnuiy rem cn be een o fll over ime, becue he AIR exceed he reurn of he underlying porfolio. The wo equiy-linked nnuiy rem hve diribuion kewed owrd higher benefi over ime, becue eing he AIR low compred o he expeced nnuiy fund reurn produce low pymen iniilly, wih higher benefi ler on. Figure here Pnel (b) of Figure illure how nnuiy pymen would vry over ime under lernive AIR of 2, 4, nd 6 percen, uming h he nnuiy fund i inveed 60 percen in equiie. I i cler h he higher he AIR, he lrger i he iniil pymen nd he lower i he ler pyou. For exmple, fer ge 75, he vrible nnuiy pymen uing he lowe AIR exceed he benefi pid uing he highe AIR. Accordingly, he reiree eeking high pyou erly in he reiremen period would elec n nnuiy wih n AIR higher hn he expeced fund reurn, nd vice ver. I i lo inereing h elecing low AIR reul in much higher diperion of pymen; neverhele, hi diperion i refleced minly in he upide poenil 0 The porfolio growh re i log-norml wih E [ R ] E[ R] ( ) R f nd STD[ R ] STD[ R]. From ln E[ R ] 0. 5( ) nd 2 he log-normliy umpion follow h 2 2 ln STD[ R ] E[ R ]. Milevky nd Young (2003) nd Koijen, Nijmn, nd Werker (2006) ue conn AIR of 4 percen; hi conform o Brown e l. (200) who noe h oberved AIR fll beween 3 nd 4 percen. Bodie nd Pendo (983) ue n AIR equl o he expeced reurn of he nnuiy fund while Milevky (2002) provide reul for vrible pyou nnuiie wih AIR rnging from 2 o 8 percen.

13 0 for pymen, while he downide rik i limied. 2 The Inveor Opimizion Problem Hving inroduced erminology, we now urn o decripion of he reiree dynmic porfolio choice problem when i include ock, bond, nd vrible life nnuiie. We ume he inveor i endowed wih iniil finncil e S 0, nd he i lo eniled o conn rel lifelong penion income of Y = Y (e.g. Socil Securiy nd/or compny penion). We poi h her preference cn be chrcerized uing conn relive rik verion (CRRA) uiliy funcion defined over ingle non-durble conumpion good. The reiree ubjecive urvivl probbiliy h he will urvive o de, given he i live, i denoed he reiree preference funcion i hen recurively defined by: p. In hi ce, C B V E p V p k, (6) wih erminl uiliy C BT V T ET k. Here C i her conumpion level ime ; i her coefficien of relive rik verion nd lo reflec her willingne o engge in ineremporl conumpion ubiuion; reflec her ime preference; nd B repreen her remining finncil welh deh. The rengh of her beque moive i repreened by he prmeer k. In hi frmework, ech yer he reiree mu decide how o lloce her curren ch on hnd, W, beween conumpion C, finncil welh S (which re her direc ock nd bond invemen), nd new purche of vrible life nnuiie PR. Hence he budge conrin ime i given by: 2 A will become ppren, he AIR h implicion for he iming of he nnuiizion nd he e llocion; below we explore how chnge in he AIR influence reul.

14 W S PR C. (7) In he nex period, ch on hnd i hen: where R R R L Y W S, (8) f denoe he hre of finncil welh S inveed in riky ock, bond growh re, nd R he riky ock reurn (umed o be i.i.d. log-normlly f R f denoe he rel diribued). 3 The um of nnul nnuiy pyou which he reiree ge from ll previouly purched nnuiie i L. If he reiree were o die +, he remining ee i given by B f f S R R R. We noe h neiher he nnuiy nor he penion income Y re vilble for beque. When he reiree purche vrible pyou life nnuiy in period = u, her immedie ingle premium i defined : 4 PR u Z T 65u 65u u ( u) nu ( u) u. (9) u p AIR In reurn for he premium pid = u, he receive for u nd unil he die, he nnuiy pymen rem repreened by Z ( u) n ( u). Thu he vlue of ll nnuiy pymen received ime + from ll nnuiie purched previouly u 0,..., L Z u) n ( u) u0 i given by: (. (0) In order o derive he ochic evoluion of L, we noe h he price of one nnuiy fund uni 3 I would be inereing o incorpore ochic inere re nd inflion in our nlyi, in Koijen, Nijmn, nd Werker (2006). However h udy ured model rcbiliy by mking nnuiizion exogenou, while our model endogenize he nnuiizion regy. Since we believe ock mrke rik nd longeviy rik re he mo imporn rik fcor in he reiree opiml porfolio choice, nd our model lredy include hree e vrible, o reduce he cure of dimenionliy we do no build in riky erm rucure nd/or inflion. 4 Lod nd dminirion fee for vrible pyou nnuiie in he US cn vry widely nd my be rher low. For exmple, Vngurd webie noe nnul dminirion co of bou 0.5 percen wih no commiion or fron lod for heir vrible pyou nnuiie. We omi modeling explici dminirion co wihou lo of generliy.

15 2 of he nnuiy purched ime u evolve : f f R R R u Z u Z ) ( ) (, wih ) ( u Z u, () where i he ock frcion ime inide he vrible nnuiie. Thu, in effec, he individul inve direcly in wo finncil e, rikle bond nd riky ock, well indirecly hrough he nnuiy wrpper. By ubiuing (9) nd () ino (0), one my verify h he recurive evoluion of L i given by: f f T R R R AIR p PR AIR L L (2) Finlly, he recurive ineremporl budge rericion cn hen be derived by ubiuing equion (7) nd (2) ino (8): Y R R AIR p PR AIR L PR C W R AIR p PR AIR L PR C W W f T f T (3) Beide he ineremporl budge rericion, he reiree i rericed from borrowing gin fuure penion income nd nnuiy pyou, well from elling hor poiion in nnuiie:.,,,,, 0 PR S (4) Her opimizion problem i hen ummrized by: 0 0 V T PR C,,, mx, (5) ubjec o rericion (3) nd (4). Numericl Soluion. Anlyicl oluion o hi ype of problem do no exi, o we olve i by bckwrd inducion hrough hree-dimenionl e pce L W,,. Alhough we ume

16 3 CRRA preference, ch on hnd W cnno be omied e vrible becue illiquid nnuiie re included in he nlyi. I i lo necery o include he um of curren nnuiy pyou L e vrible, becue once purched, nnuiie cn no longer be old. Finlly, he opiml policy depend on he reiree ge, becue he price of newly purched life nnuiie well he preen vlue of her remining penion income which below we refer o penion welh vry wih her ge. Wihou lo of generliy, we normlize he reiree penion benefi Y o, nd hen we expre ll oher ch flow vrible muliple of Y. There re hree e vrible, wo of which ch on hnd W nd nnuiy pyou L re coninuou, leving ge he only dicree e vrible. For ech grid poin, we clcule he opiml policy nd he vlue of he vlue funcion. Thereby, we compue he inegrl of he expecion operor in (6): E p V B p k p V p B k p( R ) dr (6) uing Guin qudrure inegrion. The opimizion i done by numericl conrined mximizion. We eime he policy funcion C W, L,, PR W, L,, W, L,, W, L, nd he vlue funcion V W L,, by cubic pline inerpolion. Opiml Policie for Rnge of Be Ce To illure he rnge of e locion nd llocion regie, we grph he opiml poiion for hypoheicl reiree of differen ge nd differen iniil finncil welh vlue (repreened muliple of penion income). In Figure 2, for innce, he reiree h modere relive rik verion ( 5 ), reonble re of ime preference ( 0. 96), no beque

17 4 moive (k = 0), nd no income from preexiing nnuiie (L = 0). 5 Pnel () nd (b) how h ll conumer will prilly nnuiize immediely, bu even wihou beque moive, mo will keep up o 3 percen of heir iniil e in finncil welh. Alo, mo reiree will popone full nnuiizion unil ler in life, unil round ge 88. The reon for pril nnuiizion i h our nlyi permi reiree o purche only nnuiie wih lifelong pyou. 6 Accordingly, we demonre below, full iniil nnuiizion impoed in erlier udie i ubopiml when uming reonble prmeer. 7 Figure 2 here The reon h he conumer hold on o her non-nnuiized welh erly in reiremen i h nnuiy pyou re low iniilly nd rie ler in life, becue he expeced nnuiy reurn exceed he AIR. Due o her ime preference, he reiree who eek o mooh conumpion will keep ome liquid finncil welh from which he cn wihdrw ch flexibly o compene for he low iniil nnuiy pymen. Of coure, while keeping ome finncil welh i opiml, i i lo worh noing h he morliy credi in he nnuiy lrgely compene for he rigidiy of he nnuiy pyou pern. 8 Figure 2 lo revel h he reiree iniilly hold ubnil frcion of her nnuiy in ock, nd in ome ce much 00 percen (Pnel c-d). In mo ce, he hre of her 5 Survivl probbiliie re ken from he US 996 Annuiy 2000 femle morliy ble o price nnuiie, nd from he correponding populion morliy ble o clcule uiliy. To reduce compuion ime, we reric he mximum ge o 00. The rikle re i e o 2 percen nd he AIR o 4 percen, repecively. Furher, we ume h riky ock h n expeced reurn of 6 percen nd voliliy of 8 percen. 6 Thi umpion cree mrke incompleene nnuiie h py ou only in one pecific yer, in Dvidoff e l. (2005) nd Kpur nd Orzg (999), re no vilble. Our pproch o modeling nnuiy mrke i more conien wih rel world nnuiy produc. 7 For exmple, udie by Browne, Milevky, nd Slibury (2003); Chrup nd Milevky (2002); nd Koijen, Nijmn, nd Werker (2006) ume incomplee nnuiy mrke bu exogenouly impoe immedie full nnuiizion he poin of reiremen. 8 Milevky nd Young (2006) derive he opiml iming o wich o wh hey cll n opiml money mix pln. Thi i full nnuiizion regy where reiremen, he conumer elec given frcion of fixed nd equiylinked pyou nnuiie. Poponing nnuiizion i opiml in hi eing becue he reiree i rericed o full nnuiizion nd cnno ler reblnce he frcion of fixed nd equiy-linked pyou nnuiie.

18 5 finncil welh in ock i lo high erly in reiremen, nd hen fll for older reiree. The opiml ock frcion in he vrible nnuiy decline older ge, well in he finncil porfolio. The reon i h he reiree penion welh h bond-like ri, nd i decline (in preen vlue) over ime. Accordingly, he reiree phe down her ock llocion (king ino ccoun boh finncil nd vrible nnuiy e), o o minin ol ock frcion when bond-like penion welh hrink over ge. Alo, he higher her finncil welh compred o penion welh, he higher i her deired bond expoure percen of her finncil welh if he i o rech her opimum invemen mix. Thi reul i imilr o hoe uggeed by recen lifecycle e llocion udie which omi nnuiie (Cocco, Gome, nd Menhou, 2005), nd i lo conform o finncil dvice recommending h ock frcion be decreed for older individul (Young, 2004). Figure 3 provide illurive opiml ock frcion in he combined nnuiy nd finncil welh porfolio, depiced funcion of he reiree ge nd ch on hnd (W). Pnel () confirm h he reiree overll frcion in ock decline wih ge. In Pnel (b), he opiml ock frcion vrie ccording o he reiree iniil welh, nd once gin i fll wih ge, more rpidly for higher level of iniil welh. I i inereing h hi pern implie he reiree will follow couner-cyclicl invemen regy, booing her ock frcion fer ber mrke nd decreing i in bull mrke. 9 Thi nd in hrp conr wih convenionl buynd-hold pro-cyclicl money mix regie, 20 ince in he ler, he ock frcion rie when he ock mrke ouperform he bond mrke, nd fll if he ock mrke dive. Figure 3 here 9 In mny counrie, inurnce conrc include gurnee which led o procyclicl invemen policy on behlf of he inured peron; we find he oppoie o be opiml for he CRRA inveor. 20 See Chrup nd Milevky (2002), Koijen, Nijmn, nd Werker (2006), Browne e l. (2003), nd Milevky nd Young (2003).

19 6 Sochic Anlyi. Nex we underke ochic nlyi o evlue how he reiree nnuiy purche, ock frcion, nd conumpion, would be expeced o evolve uming he followed he policy funcion derived bove. Thee compuion re bed on Mone Crlo imulion of 00,000 life cycle, nd hey ll begin by uming h her finncil welh ge 65 moun o 5 ime her penion income; ccordingly, ol ch on hnd (including penion income) i 6 ime penion income (W = 6). Furher, we ume h he hold no iniil vrible pyou nnuiie (L = 0). The rnge of imuled poibiliie pper in Figure 4, king ino ccoun he conumer opiml porfolio reblncing. The medin ph (50 h percenile) in Pnel () indice h he reiree will purche he bulk of her nnuiie ge 65 nd coninue o dd ddiionl uni herefer; ccordingly, her nnuiy welh rie wih ge nd fer ge 80 he i fully nnuiized. Ye becue oucome re ochic, he 0-90 h percenile illure likely vribiliy. In Pnel (b), we ee h he reiree permied o reblnce her vrible pyou nnuiy e llocion he ge will reduce her ock frcion long he medin ph. In ddiion, he diperion in he frcion of ock held rie over ime, rnging beween 50 nd 00 percen. Figure 4 here Pnel (c) of Figure 4 rce he diperion of conumpion ph wih ge, where conumpion i meured muliple of penion income. A he medin, he opimizing reiree boo conumpion by 40 percen over her penion income for he re of her life. The pyou could be ubnilly higher, doubling her conumpion, if he ock mrke behve exremely well when he i in her 90; converely he nnuiy would only boo conumpion by 0-20 percen, if he e in he vrible nnuiy perform very poorly (repreened in he figure by he 0 percenile probble minimum conumpion level). In oher word, wih vrible pyou

20 7 nnuiie, he reiree my deplee her finncil e no becue of longeviy rik, bu becue of ock mrke rik. Impc of he AIR. Above we noed h he AIR eleced by he inurer my hve coniderble influence on he nnuiy pyou pern, well he reiree e locion nd llocion deciion. Thi becue he reiree who buy n nnuiy i rericed o receiving cerin pyou pern deermined by he AIR. When he AIR i low, nnuiy ch flow re iniilly low bu rie ler in life; converely, high AIR produce higher iniil ch flow which cn be opimlly reinveed ccording o he relized e of nure, bu hi come he co of lower benefi le in life. I i herefore of inere o explore how chnging he AIR hpe he inveor opiml e llocion, nd o deermine how eniive he opiml oucome re o chnge in he AIR. To hi end, we reopimize he reiree e llocion policy for AIR e o 2, 4, nd 6 percen nd compre he Mone Crlo imulion bed on he opiml policie reuling in ech ce. Figure 5 repor reul, wih Pnel () providing expeced finncil welh holding. When he AIR i low, he reiree rein lrge finncil welh poiion erly in reiremen; for innce, given n AIR of 2 percen, he hold 25 percen of her ol welh in nonnnuiized form. A he AIR rie, he porion held in nnuiie lo incree; hu if he AIR i 6 percen, ll welh will be immediely nnuiized ge 65. Thi i becue nnuiy pymen re low when he AIR i low, nd o mooh conumpion, he reiree will rein more finncil welh. Converely, for higher AIR, he nnuiy benefi will be high, nd he cn inve porion of he benefi in newly purched nnuiie, if he doe no need he income. Figure 5 here Riing he AIR lo produce wi in he erm rucure of he reiree ock holding

21 8 frcion of combined finncil nd nnuiy welh (Pnel b). A reiremen, he hold ll of her e in ock, independenly of he AIR; herefer, he expeced ock frcion fll wih ge. Furhermore, if he AIR i low, hi decline in he ock frcion i lighly mller erly in reiremen, compred o higher AIR. Afer he reiree reche her mid-70, he re of decline i greer for lower AIR. Thi i becue low AIR require he reiree o reduce her ock frcion more over ime he pyou voliliy rie fer hn wih high AIR (ee Figure ). Finlly, higher AIR lo give he reiree mooher conumpion profile (Pnel c), he need no mi ou on he morliy credi by minining high level of liquid finncil welh erly in her reiremen period. Uiliy Gin from Vrible Pyou Annuiie Thu fr, we hve hown h he reiree will opimlly devoe dominn porion of her porfolio o vrible pyou nnuiie, nd he lo will hold ubnil equiie. To evlue how conequenil vrible pyou nnuiie re for he reiree wellbeing, we nex compre he reiree uiliy from hving cce o vrible nnuiie, wih hoe chievble from phed wihdrwl regy. We compre hee lernive by compuing he ddiionl moun of finncil welh he reiree would need in he rericed cenrio, h would mke her well off (provide he me expeced uiliy) in he unrericed cenrio. Techniclly, hi i ccomplihed by equing he expeced uiliy in he unrericed ce wih h of he rericed ce bu wih more iniil finncil welh. In priculr, we exmine wo phed wihdrwl regie, nmely () Pure Bond Sregy wih no Annuiie, nd (2) Sock/Bond Sregy wih no Annuiie. The fir ume h he inveor follow dynmic opiml phed wihdrwl pln, bu he cn hold no

22 9 nnuiie nd my inve only in bond. Thi clculion offer inigh ino welfre gin ocied wih cce o boh he equiy premium offered by he ock mrke nd he morliy credi offered by he nnuiy mrke. In he econd ce, he inveor i umed o opimlly inve her e in ock nd bond, bu he cnno buy nnuiie. Thi econd cenrio iole he uiliy vlue of cce o nnuiy mrke lone. The lef-hnd pnel of Tble ummrize reul for moderely rik-vere individul uch h depiced in Figure 2, for vrying iniil welh level. We lo llow beque moive in he l pnel. I i unequivoclly cler h nnuiie enhnce reiree wellbeing irrepecive of rik verion. Previouly, we hd howed h he moderely rik vere reiree wih modere iniil welh ( = 5 nd S 0 = 5) will iniilly inve ll her finncil nd nnuiy fund in ock. By conr, Column 2 indice wh hppen if h reiree were conrined o phed wihdrwl regy nd could inve only in bond. Specificlly, row 3 how h he would need o be given 34 percen more iniil finncil welh o mke her indifferen beween he pure bond phed wihdrwl pproch, veru hving n unconrined menu. Thi uiliy chnge i ubnil, nd i i comprble o vlue repored by Brown e l. (200) where heir benchmrk involved only immedie nd full nnuiizion. Vlue for oher level of rik verion re imilr (row nd 5). Tble here A reiree who lck vrible nnuiy bu h cce o boh ock nd bond mrke i relively beer off. Column 2, row 4, for innce, how h our moderely rik-vere reiree ill lloce her fund ubopimlly, bu ince he cn now benefi from he equiy premium, he require only 3 percen of her iniil finncil welh o be well off he would wih cce o vrible nnuiy. In fc, he difference beween row 3 nd 4 meure he vlue of

23 20 equiy premium which, in hi ce, exceed 20 percen of iniil welh. More generlly, he higher re iniil welh nd rik verion level, he more he reiree will vlue cce o he vrible nnuiy mrke. For exmple, he very rik-vere reiree ( = 0) wih high level of iniil finncil welh (S 0 = 0) who cn only inve in bond, would require 4 percen incree in her iniil welh o mke her well off wih he full rnge of porfolio choice. Of coure, if here i poiive beque moive, he relive ppel of nnuiie i lower. Bu even here, he vlue of cce o vrible nnuiy mrke ill rie wih iniil welh. The righ-hnd pnel of Tble provide reul for lernive AIR vrying level of rik verion, uming he iniil finncil welh i fixed (S 0 = 5). The overll concluion i h higher AIR re ppeling o more rik vere individul hving cce o vrible nnuiie. By conr, lower AIR i preferred by he more rik olern. The ce wih beque (row 7 nd 8) i comprble o he no-beque cenrio: he reiree will elec high AIR. In ny even, chnge in uiliy re mll which ugge h offering reiree choice over AIR (wihin reon) i unlikely o be conequenil. 2 The welfre gin from hi complex proce of dynmic porfolio reopimizion cn be compred o impler one-ime nnuiizion nd porfolio invemen deciion uch he conn 60/40 ock/bond mix menioned by Feldein nd Rnguelov (200). I i inereing h relively mll level of incremenl finncil welh would leve he reiree indifferen beween our opimlly dynmic reiremen regy nd imple, fixed 60/40 invemen mix. For reiree wih he rik verion nd iniil finncil welh level conidered bove, he impler rule produce uiliy loe equivlen o no more hn 8 percen of iniil finncil 2 Dellinger (2006) noe h ome vrible pyou nnuiie provider le he nnuin elec n AIR from cerin rnge he ime he nnuiy i purched. Bu uggeed by Jeffrey Brown (peronl communicion), permiing nnuiy purcher o freely chooe heir preferred AIR my foer dvere elecion: hu reiree believing hey hve inide informion bou heir own morliy propec migh elec heir AIR regiclly. I would be inereing o exmine hi hypohei empiriclly.

24 2 welh. Neverhele, oher imple rule (uch holding 00 percen of he nnuiy in ock) re much le rcive. Dicuion nd Concluion Thoe working o mnge heir money effecively in reiremen eek o benefi from cpil mrke reurn while no ouliving heir fund. Thi pper how how vrible pyou nnuiie cn enhnce he reiree porfolio choice e king ino ccoun boh ochic morliy nd riky ock reurn, priculrly if vrible nnuiie cn be bough endogenouly. Our work dvnce prior udie by llowing he inveor o nnuiize grdully; endogenizing he nnuiizion deciion men he hold nnuiie nd finncil welh imulneouly nd cn buy nnuiie incremenlly in reiremen. We lo endogenize he e llocion deciion boh inide nd ouide he vrible pyou nnuiie. And finlly, we e how pre-reiremen ving, income, nd beque moive ffec he nlyi. Regrding he e locion deciion, we find h he uiliy mximizing reiree will no nnuiize ll her fund reiremen, even if he h no beque moive. Thi reul conr wih prior udie which ugge h reiree lcking beque moive will nnuiize ll heir welh. For exmple, moderely rik vere nd moderely endowed inveor will hold up o one-qurer of her iniil welh in finncil e reiremen; herefer, he will drw from hi fund over ime o mooh her conumpion ime ph. Full nnuiizion i deferred o round ge 80 he le. Regrding her e llocion deciion, i i inereing h he reiree will opimlly inve mjoriy of her combined nnuiy nd finncil welh in ock, wih he ock frcion flling over he reiremen period. We lo how h he inveor will rec counercycliclly wih repec o ock mrke rend, o o reblnce her e llocion. By

25 22 conr, prior udie dvocing buy-nd-hold pproch o nnuiie ell reiree o follow procyclicl invemen regy. Chnge in he AIR (wihin reonble rnge) hve relively mll impc. L, our welfre nlyi indice h reiree who cn cce vrible nnuiie or ock vlue hi coniderbly, for welh equivlen mouning o much 40 percen of iniil finncil welh. Our finding re conequenil for reiree well inurer nd finncil plnner, ince we hve idenified ubnil degree of improvemen in conumer wellbeing reuling from cce o vrible nnuiie nd equiie in he reiremen porfolio. Policymker nd deigner of defined conribuion penion nd Individul Reiremen Accoun cn incorpore vrible pyou nnuiie in he reiremen mix n ppeling compromie beween he exreme of pure wihdrwl pln, on he one hnd, nd fixed nnuiy, on he oher. I i worh noing h he US Congre recenly recommended h life cycle fund become he deful for penion pricipn ccumulion phe, bu i offered no guidnce on how he fund hould be mnged fer reiremen (Purcell 2006). Likewie, he Preiden Commiion o Srenghen Socil Securiy focued on wy o help worker ccumule e, bu i remined ilen regrding wy o help reiree mnge he e decumulion proce (Cogn nd Michell, 2003). Our nlyi implie h inegred mehod o finnce reiremen offer enormou poenil for boh invemen mnger nd inurnce compnie o cpure he $5 rillion of Bby Boomer money in moion.

26 23 Reference Blke D., A. Cirn nd K. Dowd, 2003, Penionmeric 2: Sochic Penion Pln Deign during he Diribuion Phe, Inurnce: Mhemic nd Economic, Vol. 33, Iue, p Bodie, Z., nd J. Pendo, 983, Reiremen Annuiy Deign in n Inflionry Clime, in Z. Bodie nd J. Shoven, ed., Finncil Apec of he U.S. Penion Syem, Univeriy of Chicgo Pre: Chicgo, IL. Brown, J.R., O.S. Michell, nd J.M. Poerb, 200, The Role of Rel Annuiie nd Indexed Bond in n Individul Accoun Reiremen Progrm, in J.Y. Cmpbell nd M. Feldein, ed., Rik Apec of Invemen-Bed Socil Securiy Reform, Univeriy of Chicgo Pre: Chicgo, IL, p Browne, S., M.A. Milevky, T.S. Slibury, 2003, Ae Allocion nd he Liquidiy Premium for Illiquid Annuiie, Journl of Rik & Inurnce,Vol. 70, Iue 3, p Chrup, N., nd M.A. Milevky, 2002, Opiml e llocion in life nnuiie: noe, Inurnce: Mhemic nd Economic, Vol. 30, Iue 2, p Cocco, J.F., F.J. Gome, nd P.J. Menhou, 2005, Conumpion nd Porfolio Choice over he Life Cycle, Review of Finncil Sudie, 8, p Cogn, J.nd O. S. Michell Perpecive from he Preiden Commiion on Socil Securiy Reform. Journl of Economic Perpecive. 7(2). Spring. Dvidoff T., J. Brown nd P. Dimond, 2005, Annuiie nd Individul Welfre, Americn Economic Review, 95 (5), p Dellinger, J.K., 2006, The Hndbook of Vrible Income Annuiie, John Wiley & Son, Inc., Hoboken, New Jerey, USA. Du, I., R. Murer, nd O.S. Michell, 2005, Being on Deh nd Cpil Mrke in Reiremen: A Shorfll Rik Anlyi of Life Annuiie veru Phed Wihdrwl Pln, Finncil Service Review 4, 2005, p Feldein, M. nd E. Rnguelov Individul Rik in n Invemen-Bed Socil Securiy Syem. Americn Economic Review 9(4): Gerrrd, R., S. Hbermn, nd E. Vign, 2004, Opiml Invemen Choice Po-reiremen in Defined Conribuion Penion Scheme, Inurnce: Mhemic nd Economic, Volume 35, Iue 2, p Horneff, W.J., R. Murer, O.S. Michell, nd I. Du, 2006, Opimizing he Reiremen Porfolio: Ae Allocion, Annuiizion, nd Rik Averion, NBER Working Pper Horneff, W.J., R. Murer, nd M. Smo, 2006, Life-Cycle Ae Allocion wih Annuiy Mrke: I Longeviy Inurnce Good Del?, Working Pper, Goehe Univeriy, Frnkfur. Horneff, W.J., R. Murer, nd M. Smo, 2006b, Opiml Grdul Annuiizion: Qunifying he Co of Swiching o Annuiie, Working Pper, Goehe Univeriy, Frnkfur. Kpln, P.D., 2006, Ae Allocion wih Annuiie for Reiremen Income Mngemen, Journl of Welh Mngmen, Vol. 8, No. 4, (Spring 2006), p Kpur, S. nd J. Orzg, 999, A Porfolio Approch o Invemen nd Annuiizion During Reiremen, Working Pper, Birbeck College, Univeriy of London, UK.

27 24 Kingon, G. nd S. Thorp, 2005, Annuiizion nd Ae Allocion wih HARA Uiliy, Journl of Penion Economic nd Finnce, 4, p Koijen, R.S., T. Nijmn, nd B.J.M. Werker, 2006, Opiml Porfolio Choice wih Annuiizion, Working Pper, Tilburg Univeriy - Cener for Economic Reerch. McKiney, 2006, Crcking he Conumer Reiremen Code. McKiney & Co. Milevky, M.A., 998, Opiml e llocion owrd he end of he life cycle: To nnuiize or no o nnuiize?, Journl of Rik & Inurnce, Vol. 65, Iue 3, p Milevky, M.A., 2002, How o Compleely Avoid Ouliving Your Money: An Inroducion o Vrible Pyou Annuiie for Reiring Cndin, Working Pper, IFID Cenre, The Schulich School of Buine, York Univeriy, Torono, Cnd. Milevky, M.A. nd V. Young, 2002, Opiml Ae Allocion nd he Rel Opion o Dely Annuiizion, Working Pper, IFID Cenre, Schulich School of Buine, York Univeriy, Torono, Cnd. Milevky, M.A. nd V. Young, 2006, Annuiizion nd Ae Allocion, in Journl of Economic Dynmic nd Conrol forhcoming. Milevky, M.A. nd V. Young, 2006b, The Timing of Annuiizion: Invemen Dominnce nd Morliy Rik, Inurnce: Mhemic nd Economic, forhcoming. Milevky, M.A., K.S. Moore, nd V. Young, Ae Allocion nd Annuiy-Purche Sregie o Minimize he Probbiliy of Finncil Ruin, Mhemicl Finnce, 6, (4), Michell, O., J. Poerb, M. Wrhwky, nd J. Brown, 999, New Evidence on he Money Worh of Individul Annuiie, Americn Economic Review, 89, p Purcell, Prick, 2006, Summry of he Penion Proecion Ac of Congreionl Reerch Service, Domeic Socil Policy Diviion. Working Pper. US Congre. Sbile G., 2003, Opiml Timing of he Annuiy Purche: A Combined Sochic Conrol nd Opiml Sopping Problem, Working Pper, Univeri degli Sudi L Spienz di Rom, Rome, Ily. Yri M., 965, Uncerin Lifeime, Life Inurnce, nd he Theory of he Conumer, Review of Economic Sudie, 32, p Young, L., 2004, Fund Th Adju A The Yer Go By. Buine Week, July 26.

28 25 Pnel () Pnel (b) 0h/90h Percenile of Annuiy Pymen ($000) Age 0% Sock 60% Sock 00% Sock 0%/90% Percenile ($000) of Annuiy Pymen AIR = 2% Age AIR = 4% AIR = 6% Figure : Pymen Percenile for Alernive Ae Allocion in he Annuiy nd Vrying AIR. Thi illurion ume n iniil premium of $00,000 nd urvivl probbiliie from he US 996 Annuiy 2000 morliy ble for femle; compuion re bed on equion (5). Pnel : Yerly rel nnuiy fund reurn for he 60 (00) percen ock llocion re i.i.d. log-norml wih men 4.4 (6) percen nd ndrd deviion 0.8 (8) percen. The rel rikle re nd he AIR re e o 2 nd 4 percen, repecively. Pnel b: The nnuiy fund i umed o be inveed in 60/40 ock bond porfolio wih he men 4.4 percen nd ndrd deviion 0.8 percen; he AIR i e o 2, 4, nd 6 percen, repecively.

29 26 () Annuiy Welh (b) Finncil Welh (c) (d) Figure 2: Illurive Opiml Dynmic Ae Locion nd Allocion given Sock, Bond, nd Vrible Pyou Annuiie: Vriion by Age nd Ch on Hnd. Thee figure repreen opiml policie which vry funcion of ch on hnd (W) nd ge; in ll pnel, no prior nnuiizion i umed (L=0). In Pnel (), he ge-65 reiree wih ch on hnd W = 6 would pend 90 percen of her ch on nnuiie nd he re o finncil invemen (Pnel b). Her opiml ock frcion i 00 percen wihin he vrible nnuiy (Pnel c) well in finncil welh (Pnel d). Noe: Clculion re bed on bckwrd opimizion of he vlue funcion given in (5). The reiree h CRRA uiliy wih = 5, = 0.96, nd no beque moive (k = 0). Survivl probbiliie re from he US 996 Annuiy 2000 morliy ble for femle o price nnuiie nd from he correponding populion morliy ble o clcule uiliy. Yerly rel ock reurn re i.i.d. log-norml diribued wih he men 6 percen nd ndrd deviion 8 percen. The rel inere re nd he AIR re e o 2 nd 4 percen, repecively.

30 27 () How Opiml Sock Holding Frcion of Combined Annuiy nd Finncil Welh Porfolio Vry by Age nd Ch on Hnd: No Prior Annuiy Aumed (i.e. L = 0) (b) How Opiml Sock Holding Frcion of Combined Annuiy nd Finncil Welh Porfolio Vry by Age nd Iniil Welh: Single Level of Ch on Hnd Aumed (W = 6) Figure 3: Illurive Opiml Sock Frcion in he Combined Annuiy nd Finncil Welh Porfolio. Thee figure repreen he relionhip beween he reiree ge nd he opiml ock frcion in her combined nnuiy nd finncil e holding. The ler i defined he ock frcion of her expeced nnuiy welh PV (he preen vlue of remining expeced nnuiy pyou ) plu he ock frcion of her finncil welh S, percen of finncil plu nnuiy welh ( S PV ) ( S PV ). Pnel () how how hi ock frcion vrie wih ge nd ch on hnd W; Pnel (b) how how i vrie wih ge nd pre-exiing nnuiy pymen L. In boh pnel, no prior nnuiizion i umed (L=0). Noe: See Figure 2 for clculion deil.

31 28 () (b) (c) Figure 4: Diperion of Annuiy, Tol Sock, nd Conumpion Ph: Be Ce. Pnel () depic he diperion of he nnuiized frcion of welh PV/(S + PV) i vrie wih ge; Pnel (b) repor imilr ge-diperion percenile for he ock frcion of he combined nnuiy nd finncil welh porfolio [= ( S PV ) ( S PV ) ]. Pnel (c) how imilr diperion for conumpion, meured muliple of penion income. Noe: See Figure 2 for deil. Compuion re bed on 00,000 Mone Crlo imulion uing he opiml policie from Figure 3 nd iniil welh of 5 ime penion income (S 0 = 5).

32 29 () (b) (c) Figure 5: Impc of Chnging he AIR on () he Expeced Shre of Finncil Welh, (b)tol Sock Shre, nd (c)conumpion. Thi figure depic he reul from olving he opimizion problem in (5) for AIR of 2, 4, nd 6 percen repecively. Noe: See Figure 4 for deil.

33 30 Tble : Required Incree in Finncil Welh (%) o Mke Reiree Indifferen Beween he Specified Phed Wihdrwl Sregy veru Gining Acce o Vrible Pyou Annuiie: Impc of Differen Preference, Iniil Finncil Welh, nd AIR. Iniil Finncil Welh AIR Low (S0 = 2) Modere (S0 = 5) High (S0 = 0) Low (AIR = 2%) Modere (AIR = 4%) Preference Inveible Ae () (2) (3) (4) (5) (6) High (AIR = 6%) () Low Rik Averion () Bond (= 2, k = 0) (2) Bond, Sock (b) Modere Rik Averion (3) Bond (= 5, k = 0) (4) Bond, Sock (c) High Rik Averion (5) Bond (= 0, k = 0) (6) Bond, Sock (d) Modere Beque Moive (7) Bond ( = 5, k = 2) (8) Bond, Sock Noe: The ble enrie indice he moun of finncil welh he reiree would hve o be given o mke her indifferen beween hving cce o vrible nnuiie well ock nd bond, veru hving rericed choice e indiced in he Tble. For exmple, he inveor wih relive rik verion = 5, no beque moive (k = 0), nd n iniil finncil welh of S0 = 5, who cn inve ju in bond, would be indifferen beween hving cce o he vrible nnuiy mrke, veru being compened wih n ddiionl 34 percen of finncil welh (row b3, col. 2). The impc of finncil welh i evlued fixing he AIR of he vrible nnuiy o AIR = 4%. The impc of he AIR i clculed fixing iniil finncil welh S0 = 5.

34 CFS Working Pper Serie: No. Auhor() Tile 2007/20 Be Cno Romn Kräul Elecronic Trding Syem nd Inrdy Non- Liner Dynmic: An Exminion of he FTSE 00 Ch nd Fuure Reurn 2007/9 Mri Kch-Hrouounin Erik Theien Compeiion beween Exchnge: Euronex veru Xer 2007/8 Güner W. Beck Volker Wielnd 2007/7 Güner W. Beck Volker Wielnd 2007/6 Silvio Colroi Andre Zghini 2007/5 Annmri Lurdi Olivi S. Michell 2007/4 Jen Boivin Mrc P. Ginnoni Ilin Mihov Money in Monery Policy Deign under Unceriny: A Forml Chrcerizion of ECB- Syle Cro-Checking Money in Monery Policy Deign under Unceriny: The Two-Pillr Phillip Curve veru ECB-Syle Cro-Checking Grdulim, Trnprency nd Improved Operionl Frmework: A Look he Overnigh Voliliy Trnmiion Finncil Liercy nd Reiremen Prepredne: Evidence nd Implicion for Finncil Educion Progrm Sicky Price nd Monery Policy: Evidence from Diggreged U.S. D 2007/3 Virgiliu Midrign Menu Co, Muli-Produc Firm, nd Aggrege Flucuion 2007/2 Michel Woodford Robuly Opiml Monery Policy wih Ner- Rionl Expecion 2007/ Lr E. O. Svenon Noh Willim Byein nd Adpive Opiml Policy under Model Unceriny Copie of working pper cn be downloded hp://