What Will Happen to Financial Markets When the Baby Boomers Retire?

Transcription

1 Wha Will Happen o Financial Markes When he Baby Boomers Reire? Robin Brooks April 000 Absrac This paper explores he effecs of demographic change on financial asse reurns, using a calibraed overlapping generaions model in which agens hold risky socks and safe bonds. The model is solved numerically and yields porfolio behavior in which agens shif from socks o bonds as hey approach reiremen. They behave in his manner because of human capial, a non-raded asse hey implicily hold over he life cycle. Using he model o simulae a baby boom and bus shows ha a he urning poin of he demographic shif when boom urns o bus a large cohor of old boomers mus rade wih a smaller cohor of young invesors. This demographic imbalance resuls in excess demand for he safe asse, which pushes he bond reurn down relaive o he reurn on equiy o clear he bond marke. As a resul he reurn on baby boomers reiremen savings is significanly below ha of heir parens, an effec baby boomers canno diversify away. From a policy poin of view his resul is imporan because much of he debae over reform of public pension sysems akes he hisorical disribuion of asse reurns and he equiy premium as given. JEL Classificaion Numbers: E7, G, G, H55 Keywords: equiy premium, populaion aging, pension reform Inernaional Moneary Fund, Research Deparmen, h Sree, N.W., Washingon, D.C., 043; Tel: (0) ; Fax: (0) rbrooks@imf.org. Special hanks are due o Chrisopher Sims. I am also graeful o Luis Cubeddu, Wouer Den Haan, Hamid Faruqee, Peer Heller, Peer Isard, Ivailo Izvorski, Narayana Kocherlakoa, Douglas Laxon, and Eswar Prasad for many helpful discussions and commens. Any errors are mine. Since his is a preliminary draf, please do no quoe wihou permission.

2 - -. Inroducion This paper explores he effecs of demographic change on financial asse reurns. The aging of he baby boomers and speculaion over possible effecs on financial markes have raised he profile of his issue, boh in he financial press and in academic circles. Broadly speaking, here are wo opposing views. The firs holds ha reiring baby boomers will be selling heir asses o a smaller generaion of young invesors. This will drive asse prices down, leaving baby boomers wih a smaller nes-egg han anicipaed. The second mainains ha forward-looking financial markes are pricing asses o incorporae he aging of he boomer generaion. As a resul, here will no be a marke meldown when he baby boomers reire. This paper bridges he gap beween hese opposing argumens by asking he following quesion: can demographic change, which is slow-moving and predicable, have a significan impac on financial markes ha are raional and forward-looking? I presens a model in which raional, forward-looking agens of differen ages rade in financial asses and uses his framework o simulae a baby boom-baby bus of he kind observed in many developed counries over he pos-war period. The main finding of he paper is ha changes in he age disribuion have significan effecs on asse reurns, even when invesors are raional and forward-looking, and ha hese effecs have imporan implicaions for he welfare of baby boomers and surrounding cohors. The model used is a saionary overlapping generaions model wih wo asses: shares of ownership in risky capial and a riskless one-period bond in zero ne supply. Agens live for four periods: childhood, young working-age, old working-age, and reiremen. In childhood he agen relies on his paren for consumpion and is no a decision maker. In boh workingage periods he supplies labor inelasically and earns a wage. In reiremen he consumes down his savings, here being no bequess. The model has wo sources of aggregae uncerainy, a echnology shock o producion and random populaion growh, and is solved numerically using he parameerized expecaions approach. Alhough agens degree of risk aversion is consan over ime, hey inves as if increasingly risk averse wih age: young workers shor he riskless asse in order o hold equiy, while old workers hold mosly he riskless asse. This porfolio behavior sems from he risk and life cycle characerisics of a nonradable asse, human capial, which agens implicily hold. Young workers anicipae receiving wage income in old working-age, so ha nex period consumpion does no depend on savings alone. In addiion, since he reurn on capial is posiively bu imperfecly correlaed wih wage income, holding equiy will diversify he effecs of an adverse echnology shock. In conras, old workers invesmen decision reflecs he fac ha nex period consumpion is ou of savings alone. As a resul hey largely eliminae consumpion risk by invesing mosly in he riskless asse. Using he model o simulae a baby boom-baby bus calibraed o mach hisorical and projeced populaion rends yields he following effecs. Firs, here is an aggregae saving For examples of aricles in he financial press, see Passell (996) and Colvin (997).

3 - 3 - effec on asse reurns as changes in he age disribuion affec aggregae saving and herefore he real ineres rae. During he boom his effec will push up reurns on capial and he riskless asse, a resul of higher aggregae consumpion because of relaively large cohors of children. During he bus aggregae saving is relaively high, pushing reurns on boh asses down. Second, he reurn differenial beween socks and bonds changes over he demographic shif. This effec derives from he fac ha agens shif from socks o bonds as hey age. A he urning poin of he boom-bus, when a large boomer cohor of old workers rades wih a smaller cohor of young invesors, his invesmen behavior generaes excess demand for he riskless asse. As a resul he bond reurn falls sharply relaive o he reurn on capial and he reurn differenial rises. Third, wage income moves inversely wih he size of he labor force, even hough capial accumulaion in he model is endogenous. These effecs are significan. During he baby boom he expeced one-period reurns on socks and bonds rise above heir seady saes by 4.8 and 9.4 percen respecively. On an annualized basis his means ha he expeced reurn on capial rises above is seady sae of 4.54 percen by 4 basis poins, while he riskfree rae is 8 basis poins higher han is equilibrium level of 4.5 percen. During he baby bus he expeced one-period reurn on capial falls below is seady sae by up o 4.5 percen, while he riskfree rae falls by up o. percen below equilibrium. This ranslaes respecively ino up o 4 and 39 basis poins on an annual basis. The greaer sensiiviy of he riskfree rae o he boom-bus is significan for older invesors who wan o minimize consumpion risk in reiremen. These magniudes are also subsanial relaive o he impac of oher fundamenals on asse reurns. And while hey are small relaive o he recen run-up in sock prices, his simulaion exercise holds nondemographic fundamenals consan over he boom-bus and ignores he possibiliy of a speculaive bubble. These effecs go agains baby boomers, especially hose in he ail-end of he baby boom. Bu are baby boomers worse off? The simulaed boom-bus consiss of wo boom followed by wo bus periods, so ha he firs boomer cohor has relaively more children han he second. This pushes consumpion per head of he firs boomer cohor below equilibrium in parenhood. If uiliy of young workers is addiively separable in heir consumpion and ha of heir children, his effec dominaes adverse asse marke effecs. This means ha lifeime uiliy of he firs boomer cohor is below seady sae, while ha of he second is above because posiive consumpion effecs in parenhood more han offse he effecs of lower reurns on reiremen savings. In oher words, adverse asse marke effecs are second-order. The fac ha raising children is cosly is more imporan. This resul is reversed if young workers uiliy is defined over household consumpion. This specificaion reduces he effecive weigh in lifeime uiliy of consumpion when young, so ha adverse asse marke effecs dominae earlier posiive consumpion effecs. Since he asse marke implicaions of he model are qualiaively unchanged across specificaions, and since i is no obvious how o model parens uiliy, he reversal of he welfare resul is ineresing. I also poins o a weakness of he model, since humans derive uiliy no only from consumpion, bu also from having children. As such, he welfare implicaions of he model should be aken wih a grain of sal.

4 - 4 - The resul ha he reurn differenial beween socks and bonds widens a he urning poin of he boom-bus is arguably he mos ineresing resul of he paper. As noed above i derives from wo characerisics of he model: he shif from socks o bonds over he life cycle, and he limied number of agens rading a any one poin in ime. A criic of his resul migh say: if baby boomers know hey will reire in roughly 0 years, and ha hey may face adverse asse markes a ha poin, why no lock in wealh ahead of ime? Bu his is exacly wha boomer cohors do in he model. In old working-age hey inves in he riskless asse, in order o reduce consumpion risk down he road. I is precisely his behavior, and he demographic imbalance, which drives down he bond relaive o he sock reurn. A number of papers have recenly noed ha a risk associaed wih swiching o individual reiremen accouns is ha invesors may be subjec o adverse movemens in asse prices ha persis over ime. 3 This paper makes his argumen explici, by presening a framework in which invesors are subjec o cohor-specific risk ha is linked o demographic change. Furhermore he model corresponds o a world in which pay-as-you-go pension sysems have been replaced wih individual reiremen accouns, wih no governmen regulaion over agens porfolio decisions. Thus he key resul of he paper from a policy poin of view is ha he hisorical disribuion of asse reurns may be inappropriae for compuing he gains o invesors from swiching o individual accouns, because he baby boomers are only now approaching reiremen. Indeed he model suggess ha baby boomers will earn reurns ha are subsanially below reurns o previous generaions. 4 This argumen should no be undersood as an endorsemen of pay-as-you-go sysems, since a defined-benefi pension sysem can be fully funded. Indeed, he paper augmens he model wih a simple pay-as-yougo pension scheme and shows ha such a sysem does no eliminae cohor-specific demographic risk. Insead he paper underlines he role of governmen as an infiniely-lived agen, one ha can insure agens agains cohor-specific risk by adjusing governmen borrowing over ime o sabilize he riskfree rae. Of course such a move would reflec a poliical consensus, since i involves ransfers of wealh across generaions. There are a number of objecions o he approach in his paper. I presens a closed economy model, which ignores invesors abiliy o insure agains cohor-specific risk by holding an inernaionally diversified porfolio. Though his is an imporan consideraion, mos counries wih significan asse markes have experienced pos-war demographic shifs similar o he US. In effec, he model should be hough of as represening he developed world as a whole. Of course, here are regions wih very differen age disribuions, such as 3 See for example Heller (998) and Hemming (998). 4 For papers ha use he hisorical disribuion of asse reurns o compue gains from swiching o individual reiremen accouns see, for example, MaCurdy and Shoven (99) and Feldsein and Ranguelova (998).

5 - 5 - Africa or Lain America. However, i is unlikely ha asses in hese markes have risk-reurn characerisics aracive o baby boomers preparing for reiremen. 5 The model also ignores bequess. Berganino (998) finds ha, using daa from he Survey of Consumer Finances, inergeneraional ransfers are of minor imporance o households. Fewer han 5 percen of households repor having ever received a subsanial inheriance, rus, or ransfer. Of hose ha did, he median value in 995 dollars was abou $7,000 per spouse, or abou 60 percen of he median annual income. This evidence suggess ha inergeneraional ransfers are of minor imporance o mos households, especially relaive o wage income, in deermining he life-cycle pah of asse holdings. A furher shorcoming of he model is ha i does no replicae he equiy premium observed in he daa. In large par his is because wha is called equiy in he model is no a levered asse, in he sense ha here is no corporae deb. Adjusing for his he model suppors a Sharpe raio of roughly en percen, he same order of magniude as Soresleen, Telmer, and Yaron (997) whose overlapping generaions model has agen-specific, persisen income shocks. In conras Consaninides, Donaldson, and Mehra (998) generae an equiy premium in an overlapping generaions model in which young invesors are borrowing consrained, which underscores ha his model feaures no such marke imperfecion. The implici assumpion is ha he behavior of relaive reurns over a demographic shif can be adequaely characerized in he absence of such feaures. 6 The nex secion presens a brief overview of relaed papers. Secion hree presens he model, wih subsequen secions devoed o equilibrium condiions and he soluion mehod. Secion eigh discusses calibraion of he model, while secion nine characerizes he soluion. Secion en uses he model soluion o simulae he effecs of a baby boom-baby bus on asse reurns. Secion concludes.. The Paper vis-à-vis he Lieraure The raionale for agens wih consan risk aversion o subsiue from equiy o bonds as hey age has been previously explored by Jaganahan and Kocherlakoa (996). 7 They make he poin ha invesors have fewer working years ahead of hem as hey age. Assuming ha mos 5 Aanasio and Violane (000) and Brooks (000) explore he implicaions of hisorical and projeced populaion rends for inernaional capial flows. For a discussion of home counry bias in porfolio allocaion see French and Poerba (99). 6 For a review of recen papers on he equiy premium see Kocherlakoa (996). 7 A more recen discussion of he role of labor income as a non-raded asse and is impac on invesmen behavior over he life cycle can be found in Campbell, Cocco, Gomes, and Maenhou (999).

6 - 6 - invesors labor incomes are poorly correlaed wih sock reurns, hey demonsrae ha i is raional for agens o shif he composiion of heir financial wealh from socks o less risky asses as hey grow older. This paper demonsraes ha his behavior obains even when labor income and sock reurns are posiively correlaed, a resul of he ineracion of he life cycle feaures of he model wih he risk-reurn characerisics of he financial asses. 8 A number of empirical papers have recenly explored he link beween changes in he age disribuion and financial markes. Berganino (998) presens evidence linking he level of real sock prices in he US o ime series for aggregae demand of financial asses, which are derived from Survey of Consumer Finances daa on porfolio composiion by age. Brooks (998) finds ha real sock and bond prices across developed counries are posiively relaed o he share of he populaion ha is middle-aged, using he cross-secion dimension of he daa o conrol for unobserved fundamenals. In conras Poerba (998) fails o find a significan relaionship beween reurns on a range of asses and differen measures of he age disribuion. Fundamenally, he problem ha bedevils all empirical work is ha he effecive number of observaions is small. As a resul saisical ess have limied power. Perhaps his is he sronges argumen for he simulaion exercise ha follows. 3. The Model The represenaive agen lives for four periods: childhood, young working-age, old workingage, and reiremen. In childhood he agen makes no decisions of his own, wih consumpion, c 0, deermined by he paren. In young working-age he agen inelasically supplies one uni of labor and earns a wage w. Ou of wage income he consumes c for himself and assigns (+n )c 0 o his offspring where n is he period cohor growh rae. In addiion he may hold shares of ownership in risky capial, s e, and inves s b in a riskless one-period bond in zero ne supply. The budge consrain of a period young worker is hus: 0 ( + n ) c + c + se + sb = w () The agen reaches old working-age in period +. He again supplies one uni of labor inelasically, earning w +, and receives income from sock and bond holdings chosen as a young worker. Ou of oal income he consumes only for himself since his children are now self-sufficien. He consumes c +, invess s e+ in risky capial, and s b+ in he safe asse. The budge consrain of a period + old worker is herefore: ( + r ) s + ( r ) s c se+ + sb+ = w + + e+ e f b () 8 The approach in his paper conrass wih Bakshi and Chen (994) who hypohesize ha agens become more risk averse wih age. They find ha he average age of he US populaion is posiively correlaed wih fuure excess reurns on socks over reasury bills.

7 - 7 - Here r e+ is he reurn on equiy held from period ino period +, while r f is he reurn on he riskless asse, held from period ino period +. The agen reaches reiremen in period +. He receives no wage income and consumes down his savings since here are no bequess. The budge consrain of a period + reiree herefore amouns o a decision rule for consumpion: c ( + r ) ( ) + s + + r + s = e e + f b (3) Preferences are described by an addiively separable uiliy funcion. The expeced lifeime uiliy of a period young worker is given by: V = c 0 θ c θ c θ c ( ) ( ) ( ) ( ) ( ) n + + β + β θ θ E E θ θ 3 θ (4) The subjecive discoun facor is given by β where 0<β<, while θ is he coefficien of consan relaive risk aversion. The represenaive agen does no become more risk averse wih age. This raher convenional specificaion of preferences is adoped o focus aenion on he ineracion of he life cycle dimension of he model wih he represenaive agen s invesmen decision. The age disribuion in period consis of N - young workers, N - old workers, and N -3 reirees. The period cohor of children is deermined according o N =(+n )N -, where n is he realizaion of a saionary cohor growh shock. Table describes he evoluion of he age disribuion over ime: Table : The Age Disribuion over Time Period Children Young Workers Old Workers Reirees N N - N - N -3 + N + N N - N - + N + N + N N - Oupu is generaed according o a consan reurns o scale, neoclassical producion funcion: Y = K ( A L ) α α (5) where α deermines he share of oupu rewarded o capial. K - is generaed in period - by he invesmen decisions of young and old workers. A is he realizaion of a saionary laboraugmening echnology shock. L consiss of young and old workers so ha L = N - + N -. Facor markes are efficien so ha capial and labor are rewarded heir marginal producs. r e = αk α ( A L ) δ α (6)

8 - 8 - w α α α ( α ) K A L = (7) where δ is he depreciaion rae. In equilibrium he capial sock used in period + producion is deermined by share holdings of young and old workers chosen in period : K = N se + N se (8) The equilibrium condiion ha he riskless one-period bond is in zero ne supply implies ha bond holdings of young and old workers sum o zero. In oher words eiher young workers borrow from heir parens or hey lend o hem. 0 = N sb + N sb (9) Imposing hese equilibrium condiions and aggregaing across he budge consrains of young and old workers and reirees in period yields he social resource consrain: C ( ) α δ K K ( A L ) α + = K (0) Oupu in period is divided ino aggregae consumpion and gross invesmen Equilibrium Maximizing expeced uiliy period young workers choose c 0, c, s e, and s b such ha c = 0 c [ + + ] θ θ ( c ) βe ( c ) ( + r ) = e () () 9 The model has wo sources of aggregae uncerainy: he echnology shock o producion and he populaion growh rae. An invesor deciding on how much equiy o hold going forward ino period + does no know he realizaion of he period + echnology shock ha deermines r e+. In conras he reurn on he riskless asse in period + is know in period. Hence i is denoed r f, o emphasize ha i is in agens period informaion se. Similarly hough K eners producion in period +, i is deermined by invesmen decisions made in period and is herefore also in agens period informaion se. The sochasic populaion growh rae represens aggregae uncerainy over cohor size. An unexpecedly large cohor of children has repercussions for agens consumpion-invesmen decision, as hey reac o a larger consumpion requiremen in young working-age and he larger labor force in subsequen periods.

9 - 9 - θ [ + ] θ ( c ) = β ( + r ) E ( c ) f (3) 0 ( + n ) c + c + se + sb = w (4) are saisfied, aking facor reurns and he reurn on he riskless asse as given. Period old workers choose c, s e, and s b such ha [ + + ] 3 θ + rf E [ c + ] θ 3 θ ( c ) βe ( c ) ( + r ) = e θ ( c ) = β ( ) ( ) (5) (6) c ( + r ) ( ) e se + + rf s + se + sb = w + b (7) are saisfied, again aking facor reurns and he reurn on he riskless asse as given. Consumpion of he period reiree cohor is given by: c 3 ( + r ) ( ) e se + + rf s = b (8) () hrough (8) represen a sysem of eigh equaions ha characerize individual consumpion (c 0, c, c, c 3 ) and invesmen behavior (s e, s b, s e, s b ) for given wage and reurn disribuions. In equilibrium, consumpion and invesmen decision rules ha maximize expeced uiliy a he individual level mus be consisen wih condiions (8) and (9), which are he equilibrium condiions for he sock and bond markes. 5. The Sae Variables The model has only wo acive decision makers a any : young and old workers. Boh make heir consumpion-invesmen decision based on oal wealh, which for young workers is simply wage income. w = w (9) Toal wealh of old workers consiss of wage income, in addiion o sock and bond holdings and reurns. w ( + r ) s ( ) + + r s = w + e e f b (0) w and w describe he disribuion of wealh across he wo decision making cohors in period and represen endogenous sae variables. In addiion he age disribuion wih he excepion of he reiree cohor (which will no live o see he nex period) represens an

10 - 0 - exogenous sae variable. Thus he se of period sae variables ha describes he model looking ahead o period + is: Θ = [ w w, N, N, N ], 6. The Soluion Mehod The model is solved using he parameerized expecaions approach (PEA). This approach is described in deail in Den Haan and Marce (990) who solve a one-good sochasic growh model. More recenly Izvorski (997) uses he PEA o solve a model wih heergeneous agens and incomplee markes. The essence of he PEA is ha he condiional expecaions on he righ-hand sides of (), (3), (5), and (6) each represen funcions g:r + 5 R + of he sae variables in Θ. This insigh is used o subsiue each condiional expecaion wih a funcion Π(Θ,ψ), where Π (he funcional form) and ψ (he vecor of parameers) will be chosen o make Π(Θ,ψ) as close as possible o g. Using he PEA (), (3), (5), and (6) can be rewrien as: () θ ( c ) = βψ( Θ, τ ) θ ( c ) ( s ) = β ( + r ) Ω( Θ, γ ) e θ ( c ) = βλ( Θ, ξ ) θ ( c ) ( s ) = β ( + r ) Γ( Θ, ω) e f f () (3) (4) (5) Given wo sequences of lengh T for he echnology shock and he age disribuion, assuming saring values for w and w, and given values for τ, γ, ξ, and ω, i is possible o solve ou he model for T periods. The PEA begins wih exacly his sep. I draws wo sequences of lengh T for he echnology shock and he age disribuion (boh sequences are drawn only once), and solves ou he model for T periods. The PEA hen urns o fiing he condiional expecaions in (), (3), (5), and (6), finding he coefficiens in he polynomial approximaions ha minimize he mean squared error beween he acual realizaion in + and he expecaion a of ha realizaion based on Θ. 0 0 The condiional expecaions in (3) and (6) have no been parameerized in he radiional manner. Boh equaions have been muliplied by funcions of he respecive equiy holdings. This modificaion is based on Izvorski (997) and addresses an indeerminacy in he sysem of Euler equaions and aggregae equilibrium condiions ha arises in models ha solve for equilibrium holdings of wo or more asses.

11 - - The paricular version of he PEA implemened here proceeds o fi he condiional expecaions in a sep by sep approach. For he condiional expecaion in () a non-linear leas squares esimaion for τ is performed: min τ T T = θ [( c + ) ( + r e + ) Ψ( Θ, τ )] (6) A he n h ieraion a new value τ n+ is generaed according o τ n+ = λτ n + (-λ)τ e where τ e is he esimae for τ from he non-linear leas squares esimaion. Given τ n+, γ n, ξ n, and ω n, he sysem is solved ou again for T periods and he algorihm proceeds o fi he oher condiional expecaions in urn. This procedure is repeaed unil he algorihm reaches a fixed poin in τ, γ, ξ, and ω. 7. Accuracy of he Soluion Mehod The basic inuiion of he PEA is o approximae condiional expecaions of period + using he informaion se available o agens a. For a successful approximaion he PEA predicion error should herefore be orhogonal o agens informaion se a. This inuiion lies behind an accuracy es developed by Den Haan and Marce (994). The accuracy es checks for orhogonaliy beween he Euler equaion residuals and a vecor v of variables in agens period informaion se. θ * ( c+ ) ( + re+ ) Ψ( Θ, τ ) θ * ( c+ ) ( se ) Ω( Θ, γ ) 3 θ * ( c+ ) ( + re+ ) Λ( Θ, ξ ) 3 θ * ( c ) ( s ) Γ( Θ, ω ) + e = ε + (6) where τ *, γ *, ξ *, and ω * are he PEA parameer esimaes a convergence. For any m vecor v in agens period informaion se, he saisic The implemenaion of he PEA follows Den Haan and Marce (990). Raher han performing a compuaionally expensive non-linear leas squares esimaion o find τ e, i akes a firs-order approximaion of Ψ(Θ,τ n ) around τ n. Rearranging erms τ e is hen he coefficien vecor in an OLS regression.

12 - - G = ( T ) T = ' T + = ' ( ε v ) ( ε v )( ε v ) T + T + T ε + = ( v ) T (7) has an asympoic χ disribuion wih degrees of freedom given by 4 m. The vecor of sae variables Θ is chosen for v. 8. Model Parameerizaion The paper presens simulaions resuls for four versions of he model. The firs drops he riskfree asse from he analysis, focusing on he consumpion-saving decision under uncerainy. The second adds he riskfree asse back in, focusing on he full consumpioninvesmen decision. Comparing hese wo specificaions he main quesion is: how does adding he riskfree asse change agen behavior and model characerisics. The hird model augmens he second specificaion wih a simple pay-as-you-go pension sysem. This changes he represenaive agen s budge consrains in young and old working-age and reiremen o: 0 ( + n ) c + c + s e + s b = ( π ) w (8) ( ) w + ( + r ) s + ( r ) s c + π + (9) + se+ + sb+ = + e+ e f b N + N c π (30) + ( + r + ) + ( + ) e se + + rf sb+ + w = N The accuracy es is implemened in he following manner. Given τ *, γ *, ξ *, and ω * a convergence, he model is simulaed N imes, each ime for differen draws of he echnology shock and he age disribuion. For hese N simulaions, he frequency wih which he G saisic is greaer han he criical value of he 95 h percenile of a χ 0 is repored. If he percenage of G saisics above he criical value of he 95 h percenile is subsanially greaer han five percen, his is evidence agains accuracy of he soluion.

13 - 3 - where π is he payroll ax levied on young and old workers and he las erm on he righhand-side in (30) is he per capia reiremen benefi. I is worh noing ha he reiremen benefi is no a riskfree source of income as i loads on echnology via he wage. In effec his exension gives reirees access o wage income, raising he quesion wheher heir invesmen behavior changes as a resul. A furher quesion is wheher his form of pension sysem proecs invesors from cohor-specific demographic risk. The fourh specificaion is based on a modificaion of he uiliy funcion, which is addiively separable in c 0 and c. This says ha young workers care abou per capia raher han household consumpion. Since i is no obvious wheher agens uiliy is separable in heir consumpion and ha of heir children, a specificaion in which young workers care abou household consumpion is also explored. 3 V = θ ( ) ) θ + ( ) 3 n c c ( c ) θ + βe + θ + β E + θ θ (4 ) Since (4 ) leaves open how o allocae household consumpion beween young workers and heir children, i is assumed ha c 0 = c. Wih preferences represened by (4 ) equaion () drops away while () and (3) are replaced by: [ + ] θ + rf E [ c + ] θ θ ( n ) c ) = E ( c ) ( + r ) + e+ β (3) θ ( + n ) c ) = β ( ) ( ) Saring values for w and w, and for τ, γ, ξ, and ω, are chosen based on a version of he model wih perfec foresigh, in which he represenaive agen makes only a consumpionsaving decision (he porfolio problem drops away). 4 In ha framework, given he 3 The specificaion of preferences is easily exended o include a parameer a which parens discoun he uiliy of dependen offspring. In addiion he relaive weigh of consumpion over he life cycle in (4) is consan, which could be modified by giving each period is own weigh in lifeime uiliy. 4 The model is parameerized o reflec he fac ha each period corresponds o roughly weny years. The subjecive discoun rae β is se a 0.6, which corresponds o a pure rae of ime preference ρ of abou.5 percen per year where β=/(+ρ) 0. The coefficien of relaive risk aversion θ is se a so ha period uiliy akes he form u(c) = ln(c). The share of oupu ha is on average rewarded o capial α is se a 0.3. Depreciaion is assumed o occur a a rae of.5 percen per year so ha δ= =0.4. The payroll ax rae π is se a 0. in he model wih social securiy. The saionary echnology shock follows lna = φlna - + ε where φ = 0, ε ~ N(0,σ ε ) and σ ε = 0.. The period child cohor is given by lnn = ρlnn - + v wih v ~ N(0,σ v ), ρ = 0.99, and σ v = 0.0. Period populaion growh is hen backed ou according o N = (+n )N -. These parameer choices are in line wih he recen overlapping (coninued ) (3)

14 - 4 - preference and producion parameers, seady sae values of he choice variables and facor reurns are given in Table. 5 Table : Seady Sae Values for he Model wihou Uncerainy c 0 = c c c 3 s e s e K r e w w (4) (4SS ) (4 ) Wih agens preferences described by (4) young workers consume a large fracion of heir wage income, saving under five percen. Having o raise children means hey pospone saving largely unil old working-age. While he wage is he only source of income for young workers i makes up jus under 90 percen of oal income for old workers, wih he remainder generaed by savings from young working-age. Reiremen income is generaed purely ou of savings wih per capia reiremen consumpion well above ha of young and old workers. Adding pay-as-you-go social securiy o he model reduces he need o save for reiremen so ha wealh accumulaion by young and old workers falls. As a resul he equilibrium capial sock falls, pushing he reurn on capial up and he wage down (he endogenous sae variables w and w reflec afer-ax income). The seady sae reiremen benefi amouns o 0.9, corresponding o a 40 percen replacemen rae and a reurn on payroll conribuions of zero since he model is saionary. Swiching o (4 ) generaes subsanial capial deepening. This is because he savings rae of young workers increases o jus above 0 percen when hey care abou household raher han per capia consumpion, due o he concaviy of period uiliy. This capial deepening makes agens beer off, wih lifeime income subsanially above ha when preferences are represened by (4). 9. Simulaion Resuls Table 3 characerizes he soluion o he model based on (4) wih only he consumpionsaving problem, presening he descripive saisics of he model. 6 These are generaed generaions lieraure. Higgins (994) chooses α=0.33, β=0.54, δ=0.7, and θ=0.77 in he conex of a 3 period model. Higgins and Williamson (996) appear o have he same parameerizaion, wih he excepion ha θ is se a. Aanasio and Violane (000), in a model wih moraliy risk, se α=0.36, β=.0 per annum, δ=0.5 per annum, and θ=. 5 In he perfec foresigh case, he model has only one sae variable, he capial-labor raio. Imposing he equilibrium condiion ha ne saving equal ne invesmen, a non-linear equaion solver csolve.m wrien by Chrisopher Sims for Malab is used o solve for he seady sae. 6 In his specificaion he wo firs-order condiions relaing o bond holdings fall away, leaving only he condiional expecaions in () and (5) o be parameerized. The se of period sae variables remains Θ.

15 - 5 - using he coefficiens on Θ a convergence and simulaing he model for he sequence of echnology shocks and demographics used for he PEA. 7 The descripive saisics in Table 3 for he firs-order approximaion o condiional expecaions use an approximaion of he following form Ψ ( Θ τ ) = exp( τ + τ w + τ w + τ N + τ N τ N ), (33) in he case of () for example. In oher words he algorihm esimaes six coefficiens per Euler equaion. Table 3 also repors simulaion resuls based on a second-order approximaion, which is based on a reduced ensor basis ha omis erms of he hird order and higher. In ha case, he algorihm esimaes coefficiens per parameerized expecaion. Finally Table 3 repors simulaion resuls based on a hird-order approximaion, again based on a reduced ensor basis ha omis all erms of he fourh order and higher. 8 Table 3: Descripive Saisics of he Consumpion-Saving Model Firs-Order Second-Order Third-Order µ σ µ σ µ σ c c c s e s e r e w w K L Score 6.9% 5.6% 5.4% The resuls in Table 3 do no differ percepibly from he seady sae values of he perfec foresigh framework. Young workers consume a large fracion of heir wage income, saving jus under five percen on average. Saving for reiremen is posponed unil old working-age when agens save jus under fory percen of oal income. I is also clear ha he linear approximaion o expecaions performs well, wih model characerisics barely changing 7 The values for τ, γ, ξ, and ω are aken o have converged when (τ n - τ e )/ τ n, (γ n - γ e )/ γ n, (ξ n - ξ e )/ ξ n, and (ω n - ω e )/ ω n are each less han To generae he PEA soluion o he model he sequences for echnology and he age disribuion are drawn for lengh T=000. The same draws are used across specificaions for easier comparison. The resuls remain qualiaively unchanged for sequences of much greaer lengh.

16 - 6 - when moving o higher order approximaions. The score saisic, he percenage of G saisics for 500 simulaions ha are above he criical value of he 95 h percenile of a χ 0 disribuion, indicaes ha he firs order approximaion is successful in fiing expecaions and ha gains from higher order approximaions are small. The formaion of expecaions appears o be largely linear in Θ. Table 4 repors he correlaions marix associaed wih he hird-order approximaion resuls repored in able 3: Table 4: Correlaions based on Third-Order Approximaion c c c 3 r e w w c c c r e w w Table 4 shows ha he correlaion beween wage income and reurns on capial is posiive as expeced given he Cobb-Douglas producion echnology. Ye he degree of correlaion is far from perfec. Young workers would like o hold more equiy because he low degree of correlaion offers a measure of diversificaion agains echnology shocks. However heir abiliy o accumulae capial is limied so ha he correlaion beween wage income and oal income o old workers is almos perfec.

17 - 7 - Table 5 repors he descripive saisics for he full consumpion-invesmen problem. The firs, second, and hird-order approximaions o he condiional expecaions correspond o hose discussed above. Table 5: Descripive Saisics of he Consumpion-Invesmen Model Firs-Order Second-Order Third-Order µ σ µ σ µ σ c c c s e s e s b s b r e r f w w r e - r f K L Sharpe Score 39.7% 7.8% 5.9% Comparing ables 3 and 5 yields a number of insighs. Inroducing he riskless asse changes he paern of asse accumulaion in a fundamenal way. Wihou he riskless asse young workers hold almos no equiy because heir consumpion needs are large while raising children. Once he riskless asse is inroduced young workers shor he bond in order o inves in equiy. Equiy holdings of young workers rise o above fory percen of heir wage income (for he second- and hird-order approximaions). Old workers lend o young workers because his gives hem a safe income sream in reiremen. By deciding o hold almos no equiy hey largely eliminae consumpion risk in reiremen. Wih he inroducion of he riskless asse agens ac as if hey are increasingly risk averse wih age, alhough he coefficien of relaive risk aversion is consan over he life cycle. Young workers shor he riskless asse, invesing in risky equiy, while old workers hold primarily he riskfree bond. This behavior comes from he ineracion of he life cycle feaures of he model wih he risk-reurn characerisics of socks and bonds. Young workers have an addiional working period ahead of hem so ha nex period consumpion does no depend enirely on savings. This makes invesing in high yielding bu risky equiy aracive, especially since labor and ineres income are imperfecly correlaed. Bu since consumpion requiremens for young workers are high hey mus borrow o hold shares of ownership in capial, generaing supply of he riskless asse. In conras old workers reiremen consumpion depends enirely on savings. Faced wih he choice of invesing in risky equiy

18 - 8 - or safe bonds hey hold bonds almos exclusively, generaing demand for he riskless asse. Minimizing consumpion risk in reiremen dominaes old workers porfolio decision. The annualized reurn on capial in he model is abou 4.56 percen. The inroducion of he riskless asse produces sligh capial deepening because old workers do no dives hemselves enirely of equiy. The annualized reurn on he riskfree asse amouns o 4.53 percen so ha he model fails o generae an equiy premium of he order observed hisorically in he US and oher developed markes. One reason for his is ha he shares of ownership in capial do no correspond o equiy as i is commonly undersood since equiy in he model is no a levered asse (here are no corporae bonds). The reurn on capial going o shareholders in he model should be hough of as subsuming equiy, corporae bonds, and oher forms of lending and ownership. Though he implicaions for asse reurns are no of he order observed in he daa, he Sharpe raio suppored by he model is around en percen, an order of magniude comparable o Soresleen, Telmer, and Yaron (997). This resul is encouraging since he model feaures only aggregae uncerainy, while he model of Soresleen, Telmer, and Yaron includes persisen agen-specific shocks. The Sharpe raio benchmark associaed wih Mehra and Presco (985) is 37 percen. Wih agens making he full consumpion-invesmen decision expecaions become nonlinear in Θ. While he score saisic improves only marginally when going from he firs- o a second-order approximaion wihou he riskless asse, i improves by a facor of five when going from he firs- o a second-order approximaion and differences in porfolio allocaion beween young and old workers become more pronounced. Looking o he hird-order approximaion i appears ha he key non-lineariy in he formaion of expecaions is capured by second-order erms. Table 6 presens he correlaions marix associaed wih Table 5: Table 6: Correlaions based on Third-Order Approximaion using (4) c c c 3 r f r e w w r e - r f c c c r f r e w w r e - r f Table 6 illusraes he benefis o young workers of being able o borrow and hold equiy. The correlaion of consumpion wih wage income in old working-age falls relaive o he model wihou bonds. And while he correlaion of reiremen consumpion wih he reurn on capial is in he absence of he riskless asse, i is in he consumpioninvesmen framework. Essenially agens have eliminaed heir exposure o echnology shocks by holding mosly he riskless asse. Table 6 also illusraes ha he riskfree rae and

19 - 9 - he reurn on equiy end o move ogeher, because effecs relaed o changes in aggregae saving dominae effecs ha drive reurns on boh asses in opposie direcions. Table 7 repors he descripive saisics of he consumpion-invesmen model augmened wih pay-as-you-go social securiy. The firs, second, and hird-order approximaions o he condiional expecaions are as above. Table 7: Descripive Saisics of he Consumpion-Invesmen Model wih Social Securiy Firs-Order Second-Order Third-Order µ σ µ σ µ σ c c c s e s e s b s b r e r f w w b r e - r f K L Sharpe Score 9.8% 6.7% 5.5% Porfolio behavior of he represenaive agen does no change qualiaively wih he inroducion of he pay-as-you-go pension sysem. Invesors sill shif financial wealh from socks o bonds as hey grow older, possibly because he reiremen benefi is no riskfree. If his is he case i underlines how imporan consumpion risk in reiremen is for old workers. Of course his finding is condiional on he naure of he reiremen benefi. If he benefi were riskfree, here migh be subsanial repercussions for porfolio behavior. Table 8 presens he correlaions marix for he model wih social securiy:

20 - 0 - Table 8: Correlaions based on Third-Order Approximaion c c c 3 r f r e w b r e r f c c c r f r e w b r e - r f Though he per capia reiremen benefi is negaively correlaed wih he reurn on he riskless asse, i loads on wo facors: echnology and he number of workers per reiree. This makes i possible ha baby boomers who have few children may in fac receive a low per capia benefi because he workforce is relaively small when hey reire. Adding social securiy leaves he risk-reurn characerisics of he model broadly unchanged, he smaller capial sock aside. The Sharpe raio suppored by he model remains around en percen. The non-lineariy in expecaions appears o be capured by he second-order approximaion, as he score saisic improves only marginally when adding hird-order erms. Table 9 repors he descripive saisics of he consumpion-invesmen model wih young workers deriving uiliy from household consumpion. The firs, second, and hird-order approximaions o condiional expecaions are as above. Table 9: Descripive Saisics of he Consumpion-Invesmen Model using (4 ) Firs-Order Second-Order Third-Order µ σ µ σ µ σ c c c s e s e s b s b r e r f w w r e - r f K L Sharpe Score 36.4% 7.5% 6.%

21 - - Changing he way young workers hink abou heir offspring relaive o hemselves leaves agen behavior unchanged. Young workers shor he riskless asse in order o hold equiy. Old workers are willing lenders because his allows hem o almos eliminae heir exposure o echnology shocks in reiremen. The key feaure of he model, ha agens make porfolio decisions as hough hey are increasingly risk averse wih age, coninues o hold when preferences are described by (4 ). Capial deepening aside, he risk-reurn characerisics of he model are broadly sable. Though he average reurn differenial beween socks and bonds falls, he Sharpe raio is roughly unchanged, approaching en percen for he hirdorder approximaion. The non-lineariy in expecaions appears o be capured by he secondorder approximaion, as he score saisic improves only marginally when adding hird-order erms. Table 0 presens he correlaions marix associaed wih Table 9: Table 0: Correlaions based on he Third-Order Approximaion c c c 3 r f r e w w r e r f c c c r f r e w w 0.90 r e - r f Across specificaions a number of sylized facs emerge. Wih he inroducion of he riskless asse, agens shor he bond when young in order o inves in risky equiy. Old workers are willing lenders because hey wan o minimize consumpion risk in reiremen. In shor, he invesmen horizon maers, wih agens making porfolio decisions as if hey are increasingly risk averse wih age. This feaure of he model is robus o changes in preferences or adding a pay-as-you-go pension scheme. In addiion he asse marke characerisics of he model are sable across specificaions. 0. Simulaing a Baby Boom-Baby Bus This secion explores he implicaions of hisorical and projeced populaion rends for asse reurns. Figure plos annualized cohor growh raes for Norh America (NA), he European Union (EU) and a combined region for he period 950 o 50, illusraing he impac of he pos-war baby boom and bus. 9 In he 0 year period cenered around 950 annualized 9 For he period 950 o 990 he daa are aken from he Unied Naions World Populaion Prospecs: The 99 Revision, which provides daa on he age disribuion (in 5 year agegroups) for a large cross-secion of counries. The demographic daa from 995 onward are impued using counry-specific cohor growh raes (for he same 5 year age-groups) from he World Bank World Populaion Projecions: The Ediion, using he Unied Naions (coninued )

22 - - cohor growh in NA amouned o 0.5 percen, 0.6 percen in he EU, and 0.6 percen in he combined region. In he period cenered around 970, he peak of he baby boom, cohor growh rose o jus under percen per year in NA and jus under percen in he EU, for a growh rae of abou.3 percen in he combined region. In he period around 990, he peak of he baby bus, cohor growh urns negaive. I is 0.7 percen per annum in NA, 0.9 percen in he EU, and 0.8 percen in he combined region. In he period around 00 cohor growh remains negaive in he EU (-0.7 percen per year) bu rebounds o 0. percen in NA, for an average of 0. percen in he combined region. Cohor growh in he very long run is zero in boh regions, a feaure of he World Bank populaion projecions ha should be inerpreed as an equilibrium condiion raher han forecas. Figure : Cohor Growh.0%.5% Annualized Growh Rae.0% 0.5% 0.0% EU NA EU+NA % -.0% The simulaed baby boom and bus differs in several respecs from hisorical and projeced populaion rends. The simulaed ransiion assumes ha cohor growh in equilibrium is zero. The baby boom begins in 950 when cohor growh rises o hree percen per period (0.5 percen on an annualized basis) and lass unil 970 when period cohor growh is again hree percen. The baby bus begins in 990 when cohor growh is negaive a 3 percen per period and lass unil 00 when period cohor growh is again 3 percen. Thereafer cohor growh is again in equilibrium a zero. This demographic shif is saionary (and symmeric) since populaion size before and afer he ransiion are he same. As a resul he focus is purely on he ransiional dynamics, no on daa as a base. This daa splice is necessary because he Unied Naions projecions go only o 05, while he World Bank daa goes back only o 990. In 990, he base year for he projecions, he daa from boh sources are very similar. Cohor growh raes are generaed by aggregaing he 5-year age groups in he original daa ino 0-year age groups ha span 0-9, 0-39, 40-59, 60+. Cohor growh is hen he growh rae of he child cohor over he young worker cohor over a 0 year inerval.

23 - 3 - he observed feriliy decline and associaed capial deepening effecs. 0 The shock also appears small in absolue erms compared wih observed and projeced populaion rends. However he simulaed shock is large relaive o σ v =0.0, which corresponds o he sandard deviaion of US cohor growh prior o 950. Finally he boom and bus in he simulaed ransiion las for wo periods each. As a resul he simulaion will have cohors ha are early baby boomers, and cohors ha are in he ail-end of he baby boom. Figure plos cohors growh raes over he simulaed ransiion, while Figure 3 plos he age shares. Figure : Annualized Cohor Growh 0.0% 0.5% 0.0% 0.05% 0.00% gpop -0.05% -0.0% % -0.0% Figure 3: Populaion Age Shares In Percen 7% 6% 6% 5% 5% 4% 4% 3% kids young w old w reirees 0 Cohor growh raes in 90 and 930 were 6 and percen respecively in he US. Meanwhile he Wold Bank projecions sugges ha cohor growh will reach zero around 030.

24 - 4 - Beginning wih he baseline model summarized in Table 5, Figure 4 presens annualized reurns on capial and he riskless asse over he ransiion. Reurns on capial and he riskless asse move broadly ogeher, rising during he early sages of he boom when relaively large cohors of children push up aggregae consumpion, reducing aggregae saving. Once boomer cohors ener he workforce, aggregae saving rises, an effec ha is reinforced as boom urns o bus and consumpion of children becomes relaively less imporan. Figure 4: Annualized Equiy and Bond Reurns 4.8% 4.7% 4.6% 4.5% re rf 4.4% 4.3% 4.% 4.% Figure 4 hus illusraes ha demographic shifs lead o changes in aggregae saving over ime, causing he real ineres rae o vary. As he real ineres rae changes i pushes reurns on socks and bonds in he same direcion. The movemen in reurns on he risky asse are mirrored in wage income, which moves inversely wih he size of he labor force. This reflecs changes in he capial-labor raio, which falls during he boom sage of he ransiion and rises during he bus, even hough capial formaion is endogenous o he model. Figure 5 presens wage income over he shif. Boomer cohors receive lower wage income han in equilibrium. The simulaions are based on he PEA soluion, using hird-order approximaions o parameerize expecaions. Raher han allowing he echnology shock o vary randomly, as is he case when he model is solved, he echnology shock is now held fixed a one, in essence holding all non-demographic fundamenals consan. This resul is robus for θ= and θ=0.5.