Saving and Investing fo Ealy Retiement: A Theoetical Analysis Emmanuel Fahi MIT Stavos Panageas Whaton Fist Vesion: Mach, 23 This Vesion: Januay, 25 E. Fahi: MIT Depatment of Economics, 5 Memoial Dive, Cambidge MA 242. S. Panageas: The Whaton School, Univ. of Pennsylvania, Finance Depatment, 2326 Steinbeg Hall-Dietich Hall 362 Locust Walk, Philadelphia, PA 94. Contact: panageas@whaton.upenn.edu. Tel: 25 746 496. We would like to thank Andy Abel, Geoge Maios Angeletos, Olivie Blanchad, Ricado Caballeo, John Campbell, Geoge Constantinides (the discussant at Fonties of Finance 25), Domenico Cuoco, Pete Diamond, Phil Dybvig, Jey Hausman, Hong Liu, Anna Pavlova, Jim Poteba, John Rust, Kent Smettes, Nick Souleles, Dimiti Vayanos, Luis Viceia, Ivan Wening, Mak Westefield, and paticipants at the MIT Maco, Whaton Finance Faculty Lunches and paticinpants of the Fonties of Finance 25 confeence fo useful discussions and comments. We would also like to thank Jianfeng Yu fo excellent eseach assistance. All eos ae ous.
Abstact We study optimal consumption and potfolio choice in a famewok whee investos save fo ealy etiement and assume that agents can adjust thei labo supply only though an ievesible choice of thei etiement time. We obtain closed fom solutions and analyze the joint behavio of etiement time, potfolio choice, and consumption. Investing fo ealy etiement tends to incease savings and stock maket exposue, and educe the maginal popensity to consume out of accumulated pesonal wealth. Contay to common intuition, pio to etiement an investo might find it optimal to incease the popotion of financial wealth held in stocks as she ages, even when she eceives a constant income steam and the investment oppotunity set is also constant. This is paticulaly tue when the wealth of the investo inceases apidly due to stong stock maket pefomance, as was the case in the late 99 s. We also show that the model can potentially povide a ational explanation fo the paadoxical fact that some investos saving fo etiement chose to incease thei alloaction to stocks as the maket was booming and educe it theeafte. JEL Codes: G, E2 Keywods: Continuous time, Optimal Stopping, Retiement, Life Cycle Potfolio Choice, Savings, Maginal Popensity to Consume, Indivisible Labo 2
Intoduction Two yeas ago, when the stock maket was soaing, 4(k) s wee swelling and (..) ealy etiement seemed an attainable goal. All you had to do was invest that big job-hopping pay incease in a maket that poduced double-digit gains like clockwok, and you could stat taking leisuely stolls down easy steet at the ipe old age of, say, 55. (Business Week 3 Decembe 2) The damatic ise of the stock maket between 995 and 2 significantly inceased the popotion of wokes opting fo ealy etiement (Gustman and Steinmeie [22]). The above quote fom Business Week demonstates the easoning behind the decision to etie ealy: A booming stock maket aises the amount of funds available fo etiement and allows a lage faction of the population to exit the wokfoce pematuely. As a matte of fact, etiement savings seem to be one of the pimay motivations behind investing in the stock maket fo most individuals. Accodingly, thee is an inceased need to undestand the inteactions between optimal etiement, potfolio choice, and savings, especially in light of the gowing populaity of 4(k) etiement plans. These plans give individuals a geat amount of feedom when detemining how to save fo etiement. This inceased flexibility has also aised concens about the ationality in agents potfolio and savings decisions. Having a benchmak against which to check the ationality of people s choices is cucial both fom a policy pespective and in ode to fom the basis of sound financial advice. In this pape we develop a theoetical model to addess some of the inteactions between savings, potfolio choice and etiement in a utility maximizing famewok. We assume that agents ae faced with a constant investment oppotunity set and a constant wage ate while still in the wokfoce. Thei utility exhibits constant elative isk avesion and is nonsepaable in leisue and consumption. The majo point of depatue fom peexisting liteatue is that we model the labo supply choice as an optimal stopping poblem: An individual can wok fo a fixed (non-adjustable) amount of time and ean a constant wage but is fee to exit the wokfoce (foeve) at any time she chooses. In othe wods, we assume that wokes can wok eithe full time o etie. As such, individuals ae faced with thee questions to decide: ) how much to consume 2) how to invest the savings and 3) when to etie. The incentive to quit wok comes fom a discete jump in thei utility due 3
to an incease in leisue once etied. When etied, they cannot etun to the wokfoce. We also conside two extensions of the basic famewok. In the fist extension we disallow the agent to choose etiement past a pespecified deadline. In a second extension we disallow he to boow against the net pesent value (NPV) of he human capital. (i.e. we equie financial wealth to be non-negative) The majo esults that we obtain can be summaized as follows: Fist, we show that the agent will ente etiement when a cetain wealth theshold is eached, which we detemine explicitly. In this sense, wealth plays a dual ole in ou model. Not only does it detemine the esouces available fo futue consumption, it also contols the "distance" to etiement. Second, the option to etie ealy stengthens the incentives to save compaed to the case whee ealy etiement is not allowed. The eason is that saving not only inceases consumption in the futue but also bings etiement "close". Moeove, this incentive is wealth dependent: As the individual appoaches the citical wealth theshold to ente etiement, the "option" value of etiing ealy becomes pogessively moe impotant and the saving motive becomes stonge. Thid, the maginal popensity to consume (mpc) out of wealth declines as wealth inceases and ealy etiement becomes moe likely. The intuitionissimple: Aninceaseinwealthwillbing etiement close, theefoe deceasing the length of time the individual emains in the wokfoce. Convesely, a decline in wealth will postpone etiement. Thus, changes in wealth ae somewhat countebalanced by the behavio of the emaining NPV of income and thus the effect of a maginal change in wealth on consumption becomes attenuated. Once again this attenuation is stongest fo ich individuals who ae close to thei goal of ealy etiement. Fouth, the optimal potfolio is tilted moe towads stocks compaed to the case whee ealy etiementisnotallowed. Anadveseshockinthestockmaketwillbeabsobedbypostponingthe etiement time. Thus, the individual is moe inclined to take isks as she can always postpone he etiement time instead of cutting back he consumption in the event of a declining stock maket. Moeove, in ode to bing etiement close, the most effective way is to invest the exta savings in the stock maket instead of the bond maket. Fifth, the choice of potfolio ove the life cycle exhibits some new and inteesting pattens. We show that thee exist cases whee an agent might find it optimal to incease the pecent of 4
financial wealth that she invests in the stock maket as she ages (in expectation). This is tue even though he income and the investment oppotunity set ae constant. This happens, because wealth inceases ove time and hence the option of ealy etiement becomes moe elevant ("in the money" in option picing teminology). Accodingly, the tilting of the optimal potfolio towads stocks becomes stonge. Indeed, as we show in a calibation execise, the model pedicts that -pio to etiement- potfolio holdings could incease, especially when the stock maket exhibits extaodinay etuns as it did in the late 99 s when many wokes expeienced apid inceases in wealth, thus opting fo an ealie etiement date. In fact ou model suggests a possible patial ationalization fo the (appaently iational) behavio of individuals who inceased thei potfolios as the stock maket was ising and then liquidated stock as the maket collapsed. 2 We believe that this model is paticulaly succesful in accounting fo this paadoxical fact in a plausible ational way. This pape is elated to a numbe of stands in the liteatue, which is suveyed in Ameiks and Zeldes (2) 3. The pape closest to ous is Bodie, Meton, and Samuelson (992) (hencefoth BMS). The majo diffeence between BMS and this pape is the diffeent assumption about the ability of agents to adjust thei labo supply. In BMS labo can be adjusted in a continuous fashion. Howeve, thee seems to be a significant amount of evidence that labo supply is to a lage extent indivisible. In many jobs wokes wok eithe full time o they ae etied. Moeove, it appeas that most people do not etun to wok afte they etie, o if they do, they etun to less well paying jobs o wok only pat time. As BMS claim in the conclusion of thei pape Obviously, the oppotunity to vay continuously one s labo without cost is a fa cy fom the wokings of actual labo makets. A moe ealistic model would allow limited flexibility in vaying labo and leisue. One cuent eseach objective is to analyze the etiement poblem as an optimal stopping poblem and to evaluate the accompanying potfolio effects. This is pecisely the diection we take hee. Thee ae at least two majo diections in which ou esults diffe fom BMS. Fist, we show that the optimal etiement decision intoduces an option-type element in the decision of the individual, that is entiely absent if labo is adjusted continuously. Second, the hoizon and wealth effects on potfolio and consumption choice in ou 5
pape ae fundamentally diffeent than in BMS. Fo instance, the holdings of stock in BMS ae a constant multiple of the sum of (financial) wealth and human capital. This multiple is not constant in ou setup, but instead depends on wealth 4. Thid, the model pesented hee allows fo a clea way to model etiement, which is difficult in the liteatue allowing fo a continuous choice of labo/leisue. An impotant implication is that in ou setup we can calibate the paametes of the model to obseved etiement decisions. In the BMS famewok calibation to micoeconomic data is hade, because individuals do not seem to adjust thei labo supply continuously. Liu and Neiss (22) study a famewok simila to BMS, but foce an impotant constaint on the maximal amount of leisue. This howeve omits the issues elated to indivisibility and ievesibility, which lead to fundamentally diffeent implications fo the esulting potfolios as we show. In sum, the fact that labo supply flexibility is modeled in a moe ealistic way allows a close mapping of the esults to eal wold institutions than is allowed by a model exhibiting continuous choice between labo and leisue. The model is also elated to a stand of the liteatue that studies etiement decisions. A patial listing would include Stock and Wise (99), Rust (994), Laeza (986), Rust and Phelan (997), and Diamond and Hausman (984). Most of these models ae stuctual estimations that ae solved numeically. Hee ou goal is diffeent. We do not include all the ealistic amifications that ae pesent in actual etiement systems. Instead, we isolate and vey closely analyze the new issues intoduced by indivisibility and ievesibility of the labo supply / etiement decision on savings and potfolio choice. Natually, thee is a tadeoff between adding ealistic condideations and the level of theoetical analysis that can be accomplished with a moe complicated model. Sundaesan and Zapateo (997) study optimal etiement, but in a famewok without disutility of labo. Some esults of this pape shae some similaities with esults obtained in the liteatue on consumption and savings in incomplete makets. A highly patial listing would include Viceia (2), Chan and Viceia (2), Campbell et al. (24), Kogan and Uppal (2), Duffie etal. (997), Duffie and Zaiphopoulou (993), Koo (998), and Caoll and Kimball (996) on the ole of incomplete makets and He and Pages (993) and El Kaoui and Jeanblanc Pique (998) on issues elated to the inability of individuals to boow against the NPV of thei futue income. This liteatue poduces some insights on why consumption (as a function of wealth) should be 6
concave, and also has some implications on life cycle potfolio choice. Howeve, the intuitions ae quite diffeent fom the ones we obtain hee. In this pape the esults ae diven by an option component in agent s choices that is elated to thei ability to adjust thei time of etiement. In the incomplete makets liteatue esults ae diven by agents inability to effectively smooth thei consumption due to missing makets. 5 Thoughout the pape we maintain the assumption that agents eceive a constant wage. This is done not only fo simplicity, but because it makes the esults most supising: It is well undestood in the liteaue 6 that allowing fo a (positive) coelation between wages and the stock maket can poduce upwad sloping potfolio holdings ove the life cycle. What we show is that optimal etiement choice can induce simila effects even when labo income is pefectly iskless. Since the agument and the intuition fo this fact is othogonal to existing models we pefe to keep the simplest possible setup in evey othe dimension, so as to isolate the effects of optimal ealy etiement. The ole of labo supply flexibility in a geneal equilibium model with continuous labo/leisue choice is consideed in Basak (999). It is vey likely that the esults we pesent in this pape could fom the basis fo a geneal equilibium extension. It is well known in the macoeconomics liteatue that allowing indivisiblelaboisquiteimpotantifoneistoexplainthevolatilityof employment elative to wages. See fo example Hansen (985) and Rogeson (988). Technically, we extend methods poposed by Kaatzas and Wang (2) (who do not allow fo income) to solve optimal consumption poblems with discetionay stopping. The extension that we conside in section 5 uses some ideas poposed in Baone-Adesi and Whaley (987), while section 6 extends the famewok in He and Pages (993) to allow fo ealy etiement. Finally, thee papes that pesent paallel and independent wok on simila issues ae Lachance (23), Choi and Shim (24), and Dybvig and Liu (25). Lachance (23) and Choi and Sim (24) study a model with a utility function that is sepaable in leisue and consumption, but without a deadline fo etiement and / o boowing constaints. Moeove, sepaable utility does not allow consumption dops upon etiement as the ones obseved in the data. Technically, these papes use dynamic pogamming and not convex duality methods in ode to solve the poblem, which cannot be easily extended to models with deadlines, boowing constaints, etc. Ou appoach ovecomes these difficulties. Dybvig and Liu (25) study a vey simila model to 7
that in section 7 of this pape with simila techniques. Howeve they do not conside etiement pio to a deadline as we do. A deadline makes the poblem consideably hade (since the citical wealth thesholds become time dependent) and we ae able to povide a faily accuate appoximate closed fom solution fo this poblem in Section 5. One can actually pefom simple execises 7 which demonstate that in the absence of a etiement deadline the distibution of etiement times implied by the model becomes implausible. Most impotantly, compaed to the papes above we believe that the pesent pape goes into significantly geate detail in tems of the economic analysis and implications of the esults. In paticula we povide applications (like the analysis of potfolios of agents saving fo ealy etiement in the late nineties) which demonstate most clealy the eal wold implications of optimal potfolio choice in the pesence of ealy etiement. The stuctue of the pape is as follows: Section 2 contains the model setup. Section 3 pesents the solution. In section 4 we descibe the analytical esults if one places no etiement deadline. Section 5 contains an extension to the case whee etiement cannot take place past a deadline. Section 6 contains some calibation execises. Section 7 extends the model by imposing boowing constaints and section 8 concludes. Appendix. The technical details and all poofs can be found in the 2 Model Setup 2. Investment Oppotunity Set The consume can invest in the money maket, whee she eceives a fixed stictly positive inteest ate >. We place no limits on the positions that can be taken in the money maket. In addition the consume can invest in a isky secuity with a pice pe shae that evolves as dp t P t = µdt + σdb t whee µ>and σ> ae given constants and B t is a one-dimensional Bownian motion on a complete pobability space (Ω,F,P). 8 We finally define the state pice density pocess (o stochastic discount facto) as H(t) =(t)z (t), H() = 8
whee (t) and Z (t) ae defined as (t) = e t ½ Z (t) = exp Z t ¾ κdb s 2 κ2 t, Z () = and κ is the Shape atio κ = µ σ It is a standad esult, that these assumptions imply a dynamically complete maket (Kaatzas and Sheve [998] Chapte ). 2.2 Potfolio and Wealth Pocesses An agent chooses a potfolio pocess π t and a consumption pocess c t >, which ae pogessively measuable and satisfy the standad integability conditions given in Kaatzas and Sheve (998) Chaptes and 3. She also eceives a constant income steam y as long as she woks and no income steam once in etiement. Retiement is an ievesible decision. Until section 5 we will assume that an agent can etie at any time that she chooses. Theagentisendowedwithanamountoffinancial wealth W y. The pocess of stockholdings π t is the dolla amount invested in the isky asset (the stock maket ) at time t. The est, W t π t, is invested in the money maket. Shot selling and boowing ae both allowed. We will place no exta estictions on the (financial) wealth pocess W t until section 7 of the pape. Additionally in that section we will impose the estiction W t. As long as the agent is woking, the wealth pocess evolves as dw t = π t {µdt + σdb t } + {W t π t } dt (c t y ) dt () Applying Ito s Lemma to the poduct of H(t) and W (t), integating and taking expectations we get fo any stochastic time τ that is finite almost suely µ Z τ E H(τ)W (τ)+ H(s)[c(s) y ] ds W (2) This is the well known esult that in dynamically complete makets one can educe a dynamic budget constaint of the type () to a single intetempoal budget constaint of the type (2). If the agent is etied the above two equations continue to hold with y =. 9
2.3 Leisue, Income and the Optimization Poblem To obtain closed fom solutions, we assume that the consume has a utility function of the fom U(l t,c t )= α l α t c α t, > (3) whee c t is pe peiod consumption, l t is leisue and <α<. We assume that the consume is endowed with l units of leisue. l t canonlytaketwovaluesl o l. If the consume is woking, then l t = l ; when etied l t = l. We will assume that the wage ate w is constant, so that the income steam is y = w(l l ) >. We will nomalize l =. Obseve also that this utility is geneal enough so as to allow consumption and leisue to be eithe complements ( < ) o substitutes ( > ). The consume maximizes expected utility e βt U(l,c t )dt + e βτ Z Z τ max E c t,π t,τ whee β> is the agent s discount facto. 9 by solving the poblem τ e β(t τ) U(l, c t )dt (4) The easiest way to poceed is to stat backwads Z U 2 (W τ )=maxe e β(t τ) U(l, c t )dt c t,π t τ U 2 (W τ ) is the Value function once the consume decides to etie and W τ is the wealth at etiement. By the pinciple of dynamic pogamming we can ewite (4) as Z τ max E e βt U(l,c t )dt + e βτ U 2 (W τ ) c t,w τ,τ It will be convenient to define the paamete as (5) = α( ) so we can then e-expess the pe-peiod utility function as U(l, c) =l ( α)( ) c Since we have nomalized l =pio to etiement, the pe peiod utility pio to etiement is given by: U (c) =U(,c)= c (6)
Notice that > ifandonlyif > and < if and only if <. Unde these assumptions, it follows fom standad esults (See fo example Kaatzas and Sheve [998], Chapte 3), that once in etiement the Value function becomes whee U 2 (W τ )= θ = ³ µ l α Wτ θ µ + κ2 + β 2 In ode to guaantee that the Value function is well defined, we assume thoughout that θ> and β < κ2 2. It will be convenient to edefine the continuation Value function as U 2 (W τ )=K W τ whee Since l>l =we have that ³ µ K = l α (8) θ (7) K > θ K < θ if < (9) if > () 3 Solution We pesent the solution to the poblem descibed in section 2.3 and discuss the popeties of the joint etiement /consumption and potfolio choice poblem, when etiement is an ievesible discete decision. Poposition Define the constants 2 = 2 β κ 2 λ = C 2 = q ( 2 β ) κ 2 +8 β 2 κ 2 2 ( 2 )θ ³ ³ + 2 K θ " # ( 2 ) + 2 λ 2 y y () (2)
and assume that 2 ³ + 2 < θ 2 Finally let λ be the (unique) solution of Then and the optimal policy is a) If W t < W = 2 C 2 (λ ) 2 θ (λ ) + y + W t = (3) ( 2 )K θ y + 2 K θ consumption follows the pocess c s = c s = l The optimal etiement time is µ λ e C 2 >, 2 < β(s t) H(s) ( α) ( ) H(t) µ λ e {t s<τ } (4) β(s t) H(s) H(t) {s τ } (5) τ = inf{s : W s = W } = ½ ¾ = inf s : λ β(s t) H(s) e H(t) = λ (6) Moeove, the optimal consumption and the optimal stockholdings as a function of W t ae given by c t = c(w t )=(λ (W t )) (7) π t = π(w t )= κ µ σ 2 ( 2 )C 2 λ (W t ) 2 + θ λ (W t ) (8) whee the notation λ (W t ) is used to make the dependence of λ on W t explicit. b) If W t W = ( 2 )K θ y + 2 K θ the optimal solution is to ente etiement immediately (τ = t) and the optimal consumption /potfolio policy is given as in Kaatzas and Sheve (998) section 3. 2
The natue of the solution is intuitive: The agent entes etiement if and only if the level of he assets exceeds W. Up to that point he consumption is given by (4), wheeas it jumps to (5) once etied. The jump is given by: c τ + c τ = l ( ) ( α) = K / θ (9) whee the fist equality follows by dividing (5) by (4) and the second fom (8). Notice that > will imply a downwad and < an upwad jump (since l>). A key quantity in all the solutions is λ. It can be shown that λ is the deivative of the value function and is deceasing in W t. Fomally, letting J(W t ) be the value function of the poblem, J W = λ (W t ). Equation (7) suggests an altenate intepetation of λ as the maginal utility of consumption. In othe wods, (7) is the standad "Eule" equation U (c t )=J W. Moeove, an application of the implicit function theoem to (3) is sufficient to show that equation (8) leads to: π t = κ σ J W = µ J WW σ 2 J W J WW This is the familia Meton fomula fo the optimal potfolio. λ solves the simple nonlinea equation (3) and hence one can study analytically the dependence of λ on W t, something that we do epeatedly in the next section. 4 Popeties of the solution In this section we exploe some popeties of the solution. A cental theme of the analysis is the pesence of an option element in the decision of the agent, that fundamentally altes the natue of the optimal consumption / potfolio decision. The benchmak model to which we compae the esultsisamodelwheetheeisaconstantlabo income steam and no etiement (the woke woks foeve). This is the natual benchmak fo this section, since it keeps all else equal except fo the the option to etie. The esults obtained in this section allow us to isolate intuitions elated to optimal etiement, in a famewok whee solutions ae not time dependent and theefoe easie to analyze. Fotunately, all of the esults obtained in this section cay ove to the next, whee we intoduce a etiement deadline. In that section the benchmak to which we compae the optimal etiement model is moe natual: Namely, etiement is mandatoy at time T, but without the option to etie ealy. 3
4. Wealth at Retiement Wealth at etiement is given by Poposition as ( W = 2 )K θ ³ ³ y + 2 K θ As Poposition assets, fo wealth levels highe than this, it is optimal to ente etiement. Fo lowe wealth levels it is optimal to emain in the wokfoce. In the appendix we show that W is stictly positive, i.e. a consume will neve ente etiement with negative wealth since thee is no moe income to suppot post-etiement consumption. Using (9) we can ewite W as the poduct of thee tems: α W = ( 2 ) ³l ³ Ã ³ + α! y 2 l (2) The last tem shows the linea dependence of W on y. This homogeneity of degee shows that one can expess the taget wealth at etiement in tems of multiples of cuent income, and suggests the nomalization y =, which we adopt in all quantitative execises. The second tem is elated to the agent s pefeences ove consumption and leisue. Assuming >, equation (9) implies that agents who value leisue and hence expeience lage dops in consumption upon enteing etiement, will ente etiement ealie (W is lowe). An inteesting implication of (9) along with (2) is that both the second and thid tem ae in pinciple obsevable fom consumption and income data. Moeove, 2 (defined in []) only depends on paametes elated to the investment oppotunity set and not on agent pefeences. Hence, up to knowing an agents isk avesion equation (2) suggests a staightfowad way to compute W and compae it to the data. We conclude by discussing the dependence of W on κ, the Shape atio. Diffeentiating W w..t. 2 it is not had to establish that W 2 > andthendiffeentiating 2 w..t. the Shape atio (κ) and applying the chain ule gives that: W κ < (2) i.e. the etiement theshold is lowe in economies with a highe Shape atio. This is intuitive: a high Shape atio implies that the stock maket can poduce stong gains and sustain post- 4
etiement consumption, hence the agent is moe willing to etie. In othe wods the gains that can be obtained in the stock maket elative to the impotance of a fixed income steam ae moe sizeable and accodingly the agent is moe willing to go into etiement. 4.2 Optimal Consumption We concentate on a consume with wealth lowe than W, so thee is an incentive to continue woking. Optimal consumption pio to etiement is given by Poposition as c t =(λ ) whee λ solves equation (3): In the appendix we show that θ> implies that 2 C 2 (λ ) 2 θ (λ ) + y + W t = (22) 2 > (23) It is now useful to ewite (22) as 2 C 2 c ( 2 ) t + θ c t = W t + y (24) Up to the tem 2 C 2 c ( 2 ) t, this equation is the standad equation that one would obtain in a Meton-type famewok, with a constant income steam. Indeed, if one emoved the option of etiement, optimal consumption would be given by ³ c t = θ W t + y The diffeence hee is that the individual wants to etie and hence has an added incentive to save fo a given level of wealth (since 2 < and C 2 > ). Even though we cannot povide an explicit solution to this equation we can still calculate the maginal popensity to consume out of wealth and its deivative by using the implicit function theoem. We fist define the maginal popensity to consume as mpc = c t W t 5
We diffeentiate both sides of equation (24) w..t. W t, to get µ 2 ( 2 )C 2 c ( 2 ) t + mpc = (25) θ One can fist obseve fom this equation, that mpc is stictly below θ since 2 <, C 2 >. Compaed to the infinite hoizon poblem (whee one stays in the wokfoce foeve) the maginal popensity to consume out of wealth is stictly lowe due to the option value embodied in (22). One can also study the dependence of the mpc on wealth. Diffeentiating once moe and using equation (23) gives ³ mpc = mpc 3 2 ( 2 )(( 2 ) )C 2 c ( 2 ) 2 t < In othe wods, pio to etiement, the maginal popensity to consume out of wealth is a deceasing function of wealth and accodingly consumption is a concave function of wealth. To undestand this, note that the consume adopts a theshold policy fo he etiement. If wealth is high, the time to etiement is close and thus an incease in W t is countebalanced by a decease in the net pesent value of emaining income. Hence a consume eacts less to a change in wealth, the close he wealth is to the theshold level of etiement. Revesing signs in the above agument, it is also tue that the effect on consumption of a dop in pe-etiement wealth will be mitigated by an incease in the net pesent value of emaining income. Altenatively speaking, a negative shock to wealth will only patially affect consumption. A component of the dop will just postpone plans fo etiement and this will in tun incease the net pesent value of income to be eceived in the futue. Of couse once in etiement the poblem becomes a standad Meton type poblem and the usual affine elationship between consumption and wealth pevails. It is impotant to note that the key to these esults is not the pesence of labo supply flexibility pe se, but the "eal option" inheent in the etiement decision. To substantiate this claim, assume that the agent neve eties, and he leisue choice is detemined optimally on a continuum at each point in time, so that l t + h t = l 3 whee h t ae the hous devoted to wok and the instantaneous income is wh t, with w defined as in section 2.3. The solution that one obtains fo optimal consumption in such a famewok with pefect labo supply flexibility is ³ c t = C W t + y C 2 6
fo two appopiate constants C, C 2. Notice the simple affine elationship between wealth and consumption. These esults show an impotant diection in which the pesent model sheds some new insights, beyond existing famewoks, into the elationship between etiement, consumption and potfolio choice. With endogenous etiement, wealth has a dual ole. Fist, as in all consumption and potfolio poblems, it contols the amount of esouces that ae available fo futue consumption. Second, it contols the distance to the theshold at which etiement is optimal. It is this second channel that is behind the behavio of the mpc analyzed above. 4 The concavity of the consumption function is also a common esult in models combining nonspanned income and /o boowing constaints of the fom W t (e.g. Caoll and Kimball [996]). A quite impotant diffeence between these models and the one consideed hee, is that in the pesent model the effects of concavity ae most noticeable fo high levels of wealth and not fo low wealth levels. In ou model the mpc asymptotes to θ as W t y, 5 and declines fom thee to the point whee W t = W. Afte that it jumps back up to θ, eflecting the loss of the eal option associated with emaining in the wokfoce. By contast in models like Koo (998) o Duffie et al (997) the mpc is above θ fo low levels of wealth and asymptotes to θ as W t. We discuss this issue futhe in section 7, when we intoduce boowing constaints. 4.3 Optimal Potfolio By Poposition, the optimal holdings of stock ae given by π t = κ µ σ 2 ( 2 )C 2 (λ ) 2 + θ (λ ) Fom(3),wehavethat theefoe π t = κ σ 2 C 2 (λ ) 2 + y + W t = θ (λ ) ³ W t + y + κ µ σ 2C 2 (λ ) 2 ( 2 ) + The fist tem is equal to a standad Meton type stockholdings fomula fo an infinite hoizon investo with constant income but no option to etie. The second tem is positive. To see this, notice that ( 2 ) + < 7
by equation (23)and 2 <,C 2 >. Moeove, by the definitions of C 2,λ one can deive that C 2 (λ ) 2 = y µ ( 2 ) ³ λ 2 (26) + λ 2 Thus, as λ the impotance of this tem disappeas, wheeas as λ λ this tem appoaches its maximal value. It is easiest to intepet this esult by obseving that a) λ is a deceasing function of wealth (W t ) and b) by (6) λ is the lowest value that λ can attain befoe the agent goes into etiement 6. In wods, when an agent is vey poo, the elevance of ealy etiement is small and thus the stockholdings chosen esemble those of a simple Meton type setup. By contast as wealth inceases, so does the likelihood of ealy etiement and this tem becomes inceasingly impotant. In summay, the option value of wok inceases the incentive to take isk compaed to the benchmak of an infinite hoizon Meton setup with constant income but no etiement. Moeove, as wealth inceases and comes close to the etiement theshold, this incentive is maximized. The intuition fo this esult is staightfowad. As wealth inceases, it becomes moe likely that the "eal" option to etie will be execised. The most effective way to affect that likelihood is by investing in stocks. The agent is moe willing to take isks in the stock maket, because she can postpone he etiement instead of educing consumption in the event of a negative shock. It is inteesting to elate these esults to BMS. To do that, we stat by nomalizing the nominal stock holdings by W t.thisgivestheatioφ t = π t W t, o using (26) φ = κ µ + y + σ W t + κ σ y W t 2 " µλ λ 2 µ ( 2 ) + # ( 2 ) ³ + 2 The fist tem in the equation fo φ coesponds to the tem one would obtain in the absence of etiement (i.e. the Meton famewok whee a woke neve eties). The second tem is the effect of the eal option to etie. It is inteesting to note the dependence of these tems on W t. By fixing y and inceasing W t, one can obseve that the fist tem actually deceases. This is the standad BMS effect. In othe wods, ignoing the eal option to etie (which is captued in the second tem only) one would aive at the conclusion that an incease in wealth should be associated with a decline in the potfolio shae allocated to isky assets. This conclusion is not necessaily tue if 8
one consides the option to etie. To see why, compute φ W and evaluate it aound W, in ode to obtain afte some simplifications: ³ φ W (W )= µ φ(w ) κ + ³ κ 2 K θ µ ( W σ W φ(w ) σ K 2 ) + 2 θ The fist tem is clealy negative, and captues the incease in the denominato of φ = W π. The second tem though is positive and potentially lage than the fist tem, depending on paametes. This esult is diven by the eal option of wok, not labo supply flexibility pe se. Indeed one can show (using the methods in BMS) that allowing an agent to choose labo and leisue feely on a continuum would esult in: π t = µ κ W t + y σ l (l l ) This implies that φ wouldhavetobedeceasinginw t. The eason fo these diffeences is that in BMS, the amount allocated to stocks as a faction of total esouces (financial wealth + human capital) is a constant. In ou famewok this faction depends on wealth. Wealth contols both the esouces available fo futue consumption and the likelihood of "execising" the eal option of etiement. In summay, not only does the possibility of ealy etiement incease the incentive to save moe, it also inceases the incentive of the agent to invest in the stock maket because this is the most effective way to obtain this goal. Futhemoe, this incentive is stengthened as an individuals wealth appoaches the taget wealth level that tigges etiement. 4.4 The Coelation between Consumption and the Stock Maket As we showed in section 4.2 the maginal popensity to consume out of wealth (and hence the sensitivity of changes to consumption w..t. changes in the wealth) is deceased by the possibility of ealy etiement. One might also wonde whethe this also implies a deceased coelation between consumption and the stock maket (compaed to the standad Meton model). If this esult wee tue, it could then be hoped that the model can shed some light into the equity pemium puzzle, because the same fixed level of µ would be compatible with a lowe coelation between consumption and the stock maket than the standad Meton model. 7 The answe is unfotunately, that it does 9
not. The eason is quite simple and can be seen by examining fomula (6) in Basak (999) which continues to be tue in ou setup (fo agents pio to etiement) µ = cu µ cc dpt cov, dc + U µ ch dpt cov,dl (27) U c P t c U c P t U ch is the coss patial of U w..t the hous woked and dl is the vaiation in leisue. In ou setup ³ dl =pio to etiement and µ,, cucc U c ae constants in ou famewok. Accodingly, cov dpt P t, dc c is constant as well. It is impotant to note that this esult was obtained solely by the fact that dl =along with the assumption of a constant investment oppotunity set and CRRA utilities. In othe wods -pio to etiement- the consumption CAPM holds in this famewok This may seem supising in light of the esults we obtained fo the maginal popensity to consume out of wealth. One might expect that a declining mpc would be sufficienttopoducea low coelation between the stock maket and consumption. The esolution of this puzzle is that a decease in mpc in this model is accompanied by an equivalent incease in the exposue to the stock maket though a potfolio that is moe heavily tilted towads stocks. In othe wods, even though consumption becomes less esponsive to shocks in the wealth pocess, at the same time the shocks to the wealth pocess become lage because of a iskie potfolio. 8 The two effects exactly balance out. An impotant caveat is that the above discussion elies heavily on patial equilibium. To see if labo supply flexibility can indeed explain the obseved smoothness of aggegate consumption and accodingly a lage equity pemium one would have to study a geneal equilibium vesion of this model (as Basak [999] does fo continuous choice of labo/leisue). In that case a faction of the population would be enteing etiement at each instant and would expeience consumption changes due to the incease in leisue. Hence, at the aggegate the simple consumption CAPM no longe holds. µ = cu µ cc dpt cov, dc U c P t c It can be easonably conjectued that in this famewok the behavio of the inteest ate and the equity pemium would be vey diffeent than in Basak (999). Even in the base case of CRRA utilities and multiplicative technology shocks, the equity pemium and the inteest ate would exhibit inteesting dynamics. Howeve, these issues ae beyond the scope of this pape. 2
5 Retiement befoe a Deadline None of the claims made so fa elied on esticting the time of etiement to lie in a paticula inteval. The exposition was facilitated by the infinite hoizon setup because it allowed fo explicit solutions to the associated optimal stopping poblem. Howeve, the tadeoff is that in the infinite hoizon case, thee is no notion of "life" cycle, since time plays no explicit ole in the solution. Moeove, the "natual" theoetical benchmak fo the model of the pevious section is one without etiement at all. In this section we ae able to extend all the intuitions of the pevious section by compaing the ealy etiement model to a benchmak model with mandatoy etiement at time T, which is moe natual. 9 Allowing fo a etiement deadline intoduces a new state vaiable (time to etiement), which consideably complicates the analysis. Howeve, we ae able to use appoximation methods to solve the associated "finite hoizon" 2 optimal stopping poblem, that ae simple to analyze, compute and seem to wok vey well in pactice. Fomally, the only modification that we intoduce compaed to section 2 is that equation (5) becomes Z τ T max c t,w τ,τ t e β(s t) U(l,c s )ds + e β(τ T t) U 2 (W τ T ) whee T is the etiement deadline. Poposition 2 Define ev E (λ, T t) = Let e V (λ; T t) be given by: λ h³ i K θ e θ(t t) e (T t) + + λy θ (28) whee θ (T t) is given by λ (T t) = C 2(T t) λ 2(T t) + V e E (λ, T t), if λ > λ (T t) µ K (λ) if λ λ (T t) ( 2(T t) )θ (T t) ³ + 2(T t) ³ K θ (T t) θ θ (T t) = h³ i K θ e θ(t t) + y( e (T t) ) (29) 2
C 2(T t) is given by and 2(T t) is given by C 2(T t) = 2(T t) = " 2 β κ 2 ( 2(T t) ) + 2(T t) λ 2(T t) (T t) # q ( 2 β κ ) 2 +8 2 2 y ( e (T t) ) β ( e β(t t) )κ 2 (3) ev (λ, T t) is continuously diffeentiable eveywhee and ev λ (λ, T t) maps (, ) into the inteval (, y e (T t) ). Finally compute the (unique) solution of 2(T t) C 2(T t) (λ ) 2(T t) (λ ) + y ( e (T t) ) + W t = (3) θ (T t) Then an appoximate solution to (28) is given by a) If W t < W (T t) = K λ (T t) consumption follows the pocess c s = c s = l and the optimal etiement time is µ λ e β(s t) H(s) ( α) ( ) H(t) µ λ e {t s<τ } β(s t) H(s) H(t) {s τ } τ = inf{s : W s = W (T s) } = inf{s : λ β(s t) H(s) e H(t) = λ (T s)} The optimal consumption and the optimal potfolio as a function of W t ae given by c t = c(w t )=(λ (W t )) π t = π(w t )= κ µ σ 2(T t) ( 2(T t) )C 2(T t) λ (W t ) 2 + λ (W t ) θ (T t) whee the notation λ (W t ) is used to make the dependence of λ on W t explicit. b) If W t W (T t) the optimal solution is to ente etiement immediately (τ = t) and the optimal consumption /potfolio policy is given as in Kaatzas and Sheve (998) section 3. 22
The appendix discusses the natue of the appoximation and examines its pefomance against consistent numeical methods to solve the poblem. The basic idea behind the appoximation is to educe the poblem to a standad optimal stopping poblem and use the same appoximation technique as Baone-Adesi and Whaley (987). The most impotant advantage of this appoximation, is that it leads to vey tactable solutions fo all quantities involved. This can be seen most easily by obseving that equation (3) is pactically identical to equation (3). One can also check easily that the fomulas fo optimal consumption, potfolio etc. ae identical to the espective fomulas of poposition (the sole exception being that the constants ae modified by tems that depend on T t). As a esult, all of the analysis in section 4 caies though to this section. This is paticulaly tue fo the dependence of consumption, potfolio etc. on wealth. Economically, the only new dimension intoduced is that all constants depend explicitly on the distance to mandatoy etiement, and thus enable the study of inteaction effects between wealth and the an investo s age. Hee we will focus only on the implications of the model fo potfolio choice ove the life cycle. The esults fo life-cycle consumption ae simila. To povide a benchmak against which to compae the solutions, we conside the potfolio poblem of an agent with mandatoy etiement in T t peiods. Fo simplicity we also assume thee is no labo supply flexibility, i.e. standad deivations, the potfolio of an agent in this case is à π mand. t = κ e W t + y σ the agent is endowed with an income steam of y. By (T t) ³ ³ A constant faction σ κ e of the net pesent value of esouces available to the individual W + y T is invested in the stock maket iespective of he age. Since we only obseve W t and π t in the data, and not the net pesent value of futue income, it is inteesting to divide total holdings of stock by financial wealth, which gives: π mand. t W t = σ! Ã! κ + y e (T t) W t This expession captues a numbe of well undestood intuitions. Fist, the allocation towads stocks as a faction of financial wealth declines with age, fo a fixed level of W t. Second, in expectation W t will incease ove time, einfocing the fist effect. Theefoe, (in expectation) the allocation to stocks should be downwad sloping ove time. 23 (32)
Allowing fo ealy etiement consideably altes some of these conclusions. To demonstate this effect, we poceed (as in section 4.3) to aive at the optimal holdings of stock π t in the pesence of optimal ealy etiement à π t = κ e W t + y σ + κ y e (T t) à σ (T t)!! λ 2T λ (T t) µ 2(T t) +( 2(T t) ) ³ 2(T t) ³ + 2(T t) whee λ is given by (3). As in section 4.3 the second tem captues the "eal" option of ealy etiement and is stictly positive. As with most options, its elevance is lage a) the moe likely it is that it will be execised (in/out of the money) and b) the moe time is left until its expiation. Accodingly, the impotance of the second tem in (33) should be expected to decease when T t is small and/o the atio of W t to the taget wealth W (T t) is small. This now opens up the possibility of ich inteactions between "pue" hoizon effects (vaiations in T t, keeping W t constant) andwealtheffects, beyond the ones aleady pesent in (32). As an agent ages butisnotyetetied, the "pue" hoizon effects will tend to wok in the same diection as in equation (32). Howeve, in expectation wealth inceases as well and thus the option to etie ealy becomes moe and moe elevant, counteacting the fist effect. (33) Anothe popety of the optimal potfolio that is implied by the pesent famewok is a downwad jump in stockholdings, immediately afte the agent entes etiement. In that case stockholdings take the standad Meton fom π t = κ σ W t These effects ae quantitatively illustated in the next section. 6 Quantitative Implications To quantitatively assess the magnitude of the effects descibed in section 5 we poceed as follows. Fist, we fix the values of the vaiables elated to the investment oppotunity set to: =.3,µ =.,σ =.2. Fo β we choose.7 in ode to account fo both discounting and a constant pobability of death. Fo we conside a ange of values (typically 2, 3, 4). This leaves 24
one moe paamete to be detemined, namely K. K contols the shift in the maginal utility of consumption upon enteing etiement. It is a well documented empiical fact that consumption dops consideably upon enteing etiement. As such, the most natual way to detemine the value of K is to match the agent s declining consumption upon enteing etiement. Aguia and Hust (24) epot expenditue dops of 7%, wheeas Banks et al. (998) epot changes in log consumption expenditues of almost.3 in the five yeas pio to etiement and theeafte. Since these dops ae mainly calculated fo food expenditues, which ae likely to be inelastic, we also calibate the model to somewhat lage dops in consumption than that. 2 In light of (9) c τ + = K / θ c τ whee c τ is consumption immediately pio to etiement and c τ + is the consumption immediately theeafte. We theefoe detemine K so that K / θ = {.5,.6,.7}. We fix the mandatoy etiement age to be T =65thoughout, and nomalize y to be. The abbeviation "et" indicates the solution implied by a model with optimal ealy etiement (up to time T ) and "BMS" denotes the solution of a model with mandatoy etiement at time T, with no option to etie ealie o late. Figue plots the "taget" wealth that is implied by the model, i.e. the level of wealth equied to ente etiement. This figue demonstates two pattens. Fist, "theshold" wealth declines as an agent neas mandatoy etiement. This is intuitive. The option to wok is moe valuable the longe its "matuity". As a (woking) agent ages, the incentive to keep the option "alive" is educed and hence the wealth theshold declines. Second, the citical wealth implied by this model vaies with the assumptions made about isk avesion, and the disutility of wok as implied by a lowe K / θ. Risk avesion tends to shift the theshold upwads, wheeas lowe levels of K / θ (implying moe disutility of labo) bing the theshold down. These ae intuitive pedictions. An agent who is isk avese wants to avoid the isk of losing the option to wok, wheeas an agent who caes a lot about leisue will want to ente etiement ealie. Figue 2 addesses the impotance of the eal option to etie fo potfolio choice. The figue plots the second tem in equation (33) as a faction of total stockholdings π t. In othe wods, it plots the elative impotance of stockholdings due to the eal "option" component as a pecent of total stockholdings. This pecentage is plotted as a function of two vaiables: a) age and b) 25
wealth. Age vaies between 45 and 64. Wealth vaies between and x, whee x coesponds to the level of wealth that would make an agent etie (voluntaily) at 64. We nomalize wealth levels by x so that the (nomalized) wealth levels vay between and. We then plot a panel of figues fo diffeent levels of,k / θ. Figue 2 demonstates the joint pesence of "time to matuity" and "moneyness" effects in the eal option to etie. Keeping wealth fixed and vaying the time to matuity (i.e. inceasing age) the elative impotance of the eal option to etie declines. Similaly, inceasing wealth makes the eal option component moe elevant, because the eal option is moe "in the money". It is inteesting to note that the "eal option" component is lage, taking values as lage as 4% fo some paamete combinations. In Figue 3 we conside the implications of the model fo life cycle potfolio choice. We fix a path of etuns coesponding to the ealized etuns on the CRSP value weighted index between 99 and 2. We then plot the potfolio holdings (defined as total stockholdings nomalized by financial wealth) ove time fo an individual whose wealth in 99 was just enough to allow he to etie in the beginning of 2 at the age of 59. We epeat the same execise assuming vaious combinations of K / θ and. In ode to be able to compae the esults, we also plot the potfolio that would be implied if the individual had no option of etiing ealy. We label this late case as "BMS". What figue 3 shows, is that the potfolio of the agent is initially declining and then flat o even inceasing ove time afte 995. This is in contast to what would be pedicted by ignoing the option to etie ealy (the "BMS" case). This fundamentally diffeent behavio of the potfolio of the agent ove time is due to the extaodinay etuns duing the latte half of the 99 s, that makes wealth gow faste, and hence the eal option to etie vey impotant towads the end of the sample. By contast, if one assumed away the possibility of ealy etiement, the natual conclusion would be that a un-up in pices would change the composition of total esouces (financial wealth + human capital) of the agent towads financial wealth. Fo a constant income steam this would mean accodingly a decease in the potfolio chosen. Figue 4 demonstates the above effect moe clealy. In this figue we nomalize total stockholdings by total esouces (human capital + financial wealth). As aleady shown, fo the BMS case we get a constant equal to. By allowing fo an ealy etiement option we obseve that ³ κ σ the faction of total esouces invested in stock, exhibits a stak incease towads the latte half of the 99 s, because the option of ealy etiement becomes moe elevant. The incease in this 26
faction is small fo the fist half of the 99 s and lage fo the latte pat of the decade. Figue 4 is useful in undestanding the behavio of the potfolio holdings in Figue 3. In the fist half of the sample the standad BMS intuition applies. The faction of total esouces invested in the stock maket is oughly constant even afte taking the option of ealy etiement into account. Hence by the standad intuition behind the BMS esults the potfolio of the agent (total stockholdings nomalized by financial wealth) declines ove time. Howeve, in the latte half of the sample, the incease in the eal option to etie is stong enough to counteact the decline in the potfolio implied by standad BMS intuitions. Figues 5 and 6 epeat the same execise as figues 3 and 4, only now fo an agent who "came close" to etiement in 2. Howeve, we assume that he wealth at that point was slightly less than enough in ode to actually etie. To achieve this we just assume that in 99 she stated with slightly less initial wealth than necessay to etie by 2. It is inteesting to note what happens post 2. Now, the option of ealy etiement stats to become ielevant and the agent s potfolio declines. The effect of a disappeaing option magnifies the dop in the potfolio. By contast in the BMS case the abupt dop in the stock maket (and hence wealth) would be countebalanced by a change in the composition between financial wealth and human capital towads human capital. This effect tends to somehow counteact the effects of aging, and poduces a much moe modeate dop in potfolio holdings. These figues ae meant to demonstate the fundamentally diffeent economic implications that can esult once one takes into account the eal option to etie. As such they should be seen as meely an illustative application. Note howeve, that a stonge esult can be shown in the context of this execise. Fo wealth levels close to the etiement theshold, the potfolio would incease with age in expectation, and not just fo the sample path that we conside. What is special about the path that we conside is that the stong gains in the stock maket dive investo wealth close to the optimal etiement theshold. This incease of the potfolio with age (in expectation) would be impossible in the absence of an ealy etiement option. The pesent pape is theoetical in natue, and we don t claim to have modeled even a small faction of all the issues that influence eal life etiement, consumption, and potfolio decisions (like liquidity constaints, shoting and leveage constaints, tansaction costs, undivesifiable income and health shocks etc.). Howeve, note that the model does poduce "sensible" potfolios (fo the 27