Habits and Leverage. Tano Santos * Columbia University. Pietro Veronesi ** University of Chicago

Transcription

1 Habis and Leverage Tano Sanos * Columbia Universiy Piero Veronesi ** Universiy of Chicago July 13, 216 PRELIMINARY DRAFT NOT FOR CIRCULATION COMMENTS WELCOME Absrac Many sylized facs of leverage, rading, and asse prices can be explained by a fricionless general equilibrium model in which agens have heerogeneous endowmens and exernal habi preferences. Our model predics ha aggregae leverage increases in good imes when sock prices are high and volailiy is low, i should predic low fuure reurns and i is posiively correlaed wih a consumpion boom of levered agens. In addiion, negaive aggregae shocks induce levered agens o deleverage by fire selling heir risky posiions as heir wealh drops. While such agens oal leverage decreases, heir deb/wealh level increases as wealh value is especially sensiive o changes in aggregae risk aversion. * Columbia Business School, Columbia Universiy, NBER, and CEPR. js1786@gsb.columbia.edu. **The Universiy of Chicago Booh School of Business, NBER, and CEPR. piero.veronesi@chicagobooh.edu. This research has been suppored by he Fama-Miller Cener for Research in Finance and he Cener for Research in Securiy Prices, boh locaed a Chicago Booh.

2 1. Inroducion The financial crisis has elicied much research ino he undersanding of he dynamics of aggregae leverage and is impac on asse prices and economic growh. Recen empirical and heoreical research has produced a variey of resuls ha many claim should inform a reconsideraion of exising fricionless models. Amongs hese we have (i) he evidence ha excessive credi supply may lead o financial crises; 1 (ii) he growh in household deb and he causal relaion beween he deleveraging of levered households and heir low fuure consumpion growh; 2 (iii) he idea ha acive deleveraging of financial insiuions generaes fire sales of risky financial asses which furher crashes asse prices; 3 (iv) he evidence ha he aggregae leverage raio of financial insiuions is a risk facor in asse pricing; 4 (v) he view ha balance shee recessions are criical componens of business cycle flucuaions; 5 and many ohers. Mos of hese explanaions rely on some ypes of marke fricions and behavioral biases, and poin a a causal effec of leverage ono aggregae economic and financial phenomena. In his paper, we pu forward a simple fricionless general equilibrium model wih endogenous leverage ha offers a coheren explanaion of mos of hese relaions beween agens leverage, heir consumpion, and asse prices. The mechanism emphasized in his paper is sandard in he asse pricing lieraure. We posi an economy populaed wih agens whose preferences feaure exernal habis. Specifically agens uiliies are deermined by he disance beween heir own level of consumpion and he level of aggregae endowmen, appropriaely scaled; roughly agens care abou consumpion inequaliy. How much agens care abou his disance varies across agens and over he business cycle. In paricular, agens care more abou heir relaive sanding in bad imes han in good imes and here are some agens who care more han ohers abou his comparison beween heir own level of consumpion and habis. This cross secional heerogeneiy inroduces moives for risk sharing and asse rading in general. Agens also differ in heir level of wealh, which is also an imporan deerminan of heir risk bearing capaciy. The model aggregaes nicely o sandard exernal habi models such as Campbell and Cochrane (1999) and Menzly, Sanos and Veronesi (24) and hus inheris he asse pricing properies of hese models and in paricular he dynamics of risk and reurn ha were he original moivaion for hese models. 1 See for insance Jordà, Schularick and Taylor (211). 2 See Jusiniano, Primiceri and Tambaloi (213) and Mian and Sufi (215). 3 See e.g. Shleifer and Vishny (211). 4 See He and Krishnamurhy (213) and Adrian, Eula and Muir (214). 5 See Huo and Ríos-Rull (213) and Mian, Rao and Sufi (213). 1

3 Exernal habi models feaure srong discoun effecs, which, as shown by Hansen and Jagannahan (1991), are required o explain he Sharpe raios observed in financial markes. We argue ha hese srong discoun effecs are also imporan o undersand he dynamics of risk sharing. Sandard risk sharing argumens require ha agens wih large risk bearing capaciy insure hose wih low risk bearing capaciy. In models where, for insance agens have CRRA preferences, such as Longsaff and Wang (213), his means he agens who provide he insurance consume a large share of aggregae consumpion when his is large and a low share when insead aggregae consumpion is low. This is obviously also he case in our framework, bu in addiion he share of consumpion will also depend on whaever sae variable drives discoun effecs, which inroduces addiional sources of non-lineariies in he efficien risk sharing arrangemen. The reason is ha in our model risk aversion changes depending on he acual realizaion of he aggregae endowmen and hus so do he efficiency gains associaed wih risk sharing. We decenralize he efficien allocaion by allowing agens o rade in he aggregae endowmen process and deb ha is in zero ne supply and provide a full characerizaion of he corresponding compeiive equilibrium. We show ha agens wih large risk bearing capaciy provide insurance by issuing he deb ha he more risk averse agens wan o hold o insure agains flucuaions in heir marginal uiliy of consumpion. A sriking propery of he compeiive equilibrium is ha aggregae leverage is procyclical, an inuiive resul bu one ha does no obain in sandard models. The reason hinges on he decrease in aggregae risk aversion in good imes, which makes agens wih high risk bearing capaciy willing o ake on a larger fracion of he aggregae risk by issuing more risk-free deb o agens wih lower risk bearing capaciy. Thus, procyclical leverage emerges naurally as he resul of he opimal rading of uiliy maximizing agens in an equilibrium ha in fac implemens an opimal risk sharing allocaion. Besides procyclical aggregae leverage, our model has several addiional predicions ha are consisen wih numerous sylized facs. Firs, higher aggregae leverage should be correlaed wih (i) high valuaion raios, (ii) low volailiy, (iii) lower fuure excess reurns, and (iv) a consumpion boom of hose agens who lever up, who hen should experience a consumpion slump relaive o ohers, on average. The reason is ha as explained above, in good imes leverage increases as aggregae risk aversion declines. Lower risk aversion implies high valuaion raios and lower sock reurn volailiy, as well as lower fuure excess reurns, explaining (i) hrough (iii). In addiion, levered agens who ook up levered posiions do especially well when sock marke increases, implying higher consumpion in good imes. Mean reversion, however, implies ha hese same agens should also expeced a relaively lower expeced fuure consumpion growh afer heir consumpion binge, explaining (iv). 2

4 Second, our model also implies acive rading. For insance, a series of negaive aggregae shocks induces deleveraging of levered agens hrough he acive sales of heir posiions in risky socks. I follows ha sock price declines occur exacly a he ime when levered agens acively sell heir risky posiions o reduce leverage. This commonaliy of asse sales and sock price declines give he impression of a selling pressure affecing asse prices, when in fac hey are he equilibrium oucome of business cycle variaion and is differenial impac on risk aversion. Indeed, our model make ransparen he fac ha equilibrium prices and quaniies comove due o aggregae sae variables, bu here is no causal relaion beween rading and price movemens. In he special case of idenical agens, in fac, our model revers back o a sandard represenaive agen model such as in Campbell and Cochrane (1999) and Menzly, Sanos, and Veronesi (24), which feaure no rading. Ye, all of he asse pricing implicaions are idenical. Third, while our model implies ha during bad imes aggregae leverage declines, levered agens deb-o-wealh raios increase, as wealh declines faser han deb due o severe discoun-rae effecs. This implies ha while he aggregae level of deb is pro-cyclical (i.e. lower aggregae deb in bad imes), he deb-o-wealh raio of levered agens is counercyclical (i.e. higher deb-o-wealh in bad imes). Broadly inerpreing levered agens as inermediaries (hey receive funding from unlevered agens o increase heir risky posiions), he model is hus also consisen wih he recen lieraure abou he ambiguous impac of negaive aggregae shocks on he leverage of inermediaries. Specifically, we find ha he level of leverage decreases bu deb-o-equiy raios increase, as equiy drop faser han leverage due o discoun rae effecs. Moreover, because such leverage is endogenously negaively relaed o aggregae risk aversion, i would naurally become a pricing facor for asse prices, as emphasized in recen lieraure. Finally, our model has predicions abou he source of he variaion in wealh inequaliy. We show ha heerogeneiy in preferences and in endowmens work ogeher o exacerbae wealh inequaliy during good imes, bu hey work in opposie direcions in bad imes. Tha is, lower discoun raes (due o he lower aggregae risk aversion) and heerogeneiy in sock posiions (due o heerogeneous preferences) boh increase wealh dispersion when imes are good. However, when imes are bad, higher discoun raes end o decrease he wealh dispersion due o iniial differen endowmens, bu heerogeneiy in habis sill end o exacerbae hem. These effecs imply a complex dynamics of wealh dispersion ha depends boh on endowmens, bu also on opimal acions of agens ha are affeced by differenial habis. Once again, he model emphasizes ha while asse prices affec wealh inequaliy, he converse does no hold, as asse prices are idenical wih homogeneous agens, and hence in he same model wihou wealh dispersion. 3

5 Clearly many explanaions have been pu forh o explain he growh of leverage and of household deb in paricular during he run up o he crisis. For insance, among ohers, Bernanke (25) argues ha he global savings glu, he excess savings of Eas Asian naions in paricular, is o blame for he ample liquidiy in he years leading up o he Grea Recession, which reduced raes and faciliaed he remarkable rise in household leverage; Shin (212) shows how regulaory changes, he adopion of Basel II, led European banks o increase lending in he US; Pino (21), Wallison (211) and Calomiris and Haber (214) argue ha he Communiy Reinvesmen Ac played a pivoal role in he expansion of morgage lending o risky households (bu see Bhua and Ringo (215)); Mendoza and Quadrini (29) show how world financial inegraion leads o an increase in ne credi. The lis goes on. When he crisis came, he crash in prices and he rapid deleveraging of households and financial inermediaries was inerpreed appealing o classic inefficien runs argumens a la Diamond and Dybvig (1983) as in Goron and Merick (21) or conagion. He and Krishnamurhy (28) connec he fall in asse prices o he shorage of capial in he inermediaion secor. Finally, much research has focused on he impac ha he crisis had on he consumpion paerns of households. For insance Mian and Sufi (214) argue ha deb overhang is o blame for he drop in consumpion in counies where households were grealy levered. Our poin here is no o claim ha hese fricions are no imporan bu simply o offer an alernaive explanaion ha is consisen wih complee markes and ha maches wha we know from he asse pricing lieraure. We argue for insance ha when deb overhang is pu forh as an explanaion for low consumpion paerns amongs levered households he alernaive hypohesis of efficien risk sharing canno be dismissed ourigh. Boh explanaions operae in he same direcion and hus assessing he quaniaive plausibiliy of one requires conrolling for he oher. Our model has he considerable advanage of simpliciy: All formulas for asse prices, porfolio allocaion, and leverage are in closed form, no numerical soluions are required, and heir inuiion follows from basic economic principles. Moreover, because our model aggregaes o he represenaive agen of Menzly, Sanos, and Veronesi (24), excep ha we allow for ime varying aggregae uncerainy, we can calibrae is parameers o mach he properies of aggregae reurn dynamics. As such, our model has no only qualiaive implicaions as mos of he exising lieraure bu quaniaive implicaions as well. This paper is relaed o he lieraure on opimal risk sharing, saring wih Borch (1962). Our paper is closely relaed o Dumas (1989), Wang (1996), Bolon and Harris (213), and Longsaff and Wang (213). These papers consider wo groups of agens wih consan 4

6 risk aversion, and rading and asse prices are generaed by aggregae shocks hrough he variaion in he wealh disribuion. While similar in spiri, our model considers a coninuum of agens whose risk preferences are ime varying due o heir agen-specific exernal habi preferences. Our main source of variaion is aggregae economic uncerainy which is absen in hese earlier papers as i correlaes wih agens risk aversion. Our model is also relaed o Chan and Kogan (22), who consider a coninuum of agens wih habi preferences and heerogeneous risk aversion. In heir seing, however, he risk aversions of individual agens are consan, while in our seing risk aversions of individual agens are ime varying in response o variaion in aggregae uncerainy, a crucial ingredien in our model. Finally, our paper also connecs o he recen lieraure ha ries o shed ligh on he deerminans of he supply of safe asses; see for insance Barro and Mollerus (214) and Caballero and Fahri (214), hough his lieraure is more ineresed in he implicaions of he shorage of safe asses for macroeconomic aciviy. The paper is srucured as follows. The nex secion presens he model. Secion 3 characerizes he opimal risk sharing arrangemen and Secion 4 decenralizes he efficien risk sharing allocaion and characerizes he compeiive equilibrium. Secion 5 evaluaes he model quaniaively and Secion 6 concludes. All proofs are in he Appendix. 2. The model Preferences. There is a coninuum of agens endowed wih log uiliy preferences defined over consumpion C i in excess of agen-specific exernal habi indices X i : u (C i,, X i,, ) = e ρ log (C i X i ) Agens are heerogeneous in he habi indices X i, which are given by ) X i = g i (D X j dj (1) Tha is, he habi level X i of agen i is proporional o he excess aggregae oupu D over average habi X j dj, which we call excess oupu henceforh. A higher excess oupu decreases agen i s uiliy, an effec ha capures a noion of Envy he Joneses. The excess oupu ( D X j dj ) is in fac an index of he happiness of he Joneses heir uiliy is higher he higher he disance of D from average habi X j dj a fac ha makes agen i less happy as i pushes up his habi level X i and hus reduces his uiliy. Our model is hus an exernal habi model defined on uiliy as opposed o consumpion in ha oher people happiness is negaively perceived by agen i. 5

7 The sensiiviy of agen i s habi X i o aggregae excess oupu (D X j dj) depends on he agen-specific proporionaliy facor g i, which is heerogeneous across agens and depends linearly on a sae variable, o be described shorly, Y : g i = a i Y + b i (2) where a i > and b i are heerogeneous across agens and such ha a i di = 1. Endowmen. Aggregae endowmen which we also refer o as dividends or oupu follows he process dd D = µ D d + σ D (Y ) dz (3) where he drif rae µ D is consan. 6 The volailiy σ D (Y ) of aggregae endowmen which we refer o as economic uncerainy depends on he sae variable Y, which follows [ ( )] dd dd dy = k (Y Y ) d v Y E D ( Tha is, Y increases afer bad aggregae shocks, dd dd D < E D ), and i hovers around is cenral endency Y. I is useful o inerpre Y as a recession indicaor: During good imes Y is low and during bad imes Y is high. We assume hroughou ha Y is bounded below by a consan λ 1. This echnical resricion is moivaed by our preference specificaion above and i can be achieved by assuming ha σ D (Y ) as Y λ (under some echnical condiions). We oherwise leave he diffusion erms σ D (Y ) in (3) unspecified for now, alhough, o fix ideas, we normally assume ha economic uncerainy is higher in bad imes, i.e. σ D(Y ) >. A ime each agen is endowed wih a fracion w i of he aggregae endowmen process D. The fracions w i saisfy w i di = 1, and he echnical condiion w i > a i(y λ) + λ 1 Y D (4) (A1) which ensures ha each agen has sufficien wealh o ensure posiive consumpion over habi in equilibrium, and hence well defined preferences. A1 is assumed hroughou. Discussion. Our preference specificaion differs from he sandard exernal habi model of Campbell and Cochrane (1999) and Menzly, Sanos and Veronesi (24, MSV henceforh). 6 As will be shown below he drif µ D does no play any role ino any of relevan formulas, excep for he risk-free rae. The main resuls of he paper are hus consisen wih a richer specificaion of he drif µ D. 6

8 In paricular, noice ha our model is one wihou consumpion exernaliies as habi levels depend only on exogenous processes and no on consumpion choices. This, as shown below, will allow he applicaion of sandard aggregaion resuls which will considerably simplify he characerizaion of opimal sharing rules. Second our model feaures wo relevan sources of variaion across agens: Wealh, as summarized by he disribuion of ω i, and he sensiiviy of individual habis X i o excess oupu, as summarized by g i, which resuls in differences in aiudes owards risk. These wo dimensions seem a naural saring poin o invesigae opimal risk sharing as well as porfolio decisions. 7 Clearly, one could conemplae oher sources of variaion in he cross secion of households such as differences in beliefs or in invesmen opporuniy ses o which he agens have access. Noice hough ha our model feaures no idiosyncraic shocks o individual endowmen as agens simply receive a consan fracion w i of he aggregae endowmen process. Individual endowmen processes are hus perfecly correlaed and hus hey are no he driver of risk sharing moives. Insead in our model risk sharing moives arise exclusively because agens are exposed differenly o business cycle flucuaions hrough heir sensiiviy o habis. Indeed how sensiive agens are o shocks in excess oupu depend on he sae variable Y. Economically, assumpion (2) implies ha in bad imes (afer negaive oupu shocks) he habi loadings g i increase, making habi preferences become more imporan on average. However, differen sensiiviies a i imply ha changes in Y differenially impac he exernal habi index as g i increase more for agens wih high a i han for hose wih low a i. We se b i = λ(1 a i ) 1, which ensures g i > for every i and for every (as Y > λ), and allows for a simple aggregaion below. This assumpion does no affec he resuls. Finally, we noe ha he case of homogeneous preferences (a i = 1 for all i) and/or homogeneous endowmens (w i = 1 for all i) are special cases, as is he case in which habis are consan (v = in (4)). We invesigae hese special cases as well below. 7 For insance, wo recen heoreical conribuions ha consider hese wo sources of cross secional variaion are Longsaff and Wang (212) and Bolon and Harris (213). Empirically hese sources of variaion have been invesigaed by, for example, Chiappori and Paeilla (211) and Calve and Sodini (214), hough he resuls in hese wo papers are raher differen. 7

9 3. Opimal risk sharing As already menioned, markes are complee and herefore sandard aggregaion resuls imply ha a represenaive agen exiss, a planner, ha solves he program U (D, {X i }, ) = max φ i u (C i, X i, )di subjec o C i di = D (5) C i where all Pareo weighs φ i > are se a ime zero, renormalized such ha φ i di = 1 and are consisen wih he iniial disribuion of wealh in a way o be described shorly. The firs order condiion implies ha u C (C i, X i, ) = φ ie ρ C i X i = M for all i (6) where M is he Lagrange muliplier associaed wih he resource consrain in (5). 8 Sraighforward calculaions 9 show ha M = e ρ (D X j dj) ( and C i = (g i + φ i ) D ) X j dj. (7) The opimal consumpion of agen i increases if he aggregae excess oupu ( D X j dj ) increases or if he habi loading g i increases. This is inuiive, as such agens place relaively more weigh on excess oupu and hus wan o consume relaively more. In addiion, agens wih a higher Pareo weigh φ i also consume more, as such agens have a larger relaive endowmen. The opimal consumpion plans (7) seems o indicae ha as g i increases, he consumpion of all agens will increase (recall ha g i are perfecly correlaed), and hus exceed oal oupu D. This does no happen because of he effec of g i on he habi levels X i, which decreases he excess oupu D X i di. In fac, we can aggregae oal opimal consumpion and impose marke clearing o obain [ D = C i di = ] ( (g i + φ i )di D ) X i di (8) Using φ i di = 1, we can solve for he equilibrium excess oupu as D D X i di = gi di + 1 > (9) 8 This resul was firs derived by Borch (1962, equaion (1) p. 427). 9 I is enough o solve for C i in (6), inegrae across agens (recall φ i di = 1), and use he resource consrain o yield M. Plugging his expression in (6) yields C i. 8

10 This inermediae resul also shows ha individual excess consumpion C i X i is posiive for all i, which ensures all agens uiliy funcions are well defined. 1 Noice also an imporan implicaion of (9) and i is ha preferences can be expressed as u (C i,, X i,, ) = e ρ log (C i ψ i D ) wih ψ i g i gi di + 1. Individual agens compare heir own consumpion o aggregae endowmen properly scaled by ψ i, which is agen specific and dependen on Y. Roughly agens care abou heir relaive sanding in sociey, which is subjec o flucuaions. I is hese flucuaions wha inroduces moives for risk sharing. The nex proposiion solves for he Pareo weighs and he share of he aggregae endowmen ha each agen commands and illusraes he basic properies of he opimal risk sharing rules in our model. Proposiion 1 (Efficien allocaion). Le he economy be a is sochasic seady sae a ime, Y = Y, and normalize D = ρ. Then (a) he Pareo weighs are φ i = a i λ + (w i a i )Y + 1 λ (1) (b) The share of he aggregae endowmen accruing o agen i is given by C i = [a i + (w i a i ) YY ] D or s i C i D = a i + (w i a i ) Y Y (11) Pareo weighs (1) are increasing in he fracion of he iniial aggregae endowmen w i and decreasing in habi sensiiviy a i. The firs resul is sandard. To undersand he second, given opimal consumpion (7), agens wih higher sensiiviy a i have a higher habi loading g i = a i (Y λ) + λ 1 and hus would like o consume more. Given (7), for given iniial endowmen w i, he Pareo weigh φ i mus hen decline o ensure ha such consumpion can be financed by he opimal rading sraegy. Equaion (11) capures he essenial properies of he opimal risk sharing rule, ha is, agens wih high w i or low a i enjoy a high consumpion share s i = C i /D during good imes, ha is, when he recession indicaor Y is low, and vice versa. To grasp he inuiion consider firs he curvaure of he uiliy funcion of an individual agen, which we refer o as risk aversion for simpliciy: Curv i = C iu cc (C i, X i, ) u c (C i, X i, ) a i (Y λ) + λ 1 = 1 + w i Y a i (Y λ) λ + 1 (12) 1 To see his, subsiue he excess oupu ino (7) and use (1). Given g i in (2), we have g i di + 1 = Y. 9

11 I is he combinaion of wealh and sensiiviy o excess oupu wha deermines he agen s aiude owards risk: Agens wih higher wealh w i or lower habi loading a i have lower risk aversion. Moreover, an increase in recession indicaor Y increases he curvaure of every agen, bu more so for agens wih a high habi loading a i. These wo effecs combine o deermine he planner s ransfer scheme needed o suppor he opimal allocaion. Le τ i > be he ransfer received by agen i a ime above her endowmen w i D ; if insead he agen consumes below her endowmen hen τ i <. Trivial compuaions prove he nex corollary. 11 Corollary 2 The ransfers ha implemen he efficien allocaion are given by ) τ i = (w i a i ) (1 YY D. (13) Noice ha agens for whom w i a i > receive ransfers, τ i >, when Y < Y, ha is in good imes and pay τ i < in bad imes, when Y > Y. The opposie is he case for he agens for whom w i a i <. In effec, opimal risk sharing requires agens wih w i a i > o insure agens wih w i a i <. The inuiion of expression (11) is now clear. The consumpion of agen i depends on boh he aggregae consumpion, D, and he sae variable Y, our recession indicaor. There are wo effecs in equaion (11). Firs for a given Y agens wih high wealh or low sensiiviy o excess oupu, for whom w i a i >, have more risk bearing capaciy han agens for whom w i a i < and hus insure he poorer or more risk averse agens. As a resul he shares of consumpion of he agens wih large risk bearing capaciy flucuae more wih aggregae consumpion. This is he sandard resul as found for insance in Longsaff and Wang (212) (see heir Proposiion 1 as well as equaion (16) in ha paper). The second effec is due o he Joneses feaure of our preferences: As Y drops he risk bearing capaciy of he agens wih larger risk bearing capaciy (w i a i > ) increases furher and hus he flucuaions of he shares is even sronger han in he case where agens have, say, simply power uiliy funcions. This resul is reminiscen of Chan and Kogan (22, Lemma 1). An imporan propery of he share of consumpion of agen i, s i, is ha i inheris he saionariy properies of Y. Thus, unlike models in which agens differ in heir degree of consan relaive risk aversion he disribuion of wealh does no degenerae, as for example in Dumas (1989) and Wang (1996), where he proporion of wealh held by he leas risk averse agens converges o one Simply subrac from he opimal consumpion allocaion (11) he consumpion under auarchy, w i D. 12 A sandard modeling device o obain saionary disribuions and avoid degeneracy in he long run is o 1

12 We emphasize an imporan aribue of our model and ha is ha habis are key o deliver all he resuls in our paper. Indeed, assume ha Y = Y for all (i.e. v = in (4)). In his case our model collapses o an economy populaed wih agens wih log preferences, he share of consumpion of each agen is simply s i = w i and, as i will be shown below, no rading occurs amongs agens. Thus, our model does no deliver risk sharing moives beyond wha is induced by he habi feaures of our preference specificaion. 4. Compeiive equilibrium 4.1. Decenralizaion Financial markes. Having characerized he opimal allocaion of risk across agens in differen saes of naure we urn nex o he compeiive equilibrium ha suppors i. Clearly we can inroduce a complee se of Arrow-Debreu markes a he iniial dae, le agens rade and afer ha simply accep delivery and make paymens. I was Arrow s (1964) original insigh ha decenralizaion can be achieved wih a sparser financial marke srucure. There are obviously many ways of inroducing his sparser financial marke srucure bu here we follow many ohers and simply inroduce a sock marke and a marke for borrowing and lending. Specifically we assume ha each of he agens i is endowed wih an iniial fracion w i of a claim o he aggregae endowmen D. We normalize he aggregae number of shares o one and denoe by P he price of he share o he aggregae endowmen process, which is compeiively raded. Second, we inroduce a marke for borrowing and lending beween agens. Specifically we assume ha here is an asse in zero ne supply, a bond, wih a price B, yielding an insananeous rae of reurn of r. Boh P and r are deermined in equilibrium. Because all quaniies depend on one Brownian moion (dz ), markes are dynamically complee. The porfolio problem. Armed wih his we can inroduce he agens problem. Indeed, given prices {P, r } agens choose consumpion C i and porfolio allocaions in socks N i and bonds Ni o maximize heir expeced uiliies [ ] max {C i,n i,ni} E e ρ log (C i X i ) d subjec o he budge consrain equaion dw i = N i (dp + D d) + N ib r d C i d have agens die and be replaced by children wih randomly assigned coefficiens of relaive risk aversion. For a recen applicaion of his idea see Barro and Mallerus (214). 11

13 wih iniial condiion W i, = w i P. Definiion of a compeiive equilibrium. A compeiive equilibrium is a series of sochasic processes for prices {P, r } and allocaions {C i, N i, Ni } i I such ha agens maximize heir ineremporal uiliies and markes clear C i di = D, N i di = 1, and N i di =. The economy sars a ime in is sochasic seady sae Y = Y. Wihou loss of generaliy, we normalize he iniial oupu D = ρ for noaional convenience. The compeiive and he decenralizaion of he efficien allocaion. We are now ready o describe he compeiive equilibrium and show ha i indeed suppors he efficien allocaion. We leave he characerizaion of he equilibrium for he nex secion. Proposiion 3 (Compeiive equilibrium). Consider he following prices and porfolio allocaions 1. Sock prices and ineres raes ( ρ + ky Y 1) P = D (14) ρ (ρ + k) ) r = ρ + µ D (1 v)σ 2 D(Y ) + k (1 YY (15) 2. The posiion in bonds N i B and socks N i of agen i a ime are, respecively, N i B = v (w i a i )H (Y ) D (16) N i = a i + (ρ + k)(1 + v)(w i a i )H (Y ) (17) where H (Y ) = Y Y 1 ρ + k(1 + v)y Y 1 > (18) ( Then he processes P, r, N i, N ) i he efficien allocaion, (11). consiue a compeiive equilibrium which suppors We commen on hese resuls in he nex few subsecions. 12

14 4.2. Asse prices The sock price in Proposiion 3 is idenical o he one found in MSV, which was obained in he conex of a represenaive consumer model. The reason is ha our model does indeed aggregae o yield a represenaive consumer which is similar o he one posied in ha paper. Indeed, having solved for he Pareo weighs (1) and he individual consumpion allocaions we can subsiue back in he objecive funcion in (5) and obain he equilibrium sae price densiy associaed wih he represenaive agen, which we characerize in he nex Proposiion. Proposiion 4 (The sochasic discoun facor). The equilibrium sae price densiy is which follows and where r is given by (15). 13 M = e ρ D 1 Y, (19) dm M = r d σ M, dz wih σ M, = (1 + v)σ D (Y ), (2) This sae price densiy is similar o one obained in he represenaive agen, exernal habi models of Campbell and Cochrane (1999) and MSV. Indeed, we can define he surplus consumpion raio as in Campbell and Cochrane (1999) S = D X i di D = 1 Y (21) where he las equaliy sems from (9). The recession indicaor Y is hen he inverse surplus consumpion raio of MSV. Indeed, as in his earlier work, Y can be shown o be linearly relaed o he aggregae risk aversion of he represenaive agen (see foonoe 4 in MSV). As in MSV, we someimes refer o Y as he aggregae risk aversion of he economy. In wha follows, we express he resuls as funcions of he surplus consumpion raio S = 1/Y for noaional convenience. The surplus consumpion raio increases afer posiive aggregae shocks, ha is, S is high in good imes. Wih a small abuse of noaion, we denoe funcions of Y, such as oupu volailiy σ D (Y ), simply as funcions of S, σ D (S ). So, for insance, he funcion H (Y ) in (18) becomes H (S ) = Y S ρ + k(1 + v)y S > (22) 13 In wha follows and o lighen up he noaion we drop he from he compeiive equilibrium. 13

15 We are now ready o discuss he asse prices in Proposiion 3. Sar, briefly, wih he risk free rae r. The erms ρ + µ D σ 2 D(S ) in (15) are he sandard log-uiliy erms, namely, ime discoun, expeced aggregae consumpion growh, and precauionary savings. The addiional wo erms, k(1 Y S ) and v σ D (S ), are addiional ineremporal subsiuion and precauionary savings erms, respecively, associaed wih he exernal habi feaures of he model (see MSV for deails). As for he sock price, given ha S = Y 1 P = E [ M τ M D τ dτ, we can wrie ] ( ρ + ky S = ρ(ρ + k) ) D. The inuiion for his is by now sandard (Campbell and Cochrane (1999) and MSV). A negaive aggregae shock dz < decreases he price direcly hrough is impac on D, bu i also increases he risk aversion Y, and hus reduces S, which pushes down he sock price P furher. Noice ha in he seady sae, when S = Y 1, he price of he sock is P = ρ 1 D, which is he price ha obains in he sandard log economy. Exernal habi persisence models hus generae variaion in prices ha are driven no only by cash-flow shocks bu also discoun effecs. Indeed, we show in he Appendix he volailiy of sock reurns is ( σ P (S ) = σ D (S ) 1 + vky S ) ρ + ky S In addiion, as shown in (2), he marke price of risk also is ime varying, no only because of he variaion in consumpion volailiy (σ D (S )) bu also because of he variaion in he volailiy of aggregae risk aversion, given by vσ D (S ). In MSV, a lower surplus consumpion raio S increases he average marke price of risk and makes i ime varying. This generaes he predicabiliy of sock reurns. Indeed (23) E [dr P r(s )d] = σ M (S )σ P (S )d (24) where dr P = (dp + D d)/d. The risk premium E [dr P ] r(s )d increases compared o he case wih log uiliy boh because he aggregae amoun of risk σ P (S ) increases and because he marke price of risk σ M (S ) increases. An imporan propery of asse prices (P and r ) in our model is summarized in he nex Corollary. Corollary 5 Asse prices are independen of he endowmen disribuion across agens as well as he disribuion of preferences. In paricular he model has idenical asse pricing implicaions even if all agens are idenical, i.e. a i = 1 and w i = 1 for all i. 14

16 The asse pricing implicaions of our model are hus orhogonal o is cross secional implicaions: P in equaion (14) and r in (15) are independen of he disribuion of eiher curren consumpion or wealh in he populaion. This disinguishes our model from, for insance, Longsaff and Wang (212, Proposiion 2 equaion (18)) or Chan and Kogan (22, Lemma 2). For insance Chan and Kogan (22) consider caching up wih he Joneses preferences as in Abel (199), (1 γ) 1 (C/X ) 1 γ, where agens differ in he degree of curvaure γ. Pricing in ha paper depends on he cross secional disribuion of γ. Insead, he presen paper uses preferences ha are, roughly, a logarihmic version of he exernal habi model of Campbell and Cochrane (1999) and hey aggregae so as o eliminae from he Lagrange muliplier associaed wih he resource consrain, expression (6), all dependence from he disribuion of Pareo weighs φ i. 14 In boh Chan and Kogan (22) and in Longsaff and Wang (212) variaion in risk premia is driven by endogenous changes in he cross-secional disribuion of wealh. Roughly more risk-oleran agens hold a higher proporion of heir wealh in socks. A drop in sock prices reduces he fracion of aggregae wealh conrolled by such agens and hence heir conribuion o he aggregae risk aversion. The condiional properies of reurns rely hus on srong flucuaions in he cross secional disribuion of wealh. Insead in he presen paper agens risk aversions change inducing addiional variaion in premia and puing less pressure on he changes in he disribuion of wealh o produce quaniaively plausible condiional properies for reurns. Indeed, Corollary 5 assers exacly ha asse pricing implicaions are idenical even when agens are homogeneous and hus here is no variaion in cross-secional disribuion of wealh. This of course does no mean ha secular changes in he disribuion of wealh canno affec long run rends in asse prices. For insance, Hall (216) has recenly proposed ha changes in he proporion of wealh of he more risk averse agens in sociey explain he secular decline in ineres raes in he USA. 15 Corollary 5 allows us o separae cleanly he asse pricing implicaions of our model from is implicaions for rading, leverage and risk sharing, which we furher discuss below. In paricular, he corollary clarifies ha equilibrium prices and quaniies do no need o be causally relaed o each oher, bu raher comove wih each oher because of fundamenal sae variables, such as S in our model. 14 To see his noice ha he equilibrium price-dividend raio in ha paper depends on he shadow price of he resource consrain which in urn depends on he weigh he planner aaches o he agen whose aiudes owards risk are given by γ, which is f (γ) (see expression (9) and (13) in Chan and Kogan (22)). 15 Hall (216) also emphasizes differences in beliefs as a second source of heerogeneiy across agens. 15

17 4.3. Leverage and risk sharing We urn nex o he characerizaion of he porfolio sraegies in Proposiion 2. The nex Corollary follows immediaely from ha Proposiion. Corollary 6 (Individual leverage). (a) The posiion in bonds is N ib < if and only if w i a i >. Tha is, agens wih w i > a i ake on leverage. (b) The invesmen in sock of agen i in proporion o wealh is ( ρ 1 + v 1 N i P W i = 1 + v ρ+y [k+(ρ+k)(w i a i )/a i ]S ) ( 1 ρ ρ+y ks ) (25) Therefore N i P W i > 1 if and only if w i a i > Recall ha, as shown in equaion (13), opimal risk sharing requires ransfers from agens wih w i a i > o hose wih w i a i < when Y is high (or S is low) and he opposie when Y is low (or S is high). Equaions (16) and (17) show he porfolios of socks and bonds needed o implemen he efficien allocaion. This is achieved by having he agens wih large risk bearing capaciy, agens wih w i a i >, issue deb in order o insure hose agens wih lower risk bearing capaciy, w i a i <. Par (b) of Corollary 1 shows ha indeed agens wih w i a i > lever up o achieve a posiion in socks ha is higher han 1% of heir wealh. Expression (25) shows ha for given level of habi sensiiviy a i, agens wih higher wealh w i inves comparaively more in socks, a resul ha finds empirical suppor in Wacher and Yogo (21). Indeed, as in heir paper, our habi preferences imply ha uiliy is no homoheic in wealh (due o habi), hereby implying ha agens wih a higher endowmen inves comparaively more in he risky asse. Expressions (16) and (17) show ha he amoun of leverage and asse allocaion depend on he funcion H (Y ), which is ime varying as he recession indicaor Y moves over ime. We discuss he dynamics of leverage in he nex secion. 16

18 4.4. The supply of safe asses: Leverage dynamics A paricular feaure of our model is ha ha he risk aiudes of he agens in he economy flucuae wih he recession indicaor Y (see equaion (12)). As Y increases, for insance, he risk bearing capaciy of he agens for whom w i a i > decreases precisely when he demand for insurance by he agens wih w i a i < increases. The supply of safe asses, o use he erm ha has become sandard in he recen lieraure, may decrease precisely when i is mos needed. 16 This quesion has been a he hear of much discussion regarding he deerminans of he supply of safe asses. 17 In his secion we focus on he dynamics of he aggregae leverage raio, which we define as L(S ) = i:n i, < N ib di where he negaive sign is o make his number posiive and recall S = 1/Y. I is immediae o see ha aggregae leverage/oupu raio is L(S ) = vk 1 H (S ) where K 1 = (w i a i ) di > (26) D i:(w i a i )> and he funcion H (S ) is in (22). The following corollary characerizes he dynamics of aggregae leverage: Corollary 7 (Aggregae leverage). H (S ) is sricly increasing in S. Hence, aggregae leverage L(S ) is procyclical, increasing in good imes (high S ) and decreasing in bad imes (low S ). To gain inuiion on Corollary 7, we proceed in seps. Firs, if habi S = 1/Y is consan, i.e. v =, hen N ib = and N i = w i in (16) and (17). Tha is, here is no leverage and each agen i simply holds he fracion w i of shares wih which hey are iniially endowed and he model revers o he sandard log-uiliy case In our framework he deb issued by he agens wih he larges risk bearing capaciy is safe because hey delever as negaive shocks accumulae in order o mainain heir marginal uiliy bounded away from infiniy. 17 See for insance Barro and Mollerus (214), who propose a model based on Epsein-Zin preferences o offer predicions abou he raio of safe asses o oupu in he economy. Goron, Lewellen and Merick (212) and Krishnamurhy and Vissing-Jorgensen (212) provide empirical evidence regarding he demand for safe asses. In all hese papers he presence of ouside deb in he form of governmen deb plays a criical role in driving he variaion of he supply of safe asses by he privae secor, a mechanism ha is absen in his paper. 18 In his case he model is similar o a log-uiliy model wih subsisence, ln(c i X i ), wih X i = ψ i D. Given ha X i is jus proporional o aggregae oupu D, preferences are homoheic in wealh and sandard log-uiliy resuls obain. 17

19 When S is ime varying and responds o aggregae oupu shocks, leverage is procyclical. Inuiively, during good imes (S high) agens wih high endowmen w i and low habi a i have lower risk aversion compared o oher agens (see expression (12)). As a resul, hey become even more willing o ake on aggregae risk in such imes compared o bad imes, which hey do by issuing even more risk-free deb o agens wih lower risk bearing capaciy, i.e. hose wih low endowmen or a higher habi loading. Procyclical leverage hen emerges as a naural oucome in equilibrium. While an aggregae procyclical leverage may seem inuiive, i is no normally implied by, for insance, sandard CRRA models wih differences in risk aversion. In such models, less risk averse agens borrow from more risk averse agens, who wan o hold riskless bonds raher han risky asses. As aggregae wealh becomes more concenraed in he hands of less risk-averse agens, he need of borrowing and lending declines, which in urn decreases aggregae leverage. Moreover, a decline in aggregae uncerainy which normally occur in good imes acually decreases leverage in such models, as i reduces he risk-sharing moives of rade. In our model, in conras, he decrease in aggregae risk aversion in good imes make agens wih high-risk bearing capaciy even more willing o ake on risk and hence increase heir supply of risk-free asses o hose who have a lower risk bearing capaciy. Corollary 7 finally implies ha aggregae leverage L(S ) is high when S is high. However, good imes are imes when expeced excess reurns are likely low, as he marke price of risk σ M (S ) is low, and aggregae uncerainy σ D (S ) is likely low. 19 Therefore, under hese assumpions, high aggregae leverage L(S ) should predic low fuure excess reurns Individual leverage and consumpion The following corollary follows immediaely from Proposiion 1 and Corollary 6. Corollary 8 Agens wih higher leverage enjoy higher consumpion share during good imes. Afer a sequence of good economic shocks aggregae risk aversion declines. Thus, agens wih posiive (w i a i ) increase heir leverage and experience a consumpion boom. The wo effecs are no direcly relaed, however. The increase in consumpion is due o he higher invesmen in socks which payoffs in good imes. Because good imes also have lower aggregae risk aversion, moreover, hese same agens also increase heir leverage a hese 19 Noe ha we have no made any assumpions ye on σ D (S ), excep ha i vanishes for S λ 1. 18

20 imes. Hence, our model predics a posiive comovemen of leverage and consumpion a he household level. An implicaion of his resul is ha agens who ook on higher leverage during good imes are hose ha suffer a bigger drop in consumpion growh as S mean revers. In paricular, we have he following corollary abou agens consumpion growh: Corollary 9 Agen i s consumpion growh saisfies [ ] dci E /d = µ C D + (w i a i )Y S F(S ) (27) i a i + (w i a i )Y S wih F(S ) = k(1 Y S ) + v σ 2 D (S ) (28) If σ D (S ) is decreasing in S wih σ D (λ 1 ) =, hen he funcion F(S) has F (S) < and F() > and F(λ 1 ) = k(1 λ 1 Y ) <. Thus, here exiss a unique soluion S o F(S ) = such ha for all i and j wih w i a i > and w j a j < we have [ ] [ ] dci dcj E < µ C D < E for S > S (29) i C j [ ] [ ] dci dcj E > µ D > E for S < S (3) C i C j This corollary shows ha cross-secionally agens wih high w i a i > have a lower expeced growh rae of consumpion when S is high. We know ha hese are also imes when such agens are heavily leveraged. I follows hen ha agens who are heavily leveraged enjoy boh a high consumpion boom in good imes, bu a lower fuure expeced consumpion growh. These agens also expec a higher consumpion growh when S is low. Therefore, Corollaries 7 and 9 imply he following: Corollary 1 Periods wih high aggregae leverage L(S ) forecas lower consumpion growh of agens who were highly leveraged compared o hose wih lower leverage. Tha is, according o Corollary 1, periods of very high aggregae leverage should follow on average by periods in which levered agens rerench and experience consumpion growh ha is comparaively lower han hose agens who did no ake on leverage. This implicaion of our model speaks o some of he recen debae regarding he low consumpion growh of levered households following he Grea Recession. Some argue ha 19

21 he observed drop in consumpion growh was purely due o a wealh effec, as levered households end o live in counies ha experienced big drops in housing values, whereas ohers have emphasized he criical role of deb in explaining his drop. 2 Clearly hese effecs are imporan bu our conribuion is o show ha high leverage followed by low consumpion growh is precisely wha arises from risk sharing argumens in models ha can address he observed condiional properies of asse reurns, as exernal habi models do. Indeed, suppose ha such cross-secional differences have a spaial naure, e.g. counies wih richer agens or agens wih lower risk aversion. Then, such counies should experience a credi boom during good imes wih high consumpion growh, followed by a relaive consumpion slump during bad imes. Corollary 1 hen highlighs he crucial role of proper idenificaion sraegies when assering ha higher leverage casually make levered counies (or counries) suffer comparaively slower growh in he fuure. If here is any residual correlaion beween he insrumen employed in he idenificaion sraegy and S in previous equaions, hen he causaliy inerpreaion of sandard insrumenal variable or diff-in-diff esimaors is undermined Acive rading in socks and bonds The following corollary explores he implicaions of our model for sock rading. Corollary 11 (Acive rading) (a) Agens wih posiive leverage (i.e. wih w i a i > ) increase heir sock posiion in good imes (when S increases), and decrease heir sock posiion in bad imes, (when S declines.) (b) Agens wih higher absolue difference w i a i rade more in response o changes o he aggregae surplus consumpion raio S. Corollary 11-a says ha agens wih posiive leverage increase he number of unis of socks purchased in good imes, and decrease hem in bad imes. Noice ha here is an acive rading from he par of hese agens. In a model wih passive invesors, an agen who is long socks may mechanically find himself wih a higher allocaion in socks during good imes because he sock yields good reurns in good imes. Even if he unis of socks purchased o do no change, N i = consan, such an agen would sill have a higher posiion 2 See for insance Mian and Sufi (214, in paricular pages 39-45) for a nice exposiion of his debae. 2

22 in socks during good imes han during bad imes. Corollary 11-a insead says ha an agen who is leveraged (w i a i > ) acively increases leverage in good imes o buy more shares of socks in such imes. Conversely, and imporanly, Corollary 11-a also implies ha such levered raders acively deleverage as imes are geing worse (S declines) by acively selling he risky asses. The model hus implies acive deleveraging from levered invesors, which is a ypical behavior of levered insiuions a he onse of financial crisis. Because deleveraging occurs as boh he sock price plunges and in fac, also he wealh of levered invesors drops, i may appear ha a selling pressure of levered agens and who are deleveraging is he cause of he drop in he sock price. While in realiy such effecs may occur, in our model he join dynamics of deleveraging and price drop happens for he simple reason ha during bad imes aggregae risk aversion increases. Corollary 11-b predics ha some agens rade more han ohers. From formulas (16) and (17) for bond and sock posiions, we see ha agens wih w i = a i do no change heir posiion in socks and bonds a all. They have zero leverage, and hey jus purchase a i unis of sock. Everyone else no only have higher or lower sock posiion han heir preference-based posiion a i, bu hey rade in response o aggregae shocks. Those wih highes difference in w i a i rade he mos. This model herefore has specific predicions on he cross-secional difference in rading, which depend on heerogeneous endowmens and preferences. Corollary 12 ( panic deleveraging ) The funcion H (S ) in (22) is concave in S. Therefore, boh leverage and asse holdings of levered agens decrease by an increasing larger amoun as ime ge worse, i.e. as S declines. Corollary 12 shows ha H (S ) is no only increasing in S bu i is concave in i. Such concaviy has an imporan addiional economic implicaion: during good imes (S high) we should observe higher aggregae leverage and higher asse holdings of levered agens, bu less variaion of boh compared o bad imes (S low). This implies ha as S declines, levered agens decrease heir leverage by an increasingly larger amoun, giving he impression of a panic deleveraging of levered agens during bad imes. One may be emped o link asse sales by levered agens wih he decline in asse prices bu recall ha, as shown in Corollary 5, he same asse pricing implicaions obain wihou heerogeneiy and hence no rade: The asse pricing implicaions of our model are orhogonal o is cross secional implicaions. Our model hen should cauion agains he excessive 21

23 reliance on he simple inuiion of price declines due o he rading of some agens in he economy and he corresponding price pressure argumens. In our model, boh rading and asse prices are joinly deermined in equilibrium and influenced by ime varying economic uncerainy and risk preferences The dynamics of wealh and wealh dispersion We finally characerize agens wealh and he ensuing dispersion of wealh. Recall ha in our seing, all wealh is financial in naure, as i is composed by posiions in socks and bonds W i = N i P + N i B Because of dynamically complee markes, agens wealh equal he presen value of fuure consumpion We have he following proposiion: W i = E [ ] M τ C iτ dτ M Proposiion 13 The wealh-consumpion raio of agen i is given by { } W i = 1 kw i Y S 1 + C i ρ + k ρ ( ) a i + (w i a i )Y S The wealh-oupu raio of agen i is given by W i D = 1 ρ [ ] ρ ρ + k a ( ) i 1 Y S + wi Y S (31) (32) Expression (32) shows ha he wealh o oupu raio depends on agens share of aggregae endowmen w i and heir average habi sensiiviy a i. Higher w i increases agens wealh in good imes because heir higher endowmen allow hem o ake on more leverage and hus reap he gains of an increase in sock marke prices. For given w i, however, agens wih higher a i have wealh ha increases less or even decrease in good imes compared o agens wih lower a i. As discussed earlier, he laer ype of agens ends o ake on more leverage o increase heir sock holdings, which increase heir wealh when sock marke increases, and vice versa. Finally, when he economy is a he aggregae seady sae, i.e. Y S = 1, hen heerogeneiy in preference does no maer. The reason is ha we deermined he Pareo 22