Heterogeneous Households under Uncertainty

Transcription

1 Htrognous Housholds undr Uncrtainty Pitro Vronsi Univrsity of Chicago, NBER, and CEPR August 2018 Abstract I charactriz a dynamic conomy undr gnral distributions of housholds risk tolranc, ndowmnts, and blifs about long-trm growth. As th conomy xpands and th stock markt riss (a) th fraction of housholds with dclining consumption-shar incrass; (b) th walth-shar of high risk-tolrant housholds incrass; (c) richr housholds walth display a highr CAPM bta; and (d) housholds portfolios chang qualitativly. A log-utility invstor for instanc borrows in contractions but lnds in xpansions. Variations in uncrtainty and xpctd growth gnrat trading volum du to risk sharing. Highr uncrtainty incrass stock prics, risk prmiums, volatility, walth inquality and th disprsion of portfolio holdings, consistntly with th vnts in th lat 1990s. I thank John Haton and Lubos Pastor for thir commnts. I acknowldg financial support from th Fama-Millr Cntr for Rsarch in Financ and by th Cntr for Rsarch in Scurity Prics at th Univrsity of Chicago Booth School of Businss. Errors ar my own. Contact information: Pitro Vronsi: pitro.vronsi@chicagobooth.du.

2 1. Introduction Empirical studis on housholds risk prfrncs using diffrnt stimation stratgis and diffrnt sampls agr on on fact: Th distribution of risk tolranc is wid, with som housholds xtrmly risk avrs and with othrs clos to risk-nutrality. 1 In a world with larg and tim varying macroconomic uncrtainty, such larg diffrncs in risk-prfrncs must impact agnts trading bhavior and walth dynamics. For instanc, it is wll known that th lat 1990s wr charactrizd by rising pric valuation, volatility, and trading volum. Walth inquality, morovr, also trndd up strongly in that priod. 2 Raising macroconomic uncrtainty with htrognous prfrncs prdict just that. Spcifically, I mbd gnral risk-prfrnc distributions in a dynamic modl of asst pricing with macroconomic uncrtainty and obtain numrous prdictions. I first focus on a bnchmark modl without larning to obtain bnchmark rsults: In this stting, highr disprsion of risk prfrncs gnrats highr Sharp ratios and risk-fr rats in bad tims but lowr ons in good tims. As th conomy xpands and th stock markt riss (a) th fraction of housholds whos consumption-shar dclins with output incrass; (b) th walth-shar of high risk-tolrant housholds incrass; (c) richr housholds nt worth display a highr CAPM bta; and (d) housholds portfolio allocations chang qualitativly. A log-utility invstor, for instanc, borrows in bad tims but lnds in good tims. Points (a), (b), (c) ar consistnt with rcnt trnds (S Saz and Zucman (2016), Gomz (2018)). 3 Th ky drivr of ths rsults is risk sharing, as housholds with highr risk tolranc insur thos with lowr risk tolranc by slling thm risk-fr bonds and taking aggrssiv positions in stocks. Thy thus gain th most as th conomy xpands. Th xtnsion to uncrtainty about long-trm growth and larning yilds additional novl rsults. First, highr uncrtainty ntails additional motivs for risk-sharing and thus housholds with highr risk tolranc tak on vn mor xtrm positions in stocks than thos with lowr risk tolranc. In particular, an ndognous incras in uncrtainty gnrats highr disprsion in portfolio allocations across housholds which is achivd through highr trading volum, as mor risk tolrant housholds buy stocks from th lss risk tolrant ons. I show that an ndognous incras in uncrtainty thus incrass trading volum as wll as stock valuations, rturn volatility, and walth inquality. 1 S.g. Barsky, Justr, Kimball and Shapiro (1997), Kimball t al. (2008, Figur 1), Chiappori t al. (2014, Figur 2). 2 S.g. Saz and Zucman (2016). 3 In a rlatd modl with multipl countris and inquality-avrs housholds, Pastor and Vronsi (2018) show that globalization xacrbats th inquality du to risk sharing highlightd in (a) and (b) and thus lad housholds to vot for a populist who promiss to nd globalization. 1

3 Ths rsults provid thortical foundation to th vnts of th lat 1990s that saw rcord braking valuations, high volatility, trading volum, and disprsion of portfolio holdings across managrs (s.g. Grnwood and Nagl (2009)). Th xistnc of larg trading volum around bubbl vnts has bn takn as vidnc of bhavioral biass (s.g. Schinkman (2014) for a rviw), but incrasing conomic uncrtainty in th spirit of Pastor and Vronsi (2006) combind with htrognous prfrncs as in this papr provid a plausibl altrnativ story. Morovr, rising uncrtainty also incrass th walth disprsion across agnts, thrby xplaining th larg incras in walth inquality in th lat 1990s (s.g. Saz and Zucman (2016)). Undr uncrtainty and larning, housholds portfolio allocations bcom xtrm. Indd, housholds with low risk tolranc may vn short stocks to hdg against bad nws about long-trm conomic growth. Ngativ shocks to aggrgat output signal vn lowr output going forward and such housholds can hdg against this bad nws by shorting stocks and buying bonds. Indd, thir walth is ngativly corrlatd with th aggrgat conomy. Tim varying xpctd growth and uncrtainty ar thus potntially important channls for trading dynamics in a larning nvironmnt. Th focus on continuous distributions of risk tolranc also highlights th bhavior of housholds with intrmdiat risk tolranc which ar not visibl in standard two-agnts modls. For instanc, as th conomy xpands agnts with intrmdiat risk tolranc ar first borrowrs and thn lndrs, with th intnsity of borrowing and lnding dtrmind by thir risk tolranc. This bhavior is du to th chang in risk-rturn charactristics of risky stocks and intrst rats in quilibrium. As th conomy xpands, th aggrgat Sharp ratio dclins, dcrasing th attractivnss of stock invstmnt. Although also th risk-fr rat dclins, th formr ffct dominats and induc intrmdiat agnts to mov out from stocks and buy bonds. My papr is rlatd to numrous strands of litratur: Th first and mor obvious is th traditional litratur on htrognous risk prfrncs, risk sharing, and asst prics. Diffrntly from th classic paprs of Dumas (1989), Wang (1996), Longstaff and Wang (2013), and Bahmra and Uppal (2014), I considr a continuum of agnts and obtain rlations about th whol distribution of walth, and not only of two typs. This is important to discuss issus rlatd to walth distribution, for instanc, or th bhavior of housholds in th middl of th distribution, which cannot b studid in two-agnts modls. Finally, I focus on th impact of larning dynamics on portfolio holdings and asst prics, which is mostly not discussd in this arlir rsarch. 2

4 Cvitanic t al. (2011) considr N typs of htrognous agnts and obtain svral rsults that ar rlatd to som in this papr. Whil Cvitanic t al. considr multipl sourcs of htrognity (prfrncs, blifs, discounting), I rathr focus on gnric distributions of risk-tolranc and ndowmnts and larning about long-trm growth. Unlik that papr, I show that htrognity and uncrtainty xplain high valuation, trading volum, and volatility, for instanc, among othr rsults. Th paprs ar complmntary from a tchnical standpoint, but thy focus on diffrnt issus. Chan and Kogan (2002), Xiorius and Zapatro (2007) and Santos and Vronsi (2018) also considr modls with a continuum of agnts, but undr th assumption of habit formation or tim-varying risk prfrncs. My focus is instad on th impact of uncrtainty and larning on housholds consumption and trading pattrns, and thir rlation to asst prics, whn housholds hav standard prfrncs with constant rlativ risk avrsion (CRRA). From a tchnical standpoint, I xploit similar mthodologis as in Chan and Kogan (2002) and Xiorius and Zapatro (2007). Pastor and Vronsi (2018) study a two-country modl that xploits htrognous risk prfrnc within country and across countris. Unlik this papr, that papr assums that agnts ar avrs to inquality, spcially th consumption of th lits. As th global conomy xpands, intrnational risk sharing xacrbats inquality in th rich country. Aftr a long xpansion, th majority of th citizns in th rich country would prfr to consum lss but liv in a mor qual socity by limiting th consumption of th ultra rich. Thy do so by voting for a populist who promiss to nd globalization, as th mov to autarky affcts th consumption of th ultra rich th most. My papr is also rlatd to th litratur on paramtr uncrtainty and larning (.g. David (1997), Vronsi (1999, 2000), Brnnan and Xia (2001), Pastor and Vronsi (2006), Ai (2010), David and Vronsi (2013), Collin-Dufrssn t al. (2016) ). This litratur focuss on rprsntativ agnt s modls and thus dos not xplor th implications of risk sharing undr uncrtainty and larning, which this papr instad dvlops. Finally, my papr is rlatd to th litratur on bubbls and th trading pattrns that ar implid by such frnzy. For instanc, Schinkman and Xiong (2003) show that ovrconfidnc and short-sal constraints mak optimistic invstors valu stocks mor than thir tru fundamntal valu thanks to a rsal option componnt. Important, trading volum is rlatd to such rsal option and th xisting litratur mphasizs trading volum as a distinguishing pattrn of bhavioral biass (s.g. Schinkman 2014). I show that raising uncrtainty with htrognous prfrncs also inducs high valuation, stock volatility 3

5 and highr disprsion in portfolio allocations, which is in lin with th vidnc in th lat 1990s. Pastor and Vronsi (2009) show that bubbl-lik pattrns in both stock prics and volatility may b du to larning about nw tchnologis, and th bubbl bursts du to an incras in discount rats whn a nw tchnology is adoptd. That papr though dos not considr htrognity across agnts, which is th focus of this papr. Th papr dvlops as follows: Nxt sction introducs th bnchmark modl without larning, and th implications for agnts consumption, walth, trading, and asst prics. Sction 3. discusss th th implication of uncrtainty about long-trm growth for risk sharing, portfolio allocation, and trading. In sction 4. I calibrat th modl. I also put forwards an assumption on initial ndowmnts that allow for a simpl solution to th quilibrium fixd point problm and thus allows m to obtain simpl numrical solutions for th calibration. Sction 5. concluds. 2. Bnchmark Modl Tim is continuous and it spans th priod 0, T], whr T is finit but larg. A finit T hlps dal with th non-stationary natur of th conomy. Thr is a continuum of infinitsimal agnts. All agnts hav prfrncs xhibiting Constant Rlativ Risk Avrsion (CRRA) ovr thir own consumption C it : it 1 ρ i U i (C it, t) = φtc1 1 1 (1) ρ i whr ρ i 1 is agnt i s risk tolranc. If ρ i = 1 thn U i (C it, t) = φt log (C it ). Risk tolranc is boundd in th population, that is, ρ i (ρ L, ρ H ), but I mak no additional assumptions othrwis. Agnts can invst in a productiv risky opportunity or park thir savings in a risk-fr bond. At tim 0, agnts ar born with s i shars of th risky asst, with s i di = 1. 4 I dnot f(ρ, s) th joint dnsity on ρ i and s i. Th risky asst has a unit pric P t and pays a dividnd D t, whr δ t = log (D t ) follows th procss dδ t = µdt + σdz t Z t is a standard Brownian motion, and µ and σ ar constants, although this rstriction is 4 I oftn us th notation di to indicat th intgral across agnts using th rlvant dnsity on i, without spcifying th lattr to avoid notational cluttr. I mak th dnsity xplicit whn it is ncssary for conomic intrprtation or clarity. 4

6 not ncssary for th rsults. In this sction I assum µ is known to invstors. Sction 3. xamins th cas in which µ is not known but invstors larn its valu ovr tim. In addition to th risky asst, agnts can borrow or lnd at th risk-fr rat r t, whos valu is dtrmind in quilibrium. B t is th pric of a risk-fr bond at t. Both prics P t and r t ar dtrmind in quilibrium. Markts ar thus dynamically complt. Dnot ach agnt i s position in stock N it and in bonds N 0 it. Each agnt maximizs th intrtmporal utility subjct to th budgt constraint with initial condition W i,0 = s i P 0. 1 T ρ i max E 0 φtc1 it {C it,n it,nit} 0 T 0 t=0 1 1 dt ρ i dw it = N it (dp t + D t dt) + N 0 itb t r t dt C it dt (2) 2.1. Equilibrium To undrstand th logic of th quilibrium, it is usful to go through som of th stps (s Chan and Kogan (2002), Xiorious and Zapatro (2007)). Markts ar dynamically complt and thus from Cox and Huang (1989) ach agnt i quivalntly solvs th static maximization max {C it } T t=0 T E 0 0 ] T 1 1 dt subjct to E 0 M t C it dt = s i P 0 (3) ρ 0 i 1 ρ i φtc1 it whr M t is th quilibrium stat pric dnsity (normalizd to M 0 = 1). Th Lagrangan of th static optimization is 1 ( T ρ i ] ) T L i = E 0 φtc1 it dt ξ i E 0 M t C it dt s i P 0 ρ 0 i (4) whr ξ i is th Lagrang multiplir dtrmind by th static budgt constraint. Th maximization is takn stat by stat and tim by tim obtaining th first-ordr condition φt C 1 ρ i it = ξ i M t = C it = ρ i log(ξ i )+ρ i g t (5) whr g t = φt log (M t ) (6) 5

7 Aggrgat now across agnts and impos th markt claring condition D t = C it di to obtain th quilibrium condition: D t = ρ i log(ξ i )+ρ i g t di = E ] CS ρ i log(ξ i )+ρ i g t (7) whr th last stp xploits th law of larg numbrs and E CS.] dnots th cross-sctional avrag across agnts i. 5 Th quantity g t, and hnc th quilibrium stat pric dnsity M t, is th solution to quation (7), which I dnot by g t = g (δ t ). Th stat pric dnsity is thn M t = φt g(δt) (8) Givn th quilibrium g(δ t ), th constants ξ i ar dtrmind by th budgt constraints (3). Aftr substitution of M t and using P 0 = E 0 T 0 M τd τ dτ ] = E 0 T 0 φτ+δτ g(δτ) dτ ], th budgt constraint (3) can b writtn as follows: ρ i log(ξ i ) = s i λ(ρ i ) whr λ(ρ i ) = E 0 T 0 φτ+δτ g(δτ) dτ ] E 0 T 0 φτ+(ρ i 1)g(δ τ) dτ ] (9) I finally substitut this xprssion back in quilibrium condition (7) to obtain: δt = E CS s i λ(ρ i ) ρ ig(δ t) ] (10) Th normalization M 0 = 1 implis g(δ 0 ) = 0. Assuming δ 0 = 0 without loss of gnrality, I thus hav th normalization E CS s i λ(ρ i )] = 1 (11) Equations (9) and (10) highlight th natur of th functional fixd-point problm that charactrizs an quilibrium: W nd th quilibrium g(δ t ) to comput th Lagrang multiplirs ξ i and hnc λ(ρ) from (9) but w nd λ(ρ) to comput th quilibrium g(δ t ) from (10). Still, assuming that conditions ar such that th fixd point has a solution, w can charactriz th solution vn without xplicitly solving for λ(ρ) (s Chan and Kogan (2002) and Xiorius and Zapatro (2007) for rlatd rsults undr habit formation): 5 I assum throughout conditions ar such that th law of larg numbr can b applid. S Fldman and Gills (1985), Judd (1985). 6

8 Proposition 1: Lt th functional fixd-point problm in Equations (9) and (10) hav a solution for vry δ t. Thn: (a) Th solution g(δ t ) is uniqu (b) Th function g (δ t ) is globally incrasing in δ t, it is concav, and it divrgs to infinity as δ t incrass. Spcifically: g (δ t ) = 1 ] Et ρ] > 0; g (δ t ) = g (δ t ) 1 E t ρ] < 0 Et ρ] whr Et.] and E t.] dnot cross-sctional avrags computd using th following two distributions, rspctivly: f (ρ δ t ) f(ρ, s) s ds λ(ρ) ρg(δt) ; f (ρ δ t ) f (ρ δ t )ρ (c) Th function g(δ t ) > 0 if and only if δ t > 0. (d) Th function g (δ t ) is boundd abov and blow: 1 > g (δ t ) > 1, ρ L ρ H lim δ g (δ t ) = 1, t ρ L lim δ g (δ t ) = 1 t ρ H Bfor I commnt on Proposition 1, th following corollary tis it to th standard homognous cas: 6 Corollary 1: If ρ i = ρ j = ρ for all i, j, thn g (δ t ) = 1 ρ δ t and th stat pric dnsity is M t = φt 1 ρ δt = φt D 1 ρ t. Undr homognous prfrncs w hav th classic rsult that th stat pric dnsity quals th marginal utility of th rprsntativ agnt with risk avrsion 1/ρ. Evn in th homognous cas w hav g (δ t ) = 1/ρ > 0; and lim δ g(δ) = and g (δ t ) > 0 for δ t > 0. Proposition 1 confirms that such proprtis ar tru also undr htrognity. Indd, not that g (δ t ) is th invrs of th avrag risk tolranc. Such avrag, howvr, is not computd from th original distribution of ρ, f(ρ), but from a distortd distribution f (ρ δ t ) that corrcts th original on by th stat-pric dnsity factor g (δ). As δ t incrass, f (ρ δ t ) givs incrasingly mor wight to high ρ (bcaus g (δ t ) > 0) implying that th ffctiv risk avrsion g (δ t ) = 1/E ρ δ t ] dcrass as th conomy gts strongr. 6 Th rsult is asy to s: Whn all agnts ar idntical with risk tolranc ρ, λ(ρ) = 1 for all ρ and givn E CS s i ] = 1, quation (10) rducs to δ t = log ( ρg(δt)]) = ρg(δ t ). 7

9 This ffct is du to a combination of risk sharing and diffrnt lasticitis of intrtmporal substitution (EIS): Agnts with lowr risk avrsion insur agnts with highr risk avrsion, and during good tims thy nd up rprsnting mor of th aggrgat consumption of th conomy. In addition, such agnts with highr risk tolranc hav also highr EIS. Thus, thy ar happy to consum lss in bad tims but thy want to consum mor in good tims. Agnts with low risk tolranc, in contrast, want to smooth consumption ovr tim and thus thy want to consum mor in bad tims and rlativly lss in good tims. With CRRA utility th two channls ar intrchangabl. Although I did not us a rprsntativ agnt st up to obtain th quilibrium, a rprsntativ agnt xists in this conomy, as shown in th nxt proposition: Proposition 2. Th function g (δ t ) is th rlativ risk avrsion of th rprsntativ agnt in this conomy: g (δ t ) = RRA(D) = D U (D, t) U (D, t) whr U(D, t) is th utility function of th rprsntativ agnt (12) U(D, t) = φt si λ(ρ i ) ρ i g(log(d)) 1 1 ρ i di (13) Th rprsntativ agnt s risk avrsion is thus dcrasing in output drra(d) dd < 0 From Proposition 1, g(δ t ) is concav in δ t, as g (δ t ) dcrass with δ t. Th risk avrsion of th rprsntativ agnt hnc dcrass as th conomy xpands. Evn if all agnts hav constant rlativ risk avrsion, htrognity brings about a dcras in risk avrsion in good tims. As th conomy grows, mor wight is givn to agnts with lowr risk avrsion and th stat pric dnsity rflcts such lowr wightd risk avrsion. I furthr discuss this ffct in th nxt sction Optimal Consumption Th ndognous bhavior of g(δ) has implications for th quilibrium consumption shar of individual agnts, i.. thir consumption-to-output ratio, which is givn by: 7 C it D t = s i λ(ρ i ) ρ ig(δ t) δ t (14) 7 S also Chan and Kogan (2002), Xiorius and Zapatro (2008) for a similar xprssion undr habit formation. 8

10 I thn obtain th following: Corollary 2: (a) Th C/D ratio is incrasing in output for agnts with ρ i > E t ρ i ]. (b) Thus, as δ t incrass, so dos th fractions of agnts whos C/D is dclining in δ i. Corollary 2 shows that for givn δ t agnts with risk tolranc ρ i > E t ρ] hav quilibrium C/D ratios that incras as δ t incrass, and agnts with low risk tolranc hav C/D ratios that dcras. Th first typs of agnts at an incrasing largr shar of aggrgat output compard to th othr typs. Howvr, intrstingly, as output incrass so dos th ffctiv avrag risk tolranc E t ρ], as agnts with highr risk tolrant agnts wigh mor in th distribution, and thus th thrshold incrass. It follows that th fraction of agnts with dclining C/D ratio incrass as wll. That is, thr is an incrasing shar of agnts whos consumption dos not kp up with th incras in aggrgat output. This is th natur of risk-sharing with a continuum of htrognous agnts. Th xact mchanism which dlivrs this rsult (in trms of portfolio allocation) is discussd in Sction 2.5. W obsrv from xprssion (14) that th wights of th distribution f (ρ δ) in Proposition 1 dpnd on consumption shars. In particular, dnot C t (ρ, s) th tim-t consumption of an agnt born with risk tolranc ρ and an ndowmnt of s shars. Thn: f (ρ δ) = f(ρ, s) C t(ρ, s) D t ds (15) This xprssion shows that th adjustd distribution on ρ dpnds on th ndowmntwightd consumption shar of agnts with risk tolranc ρ. For instanc, vn if ρ was uniformly distributd, if agnts with high ρ consum mor, thn it is as if th avrag risk-tolranc of th conomy is highr. Corollary 3: Agnts with highr risk tolranc ρ i hav mor volatil consumption. In particular: whr µ C,i = σ C,i = dc it C it = µ C,i dt + σ C,i dz t (16) ρ i ρ i 1 + ρi Et Et ρ] ] ρ] (17) Et ρ] µ Et ρ] ρ i Et ρ] σ (18) 9

11 Risk tolranc ρ i is also th lasticity of intrtmporal substitution with CRRA utility. Highr lasticity of intrtmporal substitution thn maks agnts hav mor volatil consumption (thy ar mor lastic). Highr risk tolranc also implis a highr growth rat of consumption (from th first trm), du to th willingnss of smooth consumption ovr tim as wll and not only across stats. turns ngativ for low ρ i as Et ρ] incrass ovr tim. Th scond trm in µ C,i is positiv for high ρ i but it Corollary 4: As th conomy grows, th simpl avrag consumption volatility dclins σ C,i f (ρ) dρ = E ρ i] E t ρ] σ This simpl rsult shows that statistics that do not control for th dnsity f (ρ δ t ) would highlight proprtis of consumption volatility that ar not ral, but just an artifact of misswighting consumption growth Asst Prics Th intrst rat and risk prics only dpnd on th stat pric dnsity (8) and thus by Ito s Lmma I obtain th following proposition: Proposition 3: Th stochastic discount factor follows th procss dm t M t = r t dt σ M,t dz t (19) whr r t = φ + 1 { µ 1 Et ρ] 2 σ σ M,t = Et ρ] ( 1 + E t ρ] Et } ρ] )σ 2 Et ρ] (20) (21) In th spcial cas of ρ i = ρ = 1/γ, thn Et ρ] = Et ρ] = ρ and I obtain th standard rsult r t = φ + γµ γ2 2 σ2 and σ M,t = γσ. From Proposition 1, E t ρ] incrass with δ t, as mor of th wights gos to highr agnts with highr risk tolranc ρ i, thn risk prics σ M,t dclins as th conomy improvs (s also Chan and Kogan (2002), Xiorius and Zapatro (2007)). 10

12 Th impact of growth on th ral intrst rat is non-linar. Th last trm in th parnthsis is dclining in δ t and convrging to σ2 2ρ H. Thus, if µ is sufficintly larg intrst rats dclin as th conomy bcoms largr in siz. Intuitivly, agnts with highr risk tolranc ar also thos with highr lasticity of intrtmporal substitution (EIS). Whn th conomy has positiv growth such agnts hav lowr dsir to borrow to smooth out consumption compard to thos with lowr EIS. It follows that rats tnd to dclin as th conomy grows. A countrvailing ffct, howvr, is that as th conomy grows so dos th avrag risk tolranc, and thus prcautionary savings dmand for bonds dclins. This ffct tnds to rais intrst rats. Whn µ is sufficintly larg th first ffct vntually dominats. I nxt discuss th proprtis of th stock rturn procss. Bfor I do so, I introduc th following notation for a gnric function K (δ t+τ ) of δ t+τ = δ t + µτ + σ τx (22) whr x N (0, 1): E x,τ t K (δ t+τ )] = E = T t 0 T t 0 φ φτ ] 1 φ(t t)k (δ t+τ)dτ δ t K ( δ t + µτ + σ τx ) n(x)f τ (τ T t)dxdτ (23) whr n(x) is th standard normal dnsity, f τ (τ T t) = φ φτ 1 {τ<t t} 1 φ(t t) is truncatd random variabl τ xponntially distributd ovr 0, T t]. 8 th dnsity of a Proposition 4: Th stock pric is P t = p(t t) g(δt) E x,τ t δ t+τ g(δ t+τ) ] (24) whr p(t t) = 1 φ(t t). Thus, th stock rturn procss φ dp t + D t dt P t = µ Pt dt + σ Pt dz t whr µ Pt = r t + σ Pt σ Mt, and σ Pt = { 1 + g (δ t ) Ẽx,τ t g (δ t+τ )] } σ (25) In this formula, Ẽx,τ t 8 Indd, T t 0 f τ (τ T t)dτ =.] dnots th xpctation with rspct to th following distribution f (x, τ δ t ) n(x)f τ (τ T t) δt+τ g(δt+τ) (26) φ T t φτ dτ = 1 φ(t t) 0 11 φ 1 φ(t t) φ(t t) 1 φ = 1.

13 Pricing formula (24) is a tautology from th dfinition of E x,τ t.] and th pricing formula P t = E T t t M τ /M t D τ dτ ]. To undrstand th volatility formula, rcall that g (δ t ) = 1 E t ρ] i.. th invrs of th wightd avrag of risk tolranc. If all agnts hav ρ i = ρ = constant, thn E t ρ] = E t+τ ρ] = ρ and thus I obtain th standard rsult σ Pt = σ. Indd, in this cas, th risk prmium is simply σ 2 /ρ. For instanc, if ρ = 0.1 thn risk avrsion is 10 and th risk prmium is 10σ 2. Th xistnc of htrognity introducs tim varying risk-tolranc, which in turn gnrats tim varying volatility and xpctd rturn. Th siz and impact of th ffct dpnds on whthr th following condition is satisfid: g (δ t ) > Ẽx,τ t g (δ t+τ )] (27) Nxt corollary xplains why: Corollary 5: (a) Th pric-dividnd ratio P t = p(t t) g(δt) δt E ] t x,τ δ t+τ g(δ t+τ) (28) D t is incrasing in δ t if and only if condition (27) is satisfid. (b) Condition (27) is satisfid if ρ H > 1 and δ t is sufficintly high, or if th standardizd growth rat of th conomy µ/σ is sufficintly high. If P/D is incrasing in δ t, thn it is asy to s why th modl with htrognous risk tolranc incrass both th volatility and th risk prmium compard to a modl with homognous prfrncs. As δ t incrass, th stock pric incrass and th P/D ratio also gts a positiv kick du to a lowr discount rat. Th two ffcts compound ach othr and th stock pric is mor volatil and rquirs a highr risk prmium compard to th cas with homognous prfrncs. Howvr, th P/D ratio may not b incrasing in output δ t if an incras in δ t incrass th risk-fr rat, for instanc. If th incras in wightd risk tolranc rducs sufficintly th prcautionary savings dmand for bonds, th intrst rat may incras with δ t which in turn may dcras th P/D ratio. In this cas, th volatility of stock rturns would actually b lowr than in th cas with homognous invstors. 12

14 2.4. Walth and Walth Dynamics Dynamic markt compltnss implis that th walth of ach agnt is qual to th prsnt valu of his/hr futur consumption and thus W it = E t T t ] M s C is ds M t I now charactriz agnts walth as a function of aggrgat (log) output δ t. Proposition 5: Th walth of ach agnt i at tim t is W it = p(t t) s i λ(ρ i ) E x,τ (ρ i 1)g(δ t+τ) ] (29) and th consumption to walth ratio (C/W) is C it W it = 1 p(t t) E x,τ t (ρ i 1)(g(δ t+τ) g(δ t)) ] (30) Givn (29) and (30), I can analyz a numbr proprtis of walth and C/W ratio, vn without knowing th xact functional form of g (δ) as solution of quation (11), but only its charactrization in Proposition 1. First, th random amount of initial shars ndowmnt s i affcts th lvl of walth proportionally going forward. An agnt with 10% mor shars at tim 0 du to a lucky draw of s i will b richr by 10% going forward. Indd, th consumption/walth ratio dos not dpnd on s i. Scond, th dynamics of output δ t affcts th lvl of walth of ach agnt diffrntly dpnding on thir risk tolranc. For instanc, th log-utility agnt (ρ i = 1) has walth W ρ i=1 it t) = s i λ(1) g(δt)1 φ(t φ Thus, an incras in output incrass th log-utility agnt s walth. Howvr, if E t (ρ) > 1, th walth to output ratio (W/D) of log-utility agnt dclins with output whn δ t is larg nough. Indd, dlog ( W ρ ) i=1 it /D t = g (δ) 1 < 0 Et ρ] > 1. dδ t Assuming ρ H > 1, bcaus Et ρ] ρ H, thr is a point in which Et ρ] > 1 and W/D of log-utility invstor dclins with output. Clarly, th consumption/walth ratio of log utility agnt a myopic agnt is still dtrministic and indpndnt of output, as w would xpct C ρ i=1 it W ρ i=1 it = φ 1 φ(t t) 13

15 I obtain th following: Corollary 6: Whn δ t is sufficintly high, for givn s i, agnts with highr risk tolranc ρ i hav mor walth than agnts with lowr risk tolranc. It is asy to s how Corollary 6 stms from xprssion (29) and Proposition 3. Indd, rcall that s i λ(ρ i ) = ρ i log(ξ i ). Aftr substitution, th claim thn follows from th fact that g (δ t+τ ) log(ξ i ) > 0 for any ξ i whn δ t+τ is sufficintly high. Bcaus δ t+τ is linar in δ t, as th lattr incrass so dos th distribution of δ t+τ, yilding th rsult. Th sam argumnt implis th following: Corollary 7: Agnts with highr risk tolranc consum a lowr prcntag of thir walth than agnts with lowr risk tolranc. This is asy to s as wll: w know that (g (δ t+τ ) g (δ t )) > 0 for δ t+τ > δ t. Thrfor, in a growing conomy w hav a larg part of th distribution with g (δ t+τ ) g (δ t ) > 0. Thus, a highr ρ i implis highr E x,τ ] t (ρ i 1)(g(δ t+τ) g(δ t)) and thrfor lowr C/W. Morovr, as δ t incrass concavity implis that th diffrnc (g (δ t+τ ) g (δ t )) dcrass, and thrfor thr is a comprssion of C/W ratios as th conomy grows strongr (for givn T t). To discuss furthr proprtis, it is usful to first normaliz th walth of ach agnt by th aggrgat walth (which qual th stock markt pric, by markt claring). P t = W jt dj = p(t t) E CS s j λ(ρ j ) E x,τ ]] (ρ j 1)g(δ t+τ) (31) Corollary 8: (a) Th walth distribution is givn by wights ω it = W i,t P t = (b) Th initial wights at tim 0 ar ω i0 = s i. s i λ(ρ i ) E x,τ t E CS sj λ(ρ j ) E x,τ t (ρ i 1)g(δ t+τ) ] (c) Agnt i s walth shar ω it incrass with output if and only if (ρ i 1) Ex,τ t g (δ t+τ ) (ρ i 1)g(δ t+τ) ] E x,τ t (ρ i 1)g(δ t+τ) ] whr rcall Ẽx,τ t. ] uss dnsity (26) (ρ j 1)g(δ t+τ) ]] (32) > 1 Ẽx,τ t g (δ t+τ )] (33) 14

16 (d) In addition, for any cutoff ρ, th walth shar of agnts i with ρ i > ρ incrass as δ t incrass. Dnot ω t (ρ) = i:ρ i >ρ ω it di. Thn ω t (ρ) δ t > 0 Equation (32) shows th two sourcs of walth distribution, namly, th initial random ndowmnt s i and th componnt du to risk prfrncs (and thir implicit implication for sharing). Th scond ratio in (32) can in fact b charactrizd and it dpnds on ρ i and th lvl of currnt output δ t. Whn δ t incrass, th walth shar of th top distribution of risk tolranc ρ > ρ incrass, for any ρ (point (d)). This implis of cours that th walth shar of th ons blow th cutoff worsn. Point (c) shows th condition undr which th walth shar of agnt i incrass. Th right-hand-sid of (33) dos not dpnd on i and it is incrasing in δ t. Unfortunatly, th lft-hand-sid is also incrasing in δ t, unlss ρ = 1, in which cas this bcoms a constant. It follows that a log utility invstor has its walth shar incras for a whil and thn dclin as th conomy grows (assuming ρ H > 1). I finally charactriz th dynamics of walth: Proposition 6: Agnt i s walth dynamics is dw it + C it dt W it = µ Wit dt + σ Wit dz t whr µ Wit = r t + σ Wit σ Mt and σ Wit = g (δ t ) + (ρ i 1) Ex,τ g (δ t+τ ) ] (ρ i 1)g(δ t+τ) E x,τ (ρ i 1)g(δ t+τ) ] σ (34) Bfor discussing Proposition 6, I provid a scond charactrization to th stock pric, basd on (31): Corollary 9: Th stock rturn volatility and xpctd rturn can b xprssd as σ Pt = ω it σ Wit di µ Pt = r t + ω it µ Wit di Th volatility and risk prmium of stock rturns ar walth-wightd avrags of th volatility and risk prmium of th walth procss of individual agnts in th conomy. I thus hav th following proposition: 15

17 Proposition 7: (a) Thr is a thrshold ρ > 1 such that for all ρ i < ρ agnts with highr ρ i hav highr volatility of walth σ Wi : ρ i < ρ j σ Wit < σ Wjt (b) Lt ρ H ρ and lt th marginal distribution f(ρ) b continuous on (ρ L, ρ H ). For vry δ t thr is ρ(δ t ) such that σ Wit < σ Pt < σ Wjt for vry i and j such that ρ i < ρ(δ t ) < ρ j Th condition in Proposition 7 that thr is ρ > 1 is not ncssary but just sufficint. It could wll b that ρ = but th gnral cas provd tricky to stablish formally givn th gnrality of th distribution f(ρ). From an conomic standpoint, agnts with highr risk tolranc hav highr volatility of walth. Intuitivly, such agnts ar mor willing to hav highr consumption in good tims and lowr consumption in bad tims (bcaus thy insur th agnts with lowr risk tolranc). That is, th consumption of such agnts is mor volatil than consumption of mor risk avrs agnts, as shown in Corollary 3. It follows that thir walth th prsnt valu of such consumption strams rspond mor to aggrgat shocks as thir cash flow volatility is mor volatil. Part (b) stms from th fact that th avrag is always strictly btwn th xtrms and σ Wit is incrasing in ρ i. This rsult is important to undrstand th bta of th walth portfolio and th trading positions of agnts in this conomy, as discussd nxt. Proposition 8: Lt ρ H < ρ whr ρ is th thrshold in Proposition 7. Thn, housholds with highr risk tolranc hav a highr bta of thir walth dynamics. Lt thn β it = Cov t (dw it /W it, dp t /P t ) V ar t (dp t /P t ) ρ i < ρ(δ t ) < ρ j β it < 1 < β jt whr ρ(δ t ) is th thrshold in Proposition 7, point (b). That is, housholds with highr risk tolranc hav a highr snsitivity of thir walth to th stock markt. Bcaus such housholds also tnd to grow richr than othrs (s Corollary 6), w obtain th prdiction that richr housholds tnd to hav walth dynamics that is mor snsitiv to th stock markt. Indd, this point can also b sn from a CAPM prspctiv. Th risk prmium of houshold i s walth is in fact (35) µ Wit = σ Wit σ Mt = σ Wit σ Pt µ Pt = β it µ Pt 16

18 That is, th rat of rturn on walth is highr for housholds with highr bta and hnc, givn Proposition 8, for agnts with high risk tolranc. Bcaus vrything ls qual, agnts with high risk tolranc bcom richr in th long run (s Corollary 6), Proposition 8 lnds support to th mpirical vidnc that richr housholds hav nt worth with highr bta (s.g. Gomz (2018)). Nxt sction tis ths rsults with housholds optimal portfolio allocations Portfolio Allocation I us th rsults in Cox and Huang (1986) and th proprtis of dynamic markt compltnss to obtain th rlativ positions of agnts in stocks and bonds. Th walth of agnt i can b writtn as a portfolio of stocks and bonds: W it = N it P t + NitB 0 t (36) whr B t is th valu of th bond. W thn must hav From Ito s lmma, w thus hav dw it = N it dp t + N 0 itdb t o(dt) + σ Wit W it dz t = o(dt) + N it σ Pt P t dz t Equating th diffusion trms, th position in stocks must b givn by N it = σ Wit σ Pt W it P t (37) Givn N it, I can dtrmin th position in bonds as a rsidual ( NitB 0 t = W it N it P t = W it 1 σ ) Wit σ Pt Th modl has thus implications on th trading stratgy of agnts which dpnd on th lvl of th conomy δ t. Proposition 9: Lt ρ H < ρ and lt ρ(δ t ) b thrshold dfind in Proposition 7. Thn invstors with highr risk tolranc ρ i > ρ(δ t ) tak on lvrag: N 0 it B it < 0 if and only if ρ i > ρ(δ t ) Morovr, invstors who lvr invst mor than 100% of thir walth in risky assts N it P t W it > 1 if and only if ρ i > ρ(δ t ) 17

19 Proposition 9 provids th conomic intuition of th arlir rsults: Agnts with high risk tolranc borrow from th othr agnts to invst in th risky scurity. Th risky scurity pays off in good tims (whn δ t incrass) which in turn mak such agnts walthir and mak thm consum mor. It is all ndognous. Proposition 7 and 9 also shows that th idntity of agnts who tak on lvrag changs ovr tim. Whil gnral rsults sms difficult du to th complxity of formulas, I can provid som partial rsults around log utility ρ = 1. Proposition 10: Lt th marginal distribution f (ρ) b dfind ovr (ρ L, ρ H ) with ρ L < 1 < ρ H. A log-utility invstor lvrag if and only if 9 Ẽ x,τ t g (δ t+τ )] > 1 (38) Condition (38) is satisfid for δ t sufficintly low but it is violatd for δ t sufficintly high. Thus, th log-utility invstor movs from bing a borrowr to a lndr as th conomy grows. Th proposition shows that for sufficintly low δ t, th log-utility invstor is among thos with highr risk tolranc, and thus that invstor will lvrag its position. Howvr, as th conomy grows and th walth shifts towards th agnts with vn highr risk tolranc (ρ > 1), such an invstor nds up bing among thos that ar not lvragd. Intuitivly, th Sharp ratio kps dclining and vn log-utility invstors find it optimal to hold som bonds, vntually. Whil proving a mor gnral proposition sms hard, by continuity, th proof holds for a whol st of agnts around ρ = 1. In othr words, th agnts who tak lvrag ar diffrnt as δ t incrass. Th conjctur is that th fraction of agnts who lvr shrinks ovr tim as stock rturn volatility incrass. I find such rsult in my calibration in Sction Th Disprsion of Risk Tolranc In this sction, I study th impact of th proprtis of th distribution f(ρ) on asst prics. For simplicity, lt s i = 1 for all i, that is, all agnts hav th sam initial ndowmnt. Lt f(ρ) b th distribution of ρ. W thn hav th following: Corollary 10: Considr two distributions f 1 (ρ) and f 2 (ρ) with f 2 stochastically dominating 9 Rcall that Ẽx,τ t. ] uss distribution (26). 18

20 f 1. Thn E 2ρ δ] > E 1ρ δ]. Thrfor, risk prics σ M,t is lowr undr f 2 than undr f 1. Finally, if µ/σ is sufficintly high, thn th risk fr rat r t is lowr undr f 2 than undr f 1. That is, if f 2 (ρ) givs uniformly mor mass to highr risk tolranc agnts than f 1 (ρ), thn th wightd avrag risk tolranc incrass. Th rsult is intuitiv. I now study th impact of th disprsion of risk-tolranc on asst prics. How dos an incras in th disprsion of th distribution f(ρ) affct consumption, walth, and asst prics? I analyz th disprsion through a man-prsrving sprad to f(ρ). Proposition 11. Lt f mps (ρ) b a man prsrving sprad on f(ρ). If µ/σ is sufficintly high, thn thr ar thrsholds δ min and δ max with 0 < δ min < δ max such that th wightd risk tolranc E mpsρ δ t ] < E ρ δ t ] for δ t < δ min and E mpsρ δ t ] > E ρ δ t ] for δ t > δ max. Proposition 11 stablishs that th risk adjustd risk tolranc E ρ δ] dcrass for low δ t whn w incras th disprsion of risk tolranc, but it incrass for high δ t. This rsult lads to th following implication: Corollary 11. Undr th conditions of Proposition 11, a man prsrving sprad on th distribution of risk-tolranc incrass th markt pric of risk and th risk-fr rat for low output δ t < δ min but dcras thm for high output δ t > δ max. Essntially, an incras in th disprsion of risk tolranc maks som agnts mor risk avrs and som agnts lss risk avrs than th avrag (which is kpt constant). For low δ t, th impact of mor risk avrs ons is largr than th impact of lss risk avrs agnts and thus th consumption/adjustd avrag risk tolranc dcrass. This ffct incrass risk prics. In addition, bcaus th sam ffct dcrass th lasticity of intrtmporal substitution, mor agnts want to borrow, and hnc th intrst rat incrass. Howvr, as th conomy grows, at som point th incras in th mass of agnts with lowr risk tolranc bcoms a dominant ffct, and thus risk prics and intrst rat dclin by mor than undr th original distribution. Th impact on asst prics of a man-prsrving sprad on th distribution of risk tolranc is hardr to dtrmin. Howvr, th P/D ratio formula (28) indicats that th P/D ratio dpnds on th xpctd diffrnc E ] t x,τ g(δ t) g(δ t+τ). Whn g (δ) is highr, this xpctd diffrnc is highr, and thus w can xpct th P/D ratio to b lowr. Whil hardr to prov formally, this argumnt strongly suggsts that a man prsrving sprad on risk tolranc dcrass th P/D ratio. 19

21 2.7. Initial Endowmnts and Asst Prics I finally discuss th impact of diffrnt initial ndowmnts on assts prics. That is, what happns to asst prics whn w shift walth from.g. low risk avrs to high risk avrs agnts? This shift is likly mor prvalnt than w think in th ral world: Rdistributiv policis that shift walth from rich to poor likly shift it from agnts with low risk avrsion to thos with high risk avrsion, as th formr ar thos that bcom richr in good tims. Indd, th prvious sctions show that agnts with high risk tolranc ρ vntually bcom richr than agnts with low risk tolranc. I now study th impact of initial ndowmnts that dpnd on risk tolranc ρ: Proposition 12: Lt s i = s(ρ) s i whr s(ρ) is monotonic in ρ such that E CS s(ρ)] = 1 and s i is a unit-man positiv random nois trm uncorrlatd with ρ i. Thn if µ/σ is sufficintly high, for any two functions s 1 (ρ) and s 2 (ρ), s 1 (ρ) > s 2 (ρ) = E 1ρ δ] > E 2ρ δ] Proposition 12 vrifis that if w giv a highr ndowmnt to low risk tolrant agnts at tim zro, thn th adjustd avrag risk-tolranc in th conomy dcrass. That is, rdistributiv policis that transfr funds from high risk tolrant agnts to low risk tolrant agnts hav th implicit impact of dcrasing th wightd avrag risk tolranc, and hnc risk prics, as implid by th following corollary: Corollary 12: Undr th conditions of Proposition 12: s 1 (ρ) > s 2 (ρ) = σ Mt,1 < σ Mt,2 Most dvlopd conomis hav tax and subsidy policis that implicitly transfr walth from rich agnts to poor agnts. Givn that walth is ndognous, howvr, in my modl th rich agnts ar thos that likly ar lss risk avrs than th poor agnts. Consquntly, th transfr of walth from rich to poor amount to an incras in walth of mor risk avrs agnts and thus an incras in th risk prmium. 20

22 3. Macroconomic Uncrtainty and Larning I now introduc macroconomic uncrtainty and larning. Th additional assumption is simpl: Lt s assum that th drift rat of output growth µ is not known but distributd according to a gnric prior distribution π 0 (µ): µ π 0 (µ) Bcaus I am intrstd in studying th possibility of an incras in uncrtainty, I do not tak π 0 (µ) to b normally distributd bcaus th proprtis of th Kalman-Bucy filtr would imply that uncrtainty dcrass dtrministically. Othr non-normal prior distributions π 0 (µ) may gnrat an incras of uncrtainty for som priods of tim. Evn if π 0 (µ) is a gnral prior distribution ovr th paramtr µ, I can still charactriz th dynamics of th postrior dnsity of π t (µ). Lmma: Lt π 0 (µ) b such that π 0 (µ)dµ = 1 and π 0 (µ) > 0. Thn for vry µ, th dnsity π t (µ) follows th martingal procss: dπ t (µ) = π t (µ)(µ E t µ])σ 1 dẑt (39) whr E t µ] = µ π t (µ) dµ (40) dẑt = σ 1 (dδ t E t µ] dt) (41) In addition, π t (µ) > 0 and π t (µ)dµ = 1 almost surly. That is, w can asily track th whol distribution π t (µ) ovr tim. From (41) w can rwrit th dividnd procss undr th information filtration as dδ t = E t µ] dt + σ dẑt It follows that all stat variabls dpnd only on th innovation procss dẑt and thus markts ar dynamically complt. Thrfor, th rsults in th prvious sction xtnd immdiatly to this cas, as w show nxt. Th bnfit of this stting is that w can thn invstigat th implications of larning and stochastic changs in uncrtainty and asst prics and trading Asst Prics undr Paramtr Uncrtainty Th drivations in Sction 2.2. highlight that th rsults in Propositions 1 and 2, and Corollaris 1 and 2 also hold undr uncrtainty and larning about µ. Indd, µ dos not ntr anywhr in thos drivations or xprssions bcaus of tim-sparabl prfrncs and 21

23 markt compltnss. Th consumption dynamics in Corollary 3, Equation (16), is almost idntical, xcpt that E t µ] rplacs µ (which is not obsrvabl) in th formula for xpctd consumption growth (17), and th shock to housholds consumption growth is th Brownian motion dẑt in (41) instad of th unobsrvabl dz t. Th stat pric dnsity M t in Proposition 3 also only dpnds on δ t, which is obsrvabl, and not on th paramtr µ. Thus, paramtr uncrtainty dos not affct M t ithr. Th stochastic discount factor dm t /M t in Proposition 3, Equation (19), is only mildly affctd as th intrst rat r t in (20) now dpnds on E t µ] instad of µ and th shock to th SDF is now dẑt instad of dz t. Th risk pric σ Mt in (21) is unaffctd. In contrast to housholds consumption and th stat-pric dnsity, th stock pric is affctd by th unobsrvabl µ, as th lattr affcts th xpctation Et x,τ.]. Howvr, it only amounts to a chang of th dfinition of th xpctation itslf. For any function K(δ t+τ ) whr δ t+τ = δ t + µτ + σ τx w can dfin E µ,x,τ t K (δ t+τ )] as in (23) xcpt that th xpctation is takn also with rspct ovr µ π t (µ) in addition to th normal dnsity n(x) and th dnsity f τ (τ τ < T). 10 W thus obtain th following modification to Proposition 4: Proposition 13: Th stock pric is whr p(t t) = whr µ Pt = r t + σ Pt σ Mt and P t = p(t t) g(δt) E µ,x,τ t 1 φ(t t). Thus, th stock rturn procss φ dp t + D t dt P t = µ Pt dt + σ Pt dẑt δ t+τ g(δ t+τ) ] (42) σ Pt = { 1 + g (δ t ) Ẽµ,x,τ t g (δ t+τ )] } σ + (Êtµ] E t µ] ) σ 1 (43) In this formula, Ẽµ,x,τ t and Êt µ] uss th modifid distribution.] dnots th xpctation with rspct to th distribution f (µ, x, τ δ t ) π t (µ) n(x) f τ (τ) δt+τ g(δt+τ) (44) π t (µ) π t (µ) E x,τ t δ t+τ g(δ t+τ) µ ] (45) Th pric formula (42) is th dirct xtnsion of th on undr crtainty in xprssion (24). Th nxt corollary follows: 10 As shown in Sction 4., such xpctation is straightforward to comput by Mont Carlo simulations. 22

24 Corollary 13: A man-prsrving sprad on th distribution π t (µ) incrass th stock pric P t. That is, highr macroconomic uncrtainty incrass stock valuations. It is asy to s that th paramtr µ ntrs linarly into δ t+τ and δt+τ g(δt+τ) is convx in δ t+τ and hnc in µ. A man prsrving sprad ovr a convx function incrass its xpctd valu. This rsult is rlatd to th analogous rsult undr homognous prfrncs (s Vronsi (2000, proof of Lmma 3) and also Appndix to Pastor and Vronsi (2006)). In particular, Pastor and Vronsi (2006) us a similar convxity rsult to xplain th high valuations in th lat 1990s, a point I will rturn to in Sction 4. Th volatility formula (43) undr larning is similar to th on undr crtainty in Proposition 4, xcpt that w hav th additional larning-rlatd trm (Êtµ] E t µ] ) σ 1. Fortunatly, as in Vronsi (2000), I can provid a sign to this trm dpnding on whthr δ t+τ g(δ t+τ) µ ] is incrasing or dcrasing in µ. This in turn dpnds on th distribu- E x,τ t tion of risk tolranc ρ and th currnt output δ t. Corollary 14: (a) If ρ H < 1, thn Êtµ] < E t µ]; (b) If ρ H > 1 and δ t is sufficintly larg, thn Êtµ] > E t µ] Charactrizing th sign of Êtµ] E t µ] is not sufficint to also sign th full impact of larning on volatility and risk prmium, howvr, bcaus uncrtainty also impacts th trm Ẽ µ,x,τ t g (δ t+τ )]. W will s in th numrical xampl that th scond trm dominats th impact of larning on asst rturns. Moving to conomics, as in Vronsi (2000), if all agnts hav EIS< 1 (part (a)), thn highr growth µ always implis a lowr P/D ratio. Thus, th diffusion of stock rturns is smallr undr larning bcaus a positiv shock to dividnds incrass th xpctd futur growth which in turn rducs th P/D ratio. Consquntly, whil th positiv shock incrass th pric dirctly bcaus of highr dividnd rat, th incras in xpctd growth rducs th P/D ratio (for givn discount) and thus rducs th impact of th positiv shock. If ρ H > 1, howvr, and th conomy is growing, at som point th wight on high EIS agnts is sufficint to imply that an incras in xpctd growth rat incrass th P/D ratio. This additional kick to prics will incras th volatility of stocks. Unfortunatly, th risk prmium at this point would still b low bcaus risk prics ar low A popular xpdint to obtain highr volatility and thus highr risk-prmium is to assum that quity is lvragd. Following Abl (1990), th pric of a scurity that pays D L,t = D α t for α > 0 not only has highr volatility on its own, but it gnrat a P/D ratio that is incrasing in µ if α is sufficintly high. As such, th volatility and quity prmium of such scurity would b nhancd substantially. 23

25 3.2. Walth Dynamics undr Uncrtainty Undr paramtr uncrtainty, th walth lvl and its dynamics ar immdiatly rlatd to thir countrparts in Propositions 4 and 5: Proposition 14: Th walth of ach agnt i at tim t is W it = p(t t) s i λ(ρ i ) g(δt) E µ,x,τ ] t (ρ i 1)g(δ t+τ) (46) and th walth dynamics is dw it + C it dt W it = µ Wit dt + σ Wit dẑt (47) whr µ Wit = r t + σ Wit σ Mt and σ Wi,t = g (δ t ) + (ρ i 1) Eµ,x,τ t g (δ t+τ ) ] (ρ i 1)g(δ t+τ) E t µ,x,τ (ρ i 1)g(δ t+τ) ] σ + (Êi t µ] E tµ] ) σ 1 whr Êi.] uss th distribution π i t(µ) π t (µ)e x,τ (ρ i 1)g(δ t+τ) µ ] Th volatility of walth (48 ) is similar to th countrpart in quation (34) without larning, xcpt for th last trm, which dpnds on paramtr uncrtainty. Fortunatly, w can asily s whthr this trm is positiv or ngativ, and it dpnds on risk tolranc: Corollary 15: Êi tµ] > E t µ] if and only if ρ i > 1. Corollary 15 suggsts that paramtr uncrtainty incrass walth volatility of agnts that ar mor risk-tolrant than log-utility, whil it dcrass walth volatility of invstors that ar lss risk-tolrant than log. It is hardr to prov th gnral statmnt bcaus paramtr uncrtainty also affcts th scond trm in quation (48), which is hardr to sign. Still, to first ordr, it sms that introducing larning incrass th volatility of walth for agnts that ar mor risk tolrant than log. Indd, w can instad prov th following: Corollary 16: Th walth volatility of log-utility invstors is not affctd by paramtr uncrtainty. If all invstors in th conomy ar lss risk tolrant than log (i.. ρ H < 1), th all housholds hav lowr walth volatility undr larning than undr crtainty. Although such rsult appars paradoxical, it is of cours rlatd to th sam rsult alrady highlightd for stocks and discussd in Vronsi (2000). 24

26 3.3. Portfolio Allocation undr Uncrtainty Bcaus markts ar still dynamically complt undr larning, th allocations to stocks and bonds ar still givn by xprssions (37) and (38). To bttr charactriz how th chang in uncrtainty affcts trading bhavior, I considr again a man-prsrving sprad on th distribution π t (µ). Gnral rsults ar hard to obtain in this cas, but I can prov th following: Proposition 15: Lt π t (µ) b a man-prsrving-sprad on th distribution π t (µ). If σ is sufficintly small, thn a man-prsrving sprad on π t (µ) incrass th rlativ position in stocks of agnts with risk-tolranc highr than on, and dcrass it for thos with risktolranc lowr than on. That is Ñ ρ i>1 it Ñ ρ i<1 it > Nρ i>1 it N ρ i<1 it Hnc, highr uncrtainty on th drift rat of consumption incrass th rlativ position in stocks of agnts with risk tolranc highr than 1 and dcrass it for agnts with risk tolranc lowr than 1. This implis that whn htrognity of risk avrsion ncompasss agnts with risk tolranc abov and blow on, thr is a clar incras in th disprsion of stock positions, and hnc of trading across agnts. Th proposition can only b shown whn σ is sufficintly small by a continuity argumnt, but simulations blow show that it is rathr gnral fact. Corollary 13 and Proposition 15 show that an incras in uncrtainty brings about an incras in stock prics and an incras in th disprsion of portfolio holdings, which can only b achivd through trading. Bcaus my modl with gnral priors implis that uncrtainty may ndognously incras, w can xpct that th pric of th stock to incras and th trading volum incras at th sam tim. Th rason is that highr macroconomic uncrtainty incrass risk sharing opportunitis across htrognous agnts and hnc it gnrats additional trading. Morovr, highr uncrtainty also implis that blivs fluctuat mor wildly, gnrating trading volum. 25

27 4. A Calibration To calibrat th modl w nd to first rturn to th issu of solving th fixd-point problm in Equations (9) and (10), that is: ] δt = E CS s i λ(ρ i ) ] ρ ig(δ t) ; λ(ρ i ) = Eµ,x,τ 0 δ τ g(δ τ) E µ,x,τ 0 (ρ 1)g(δτ) ] For gnral ndowmnt distributions s i, th functional fixd point is hard to solv. I rathr procd as follows: (48) 1. Lt th ndowmnt b givn by s i = s(ρ i ) s i whr s i > 0, E s i ] = 1, Cov s i, ρ i ] = 0 ; 2. Assum th product s(ρ)λ(ρ) has th functional form for som constant y; s(ρ)λ(ρ) = ρy E CS ρy ] 3. Comput g(δ t ) by solving th quation ] δt = E CS s(ρ)λ(ρ) ρg(δt)] = ECS ρ(g(δ t) y) E CS ρy ] ; (49) 4. Comput 5. Comput th initial ndowmnt λ(ρ) = Eµ,x,τ 0 Som algbra shows that E CS s(ρ)] = 1. E µ,x,τ δ τ g(δ τ) ] 0 (ρ 1)g(δτ) ] ; s(ρ) = 1 ρy λ(ρ) E CS ρy ] ; Whn µ/σ is high, λ(ρ) is dcrasing in ρ, which in turn maks s(ρ) incrasing in ρ. This is a problm as th procdur abov naturally givs mor ndowmnt to high risk tolrant agnts. Howvr, by judiciously choosing th constant y I can undo th impact of λ(ρ) on s(ρ) and obtain initial distribution for s(ρ) that is incrasing or dcrasing in ρ (or U- shapd). Unfortunatly, th simpl mthodology abov to numrically solv th modl dos 26