The Great Recession gave way to a period of very low short-term

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1 Economic Quarerly Volume 00, Number 3 Third Quarer 204 Pages How Can Consumpion-Based Asse-Pricing Models Explain Low Ineres Raes? Felipe Schwarzman The Grea Recession gave way o a period of very low shor-erm nominal and real ineres raes. As he recovery proceeds and he Federal Reserve sars o decide he rhyhm wih which i inends o raise policy raes, one fundamenal quesion is wheher he low ineres raes are jus a sympom of a recessionary period (even if prolonged) in which he Federal Reserve chose o ake a deliberaely expansionary sance, or if hey re ec longer-run fundamenal forces ha may no dissipae easily. In he laer case, opimal policy may warran a slow increase of he policy ineres rae, so ha i remains low by hisorical sandards even when in aion and he labor marke are close o heir long-run levels. Currenly, Federal Open Marke Commiee members appear o forecas such a slow increase, as documened in he Summary of Economic Projecions. The purpose of his aricle is o use consumpion-based asse-pricing models o gain some insigh ino he deerminans of he naural ineres rae, ha is, he ineres rae ha would prevail in he absence of nominal rigidiies. Since his naural rae is no iself a funcion of cenral bank decisions, i can be used as a yardsick for he sance of moneary policy. In paricular, in erms of modern moneary heory (Woodford 2003), one can say ha he policy sance is expansionary The auhor hanks Alex Wolman, Marianna Kudlyak, Pierre-Daniel Sare, and Seven Sabol for valuable commens ha helped improve his aricle. The views expressed in his aricle are hose of he auhor and do no necessarily re ec hose of he Federal Reserve Bank of Richmond or he Federal Reserve Sysem. Felipe.Schwarzman@rich.frb.org.

2 20 Federal Reserve Bank of Richmond Economic Quarerly if he ineres rae is below he naural rae of ineres and conracionary oherwise. The quesion abou he opimal pace of ineres rae lifo can hus be recas in erms of he speed wih which he naural rae of ineres is likely o increase. Consumpion-based asse-pricing models are a naural saring poin for he discussion of he fundamenal deerminans of ineres raes for macroeconomiss since hey share convenional assumpions of mos workhorse macroeconomic models: raional expecaions, fricionless asse markes, and a represenaive household. This conrass wih behavioral economics models, which emphasize deparures from raional expecaions, and wih segmened markes models, in which asse prices are deermined by only a subse of households. 2 While hese alernaives are cerainly worhy of furher discussion, he purpose of his aricle is o provide a rs look a he progress ha one can make wih his more familiar baseline. 3 I will review hree main srands wihin he consumpion-based asse-pricing lieraure: habi formaion, long-erm risk, and disaser risk. Raher han provide a comprehensive review of he lieraure wihin each of hose srands, I will discuss some of he main ideas based on a small number of in uenial aricles. 4 A he end of each secion I include a shor discussion of how he model could be used o explain low ineres raes. Those discussions are mean o be illusraive raher han conclusive, in ha hey delimi promising areas for furher research raher han provide a complee answer o how well consumpion-based asse-pricing models can explain currenly low ineres raes. As we will see in he models reviewed, ineres raes can be low eiher because marke paricipans expec consumpion growh o be low, because hey perceive consumpion risk o be high, or because Naurally, he cenral bank chooses he nominal ineres rae, wih he real ineres rae being deermined endogenously, whereas he naural rae of ineres is ypically undersood o be a real rae. For more on he link beween real and nominal ineres raes from a consumpion-based asse-pricing perspecive, see Sare (998) and Wolman (2006). 2 In paricular, Mehra and Presco (2008) quesion he assumpion abou wheher he highly liquid Treasury bill rae is an appropriae measure of he ineres rae ha households use o save for reiremen and smooh consumpion. 3 For examples of aricles relying on segmened markes o accoun for he reducion in ineres raes pos-2008, see Del Negro e al. (200), Guerrieri and Lorenzoni (20), and Eggerson and Krugman (202), and more generally, Vissing-Jørgensen (2002) and Vissing-Jørgensen and Aanasio (2003) for a discussion of how marke segmenaion a ecs ineres raes. Seminal aricles in he behavioral nance lieraure are Barberis, Shleifer, and Vishny (998); Daniel, Hirshleifer, and Subrahmanyam (998); and Hong and Sein (999). See also Shleifer (2000) and Barberis and Thaler (2003) for reviews. 4 In fac, o a large exen he maerial in his aricle is a reorganizaion of maerial in more deailed reviews by Campbell (2003), Barro and Ursúa (20), and Cochrane (20). While his aricle is wrien so as o be largely self-conained, he reader is referred o hose exs for many of he deails (including some of he derivaions).

3 F. Schwarzman: Consumpion-Based Asse-Pricing Models 2 Figure The Equiy Premium and he Risk-Free Rae hey have low risk olerance. In conras, equiy risk premia do no depend on expeced consumpion growh. Hence, one can gain some insigh ino he driving force behind low ineres raes by examining he behavior of he risk premium. The evoluion over ime in he wo variables can be seen in Figure. I depics he poswar values of he real ineres rae, measured by he 30-day Treasury bill rae de aed by he consumer price index, and of he equiy risk premium, boh of which averaged over various ve-year periods. 5 The ve years since he onse of he Grea Recession sand ou no only because of he excepionally low real rae of ineres, bu also because of a hisorically high equiy risk premium. Given he models reviewed, he high risk premium suggess ha low ineres raes in he recen period are likely o be eiher a consequence of a percepion ha consumpion risk is paricularly high, or of very low risk olerance. The aricle is srucured as follows: In he following secion, I lay ou he noaion used in he aricle as well as common convenions, simpli caions, and approximaions. Each subsequen secion discusses 5 To calculae he equiy risk premium, I use he value weighed equiy reurns index from he Cener for Research in Securiy Prices.

4 22 Federal Reserve Bank of Richmond Economic Quarerly one variey of consumpion-based asse-pricing models: he Mehra and Presco (985) benchmark, he recursive uiliy and long-run risk exensions of Weil (989) and Bansal and Yaron (2004), he disaser-risk model of Riez (988) and Barro (2006), and Campbell and Cochrane s (999) habi-formaion models. The nal secion concludes.. NOTATION, CONVENTIONS, SIMPLIFICATIONS, AND APPROXIMATIONS Asses are claims on sreams of dividends. In paricular, purchasing some asse, i, provides an economic agen wih a sochasic sream of dividends D+s i for as long as he agen holds i. In consumpionbased asse-pricing models here are no liquidiy consrains or oher s=0 ransacion coss, so agens can rade asses freely a each period. If he price of asse i is given by P i, hen we can de ne is reurn beween periods and + as R i;+ P i;+ + D i;+ P i; : () Asse pricing concerns iself eiher wih deermining he pricedividend raio for an asse, P i ; or is expeced reurns, E D i [R i;+ ]. Typically, higher reurns are associaed wih lower price-dividend raios. While he lieraure discusses he pricing of many kinds of asses, he hree main ones are he risk-free asse, a marke porfolio of equiies, and oal wealh. The risk-free asse (denoed by i = f) is exacly wha he name implies: an asse ha pays he same dividend in all saes of naure. As an empirical maer, he asse-pricing lieraure ideni es he risk-free asse wih shor-erm Treasury bills. Thus, he predicions of he models under review for he risk-free rae are going o be he mos relevan ones for he purpose of moneary policy analysis. The marke porfolio of equiies (i = e) refers o a well-diversi ed porfolio of shares issued by rms and raded in sock markes wih prices summarized by indices such as he S&P 500. This is, in urn, di eren from oal wealh (i = w), which is a ciious asse (in he sense ha here are no formal markes for i) ha pays ou aggregae consumpion as dividends. I includes equiy, bonds, housing, and human capial. Ofenimes sudies of equiy pricing a rs idenify equiy wih he wealh porfolio and hen in re nemens rea he wo as disinc. The disincion beween equiy and he wealh porfolio normally focuses on he fac ha rms are leveraged, boh because hey issue bonds and because salaries are normally insulaed from

5 F. Schwarzman: Consumpion-Based Asse-Pricing Models 23 high-frequency ucuaions in oupu. Therefore, for any change increase in aggregae endowmen, dividends should change by a greaer amoun. The simples way of modeling his leverage is o assume ha aggregae dividends on equiy are a deerminisic funcion of consumpion, wih D e = (D w ) = C, for some >. One simpli caion used by he asse-pricing lieraure o obain analyical resuls is o rely on log normaliy assumpions. If he log of asse reurns is normally disribued, one can use he fac ha for any normally disribued x, E [e x ] = e E[x]+ 2 V ar[x]. Thus, if reurns R i;+ are log-normally disribued, ln (E [R i;+ ]) = E [r i;+ ] + 2 V ar [r i;+] ; where we use small leers o denoe he naural logarihm. A furher simpli caion, used in disaser models, is he use of a coninuous ime formulaion o sudy disaser risk. Denoe by d he lengh of a period of ime. Le e r i;+d be he gross reurn per period of ime of ha asse. Suppose he reurn on some asse i is eiher e rd wih probabiliy e pd or ( b) e rd wih probabiliy e pd. Then h i h E e r i;+dd = e pd + e pd i ( b) e r : Taking logs and dividing by d yields ln E e r i;+dd d = r + ln e pd + e pd ( b) Taking he limi as d! 0 and applying l Hopial s rule, d : E [r i;+d ] = r pb: The coninuous ime approximaion yields an inuiive expression for expeced log reurns. Those are equal o r, excep ha wih probabiliy p hey fall by b. Finally, a common approximaion used in he analyical lieraure is o log-linearize equaion () o obain r i;+ = p i;+ + ( ) d i;+ p i; ; P where is he average P +D raio and is ypically calibraed o some value close o. Rearranging and ieraing forward up o some ime + T wih T > 0 yields

6 24 Federal Reserve Bank of Richmond Economic Quarerly p i; d i; = TX s d ++s s=0 T X s=0 s r i;++s + T + p i;t + : The expression is useful in ha i breaks down hree di eren deerminans of he price-dividend raio. The rs erm on he righ-hand side is a discouned sum of fuure dividends growh. The faser dividends are expeced o grow, he more a porfolio ha pays o he consumpion good as dividends is worh. The second erm is a discouned sum of reurns. All else consan, if prices are low in spie of high dividend growh, hen he reurns will be high as prices cach up wih dividends. The hird erm is a bubble erm. In mos assepricing applicaions, one assumes ha he bubble erm goes o zero almos as surely as T increases. Given he no-bubble condiion, X p i; d i; = s d ++s s=0 X s=0 s r i;++s : The equaion highlighs ha a high price-dividend raio can forecas eiher a high growh in dividend paymens or low fuure raes of reurns. Taking expecaions and rearranging, (E + E ) r i;+ = (E + X E ) s d ++s (E + X E ) s r i;++s ; (2) s=0 where (E + E ) r i;+ r i;+ E r i;+ denoes he surprise in reurns. The laer equaion is useful o assess he sources of volailiy in an asse reurn. I emphasizes ha he volailiy in reurns for any asse can be a funcion of eiher he volailiy of news concerning is fuure dividend ows or news concerning is fuure reurns. s= 2. THE MEHRA AND PRESCOTT BENCHMARK We sar by examining a simpli ed version of he power uiliy benchmark case examined by Mehra and Presco (985). This corresponds o he common seup in macroeconomic models in which households are endowed wih a separable power uiliy of consumpion. As commonly done in he nance lieraure, Mehra and Presco follow Lucas (978) and focus on he case of an endowmen economy in which households

7 F. Schwarzman: Consumpion-Based Asse-Pricing Models 25 consume and rade claims on immediaely perishable fruis ha fall from an in niely lived ree. 6 Individual households deermine how much o consume in each period of ime and how much o inves in a porfolio of asses ha i has available. We assume ha here are N di eren asses, indexed i 2 f; :::; Ng, and ha hose asses compleely span he shocks ha he households are subjec o so ha markes are complee. The problem of he household is " X max E 0 fx i g (;N) (;i)=(0;) =0 NX s:: : C + P i; x i; = i= C # NX x i; (P i; + D i; ) ; where x i is he amoun of shares of asse i held by he household a ime and, as before, P i is he realized price and D i is is realized dividend. The parameer is he coe cien of relaive risk aversion and governs he olerance ha households have for risk. I is also he inverse of he ineremporal elasiciy of subsiuion, governing he household s desire o smooh consumpion over ime. The opimaliy condiion for he household is P i; C i= h i = E C + (P i;+ + D i;+ ) : Le R i;+ P i;++d i;+ P i; be he reurn on asse i. Reurns, like prices, are equilibrium objecs deermined endogenously. Given expeced fuure prices and dividends, higher reurns are ied o lower prices a. Given he de niion of reurns and he opimaliy condiion, we have ha = E " C+ R i;+# : (3) C C+ The raio of marginal uiliies C is he pricing kernel in his economy. In order o hold a posiive and nie amoun of an 6 The analysis of asse-pricing models o environmens wih producion ( Producion Based Asse Pricing ) is iself an acive area of research ha we will leave undiscussed. For imporan conribuions in ha lieraure, see Cochrane (99); Jermann (998); Boldrin, Chrisiano, and Fisher (200); and Gomes, Kogan, and Zhang (2002), among many ohers.

8 26 Federal Reserve Bank of Richmond Economic Quarerly asse, a risk-neural household ( = 0) requires ha he reurn of he asse i be, on average, equal o irrespecive of is variance. If > 0, he household insead requires o be equal o a weighed average of reurns, giving more weigh o saes of he world where is consumpion growh is lowes. The implicaion of his weighing is easies o see if one rewries equaion (3) as " C+ # = E E R i + cov C C+ C ; R i Suppose here is a risk-free asse, denoed by i = f, so ha var 0. Then and E R i R f R f = R f! : R f = E C+ C ; (4) = cov C+ C ; R i! ; (5) so ha households reques a higher premium over he risk-free rae for asses in which he covariance beween he pricing kernel and he rae of reurns is negaive. I is possible o express equaions (4) and (5) in log-linear form if one is willing o assume ha he logs of consumpion growh and asse reurns are normally disribued. Then, wih E [r i;+ ] r f;+ + 2 i 2 = ic; (6) 2 2 c r f;+ = log + E c + 2 ; (7) where r i;+ are he log reurns on asse i, r f;+ are he log reurns on he risk-free asse, 2 i is he variance of he logarihm of he reurns on asse i, 2 c is he variance on he logarihm of consumpion growh, and ic is he covariance beween log reurns and log consumpion growh. The rs wo erms on he lef-hand side of equaion (6) are jus he di erences beween he expeced reurn on some asse i and he riskfree asse. The hird erm is a Jensen s inequaliy adjusmen erm, accouning for he fac ha, since logarihm is a concave funcion, he

9 F. Schwarzman: Consumpion-Based Asse-Pricing Models 27 logarihm of an expeced variable is always larger han he expecaion of he logarihm. 7 The erm on he righ-hand side has wo componens. The second, ic, is he covariance beween he asse reurn and consumpion growh and can be inerpreed as he quaniy of risk in he asse. The rs,, is he coe cien of relaive risk aversion and i can be inerpreed as he price of risk. Under power uiliy, he price of risk is consan, and asse prices only depend on he risk one period ahead. As famously demonsraed by Mehra and Presco (985), he model performs poorly in quaniaive erms. In heir baseline exercise, hey equae equiy wih he wealh porfolio, i.e., an asse ha pays ou aggregae consumpion as dividends. 8 Given ha consumpion growh does no vary much, he quaniy of risk ic is very low. Because of ha, Mehra and Presco nd ha for reasonable values of (0 and under), he equiy risk premium implied by he righ-hand side of equaion (6) is an order of magniude smaller han he one found in he daa. This observaion has spurred a very large lieraure and is a cornersone of modern asse-pricing research. For a large enough ; he model is of course able o mach he equiy premium. However, seing o a very large number also has implicaions for he risk-free rae ha do no he daa. In an average quarer, consumpion growh E [c + ] is close o 2 percen in yearly erms and he sandard deviaion has a similar magniude. If we ake he coe cien of risk aversion o be = 0, close o Mehra and Presco s upper bound, hen maching he risk-free rae of percen in yearly erms would require a discoun rae of close o 9 percen per year. In a period of ime where expeced consumpion is percen insead of 2 percen, he ineres rae would fall from percen o 9 percen. Inuiively, he reason for he radeo beween maching he high risk premium and he low ineres rae is ha capures how unwilling households are o le consumpion vary, be i over ime or beween saes of naure. The higher, he more households dislike variaion in consumpion along eiher dimension. Hence, if a household wih a high foresees ha is consumpion will grow slower, i will be very willing o borrow in order o keep consumpion smoohed ou over ime. In equilibrium, his leads o a sharp reducion in he ineres rae. 7 Formally, Jensen s inequaliy saes ha if g is a concave (convex) funcion, h i hen Ri;+ g (E [x]) > (<) E [g (x)]. In ha speci c case, he lef-hand side is log E > R f;+ E h log h ii Ri;+ = E R [r i;+ ] r f;+. f;+ 8 As a robusness, hey also consider he case where leverage increases he volailiy of equiies.

10 28 Federal Reserve Bank of Richmond Economic Quarerly Figure 2 Consumpion Growh and Risk Implicaions for he Ineres Rae in he Recen Period While, in quaniaive erms, he Mehra and Presco benchmark fails as an explanaion of asse pricing, i is sill a useful benchmark in ha i highlighs which facors are likely o maer for ineres raes in consumpion-based asse-pricing models. In wha follows, I use his benchmark as a qualiaive guide o he facors driving he risk-free ineres rae and show how hey have evolved in he curren recession. For convenience, I resae equaion (6) for he risk-free rae below: r f;+ = log + E c c 2 : (8) As equaion (8) makes clear, ineres raes can eiher be low because marke paricipans expec consumpion growh o be low or because hey perceive consumpion risk o be high. Figure 2 shows he average and sandard deviaions of quarerly consumpion growhs, boh expressed in annualized erms and averaged

11 F. Schwarzman: Consumpion-Based Asse-Pricing Models 29 over various ve-year periods. 9 While does feaure excepionally low consumpion growh for hisorical sandards, i also feaures excepionally low consumpion variance. Hence, in qualiaive erms, he model would have o accoun for he low ineres raes hrough low expeced consumpion growh. I is worh highlighing ha, given he Mehra and Presco (985) benchmark, here is a ension beween Figures and 2, since equaion (6) implies ha, if consumpion is correlaed wih dividends, a high variance of consumpion growh ough o be associaed wih a high equiy premium. 0 In conras, we observe a low variance of consumpion growh and a high equiy premium. As we will see, alernaive consumpion-based asse-pricing models can provide poenial resoluions o his inconsisency, as hey allow eiher for he possibiliy ha he price of risk may be changing (as in habi formaion models) or ha he kind of shor-erm consumpion risk depiced in Figure 2 may no be he bes measure of he kind of risk ha asse holders are mosly concerned wih when making heir porfolio decisions. 3. RECURSIVE UTILITY AND LONG-RUN RISK As discussed above, a major challenge facing common power-uiliy models is he di culy in maching boh households willingness o le heir consumpion change over ime (capured by a low ineres rae) and heir unwillingness o le i vary across saes of naure (capured by he high equiy risk premium). One possible soluion o his ension is o allow for he possibiliy ha he desire for ineremporal smoohing is governed by a di eren parameer han he desire for insurance. This is provided by he recursive uiliy funcion proposed by Epsein and Zin (989) and Weil (989, 990), based on prior work by Kreps and Poreus (978). In paricular, he recursive uiliy funcion 9 The consumpion series is aken from Marin Leau s websie and is de ned in Leau and Ludvigson (200). In paricular, i excludes durable goods, shoes, and clohing. 0 Consumpion variance is an imporan facor in explaining he equiy risk premium under he assumpion ha ha consumpion growh is i.i.d. and ha growh in sock dividends is perfecly correlaed wih consumpion growh, so ha d e; = c e;. Then, if we guess ha equiy reurns are also i.i.d., from equaion (2) we have ha (E + E ) r e;+ = (E + E ) d + : Since dividend growh is i.i.d., he guess ha equiy reurns are i.i.d. is veri ed. In his case, he covariance beween consumpion and equiy reurns ec is simply var (c e;). Hence, from equaion (6), higher consumpion variance is associaed wih a higher equiy risk premium.

12 220 Federal Reserve Bank of Richmond Economic Quarerly provides for he represenaion of preferences over loeries in which agens rank hem in erms of he ime in which uncerainy is resolved. For example, an agen may face wo di eren loeries ha pay he same amouns a some disan dae depending on he ip of a coin, bu in one loery he coin ip akes place immediaely, whereas in he oher i only akes place much laer. Under his kind of preference, agens may prefer he rs loery o he second even hough he disribuion of oucomes is idenical. The Epsein-Zin-Weil (EZW) uiliy funcion can be wrien as U = ( ( ) [C ] + E U + ) ; (9) where U is he uiliy a ime. Preferences for early resoluion of uncerainy emerge if <. The parameer can be inerpreed as he ineremporal elasiciy of subsiuion. This inerpreaion becomes mos clear in he deerminisic case. Wihou uncerainy, he exponens in around U + cancel ou and, wih a sligh rearrangemen, equaion (9) collapses o he usual Bellman equaion forma, wih period uiliy of consumpion given by C. The parameer can be inerpreed as a risk-aversion parameer. Heurisically, his can be seen in a version of he problem where he household only consumes in = 2 so ha here are no ineremporal choices o be made. Then, U = 0 for > 2, U 2 = C 2 ; and U = i E hc 2 h o maximizing expeced uiliy E, so ha he problem of he household is equivalen i. 2 C 2 Finally, i is also sraighforward o check ha, if =, equaion (9) collapses back o a recursive version of he benchmark power-uiliy case, in which he coe cien of relaive risk aversion is equal o he inverse of he ineremporal elasiciy of subsiuion. Given his uiliy funcion, one can derive he following Euler equaion for porfolio decisions: 3 This is a violaion of he independence axiom for preferences so ha wih Epsein-Zin-Weil preferences, uiliy will no necessarily be separable across saes of naure. 2 Sicly speaking, for his example we would need <, so ha he uiliy funcion is sill well de ned for C = 0. 3 See he Appendix for a derivaion.

13 F. Schwarzman: Consumpion-Based Asse-Pricing Models 22 2( = E 4 C+ ) C R w;+ R i;+ 3 5 ; (0) where R w;+ is he reurn on oal household wealh and, so ha in he benchmark power-uiliy case, =. The pricing kernel is C+ C R w;+ and is a weighed average of he pricing kernel obained in he benchmark separable uiliy case and he reciprocal of he reurn on wealh, R w;+.the reurn on wealh in he pricing kernel capures he impac of news abou fuure consumpion on agen s marginal uiliy. To see his, recall ha, from equaion (2), surprises in he reurns o he wealh porfolio saisfy (E + E ) r w;+ = (E + X E ) s c ++s (E + X E ) s r w;++s ; () s=0 where we use he fac ha, by de niion, he dividends on he wealh porfolio are equal o aggregae consumpion. Thus, surprises o he reurns on wealh re ec surprises in fuure consumpion growh, discouned by surprises o he fuure reurns on wealh iself. The reason why reurns on wealh are facors in he pricing kernel under EZW preferences is because of he nonseparabiliy beween uiliy for curren and fuure consumpion. Wih power uiliy, preferences are separable. Given ha agens are able o compleely change heir porfolio each period, hey need no concern hemselves wih consumpion ows in he far fuure when evaluaing which porfolio o hold beween wo adjacen periods. This is no longer rue wih EZW preferences. If he logs of consumpion growh and reurns are normally disribued, we can wrie he following expression for he risk premium associaed wih any given asse i: s= E r i;+ and for he risk-free rae, r f;+ + 2 i 2 = ic + ( ) iw ; (2) r f;+ = log + E [c + ] w c; (3)

14 222 Federal Reserve Bank of Richmond Economic Quarerly where now iw is he covariance beween he reurns on asse i and he reurn on oal household wealh, and 2 w is he variance of he oal reurns on wealh. Recursive preferences allow one o accoun for he equiy premium puzzle in wo ways. Firs, as highlighed by Weil (989), here is no longer a radeo beween maching he equiy risk premium and he risk-free rae, as here is an addiional parameer o be calibraed. Furhermore, as explored in deail by Bansal and Yaron (2004), wih 6= ; he covariance of he asse reurn wih he reurn on oal wealh iw becomes an addiional facor in deermining he equiy premium. Thus, if, for example, he variaion in oal reurn on wealh is similar o he variaion in equiy reurns, hen reurns on oal wealh are clearly much more volaile han consumpion, so ha iw is poenially much larger han ic. One problem wih evaluaing equaions (2) and (3) is ha he variance of oal wealh is hard o measure since oal wealh includes human capial. One can make some progress by imposing srucure on he process for consumpion. In paricular, suppose ha he consumpion growh c + is he sum of a predicable componen z and an unpredicable one c;+ as in c + = z + c c;+ z + = ( ) g + z + z z;+ ; wih c;+ and z;+ i.i.d. sandard normal variables. 4 Wih close o and high c, his srucure allows for consumpion growh o have a predicable, sochasic, long-erm componen, even if a high frequencies overall consumpion growh is hard o predic. 5 For he wealh porfolio, he dividends are equal o aggregae consumpion, so ha R w; = P w;++c + P w;. From equaions (2) and (3) we have ha, if risk doesn vary over ime (so ha i is homoscedasic), hen E r w;+ = + E [c + ] ; where is a consan ha depends on he variances. This allows us o subsiue ou he reurns from he righ-hand side of equaion (2) o obain 4 I is sraighforward bu edious o allow for correlaion beween c;+ and z;+, so we will assume ha hey are uncorrelaed. 5 The quesion of wheher or no consumpion growh rae has a persisen componen is hard o sele, since is hard o esimae in small samples. Bansal and Yaron (2004) show ha a large is no inconsisen wih observed auocovariance of consumpion growh and observed variances of consumpion growh a di eren horizons.

15 F. Schwarzman: Consumpion-Based Asse-Pricing Models 223 (E + E ) r+ w = (E + E ) c ++s X + (E + E ) s c ++s : (4) We can now use he expression jus derived o describe he sources of one-sep-ahead variaion in reurns o he wealh porfolio. The rs componen on he righ-hand side is he innovaion in consumpion growh, wih variance 2 c. The second componen is a discouned sum of fuure consumpion growh. I changes as news abou fuure consumpion growh arrives, in he form of innovaions o z +. This second componen incorporaing news abou fuure consumpion is wha allows reurns on he wealh porfolio, and hence he pricing kernel, o be signi canly more volaile han consumpion growh. If, insead, consumpion growh were i.i.d. so ha his componen would be equal o zero, he variance of reurns on wealh would be as small as he variance of consumpion growh. The higher variance of he pricing kernel associaed wih persisence in consumpion growh is wha allows models wih EZW preferences o imply subsanially larger risk premia han models wih power uiliy for a given value of he risk-aversion parameer, as one can see from equaions (2) and (3) deermining he risk premium and he risk-free rae. Bansal and Yaron (2004) emphasize ha a reasonable parameerizaion of he model requires boh > and >. They choose = 0, a he upper bound of Mehra and Presco s (985) exercise, and = :5. The choice of is subjec o debae, as many empirical sudies of consumpion behavior over ime poin o very low values for. Bansal and Yaron (2004) couner ha sochasic variance in consumpion inroduces a downward bias in esimaes of and ha, furhermore, sudies wih more disaggregaed consumpion daa suppor lower. Imporanly, hey also poin ou ha one can discipline he value of hrough he correlaion beween asse prices and news abou consumpion growh and consumpion volailiy. This can be seen in equaion (4), where, wih >, news abou fuure consumpion growh leads o an increase in he reurns on wealh, bu wih < such news leads o a reducion. As emphasized by Bansal and Yaron (2004), recursive preferences imply ha risk premia vary no only wih news abou fuure consumpion growh, bu also wih news abou is variance. A higher variance of innovaions o fuure consumpion growh increases he variance of reurns on he wealh porfolio and, hence, of he pricing kernel, leading o a higher equiy premium and lower risk-free raes. Therefore, s=

16 224 Federal Reserve Bank of Richmond Economic Quarerly ime variaion in he variance of long-run growh ( long-run risk ) can be an imporan facor explaining he variance in risk premia observed in he daa. Implicaions for he Ineres Rae in he Recen Period For convenience, I resae he equaion describing he deerminans of he risk-free rae: r f;+ = log + E [c + ] w c: Noe ha under he calibraion adoped by Bansal and Yaron (2004), = 2 3, 2 = 4, and = 6, so ha he weigh placed 2 2 on he wo risk facors is comparaively large. This equaion holds for he case of homoscedasic risk. Bansal and Yaron (2004) also provide a derivaion of he risk-free rae when risk is ime varying so ha 2 w and 2 c are funcions of ime. In ha case, he coe ciens change bu he essenial facors deermining he risk-free rae remain he same. 6 The recursive preferences model implies ha he risk-free rae changes no only wih he expeced growh rae of consumpion or wih he variance of ha growh rae, bu also wih changes in he mean and variance of reurns on wealh, 2 w. As previously discussed, hese are, in urn, funcions of he variance of he long-erm componen of consumpion growh. Given he calibraion advocaed by Bansal and Yaron (2004), a reducion in he ineres rae could hus sem no only from he same facors ha explain he reducion in ineres raes in he benchmark ime-separable model, bu also from an increase in he variance of he long-run componen of consumpion growh. Toal wealh in he economy includes no only equiy in rms, bu also housing and human capial. Figure 3 depics he volailiies of equiy reurns and house price increases over ve-year periods. 7 Boh volailiies were high by hisorical sandards in he period, mos noably he volailiy of housing reurns. Thus, long-run risk could, in principle, help explain he low ineres raes while accouning for he disconnec beween high risk premia and he low volailiy of consumpion growh in ha period. More generally, however, he 6 Bansal and Yaron consider a case in which here is only one sochasic risk facor so ha 2 w and 2 c co-move perfecly. 7 The housing price daa is from Shiller (205). House price increases are a good approximaion for housing reurns so long as rens are sable.

17 F. Schwarzman: Consumpion-Based Asse-Pricing Models 225 Figure 3 Wealh Risk correlaion beween hese volailiies and he equiy premium is quesionable. For example, he period exhibis very high house price volailiy even as he equiy risk premium is very low (see Figure ). Likewise, he period exhibis very low equiy risk premia ogeher wih a very volaile equiy premium. Naurally, hese are only rough correlaions based on period averages using arbirary cuo s, so his should no be seen as grounds for rejecing he long-run risk model. Also, we have ignored he hard o measure conribuion of volailiy in reurns o human capial. 4. DISASTER RISK One early reacion o Mehra and Presco s (985) equiy premium puzzle is ha he disribuion of asse reurns and consumpion growh is prone o rare bu large disasers. If hose disasers are likely o have a larger impac on he dividends paid ou by equiies han on he reurn on sovereign bonds, hey can generae a large premium beween socks and bonds as privae agens seek o insure hemselves agains hose rare occurrences. The argumen was rs pu forward by Riez (988). Barro (2006) makes a case for he argumen by using inernaional daa o

18 226 Federal Reserve Bank of Richmond Economic Quarerly calculae he probabiliies and magniudes of large disasers, puing a.7 percen probabiliy of a collapse in consumpion of, on average, 30 percen. 8 He also calculaes he probabiliy of sovereign defaul in he even of a disaser and he recovery rae ha invesors can expec in hose evens. He nds ha wih a coe cien of relaive risk aversion as small as four and a discoun rae of 3 percen per year, i is possible o obain equiy premia and risk-free raes ha are closer o he daa. Barro (2006, 2009) considers an environmen where he aggregae endowmen follows a random walk wih drif g and variance 2 c mos of he ime, bu wih probabiliy p i collapses permanenly o a fracion b of is value, where b is iself a random variable drawn from he empirical disribuion of disasers ha he documens. Taking a coninuous-ime limi, Barro (2009) arrives a expressions ha, afer a subsiuion, yield he following expressions for he risk-free rae: 9 r f = log + g p E ( b) ; and, if one akes, as he does, equiy o incorporae all of he wealh porfolio, for he risk premium: r e h i r f = 2 + p E ( b) E ( b) E [b] : Thus, an increase in he probabiliy of disasers leads o a reducion in he riskless rae and an increase in he equiy risk premium. One imporan resul is ha asse reurns are nonlinear funcions of he size of disasers b. This enhances he abiliy of disasers generaing large risk premia and low ineres raes since, as b approaches ; he marginal uiliy of consumpion in he disaser sae approaches in niy. Furhermore, as emphasized by Barro, he model can accommodae bonanzas, which are as large as he disasers and sill generae large risk premia, since households will be much more concerned wih he disaser saes (in which hey have high marginal uiliy) han wih he bonanza saes (in which heir marginal uiliy is low). Barro and Ursúa (20) provide a comprehensive review of he small lieraure ha has emerged around he noion of disaser risk being a key driver of asse-pricing daa. This lieraure has expanded he model 8 In paricular, Barro de nes a disaser as an even in which gross domesic produc (GDP) drops by 5 percen or more, and equae he change in consumpion wih he observed change in GDP. h 9 The isubsiuion in quesion is from he expeced consumpion growh C+ C E (denoed g C in Barro [2009]) for is deerminans, g p E [b]. The subsiuion singles ou g since i is likely o be closer o observed average log consumpion growh hen g.

19 F. Schwarzman: Consumpion-Based Asse-Pricing Models 227 o allow for ime-varying disaser risk, hus allowing i o explain imevarying risk premia (Gourio 200), and disasers ha are correlaed across counries and happen slowly raher han quickly (Nakamura e al. 200), as well as o evaluae implicaions of he model for addiional asse pricing facs (Gabaix 2008). Implicaions for he Ineres Rae in he Recen Period For convenience, I resae he equaion describing he deerminans of he risk-free rae: r f = log + g p E ( b) : In addiion o he deerminans of ineres raes in he oher models (expeced growh and one-sep-ahead volailiy of consumpion), models wih economic disasers imply ha ineres raes ough o change in response o changes in he probabiliy of disaser or o changes in he expeced size of disasers. I is plausible ha, in he afermah of he Grea Recession, economic agens have updaed upward heir subjecive probabiliies of such an episode occurring again. This could go some way in explaining he smaller ineres rae observed in he recen period. In paricular, consumpion dropped 2.7 percen beween Q2:2007 and Q4:2009. Relaive o a 2 percen per year rend, he reducion was 7.9 percen. Suppose ha, given ha observaion, agens assign a probabiliy of 5 percen o a drop in consumpion of 5 percen in any given period, so ha such a disaser occurs on average every 20 years. 20 Then, if hey have a risk aversion of four, hey will reques a risk-free rae ha is 0:05 0:95 4 = :4 percen in yearly erms smaller han before. This revision is unlikely o dissipae very quickly since, given he small probabiliies of a disaser occurring, he fac ha anoher one hasn come o fruiion should weigh lile on he probabiliy assessmen. Imporanly, apar from helping explain he lower ineres rae, he disaser risk could allow one in principle o reconcile he low volailiy of 20 A 5 percen probabiliy would be high compared o he.7 percen calculaed by Barro (2006), bu, in conras, he reducion in consumpion of 5 percen is less exreme han he average 30 percen reducion found in ha sudy. One obvious cavea is ha he drop in consumpion occurred smoohly, over wo and a half years, whereas he model assumes ha he whole change occurs insananeously. Nakamura e al. (200) show how he rare disaser model can accomodae slow disasers if agens have EZW preferences.

20 228 Federal Reserve Bank of Richmond Economic Quarerly consumpion wih he high equiy premium in he pos-grea Recession era. 5. HABITS In boh he discussion of rare disasers and long-erm risk, he ime variaion in expeced risk premia is undersood primarily as semming from ime variaion in he quaniy of risk ha households face. Under habi formaion, his same ime variaion is explained as semming from variaion in he risk olerance of households, which deermines he price of risk. In habi-formaion models, he marginal uiliy of consumpion depends on a ime-varying sae variable ha evolves as a funcion of pas consumpion decisions. The key idea is ha as households become habiuaed o cerain consumpion levels, heir marginal uiliy of consumpion becomes higher for a given level of consumpion. Habiformaion models di er along several dimensions, including wheher habis are inernal (where habi depends on individual household consumpion) or exernal (where habi depends on aggregae consumpion), wheher habis ener in he uiliy funcion muliplicaively or addiively, and wheher habis change more or less quickly wih consumpion. 2 In wha follows, we discuss he model by Campbell and Cochrane (999). Campbell and Cochrane (999) poin ou ha habi models are successful in generaing volaile ime-varying risk premia because hey increase he volailiy of he marginal uiliy of consumpion. They assume habis ener addiively and are exernal so ha u (C ; X ) = (C X ) ; where X is he sock of habis. Then he curvaure of he uiliy funcion wih respec o C is given by u cc (C ; X ) C u c (C ; X ) = S ; where S C X C is surplus consumpion, he gap beween consumpion and he habi. I follows ha he curvaure is higher in absolue 2 See Campbell (2003) for a more deailed discussion. Models of habis in nance include Sundaresan (989), Abel (990, 999), Consaninides (990), and Campbell and Cochrane (999), among ohers.

21 F. Schwarzman: Consumpion-Based Asse-Pricing Models 229 erms when consumpion is closes o is habi level X. This imevarying curvaure implies ha he pricing kernel uc(c +;X + ) u c(c ;X ) is also likely o vary wih S. To complee he speci caion of preferences, Campbell and Cochrane (999) need o specify how habis evolve over ime. Raher han using a more convenional speci caion in which he habi sock X evolves as a log-linear funcion of C, hey recur o a nonlinear speci caion in which S is a log-linear funcion of changes in log C. One advanage of his speci caion is ha i ensures ha surplus consumpion S is always posiive, which is necessary for he uiliy o be well de ned. They de ne he evoluion of surplus consumpion o be given by s + = s + ( ) s + (s ) (c + c E [c ]) ; where S is he seady-sae level of he habi (and s is log), and (s ) is a nonlinear funcion of s. The nonlinear erm (s ) helps hem deal wih one imporan di culy wih habi-formaion models. This is ha, while a ime-varying pricing kernel helps generae volaile expeced risk premia, i can also give rise o counerfacually volaile ineres raes. In Campbell and Cochrane s speci caion, he risk-free rae is given by r f = ln () + E 2 2 [c + ] ( ) (s s) [ + (s )] 2 : 2 The rs wo erms are he ones obained in a model wihou habis. The following wo include he e ec of habis. The hird erm summarizes he e ec of habis on ineremporal subsiuion. Surplus consumpion is expeced o mean rever a he rae. If i is above is seady-sae levels, hen households expec i o become smaller over ime, which is o say ha hey expec heir marginal uiliy o become smaller. Thus, hey become more paien, leading o a smaller equilibrium risk-free rae. The las erm on he righ-hand side capures he e ec of consumpion risk on he risk-free rae. Now, apar from he usual reason hrough which consumpion risk generaes precauionary savings, households also seek o keep heir consumpion risk low because i is correlaed wih heir habi formaion. In periods in which realizaions of consumpion are high, surplus consumpion also increases. Campbell and Cochrane (999) discipline heir choice of (s ) by adding hree requiremens. Two of hem are echnical. They impose ha X is pre-deermined in seady sae and ha i is always increasing in shocks o c. These condiions ensure ha, close o seady sae, heir process for habis resembles more common speci caions. The

22 230 Federal Reserve Bank of Richmond Economic Quarerly hird requiremen is ha risk-free raes do no vary wih habis. Thus, by consrucion, heir model delivers a low volailiy for he risk-free raes, as in he daa. This allows hem o focus more sharply on he variaion in risk premia. Given Campbell and Cochrane s (999) calibraion, he ineres rae is r f + = log () + E [c + ] S 2 2 c 2 : Noe ha S is he curvaure of he uiliy funcion wih respec o consumpion in seady sae and is hus a measure of he risk olerance of households. If S <, i is possible for he model o have a large seady-sae curvaure wih respec o consumpion ( S ), leading o high risk premia, even if i has a relaively low. This, in urn, allows i o admi more moderae ineres raes. Speci cally, Campbell and Cochrane calibrae = 2:372 and S = 0:049, so ha he curvaure of he uiliy funcion close o seady sae is approximaely equal o 48. Campbell and Cochrane (995) also consider an exension of he model in which hey choose (s ) o ensure ha risk-free raes are a linear funcion of log habis, decreasing when surplus consumpion is high. They pick he inercep o correspond o a percen real ineres rae and he slope so ha he lower bound for he real ineres rae is zero. Implicaions for he Ineres Rae in he Recen Period For convenience, I resae he equaion describing he deerminans of he risk-free rae: r f = log () + E [c + ] c S 2 : As calibraed by Campbell and Cochrane, he facors deermining he real ineres rae in he model wih habis are he same as in he Mehra and Presco (985) benchmark, he only di erence being ha he model wih habis assigns a greaer weigh o consumpion volailiy. The models wih long-run or disaser risk are able o explain he reduced ineres rae wih he inroducion of risk facors ha canno be easily discerned by measured consumpion volailiy. The model wih habis sands in conras o ha. Thus, like he Mehra and Presco (985) benchmark, i needs o rely on he hisorically low consumpion growh rae o accoun for he low ineres raes. However, for any

23 F. Schwarzman: Consumpion-Based Asse-Pricing Models 23 choice of, he habi model also pus a greaer weigh on he variance of consumpion growh c (since S < ), which was also low by hisorical sandards in he pos-2009 period. Therefore, for any choice of, he habi model would imply ha he risk-free rae should have fallen by less han wha is implied by he Mehra and Presco (985) benchmark. One signi can advanage of he habi formaion model over he Mehra and Presco model is ha i can also accommodae he hisorically high equiy risk premium, since he reducion in consumpion in he afermah of he Grea Recession would have mean ha surplus consumpion S would be paricularly low, leading o increased risk aversion. 6. SUMMARY AND CONCLUSION The large drop in ineres raes following he 2008 recession has given rise o discussions abou wheher he reducion was mainly due o policy or wheher policy was following as bes i could he naural rae and, in he laer case, wha he deerminans of ha reducion could be. While explanaions focusing on marke segmenaion have gained prominence, asse-pricing models in fricionless environmens migh also be able o provide sensible explanaions for ha drop. In he ex above, I discussed, on op of he benchmark power uiliy of Mehra and Presco (985), hree leading varieies of consumpionbased asse-pricing models wih special focus on he deerminans of he risk-free rae: long-run risk, disaser risk, and habi formaion. All varians sugges ha ineres raes ough o be a funcion of expeced consumpion growh. This implicaion is consisen wih he fac ha consumpion growh was low by hisorical sandards in he period. A he same ime, wihin his period here was a reducion in he volailiy of consumpion growh, which could enhance he e ec of he reduced growh rae. The challenge for he benchmark Mehra and Presco (985) framework is ha his period also exhibis an equiy premium ha is high by hisorical sandards, bu consumpion volailiy is small. The hree varians discussed are able o resolve ha ension in di eren ways. Under long-run risk and disaser-risk models, agens risk percepion would increase because of, respecively, higher variance in he longrun componen of consumpion growh or a perceived increase in he probabiliy of a large consumpion decrease. The former is consisen wih hisorically high equiy marke volailiy, and he laer wih an upward revision of he probabiliy of disaser following he Grea Recession. Under he habi-formaion model, he ension can poenially be resolved by he observaion ha he reducion in consumpion

24 232 Federal Reserve Bank of Richmond Economic Quarerly following he Grea Recession led o increased risk aversion as households found hemselves closer o heir subsisence level of consumpion. The explanaions based on increased risk diverge from he habi formaion in ha he same increase in perceived risk ha leads o an increased equiy risk premium can also be an added facor explaining he reduced ineres rae. In conras, in he benchmark calibraion adoped by Campbell and Cochrane (995) for he habi-formaion model, he presence of habis have no direc impac on how ineres raes change over ime bu could reinforce he dampening e ecs of reduced one-sep-ahead consumpion volailiy. A priori, here is no reason why he di eren models canno be combined. In paricular, Nakamura e al. (200) invesigae assepricing implicaions of disasers ha ake muliple quarers o unfold when households have EZW preferences. Such disasers can be viewed as an inermediae case beween he one-o disaser risk in Barro (2006) and he consumpion growh rae uncerainy in Bansal and Yaron (2004). I is unclear wheher exending a habi-formaion model o allow for disaser risk would yield any addiional insigh. Combining habi formaion wih long-run risk would presen a challenge since i would involve combining wo forms of nonseparabiliy in preferences. APPENDIX: EULER EQUATION UNDER EZW PREFERENCES The Euler equaion under EZW preferences is obained from he rsorder condiions of he household subjec o he budge consrain: C + NX P i; x i; = i= NX x i; (P i; + D i; ) : i= To derive he Euler equaion under EZW preferences, we de ne household wealh as W + NX x i; (P i; + D i; ) = i= NX x i; P i; R i; : Given ha de niion, we can rewrie he budge consrain as C + i= NX P i; x i; = W : i=

25 F. Schwarzman: Consumpion-Based Asse-Pricing Models 233 Given ha resaed budge consrain, sar wih he guess ha we can express he uiliy funcion as a linear funcion of wealh: U = A W ; for some A o be deermined. Noe ha A is ime-varying, re ecing he fac ha, if reurns are no i.i.d., he uiliy of he household will vary as a funcion of he sae of he economy. This is a reasonable guess since realized wealh is he only sae variable in he household s problem and he uiliy funcion is homogeneous of degree in U and C. Given ha rede niion and ha guess, he household s problem becomes A W = max W + ;C ;fx i; g I i= 8 >< >: ( ) [C ] ( + E h(a + W + ) )i 9 >= >; ; s:: : C + X P i; x i; = W NX W + = P i; x i; R i;+ : i= The rs-order condiions are for W + :! + = U for C : = ( ) U E h (A + W + ) ( )i A + (W +) C for x i; : = E [! + R i;+ ] : Noe ha here are in fac muliple rs-order condiions for W + since W + will vary as a funcion of he ex-pos realized sae. There are accordingly muliple! +. The pricing kernel is given by h! E + = which can be rearranged as (A + W + ) ( )i ( ) C A + (W +) ;

26 234 Federal Reserve Bank of Richmond Economic Quarerly! + = E h(a + W + ) (A + W + ) ( )i A A + (W +) ( ) C : is Given he guess for he funcional form of U, he envelope condiion A = = ( ) U C : Subsiuing ino he rs-order condiion for C and using he guess ha U = A W, we can wrie he envelope condiion as So ha, rearranging A = ( ) A W C : A = ( ) W Lead his expression one period and use subsiue ou A + from he second erm in he pricing kernel: C :! + = E h(a + W + ) (A + W + ) ( )i A The expression hen simpli es o ( ) W+ C + (W + ) ( ) C :! + 0 E h(a + W + ) (A + W + ) ( )i A C+ C : We can obain he policy funcion for consumpion by rearranging he envelope condiion o obain C = ( ) A W W ; so ha consumpion is linear in wealh. To obain an expression for nex-period wealh as a funcion of curren wealh, we can wrie he second consrain alernaively as

27 F. Schwarzman: Consumpion-Based Asse-Pricing Models 235 W + = R w;+ NX P i; x i; ; where R w;+ P N P i; x i; i= P N R i= P i; is he reurn on oal wealh. i;x i; Since, in equilibrium, x i; equals he supply of di eren asses i, R w;+ can be aken as exogenous o he household s problem. Wih his change in noaion, we can combine he wo consrains on he household s problem o obain i= W + = R w;+ (W C ) R w;+ ( ) W : Finally, one can use he envelope condiion o wrie A as a funcion of : A = ( ) : Wih hese wo expressions, we can verify he guess ha uiliy is linear in wealh. Subsiue hem ino he uiliy funcion o obain = A W 8 >< ( ) [ ] ( >: + E h(a + R w;+ ( )) )i 9 >= >; W : We can hen cancel ou W from boh sides, o obain an expression relaing A and : 8 >< A = >: Rearranging, ( ) [ ] ( + E h(a + R w;+ ( )) )i 9 >= >; : A = ( ) [ ] + Subsiuing in A ( E h(a + R w;+ ) )i ( ) : = ( ),