Inflation and the Stock Market: Understanding the Fed Model

Transcription

1 Inflaion and he Sock Marke: Undersanding he Fed Model Geer Bekaer Columbia Universiy and NBER Eric Engsrom Federal Reserve Board of Governors This Draf: Sepember 2008 JEL Classificaions G12, G15, E44 Keyphrases Money illusion, Equiy premium, Counercyclical risk aversion, Fed model, Inflaion, Economic Uncerainy Dividend yield, Sock-Bond Correlaion, Bond Yield Absrac: The so-called Fed model posulaes ha he dividend or earnings yield on socks should equal he yield on nominal Treasury bonds, or a leas ha he wo should be highly correlaed. In US daa here is indeed a srikingly high ime series correlaion beween he yield on nominal bonds and he dividend yield on equiies. This posiive correlaion is ofen aribued o he fac ha boh bond and equiy yields comove srongly and posiively wih expeced inflaion. While inflaion comoves wih nominal bond yields for well-known reasons, he posiive correlaion beween expeced inflaion and equiy yields has long puzzled economiss. We show ha he effec is consisen wih modern asse pricing heory incorporaing uncerainy abou real growh prospecs and also habi-based risk aversion. In he US, high expeced inflaion has ended o coincide wih periods of heighened uncerainy abou real economic growh and unusually high risk aversion, boh of which raionally raise equiy yields. Our findings sugges ha counries wih a high incidence of sagflaionshouldhaverelaively high correlaions beween bond yields and equiy yields and we confirm ha his is rue in a panel of inernaional daa. This work does no necessarily reflec he views of he Federal Reserve Sysem or is saff. In paricular, our use of he erm "Fed Model" reflecs only convenional parlance among finance pracicioners for his common valuaion model. Columbia Business School, 802 Uris Hall, 3022 Broadway, New York New York, 10027; ph: (212) ; fx: (212) ; gb241@columbia.edu. Board of Governors of he Federal Reserve Sysem, Washingon, DC, 20551, ph: (202) ; fx: (202) ; eric.engsrom@frb.gov. We hank he paricipans of seminars a Columbia Universiy, he Federal Reserve Board, Tilburg Universiy, and he Fourh Annual Asse Pricing Rerea for helpful commens. All errors are he sole responsibiliy of he auhors.

2 1 Inroducion The so-called Fed model 4 posulaes ha he dividend or earnings yield on socks should equal he yield on nominal Treasury bonds, or a leas ha he wo should be highly correlaed. Boh invesmen professionals (see for insance Asness (2003)) and academics (see for insance Thomas (2005)) have long been sruck by he srengh of he empirical regulariy. Figure 1 shows a graph of he yield on a long-erm nominal bond and he equiy yield (using dividends) for he US aggregae sock marke. While some invesmen professionals are using he Fed model as a model of equiy valuaion (see references in Esrada (2005)), boh praciioners and academics have concluded ha he model is inconsisen wih a raional valuaion of he sock marke (see for insance, Asness (2003), Feinman (2005), Campbell and Vuoleenaho (2004), Cohen, Polk and Vuoleenaho (2005), Rier and Warr (2002) and Sharpe (2002)). The difficuly in squaring he model wih raional valuaion can be illusraed using a simple decomposiion of he dividend yield and he nominal bond yield. Using he Gordon model, we can wrie he equiy cash yield, EY, on he aggregae sock marke as consising of hree componens: EY = EDIV + RRF + ERP (1) where EDIV is he expeced growh rae of real equiy dividends, RRF is he real risk free rae of ineres and ERP is he equiy risk premium. Similarly, he yield on a nominal bond is: BY = EINF + RRF + IRP (2) where EINF is expeced inflaion, RRF is again he real ineres rae, and IRP is he inflaion risk premium. The high correlaion beween dividend yields and nominal bond yields is difficul o reconcile wih raional models because expeced inflaion is a dominan source of variaion in nominal yields and he exan lieraure seems o have concluded ha i is impossible for expeced inflaion o have a large (raional) effec on any of he real componens ha drive he equiy cash yield. In fac, he aforemenioned auhors all resor o he simple behavioral model proposed by Modigliani and Cohn in 1979 o explain he empirical regulariy: inflaion (or money) illusion. Inflaion illusion suggess ha when expeced inflaion increases, bond yields duly increase, bu because equiy invesors incorrecly discoun real cash flows using nominal raes, he increase in 4 The Fed Model may have gained is moniker from Prudenial Securiies sraegis Ed Yardeni in 1997 who noed ha in he Federal Reserve Humphrey Hawkins Repor for July 1997, a char ploed he ime series for he earnings-price raio of he S&P 500 agains he 10-year consan-mauriy nominal reasury yield. 1

3 nominal yields leads o equiy underpricing (he equiy yield rises wih bond yields and is now oo high) and vice versa. Alernaively, one can view equiy invesors as correcly discouning nominal cash flows and using nominal discoun raes, bu failing o increase expeced nominal cash payous in response o increases in expeced inflaion. The imporance of his conclusion exends beyond he narrow confines of esing he Fed model. If behavioral biases induced by inflaion cause misvaluaion in he equiy marke, hen he poenial exiss for informed praciioners o devise rading sraegies o ake advanage of he mispricing. For policy makers, if money illusion causes undue variaion in equiy prices during periods of inflaion uncerainy, his suggess anoher moive for inflaion sabilizaion policies, as Campbell and Vuoleenaho (2004) poin ou. In his aricle, we carefully re-examine he evidence consrucing dynamic versions of Equaions (1) and (2) in a vecor auoregressive (VAR) framework, building on Campbell and Shiller s (1988) seminal work. In hese compuaions, we consruc he risk premium componens of yields as residuals since hey are no direcly measurable. We find ha bond yields are indeed highly posiively correlaed wih he dividend yield and ha expeced inflaion is he primary bond yield componen responsible for he high sock-bond yield correlaion. In he conex of a raional model, expeced inflaion mus be posiively correlaed wih he dividend yield hrough some combinaion of posiive correlaion wih he real rae and he equiy risk premium, or a negaive correlaion wih expeced cash flow growh. We find ha only a relaively small porion of he overall comovemen beween expeced inflaion and he dividend yield can be ascribed o he correlaion beween expeced inflaion and real raes. A somewha larger bu sill no dominan piece is due o a negaive covariance beween expeced inflaion and expeced cash flows. 5 The bulk of he posiive covariance beween he dividend yield and expeced inflaion comes from posiive comovemen beween expeced inflaion and he equiy risk premium. Imporanly, because we measure he equiy premium as a residual, hese iniial resuls do no idenify wheher money illusion-induced misvalauion or raional equiy risk premiums are responsible for he high correlaion expeced inflaion. Our subsequen analysis srongly suppors he laer explanaion. We demonsrae ha he high correlaion beween expeced inflaion and he dividend yield is almos enirely due o he posiive correlaion beween expeced inflaion and wo plausible proxies for raional ime-varying risk premiums: a measure of economic uncerainy (he uncerainy among professional forecasers regarding real GDP growh) and a consumpionbased measure of risk aversion. These measures of raionally ime-varying risk premiums feaure prominenly in recen asse pricing aricles showing ha hey help o explain a number of salien asse reurn feaures. 5 This confirms Modigliani and Cohn s careful work ha he effec is no due o expeced real cash flow growh raes being adversely affeced by expeced inflaion. 2

4 Bansal and Yaron (2004, BY henceforh) have sressed he imporance of economic uncerainy and Campbell and Cochrane (1999, CC henceforh) have buil a model of exernal habi, leading o a measure of ime-varying risk aversion ha can be consruced from curren and pas consumpion daa and is couner-cyclical. Bekaer, Engsrom and Xing (2008) combine boh measures in one model. Consequenly, a raional channel seems a work in explaining why he Fed model works: high expeced inflaion coincides wih periods of high risk aversion and/or economic uncerainy. We also provide an ou-of-sample es of our inerpreaion of he US daa. Specifically, our resuls sugges ha he correlaion beween equiy and bond yields ough o be higher in counries wih a higher average incidence of sagflaion. We confirm ha his is he case. We also make sure ha our US resuls are robus, invesigaing a wide variey of alernaive VAR specificaions. The concluding secion ies our findings o an older lieraure on inflaion-sock marke linkages and discusses some issues for fuure invesigaion. 2 Empirical Mehodology 2.1 Yield Decomposiions Our goal is o consruc dynamic versions of Equaions (1) and (2). Beginning wih he laer ask, we simply assume he nominal yield decomposiion relaionship holds a each poin in ime using coninuously compounded raes, which we denoe using he lower case. In paricular, we model by, he coninuously compounded bond yield, as, by = einf + rrf + irp. (3) where rrf is a real risk free rae assumed o have mauriy equal o ha of he nominal bond, einf is he average (annualized) expeced inflaion over he life of he bond, and irp is he inflaion risk premium associaed wih he bond. In principle, all hree componens are unobserved. We achieve idenificaion by finding observable proxies for he real rae and expeced inflaion, and use equaion (3) o infer he inflaion risk premium. We describe all empirical variable definiions and daa sources briefly in he nex secion and in more deail in Appendix To decompose he equiy yield ino is componens, we use he Campbell-Shiller (1988, CS henceforh) 6 In a robusness exercise, we also conduc our main analysis using a differen idenificaion scheme for real raes ha assumes we can measure he inflaion risk premium more direcly as a funcion of inflaion uncerainy. See Secion 5 for deails. 3

5 decomposiion. CS arrive a he following formula for he equiy yield, ey : ey = k X 1 ρ + E ρ j (r +j+1 d +j+1 ). (4) j=0 where k and ρ are linearizaion consans, r is he one-period real reurn o holding equiy, and d is oneperiod real dividend growh. Wihou loss of generaliy, we can spli he expeced rae of reurn on equiy ino risk-free and risk premium componens, E [r +1 ]=rrf + erp (5) where erp is he coninuously compounded one-period equiy risk premium. Equaion (3), his premium is defined relaive o a long-erm real risk free rae. Given he definiion of rrf in Subsiuing, ey = k X 1 ρ E ρ j d +j+1 + E j=0 X ρ j rrf +j + E X j=0 j=0 ρ j erp +j (6) which is he dynamic version of Equaion (1). Here oo, he risk premium componen will be reaed as he residual, wih he wo oher componens consruced empirically using our assumed daa generaing process, described nex. 2.2 Empirical Model: VAR To model he join dynamics of sock and nominal bond yields and heir componens, we sack he following variables ino a vecor, Y, Y =[einf,rrf, d,erp,irp,x 0 ] 0, (7) wih x denoing a vecor of ime- observable informaion variables ha will be useful in inerpreing he resuls: x =[ra,vr, ern,gern su ] 0. (8) Hence, here are a oal of nine variables in he VAR 7. The firs wo elemens of he informaion vecor, x, are designed o capure raional componens of he equiy risk premium, erp. Firs, ra, is a measure of raional risk aversion based on he specificaion of exernal habi persisence in CC. Second, vr is a measure of uncerainy abou real economic growh. BY use uncerainy in he conex of a daa generaing process for 7 Secion 5 considers some robusness checks wih respec o he variables included in he VAR. 4

6 dividend and consumpion growh and demonsrae ha a modes amoun of ime-varying uncerainy abou real growh can, under some assumpions abou invesor preferences, generae nonrivial variaion in he equiy risk premium. The oher wo variables in x represen conemporaneous realized real earnings growh, ern, and a subjecive measure of expeced earnings growh, gern su. These variables allow us o compare objecive and subjecive forecass of profi growh, which is useful for assessing he possible impac of subjecive biases abou profi prospecs, such as money illusion, in he comovemen beween sock and bond yields. They may also be useful predicors of fuure dividends, and hus imporan for undersanding he dynamics of ey. We proceed by assuming a simple daa generaing process for Y, and using he fully observable vecor, W =[einf,rrf, d,ey,by,x 0 ] 0, (9) o idenify he dynamics of Y. Specifically,weassumeafirs-order VAR for Y, Y = μ + AY 1 + Σε (10) where μ is a vecor of consans wih he same dimension as Y, A is a square marix of parameers governing he condiional mean of Y, Σ is a lower riangular square marix of parameers governing he covariance of shocks o elemens of Y (ha is, Σ is he he Cholesky decomposiion of he covariance marix of shocks) and ε is a vecor of i.i.d. shocks. Once he Y dynamics are specified o ake his form, a simple linear ranslaion beween Y and he observable vecor, W is available. Moreover, we can obain esimaes of μ, A and Σ by firs esimaing avaronw. More concreely, we firs esimae W = μ w + A w W 1 + Σ w ε (11) n o obain esimaes of μ c w, A cw, Σ c o n o n w andhencalculae bθ, Θ b = F 1 μ c w, A cw, Σ c o w where F 1 is a marix funcion such ha by = b θ + b ΘW (12) Nex, we can idenify he VAR parameers of Y as n o n o bμ, ba, bσ = F 2 μ c w, ca w, cσ w (13) where F 2 is a second marix funcion. Appendix 7.1 explains, in deail, how hese calculaions are made. 5

7 2.3 Decomposing Yields under he VAR As saed above, he nominal bond yield is rivially affine in componens of Y, as he righ hand side erms of Equaion (3) are direc elemens of Y. We can also now more explicily describe our decomposiion of he equiy yield ino hree componens, ey = cons + ey d + ey rrf + ey erp (14) where ey d = E P j=0 ρj d +j+1 represens he oal effec of cash flow expecaions, ey rrf = E P j=0 ρj rrf +j, represens he oal effec of real ineres raes, and ey erp P = E j=0 ρj erp +j represens he oal effec of equiy risk premiums. We use objecive condiional expecaions under he VAR o calculae each of hese quaniies, and because of he simple VAR srucure, he hree equiy yield componens are affine in Y. For example, ignoring consan erms, and defining e 0 d such ha d = e 0 d Y, ey d = e 0 de X ρ j Y +j+1 = e 0 dρa (I ρa) 1 Y j=0 which is indeed a linear funcion of Y. To deermine he source of he high covariance beween sock and bond yields, we decompose i ino is nine componens: COV (ey,by ) = COV ey d +COV +COV (ey erp,einf + COV ey d,rrf + COV ey d,irp ³ ey rrf,rrf + COV ³ ey rrf,einf + COV ³ ey rrf,irp,einf )+COV (ey erp,rrf )+COV (ey erp,irp ) (15) Each of hese covariances is readily calculaed using VAR arihmeic. For insance, COV ey d,einf = e 0 d ρa (I ρa) 1 COV (Y ) e 0 einf (16) where vec [COV (Y )] = (I A A) 1 vec [ΣΣ 0 ]. Noe ha every elemen of COV (ey,by ) is ulimaely a n funcion of he parameers of he observable VAR, μ c w, A cw, Σ c o w. 6

8 2.4 Orhogonalizing he Equiy Risk Premium The equiy risk premium componen of equiy yields in our decomposiions above, ey erp, is essenially a residual, he difference beween he observed equiy yield and he summed presened values, calculaed under he VAR, of fuure cash flows and real risk free raes. A disadvanage of his approach is ha model misspecificaion could conaminae he equiy risk premium esimaes. To ry o isolae he componen of he equiy risk premium ha is consisen wih raional pricing, we draw on recen heoreical advances in he empirical asse pricing lieraure. CC and BY sugges ha erp is approximaely linear in risk aversion, ra, or real uncerainy, vr respecively. In he model of Bekaer, Engsrom and Xing (2008), he equiy risk premium is a funcion of risk aversion and real economic uncerainy. We parse ey erp ino wo componens: one spanned-by and one orhogonal-o he vecor [ra,vr ]. Figure 2 plos he wo series. Because his vecor is a subse of he informaion variable vecor in he VAR, x, we can easily decompose ey erp ino hese wo componens wihou any furher esimaion. Concepually, he process is analogous o running a regression of ey erp on ra and vr and inerpreing he regression residual as he orhogonal componen, which we denoe ey erp re. For example, we calculae ey erp sp = β erp0 [1,ra,vr ] ey erp re = ey ey erp sp (17) where he coefficiens, β erp are given under OLS as, E [1,ra,vr ][1,ra,vr ] 0 1 E ey erp [1,ra,vr ] 0 and he wo uncondiional expecaions ha comprise he coefficiens are readily calculaed from he VAR. Wih his addiional decomposiion, here are welve poenial componens o he covariance beween sock and bond yields, COV (ey erp,by ) = COV ey d +COV,einf + COV ey d ³ ey rrf,einf + COV +COV ey erp sp +COV ey erp re,rrf + COV ey d,irp ³ ey rrf,rrf + COV ³ ey rrf,irp,einf + COV ey erp sp,rrf + COV ey erp sp,irp,einf + COV ey erp re,rrf + COV ey erp re,irp (18) If money illusion were presen in he daa, we would expec o findaposiivecovariancebeweenheresidual equiy yield and expeced inflaion, COV ey erp re,einf as all he oher covariances wih expeced inflaion are consruced in a manner consisen wih raional pricing. 7

9 2.5 Calculaing he Subjecive Bias in Profi Expecaions We compue he equiy premium residual assuming ha agens use correc cash flow forecass. However, some descripions of money illusion sugges ha he effec comes hrough incorrec subjecive cash flow predicions by marke paricipans which are correlaed wih inflaion expecaions. Of course, in our VAR sysem, subjecive errors in cash flow forecass would end up in he residual, he equiy premium, and if no relaed o ra and vr, hey will sill be aribued o he residual componen of he equiy premium, ey erp re. To shed ligh on wheher a subjecive bias in cash flow expecaions is relaed o he variaion in equiy yields and expeced inflaion, we use our VAR o esimae he bias and hen check for comovemen of he bias wih inflaion and equiy yields. Specifically, we calculae he subjecive bias in profi expecaions as he difference beween hesubjecivemeasureofrealprofi expecaions and an objecive growh esimae under he VAR, gern ob,a he same horizon (four quarers). The laer is readily calculaed using VAR mahemaics because we include realized real earnings growh, ern, as an elemen of he informaion vecor in he VAR, x. Because he subjecive earnings expecaions measure predics annual earnings, and we use quarerly daa, we compue (ignoring consan erms): gern ob = e 0 ern A + A 2 + A 3 + A 4 Y. (19) We define he subjecive bias as bias = gern su which is clearly affine in Y given ha gern su is also in he informaion vecor, x. gern ob (20) 3 VAR Resuls 3.1 Daa and Empirical Mehods We esimae he VAR using quarerly daa, exending from he 4h quarer, 1968 hrough he end of The daa are described in deail in Appendix 7.2. Here we give a shor overview. The bond yield is he yield o mauriy on a nominal 10 year US Treasury bond 8. As a proxy for he real rae, we use he esimae for he 5 year zero coupon real rae provided in Ang, Bekaer and Wei (2008). As is well known, real erm srucures are relaively fla a longer mauriies so ha his mauriy is a reasonable proxy for a coupon bond wih duraion 8 While he coupon bonds on which hese yields are based have a roughly sable mauriy, heir duraion naurally varies over ime. We can roughly gauge he degree of his variaion under some simplifying assumpions: If (1), he bonds pay semi-annual coupons, and (2) rade a par, hen he bonds duraion is funcion of yield alone. These calculaions yield a Macaulay duraion series for he bonds ha has a mean of around 7.5 years and a sandard deviaion of abou 0.8 years. 8

10 significanly lower han 10 years. There is a voluminous lieraure on inflaion forecasing, bu recen work by Ang, Bekaer and Wei (2007) srongly suggess ha professional surveys provide he bes ou-of-sample forecass of inflaion. Therefore, we use a proxy for inflaion expecaions from he Survey of Professional Forecasers (SPF). The equiy daa we use are sandard and represen informaion on he S&P500 Index. In our base resuls we use dividends no accouning for repurchases, bu we discuss resuls wih an adjused measure in Secion 5. Consequenly, real earnings, dividend growh and he equiy yield all refer o he S&P500 Index. Subjecive expecaions regarding earnings growh are also exraced from he SPF. Finally, we need empirical proxies for fundamenal risk aversion and for economic uncerainy. Our proxy for (he log of) risk aversion akes he specificaion for local risk aversion in CC based on an exernal habi model. In his model, risk aversion is negaively relaed o he consumpion surplus raio, which is he raio of he surplus of consumpion over he habi sock divided by consumpion. The sochasic process is auoregressive and he shocks are derived from US consumpion daa. By saring he process in 1947, he effec of iniial condiions has died ou by he ime our sample sars. real uncerainy is also based on SPF daa. The resuling measure is clearly couner-cyclical. Our measure for We combine informaion from a survey abou he probabiliy of a recession he nex quarer and from he dispersion across respondens abou nex year s real GDP growh. The Appendix has all he deails. We esimae he VAR on W using OLS. Deailed VAR resuls are available on reques. Our daa sample is comprised of 157 quarerly observaions of a nine-variable vecor. In addiion o he 9 uncondiional means, he firs-order VAR ransiion marix, A w, has 81 elemens and he innovaion covariance marix, Σ w Σ w0,has 45 disinc elemens. The "sauraion raio," or he raio of he number of he oal number of daa poins o he number of esimaed parameers, is hus (157 9)/( ) = This is saisfacory bu suggess many VAR coefficiens may no be saisically significan. To make sure our resuls are no due o over-fiing he robusness secion will consider VARs wih insignifican coefficiens zeroed ou and smaller VARs. In he resuls discussion, we immediaely focus on he comovemens saisics derived from he VAR. Because all of hese saisics are funcions of he VAR parameers, i is possible o derive sandard errors for hem using he parameer sandard errors and he dela mehod. However, here are many reasons o suspec asympoic heory may no work well in his conex: some of he variables are persisen, he sauraion raio is no exceedingly large and he residuals are likely fa-ailed. Therefore, we use sandard errors derived from a boosrap procedure, which is described in Appendix

11 3.2 Main Resuls Table 1 conains he main resuls. In Panel A, he op line simply repors he variance of he bond and equiy yields, heir covariance and heir correlaion. The hear of he puzzle is ha he correlaion beween ey and by is 78 percen. Under he VAR poin esimaes, a (boosrapped) 90 percen confidence inerval for his correlaion ranges from 34 o 90 percen. This is puzzling because, as shown under he variance decomposiions for he wo yields, 55 percen of he variance of he bond yield is driven by expeced inflaion, whereas 78 percen of he variaion of he equiy yield is driven by he equiy risk premium. For he yields o comove so srongly, expeced inflaion, a nominal concep, mus correlae highly wih he equiy premium, a real concep. This is confirmed in he covariance decomposiion on he righ side of Panel A. More han half of he comovemen comes from he correlaion beween expeced inflaion and he equiy premium. The oher wo relaively large conribuors are he covariance beween he real rae and he equiy premium, which is posiive and conribues 16 percen o he ey by covariance, and he covariance beween expeced inflaion and he cash flow componen of he equiy yield, which conribues 12 percen. The laer effec implies ha expeced inflaion is on average posiively correlaed wih periods of low cash flow expecaions, as he cash flow componen of he equiy yield is negaively relaed o cash flow projecions. This in iself already suggess ha above-average inflaion in he US has occurred ofen a imes of depressed earning (and dividend) expecaions. Finally, expeced inflaion and he real rae are posiively correlaed, which conribues 7 percen o he comovemen beween he bond and equiy yield. While his number is small, i is relaively precisely esimaed. This resul is inconsisen wih he well-known Mundell-Tobin effec ha suggess a negaive relaion. However, our measures here are long-erm (proxying for a 5 o 10 year horizon) and Ang, Bekaer and Wei (2008) also find a posiive correlaion beween expeced inflaion and long-erm real raes. Looking a he las row of he covariance decomposiion marix, we noe ha 79 percen of he comovemen beween equiy yields and bond yields comes hrough he equiy premium, a residual in he equiy yield decomposiion. While i is emping o conclude ha irraional forces are a work, he nex panel proves oherwise. In Panel B, we decompose he equiy yield ino a par spanned by risk aversion and uncerainy and an unspanned par; 80 percen of wha he equiy premium explains of he oal ey by covariance comes from he spanned, raional par 9. If we focus on COV (ey erp,einf ), he expeced inflaion componen, abou 86 percen can be ascribed o he raional componen, COV ey erp sp,einf wih he res, poenially, coming from money illusion. In panel C, we explore he comovemens among equiy yields, expeced inflaion, and he subjecive earnings 9 Calculaed as he sum of he firs line in Panel B divided by he sum of he las line in Panel A (0.64/0.81). 10

12 bias. On he lef side, we see ha he subjecive earnings bias is barely correlaed wih eiher he equiy yield or expeced inflaion. This suggess ha subjecive bias in cash flow expecaions (1) is no an imporan driver of he equiy yield and (2) does no comove srongly wih expeced inflaion. Boh of hese effecs are in sharp conras wih he assumpions of money illusion. Sill, equiy yields are highly correlaed wih expeced inflaion. On he righ hand side of Panel C, we decompose his comovemen because he Fed model puzzle essenially is due o he high correlaion beween expeced inflaion and equiy premiums. The Panel shows ha abou 10 percen of heir comovemen comes from he posiive comovemens of real raes and expeced inflaion, 16 percen of he comovemen can be ascribed o he negaive correlaion beween expeced inflaion and cash flow expecaions, bu 63 percen can be ascribed o he fac ha risk aversion and uncerainy are high in imes of high expeced inflaion. The unexplained residual is a palry 10%, which severely limis he poenial role of money illusion. 4 Inernaional Resuls 4.1 Moivaion Given previous resuls in he lieraure, our findings are perhaps surprising. For example, Campbell and Vuoleenaho (2005, CV henceforh) perform a closely relaed VAR-based analysis and inerpre heir findings as clearly suggesive of money illusion. How can heir resuls be so differen from ours? We believe here are four main reasons. Firs, CV rea cash flows as residuals. All unexplained variaion is hence assigned o cash flow variaion. In conras, we aemp o measure cash flows direcly and leave he equiy premium as he residual componen. are clearly measurable. We prefer he laer mehod because, alhough hey are highly seasonal cash flows Second, CV measure he equiy risk premium wih a variable due o Cohen, Polk and Vuoleenaho (2005) ha may be subjec o considerable measuremen error and i no, o dae, widely used in he lieraure. Third, CV work direcly in erms of excess reurns, and herefore ignore one poenially imporan raional source of common variaion in he wo yield variables: real raes. Our resuls in Table 1 indicae ha hey herefore miss abou 20 percen of he comovemen beween equiy and bond yields. Finally, subsequen research has found ha Campbell and Vuoleenaho s resuls are no robus o he pos-war subsample on which we focus (Wei and Jouz, 2007). Neverheless, boh heir work and ours analyzes one US based daa se, wih one hisory of inflaion, bond yields and equiy yields. Using his daa se alone, i is likely hard o definiively exclude he money illusion sory in favor of our sory. We believe ha inernaional daa offer an ineresing ou-of-sample es of our hypohesis. 11

13 Essenially, we argue ha he US experienced high correlaions beween equiy yields and bond yields because higher inflaion happened o occur during recessions, so ha in recessions equiy and bond premiums are boh relaively high. In oher words, he Fed model works in counries wih a high incidence of sagflaion. Esrada (2006) shows ha here is indeed subsanial cross-secional variaion in he srengh of he correlaion beween bond and equiy yields across counries. He focuses on saisical problems in inerpreing he correlaions in a panel of inernaional daa. We now explore he possibiliy ha sagflaion incidence accouns for par of he cross-secional variaions in sock-bond yield correlaions using daa similar o he Esrada sample. Specifically, we collec four variables for 20 counries over he period from December 1987 o June Firs, we use he dividend yield, ey i,, provided by Thomson for each counry s equiy index. The measure is no perfecly available, bu 97% of all possible counry-monhs are populaed. We also use a long erm risk free local currency nominal bond yield, by i,, from Thomson. repored by he local governmens, infl i,. Third, we measure he inflaion rae for each counry-monh as Where available, we use he coninuously compounded change in he CPI index. If no such series is available for a paricular counry, we use he GDP deflaor. If his variable is available only quarerly, we divide he quarerly inflaion rae by hree and use repeaed values for monhs in ha quarer. Finally, we measure real aciviy using he recession indicaor recess i, published by he Economic Cycle Research Insiue, which provides monhly indicaor series for he incidence of recession. Where recession indicaors are no available (8 counries and in 2005 for all counries), we define recessions as wo consecuive quarers of negaive real GDP growh. 4.2 Cross-Counry Analysis We sar wih a heurisic analysis of he cross-secional associaion beween Fed Model effec inensiy and sagflaion inensiy. To capure he inensiy of he Fed model effec, we compue he ime series correlaion beween he dividend yield and he nominal long bond yield for each counry. To measure he inensiy of sagflaion for a counry, we similarly compue he ime series correlaion of he recession indicaor wih inflaion for each counry. Figure 3 plos each counry along hese wo dimensions. Alhough here are only 20 counry observaions, a posiive relaionship seems eviden. In fac, he cross-secional correlaion beween fed model inensiy and sagflaion inensiy on his plo is 0.50, and significan a he 5 percen level (no accouning for he sampling uncerainy in he ime series correlaions). Moreover, a cross secional OLS regression of fed model inensiy on sagflaion inensiy produces a posiive slope coefficien of 1.35 which is also significan a he 5 percen level (again, no accouning for he sampling uncerainy in he ime series correlaions). The significance of he slope coefficien is robus o he (sequenial) exclusion of Japan and Ausria, poenial ouliers. 12

14 We inerpre hese resuls as supporive of a posiive relaionship. The relaionship exiss even hough he U.S. iself has no exhibied sagflaion in he pos-1987 sample while reaining a high by ey correlaion. We add more saisical formaliy o his analysis by esimaing wo ses of cross-secional regressions wih he cross-secion of counries sock-bond yield correlaions as he dependen variable. The resuls for boh ses of regressions are repored in Table 2. The firs regression se (numbers on he lef of he able) focuses on he incidence of sagflaion, defined as he percen of observaions where a recession occurs simulaneously wih high inflaion. Our cu off value for high inflaion is 10%, bu we also conduced he analysis using an inflaion level of 5% as he cu-off wih largely similar resuls. Regression (3) shows ha sagflaion by iself has a huge effec on he equiy bond yield correlaion: a counry wih 1% higher sagflaion incidence han he average has a 21 percenage poin higher equiy-bond yield correlaion. Of course, he sagflaion effec could be due o is separae componens, recession or simply inflaion. Regressions (1) and (2) show ha he percen of high inflaion monhs by iself does increase he equiy yield-bond yield correlaion whereas a high frequency of recessions acually reduces i, bu he laer effec is no significan. Regression (4) includes all hree dependen variables in one regression. This regression provides a nice es of our sagflaion sory versus jus money illusion. If money illusion drives he correlaion, he coefficien on inflaion should be significan, and here is lile reason for sagflaion o have a paricular effec on he bond-equiy yield correlaion. However, we find ha inflaion has an insignifican effec on he correlaion. The recession effec is sill negaive bu no significan, and he sagflaion effec is large and significanly differen from zero. While he associaed -saisic is large, he regression suffers from hree economeric problems. Firs, he sample is small (20 observaions). Second, he regressors and regressands involve pre-esimaed saisics. Third, he differen observaions arise from correlaed ime series. Therefore, we conduc a Mone Carlo analysis, described in deail in he Appendix 7.4, and generae a small sample disribuion for he -saisics in he regressions. Significan -saisics according o he small sample disribuion are indicaed wih aserisks. The sagflaion coefficien remains significan when using he small sample disribuion for he -saisics. The second se of regressions, replace high inflaion incidence by average inflaion, and sagflaion by he ineracion of inflaion and he recession indicaor. The univariae regression, Regression (5), reveals ha counries wih high average inflaion do have significanly higher equiy yield-bond yield correlaions, bu when his variable is added o a regression ha includes he inflaion-recession ineracion, Regression (7), he direc effec of inflaion disappears. The inflaion-recession ineracion comes in very significanly and he significance survives a he 5% level under he small sample disribuion. The direc effec of he frequency of recessions coninues o be negaive bu insignifican. 13

15 5 Robusness Checks This secion describes he se of robusness exercises agains which we have esed our main resuls. 5.1 VAR Specificaion Tess Table 3 repors a few specificaion ess on he VAR residuals. In Panel A, we repor he sandard Schwarz (BIC) and Akaike (AIC) crieria. The BIC crierion clearly selecs a firs-order VAR whereas he AIC crierion selecs a second-order VAR. In he second panel, we repor Cumby-Huizinga (1987) ess on he residuals of a firs and second-order VAR for each variable separaely. We use 4 auocorrelaions. While he selecion crieria in Panel A sugges ha a VAR(1) adequaely describes he dynamics of he daa, he Cumby-Huizinga ess in Panel B sugges some serial correlaion remains wih a firs-order VAR and ha a second-order VAR is more appropriae. Therefore, we repea he analysis using a VAR(2) daa generaing process. All he resuls in Table 1 are essenially unchanged. 5.2 VAR Robusness Exercises Table 4 summarizes a number of robusness exercises. We only focus on he criical saisics: he percen conribuion of he covariance beween expeced inflecion and he equiy premium o he oal yield covariaion, and he percen conribuion of he covariance beween expeced inflaion and he non-spanned, residual par of he equiy premium, erp re. For ease of comparison, he firs line repeas he resuls from he main VAR repored in Table 1, and he second line repors he resuls from a VAR(2). 1. The use of a large VAR (9 variables) may imply ha many coefficiens are insignifican, ye sill influence he saisics of ineres. Our boosrapping procedure for calculaing sandard errors should address his issue o a large exen, bu we also conduc wo exercises o direcly verify he robusness of he poin esimaes: (a) We calculae he resuls presened in Table 1 afer zeroing-ou any elemen of A which has an OLS -saisic less han one. The resuls are largely unchanged. (b) We also repea he calculaions using a smaller VAR excluding he informaion variables, ha is dropping x. While his precludes us from decomposing he equiy risk premium and calculaing he subjecive earnings bias, all he resuls in Panel A of Table 1 are essenially unchanged. 2. We conduc hree exercises o check he robusness of resuls o alernaive bond yield decomposiions. 14

16 (a) We add an addiional informaion variable o he VAR, a measure of inflaion uncerainy based on SPF daa (using a procedure similar o ha which we used for real uncerainy). The VAR resuls remain robus. (b) We subsiue a longer-erm measure of survey-based inflaion expecaions (our sandard measure looks ahead only four quarers) as our measure of expeced inflaion. The longer-erm measure is no available early in he sample, so we mus firs filer is early values (see daa appendix for a descripion of his procedure). Our resuls are robus o his change. (c) We use a compleely differen measure of he real rae, by assuming we can measure he inflaion risk premium direcly as proporional o inflaion uncerainy. Specifically, we subrac long-erm inflaion expecaions and a consan imes inflaion uncerainy from nominal raes. We use he residual as an alernaive real rae measure. We choose he consan of proporionaliy o mach he uncondiional mean of he real rae o ha of our sandard measure from Ang, Bekaer, and Wei (2008). Our main resuls are no maerially affeced by his change. 3. We also conduc he analysis presened in Table 1 using wo alernaive measures of he cash flow from equiy. (a) Firs, we use earnings insead of dividends, boh for consrucing cash flow growh and calculaing he equiy yield. Tha is, we now invesigae he earnings yield. We are moivaed o do his, in par, because praciioners overwhelmingly focus on earnings as he uni of fundamenal analysis for equiy valuaion. However, o do formal analysis using earnings in he CS framework, we make he no-enirely saisfacory assumpion of a consan payou raio. The resuls for earnings-based equiy yields are largely consisen wih our main resuls. (1) The sock-bond yield covariance is very high, (2) he majoriy of he comovemen comes hrough he covariance of he equiy yield wih expeced inflaion, and (3) very lile of he covariance involves he ey erp re componen of he equiy yield. One difference from our main resuls is ha he conribuion of COV ey d,einf o he oal ey by covariance is subsanially larger when using earnings raher han dividends, accouning for 41 percen of he covariance versus jus 12 percen under our baseline VAR as repored in Table 1. Hence, raher han he covariance beween expeced inflaion and cash flow growh uncerainy being he driver for he sock-bond yield covariance, i is now comovemen beween expeced inflaion and expeced cash flow growh. Neverheless, even if his is he correc inerpreaion of he daa, sagflaion remains a criical ingredien: Inflaion happens o occur a imes of depressed earnings 15

17 expecaions. Noe ha we use objecive, no subjecive, earnings forecass, so ha his canno be caused by money illusion. (b) Second, we add repurchases o dividends in calculaing cash flow, because repurchases have been an imporan channel by which companies have reurned cash o shareholders in he pas few decades, and his can have imporan asse pricing implicaions (see Boudoukh e al, 2007). The correlaion of he resuling equiy yield measure wih he bond yield remains posiive bu no saisically significan. This owes o he fac ha repurchases have, on a quarerly basis, been exremely volaile, especially over he pas few years. The poin esimaes of our main resuls are broadly similar o hose presened in Table 1, bu he esimaes of all he ey by covariance componens are very imprecisely esimaed and none are individually saisically differen from zero. While his is a disappoining resul, i is likely similarly due o he excessive volailiy of repurchases. 5.3 Inernaional Resuls For robusness o our use of dividends as he relevan equiy cash flow in he inernaional daa, we also conduc he analysis using year-ahead analys-expeced earnings in calculaing he equiy yield. This change does no affec he resuls of Table 2 very much. Finally, because he dependen variable in he cross-secional regressions are correlaions and hus limied o he inerval [0, 1], we conduced he OLS regressions using a ransformaion of he correlaion, ln (1 + corr.) / ln (2 corr.), whicheffecively spreads he range of he dependen variable o (, + ). The OLS -saisics using his ransformaion are very similar o hose repored in Table 2. 6 Conclusion In his aricle, we re-examine poenial explanaions for he surprisingly high correlaion beween he real equiy yields and nominal bond yields in US pos-war daa. We show ha he prevailing explanaion, money illusion, acually has raher limied explanaory power. We ascribe a large par of his covariaion o he raher high incidence of sagflaions in he US daa. We posulae ha in recessions economic uncerainy and risk aversion may increase leading o higher equiy risk premiums, increasing yields on socks. If expeced inflaion happens o also be high in recessions, bond yields will increase hrough heir expeced inflaion and, poenially, heir inflaion risk premium componens, and posiive correlaions emerge among sock and bond yields and inflaion. We esablish his resul using a VAR mehodology ha uses measures of inflaion expecaions and wo proxies for raional variaion in risk premiums, one based on economic uncerainy, one based on he 16

18 habi model formulaed by Campbell and Cochrane (1999). Our confidence in hese findings is bolsered by a cross-counry analysis ha demonsraes ha sagflaion incidence accouns for a significan fracion of he cross-secional variaion in equiy - bond yield correlaions. Our findings have poenially imporan policy implicaions. If money illusion afflics pricing in he sock marke, inflaion sabilizaion also helps preven disorions and mis-pricing in he sock marke. If money illusion does no affec he sock marke, he Federal Reserve s inflaion policy has no bearing on he equiy marke beyond is implicaions for real economic growh. Finally, our work is relaed o bu disinc from anoher old hypohesis regarding he relaionship beween inflaion and he sock marke: Fama s (1981) proxy hypohesis. Fama argues ha he srong negaive relaionship beween sock reurns and inflaion is due o inflaion acing as a proxy for expeced real aciviy. Hence, he hypohesis also relies on sagflaion being an imporan par of US daa. Because our VAR allows us o compue cash flow expecaions we can direcly measure he imporance of he proxy hypohesis. According o Table 1, we find ha only 14 percen of he equiy-bond yield covariance can be ascribed o he covariance beween einf and ey d dominaes. So, while he proxy hypohesis is par of he explanaion, our risk-based sory clearly 17

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23 7 Appendix 7.1 Recovering he Dynamics of he Laen Facors By assumpion, he parially unobserved sae (we denoe is dimension as l) vecor follows a VAR(1), Y = μ + AY 1 + Σε. (21) The observable vecor, W, (also of dimension l) is a linear combinaion of concurren values of Y as well as expecaions of fuure values of Y. The mos general relaion we consider is W = M 0 + M 1 Y + M 2 E X j=0 ρ j Y +j+1 (22) where he marices M 0 (l 1), M 1 (l l) andm 2 (l l) are comprised of known consans. Under he VAR(1) srucure, his has he implicaion ha Y and W are relaed by a linear ransformaion, which we denoe as Y = θ + ΘW (23) and we mus solve for θ and Θ. Once we have done so, we can use he empirical esimaes of he VAR parameers for W from he daa: W = μ w + A w W 1 + Σ w ε o recover esimaes of μ, A, andσ as: μ = θ + Θμ w ΘA w Θ 1 θ A = ΘA w Θ 1 Σ = chol (ΘΣ w Σ w0 Θ 0 ). (24) where chol ( ) denoes he Cholesky decomposiion. We use he mehod of undeermined coefficiens. Specifically, combining Equaions (22) and (23), we obain, W = M 0 + M 1 (θ + ΘW )+M 2 E X j=0 ρ j (θ + ΘW +j+1 ) (25) 22

24 Solving for he firs erm of he above summed expecaions, we obain, X E j=0 ρ j θ =(1 ρ) 1 θ (26) For he second piece, we firs noe ha E W +j = W +(A w ) j W W defining W =(I A w ) μ w. Using his o expand he second erm in he summed expecaions, X E j=0 ρ j ΘW +j+1 = = X j=0 j=0 ρ j Θ ³W +(A w ) j+1 W W X ρ j ΘW + 1 X Θ (ρa w ) j+1 W W ρ j=0 = (1 ρ) 1 ΘW + 1 ρ hθ (ρa w )(I ρa w ) 1 W W i = (1 ρ) 1 ΘW + ΘA w (I ρa w ) 1 W W (27) Puing his all ogeher, we obain, W = M 0 + M 1 (θ + ΘW ) +M 2 θ + ΘW φ1 ΘΦ 1 W + ΘΦ 1 W (28) where Φ 1 = A w (I ρa w ) 1 and φ 1 =(1 ρ) 1. Equaing W coefficiens on boh sides of he equaions yields a soluion for Θ: I = M 1 ΘI + M 2 ΘΦ 1 vec (Θ) = (I 0 M 1 + Φ 0 1 M 2 ) 1 vec (I) (29) Using Equaions (24) and (29), we can compleely specify he dynamics of Y in erms of esimaed parameers. The models considered in his aricle are all special cases of Equaion (25). For he baseline model, M ey 0 = k 1 ρ,mey 1 = e 0 rrf + e 0 erp, M ey 2 = e 0 d + ρe 0 rrf + ρe 0 erp M by 0 = 0, M by 1 = e 0 einf + e 0 rrf + e 0 irp, M by 2 =0 (30) 23