MONEY ILLUSION IN THE STOCK MARKET: THE MODIGLIANI-COHN HYPOTHESIS*

MONEY ILLUSION IN THE STOCK MARKET: THE MODIGLIANI-COHN HYPOTHESIS* RANDOLPH B. COHEN CHRISTOPHER POLK TUOMO VUOLTEENAHO Modigliani and Cohn hypothsiz that th stock markt suffrs from mony illusion, discounting ral cash flows at nominal discount rats. Whil prvious rsarch has focusd on th pricing of th aggrgat stock markt rlativ to Trasury bills, th mony-illusion hypothsis also has implications for th pricing of risky stocks rlativ to saf stocks. Simultanously xamining th pricing of Trasury bills, saf stocks, and risky stocks allows us to distinguish mony illusion from any chang in th attituds of invstors toward risk. Our mpirical rsults support th hypothsis that th stock markt suffrs from mony illusion. I. INTRODUCTION Do popl suffr from mony illusion, confusing nominal dollar valus with ral purchasing powr? Whn th diffrnc btwn ral and nominal quantitis is small and staks ar rlativly low, quating th nominal dollar amounts with ral valus provids a convnint and ffctiv rul of thumb. Thrfor, it sms plausibl that popl oftn ignor th rat of inflation in procssing information for rlativly small dcisions. 1 Modigliani and Cohn [1979] hypothsiz that stock markt invstors may also suffr from a particular form of mony illusion, incorrctly discounting ral cash flows with nominal discount rats. An implication of such an rror is that tim variation in th lvl of inflation causs th markt s subjctiv xpctation of th futur quity prmium to dviat systmatically from th * An arlir draft of th papr was circulatd undr th titl How Inflation Illusion Killd th CAPM. W would lik to thank Clifford Asnss, John Campbll, Edward Glasr, Jussi Kppo, Stfan Nagl, Andri Shlifr, Jrmy Stin, and thr anonymous rfrs for hlpful commnts. 1. Th trm mony illusion was coind by John Maynard Kyns arly in th twntith cntury. In 1928 Irving Fishr gav th subjct a thorough tratmnt in his book Th Mony Illusion. Sinc thn, numrous paprs hav dscribd implications of mony illusion to tst for its xistnc. Th most widly discussd of ths implications is stickinss in wags and prics (s Gordon [1983] for a rviw of th vidnc on this topic). Although mony illusion can xist vn in th absnc of inflation, inflation is cntral to most mony illusion storis. Fishr and Modigliani [1978] catalog th ways in which inflation could affct th ral conomy, with mony illusion as on important sourc of ral ffcts. Shafir, Diamond, and Tvrsky [1997] xamin in dtail potntial ffcts of mony illusion and prsnt vidnc on ths ffcts along with a thory of th psychological undrpinnings of th illusion. 2005 by th Prsidnt and Fllows of Harvard Collg and th Massachustts Institut of Tchnology. Th Quartrly Journal of Economics, May 2005 639

640 QUARTERLY JOURNAL OF ECONOMICS rational xpctation. Thus, whn inflation is high (low), th rational quity-prmium xpctation is highr (lowr) than th markt s subjctiv xpctation, and th stock markt is undrvalud (ovrvalud). Th claim that stock markt invstors suffr from mony illusion is a particularly intriguing and controvrsial proposition, as th staks in th stock markt ar obviously vry high. Nvrthlss, rcnt tim-sris vidnc suggsts that th stock markt dos suffr from mony illusion of Modigliani and Cohn s varity. Sharp [2002] and Asnss [2000] find that stock dividnd and arnings yilds ar highly corrlatd with nominal bond yilds. Sinc stocks ar claims to cash flows from ral capital and inflation is th main drivr of nominal intrst rats, this corrlation maks littl sns, a point mad rcntly by Rittr and Warr [2002], Asnss [2003], and Campbll and Vuoltnaho [2004]. Ths aggrgat studis suffr from on srious waknss, howvr. Inflation may b corrlatd with invstors attituds toward risk, which dirctly influnc stock prics vn if invstors do not suffr from mony illusion. To th xtnt that ths aggrgat studis fail to fully control for risk, th rsults may confound th impact of risk attituds and mony illusion. Our novl tsts xplor th cross-sctional asst-pricing implications of th Modigliani-Cohn mony-illusion hypothsis. Simultanously xamining th pricing of Trasury bills, saf stocks, and risky stocks allows us to distinguish mony illusion from changing attituds of invstors toward risk. Th ky insight undrlying our tsts is that mony illusion will hav a symmtric ffct on all stocks yilds, rgardlss of thir xposur to systmatic risk. In contrast, th impact of invstor risk attituds on a stock s yild will b proportional to th stock s risk, as risky stocks yilds will b affctd much mor than saf stocks yilds will b. This insight allows us to clanly sparat th two compting ffcts. Spcifically, w assum that invstors us th logic of th Sharp-Lintnr capital asst pricing modl (CAPM) [Sharp 1964; Lintnr 1965] to masur th riskinss of a stock and to dtrmin its rquird risk prmium. According to th CAPM, a stock s bta with th markt is its sol rlvant risk masur. In th absnc of mony illusion (and othr invstor irrationalitis), th Sharp-Lintnr CAPM prdicts that th risk compnsation for on unit of bta among stocks, which is also calld th slop of th scurity markt lin, is always qual to th rationally x-

MONEY ILLUSION IN THE STOCK MARKET 641 pctd prmium of th markt portfolio of stocks ovr short-trm bills. For xampl, if a risky stock has a bta of 1.5 and th rationally xpctd quity prmium is 4 prcnt, thn that stock should hav a rationally xpctd rturn of th Trasury-bill yild plus 6 prcnt. Convrsly, a saf stock with a bta of 0.5 should only arn a 2 prcnt prmium ovr Trasury bills, and th risky stock will thrfor rturn a prmium of 4 prcnt ovr th saf stock. Th joint hypothsis of mony illusion and th CAPM offrs a sharp, quantitativ prdiction. W show that mony illusion implis that, whn inflation is low or ngativ, th compnsation for on unit of bta among stocks is largr (and th scurity markt lin stpr) than th rationally xpctd quity prmium. Convrsly, whn inflation is high, th compnsation for on unit of bta among stocks is lowr (and th scurity markt lin shallowr) than what th ovrall pricing of stocks rlativ to bills would suggst. Suppos that, in our abov xampl, high inflation lads mony-illusiond invstors, who still dmand a 4 prcnt quity prmium, to undrvalu th stock markt to th xtnt that th rational xpctation of th quity prmium bcoms 7 prcnt. Thn ths invstors will pric th risky stock to yild only a 4 prcnt rturn prmium ovr th saf stock. Consquntly, whn inflation is high, th avrag ralizd quity prmium (7 prcnt) will b highr than th avrag rturn prmium of th risky stock ovr th saf stock (4 prcnt). Our mpirical tsts support this hypothsis. First, as an illustration, w sort th months in our 1927 2001 sampl into quartils basd on laggd inflation and xamin th pricing of bta-sortd portfolios in ths quartils. Th slop of th solid lin in Figur I dnots th pric of risk implid by th pricing of th ovrall stock markt rlativ to that of short-trm bills, i.., th quity prmium that a rational invstor should hav xpctd. Th dashd lin is th scurity markt lin, th slop of which is th pric of risk implid by th pricing of high-risk stocks rlativ to that of low-risk stocks. As prdictd by th mony-illusion hypothsis, th figur shows that during months that ar prcdd by inflation in th lowst quartil of our sampl, th rlation btwn avrag rturns and CAPM btas is stpr than th slop prdictd by th Sharp-Lintnr CAPM and no mony illusion. Convrsly, during months that ar prcdd by inflation in th highst quartil of our sampl, th scurity markt lin

642 QUARTERLY JOURNAL OF ECONOMICS FIGURE I Avrag Excss Rturns and Bta in Diffrnt Inflation Environmnts W first crat tn portfolios by sorting stocks on thir past stimatd btas. W thn rcord th xcss rturns on ths portfolios. Nxt, w sort months in our 1927:06 2001:12 sampl into four groups basd on laggd inflation (dfind as th smoothd chang in th producr pric indx). For ach group, w thn stimat th postformation btas and avrag xcss rturns. Th avrag annualizd xcss rturns ( y-axis) and btas ( x-axis) of ths portfolios form th graphs. Th solid lin (drawn from th [0,0] to [1, avrag markt s xcss rturn in this subsampl]) is th rlation prdictd by th Sharp-Lintnr CAPM. Th dashd lin is th fittd lin computd by rgrssing th avrag rturns on btas in ach subsampl. stimatd from th cross sction of bta-sortd portfolios is much shallowr than th xpctd quity prmium. Scond, w introduc a nw mthod for stimating th xcss slop and xcss intrcpt of th scurity markt lin among stocks, rlativ to th prdictions of th Sharp-Lintnr CAPM. Our statistical tst combins Fama-MacBth [1973] cross-sctional and Black-Jnsn-Schols [1972] tim-sris rgrssions to solv for th xcss slop and xcss intrcpt as a function of th btas and conditional alphas from th tim-sris rgrssion s paramtrs. Th ida bhind this statistical tst is xactly th

MONEY ILLUSION IN THE STOCK MARKET 643 sam as th on illustratd in Figur I, but allows for a convnint and powrful statistical hypothsis tst. Our tsts indicat that th xcss intrcpt of th scurity markt lin comovs positivly and th xcss slop ngativly with inflation, as prdictd by th Modigliani-Cohn mony-illusion hypothsis. At first, it may sm incrdibl that stock markt invstors, with trillions of dollars at stak, mak such a pdstrian mistak. Fortunatly, or prhaps unfortunatly, w nd not look any furthr than to th lading practitionr modl of quity valuation, th so-calld Fd modl, 2 to find corroborating vidnc of stock markt invstors falling pry to mony illusion. Th Fd modl rlats th yild on stocks to th yild on nominal bonds. Practitionrs argu that th bond yild plus a risk prmium dfins a normal yild on stocks, and that th actual stock yild tnds to rvrt to this normal yild. Consistnt with this practitionr argumnt, Sharp [2002], Asnss [2000], and Campbll and Vuoltnaho [2004] find that th Fd modl is quit succssful as an mpirical dscription of aggrgat stock prics prics ar st as if th markt usd th Fd modl to pric stocks. Logically, howvr, th Fd modl is on wak grounds, as it is basd on prcisly th mony-illusion rror notd by Modigliani and Cohn. Evn if most stock-markt invstors confus nominal and ral quantitis, could a small numbr of walthy and rational arbitragurs still liminat any potntial mispricing? W bliv that rational arbitragurs would b vry consrvativ in accommodating supply and dmand du to mony illusion. Th Sharp ratio (th xpctd xcss rturn dividd by th standard dviation of xcss rturn) of a bt against th mony-illusion crowd is likly to b rlativly low, bcaus on can only mak a singl bt at a tim and bcaus th mispricing may b corrctd vry slowly. This potntial slow corrction of mispricing is a particularly important limiting factor of arbitrag, as any attmpt to corrct th inflation-rlatd mispricing xposs th arbitragur to th uncrtain dvlopmnt of th stock markt s fundamntals. Mispricing that corrcts slowly ncssarily rquirs long holding priods for arbitrag positions along with significant xposur to volatility, as th varianc of fundamntal risk grows linarly in tim. In fact, if a rational arbitragur had bt against mony illusion by buying stocks on margin in th arly 1970s, his profits 2. Dspit this nam, th modl has absolutly no official or spcial status within th Fdral Rsrv systm.

644 QUARTERLY JOURNAL OF ECONOMICS would hav bn ngativ for mor than a dcad. As Modigliani and Cohn notd in 1979: On th othr hand, thos xprts of rational valuation who could corrctly assss th xtnt of th undrvaluation of quitis, had thy actd on thir assssmnt in th hop of acquiring richs, would hav mor than likly ndd up with substantial losss. In summary, mispricing causd by mony illusion has prcisly thos charactristics that Shlifr and Vishny [1997] suggst ffctivly prvnt arbitrag activity. II. MONEY ILLUSION AND ITS IMPLICATIONS II.A. Modigliani and Cohn s Mony-Illusion Hypothsis Th corrct application of th prsnt-valu formula discounts nominal cash flows at nominal discount rats or ral cash flows at ral discount rats. Modigliani and Cohn [1979] propos that stock markt invstors, but not bond markt invstors, suffr from mony illusion, ffctivly discounting ral cash flows at nominal rats. What mchanism could caus th bond markt to corrctly rflct inflation, whil th stock markt suffrs from mony illusion? According to th Modigliani-Cohn hypothsis, mony illusion is du to th difficulty of stimating long-trm futur growth rats of cash flows. Considr an invstor who thinks in nominal trms. Sinc nominal bonds hav cash flows that ar constant in thos trms, stimating a growth rat for bonds is not difficult. In contrast, th task of stimating th long-trm xpctd cash-flow growth for stocks is far from trivial. For xampl, suppos that this invstor rronously assums that long-trm arnings and dividnd growth ar constant in nominal trms, and uss all past historical data to stimat a long-trm growth rat for a stock. Of cours, a mor rasonabl assumption would b that xpctd long-trm growth is constant in ral trms. If xpctd long-trm growth is constant in ral trms, yt th invstor xpcts it to b constant in nominal trms, thn in quilibrium stocks will b undrvalud whn inflation is high and ovrvalud whn inflation is low. Th basic intuition of th Modigliani-Cohn hypothsis can asily b capturd by xamining a mony-illusiond invstor s approach to stock valuation. Considr th classic Gordon growth modl [Williams 1938; Gordon 1962] that quats th dividnd-

MONEY ILLUSION IN THE STOCK MARKET 645 pric ratio with th diffrnc btwn th discount rat and xpctd growth: (1) D t /P t 1 R G, whr R is th long-trm discount rat and G is th long-trm growth rat of dividnds. R and G can b ithr both in nominal trms or both in ral trms, but th Gordon growth modl dos not allow mixing and matching nominal and ral variabls. If th xpctd rturns ar constant, th discount rat is xactly qual to th xpctd rturn on th asst. If conditional xpctd rturns vary ovr tim, howvr, th discount rat is only approximatly qual to th long-horizon xpctd holding priod rturn on th asst. Th Gordon growth modl can also b thought of in trms of th invstor s first-ordr condition. If an invstor is at th optimum portfolio allocation, thn th discount rat or xpctd rturn R on stocks must qual th yild on bonds plus a prmium du to th highr covarianc of stock rturns with th invstor s consumption. If an othrwis optimizing invstor suffrs from mony illusion of Modigliani and Cohn s varity, thn h thinks of R in nominal trms and xpcts G to b constant in nominal trms. If th inflation is tim varying, howvr, th assumption of constant nominal G dos not mak any sns, as it would imply a wildly variabl ral G. In ral trms, thr is no obvious rason why ithr R or G should chang mchanically with xpctd inflation, if th consumr is rational. 3 If stock markt invstors suffr from mony illusion and xpct constant long-trm growth in nominal trms, what will happn whn inflation riss? Highr nominal intrst rats rsulting from inflation ar thn usd by stock markt participants to discount unchangd xpctations of futur nominal dividnds. Th dividnd-pric ratio movs with th nominal bond yild bcaus stock markt invstors irrationally fail to adjust th nominal growth rat G to match th nominal discount rat R. From th prspctiv of a rational invstor, stocks ar thus undrvalud whn inflation is high and ovrvalud whn inflation is low. A singl small rational invstor, facing a markt populatd by mony-illusiond invstors, would thn tilt his portfolio toward 3. Som businss-cycl dynamic (such as Fama s [1981] proxy hypothsis) might crat a corrlation btwn inflation and ithr nar-trm discount rats or nar-trm growth rats. Howvr, such movmnts ar a priori unlikly to mov long-trm discount rats or growth rats much.

646 QUARTERLY JOURNAL OF ECONOMICS stocks whn inflation is high and away from stocks whn inflation is low, so that th quilibrium risk prmium of stocks would b justifid by stock rturns covarianc with his consumption. To adapt th notation to conform with our subsqunt mpirical tsts, first subtract th risklss intrst rat from both th discount rat and th growth rat of dividnds. W dfin th xcss discount rat as R R R f and th xcss dividnd growth rat as G G R f, whr all quantitis should again b ithr nominal or ral. As w ar considring th possibility that som invstors ar irrational, w follow Campbll and Vuoltnaho [2004] and distinguish btwn th subjctiv xpctations of irrational invstors (suprscript SUBJ) and th objctiv xpctations of rational invstors (suprscript OBJ). As long as irrational invstors simply us th prsnt valu formula with an rronous xpctd growth rat or discount rat, both sts of xpctations must oby th Gordon growth modl: (2) D/P R,OBJ G,OBJ R,SUBJ G,SUBJ G,OBJ R,SUBJ G,OBJ G,SUBJ. In words, th dividnd yild has thr componnts: (1) th ngativ of objctivly xpctd xcss dividnd growth, (2) th subjctiv risk prmium xpctd by irrational invstors, and (3) a mispricing trm du to a divrgnc btwn th objctiv (i.., rational) and subjctiv (i.., irrational) growth forcast, ε G,OBJ G,SUBJ. Notic that mispricing ε is spcifid in trms of xcss yild, with ε 0 indicating ovrpricing and ε 0 undrpricing. Notic also that th Gordon growth modl rquirs that th xpctational rror in long-trm growth rats, G,OBJ G,SUBJ, b qual to th xpctational rror in long-trm xpctd rturns, R,OBJ R,SUBJ. Campbll and Vuoltnaho [2004] formaliz th Modigliani- Cohn mony-illusion story by spcifying that mispricing or xpctational rror is a linar function of past smoothd inflation: (3) ε G,OBJ G,SUBJ R,OBJ R,SUBJ 0 1, whr is th xpctd inflation and 1 0. If on taks th Modigliani-Cohn hypothsis litrally, on could argu that 1 1, i.., inflation is (irrationally) fully pricd into stock yilds. Th cas in which 1 1 is consistnt with th simpl form of mony illusion in which invstors assum that futur xpctd cash-flow growth is constant in nominal trms.

MONEY ILLUSION IN THE STOCK MARKET 647 II.B. Cross-Sctional Implications of Mony Illusion Whil prvious rsarch has tstd th aggrgat tim-sris prdictions of th Modigliani-Cohn mony-illusion hypothsis, th cross-sctional implications of this hypothsis hav bn largly unxplord in ithr th litratur on bhavioral financ thory or th mpirical litratur in gnral. (Th main xcption is Rittr and Warr s [2002] study, which xamins th diffrntial impact of inflation on a firm s stock pric as a function of its financial lvrag.) W fill this gap in th litratur by dvloping and tsting cross-sctional prdictions rsulting from th original Modigliani-Cohn hypothsis. W bas our cross-sctional prdictions on thr substantiv assumptions. First, w assum that th markt suffrs from mony illusion of th typ dscribd by quation (3). Scond, w assum that th markt maks no othr typ of systmatic mistak in valuing stocks. Togthr, ths two assumptions imply that quation (3) holds not only for th markt but also for ach individual stock: (4) ε i G i,obj G i,subj R i,obj R i,subj 0 1. An important rsult of ths assumptions is that mony illusion s influnc on mispricing is qual across stocks, i.., ε i ε M 0 1. Our final assumption is that invstors bhav according to modrn portfolio thory in valuating risks; that is, thy us th Sharp-Lintnr CAPM to st rquird risk prmiums. This implis that th slop of th rlation btwn th subjctiv rturn xpctation on an asst and that asst s CAPM bta is qual to th subjctiv markt prmium: (5) R i,subj i R M,SUBJ. This is in contrast with th usual, rational-xpctations spcification of th CAPM: R i,obj i R M,OBJ. Not that w implicitly assum that btas ar known constants so that subjctiv and objctiv xpctations of btas ar thus qual. Ths assumptions allow us to driv th cross-sctional implication of th Modigliani-Cohn [1979] mony-illusion hypothsis. Substituting th subjctiv Sharp-Lintnr CAPM into (4) yilds (6) ε i R i,obj i R M,SUBJ.

648 QUARTERLY JOURNAL OF ECONOMICS Rcognizing that markt mispricing ε M quals th wdg btwn objctiv and subjctiv markt prmiums rsults in (7) ε i R i,obj i R M,OBJ ε M N i OBJ R i,obj i R M,OBJ ε i i ε M. Abov, i OBJ is an objctiv masur of rlativ mispricing, calld Jnsn s [1968] alpha in th financ litratur. Sinc mispricing for both th markt and stock i is qual to th sam linar function of xpctd inflation, 0 1, w can writ (8) i OBJ 0 1 i 0 1. Equation (8) prdicts that th (conditional) Jnsn s alpha of a stock is a linar function of inflation, th stock s bta, and th intraction btwn inflation and th stock s bta. If th markt suffrs from mony illusion, thn whn inflation is high a rational invstor would prciv a positiv alpha for low-bta stocks and a ngativ alpha for high-bta stocks. Convrsly, whn inflation is low (or ngativ), a rational xpctation of a stock s alpha is ngativ for low-bta stocks and positiv for high-bta stocks. Rcall that th scurity markt lin is th linar rlation btwn a stock s avrag rturn and its bta. Equivalntly, quation (8) stats that both th intrcpt and th slop of th obsrvd scurity markt lin dviat systmatically from th rational-xpctation Sharp-Lintnr CAPM s prdiction. Morovr, this dviation is a function of inflation. Dfin th xcss slop of th scurity markt lin as th cross-sctional slop of (objctiv) alpha on bta. Dfin th xcss intrcpt of th scurity markt lin as th (objctiv) alpha of a unit-invstmnt stock portfolio that has a zro bta. Equation (8) prdicts that th xcss intrcpt of th scurity markt lin quals 0 1 and th xcss slop quals ( 0 1 ) undr th joint hypothsis of mony illusion and th Sharp-Lintnr CAPM. Th abov rasoning assums that prics ar xclusivly st by invstors who suffr from mony illusion. What happns if som invstors suffr from mony illusion whil othr invstors do not, and th two groups intract in th markt? In th Appndix w dscrib a vry stylizd quilibrium modl, in which a fraction of th risk-baring capacity in th markt suffrs from mony illusion. This stylizd modl givs an intuitiv prdiction: th xcss slop of th scurity markt lin is dtrmind by th

MONEY ILLUSION IN THE STOCK MARKET 649 product of inflation and th fraction of th markt s risk-baring capacity controlld by mony-illusiond invstors. Th abov hypothss ti in closly with rcnt rsarch on quity-prmium prdictability and inflation. A papr by Polk, Thompson, and Vuoltnaho [forthcoming] assums that th CAPM holds in trms of invstors subjctiv xpctations, and uss th rlativ prics of high and low bta stocks to driv an stimat of th subjctiv quity prmium. Polk, Thompson, and Vuoltnaho find that this stimat corrlats wll with proxis for th objctiv quity prmium such as th dividnd yild, and also has prdictiv powr for th futur quity prmium. Th major xcption to thir finding occurs in th arly 1980s, whn thir subjctiv quity prmium masur is low but th dividnd yild, as wll as th subsqunt aggrgat stock markt rturn, is high. It is notworthy that this priod was also th pak of U. S. inflation. Campbll and Vuoltnaho [2004] assum th validity of Polk, Thompson, and Vuoltnaho s [2004] masur of th subjctiv quity prmium. Campbll and Vuoltnaho combin this masur with th Gordon growth modl for th aggrgat markt to stimat th subjctivly xpctd growth rat of aggrgat cash flows. It appars that inflation drivs a wdg btwn th subjctiv and objctiv stimats of aggrgat growth, just as prdictd by th Modigliani-Cohn hypothsis. In contrast, w ssntially circl back to ask how mony illusion affcts th objctiv validity of th CAPM. Evn if invstors subjctivly us th CAPM, dos th CAPM dscrib th pattrn of objctiv rturns in th cross sction? Th answr is that thr should b an objctiv scurity markt lin, but it can b stpr or flattr than th prdiction of th Sharp-Lintnr CAPM, i.., th rational xpctation of th quity prmium. III. EMPIRICAL METHODOLOGY AND RESULTS Our main tsts xamin tim variation in th xcss intrcpt and slop of th scurity markt lin, and th rlation of this tim variation to inflation. Our stimation stratgy is th following. First, w construct dynamic stock portfolios that ar likly to show a larg and consistnt cross-sctional sprad in thir CAPM btas. Th natural way to construct such portfolios is to sort stocks into portfolios ach month on thir past stimatd stock-

650 QUARTERLY JOURNAL OF ECONOMICS lvl btas. W rcord th rturns on ths valu-wight portfolios, which bcom our basis assts. Spcifically, w gnrat our basis asst rturns from th Cntr for Rsarch in Scuritis Prics (CRSP) monthly stock fil, which provids monthly prics; shars outstanding; dividnds; and rturns for availabl NYSE, AMEX, and NASDAQ stocks. W masur btas, ˆ i,t, for individual stocks using at last on and up to thr yars of monthly rturns in a markt-modl OLS rgrssion on a constant and th contmporanous rturn on th valu-wight NYSE-AMEX-NASDAQ portfolio. 4 As w somtims stimat bta using only twlv rturns, w cnsor ach firm s individual monthly rturn to th rang ( 50 prcnt, 100 prcnt) in ordr to limit th influnc of xtrm firm-spcific outlirs. W us ths stock-lvl stimats to form bta-sortd portfolios. Th portfolios ar valu-wight and r-formd ach month using th most rcnt availabl btas. W considr sorts into 10, 20, and 40 portfolios. Th rsults ar not snsitiv to th numbr of portfolios, and w thus concntrat on th twntyportfolio data st for most tsts. Ths portfolio-rturn sris span th 895-month priod, 1927:06 2001:12. Scond, w stimat rolling btas on ths 20 bta-sortd portfolios using a trailing window of 36 months. (W hav rplicatd our rsults using 24- and 48-month bta-stimation windows, and th rsults ar robust to variation in window lngth.) W dnot th tim sris of ths rolling btas as th postformation btas of th basis assts. Ths postformation bta sris span th 860-month priod, 1930:05 2001:12. At this stag of th analysis, it is important to vrify that our stock-lvl bta stimats ar actually usful and rsult in crosssctional sprad in th avrag postformation btas. W find that thy ar, as th avrag postformation bta of th lowst bta portfolio is 0.63 whil th postformation bta of th highst bta portfolio is 1.77. Howvr, th stimatd postformation btas for a particular portfolio ar not constant through tim. For th lowst bta portfolio, th postformation bta varis from 0.35 to 1.92, whil th highst bta portfolio s postformation bta varis from 0.59 to 3.63. Of cours, most of this tim-sris variation in th postformation btas is simply du to sampling variation. Third, w form two portfolios from ths 20 basis assts using 4. W skip thos months in which a firm is missing rturns. Howvr, w rquir all obsrvations to occur within a four-yar window.

MONEY ILLUSION IN THE STOCK MARKET 651 Fama and Macbth s [1973] cross-sctional rgrssion tchniqu. Th purpos of this stp is to dirctly control for th tim variation in postformation btas documntd abov. Spcifically, for ach cross sction, w rgrss th futur xcss rturn on th 20 basis assts on a constant and th portfolios trailing-window postformation bta. As shown by Fama and Macbth, th tim sris of ths cross-sctional rgrssion cofficints ar xcss rturns on portfolios as wll: (9) r intrcpt,t r slop,t 1 ˆ t 1 1 ˆ t 1 1 1 ˆ t 1 r t. Abov, 1 is a vctor of constants and ˆ t 1 a vctor of postformation btas of bta-sortd portfolios stimatd using a trailing window that nds at t 1. r t is th vctor of xcss rturns on th bta-sortd portfolios. W prsnt th rgrssion cofficints in matrix notation in quation (9) to highlight th fact that th cross-sctional rgrssion cofficints ar portfolios. As long as th trailing postformation btas ar accurat forcasts of futur postformation btas, th intrcpt portfolio rturn will b th xcss rturn on a unit-invstmnt zro-bta stock portfolio and th slop portfolio rturn will b th xcss rturn on a unit-bta zro-invstmnt portfolio. Furthrmor, ths portfolio stratgis ar implmntabl as long as th xplanatory variabls (i.., th btas) ar known in advanc of th dpndnt variabls (i.., th basis-asst xcss rturns). Th intrcpt and slop portfolio hav avrag rturns of 44 and 19 basis points pr month rspctivly, though only th intrcpt portfolio s man rturn is statistically significantly diffrnt from zro. Ths two xcss-rturn sris span th 859-month priod, 1930:06 2001:12. Though th stps takn so far ar complicatd, ths complications ar justifid as thy will produc two portfolio rturn sris with rlativly constant, prcisly masurd btas of zro and on for th intrcpt and slop portfolios, rspctivly. This is dsirabl, as th tim-sris rgrssions in th nxt stag critically rquir that th portfolios w us hav constant btas. Fourth, w rgrss th intrcpt and slop portfolio s xcss rturns on a constant, th contmporanous markt xcss rturn, and laggd inflation. As abov, w us th valu-wight NYSE-AMEX-NASDAQ portfolio as our proxy of th markt portfolio. Th xcss rturn is computd by subtracting th thr-

652 QUARTERLY JOURNAL OF ECONOMICS month Trasury-bill rat from CRSP. Our masur of inflation is th sris usd by Campbll and Vuoltnaho [2004] in thir study invstigating aggrgat markt valuations and inflation. W first comput log growth rats on th producr pric indx. As ths growth rats ar vry noisy spcially in th first part of our sampl, w smooth ths log growth rats by taking an xponntially wightd moving avrag with a half-lif of 36 months (i.., monthly dcay to th powr of 0.9806). Not that th xponntially wightd moving avrags us trailing inflation data, so thr is no look-ahad bias in our smoothing. W also dman this inflation sris using its full sampl man in ordr that th subsqunt rgrssion paramtrs ar asir for th radr to intrprt. Th two tim-sris rgrssions (10) ar analogous to Black, Jnsn, and Schols [1972] and Gibbons, Ross, and Shankn [1989] tim-sris rgrssions with tim-varying Jnsn s [1968] alphas: (10) r intrcpt,t r slop,t a 1 b 1 r M,t c 1 t 1 u 1,t a 2 b 2 r M,t c 2 t 1 u 2,t. Th mpirical stimats of th two rgrssion quations in (10) show that both portfolios hav vry prcisly masurd btas. Tabl I shows that for our prfrrd spcification (20 bta-sortd portfolios whr postformation btas ar stimatd using a 36- month trailing window), th intrcpt portfolio has a bta of 0.0041 with a t-statistic of 0.14, whil th slop portfolio has a bta of 1.0205 with a t-statistic of 34.38. W also find that th conditional alpha of th intrcpt portfolio varis positivly with laggd inflation as th stimat of c 1 is 1.50 with a t-statistic of 2.41. Our stimat of c 2 is rliably ngativ (valu of 1.48, t-statistic of 2.35) indicating that inflation tracks th conditional alpha of th slop portfolio in an opposit fashion. Bcaus of our novl mthodology, w now hav idntifid two portfolios with rlativly stabl btas. If w could b confidnt that th trailing-window postformation bta stimats ar prfct forcasts of th futur basis-asst btas, th xcss intrcpt and xcss slop of th scurity markt lin would b givn by a 1 c 1 t 1 and a 2 c 2 t 1. In that hypothtical cas, th timsris rgrssion cofficints b 1 and b 2 would b xactly qual to zro and on. Dspit th usfulnss of our nw approach, ralistically spaking, th trailing-window btas w us as inputs of

MONEY ILLUSION IN THE STOCK MARKET 653 TABLE I TIME-SERIES REGRESSIONS OF INTERCEPT AND SLOPE PORTFOLIOS 2 K N a 1 a 2 b 1 b 2 c 1 c 2 R intrcpt 2 R slop 36 20 0.0008 0.0009 0.0041 1.0205 1.5048 1.4784 0.44% 58.03% (0.36) ( 0.40) (0.14) (34.38) (2.41) ( 2.35) 36 10 0.0008 0.0009 0.0344 1.0558 1.4588 1.4151 0.49% 57.69% (0.33) ( 0.40) ( 1.12) (34.15) (2.23) ( 2.16) 36 40 0.0007 0.0004 0.0575 0.9732 1.5099 1.5722 1.02% 58.58% (0.33) ( 0.21) (2.07) (34.75) (2.57) ( 2.65) 24 20 0.0001 0.0001 0.0589 0.9641 1.6340 1.5561 1.12% 57.19% ( 0.06) ( 0.03) (2.10) (34.02) (2.73) ( 2.57) 48 20 0.0016 0.0017 0.0163 1.0394 1.2269 1.2224 0.22% 57.23% (0.70) ( 0.72) ( 0.53) (33.55) (1.86) ( 1.85) Th tabl shows OLS rgrssions of th intrcpt portfolio s (r intrcpt,t ) and th slop portfolio s (r slop,t ) xcss rturn on a constant, contmporanous xcss markt rturn (r M,t ), and dmand laggd inflation ( t 1 ): r intrcpt,t r slop,t a 1 b 1 r M,t a 2 b 2 r M,t c 1 t 1 u 1,t c 2 t 1 u 2,t. Th intrcpt and slop portfolios ar constructd using Fama-Macbth [1973] rgrssions of xcss rturns on N bta sortd portfolios on a constant and th portfolios laggd K-month postformation btas. t-statistics ar in parnthss. R 2 is adjustd for dgrs of frdom. Th rgrssions ar stimatd from th sampl priod 1930:06 2001:12, 859 monthly obsrvations. th Fama-MacBth stag will nvr b prfct forcasts of futur btas; thr is no guarant that b 1 0 and b 2 1 xactly. Sinc th point stimats ar always clos for th basis assts w considr, our mthod is informativ nough to allow us to simply modify th formulas for th conditional xcss intrcpt and xcss slop of th scurity markt lin to tak ths small dviations into account. As w can confidntly rjct th hypothss that b 2 0 and b 2 b 1 for all sts of basis assts, straightforward algbra provids th alphas of a zro-bta and a unit-bta stock portfolio implid by th stimats of quation (10). Th functions that map stimats of th paramtrs in rgrssion (10) into th paramtrs of quation (8) ar as follows. Th xcss slop of th scurity markt lin is g 0 g 1 t 1 (11) g 0 a 2 /b 2 g 1 c 2 /b 2. Th xcss intrcpt of th scurity markt lin is givn by th function,

654 QUARTERLY JOURNAL OF ECONOMICS h 0 h 1 t 1 (12) h 0 a 1 a 2 b 1 /b 2 h 1 c 1 c 2 b 1 /b 2. To summariz, ths two formulas ar th rsult of solving for th conditional alpha of a zro-bta and a unit-bta portfolio implid by stimats of systm (10). It is important to not that quations (11) and (12) also provid a corrction for any potntial masurmnt rror problm causd by th us of stimatd btas at th Fama-Macbth stag. Evn if btas ar stimatd with rror in arlir stags, our final stimats of th xcss slop and th xcss intrcpt of th scurity markt lin ar consistnt. Tabl II rports th point stimats of th xcss slop of th scurity markt lin. W focus on th spcification using 20 portfolios and a 36-month bta-stimation window in th Fama-Mac- Bth stag, but as th tabl shows, th rsults ar robust to small TABLE II EXCESS INTERCEPT AND SLOPE OF THE SECURITY MARKET LINE K N g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 20 0.0009 1.4487 0.0008 1.5108 5.82 0.00 ( 0.40) ( 2.35) (0.36) (2.40) [0.05] [0.96] 36 10 0.0009 1.3403 0.0007 1.4126 5.20 0.00 ( 0.40) ( 2.16) (0.33) (2.23) [0.07] [0.95] 36 40 0.0004 1.6155 0.0007 1.6029 7.12 0.00 ( 0.21) ( 2.64) (0.32) (2.57) [0.03] [0.99] 24 20 0.0001 1.6140 0.0001 1.7290 8.37 0.01 ( 0.06) ( 2.56) ( 0.05) (2.71) [0.02] [0.93] 48 20 0.0016 1.1761 0.0016 1.2077 3.46 0.00 ( 0.72) ( 1.85) (0.71) (1.86) [0.18] [0.98] Th tabl shows th stimatd function that maps inflation into th xcss slop and intrcpt of th scurity markt lin. First, w rgrss th intrcpt portfolio s (r intrcpt,t ) and th slop portfolio s (r slop,t ) xcss rturn on a constant, contmporanous xcss markt rturn (r M,t ), and laggd inflation ( t 1 ): r intrcpt,t a 1 b 1 r M,t c 1 t 1 u 1,t r slop,t a 2 b 2 r M,t c 2 t 1 u 2,t. Th intrcpt and slop portfolios ar constructd using Fama-Macbth [1973] rgrssions of xcss rturns on N bta sortd portfolios on a constant and th portfolios laggd K-month postformation btas. Scond, w comput th functions that map th rgrssion paramtrs to th xcss slop and intrcpt of th scurity markt lin. Th xcss slop is dfind as g 0 g 1 t 1, whr g 0 a 2 /b 2 and g 1 c 2 /b 2. Th xcss intrcpt is computd as h 0 h 1 t 1, whr h 0 a 1 a 2 b 1 /b 2 and h 1 c 1 c 2 b 1 /b 2. t-statistics computd using th dlta mthod ar in parnthss. W also rport th tst statistic and th two-sidd p-valus [in brackts] for th hypothss that [g 1,h 1 ] [0,0] and g 1 h 1 0. Th rgrssions ar stimatd from th sampl priod 1930:06 2001:12, 859 monthly obsrvations.

MONEY ILLUSION IN THE STOCK MARKET 655 variations in ths choics. Incrasing th numbr of basis-asst portfolios in th tsts typically strngthns our rsults. W stimat g 0 as 0.0009 with a t-statistic of 0.40 and h 0 as 0.0008 with a t-statistic of 0.36. Th intrprtation of ths nar-zro intrcpt stimats is that whn inflation is at its man, th mpirical bta slop and th zro-bta rat among stocks ar consistnt with th prdiction of th Sharp-Lintnr CAPM. In othr words, whn inflation is at its tim-sris avrag, th Sharp-Lintnr CAPM works. This is consistnt with a form of mony illusion in which popl us historical avrag nominal growth rats to valu th stock markt, ignoring th currnt lvl of inflation which may b vry diffrnt from inflation s historical avrag. Our stimat for g 1 is 1.4487 with a t-statistic of 2.35. As prdictd by th Modigliani-Cohn hypothsis, th xcss slop of th scurity markt lin comovs ngativly with inflation. Our point stimats for th xcss-intrcpt function ar also consistnt with th prdictions of th thory: th stimat of h 1 is 1.5108 with a t-statistic of 2.40, which is statistically significantly diffrnt from zro but not from on. In words, w can rjct th hypothsis that th markt dos not suffr from mony illusion, but w cannot rjct th hypothsis that inflation is (irrationally) fully pricd into ral stock yilds. Furthrmor, g 1 is conomically and statistically vry clos to h 1, as prdictd. Finally, w can rjct th joint hypothsis that both g 1 0 and h 1 0 against th two-sidd altrnativ at th 5 prcnt lvl of significanc. III.A. Additional Robustnss Chcks Our rsults ar not snsitiv to small variations in th inflation masur. For xampl, all of our conclusions rmain valid if w us as our masur of inflation th fittd valu from a rgrssion of monthly (unsmoothd) inflation on its laggd valu, th thr-month Trasury-bill yild, and th tn-yar Trasury-bond yild. W hav also rplicatd our rsults with xpandd sts of basis assts, prsntd in Tabl III. Th first panl uss 20 bta-sortd and 10 siz-sortd portfolios as basis assts. Th scond panl uss 20 bta-sortd and 10 book-to-markt-sortd portfolios as basis assts. Th third and final panl uss 20 bta-sortd, 10 siz-sortd, and 10 book-to-markt-sortd portfolios as basis assts. Th siz-sortd and book-to-markt-sortd portfolios ar providd by Knnth Frnch on his Wb sit. Add-

656 QUARTERLY JOURNAL OF ECONOMICS TABLE III RESULTS FROM EXPANDED ASSET SETS 20 bta-sortd and 10 ME-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0005 1.6660 0.0002 1.5911 6.94 0.00 (0.22) ( 2.55) (0.07) (2.41) [0.03] [0.95] 20 bta-sortd and 10 BE/ME-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0001 1.9251 0.0000 2.0335 10.90 0.01 (0.03) ( 3.12) ( 0.02) (3.23) [0.00] [0.93] 20 bta-sortd, 10 ME-sortd, and 10 BE/ME-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0012 2.0503 0.0007 2.0644 9.86 0.00 (0.52) ( 3.13) ( 0.30) (3.13) [0.01] [0.99] Th tabl shows th stimatd function that maps inflation into th xcss slop and intrcpt of th scurity markt lin, stimatd from xpandd asst sts. First, w rgrss th intrcpt portfolio s (r intrcpt,t ) and th slop portfolio s (r slop,t ) xcss rturn on a constant, contmporanous xcss markt rturn (r M,t ), and laggd inflation ( t 1 ): r intrcpt,t a 1 b 1 r M,t c 1 t 1 u 1,t r slop,t a 2 b 2 r M,t c 2 t 1 u 2,t. Th intrcpt and slop portfolios ar constructd using Fama-Macbth [1973] rgrssions of xcss rturns on basis-asst portfolios on a constant and th portfolios laggd K-month postformation btas. Scond, w comput th functions that map th rgrssion paramtrs to th xcss slop and intrcpt of th scurity markt lin. Th xcss slop is dfind as g 0 g 1 t 1, whr g 0 a 2 /b 2 and g 1 c 2 /b 2. Th xcss intrcpt is computd as h 0 h 1 t 1, whr h 0 a 1 a 2 b 1 /b 2 and h 1 c 1 c 2 b 1 /b 2. t-statistics computd using th dlta mthod ar in parnthss. W also rport th tst statistic and th two-sidd p-valus [in brackts] for th hypothss that [g 1,h 1 ] [0,0] and g 1 h 1 0. Th rgrssions ar stimatd from th sampl priod 1930:06 2001:12, 859 monthly obsrvations. ing ths charactristics-sortd portfolios to th st of basis assts dos not altr our basic conclusions, as th point stimats rmain clos to thos obtaind in th arlir tsts. Thus, w argu that our main conclusions ar not snsitiv to small changs in th st of basis assts. In unrportd tsts, w also xamin th Modigliani-Cohn hypothsis using long-horizon rturns. W us th sam portfolios as in our prvious tsts, xcpt w hold th stocks for horizons ranging from 3 to 60 months. Our markt rturn is also compoundd in th sam way, and thn th compoundd thr-month Trasury-bill intrst rat is subtractd. Smoothd inflation is scald to th sam tim units as th rturns. Othr than th chang in th holding priod, th tst procdur is xactly th sam as in th prvious tsts. W find point stimats consistnt

MONEY ILLUSION IN THE STOCK MARKET 657 with th joint hypothsis of mony illusion and th CAPM at th quartrly horizons and at horizons of thr yars and fiv yars. Howvr, for intrmdiat horizons (12 24 months), any ffct is small, with point stimats occasionally having th wrong sign. Though unfortunat, th low powr and larg standard rrors of ths long-horizon tsts ar at last partially to blam, as th Modigliani-Cohn hypothsis is nvr rjctd statistically. As part of our long-horizon tsts, w also chck to s whthr our point stimats of th cross-sctional ffct of mony illusion ar consistnt with th aggrgat mispricing of stocks vrsus bonds by Campbll and Vuoltnaho [2004]. In particular, w stimat a rgrssion forcasting th xcss markt rturn with smoothd inflation, whil controlling for th subjctiv riskprmium masur SRC of Polk, Thompson, and Vuoltnaho [2004]. As prdictd by th Modigliani-Cohn hypothsis, th partial rgrssion cofficint on inflation is positiv, significant, and similar to our short-horizon cross-sctional stimat at all horizons. Though w find vidnc of Modigliani and Cohn s mony illusion, our tsts so far hav only considrd th Sharp-Lintnr vrsion of th CAPM. Howvr, it is thortically possibl that our rsults ar simply du to an incorrct rstriction on th intrcpt of th scurity markt lin implicit in that vrsion of th CAPM. Black [1972] considrs th possibility that invstors cannot borrow at th Trasury-bill rat. If so, th likly ffct of such inability to borrow is that th zro-bta rat among stocks dviats from th Trasury-bill rat. In othr words, th Black CAPM allows th xcss intrcpt and slop of th scurity markt lin to b nonzro. Thrfor, an altrnativ xplanation for our findings is that th sprad btwn th tru borrowing rat facing invstors and th Trasury-bill rat comovs with inflation. Fortunatly for our conclusions, data on actual borrowing rats indicat that th sprad dos not comov positivly with inflation. Our thr mpirical proxis for th tru borrowing rat ar car-loan rats from commrcial banks, prsonal-loan rats from commrcial banks, and crdit-card intrst rats. W obtain ths quartrly data from th Fdral Rsrv s Wb sit. Th data from commrcial banks bgin 1972:02, whil th crdit-card rat data bgin 1994:11. W first comput th yild sprad btwn ths loans and maturity-matchd Trasury yilds. W

658 QUARTERLY JOURNAL OF ECONOMICS thn rgrss ths sprads on smoothd inflation (in th sam annualizd units). Th rgrssion rsults in Tabl IV show that th yild sprad btwn individuals borrowing rats and Trasury rats comovs ngativly, not positivly, with laggd inflation. This rsult is not surprising, as Ausubl [1991] finds that crdit-card intrst rats appar sticky in rsponding to changs in markt intrst rats. Thus, w rjct th Black CAPM as an altrnativ xplanation for th obsrvd tim-variation in th xcss slop of th scurity markt lin. W also considr subjctiv risk prmiums dtrmind in a world whr multipl risk factors dtrmin th cross sction of subjctiv xpctd rturns. That is, w assum a world in which invstors mistaknly misstimat ral cash-flow growth of (and thus xpctd rturns on) all stocks du to mony illusion, but othrwis pric stocks corrctly in accordanc with a multifactor modl. Furthrmor, w assum that masurd btas ar not matrially affctd by this mispricing. TABLE IV INFLATION AND THE SPREAD BETWEEN BORROWING AND TREASURY RATES 48-month car loans from commrcial banks, sprad ovr th 48-month T-not yild constant (t-statistic) slop on (t-statistic) Adj. R 2 N 3.4327 (13.9) 0.0489 ( 0.9) 0.01 124 24-month prsonal loans from commrcial banks sprad ovr th 24-month T-not yild constant (t-statistic) slop on (t-statistic) Adj. R 2 N 9.2135 (29.4) 0.5596 ( 7.9) 0.58 124 Crdit card accounts (intrst rats), sprad ovr th 90-day T-bill yild constant (t-statistic) slop on (t-statistic) Adj. R 2 N 11.4532 (22.8) 0.4270 ( 2.0) 0.15 33 Crdit card accounts (assssd intrst), sprad ovr th 90-day T-bill yild constant (t-statistic) slop on (t-statistic) Adj. R 2 N 11.1719 (21.1) 0.4547 ( 2.0) 0.12 33 Th tabl rgrsss proxis for th sprad btwn borrowing rats that individuals fac and Trasury rats on laggd inflation. Th inflation sris ( ) is th smoothd inflation usd in arlir tsts, annualizd by multiplying th sris by twlv. Th t-statistics ar basd on Nwy-Wst standard rrors computd using four lags and lads. Th hading of ach panl spcifis th sprad masur bing usd as th dpndnt variabl. Data ar quartrly.

MONEY ILLUSION IN THE STOCK MARKET 659 In our robustnss chcks blow, w mploy th wll-known thr-factor modl of Fama and Frnch [1993], but th stps blow will asily gnraliz to any multifactor modl for which th additional factors ar xprssd as long-short stock portfolios. Th quations givn abov for th Sharp-Lintnr CAPM cas, and thrfor th rgrssions w will run to tst th modl and th Modigliani-Cohn hypothsis, asily gnraliz to this cas. W bgin by rplacing quation (5) with th multifactor bta rprsntation of assts subjctiv risk prmiums: (13) R i,subj i R M,SUBJ i f. f is a column vctor of factor ralizations for th givn priod and i is a column vctor of asst i s multipl-rgrssion loadings on thos factors. Hr w assum that th factor-mimicking portfolios ar long and short stocks in qual dollar amounts. Undr ths conditions thr is no nd for SUBJ suprscripts, as th inflation-rlatd mispricing affcts th yilds of all stocks idntically so that th xpctd rturn of any long-short stock portfolio is unaffctd. Thus, (14) ε i R i,obj i R M,SUBJ i f, and thrfor, (15) ε i R i,obj i R M,OBJ i f ε M i OBJ N i OBJ R i,obj i R M,OBJ i f ε i i ε M. dnots th Jnsn s alpha rlativ to th multifactor modl, and is almost idntical to th xprssion drivd in th CAPM cas, xcpt that is a multifactor snsitivity on th markt rturn: (16) i OBJ 0 1 i 0 1. In th Fama-MacBth rgrssions w now includ as xplanatory variabls th stimatd loadings on all thr factors, including multifactor markt btas. Lt th additional nonmarkt factor loadings b dnotd by ˆ t 1 whr ˆ t 1 has on row for ach asst and on column for ach nonmarkt factor. Th rturns on th intrcpt and (all) slop portfolios ar thn givn by

660 QUARTERLY JOURNAL OF ECONOMICS (17) r intrcpt,t r intrcpt,t r allslops,t 1 ˆ t 1 ˆ t 1 1 ˆ t 1 ˆ t 1 1 1 ˆ t 1 ˆ t 1 r t. rprsnts th rturn (in xcss of th risklss rat) on a portfolio anticipatd to hav zro loadings on all factors (including th markt) and a unit nt invstmnt in stocks. r slop,t, which is dfind as th first lmnt of r allslops,t, is th rturn on a portfolio anticipatd to hav a unit markt loading and a zro loading on th othr factors. Th rmaining lmnts of r allslops,t, ar rturns on portfolios with unit loadings on th othr factors; thy ar not usd in our subsqunt analysis. Th actual factor loadings of th r intrcpt,t and r slop,t portfolios ar again rasonably clos to thir hypothtical valus. W obsrv this by rgrssing th tim sris of rturns on th factors, as wll as on, our inflation variabl: (18) r intrcpt,t r slop,t a 1 b 1 r M,t B 1 f t c 1 t 1 u 1,t a 2 b 2 r M,t B 2 f t c 2 t 1 u 2,t. f t is a vctor of factor ralizations at tim t. B 1 and B 2 ar rgrssion cofficints on th nonmarkt factors. As abov, in ordr to stimat th slop of th scurity markt lin, w nd to adjust th intrcpt and slop portfolios slightly to gt portfolios that (in sampl) actually hav th ncssary loadings. Again, th procss of claning out any xtranous loadings on othr factors convnintly lavs us with scurity markt lin quations that ar virtually idntical to thos in th CAPM cas (xcpt that th b 1 and b 2 now com from th rgrssion that includs th othr factors (i.., thy ar multifactor btas). Th xcss slop and xcss intrcpt of th scurity markt lin ar again givn by quations (11) and (12). Tabl V contains our stimats for th Fama and Frnch [1993] multifactor modl, which contains two factors in addition to th markt factor. Th factor sris ar providd by Knnth Frnch on his Wb sit. Th first is SMB, th diffrnc btwn th rturn on small and big markt-capitalization stocks. Th scond is HML, th diffrnc btwn th rturn on high and low book-to-markt ratio stocks. In Tabl V w find that th stimatd g 1 is clos to 1, th stimatd h 1 is clos to 1, and th two ar clos to qual in absolut valu but opposit in sign, just

MONEY ILLUSION IN THE STOCK MARKET 661 TABLE V RESULTS FOR THE FAMA-FRENCH THREE-FACTOR MODEL 20 bta-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0003 1.2788 0.0003 1.2787 3.13 0.00 (0.11) ( 1.75) ( 0.13) (1.71) [0.21] [1.00] 20 bta-sortd and 10 ME-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0018 0.9980 0.0018 1.0099 2.30 0.00 ( 0.76) ( 1.51) (0.75) (1.52) [0.32] [0.99] 20 bta-sortd and 10 BE/ME-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0014 0.9099 0.0009 0.9615 2.28 0.00 ( 0.40) ( 3.12) (0.35) (1.36) [0.32] [0.99] 20 bta-sortd, 10 ME-sortd, and 10 BE/ME-sortd portfolios K g 0 g 1 h 0 h 1 [g 1,h 1 ] 0 g 1 h 1 0 36 0.0014 1.1630 0.0012 1.1950 3.54 0.00 ( 0.62) ( 1.80) (0.60) (1.84) [0.17] [0.98] Th tabl rpats th tsts of Tabl II using th Fama-Frnch [1993] thr-factor modl. First, w rgrss th xcss rturns on th basis-asst portfolios on a constant and th portfolios laggd K-month postformation factor loadings. Th intrcpt portfolio s (r intrcpt,t ) and th slop portfolio s (r slop,t ) xcss rturns ar th cofficint tim sris corrsponding to th intrcpt and th thr-factor modl s markt loading, rspctivly. Scond, w rgrss ths rturns on a constant, contmporanous factor rturns, and laggd inflation ( t 1 ): r intrcpt,t a 1 b 1,1 r M,t b 1,2 r SMB,t b 1,3 r HML,t c 1 t 1 u 1,t r slop,t a 2 b 2,1 r M,t b 2,2 r SMB,t b 2,3 r HML,t c 2 t 1 u 2,t. Th xcss slop is dfind as g 0 g 1 t 1, whr g 0 a 2 /b 2,1 and g 1 c 2 /b 2,1. Th xcss intrcpt is dfind as h 0 h 1 t 1, whr h 0 a 1 a 2 b 1,1 /b 2,1 and h 1 c 1 c 2 b 1,1 /b 2,1. t-statistics computd using th dlta mthod ar in parnthss. W also rport th tst statistic and th two-sidd p-valus [in brackts] for th hypothss that [g 1,h 1 ] [0, 0] and g 1 h 1 0. Th rgrssions ar stimatd from th sampl priod 1930:06 2001:12, 859 monthly obsrvations. as prdictd by th Modigliani and Cohn hypothsis (and just as w found using th CAPM as th risk modl). For our prfrrd spcification (36 months in postformation loading rgrssions, 20 tst asst portfolios) w obtain point stimats of 1.28 for g 1 (t-statistic of 1.75) and 1.28 for h 1 (t-statistic of 1.71). Th tsts using th multifactor modl hav lss powr, but w can still rjct at th 10 prcnt lvl th hypothsis that inflation plays no rol in th dtrmination of th cross-sctional bta prmium. Th rsults for othr spcifications ar qualitativly similar, as can b sn in Tabl V.

662 QUARTERLY JOURNAL OF ECONOMICS IV. CONCLUSIONS Do popl suffr from mony illusion? Whil w may disagr on th answr to this qustion, its importanc is indisputabl. Many important dcisions, such as choosing btwn buying and rnting a hom or allocating on s portfolio btwn stocks and nominal bonds, dpnd critically on th dcision-makr s ability to distinguish btwn nominal and ral quantitis. Modigliani and Cohn [1979] suggst that stock markt invstors suffr from mony illusion. Consistnt with this hypothsis, prvious tim-sris studis hav found that high inflation coincids with low prics for stocks rlativ to bonds. This obsrvd rlation may b causd by mony illusion; howvr, as notd by Campbll and Vuoltnaho [2004] and othrs, it may also b causd by th ral discount rats usd by invstors bing positivly corrlatd with inflation. W prsnt novl cross-sctional vidnc supporting Modigliani and Cohn s hypothsis. Simultanously xamining th futur rturns of Trasury bills, saf stocks, and risky stocks allows us to distinguish mony illusion from any chang in th attituds of invstors toward risk. Th ky insight undrlying our tsts is that mony illusion will hav a constant additiv ffct on all stocks futur rturns, rgardlss of thir xposur to systmatic risk. This constant ffct is in contrast to th impact of invstor risk attituds on futur stock rturns, which is proportional to th stock s risk, as risky stocks futur rturns will b affctd much mor than saf stocks futur rturns. Our mpirical tsts indicat that whn inflation is high (low), stock rturns ar highr (lowr) than justifid by an amount that is constant across stocks, irrspctiv of th riskinss of th particular stock. A critical assumption in our tsts is that invstors us th Sharp-Lintnr capital asst pricing modl (CAPM) to valuat th risk of a stock. Our cross-sctional tsts lvrag this Sharp-Lintnr CAPM assumption by noting that any ffct of risk attituds on a stock s pric would hav to rsult in an xpctd-rturn ffct that is proportional to th stock s CAPM bta. Consquntly, on may nvr compltly rul out th possibility of our rsults bing du to a misspcifid modl of risk. In addition to suggsting that stock markt invstors suffr from mony illusion, our rsults offr a partial xplanation for th Sharp-Lintnr CAPM s poor mpirical prformanc in rcnt

MONEY ILLUSION IN THE STOCK MARKET 663 sampls. In an influntial papr Fama and Frnch [1992] fail to find support for th cntral prdiction of [th CAPM], that avrag rturns ar positivly rlatd to markt [btas]. Curiously, this ngativ rsult is primarily drivn by thir 1951 1960 and 1981 1990 subsampls, both of which wr prcdd by high inflation. Th cross-sctional implication of th Modigliani-Cohn hypothsis is that th slop of avrag rturns on bta should b much lowr than th quity prmium in prcisly thos subsampls. In a sns, mony illusion may hav killd th Sharp- Lintnr CAPM. Although w do not xplicitly considr mony illusion s ffct on invstor wlfar, w bliv that our rsults may nvrthlss hav som policy implications, howvr spculativ. First, if invstors suffr from mony illusion, stabl and low inflation is likly to rsult in a lss mispricd stock markt than volatil and high inflation. To th xtnt that ral invstmnt dcisions ar influncd by stock markt (mis)valuations, on would xpct low and stabl inflation to b bnficial to socity. 5 Scond, if govrnmnt borrowing shifts from nominal bonds to inflation-indxd or ral bonds, it is possibl that th stock markt will valu stocks rlativ to ral (instad of nominal) bonds, liminating th ffct of mony illusion on stock prics. Third, and most importantly, to th xtnt that invstors prciv a bnfit from valuing stocks using nominal quantitis, thy should pay mor attntion to xpctd inflation whn forcasting futur nominal cash flows. APPENDIX In this Appndix w prsnt a stylizd modl of markt quilibrium, whr som invstors suffr from mony illusion and othrs do not. Th modl givs a simpl prdiction: th diffrnc btwn th quity prmium and th cross-sctional bta prmium (i.., th slop of th scurity markt lin) is qual to c /(1 ), whr is th rat of inflation and c is a masur of th risk-baring capacity of mony-illusiond invstors rlativ to 5. Dow and Gorton [1997] modl th connction btwn stock markt fficincy and conomic fficincy. Stin [1996] focuss on th link btwn markt infficincy and firms ral invstmnt policis, modling how an infficint capital markt can rsult in managrs catring to markt mispricing. Polk and Sapinza [2004] documnt catring ffcts in firms capital xpnditurs rlatd to mispricing.

664 QUARTERLY JOURNAL OF ECONOMICS that of rational invstors. On th on hand, as xpctd, if monyillusiond invstors hav all of th risk-baring capacity in th conomy, inflation translats almost on-for-on into th xcss slop of th scurity markt lin. On th othr hand, also as xpctd, if rational invstors hav all of th risk-baring capacity, thn th Sharp-Lintnr CAPM holds. For intrmdiat cass, th xcss slop of th scurity markt lin dpnds linarly on th shar of risk-baring capacity of th two invstor groups. Th modl has two priods, dnotd by 0 and 1. At tim 0, th invstors trad but do not consum. At tim 1, invstors consum th payoffs from thir portfolios, and thr is no trading. Thr ar thr tradd assts, on nominal bond (asst indxd 1) in zro nt supply and two stocks (assts indxd 2 and 3) both with on shar outstanding. Th prics of ths assts at tim 0 ar p 1, p 2, and p 3. Bcaus w hav no consumption at tim 0, asst prics hav maning only in rlativ trms. W st p 1 1 as an arbitrary normalization, intrprting all rlativ to th nominal bond s pric. Th asst payoffs ar as follows. Th nominal dbt has no dfault risk. Th rat of inflation,, is known, and th nominal dbt is thus risk-fr in ral trms as wll. Th ral payoff to th dbt is X 1 F/(1 ), whr F is th nominal fac valu of th bond. St F 1 without any loss of gnrality, and thus th ral dbt payoff is X 1 1/(1 ). Th scond asst is th stock of a rlativly saf company, and th third asst is a rlativly risky company s stock. Th ral payoffs to th scond asst and th third asst ar X 2 and X 3. For simplicity, w mak th xpctd payoffs for both assts qual; that is, E(X 2) E(X 3) X. Th uncrtain futur valus of ral assts ar indpndnt random variabls with known variancs var(x 2) 2 and var(x 3) k 2, k 1. In particular, ths ral payoffs do not dpnd on th rat of inflation. Th ral payoff to th markt portfolio of all assts is thus X 2 X 3, and th ral rturn on th markt portfolio is (X 2 X 3)/(p 2 p 3 ). Th nominal rturn on th markt portfolio is (1 )(X 2 X 3)/ (p 2 p 3 ). Suppos that two invstors (or groups of invstors), dnotd by A and B, hav man-varianc prfrncs ovr tim 1

MONEY ILLUSION IN THE STOCK MARKET 665 consumption and bhav comptitivly as pric takrs. Initially, th assts ar ndowd vnly btwn th two groups of invstors. Both invstors maximiz a man-varianc objctiv function, but in addition invstor B suffrs from mony illusion. Invstor A has an absolut risk-baring capacity of (1 c), 0 c 1. Invstor A dos not suffr from mony illusion, prcivs th abov ral payoffs corrctly, and maximizs th man-varianc prfrnc of tim 1 consumption: (19) w 1 X 1 w 2 E X 2 w 3 E X 3 1 2 1 c w 2 2 var X 2 1 2 1 c w 3 2 var X 3 1 1 w 1 w 2 w 3 X 1 2 1 c w 1 2 2 2 2 1 c w 3 2 k 2, subjct to th budgt constraint.5( p 2 p 3 ) w 1 p 2 w 2 p 3 w 3 by choosing his portfolio allocations w. Th first-ordr conditions yild dmand curvs for risky assts: (20) w 2 1 c X 2 w 3 1 c X k 2 1 c p 2 1 2 1 c p 3 1 k 2. Th ida that invstor B suffrs from mony illusion manifsts itslf in two ways. First, invstor B maximizs man-varianc prfrncs ovr nominal walth. Scond, invstor B blivs that th nominal growth in th valu of corporat assts is a random variabl with a distribution that dos not dpnd on th rat of inflation. In othr words, invstor B prcivs nominal payoffs, X B1 F 1, X B2 X 2 and X B3 X 3, irrspctiv of th inflation nvironmnt. W st th risk-baring capacity of invstor class B to c, which has th advantag of kping th riskbaring capacity of th conomy as a whol constant as c changs. Thus, invstor B maximizs (21) w B1 X B1 w B2 E X B2 w B3 E X B3 1 2c w 2 B2 var X B2 1 2c w 2 B3 var X B3 w B1 w B2 w B3 X 1 2c w B2 2 2 1 2c w B3 2 k 2,

666 QUARTERLY JOURNAL OF ECONOMICS subjct to th budgt constraint.5( p 2 p 3 ) w A1 p 2 w A2 p 3 w A3 by choosing his portfolio allocations. Th first-ordr conditions yild dmand curvs for risky assts: (22) w B2 cx / 2 cp 2 / 2 w B3 cx / 2 cp 3 / 2. In th quilibrium th asst markt clars, and on shar of both assts must b hld by both invstors: (23) 1 w 2 w B2 X 2 1 c p 2 1 2 1 w 3 w B3 X 1 c p 3 k 2 1 k 2. Solving for prics yilds (24) p 2 1 1 c X 2 p 3 1 k 1 c X 2. Ths prics maks sns: if mony-illusiond invstors dominat th markt (i.., c 1), thn th pric of a ral asst in rlation to th pric of a nominal asst dos not chang with th rat of inflation. Th xpctd rturn prmium of asst 3 ovr that of asst 2 is (25) E X 3 p 3 E X 2 p 2 1 c 1 X X k 2 X X 2. Th rturn on th markt portfolio is (X 2 X 3/p 2 p 3 ), and th CAPM btas of th two stocks ar (26) 2 2/ 1 k X 2 X 2 3 2k/ 1 k X k 2 X k 2. Th diffrnc btwn th btas of th two assts is

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