Online Appendix Stock Price Booms and Expected Capital Gains Klaus Adam, Albert Marcet and Johannes Beutel

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1 VOL. VOL NO. ISSUE STOCK PRICE BOOMS Online Appendix Sock Price Boom and Expeced Capial Gain Klau Adam, Alber Marce and Johanne Beuel APPENIX A. aa Source Sock price daa: our ock price daa i for he Unied Sae and ha been downloaded from The Global Financial aabae (hp:// The period covered i Q:946-Q:0. The nominal ock price erie i he SP 500 Compoie Price Index (w/gf exenion) (Global Fin code _SPX ). The daily erie ha been ranformed ino quarerly daa by aking he index value of he la day of he conidered quarer. To obain real value, nominal variable have been deflaed uing he USA BLS Conumer Price Index (Global Fin code CPUSAM ). The monhly price erie ha been ranformed ino a quarerly erie by aking he index value of he la monh of he conidered quarer. Nominal dividend have been compued a follow I./=I. / I N./ I N./=I N. / where I N denoe he SP 500 Compoie Price Index (w/gf exenion) decribed above and I i he SP 500 Toal Reurn Index (w/gf exenion) (Global Fin code _SPXTR ). We fir compued monhly dividend and hen quarerly dividend by adding up he monhly erie. Following Campbell 003, dividend have been deeaonalized by aking average of he acual dividend paymen over he curren and preceding hree quarer. Inere rae daa: A nominal inere rae we ue he 90 ay T-Bill Secondary Marke (Global Fin code ITUSA3S). The weekly (o he end of 953) and daily (afer 953) erie ha been ranformed ino a quarerly erie uing he inere rae correponding o he la week or day of he conidered quarer and i expreed in quarerly rae (no annualized). To obain real value, nominal variable have been deflaed uing he USA BLS Conumer Price Index (Global Fin code CPUSAM ). Sock marke urvey daa: The UBS urvey i he UBS Index of Inveor Opimim, which i available (again a fee) a hp:// daa_acce/daa/daae/ub_inveor.hml. The quaniaive queion on ock marke expecaion ha been urveyed over he period Q:998-Q4:007 wih 70 repone per monh on average and ha hereafer been upended. For each quarer we have daa from hree monhly urvey, excep for he fir four quarer and he la quarer of he urvey period where we have only one monhly urvey per quarer. The Shiller urvey daa cover individual inveor over he period Q:999Q-Q4:0 and ha been kindly made available o u by Rober Shiller a Yale Univeriy. On average 73 repone per quarer have been recorded for he queion on ock price growh. Since he Shiller daa refer o he ow Jone, we ued he P raio for he ow Jone, which i available a hp:// o compue correlaion. The CFO urvey i colleced by uke Univeriy and CFO magazine

2 THE AMERICAN ECONOMIC REVIEW MONTH YEAR and collec repone from U.S. baed CFO over he period Q3:000-Q4:0 wih on average 390 repone per quarer, available a hp:// Inflaion expecaion daa: The Survey of Profeional Forecaer (SPF) i available from he Federal Reerve Bank of Philadelphia a hp:// The Michigan Survey of Conumer are colleced by Thomon Reuer/Univeriy of Michigan (hp:// A. Correlaion beween P Raio and Acual/Survey Reurn Thi appendix documen for he UBS Survey, he CFO urvey and he Shiller individual inveor urvey ha here exi a poiive correlaion beween he P raio and urvey expeced reurn (or capial gain), bu a negaive correlaion beween he P raio and acual reurn (or capial gain). Table A documen he poiive correlaion for he UBS urvey. Reul are independen of how one exrac expecaion from he urvey (uing he median or he mean expecaion, uing inflaion expecaion from he Michigan urvey or he Survey of Profeional Forecaer (SPF) o obain real reurn expecaion, uing plain nominal reurn inead or real reurn, or when rericing aenion o inveor wih more han US$ in financial wealh). The number repored in bracke in able A (and in ubequen able) are auocorrelaion robu p-value for he hypohei ha he correlaion i maller or equal o zero. 98 Thee p-value are no adjued for mall ample bia, a here i no generally acceped approach for how o perform uch adjumen. Thi aid, he p-value for he null hypohei are all below he 5% ignificance level and in many cae below he % level. A poiive correlaion i equally obained when conidering oher urvey daa. Table A repor he correlaion beween he P raio and he ock price growh expecaion from Bob Shiller Individual Inveor Survey. 99 The able how ha price growh expecaion are alo rongly poiively correlaed wih he P raio, uggeing ha he variaion in expeced reurn oberved in he UBS urvey i due o variaion in expeced capial gain. Table A alo how ha correlaion eem o become ronger for longer predicion horizon. Table A3 repor he correlaion for he ock reurn expecaion repored in he Chief Financial Officer (CFO) urvey which urvey chief financial officer from large U.S. corporaion. Again, one find a rong poiive correlaion. Table A4 repor he correlaion beween he P raio and he realized real reurn (or capial gain) in he daa, uing he ame ample period a are available for he urvey conidered in able A o A3, repecively. The poin eimae for he correlaion i negaive in all cae, alhough he correlaion fall hor of being ignifican he 5% level due o he hor ample lengh for which he urvey daa i available. 98 The ampling widh i four quarer, a i andard for quarerly daa, and he e allow for conemporaneou correlaion, a well a for cro-correlaion a lead und lag. The p-value are compued uing he reul in Roy Shiller price growh daa refer o he ow Jone Index. The able hu repor he correlaion of he urvey meaure wih he P raio of he ow Jone.

3 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 3 UBS Gallup Own porfolio, >00k US$ Own porfolio, all inveor Sock marke, >00k US$ Sock marke, all inveor Nominal Real Re. Exp. Real Re. Exp. Reurn Exp. (SPF) (Michigan) Average Median Average Median Average Median (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.03) (0.04) (0.03) (0.03) (0.03) (0.03) (0.03) (0.04) (0.03) (0.04) (0.03) (0.03) Table A: Correlaion beween P raio and -year ahead expeced reurn meaure (UBS Gallup Survey, robu p-value in parenhee, wihou mall ample correcion for p-value ) Shiller Nominal Real Capial Gain. Real Capial Gain Survey Capial Gain Exp. Exp. (SPF) Exp. (Michigan) Horizon Average Median Average Median Average Median monh 0.46 (0.0) 0.48 (0.0) 0.45 (0.0) 0.47 (0.0) 0.46 (0.0) 0.49 (0.0) 3 monh 0.57 (0.0) 0.64 (0.00) 0.54 (0.0) 0.6 (0.00) 0.56 (0.0) 0.6 (0.0) 6 monh 0.58 (0.0) 0.75 (0.0) 0.54 (0.0) 0.70 (0.0) 0.56 (0.0) 0.7 (0.0) year (0.03) 0 year 0.74 (0.0) (0.0) 0.75 (0.0) (0.05) 0.66 (0.0) (0.0) 0.7 (0.0) (0.04) 0.7 (0.0) (0.0) 0.75 (0.0) Table A: Correlaion beween P raio and expeced ock price growh (Shiller Individual Inveor Survey, robu p-value in parenhee, wihou mall ample correcion for p-value ) CFO Nominal Real Reurn Real Reurn Survey Reurn Exp. Exp. (SPF) Exp. (Michigan) Average Median Average Median Average Median year 0.7 (0.00) 0.75 (0.00) 0.6 (0.00) 0.69 (0.00) 0.67 (0.00) 0.7 (0.00) Table A3: Correlaion beween P raio and -year ahead expeced ock reurn meaure (CFO Survey, robu p-value in parenhee, wihou mall ample correcion for p-value )

4 4 THE AMERICAN ECONOMIC REVIEW MONTH YEAR Variable Time Period Sock Index Correlaion P, year-ahead real reurn UBS Gallup ample 0:66 S&P 500 (ock marke exp.).0:08/ P, year-ahead real price growh Shiller year ample ow Jone 0:4.0:06/ P, 0 year-ahead real price growh Shiller 0 year ample ow Jone 0:88.0:6/ P, year-ahead real reurn CFO ample S&P 500 0:46.0:06/ Table A4: Correlaion beween P and acual real reurn/capial gain (robu p-value in parenhee, wihou mall ample correcion for p-value ) A3. Proof of Propoiion Proof of par a) Under he null hypohei of raional expecaion (E E ) equaion () implie (A) R ;CN a N C c N C u N C " N ; where " N i he predicion error R ;CN E R ;CN from he rue daa-generaing proce, he condiional expecaion i aken wih repec o inveor informaion a. Since = i in hi informaion e under RE and given () we have (A) E x u N C " N 0: Therefore, u N C " N u N and he null hypohei of raional expecaion implie (A3) c N c N : Equaion (3) (5) define a SUR yem of equaion wih dependen variable E N and R ;CN ; and explanaory variable in boh equaion x.; / 0 : Under he null hypohei he error erm aify he orhogonaliy condiion (4) and (A). For par a) of Propoiion we define he OLS eimaor equaion by equaion b T a 6 b T 4 ba T N bc T N ba N T bc N T I! TX X T x x 0 E N R ;CN x ; where I i a ideniy marix. A andard reul enure ha under he aumpion OLS equaion by equaion i conien and efficien among he e of eimaor ha ue orhogonaliy condiion (4) and (A).

5 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 5 A i well known, a T! we have p (A4) T b T 0! N 0; I E.x x 0 / Sw I E.x x 0 / ; in diribuion, where X S w 0 0 C 0 k C 0k 0 k u u 0 k E u" k ; u" k x x 0 k ; where u u N C N and u" u N C " N. The foonoe of he propoiion conain all boundedne condiion required o enure validiy of aympoic diribuion, E.x x 0/ i inverible becaue var. = / > 0. To build he e-aiic, we only need o find an eimaor for var-cov marix in (A4). We eimae E x x 0 by P T T x x 0. Since u and u" are no forecaing error, here i no reaon why 0 k hould be zero for any k, o we ue a Newey-We eimaor bs w. Therefore, pomuliplying T 0 by [0; ; 0; ] 0 in (A4), and leing 3 " # (A5) b c c T [0; ; 0; ] I " # TX x x 0 bs w I 0 TX x x T T 0 5 we have ha under he null hypohei p bc N T T bc T N! N.0; / in diribuion. b c c Proof of par b) Equaion (3) (5) in he curren paper are each of he form of equaion () in Sambaugh 999. Focuing fir on (3), our (E N ; u C ; P ; " C P / play he role of.y ; u ; x ; v / in Sambaugh. Noe, in paricular, ha o mach hi framework we need o have P play he role of x ; implying ha our " C P play he role of Sambaugh v : Therefore, aumpion a he boom of page 378 in Sambaugh require ha (u N C N ; " C P / i joinly normal. Under normaliy, uing orhogonaliy of meauremen error, i follow from propoiion 4 in Sambaugh 999 ha (A6) E.bc T N c N / cov." C P ; u N / var." P E.b T / / where b T i he OLS eimaor of. Since u conain informaion ha under he null i ueful for predicing fuure reurn we could expec cov." C P ;u N / 6 0 and he bia o be var." P /

6 6 THE AMERICAN ECONOMIC REVIEW MONTH YEAR non-zero. Similarly, we have, (A7) (A8) E.bc T N c N / cov." C P ; u N C " N / var." P E.b T / / E.bc T N c N / C cov." C P ; "N / var." P E.b T / A4. Parameerizaion of he Wage Proce /: We e C equal o he average conumpion-dividend raio in he U.S. over he period 946-0, uing he Peronal Conumpion Expendiure and Ne Corporae ividend erie from he Bureau of Economic Analyi. Thi deliver. The conumpion-dividend raio flucuae coniderably over ime and diplay a cloe o uni roo behavior, wih he quarerly ample auocorrelaion being equal o 0.99, promping u o conider only value cloe o one for he perience parameer p. Following Campbell and Cochrane 999, our remaining calibraion arge are (A9) c; 7 and (A0) c;d; 0:; where c; denoe he condiional andard deviaion of log conumpion growh and c;d; he condiional correlaion beween log conumpion growh and log dividend growh. Aggregae conumpion i given by C C W o ha C W log C log C C W c C log " C log " W. p/ log C W ; where c ummarize conan erm. Condiional variance of log conumpion growh i hu equal o (A) c; C W C W and condiional covariance beween log conumpion and dividend growh given by cov log C ; log cov log " C log " W ; log " C cov log " C log " W ; log " C W :

7 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 7 The condiional correlaion i c;d; corr log C ; log C q C W C W C W 7 C W ; where he la line ue (A9) and (A). Targeing a correlaion of 0. hu deliver 0: (A) W 7 : From (A) we hen ge (A3) W 49 0: 7 : Equaion (A) and (A3) deliver our calibraion arge. A i eaily verified, he implied covariance marix for he innovaion log " ; log " W ha a poiive deerminan. We now check wha he calibraion implie for he variance of uncondiional log conumpion growh C. Uing he fac ha ime hock are independen of log C W and leing C denoe he andard deviaion of uncondiional log conumpion growh, we ge C C W C W C. p/ var log C W : The uncondiional variance of log C W i given by hence var log C W C C. p/ W C p W 0: C. p/ C 7 p W p ; 49 0: 7 For p!, we hu have he reul ha uncondiional and condiional conumpion growh volailiy are idenical ( C c; =7). For lower value of he perience parameer p, uncondiional conumpion volailiy decreae omewha relaive o condiional conumpion volailiy. For he parameer value conidered in he main ex, we

8 8 THE AMERICAN ECONOMIC REVIEW MONTH YEAR have C 7:0 for p :0 4:7 for p 0:95 : A5. Exience of Opimum, Sufficiency of FOC, Recurive Soluion Subiuing ou C uing he budge conrain (4a) in problem (4), one obain a problem ha involve conumpion choice only. Given he ock holding limi, he choice e i compac. I i alo non-empy ince S for all i feaible. The following condiion hen inure exience of opimal plan: Condiion The uiliy funcion u./ i bounded above and for all i [0; ] (A4) E Pi 0 X u.w C / > : 0 The expreion on he lef-hand ide of condiion (A4) i he uiliy aociaed wih never rading ock (S i for all ). Since hi policy i alway feaible, condiion (A4) guaranee ha he objecive funcion in (4a) i alo bounded from below, even if he flow uiliy funcion u./ i ielf unbounded below. From >, ee equaion (0), we have u.c / C 0 and hu a uiliy funcion ha i bounded above. Provided (A4) hold, he opimizaion problem (4a) hu maximize a bounded coninuou uiliy funcion over a compac e, which guaranee exience of a maximum. Under he aumpion made in he main ex (uiliy funcion given by (0), knowledge of (7) and ()), condiion (A4) indeed hold, a can be een from he following derivaion: Uing E Pi 0 X u.w C / 0 X E 0 u.w C / 0 E 0 X 0 C W 0 0. C W / 0 E 0 X 0 C W C W 0 0 0! : C W C W 0 0 C C W 0 0 Y 0! p " j j0 Y j0 " W p j j

9 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 9 we ge E Pi 0 X u.w C / 0 C W E 0 X 0 C C W 0 0!. p /. /. / The infinie um i bounded if E[ " W p j " ] i bounded below one for all j > 0. The following derivaion eablih hi fac: E " W p j " h E " i W p j. / C E[ " ] e p j. /.Cp j. // W C e. /! Y " j " W p j j j0 e. /.C. // W C e. / h E " W " i < ; where he weak inequaliy follow from > and p [0; ] and he ric inequaliy from (). Thi eablihe exience of opimal plan. Since (4a) i a ricly concave maximizaion problem he maximum i unique. Wih he uiliy funcion being differeniable, he fir order condiion (A5) u 0.C i / EPi u 0.C i C / C C C plu a andard ranveraliy condiion are neceary and ufficien for he opimum. Recurive Formulaion. We have a recurive oluion whenever he opimal ockholding policy can be wrien a a ime-invarian funcion S i S i.x / of ome ae variable x. We eek a recurive oluion where x conain appropriaely recaled variable ha do no grow o infiniy. Wih hi in mind, we impoe he following condiion: Condiion The flow uiliy funcion u./ i homogeneou of degree 0. Furhermore, he belief P i imply ha ; ; W ha a ae pace repreenaion, i.e., he condiional diribuion P i. C j! / can be wrien a (A6) (A7) P i. C j! / F i.m i / m i R i.m i ; / for ome finie-dimenional ae vecor m i F i and R i. and ome ime-invarian funcion

10 0 THE AMERICAN ECONOMIC REVIEW MONTH YEAR Under Condiion, problem (4a) can hen be re-expreed a (A8) max E fs i 0 Pi [S;S]g 0 X 0 u S i P C S i C W ; given S i ; where i a ime-varying dicoun facor aifying and " : The reurn funcion in (A8) depend only on he exogenou variable conained in he vecor. Since he belief P i are aumed o be recurive in, andard argumen in dynamic programming guaranee ha he opimal oluion o (A8) ake he form (). Thi formulaion of he recurive oluion i ueful, becaue caling and W by he level of dividend eliminae he rend in hee variable, a deired. Thi will be ueful when compuing numerical approximaion o S i./. The belief yem P i inroduced in ecion V will aify he requiremen aed in condiion. A6. Proof of Propoiion For general p we have. C W /. C /. p/. C W / p ln " W " o ha for S i for all 0, he budge conrain implie C i CW.C W /. Subiuing hi ino he agen fir order condiion deliver (A9) E 4. C W! C C / C. C W.P C C C/3 5 : / Auming ha he following ranveraliy condiion hold 80 9 < (A0) lim E 4 C W C j C j A = C j j! : C W ; C j 3 5 0; one can ierae forward on (A9) o obain 0 X E 4 C W C j C j A j C W C j 3 5 :

11 VOL. VOL NO. ISSUE STOCK PRICE BOOMS Uing C j =. / j Q j k " Ck and one ha (A) E X j C W C j C j C W C C W. / 4 X j j j E! p j Y i0 " C W p i j i ; C W C j 3 C j A C j 5 C W! 4 C. p j / jy C W " Ck " Ck W k The infinie um in he previou expreion i bounded, if E h p i " C W p j 3 5 " C remain bounded away from one for all j >. Thi follow from he following derivaion: i E h E h " C W p j " C i " C W p j i h C E " C i e p j W C. p j / W C e. / C. / e.c / W C e. / h E " C W " C i < ; where he inequaliy in he hird o la line follow from > and p [0; ] and he la inequaliy from aumpion (). For he pecial cae p and p 0, equaion (A) deliver he expreion aed in propoiion. A7. Bayeian Foundaion for Lagged Belief Updaing We now preen a lighly modified informaion rucure for which Bayeian updaing give rie o he lagged belief updaing equaion (39). Specifically, we generalize he perceived price proce (6) by pliing he emporary reurn innovaion ln " C ino wo independen ubcomponen: ln C ln ln C C ln " C C ln " C wih ln " C ii N. "; ; " /, ln " C ii N. " ; " / and " " C " :

12 THE AMERICAN ECONOMIC REVIEW MONTH YEAR We hen aume ha in any period agen oberve he price, dividend and wage up o period, a well a he innovaion " up o period. Agen ime informaion e hu coni of I f ; ; W ; " ; ; ; W ; " ; :::g. By oberving he innovaion ", agen learn - wih a one period lag - omehing abou he emporary componen of price growh. The proce for he perien price growh componen ln remain a aed in equaion (7), bu we now denoe he innovaion variance by ev inead of v. A before, ln m denoe he poerior mean of ln given he informaion available a ime. We prove below he following reul: PROPOSITION 4: Fix " > 0 and conider he limi "! 0 wih ev " g =. g/. Bayeian updaing hen implie (A) ln m ln m C g.ln ln ln m / g ln " The modified informaion rucure hu implie ha only lagged price growh rae ener he curren ae eimae, o ha belief are predeermined, preciely a aumed in equaion (39). Inuiively, hi i o becaue lagged reurn become infiniely more informaive relaive o curren reurn a "! 0, which eliminae he imulaneiy problem. For non-vanihing uncerainy " he weigh of he la obervaion acually remain poiive bu would ill be lower han ha given o he lagged reurn obervaion, ee equaion (A5) in he proof below and he ubequen dicuion for deail. We now provide he proof of he previou propoiion. Le u define he following augmened informaion e ei I [ f" g. The poerior mean for given ei, denoed ln m j ei i readily recurively deermined via (A3) ln m j ei ln m j ei ev Ceg ln ln ln " C ev C " and he eady ae poerior uncerainy and he Kalman gain by q e C e C 4 e " eg (A4) We furhermore have " ln m j ei and E[ln ln jei ] ln m j ei " C ".ln ln / E[ln ln jei ] ln ln m j ei Cln " C Cln " C " C " o ha

13 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 3 ln ln ln P jei N ln m j ei ln m j ei " C "! ; C " C "! ; where he covariance beween ln jei and ln ln jei can be compued by exploiing he fac ha ln ln m j ei and ln " C C ln " are independen and uing ln E [ln jei ] ln ln m j ei. Uing andard normal updaing formulae, we can hu compue ln m ji E[ln ji ] E[ln jei ; ln ln ] ln m j ei C C " C ln ln C " C " (A5) " ln m j ei ; where he econd equaliy exploi he fac ha ; W conain no informaion abou ln ; and he econd inequaliy follow from he fac ha ei conain informaion up o and including " ; ince he laer i independen of (ln ln /. Since C " C " < " eg, he weigh of he price obervaion daed i reduced relaive o he earlier obervaion daed becaue i i noiier. Now conider he limi "! 0 and along he limi chooe " " " and ev g g ", a aumed in he propoiion. From! 0 and equaion (A5) i hen follow ha ln m ji ln m j ei, i.e., he weigh of he la obervaion price converge o zero. Moreover, from ev g and (A4) we ge eg g. Uing hee reul, equaion (A3) implie g " equaion (A). A8. Proof of Propoiion 3 The proof ue he aumpion of no uncerainy o ha for any funcion f we have E P f.x C j ; Y C j / f.e P X C j ; E P Y C j /. Simplifying noaion (and lighly abuing i) in hi appendix we le X C j E P X C j for all j, o ha X C j below denoe he ubjecive expecaion condiional on informaion a ime of he variable X a ime C j. The fir order condiion (A5) can hen be wrien a (A6) C C C R C H) C j jy R C C C j for all ; j 0; auming he ock limi are no binding in period ; C; :::; C j. Ieraing forward N period on he budge conrain of he agen and uing he fac ha

14 4 THE AMERICAN ECONOMIC REVIEW MONTH YEAR eiher NY R C! 0 or S CN! 0 a N! we have. C / S X j0 jy R C Uing equaion (A6) o ubiue ou C C j give. C / S X j0 jy R C!! " W C C C j j W C j jy # RC W C j auming he ock limi are large enough no o be binding. Impoing on he previou equaion S (he marke clearing condiion for S if 0, or he iniial condiion for period 0) and C C W (he marke clearing condiion for conumpion) one obain C NX j0 jy R C! " W C j jy RC W C j Cancelling he erm for j 0 in each ummaion give for he marke-clearing price! " NX jy # RC W jy C j W C j RC Uing (40) give (4). j A9. Verificaion of Condiion (40) For he vanihing noie limi of he belief pecified in ecion V we have E [C j ].m / j E [ C j ] j E [W C j ] j W where we have abraced from raniional dynamic in he W = raio and aume W =, a raniional dynamic do no affec he limi reul. We fir verify he inequaliy on he l.h.. of equaion (40). We have # lim T! E [R T ] m C lim T! m T ; o ha for m > he limi clearly aifie lim T! E [R T ] > due o he fir erm on he r.h..; for m < he econd erm on he r.h.. increae wihou bound, due o >, o ha lim T! E [R T ] > alo hold.

15 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 5 In a econd ep we verify ha he inequaliy condiion on he r.h.. of equaion (40) hold for all ubjecive belief m > 0. We have lim T! E (A7) TX Y j i j R Ci W C j! lim T! W E lim T! W TX j X j TX j j Y j i R Ci! where (A8) X j j Q j i.m C i m 0 / A ufficien condiion for he infinie um in (A7) o converge i ha he erm X j are bounded by ome exponenially decaying funcion. The denominaor in (A8) aifie (A9) Y j.m i C.m / j C m i m j j / ; where he fir erm capure he he pure produc in m, he econd erm he pure produc in m i, and all cro erm have been dropped. We hen have X j j Q j i.m C i m j.m / j C j m / j m j C m j j ;. / j where all erm in he denominaor are poiive. For m > we can ue he fir erm in he denominaor o exponenially bound X j, a X j m j; for m < we

16 6 THE AMERICAN ECONOMIC REVIEW MONTH YEAR can ue he econd erm: X j m j j. / j j j m Since m < here mu be a J < uch ha m j m > for all j J, o ha he X j are exponenially bounded for all j J. A0. Proof of Lemma Proof of lemma : We ar by proving he fir poin in he lemma. The price, dividend and belief dynamic in he deerminiic model are decribed by he following equaion (A30) ln m ln m C g.ln ln ln m / ln ln f.ln m / ln ln ; where f./ i a coninuo funcion, implicily defined by he log of he = oluion o equaion (4). 00 Subiuing he laer wo equaion ino he fir deliver ln m ln m g f.ln m / f.ln m / C ln ln m. If ln m converge, hen he l.h.. of he previou equaion mu converge o zero. Since f./ i coninuo, hi mean ha m mu converge o, a claimed. We now prove he econd poin in he lemma. The belief dynamic implied by he econd order difference equaion (A30) can expreed a a wo-dimenional fir order difference equaion uing he mapping F : R! R, defined a x C g.ln P.e F.x/ x / ln P.e x / C ln x / ; o ha ln m ln m ln m F ln m : Clearly, F ha a fixed poin a he RE oluion, i.e.,.ln ; ln / 0 F.ln ; ln / 0. Moreover, m locally converge o he RE belief if and only F.ln ; ln / C g. / g 0 0 x 00 Since we are inereed in aympoic reul and ince W =!, pricing i aympoically given by equaion (4).

17 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 7 ha all eigenvalue le han one in abolue value, ln P.eln m ln m m wih jj <. The eigenvalue of he marix in equaion (A3) are P. m C g. q /. C g. // 4g From jj < and g < follow ha. C g. // 4g > g g 0, o ha all eigenvalue are real. A i eaily verified, we have C < becaue q C g. / <. C g. // 4g, q. C g. // 4g < g. /,. C g. // 4g <. g. //, g. / 4g < g. /, g < 0 : and < becaue C g. / < q. C g. // 4g where he l.h.. i negaive and he r.h.. poiive. We have C > if and only if q C g. / >. C g. // 4g From jj < and g < he l.h.. i weakly poiive, while he r.h.. i ricly negaive. We have > if and only if q 3 C g. / >. C g. // 4g,. C. C g. /// >. C g. // 4g, C. C g. // > g, C g. / > 0 The la equaion hold ince jj < and g <. Thi how ha he eigenvalue of (A3) are all inide he uni circle. A. Capial Gain Expecaion and Expeced Reurn: Furher eail Figure A depic how expeced reurn a variou horizon depend on agen expeced price growh expecaion uing he ame parameerizaion a ued in figure 5. I how ha expeced reurn covary poiively wih capial gain expecaion for m, a ha been claimed in he main ex. The fla par a around m 0:0 arie becaue in ha area he P raio increae rongly, o ha he dividend yield fall. Only

18 8 THE AMERICAN ECONOMIC REVIEW MONTH YEAR Expeced reurn quarer ahead 5 quarer ahead 0 0 quarer ahead 40 quarer ahead β Expeced capial gain (m ), per quarer FIGURE A. EXPECTE RETURN AS A FUNCTION OF EXPECTE CAPITAL GAIN for peimiic price growh expecaion (m < ) and long horizon of expeced reurn we find a negaive relaionhip. The laer emerge becaue wih price expeced o fall, he dividend yield will rie and evenually reul in high reurn expecaion. A. Numerical Soluion Algorihm Algorihm: We olve for agen ae-coningen, ime-invarian ockholding (and conumpion) policy () uing ime ieraion in combinaion wih he mehod of endogenou grid poin. Time ieraion i a compuaionally efficien, e.g., Aruoba e al. 006, and convergen oluion algorihm, ee Rendahl 03. The mehod of endogenou grid poin, ee Carroll 006, economize on a coly roo finding ep which peed up compuaion furher. Evaluaion of Expecaion: Imporanly, agen evaluae he expecaion in he fir order condiion (A5) according o heir ubjecive belief abou fuure price growh and heir (objecive) belief abou he exogenou dividend and wage procee. Expecaion are approximaed via Hermie Gauian quadraure uing hree inerpolaion node for he exogenou innovaion. Approximaion of Opimal Policy Funcion: The conumpion/ockholding policy i approximaed by piecewie linear pline, which preerve he nonlineariie ariing in paricular in he P dimenion of he ae pace. Once he ae-coningen conumpion policy ha been found, we ue he marke clearing condiion for conumpion good o deermine he marke clearing P raio for each price-growh belief m. Accuracy: Carefully chooing appropriae grid for each belief i crucial for he accuracy of he numerical oluion. We achieve maximum (relaive) Euler error on he order of 0 3 and median Euler error on he order of 0 5 (average: 0 4 ). Uing our analyical oluion for he cae wih vanihing noie, we can ae he accuracy of our oluion algorihm more direcly. Seing he andard deviaion of

19 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 9 exogenou diurbance o 0 6 he algorihm almo perfecly recover he equilibrium P raio of he analyical oluion: he error for he numerically compued equilibrium P raio for any price growh belief m on our grid i wihin 0.5 % of he analyical oluion. A3. Model wih Sock Supply Shock Wih he upply hock pecified in he main ex, he individual opimizaion problem remain unchanged. The only poin where he model change i when we compue marke clearing price. Thee now have o aify he relaion where S.e " ; ; W ; m / e " ; " " ii N. ; " /: S To illurae model performance in he preence of upply hock and o compare i o he model wihou upply hock, we coninue o keep he parameer value for. ; ; p/ equal o he eimaed one for he model from able 3 (diagonal marix) and only reeimae he gain value g, o a o fi he ae pricing momen repored in able 3, uing a diagonal weighing marix. Clearly, an even beer mach wih he daa momen can be achieved by reeimaing all parameer. For :5 0 3, he new gain eimae i " bg 0:035. S Table A5 below repor he momen and -raio for he model wih upply hock. The model implied andard deviaion for he rik free rae and he auocorrelaion of he exce ock reurn and ock reurn are repored in able 6 in he main ex. In erm of he -raio in able A, he model perform equally well a he baeline model from able 3, excep for he auocorrelaion of he P raio which i now omewha lower. The fi wih he laer momen could be improved furher by relaxing he aumpion ha "

20 0 THE AMERICAN ECONOMIC REVIEW MONTH YEAR i iid. Subj. Belief Model wih Supply Shock Momen -raio E[P] Sd[P] Corr[, ] Sd[R] c R E[R] E[R b ] UBS Survey aa: Corr[,E R ;C4 ] Table A5: Ae pricing momen, eimaed model wih upply hock from able 6 in he main ex A4. erivaion of Approximae Sharpe Raio Under raional and ubjecive price belief, he following wo fir-order-condiion hold, namely he fir order condiion for ock " # (A3) E P CC. C rc / ; C where C rc CC C i he gro real ock reurn, and he fir order condiion for bond (A33) C r b E : CC C Equaion (A3) can be wrien a " # (A34) E P CC E C r C C cov P " CC C ; r C # :

21 VOL. VOL NO. ISSUE STOCK PRICE BOOMS ividing he previou equaion by E (A35) E r C r b. C r b / cov CC C and uing (A33) one obain " CC C ; C r C # : ividing by Sd r C deliver (A36) E P r C r b Sd P. C r r b / Sd C.CC =C / corr " CC C ; C r C # ; where he lef-hand ide i he (ubjecive) condiional Sharpe raio and corr P (ubjecive) condiional correlaion. Uing he fac ha [] he corr " CC C C r b # ; C rc ; where he laer follow from he fir order condiion (A3), we have under he addiional aumpion of log-normal conumpion growh (which hold exacly in he cae wih raional price expecaion): E P r (A37) C r b Sd P Sd P r CC =C : C For he cae wih raional price expecaion (E P [] E [], Sd P [] Sd []), i hen follow from (4) ha Sd r C Sd r C, o ha by uing hi relaionhip o ubiue Sd r C in (A37) and by applying he uncondiional raional expecaion operaor on boh ide of he equaion, one obain equaion (45) in he main ex. For he cae wih ubjecive price expecaion, we have Sd P r C Sd P rc (A38) Sd r ; where he fir approximaion i a feaure of he ubjecive price belief yem 0 and 0 According o agen ubjecive belief (A39) C r C C C C C v C " C C " : Since i mall, C v C, and he andard deviaion of " i mall relaive o he andard deviaion of " C, he la erm in (A39) conribue lile o he andard deviaion of r C. I hen follow from Cv C ha

22 THE AMERICAN ECONOMIC REVIEW MONTH YEAR he econd approximaion due o he way we calibraed he andard deviaion " of he raniory price hock " in able. Uing (A38) o ubiue Sd P r C in (A37) and applying he uncondiional expecaion operaor on boh ide of he equaion deliver E E P r C E r b E Sd P Sd [r CC =C ; ] which implie (46). A5. Simulaneou Belief Updaing Thi appendix provide furher informaion abou he exended model wih a hare of imulaneou belief updaer conidered in ecion X.C. Table A6 repor for differen value of he model momen when he elecion rule chooe he cloe marke clearing price. 0 The able how ha mo ae pricing implicaion of he ubjecive belief model are raher robu o allowing for conemporaneou belief updaing. The main quaniaive effec of inroducing curren updaer coni of increaing he volailiy of ock reurn, he volailiy of he P raio and he equiy premium. Thee effec become more pronounced a he hare of curren updaer increae, a hi lead o an increae in he percenage hare of period wih muliple marke clearing price. Imporanly, however, he objecively expeced dicouned uiliy of agen ha ue lagged belief updaing exceed - for all value of - ha of agen who ue curren belief updaing, ee he la wo row in able A6. 03 Curren updaer would hu have an incenive o wich o lagged updaing, i.e., o he eing conidered in our baeline pecificaion. Table A7 repor he model momen when he equilibrium elecion rule chooe inead he marke clearing price ha i furhe away from he previou period price. For 0:7 he ame phenomena occur a for he alernaive elecion rule conidered before, i.e., volailiy and rik premia increae, wih he quaniaive effec being now more pronounced. For 0:8, he hare of period wih muliple equilibria increae ubanially, o ha here are more ofen wo conecuive period wih muliple equilibria. The elecion rule hen creae an ocillaing paern beween high and low marke clearing price. Thi manife ielf in a reduced auocorrelaion for he P raio, which become even negaive for 0:9. A before, curren forecaer (objecively) expeced uiliy alway fall hor of ha experienced by forecaer uing only lagged price growh. For large value of, he uiliy gap widen ignificanly becaue he foreca qualiy of forecaer uing curren price informaion deeriorae ignificanly in he preence of ocillaing price paern. Overall, we find ha - in line wih he poulaed ubjecive belief rucure in our baeline eing - agen will find i opimal o ue only lagged price obervaion o updae belief. Even if ome agen would ue curren price informaion for updaing belief, Sd h i rc Sd P " C, which i ime invarian, a claimed. 0 The parameer in able A6 and A7 are hoe given by he eimaed model from able 3 (diagonal marix). To compue Corr[P ; E [R C ]] we le E [R C ] denoe he average reurn expecaion acro agen, in line wih how he reurn expecaion are compued in he urvey daa. 03 The able repor he uncondiional expecaion of dicouned conumpion uiliy uing he objecive diribuion for conumpion, a realized in equilibrium.

23 VOL. VOL NO. ISSUE STOCK PRICE BOOMS 3 he model coninue o produce high amoun of ock price volailiy and alo end o deliver a poiive correlaion beween he P raio and ubjecive expeced reurn.

24 4 THE AMERICAN ECONOMIC REVIEW MONTH YEAR U.S. aa Share of curren updaer () 946:-0: E[P] Sd[P] Corr[P, P ] Sd[R] c R E[R] E[R b ] UBS Survey aa: Corr[P,E R;C4] Percen of period w muliple equilibria: Expeced uiliy: Lagged updaing Curren updaing Table A6: Ae pricing momen wih imulaneou belief updaing (marke clearing price cloe o previou period price) The able repor U.S. ae pricing momen (econd column) uing he daa ource decribed in Appendix A.A, ee able 3 for a decripion of he label ued in he fir column, and he ae pricing momen for he eimaed model from able 3 (diagonal marix), conidering differen value for he hare of belief updaer uing curren price growh obervaion (column 3 o 3). The able alo repor he percenage hare of period in which muliple equilibria are encounered along he equilibrium (fifh o la line) and he uncondiional expeced uiliy of agen uing curren or lagged price growh obervaion for updaing belief (la wo line). In period wih muliple equilibria, he equilibrium price cloe o he previou marke clearing price i eleced.

25 U.S. aa Share of curren updaer () 946:-0: E[P] Sd[P] Corr[P, P ] Sd[R] c R E[R] E[R b ] UBS Survey aa: Corr[P,E R;C4] Percen of period w muliple equilibria: Expeced uiliy: Lagged updaing Curren updaing VOL. VOL NO. ISSUE STOCK PRICE BOOMS 5 Table A7: Ae pricing momen wih imulaneou belief updaing (marke clearing price furhe away from previou period price) The able repor U.S. ae pricing momen (econd column) uing he daa ource decribed in Appendix A.A, ee able 3 for a decripion of he label ued in he fir column, and he ae pricing momen for he eimaed model from able 3 (diagonal marix), conidering differen value for he hare of belief updaer uing curren price growh obervaion (column 3 o 3). The able alo repor he percenage hare of period in which muliple equilibria are encounered along he equilibrium (fifh o la line) and he uncondiional expeced uiliy of agen uing curren or lagged price growh obervaion for updaing belief (la wo line). In period wih muliple equilibria, he equilibrium price furhe away from he previou marke clearing price i eleced.