Revisiing he Fama and French Valuaion Formula Absrac Using he dividend discoun model Fama and French (2006) develop a relaion beween expeced profiabiliy, expeced invesmen, curren BM and expeced sock reurns. They examine hese predicions in he daa wih limied success. Specifically, he relaionship beween expeced invesmen and reurns ha is supposed o be negaive urns ou o be posiive and insignifican. We argue ha he above relaionships should be esed a he firm level and no a he per share level as done in he original paper. We examine his argumen empirically and find ha once variables are measured a he firm level he predicions of Fama and French's paper hold. 1. Inroducion The relaionships beween profiabiliy, invesmen, BM, and expeced reurns have been sudied exensively in he las wo decades. 1 Fama and French (2006) (henceforh F&F) provide an imporan insigh ino hese relaionships by examining hese reurn regulariies ogeher. Using he dividend discoun model, F&F sugges ha, ceeris paribus, fuure sock reurn should be posiively correlaed wih curren BM and expeced profiabiliy, and negaively correlaed wih expeced invesmen. F&F poin ou anoher ineresing aspec of working wih he valuaion formula. The above predicions should hold, regardless of wheher behavioral or raional financing prevail in he markes. In oher word, he relaionships beween expeced profiabiliy, expeced invesmen, curren BM, and sock reurns should hold independenly of he mechanism used by invesors o price firms. Therefore, he failure of F&F o find he above relaionship in he daa (F&F repor a posiive, insignifican coefficien for expeced invesmens) seems puzzling. 1 For example, Rosenberg e al. (1985) and Fama and French (1992) documen posiive relaionships beween curren book o marke and fuure sock reurns. Timan e al. (2004) and Cooper e al. (2008) documen a negaive relaionship beween curren invesmen and fuure sock reurns; Haugan and Baker (1996) documen ha profiabiliy is posiively correlaed wih fuure reurns, even afer conrolling for book o marke. Chen e al. (2010) sugges a new hree-facor model in which invesmen and profiabiliy are he main explanaory variables.
In his paper we revisi he work of F&F. We argue ha he failure of he F&F empirical work is largely relaed o heir choice of variables. We show ha he relaionship beween he four variables (expeced profiabiliy, expeced invesmen, expeced reurn, and curren BM) in he F&F valuaion formula is bes capured a he firm level, and no a he per-share level as in he F&F work. This is because changes in he number of shares in he firm, eiher by issuing or repurchasing, are likely o miigae he relaionship beween expeced invesmen and expeced reurns. For example, consider a firm ha increases is invesmens and finances hese new invesmens by issuing equiy. 2 When variables are measured a he firm level, he firm will regiser a large increase in invesmen. However, once we scale he invesmen by he oal number of shares (as done in F&F), he increase in invesmen per share is likely o be much less, if any a all. Indeed, our empirical work demonsraes ha he coefficien of expeced invesmen shows he greaes difference beween he firm and he per-share level. As in F&F, we find ha he coefficien of expeced invesmen is small and insignifican a he per-share level. However, a he firm level we find ha he coefficien becomes negaive and significan, as suggesed by he valuaion formula. Consisen wih our argumen, we also find ha he difference beween he wo coefficiens is driven by firms ha regiser an increase in invesmen and an increase in he number of shares ousanding. Valuaion formula a he per-share and he firm levels In his par we highligh he differences beween he F&F valuaion formula in he per-share level and he firm level. We use lower case for variables ha are in he per-share level and upper case for variables in he firm level (see he appendix for a deailed analysis). F&F use he dividend discoun model o derive he relaionship beween expeced profiabiliy, expeced invesmen, curren BM, and fuure reurn. According o he dividend discoun model: 2 Fama and French (2005) and Lyandres a al. (2008) boh show ha firms which increase heir invesmens are likely o be large issuers. 1
p E[ div ] (1) 1 1 r We can represen he equiy value a he firm level. Using clear surplus accouning, he value of he firm can be calculaed as he discouned value of he sum of fuure dividends minus he issuing in each period, so ha: M E ( Y db ), (2) 1 1 r where M is he curren marke value of he firm s equiy, Y is he firm s earnings a ime, db B B is he change in book equiy of he firm, and r is he required rae of equiy 1 reurns. Dividing boh sides by he book equiy: M B [ ] / (1 ) E Y 1 db r (3) B Equaion (3) is used o derive he relaionship a he firm level. As in F&F, he equaion suggess ha, ceeris paribus, here should be a posiive relaionship beween expeced profiabiliy and fuure sock reurns, a negaive relaionship beween expeced invesmens and fuure sock reurns, and a posiive relaionship beween curren BM and fuure sock reurns. Nex we examine hese relaionships in he per-share level. In order o do so we divide boh numeraor and denominaor by he number of shares a ime (n ) and muliply and divide he numeraor by he number of shares in each period (n +τ ), so ha: M B n Y db n [ ] / (1 ) E r E[ y ] / (1 ) 1 1 b r n n n n B n b (4) 2
Here, y Y is he per-share earnings a ime n, and B B 1 b is he per-share n change in book value. As one can see from equaion (4), when here is no share issuance, he equaion simplifies o: M B E [ y ] / (1 ) 1 db r, (5) b B B and db +τ. Equaion (5) leads o he same relaionship beween he variables a he n n per-share level as a he firm level and is similar o he one used by F&F. However, when here is sock issuance hese relaionships are also dependan on he expeced sock issuance as expressed in equaion (4). We noe ha a simple correcion by rying o accoun for E ( n / n ) is no enough; expeced sock issuance is likely o be correlaed wih expeced invesmen and migh be correlaed wih expeced profiabiliy. To illusrae, consider wo firms. Boh firms have he same BM and expeced profiabiliy bu he expeced invesmen of firm A is higher han firm B. Invesors also expec ha he firm will finance he new invesmen by raising equiy. A he firm level, equaion 3 suggess ha firm A should have lower reurns han Firm B does. However, if he researcher uses per-share analysis wihou accouning for sock issuance (equaion 5), i is no clear wheher he expeced invesmen increases or decreases. This is because he numeraor (invesmen) and he denominaor (number of shares) boh increase. As a resul is no clear which firm should have he higher expeced reurn. Fama and French (2008a) argue ha he limied success of he valuaion formula is because BM capures informaion abou boh expeced cash flows and discoun raes. They use wo variables in order o disenangle informaion abou expeced cash flows and expeced discoun raes. These variables are he evoluion of BM and ne issuing. However, hey do no use hese variables o improve he predicions abou expeced invesmen and expeced reurns, bu hey use heir curren 3
values as is done in mos of he asse-pricing lieraure. Novy-Marx (2010) argues ha using a profiabiliy measure higher up in he financial saemen leads o higher predicive abiliy of fuure profiabiliy. He shows ha gross profi is more correlaed wih fuure profiabiliy han is he commonly used ne income and ha gross profi is posiively correlaed wih fuure reurns. However, as wih Fama and French (2008a), Novy-Marx uses curren gross profiabiliy, raher han using i in he conex of he valuaion formula. Our paper differs from hese papers, as i examines he performance of he valuaion formula direcly. The res of he paper in organized as follows. Secion 2 describes he daa, mehodology, and summary saisics. Secion 3 presens our empirical resuls, and Secion 4 concludes. 2. Daa, Mehodology and Summary Saisics Our daa are drawn from he crsp and compusa annual daabases. Our sample period is 1963 2009. Following F&F, we exclude firms from he sample wih he following characerisics: Financial firms (SIC code 6000 6999); Firms ha do no have all he accouning and marke informaion necessary o calculae all he variables used in order o predic invesmen and profiabiliy; All firms wih oal asses less han 25 million dollar or book equiy less han 12.5 million dollar Our final sample includes 85,680 firm years. To reduce he effec of exreme observaions, we winsorize all independen variables in he 99.5% and 0.5% level. Our mehodology is similar o F&F. We firs esimae he relaionship beween various accouning and marke variables and fuure invesmen and profiabiliy. We refer o his par as firs pass regression. Using he esimaions from he firs pass regression, we calculae for each firm he expeced profiabiliy and invesmen. Finally, we examine he relaionship beween expeced 4
invesmen and profiabiliy and fuure sock reurns, while conrolling for size and book o marke. We refer o his sage as second pass regression. We use his procedure boh a he firm level and a he per-share level. In some regressions, we replace he measure of invesmen from growh in oal asses o growh in book equiy. We define growh in book equiy as he percenage change in book equiy beween year and year -1. In Table 1 we presen he main variables ha are used in he firs pass regressions. For each variable we presen he iner-quarile break poin and he spread beween he values of he firms in he 90 h percen, o he value of he firm in he 10 h percen. We laer use his spread o demonsrae he imporance of various variables o he expeced invesmen and profiabiliy calculaions. In he firs wo rows we presen he wo main variables in our research: profiabiliy and invesmen. Profiabiliy is measured as income before exraordinary iems, scaled by oal asses. Invesmen is defined as he percenage change in oal asses beween he curren and previous years. TA inv TA 1 1 In Row 3 we presen invesmen per share, which is defined as invesmen divided by he number of shares ousanding in he same period. Resuls show ha invesmen per share has a much smaller spread han invesmen iself does (0.38 compared wih 0.59). Fama and French (2005) documen ha firms ha issue equiy end o increase heir invesmen before or afer he issuance. This suggess ha firms wih high asse growh are likely also o be large issuers leading o smaller spreads in invesmen per share. In unrepored resuls, we confirm his in our sample by documening ha high invesing firms also end o have high ne issuance. The res of Table 1 presens he oher variables used in he firs pass regression, excluding binary variables. The nex wo variables (P_acc and N_acc) measure he accruals of he firm. Posiive (negaive) accruals are accruals scaled by book equiy for all firms wih posiive (negaive) accruals 5
and zero oherwise. Div_be is oal dividends scaled by book equiy, whereas ne issuing (ns) is he change in he number of shares ousanding beween he curren and previous years. 6
3. Empirical Resuls 3.1. Valuaion formula in he firm level F&F use wo-sage regression o examine he predicions of he valuaion formula (equaion 3). Firs, hey esimae a regression ha examines he relaionships beween various accouning and marke variables and fuure invesmen and profiabiliy. They hen use he coefficiens from his regression o esimae he expeced invesmen and profiabiliy for each firm-year. In he second pass regression, hey use he expeced invesmen and profiabiliy, along wih he curren BM of he firm, and regress hem ogeher as predicors of fuure reurn. We use he same procedure in our paper, excep for one major difference. Our analyical derivaions sugges ha he relaionships beween expeced profiabiliy, expeced invesmen, BM, and reurn are bes expressed a he firm level. Therefore, we use all variables a he firm level and no a he per-share level as underlies he work of F&F. Resuls of he firs pass regressions are presened in he upper par of Table 2. For comparison reasons, our firs specificaion (Model 1) uses he same nine variables as F&F use. These variables include seven of he variables ha are described in Table 1 (excluding invesmen per share and ne issuing). Addiionally, we use wo binary variables: N_prof o which we assign he value of 1 if he firm regisers negaive earnings a ime ; and No_div o which we assign he value of 1 if he firm did no pay dividends in ha year. We use hese nine variables as independen variables in wo regressions. In he firs regression, he dependan variable is he invesmen (measured as change in oal asses) in he nex year. In he second regression he dependan variable is profiabiliy (measured as income before exraordinary iems, scaled by book value) in he nex year. Resuls for boh regressions are similar o hose repored by F&F. There is a srong negaive relaionship beween BM and expeced invesmen, and a posiive relaionship wih profiabiliy. These findings are consisen wih he F&F findings and he noion of low-bm firms as growh and glamour socks (e.g. Fama and French 1995). The effec of BM on fuure invesmen is he greaes 7
of all he variables used. For example, a firm in he 90 h percenile of BM is expeced o have 0.275 higher invesmen han a firm in he 10 h percenile of BM. The coefficien of BM on profiabiliy is less han half han of ha on invesmen, suggesing ha BM will have a much lower influence on expeced profiabiliy. Size is also negaively correlaed wih fuure invesmen and is no correlaed wih fuure profiabiliy. The effec of size on invesmen is much less han BM, as he spread is roughly 0.05. This small effec of size is in par because of he censoring of all iny firms. The coefficien of he binary variable N_prof is negaive in boh regressions, suggesing ha unprofiable firms have lower fuure invesmen and lower fuure profiabiliy. The coefficien of curren profiabiliy (ibbe) is posiive in boh regressions, suggesing ha curren profiabiliy is posiively correlaed o boh fuure invesmen and fuure profiabiliy. The effec of curren profiabiliy on fuure profiabiliy is he larges of all variables. A firm in he 90 h profiabiliy percenile is expeced o have a fuure profiabiliy ha is 0.22 higher han for a firm in he 10 h profiabiliy percenile. The effec of expeced invesmen is much smaller, and he spread is only 0.05. The accruals variables (P_acc and N_acc) are significan boh in he invesmen and profiabiliy regressions. However, heir economic significance is very sligh. For example, he spread in expeced invesmen beween a firm in he 90 h percenile and 10 h percenile of posiive accruals is only 0.02. Our resuls sugges a negaive relaionship beween dividends and expeced invesmen, and a posiive relaionship wih expeced profiabiliy. The binary variable No_div has lile effec on eiher expeced invesmen or expeced profiabiliy. Our resuls sugges a negaive relaionship beween curren dividends and fuure invesmen, and a posiive relaionship beween curren dividends and fuure profiabiliy. The variable oal dividends scaled by book equiy (Div_be) has a relaively large effec on expeced invesmen (spread of 0.06), and a very small effec on expeced profiabiliy (spread of 0.01).. Finally, curren invesmen is posiively correlaed wih fuure invesmen and is negaively correlaed wih fuure profiabiliy. The coefficien of curren 8
invesmen is more han double he one repored by F&F. The economic significance of curren invesmen is relaively low, and he difference in expeced invesmen beween he 90 h and he 10 h percenile firms is roughly 0.05 or less han one fifh of he effec of BM. In he second sage regression, we use he coefficien obained from he firs pass regressions o calculae he expeced invesmen and expeced profiabiliy. Then we esimae he following regression for each firm year in our sample. r ln _ BM Ln _Size Exp _ Inv Exp _ Prof ε i, 1 i, 1 1 Resuls of his esimaion are presened in he lower par of Table 2. Resuls show ha he coefficiens of he second pass regression are all consisen wih heir predicive signs. The coefficien of BM and expeced profiabiliy is posiive and significan, as prediced by he valuaion formula. Mos imporanly, he coefficien of expeced invesmen is negaive and significan ( 1.57; = 2.17). This resul is in sharp conras o F&F who repor a posiive and insignifican coefficien for expeced invesmen. Our sample period is five years longer han he one used by F&F. These five years (2005 09) include he global financial crises. Thus i may be ha he difference in resuls beween our and F&F esimaes are no relaed o he difference beween per-share and firm-level analysis bu o he end years in our sample. To examine his possibiliy, we censor from our sample he las five years and re-esimae boh firs and second pass regressions. Resuls for he esimaion (Model 2) show ha here are hardly any differences in he coefficiens of he firs pass regressions. The larges difference is in he coefficien of negaive accruals (N_acc) in he expeced profiabiliy regression. I changes from 0.10 o 0.19. However, given he small influence of accruals on boh expeced invesmen and profiabiliy, his difference has lile economic implicaion. Resuls of he second pass regression confirm ha he difference in he sample periods is no driving he difference in 9
resuls beween he F&F work and ours. All he coefficiens in Model 2 are similar o hose of Model 1. Specifically, he coefficien of expeced invesmen remains negaive and significan. In Model 3 we add ne issuing (ns) o he variables used o predic invesmen and profiabiliy. Previous research shows ha firms ha issue equiy end o increase heir invesmen in he subsequen period (e.g. Fama and French (2005) Lyandres e al. (2008)). Loughran and Rier (1997) documen ha firms ha issue equiy suffer from a decrease in profiabiliy afer he issuance. A vas body of lieraure shows ha sock issuance is followed by subsequen low reurns. 3 Fama and French (2008a) argue ha he failure of heir valuaion formula is due o he fac ha book o marke is unable o separae an increase in cash flows and a decrease in he discoun rae. They argue ha ne issuing can serve as a variable ha helps o separae he wo effecs. Their resuls show ha ne issuing is negaively correlaed wih sock reurns, even afer conrolling for book o marke. We add o his lieraure by examining he effec of including ne issuing as a predicor in he valuaion formula. Resuls of Model 3 show ha, as expeced, ne issuing and fuure invesmen are posiively correlaed. The coefficien of ne issuing is posiive and highly significan. The economic effec of ns is relaively low (a spread of 0.027 in expeced invesed). Ineresingly, he coefficien of invesmen is maerially decreasing once ne issuing is added as a predicor (from 0.09 o 0.06). The oher coefficiens are barely affeced by he inclusion of ne issuing. The coefficien of ne issuing in he expeced profiabiliy equaion is only slighly negaive ( 0.01) and saisically insignifican. Resuls for he second pass regression show ha he coefficien of expeced invesmen becomes more negaive compared wih Model 1 (from 1.53 in Model 1 o 1.94) and i is now significan a he 1% level. The increase in he predicive power of expeced invesmen is accompanied by a decrease in he predicive power of BM. The coefficien of BM falls from 0.22 in 3 See for example Loughran and Rier (1995); Brav e al (2000); Lyandres e al. (2008); Poniff and Woodgae (2008); Fama and French (2008b). 10
Model 1 o 0.16 in Model 3 and i is only significan a he 10% level. This may sugges ha he predicive powers of BM and expeced invesmen are driven by similar economic forces. Thus, an improvemen of one is likely o be a he expense of he oher. Finally, we noe ha he valuaion formula suggess ha here should be a relaionship beween change in book equiy and fuure sock reurns. However, in heir empirical work, F&F use he change in oal asses raher han he change in book equiy. F&F argue ha he use of asse growh raher han change in book equiy provides a beer predicor of oal invesmen, as i is less noisy. While we agree wih his saemen, we sill believe i is of ineres o examine he valuaion formula in using he change in book equiy, raher han he change in oal asses, as his reflecs he closes relaionship beween equaion (3) and he empirical procedures. Accordingly, in Model 4 we replace he change in oal asses wih he change in book equiy in boh he firs pass and second pass regressions. The resuls of he expeced invesmen regression sugges ha here are some major differences compared wih Model 3. Firs, he coefficien of profiabiliy is almos riple he 0.15 in Model 3 being 0.42 in Model 4. In conras, he coefficien of invesmen is reduced by wo-hirds from 0.06 o 0.02. The coefficien of he binary variable No_div changes sign from negaive in Model 3 o posiive in Model 4. Similarly, boh coefficiens of accruals also change signs beween Model 3 and Model 4. Oher coefficiens remain largely he same. In conras o hese large changes in he expeced invesmen regression, he coefficien hardly changes in he expeced profiabiliy regression. The large changes in he firs pass regression lead also o changes in he second pass regression. The coefficien of expeced invesmen furher increases o 2.23. However, consisen wih he F&F argumen, here is an increase in he sandard deviaion. As a resul, he -saisic falls slighly from 2.79 o 2.52. Ineresingly, he increase in he coefficien of expeced invesmen is again a he expense of book o marke. The coefficien of BM is reduced by almos a half and i is saisically insignifican. This resul suggess ha BM may be relaed o fuure sock reurns, 11
because i is correlaed wih expeced changes in book equiy ha in urn are associaed wih fuure low reurns. 3.2 Sample-specific? Our resuls so far indicae ha he valuaion formulae of F&F are rue in he daa. While we aribue he success of he valuaion formula o he examinaion a he firm level, raher han in he pershare level, anoher possibiliy is ha our sample differs from he one used by F&F. To explore his possibiliy we examine he resuls in he per-share level by examining he predicion equaion (5). We use he wo pass regression as in previous ess and repor he resul in Table 3. Model 1 examines he per-share analysis for he enire sample period (1965 2009). Resuls of he firs pass regression show ha he coefficiens are largely of he same sign as in Table 2. However, in he expeced invesmen regression here are some imporan differences in he values of he coefficiens. The coefficien of invesmen is roughly halved when compared wih he Table 2 resuls (0.04 o 0.09 respecively). The coefficien of profiabiliy is roughly 50 per cen larger compared wih he firm-level analysis. The coefficien of BM is slighly smaller in absolue erms han he coefficien of BM in Table 2 is ( 0.10 compared wih 0.13). The res of he coefficiens are similar o hose of Table 2. In conras, he coefficiens in he expeced profiabiliy regression are almos idenical in Table 2 and Table 3. In he lower par of Table 3 we presen he resuls for he second pass regressions. The resuls are considerably differen o hose of Table 2 and are much closer o he ones repored by F&F (see column 3, Table 3). Alhough he coefficien of expeced invesmen is negaive, i is a quarer of is value in Table 2 ( 0.39 compared wih 1.57) and saisically insignifican (= 0.48). In conras, he coefficien of BM is almos double he one repored in Table 2 (0.39 compared wih 0.21). The coefficien of expeced profiabiliy is almos idenical suggesing again ha profiabiliy is no affeced, wheher he variables are measured a he per-share level or he firm level. 12
In Model 2 we esimae he same regression afer censoring he las five years of our sample period. Our resuls show ha his leads o only small differences in he coefficiens of he firs pass regression. This model is of ineres, as i allows us o have direc comparison wih he F&F resuls. In he expeced invesmen regression, we noe ha all coefficiens have he same sign as in he F&F work. In eigh of he nine variables he magniude of he coefficien is similar. Only he coefficien of size is much larger in he F&F work han in ours ( 0.53 compared wih 0.01). 4 The second pass regression shows ha he coefficien of expeced profiabiliy is higher han in Model 1. All of he oher variables are of he same magniude as in Model 1. Specifically, he coefficien of expeced invesmen coninues o be small and insignifican. Again, comparing hese resuls o F&F, we find very similar resuls. The larges difference beween heir resuls and ours is in he expeced invesmen variables ( 0.43 compared wih 0.04 in F&F). However, his difference is relaively small and insignifican. The similariy beween our resuls and hose of F&F suggess ha our main resul he validaion of he valuaion formula in Table 2 is because of he differences in firm and per-share level analysis and is no due o differences in sample characerisics beween our work and heirs. To summarize, our resuls sugges ha he predicion of he valuaion formula does no hold when variables are measured in he per-share level. This is consisen wih boh our analysis and he F&F empirical resuls. 3.3 Valuaion formula across BM quiniles In he las par of our empirical work we examine he robusness of our resuls. The valuaion formula suggess ha here should be a relaionship beween expeced invesmen, expeced profiabiliy, and expeced reurn afer conrolling for curren BM. In Table 2 we esimae a linear 4 We noe ha he spread in he naural log of size beween firms in he 90 h and 10 h perceniles is 4.72. The coefficien of F&F of 0.53 suggess ha small firms will have an invesmen of close o 250 per cen higher han a large firm. This number is oo high, as he spread beween firms in 90 h percenile of expeced invesmen and he 10 h percenile of expeced invesmen is only 0.59. Furhermore, he -saisics of F&F and ours are of he same magniude ( 5 and 3). Since we use he naural log of size in he empirical research i canno be a simple magniude effec. 13
approximaion of hese relaionships by enering in he righ hand side expeced invesmen, profiabiliy, and curren BM. In he nex es we conrol for BM by dividing he sample ino quiniles of BM and hen esimaing in each BM quinile he following regression. r Ln(Size ) E(Inv ) E(Prof ) ε i, 1 i, 1 1 This es has wo main objecives. Firs, he es serves as a robusness es for our resuls in Table 2 in order o examine wheher expeced profiabiliy and, especially, expeced invesmen have predicive power afer conrolling for boh size and BM. Second, he es provides an insigh regarding he relaionship beween expeced invesmen and reurns. Specifically, he es can demonsrae wheher he negaive relaionship beween expeced invesmen and reurns is driven by a few firms wih very large expeced invesmen, or wheher he relaionship is more cenral. Resuls of he firs pass regression show ha BM is he mos significan variable in deermining expeced invesmen, suggesing a negaive correlaion beween he wo variables. Thus, if expeced invesmen will show up only among low-bm porfolios, hen i is likely ha only exreme values of expeced invesmen are followed by low realized reurns. In conras, if he negaive correlaion exiss for all BM quiniles, hen he relaionship is much more cenral. Resuls in Table 4 Column 1 confirm he above observaions by showing ha average expeced invesmen in he lowes BM quinile is more han wice as large as he average invesmen in he median quinile is. In he highes BM quinile, boh he average and he median expeced invesmen are negaive. Addiionally, we can repor ha less han 0.2 per cen of he firms in he high-bm porfolio have higher expeced invesmen han he average expeced invesmen in he low-bm porfolio. The nex column presens he average and median expeced profiabiliy across BM quiniles. There is a negaive monoonic relaionship beween BM and expeced invesmen. However, his spread in expeced profiabiliy is much lower han expeced invesmen. 14
The nex four columns presen he resuls of equaion (6) for each BM quinile. Resuls show ha he coefficien of expeced invesmen is negaive in four of he quiniles and is significan in hree of hem. The coefficien of expeced profiabiliy is posiive in all quiniles and is significan in wo of hem. Imporanly, we find ha he coefficien of expeced invesmen is negaive and significan among high-bm firms. This suggess ha he success of he expeced invesmen is no driven by a few firms wih high-value expeced invesmen bu exiss among low and high-bm firms. 4. Conclusions The Fama and French valuaion formula suggess a basic relaionship beween four variables: curren BM, expeced profiabiliy, and expeced invesmen, and fuure sock reurns. However, he F&F empirical work had limied success, paricularly in failing o find a relaionship beween expeced invesmen and sock reurns. Our analyical derivaions imply ha he relaionships suggesed in he F&F valuaion formula do no necessarily hold in he per-share analysis. This is because issuing of new equiy changes he number of socks. The change in he number of socks can lead o miigaion of he relaionship suggesed in he valuaion formula. In conras, variables ha are measured a he firm level are no affeced by he number of socks, and hence he relaionship suggesed in F&F should hold. Our empirical work suppors he above argumen. We find ha he empirical predicions of he valuaion formula are rue in he daa, once variables are measured a he firm level. As prediced by he valuaion formula, we find a posiive relaionship beween BM and reurns, a posiive relaionship beween expeced profiabiliy and reurns, and a negaive relaionship beween expeced invesmen and reurns. Consisen wih he F&F empirical findings, our resuls show ha once variables are measured a he per-share level here is no significan relaionship beween expeced invesmen and reurns. We furher sudy he robusness of hese resuls. Our findings sugges ha hese resuls are no driven by he global financial crises. Furhermore, our resuls show ha he predicions of he 15
valuaion formula hold in mos BM quiniles. This ensures ha he success of he valuaion formula is no relaed o a few small socks wih exreme values of expeced invesmen. Moreover, he fac ha iny socks are censored from he sample furher emphasizes he cenraliy of our resuls. One of he ineresing feaures of he valuaion formula is he predicion ha he formula should hold regardless of he mechanism ha researchers use o price firms. This is boh a weakness and srengh of he valuaion formula. On he downside, working in he framework of he valuaion formula canno explain he cross-secional differences in sock reurns. Indeed, as F&F noed, he valuaion formula canno even disinguish wheher behavioral or risk-based explanaions lead o he differences in he cross-secional reurns. However, he upside is ha he basic naure of he predicions in he valuaion formula can provide researchers wih a guide o wha reurns should be, wihou going ino he ongoing debae abou invesors raionaliy. 16
References Anderson, Chrisopher W and Luis Garcia-Feigoo, 2006, Empirical evidence on capial invesmen, growh opions, and securiy reurns, Journal of Finance 61, 171-94 Brav, Alon, Geczy, Chrisopher, and Paul Gompers (2000), Is he Abnormal Reurn Following Equiy Issuances Anomalous? Journal of Financial Economics 56, 209-249 Chen, Long, Rober Novy-Marx, and Lu Zhang, 2010, An alernaive hree-facor model, Working Paper series Cooper Michael J, Gulen Huseyin and Michael J Schill, 2008, Asse growh and he cross-secion of sock reurns, Journal of Finance 63, 1609-51 Fama, Eugene F and James D MacBeh, 1973, Risk, reurn, and equilibrium: empirical ess, Journal of Poliical Economy 81, pp. 607-36 Fama, Eugene F and Kenneh R French, 1992, The cross-secion of expeced sock reurns, Journal of Finance 47, 427-65 Fama, Eugene F and Kenneh R French, 1995, Size and book-o-marke facors in earnings and reurns, Journal of Finance 50, 131-56 Fama, Eugene F and Kenneh R French, 2005, Financial decision: Who issues socks? Journal of Financial Economics 76, 549-82 Fama, Eugene F. and Kenneh R. French, 2006, Profiabiliy, invesmen, and average reurns, Journal of Financial Economics 82, 491-518 Fama, Eugene F and Kenneh R French, 2008a, Average Reurns, B/M, and Share Issues, The Journal of Finance 63, 2971-95 Fama, Eugene F and Kenneh R French, 2008b, Dissecing anomalies, The Journal of Finance 63, 1653-78 Haugen, R A and N L Baker, 1996. Commonaliy in he deerminans of expeced sock reurns. Journal of Financial Economics 41, 401-39 Loughran Tim and Jay Rier, 1995, The new issue puzzle, Journal of Finance 50, 23-42 Loughran Tim and Jay Rier, 1997, The operaing performance of firms conducing seasoned equiy offerings, Journal of Finance 52, 1823-50 Lyandres, Evgeny, Le Sun, and Lu Zhang, 2008, The new issues puzzle: Tesing he invesmenbased explanaion, Review of Financial Sudies 21, 2825 55. Novy-Marx Rober, 2010, The oher side of value, Working Paper series Poniff Jeffrey and Aremiza Woodgae, 2008, Share issuance and cross-secional reurns, Journal of Finance 63, 921-44 Rosenberg, Barr, Kenneh Reid and Ronald Lansein, 1985, Persuasive evidence of marke inefficiency, Journal of Porfolio Managemen 11, 9 17 Timan, S, K C J Wei, and F Xie, 2004, Capial invesmens and sock reurns. Journal of Financial and Quaniaive Analysis 39, 677 700 17
Appendix: Derivaion of Valuaion Formula Accouning for Share Issuance In his appendix we derive he rivial bu suble valuaion formula wih he sock issuance. Noaion: M = value of firm's equiy a period ex any dividend a he end of period -1. p = ex - dividend price per share a he end of period. n = number of shares of record a he period. M = n p. d = dividend per share paid a he end of period o holders of record a he sar of period. D = oal dividend paid a he end of period -1/ begining of period o he holders of record a he sar of period 1 n d. 1 There are wo ways o price he sock value oday: a he firm level and a he per share level. We can value each share as he presen value of he fuure dividends payable o an invesor who buys and holds one share (for simpliciy of noaion, in he appendix all he fuure variables are expeced values). p 1 d 1 r, where r = he required reurn on equiy. M n p 1 nd 1 r. Noe ha he oal equiy value is only equal o he presen value of all fuure dividends paid by he firm if he firm is never o issue or repurchase any shares. A he firm level, noe ha if here is any equiy issuance over ime hen D n d. To link he curren oal equiy value o he expeced oal dividend, we need o consider he effec of ne issuance of equiy NE : 18
M 1 D NE 1 r. One can look a his as follows. Consider he shareholders as one group, hen he value of he oal equiy oday is equal o he presen value of he sum of expeced fuure cash inflow (oal dividend paid) and cash ouflow (oal equiy raised). To link he dividend o he book value of equiy, and o earnings when here is equiy issuance, noe ha: B book value of equiy a he sar of period B Y D NE B Y D m p 1 1. I follows ha a he firm level, M 1 D NE Y B B Y B 1 r 1 r 1 r 1 1 1. To conver his o per-share iems, we can divide he oal number of shares oday on boh sides: p =1 n n y 1 r b, where y Y is he period n equiy earnings per beginning of period' share ousanding, B B 1 and b is period change in book equiy per beginning of period share n ousanding. Noe ha when here is a change in number of shares ousanding, B b n B n 1 1 One can see ha a he firm level, he relaionship beween curren marke value and expeced fuure earnings, change in book value, and required rae of reurn are similar o hose in F&F. Thus 19
all he predicions among he four variables will hold. Ye a he per-share level, one mus accoun for he change in he fuure shares. Furhermore, i is he sraigh forward expeced change in he number of shares ousanding, because he expecaion is aken wih respec o he whole of he erms in he sum. 20
Table 1: Disribuion of he Main Variables In his able we presen he disribuion of he main variables in our research. For each variable we presen he quadruple break poins and he spread beween he sock ha is ranked in he 90 h percenile and he socks ha is ranked in he 10 h percenile. The variables are: ibbe income before exraordinary iems scaled by oal asse value Inv defined as he percenage change in oal asse beween he curren and previous year Inv_per invesmen a ime divided by he number of shares a he same period log_bm he naural log of he book value of he previous fiscal year divided by he marke value of he same period Log_me he naural log of marke value a he end of he previous fiscal year N_prof a dummy variable o which we assign he value of 1 if he firm repors a negaive income before exraordinary variables P_acc posiive accruals scaled by oal asses N_acc negaive accruals scaled by oal asses Div_be is dividend divided by oal asses ns ne issuing of he firm defined as he number of socks 25% Median 75% Spread 90 10% Ibbe 0.031 0.096 0.149 0.348 Inv 0.006 0.091 0.219 0.590 Inv_per 0.010 0.746 0.183 0.385 Log_bm 0.971-0.399 0.129 2.117 Log_mef 3.983 5.181 6.554 4.720 P_acc 0.000 0.019 0.088 0.192 N_acc 0.032 0.000 0.000 0.116 Div_be 0.000 0.006 0.038 0.063 ns 0.000 0.006 0.032 0.183 21
Table 2: Firs and Second Pass Regressions: Firm level This able presens he resuls of he firs and second pass regressions in order o examine wheher he prediced relaionships in he valuaion formula are rue in he daa. The firs pass regressions examine he relaionship beween invesmen and profiabiliy in he nex year (+1) o various accouning and marke variables in he curren period (year ). We esimae he Fama-MacBeh regression for he 40 years in our sample. The independen variables ha are used are: log_bm, Log_mef, ibbe, P_acc, N_acc, Div_be, Inv, ns (see he definiions in Table 1). Addiionally we use wo binary variables: N_prof o which we assign he value of 1 if he firm repors negaive earnings in he curren fiscal year, and No_Div o which we assign he value of 1 if he firm didn pay dividends in he curren fiscal year. Thus we esimae in Model 1 he following wo regressions: INV Ln _ BM Ln N ibbe P N No Div inv 1 mef Prof acc acc Div be Prof L n( BM ) Ln( mef ) NProf ibbe P N NoDiv Div / be inv 1 acc, acc, Model 1 examines hese variables for he enire sample period; Model 2 examines he same regression equaions bu wihou he las five years in he sample. In Model 3 we add ne issuing (ns) as a predicor. In Model 4 we replace invesmen from change in oal asses o change in book equiy. *; ** denoe ha he coefficien is significan a he 5% and 1% levels respecively. In he second pass regression we esimae he following regression: r ln _ BM Ln _Size Exp _ Inv Exp _ Prof ε i, 1 i, 1 1 Sock reurns are from July of year +1 o June of year +2. Size is he marke value a he end of June of year +1 and book o marke is he book value a he end of he previous fiscal year divided by he marke value of December of year. Exp_Inv and Exp_Prof are calculaed using he coefficiens obained from he firs pass regressions. The numbers in brackes are he -saisics. 22
Firs Pass Regression Model 1 Model 2 Model 3 Model 4 Ex_Inv Ex_prof Ex_Inv Ex_prof Ex_Inv Ex_prof Ex_Inv Ex_prof Ln BM 0.13 ** 0.05 ** 0.13 ** 0.05 ** 0.13 ** 0.05 ** -0.15 ** 0.05 ** Ln mef 0.01 ** 0.00 0.01 ** -0.00 0.01 ** 0.00-0.01 ** 0.00 N_Prof 0.07 ** 0.03 ** 0.07 ** 0.02 * 0.07 ** 0.03 ** -0.05 ** 0.03 ** Ib_be 0.13 ** 0.64 ** 0.14 ** 0.66 ** 0.15 ** 0.64 ** 0.42 ** 0.63 ** P_acc 0.10 ** 0.03 * 0.10 ** 0.05 ** 0.10 ** 0.03 * 0.03 0.04 * N_acc 0.07 ** 0.10 ** 0.05 * 0.19 ** 0.06 ** -0.10 ** -0.09 ** 0.11 ** No_Div 0.00 0.02 ** 0.01 * 0.02 ** 0.01 * 0.02 ** 0.02 ** 0.02 ** Div_be 0.99 ** 0.20 ** 1.05 ** 0.18 ** 1.01 ** 0.20 ** -1.11 ** 0.20 ** Inv 0.09 ** 0.03 ** 0.09 ** 0.02 ** 0.06 ** 0.03 ** 0.02 * 0.01 Ns 0.15 ** 0.01 0.15 ** 0.03 * R 2 0.193 0.453 0.202 0.449 0.211 0.454 0.213 0.453 Second Pass Regression Model 1 Model 2 Model 3 Model 4 Inercep 1.67 (4.25) 1.72 (4.13) 1.73 (4.41) 1.71 (4.71) Log_BM 0.21 (2.29) 0.22 (2.17) 0.16 (1.80) 0.09 (1.11) Log_Size 0.07 ( 1.77) 0.07 ( 1.60) 0.08 ( 1.93) 0.09 ( 2.44) Exp_Prof 1.47 1.65 1.48 2.25 (2.36) Exp_Inv 1.57 ( 2.17) (2.37) 1.64 ( 2.09) (2.39) 1.93 ( 2.79) (2.52) 2.23 ( 2.52) 23
Table 3: Per Share Analysis In his able we esimae firs and second pass regressions in he per-share level as is done in he F&F work. In he firs pass regression we esimae he same variables as in Table 2. In Model 1 we use he enire sample period (1963 2009), whereas in Model 2 we censored he las five years from he sample. In he hird column we presen he original resuls repored in he F&F research resuls. Finally in Model 4 we use he firs pass coefficiens obained by F&F and esimae he second pass regression on our sample. Firs Pass Regression Model 1 Model 2 F&F Ex_Inv Ex_prof Ex_Inv Ex_prof Ex_Inv Ex_prof Ln BM 0.10 ** 0.05 ** 0.10 ** 0.05 ** 0.12 ** 0.04 ** Ln me 0.01 ** 0.00 0.01 ** 0.00 0.53 ** 0.00 N_Prof 0.08 ** 0.03 ** 0.07 ** 0.03 * 0.09 ** 0.07 ** Ib_be 0.17 ** 0.60 ** 0.18 ** 0.62 ** 0.19 ** 0.78 ** P_acc 0.12 ** 0.04 * 0.12 ** 0.06 ** 0.09 ** 0.07 * N_acc 0.06 ** 0.10 ** 0.05 * 0.09 ** 0.03 0.03 ** No_Div 0.01 ** 0.02 ** 0.02 ** 0.02 ** 0.01 * 0.03 ** Div_be 0.94 ** 0.20 ** 1.00 ** 0.20 ** 1.13 ** 0.02 Inv 0.04 ** 0.03 ** 0.05 ** 0.03 ** 0.05 ** 0.00 R 2 0.188 0.457 0.189 0.453 0.12 0.39 Second Pass Regression Model 1 Model 2 F&F Inercep 1.87 (2.19) 1.94 (2.07) 1.61 (2.06) Log_BM 0.39 (4.21) 0.41 (4.05) 0.37 (3.42) Log_Size 0.05 ( 1.24) 0.05 ( 1.09) 0.08 ( 1.92) Exp_Prof 1.48 1.70 1.58 (2.05) Exp_Inv 0.39 ( 0.48) (2.07) 0.43 ( 0.47) (2.03) 0.04 (0.05) 24
Table 4: The Valuaion Formula across BM Porfolios In his able we examine he performance of he valuaion formula across BM quiniles. For each BM quinile we esimae he following regression: r Ln _Size Exp _ Inv Exp _ Prof ε i, 1 i, 1 1 We use he coefficiens obained from Table 2, Model 1 o calculae expeced invesmen and expeced profiabiliy. The firs column is he BM quinile; Column 2 and Column 3 presen he average and median (in brackes) of expeced invesmen and expeced profiabiliy in each of he BM quiniles. Columns 4 o 7 presen he inercep and he coefficiens of size, expeced invesmen, and expeced profiabiliy for he above regressions. Numbers in brackes are he -saisics. Characerisics Regression esimaion qbm Exp_Inv Exp_Prof Inercep Ln_size Exp_Inv Exp_prof 1 0.27 (0.26) 0.13 (0.18) 1.15 (2.248) 0.01 ( 0.282) 2.43 ( 3.743) 1.87 (3.342) 2 0.18 (0.17) 0.09 (0.13) 1.58 (3.435) 0.06 ( 1.471) 1.98 ( 2.156) 1.58 (3.436) 3 0.12 (0.12) 0.07 (0.10) 1.51 (3.584) 0.05 ( 1.198) 0.83 ( 0.734) 1.10 (1.214) 4 0.07 (0.07) 0.03 (0.07) 1.58 (4.04) 0.06 ( 1.417) 0.410 (0.329) 0.76 (0.671) 5 0.01 ( 0.00) 0.03 (0.01) 2.20 (5.824) 0.15 ( 3.237) 2.23 ( 2.296) 0.83 (0.839) 25