cost of equity and long-term growth Alexander Nekrasov University of California, Irvine

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1 Usng earnngs forecasts to smultaneously estmate frm-specfc cost of equty and long-term growth by Alexander Nekrasov Unversty of Calforna, Irvne Mara Ogneva Stanford Unversty Ths verson: October 8, 00 Keywords: Cost of equty, expected return, expected earnngs growth rate, resdual ncome model JEL Classfcaton:, 4, 7, 3, M4 We are thankful to James Ohlson (the edtor), an anonymous revewer, Pervn Shroff, and K.. Subramanyam for ther nsghtful comments and suggestons. The paper also beneftted from the comments receved from the semnar partcpants at UC Irvne, Santa Clara Unversty, and Southern Methodst Unversty. Mara Ogneva acknowledges fnancal support from the Mchelle. Clayman Faculty Scholar endowment. We are grateful to Suhas Srdharan for her excellent research assstance.

2 Usng earnngs forecasts to smultaneously estmate frm-specfc cost of equty and long-term growth Abstract A growng body of lterature n accountng and fnance reles on mpled cost of equty (COE) measures. Such measures are senstve to assumptons about termnal earnngs growth rates. In ths paper we develop a new COE measure that s more accurate than exstng measures because t ncorporates endogenously estmated long-term growth n earnngs. Our method extends Easton, Taylor, Shroff, and Souganns (00) method of smultaneously estmatng sample average COE and growth. Our method delvers COE (growth) estmates that are sgnfcantly postvely assocated wth future realzed stock returns (future realzed earnngs growth). The predctve ablty of our COE measure subsumes that of other commonly used COE measures and s ncremental to known rsk determnants.

3 . Introducton In ths study, we propose a new frm-specfc measure of mpled cost of equty captal (COE) that s more accurate than exstng measures because t ncorporates endogenously estmated long-term growth n earnngs. Impled COE measures are nternal rates of return that equate a frm s current stock prce to the sum of dscounted future payoffs. Payoffs beyond the short-term horzon are assumed to grow at a certan constant long-term growth rate, whch makes growth an mportant nput n COE estmaton. Any error n the growth estmate feeds drectly nto the mpled COE. In partcular, the more postve (negatve) s the error n the long-term growth rate, the more upwardly (downwardly) based s the mpled COE. Extant mpled COE measures assume the same long-term growth rate across all frms (Claus and Thomas 00; ode and Mohanram 003). 3 Ths assumpton s unlkely to hold n practce, however, because a number of factors nfluence a frm s termnal growth rate, such as the frm s degree of accountng conservatsm and expected growth n nvestment (Feltham and Ohlson 995; Zhang 000). Exstng measures of mpled COE therefore systematcally over- or understate growth, whch can lead to spurous nferences Ths growth rate s often referred to as the termnal growth rate or the growth rate n perpetuty. Throughout the paper we use the terms long-term growth, termnal growth, and growth n perpetuty nterchangeably. Valuaton textbooks emphasze that frm valuaton can be hghly senstve to the assumed termnal growth rate of earnngs (Penman 009; Whalen et al. 00). For example, Damodaran (00) states that of all the nputs nto a dscounted cash flow valuaton model, none can affect the value more than the stable growth rate. 3 Another commonly used COE measure developed by ebhardt et al. (00) assumes a convergence n proftablty to an ndustry benchmark over twelve years wth constant zero growth thereafter. But as Easton (006) pont out, ths approach creates systematc bases to the extent that frms wth certan characterstcs have other expected growth patterns. 3

4 (Easton 006, 007). Our measure of COE helps avod such spurous nferences by takng nto account a frm s growth rate as mpled by the data. 4 Our estmaton method bulds upon the poneerng work of Easton, Taylor, Shroff, and Souganns (00) (hereafter, ETSS). ETSS develop a method to smultaneously estmate the average COE and average earnngs growth rate for a gven portfolo of frms. Despte ths method s conceptual and practcal appeal, however, t cannot be used n many research settngs because t only allows one to estmate the average COE and growth rate for a gven sample of frms. In ths paper we extend the ETSS approach to allow for estmaton of COE and expected earnngs growth for ndvdual frms. Our approach s motvated by the ndustry practce of usng frm peers when valung prvately-held companes. Practtoners often compare a gven frm aganst frms wth smlar characterstcs to determne an approprate COE and/or growth rate (Pratt and Nculta 007; Damodaran 00). Accordngly, our method estmates a frm s COE (growth) as the sum of the COE (growth) typcal of frms wth the same rsk-growth profle plus a frm-specfc component. Emprcally, COE and growth are estmated by 4 Developng a more accurate and less based mpled COE measure s mportant gven the ncreasng use of mpled COE measures n accountng and fnance lterature. Impled COE measures have been used to shed lght on the equty premum puzzle (Claus and Thomas 00; Easton et al. 00), the market s percepton of equty rsk (ebhard et al. 00), rsk assocated wth accountng restatements (Hrbar and Jenkns 004), dvdend taxes (Dhalwal et al. 005), accountng qualty (Francs et al. 004), legal nsttutons and regulatory regmes (Hal and Leuz 006), and qualty of nternal controls (Ogneva et al. 007), as well as to test ntertemporal CAPM (Pastor et al. 008), nternatonal asset prcng models (Lee et al. 009), and the prcng of default rsk (Chava and Purnanandam 00). 4

5 regressng the rato of forecasted earnngs to book value of equty on the market-to-book rato and a set of observable rsk and growth characterstcs. 5 We test the accuracy of our COE estmates by examnng ther ablty to explan future stock returns for a sample of 7,63 frms wth I/B/E/S forecasts over the 980 to 007 perod. Ths analyss uses unadjusted earnngs forecasts as well as forecasts adjusted for predctable analyst bases as n ode and Mohanram (009). We fnd that usng ether raw earnngs forecasts (unadjusted measure) or earnngs forecasts adjusted for predctable bases (adjusted measure), our mpled COE measure surpasses conventonal mpled COE measures (hereafter referred to as benchmark COE measures) n predctng future realzed stock returns. More specfcally, when we sort stocks nto quntles based on COE, the return spread between the top and bottom quntles s 6.5% (9.3%) per annum usng our unadjusted (adjusted) measure, whereas ths spread does not exceed 5.0% (7.%) per annum usng the benchmark unadjusted (adjusted) COE measures. Multvarate regresson analyses further suggests that our mpled COE measure has ncremental predctve ablty relatve to the benchmark COE measures and commonly used rsk proxes (CAPM beta, sze, book-to-market, and past twelve-month stock returns). Specfcally, our measure remans sgnfcantly postvely related to future realzed stock returns even after controllng for ether the benchmark COE measures or 5 Specfcally, we use the CAPM beta, sze, book-to-market, and momentum as the observable rsk characterstcs, and we use analysts long-term growth forecast, the dfference between the frm s and the ndustry s target OE, and the rato of &D expenses to sales as the observable growth characterstcs. We take the part of COE (growth) that s not explaned by these observable rsk (growth) characterstcs to be due to unobservable rsk (growth) factors. Examples of undentfed rsk factors may nclude the rsk of ncreased competton and extreme weather, credt rsk, and ltgaton rsk as perceved by market partcpants but not fully captured by the set of observable rsk characterstcs that we consder. 5

6 the commonly used rsk proxes. In contrast, none of the benchmark COE measures s sgnfcantly related to future stock returns after controllng for our measure. Addtonal tests that rely on Easton and Monahan s (005) methodology suggest that our mpled COE measure delvers the lowest measurement error compared to the benchmark COE estmates. To examne the accuracy of our long-term growth estmates, we test ther predctve ablty wth respect to future earnngs growth rates. Specfcally, we estmate the realzed growth n aggregate four-year cum-dvdend earnngs from years t to t4, to years t5 to t8. We fnd that our mpled growth estmates are sgnfcantly assocated wth future earnngs growth: when we sort stocks nto quntles based on predcted growth, the annualzed growth spread between the top and bottom quntles s between.5% and 0.4% (5.5% and 8.6%) per annum usng our unadjusted (adjusted) measure. In multvarate regresson analyss, however, the growth spread s no longer statstcally sgnfcant after we control for analysts long-term growth forecasts, 6 the dfference between the frm s and the ndustry s return on equty, and the rato of &D expenses to sales. We therefore conclude that the sgnfcant predctve of our mpled growth measure can be attrbuted to these observable growth characterstcs. 7 6 Note that analysts long-term growth forecasts are based on 3- to 5-year forecasts, whle our mpled longterm growth rate corresponds to the rate of growth n perpetuty. 7 Although observable growth characterstcs appear to explan the predctve ablty of our growth measure, our measure contans nformaton that s ncremental to a smple statstcal forecast based on such characterstcs. To demonstrate, we use a smple cross-sectonal predcton model that frst regresses past growth on past growth characterstcs and then apples the resultng coeffcents to current growth characterstcs to arrve at a growth forecast. We fnd that such smple forecasts, even when postvely assocated wth future realzed growth rates, never fully subsume the predctve ablty of our mpled growth measure. 6

7 In addton to examnng COE and growth rates for ndvdual frms, we revst ETSS fndngs wth respect to the market-wde levels of COE and earnngs growth. As we dscuss n detal n Secton, explctly ncorporatng rsk and growth characterstcs as we do n our model may mprove estmates of sample average COE and growth rates. For nstance, usng our method, we obtan estmates of average mpled COE and equty rsk prema that are sgnfcantly lower than those obtaned from the ETSS model. Further, unlke ETSS, our method also delvers rsk prema that are consstent wth theoretcally derved equty rsk prema (Mehra and Prescott 985). 8 Our paper s related to earler work by Huang et al. (005), who use ETSS method to estmate frms COE and growth based on the tme seres of monthly stock prces and earnngs forecasts. However, our method dffers from that proposed by Huang et al. along several dmensons. Frst, ther method assumes that a frm s rsk exposure and expected earnngs growth do not change over the estmaton perod (36 months), whch lmts the practcal appeal of the resultng measures (.e., they cannot be used to examne changes n rsk over short horzons). In contrast, we provde pont-n-tme COE estmates. Second, ther estmaton pars monthly stock prces wth annual book values of equty, whch mplctly assumes that the book value of equty does not change wthn a gven fscal year. Our method reles on annual stock prces correspondng to annual book values of equty. Fnally, by usng monthly analyst forecasts and stock prces, ther method assumes that forecasts and prces are smultaneously updated to reflect new 8 Note that ths market-wde analyss s subject to an mportant caveat: the actual expected returns may be somewhat hgher than the mpled COE estmates when nterest rates are stochastc (Hughes et al. 009). 7

8 nformaton on a tmely bass, whch s nconsstent wth pror research documentng sgnfcant sluggshness n analyst forecasts (uay et al. 005). Our paper contrbutes to the lterature n three ways. Frst, we expand the lterature on COE estmaton by developng an mpled COE measure that reles on endogenously determned long-term earnngs growth rates. By takng nto account a frm s growth rate as mpled by the data, our mpled COE measure mtgates the potental bas that arses due to ncorrect assumptons about growth rates. Second, we complement the lterature on forecastng long-term frm performance by developng and valdatng an mpled earnngs growth measure that has economcally sgnfcant predctve ablty wth respect to future long-term earnngs growth. 9 To the best of our knowledge, ths s the frst paper to propose an mpled growth measure that can predct future long-term earnngs growth. Fnally, we contrbute to the equty premum lterature by provdng a measure that delvers frm-level equty rsk prema consstent wth theoretcally justfed low mpled market-wde rsk prema. The rest of the paper s organzed as follows. Secton dscusses our estmaton of frm-level COE and growth. Secton 3 descrbes the data and varable estmaton. In Secton 4 we present the emprcal results. Secton 5 provdes concludng remarks. 9 Whle we are not aware of any papers that estmate termnal growth rates, a few papers forecast earnngs over horzons beyond two years. For example, Chan et al. (003) forecast earnngs growth over the next fve years, whle Hou et al. (00) forecast three-year-ahead earnngs. 8

9 . Estmaton of frm-level cost of equty and growth In ths secton, we develop a method to smultaneously estmate frms cost of equty (COE) and expected earnngs growth rate usng stock prces, book values of equty, and earnngs forecasts. Our method extends Easton, Taylor, Shroff, and Souganns (00) (ETSS), who smultaneously estmate average COE and expected earnngs growth for a gven sample of frms. Smlar to ETSS, our approach s based on the resdual ncome model (e.g. Ohlson 995), whch expresses frm value as the reported book value plus the dscounted sum of expected resdual earnngs: 0 Et r Bt P 0 = B0 t () ( r ) t= where P 0 s the market value of equty, B 0 s the book value of equty, E t s expected earnngs for year t gven nformaton at t=0, and r s the COE (unless specfcally stated otherwse, the terms COE and expected return are used nterchangeably throughout the paper). Followng ETSS, we re-wrte the valuaton equaton usng fnte (four-year) horzon forecasts and defne g as the perpetual annual growth rate such that: P 0 X ( ) B ct 0 = B0 () 0 The resdual ncome model s equvalent to the dscounted dvdend model assumng the clean surplus relaton,.e. the book value of equty at the end of year t s equal to the book value of equty at the end of year t plus net ncome for year t mnus dvdends for year t. 9

10 where g 4 = ( ) s one plus the expected rate of growth n four-year resdual ncome, r 4 = ( ) s one plus the four-year expected return, X ct = 4 t= 3 E t (( r) 4 t )d t s expected aggregate four-year cum-dvdend earnngs, and d t t= s expected dvdends n year t gven nformaton at t=0. as: In order to estmate COE and growth, ETSS re-arrange the valuaton equaton () X / B = ( ) (3a) ct 0 ETSS further observe that the sample average and n equaton (3a) can be estmated from the ntercept and the slope n a cross-sectonal regresson of the rato of cumulatve earnngs to book value on the market-to-book rato: X / B = γ γ ε (3b) ct 0 0 where γ =, γ =, and 0 ε = ε ( ) ε. The and are the sample means of and respectvely, and ε = and ε = are the frmspecfc devatons of and from ther sample means. Estmatng regresson (3b) usng OLS obtans the sample means of COE and growth as = γ 0 γ and = γ 0, but leaves the frm-specfc components of and undentfed. Our approach to estmatng frm COE and growth rates ntroduces two nnovatons to the ETSS method. Frst, consstent wth a large body of extant research, we explctly recognze that COE and growth rates are assocated wth certan frm 0

11 characterstcs. Each ndvdual ( ) can therefore be expressed as the average ( ) plus components due to observable and unobservable rsk (growth) factors: = λ ' x ε = λ ' x ε where ( ) s the sample mean of ( ) n the estmaton year t, x ( x ) s a vector of observable frm rsk (growth) drvers (the drvers are demeaned to ensure that and can be nterpreted as sample means), λ ( λ ) s a vector of prema on the observable rsk (growth) drvers, and ε ( ε ) s a frm-specfc component of ( ) that s due to some unobservable rsk (growth) factors. Incorporatng the observable rsk and growth drvers serves two purposes. Frst, t provdes estmates of frm COE and growth rate condtonal on the observable frm characterstcs related to rsk and growth. Second, t helps to obtan more accurate estmates of average COE and growth rates. To see ths, note that the estmates of average COE and growth rate ( and ) are derved from the ntercept and slope estmates n (3b). The resduals n (3b) are a lnear functon of frm-specfc components of COE and growth rate ( ε = ε ( ) ε ). The resduals are therefore lkely to be correlated wth frm COE and growth rates, whch are n turn correlated wth the ndependent varable n regresson (3b) the market-to-book rato (e.g. Fama and French Emprcally, we use the CAPM beta, sze, book-to-market rato, and momentum as observable rsk drvers, and we use the analyst long-term growth forecast and the dfference between the frm and ndustry target OE as observable growth drvers. The component due to unobservable rsk (growth) factors s defned as the part of COE (growth) that s not explaned by the observable rsk (growth) drvers. For example, undentfed rsk factors may nclude the rsk of ncreased competton, lqudty rsk, credt rsk, ltgaton rsk, and poltcal rsk as perceved by market partcpants but not fully captured by the above observable rsk drvers.

12 993; Penman 996). Note, that because the resduals n (3b) are a complex functon of the frm-specfc COE, growth rate, and market-to-book rato, t s unclear whether such correlatons represent a source of bas n the regresson coeffcents. Explctly ncorporatng observable rsk and growth factors n equaton (3b) mtgates any concerns regardng the possble bas and may lead to more accurate estmates of average COE and growth rates. 3 As our second nnovaton, we decompose the resduals ε n the cross-sectonal regresson (3b) nto the COE component ( ε ) and expected growth component ( ε ). We perform ths decomposton by jontly mnmzng the components due to unknown rsk and growth factors, ε and ε. For ths purpose we set up the followng mnmzaton program: Mn w ( ε ) w ( ε ),, λ, λ, ε, ε = λ ' x ε = λ ' x ε (4) where w and w are some predetermned non-negatve weghts (wth at least one of the two weghts beng postve), and the other varables are as defned above. Intutvely, the mnmzaton functon n (4) represents a loss (cost) functon that ncreases wth the magntude of the unexplaned components of COE and growth. Introducng cost due to unexplaned components reflects the dea akn to Occam's razor prncple everythng else beng equal, estmates that can be explaned by known factors 3 The analyss that follows leads to such extended regresson relaton (equaton (5a) below).

13 are preferred to estmates that appeal to some unobservable factors. The weghts w and w reflect the relatve cost of the components due to unobservable rsk and growth factors respectvely. For example, the assumpton that the growth rate does not vary across frms beyond the varaton mpled by the known growth factors, = λ ' x, s a specal case wth w 0. = The appendx shows that our mnmzaton program (4) s equvalent to the followng mnmzaton program that can be estmated usng a weghted least squares (WLS) regresson: 4 Mn w ( v ) ε, γ0, γ, λ, λ s.t. X / B = γ γ λ ' x λ ' x ( ) v ct 0 0 (5a) where the weghts w are equal to ( ww / w( ) w( ) ). 5 Usng the coeffcent and resdual estmates (γ 0, γ, λ, λ, and ε ) from the WLS regresson (5a), frm COE ( ) and growth rate ( ) are determned as follows (see the dervaton n the appendx): = λ ' x ε = λ ' x ε (5b) where 4 egresson (5a) assumes that ndependent varables are exogenous,.e. E[ ε, x, ( ) x ] = 0. A suffcent but not necessary condton for the exogenety s the assumpton that ε and ε are ndependent of, x, and x. 5 Note that the WLS regresson restrcts nether the magntudes nor the sgns of rsk and growth prema, λ and λ, whch are determned endogenously based on earnngs forecasts and stock prces. 3

14 = γ 0 = γ γ ε ε 0 w = v w ( ) w ( ) w ( ) = v w ( ) w ( ) (5c) To summarze, our method allows smultaneously estmatng mpled COE and growth rates by ncorporatng known rsk and growth drvers n the estmaton procedure and by mnmzng the components due to unknown rsk and growth factors. Estmaton procedure We estmate frms COE and growth rates n two steps detaled below. Step : In each sample year, we estmate the followng cross-sectonal regresson usng WLS wth the weghts equal to /(( ) ( ) ) : 6 X / B = γ γ ( λ Beta λ LogSze λ λ ret ) ct 0 0 Beta Sze ret λ 'x ( λltg Ltg λdoedindoe λdsalesdsales )( ) v λ ' x (6) where the vector of rsk characterstcs, x, corresponds to the three-factor Fama-French model augmented wth Carhart (997) momentum factor: the CAPM beta (Beta), market value of equty (LogSze), market-to-book rato (), and past twelve months stock 6 These weghts assume equal weghtng of the COE and growth components due to unknown factors n (4), that s = w w =. As a robustness check, we vary the rato of the weghts ( w / w ) from 0.5 to. Our nferences are robust to these varatons. 4

15 return (ret ). 7 The vector of growth characterstcs, x, conssts of the analysts longterm growth forecast (Ltg), the dfference between the ndustry OE and the frm forecasted OE (dindoe), whch serves as a proxy for the mean-reverson tendency n OEs, and the rato of &D expenses to sales (dsales). The latter characterstc serves a dual purpose as a proxy for the extent of accountng conservatsm, whch affects termnal growth rate n resdual ncome (Zhang 000), and as one of the known predctors of the long-term growth n earnngs (Chan et al. 003). 8 The calculaton of X ct requres a COE estmate, r, whch s not known. We use an teratve procedure smlar to that descrbed n ETSS to estmate both X ct and r. Namely, we frst set r equal to 0% for all frms and calculate the ntal values of X ct. We then use obtaned X ct to estmate the WLS regresson, whch produces revsed estmates of r. We then re-calculate X ct usng the revsed estmates of r and agan reestmate the WLS regresson. The procedure s repeated untl the mean (across all frms) of absolute change n r from one teraton to the next s less than 0-7. The estmaton s robust to usng other ntal values of r and n most cases nvolves less than 0 teratons. 9 7 Leverage s another characterstc assocated wth equty rsk. We do not nclude leverage n the estmaton because Fama and French (99) show that the predctve power of leverage wth respect to future stock returns s subsumed by the beta, sze, and book-to-market rato. 8 In our senstvty tests, we have also ncluded other growth predctors suggested n Chan et al. (003), ncludng past sales growth, earnngs-to-prce rato, and alternatve conservatsm proxes used n Penman and Zhang (000). Our results are not senstve to ncludng them n the estmaton, and we opt for a parsmonous set of varables to avod addtonal sample restrctons. 9 Note that numercal estmaton of mpled COE s typcal n models that assume dfferent short-term and long-term growth rates n earnngs (e.g. ebhardt et al. 00, Claus and Thomas 00). The method proposed here s not more computatonally complex than the extant COE estmaton methods. 5

16 Step : Usng the ntercept and the slope on the market-to-book rato from the regresson estmated n Step, we calculate the mean and as = γ γ and 0 = γ 0. We use the resduals from the same regresson to calculate the frm-specfc components of and, as ε = /(( ) ( ) ) and v ε = ( ) / (( ) ( ) ). The total frm and are calculated by v combnng the mean and estmates and the resduals ε and ε wth the estmates of λ ' x and λ' x from regresson (6), so that = λ ' x ε and = λ ' x ε. 3. Data and Varable Estmaton Our sample conssts of December fscal-year end frms avalable n I/B/E/S, Compustat, and CSP from 980 to 007. The one- and two-year-ahead analyst earnngs forecasts, long-term growth forecasts, realzed earnngs, stock prces, dvdends, and number of shares outstandng are obtaned from I/B/E/S; book values of common equty are obtaned from Compustat; CAPM betas, as well as past and future buy-and-hold stock returns are estmated usng monthly stock returns from CSP. We exclude frm-years wth negatve two-year-ahead earnngs forecasts, book-to-market ratos less than 0.0 or greater than 00, or stock prces below one dollar. Our fnal sample ncludes 7,63 frms, wth the by-year number of observatons rangng from 76 n 980 to,447 n

17 Inputs to Smultaneous Estmaton of COE and rowth As dscussed n detal n the prevous secton, our method requres estmatng the followng cross-sectonal regresson usng WLS: X / B = γ γ ( λ Beta λ LogSze λ λ ret ) x ct 0 0 Beta Sze ret ( λ Ltg λ dindoe λ dsales )( ) x v Ltg doe dsales (6) where X ct = expected four-year cum-dvdend earnngs, E t (( r) 4 t )d t, where E and E are one- and two-year-ahead consensus earnngs per share forecasts from I/B/E/S reported n June of year t; E 3 and E 4 are three- and four-year-ahead earnngs per share forecasts computed usng the long-term 4 t= 3 t= growth rate from I/B/E/S as: E 3 = E (Ltg) and E 4 = E 3 (Ltg); 0 d to d 3 are expected dvdends per share calculated assumng a constant dvdend payout rato from fscal year t; B 0 Beta = book value of equty from Compustat at the end of year t dvded by the number of shares outstandng from I/B/E/S; = market-to-book rato, calculated as stock prce from I/B/E/S as of June of year t, dvded book value of equty per share estmated as prevously descrbed; = CAPM beta estmated usng sxty monthly stock returns precedng June of year t (wth at least twenty four non-mssng returns requred); LogSze = the log of the market value of equty calculated as stock prce from I/B/E/S as of June of year t multpled by shares outstandng from I/B/E/S; ret - = the twelve-month buy-and-hold stock return precedng June of year t; 0 When the long-term growth forecast s mssng, we use the growth mpled by E and E. Values of Ltg greater than 50% are wnsorzed. 7

18 Ltg = the long-term growth consensus forecast from I/B/E/S as of June of year t; dindoe = the ndustry OE (ncome before extraordnary tems dvded by the average book value of equty) mnus the frm average forecasted OE over fscal years t to t4. Industres are defned usng the Fama and French (997) 48- ndustry classfcaton. Industry OE s calculated as a ten-year movng medan OE after excludng loss frms (ebhardt et al. 00); dsales = the rato of &D expenses to sales. All varables are calculated as the dfference between the frm value and the sample mean n the estmaton year. COE from Benchmark Models We compare performance of our COE measure wth three COE measures derved usng an assumed long-term earnngs growth rate. The frst mpled COE measure, r zero, s derved as an nternal rate of return from the resdual ncome valuaton model assumng a zero resdual earnngs growth rate after year t=4: P 0 3 = B0 τ = Eτ r B E r B (r zero ) zero τ 4 zero 3 t 3 ( rzero) rzero( rzero) where P 0 s the stock prce as of June of year t= from I/B/E/S; B 0 s the book value of equty at the end of year t=0 from Compustat dvded by the number of shares outstandng from I/B/E/S; E τ s expected earnngs for year τ; B τ s the expected per-share book values of equty estmated usng the clean surplus relaton (B t = B t E t d t ). The second mpled COE measure, r LS, s developed by ebhardt et al. (00) and s frequently used n both accountng and fnance studes. It s derved usng explct earnngs forecasts for years t= and t=, and assumes that return on equty fades to the ndustry medan OE from year t=3 to year t=. A zero growth n resdual earnngs s 8

19 assumed afterwards. The measure s estmated as an nternal rate of return from the followng valuaton equaton: P 0 = B0 τ = ( OEτ rls ) Bτ ( IndOE r ) B t (r LS ) ( r ) r ( LS LS LS rls ) where OE τ s expected future return on equty calculated as earnngs per share forecast (E τ ) dvded by the book value of equty per share at the end of the prevous year (B τ- ); OE and OE are calculated usng one- and two-year-ahead consensus earnngs per share forecasts from I/B/E/S reported n June of year t; OE 3 s computed by applyng the long-term growth rate from I/B/E/S to the two-year-ahead consensus earnngs per share forecast; beyond year t3 OE are assumed to fade to ndustry medan OE (IndOE) by year t. The thrd mpled COE measure, r PE,, s taken from ode and Mohanram (009). It s based on the abnormal earnngs growth model (Ohlson and Juettner-Nauroth 005) and assumes a zero abnormal earnngs growth beyond year t. The measure s calculated as: r PE = E g, g = (E / E ) Ltg P 0 (r PE ) where P 0 s the stock prce as of June of year t from I/B/E/S; E and E are oneand two-year-ahead consensus earnngs per share forecasts from I/B/E/S reported n June of year t; Ltg s the long-term earnngs growth forecast from I/B/E/S reported n June of year t. Ths measure s a modfed verson of the Easton (004) PE measure, whch assumes g =E /E. 9

20 Adjustng Analysts Forecasts for Predctable Errors Pror lterature shows that analyst short-term earnngs forecasts are systematcally based, wth the drecton and the extent of the bas correlated wth varous frm-year characterstcs (e.g. uay et al. 005, Hughes et al. 008). Usng based earnngs forecasts as nputs n the valuaton equaton nevtably produces based mpled COE estmates (Easton and Sommers 005). To mtgate the effect of the bas, we follow ode and Mohanram (009) and frst adjust analyst forecasts for predctable errors and then recompute the mpled COE measures usng the adjusted forecasts., We obtan predctable errors n earnngs forecasts by frst regressng realzed forecast error n k-year-ahead earnngs scaled by prce (FE k, k =,, 3, and 4) on the forward earnngs-to-prce rato, EP, long-term growth forecast, Ltg, change n gross PP&E, CH_PPE, tralng twelve-month stock return, et -, and the revson of one-yearahead earnngs forecast from the forecast made three months earler, EV. The regressons are estmated annually based on the hold-out sample lagged by k years. The obtaned coeffcents are combned wth varables n year t to estmate the predctable bas n k-year-ahead earnngs forecasts. We then correct earnngs forecasts for the We would lke to thank Partha Mohanram for sharng hs forecast error adjustment codes wth us. Hughes et al. (008) suggest that the tradng strategy based on explotng predctable analyst forecast errors does not produce statstcally sgnfcant returns, whch s consstent wth the market not beng subject to the same bases as analysts. However, t s possble that n some nstances the stock prces may ncorporate earnngs expectatons based n the same drecton as analyst earnngs forecasts. If ths s the case, adjustng earnngs forecasts for such predctable errors leads to mpled COE estmates that do not represent the market s expectatons of future returns, but nstead are equal to the market s expectaton of future returns plus the predctable return due to subsequent correcton of the msprcng. The adjusted COE measure then represents the total COE that the frm faces due to both rsk and msprcng. In our emprcal analyses, we do not dstngush between the two nterpretatons of the mpled COE. 0

21 predctable bas and calculate the adjusted COE and growth rate based on the corrected forecasts. The obtaned COE and growth rates are labeled as adjusted. 4. Emprcal Analyses Descrptve Statstcs Table, Panel A, reports descrptve statstcs for our sample frms. 3 Consstent wth other studes that use I/B/E/S analyst earnngs forecasts, the frms n our sample are relatvely large wth the mean (medan) market captalzaton of $3,63 (54) mllon. The mean CAPM beta s.04 whch s comparable to the beta of one for the market value-weghted portfolo. The mean (medan) of our COE estmate, r SE (where SE stands for smultaneous estmaton), s 8.4% (8.9%) and the mean (medan) of our growth estmate, g SE, s 0.5% (0.3%). Our COE estmates are somewhat lower than those based on the model wth zero growth rate, LS model, and PE model (means 9.5%, 0.4%, and.% respectvely). When earnngs forecasts are corrected for analyst forecast bases, COE estmates for all models declne suggestng that earnngs forecasts were on average adjusted downwards to correct for the overall optmstc forecast bas. Panel B of Table presents means of by-year correlatons among the COE estmates. The average correlatons between unadjusted (adjusted) r SE and r zero, r LS, and r PE are 0.84, 0.7, and 0.53 (0.79, 0.6, and 0.43), respectvely. The hgh correlaton 3 To avod the nfluence of extreme observatons, we wnsorze all varables except future realzed returns at the st and 99 th percentles.

22 between r SE and r zero reflects the fact that these COE measures are based on the same valuaton model wth dentcal nputs, except the termnal growth rate. Overall, correlatons among all COE measures are postve and sgnfcant n all sample years, suggestng that they capture the same underlyng construct. Correlatons between Impled COE and Frm Characterstcs In the approach followng Botosan and Plumlee (005) we verfy that COE measures exhbt predctable assocatons wth rsk-related frm characterstcs. Pror lterature (e.g. Fama and French 99, Carhart 997) proposes several proxes for frm rsk, ncludng the CAPM beta (Beta), debt-to-equty rato (Leverage), frm sze (LogSze), book-to-market rato (B/M), and momentum (pror twelve-month return, et - ). It s worth notng that the valdty of most of these rsk characterstcs s tself derved from ther emprcal assocatons wth future realzed returns, wth ex-post ratonalzatons relyng on modfed asset-prcng models. 4 Therefore, we treat the evdence presented n ths subsecton as more descrptve n nature. In addton to rsk proxes, we examne the mpled COE's assocaton wth the volatlty of expected returns and cash flows. Hughes et al. (009) show that, when expected returns are stochastc, the mpled COE estmates devate from expected returns. These devatons depend on the volatlty of expected returns and cash flows. The lower s COE estmate s correlaton wth these volatltes the lower the estmate s devaton from expected returns. We use analyst forecast dsperson n one-year-ahead earnngs 4 The exceptons are the CAPM beta, whch s derved from the equlbrum asset-prcng theory, and debtto-equty rato, whch proxes for the dfferences s equty rskness due to the degree of fnancal leverage (e.g. Hamada 97). Note that f the CAPM represents a vald descrpton of the world and the CAPM beta s measured wthout error, the explanatory power of leverage should be subsumed by the CAPM beta.

23 scaled by stock prce, Dsp, as a proxy for expected cash flow volatlty, and the standard error of the estmate of the CAPM beta, σ β, as a proxy for expected return volatlty. 5 Table reports the results of pooled regressons of COE estmates on frm characterstcs. The regressons nclude year fxed effects, and standard errors are clustered by frm and year (Petersen 009). The assocatons between the benchmark COE measures and Beta, Leverage, Sze, B/M, and Dsp are generally consstent wth those reported by Botosan and Plumlee (005). Notably, all unadjusted COE measures are negatvely related to past returns consstent wth the sluggshness n analyst forecasts (uay et al. 005). 6 The adjusted COE measures, on the other hand, correlate postvely wth past returns reflectng the momentum effect n stock returns. The relaton between our COE measure and frm characterstcs s smlar to those of other COE measures wth the followng exceptons. The relaton between r SE and Beta s negatve, whch s nconsstent wth the sngle-perod CAPM, but s n lne wth the negatve nsgnfcant coeffcent n asset-prcng tests that use realzed returns (Fama and French 99; Petkova 006; Core et al. 008). 7 Both adjusted and unadjusted mpled COE measures are correlated ether wth our proxy for the volatlty of future cash flows, Dsp, or wth the proxy for the varablty of future expected returns, σ β. Ths result confrms the cautonary 5 In Hughes et al. (009), the volatlty of expected returns s reflected by the volatlty of the frm s forward-lookng beta, whch we proxy by the standard error of the frm's hstorcal beta. 6 When analyst forecasts are sluggsh, they do not ncorporate the recent postve (negatve) earnngs news and are therefore based downward (upward) followng recent postve (negatve) stock returns. The bas n forecasts mechancally leads to downwardly (upwardly) based mpled COE estmates followng postve (negatve) stock returns. 7 The nsgnfcant relaton between the CAPM beta and stock returns was a key motvaton for more elaborate asset-prcng models (Merton 973; Jagannathan and Wang 996; Lettau and Ludvgson 00). 3

24 statement n Hughes et al. (009) that t s mportant to control for the volatlty measures n the emprcal studes that use mpled COE measures. Overall, we fnd that our measure, r SE, generally exhbts smlar correlatons wth the rsk characterstcs when compared to benchmark mpled COE measures. The results however need to be nterpreted wth cauton because they depend on the construct valdty of the rsk characterstcs. Impled COE and Future ealzed eturns In ths subsecton, we employ a dfferent approach to valdatng the mpled COE measures by documentng ther assocaton wth future realzed returns (uay et al. 005; Easton and Monahan 005; ode and Mohanram 009). We frst verfy the out-of-sample predctve ablty of the COE measures wth respect to future stock returns by sortng sample frms nto quntles of the mpled COE dstrbuton at the end of June of each year. For each portfolo, we calculate the mean of buy-and-hold returns for the next twelve months. We then calculate hedge returns - the dfference n returns between the top (Q5) and bottom (Q) quntles of mpled COE. Ther sgnfcance s evaluated usng Fama-MacBeth t-statstcs wth the Newey-West autocorrelaton adjustment. Fgure plots the tme-seres means of portfolo returns. The numbers next to the Q5-Q labels refer to the magntude of hedge returns. Panel A reports returns by portfolos based on unadjusted COE measures. Our measure, r SE, exhbts a strong monotonc relaton wth future realzed returns. The dfference n returns between the top and bottom quntles, Q5-Q, s equal to 6.5% (statstcally sgnfcant at the 5% level). In 4

25 contrast, the predctve ablty of r LS and r PE s weak. The hedge returns, Q5-Q, are only 3.8% (0.0%) for r LS (r PE ) and not statstcally sgnfcant. Predctve ablty of the benchmark zero-growth measure, r zero, s weaker than that of our measure, producng a margnally sgnfcant hedge return of 5%. Panel B reports returns by portfolos based on COE measures adjusted for forecast errors. Performance of all COE measures s markedly mproved, wth our measure stll performng best. The hedge return, Q5-Q, ncreases to 9.3%, 7.%, 6.8%, and 4.5% for r SE, r zero, r LS, and r PE respectvely, and s unformly statstcally sgnfcant at least at the 5% level. Overall, our COE measure produces more sgnfcant return dfferences than the benchmark models both wth and wthout adjustments for analyst forecast errors. Havng establshed that our measure has sgnfcant predctve ablty wth respect to future stock returns at the portfolo level, we turn to regresson analyss to, frst, explore ts predctve ablty at the ndvdual frm level, second, establsh whether ts predctve ablty s ncremental to that of other COE measures, and thrd, shed some lght on the source of ts predctve ablty. Frst, we nvestgate return predctve ablty of ndvdual COE measures at the frm level. Panel A of Table 3 reports the results of cross-sectonal regressons of future one-year stock returns on the COE measures. Each slope coeffcent has two correspondng t-statstcs reflectng how sgnfcantly dfferent the coeffcent s from zero and one. The slope on a vald COE measure should be sgnfcantly dfferent from zero, and not sgnfcantly dfferent from one. Consstent wth the evdence from Fgure, our measure, r SE, s sgnfcantly related to future stock returns, and the regresson coeffcent s statstcally ndstngushable from one. None of the other measures 5

26 unadjusted for analyst forecast bases performs as well ther slopes are not statstcally sgnfcant. After the forecast bas adjustment, the slopes ncrease for all measures and become (reman) sgnfcantly postve and ndstngushable from one for r zero and r LS (r SE ). The slope on r PE, although postve, s stll not sgnfcant, whch s n contrast to portfolo-level tests from Fgure. Next, we examne the ncremental explanatory power of r SE and the benchmark COE measures relatve to each other by regressng future realzed returns on the pars of COE measures. Although we contnue to report two sets of t-statstcs for the slope coeffcents, we no longer expect the coeffcent on r SE to be statstcally ndstngushable from one. The results are reported n Panel B of Table 3. Both unadjusted and adjusted r SE have sgnfcant explanatory power after controllng for r zero, r LS, or r PE. However, nether of the benchmark COE s sgnfcant after controllng for r SE. Therefore, t appears that r SE subsumes the predctve power of other COE measures. Fnally, we provde evdence on the relatve mportance of the two nformaton sources underlyng our measure, r SE. ecall that r SE s estmated to () most accurately ft the rsk profle (.e. rsk characterstcs) of the company, and () mnmze the resdual COE that s due to unobservable rsk sources. Next, we control for the observable rsk characterstcs to assess the valdty of our measure beyond proxyng for the rsk profle of the company. 8 Panel C of Table 3 regresses realzed returns on COE proxes after 8 In supplementary analyses, we verfy that the rsk and growth drvers combned n a smple statstcal predcton model do not predct future returns. Specfcally, we estmate cross-sectonal regressons of oneyear stock returns on the lagged rsk and growth drvers for a one-year holdout perod t-. The obtaned coeffcents are combned wth rsk and growth drvers at tme t to come up wth a statstcal forecast of t realzed returns. The results (untabulated) suggest that such smple statstcal return predcton model s unable to forecast future returns. When realzed returns are regressed on the statstcal forecasts, the coeffcents on the statstcal forecasts are not statstcally sgnfcant. 6

27 controllng for Beta, Sze, B/M, and past stock returns. The results show that the slopes on both adjusted and unadjusted r SE reman statstcally sgnfcant. That confrms the construct valdty of our measure beyond smply capturng the observable rsk profle of the company. 9 Overall, the results n Fgure and Table 3 demonstrate that our COE measure s sgnfcantly postvely assocated wth future realzed returns. Furthermore, t contans nformaton about frms expected returns that s not captured by the CAPM beta, frm sze, book-to-market rato, past stock returns, as well as other mpled COE measures. Impled rowth ates and Future ealzed Earnngs rowth In ths subsecton, we valdate the mpled growth rates by documentng ther assocaton wth future realzed growth n earnngs. From Secton 3, the mpled growth rate captures expected growth n four-year cum-dvdend resdual earnngs from perod t4 onwards. The drect valdaton test would nvolve correlatng the mpled growth rates wth earnngs growth from t4 to perpetuty. Such test s nfeasble n practce. Accordngly, we estmate growth n fouryear cum-dvdend earnngs 30 from [t, t4] to [t5, t8] as: 9 In addtonal analyses (untabulated), we fnd that our COE measure, both before and after adjustment for analyst forecast errors, remans sgnfcantly postvely assocated wth future realzed returns even when both the benchmark COE measures and rsk characterstcs are ncluded as control varables.. 30 A more drect valdaton requres estmatng realzed growth n resdual earnngs. We choose not to use growth n resdual earnngs n our man tests for two reasons. Frst, f our mpled growth and COE estmates are correlated, usng our COE estmate to calculate realzed resdual earnngs may cause the latter to be spurously correlated wth our mpled growth estmate. Second, when we use rsk-free rates to calculate realzed resdual earnngs, over 50% of cumulatve resdual earnngs before extraordnary tems (EBEI) over the frst four years are negatve and thus cannot be used as a base to estmate growth. Percentage of negatve observatons s lower when operatng ncome before deprecaton (OI) s used to 7

28 = X / X, cumd cumd t 4, t 8 t 8 t 4 where cumd X T = T t= T 3 E t T 4 t (( r) ) dt, t t= T 3 E s realzed earnngs for year t, d t s dvdends declared n year t, and r s the rate of return at whch the dvdends are renvested. For the rate of return, we use the rsk-free rate at perod t. 3 The realzed earnngs are ether earnngs before extraordnary tems (EBEI), or operatng ncome before deprecaton (OI). Earnngs before extraordnary tems corresponds more drectly to earnngs underlyng our predcted long-term growth, however t s frequently negatve or small causng problems when used n the denomnator. Calculatng growth usng operatng ncome before deprecaton mtgates ths problem. Table 4, Panel A contans the descrptve statstcs for the growth rates n fouryear cum-dvdend earnngs growth rates. The mean (medan) growth rates are 0.48 (0.30) for EBEI and 0.5 (0.3) for OI. These growth rates can be nterpreted as a geometrc average growth over four years, and they correspond to annualzed rates of 0% (7%) for EBEI and % (7%) for OI. 3 Fgure, Panel A plots mean growth rates by quntles of mpled growth measure. Casual observaton suggests a postve assocaton between the mpled and estmate resdual earnngs. Accordngly, we replcate analyses presented n ths subsecton usng growth n resdual OI, and produce a smlar set of results (untabulated). 3 By usng a rsk-free rate we avod spurous correlatons wth mpled growth rates that could arse had we used prevously estmated mpled COE estmates. The results are robust to usng a unform 0% rate as n Penman (996), or a 0% rate that assumes no renvestment of dvdends. 3 We do not use annualzed growth rates n the analyss because we cannot annualze four-year growth rates that are less than negatve 00%. 8

29 realzed growth rates, wth the excepton of unadjusted mpled growth beng used to predct growth n OI. Ths observaton s formally confrmed n regresson analyss. Specfcally, we regress realzed growth rates on the quntle rank of unadjusted (adjusted) mpled growth rate. The regressons use a pooled sample, wth tme fxed effects and standard errors clustered by frm and year. The results are reported n Panels B and C of Table 4. The coeffcents on the ranks of unadjusted (adjusted) mpled growth rate are 0. (0.098) and 0.06 (0.060) when predctng growth n EBEI and growth n OI, respectvely. These slope coeffcents multpled by four can be nterpreted as average dfferences n fouryear earnngs growth between the extreme quntles of mpled growth. The dfferences n four-year growth rates are, therefore, 48.8% (39.%) and 0.4% (4%). On the annualzed bass, these dfferences n growth rates correspond to 0.4% (8.6%) and.5% (5.5%), respectvely. All slope coeffcents, except the slope n the regresson of OI growth on unadjusted mpled growth, are statstcally sgnfcant at the % level. Overall, we fnd that our mpled growth measure s a statstcally and economcally sgnfcant predctor of future growth n earnngs. Next, we nvestgate whether the mpled growth rates retan the ablty to predct future realzed growth after controllng for the growth drvers used n the estmaton of mpled growth. For that purpose, we regress future realzed growth rates on the quntle rank of mpled growth estmates, (g SE ), and control varables analysts predcted earnngs growth, Ltg, devaton of frm s OE from ndustry s OE, dindoe, and a rato of &D expenses to sales, dsales. The results reported n Panels B and C of Table 4 suggest that predctve ablty of the mpled growth measure derves entrely from the 9

30 growth drvers none of the coeffcents on ranked mpled growth remans statstcally sgnfcant after controllng for growth characterstcs. ven that the growth drvers (ex post) fully explan the mpled growth s predctve ablty, we next nvestgate whether a smple statstcal model based on the same growth drvers s equally successful n predctng earnngs growth. Each year t, we use a hold-out sample lagged by eght years to regress past realzed four-year cumdvdend earnngs growth rates ( t-4,t ) on the earnngs growth drvers (Ltg, dindoe, and dsales) from year t-8. We then combne the obtaned coeffcents wth growth drvers from year t to calculate a statstcal predctor of future growth n four-year cumdvdend earnngs (p t4, t8 ). Panels D and E of Table 4 report regressons of realzed growth rates on the quntle ranks of both mpled and statstcally predcted growth. Fgure 3 plots average realzed growth rates by quntles of statstcally predcted growth. Due to the hold-out sample requrements, these regressons are based on the sample perod. For ths perod, the mpled growth measure exhbts a stronger predctve ablty the coeffcents on (g SE ) are hgher than n Panels B and C of Table 4, and sgnfcant at least at the % level. The mpled growth measure retans ncremental predctve ablty after controllng for the statstcal predctors. Moreover, t subsumes predctve ablty of the latter wth respect to future growth n EBEI. Importantly, statstcal predctors of growth seem to be ftted to a specfc earnngs measure. Namely, statstcally predcted growth n OI (EBEI) has no power n predctng growth n EBEI (OI). The above evdence, combned, suggests that whle t s possble to predct future realzed growth n earnngs statstcally, the statstcal growth measures need to be ftted to a specfc 30

31 earnngs metrc and they do not perform as well as the mpled growth at predctng growth n bottom-lne earnngs. The mpled growth measure, on the other hand, provdes unversal predctve ablty, regardless of earnngs defnton, and contans nformaton beyond smple statstcal predctors. Overall, the mpled growth rates are predctve of future long-term growth n earnngs. Although the predctve ablty of mpled growth s derved entrely from the growth drvers, estmatng mpled growth rate from our model s superor to combnng growth drvers usng a smple statstcal model. The analyses n ths subsecton are, however, subject to an nherent survvorshp bas, whch s unavodable when measurng growth over long horzons. We further nvestgate the effects of the bas n our robustness check secton (Secton 5). Comparson between Our Measure and LS Measure esults n Tables 3 and 4 show that our COE measure surpasses the benchmark COE measures n predctng future returns over a broad cross-secton of frms. A natural queston then arses: when does our measure perform especally well relatve to commonly used measures, such as r LS? 33 In other words, when does r LS fal to predct future returns but our measure succeeds. A key assumpton of the LS model s that the frm s OE converges to the ndustry average OE. For frms where ths assumpton s a good approxmaton of expected growth, the LS model s expected to perform well. In contrast, for frms that are so dfferent from other frms n the ndustry that ther OE s not expected to converge to the ndustry OE, the LS model s expected to perform 33 Although, both r LS and r PE are wdely used n the lterature, we chose to examne r LS because the predctve ablty of ths measure sgnfcantly exceeds that of the r PE n our sample. 3