o Extended Dividend, Cash Flow and Residual Income Valuation Models - Accounting for Deviations from Ideal Conditions

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1 o xended Dividend, Cash Flow and Residual Income aluaion Models - Accouning for Deviaions from Ideal Condiions Dieer Hess, Carsen Homburg, Michael Lorenz, Soenke Sievers Corporae Finance Deparmen Accouning Deparmen Universiy of Cologne Alberus-Magnus Plaz, D Cologne, Germany Correspondence: sievers@wiso.uni-koeln.de lorenz@wiso.uni-koeln.de We hank seminar paricipans a he 2007 GOR Conference, he 2008 WHU Campus for Finance Research Conference, he Midwes Finance Associaion 57 h Annual Meeing, he 2008 asern Finance Associaion Annual Meeing, he I Workshop on mpirical Research in Financial Accouning, he 2008 uropean Accouning Associaion Doal Colloquium, he 2008 uropean Accouning Associaion Conference, he 2008 German Academic Associaion for Business Research Meeing and he Universiy of Cologne for helpful commens. We are especially graeful o Joachim Gassen, Wayne Landsman, Chrisian Leuz, Sephen Lin and Peer Pope.

2 Absrac Sandard equiy valuaion approaches (i.e., DDM, RIM, and DCF) are based on resricive assumpions regarding he availabiliy and qualiy of payoff daa. herefore, we provide exensions of he sandard approaches being suiable under less han ideal condiions (e.g. diry surplus accouning and inconsisen seady sae growh raes). mpirically, our exended models yield considerably smaller valuaion errors, suggesing ha he models are worhwhile o implemen. Moreover, obaining idenical value esimaes across he exended models, our approach provides a benchmark implemenaion. his allows us o quanify he magniude of errors resuling from individual violaions of ideal condiions in he sandard approaches. JL Classificaion: G10, G12, G34, M41. Keywords: Dividend Discoun Model, Residual Income Model, Discouned Cash Flow Model, Ideal Condiions, Diry Surplus, erminal alue, Seady Sae, aluaion rror.

3 1 Inroducion he mos widely used equiy valuaion approaches, i.e., he dividend discoun model (DDM), he residual income model (RIM) and discouned cash flow model (DCF), res on raher resricive assumpions. In paricular, hey require clean surplus accouning and payoff projecions for infinie horizons. Such ideal valuaion condiions are almos never given, neiher in pracice nor in empirical research. herefore, we exend he hree sandard models o accoun for ypically encounered deviaions from ideal valuaion condiions. Basically, we derive recion erms, which capure differences beween ideal and real daa. he exended models yield wo major advanages: Firs, he proposed models generae considerably smaller valuaion errors, suggesing ha marke prices are explained much beer if deviaions from ideal condiions are aken ino accoun. Second, he exended models provide a benchmark since hey yield idenical valuaion resuls under ideal as well as non-ideal condiions. his benchmark allows us o analyze o wha exen specific violaions disor he valuaion resuls of sandard models. herefore, i explains, for example, previous sudies findings ha RIM yields remarkably robus resuls. o circumven he lack of infinie payoff forecass, mos frequenly, so-called wo-sage models are implemened assuming a cerain seady sae payoff growh rae for he second phase. As poined ou by Penman [1998], hen a paricularly imporan problem can arise from inconsisencies beween he assumed growh rae and he payou raio. Because being affeced differenly, valuaion resuls of he sandard models may hen diverge. In conras, Levin and Olsson [2000] and Lundholm and O Keefe [2001a], for example, show ha he hree sandard models yield idenical value esimaes if in seady sae all iems on he balance shee and income saemen grow a he same rae. aking ino accoun differences in he underlying seady sae assumpions of he sandard models, we analyze he impac of inconsisen erminal value calculaions and derive an appropriae recion. Diry surplus accouning, narrow dividend measures and differences beween book and marke values of deb pose addiional challenges for he sandard models (see e.g. Lo and Lys [2000], Fama and French [2001] and Sweeney, Warga, and Winers [1997]). o rec for diry surplus accouning, we simply include differences beween he saed (diry) income and he income derived under clean surplus. o adjus for narrow dividend definiions, we include oher capial ransacions beween owners and he firm. o rec for violaions of he assumpion ha deb is marked 1

4 o marke (i.e., he ne ineres relaion required for weighed average cos of capial (WACC) versions of he DCF model), we rec for differences beween ineres expenses acding o he so called ne ineres relaion and ineres expenses as repored in he income saemen. While he las hree recions are easily obained one by one, hey affec boh he explici forecas period and he erminal period, and hus, inerac wih he erminal value recion. Our analysis shows how o accoun for hese ineracions. Inuiively, he general principle o derive our adjusmens is ha we mimic an inegraed financial planning approach. herefore, our exended valuaion equaions are based on comprehensive (i.e., all-inclusive) payou measures and seady sae growh raes ha are consisen wih given payou raios. Using a porfolio approach wih realized daa from 1987 o 2004, we adop a perfec foresigh seing wih unbiased and consisen analyss forecass. his approach yields he following main empirical resuls: Firs, bias and inaccuracy decrease remarkably suggesing ha he proposed models are worhwhile o implemen. For example, he exended DCF model has a 62 percenage poins smaller bias compared o is sandard counerpar. Second, we obain idenical value esimaes for he exended DDM, RIM and DCF model, even under non-ideal valuaion condiions. hus, he exended models provide a benchmark valuaion. his allows us o measure o wha exen he sandard models are affeced by individual violaions of ideal condiions. By quanifying he magniude of hese violaions in a unified framework, our findings add o he explanaion of previous horse race lieraure resuls, where various model specificaions haven analyzed separaely (see e.g. Penman and Sougiannis [1998], Francis, Olsson, and Oswald [2000] and Coureau, Kao, and Richardson [2001]). In paricular, we find ha he ranking of he hree models depend on he number of considered recion erms. For example while RIM is generally more robus agains deviaions from ideal condiions and hus ranked firs wihou any recions, DCF is ranked hird, respecively. However, if one inroduces only a recion erm for inconsisen growh raes he bias is reduced by around 46% percenage poins and furhermore he ranking of hese models changes. Moreover, our empirical resuls highligh he imporance of reasonable seady sae assumpions, recions for diry surplus accouning, and a wide dividend definiion. Due o hese findings, his sudy gives guidance for analyss and sandard seers alike. On he one hand, we would recommend ha analyss should forecas all he componens necessary o derive all-inclusive payoff measures, in order o faciliae a beer esimaion of socks inrinsic values. On he oher hand, our resuls have broad implicaions for he sandard seers, since he derivaion of fair value esimaes are encounered in many circumsances under US- 2

5 GAAP. 1 Moreover, he resuls are imporan for researchers and praciioners in order o assess he relaive impac of deviaions from ideal condiions encounered in pracice. While projecing (pro-forma) company accouns, i.e., balance shees and income saemens, deviaions from ideal condiions should be considered. In paricular, company valuaion based on hese projeced company accouns should be carried ou by inporaing he proposed recions. he remainder of his sudy is organized as follows. Secion 2 discusses he relaed lieraure. Secion 3 briefly reviews he sandard models and inroduces he exended DDM, RIM and DCF model. Secion 4 describes he daa and conains he empirical resuls. specially, we repor he valuaion errors for he sandard and he exended models and quanify he magniude of each recion erm separaely. Secion 5 summarizes he resuls and concludes. 2 Relaed Lieraure Obviously, here are quie a few sudies concerned wih eiher company valuaion or non-ideal valuaion condiions such as diry surplus accouning. However, his is he firs sudy we are aware of ha direcly inporaes recions of deviaions from ideal condiions ino valuaion models. hereby hese wo branches of lieraure are combined in an innovaive way. arious sudies deal wih measuring he magniude and he value relevance of diry surplus accouning flows. Alhough, comprehensive income as defined in SFAS 130 is no an allinclusive income measure ha compleely saisfies he clean surplus relaion, oher comprehensive income (OCI) is a raher good proxy for diry surplus flows (e.g. Chambers e al. [2007]). he resuls on he imporance of diry surplus flows are mixed. For example, O'Hanlon and Pope [1999], Dhaliwal, Subramanyam, and rezevan [1999] documen a median of diry surplus flows deflaed by marke value of shareholders equiy of 0.4% in he Unied Kingdom and 0% in he US, respecively. In conras, Lo and Lys [2000] find ha firms are comparaively srongly affeced by diry surplus flows under US-GAAP. In paricular, 14% of heir observaions repor diry surplus flows ha are larger han 10% of he clean surplus income. Similar resuls are found for several oher counries. 2 For our sample 1 2 An imporan case, where calculaions of inrinsic company values migh be necessary in order o derive fair value esimaes, is e.g. he impairmen es. Moreover, SFAS 157 par. 18 saes ha he income approach uses valuaion echniques o conver fuure amouns (for example, cash flows or earnings) o a single presen amoun (discouned). he measuremen is based on he value indicaed by curren marke expecaions abou hose fuure amouns. hose valuaion echniques include presen value echniques. Cahan e al. [2000] analyze he imporance of diry surplus flows for New Zealand, Isidro, O Hanlon, and Young [2006] for France, Germany, he U.K. and he US, Wang, Buijink, and ken [2006] for he 3

6 using our diry surplus measure even 45% of observaions repor diry surplus flows ha are larger han 10% of clean surplus income. For furher deails, see Appendix 1. In addiion, also he resuls on he value relevance of diry surplus accouning flows are mixed. Dhaliwal, Subramanyam, and rezevan [1999] find no evidence for he US ha comprehensive income is more srongly associaed wih reurns/marke values or beer predics fuure cash flows/income han ne income. hey find some evidence beween reurns and unrealized gains on markeable securiies. Overall, heir resuls do no suppor he claim ha comprehensive income is a beer measure of firm performance han ne income. In conras, Kanagarenam, Mahieu, and Shehaa [2005] using more recen daa find a sronger associaion beween diry surplus and share reurns. Biddle and Choi [2006] repor ha comprehensive income as defined in SFAS 130 dominaes ne income in explaining equiy reurns. Chambers e al. [2007] find ha OCI is value relevan. Invesors price especially wo componens of OCI, foreign currency ranslaion adjusmen and unrealized gains/losses on available-for-sale securiies. Ineresingly, hey find ha markeable securiies adjusmens are valued a a rae greaer han dollar-for-dollar, alhough heory predics, ha hese componens should be purely ransiory (see Ohlson [1999]). Summing up, Chambers e al. [2007] aribue he lack of consisen resuls in research amongs ohers o he differen employed research designs. Concerning diry surplus flows our sudy is mos relaed o Isidro, O Hanlon, and Young [2006]. heir sudy explores he associaion beween valuaion errors from he sandard RIM and violaions of he clean surplus relaion. For he US, hey find weak evidence of he relaionship beween valuaion errors and diry surplus flows by using a wo-sep approach. Firs, a clean surplus RIM based on IBS forecass is employed and second he impac of diry surplus on valuaion errors using realized daa is analyzed. In conras, we follow a one-sep approach by inegraing a diry surplus recion direcly ino he RIM and find ha his significanly increases he fi of he model. In addiion, we inroduce furher recion erms and analyze he impac on several models (i.e., DDM and DCF) as well. Besides diry surplus, previous sudies have poined ou oher violaions of ideal condiions. ransacions wih he equiy owners via capial increases and share repurchases have dramaically increased in he recen pas (see e.g. Fama and French [2001], Grullon and Michaely [2002]). herefore, marke paricipans have o be aware of hese cash disribuions. Neherlands, Kanagarenam, Mahieu, and Shehaa [2005] for Canadian and US firms or Biddle and Choi [2006] and Chambers e al. [2007] for U.S. daa. 4

7 Also our empirical resuls confirm ha an inclusion of hese cash ransfers enhance he precision of he inrinsic value esimaes obained from he DDM. In addiion, our research conribues indirecly o he analysis wheher marke or book values of deb should be used in empirical research. Alhough heory is normally derived in erms of marke values of deb, 3 empirical research ypically relies on book values raher han on marke values (see e.g. Bowman [1979]). 4 his holds rue for he DCF model as well, since i is assumed ha deb is marked o marke under ideal condiions and hus he ne ineres relaion holds. Acding o he ne ineres relaion, he ineres expense can be calculaed by muliplying he ineres bearing deb wih he cos of deb. herefore, we exend he DCF model by inporaing deviaions of accouning cos of deb, i.e., he observed ineres expense on he income saemen from he esimaed cos of deb acding o he ne ineres relaion. Sweeney, Warga, and Winers [1997] provide srong empirical evidence ha book values are a good proxy for he marke values of deb if long-erm bond yields remain raher sable over ime bu can diverge largely during imes of relaively fas ineres rae changes. Since ineres raes are raher sable over our sample period, we expec ha he deviaions of boh measures (accouning vs. economic cos of deb) will be raher small. Finally bu mos imporanly, our sudy is relaed o research on company valuaion, especially o inermodel evaluaions of he DDM, RIM and DCF model. he heoreical equivalence of valuaion echniques has been esablished by differen sudies (e.g. Ohlson [1995], Felham and Ohlson [1995], Penman [1998], Levin and Olsson [2000]), however, primarily by (implicily) assuming ideal condiions. Felham and Ohlson [1995] show ha he DDM, RIM and DCF are equivalen if payoff daa for an infinie horizon are available. Penman [1998] shows ha he RIM and DCF model can be reformulaed in a finie valuaion conex as he DDM, given appropriae erminal value calculaions. Levin and Olsson [2000] and Lundholm and O Keefe [2001a] analyze differen seady sae condiions and heir impac on he valuaion equivalence. Given he heoreical equivalence under resricive assumpions, oher sudies have invesigaed he abiliy of valuaion echniques o obain reasonable esimaes of marke values (e.g. Bernard [1995], Kaplan and Ruback [1995], Frankel and Lee [1998], Sougiannis and Yaekura [2001]). Kaplan and Ruback [1995] explore he abiliy of DCF value esimaes o explain ransacion values of firms engaged in highly leveraged ransacions. hey find ha 3 4.g. mos of he lieraure on capial srucure research saring wih Modigliani and Miller [1958, 1963]. In addiion, Coureau, Kao, and Richardson [2001] menion ha he financial asses are marked o marke is a crucial assumpion in DCF valuaions. 5

8 DCF esimaes significanly ouperform esimaes based on comparable approaches. Penman and Sougiannis [1998] are concerned wih he imporan pracical quesion how he hree inrinsic value mehods perform if hey are applied o a runcaed forecas horizon arising naurally in pracice. Based on an ex-pos-porfolio approach wih realized payoff daa, hey find evidence ha RIM yields he lowes valuaion errors followed by he DDM and DCF model. mploying an ex-ane approach based on analyss forecass Francis, Olsson, and Oswald [2000] provide supporing evidence for he RIM ouperformance. In addiion, Coureau, Kao, and Richardson [2001] compare he DCF model o he RIM approach. Using alue Line (L) daa, hey find ha DCF and RIM do no differ significanly neiher for price nor for non-price based erminal values. Finally, Lundholm and O Keefe [2001a] poin ou ha he empirical findings in he afore menioned sudies are driven by he paricular implemenaion. 5 hey aribue hese mixed findings paricularly o hree reasons: Firs, differen seady sae assumpions in he hree models lead o differen value esimaes. Second, circulariy difficulies occur when he cos of equiy and he weighed average cos of capial (WACC) are independenly deermined in he valuaion process. hird, diry surplus accouning impairs valuaion equivalence. In our sudy we provide amongs oher hings a soluion for he problems menioned by Lundholm and O`Keefe [2001a]. Analyzing individual deviaions from ideal condiions, we derive appropriae adjusmen erms for he hree models, and hus, resore heir empirical equivalence. Beside his, he appealing characerisic of our exended models is ha hey lead o significanly smaller valuaion errors compared o heir sandard counerpars. 3 aluaion Mehods 3.1 aluaion Mehods under Ideal Condiions We consider he hree mos commonly used equiy valuaion echniques, which all are based on he idea ha he value of a share is given by is discouned expeced fuure payoffs. Acding o he firs model, he Dividend Discoun Model (DDM) 6, he marke value of equiy a ime is obained by discouning expeced fuure ne dividends d o shareholders a he cos of equiy r : See also he discussion beween Penman [2001] and Lundholm and O Keefe [2001b] in he Conemporary Accouning Research. his is he sandard model for firm valuaion which is commonly aribued o Williams [1938], Gordon [1959], Gordon and Shapiro [1956]. For ease of noaion, he following valuaion formulas conain only a ime invarian discoun rae and we suppress he (condiional) expecaion operaor in he numeraor. However, in our empirical implemenaion, we employ ime varian discoun raes. 6

9 (DDM) = d +. (1) = 1 ( 1+ r ) Ne dividends include all posiive cash ransfers o shareholders, such as cash dividends or share repurchases, as well as negaive cash ransfers, e.g. due o capial increases. Assuming compliance wih clean surplus accouning he DDM can be ransferred o a second approach, he Residual Income Model (RIM) 8. Boh DDM and RIM yield idenical value esimaes, if he clean surplus relaion holds. he clean surplus relaion (CSR) posulaes ha changes in book value of equiy bv beween wo periods resul exclusively from differences beween earnings x and ne dividends d : (CSR) bv bv 1 x d = +. (2) In oher words, equiy changes can arise exclusively from reenions of earnings or ransacions wih equiy holders. Solving for d in equaion (2) and subsiuing ino he DDM leads under he ransversaliy condiion 9 o he RIM: (RIM) a x + = bv +, (3) ( 1+ r ) = 1 where residual income, also referred o as abnormal earnings, i.e., regular earnings minus a charge for equiy employed. a x, is given by a x = x r bv 1, he hird heoreically equivalen valuaion approach is he Discouned Cash Flow (DCF) 10 model. In order o deermine he marke value, forecass of free cash flows are discouned a an appropriae risk-adjused cos of capial. he DCF approach can be derived from he DDM by combining he CSR and he free cash fcf = oi oa oa. Free Cash Flow fcf is he afer-ax cash flow flow definiion ( ) 1 available o all invesors, i.e., deb and equiy holders. oa denoes ne operaing asses (oal asses minus all non-ineres-bearing liabiliies), oi is he operaing income, defined as all income excep ineres expense x on he ineres-bearing liabiliies in, ne of ax s, i.e. ( ) oi = x + in 1 s See e.g. Preinreich [1938], dwards and Bell [1961], Peasnell [1982]. I.e., he assumpion ( ) lim 1+ r bv 0. + See e.g. Rappapor [1986], Copeland, Koller, and Murrin [1990] and he laes ediion of Koller, Goedhar, and Wessels [2005]. See Lundholm and O Keefe [2001a], pp and p. 333 endnoe 8. 7

10 From he CSR and he free cash flow definiion, he financial asse relaion (FAR) is obained: 12 deb = deb + in 1 s + d fcf, (4) (FAR) ( ) 1 where deb is he sum of ineres-bearing liabiliies and preferred sock. 13 By furher assuming he validiy of he ne ineres relaion (NIR): (NIR) in = rddeb 1, (5) where r D denoes he cos of deb, a DCF model varian, i.e., he well-known ex book WACC approach can be obained (see Appendix 2): fcf + = deb. (6) = 1 ( 1+ rwacc ) Alhough inuiively appealing, he WACC model in equaion (6) is difficul o apply because i requires he esimaion of he weighed average cos of capial r WACC. Since capial weighs have o be derived from marke values, his approach encouners circulariy problems. hese difficulies are avoided by he feasible (implici) WACC model, which is used, for example, by Coureau, Kao, and Richardson [2001] (see Appendix 2): (DCF) ( ) ( 1+ r ) fcf r 1 s deb + r deb = + D deb. (7) = 1 While quaion (7) sill assumes ha deb is marked o marke, i.e., he ne ineres relaion equaion (5) mus hold, i is advanageous since i employs only he equiy cos of capial r, which is also used in he DDM and RIM. hus, all hree models are direcly comparable. 3.2 xended aluaion Mehods under Non-Ideal Condiions he models presened above are based on raher resricive assumpions. In pracice however, we are confroned wih less han ideal condiions, in paricular diry surplus accouning or deviaions from he ne ineres relaion. In addiion, differen seady sae assumpions lead o inconsisencies and can have a remarkable impac on valuaions. herefore, i is necessary o inroduce several recions in order o guaranee ha he hree valuaion mehods remain applicable under less han ideal condiions. Specifically, we derive adjusmens for diry 12 he FAR can be derived by subsiuing bv = oa deb in he clean surplus relaion, subracing his resaed clean surplus relaion oa deb = oa 1 deb 1 + x d from he free cash flow definiion oa oa oi fcf oi x = in 1 s. 13 = + and assuming ( ) 1 In our analysis, we absrac from a disincion beween operaing and financial asses (i.e., rade securiies). See for insance Felham and Ohlson [1995], where financial asses are defined as cash and markeable securiies minus deb. For he reamen of preferred sock as deb, see e.g. Penman [2006]. 8

11 surplus accouning, narrow dividend definiions and ne ineres relaion violaions. In addiion, he exended models allow us o analyze differen seady sae assumpions simulaneously Seady Sae Assumpions and he Calculaion of a erminal alue he DDM, RIM and DCF model equaions in he preceding secion require projecing all fuure payoffs o infiniy, which is impossible in pracice. o circumven his problem, he fuure is ypically divided ino wo periods: an explici forecas period where payoffs are projeced explicily for a limied number of years and a erminal period. he erminal period capures he value beyond he explici forecas period by a erminal value, which is ofen calculaed based on (growing) perpeuiies. 14 I is well known ha an inadequae shor explici forecas horizon, and hus an early erminal value calculaion, leads o inaccurae value esimaes. 15 Our sudy does no focus on his quesion, when seady sae is achieved, alhough by applying differen forecas horizons, we provide some indicaive resuls. Acding o Levin and Olsson [2000] he noion of seady sae can be separaed ino necessary and sufficien condiions. While he former posulaes ha he qualiaive behavior of he company remains consan in he erminal period, i.e., valuaion aribues can be expeced o grow a a consan rae g, he laer condiion focuses on he ineracions of he balance shee and income saemen iems, which boh have o be modeled in a consisen manner. Regarding his issue, Levin and Olsson [2000] and Lundholm and O Keefe [2001a] focus on differen seady sae condiions in heir derivaions. he following seady sae condiions are defined by Levin and Olsson [2000]: 16 (BSS) Balance shee seady sae: BS = (1 + g)bs i,, iem,i iem,i d = 1+ g d, DSS (DSS) Dividend seady sae: ( ) (RSS) Residual income seady sae: x = (1 + g)x, a,rss a (CSS) Cash flow seady sae: 17 CSS cf = (1 + g)cf he erm coninuing value or horizon value is someimes used insead of erminal value in he lieraure. See e.g. Sougiannis and Yaekura [2001]. Saring in period + he responding payoff, i.e., dividend, residual income, cash flow or all iems on he balance shee and income saemen are assumed o grow beyond he explici forecas horizon a he rae g up o infiniy. In conras o Levin and Olsson [2000] we exrapolae he oal numeraor of he DCF model in equaion (7) denoed by cf beyond he explici forecas horizon a (1+g). 9

12 he balance shee seady sae (BSS) definiion responds o he implemenaion in Lundholm and O Keefe [2001a]. I is shown ha his assumpion assures ha he forecased balance shees and income saemens are inernally consisen o each oher. his assumpion implies, ha he reurn on equiy Ro + + (1 g) x + / ((1 g) bv+ ) + + (i.e., for 0 ) and all oher relevan parameers remain consan in he erminal period. In conras, he use of he oher hree seady sae conceps (DSS, RSS and CSS) leads o inconsisencies and consequenly o differen value esimaes. We expand he work of Lundholm and O Keefe [2001a] and Levin and Olsson [2000], by combining eiher DSS, RSS or CSS wih he BSS assumpion in each valuaion formula. his allows us o analyze he impac of differen seady sae assumpions simulaneously and o derive appropriae recion erms. Firs, spliing he infinie forecas horizon ino wo sages leads o he following DDM: d d. (8) DDM = + = 1 ( 1+ r ) ( 1+ r ) ( r g) he firs years represen he explici forecas period and consis of explici and exogenous dividend forecass. In he following erminal period, he dividend is assumed o grow a a consan growh rae g. he esimaion of d is crucial, since a leas wo differen seady sae assumpions can be employed. Acding o he balance shee seady sae (BSS) assumpion, BSS d is obained by leing each line iem on he balance shee (operaing asses, deb, shareholders equiy ec.) and he income saemen (ne income, operaing income, ineres expense ec.) grow a he rae g. his seady sae growh has o be applied firs for period o +1 as well as all subsequen periods. Hence, under ideal condiions (e.g. clean surplus accouning), he DDM saring value of he perpeuiy, which guaranees consisency across he hree approaches, is given by: 18 ( ) ( ) ( ) d = 1+ g x 1+ g bv + bv = 1+ g x g bv. (9) BSS Alernaively, acding o he dividend seady sae assumpion (DSS) he payoff in period ++1 is deermined by: ( ) d = 1+ g d. (10) DSS Combining expressions (9) and (10) as ( ) d = d + d d DSS BSS DSS and insering ino equaion (8) leads o: 18 See Lundholm and O Keefe [2001a]. 10

13 where DDM ( 1+ g) d+ + v+ + 1 ( ) ( ) + = + = 1 ( ) BSS,DDM d (11) 1+ r 1+ r r g ( ) ( ) wih v = 1+ g x g bv 1+ g d, BSS,DDM BSS,DDM v capures he difference beween hese wo seady sae calculaions. Noe, ha his approach means ha all models are implicily based on he balance shee seady assumpion. Sill, our approach is advanageous since i allows analyzing boh seady sae assumpions simulaneously. his procedure is applied o he oher wo models in a similar manner. urning o he RIM, he infinie forecas horizon model (equaion (3)) is divided ino he wo periods as: x x (12) a a RIM = bv + + = 1 ( 1+ r ) ( 1+ r ) ( r g) Under he balance shee seady sae (BSS) assumpion, he final payoff in he RIM is calculaed as: ( ) x = 1+ g x r bv. (13) a,bss Alernaively, assuming residual income seady sae (RSS), he numeraor of he erminal value is given by: ( ) x = (1 + g)x = (1 + g) x r bv. (14) a,rss a Insering hese wo expressions ((13) and (14)) in he same way as above ino equaion (12) resuls in: a ( 1+ g) x + + v+ + 1 ( ) ( ) a RIM + = bv + + = 1 ( ) BSS,RIM x (15) 1+ r 1+ r r g ( ( ) ) wih v = r bv 1+ g bv. BSS,RIM he erminal value adapaion erm represens again he difference beween he wo seady sae assumpions. Finally, he wo-sage version for he DCF model is given by: cf cf DCF = + deb (16) = 1 ( 1+ r ) ( 1+ r ) ( r g) ( ) wih cf = fcf r 1 s deb + r deb, and + + D fcf = oi (oa oa )

14 Again referring o BSS, assuming clean surplus accouning and compliance wih he ne ineres relaion, he numeraor of he perpeuiy in he DCF model is calculaed as: ( ) ( ) ( ) ( ) cf = 1+ g oi 1+ g oa + oa 1+ g r 1 s deb + r deb. (17) BSS D In conras, he exrapolaion of he las payoff acding o he cash flow seady sae (CSS) assumpion resuls in: ( ) ( ) ( ) CSS cf = 1+ g cf+ = 1+ g fcf + rd 1 s deb rdeb + 1 (18) Using he same subsiuions as in he oher wo models yields: DCF ( 1+ g) cf + + v + + ( ) ( ) BSS,DCF cf+ 1 = + deb (19) = 1 ( 1+ r ) 1+ r r g ( ) wih cf = fcf r 1 s deb + r deb, + + D fcf = oi (oa oa ), and ( ) ( ) v = oa 1+ g oa + r deb 1+ g r deb. BSS,DCF Summarizing, his exended approach yields wo sage valuaion formulas for he DDM, RIM and DCF model. Mos imporanly, i is advanageous o oher model specificaions since each model ness boh seady sae assumpions (i.e., he respecive model specific seady sae formula (DSS, RSS or CSS) and in addiion he BSS). In conclusion, implemening he BSS assumpion assures idenical value esimaes and allows us o analyze he impac of oher seady sae assumpions on he accuracy of he value esimaes. Noe ha hese derivaions are obained under ideal condiions, i.e., clean surplus accouning, compliance wih he ne ineres relaion and full payoff informaion like share repurchases and capial conribuions. In order o relax hese resricive consrains, all hree models are nex enhanced o deal wih deviaions from ideal condiions. Specifically, we derive adjusmens for diry surplus accouning, narrow dividend definiions and ne ineres relaion violaions Addiional Model Specific Correcions Dividend Discoun Model Noice ha he dividend d in equaion (11) mus include all cash ransfers beween owners and he firm. If, for simpliciy, only cash dividends are used (as, for example, in Francis, Olsson and Oswald [2000]) a subsanial par of cash ransfers is negleced. 19 o accoun for his, we 19 Noe again ha his is no a criicism, since firs unforunaely alue Line does no provide easily accessable forecass of share repurchase volumes and prices and second he purpose of Francis, Olsson, and Oswald (2001) was o provide evidence how he models perform under common pracice. 12

15 subsiue d = d + d where cash increases and share repurchases. d conains all negleced cash componens, namely capial Moreover, he valuaion equaion (11) requires clean surplus accouning as assumed in equaion (9). Since his relaion is usually violaed under US-GAAP accouning, i is necessary o inporae a diry surplus recion in he erminal period of he DDM. 20 o accoun for diry surplus elemens, we subsiue which is affeced by diry surplus accouning. 21 x = x + dir, where dir dir x denoes he ne income, he diry surplus recion erm dir capures any differences beween he earnings number x, which is calculaed from he clean surplus relaion and he income measure dir x, observed from he income saemen. he clean surplus income x conains all changes in book value of equiy no resuling from ransacions wih he owners. 22 hus, he diry surplus amoun is calculaed as: 23 ( ) dir = x x = bv bv + d + d x. (20) dir cash dir Hence, subsiuing d = d + d for all and x = x + dir leads o he final exended DDM valuaion equaion, (DDM exended ) cash dir ( )( ) 1+ g d + d + dir + v (21) cash BSS,DDM cash DDM d+ + d dir,+ + 1 = + = 1 ( 1+ r ) ( 1+ r ) ( r g) ( ) ( )( ) wih v = 1+ g x g bv 1+ g d + d, BSS,DDM dir cash dir, dir = x x, and + dir d = share repurchases in + capial increases in Clean surplus violaions are e.g. unrealized gains and losses on securiies available for sale, on foreign currency ranslaions or on derivaive insrumens. Alernaive specificaions of diry surplus income can be earnings measures such as comprehensive income acding o SFAS No. 130, ne income before exraordinary iems or ne income before exraordinary iems dir and special iems. In our sudy, we employ ne income as he x measure, because SFAS 130 Reporing of Comprehensive Income became effecive in 1997 and hus is no compleely available for our sample period. For empirical evidence on diry surplus accouning see Appendix 1. Noe ha our approach only implicily deals wih sock opions, since hey are conained in our clean surplus calculaion of x. Since here is no furher daa breakdown in Compusa, we are no capable of disenangling he effecs of sock opion accouning furher. Alernaively, acding o Lo and Lys [2000] he clean surplus earnings can be esimaed as he change of reained earnings afer cash dividends. Alhough, his definiion has o be reaed wih care, since sock dividends, ha are disribuions o shareholders in addiional shares, lead o an increase of paid-in capial and a decrease of reained earnings and hence o a biased disclosure of clean surplus income. Moreover, his approach causes biases by neglecing capial increases. 13

16 Noe ha he dir + erm is necessary only because we need a (diry) income measure o calculae he saring dividend in he erminal period if BSS is assumed. herefore, he diry surplus recion affecs only he erminal value expression and we ge a slighly differen erminal recion as opposed o equaion (11). Boh recions required simulaneously in he erminal period. dir + and BSS,DDM v dir are So far, we have employed a simple perpeuiy wih growh in he erminal value expression. Alernaively, acding o Penman [1998] a discouned -year ahead sock price forecas 24 could be employed o subsiue he erminal value calculaion. Using his price-based erminal value insead of he growh rae based perpeuiy (i.e., he so called non-price-based erminal value ) he exended DDM is equal o: (DDM exended-price ) d + d P. (22) cash DDM,Price = + = 1 ( 1+ r ) ( 1+ r ) In conras o he DDM exended implemenaion, he recion erms obviously unnecessary, if such a price-based valuaion is employed. dir and BSS,DDM v dir are Residual Income Model If he clean surplus relaion is violaed under US-GAAP accouning i can be seen from equaion (23) ha a diry surplus recion should also be inporaed in he RIM approach. Noe ha ( ) ( ) x = x r bv = x + dir r bv a dir 1 1 = x r bv + dir = x + dir dir a,dir 1 a x is calculaed on he supposiion ha he clean surplus relaion holds. Hence, consiss of a residual income resuling from he usage of an acually observed income measure dir x and a diry surplus recion (23) dir. In conras o he DDM, clean surplus violaions have o be inporaed during he explici forecas period as well as he erminal period. he exended RIM implemenaion, which capures he difference beween he seady sae assumpions and he diry surplus recion, consequenly resuls in: (RIM exended ) ( )( ) 1+ g x + dir + v (24) a,dir a,dir BSS,RIM RIM x+ + dir = bv + + = 1 ( 1+ r ) ( 1+ r ) ( r g) ( ( ) ) wih v = r bv 1+ g bv, and BSS,RIM dir = x x. dir a x 24 Providers of long-erm price forecass are e.g. alueline. 14

17 If a erminal sock price forecas is available, he exended RIM employing a price-basederminal value is given by: (RIM exended-price ) x + dir P bv. (25) a,dir RIM,Price = bv + + = 1 ( 1+ r ) ( 1+ r ) he ideal price-based erminal value is he difference beween he forecased marke price of he sock and he book value of equiy a he horizon +. A posiive premium [ P bv ] indicaes accouning conservaism or posiive ne presen value projecs in he fuure. + + Discouned Cash Flow Model In line wih he DDM and RIM, diry surplus accouning necessiaes he inclusion of an appropriae recion erm in he DCF approach: fcf = fcf + dir = oi (oa oa ) + dir. (26) dir dir 1 quaion (26) saes ha he free cash flow calculaed on he assumpion of clean surplus accouning consiss of he diry surplus free cash flow dir fcf, which is calculaed indirecly saring from he ne income dir x, and he dir erm. By inporaing equaion (26) ino equaion (19), he modified DCF model, which explicily regards diry surplus accouning, is given by: ( )( ) 1+ g cf + dir + v dir BSS,DCF dir DCF cf+ + dir = + deb (27) = 1 ( 1+ r ) ( 1+ r ) ( r g) ( ) wih cf = fcf r 1 s deb + r deb, dir dir + + D fcf = oi (oa oa ), dir dir ( ) ( ) v = oa 1+ g oa + r deb 1+ g r deb, and BSS,DCF dir = x x. dir Nex, if deb is no marked o marke he ineres expense of a paricular period canno be deermined acding o he ne ineres relaion (NIR) in = r deb and hus one NIR D 1 assumpion of he WACC model is violaed. o accoun for he possible deviaion beween ineres expense from he income saemen NIR in, a las new recion erm, namely IS in and ineres expense acding o he NIR nir, is inporaed ino he DCF model: ( )( ) ( )( ) nir = in in 1 s = r deb in 1 s. (28) NIR IS IS D 1 Accouning for he ne ineres relaion adjusmen nir in equaion (27) leads o he following final exended DCF model: 15

18 (DCF exended ) cf + dir + nir = + dir DCF = 1 ( 1+ r ) ( )( ) dir BSS,DCF 1+ g cf+ + dir + + nir + + v deb ( 1+ r ) ( r g) (29) ( ) wih cf = fcf r 1 s deb + r deb, dir dir + + D fcf = oi (oa oa ), dir dir ( ) ( ) v = oa 1+ g oa + r deb 1+ g r deb, BSS,DCF dir = x x, and dir IS ( )( ) = ( D )( ) nir = in in NIR IS s r deb in 1 s. Again, if a erminal sock price forecas for ime + is available, he ideal price-based erminal value is he discouned sum of [ P deb ] coninuing value is hen given by: +. he DCF model using a price-based + + (DCF exended-price ) cf + dir + nir P + deb dir DCF = + deb. (30) = 1 ( 1+ r ) ( 1+ r ) 3.3 Special Cases of he xended aluaion Mehods: he Sandard Models As already menioned in he inroducion each exended valuaion model ness is sandard model counerpar. herefore as a saring poin for model evaluaion purposes, we inroduce he sandard models, where all he above given recions are negleced. he sandard DDM considers only cash dividends, i.e., a narrow dividend definiion, by leaving ou he (DDM sandard ) d, (DDM sandard-price ) dir and DDM BSS,DDM v erms. cash ( 1+ g) d+ ( ) ( ) d = + (31) 1+ r 1+ r r g cash + = 1 ( ) d P (32) cash DDM,Price + + = + = 1 ( 1+ r ) ( 1+ r ) he sandard RIM implemenaion absracs from he diry surplus and erminal value adjusmen. (RIM sandard ) a,dir ( 1+ g) x + ( ) ( ) x (33) 1+ r 1+ r r g a,dir RIM + = bv + + = 1 ( ) 16

19 (RIM sandard-price ) x P bv (34) a,dir RIM,Price = bv + + = 1 ( 1+ r ) ( 1+ r ) Finally, he sandard DCF model disregards he recions for diry surplus, violaions of he ne ineres relaion and he erminal value calculaion. (DCF sandard ) (DCF sandard-price ) DCF dir ( 1+ g) cf+ ( ) ( ) cf = + dir + deb (35) = 1 ( 1+ r ) 1+ r r g cf P + deb dir DCF,Price = + deb (36) = 1 ( 1+ r ) ( 1+ r ) For convenience, able 1 summarizes he differen recion erms used in he DDM, RIM and DCF model. [Inser able 1 abou here] Furhermore, Appendix 3 provides an illusraive example where he recion erms are calculaed for a specific firm, namely he 3M Corporaion. 4 mpirical Analysis 4.1 Research Design and Daa Descripion We use daa from COMPUSA Annual and Research Files conaining companies lised a he New York Sock xchange (NYS), he American xchange (AMX) and he Naional Associaion of Securiies Dealers Quoaions (NASDAQ) marke. Our sudy comprises he ime period from 1987 o 2004, mainly because credi raings needed o calculae cos of deb are no available before In line wih oher sudies, financial companies (SIC codes 6000 o 6999) are excluded from he sample due o heir differen characerisics. Furhermore, we exclude companies wih negaive equiy book values, share values smaller han $1.00 and fewer han 1 million shares ousanding. his selecion procedure avoids largely disorions due o ouliers and hus yields more robus model esimaes. 26 In oal, we obain 36,112 company years consising of 4,285 differen companies. he number of companies ranges Credi raings are only sparsely available for firms in COMPUSA beginning in 1985/1986 and more reliably available saring in Overall, our daa selecion procedure is comparable o mos oher sudies (e.g. Frankel and Lee [1998]). However, Bhojraj and Lee [2002] or Liu, Nissim, and homas [2002] impose more severe resricions wih regard o COMPUSA daa. 17

20 from 1,530 companies in o 2,335 companies in 1996 (see able 2 for addiional deails). 28 he payoff definiions and heir implemenaion wih COMPUSA daa are given in Appendix 4. Following Penman and Sougiannis [1998] we use realized payoff daa (i.e., a perfec foresigh seing) in connecion wih a porfolio approach. Using realized daa insead of forecass is advanageous for several reasons. Firs, i leads o a larger daabase. For example, we can analyze four imes more companies per year in conras o he sudies of Francis, Olsson and Oswald [2000] and Coureau, Kao, and Richardson [2001] who use analys forecass from alue Line. 29 Second, i is well known ha forecass for several iems (e.g. dividends, book values of equiy and earnings, ec.) are no necessarily consisen o each oher (see e.g. Coureau, Kao, and Richardson [2001]). Moreover, analyss forecass can be biased (see e.g. Chan, Karceski, and Lakonishok [2003]). Biases and inconsisencies in analyss forecass, however, are problems we do no wan o address here since hey add unnecessary complexiy o he comparison of he hree models. hird, using realized daa allows exac measuremen of diry surplus amouns, growh raes of payoffs and oher imporan inpu variables such as capial expendiures, free cash flows, dividends, capial increases, share repurchases and earnings. Neverheless, realized daa do no perfecly mach expecaions. However, he use of realized daa is jusifiable as long as he ex pos observed payoffs mach expeced values. Presuming ha deviaions of realized from expeced values cancel ou on average, he companies are grouped ino 20 porfolios. his resuls in an average number of firms of 69 per porfolio (over all years). Companies are assigned randomly o individual porfolios in order o calculae he presen value for a paricular year. he porfolio composiion is mainained hroughou all periods associaed wih a single valuaion. o compue presen values for subsequen years, he evaluaion window is moved ahead and companies are assigned randomly o porfolios, again. For each porfolio he average relevan figure (cash flows, earnings, dividends, ec.) is compued for each horizon up o 10 subsequen years ( +, = 2,...,10) and discouned a he average coss of equiy capial in order o obain an average presen value per porfolio ven if no daa requiremens are made, he number of observable firms from he COMPUSA Annual and Research Files has decreased in he las years of our sample period. Compared o Penman and Sougiannis [1998] our sample conains fewer companies. his is aribuable o he fac, ha more COMPUSA iems are used han in heir sudy. alue Line forecass abou 1,600 US companies. Francis, Olsson, and Oswald [2000] examine abou 600, Coureau, Kao, and Richardson [2001] examine 422 companies per year for a five year evaluaion period. 18

21 We esimae cos of equiy r using he annualized one-year reasury bill rae as he risk-free rae and hen adding Fama and French s [1997] indusry specific risk premiums (48 indusry code). his resuls in a ime invarian risk premium of 6.60% on average, ranging from 1.5% for Drugs o 12.2% for Fabricaed Producs. he average median cos of equiy is 11.36%. 30 For he cos of deb r D we use Reuers indusrial porae spread daa. Unforunaely, he firm specific raing informaion can be obained from COMPUSA only for a sub-sample of 14,675 firm years. We replace missing raing informaion by he median raing of firms in responding indusries using he Fama/French 48 indusry classificaion. We hen calculae cos of deb by adding Reuers 5 year spreads o he risk free rae. 31 As shown in able 2 average median cos of deb over all years is approximaely 6.5% and he median company raing is BBB. In line wih Kaplan and Ruback [1995], Penman and Sougiannis [1998], Francis, Olsson, and Oswald [2000] and ohers, we evaluae he valuaion echniques by comparing acual raded prices wih inrinsic values calculaed from payoffs prescribed by he echniques. Assuming marke efficiency he marke capializaion is an appropriae crierion o evaluae he model performance. he signed predicion error (bias) denoes he deviaion of inrinsic value esimae a from share price a. his error is defined as bias = (price inrinsic value esimae ) / price. he absolue predicion error is calculaed as accuracy = price inrinsic value esimae / price. Noe ha a posiive bias indicaes ha he inrinsic value is smaller han he marke price. [Inser able 2 abou here] Furhermore, summary saisics on he mos imporan inpu variables are given in able 2. For example, he companies equiy book value is $6.66 per share compared o an average median deb level of $2.90 per share. hus firms are mainly equiy financed (median leverage raio based on book values amouns o 0.44 = 2.90/6.66). he median marke value of equiy varies beween $96.85 million in 1987 and $ million in Several sensiiviy ess of our resuls are performed. he coss of equiy were also compued based on a 10- year -bond rae as risk-free ineres rae and an alernaive risk premium in erms of a marke premium of 6% (see Ibboson and Sinquefield [1993]) in conjuncion wih a CAPM firm specific risk componen (rolling bea-esimaion). Moreover, an analysis was performed wih a uniform cos of equiy rae for all companies and years of 10%, 11%, 12% and 13%. he empirical resuls (no repored) for our sample do no reac sensiively o he choice of he coss of capial, alhough some minor level effecs concerning he bias and inaccuracy are obviously observed. 5 years is a reasonable assumpion acding o he findings of Sohs and Mauer [1996]. 19

22 We observe an average median book o marke raio of 57% suggesing ha he sample firms follow conservaive financial reporing. Median cash dividend paymens per share of dividend paying firms range from $0.30 in 1987 o $0.46 in 2004 wih a dividend payou raio varying beween 42% (in 1991) o 30% (in 2000 and 2004). Average median ne dividends per share ($0.41) urn ou o be higher han cash dividends per share ($0.37) because share repurchases exceed capial increases. Median free cash flow is equal o $0.12 per share and median residual income is $0.04 on average. As expeced, in he years of he echnology bubble (especially 2001 and 2002) he residual income per share is negaive and he reurn on asses (ROA) is comparaively small. Overall, our sample has similar characerisics as in oher sudies (see e.g. Frankel and Lee [1998]). 4.2 mpirical Resuls aluaion rrors able 3 repors average valuaion errors bias (Panel A) and inaccuracy (Panel B) for he hree exended valuaion approaches in comparison o he sandard model implemenaion for a +6 forecas horizon. [Inser able 3 abou here] Mos imporanly, all hree exended models perform subsanially beer han heir sandard counerpars in erms of bias and inaccuracy. In paricular, he huge average bias associaed wih he sandard DCF model can be reduced remarkably by implemening our exended model version (from 78% o 17% for a seady sae growh rae of 2%). Similarly, large gains wih respec o bias are observed for he DDM (from 54% o 17%) and even for RIM bias is reduced by half (35% o 17%). Clearly, gains in bias are less pronounced for he price-based models, which is consisen wih Coureau, Kao, and Richardson [2001]. A similar picure is observed regarding absolue valuaion errors (Panel B), alhough inaccuracy is of greaer magniude as compared o bias and differences beween he models are less pronounced. Overall, implemening our exended valuaion models yields idenical valuaion resuls (e.g. a bias of 17% for 2% growh) being associaed wih subsanial reducions in mean valuaion errors even when compared o he bes sandard model. o evaluae he robusness ha he exended models perform beer han heir sandard counerpars, we repea he above analysis on year by year basis (see Figure 1). In general, as 20

23 in able 3 he exended models provide considerably smaller valuaion errors han heir sandard counerpars even on a year-by-year comparison. he only excepion is observed for he DCF model in 1996 when he sandard DCF model slighly ouperforms he exended model version in erms of bias. Ineresingly, sandard DDM produces he mos sable errors, underesimaing marke values by an almos consan 60%. his resul is in line wih expecaions since cash dividends are generally smoohed over ime. In conras, sandard DCF produces more volaile average valuaion errors, exceeding he oher models errors in every single year, whereas sandard RIM comes closes o he exended model. [Inser Figure 1 abou here] As a furher robusness check, we analyze wheher he observed improvemen in valuaion errors depends on forecas horizons. able 4 provides bias (Panel A) and inaccuracy (Panel B) for differen forecas horizons (+2, +4, +8, +10). In line wih he monooniciy-propery developed by Ohlson and Zhang [1999], we observe ha valuaion errors for he sandard as well as he exended models decline seadily wih a longer finie forecas horizon. For example, employing a non-price-based erminal value wih a 2% growh rae he bias of he exended models declines seadily from 29% o 7% wih an increasing forecas horizon (+2 o +10, respecively). A similar seady decline, alhough on a higher level, is observed for he hree sandard models. [Inser able 4 abou here] Again, a similar picure is obained regarding mean inaccuracy (Panel B). Concerning he exended models, inaccuracy for he non-price-based approaches again declines from 52% o 39% (g=2%). In conras, while also declining, he inaccuracy of sandard DCF is abou wice as high, of sandard DDM abou 10 percenage poins higher, whereas almos no difference is observed for RIM. Overall, he above resuls sugges ha he exended models provide considerable advanages. hey lead o more precise valuaion esimaes and hus smaller valuaion errors. his resul is robus for differen sampling periods and differen forecas horizons. Moreover, relaive and 21

24 absolue valuaion errors are for our exended models (especially DDM and DCF) considerably smaller han previously repored Robusness of Sandard Models agains iolaions of Ideal Condiions Besides yielding lower valuaion errors, a second major advanage of he exended models is he resored valuaion equivalence. his provides a benchmark for analyzing o which exen he sandard models are affeced by specific violaions of he underlying assumpions. he resuls of such an analysis are given in able 5, which provides an assessmen of he relaive imporance of each valuaion componen. Firs, we analyze he absolue Dollar amoun and heir respecive percenage share of inrinsic value esimae for each componen in able 5 (Panel A). Second, heir responding impac on he valuaion bias is evaluaed in Panel B. As before, calculaions are based on a 6-year explici forecas horizon wih subsequen erminal value. [Inser able 5 abou here] In he DDM he hree recions d, dir and BSS v (capuring he difference beween he seady sae assumpions BSS and DSS) are nearly equally imporan. he dividend recion alone accouns for 16% of he inrinsic value (for g = 2%). In conras, in he RIM he book value of equiy and he presen value of residual income accoun wih 76% for a very large fracion of he inrinsic value. In he RIM and he DCF model he diry surplus recion dir is idenical and represens nearly a quarer of he inrinsic value. In comparison, he respecive diry surplus recion in he DDM is smaller since his recion occurs only in he erminal period. herefore, i is paricularly imporan o rec for diry surplus in he RIM and DCF model. In he DCF model he presen value of he recion componens ogeher (i.e., he sum of dir, nir, and BSS v ) accouns for 75% of he mean inrinsic value esimae. By far he larges recion erm is BSS v wih 55% of he inrinsic value esimae. his resul highlighs he imporance of a reasonable seady sae assumpion wihin he erminal value calculaion of he DCF model and demonsraes ha he CSS condiion leads o heavy disorions of he inrinsic value esimae. Whereas he recion for violaions of he ne ineres relaion nir in he DCF model is wih -5% raher small (he negaive sign 32 For insance, Penman and Sougiannis [1998] repor a bias for he DDM of 31.4%, for he RIM of 8.3% and he DCF model of 111.2% assuming a +4 forecas horizon and no growh in he erminal period. Francis, Olsson, and Oswald [2000] repor a bias (inaccuracy) based on analys forecass of 75.5% (75.8%), 20.0% (33.1%), 31.5% (48.5%) for he DDM, RIM and DCF model, respecively. 22