Equity Valuation Employing the Ideal versus Ad Hoc - Berkeley-Haas

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Equity Valuation Employing the Ideal versus
Ad Hoc Terminal Value Expressions*
LUCIE COURTEAU, Université Laval
JENNIFER L. KAO, University of Alberta
GORDON D. RICHARDSON, University of Waterloo
Abstract
Recently, Penman and Sougiannis (1998) and Francis, Olsson, and Oswald (2000) compared
the bias and accuracy of the discounted cash flow model (DCF) and Edwards-Bell-Ohlson
residual income model (RIM) in explaining the relation between value estimates and
observed stock prices. Both studies report that, with non–price-based terminal values, RIM
outperforms DCF.
Our first research objective is to explore the question whether, over a five-year valuation horizon, DCF and RIM are empirically equivalent when Penman’s (1997) theoretically
“ideal” terminal value expressions are employed in each model. Using Value Line terminal
stock price forecasts at the horizon to proxy for such values, we find empirical support for
the prediction of equivalence between these valuation models. Thus, the apparent superiority of RIM does not hold in a level playing field comparison.
Our second research objective is to demonstrate that, within each class of the DCF and
RIM valuation models, the model that employs Value Line forecasted price in the terminal
value expression generates the lowest prediction errors, compared with models that employ
non–price-based terminal values under arbitrary growth assumptions. The results indicate
*
Accepted by Jerry Feltham. This paper was presented at the 2000 Contemporary Accounting
Research Conference, generously supported by the CGA-Canada Research Foundation, the
Canadian Institute of Chartered Accountants, the Society of Management Accountants of
Canada, the Certified General Accountants of British Columbia, the Certified Management
Accountants Society of British Columbia, and the Institute of Chartered Accountants of
British Columbia. We would like to thank workshop participants at the 2000 American Accounting Association meetings; 2000 Canadian Academic Accounting Association Conference; 2000
Contemporary Accounting Research Conference; 2000 European Accounting Association Conference; HEC, Laval; University of Queensland, University of Technology–Sydney; and University
of Waterloo for their comments.
Special thanks are extended to Sati Bandyopadhyay, Joy Begley, Brian Bushee, Peter Clarkson, Steve Fortin, Kin Lo, Russell Lundholm (the discussant), Pat O’Brien, Terry O’Keefe, Steve
Penman, Ranjini Sivakumar, Theodore Sougiannis, Ken Vetzal, and especially Jerry Feltham (the
editor) for their helpful comments and suggestions on earlier versions of the paper; Kendrick
Fiorito and Mort Siegel at Value Line for their advice on the project; Nick Favron for programming assistance; and Daniel Roy and Nicole Sirois for their excellent research assistance.
The research is supported by the Social Sciences and Humanities Research Council of Canada
and the Canadian Academic Accounting Association. Jennifer Kao also receives financial support
from Canadian Utilities Fellowship for this project. All remaining errors are the authors’ sole
responsibility.
Contemporary Accounting Research Vol. 18 No. 4 (Winter 2001) pp. 625–61 © CAAA
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Contemporary Accounting Research
that, for both DCF and RIM, price-based valuation models outperform the corresponding
non – price-based models by a wide margin. These results imply that researchers should
exercise care in interpreting findings from models using ad hoc terminal value expressions.
Keywords Financial information; Residual income model; Terminal values; Valuation
Condensé
Penman et Sougiannis (1998, ci-après P&S) comparaient récemment la distorsion et la
précision du modèle d’actualisation des flux de trésorerie (DCF) et du modèle des bénéfices
résiduels d’Edwards, Bell et Ohlson (RIM) dans l’explication de la relation entre les
estimations de valeur et le cours observé des actions. Utilisant les bénéfices futurs réels
comme mesure des bénéfices attendus, P&S (1998) constatent que les erreurs d’évaluation
du modèle DCF sur un horizon de 10 ans excèdent largement celles du modèle RIM. Ils
attribuent ce résultat au fait que des montants comptabilisés conformément aux PCGR dans
le modèle RIM permettent une prise en compte plus rapide de flux de trésorerie futurs, de
sorte que leur pertinence à l’égard de la valeur est plus grande que celle des flux de trésorerie ou des dividendes. Francis, Olsson et Oswald (2000) jettent un nouveau regard sur ce
parallèle, en recourant à une méthode ex ante et aux prévisions de Value Line (VL), pour
conclure à leur tour que, lorsque les valeurs finales ne sont pas fondées sur les prévisions du
cours des actions, l’efficacité du RIM est supérieure à celle du DCF.
Le premier objectif des auteurs est de vérifier si, sur un horizon prévisionnel de cinq
ans, le DCF et le RIM sont empiriquement équivalents, lorsqu’on utilise les expressions de
valeur finale théoriquement « idéales » de Penman (1997), dans l’application de chacun des
modèles. Ces expressions de valeur nécessitent le cours du marché prévu (P ) au terme de
l’horizon prévisionnel et l’excédent de ce cours sur la valeur comptable, pour un système
comptable donné. L’équivalence des modèles DCF et RIM pour des horizons finis et dans
des conditions idéales, malgré qu’elle soit bien établie en théorie, n’a pas été démontrée
dans les études empiriques. Au premier abord, les arguments semblent circulaires : si des
prévisions fiables de cours sont disponibles, le modèle d’actualisation des dividendes
(DDM) devrait suffire, et il n’est pas nécessaire de recourir au DCF ou au RIM. La chose
n’est cependant pas évidente, du fait que les prévisions de cours formulées par le marché
ne sont pas observables ; les auteurs utilisent donc les prévisions de cours final de VL comme
substitut. Bien que ces prévisions soient loin d’être idéales et qu’elles puissent contenir des
erreurs de distorsion ou de mesure (voir Abarbanell et Bernard, 2000), les auteurs font
l’hypothèse que toute erreur de distorsion ou de mesure serait un facteur constant dans les
comparaisons entre DCF et RIM. Ils supposent également, comme P&S (1998) et Francis et
al. (2000), que le marché est efficient.
Le deuxième objectif des auteurs consiste à démontrer que les valeurs intrinsèques
calculées à l’aide des prévisions de cours final de VL donnent lieu à des erreurs d’évaluation
plus modestes que les valeurs intrinsèques déterminées en fonction des expressions de
valeur finale improvisées. Les expressions simples de perpétuité, qui supposent que les
bénéfices anormaux postérieurs à l’horizon prévisionnel croîtront soit au taux de 0 pour cent,
soit au taux nominal d’inflation, ont été amplement utilisées dans les recherches empiriques
(par Francis et al., 2000, et par Frankel et Lee, 1998, entre autres). Gebhardt, Lee et Swaminathan (2001) utilisent un procédé de taux de décroissance qui est aussi problématique que
CAR Vol. 18 No. 4 (Winter 2001)
Equity Valuation Employing Terminal Value Expressions
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les expressions basées sur la perpétuité, étant donné qu’il suppose que le rendement anormal
du capital investi postérieur à l’horizon prévisionnel convergera vers la moyenne du secteur
d’activité, sur une période de sept ans. Les auteurs constatent que les valeurs finales sont en
moyenne sensiblement sous-évaluées lorsque des estimations improvisées de l’achalandage
au terme de l’horizon prévisionnel sont utilisées, ce qui suppose que les estimations de la
valeur intrinsèque (Frankel et Lee, 1998) ou du coût du capital ex ante (Gebhardt et al.,
2001) sont sous-évaluées lorsque de telles expressions de valeur finale sont employées. Ces
inférences peuvent être importantes, selon le but visé par la recherche, et elles demeurent
valides même dans les dernières années de la période étudiée, une fois que s’est atténué
l’optimisme de VL dans les prévisions de cours.
L’échantillon des auteurs compte 422 sociétés (ou 2 110 exercices-société), à l’égard
desquelles ils disposent de données prévisionnelles et historiques complètes pour toute la
durée de la période couverte. Le fait que l’échantillon soit constant et que les mesures se
répètent au fil d’une période donnée pour les mêmes sociétés retient l’attention des auteurs
qui empruntent la méthodologie de l’échantillon constant (voir Kmenta, 1986) exploitant les
autocorrélations dans les données pour réaliser certains tests statistiques. Le coût des capitaux propres est calculé à l’aide du modèle d’évaluation des actifs financiers. Le taux sans
risque est mesuré comme étant le taux constant à l’échéance des bons du trésor de cinq ans,
au début du mois de prévision, provenant de la base de données de la Chicago Federal
Reserve Bank, et la prime de risque est mesurée comme étant le produit du bêta de l’entreprise
fourni par VL et de la prime de marché historique approximative de 6 pour cent. Les auteurs
utilisent les premières prévisions complètes de VL, habituellement publiées au troisième
trimestre de l’exercice de l’entreprise. À l’instar de Francis et al. (2000), les auteurs
actualisent les prévisions de VL pour les attributs d’évaluation du n e exercice en utilisant un
facteur de (n − 1 + f ), où f représente la portion d’exercice se situant entre la date où sont
faites les prévisions et la clôture du premier exercice. Étant donné que tous les modèles
d’évaluation exigent des valeurs comptables à la date de la prévision, ce que VL ne fournit
pas directement, il faut intrapoler les valeurs comptables (les actifs financiers nets) à cette
date pour le RIM (le DCF), à partir de leur valeur au début de l’exercice de prévision et des
prévisions de VL relatives aux variables de l’exercice courant. VL publie des prévisions
pour trois horizons : l’exercice en cours (soit l’exercice 1), l’exercice suivant (soit l’exercice
2) et le long terme (soit l’exercice 5). Étant donné que les prévisions annuelles des attributs
d’évaluation pour les exercices 3 et 4 ne sont pas publiées dans le Value Line Investment
Survey, à la suggestion des analystes de VL, les auteurs intrapolent de façon linéaire les
données relatives à ces deux exercices, en fonction de la croissance prévue entre l’exercice 2
et l’exercice 5. Le cours le plus récent rapporté par VL est utilisé dans cette étude comme
variable dépendante. Dans le cas du RIM comme dans celui du DCF, le chercheur doit parfois
faire face à la situation où les valeurs finales sont négatives. Le cas peut se produire si l’un
ou l’autre des excédents prévus du cours sur la valeur comptable, compte tenu des prévisions de cours final ou des attributs de l’évaluation au terme de l’horizon prévisionnel, est
négatif. Les auteurs choisissent de ne pas plafonner les valeurs finales négatives à zéro parce
que tout attribut négatif au terme de l’horizon prévisionnel devrait être intégré dans le cours
du marché en vigueur.
Les tests statistiques révèlent que, pour l’ensemble de l’échantillon, les erreurs
prévisionnelles absolues médianes sont de 13,71 pour cent et de 14,18 pour cent respectivement
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pour le DCF et le RIM. Ainsi, en mettant l’accent sur la précision, on ne constate pas de
supériorité du RIM sur le DCF lorsque les valeurs finales idéales sont employées. Toute
« mise à l’épreuve » du RIM et du DCF risque de placer ce dernier en situation défavorable
en raison des limites inhérentes à l’estimation de la version de Copeland et al. (1995) du
modèle financier des flux de trésorerie disponibles, une méthode employée par Francis et al.
(2000) qui fait appel aux données de VL. Pour résoudre ce problème, les auteurs estiment la
version du DCF proposée par Penman (1997). Cette spécification particulière convient
mieux aux données de VL parce que l’attribut d’évaluation — les flux de trésorerie disponibles aux actionnaires ordinaires — peut être tiré directement des données VL et qu’il
n’est pas nécessaire d’estimer le coût moyen pondéré du capital. Les auteurs formulent
néanmoins une mise en garde : l’application de la version du DCF proposée par Penman,
qui fait un usage empirique des données de VL, peut encore contenir des erreurs de mesure
étant donné que VL ne fournit pas de prévisions relatives aux impôts sur le revenu reportés ou
aux sommes immobilisées dans le fonds de roulement. Compte tenu de ces limitations
pratiques de l’estimation du DCF, il est assez remarquable d’avoir pu établir une équivalence
approximative entre le DCF et le RIM.
Mettant en opposition les valeurs intrinsèques des modèles qui utilisent les prévisions
de cours final et les valeurs intrinsèques des modèles qui ne le font pas, les auteurs constatent
que, tant pour le DCF que pour le RIM, l’efficacité des modèles d’évaluation basés sur les
prévisions de cours surpasse de beaucoup celle des modèles correspondants qui ne
s’appuient pas sur les cours. Bien sûr, l’utilisation des prévisions de cours VL comme point
de repère n’est pas valide si ces prévisions sont optimistes. Toutefois, même pour les deux
dernières années incluses dans l’étude (1995 – 1996), au moment où l’optimisme des prévisions de cours de VL se trouve ramené à un niveau négligeable, l’erreur prévisionnelle
médiane continue d’indiquer une importante distorsion à la baisse lorsque les expressions
improvisées de valeur finale sont utilisées. Ces résultats donnent à penser que les chercheurs
qui étudient le coût du capital ex ante ou les stratégies d’investissement faisant appel aux
expressions simplifiées de valeur finale pour le RIM devraient interpréter leurs résultats
avec une certaine prudence.
En remplaçant les prévisions de cours final de VL par des expressions traditionnelles
de valeur finale à l’aide d’estimations de croissance simples, semblables à celles qu’emploient
P&S (1998) et Francis et al. (2000), les auteurs sont en mesure de reproduire les constatations
précédentes selon lesquelles le RIM surpasse le DCF en efficacité. Ainsi, dans la régression
des cours en vigueur en fonction des valeurs intrinsèques, la valeur de R2 est supérieure dans
le cas du RIM comparativement au DCF (par exemple, 79,65 pour cent contre 67,95 pour
cent, et 77,02 pour cent contre 60,46 pour cent, lorsque les hypothèses de croissance sont
respectivement de 0 et de 2 pour cent). La supériorité du RIM sur le DCF lorsque la valeur
finale idéale n’est pas disponible est expliquée par P&S (1998) dans les termes suivants : la
valeur comptable actuelle des capitaux propres inclut déjà une partie des flux de trésorerie
futurs et laisse relativement peu de valeur à encaisser au terme de l’horizon prévisionnel. Par
exemple, selon l’hypothèse de croissance de 0 pour cent (2 pour cent), 20,77 pour cent (25,53
pour cent) de la valeur intrinsèque dans le cas du RIM est dérivée de la valeur finale actualisée,
le chiffre correspondant dans le cas du DCF étant de 91,81 pour cent (93,19 pour cent).
Lundholm et O’Keefe (2001, ci-après L&O) relèvent des erreurs dans l’application des
modèles du RIM et du DCF qui se soldent par des estimations incohérentes de la valeur
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intrinsèque. Bien que leurs travaux mettent surtout en relief l’importance d’éviter ces
écueils, L&O affirment que l’équivalence du RIM et du DCF est « assurée ». L’équivalence
du RIM et du DCF lorsqu’on emploie les expressions de valeur finale idéale, ne peut se confirmer que si 1) la même prévision des cours de VL est utilisée dans les deux modèles et si
2) le chercheur évite les écueils évoqués par L&O. L’un des écueils qu’évoquent L&O, sans
proposer de solution pratique, est celui de la difficulté de l’estimation du coût moyen pondéré
du capital pour le modèle DCF. La version Penman du modèle DCF utilisée par les auteurs
minimise le risque d’erreurs dans l’application du modèle DCF et n’exige pas l’estimation
du coût moyen pondéré du capital. Fait d’égale importance, elle ne fait pas intervenir
l’hypothèse traditionnelle des versions empiriques précédentes du modèle des flux de
trésorerie disponibles selon laquelle les actifs financiers nets sont comptabilisés à la valeur
du marché. S’ils ne le sont pas, le chercheur doit enrichir le modèle traditionnel des flux de
trésorerie disponibles en y ajoutant un terme qui représente la différence entre la juste valeur
et la valeur comptable des actifs financiers nets. Sans ce terme, l’équivalence entre le DCF
et le RIM ne peut être démontrée. Un autre des apports de cette étude est qu’elle indique aux
chercheurs comment réduire au minimum les incohérences dans l’estimation des valeurs
intrinsèques grâce à l’utilisation des données de VL dans l’application du modèle DCF.
Les questions de recherche de cette étude ont une pertinence pratique. L’équivalence
du DCF et du RIM est tenue pour acquise par ceux et celles qui étudient l’analyse des états
financiers, et la démonstration empirique de cette équivalence est utile. Bien sûr, dans le cas
d’un horizon fini, l’équivalence n’est possible qu’avec des prévisions de cours final, faute de
quoi il convient de déterminer avec soin la longueur de l’horizon prévisionnel et la forme
de l’expression de la valeur finale axée sur les cours, ainsi que l’ont démontré les travaux
antérieurs.
1. Introduction
Recently, Penman and Sougiannis (1998; hereafter P&S) compared the bias and
accuracy of the discounted cash flow model (DCF) and Edwards-Bell-Ohlson
residual income model (RIM) in explaining the relation between value estimates
and observed stock prices. Using a perfect foresight approach, P&S find that valuation errors for DCF over a 10-year horizon exceed those of RIM by a substantial
margin. They attribute this result to generally accepted accounting principles
(GAAP)-based accounting accruals under RIM, which bring future cash flows forward and hence are more value-relevant than either cash flows or dividends. Francis, Olsson, and Oswald (2000) take a second look at that comparison using an ex
ante approach and Value Line (VL) forecasts and also conclude that with non –
price-based terminal values RIM outperforms DCF.
Our first research objective is to explore whether, over a five-year valuation
horizon, DCF and RIM are empirically equivalent using Penman’s (1997) theoretically “ideal” terminal value expressions in each model. These expressions require
the market’s expected stock price (P) at the horizon and the premium of that price
over book value for a particular accounting system. The equivalence of DCF and
RIM for finite horizons under ideal conditions, though well established theoretically, has not been demonstrated in the empirical literature. At first glance, the
arguments seem circular: if one has reliable price forecasts, the dividend discount
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model (DDM) should suffice and one does not need DCF or RIM. However, the
point is not obvious because the market’s stock price expectations are not observable, and we use VL’s terminal stock price forecasts as a surrogate. Although VL
terminal price forecasts are far from ideal and may contain bias/measurement error
(see Abarbanell and Bernard 2000), we invoke the assumption that any bias/
measurement error will be a constant factor in comparisons across DCF and RIM.
Market efficiency is a maintained assumption in our study, as it is in P&S 1998 and
Francis et al. 2000.
Our second research objective is to demonstrate that intrinsic values calculated using VL terminal stock price forecasts produce smaller valuation errors than
intrinsic values employing ad hoc terminal value expressions. Simple perpetuity
expressions that assume that post-horizon abnormal earnings will grow at either a
rate of 0 percent or the rate of nominal inflation have been widely employed by
empirical researchers (e.g., Francis et al. 2000 and Frankel and Lee 1998). Gebhardt,
Lee, and Swaminathan (2001) use a fade rate procedure that is also ad hoc in that it
assumes that post-horizon abnormal return on equity will converge to the industry
average over a seven-year period beyond the forecast horizon. We find that terminal
values are on average substantially understated using ad hoc estimates of horizon
goodwill, implying that estimates of intrinsic value (Frankel and Lee) or the ex
ante cost of capital (Gebhardt et al.) are understated when ad hoc terminal value
expressions are used. These inferences can be important depending on the intended
research purpose, and hold even in the latter years of our sample when the optimism in VL stock price forecasts has abated.
Lundholm and O’Keefe (2001, hereafter L&O) identify errors in application
of the RIM and DCF models that lead to inconsistent estimates of intrinsic value.
While much of their paper emphasizes the importance of avoiding these pitfalls,
L&O assert that the equivalence of RIM and DCF is “guaranteed” to hold. The
equivalence of RIM and DCF employing ideal terminal value expressions will only
hold if (1) the same VL price forecast is used in RIM and DCF; and (2) the
researcher avoids the pitfalls discussed in L&O. One pitfall that L&O refer to, but
offer no practical suggestion for, is the conundrum of estimating the weighted
average cost of capital for the DCF model. We introduce into the empirical literature a version of DCF derived by Penman 1997 that minimizes the potential for
errors in applying the DCF model. Specifically, our version of DCF does not
require estimates of the weighted average cost of capital (i.e., WACC) and, just as
important, it does not invoke the assumption typical of prior empirical versions of
the free cash flow model that net financial assets are marked to market. If financial
assets are not marked to market, then the researcher must augment the traditional
free cash flow model by incorporating a term representing the fair value increment
on current net financial assets. If this term is missing, the equivalence across DCF
and RIM cannot be demonstrated. Thus, another contribution of our study is to show
researchers how to minimize inconsistent estimates of intrinsic values using VL data
to estimate the DCF model.
Our research questions have practical importance. The equivalence of DCF
and RIM is something students of financial statements analysis take “on faith” and
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the empirical demonstration of this equivalence is useful. Of course, for a finite
horizon, the equivalence is possible only with terminal stock price forecasts. Without such forecasts, careful attention must be paid to the length of the forecast horizon
and the form of the non–price-based terminal value expression, as prior literature
has shown.
We inherit the focus on pricing errors and the maintained assumption of market
efficiency from the “horse race” conducted by P&S 1998 and Francis et al. 2000.
Our aim is to revisit the setting of these studies and set the record straight on the
apparent superiority of RIM over DCF, employing a level playing field where both
models use an approximation of ideal terminal values. The pricing error approach
seemingly conflicts with the rationale for VL and fundamental analysis — that is,
to spot mispriced securities. In section 6, we provide suggestions for further
research focusing on future excess returns.
The remainder of this study is organized as follows. A literature review is provided in section 2. Section 3 lays out the research methodology and hypotheses.
The sample selection and measurement issues are discussed in section 4, and the
empirical results are presented in section 5. Finally, our summary and conclusions
appear in section 6.
2. Literature review
The DDM and DCF models are well-known approaches to valuation in the finance
literature (see Cornell 1993; Copeland, Koller, and Murrin 1995). RIM is discussed
extensively by Ohlson 1995, who shows that theoretically GAAP book values and
earnings are valid valuation attributes. Feltham and Ohlson (1995) establish the
theoretical equivalence of DDM, DCF, and RIM for infinite valuation horizons. All
three models follow from the familiar present value of expected dividends (PVED)
expression for value, and the last two models substitute out dividends in PVED for
relevant valuation attributes.1
Penman (1997) establishes the theoretical equivalence of DDM, DCF, and
RIM for finite valuation horizons, provided one that has access to data necessary to
estimate the following “ideal” terminal values at the end of forecast horizon T:
E t (Pt + T) for DDM; E t (P − B) t + T for RIM; and E t (P − FA) t + T for DCF. In the
above, E t (·) denotes market expectations at time t; Pt + T and Bt + T denote forecasted stock price and book value of owner’s equity at the horizon T periods hence;
and FA t + T denotes forecasted net financial assets at the horizon. In his paper, Penman does not anticipate that the researcher would have access to forecasts of stock
price at the horizon, and hence much of his paper discusses possible estimates of
terminal values for each of the DDM, DCF, and RIM models when forecasted
stock price is unavailable. One of the objectives of our research is to establish Penman’s hypothesized equivalence over a five-year forecast horizon using VL proxies
for the above “ideal” terminal values. A potential limitation of this approach is that
market expectations can be measured with error using VL forecasts of future stock
price and other valuation attributes. An extensive literature exists that suggests that
VL earnings forecasts are biased and/or inefficient (see Abarbanell 1991; Abarbanell and Bernard 1992; and Debondt and Thaler 1990). In a similar vein, Botosan
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(1997) observes that the optimism in VL’s terminal price forecasts yields implausibly high estimates of the equity cost of capital. On the other hand, other researchers have shown that the accuracy of VL forecasts and their association with stock
price changes are comparable to those of other analysts, such as I/B/E/S and Zacks
(see Abarbanell 1991; Bandyopadhyay, Brown, and Richardson 1995; Philbrick
and Ricks 1991; and Stickel 1992). The advantages of VL forecasts over I/B/E/S
are that VL’s long-run “target price range” yields forecasts of stock price at the
horizon five years hence, and no similar price forecasts exist in I/B/E/S. Moreover,
VL forecasts dividends, earnings, book values, and future cash flows separately;
whereas only forecasts of earnings are available in I/B/E/S.2 Forecasts of dividends
and book values are required by RIM, and forecasts of future free cash flows are
required by DCF.
Although VL provides estimates of long-run target price range, neither point
nor range estimates of short-run stock price are made. However, in the Value Line
Investment Survey, VL publishes a timeliness rank, which is based on a mechanical
model (see Foster 1986, 430–2, for details) and represents expected price appreciation over the next 12 months. Many studies have shown that, using VL’s timeliness
measure, investors can earn abnormal returns around a three-day publication period
(e.g., Copeland and Mayers 1982; Huberman and Kandel 1987). Similarly, Peterson (1995) finds that publication of “stock highlights” by VL elicits positive abnormal
returns. Since VL’s long-run target price ranges come from the same underlying data
set that generates timeliness ranks and stock highlights, they can be taken seriously
even though their usefulness has not been established in the prior literature. In section 4, we elaborate on how VL constructs these long-run target price ranges.
In the empirical domain, P&S (1998) and Francis et al. (2000) are the direct
antecedents of our work. P&S use a perfect foresight approach and find that valuation errors for DCF over a 10-year horizon are often in excess of 100 percent and,
moreover, these errors consistently exceed those of RIM by a substantial margin.
This result, according to P&S, may be due to GAAP-based accounting accruals
under RIM, which bring future cash flows forward, compared with the DCF model,
which “expenses” investment outlays. Francis et al. revisit the issue of model comparison from an ex ante perspective using VL forecasts over a 5-year horizon, and
similarly conclude that RIM dominates over DCF. Like Francis et al., we also take
an ex ante approach to study the relative performance of RIM and DCF. However,
in contrast to Francis et al., we make use of VL terminal stock price forecasts in
calculating terminal values for each model, thus avoiding the need either to extrapolate such values from near-term valuation attributes or to assume an ad hoc
growth rate, which may or may not correspond to market expectations.
Three other extant empirical studies have also employed an ex ante approach
to explore the valuation errors associated with RIM. Bernard (1995) uses VL forecasts of (P − B) at the horizon five years hence to measure terminal value, and
shows that the intrinsic values for RIM explain 80 percent of the cross-sectional
variation in the level of current stock price. Abarbanell and Bernard (2000) examine
the importance that the market attaches to the present value of terminal forecasts of
(P − B), and find its regression coefficient to be around 0.67, considerably below
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the predicted value of unity.3 This result is consistent with the notion that VL terminal price forecasts contain bias/measurement error. Finally, Sougiannis and
Yaekura (2001) explore the valuation errors associated with several GAAP-based
RIM valuation models using I/B/E/S rather than VL forecast data. For RIM with
conventional non – price-based terminal value expressions at a horizon five years
hence, they obtain median signed (absolute) valuation errors of − 26 (35) percent.
As we will see later, the magnitudes of these errors are comparable to our RIM errors
for valuation estimates that do not employ VL terminal stock price forecasts.4
3. Research methodology and hypotheses development
Valuation models
In this section, we develop the two major classes of valuation models to be tested
in the paper, namely, DCF and RIM, by appealing to Penman 1997. These models
are based on the following well-known present value of expected dividend model
(PVED) for an infinite horizon:
∞
Pt =
∑R
–τ
Et ( d t + τ )
(1),
τ=1
where Pt is the current market price at time t; R denotes one plus the cost of equity
capital; dt + τ denotes dividends for each future period, t + τ ; and Et indicates an
expectation conditional upon information available at time t . Penman (1997)
shows that when the horizon is finite, the intrinsic value, denoted as Wt, under
DDM for T periods hence is given by:
T
Wt (DDM) =
∑R
–τ
E t ( d t + τ ) + R −T Et(Pt + T)
(2),
τ=1
where Pt + T is the market’s forecasted price at the horizon, t + T. It is an ideal terminal
value for DDM.
DCF model
The following clean surplus relation (CSR) is assumed to hold for net financial
assets at time t + τ, t = 1, 2, … , T:
FA t + τ = FA t + τ
−1
+ Ct +τ − It +τ + it +τ − dt +τ
(3),
where for each future period t + τ, FA denotes net financial assets (i.e., cash and
marketable securities minus debt and preferred equity) and is negative if there is
net debt; (C − I ) is operating cash flows minus capital expenditures (i.e., free cash
flows generated by operating assets); and i is interest flow from net financial assets,
634
which represents interest paid (earned), including preferred dividends, if net financial
assets are negative (positive). Bringing d t + τ to the left-hand side of (3) and all other
expressions to the right, and substituting for E t (d t + τ ) in equation (1), Penman
(1997) derives the following version of the DCF model for an infinite horizon:
∞
Wt (DCF) = FA t +
∑R
–τ
E t [C t + τ − I t + τ + i t + τ − (R − 1)FA t + τ
− 1]
(4).
τ =1
It is important to note that (4) assumes only PVED and CSR for net financial
assets, and one can easily revert back to PVED by substituting in the opposite
direction. As explained by Feltham and Ohlson 1995, (4) represents the cash
accounting model where operating assets are expensed; book value is represented
by current net financial assets; and “earnings” are represented by free cash flows
from operations, (C − I ), plus (minus) interest income (expense), i.
Penman (1997) shows that if one relaxes the assumption of risk neutrality but
assumes that net financial assets are marked to market at all times, (4) becomes the
familiar free cash flow model:
∞
Wt (DCF) = FA t +
∑ RW Et (C t + τ − I t + τ )
–τ
(5),
τ =1
where RW denotes one plus the weighted average cost of capital (i.e., WACC). The
equation states that the value of owner’s equity equals the sum of the fair values of
net financial assets and net operating assets, with the latter represented by the
present value of expected future free cash flows.5
Francis et al. (2000) estimate variations of (5). We prefer Penman’s version of
the DCF (i.e., (4)) for several reasons. First, (5) requires the estimation of WACC,
where the weights must be based on the estimated value of equity and debt, not on
either their book value or a target capital structure. On the other hand, (4) requires
the equity cost of capital, thus placing DCF on an equal footing with RIM. Second,
(5) assumes that FAs are marked to market and, to the extent that fair value does
not equal book value, it introduces noise in the intrinsic value expressions. By
comparison, (4) does not make that assumption, and implicitly corrects for any fair
value increment on debt.6 Finally, (5) requires forecasted operating cash flows (i.e.,
C − I ), which, unlike forecasts of free cash flows to common (i.e., C − I + i), are
not provided by VL. To derive C − I, one needs to remove the effects of interest
expense (income), i. This is problematic because VL does not provide forecasts of
“i ” to the horizon for either debt or preferred shares.
For an arbitrary finite horizon T, Penman (1997) shows that the ideal terminal
value for a finite horizon version of (4) is the market’s expected premium, (P − FA),
at the forecast horizon. The following equation represents our DCF model employing VL forecasted price in the terminal value expression:
T
Wt (DCF1) = FA t +
∑R
–τ
τ =1
635
− 1]
+ R −T Et (Pt + T − FA t + T)
(6a).
The expression (Pt + T − FAt + T) at the horizon captures the present value of posthorizon operating cash flows because the cash accounting model expenses operating assets. It also captures any post-horizon fair value increment, if net financial
assets are not marked to market.
When a terminal stock price forecast is not available, it is of interest to explore
other non–price-based expressions. Following Frankel and Lee 1998, P&S 1998,
and Francis et al. 2000, we employ two expressions: one assumes a simple perpetuity without growth and the other assumes a perpetuity with constant growth rate
g = 2 percent.7 Modifying (6a) accordingly results in:
T
Wt (DCF0) = FA t +
∑R
–τ
− 1]
τ =1
+ R−T (R − 1) −1E t [C t + T + 1 − I t + T + 1 + i t + T + 1 − (R − 1)FA t + T ]
T
Wt (DCF2) = FA t +
∑R
–τ
(6b);
− 1]
τ =1
+ R−T (R − 1 − g) −1E t [C t + T + 1 − I t + T + 1 + i t + T + 1 − (R − 1)FA t + T ]
(6c).
We compute the numerator of the ad hoc terminal value expression as
[C t + T + 1 −I t + T + 1 + i t + T + 1 − (R − 1)FA t + T] =
(1 + g) (C t + T − I t + T + i t + T) − (R − 1)FA t + T.
For the purposes of estimating (6a)–(6c), and (7a)–(7c) discussed next, all variables
are deflated by the number of shares outstanding at the end of forecast year.8
RIM model
Penman (1997) shows that, for a finite horizon T, the ideal terminal value for RIM
is the market’s expected premium, (P − B), at the forecast horizon. This expression
represents the present value of post-horizon abnormal earnings (i.e., subjective
goodwill) and reflects the joint effects of positive net present value projects and
accounting conservatism. Since VL explicitly forecasts book value five years
hence, the expected premium can be calculated to yield the following “best” contender from the RIM family of valuation models:
636
T
Wt (RIM1) = Bt +
∑R
–τ
a
E t ( X t + τ ) + R −T E t (Pt + T − B t + T )
(7a),
τ =1
a
where X t + τ denotes abnormal income for forecast year t + τ, measured as VL’s
forecasted net income minus a charge on the capital employed (i.e., R − 1 times
opening B ). The corresponding expressions that do not employ terminal price
forecasts are:
T
Wt (RIM0) = B t +
∑R
–τ
a
a
E t ( X t + τ ) + R −T(R − 1) −1 Et( X t + T + 1)
(7b),
τ =1
T
Wt (RIM2) = Bt +
∑R
–τ
a
a
E t ( X t + τ ) + R −T(R − 1 − g) −1 Et( X t + T + 1)
(7c),
τ =1
a
where X t + T + 1 = (1 + g)Xt + T − (R − 1)Bt + T , under the assumptions of a simple
perpetuity with constant growth.
Hypotheses development
Following P&S 1998 and Francis et al. 2000, we focus on comparing the signed
and absolute prediction errors across DCF and RIM. We do not consider DDM in
this paper because our price-based DCF model (i.e., (6a)) is developed directly
from the corresponding DDM (i.e., (2)) given the financial assets continuity
account (i.e., (3)) for each pre-horizon year, thus guaranteeing their theoretical and
empirical equivalence.9 The empirical comparison between DCF and RIM that
employ Penman’s 1997 ideal price-based terminal value expressions (i.e., (6a) and
(7a)) is, however, complicated by the presence of several additional sources of
inconsistency, which may or may not be easily avoided by the researcher, as discussed in the introduction and elaborated further in section 5. The equivalence of
these two models is not guaranteed empirically unless errors in implementing each
model are carefully considered and minimized. We conjecture that, in the absence
of implementation errors, the choice between DCF and RIM should be a matter of
indifference. This is formalized in our first hypothesis (stated in the null form):
HYPOTHESIS 1. Across the versions of DCF and RIM that employ VL forecasted
price in the terminal value expression, there is no difference in prediction
errors.
We next compare non – priced-based models under 0 percent and 2 percent
constant growth assumptions with the corresponding price-based models within the
same class of DCF and RIM. We expect the model that uses VL’s price forecasts in
637
the terminal value expressions (i.e., (6a) and (7a)) to beat other contenders within
their class (i.e., (6b) – (6c) for DCF; and (7b) – (7c) for RIM). Intuitively, if the
researcher cannot observe VL’s post-horizon expectations, any non – price-based
terminal value calculated under an arbitrary growth assumption is at best ad hoc.
The above discussion leads to our second hypothesis (stated in the alternative form):
HYPOTHESIS 2. Within each class of the DCF and RIM valuation models, the
model that employs VL forecasted price in the terminal value expression
generates the lowest prediction errors, compared with models that employ
non – price-based terminal value under arbitrary (0 percent and 2 percent) growth assumptions.
It is worthwhile pointing out that Hypothesis 2 is also not “guaranteed” empirically. For example, if VL’s price forecasts are pure noise or if VL simply applies
the reciprocal of the equity cost of capital at the horizon and multiplies this by
forecasted earnings five years hence, then the forecasted price would fail to capture
subjective goodwill beyond the horizon. In this case, (7a) will have no edge over
(7b) or (7c),10 and the superiority result predicted in Hypothesis 2 is generally not
assured.
4. Data description and measurement issues
Data description
Our initial sample consists of 500 firms (or 2,500 firm-year observations), which
were followed by VL over a five-year period, 1992–96, and were on both CRSP
and COMPUSTAT during that time. This sample size is chosen for practical considerations because the forecasts of prices, book values, dividends, cash flows, and
other relevant accounting valuation attributes are not available from machinereadable sources and must be hand-collected from the archived Value Line Investment Survey. To draw the sample, we first obtain an intersection of 1,089 firms,
excluding those in the financial services sector, from the 1996 coverage of VL,
CRSP, and COMPUSTAT, and then apply a random number generating procedure.
We require five years of forecast data for all firms included in the sample. Nonforecast-related historical data are extracted from the Value Line Data File. Due to
missing data in the data file, 36 firms are eliminated, 41 firms are dropped because
VL’s annual capital investment estimates are not provided for some industries,11
and another firm is deleted because VL did not provide price forecasts for one of
the years (i.e., 1996). This leaves us with a final sample of 422 firms (or 2,110
firm-years), each with complete forecasted and historic data over the entire sample
period under investigation. The panel nature of the data with repeated measures
over calendar time for the same firms appeals to us, and panel data methodology
(see Kmenta 1986) exploiting autocorrelations in the data will be used for formal
statistical tests.
Our sample firms are large, with mean (median) market capitalization of $4.95
($1.18) billion. The minimum market capitalization is $25.80 million, and the
638
maximum is $16.87 billion. Firm-specific betas provided by VL range from a low
of 0.05 to a high of 2 with the mean beta given by 1.02. The equity cost of capital is
computed based on the CAPM. The riskless rates are measured as the five-year treasury constant maturity rates at the beginning of the forecast month from Chicago
Federal Reserve Bank data base and the risk premiums are measured as the product
of VL firm-specific betas and the approximate historical equity premium of 6
percent.12 The cost of equity has a mean of 12.28 percent and the minimum and
maximum of 5.69 percent and 18.64 percent, respectively.
Measurement issues
To measure the variables described in (6a)–(6c) and (7a)–(7c) for each of the five
sample years, we take the first complete published VL forecasts, which typically
appear in the third quarter of the firm’s fiscal year. Following Francis et al. 2000,
we discount VL forecasts of the nth year’s valuation attributes by a factor of (n − 1 + f ),
where f reflects the fraction of year between the market valuation date and the first
fiscal year-end.13 Since all the valuation models require book values at the date of
forecast, which VL does not directly provide, we need to interpolate book values
(net financial assets) to the forecast date for the RIM (DCF) models based on their
values at the beginning of forecast year and the VL forecasts of related current fiscal year’s variables.14
Our conversations with VL personnel indicate that VL analysts’ predictions of
the target stock price range, three to five years ahead, are not mechanical. Judgement is required of the analyst at three stages in constructing the target price range.
First, judgement is used to forecast projected earnings three to five years ahead.
Second, judgement is used in applying the appropriate P/E ratio to forecasted earnings, and the P/E ratio can deviate from the projected average P/E ratio for the
market according to the analysts’ long-term growth projections for the stock in
question. Third, judgement is applied in selecting one of five possible range categories to surround a point estimate of forecasted price, with smaller ranges associated
with greater financial strength/safety as assessed by the VL analyst and her supervisors. We conclude from these conversations that the width of target stock price
ranges reflects uncertainty, and that such price forecasts impound the VL analysts’
growth assumptions beyond the horizon. For our purposes, we define the terminal
price forecasts, Pt + T , as the mid-point of the target price range.
VL publishes forecasts for three horizons: current fiscal year (i.e., year 1), the
following fiscal year (i.e., year 2), and long run (i.e., year 5). Since the internal yearby-year forecasts of valuation attributes for years 3 and 4 are not published in the
Value Line Investment Survey, at the suggestion of VL analysts, we interpolate data
for these two years based on implied straight-line growth from year 2 to year 5.15 The
most recent stock price reported by VL is used as the dependent variable in our study.16
For both RIM and DCF, the researcher must confront the occasional existence
of negative terminal values. Such values can arise if either forecasted premiums
given terminal price forecasts or valuation attributes at the horizon are negative.
We choose not to cap negative terminal values at zero because any negative
attribute at the horizon is expected to be impounded in current market price.
639
5. Empirical results
Descriptive statistics
Table 1 reports the relative importance of the various components of DCF and
RIM. Discounted terminal value (DTV) accounts for the majority of intrinsic value
in all three versions of the DCF model (i.e., 95.38 percent, 91.81 percent, and
93.19 percent for DCF1, DCF0, and DCF2, respectively).17 The importance of
DTV in the DCF models is consistent with prior literature. Copeland, Koller, and
Murrin (1995), for instance, document that, for a sample of companies appraised
by McKensey and Company, non–price-based DTV amounts to 56 to 125 percent
of intrinsic value. The remaining two components of DCF models tend to have
opposite signs in our data. Current book value (i.e., FA t ) is, on average, negative,
reflecting current net debt; whereas the present value of pre-horizon “free cash
flows” coming to treasury from operations (i.e., PV) is positive.
Turning to the RIM models, DTV makes up of 52.23 percent, 20.77 percent,
and 25.53 percent of intrinsic value for RIM1, RIM0, and RIM2, respectively.
These percentages are considerably lower than the corresponding figures for DCF.
Conversely, current book value (i.e., B t ) takes on a more significant role in all three
versions of RIM than DCF (e.g., 41.04 percent for RIM1 versus −25.71 percent for
DCF1), confirming the findings of prior research that more wealth is captured in
valuation attributes to the horizon under RIM than under DCF. Within the family
of RIM models, Bt is least important and DTV most important when the “ideal”
terminal value is employed. For example, Bt (DTV) accounts for 41.04 percent
(52.23 percent) of the intrinsic value under RIM1, compared with 68.06 percent
(20.77 percent) and 63.97 percent (25.53 percent) under RIM0 and RIM2, respectively. However, even for RIM0 and RIM2, DTV represents more than 20 percent
of the intrinsic value estimates, implying that post-horizon forecasts of abnormal
earnings remain crucial to firm valuation under RIM even though, as pointed out
by P&S 1998 and Francis et al. 2000, current book value brings future cash flows
forward.
Two sets of analyses are performed in this study, one based on prediction error
defined as the difference between model intrinsic value estimate and current stock
price, scaled by current stock price, and the other based on regression analysis.
Panels A and B of Table 2 present the distribution of signed and absolute prediction errors, respectively, for the overall sample period. As is evident from the
fourth column, the skewness measures are uniformly positive across all valuation
models, implying that our data are positively skewed. For the purpose of testing the
predictions of Hypotheses 1 and 2, we therefore focus on the median, as opposed
to the mean, signed and absolute prediction errors, and use non-parametric Wilcoxon
signed rank tests.
Some unusually large prediction errors are evident at both ends of the distributions, especially in models for which non–price-based terminal value expressions
are employed. The prediction error analysis, in particular that involving Hypothesis 2, may be affected because outliers can come from different firms depending on
whether price- or non – price-based models are used.18 To assess potential probCAR Vol. 18 No. 4 (Winter 2001)
640
TABLE 1
Relative importance of components of valuation models*
DCF1
DCF0
DCF2
RIM1
RIM0
RIM2
Mean opening
FA or B†
(% of mean IV)
Mean PV ‡
(% of mean IV)
Mean DTV §
(% of mean IV)
Mean IV #
(% of mean IV)
−8.46
(−25.71%)
−8.46
(−45.58%)
−8.46
(−37.90%)
13.53
(41.04%)
13.53
(68.06%)
13.53
(63.97%)
9.98
(30.33%)
9.98
(53.77%)
9.98
(44.71%)
2.22
(6.73%)
2.22
(11.17%)
2.22
(10.50%)
31.39
(95.38%)
17.04
(91.81%)
20.80
(93.19%)
17.22
(52.23%)
4.13
(20.77%)
5.40
(25.53%)
32.91
(100.00%)
18.56
(100.00%)
22.32
(100.00%)
32.97
(100.00%)
19.88
(100.00%)
21.15
(100.00%)
Notes:
*
See the appendix for a description of DCF1, DCF0, DCF2, RIM1, RIM0, and RIM2.
†
FA is the net financial assets per share (i.e., cash and marketable securities minus debt
and preferred equity), and B is the book value of owner’s equity per share;
‡
PV is the present value of operating cash flows to common shareholders or abnormal
earnings to the horizon, on a per-share basis, under the discounted cash flows
(DCF) or residual income model (RIM), respectively.
§
DTV is the discounted terminal value per share given by the last component under
DCF1, DCF0, DCF2, RIM1, RIM0, and RIM2.
#
IV is the intrinsic value estimates per share under DCF1, DCF0, DCF2, RIM1, RIM0,
and RIM2.
lems associated with outliers, we conduct the prediction error analysis with and
without winsorizing at the 1st and 99th percentile. The results are very similar
qualitatively speaking. Thus, only one set of results based on data before applying
the winsorization procedure will be reported in the paper.
Prediction-error analyses
Results from tests of Hypothesis 1
Panel A of Table 3 reports the median intrinsic value estimates, median signed
prediction errors, and pair-wise comparisons of these figures for the price-based
valuation models over the entire sample period (1992 – 96) and by year. At the
overall level, the median intrinsic values of $29.181 and $29.104 for DCF1 and
641
TABLE 2
Distribution of prediction errors
Panel A: Signed prediction errors (bias) — Overall (1992–96)*
Standard
Mean (%)
Median (%)
deviation
Price-based models
DCF1
RIM1
Skewness
Kurtosis
8.42%
8.39%
4.82%
4.73%
0.261
0.269
1.624
1.557
8.199
8.094
Non–price-based models
DCF0
−37.76%
RIM0
−34.18%
DCF2
−24.18%
RIM2
−30.26%
−41.34%
−37.95%
−30.50%
−34.36%
0.336
0.275
0.427
0.306
1.454
1.613
1.771
1.623
6.512
9.370
7.389
8.105
Panel B: Absolute prediction errors (accuracy) — Overall (1992–96)†
Standard
Mean (%)
Median (%)
deviation
Skewness
Kurtosis
Price-based models
DCF1
RIM1
19.11%
19.54%
13.71%
14.18%
0.197
0.204
3.203
3.209
21.069
20.109
Non–price-based models
DCF0
43.90%
RIM0
38.73%
DCF2
39.68%
RIM2
37.19%
42.81%
38.80%
35.48%
36.42%
0.250
0.207
0.288
0.217
0.757
0.873
2.034
1.156
3.151
7.409
11.180
8.541
Notes:
*
†
Signed prediction errors are calculated as ( IV itM − Pit )/Pit .
Absolute prediction errors are calculated as | IV itM − Pit |/Pit , where Pit is the
recent stock price published in the VL forecast report; and IV itM is the
intrinsic value estimate per share for security i in year t calculated under M =
DCF1, DCF0, DCF2, RIM1, RIM0, and RIM2, described in the appendix.
RIM1, respectively, overestimate current median stock price of $28, reflecting
optimism in VL forecasts noted by Botosan 1997. For both models, the median
signed prediction errors decline steadily over time. For example, the median
signed prediction errors for DCF1 reduce to 1.43 percent and − 1.95 percent in
1995 and 1996, respectively, from the peak of 11.95 percent in 1992. Thus, VL
optimism appears to have completely abated toward the end of our sample period.
These patterns are depicted in Figure 1.
For the pooled 1992–96 data, the median intrinsic value estimates and median
signed prediction errors for DCF1 and RIM1 are all within a very small neighborhood of one another (i.e., $29.181 versus $29.104; 4.82 percent versus 4.73 percent).
−0.14‡
4.82
4.73
26
27.873
28.126
−0.04
11.95
11.72
Median, Median,
$
%
26
27.574
27.443
−0.07
5.86
5.84
Median, Median,
$
%
DCF1
29.181
RIM1
29.104
Test of Hypothesis 1
DCF1 − RIM1
−0.06‡
13.71
14.18
Median, Median,
$
%
27.873
28.126
−0.08§
14.77
15.56
Median, Median,
$
%
27.574
27.443
−0.15§
14.24
14.82
Median, Median,
$
%
Panel B: Median absolute prediction errors (accuracy) — Overall and by year
1992–96
1992
1993
Current stock price
28
DCF1
29.181
RIM1
29.104
Test of Hypothesis 1
DCF1 − RIM1
Median, Median,
$
%
Panel A: Median signed prediction errors (bias) — Overall and by year
1992–96
1992
1993
5.19
4.60
−0.07
29.274
28.994
−0.00
12.49
13.08
Median, Median,
$
%
1994
28
29.274
28.994
Median, Median,
$
%
1994
TABLE 3
Tests of hypothesis 1*: Median signed and absolute prediction errors for the price-based valuation models†
1.43
2.89
−0.23‡
−0.09
13.56
14.04
−0.27‡
−1.95
−1.41
31.513
31.479
0.00
13.82
13.42
Median, Median,
$
%
1996
32
31.513
31.479
Median, Median,
$
%
1996
(The table is continued on the next page.)
30.112
30.330
Median, Median,
$
%
1995
29.5
30.112
30.330
Median, Median,
$
%
1995
642
643
TABLE 3 (Continued)
Notes:
*
Hypothesis 1 states that there is no difference in prediction errors across DCF1 and
RIM1. Wilcoxon signed rank tests are employed to test the difference in median
signed/absolute prediction errors.
†
Signed prediction errors are calculated as ( IV itM − Pit )/Pit and absolute prediction
errors are calculated as | IV itM − Pit |/Pit , where Pit is the recent stock price
published in the VL forecast report; and IV itM is the intrinsic value estimate per
share for security i in year t calculated under M = DCF1, DCF0, DCF2, RIM1,
RIM0, and RIM2, described in the appendix.
‡
Significant at the 1 percent level.
§
Nevertheless, the Wilcoxon signed rank test performed on percentage errors rejects
the prediction of Hypothesis 1 at the 1 percent level. The sensitivity of small
median difference to statistical test may be partially explained by the power of test
given the large number of observations overall, and by the fact that most (80 percent) of the differences are in the same direction. The support for Hypothesis 1 is
considerably stronger when a separate analysis is performed for each of the sample
years. In particular, none of the pair-wise comparisons in years 1992 to 1994
rejects the null of no difference (i.e., Hypothesis 1) at the conventional levels of
significance.
The median absolute prediction errors and pair-wise comparisons based on
these errors are presented in panel B of Table 3. For both DCF1 and RIM1, the
annual median absolute prediction errors decline from their highest levels in 1992
to the lowest levels by the mid-point of our sample period (i.e., 1994). The trend
reverses itself in the second half of the sample period. The prediction of Hypothesis 1 is not rejected in the last three years of the sample period (1994 – 96), but
rejected in the first two years (1992–93). Overall, the median absolute prediction
errors are 13.71 percent and 14.18 percent for DCF1 and RIM1, respectively. The
Wilcoxon signed rank test rejects Hypothesis 1 at the 1 percent level. In short, the
evidence presented is largely consistent with the prediction that there is no difference
in the median signed or absolute prediction errors across DCF and RIM models
(Hypothesis 1), especially when analysis is conducted at the year-by-year level.
In testing Hypothesis 1, the researcher is confronted with several potential
sources of inconsistency that can lead to differences across valuation models. We
now discuss three sources that we avoid. First, we employ Penman’s version of
DCF model (i.e., (4)) to avoid L&O’s “inconsistent discount rate error” referred to
in the introduction. Thus, even if the equity cost of capital were incorrectly measured, both DCF1 and RIM1 would produce the same incorrect intrinsic value so
as not to affect the test of Hypothesis 1. Second, our conversations with VL personnel indicate that, in the Value Line Investment Survey, forecasts of “free cash
644
Figure 1
Median signed prediction errors for price-based DCF and RIM: Years 1992 to 1996*
0.14
Median signed prediction error
0.12
0.10
DCF1
RIM1
0.08
0.06
0.04
0.02
0.00
−0.02
1992
1993
1994
1995
1996
Year
*
Signed prediction errors are calculated as ( IV itM − Pit )/Pit , where Pit is the recent
stock price published in the VL forecast report; and IV itM is the intrinsic value
estimate per share for firm i in year t calculated under M = DCF1 and RIM1,
described in the appendix. The graph depicts the median of signed prediction
errors for all sample firms in each of the five years under investigation (1992–96).
flows” typically increase working capital unless the VL analyst anticipates some
other uses for the cash, such as share repurchases or retirement of long-term debt.
Value Line ensures internally in their spreadsheets that all projected sources and
uses of cash are reconciled through a projected statement of funds, and that CSR
holds for FA. Since it is not easy for the researcher to observe where VL has
applied funds for a particular firm-year, we create our own FA continuity schedule,
starting with opening net FA and building up the next period FA using VL’s forecasts of Ct + τ , It + τ , it + τ , and dt + τ in (3). Effectively, we generate our own forecasts of future FA and do not use VL’s forecasted long-term debt. This procedure
guarantees that CSR will hold for our version of forecasted FA, just as it does internally for VL based on our discussions with VL analysts. This adjustment minimizes
L&O’s so-called missing cash flows problem. Third, 301 to 770 firm-year observations had negative opening book value of owners’ equity, and hence larger abnormal
earnings than reported earnings under RIM1 during the forecast periods t + 1 to
645
t + 5. However, capping abnormal earnings at forecasted earnings for these firms
would introduce inconsistencies in comparisons with DCF. For that reason, we do
not impose an upper cap on abnormal earnings. Results reported in this and the
next sections are nonetheless essentially the same qualitatively, with or without the
capping requirement. For the same reason, we also do not impose a lower bound
on the two firm-year observations where RIM1 intrinsic value is negative. Once
again, our results are not sensitive to this treatment.
We next turn to two sources of inconsistency that we do not avoid. First, for
the DCF model, it is not practical for the researcher to adjust forecasted “free cash
flows” for sources or uses of cash due to working capital requirements and
deferred taxes. Regarding working capital, the cash versus non-cash components
of forecasted working capital are not available from VL. For a typical firm in our
sample, ignoring cash tied up in working capital overstates free cash flows,
whereas not recognizing deferred taxes understates free cash flows. Second, for
RIM, we do not force the CSR to hold when forecasted violations of CSR exist
because violations of CSR do exist in U.S. accounting principles and, therefore,
should exist in expectation.19 If one were to force CSR to hold, the plug would
have to go to either forecasted earnings or forecasted equity infusions (i.e., negative dividends). The latter treatment implies that the corresponding DCF model
valuation attributes would have to change, leading to an unknown degree of error
in the DCF model. Since we do not adjust for these two sources of inconsistency,
the empirical equivalence of DCF and RIM is not “guaranteed” to hold. Nevertheless, even without attempting adjustments in these areas, the differences in errors
across valuation models, reported in Table 3, are small.
Results from tests of Hypothesis 2
Panel A (B) of Table 4 presents results from testing the prediction of Hypothesis 2
that the median signed (absolute) prediction errors are smaller for the price-based
valuation models than for the corresponding non–price-based models.
For the pooled 1992–96 data, the median signed prediction errors of 4.82 percent (4.73 percent) for DCF1 (RIM1) are considerably closer to zero than −41.34
percent (−37.95 percent) and −30.50 percent (−34.36 percent) for DCF0 (RIM0)
and DCF2 (RIM2), respectively (see panel A).20 Wilcoxon signed rank tests of
pair-wise differences between the price-based and the non – price-based models
within the same family are all significant at the 1 percent level, supporting the prediction of Hypothesis 2 at the overall level. Results are similar and uniformly in
support of Hypothesis 2 when analysis is extended to each sample year. For example, focusing on the last two years when VL optimism is minimal and for which
use of price-based models as a benchmark is most appropriate, the median signed
prediction errors of 2.89 percent and −1.41 percent for RIM1 are smaller than −40.06
percent and − 43.32 percent (− 36.27 percent and − 39.98 percent) for RIM 0
(RIM 2), respectively, again significant at the 1 percent level. Over the same two
years, DCF1 is also associated with considerably smaller median signed prediction
errors than DCF0 and DCF2 (1.43 percent versus −42.48 percent and −31.07 percent
in 1995; −1.95 percent versus −47.81 percent and −37.86 percent in 1996).
Current stock price
28
DCF1
29.181
DCF0
15.768
DCF2
19.062
Tests of Hypothesis 2
DCF1-DCF0
DCF1-DCF2
DCF0-DCF2
RIM1
29.104
RIM0
17.042
RIM2
17.918
RIM1-RIM0
RIM1-RIM2
RIM0-RIM2
28.126
17.998
19.170
44.00‡
33.17‡
−10.67‡
4.73
−37.95
−34.36
40.84‡
37.38‡
−3.37‡
26
27.873
16.478
19.880
38.42‡
32.67‡
−5.23‡
43.24‡
29.19‡
13.92‡
11.72
−27.17
−21.70
11.95
−33.84
−18.44
Median, Median,
$
%
4.82
−41.34
−30.50
Median, Median,
$
%
27.443
15.212
15.775
26
27.574
14.303
16.683
43.42‡
40.79‡
−3.07‡
45.68‡
34.50‡
−10.28‡
5.84
−37.52
−34.58
5.86
−40.25
−30.74
Median, Median,
$
%
Panel A: Median signed prediction errors (bias) — Overall and by year
1992–96
1992
1993
28.994
16.647
17.460
28
29.274
15.552
18.382
42.21‡
39.04‡
−2.91‡
44.59‡
35.28‡
−10.06‡
4.60
−39.04
−35.80
5.19
−42.77
−32.62
Median, Median,
$
%
1994
39.53‡
36.89‡
−3.11‡
42.42‡
31.71‡
−10.25‡
2.89
−40.06
−36.27
1.43
−42.48
−31.07
31.479
18.347
19.636
32
31.513
17.084
20.348
40.72‡
37.23‡
−3.26‡
42.55‡
33.32‡
−9.52‡
−1.41
−43.32
−39.98
−1.95
−47.81
−37.86
Median, Median,
$
%
1996
30.330
17.238
18.189
29
30.112
16.103
19.283
Median, Median,
$
%
1995
TABLE 4
Tests of Hypothesis 2*: Median signed and absolute prediction errors for the price-based and non–price-based valuation models†
646
Current stock price
28
DCF1
29.181
DCF0
15.768
DCF2
19.062
DCF1-DCF0
DCF1-DCF2
DCF0-DCF2
RIM1
29.104
RIM0
17.042
RIM2
17.918
RIM1-RIM0
RIM1-RIM2
RIM0-RIM2
28.126
17.998
19.170
−28.11‡
−20.32‡
7.96‡
14.18
38.80
36.42
−25.55‡
−22.62‡
2.61‡
26
27.873
16.478
19.880
−13.09‡
−10.90‡
3.22‡
−21.25‡
−13.97‡
7.83‡
15.56
29.88
29.11
14.77
36.87
30.15
Median, Median,
$
%
13.71
42.81
35.48
Median, Median,
$
%
27.443
15.212
15.775
26
27.574
14.303
16.683
−25.97‡
−21.67‡
2.39‡
−26.41‡
−17.66‡
7.93‡
14.82
38.24
36.25
14.24
41.19
34.29
Median, Median,
$
%
Panel B: Median absolute prediction errors (accuracy) — Overall and by year
1992–96
1992
1993
TABLE 4 (Continued)
28.994
16.647
17.460
28
29.274
15.552
18.382
−27.18‡
−24.12‡
2.36‡
−30.75‡
−22.41‡
7.98‡
13.08
39.24
37.10
12.49
43.60
35.38
Median, Median,
$
%
1994
−26.58‡
−25.32‡
2.47‡
−28.21‡
−21.04‡
7.67‡
14.04
40.72
38.63
13.56
42.87
35.61
31.479
18.347
19.636
32
31.513
17.084
20.348
−29.91‡
−26.79‡
2.76‡
−33.20‡
−24.91‡
8.15‡
13.42
43.39
40.11
13.82
48.64
40.02
Median, Median,
$
%
1996
30.330
17.238
18.189
29
30.112
16.103
19.283
Median, Median,
$
%
1995
647
648
TABLE 4 (Continued)
Notes:
*
Hypothesis 2 states that the terminal value expression that employs VL’s forecasted
price will have the lowest prediction errors within each class of the DCF and RIM
models. Wilcoxon signed rank tests are employed to test the difference in median
signed/absolute prediction errors.
†
Signed prediction errors are calculated as ( IV itM − Pit )/Pit and absolute prediction
errors are calculated as | IV itM − Pit |/Pit , where Pit is the recent stock price
published in the VL forecast report; and IV itM is the intrinsic value estimate per
share for security i in year t calculated under M = DCF1, DCF0, DCF2, RIM1,
RIM0, and RIM2, described in the appendix.
‡
Turning to accuracy, for the entire sample period, the median absolute prediction
errors for DCF1 and RIM1 are significantly lower than those for the corresponding
non–price-based models. The contrast for DCF is 13.71 percent versus 42.81 percent and 35.48 percent, and that for RIM is 14.18 percent versus 38.80 percent and
36.42 percent (see panel B), lending strong support for Hypothesis 2. The results
are equally strong at the year-by-year level.
Evidence on the relative performance of RIM versus DCF with ad hoc growth
assumptions in the terminal value expression is mixed. At an assumed growth rate
of 0 percent, RIM is less biased and more accurate than DCF overall (i.e., −37.95
percent versus −41.34 percent; 38.80 percent versus 42.81 percent); whereas the
converse is true when the growth rate is assumed to be 2 percent (i.e., −34.36 percent versus −30.50 percent; 36.42 percent versus 35.48 percent). Similar patterns
can also be found in each of the five sample years. These results are to be contrasted with analogous pair-wise comparisons between RIM and DCF when the
ideal terminal price forecasts are employed. As reported under the heading “Results
from tests of Hypothesis 1”, above, RIM1 does not dominate, nor is it dominated
by DCF1 for most of the annual comparisons of median signed and absolute prediction errors. Thus, the conclusion by P&S 1998 and Francis et al. 2000 that RIM
outperforms DCF would appear to be quite sensitive to the growth assumption
made about valuation attributes and the way terminal values are measured.
As sensitivity tests of Hypothesis 2, we repeat the analysis under the alternative, albeit similarly ad hoc, growth assumptions of 4 percent, 6 percent, 8 percent,
and 10 percent for DCF employed in Francis et al. 2000. The results indicate that the
median absolute prediction errors for DCF1 continue to be lower than the corresponding ad hoc growth models. The median differences are −17.41 percent, −24.20
percent, −58.98 percent, and −154.40 percent, respectively, all significant at the 1
percent level. For RIM, we use Gebhardt, Lee, and Swaminatham’s 2001 fade-rate
procedure to generate the alternative ad hoc growth assumption. For each sample
firm-year observation, the linear fade rate is defined as the rate at which a firm’s
abnormal return on equity at the horizon will converge to the industry average evenly
649
over a seven-year period beyond the forecast horizon. The median difference in
absolute prediction errors between RIM1 and the fade-rate-based RIM model
(denoted RIMf) is −35.82 percent, in favor of RIM1. The corresponding median
signed difference (i.e., RIM1 − RIMf) is 52.07 percent, implying that RIMf seriously understates the post-horizon goodwill projected by VL.
Taken together, these results are strongly in support of the prediction of
Hypothesis 2, and suggest that the researcher should exercise care in interpreting
results based on models that employ ad hoc terminal value expressions, because
intrinsic value estimates in these models can be severely downward biased. This
observation applies to studies such as Frankel and Lee 1998, who use a simple perpetuity expression for their RIM terminal values to identify mispriced securities
and profitable trading strategies; and Gebhardt et al. 2001, who use a simple faderate procedure for their RIM terminal values in order to solve for a firm’s ex ante
cost of capital implied by the RIM intrinsic value estimates and current stock price.
As with tests of Hypothesis 1, the researcher needs to deal with several potential sources of inconsistency in testing Hypothesis 2. First, we adopt the approach
recommended by L&O 2001 to extrapolate valuation attributes in the first year
beyond the forecast horizon (see section 3, above). However, we also repeat the
analysis using the approach commonly employed in the “horse race” literature by
redefining year t + 6 abnormal earnings as year t + 5 abnormal earnings multiplied by
(1 + g) (see P&S 1998 and Francis et al. 2000). The results (not reported in a table)
are qualitatively the same across these two extrapolation methods. For example,
focusing on accuracy, the median absolute prediction errors for RIM2 and DCF2
now become 37.25 percent and 37.42 percent, compared with the corresponding
figures of 36.42 percent and 35.48 percent reported previously in panel B of Table
4. The prediction of Hypothesis 2 is once again strongly supported at the 1 percent
level, implying that our earlier conclusion about the superiority of models using
VL terminal price forecasts over non–price-based models within the same family
are robust to the manner in which ad hoc growth rates are applied to the terminal
value expressions. Second, 8 (49) firm-year observations have negative intrinsic
values under RIM0 (DCF0) and 9 (34) under RIM2 (DCF2). For these firms,
unless negative intrinsic values are capped at zero, comparing across the pricebased and non–price-based models would overstate the difference due to limited liability constraints. Notwithstanding this capping requirement, it should be noted
that all the results continue to hold when the restriction is relaxed. Third, firms
might have reached a steady state prior to the horizon potentially affecting tests of
Hypothesis 2. To rule out this possibility, we delete 202 observations whose abnormal earnings change signs from positive to negative before the horizon and repeat
the analysis presented in Table 4. The results (not reported in a table) are similar
qualitatively. For example, differences in the median absolute prediction errors are
−27.09 percent, −18.93 percent, −24.65 percent, and −21.90 percent for DCF1 −
DCF0, DCF1 − DCF2, RIM1 − RIM0, and RIM1 − RIM2, respectively. Wilcoxon
signed rank tests again all support Hypothesis 2 at the 1 percent level.
650
Regression analyses
Panels A and B of Table 5 report results from the pooled GLS panel regressions of
contemporaneous stock prices on intrinsic value estimates for DCF and RIM models,
respectively. For this analysis and that reported in Table 6, we use the GLS procedure because standard tests reject homoscedasticity of model residuals obtained
from OLS regressions using share-deflated variables.21 The GLS procedure transforms the data to correct for heteroscedasticity and removes autocorrelation from the
residuals (Kmenta 1986). While GLS does not correct for cross-sectional correlation
in residuals, any such correlation is unlikely to represent a serious departure from
standard assumptions, given modest calendar and industry clustering in our sample.
Several results from Table 5 confirm the impressions obtained previously.
First, as a benchmark model, the results for DCF1 are striking. While the slope
coefficient of 0.966 differs significantly from a theoretical prediction of unity at the
1 percent level, the R2 of 93.71 percent suggests that the measurement error resulting from using VL terminal price forecasts as a proxy for market expectations is
modest. Second, the R2s of 93.71 percent and 93.04 percent for the DCF1 and
RIM1 models are quite close, implying that these models apparently have similar
ability to explain cross-sectional variation in current stock price. To test for the
pair-wise difference in R2s, we compute the Vuong Z-statistic (Dechow 1994) for
DCF1 versus RIM1 and cannot reject the null of no difference at the 5 percent
level.22 This result lends further support for the prediction of Hypothesis 1 that,
with ideal terminal value expressions, the choice of valuation models is a matter of
indifference. Third, the R2s are considerably higher for the price-based models,
compared with their non – price-based counterparts. For the DCF models, the
former is 93.71 percent, and the latter are 67.95 percent and 60.46 percent for
DCF0 and DCF2, respectively. The corresponding figures for RIM are 93.04 percent versus 79.65 percent and 77.02 percent, respectively. Thus, both price-based
valuation models appear to be far more successful in explaining the variability of
current stock price than the non – price-based models within the same family, a
result consistent with the prediction of Hypothesis 2. Fourth, the R2s for the non–
price-based RIM models are considerably higher than those for the non–price-based
DCF models (i.e., 79.65 percent versus 67.95 percent and 77.02 percent versus 60.46
percent, based on the 0 percent and 2 percent growth assumptions, respectively),
implying that RIM is superior to DCF in situations where terminal price forecasts
are not available.
Panels A and B of Table 6 present evidence on the incremental explanatory
power of various components of intrinsic value estimates for the DCF and RIM
models, respectively. Comparing across variants within the same family of DCF
and RIM valuation models, we find that the overall R2 is at its highest level in
DCF1 and RIM1 (i.e., 93.40 percent versus 67.68 percent and 63.66 percent for
DCF; 92.79 percent versus 81.11 percent and 80.52 percent for RIM), establishing
once again the superiority of price-based models over their non–price-based counterparts (i.e., Hypothesis 2).
8.753‡
11.341
0.971
77.02 %
38.305‡
83.851‡
−2.478‡
t-statistics
H0: aj = 1
90.630‡
RIM2
t-statistics
H0: aj = 0
‡
9.943
1.107
79.65 %
Coefficient
DTV is the discounted terminal value per share given by the last component of valuation models described above.
−8.741‡
t-statistics
H0: aj = 1
†
167.520‡
34.796‡
t-statistics
H0: aj = 0
RIM0
Coefficient
3.886‡
t-statistics
H0: aj = 1
−23.643‡
t-statistics
H0: aj = 1
Non–price-based DTV†, g = 2%
16.161
0.706
60.46 %
50.076‡
56.638‡
−7.485‡
t-statistics
H0: aj = 0
52.272‡
66.692‡
Coefficient
DCF2
t-statistics
H0: aj = 1
DCF0
t-statistics
H0: aj = 0
15.604
0.899
67.95 %
Coefficient
RIM1
t-statistics
H0: aj = 0
−6.245‡
t-statistics
H0: aj = 1
Model: Pit = a0 + a1 IV itM + eit . The slope coefficient on each of the intrinsic value estimates is predicted to be 1, where IV itM is the intrinsic
value estimate per share for firm i in year t calculated under M = DCF1, DCF0, DCF2, RIM1, RIM0, and RIM2, described in the appendix.
0.598
0.950
93.04 %
1.101
176.740‡
DCF1
t-statistics
H0: aj = 0
Price-based DTV†
Coefficient
0.162
0.966
93.71 %
Coefficient
Price-based DTV†
*
Notes:
Intercept
IV
R2
Panel B
Intercept
IV
R2
Panel A
TABLE 5
Pooled GLS panel regression of contemporaneous stock prices on intrinsic value estimates*
651
Intercept
B
PV
DTV
R2
Panel B
Intercept
FA
PV
DTV
R2
Panel A
0.521
0.962
0.909
0.945
92.79%
9.66%
0.73%
15.38%
8.625
1.249
1.679
0.630
81.11%
31.65%
3.61%
3.70%
RIM0
30.749‡
64.622‡ 12.867‡
28.865‡ 11.675‡
19.143‡ −11.257‡
17.13%
15.49%
20.09%
RIM1
3.143‡
73.233‡ −2.863‡
25.812‡ −2.592‡
82.589‡ −4.780‡
DCF0
49.050‡
52.254‡ −5.683‡
29.016‡
3.290‡
‡
27.719
−9.401‡
Non–price-based DTV, g = 0%
t-stat.
t-stat.
H0:
H0:
Incremental
Coefficient αj = 0
αj = 1
R2†
15.61%
7.82%
45.81%
15.443
0.902
1.128
0.760
67.68%
t-stat.
t-stat.
H0:
H0:
Incremental
Coefficient αj = 0
αj = 1
R2†
Price-based DTV
t-stat.
t-stat.
H0:
H0:
Incremental
Coefficient αj = 0
αj = 1
R2†
DCF1
0.038
0.257
0.951
86.977‡ −4.476‡
2.396§
1.043
58.253‡
0.942 133.250‡ −8.215‡
93.40%
Price-based DTV
t-stat.
t-stat.
H0:
H0:
Incremental
Coefficient αj = 0
αj = 1
R2†
DCF2
51.914‡
40.478‡ −9.767‡ 12.76%
26.114‡ 3.325‡ 12.50%
24.023‡ −21.277‡ 16.07%
RIM2
30.976‡
62.331‡ 11.328‡ 31.62%
30.190‡ 13.084‡ 4.29%
17.604‡ −20.918‡ 3.11%
8.990
1.222
1.765
0.457
80.52%
t-stat. t-stat.
H0:
H0: Incremental
Coefficient αj = 0 αj = 1
R2†
16.126
0.806
1.146
0.530
63.66%
t-stat. t-stat.
H0:
H0: Incremental
Coefficient αj = 0 αj = 1
R2†
TABLE 6
Pooled GLS panel regression of contemporaneous stock prices on the components of intrinsic value estimates*
652
653
TABLE 6 (Continued)
Notes:
*
Model for DCF is Pit = α 0 + α1FAit + α 2 PVit + α 3 DTVit + εit and model for RIM is
Pit = α 0 + α1Bit + α 2 PVit + α 3 DTVit + εit , where Pit is the recent stock price
published in the VL forecast report; FA is the net financial assets per share (i.e.,
cash and marketable securities minus debt and preferred equity); B is the book
value of owner’s equity per share; PV is the present value of operating cash flows
to common shareholders or abnormal earnings to the horizon, on a per-share
basis, under the discounted cash flows (DCF) or residual income model (RIM),
respectively; and DTV is the discounted terminal value per share given by the last
component under DCF1, DCF0, DCF2, RIM1, RIM0, and RIM2.
†
Incremental R2 is calculated as the difference between R2 for the full model and R2
for the model excluding the variable in question.
‡
§
At the component level, the incremental explanatory power of DTV is highest
when VL price forecasts are used in the terminal value calculations. For instance,
DTV explains 45.81 percent (15.38 percent) of the cross-sectional variations in
contemporaneous stock prices in DCF1 (RIM1), but only 20.09 percent (3.70 percent) and 16.07 percent (3.11 percent) for DCF0 (RIM0) and DCF2 (RIM2),
respectively. These results imply that VL forecasts of (P − FA) or (P − B) are far
less noisy than the non – price-based DTVs in capturing post-horizon goodwill,
confirming the impression from Table 1 that DTV accounts for the lion’s share of
intrinsic value estimates in the price-based valuation models. In both DCF and
RIM, the coefficients on DTV are much closer to the theoretical prediction of unity
in the price-based models than in the non–price-based models (i.e., 0.942 versus
0.760 and 0.530 for DCF; 0.945 versus 0.630 and 0.457 for RIM). However, the
theoretical prediction of unity is rejected for all the slope coefficients at the 1 percent
level.
When the comparisons are made across families of valuation models, the
overall R2s appear to be similar when VL terminal price forecasts are employed
(i.e., 93.40 percent and 92.79 percent for DCF1 and RIM1, respectively). A Vuong
test fails to reject the null of no difference in R2 between DCF1 and RIM1 at the
conventional levels of significance. This result is consistent with the prediction of
Hypothesis 1. Similar to the evidence presented in Table 5, the non – price-based
RIM models continue to have higher overall R2 than the corresponding DCF,
regardless of the growth assumption (i.e., 81.11 percent versus 67.68 percent given
0 percent growth; 80.52 percent versus 63.66 percent given 2 percent growth).
654
Sensitivity analyses
Analysis based on VL’s uncertainty about future stock prices
As discussed in section 4, the target price range at the forecast horizon reflects
VL’s uncertainty about future stock prices. In particular, uncertainty is directly
related to the width of the range. A priori, one would expect to see larger prediction
errors and a diminished edge of the price-based valuation models over those that
employ non–price-based terminal value expressions, when VL is less certain about
the future. To gain some insight into this issue and provide indirect evidence that
VL target price forecasts are judgemental in nature, we rank our sample observations pooled over the 1992–96 period in ascending order according to the degree
of uncertainty facing VL, where uncertainty is measured as the difference between
the two endpoints of the range, scaled by the mid-point.
Focusing on accuracy, we first compute the median absolute prediction errors
for the top (i.e., most certain) and bottom (i.e., least certain) quartiles. The figures
for DCF1 and RIM1 are 12.40 percent versus 16.22 percent and 12.43 percent versus
16.64 percent, respectively. Wilcoxon two-sample tests of pair-wise comparisons
across the top and bottom quartiles for each of the two price-based models are all
significant at the 1 percent level, implying that VL target price forecasts are less
representative of the investor beliefs impounded in the current market price as
uncertainty increases.
Next, we compute the difference in the median absolute prediction errors
between DCF1 and DCF0, and RIM1 and RIM0 within the same quartile. The
differences for the top and bottom quartiles are − 29.14 percent and − 22.99 percent (− 27.18 percent and − 18.23 percent) for DCF1 − DCF0 (RIM1 − RIM0),
respectively. A negative difference reflects greater representativeness of the pricebased valuation model, compared with the corresponding ad hoc 0 percent growth
model. We then contrast the pair-wise differences for the same family across the
top and bottom quartiles using a Wilcoxon two-sample test, and find it to be significant for both DCF and RIM models at the 1 percent level. The result is similar for
interquartile comparison of RIM1 − RIM2, whereas an analogous comparison of
DCF1 − DCF2 is not statistically significant at any conventional level. These
results once again suggest a generally declining representativeness edge of pricebased valuation models over their non – price-based counterparts when VL faces
greater uncertainty about the future, confirming our earlier understanding based on
conversations with VL analysts that VL price forecasts are judgemental, rather than
mechanical, in nature.
Analysis based on Fama-French industry cost of equity
Following the convention in the literature, we have addressed the issue of risk by
discounting a stream of future flows of valuation attributes (i.e., cash flows and
abnormal earnings) at a risk-adjusted discount rate. As discussed in section 4, each
firm’s risk premium is calculated as the product of VL firm-specific beta and an
assumed market premium. Feltham and Ohlson (1999) point out that this approach
is ad hoc and lacks theoretical foundation. The conceptually preferred method
655
would involve adjusting valuation attributes for risk and then discounting the
resulting expressions at a riskless rate. However, implementing the certainty equivalence approach advocated by Feltham and Ohlson is difficult in practice.
As an alternative, we reperform the analysis using Fama and French’s 1997
industry costs of equity. This requires that we first classify our sample firms into
one of the 48 industries described in Appendix A of Fama and French (179–81),
and then use industry risk premiums from the five-year rolling three-factor model
(see last column, Table 7 of Fama and French, 172–3). The results (not reported in
a table) are very similar to those reported previously in Tables 3 – 6, and provide
comfort that the findings of this study are unlikely to be driven by the way we capture
risk.
6. Conclusion
Prior studies by Penman and Sougiannis 1998 and Francis et al. 2000 have compared the bias and accuracy of the non–price-based DCF and RIM models, measured in terms of the signed and absolute prediction errors, respectively, and
concluded that RIM outperforms DCF. In this study, we provide evidence to show
that these findings need not hold when Penman’s 1997 theoretically “ideal” terminal value for each model is employed. Using Value Line terminal stock price forecasts at the horizon to proxy for such values, we explore the empirical equivalence
of DCF and RIM over a five-year valuation horizon under the assumptions that any
measurement error in VL price forecasts is “neutral” across these valuation models, and that we have avoided the errors that can impede a comparison of such
models. For the overall sample, the median absolute prediction errors are 13.71
percent and 14.18 percent for DCF1 and RIM1, respectively. Thus, focusing on
accuracy, RIM does not dominate DCF when the ideal terminal values are employed.
Contrasting intrinsic values for models employing terminal price forecasts with
those that do not, we find that, for both DCF and RIM, the price-based valuation
models outperform the corresponding non–price-based models by a wide margin.
Of course, using VL price forecasts as the appropriate benchmark is invalid if such
forecasts are optimistic. However, even for the last two years of our sample (1995–96)
when the optimism in VL price forecasts has abated to a negligible level, our median
signed prediction error evidence continues to indicate a serious downward bias
when ad hoc terminal value expressions are used. These results imply that researchers who study the ex ante cost of capital or trading strategies using ad hoc terminal
value expressions for RIM should exercise care in interpreting their results.
Replacing VL terminal price forecasts with conventional terminal value
expressions using ad hoc growth estimates, similar to those employed by P&S
1998 and Francis et al. 2000, we are able to replicate their findings that RIM outperforms DCF. For example, when regressing current stock prices on intrinsic values, the R2 is highest in RIM, compared with DCF (e.g., 79.65 percent versus
67.95 percent, and 77.02 percent versus 60.46 percent under 0 percent and 2 percent growth assumptions, respectively). The superiority of RIM over DCF when
the ideal terminal value is not available is explained by P&S 1998 as follows: current book values of owner’s equity bring future cash flows forward and leave relaCAR Vol. 18 No. 4 (Winter 2001)
656
tively little value to be captured in the conventional terminal value expressions;
whereas the DCF model expenses operating assets and defers most of the value to
be captured at the horizon. For example, under a 0 percent (2 percent) growth
assumption, 20.77 percent (25.53 percent) of intrinsic value for RIM is derived
from discounted terminal value, and the corresponding figure for DCF is 91.81
percent (93.19 percent).
Any “horse race” between RIM and DCF may be biased against DCF because
of the inherent limitations in estimating the Copeland et al. 1995 version of the
finance free cash flow model, an approach employed by Francis et al. 2000 using VL
data. To address this issue, we estimate a version of DCF introduced by Penman
1997. This alternative specification is better suited to VL’s data because the valuation
attribute, free cash flows to common, is available from VL and there is no need to
estimate the WACC. Nevertheless, we caution that implementing Penman’s version
of DCF empirically using VL data may still contain measurement errors because VL
does not provide forecasts of either deferred income taxes or cash tied up in working
capital. Given these practical limitations of estimating DCF, it is quite remarkable
that we were able to establish the approximate equivalence between DCF1 and
RIM1.
For students of financial statement analysis, our paper contains several important messages. We agree with P&S 1998 and Francis et al. 2000 that RIM outperforms DCF when ideal terminal values are not available. However, we also show
that these models are empirically equivalent given ideal terminal values. Thus, in
our view, the main focus of valuation analysis should be improving forecasts of
attributes beyond a finite horizon. Post-horizon forecasts of free cash flows or
abnormal earnings can easily articulate across valuation approaches and ultimately
articulate back to the benchmark model of forecasting the present value of
expected dividends. A reliable “set” of post-horizon forecasts of valuation
attributes for DCF and RIM gives the analyst the key to valuation, namely, a reliable price forecast at the horizon. Hence, the dilemma over which valuation model
to use is replaced by the challenge of forecasting post-horizon valuation attributes.
This study has focused on the pricing errors and viewed market efficiency as a
maintained assumption. As an avenue for future research, tests for equivalence
could be conducted with future excess returns. Under the null of equivalence, valuation models using ideal terminal value expressions should generate mispricing
signals (i.e., intrinsic value minus price) that yield identical excess returns when
trading strategies are implemented.
657
Appendix
Summary of valuation models and notations
Valuation models
T
DCF1: Wt (DCF1) = FA t +
∑R
–τ
E t [C t + τ − I t + τ + i t + τ − (R − 1)FA t + τ − 1]
τ=1
+ R −T E t (Pt + T − FA t + T)
T
∑R
–τ
τ =1
+ R−T(R − 1) −1 E t [C t + T + 1 − I t + T + 1 + i t + T + 1 − (R − 1)FA t + T]
T
∑R
–τ
τ =1
+ R−T(R − 1 − g) −1 E t [C t + T + 1 − I t + T + 1 + i t + T + 1 − (R − 1)FA t + T]
T
RIM1: Wt (RIM1) = Bt +
∑R
–τ
a
E t ( X t + τ ) + R −T E t (Pt + T − Bt + T )
τ =1
T
∑R
–τ
a
a
E t ( X t + τ ) + R −T(R − 1)−1 E t ( X t + T + 1)
τ =1
T
∑R
–τ
a
a
E t ( X t + τ ) + R −T(R − 1 − g)−1 E t ( X t + T + 1)
τ =1
Notations
Wt
=
Intrinsic value estimates per share under DCF1, DCF0, DCF2, RIM1, RIM0,
and RIM2.
R
=
One plus the cost of equity.
Pt + T =
VL’s forecasted price at the horizon, t + T.
Ct +τ
=
Operating cash flows, on a per-share basis, for forecast year t + τ.
It +τ
=
Capital expenditures, on a per-share basis, for forecast year t + τ.
it +τ
=
Interest flow from net financial assets for forecast year t + τ, on a per-share
basis. It represents interest paid (earned), including preferred dividends, if net
financial assets are negative (positive).
658
a
Xt + τ =
Abnormal income, on a per-share basis.
FAt
=
Net financial assets per share (i.e., cash and marketable securities minus debt
and preferred equity).
Bt
=
Book value of owner’s equity per share.
In addition, the numerator of the terminal value expression for the DCF0/
DCF2 and RIM0/RIM2 models is given by:
Ct + T + 1 − It + T + 1 + it + T + 1 − (R − 1)FA t + T = (1 + g)(C t + T − I t + T + i t + T)
− (R − 1)FA t + T ,
and
a
X t + T + 1 = (1 + g) X t + T − (R − 1)Bt + T ,
respectively.
Endnotes
1. The substitution “works” because both DCF and RIM are accounting systems (cash
versus accrual accounting) that obey a clean surplus relation. However, the substitution
can just as easily occur in the opposite direction such that one ends up back with
PVED. The choice between these three models, for infinite valuation horizons, is a
matter of indifference.
2. Some caveats are in order. First, as discussed in section 3, VL forecasts cash flows to
common, not operating, cash flows. Second, VL’s definition of cash flows ignores
deferred income taxes and changes in working capital. These limitations could bias
model comparisons against the DCF model.
3. The 0.67 estimate pertains to the sum of present value of forecasted (P − B) and
forecasted abnormal earnings for the last three years of the VL forecast horizon. When
the second component is separated, the (unreported) valuation coefficient on terminal
forecasts of (P − B), according to Abarbanell and Bernard 2000, is only “slightly
higher”.
4. The version of our RIM models that is closest to Sougiannis and Yaekura 2001 is
RIM2, which assumes a 2 percent growth rate. As reported in section 5, the median
signed and absolute prediction errors for our RIM2 model are −34.36 percent and
36.42 percent, respectively, comparable to those documented by Sougiannis and
Yaekura using I/B/E/S data.
5. See Proposition 1 of Feltham and Ohlson 1995 for a reconciliation of the RIM and
DCF models when infinite horizons and risk neutrality are assumed.
6. The correction is evident since “i” in (4) denotes the interest expense (income) that
will be reported at date t + τ .
7. In both cases, the firm is assumed to have reached a steady state at the horizon. The 2
percent growth rate approximates the rate of inflation during our sample period.
659
8. Implicitly, we assume that the residual cash flows from financial assets grow at the
same rate as that generated by operating assets. If financial assets were marked to
market, expected financial residual cash flows would be zero, and the terminal value
would depend only on operating cash flows.
9. The results for the DDM model (not reported in a table) are in fact exactly the same as
those under DCF1, which appear in Tables 3–6. See section 5 for an elaboration.
10. To see this, note that if VL simply employs the reciprocal of the equity cost of capital as a
forecasted P/E ratio at the horizon, (7a) will collapse to (7b) because (Pt + 5 − Bt + 5) =
Xt + 6 /re − Bt + 5 = (Xt + 6 − re Bt + 5 )/re = X ta+ 6 /re, under the assumption that Pt + 5 =
Xt + 6 /re.
11. Sectors affected include retail stores and airlines.
12. When the VL firm-specific beta is missing, the average VL industry beta at the twodigit SIC level is used. The historic market premium of 6 percent is appropriate,
according to Ibbotson and Sinquefield 1983.
13. For most of our sample firms, f ’s are given by (1/4), implying that the forecast is
generally made in the third quarter.
14. Specifically, the first period abnormal earnings for RIM, covering a fraction of the
year from forecast (i.e., evaluation) date to the end of forecast year, are X ta+ 1 =
f X t + 1 − [(1 + r) f − 1]Btq , where Btq = Bt + (1 − f )(Xt + 1 − Dt + 1). For DCF, the first
period residual financial income is Y ta+ 1 = f (Ct + 1 − It + 1) − [(1 + r) f − 1]FAtq, where
FAtq = FAt + (1 − f )(Ct + 1 − It + 1 − Dt + 1). The notation is as defined in the text and
summarized in the appendix.
15. VL forecasts to be interpolated using this procedure include cash flows, capital
spending, number of common shares outstanding, dividends, and tax rates. In order to
preserve CSR in years 3 and 4, we assume that earnings for these two years are equal.
Appealing to CSR for years 3 to 5 and solving for Xt + 3 (or equivalently Xt + 4 ), we
get Xt + 3 = Xt + 4 = 1⁄ 2 (Bt + 5 − Xt + 5 + dt + 5 − Bt + 2 + dt + 3 + dt + 4 ).
16. We use the most recent VL stock price prior to the forecast date as the dependent
variable because it represents the market’s evaluation of the firm at the time when VL
generates its forecasts. At that time, the conditioning information set of the market and
that of VL are approximately synchronous in time. Allowing the passage of time so that
market price impounds VL forecasts would introduce a price influenced by subsequent
information not available to VL at the forecast date and hence confound inferences.
17. These percentages are very similar to that for DDM (not reported in a table) where
92.28 percent of intrinsic value comes from DTV, implying that dividend payments to
the horizon per se are only value relevant at the margin because they represent wealth
distribution rather than wealth creation.
18. The impact on the test of Hypothesis 1 is minimal because there is substantial (i.e., 76
percent) overlap in the firms falling in the extreme distributions of DCF1 and RIM1
models.
19. Note that 258 (or 12.27 percent) of our 2,110 sample observations do not satisfy CSR
for book values within ±5 percent of book values. This is consistent with Bushee 2001,
who reports that for his sample VL’s expectational data “satisfy” a similar CSR
condition about 90 percent of the time.
660
20. The corresponding median signed prediction errors in Francis et al. 2000 are −42.7
percent and −28.2 percent for DCF0 and RIM0, respectively. They do not report results
for DCF2 and RIM2.
21. The results presented in both tables are based on a subset of the original sample, after
deleting 35 observations (or seven firms) with studentized residuals exceeding an
absolute value of 2.5.
22. As discussed earlier, the Kmenta 1986 GLS procedure is essentially OLS after
transforming the data to remove autocorrelation and heteroscedasticity. Since the
Vuong test is appropriate for OLS, it is also appropriate for residuals arising from the
final-stage regression.
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