∗
‡
‡
∗
∗
Corporate Finance Department
‡ Accounting Department
University of Cologne
Albertus-Magnus Platz, D-50923 Cologne, Germany
Correspondence: sievers@wiso.uni-koeln.de; +49 221 4707882 lorenz@wiso.uni-koeln.de +49 221 4704452

Abstract
From a theoretical point of view the three standard equity valuation approaches, i.e. the
Dividend Discount Model (DDM), the Residual Income Model (RIM) and the Discounted
Cash Flow Model (DCF) must yield identical equity values if ideal valuation conditions are given. However, under less than ideal conditions, practitioners as well as researchers report different value estimates, especially if dirty surplus accounting prevails or if payoff data for an infinite horizon are unavailable. As result, several recent studies (e.g. Penman/Sougiannis
(1998), Francis/Olsson/Oswald (2000), Courteau/Kao/Richardson (2001)) find that RIM tends to outperform DCF and DDM.
This paper demonstrates that even under less than ideal payoff conditions and different steady state assumptions identical value estimates can be achieved with the three models if they are properly adjusted. Specifically, we derive adjustments for dirty surplus accounting, narrow dividend definitions, net interest relation violations and model specific growth rates for the terminal value calculation.
Based on this comprehensive decomposition approach we demonstrate for a broad US data set of realized payoffs from 1987-2004 that valuation errors including all necessary adjustment terms are the same across the three models and smaller than neglecting these correction terms.
Moreover, we provide empirical evidence on the magnitude of errors resulting from non-ideal conditions and different steady state assumptions. Finally, by leaving out the different correction terms we are able to approximate the findings of previous studies.
JEL Classification: G10, G12, G34, M41.
Keywords : Valuation Equivalence, Dividend Discount Model, Residual Income Model,
Discounted Cash Flow Model, Dirty Surplus, Terminal Value, Steady State, Valuation Error.

In general, intrinsic company values are calculated by discounting expected future payoffs with appropriate risk adjusted cost of capital. Different specific approaches such as the dividend discount model (DDM), the various versions of the discounted cash flow model
(DCF) and the residual income model (RIM) focus on alternative payoff definitions.
Nevertheless, under consistent payoff assumptions, all models must yield identical firm values.
1 In particular, the equivalence holds under clean surplus accounting and if (estimated) payoffs for an infinite horizon are available. However, such ideal conditions are almost never given in practice and therefore, empirical studies find different results.
2 Consequently, they yield different assessments regarding the performance of the three approaches.
3
In our model implementation, we account for deviations from ideal payoff conditions and different steady state assumptions.
4 Corresponding to the selected steady state version we derive model specific growth rates for the terminal value calculation. This approach allows us then to address the following empirically important research questions. First, we want to measure the magnitude of the violations of ideal conditions. Second, we verify whether it is worthwhile to implement a consistent model in order to yield lower valuation errors. Third, we calculate the different model specific growth rates for the terminal value calculation given realistic conditions. Fourth, we want to reconcile the findings of other studies and explain the differences in valuation estimates. Finally, our model provides guidance for further research since the question shifts from what is the best model to the question how forecasts can be improved. The evaluator can assess the consequences of different steady state assumptions and distortions of ideal conditions concerning the accuracy of intrinsic value estimates.
Thus, we demonstrate for a broad sample and not only for a single firm that even under less than ideal conditions and different steady state assumptions identical value estimates can be
1 See Penman (1998), Penman/Sougiannis (1998), Levin/Olsson (2000), Lundholm/O'Keefe (2001) and the
2 discussion between Penman (2001) and Lundholm/O’Keefe (2001) in Contemporary Accounting Research.
We stress that this is not a criticism since the purpose of e.g. Penman/Sougiannis (1998) and
Francis/Olsson/Oswald (2001) was to analyze the performance of these valuation methods in common practice.
3 This line of research is known as „Horse Race“ literature, since it tests valuation approaches, similar to contestants in a race, on one dataset in order to determine which model explains best observed stock market prices.
4 However, we do not focus on cost of capital adjustments in order to achieve the equivalence since these issues have been extensively explored in other studies. See Levin (1998), Plenborg (2002) or Fernández (2002) .
1

achieved with the three models if they are properly adjusted.
5 Specifically, we derive adjustments for dirty surplus accounting, narrow dividend definitions and net interest relation violations. Moreover, for different steady state assumptions we derive model specific growth rates for the terminal value calculation, which retain valuation equivalence. If all items on the balance sheet and income statement are assumed to grow after the explicit forecast horizon at the same growth rate g (Levin/Olsson (2000) denote this as the balance sheet steady state
(BSS)) the resulting model specific payoffs i.e. dividends, residual income and free cash flows yield equal value estimates.
6 Otherwise, if in contrast to the balance sheet steady state assumption, the company’s predicted dividends, residual income and free cash flows are assumed to grow independently from each other at the same constant rate, valuation equivalence does not hold anymore.
7 Thus, the derived model specific growth rates implemented in the terminal value calculation make sure that valuation equivalence remains further on. In addition, their empirical quantification provides important guidance if one wants to employ different steady state assumptions.
Based on a broad US dataset of realized payoffs from 1987-2004, our valuation errors become as low as 13%, if we apply a two stage valuation process with a ten year explicit forecast period. We show that it is worthwhile to incorporate the proposed correction terms in the model implementation since the valuation errors are significantly smaller than neglecting these corrections. Furthermore, we quantify the magnitude of each valuation correction term and the resulting valuation errors if the correction terms are neglected. This is important for practitioners and researchers alike to avoid valuation errors caused by non-ideal conditions encountered in practice. For example, we find that disregarding the dirty surplus correction alone amounts already to an underestimation of 23% in the no growth case and 25% when a growth rate of 2% is assumed. In addition, the valuation models imply a broad classification of dividends including share sales and repurchases (see e.g. Jiang/Lee (2005)). The consideration of this component is of major importance since there is ample evidence that the significance of transactions with equity owners has increased over time (see e.g. Fama/French (2001),
Grullon/Michaely (2002)). Indeed, we find that the present value of net contributions (share sales minus share repurchases) sums to nearly 16% of the intrinsic value estimate regarding the DDM.
5 For example, Lundholm/O’Keefe (2001) already state: „But we argue that even in a practical implementation or large sample study, the models should still be equivalent – for every firm in every year.”
Lundholm/O'Keefe (2001), p. 315.
6 See Lundholm/O’Keefe (2001), Levin (1998) or Levin/Olsson (2000).
7 These steady state assumptions are implemented for instance in the empirical studies of Penman/Sougiannis
(1998) and Francis/Olsson/Oswald (2000).
2

Due to the different modeling options for steady state the growth rates can differ across the three models. We find that the model specific growth rates for the RIM are about 4% points smaller than in the DDM and the DCF model. Thus, the steady state assumption for the constant growth of the residual earnings is more reasonable and results in a more accurate value estimate than the steady state assumption of dividends and free cash flows. Further, it implies that the terminal value has only an inferior importance in the RIM.
Moreover, we are able to approximate the findings of previous studies. Thus, we disclose the reasons why
previous studies get different equity value estimates by disentangling ceteris paribus the effect of different steady state assumptions. It is consistent with previous studies concerning the robustness of the RIM to deviations from ideal conditions, e.g. if truncated forecast horizons are considered etc., but argues that it is worthwhile to enhance even the RIM with the proposed dirty surplus correction to yield more accurate value estimates.
The proposed decomposition approach provides a guideline to future studies concerning the accuracy and consistency of analyst forecasts since it allows to create a benchmark environment which is under control of the evaluator. Therefore, it shifts the focus from asking which model is preferable to the question how forecasted payoffs could be improved, since the evaluator can assess the consequences of simplifications and distortions of ideal conditions concerning the accuracy of the intrinsic value estimation.
The remainder of this study is organized as follows. Section 2 reviews the related literature.
Section 3 demonstrates the theoretical equivalence of the valuation approaches DDM, RIM and DCF model for an infinite forecast horizon and then possible steady state assumptions are discussed. Further, the empirically implemented models including our comprehensive decomposition approach with the new correction terms are introduced and the model specific growth rates are derived. Section 4 describes the data and shows the empirical results based on the models from the previous section. In addition, the valuation errors are reported for a consistent implementation. Subsequently, we quantify the magnitude of the different correction terms and the model specific growth rates. Finally, our approach allows us to approximate the findings of previous studies. Section 5 summarizes the results and concludes.
Several studies investigate the ability of valuation techniques to obtain reasonable estimates of market values.
8
Empirically
, our study is most closely related to the work of
Penman/Sougiannis (1998), which is concerned with the important practical question how the
8 See e.g. Bernard (1995), Kaplan/Ruback (1995), Frankel/Lee (1998), Sougiannis/Yaekura (2001).
3

three intrinsic value methods perform if they are applied to a truncated forecast horizon arising naturally in practice. Based on an ex-post-portfolio approach with realized payoff data, they are the first to provide evidence that RIM yields the lowest valuation errors followed by the
DDM and DCF model. For example, RIM shows the smallest valuation errors for a 10 year detailed planning period followed by an ad-hoc perpetuity payoff (RIM: -16% vs. DCF: 82%).
These errors can be compared to our result (valuation error of 13%) recognizing the important difference that under our model implementation all equity value estimates are the same across the three models. In contrast to Penman/Sougiannis (1998) we allow for different steady state assumptions and deviations from ideal conditions.
As in Penman/Sougiannis (1998), we use realized payoffs based on a portfolio approach instead of forecasts for several reasons. First, realized data allows us to address the above mentioned issues for a much broader data set, since forecasted data for all three models would be available only for a limited number of firms. Second, it is well known that forecasts for several items (e.g. dividends, book values of equity and earnings, etc.) are not necessarily consistent to each other (see e.g. Courteau/Kao/Richardson (2001)). Moreover, analysts’ forecasts can be biased (see e.g. Chan/Karceski/Lakonishok (2003)). Biases and inconsistencies in analysts’ forecasts, however, are problems we do not want to address here since they add unnecessary complexity to the comparison of the three models. Finally, we use realized data for our research design, since it allows exact measurement of dirty surplus amounts, growth rates of pay offs and corresponding financial statement data.
9
Nevertheless, there are some other studies related to ours that use forecasts instead of realized data. For example, Francis/Olsson/Oswald (2000) employ an ex-ante approach based on analyst forecasts and support the findings of Penman/Sougiannis (1998) that RIM is superior to DDM- and DCF approaches. In addition, Courteau/Kao/Richardson (2001) compare the
DCF model to the RIM approach. Using Value Line (VL) price forecasts instead of perpetuity terminal values the authors find that DCF and RIM do not differ significantly, thus valuation equivalence cannot be rejected. Furthermore, it is shown that valuation errors of RIM and
DCF are close to each other in the case of non-price terminal values.
10 However, valuation equivalence is again not fully established and the ranking performance of the three valuation models is mixed (mostly RIM tends to outperform but there are cases where DCF is preferable). Finally, Dittmann/Maug (2007) analyze, if the ranking of several valuation methods (multiples and the three intrinsic value approaches DDM, RIM and DCF) depends on
9 E.g. capital expenditures for free cash flow calculations, dividends, capital increases, share repurchases and
10 earnings.
See Courteau/Kao/Richardson (2001).
4

the employed error measure (relative, absolute or log errors). Their findings suggest that the chosen error measure has a crucial impact and RIM is preferable to DCF and DDM. However, like previous studies, they employ single payoff steady state assumptions. Essentially, their implementation corresponds to our “basic model” in which all necessary correction terms are neglected.
From the theoretical
perspective our work is related to Penman (1997), Levin/Olsson (2000) and Lundholm/O'Keefe (2001). While Levin/Olsson (2000) analyze different steady state conditions and their effect on the valuation equivalence, Lundholm/O’Keefe (2001) point out that implementation approximations employed by Penman/Sougiannis (1998),
Francis/Olsson/Oswald (2000) and Courteau/Kao/Richardson (2001) drive the studies’ results.
11 In particular, three common problems concerning inconsistent forecasts, inconsistent discount rates and missing cash flows are identified. The first implementation problem occurs when the same growth rate is applied for different model-specific payoff definitions (e.g. dividends and free cash flows) due to different steady state assumptions. The second difficulty is due to circularity problems when the cost of equity and the weighted average cost of capital
(WACC) are independently determined in the valuation process (see Modigliani/Miller (1963).
The third potential pitfall concerns missing cash flows, if deviations from clean surplus accounting are given. Interestingly, Lundholm/O’Keefe (2001) suspect the present value of dirty surplus to be a substantial component and claim that missing cash flows affect the results of Courteau/Kao/Richardson (2001) and Francis/Olsson/Oswald (2000) since both rely on VL forecasts without correcting for dirty surplus. Further, they show that the studies of
Francis/Olsson/Oswald (2000) and Penman/Sougiannis (1998) may suffer from inconsistent discount rates since both employ the WACC independent from the other models’ discount rates.
Beside the importance of consistent terminal value calculations and the necessity to control for dirty surplus in valuation, Courteau/Kao/Richardson (2001) mention that the assumption that financial assets are marked to market is crucial in DCF valuation.
12 Thus, while
Lundholm/O’Keefe (2001) and Courteau/Kao/Richardson (2001) only note that these components are important to regard they do not provide corresponding correction terms.
Moreover, empirical evidence is still missing on how large the errors are if these terms are neglected. These two important issues are addressed here. We derive appropriate adjustment
11 It should be noted that Lundholm/O’Keefe’s (2001) critique is also applicable to Dittmann/Maug (2007),
12 since the same inconsistent approximations are employed.
See Courteau/Kao/Richardson (2001), p. 630.
5

terms and demonstrate that their implementation yields identical value estimates for the three models even under non-ideal valuation conditions and different steady state assumptions.
Most importantly, the proposed decomposition approach allows to create a controllable environment, which allows us amongst others to approximate the findings of previous studies by consciously omitting the proposed correction terms.
We consider the three most commonly used equity valuation techniques, which all are based on the idea that the value of a share is given by its discounted expected future payoffs.
According to the first model, the
Dividend Discount Model (DDM)
13 , the market value of equity V at time t is obtained by discounting expected future net dividends d to shareholders at the cost of equity r : 14
(DDM) V t
=
∞
∑
τ= 1
( d t +τ
+
E
) τ
. (1)
Net dividends include all positive cash transfers to shareholders, such as cash dividends or share repurchases, as well as negative cash transfers, e.g., due to capital increases.
Assuming compliance with clean surplus accounting the DDM can be transferred to a second approach, the
Residual Income Model (RIM)
.
15 Both DDM and RIM yield identical value estimates, if the clean surplus relation holds. The clean surplus relation (CSR) postulates that changes in book value of equity bv between two periods result exclusively from differences between earnings x and net dividends d :
(CSR) bv t
= bv + x t
− d t
. (2)
In other words, equity changes can arise exclusively from retentions of earnings or transactions with equity holders. Solving for d in equation (2) and substituting into the DDM leads under the transversality condition 16 to the RIM:
13 This is the basic model for firm valuation which is commonly attributed to Williams (1938) and Gordon
(1959).
14 For ease of notation the expectation (conditional) operator E t
in the numerator is suppressed in the following
15 analysis.
See e.g. Preinreich (1938), Edwards and Bell (1961), Peasnell (1982).
16 I.e. the assumption
τ→∞
( +
E
) −τ bv t +τ
→ 0
6

(RIM) V t
= bv t
+
∞
∑
τ= 1
( x a t +τ
+
E
) τ
, (3) where residual income, also referred to as abnormal earnings, x t a = x t
− r bv , i.e., regular earnings minus a charge for equity employed.
The third theoretically equivalent valuation approach is the Discounted Cash Flow (DCF)
Model .
17 In order to determine the market value, forecasts of free cash flows are discounted at an appropriate risk-adjusted cost of capital.
The DCF approach can be derived from the DDM by combining the CSR and the free cash flow definition fcf t
= oi t
− ( oa t
− oa
)
.
Free Cash Flow fcf is the after-tax cash flow available to all investors, i.e. debt and equity holders. oa denotes net operating assets (total assets minus all non-interest-bearing liabilities), oi is the operating income, defined as all income except interest expense on the interest-bearing liabilities, net of tax.
18
From the CSR and the free cash flow definition, the financial asset relation (FAR) is obtained: 19
(FAR) debt t
= debt + ( ) d fcf t
, (4) where debt is the sum of interest-bearing liabilities and preferred stock, 20 int denotes the interest expense and s represents the tax rate. By further assuming the validity of the net interest relation (NIR):
(NIR) int t
= r debt , (5) where
WACC-Approach can be obtained (see Appendix 2):
V t
=
∞
∑
τ= 1
( fcf t +τ
+
WACC
) τ
− debt t
. (6)
Although intuitively appealing, the WACC-Model in equation (6) is difficult to apply because it requires the estimation of the weighted average cost of capital r
WACC
. Since capital weights have to be derived from market values, this approach encounters circularity problems. These
17 See e.g. Rappaport (1986), Copeland/Koller/Murrin (1990) and the latest edition of Koller/Goedhart/Wessels
(2005).
18 See Lundholm/O'Keefe (2001), pp. 324-325 and p. 333 endnote 8.
19 The FAR can be derived by substituting bv t
= ( oa t
− debt t
)
in the clean surplus relation and subtracting this restated clean surplus relation oa t
− debt t
= oa − debt + x t
− d t
from the free cash flow definition oa t
= oa + oi t
− fcf t
.
20 In our analysis, we abstract from a distinction between operating and financial assets (i.e. trade securities).
See for instance Feltham/Ohlson (1995), where financial assets are defined as cash and marketable securities minus debt.
7

difficulties are avoided by the feasible (implicit) WACC-Model, which is used, for example, by Courteau/Kao/Richardson (2001):
(DCF) V t
=
∞
∑
τ= 1
( fcf t +τ
− ( − ) t +τ− 1
( +
E
) τ
+ t +τ− 1
)
− debt t
. (7)
While Equation (7) still assumes that debt is marked to market, i.e. the net interest relation equation (5) must hold, it is advantageous since it employs only the equity cost of capital which are also used in the DDM and RIM. Thus, all three models are directly comparable.
The theoretical equivalence of the models presented above is based on rather restrictive assumptions. In practice however, we are confronted with less than ideal conditions, i.e., dirty surplus accounting or deviations from the net interest relation. In addition, different steady state assumptions lead to inconsistencies and violations of the valuation equivalence.
Therefore, it is necessary to introduce several corrections in order to guarantee the theoretical equivalence of the three valuation methods under less than ideal conditions. Specifically, we derive adjustments for dirty surplus accounting, narrow dividend definitions and net interest relation violations. In addition, the proposed decomposition approach allows us to analyze different steady state assumptions and its effect on the growth rates.
The DDM, DCF and RIM equations in the preceding section require projecting all future payoffs to infinity, which is impossible in practice. Thus, one often makes explicit forecasts for only a limited number of years and accounts for the time after that with a terminal value.
21
The future is thus divided into two periods: the explicit forecast period and the terminal period.
Indeed, an inadequate short explicit forecast horizon and a thus an early terminal value calculation leads to inaccurate value estimates. Our study does not focus on this question, when steady state
is achieved, although by applying different forecast horizons, we provide some indicative results.
According to Levin/Olsson (2000) the notion of steady state can be separated into necessary and sufficient conditions. While the former postulates that the qualitative behavior of the company remains constant in the terminal period i.e. valuation attributes can be expected to
21 The term continuing value or horizon value is sometimes used instead of terminal value in the literature.
8

grow at a constant rate, the latter condition focuses on the interactions of the balance sheet and income statement items, which both have to be modeled in a consistent manner.
While it is very interesting to determine the point in time, when steady state is achieved, 22
Levin/Olsson (2000) and Lundholm/O’Keefe (2001) focus on different steady state conditions in their derivations. The following steady state definitions are defined by Levin/Olsson (2000):
(DSS) Dividend steady state: d = ( )
,
(RSS)
(FSS)
Residual income steady state:
Free cash flow steady state: x a t T 1 fcf a t T
,
,
(BSS) Balance Sheet steady state: BS item,i
+ + item,i
+
∀ i .
It is shown that the assumption of balance sheet steady state (BSS) assures that the forecasted balance sheets and income statements are internally consistent to each other. Based on this information set the calculation of all three payoff figures is coherent to each other. This assumption implies, that the return on equity RoE ≡ +
(
+
)
(i.e. for
τ ≥ 0 ) and all other relevant parameters remain constant in the terminal period. In contrast, it is shown that the first three steady state concepts (DSS, RSS and FSS) lead to different value estimate if employed independently. It should be noted, that Levin/Olsson’s (2000) balance sheet steady state definition corresponds to the implementation in Lundholm/O’Keefe (2001).
We expand these findings, by combining either DSS, RSS or FSS with the BSS assumption in each valuation formulae.
First, splitting the infinite forecast horizon into two stages leads to the following DDM:
V t
DDM =
T
∑
τ= 1
( d t +τ
+
E
τ
+
+
E d
) ( r
E
− g
)
. (8)
The first T years represent the finite forecasting period and consist of explicit and exogenous dividend forecasts. In the following terminal period, the dividend is assumed to grow at the constant growth rate g. The estimation of d is crucial, since at least two different steady state assumptions can be employed. According to the balance sheet steady state (BSS) assumption, d is obtained by letting each line item on the balance sheet (operating assets, debt, shareholders’ equity etc.) and the income statement (net income, operating income, interest expense etc.) grow at the rate g
. This steady state growth has to be applied first for period T to T+1 as well as all subsequent periods. Hence, under ideal conditions (e.g. clean
22 See. e.g Sougiannis/Yaekura (2001).
9

surplus accounting), the DDM starting value of the perpetuity, which guarantees valuation equivalence across the three approaches, is given by: 23 d = ( ) ( ) + bv = ( )
.
(9)
Alternatively, according to the dividend steady state assumption (DSS) the payoff in period t+T+1 is determined by: d = ( )
. (10)
Inserting expressions (9) and (10) into equation (8) leads to:
V t
DDM =
T ∑
τ= 1
( d t +τ
+
E
) τ
+
( ) + tv BSS,DDM t T 1
( +
E
) ( r
E
− g
) 1
(11) with tv BSS,DDM t T 1
( ) ( )
, where tv BSS,DDM captures the difference between these two steady state calculations. This procedure has to be applied to the other two models in a similar manner.
Turning to the RIM, the infinite forecast horizon model (equation (3)) is divided into the two periods as:
V t
RIM = bv t
+
T ∑
τ= 1
( x t a
+τ
+
E
) ( +
E x a t T 1
) ( r
E
− g
)
(12)
Under the balance sheet steady state (BSS) assumption, the final payoff in the RIM is calculated as: x a t T 1
( ) − . (13)
Alternatively, assuming residual income steady state (RSS), the numerator of the terminal value is given by: x a t T 1 a t T
( − )
. (14)
Inserting these two expressions ((13) and (14)) into equation (12) results in:
V t
RIM = bv t
+
T ∑
τ= 1
( x t a
+τ
+
E
) τ
+
( ( ) a t T
+ tv BSS,RIM t T 1
( +
E
) ( r
E
− g
)
)
(15) with tv BSS,RIM = −
( ( ) )
.
The terminal value adaptation term represents again the difference between the two steady state assumptions.
Finally, the two-stage version for the DCF model is given by:
23 See Lundholm/O’Keefe (2001).
10

V t
DCF =
T
∑
τ= 1
( cf t +τ
+
E
τ
+
+
E cf
) ( r
E
− g
)
− debt t
(16) with cf t +τ fcf t +τ
= fcf t +τ
−
D
( − ) t +τ− 1
+ r debt t +τ− 1
, and
= oi t +τ
− (oa t +τ
− oa t +τ− 1
).
Again referring to BSS, assuming clean surplus accounting and compliance with the net interest relation, the numerator of the perpetuity in the DCF model is calculated as: cf
( ) ( ) + oa
( ) ( ) + . (17)
In contrast, the extrapolation of the last payoff according to the free cash flow steady state
(FSS) assumption results in: cf
( ) ( ) − ( − ) + (18)
Using the same substitutions as in the other two models yields:
V t
DCF =
T
∑
τ= 1
( cf t +τ
+
E
) τ
+
( ( ) + tv BSS,DCF t T 1
( +
E
) ( r
E
− g
)
)
− debt t
(19) with cf t +τ
= fcf t +τ
−
D
( − ) t +τ− 1
+ r debt t +τ− 1
, fcf t +τ
= oi t +τ
− (oa t +τ
− oa t +τ− 1
), and tv BSS,DCF t T 1
= oa
( ) + ( )
.
Summarizing, this extended approach yields two stage valuation formulas for the DDM, RIM and DCF model. Most importantly, it is advantageous to other model specifications since each model nests both steady state assumptions (i.e. the respective model specific steady state formula (DSS, RSS or FSS) and in addition the BSS).
Note that these derivations are obtained under ideal conditions, i.e. clean surplus accounting, compliance with the net interest relation and full payoff information like share repurchases and capital contributions.
In order to relax these restrictive constrains, all three models are next enhanced to deal with deviations from ideal conditions. Specifically, we derive adjustments for dirty surplus accounting, narrow dividend definitions and net interest relation violations.
Notice that the dividend d in equation (11) must include all cash transfers between owners and the firm. If, for simplicity, only cash dividends are used (as, for example, in
Francis/Olsson/Oswald (2000)) a substantial part of cash transfers is neglected. To account for
11

this, we substitute d t
= d cash t
+ d t cor where capital increases and share repurchases.
Moreover, the valuation equation (11) requires clean surplus accounting as assumed in equation (9). Since this relation is usually violated under US-GAAP accounting, it is necessary to incorporate a dirty surplus correction in the terminal period of the DDM.
24 To account for dirty surplus elements, we substitute x = x dirt t T
+ dirt cor t T
, where which is affected by dirty surplus accounting.
25
The dirty surplus correction term dirt cor captures any differences between the earnings number x , which is calculated from the clean surplus relation and the income measure from the income statement. The clean surplus income x contains all changes in book value of equity not resulting from transactions with the owners. Thus, the dirty surplus amount is then calculated as: 26 dirt cor
+
= x − x dirt
+
=
( bv − bv +
( d cash
+
+ d cor
+
) )
− x dirt
+
. (20)
Hence, substituting d t
= d t cash + d t cor for all t and x consistent DDM valuation equation,
(DDM consistent ) V t
DDM =
T ∑
τ= 1
⎣ d t cash
+τ
+ d cor t +τ
( +
E
) τ
⎤
= x dirt t T
+ dirt cor t T
leads to the final
( + ) ( cash + d cor + dirt cor
( +
E
) ( r
E
− g
)
)
+ tv BSS,DDM
(21) with tv BSS,DDM dirt,t T 1 dirt cor
+ d t cor
+τ
=
( ( ) dirt t T
) ( ) ( cash t T
+ d cor t T
)
,
=
= ( x − x dirt
+
, and share repurchases
) ( )
Note that the dirt cor
+
term is necessary only because we need a (dirty) income measure to calculate the starting dividend in the terminal period. Therefore, the dirty surplus correction affects only the terminal value expression and we get a slightly different terminal correction as opposed to equation (19).
24 Clean surplus violations are e.g. unrealized gains and losses on securities available for sale, on foreign
25 currency translations or on derivative instruments.
Alternative specifications of dirty surplus income can be earnings measures such as comprehensive income according to SFAS No. 130, net income before extraordinary items or net income before extraordinary items and special items. In our study, we employ net income as the x dirt measure, because SFAS 130 “Reporting of
Comprehensive Income” became effective in 1997 and thus is not completely available for our sample period.
26
For empirical evidence on dirty surplus accounting see Appendix 5.
Alternatively according to Lo/Lys (2000) the clean surplus earnings can be estimated as the change of retained earnings after cash dividends. Although, this definition has to be treated with care, since stock dividends, that are distributions to shareholders in additional shares, lead to an increase of paid-in capital and a decrease of retained earnings and hence to a biased disclosure of clean surplus income.
Moreover, this approach causes biases by neglecting capital increases.
12

So far, we have employed a simple perpetuity with growth in the terminal value expression.
Alternatively, according to Penman (1997) a discounted T-year ahead stock price forecast 27 could be employed to substitute the terminal value calculation. Using this “price-based terminal value” instead of the growth rate based perpetuity (i.e., the so called “non-price-based terminal value”) the consistent DDM is equal to:
(DDM consistent-price ) V t
DDM,Price =
T
∑
τ= 1
( d
( t cash
+τ
+
+
E d
) cor t +τ
τ
)
+
(
P
+
E
) T
− debt t
. (22)
In contrast to the DDM consistent implementation, the correction terms obviously unnecessary, if such a price-based valuation is employed. tv cor,DDM are
If the clean surplus relation is violated under GAAP accounting it can be seen from equation
(23) that a dirty surplus correction should also be incorporated in the RIM approach. x t a = x t
− r bv =
( x t dirt + dirt t cor
)
− r bv = x t a,dirt + dirt cor t
(23)
Note that t t consists of a residual income resulting from the usage of an actually observed income measure have to be incorporated during the finite forecast period as well as the terminal period.
A consistent RIM implementation, which captures the difference between the steady state assumptions and the dirty surplus correction, consequently results in:
(RIM consistent ) V t
RIM = bv t
+
T
∑
τ= 1
( x a,dirt t +τ
+ dirt t cor
+τ
( +
E
) τ
)
+
( ( ) ( a,dirt t T
+ dirt cor t T
)
+
( +
E
) ( r
E
− g
) tv cor,RIM t T 1
)
(24) with tv BSS,RIM t T 1 dirt t cor
= −
(
= x t
− x .
( ) )
, and
When a terminal stock price forecast is available the consistent RIM employing a price-basedterminal value is given by:
(RIM consistent-price ) V t
RIM,Price = bv t
+
T
∑
τ= 1
( x t a,dirt
+τ
+ dirt t cor
+τ
( +
E
) τ
)
+
(
P − bv
( +
E
) T
)
. (25)
27 Providers of price forecasts are e.g. ValueLine.
13

The ideal price-based terminal value is the difference between the forecasted market price of the stock and the book value of equity at the horizon t+T. A positive premium
[
P − bv
] indicates accounting conservatism or positive net present value projects in the future.
In addition, dirty surplus accounting necessitates the inclusion of an appropriate correction term in the DCF approach as can be seen from the following equation: fcf t
= fcf t dirt + dirt t cor =
( oi t dirt
+τ
− (oa t +τ
− oa t +τ− 1
)
)
+ dirt t cor . (26)
Equation (26) states that the free cash flow calculated on the assumption of clean surplus accounting consists of the dirty surplus free cash flow fcf dirt , which is calculated indirectly starting from the net income equation (19), the modified DCF model which explicitly regards dirty surplus accounting is given by:
V t
DCF =
T ∑
τ= 1
( cf t dirt
+τ
+ dirt cor t +τ
( +
E
) τ
)
+
( ( ) ( dirt t T
+ dirt t cor
+τ
)
+
( +
E
) ( r
E
− g
) tv BSS,DCF t T 1
)
− debt t
(27) with cf t dirt
+τ
= fcf t dirt
+τ
− dirt cor t T
= x −
( − ) t +τ− 1
+ r debt t +τ− 1
, fcf t dirt
+τ
= tv BSS,DCF t T 1 oi t dirt
+τ
= oa
− (oa t +τ
− oa t +τ− 1
),
( ) + ( )
, and
Next, if debt is not marked to market the interest expense of a particular period cannot be determined according to the net interest relation (NIR) int t
NIR = and thus one assumption of the WACC-Model (equation (6)) is violated. To account for the possible deviation between interest expense from the income statement t according to the NIR int t
NIR , a last correction term, namely model: nir t cor
+τ
=
( int t
NIR
+τ
− int t
IS
+τ
) ( ) ( r debt t +τ− 1
− int t
IS
+τ
) (
1 s
)
. (28)
Accounting for the net interest relation adjustment following final consistent DCF model:
14

(DCF consistent )
V t
DCF =
+
(
T ∑
τ= 1
( cf t dirt
+τ
( ) (
+ dirt cor t +τ
( +
E
) τ
+ dirt t T
+ nir t cor
+τ dirt cor t T
+
)
+ nir cor t T
( +
E
) ( r
E
− g
)
)
+ tv BSS,DCF t T 1
)
− debt t
(29) with cf t dirt
+τ
= fcf t dirt
+τ
− ( − ) t +τ− 1
+ r debt t +τ− 1
, fcf t dirt
+τ
= oi t dirt
+τ tv BSS,DCF
+ +
= oa
− (oa t +τ
− oa t +τ− 1
),
( ) + ( )
, dirt t cor
+τ nir t cor
+τ
=
=
( x t +τ int
− t
NIR
+τ
− int t
IS
+τ
) ( ) ( r debt t +τ− 1
− int t
IS
+τ
) ( )
Again, if a terminal stock price forecast for time t+T is available, the ideal price-based terminal value is the discounted sum of
[
P + debt
]
. The DCF model using a price-based continuing value is then given by:
(DCF consistent-price ) V t
DCF =
T
∑
τ= 1
( cf t dirt
+τ
+ dirt cor t +τ
+ nir t cor
+τ
( +
E
) τ
)
+
(
P + debt
( +
E
) T
)
− debt t
. (30)
As demonstrated in previous sections, the three consistent standard valuation techniques employing a non-price-based terminal value (see equations (21), (24) and (29)) use the same growth rate g in the terminal value calculation to yield equal value estimates. Thus, the valuation estimates will not depend on the specific valuation model being used to discount the relevant payoffs. Indeed, an inadequate short explicit forecast horizon and a thus an early terminal value calculation leads to inaccurate value estimates, but the inaccuracies will be the same for all models. However, valuation estimates would diverge, if DSS, RSS and FSS are assumed and thus interrelation between the payoffs are disregarded.
Thus, to retain valuation equivalence we derive model specific growth rates g DDM , g RIM and g DCF for the DDM, RIM and DCF model, respectively. These growth rates consist of the adhoc assumed growth rate g and model-specific growth rate corrections ∆ DDM , ∆ RIM and ∆ DCF .
This exercise is worthwhile since it is of greatest importance how different steady state assumptions effect the models theoretically as well as empirically. However, the different correction terms during the finite forecast horizon have to be considered further on since they are not captured in the terminal period growth rate correction.
The consistent DDM in equation (21) can be reformulated to the following expression:
15

V t
DDM =
T ∑
τ= 1
⎣ d t cash
+τ
+ d cor t +τ
( +
E
) τ
+
(
(
DDM
( +
E
) T r
E
(
) d cash t T
DDM
) ) (31)
Obviously, no correction terms emerge explicitly in the terminal value calculation any more.
The dividend correction d co r t T
, the adjustments for dirty surplus accounting dirt co r t T
and the terminal value calculation tv BSS,DDM t T
are now implicitly included in the growth rate correction
∆ DDM . By equating the terminal value expressions in equation (21) and (31), the growth rate adjustment is given after some simple rearrangements by: 28
∆ DDM =
( +
E
) d cash t T
( r
E
)
( ( d
( cash
+
(
+ d cash t T d cor
+
+ d
) (
1 g cor t T
) (
) −
1 g bv
) − bv
(
1 g
) + bv
T
)
(
1 g
) + bv
T
)
( )
(32)
∆ DDM is a function of g, d cash
+
, d co r
+
, dirt co r
+
and book value of equity bv . The DDM growth correction is positive if the quotient is greater than one plus the ad hoc growth rate g. The sum of ∆ DDM and growth rate g indicates the consistent growth rate for the DDM g DDM .
Similarly to the DDM, a consistent model implementation for the RIM employing a model specific growth rate is given by:
V t
RIM = bv t
+
T ∑
τ= 1
( x a,dirt t +τ
+ dirt t cor
+τ
( +
E
) τ
)
+
(
(
RIM
( +
E
) T r
E
(
) x a,dirt t T
RIM
) ) (33)
The dirty surplus correction and the difference between the steady state assumptions tv BSS,RIM t T are contained in ∆ RIM . Solving for the growth rate adjustment, results in:
∆ RIM =
( +
E
) ( x a,dirt
+
( + ) + dirt cor
+
( + ) + tv BSS,RIM
+ + x a,dirt t T
( r
E
) ( x a,dirt t T
( + ) + dirt cor t T
( + ) + tv
)
BSS,RIM t T 1
)
( )
(34)
The structure of ∆ RIM is similar to ∆ DDM . A positive ∆ RIM increases g RIM and results in a relatively more important terminal value.
Finally, we derive the corresponding growth rate correction ∆ DCF for the DCF model. An equivalent implementation to our DCF consistent model (equation (29)) employing ∆ DCF is given by:
V t
DCF =
T
∑
τ= 1
( cf t dirt
+τ
+ dirt cor t +τ
+ nir t cor
+τ
( +
E
) τ
)
+
( (
(
DCF
( +
E
) T r
E
(
) cf dirt t T
)
DCF
) ) (35)
28 For derivations of the different growth rate adjustments see Appendix 3.
16

∆ DCF captures violations of the clean surplus relation and the net interest relation as well as the difference between the steady state assumptions. The growth rate adjustment for the DCF model can be calculated as:
∆ DCF =
( +
E
)
⎣
− oa
( + ) + x
( + ) + oa cf dirt t T
( r
E
− g
) + − oa
(
1 g
) + x
( + ) + oa
+
+
( )
(36)
The three growth rate adjustments have the same composition and look very similarly. An analyst should be aware that depending on the selected steady state assumption either the same growth g can be use or model specific growth should be applied in the terminal value calculation.
Table 1 provides an overview of the different correction terms used in the DDM, RIM and
DCF model.
[Insert Table 1 about here]
We show that even under less than ideal conditions identical value estimates can be achieved with the three models if they are properly adjusted. Appropriate adjustment terms for the models are derived in order to account for deviations from ideal conditions and different steady state assumptions. Specifically, we derive adjustments for dirty surplus accounting, net interest relation violations and narrow dividend definitions. Furthermore, we analyze the consequences of the DSS, RSS and FSS assumptions concerning the valuation equivalence.
Already Levin/Olsson (2000) and Lundholm/O´Keefe (2001) stated that the BSS assumption is necessary within the terminal calculation to retain value equivalence. We expand these findings, by combining the DSS, RSS and FSS with the BSS assumption. Although,
Lundholm/O’Keefe (2001) discuss theoretically the problem that occurs if the financial statement forecasts do not satisfy the clean surplus relation, we propose a correction for dirty surplus. Moreover, we take into account the interdependency between the terminal value calculation and the dirty surplus adjustment in the terminal value period. Both corrections are required simultaneously in the terminal period. In addition, with our dividend correction we point out that it is necessary to regard both stock repurchases and capital increases – i.e. a wide dividend definition – to guarantee valuation equivalence.
29 The
29 Penman/Sougiannis (1998) regard cash dividends and stock dividends in their analysis but they do not include capital increases in their dividend definition. See Penman/Sougiannis (1998), p. 377.
17

correction, which is necessary to incorporate in the DCF model if the interest expense calculation based on the net interest relation deviates from interest expense from the income statement.
In Appendix 4 we provide an illustrative example in which the correction terms are calculated for a specific firm.
In contrast to previous studies, we demonstrate that equivalent equity values can be achieved if the proposed decomposition approach is employed. In this case, valuation errors must be identical as well. Interestingly, these valuation errors are significantly reduced by including the proposed adjustment terms. Furthermore, we evaluate the relative importance of the forecasting and the terminal period. Thereafter we measure the model specific growth rates employing the DDS, RSS and FSS. Moreover, we quantify the magnitude of the individual correction terms. Because individual components translate directly into valuation errors, this allows us to derive conclusions about the models’ robustness against violations of the underlying assumptions. The quantification of errors enables us to resolve the puzzling and seemingly robust findings of previous studies that certain models outperform in terms of valuation errors. For example, we can approximate the findings of Penman/Sougiannis (1998),
Francis/Olsson/Oswald (2000) and Dittmann/Maug (2007) that RIM tends to outperform the
DDM and DCF model.
We use Compustat North American Annual Active and Research Files, which contain companies that are listed at the New York Stock Exchange (NYSE), the American Exchange
(AMEX) and at the National Association of Securities Dealers Quotations (NASDAQ) market.
For the cost of debt calculation credit ratings are necessary, but these are only sparsely available for firms in COMPUSTAT beginning in 1985/1986 and more reliably available starting in 1987. Financial companies (SIC codes 6000 to 6999) are excluded from the sample due to their different characteristics. Furthermore, companies with negative equity book values, share values smaller than $1.00 and fewer than 1 million shares outstanding are excluded from the analysis. This selection procedure avoids largely distortions due to outliers
18

and thus yields robust model estimates.
30 In total, we obtain 36,112 company years consisting of 4,285 different companies for the time period from 1987 to 2004. The number of companies ranges from 1,530 companies in 2004 to 2,335 companies in 1996 (see Table 2 for additional details).
31
Following Penman/Sougiannis (1998) we use realized payoff data in connection with a portfolio approach. Using realized data instead of forecasts is advantageous for several reasons. First, it leads to a larger database. For example, we can analyze four times more companies per year in contrast to the studies of Francis/Olsson/Oswald (2000) and
Courteau/Kao/Richardson (2001) who use analyst forecasts from Value Line.
32 Explicit dividends and cash flow forecasts are not widely available. Earnings forecasts are, but only for quite short horizons. Second, realized data allows exact measurement of dirty surplus amounts, growth rates of pay offs and other important input variables such as capital expenditure for free cash flow calculations, dividends, capital increases, share repurchases and earnings.
Nevertheless, realized data do not perfectly match expectations. However, the use of realized data is justifiable as long as the ex post observed payoffs match expected values. Presuming that deviations of realized from expected values cancel out on average, the companies are grouped into 20 portfolios. Companies are assigned randomly to individual portfolios in order to calculate the present value for a particular year.
The portfolio composition is maintained for all periods needed. To compute present values for subsequent years, the evaluation window is moved ahead and companies are assigned randomly to portfolios, again. For each portfolio the average relevant figure (cash flows, earnings, dividends, etc.) is computed for each horizon up to 10 subsequent years (t + T, T =
2,...,10) and discounted at the average costs of equity capital in order to obtain an average present value per portfolio. The randomization produces portfolios with similar characteristics, including risk and cost of capital estimates.
Cost of equity rate and then adding Fama/French’s (1997) industry specific risk premiums (48 industry code).
30 Overall, our data selection procedure is comparable to most other studies (e.g. Frankel/Lee (1998)). However,
Bhojraj/Lee (2002) or Liu/Nissim/Thomas (2002) impose more severe restrictions with regard to Compustat data.
31 Compared to Penman/Sougiannis (1998) our sample contains fewer companies. This is attributable to the fact, that more Compustat items are used than in their study.
32 Value Line forecasts about 1,600 US companies. Francis/Olsson/Oswald (2000) examine about 600,
Courteau/Kao/Richardson (2001) examine 422 companies per year for a five year evaluation period.
19

This results in a time invariant risk premium of 6.60% on average, ranging from 1.5% for
Drugs to 12.2% for Fabricated Products. The average cost of equity is 11.21%.
33
For the cost of debt specific rating information can be obtained from Compustat only for a sub-sample of 14,675 firm years. Missing rating information is replaced by the median rating of firms in corresponding industries using the Fama/French 48 industry classification. Cost of debt are then calculated by adding Reuters 5 year spreads to the risk free rate.
34 As shown in Table 2 mean cost of debt over all years is approximately seven percent and the median company rating is BBB.
Valuation techniques are evaluated by comparing actual traded prices with intrinsic values calculated from payoffs prescribed by the techniques. Assuming market efficiency the market capitalization is an appropriate criterion to evaluate the model performance. The signed prediction error (bias) denotes the deviation of intrinsic value estimate from share price. This
. The absolute prediction error is calculated as accuracy = price intrinsic value estimate / price . Note that a positive bias indicates that the intrinsic value is lower than the market price.
[Insert Table 2 about here]
Furthermore, summary statistics on the most input variables are given in Table 2. For example, the companies’ equity book value is $8.96 per share compared to an average debt level of
$7.20 per share. Thus firms are mainly equity financed (average leverage ratio based on book values 0.80). The average market value of equity varies between $798.89 million in 1987 and
$4,153.69 million in 2004. Regarding the average book to market ratio of 71% it becomes clear that equity book values are lower compared to the market capitalization indicating a conservative financial reporting. The average cash dividends per share range from $0.26 to
$0.40. The dividend payout ratio varies from a maximum of 23% in 1990 to a minimum of
33 Several sensitivity tests of our results are performed. The costs of equity were also computed based on a 10year T-bond rate as risk-free interest rate and an alternative risk premium in terms of a market premium of 6%
(see Ibbotson/Sinquefield (1993)) in conjunction with a CAPM firm specific risk component (rolling betaestimation). Moreover, an analysis was performed with a uniform cost of equity rate for all companies and years of 10%, 11%, 12% and 13%. The empirical results (not reported) for our sample do not react sensitively to the choice of the costs of capital, although some minor level effects concerning the bias and accuracy are
34 obviously observed.
5 years is a reasonable assumption according to the findings of Stohs/Mauer (1996).
20

14% in the years 2000 to 2002.
35 It can be seen that, except for 2004, the payout ratio has continuously declined since 1991. Average net dividends ($0.15 per share) turn out to be lower than the cash dividends because capital increases exceed share repurchases by a substantial amount. Free cash flow is equal to $0.04 per share on average and mean residual income per share is $-0.15. Interestingly, the standard deviation of net dividends 0.14 is considerably smaller as compared to free cash flows 0.33 or residual income per share 0.36.
Overall our sample has similar characteristics as in other studies (see e.g. Frankel/Lee (1998)).
Table 3 depicts average valuation errors – bias and accuracy – for the three consistent valuation approaches depending on the forecast horizon (t+2 to t+10) and the chosen terminal value calculation (non-price-based vs. price-based). The mean valuation errors are calculated over years of means for 20 portfolios to which firms were randomly assigned in each year.
36
Standard deviations of portfolio errors (given in brackets below valuation errors) are calculated over all portfolio values. The mean number of firms in portfolios (over all years) was 69.
[Insert Table 3 about here]
First of all, note that all three approaches yield identical valuation errors under our consistent model application by applying appropriate adjustment terms to account for deviations from ideal conditions. Using a non-price-based terminal value the relative valuation errors (bias) decline from 33% to 13% with an increasing forecast horizon. The steady decline of valuation errors support the monotonicity-property developed by Ohlson/Zhang (1999). Assuming a 2% growth rate in the terminal period the bias declines from 29% to 7%, thus smaller than the no growth case. The standard deviation declines with an increasing forecast horizon for the nonprice-based model (with growth) from 17% to 4% (from 37% to 25%). The variation is higher for the non-price-based-model with growth, since discounting with
( r
E
− g
)
leads to a higher proportion of the terminal value resulting in a higher volatility of the estimated value.
35 Companies with a payout ratio higher than 100% are assigned a payout ratio of 100% (see e.g. Frankel/Lee
36
(1998)).
Consequently, by forming 20 portfolios per year and assuming for example a two year forecast horizon and a terminal value estimate in t+3 the valuation models are only computable from 1987 to 2001. Thus, the mean valuation error is calculated over 300 portfolio values.
21

However, the results indicate that forecasts at each horizon convey new value relevant information and lead to more accurate and precise value estimates. The model employing a price-based terminal value possesses by far the smallest valuation errors and standard deviations. The bias is very close to zero and varies with increasing forecast horizon between
2% and -5%. The standard deviation decreases from 0.11 to 0.08.
In contrast, absolute valuation errors are of greater magnitude. Valuation errors for the nonprice-based approaches again decline from 48% to 35% and with a 2% growth rate from 52% to 39%. The absolute error for the price-based terminal value increases from 11% to 14%. The results concerning both signed and absolute valuation errors are similar to the RIM valuation errors in previous studies. However, bias and accuracy for the DDM and DCF model are considerably lower than previously reported.
37
The relative importance of the forecast period and the terminal period of the DDM, RIM and
DCF model based on a 6-year forecast horizon are analyzed in Table 4.
[Insert Table 4 about here]
For the DDM the equity value generated in the terminal period (DTV) amounts to approximately three quarters (72.79% for g=0%). A similar result is obtained for the DCF model, however in terms of the entity value: slightly more than three quarters of the combined value of equity and debt is determined by the terminal value (i.e. 124.46% of 164.05%). In relation to the intrinsic value (IV), the terminal value of the DCF model accounts for an extremely high fraction of intrinsic value. Consistent with prior literature,
Copeland/Koller/Murrin (1995) or Courteau/Kao/Richardson (2001), for instance, report DTV amounts to 125% and 93%, respectively. While in the DDM and DCF model a similarly large fraction of value is allocated in the terminal period, a completely different picture is obtained under the RIM. Here, around two thirds (63.08% for g=0%) of the market value of equity is determined by its book value bv, while the terminal period accounts only for slightly less than a quarter (23.89%). These findings confirm prior literature that more wealth is captured in valuation attributes to the horizon under RIM than under DDM and DCF. For the DDM and the DCF model, these results seem to be quite robust for alternative assumptions regarding the terminal value calculations (i.e. a growth of 2% or a price-based calculation). For the RIM,
37 For instance, Penman/Sougiannis (1998) report a bias for the DDM of 31.4%, for the RIM of 8.3% and the
DCF-Model of 111.2% assuming a t+4 forecast horizon and no growth in the terminal period.
Francis/Olsson/Oswald (2000) report a bias (accuracy) based on analyst forecasts of 75.5% (75.8%), 20.0%
(33.1%), 31.5% (48.5%) for the DDM, RIM and DCF-Model, respectively.
22

however, the percentage of value determined by the book value of equity is sensitive to terminal assumptions. Thus, DTV becomes more important and bv is least important. For example, bv (DTV) accounts for 48.97% (40.92%) of intrinsic value estimate under
RIM consistent-price and 58.48% (29.45%) under RIM consistent (g=2%) compared to 63.08%
(23.89%) under RIM consistent (g=0%).
As introduced in section 3.2.5, we derive model specific growth rates, which are consistent with the dividend steady state (DSS), residual income steady state (RSS) and free cash flow steady state (FSS) assumptions. These growth rates guarantee that all three valuation models obtain equal value estimates although steady state assumptions are made for each payoff separately. This analysis is very important from a practical point of view, since it allows us to assess the differences in growth rates, which hold up valuation equivalence across the models.
[Insert Figure 1 about here]
In Figure 1 mean growth rate adjustments are depicted for varying forecast horizons. In Panel
A an ad-hoc growth rate g of 0% is assumed. The growth rate adjustment for the RIM ∆ RIM is with 2% to 3% considerably small compared to the DDM ∆ DDM and the DCF model ∆ DCF .
This is in line with above results that the terminal value has only an inferior importance in the
RIM and thus the RSS assumption in the terminal value calculation leads only to smaller distortions in the value estimate. For the DDM and DCF model an approximately 6% growth rate adjustment is necessary, i.e. a 4% higher growth rate adjustment than in the RIM. In addition, this confirms the findings from above that the terminal value accounts for the major part in these two models. In Panel B growth rate adjustments are represented for the 2% adhoc growth rate. Indeed, the growth rate adjustments have a different level but also here the
RIM requires only a slightly growth rate adjustment compared to the DDM and DCF model.
We can learn from the analysis that if we use exogenous analyst forecasts and implement a terminal value according to the DSS, RSS and FSS assumption, we should at least apply three different ad-hoc growth rates. With these different growth rates, similar equity estimates can be achieved.
In this section the relative importance of the valuation components including the correction terms are analyzed. It is of high interest to regard the present value of each component
23

separately in order to evaluate their relevance. First, we analyze the absolute Dollar amounts of each component. Second, we translate these Dollar values to valuation errors.
The importance of the valuation models’ particular components are analyzed on a disaggregated level in Table 5. Again, the calculations are based on a 6-year detailed forecast horizon with subsequent terminal value. As already shown in Table 4 a consistent implementation of the DDM, RIM and DCF model yields identical estimates of the market value of equity ($11.55 for g=0%, $11.95 for g=2% and $14.88 for a price based terminal value).
[Insert Table 5 about here]
The DDM comprises the present value of cash dividends d cash as well as the difference between stock repurchases and capital contributions d cash and approximately 65% of the estimated market value of equity, presuming a non-price-based terminal value without growth. Compared to
50% for a higher share of the market value of equity. In addition, the DDM consists of the dirty surplus correction tv BSS and they are nearly equally important to consider, since the difference between the steady state assumptions (BSS and DSS) accounts for nearly 20% of the share price while the dirty surplus correction represents 15% of the present value. By introducing growth in the terminal period the share of tv BSS reduces to 12% while the other three DTV-components’ shares increase accordingly.
In the RIM employing a non-price-based terminal value without growth the sum of book value of equity bv and the present value of residual income accounts with 82% for an important fraction of the estimated market value of equity. While it is of highest importance to correct for dirty surplus accounting in the RIM since it accounts for 23% of the share price, the discounted difference between the steady state assumptions (BSS and RSS) amounts to a relatively small fraction of -2% compared to the other valuation approaches. Neglecting this correction would lead to a slight overestimation of the market value of equity for our data. In contrast, omitting the dirty surplus correction would result in an underestimation of the market value of equity amounting up to $2.60. Assuming a 2% growth rate yields slightly larger values for the sum of book value of equity and present value of abnormal earnings as well as the tv BSS increases to -1% of the share price. If a price-based terminal value is employed in the RIM, only the dirty surplus correction is necessary. The amounts to 6%, so even with the price-based terminal value it is worthwhile to consider.
24

Turning to the consistent DCF model, this entity approach comprises the present value of cash flows cf from which the present value of debt is subtracted (cf debt).
Furthermore, the
DCF model requires three additional correction terms, the net interest relation nir cor
, the dirty surplus correction tv BSS . The present value of the correction components together accounts for approximately 86% of the mean intrinsic value estimate presuming a non-price-based terminal value without growth. The by far largest correction term is tv BSS with 68% or $7.86 of the total share price of $11.55.
While the dirty surplus correction is of same importance as in the RIM (23%),
4.36% rather small (the negative sign indicates that
( int t+T
> )
).
These results point out the importance of the terminal value in the DCF model and the need for the consideration of the difference between the steady state assumptions (BSS and FSS) to guarantee the valuation equivalence. While these findings are similar for the 2% growth case, they are different for the price-based terminal value case. In this setting, no terminal value adjustment is necessary and the present value of free cash flows and the discounted pricebased terminal value minus debt account for 95% of the share price. Obviously, the remaining correction terms (
In a next step, we translate the Dollar values to valuation errors, in order to analyze the importance of each correction term regarding the valuation errors. Therefore as starting point we introduce a basic model in which all different corrections are neglected.
The basic DDM considers only cash dividends - i.e. a narrow dividend definition - by leaving out the
(DDM basic ) tv BSS terms.
V t
DDM =
T
∑
τ= 1
( d cash t +τ
+
E
) τ
+
(
( )
) ( cash
+
+
E
T r
E
− g
)
(37)
(DDM basic-price ) V t
DDM,Price =
T
∑
τ= 1
( d cash t +τ
+
E
) (
P
+
E
) T
(38)
The basic RIM implementation abstracts from dirty surplus and the terminal value adjustment.
(RIM basic ) V t
RIM = bv t
+
T
∑
τ= 1
( x t a,dirt
+τ
+
E
) τ
+
(
( )
) ( a,dirt t T
+
E
T r
E
− g
)
(39)
(RIM basic-price ) V t
RIM,Price = bv t
+
T
∑
τ= 1
( x t a,dirt
+τ
+
E
) τ
+
(
P − bv
( +
E
) T
)
(40)
The basic DCF model disregards dirty surplus, net interest relation and the adjustments for
BSS assumption.
25

(DCF basic )
(DCF basic-price )
V t
DCF =
T ∑
τ= 1
( cf t dirt
+τ
+
E
) τ
+
(
( )
) ( dirt t T
+
E
T r
E
− g
)
(41)
V t
DCF,Price =
T ∑
τ= 1
( cf t dirt
+τ
+
E
) τ
+
(
P + debt
( +
E
) T
)
(42)
These basic models allow us to analyze the joint effect on the valuation errors, i.e. bias and accuracy by leaving out the different correction terms.
[Insert Table 6 about here]
Table 6 depicts the bias (Panel A) and the accuracy (Panel B) of the basic valuation models
DDM, RIM and DCF for different terminal value calculations and forecast horizons. As in the consistent application, valuation errors decrease with an increasing forecast horizon and the employment of a terminal value with growth also reduces valuation errors.
The basic RIM turns out to be the least affected valuation method according to bias and accuracy, followed by DDM and DCF model.
38 Especially, the DSS, RSS and FSS assumptions affect firm estimates based on the DDM, RIM and DCF model. We can see that if the steady state condition is not reached when the terminal value is calculated, the RIM yields more accurate firm value estimates than the other two approaches. The bias of the RIM decreases from 44.70% with a two-year finite forecast horizon to 30.74% with a ten-year finite forecast horizon. For the basic DDM (basic DCF model) the bias decreases from 66.11%
(95.84%) to 53.81% (79.91%) with an increasing forecast horizon assuming no growth. The bias resulting from the application of a price-based terminal value is close to zero and differs only insignificantly employing the Wilcoxon signed rank test (not reported) among the approaches. The volatility of the valuation errors is the lowest for DDM (5%), supposedly caused by commonly practiced dividend smoothing. In contrast, the DCF model possesses a considerably high variation up to 34%.
Compared to the consistent model implementation (Table 3), the valuation errors for the basic valuation models are significantly higher for all three approaches. The RIM shows thereof the lowest increment with regard to bias. The DCF model and the DDM show heavily biased market value estimates due to the simplified implementation. Thus, the RIM seems to be more robust to simplifications with regard to the bias since there is less room for mistakes if the book value is included. The deviations from the benchmark model (Table 3) using the
38 Moreover, we calculate logarithmic valuation errors (not reported) as proposed by Dittmann/Maug (2007). In contrast to Dittmann/Maug (2007) our ranking of the three valuation models does not depend on the error measure.
26

accuracy criterion are smaller for the DDM, RIM and the DCF model. The change in accuracy is the lowest for RIM and the highest for the basic DCF model. Moreover, the RIM occasionally shows lower accuracy for the simplified model than for the consistent implementation.
Finally, in Figure 1 we illustrate exemplarily the importance of the different correction terms for a t+6 forecast horizon concerning bias. Starting with the basic model (left bar) and then introducing our correction terms step-by-step leads to the consistent model implementation
(right bar).
[Insert Figure 1 about here]
As can be seen from this figure, the DDM needs three correction terms in order to yield a consistent valuation approach and all of them are nearly equally important. Furthermore, the
RIM is most heavily affected by dirty surplus accounting and the terminal value calculation is only of minor importance.
The DCF model requires three adjustment terms, from which the difference between the steady state assumptionss is by far the most important, but also dirty surplus accounting should be considered in order to yield meaningful results based on this technique.
In the end, it can be stated that all three equity valuation techniques yield the same intrinsic value estimates, if there are properly adjusted, which is required if realistic, i.e. non-ideal, valuation conditions are given. The RIM is more robust against valuation approximations and if less than ideal conditions are given.
The findings outlined in the preceding section allow us to resolve the puzzling findings of different equity value estimates obtained in previous studies.
39 Especially, we are able to approximate the results of Penman/Sougiannis (1998), Francis/Olsson/Oswald (2000) and
39 Although we are well aware that the studies of Penman/Sougiannis (1998) and Francis/Olsson/Oswald (2000) have another purpose by trying to replicate the situation of a typical investor facing an investment decision.
However, it is questionable if the outperformance of the RIM is driven by common practice. As shown in other studies (e.g. Block (1999), Bradshaw (2002, 2004), Demirakos/Strong/Walker (2004)) the RIM – although theoretically desirable- is not the most often applied valuation method and thus observed market prices cannot reflect RIM estimates. Given these results, one could argue that investment professionals are more sophisticated users of the DCF model, since the small valuation errors reflect a consistent model specification.
27

Dittmann/Maug (2007).
40 Most importantly, by employing the above outlined basic model implementations we expect to find the same results as in Penman/Sougiannis (1998),
Francis/Olsson/Oswald (2000) and Dittmann/Maug (2007), i.e. RIM outperforms DDM and
DCF.
[Insert Table 7 about here]
As shown in Table 7 the ranking of our basic model implementation corresponds entirely with the findings in Penman/Sougiannis (1998) since RIM performs best, second DDM and third
DCF regarding bias as performance criterion. The other two studies by Francis/Olsson/Oswald
(2000) and Dittmann/Maug (2007) employ forecasts instead of realized values, but again, the
RIM performs best independent of the chosen evaluation measure. Only minor differences are observed with regard to the second and third placement depending on the chosen performance measure. While the implementations of Francis/Olsson/Oswald (2000) and Dittmann/Maug
(2007) result in a second place for the DCF model with regard to bias, it is ranked third in our study. However, regarding accuracy, the placements of Dittmann/Maug (2007) and our results are again completely consistent to each other (1st RIM, 2nd DDM, 3rd DCF). Only
Francis/Olsson/Oswald (2000) place the DCF always second and the DDM third, which might be due their specific DCF model implementation and necessary approximations.
41
Overall, our hypothesis is confirmed that the ranking of the three intrinsic value methods depends on the violations of ideal conditions and the different steady state assumptions.
Finally, we have shown that controlled deviations from our consistent model implementations allow us to reconcile the empirical rankings of the above mentioned studies. Thus the newly proposed approach to implement the three most common equity valuation techniques could be used as benchmark to measure deviations from ideal conditions and most important to analyze their impact in a systematic manner instead of ad-hoc assumptions.
This study is motivated by empirical results for the three standard valuation approaches DDM,
RIM and DCF model based on realized values or forecasts. Although the theoretical
40 Courteau/Kao/Richardson (2001) implement the terminal value of the RIM and DCF-Model consistently. We suppose that the differences in bias and accuracy in their study are amongst others mainly caused by the missing dirty surplus correction.
41 Among these approximations are the use of an explicit WACC-approach with target weights inferred from
Value Line instead of the consistent cost of equity approach, adding excess cash and marketable securities to the firm value, setting FCF equal to zero in the terminal period if FCF is smaller than zero etc.
28

equivalence of these three valuation techniques is indisputable, several studies show that RIM tends to outperform DCF and DDM concerning bias and accuracy.
This paper demonstrates that even under less than ideal conditions identical value estimates can be achieved with the three models if they are properly adjusted. Within a comprehensive decomposition approach, we derive appropriate adjustment terms for the models in order to account for deviations from ideal conditions and different steady state assumptions.
Specifically, we derive adjustments for dirty surplus accounting, narrow dividend definitions and net interest relation violations. Furthermore, we derive model specific growth rates for the terminal value calculation consistent with different payoff steady state assumptions.
Valuation errors steadily decline with an increasing forecast horizon if our comprehensive decomposition approach including all correction terms is implemented. It is worthwhile to apply this approach with all corrections since the obtained value estimates are more accurate concerning bias and accuracy than neglecting these correction terms. Assuming a 2% growth rate in the terminal period, valuation errors are smaller than in the no growth case. The models employing a price-based terminal value possesses a valuation bias close to zero.
Moreover, we provide empirical evidence on the magnitude of errors resulting from non-ideal conditions and approximations employed by researchers and practitioners. This allows us to analyze each valuation component separately and disclose reasons for valuation differences obtained in other studies. Compared to the DDM and DCF model, the RIM is more robust against violations from ideal conditions. More wealth is captured in valuation attributes to the horizon under RIM and thus, the selected steady state assumption is less important. However, in the DDM and DCF model different steady state assumptions result in completely different estimates and thus are more severe. This different importance of the steady state assumption is also pointed out by the higher growth rate adjustments.
Neglecting the clean surplus correction term leads to considerably higher valuation errors for all models. The bias resulting from dirty surplus accounting is smallest for the DDM since the corresponding correction is only necessary in the terminal period. Regarding the RIM and
DCF model the correction for dirty surplus accounting accounts for 23% of the share price.
Moreover, we point out the importance of a broad dividend definition in the DDM since the significance of this component has increased remarkably over time. This dividend correction
(share sales minus share repurchases) leads to significantly smaller valuation errors. The by far largest correction term in the DCF model is the difference between the steady state assumptions (BSS and FSS) with around 68% of the equity value estimate. However, the adjustment for violations of the net interest relation is negligible small.
29

Finally, by leaving out the different correction terms we are able to approximate the findings of previous studies. The ranking of our basic model implementation corresponds entirely with the findings in Penman/Sougiannis (1998) - RIM performs best, second DDM and third DCF regarding bias as performance criterion. Moreover, we confirm the studies employing analyst forecasts that RIM outperforms DDM and DCF model.
For practitioners and researchers dealing with financial statement analysis, our paper contains several important messages. We agree with Penman/Sougiannis (1998), Levin/Olsson (2000) and Francis/Olsson/Oswald (2001) that RIM is more robust against different steady state assumptions and if less than ideal conditions are given. Furthermore, it is worthwhile to enhance even the RIM with the proposed dirty surplus correction to yield smaller valuation errors. However, we also show that these models are equivalent if they are properly adjusted given complete forecasted financial statements. Based on our theoretical and empirical findings, we suggest in line with other researchers, that subsequent efforts should be directed to the important issue how future payoff forecasts can be improved. Finally, the here proposed model provides a guideline and starting point for this issue since it shifts the research question from “what is the best model” to “how could valuation be improved”.
30

Table 1
The Different Correction Terms – an Overview
DDM consistent d cor
T
∑
τ= 1
( d t cor
+τ
+
E
) τ dirt cor nir cor tv
BSS explicit forecasting period
RIM consistent
DCF consistent
T
∑
τ= 1
( x t +τ
− x t dirt
+τ
( +
E
) τ
)
T
∑
τ= 1
( x t +τ
− x t dirt
+τ
( +
E
) τ
)
T
∑
τ= 1
( t +τ− 1
− int t
IS
+τ
) (
1 s
)
( +
E
) τ
Terminal
Value period
DDM consistent
RIM consistent
(
(
) (
) cor t T
+
E
T r
E
− g
)
( ) (
− x dirt t T
( +
E
) ( r
E
− g
)
)
( ) (
− x dirt t T
( +
E
) ( r
E
− g
)
)
(
−
( ) dirt t T
( +
E
) ( r
E
− g
)
( ( )
( +
E
) ( r
E
− g
)
) (
)
) ( cash t T
+ d cor t T
)
DCF consistent
( ) (
− x dirt t T
( +
E
) ( r
E
− g
)
) (
− int IS t T
) (
1 s
)
( +
E
) ( r
E
− g
)
( oa
( ) +
( +
E
) ( r
E
− g
)
( ) )
Notes:
DDM consistent
denotes the consistent dividend discount model according to equation (21). RIM consistent
represents the residual income model from equation (25). DCF consistent
is the discounted cash flow model in equation (29). The different correction terms in the terminal period are not required if a price-based terminal value is employed (see equations
(22), (25), (30)). d cor indicates the dividend correction which comprises the difference between share repurchases and capital increases. dirt symbolizes the correction of the net interest relation violation and tv BSS the clean surplus relation, x dirt
represents the difference between the steady state assumptions. x is the calculated income derived from
is an actually observed income measure, bv indicates book value of equity, oa is operating assets, int IS cor is the dirty surplus correction, nir cor
is interest expense from the income statement and s is the tax rate. r
E
refers to the cost of equity, r
D
denotes the cost of debt and g is the constant growth rate beyond the horizon.
31

Table 2
Summary Statistics by Year
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
No. firms
2,166
2,033
1,934
1,933
1,980
2,085
2,173
2,210
2,254
2,335
2,272
2,129
2,006
1,873
1,761
1,737
1,701
1,530
Mean Cost of Equity
0.1219
0.1304
0.1509
0.1453
0.1231
0.1016
0.0952
0.1054
0.1222
0.1174
0.1178
0.1136
0.1118
0.1236
0.1037
0.0816
0.0757
0.0773
Mean Cost of Debt
0.0723
0.0806
0.1005
0.0951
0.0729
0.0524
0.0468
0.0573
0.0743
0.0703
0.0702
0.0663
0.0649
0.0772
0.0573
0.0364
0.0309
0.0320
Mean Book
Value per
Share
7.8773
7.9317
8.1032
8.2428
8.1361
7.9319
7.8579
8.0460
8.1772
8.5098
8.7731
9.1757
9.6548
10.2666
10.0415
9.8903
10.6737
11.9392
Mean Debt per Share
6.7040
6.9469
7.2818
7.5124
6.9258
6.4013
6.1380
6.0681
6.0813
6.1315
6.4881
7.2726
7.9189
8.6849
8.3560
8.2843
8.1363
8.1843
Mean
Market
Value of
Equity
798.8913
898.4337
1,130.5620
1,090.6720
1,328.8250
1,319.5330
1,382.3240
1,335.0620
1,648.2170
1,839.0620
2,365.0060
2,836.6580
3,203.9340
3,518.7010
3,384.5370
2,758.6680
3,462.5950
4,153.6860
Mean Book to Market
Ratio
0.7420
0.6951
0.6670
0.8839
0.7346
0.6784
0.5949
0.6437
0.5941
0.5716
0.5243
0.6939
0.7717
0.9468
0.8415
0.9401
0.6456
0.5547
Mean Cash
Dividends per Share
0.3300
0.3700
0.3847
0.3955
0.3482
0.3321
0.3220
0.3188
0.3297
0.3481
0.2938
0.3084
0.3083
0.2763
0.2658
0.2625
0.2692
0.3561
Mean ROA
0.0411
0.0438
0.0432
0.0373
0.0261
0.0305
0.0268
0.0338
0.0334
0.0351
0.0301
0.0233
0.0306
0.0213
-0.0219
-0.0287
0.0005
0.0190
Mean
Payout
Ratio
0.2056
0.2053
0.2192
0.2289
0.2301
0.2192
0.2092
0.1960
0.1803
0.1688
0.1591
0.1519
0.1503
0.1397
0.1393
0.1425
0.1371
0.1574
Mean Net
Dividend per Share
0.1837
0.4185
0.3011
0.3772
0.0842
-0.0001
-0.0354
0.1137
0.0897
-0.0023
0.0972
0.3395
0.3231
0.0494
0.0575
0.0991
0.0904
0.0897
Mean FCF per Share
0.0207
-0.1504
-0.0038
0.3289
0.5072
0.3447
-0.0061
0.1406
-0.1687
-0.2174
-0.4086
-0.4947
-0.3065
-0.2877
0.0971
0.7025
0.4000
0.2463
Mean 2,006 0.1121
0.0643
8.9572
7.1954
2,136.4093
0.7069
0.3233
0.0236
0.1800
0.1487
0.0413
-0.1504
Std. Dev.
222 0.0209
0.0195
1.1931
0.8918
1,075.1199
0.1279
0.0399
0.0205
0.0340
0.1402
0.3329
0.3605
Notes:
Table values represent annual, equally-weighted average statistics for the sample firms. Mean market value of equity is measured in millions of dollars. All other values are in
US$ except for percentages. Averages and standard deviations reported in the bottom row represent time series means of the annual statistics. The cost of equity are calculated by using the one-year Treasury bill rate as the risk free rate and then adding Fama/French’s (1997) industry specific risk premiums (48 industry code). Industry specific cost of debt are calculated by adding Reuters 5 year spreads to the risk free rate. Book value per share denotes book value of equity (#70) divided by common shares outstanding (#25). Debt per share is calculated as the sum of long-term debt (#9), debt in current liabilities (#34) and preferred stock (#130) divided by common shares outstanding. Cash dividends per share are computed as common stock dividends (#21) divided by common shares outstanding. Dividend payout ratio is estimated as common stock dividends (#21) divided by net income (#172). For firms with negative earnings, payout ratio is computed as common stock dividends divided by (total assets x 0.024). The figure 0.024 is derived from mean ROA over all years. Companies with a payout ratio higher than 100% are assigned a payout ratio of 100%. ROA is the return on total assets and is estimated as net income divided by total assets (#6). In addition to cash dividends, net dividends include the purchase (#115) and sale (#108) of stocks. Free cash flow per share is estimated as operating income minus the change in net operating assets divided by common shares outstanding. Operating income denotes net income (#172) plus net interest, net of tax (#15 x (1-s)).
Net operating assets are defined as assets total (#6) minus liabilities total (#181) plus long term debt total (#9) plus debt in current liabilities (#34). Residual income is calculated as net income (#172) minus a charge for cost of equity (r
E
x (#60)).
Mean
Residual
Income per
Share
-0.1413
-0.1862
-0.3877
-0.4404
-0.5575
-0.3973
-0.2434
0.0074
-0.0376
0.0520
-0.0415
-0.0130
0.0933
-0.0719
-0.7434
-0.6760
0.3395
0.7381
32

Table 3
Panel A: Mean Signed Prediction Errors (Bias) using a Consistent Application t+2 t+4
Horizon (t+T) t+6
DDM consistent ; RIM consistent ; DCF consistent (g = 0%)
DDM consistent ; RIM consistent ; DCF consistent (g = 2%)
DDM consistent-price
; RIM consistent-price
; DCF consistent-price
0.3318
(0.1684)
0.2863
(0.3667)
0.0168
(0.1077)
0.2931
(0.1712)
0.2457
(0.3430)
0.0053
(0.1009)
0.2245
(0.1044)
0.1674
(0.2462)
-0.0175
(0.1194) t+8
0.1797
(0.1261)
0.1214
(0.2744)
-0.0477
(0.0937) t+10
0.1336
(0.0428)
0.0709
(0.2545)
-0.0530
(0.0801)
Panel B: Mean Absolute Prediction Errors (Accuracy) using a Consistent Application t+2 t+4
Horizon (t+T) t+6 t+8 t+10
DDM consistent
; RIM consistent
; DCF consistent
(g = 0%) 0.4827
(0.1077)
0.4472
(0.1009)
0.3875
(0.1194)
0.3683
(0.0937)
0.3458
(0.0801)
DDM consistent
; RIM consistent
; DCF consistent
(g = 2%) 0.5218
(0.2165)
0.4834
(0.2084)
0.4205
(0.1366)
0.4085
(0.1817)
0.3916
(0.0778)
DDM consistent-price ; RIM consistent-price ; DCF consistent-price 0.1095
0.1189
0.1413
0.1343
0.1411
(0.0503) (0.0366) (0.0368) (0.0515) (0.0415)
Notes:
DDM consistent consistent-price model in equation (29). DCF
is the consistent RIM in equation (25). DCF consistent is the discounted cash flow consistent-price is the DCF employing a price-based terminal value according to equation
denotes the consistent dividend discount model according to equation (21),
DDM consistent-price is the consistent model in equation (22). RIM consistent represents the residual income model from equation (24). RIM
(30). Signed prediction errors (bias) are calculated as (price - intrinsic value estimate) / price . Absolute prediction errors (accuracy) are calculated as (price - intrinsic value estimate) / price . Means are means over years of means for 20 portfolios to which firms were randomly assigned in each year. Standard deviation in parentheses is the standard deviation over all portfolio valuation errors for a given horizon.
33

Table 4
Relative Importance of the Forecast Period and the Terminal Period
Horizon (t+6)
Mean opening bv or debt
Mean PV
(% of mean IV)
Mean DTV
(% of mean IV)
Mean Intrinsic
Value (IV)
(% of mean IV)
DDM consistent (g = 0%)
(% of IV)
RIM consistent
(g = 0%)
(% of IV)
DCF consistent
(g = 0%)
(% of IV)
7.29
63.08%
-7.40
-64.05%
3.14
27.21%
1.51
13.03%
4.57
39.59%
8.41
72.79%
2.76
23.89%
14.38
124.46%
11.55
100.00%
11.55
100.00%
11.55
100.00%
DDM consistent
(g = 2%)
(% of IV)
RIM consistent
(g = 2%)
(% of IV)
DCF consistent
(g = 2%)
(% of IV)
7.29
58.48%
-7.40
-59.38%
3.14
25.23%
1.51
12.08%
4.57
36.70%
9.32
74.77%
3.67
29.45%
15.29
122.68%
12.46
100.00%
12.46
100.00%
12.46
100.00%
DDM consistent-price
(% of IV)
RIM consistent-price
(% of IV)
DCF consistent-price
(% of IV)
7.29
48.97%
3.14
21.13%
1.51
10.11%
11.74
78.87%
6.09
40.92%
14.88
100.00%
14.88
100.00%
-7.40
-49.72%
4.57
30.73%
17.71
118.99%
14.88
100.00%
Notes:
Calculations are based on a t+6 year forecast horizon. All reported components represent present values on a per share basis and are in US$. DDM consistent-price consistent denotes the consistent dividend discount model according to equation
is the consistent model in equation (22). RIM (21), DDM from equation (24). RIM consistent-price consistent represents the residual income model
is the consistent RIM in equation (25). DCF flow model in equation (29). DCF consistent-price consistent is the discounted cash
is the DCF employing a price-based terminal value according to equation (30). bv denotes the book value of equity per share, PV represents the present value of net dividends, abnormal earnings, free cash flow and all correction terms of the finite forecast horizon on a per share basis.
DTV are the discounted terminal value components per share. IV denotes the intrinsic value estimate per share.
34

Figure 1
Panel A: Mean Growth Rate Adjustments for the DDM, RIM and DCF model assuming an Ad-Hoc
Growth Rate of g=0% growth rate adjustme nt
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
2 4 6 8 10 fore cast horiz on t+T
∆ DDM (0%) ∆ RIM (0%) ∆ DCF (0%)
Panel B: Mean Growth Rate Adjustments for the DDM, RIM and DCF model assuming an Ad-Hoc
Growth Rate of g=2%
0.06
0.05
0.04
0.03
growth rate adjustme nt
0.08
0.07
0.02
0.01
0
2 4 6 8 10 fore cast horiz on t+T
∆ DDM (2%) ∆ RIM (2%) ∆ DCF (2%)
Notes:
Panel A and panel B depict mean growth rate corrections ∆ DDM , ∆ RIM and ∆ DCF assuming an ad-hoc growth rate g of 0% and 2%, respectively. These corrections occur in equations (43), (44) and (45). The consideration of these growth rate corrections guarantees that all value estimates are equal although a terminal value calculation according to the DSS, RSS, FSS assumptions (see equations (10), (14) and (18)) in combination with BSS is applied.
35

Table 5
Relative Importance of the Correction Terms
DDM consistent
(% of IV)
(g = 0%)
RIM consistent (g = 0%)
(% of IV)
DCF consistent
(% of IV)
(g = 0%)
DDM consistent (g = 2%)
(% of IV)
RIM consistent
(% of IV)
(g = 2%)
DCF consistent (g = 2%)
(% of IV) d cash
5.81
50.32%
6.75
54.13% bv+x a,dirt
9.18
79.47%
9.51
76.32% cf-debt
1.59
13.80%
3.09
24.77%
Horizon (t+6) d cor
1.74
15.03% dirt cor
1.73
14.95%
2.60
22.50%
2.60
22.50% nir cor
-0.50
-4.36% tv
BSS
Intrinsic
Value (IV)
2.28
19.69%
-0.23
-1.97%
7.86
68.06%
11.55
100.00%
11.55
100.00%
11.55
100.00%
2.00
16.08%
2.21
17.73%
3.08
24.72%
3.08
24.72%
-0.62
-4.94%
1.50
12.06%
-0.13
-1.04%
6.91
55.45%
12.46
100.00%
12.46
100.00%
12.46
100.00%
DDM consistent-price
(% of IV)
14.20
95.38%
0.69
4.62%
14.88
100.00%
RIM consistent-price
(% of IV)
14.01
94.14%
0.87
5.86%
14.88
100.00%
DCF consistent-price
(% of IV)
14.18
95.25%
0.87
5.86%
-0.17
-1.11%
14.88
100.00%
Notes:
Calculations are based on a t+6 year forecast horizon. All reported components represent mean present values on a per share basis and are in US$. DDM consistent denotes the consistent dividend discount model according to equation (21), DDM consistent-price from equation (24). RIM
is the model in equation (22). RIM consistent-price flow model in equation (29). DCF consistent-price consistent represents the residual income model
is the consistent RIM in equation (25). DCF consistent is the discounted cash
is the DCF employing a price-based terminal value according to equation (30). d cash are cash dividends, bv denotes book value of equity, x a,dirt is residual income, cf indicates the cash flow, d cor is the difference between stock repurchases and capital contributions, dirt cor is the correction for dirty surplus accounting, nir cor is the correction for violations of the net interest relation and the difference between the steady state assumptions. IV denotes the intrinsic value estimate. tv BSS denotes
36

Table 6
Panel A: Mean Signed Prediction Errors (bias) using the Basic Model Implementation t+2 t+4
Horizon (t+T) t+6 t+8 t+10
DDM basic (g = 0%)
RIM basic
(g = 0%)
DCF basic
(g = 0%)
DDM basic (g = 2%)
RIM basic (g = 2%)
DCF basic
(g = 2%)
DDM basic-price
RIM basic-price
DCF basic-price
0.6611
(0.0509)
0.4470
(0.1653)
0.9584
(0.3431)
0.5872
(0.0707)
0.4405
(0.2154)
0.8311
(0.4415)
0.0281
(0.1078)
0.0356
(0.1071)
0.0305
(0.1083)
0.6266
(0.0405)
0.4254
(0.1790)
0.9168
(0.3128)
0.5576
(0.0501)
0.4155
(0.2239)
0.7996
(0.3893)
0.0309
(0.1014)
0.0440
(0.1119)
0.0350
(0.1125)
0.6015
(0.0346)
0.3695
(0.1108)
0.8824
(0.2345)
0.5402
(0.0419)
0.3480
(0.1326)
0.7848
(0.2914)
0.0257
(0.1198)
0.0365
(0.1276)
0.0245
(0.1293)
0.5729
(0.0344)
0.3333
(0.1104)
0.8274
(0.2018)
0.5151
(0.0504)
0.3093
(0.1413)
0.7322
(0.2690)
0.0126
(0.0938)
0.0127
(0.0982)
-0.0014
(0.1010)
0.5381
(0.0373)
0.3074
(0.0843)
0.7991
(0.2670)
0.4808
(0.0509)
0.2813
(0.1076)
0.7114
(0.3460)
0.0226
(0.0844)
0.0150
(0.0825)
-0.0029
(0.0850)
Panel B: Mean Absolute Prediction Errors (Accuracy) using the Basic Model Implementation t+2 t+4
Horizon (t+T) t+6 t+8 t+10
DDM basic
(g = 0%)
RIM basic (g = 0%)
DCF basic
(g = 0%)
0.6611
(0.0509)
0.4715
(0.1432)
1.0494
(0.3036)
0.6266
(0.0405)
0.4607
(0.1515)
1.0167
(0.3030)
0.6015
(0.0346)
0.3891
(0.1018)
0.9459
(0.2261)
0.5729
(0.0344)
0.3725
(0.0827)
0.9435
(0.1503)
0.5381
(0.0373)
0.3398
(0.0589)
0.9028
(0.2034)
DDM basic (g = 2%)
RIM basic (g = 2%)
DCF basic
(g = 2%)
0.5872
(0.0707)
0.4862
(0.1713)
1.0092
(0.3567)
0.5583
(0.0499)
0.4703
(0.1787)
0.9778
(0.3511)
0.5410
(0.0420)
0.3909
(0.1211)
0.9157
(0.2569)
0.5166
(0.0483)
0.3761
(0.0998)
0.9454
(0.1833)
0.4808
(0.0509)
0.3431
(0.0667)
0.9110
(0.2207)
DDM basic-price
RIM basic-price
DCF basic-price
0.1114
(0.0503)
0.1136
(0.0545)
0.1144
(0.0522)
0.1208
(0.0398)
0.1295
(0.0554)
0.1283
(0.0532)
0.1438
(0.0350)
0.1483
(0.0386)
0.1472
(0.0403)
0.1240
(0.0341)
0.1321
(0.0436)
0.1321
(0.0485)
0.1364
(0.0264)
0.1384
(0.0262)
0.1393
(0.0276)
Notes:
DDM basic , DDM basic-price , RIM basic , RIM basic-price , DCF basic and DCF basic-price are implemented according to equations
(37) - (42). Signed prediction errors (bias) are calculated as (price - intrinsic value estimate) / price . Absolute prediction errors (accuracy) are calculated as (price - intrinsic value estimate) / price . Means are means over years of means for 20 portfolios to which firms were randomly assigned in each year. Standard deviation in parentheses is the standard deviation over all portfolio valuation errors.
37

Figure 2
Magnitude of the different Correction Terms – an Illustration
1.00
bias
0.80
0.60
0.40
0.20
0.00
DDM basic
DDM prediction errors for a t+6 forecast horizon g = 0% d cor dirt cor nir cor tv BSS,DDM
RIM prediction errors for a t+6 forecast horizon g = 0%
DDM consi stent
1.00
bias
0.80
0.60
0.40
0.20
0.00
1.00
bias
RIM basic
0.80
0.60
0.40
d cor dirt cor nir cor tv BSS,RIM
DCF prediction errors for a t+6 forecast horizon g = 0%
RIM consi stent
0.20
0.00
DCF basic d cor dirt cor nir cor tv BSS,DCF DCF consi stent
Notes:
Calculations are based on a t+6 year forecast horizon. The mean bias of the consistent and basic valuation models is calculated as (price - intrinsic value estimate) / price . The mean bias of the correction terms is determined as the difference between the mean price and the mean present value of the correction terms divided by the mean price. DDM
(22). RIM consistent consistent is the model according to equation (21), DDM
represents the model in (24). RIM consistent-price consistent-price is the model in equation
is the RIM in equation (25). DCF consistent is model in equation (29). DCF consistent-price is the DCF employing a price-based terminal value according to equation (30).
The basic version of the DDM, RIM and DCF are given by neglecting all different correction terms in the model implementation (see also Table 6 and its notes, respectively). d cor is the difference between stock repurchases and capital contributions, dirt cor is the correction for dirty surplus accounting, violations of the net interest relation and tv BSS denotes the difference between the steady state assumptions.
38

Table 7
Comparison of Valuation Studies
This Study
Table 6, basic model t+6, (g=0%)
Penman/Sougiannis
(1998), Table 1, p. 356 t+6, (g=0%)
Francis/Olsson/Oswald
(2000), Table 1, p. 55
t+5, (g=0%)
Dittmann/Maug
(2007), Table, 6 p. 36 t+5, (g=3%)
DDM
RIM
Mean Bias
Mean Accuracy
Median Accuracy
Mean Bias
Mean Accuracy
Median Accuracy
60.15%
60.15%
61.64%
36.95%
38.91%
37.32%
42.60% n/a n/a
33.10% n/a n/a
75.50% n/a
75.80%
20.00% n/a
33.10%
-43.50%
53.90%
55.20%
23.10%
43.20%
25.40%
DCF-Model
Mean Bias
Mean Accuracy
Median Accuracy
88.24%
94.59%
81.24%
124.00% n/a n/a
31.50% n/a
41.00%
31.50%
166.10%
107.60%
Notes:
The first column shows valuation errors for the three models as implemented in this study according to equations
(37), (39) and (41) based on a t+6 year forecast horizon. Columns 2 to 4 reprint valuation errors as given by the respective studies. Note that we adjust the errors reported by Penman/Sougiannis (1998) (column 2) by adding their price model errors to their respective models’ results. This is reasonable since they argue that valuation errors should be compared to their so named price model. This price model is equivalent to our consistent DDM employing a price-based terminal value. Since the valuation errors of this model are negligible in our study, no adjustment is necessary. Further, since the employed growth rates in the terminal period differ from study to study, the most common base line is chosen, namely a growth rate of 0%. An exception is Dittmann/Maug
(2007) since they report only a growth rate of 3%. Consistently in all studies bias is calculated as
(price - intrinsic value estimate) / price and accuracy is given by (price - intrinsic value estimate) / price .
39

Appendix
Appendix 1
Variable Definitions
Label bv t
Description
= common equity total at date t d t cash = common cash dividends at date t d t cor
= difference between stock repurchases and capital contributions at date t debt t
= debt at date t dirt t cor = dirty surplus at date t fcf t dirt g
= free cash flow at date t
= growth rate int t
IS
Measurement
= #60
= #21
= #115 - #108
= (#9 + #34 + #130)
= x t clean − x t
= oi t dirt − (oa t
− oa )
= interest expense from the income statement at date t = #15 int t
NIR oa t oi t dirt
= interest expense derived from the net interest relation at date t
= net operating assets at date t
= operating income at date t
= debt t 1
⋅ r
D
= #6 - (#181 - #9 - #34)
= #172 + (1-s) · #15
P t
= price for a company's stock at date t = #199 r
D r
E
= cost of debt
= cost of equity capital
= 1 year T-Bill rate + industry specific premium depending on the credit rating by Reuters
= 1 year T-Bill rate + industry specific risk premium by
Fama/French
= 1 year T-Bill rate r
F s
V t
= risk free rate
= constant tax rate
= estimate of the market value of equity at date t
= 0.39 x t x t dirt
= clean surplus income at date t
= net income at date t
= #60 t
- #60 t-1
+ d t cash
= #172
+ d t cor
Notes:
We obtain the following items from COMPUSTAT [data item number (if available), (mnemonic), description]:
#6
#9
(AT):
(DLTT):
#15 (XINT):
Assets Total
Long Term Debt Total
Interest Expense
#17 (SPI):
#18 (IB):
#21 (DVC):
#25 (CSHO):
Special Items
Income Before Extraordinary Items
Common Cash Dividends
Common Shares Outstanding – Company
#34 (DLC):
#60 (CEQ):
Debt in Current Liabilities
Common Equity Total
#108 (SSTK): Sale of Common and Preferred Stock
#115 (PRSTKC): Purchase of Common and Preferred Stock
#130 (PSTK):
#172 (NI):
#181 (LT):
#199 (PRCCF):
Preferred Stock
Net Income (Loss)
Liabilities Total
Price - Fiscal Year – Close n.a. (MKVAL): Market Value - Total
40

Appendix 2
Derivation of the DCF model for a company with an infinite life-span
Propostion:
V t
=
∞
∑
τ= 1
[ t +τ
]
( +
E
) τ
−
∞
∑
τ= 1
E int t +τ
( ) debt t +τ
( +
E
) τ
Proof:
It is sufficient to show that
∞
∑
τ= 1
(
∆ debt
E t
)
+τ
τ
Let D T
=
=
∞
∑
τ= 1
T
∑
τ= 1
( +
E t +τ− 1
) τ
∆ debt
+
E t
( )
+τ
τ
Hence it follows:
− deb t t
⇔ V t
=
∞
∑
τ= 1
E fcf t +τ
− int t +τ
( ) r debt t
( +
E
) τ
+τ− 1 − debt t
D T =
T
∑
τ= 1
( debt
+
E t +τ
) τ
−
T
∑
τ= 1
( debt t
+
E
+τ− 1
) τ
=
T
∑
τ= 0
( debt
+
E t +τ
) τ
−
T
∑
τ= 1
( debt
+ t +τ− 1
E
) τ
=
T
∑
τ= 0
( +
E
) t +τ
τ+ 1
+ ∑
τ= 1
( debt
+ t +τ− 1
E
) τ
−
T
∑
τ= 1
( debt t
+
E
+τ− 1
) τ
− debt t and consequently by assuming the standard transversality condition:
= ∑
τ= 1
− debt t
( +
E t +τ− 1
) τ
+
( debt
+
E
) T
∞
∑
τ= 1
∆ debt
+
E t
( )
+τ
τ
= lim
T →∞
T
∑
τ= 1
∆ debt
+
E t
( )
+τ
τ
=
T lim
→∞
∑
τ= 1
( +
E t +τ− 1
) τ
+
T lim
→∞
( debt
+
E t +
)
T
T
− debt t
=
∞
∑
τ= 1
− debt t
( +
E t +τ− 1
) τ
− debt t
Derivation of the WACC-Model for a company with an infinite life-span
Proposition:
V t
=
∞
∑
τ= 1
E fcf t +τ
− int t +τ
(
1 s
) r debt t +τ− 1
( +
E
) τ
− debt t
=
∞
∑
τ= 1
( t +τ
) τ
Proof:
Knowing that int = t
V t
=
E fcf − r debt t
( )
( +
E
) r debt t
+ E [V ]
− debt t
⇔ (
V t
+ debt t
)( +
E
) = − ( ) r debt t
+ E [V ]
⇔ E [fcf ] = (
V t
+ debt t
)( +
E
) + ( ) r debt t
− E [V ]
⇔
V t
=
(
V t
+ debt t
)
+ t
(
V t r V t
+ debt t
− debt t
)
+
( t
( −
V t
+ debt t
)
)
−
]
(
V t
+ debt t
)
− debt t
−
(
V t
+ debt t
) and hence via induction on k:
V t
= k
∑
τ= 1
( t +τ
⎤
τ ) k
− debt t
Since this relation holds for all k ∈ ` and assuming tra nsversality again yields:
V t
=
∞
∑
τ= 1
( t +τ
) τ
− debt t
41

Appendix 3
Derivation of the model specific growth rates
By equating the terminal value expressions in equation (21) and (31), the growth rate adjustment for the DDM is given after some simple rearrangements by: d cash t T
( +
E
) T
(
(
1 +
( g + ∆ DDM r
E
−
( g + ∆
)
DDM
) replacing dirt cor t T
=
( x −
⇒ d
( t cash
+ T
(
1 +
( g + ∆ r
E
−
( g + ∆
DDM
DDM
) )
) )
=
) ) = d cash t T
(
1 g
) + d cor t T
( + ) + dirt cor t T
( + ) + tv BSS,DDM t T 1
( +
E
) ( r
E
− g
) x dirt t T
)
and tv BSS,DDM t T 1
=
( x dirt t T
(
1 g
) g bv
⎛
⎜ d cash t T
(
1 g
) + d cor t T
( + ) ( )
+
( x dirt t T
(
1 g
) − bv
( + ) + bv
( r
E
− g
)
(
− x dirt t T d cash t T
)
+
+ d cash t T d cor t T
+ d cor t T
) ( )
) ( )
⎞
⎟
replacing x
⇔ d
( cash t T r
E
(
1 +
( g
=
+ ∆
−
( g + ∆ bv
DDM
DDM
) )
)
−
)
= bv + d cash t T
+ d cor t T
results in: d cash t T
(
1 g
) + d cor t T
(
(
+ ) − r
E
− g
) bv
(
1 g
) + bv
T
⇔
⇔ d cash t T d cash
+
(
=
1
( g + ∆ DDM
) )
= d cash
+
(
1 g
) + d cor
+
(
(
+ ) − r
E
− g
) bv d cash
+
(
1 g
) + d cor
+
(
(
+ ) − r
E
− g
) bv
(
1 g
) + bv
T
(
( + ) + bv
T r
E
−
( g + ∆ DDM
) )
(
− r
E
−
( g + ∆ DDM d cash
+
( g
) )
+ ∆ DDM
( r
E
− g
)
) ( r
E
− g
)
⇔ d
⇔ ∆
⇔ ∆ cash
+
DDM
DDM
=
= −
= d
(
( cash t T
( + ) + d cor t T
( + ) − bv d cash
+
( r
E
− g
) −
( cash
+
( + ) + d cor
+
( + ) − bv d cash t T
+ cash t T
E
(
) r
⋅
E
( d
− g cash t T
⋅ ( r
E
− g
) +
)
(
(
+
(
(
+ d d cash t T cash t T
( cash
+
(
1 g
) + d cor
+
( + ) − bv
) +
+
(
1 g
) + d cor t T
( + ) − bv d d cor t T cor t T
( + ) −
(
1 g
) + bv
( + ) + bv
T bv
) (
1 g
) − bv
T
)
≠ 0
) ( g + ∆
( r
E
− g
)
)
DDM
)
⎛
⎜
( + ) + bv
T
(
1 g
) + bv
( + ) + bv
T
( + ) + bv
T
)
T
) d cash
+
(
1 g
) + d cor
+
( + ) − bv
+ bv
T
+ d
)
( )
− g cash
+
( r
E
− g
)
⇒ ∆ DDM
1 +
( d cash
+
( +
E
(
)
(
1 g
) + d cor t T d cash
+ r
E
− g
( + ) − bv
)
(
1 g
) + bv
T
)
( )
(
1 g
) + ⎞
⎟
42

By equating the terminal value expressions in equation (24) and (33), the growth rate adjustment for the RIM is given after some simple rearrangements by: x a,dirt t T
( +
E
) T
(
(
1 g RIM r
E
−
( g + ∆
)
RIM replacing dirt
⇒ x
( a,dirt
+ cor
+
(
1 g RIM r
E
−
( g + ∆
=
RIM
(
) ) x
)
)
=
)
− x
=
( ( x a,dirt
+
+ dirt cor
+
) (
1 g
) + tv BSS,RIM
+ +
( +
E
) ( r
E
− g
) x dirt
+
)
and tv BSS,RIM
+ + a,
+ dirt t T
(
1 g
) +
( x −
= − x dirt
+
)
( ( )
) ( + ) − r bv
( r
E
− g
)
+
)
+
leads to:
⇔
⇔ x x a,dirt t T a,dirt t T
(
1 g RIM
= x a,dirt t T
)
= x
(
1 g
) +
( x a,dirt t T
(
1 g
− x dirt t T
) +
( x − x dirt t T
) ( + ) − r bv
( r
E
− g
)
) ( + ) − r bv
( r
E
− g
)
+ +
+
− 1
(
+ r
E
−
( g + ∆ RIM
(
) r
E
−
( g + ∆ RIM
)
− x a,dirt t T
⇔ x a,dirt
+
=
⇔ ∆ RIM = − r
E x
⎛
⎜ x x
− a,dirt
+ a,dirt t T a,dirt t T
( r
E
(
1 g
− g
+
)
)
−
+
( x
+ r x
( r
E
− g
) + x
− a,dirt
+ a,dirt t T
(
( x dirt t T
1 g
1 g
)
) (
1 g
) +
)
+
+
(
( x x
−
⎞
⎟
− x
−
( g + ∆ RIM
)
⎛
⎜ x
+
( a,dirt t T x x
( r
E
− g
) dirt
+ dirt t T
)
)
(
(
+
+
)
)
−
− r bv r bv
( r
E
− g
) + x
− x dirt t T
) ( a,dirt t T
+ )
(
1 g
) +
− r bv
+
+
+
+
)
)
−
(
⇔ ∆ RIM = −
⇔ ∆ RIM = a,dirt t T x x x a,dirt t T a,dirt
+
( r
E
− g
) − r x
( r
E
− g
) +
(
( x a,dirt
+
(
(
+ r
E
E
−
) ( g
) x a,dirt
+
+
( x a,dirt t T a,dirt
+
(
1 g
) + dirt cor
T
( + ) + tv a,dirt
+
( + ) + dirt cor
T
( + )
(
1 g
) + dirt cor
T
( + ) + tv BSS,RIM
+ +
(
1 g
) + di rt cor
T
( + ) + tv BSS,RIM t T 1
)
≠ 0
+
(
1 g
) + dirt cor
T
( + ) + tv BSS,RIM
+ +
BSS,RIM
+ +
) tv BSS,RIM t T 1
)
)
)
− g
( ) g
( g
+
) )
+ ∆ RIM
( r
E
− g
)
) ( r
E
− g
)
+
⎞
⎟
⇒ ∆ RIM =
1 +
( x a,dirt
+
(
1 g
) x
( a,dirt t T
+
E
)
( )
+ dirt cor
T r
E
− g
( + ) + tv BSS,RIM
+ +
)
( )
43

Finally, by equating the terminal value expressions in equation (29) and (35), the growth rate adjustment for the
RIM is given after some simple rearrangements by:
( cf
+ dirt t T
E
)
(
T
(
1 r
+
E
(
− g
(
+ ∆ g
DCF
+ ∆
)
DCF
) replacing nir cor t T
=
( cf dirt t T
= fcf dirt t T
− ( − )
) ) = cf dirt t T
(
1 g
) + dirt cor t T
( + ) + nir cor t T
( + ) + tv BSS,DCF t T 1
( +
E
) ( r
E
− g
)
− int IS t T
+
) ( ) r debt on the r ight hand side leads to:
⇒ cf
( dirt t T
(
1 +
( g r
E
−
( g + ∆
+ ∆ DCF
DCF
) )
) )
=
⎛
⎜
( fcf dirt t T
+
) (
1 g
) +
− int IS t T
( − )( +
(
) + tv r
E
− g
BSS,DCF t T 1
) dirt cor t T
( + ) + ⎞
⎟
⇔ cf dirt t T
(
1 +
( g replacing dirt cor t T
+ ∆
=
DCF
( x
) )
=
−
⎛
⎜
( fcf
− int dirt t T
IS x dirt t T
)
, fcf
+ dirt t T
=
) (
1 g
) +
( )( ) + tv r
E
− g
BSS,DCF t T 1
( ) dirt cor t T
( oi t dirt
+τ
− (oa t +τ
− oa t +τ− 1
),
+ ) + ⎞
⎟
( r
E
−
( g + ∆ DCF
) ) tv BSS,DCF t T 1
= oa
( ) + ( )
, and oi dirt t T
⇔
⇔ cf
= dirt t T cf dirt t T x dirt t T
(
1
=
(
+
(
+ g int IS t T
+ ∆
( )
DCF
) )
=
(
− oa
( + ) + x
( + ) + oa
( r
E
− g
)
− oa
( + ) + x
( + ) + oa
( r
E
− g
)
+
)
(
+ r
E
−
( g + ∆ DCF
)
) )
( r
E
−
( g + ∆ DCF
− cf dirt t T
( g
) )
+ ∆ DCF
( r
E
− g
)
) ( r
E
− g
) r
E
(
− oa
( + ) + x
( + ) + oa +
) ( )
⎛
− g + ∆ DCF ⎜
− oa
+ oa
(
+
+ ) + x
+
( + ) + cf dirt t T
( r
E
− g
)
⎞
⎟
⇔ cf dirt t T
=
⇔ ∆ DCF = −
⇔ ∆ DCF = cf dirt t T
( r
E
− g
) (
(
( r
E
− g
) cf dirt t T
( r
E
− g
) − r
E
− oa
( + ) + x
( + ) + oa oa
(
1 g
) + x
( + ) + oa
( +
E
) (
− oa cf dirt
+
( r
E
− g
) ( t + T
(
1 g
) + x
( + ) + oa oa
(
1 g
) + x
( + ) + oa
+
+
+
+
)
)
)
)
− g
( )
⎣
− ( + ) + x
( + ) + oa
⇒ ∆ DCF =
1 +
(
+
− oa
( + ) + x cf
( dirt
+
+
E
(
( r
+
E
)
−
) g
+
) oa +
≠ 0
)
( )
44

Appendix 4
Excerpt of the financial statement for the 3M Corporation between 1998 and 2003 from Compustat in millions of US-Dollars net operating assets (oa) debt debt in current liabilities long term debt preferred stock stockholders' equity (bv) net income (x dirt
) interest expense (int
IS
) cash dividends (d cash
)
1998
9,042
3,106
1,492
1,614
0
5,936
1,175
139
887
1999
8,899
2,610
1,130
1,480
0
6,289
1,763
109
901
2000
9,368
2,837
1,866
971
0
6,531
1,782
111
918
2001
8,979
2,893
1,373
1,520
0
6,086
1,430
124
948
2002
9,370
3,377
1,237
2,140
0
5,993
1,974
80
968
2003
10,892
3,007
1,202
1,805
0
7,885
2,403
84
1,034
Calculated input parameters r
E r
D net dividends (d cash
+ d cor
) x x a,dirt fcf dirt cf dirt int
NIR d cor dirt cor nir cor
1998
0.1176
0.0679
1,213
326.00
1999
0.1160
0.0663
1,336
1,689
1,074.42
1,972.49
2,207.13
205.99
435.00
-74.00
59.17
2000
0.1279
0.0784
1,307
1,549
977.64
1,380.71
1,589.79
204.50
389.00
-233.00
57.03
2001
0.1077
0.0585
1,808
1,363
726.61
1,894.64
2,098.95
165.96
860.00
-67.00
25.59
2002
0.0854
0.0365
1,388
1,295
1,454.26
1,631.80
1,814.40
105.67
420.00
-679.00
15.66
2003
0.0793
0.0309
1,164
3,056
1,927.76
932.24
1,136.42
104.29
130.00
653.00
12.38
Intrinsic Values estimated by the DDM, the RIM and the DCF model
Dividend Discount Model d cash d cor dirt cor tv
BSS,DDM dirt
Sum of Present Value components
Intrinsic Value per share
Present Value
11,472.09
2,676.38
5,041.40
9,565.53
28,755.40
35.77
1999
901.00
435.00
2000
918.00
389.00
2001
948.00
860.00
2002
968.00
420.00
2003
1,034.00
130.00
2004
1,034.00
130.00
653.00
1,239.00
bv in 1998 x a,dirt dirt cor tv
BSS,RIM
Sum of Present Value components
Intrinsic Value per share
Present Value
5,936.00
19,284.68
4,693.05
-1,158.33
28,755.40
35.77
1999
1,074.42
-74.00
Residual Income Model
2000 2001
977.64
-233.00
726.61
-67.00
2002
1,454.26
-679.00
2003
1,927.76
653.00
2004
1,927.76
653.00
-150.04
cf dirt dirt cor nir cor tv
BSS,DCF debt in 1998
Present Value
15,414.30
4,693.05
230.18
11,523.87
3,106.00
1999
2,207.13
Discounted Cash Flow Model
2000 2001
1,589.79
2,098.95
-74.00
-233.00
-67.00
59.17
57.03
25.59
2002
1,814.40
-679.00
15.66
2003
1,136.42
653.00
12.38
2004
1,136.42
653.00
12.38
1,492.66
Sum of Present Value components 28,755.40
Intrinsic Value per share 35.77
Notes:
Calculations are based on a five year forecast horizon (t+5), no growth (g=0%) and a tax rate of 39%. Reported figures are in millions of US-Dollars. Implemented models are given in equation (21), (24) and (29).
45

The growth rate adjustments for the three models are calculated as follows:
DDM
∆ DDM =
( +
E
) ( d cash t T
( + ) + d cor t T
( + ) − bv d cash t T
( r
E
− g
) +
( ( d cash t T
+ d cor t T
) (
1 g
) − bv
( + ) + bv
T
( + ) + bv
T
)
)
( )
∆ DDM =
(
1, 034 0.0793
+ ( + − +
)
)
3, 298.3408
3,137.9962
=
V t
DDM =
T
∑
τ= 1 d cash
( t +τ
+
E
) τ
+
T
∑
τ= 1
( d t cor
+τ
+
E
) τ
+
( d
+ cash
E
)
(
T
(
1 + r
E
( g
−
(
+ ∆ g
DDM
+ ∆
)
DDM
)
(
) )
)
+
( (
RIM
∆ RIM = x a,dirt
+
( +
E
) ( x a,dirt t T
(
( + ) + dirt cor
T
( + ) + tv BSS,RIM t T 1
)
( r
E
− g
) + x a,dirt
+
( + ) + dirt cor
T
( + ) + tv BSS,RIM
+ +
)
( )
∆ RIM = 0.015438
V t
RIM = bv t
+
T
∑
τ= 1
( x t a,dirt
+τ
+ dirt t cor
+τ
(
1 r
E
) τ
)
+
(
(
RIM
) x a,dirt
+
(
1 r
E
) T r
E
−
( g + ∆ RIM
) ) = 28,755.40
DCF
∆ DCF =
( +
E
)
⎣
− oa
( + ) + x
( + ) + oa cf dirt t T
( r
E
− g
)
⎣ oa
( + ) + x
( + ) + oa
+
+
( )
∆ DCF = 0.0505624953
42
V t
DCF =
T
∑
τ= 1
( cf t dirt
+τ
+ dirt t cor
+τ
(
1 r
E
) τ
+ nir t cor
+τ
)
+
(
( (
1 r
E
) T
( r
E
−
DCF
( g
) cf
+ ∆ dirt
+
DCF
)
) )
= 28,755.40
) )
= 28,755.40
42 We report ten digits for this adjustment, because the result is quite sensitive in order to retain the exact equivalence.
46

Appendix 5
Descriptive Statistics on Dirty Surplus
Absolute dirty surplus as % of the absolute clean surplus income
Absolute dirty surplus as % of equity book value
Absolute dirty surplus as % of total assets
Mean
Median
% of obs > 10 %
Mean
Median
% of obs > 1 %
Mean
Median
% of obs > 2 %
Net Income
22.78%
7.68%
45.13%
5.28%
0.95%
36.02%
2.44%
0.43%
33.96%
Income Before
Extraordinary Items
26.60%
10.29%
50.55%
6.26%
1.23%
53.96%
2.82%
0.57%
26.24%
Income Before
Extraordinary Items and Special Items
38.25%
20.77%
62.87%
12.82%
2.49%
65.95%
8.16%
1.14%
39.44%
Number of firm years 36,112 36,112 36,112
Notes:
Dirty surplus is the absolute value of the difference between the clean surplus income and a particular measure of income. Used income measures (Compustat item numbers in parentheses) are GAAP net income (#172), income before extraordinary items (#18), income before extraordinary and special items (#18+#17). Dirty surplus of more than 100% is included with a maximum of 100% in order to mitigate the effect of random outliers.
47

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