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Selecting an Accounting-based Valuation Model*

Woo-Jin Chang INSEAD

Boulevard de Constance 77305, Fontainebleau

France woo-jin.chang@insead.edu

+33 (0)1 60 72 92 48 (Telephone) +33 (0)1 60 72 92 53 (Facsimile)

Wayne R. Landsman

University of North Carolina at Chapel Hill CB# 3490 McColl Building

Chapel Hill, NC 27599-3490

wayne_landsman@unc.edu +1 919-962-3221 (Telephone) +1 919-962-4727 (Facsimile)

Steven J. Monahan

INSEAD Boulevard de Constance 77305, Fontainebleau

France steven.monahan@insead.edu

+33 (0)1 60 72 92 14 (Telephone) +33 (0)1 60 72 92 53 (Facsimile)

January, 2012

Keywords: Abnormal earnings growth valuation, accounting-based valuation, non-steady-state

forecasting, residual income valuation.

JEL classification codes: G11, G12, G17, and M41.

* We benefited from the comments of Hubert Gatignon and workshop participants at INSEAD, Nanyang Technological University, National University of Singapore, and Singapore Management University.

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Selecting an Accounting-based Valuation Model

Abstract

We evaluate the relative valuation accuracy of six accounting-based valuation models. We begin

by demonstrating that there is no single best model. The model with the lowest mean unsigned

valuation error does not have the lowest median unsigned valuation error; and, the model with

the lowest mean (median) unsigned valuation error is the most accurate for only 10.41 (14.41)

percent of our sample observations. In light of these facts, we evaluate whether firm attributes

can be used to identify the most accurate model ex ante. Using fitted probabilities from a set of

ordered logistic regressions we are able to identify the most accurate model for 35.80 percent of

the observations in our sample. Moreover, our model-selection algorithm is superior to several

alternative algorithms. We also provide evidence regarding the factors underlying the firm

attributes that we use to predict relative accuracy; and, we show that a lifecycle factor has strong

predictive power. This result is important as it suggests a parsimonious and practical criterion

that researchers and practitioners can use to choose among valuation models.

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1. Introduction

Accounting-based valuation models are popular because they can be used to develop

rigorous estimates of value that do not rely on arbitrary assumptions about dividend policy.

As a result, equity analysts often rely (directly or indirectly) on these models when

developing price estimates. In addition, academic researchers and practitioners frequently

use accounting-based valuation models to estimate the expected rate of return on equity

capital (see Easton [2007] for a review of the literature).

An issue that users of accounting-based valuation models confront is: In terms of

accuracy, which model is best? Clearly, this is a moot issue if it is possible to develop a set

of unbiased, steady-state forecasts of future accounting numbers. However, this is often

infeasible, particularly for young, high-growth firms and firms that are undergoing structural

or life-cycle changes. Hence, users have to make assumptions about the forecast horizon and

the type of model (i.e., residual income valuation or abnormal earnings growth valuation) to

employ.

We evaluate the relative unsigned valuation errors of six accounting-based valuation

models that are representative of the models commonly used by practitioners and academics.

The models we evaluate differ along two dimensions: type and forecast horizon. Regarding

model type, we consider price estimates derived from either the residual income valuation

(RIV) model or the abnormal earnings growth valuation (AEGV) model. For each of these

two model types, we consider three different forecast horizons: (1) a naïve horizon in which

our RIV (AEGV) estimate equals contemporaneous equity book value per share (capitalized

forward earnings per share); (2) a short horizon of five years; and, (3) a long horizon of

fifteen years.

We begin our analyses by determining the mean and median unsigned valuation error

of each model. For a particular model and observation, we define the unsigned valuation

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error as the ratio of the absolute value of the difference between the estimated share price and

the contemporaneous share price to contemporaneous share price. We demonstrate that the

RIV model assuming a short forecast horizon, RIV_S, and the AEGV model assuming a

short forecast horizon, AEGV_S, have the lowest mean and median unsigned valuation

errors, respectively. Hence, the model that is best for the typical firm, AEGV_S, is not best

on average, RIV_S. Moreover, and perhaps more importantly, for many of our sample

observations neither of these models is best. In particular, RIV_S (AEGV_S) has the lowest

unsigned valuation error for only 10.41 (14.41) percent of our sample observations.

The results described above lead us to conclude that there is no single best model.

Although this conclusion may be uncontroversial, it goes against an implicit assumption

made by many researchers. For example, when evaluating accounting-based estimates of the

expected rate of return on equity capital, Easton and Monahan [2005] implicitly assumes

there is a single model that generates the most reliable estimate. More recently, Jorgensen,

Lee, and Yoo [2011] (hereafter JLY) evaluates several accounting-based valuation models.

However, because the study primarily focuses on mean and median valuation errors, JLY

implicitly assumes there is a single, best model.

With the above in mind, we develop and evaluate an algorithm that uses observable

firm attributes to select a valuation model for a particular observation. Our algorithm

involves six steps. First, we create two random samples, which we refer to as the estimation

sample and the holdout sample. Second, for each observation in the estimation sample we

assign a rank to each of the six valuation models. This rank corresponds to the relative

magnitude of the unsigned valuation error obtained when we use that particular model to

estimate the price for the observation. The Nth most accurate model is assigned a rank of N -

i.e., the most accurate model is assigned a rank of one; the second most accurate model is

assigned a rank of two; etc.

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Third, using the estimation sample we estimate six ordered logistic regressions, i.e.,

one regression for each valuation model. The dependent variable in each regression is the

rank described above. The independent variables are 23 firm attributes motivated by extant

analytical and empirical research. Fourth, for each holdout-sample observation and each

valuation model, we use the estimated coefficients from the relevant ordered logistic

regression to estimate the fitted probability that the valuation model in question will have

rank N. Fifth, for each holdout-sample observation and each valuation model, we create the

variable FIT_RANK, which equals the rank with the highest fitted probability. Finally, for

each holdout-sample observation, we select the valuation model with the lowest value of

FIT_RANK.

The above algorithm selects the AEGV model assuming a long forecast horizon,

AEGV_L, for 51.42 percent of the holdout-sample observations, the AEGV model assuming

a short forecast horizon, AEGV_S, for 28.27 percent of the holdout-sample observations, and

the naïve residual income model, RIV_N, for 17.82 percent of the holdout-sample

observations. Hence, on an ex ante basis, the algorithm tends to select one of two AEGV

models or a naïve RIV model. Even though the algorithm leads to a fairly distinct

categorization of observations, it still performs well. For 35.80 percent of the holdout-sample

observations the selected valuation model per the algorithm generates the most accurate

estimate of price. Moreover, it performs better than several alternative algorithms. For

example, for 70.70 (48.16) percent of the holdout-sample observations, the model selected by

algorithm is no less accurate (is more accurate) than a model selected randomly after

assigning each valuation model a selection probability equal to the actual percentage of the

estimation-sample observations for which that model is the most accurate.

A shortcoming of the algorithm described above is that it only provides indirect

evidence regarding the types of firms for which a particular accounting-based valuation

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model should be used. To address this shortcoming, we factor analyze the independent

variables used in the ordered logistic regressions; and, we use the results of these analyses to

estimate factor scores. When doing this we only use observations from the estimation

sample. The analysis reveals that there are three latent factors, which relate to: (1)

investment opportunities; (2) lifecycle stage; and, (3) net operating asset turnover.

After performing the factor analysis, we modify the selection algorithm. First, for

each estimation-sample observation we replace the original independent variables used in the

ordered logistic regressions with the factor scores; and, we re-estimate the ordered logistic

regressions. We show that the lifecycle factor has strong explanatory power. In particular,

we show that for large, mature, stable firms: (1) AEGV models are superior to RIV models,

and (2) models that assume relatively long forecast horizons are more accurate.

Second, we use the factor patterns obtained from the estimation sample to estimate a

factor score for each holdout-sample observation. We combine this factor score with the

coefficients from the modified ordered logistic regressions to arrive at a fitted probability for

each holdout-sample observation and valuation model. We then recalculate the variable

FIT_RANK and we select the valuation model with the lowest value of FIT_RANK. Similar

to the original algorithm, the modified algorithm tends to select the AEGV_L (48.37 percent

of the holdout-sample observations), AEGV_S (39.49 percent of the holdout-sample

observations), or RIV_N (12.15 percent of the holdout-sample observations) model. It also

performs well. For 30.32 percent of the holdout-sample observations the selected valuation

model per the modified algorithm generates the most accurate estimate of price. Moreover,

for 67.80 (46.32) percent of the holdout sample observations, the predicted model from the

modified algorithm is no less accurate (is more accurate) than a model selected randomly

after assigning each valuation model a selection probability equal to the actual percentage of

the estimation-sample observations for which that model is the most accurate. Additional

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analyses reveal that the lifecycle factor is the key determinant of the identity of the model

selected by the modified algorithm.

Our study extends the literature in three ways. First, we demonstrate that there is no

single, best valuation model. In particular, models that are considered “best” per extant

research frequently perform poorly for many firms. Second, we show that relative valuation

accuracy is predictable ex ante based on knowledge of a firm’s attributes. Finally, we show

that where a firm is its lifecycle is a key determinant of the identity of the appropriate

valuation model for that firm. This final result is important as it suggests that researchers and

practitioners should evaluate a firm’s lifecycle stage when deciding on which approach to use

to estimate the value of the firm’s equity.

The remainder of the paper proceeds as follows. In section two we briefly discuss the

related literature. In section three we describe our six accounting-based valuation models,

how we estimate the inputs to these models, and how we construct our samples. In sections

four and five we discuss our main empirical results and our sensitivity analyses. We

conclude in section six. Finally, there is one appendix.

2. Related Literature

A key analytical result in the literature is provided by Ohlson [1995], who solves a

conundrum: While value equals the present value of expected future dividends (i.e., the

dividend discount model or DDM), dividend policy is irrelevant from a first-order

perspective (Modigliani and Miller [1961]). Hence, to estimate value via the DDM, arbitrary

assumptions about future dividend policy must be made. Ohlson [1995] solves this

conundrum by showing that accounting-based valuation models can be used to develop

rigorous estimates of value that do not rely on assumptions about future dividend policy.

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Penman and Sougiannis [1998] (hereafter PS) is the first study to provide empirical

support for the analytical results developed by Ohlson [1995]. PS shows that price estimates

obtained from accounting-based valuation models are less biased (i.e., have lower signed

errors) than estimates of value obtained from either the DDM or a cash-flow based model.

PS also performs conditional tests in which they evaluate how bias varies with certain

portfolio-level attributes. Although PS makes a clear contribution, the study is not definitive.

PS focuses on bias rather than accuracy, performs its analyses at the portfolio level, and uses

ex post realizations of the relevant payoffs to generate their value estimates. Hence, the PS

results do not shed light on either the firm-level determinants of relative accuracy or the

tradeoff between forecast accuracy and horizon length. Finally, when conducting the

conditional analyses, PS uses several conditioning variables that are a function of

contemporaneous price; hence, there is an issue of circularity.

Francis, Olsson, and Oswald [2000] (hereafter FOO) builds on the PS study in several

ways. FOO focuses on accuracy (i.e., unsigned valuation errors) and associations, uses Value

Line forecasts, and conducts analyses at the firm level. FOO shows that accounting-based

models are more accurate; however, they only consider one accounting-based model. Hence,

although FOO makes a clear contribution to the literature, it neither sheds light on the relative

accuracy of different accounting-based models nor attempts to identify firm-level attributes

that are associated with relative accuracy. Similarly, as discussed above, JLY evaluates

several accounting-based models; however, JLY primarily focuses on mean and median

unsigned errors and does not attempt to develop an approach for selecting between models.

Finally, a related stream of research pertains to studies that evaluate the reliability of

accounting-based measures of the expected rate of return on equity capital (Easton and

Monahan [2010] provides a review and discussion). Although these studies focus on

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different phenomena such as the relation between accounting-based proxies and realized

returns, they also shed light on the relative merits of the underlying valuation models.

3. Empirical Proxies and Sample Construction

In section 3.1, we provide a general description of the six accounting-based valuation

models that we evaluate. In section 3.2, we provide details about how we estimate model

inputs. In section 3.3, we describe our sample construction algorithm.

3.1 Accounting-based Valuation Models

We evaluate three versions of the RIV model and three versions of the AEGV model.

We begin by describing our RIV models. To do this we refer to equation (1), which is shown

below.

RIV_ INFi ,0 = Bi ,0 +ROEi ,t − ri( ) × Bi ,t −1

1+ ri( )tt =1

∞∑ (1)

In equation (1), RIV_INFi,0 is the estimate of firm i’s share price at time zero obtained from a

residual income valuation model in which an infinite forecast horizon is adopted; Bi,t is firm

i’s expected equity book value per share at the end of time t; ROEi,t is firm i’s expected

accounting return on equity for time t, which equals the ratio of firm i’s expected earnings per

share for year t (i.e., epsi,t) to Bi,t-1; and, ri is the expected rate of return on equity capital for

firm i. All expectations are formed at time zero.

As discussed in Ohlson [1995], equation (1) is identical to the DDM as long as there

is clean surplus accounting and the discount rate is neither stochastic nor time varying.

Equation (1) is not directly implementable as it requires an infinite series of forecasts.

Hence, a finite forecast horizon must be adopted and a terminal value correction must be

made. With this in mind we consider the following models.

0,0,_ ii BNRIV = (1.A)

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RIV_Si ,0 = Bi ,0 +ROEi ,t − ri( ) × Bi ,t −1

1+ ri( )tt=1

5∑ (1.B)

RIV_Li ,0 = Bi ,0 +ROEi ,t − ri( ) × Bi ,t−1

1+ ri( )tt =1

15∑ (1.C)

RIV_Ni,0 is the estimate of firm i’s share price at time zero after making a naïve assumption

about the forecast horizon; RIV_Si,0 is the estimate of firm i’s share price at time zero after

assuming a short horizon of five years; and, RIV_Li,0 is the estimate of firm i’s share price at

time zero after assuming a long horizon of fifteen years. In all cases we assume a terminal

value of zero.

The different versions of equation (1) reflect a tradeoff. Ceteris paribus, lengthening

the forecast horizon improves a model’s accuracy. Specifically, economic rents and the bias

in ROE attributable to conservative accounting are both expected to dissipate over time.

Hence, as discussed in Zhang [2000], in the long run the expected growth rate in residual

income converges to a constant. However, lengthening the forecast horizon is not costless as

long-horizon forecasts tend to be less precise than short-horizon forecasts. Hence, there is a

tradeoff and the issue of which forecast horizon to adopt is an empirical question.

A second tradeoff relates to the choice of the valuation anchor. In particular, rather

than using RIV and anchoring on contemporaneous equity book value per share, we can use

AEGV and anchor on forward earnings per share.

AEGV_ INFi ,0 =epsi ,1

ri+

RONECi ,t +1 − ri( ) × epsi ,t − dpsi ,t( )ri × 1+ ri( )t

t =1

∞∑ (2)

In equation (2), AEGV_INFi,0 is the estimate of firm i’s share price at time zero obtained

from an abnormal earnings growth valuation model in which an infinite forecast horizon is

adopted; epsi,t is firm i’s expected earnings per share for time t; RONECi,t+1 is firm i’s

expected accounting return on new equity capital for time t+1, which equals the ratio of the

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expected change in firm i’s earnings per share for year t+1 (i.e., ∆epsi,t+1) to expected

reinvested earnings in time t (i.e., epsi,t-dpsi,t, dpsi,t denotes expected dividends per share for

firm i in time t); and, ri is firm i’s expected rate of return on equity capital. All expectations

are formed at time zero.

Equation (2) has two potential advantages vis-à-vis equation (1). First, ignoring

unconditional conservatism and assuming economic rents are expected to dissipate over time,

expected future RONEC’s, which primarily relates to the expected performance of future

investments, will converge towards ri faster than expected future ROE’s, which reflect the

expected performance of both past and future investments. Second, unconditional

conservatism only affects earnings per share and RONEC when there is non-zero growth in

assets accounted for conservatively. On the other hand, equity book value per share and ROE

are always affected by unconditional conservatism.

Despite its advantages, it is not obvious that the AEGV model dominates the RIV

model. Because the RIV model is anchored on equity book value per share it reduces the

importance of forecasting and, consequently, mitigates the impact of forecast errors.

Moreover, equity book value per share is a function of the firm’s past investments; hence, it

reflects management’s expectations regarding future growth and profitability.

In light of the above, the relative accuracy of equation (1) and equation (2) is an

empirical question. To address this question, we adjust equation (2) so that it can be

implemented. In particular, as with equation (1), equation (2) requires that an infinite

sequence of forecasts be developed, which is infeasible. Hence, we consider the following

versions of equation (2).

i

ii r

epsNAEGV 1,

0,_ = (2.A)

( ) ( )( )∑

=

+

+×−×−

+=4

1

,,1,1,0,

1_

tt

ii

titiiti

i

ii

rr

dpsepsrRONEC

r

epsSAEGV (2.B)

10

( ) ( )( )∑

=

+

+×−×−

+=14

1

,,1,1,0,

1_

tt

ii

titiiti

i

ii

rr

dpsepsrRONEC

r

epsLAEGV (2.C)

AEGV_Ni,0 is the estimate of firm i’s share price at time zero after making a naïve

assumption about the forecast horizon; AEGV_Si,0 is the estimate of firm i’s share price at

time zero after assuming a short horizon of five years; and, AEGV_Li,0 is the estimate of firm

i’s share price at time zero after assuming a long horizon of fifteen years. In all cases we

assume a terminal value of zero.

As discussed above, the different versions of equation (2) reflect a tradeoff between

the costs and benefits of lengthening the forecast horizon. Again, the nature of this tradeoff is

an empirical question.

3.2 Estimation of Model Inputs

Price at time zero, Pi,0, equals the share price per I/B/E/S on the date immediately

after the date that the last consensus I/B/E/S forecast was made available in fiscal year zero.

Equity book value per share at time zero, Bi,0, equals the ratio of COMPUSTAT data item

CEQ to COMPUSTAT data item CSHO (we adjust for stock splits and dividends). Earnings

per share at time zero, epsi,0, equals the ratio of COMPUSTAT data item IB to

COMPUSTAT data item CSHO (we adjust for stock splits and dividends). Dividends per

share at time zero, dpsi,0, equals COMPUSTAT data item DVPSX_F. Expected earnings per

share for years one through five (i.e., epsi,t ∀ t ∈ [1,5]) equals the median analysts’ consensus

forecast of earnings per share. We determine median forecasts from the available analysts’

forecasts in the last I/B/E/S file released prior to the end of fiscal year zero. If there is no

explicit consensus forecast of epsi,t, we set it equal to epsi,t-1×(1+gi,ltg). gi,ltg equals the

analysts’ consensus long-term growth rate in earnings per share. Expected dividends per

share, dpsi,t ∀ t > 0, equals to Ki,0×epsi,t. The variable Ki,0 is determined in the following

manner. If epsi,0 > 0, Ki,0 = MAX(0,MIN(dpsi,0÷epsi,0,1)). However, if epsi,0 ≤ 0, Ki,0 =

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MAX(0,MIN(dpsi,0÷(0.06×Bi,0),1)). We use the clean surplus relation to determine expected

equity book value per share (i.e, Bi,t ∀ t ∈ [1,5]); in particular, Bi,t = Bi,t-1+epsi,t-dpsi,t.

Expected return on equity for years one through five (i.e., ROEi,t ∀ t ∈ [1,5]) equals

epsi,t÷Bi,t-1. Expected return on new equity capital for years two through five (i.e.,

RONECi,t+1 ∀ t ∈ [2,5]) equals ∆epsi,t+1÷(epsi,t-dpsi,t).

Based on results in Nissim and Penman [2001], we assume that during years six

through fifteen expected ROE fades to a constant. We implement this assumption in the

following manner. First, for each year between 1963 and 1999 we identify all COMPUSTAT

firms with the requisite data to compute ROE. Second, for each year we sort firms on the

basis of ROE and we form 20 portfolios. Third, for each annual portfolio we calculate the

median ROE for the year in which the portfolio was formed and the subsequent ten years.

We refer to these ROE’s as MROEk,p,t (i.e., the median ROE in year t+k of firms in portfolio

p formed in year t). Fourth, for each k and each p, we calculate the grand mean of the

portfolio medians. We refer to this variable as ∑=

=37

1

,,, 37

_t

tpkpk

MROEMROEGM . Finally, for

the i’th observation in our sample, we identify the portfolio p with the value GM_MROE0,p

that is closest in absolute value to the expected ROE for observation i in year 5 (i.e., ROEi,5)

and we set ROEi,t equal to GM_MROEt-5,p ∀ t ∈ [6,15].

In Figure One we graph GM_MROEk,p for the twenty portfolios. The value of k,

which varies between zero and ten, is shown on the x-axis. The value of GM_MROE0,p for

each value of p ∈ [1,20] is shown on the y-axis. As shown in the graph, ROE mean reverts

rapidly and begins to level off by year eight; hence, our assumption about the evolution of

ROE between years six and 15 appears reasonable.

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Expected RONEC for years six through 15 is calculated as follows: RONECi,t =

RONECi,t-1-((RONECi,5-ri)÷10) ∀ t ∈ [6,15]. Hence, we assume that RONEC fades at a

constant rate until it ultimately equals ri in year 15.

To implement the RIV_L we need expected equity book value per share (i.e., Bi,t) for

years five through 14. As discussed above, we obtain Bi,5 via the clean surplus relation, the

analysts’ consensus forecast of epsi,5, and our assumption about dividends. To obtain Bi,t for t

> 5 we continue to use the clean surplus relation; however, epsi,t = ROEi,t×Bi,t-1 (the

determination of ROEi,t ∀ t > 5 is described above). We continue to set dpsi,t ∀ t > 5 equal to

Ki×epsi,t. To implement AEGV_L we need expected reinvested earnings per share (i.e.,

epsi,t-dpsi,t) for years five through 14. We use the analysts’ consensus forecast of epsi,5 and

our assumption about the dividend payout ratio to calculate the reinvested earnings per share

for year five. For years six through 14, (epsi,t-dpsi,t) = epsi,t-1×(RONECi,t×(1-Ki)+1)×(1-Ki).

Finally, we estimate ri as follows. First, for each industry j we estimate the following

weighted least squares regression for each month m between January, 1980 and December,

2010. The weights, which are not shown in the equation, equal the contemporaneous equity

market values of the observations. We use the industry definitions shown in Fama and

French [1997].

reti ,τ = α j ,m + βLEV_ IND j ,m × rM ,τ − rf ,τ( ) + εi ,τ (3)

In equation (3), reti,τ is realized stock return for firm i in month τ; αj,m is the estimated

intercept for industry j in month m; βLEV_IND j,m is the estimated levered beta for industry j

in month m; rM,τ is the realized return for month τ on the value-weighted market portfolio; rf,τ

is the market yield on ten-year, constant-maturity, U.S. treasury bonds issued in month τ;

and, εi,τ is an error term. We estimate the regression using all members of industry j in the

CRSP monthly files that have at least 30 monthly returns between month m-60 and month m.

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Second, because βLEV_IND j,m reflects industry leverage, which likely differs from

firm-level leverage, we unlever it using the industry leverage ratio in month m. We refer to

this measure of beta as βUNLEV_IND j,m. Third, for sample observation i, we select the

month m that corresponds to the year and month in which our value estimate pertains and we

use i’s leverage ratio to re-lever the unlevered industry beta. We refer to this estimated firm-

level beta as βLEV_FIRMi,m. Finally, to determine ri, we multiply βLEV_FIRMi,m by five

percent (i.e., our assumed equity premium) and then add the contemporaneous market yield

on ten-year, constant-maturity, U.S. treasury bonds.1

3.3 Sample Construction

We begin by combining the 2010 COMPUSTAT annual file with the summary 2010

I/B/E/S tapes. We eliminate all observations with a fiscal year of 1979 or earlier because the

I/B/E/S data are sparse prior to 1980. We also eliminate observations with a standard

industrial classification (i.e., SIC) code that does not belong to any of the 48 industry groups

described in Fama and French [1997]. We do this because we cannot estimate

βLEV_FIRMi,m for these observations. We also eliminate observations for which

βLEV_FIRMi,m ∉ [0,2].

Finally, for each industry-year we randomly assign half of the observations to the

estimation sample and the remaining half to the holdout sample, which we use to evaluate the

out-of-sample performance of our selection algorithm. This leaves us with 35,473

observations for the estimation sample; however, because of data limitations, some of our

tests are based on smaller samples. The tenor of our results does not change if we estimate

all of our tests on a common sample.

1 We evaluate the sensitivity of our results to the assumptions underlying our estimate of ri. First, we assume four alternative values of the equity premium: three percent, four percent, six percent, and seven percent. Second, we consider alternative approaches for estimating beta. For example, we estimate firm-specific betas and we calculate the industry-level beta by value weighting the betas of the firms in the industry. Finally, we assume ri equals 10 percent for all firm-years. None of these alternative approaches for estimating ri affect the tenor of our results.

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4. Main Empirical Results

4.1 Descriptive Statistics

Panel A of Table One contains descriptive statistics on key attributes of the

observations in the estimation sample. Two comments are warranted. First, the estimation

sample consists of large firms most of which are expected (at least by analysts who provide

forecasts to I/B/E/S) to perform well. In particular, the mean (median) of ROE5 is 0.199

(0.160) and the mean (median) of RONEC5 is 207.066 (0.175). These numbers are

considerably higher than the mean (median) of r, which equals 0.113 (0.111). Second, both

ROE and RONEC take on extreme values. For example, the first and 99th percentiles of

ROE5 are −0.379 and 1.028; moreover, the first and 99th percentiles of RONEC5 are 0 and

1.173. These extreme values suggest we will obtain extreme valuation errors for some of our

sample observations. However, because we focus on relative ranks, these extreme errors

should not confound our statistical tests.

In Panel B of Table One we provide descriptive statistics on the signed valuation

errors of each of our six valuation models. These statistics pertain to estimation-sample

observations. To determine the signed valuation error for particular observation and

valuation model we divide the difference between the model’s value estimate and the

contemporaneous stock price by the contemporaneous stock price. All of our models tend to

underestimate equity value. Moreover, with the exception of AEGV_L, which has mean

signed error of 0.941, the mean signed errors are also negative. Not surprisingly, models

with longer forecast horizons exhibit mean (median) errors that are less negative than models

with short forecast horizons. Nonetheless, even RIV_L, which assumes a fifteen-year

forecast horizon, has a median (mean) signed error of −0.310 (−0.258). Finally, consistent

with the extreme values of ROE and RONEC shown in Panel A of Table One, the absolute

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magnitudes of some of the signed errors are large. For example, the first and 99th percentiles

of the signed errors obtained from the AEGV_L model are -4.506 and 20.919.

Finally, in Panel C of Table One we show correlations between contemporaneous

price and each of our value estimates as well as the correlations between the value estimates.

Pearson product moment (Spearman rank order) correlations are shown above (below) the

diagonal. The correlations and t-statistics shown in the table relate to the estimation sample.

The correlations equal the mean of the annual correlations and each t-statistic equals the

relevant mean correlation divided by its temporal standard error. Because of the extreme

observations noted above, we focus on the Spearman rank order correlations. Two comments

are warranted. First, all of the value estimates have a significant, positive relation with price.

Second, the correlations between our naïve value estimates and price are roughly equal to, or

larger than, the correlations between price and the value estimates that rely on more

sophisticated assumptions about the forecast horizon. For example, the correlation between

price and RIV_N is 0.71, which is larger than the correlation between price and AEGV_L,

0.70. Taken together these results suggest that: (1) all of the value estimates are associated

with price and (2) naïve valuation models perform roughly as well as more sophisticated

approaches. As shown below, this second result is knife-edged.

4.2 Relative Unsigned Valuation Errors

Panel A of Table Two contains descriptive statistics on unsigned valuation errors.

These statistics pertain to the estimation sample. To determine the unsigned valuation error

for a particular observation and valuation model we divide the absolute value of the

difference between the model’s value estimate and the contemporaneous stock price by the

contemporaneous stock price. We also provide descriptive statistics for the best model,

which, for a particular observation, we define as the model with the lowest unsigned

valuation error for that observation. Several comments bear mentioning.

16

First, ignoring the best model, the mean and median unsigned valuation errors are

large for all of the models. For example, the mean (median) unsigned valuation error for

AEGV_L is 2.195 (0.465). Second, and related to the first point, some of the unsigned

valuation errors are extremely large. For example, ignoring the best model, the unsigned

valuation errors of RIV_N have the lowest 99th percentile. Nonetheless, the 99th percentile of

RIV_N’s unsigned valuation errors exceeds one (i.e., 1.337). Moreover, the 75th percentiles

of the valuation errors for AEGV_S and AEGV_L are 0.633 and 0.875, respectively. Third,

the model with the lowest median error (i.e., AEGV_S with a median unsigned valuation

error of 0.376) does not have the lowest mean unsigned valuation error. In particular,

AEGV_S has a mean unsigned valuation error of 0.662 whereas RIV_S has the lowest mean

unsigned valuation error (i.e., 0.470). Finally, ex post using the best model for each

observation is clearly superior to using the same model for each observation. Specifically,

the mean (median) unsigned valuation error for the best model is 0.203 (0.124), which is

orders of magnitude smaller than the mean (median) unsigned valuation errors of each of the

other models.

In Panel B of Table Two we further explore the potential improvement in accuracy

obtained from using the most accurate model for each estimation-sample observation. In

particular, for each observation we compute the difference between the unsigned valuation

error of each model and the unsigned valuation error of the best model. In Panel B of Table

Two, we provide descriptive statistics for these differences. These results illustrate that there

is considerable potential for improvement. For example, the AEGV_S (RIV_S) has the

smallest median (mean) difference, which equals a nontrivial 0.162 (0.267).

Finally, in Table Three we show the percentage of estimation-sample observations for

which each valuation model ranks first, second, third, etc., in terms of relative unsigned

valuation errors. Note that lower rankings imply higher relative accuracy. First, only three

17

models (i.e., AEGV_L, RIV_N, and AEGV_N) rank first for more than 16.67 percent (i.e.,

one-sixth) of the observations. The one-sixth cutoff is relevant because it represents the

frequency of observations for which each model would rank first if relative accuracy was

purely random. Second, AEGV_L ranks first most often. It is the best model for 28.53

percent of the observations; whereas, RIV_N and AEGV_N are the most accurate for 19.10

and 17.06 percent of the observations, respectively. Finally, although AEGV_L, RIV_N, and

AEGV_N are the most accurate, they are also the least accurate. In fact, for a randomly

chosen observation each of these models is more likely be the least accurate and not the most

accurate. In particular, for 37.58, 39.73, and 18.06 percent of the observations AEGV_L,

RIV_N, and AEGV_N ranks last in terms of valuation accuracy.

Based on the results shown in Table Two and Table Three we conclude that there is

no single, best model. The model that is best for the typical firm is not best on average.

Moreover, models that have the highest frequency of being the most accurate also have the

highest frequency of being the least accurate. In fact, these models are more likely to be the

least accurate not the most accurate.

4.3 Predicting Ranked Accuracy

As discussed in section 4.2, there is considerable potential for improvement vis-à-vis

an algorithm of using the same valuation model for each observation. Whether this potential

can be realized ex ante, however, is not obvious. With this in mind we develop an algorithm

for predicting model ranks. In particular, for each valuation model we estimate an ordered

logistic regression in which for each observation the dependent variable is the model’s rank

in terms of accuracy and the independent variables are listed below. We estimate the ordered

logistic regression using observations from the estimation sample.

18

Variable Description LOGSALE Natural log of contemporaneous sales AGE Duration in months since CRSP initiated coverage of the firm VOL Volatility of market-adjusted stock returns RET Market-adjusted, buy-hold stock return ROE Contemporaneous return on equity LOSS Equals one if contemporaneous net income is negative, zero otherwise NOATO Contemporaneous net operating asset turnover

∆NOATO Change in net operating asset turnover OPM Contemporaneous operating profit margin

∆OPM Change in operating profit margin SGROW Contemporaneous sales growth CASHFLOW Contemporaneous cash flow to sales (cash flow equals the sum of GAAP cash

flow from operations and research and development expenditures) INVEST Contemporaneous investment to sales (investment equals the sum of research

and development expenditures and capital expenditures) PAYER Equals one if the firm paid dividends in the current year, zero otherwise LEV Contemporaneous ratio of financial obligations to total assets

∆LEV Change in LEV CASH Contemporaneous ratio of financial assets to total assets

∆CASH Change in CASH ACCRUALS Contemporaneous ratio of working-capital accruals to sales C-SCORE Contemporaneous C-score per Penman and Zhang [2000] REV Ratio of most recent forecast revision to equity book value per share LTG The contemporaneous analysts' consensus long-term growth rate in earnings

per share ∆FOLLOW Change in analysts' following during the last three months before the end of

year zero.

Table A1 of Appendix A provides descriptive statistics on the variables shown above. We

motivate our choice of variables as follows.

First, the variables LOGSALE, AGE, and VOL relate to firm lifecycle. In particular,

larger, older firms with low return volatility are more established; hence, ceteris paribus,

forecast precision is higher for these firms. Second, to the extent current stock returns and

accounting performance is indicative of future performance, variables such as RET, ROE,

LOSS, NOATO, ∆NOATO, OPM, ∆OPM, SGROW, and CASHFLOW are relevant. In

particular, higher performance implies higher economic rents, which are not reflected in

either contemporaneous equity book value per share or expected earnings per share in year

19

+1. Similarly, to the extent current investment choices reflect manager’s expectations about

future performance, INVEST is relevant. Third, the variables PAYER, LEV, ∆LEV, CASH,

and ∆CASH relate to financial policy, which is linked to investment policy as well as

performance. Fourth, ACCRUALS and C-SCORE relate to earnings quality (e.g., Sloan

[1996]) and the effect of unconditional conservatism on contemporaneous equity book value

per share (e.g., Feltham and Ohlson [1996]). Finally, extant research (e.g., Hughes, Liu, and

Su [2008]) shows that REV and LTG are related to analysts’ forecast errors, and we add the

variable ∆FOLLOW as changes in coverage may affect the quality of the consensus forecast.

Although our variables are motivated by analytical and empirical results, they are ad

hoc; and, each variable relates to several phenomena. For example, Hughes, Liu, and Su

[2008] shows that RET is related to future analysts’ forecast errors; firms that pay dividends

are typically more stable and mature; firm lifecycle and investment opportunities are related;

etc. Moreover, the a priori relation between the independent variables and the relative

accuracy of our accounting-based valuation models is unclear. For example, the forecasts of

early-stage firms’ earnings are likely less precise. However, these firms are also likely to

have relatively valuable investment opportunities. Hence, earnings forecasts are more

relevant.

In light of the above, we consider the relation between the independent variables and

the relative valuation accuracy of the different accounting-based valuation models to be an

empirical issue. Hence, we estimate the ordered logistic regressions and evaluate their

predictive power. Given the degree of complexity involved (i.e., six regressions with 23

independent variables each), we do not discuss the estimated coefficients for each of the

ordered logistic regressions in the main text. Rather, in the main text we focus on their

predictive power. We relegate the discussion of the details regarding how we estimate each

regression, the estimated coefficients, etc. to Appendix A and Table A2.

20

For each observation in the holdout sample, we use the following approach to arrive

at the valuation model that is predicted to be the most accurate ex ante. First, for each

observation and each valuation model, we determine the fitted probability of each rank (i.e.,

one through six). Second, for each observation and each valuation model we create the

variable FIT_RANK that equals the rank with the highest fitted probability for that

observation and valuation model. Finally, for each observation we select the valuation model

that has the lowest value of FIT_RANK (i.e., lower ranks imply greater accuracy). If, for a

particular observation, two or more valuation models have the lowest value of FIT_RANK,

we choose the valuation model with the highest cumulative probability assigned to

FIT_RANK per the fitted probabilities from the estimated ordered logistic regressions.

We provide the following hypothetical example to clarify our selection algorithm.

For purposes of exposition but without loss of generality we assume there are only three

valuations models: RIV_N, RIV_S, and RIV_L. Hence, there are only three possible ranks.

For a hypothetical observation in the holdout sample assume the fitted probabilities implied

by the estimated coefficients from the ordered logistic regressions are as follows:

Fitted Probability that the Model will have Rank:

One Two Three RIV_N 0.25 0.00 0.75 RIV_S 0.21 0.79 0.00 RIV_L 0.00 0.80 0.20

To choose the valuation model for the hypothetical observation we begin by

determining the value of FIT_RANK for each valuation model. These values are three, two,

and two for the RIV_N, RIV_S, and RIV_L models, respectively. Next, we select the

valuation model with the lowest value of FIT_RANK. However, there is a tie – i.e., both

RIV_S and RIV_L have a FIT_RANK of two. Hence, we select RIV_S because the

21

cumulative probability associated with its FIT_RANK is 1.00, which exceeds the cumulative

probability of 0.80 associated with the FIT_RANK of RIV_L.

As the above example illustrates, our algorithm reduces the likelihood of choosing a

model that will exhibit extreme performance. In particular, the RIV_N model is not chosen

even though it has the highest probability of ranking first (i.e., 0.25). The reason for this is

that it has an even higher probability of ranking last (i.e., 0.75). Moreover, even though

RIV_L model has a higher likelihood of ranking second than the RIV_S model, it is not

chosen because the cumulative probability of RIV_S ranking second is higher.

In Panel A of Table Four we show the percentage of holdout-sample observations for

which each of the six valuation models is selected per the algorithm described above. Two

AEGV models dominate: AEGV_L and AEGV_S, which are selected for 51.42 and 28.27

percent of the observations. Only one RIV model, RIV_N, is selected for more than one-

sixth of the observations (it is selected for 17.82 percent of the observations).

In Panel B of Table Four we evaluate the effectiveness of our selection algorithm. In

particular, we show the percentage of holdout-sample observations for which the selected

model has realized rank one, two, etc. Our algorithm performs well. For 35.80 and 16.56

percent of the observations, the selected model is the best and second best model. Moreover,

although the selected model is the worst model for 14.34 percent of the observations, this is

an improvement vis-à-vis an approach in which one of the most accurate models per Table

Three is used exclusively (i.e., only AEGV_L, only RIV_N, or only AEGV_N). In particular,

as shown in Table Three, these three models are the most inaccurate for 37.58, 39.73, and

18.06 percent of the observations, respectively.

We further evaluate our model-selection algorithm in Panel C of Table Four. We

begin by providing descriptive statistics for the unsigned valuation errors obtained assuming

that for each observation we use the model selected by the algorithm. We refer to this as SM

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(i.e., the selected model). The median unsigned valuation error for SM is 0.354, which is

smaller than the median unsigned valuation errors of each of the separate models; however,

the mean unsigned valuation error for SM is 0.502. This exceeds the mean unsigned

valuation error for RIV_S (i.e., 0.470), which per Panel B of Table Two is the model with

lowest mean unsigned valuation error.

Given the above results are not clear-cut, we conduct further analyses. In particular,

in the remaining rows we provide descriptive statistics for the difference in the unsigned

valuation error of SM and the unsigned valuation errors obtained from a model selected via

one of five alternative algorithms.

1. An algorithm in which we select the best ex post model for each observation. We

refer to the valuation model selected using this algorithm as BEST. This algorithm

provides us with the lower bound of the valuation error.

2. A random algorithm in which, for each observation, each valuation model has a 1/6th

probability of being selected. We refer to the valuation model selected using this

algorithm as RAN1. This algorithm embeds the assumption that relative accuracy of

each of the valuation models does not depend on firm-level attributes.

3. A random algorithm in which, for each observation, each valuation model has a

probability of being selected that equals the percentage of estimation-sample

observations for which that model is best per the results shown in Table Three. We

refer to valuation the model selected using this algorithm as RAN2. This algorithm

embeds the assumption that the percentage of observations for which a particular

valuation model is best is known ex ante but that the identity of the specific

observations for which the model is best is unknown.

4. A random algorithm in which, for each observation, each valuation model has a

probability of being selected that equals the percentage of estimation-sample

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observations for which that model is selected by the algorithm that is based on the

results of the ordered logistic regressions. We refer to the valuation model selected

using this algorithm as RAN3. This algorithm embeds the assumption that the

percentage of observations for which a particular valuation model is selected by our

algorithm is known ex ante but that the identity of the specific observations for which

the model is selected is unknown.

5. A random algorithm in which we always use the AEGV_L model, which, per Table

Three, is the valuation model that is most frequently ranked first. This algorithm

embeds the assumption that the model that is most frequently the best ex post is the

best choice for each observation ex ante.

Ignoring the best model, SM (i.e., the selected model per our algorithm) outperforms

all of the models selected via the alternative algorithms. It is more accurate (no less accurate)

than: (1) RAN1 for 50.89 (67.36) percent of the observations; (2) RAN2 for 48.16 (70.70)

percent of the observations; (3) RAN3 for 38.96 (76.86) percent of the observations; and, (4)

AEGV_L 32.56 (83.98) percent of the observations. Moreover, the median difference

between the unsigned valuation error of SM and the BEST model is 0.104. This is roughly

64 percent of the smallest median shown in Panel B of Table Two (i.e., the median of

(|AEGV_S-P|−|BEST-P|)÷P, which equals 0.162).

4.4 Factor Analysis

The algorithm discussed above performs well; however, because of the large number

of independent variables in the logistic regressions, it is difficult to use the results shown in

Table A2 to draw conclusions about the types of firms that are matched to a particular

valuation model. With this in mind we perform exploratory factor analysis on the 23

independent variables. When conducting our factor analysis we only use observations from

the estimation sample. We retain factors with eigenvalues greater than one and we find three

24

factors. The coefficients associated with the raw factor patterns are presented in the first

three columns of Table Five. To clarify our interpretation of the factors we rotate them using

varimax rotation. Factor patterns after rotation are presented in the last three columns of

Table Five. Numbers shown in bold font relate to coefficients that: (1) have a correlation of

0.25 or higher with the factor in the column heading and (2) are not highly correlated with

either of the remaining factors.

First, regarding factor one, OPM, ∆OPM, and CASHFLOW have strong positive

loadings and INVEST has a strong negative loading. Hence, we interpret factor one as

relating to investment opportunities and we refer to it as INV_OPS. This interpretation is

based on the logic that firms with high (low) profits and cash flow but that aren’t (are)

investing arguably have poor (good) investment opportunities. Second, we interpret factor

two as relating to firm lifecycle and we refer to it as LIFECYCLE. This interpretation

reflects the fact that factor two has high positive loadings on LOGSALE, AGE, PAYER, and

LEV but high negative loadings on VOL, LOSS, CASH, and LTG. Finally, factor three loads

positively on NOATO and ∆ΝΟΑΤΟ; hence, we refer to it as NOA_TURN.

To evaluate the factors we modify the selection algorithm described in Section 4.3.

We refer to this as the modified algorithm. First, for each observation in the both the

estimation and holdout sample and each factor we compute a factor score. Second, for each

valuation model we estimate an ordered logistic regression in which the dependent variable is

the rank of the valuation model and the independent variables are the factor scores related to

INV_OPS, LIFECYCLE, and NOA_TURN. When estimating these regressions we only use

observations from the estimation sample. Third, as discussed in section 4.3, we compute

FIT_RANK for each holdout-sample observation and valuation model. Finally, for each

holdout-sample observation, we select the valuation model with the highest value of

FIT_RANK.

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Table Six shows the results obtained from estimating the modified ordered logistic

regressions on the estimation sample. For each of the three factors we show the estimated

coefficient and two marginal effects. ME_1 (ME_6) is the marginal effect on the probability

that the related valuation model will rank first (sixth) assuming a change of one standard

deviation in the relevant factor and setting the remaining two factors equal to their mean

values. Note that, as show in Table Three, RIV_S and AEGV_S rank sixth for only 0.01 and

0.74 percent of the observations; hence, it is not feasible to calculate the marginal effects of

the changes in the factors on the probability that these models will rank sixth.

The results shown in Table Six illustrate that the factor LIFECYCLE has strong

explanatory power regarding the relative rankings of the different models, and it is the only

factor for which this is the case. Two observations are noteworthy. First, for all of the RIV

models, an increase in LIFECYCLE leads to a significant reduction in the probability that the

model will rank first and a significant increase in the probability that a particular RIV model

will rank last. On the other hand, for all of the AEGV models, an increase in LIFECYCLE

leads to a significant increase in the probability that the model will rank first and, with the

exception of AEGV_S, a significant decrease in the probability that a particular AEGV model

will rank last. Second, differences in LIFECYCLE are also related to differences in forecast

horizon. For example, a one standard deviation change in LIFECYCLE changes the

probability of AEGV_L ranking first (last) by 0.089 (−0.091). However, a one standard

deviation change in LIFECYCLE changes the probability of AEGV_N ranking first by only

0.055 (−0.068).

Finally, the pseudo r-squared of each of the ordered logistic regressions is less than

five percent. Hence, whether the factors, particularly LIFECYCLE, provide useful

information about the relative accuracy of the valuation models is unclear. With this in mind,

we evaluate the accuracy of the modified algorithm. To do this we follow the same

26

procedure we used to evaluate the accuracy of the original algorithm based on the full set of

23 variables.

In Panel A of Table Seven we show the percentage of holdout-sample observations

for which each of the valuation models is selected by the modified algorithm. Similar to the

original algorithm, the modified algorithm selects either the AEGV_S and AEGV_L models

(39.49 and 48.37 percent of the observations) or the RIV_N model (12.15 percent of the

observations). However, as shown in Panel B of Table Seven, the modified algorithm does

not perform as well as the original algorithm, which is not surprising given there is a loss of

information when the 23 independent variables are reduced to three factors. Nonetheless, the

modified algorithm performs better than a simple, one-size-fits-all approach. For example,

the modified algorithm selects the best model for 30.32 percent of the observations, which, as

shown in Table Three, is higher than the percentage related to any specific model. Moreover,

the modified algorithm selects the worst model much less frequently than any of the three

best models per Table Three (i.e., AEGV_L, RIV_N, and AEGV_N). In particular, the

modified algorithm selects the worst model for 13.93 percent of the observations; whereas, as

shown in Table Three, AEGV_L, RIV_N, and AEGV_N is the worst model for 37.58, 39.73,

and 18.06 percent of the estimation-sample observations.

In Panel C of Table Seven we provide additional results regarding the performance of

the modified algorithm. In particular, we provide descriptive statistics regarding the

difference between the unsigned valuation errors of the selected model via the modified

algorithm (i.e., SM) and the unsigned valuation errors of models selected by various

alternative criteria, which we describe in Section 4.3. The findings are similar to those based

on original algorithm reported in Panel D of Table Four. In particular, the modified

algorithm is more accurate (no less accurate) than: (1) RAN1 for 48.28 (64.89) percent of the

observations; (2) RAN2 for 46.32 (67.80) percent of the observations; (3) RAN3 for 34.33

27

(74.93) percent of the observations; and, (4) AEGV_L for 32.22 (80.59) percent of the

observations. Moreover, the median difference between the unsigned valuation error of SM

and the BEST model is 0.144.

Finally, in Table Eight we present analyses that are analogous to those in Table

Seven. The key difference is that in Table Eight we only use the LIFECYLCE factor to

determine the selected model.2 The findings are virtually identical to those in Table Seven.

Hence, these results demonstrate a key result: Where a firm lies in its lifecycle is the primary

determinant of the best valuation model for that firm.

5. Sensitivity Analyses

5.1 Alternative Approaches to Forecasting Earnings

As discussed in section 3.2, we obtain our earnings forecasts for years t+1 through t+5

from I/B/E/S. This is a common approach. Nonetheless, recent results in Hou, van Dijk, and

Zhang [2011] (HVZ hereafter) show that (abstract): “earnings forecasts generated [from a]

cross-sectional model are superior to analysts’ forecasts in terms of coverage, forecast bias,

and earnings response coefficient. Moreover, the model-based ICC [(i.e., implied cost of

capital)] is a more reliable proxy for expected returns than the ICC based on analysts’

forecasts.”

The above quotation implies that it is possible that our results lack generality. In

particular, if I/B/E/S-based forecasts are not representative of the earnings expectations

embedded in price and the effect that these errors have on valuation accuracy varies across

the valuation models we evaluate, our results may be an artifact of using I/B/E/S data.

2 In particular, using the estimation sample we estimate univariate ordered logistic regressions for each valuation model in which LIFECYCLE is the only independent variable. Next, we use the estimated coefficients from the univariate ordered logistic regressions to determine FIT_RANK for each holdout-sample observation and valuation model. Finally, for each holdout-sample observation we select the valuation model with the lowest value of FIT_RANK.

28

To address this potential concern, we compute six alternative value estimates based

on one of the models described in section 3.1, however, using earnings forecasts implied by a

modified version of the cross-sectional forecasting model described in HVZ.3 We refer to

these value estimates as the HVZ-based estimates and we refer to our original value estimates

as the I/B/E/S-based estimates. Next, we compare the valuation accuracy of each HVZ-based

estimate to its I/B/E/S-based counterpart. Untabulated results show that the I/B/E/S-based

estimate per AEGV_N, AEGV_S, AEGV_L, RIV_N, RIV_S, RIV_L is more accurate than

the HVZ-based estimate for 81.2, 75.4, 67.9, 0.0, 81.2, and 77.1 percent of the estimation-

sample observations.4 Hence, we conclude that the I/B/E/S-based value estimates are more

reliable than HVZ-based value estimates, and that our results do not appear to be an artifact

of using I/B/E/S data.

5.2 Potential Hindsight Bias

As discussed in section 3.3, the estimation sample is formed by randomly selecting

half of the observations from each industry-year. This implies that the coefficients estimated

via the logistic regressions reflect information from all the years between 1980 and 2010

inclusive. These coefficients are then used to determine the selected model for each

observation in the holdout sample regardless of the date of the observation. Hence, our

results may be affected by hindsight bias.

To determine whether hindsight bias is an issue we compare the selected model per

our algorithm (i.e., the original algorithm) to the selected model from an alternative algorithm

that is based on “rolling” regressions (i.e., the rolling algorithm). In particular, for each year t

∈ [1990, 2010] we estimate logistic regressions that are identical to the regressions discussed

3 In particular, we use a modified version of equation (1) in HVZ. Our modification is to add an additional regressor that equals the interaction of Ei,t (i.e., earnings of firm i in year t) and NegEi,t (an indicator that equals one if Ei,t < 0 and zero otherwise). 4 The RIV_N model does not rely on earnings forecasts; hence, the HVZ-based estimate is identical to the I/B/E/S-based estimate. Consequently, the I/B/E/S-based estimate is neither more nor less accurate than the HVZ-based estimate.

29

in section 4.3 with the exception that we only use data from year t−10 through year t−1

inclusive.5 Hence, the estimates obtained from the rolling regressions are not affected by

hindsight bias.

Untabulated results show that the rolling algorithm yields very similar results as the

original algorithm. When the rolling algorithm is used the RIV_N, RIV_S, RIV_L,

AEGV_N, AEGV_S, and AEGV_L model is selected for 18.01, 1.24, 1.62, 1.11, 22.65, and

55.37 percent of the observations that relate to the years occurring between 1990 and 2010

inclusive. These percentages are very similar to the percentages shown in Panel A of Table

Four, which are 17.83, 1.73, 0.36, 0.39, 28.87, and 51.42 for the RIV_N, RIV_S, RIV_L,

AEGV_N, AEGV_S, and AEGV_L model, respectively.6 Moreover, the selected model per

the rolling algorithm has rank one, two three, four, five, or six for 36.39, 14.00, 9.29, 9.92,

13.63, and 16.78 percent of the observations that relate to the years occurring between 1990

and 2010 inclusive. These percentages are very similar to the percentages shown in Panel B

of Table, which are 35.80, 16.56, 9.23, 10.37, 13.72, and 16.78 for rank one, two, three, four,

five, and six, respectively. In light of these facts we conclude that our results based on the

original algorithm are not attributable to hindsight bias.

6. Conclusion

Valuation is ultimately a practical endeavor. Accounting-based valuation models are

practical because they can be used to develop rigorous estimates of value without having to

make impractical and arbitrary assumptions about dividend policy. Nonetheless, even when

accounting-based models are used, another practical question remains: When steady-state 5 Because we require ten years of historical data for the rolling regressions, we can only use the rolling algorithm to select a valuation model for observations that relate to the year 1990 or later. 6 We also identify the observations that have a selected model per both original and rolling algorithm. Untabulated results show that for 79.04 percent of these observations the selected model per the rolling algorithm is identical to the selected model per the original algorithm.

30

forecasting is infeasible, which model is most appropriate? To address this question, we

evaluate two widely used classes of models, residual income and abnormal earnings growth,

and three forecast horizons for each model class.

Our study has three primary findings that are relevant to both academics and

practitioners who use accounting-based valuation models. First, we demonstrate that there is

no single, best valuation model; and, that models that are considered “best” per extant

research frequently perform poorly. Second, we develop a model selection algorithm and we

show that relative valuation accuracy is predictable ex ante based on knowledge of a firm’s

attributes. Third, we demonstrate that where a firm is its lifecycle is a key determinant of the

identity of the appropriate valuation model for that firm. This final result is important as it

suggests a parsimonious and practical criterion that researchers and practitioners can use to

choose among valuation models.

31

Appendix A

In this appendix we provide descriptive statistics and results of estimating the ordered

logistic regressions that we use in our original selection algorithm (i.e., the algorithm that is

described in section 4.3). Descriptive statistics are shown in Table A1. The results of

estimating the ordered logistic regressions are shown in Table A2.

Variables are defined as follows. Please note that in the definitions, we refer to year

zero, which is pertinent because our value estimates are made as of the end of year zero. In

addition, we omit firm subscripts. LOGSALE is the log of COMPUSTAT data item SALE

for year zero. AGE is the number of months between the date the observation first appears

on CRSP and the last fiscal month of year zero. VOL is the volatility of daily, market-

adjusted stock returns for year zero. RET is the buy-hold, market adjusted stock return for

year zero. The data for both VOL and RET are from the CRSP daily stock return files. ROE

is return on equity and equals the ratio of the year zero value of COMPUSTAT data item IB

(i.e., income before extraordinary items) to the year -1 value of COMPUSTAT data item

CEQ (i.e., common equity). LOSS is an indicator variable that equals one if IB in year zero

is negative and zero otherwise. The definitions NOATO, ∆NOATO, OPM, and ∆OPM are

provided in the appendix on pp. 149-151 of Nissim and Penman [2001]. SGROW equals the

annual growth rate in year zero of the COMPUSTAT data item SALE. CASHFLOW equals

the sum of COMPUSTAT data item OANCFC in year zero (i.e., net cash flow from

continuing operations) and COMPUSTAT data item RDIP in year zero (i.e., research and

development expense) divided by COMPUSTAT data item SALE in year zero. INVEST

equals the sum of COMPUSTAT data item RDIP in year zero and COMPUSTAT data item

PPENT in year zero (i.e., net property, plant, and equipment). PAYER equals one if

COMPUSTAT data item DVPSX_F (dividends per share – ex date – fiscal) is positive in

year zero, and zero otherwise. LEV (∆LEV) equal the ratio of (the change in) financial

32

obligations for year zero to COMPUSTAT data item AT (i.e., total assets) in year zero.

CASH, and ∆CASH equal the ratio of (the change in) financial assets for year zero to

COMPUSTAT data item AT in year zero. The definitions financial obligations and financial

assets are provided in the appendix on pp. 149-151 of Nissim and Penman [2001].

ACCRUALS equals the ratio of accruals in year zero, which are measured via the balance-

sheet method described in Sloan [1996], to COMPUSTAT data item SALE in year zero. We

use the definition of C-score shown in Penman and Zhang [2002]. REV equals the change in

the analysts’ consensus forecast of earnings per share in year plus one. The change is

measured by subtracting the value at the end of third fiscal quarter of year zero from the

value at the end of the fourth fiscal quarter of year zero. We deflate REV by COMPUSTAT

data item CEQ as of year -1. LTG is the analysts’ consensus long-term growth rate in

earnings per share as of the last fiscal month of year zero. ∆FOLLOW is the change in

analyst following during year zero.

33

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empirical implementations of the abnormal earnings growth model: U.S. evidence.

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Modigliani, F., and M. H. Miller. 1961. Dividend policy, growth, and the valuation of shares.

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Ohlson, J. 1995. Earnings, book values, and dividends in equity valuation. Contemporary

Accounting Research 11: 661-687.

34

Penman, S. H., and T. Sougiannis, 1998. A comparison of dividend, cash flow, and earnings

approaches to equity valuation. Contemporary Accounting Research 15 (3): 343-383.

Penman, S. H., and X. Zhang. 2002. Accounting conservatism, quality of earnings, and stock

returns. The Accounting Review 77 (2): 237-264.

Sloan, R. 1996. Do stock prices fully reflect information in accruals and cash flows about

future earnings? The Accounting Review 71: 289-315.

Zhang, X. 2000. Conservative accounting and equity valuation. Journal of Accounting and

Economics 29 (1): 125-149.

35

Figure One

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36

Table One – Descriptive Statistics and Correlations (Estimation Sample) Panel A – Descriptive Statistics for Model Inputs and Key Firm Attributes

MEAN STD MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAXMVE 3,238.6 14,034.2 0.6 11.4 28.3 45.5 113.1 371.0 1,429.8 5,341.4 12,627.9 56,856.0 460,767.9ASSET 5,062.7 38,945.0 0.9 9.8 24.9 41.8 107.4 384.8 1,762.07,018.6 16,339.0 75,889.0 1,916,658.4SALE 2,604.4 11,076.8 -299.3 0.8 13.5 29.6 91.9 330.2 1,307.6 4,887.3 10,467.0 42,539.6 425,071.0BM 0.575 0.455 -11.961 -0.064 0.105 0.169 0.301 0.494 0.750 1.068 1.334 2.051 11.633EP 0.063 0.115 -5.153 -0.291 -0.052 0.013 0.046 0.070 0.098 0.134 0.162 0.232 5.331βLEV_FIRM 0.962 0.446 0.000 0.065 0.267 0.400 0.632 0.933 1.269 1.597 1.763 1.944 2.000r 0.113 0.036 0.024 0.046 0.058 0.068 0.087 0.111 0.136 0.1610.178 0.210 0.249

ROE1 0.130 3.987 -300.403 -1.200 -0.174 0.024 0.094 0.144 0.204 0.291 0.386 0.962 519.786

ROE2 0.169 4.319 -250.557 -1.273 -0.042 0.052 0.107 0.152 0.208 0.291 0.386 1.402 426.775

ROE3 0.146 16.387 -2,368.667 -0.932 0.012 0.064 0.113 0.156 0.211 0.295 0.398 1.382 1,687.318

ROE4 0.174 3.345 -529.862 -0.645 0.028 0.070 0.115 0.158 0.214 0.297 0.404 1.266 161.842

ROE5 0.199 3.392 -87.718 -0.379 0.039 0.074 0.116 0.160 0.216 0.299 0.400 1.028 560.541

RONEC2 328.739 34,750.513 -4,866.412 -1.846 -0.111 0.000 0.062 0.177 0.275 0.500 0.894 3.923 6,223,529.409

RONEC3 267.489 22,325.700 -4,866.412 -0.759 0.000 0.000 0.097 0.175 0.250 0.380 0.566 2.130 3,620,370.360

RONEC4 400.836 48,095.302 -4,866.412 -0.356 0.000 0.000 0.100 0.173 0.250 0.355 0.500 1.729 8,831,999.990

RONEC5 207.066 14,556.055 -4,866.412 0.000 0.000 0.000 0.100 0.175 0.250 0.350 0.500 1.173 1,839,999.997

37

Panel B – Descriptive Statistics for Signed Valuation Errors

MEAN STD MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX(RIV_N-P)÷P -0.415 0.523 -10.023 -1.064 -0.893 -0.829 -0.696 -0.502 -0.243 0.084 0.351 1.075 32.822(RIV_S-P)÷P -0.374 0.572 -32.398 -1.621 -0.888 -0.748 -0.587 -0.400 -0.180 0.078 0.273 0.913 23.541(RIV_L-P)÷P -0.258 0.983 -81.628 -2.109 -0.924 -0.743 -0.541 -0.310 -0.026 0.358 0.721 1.864 19.988(AEGV_N-P)÷P -0.452 1.742 -192.344 -4.813 -1.546 -0.876 -0.581 -0.348 -0.077 0.245 0.530 1.427 35.340(AEGV_S-P)÷P -0.156 2.092 -152.269 -3.604 -1.068 -0.743 -0.479 -0.213 0.126 0.655 1.204 3.423 69.589(AEGV_L-P)÷P 0.941 22.897 -1993.076 -4.506 -1.058 -0.806 -0.451 -0.063 0.497 1.597 2.937 20.919 1570.048 Panel C – Correlations

P RIV_N RIV_S RIV_L AEGV_N AEGV_S AEGV_LP CORR 1.00 0.88 0.74 0.73 0.58 0.61 0.51

T-STAT 24.18 7.53 7.46 4.61 5.31 5.88RIV_N CORR 0.71 1.00 0.84 0.81 0.65 0.66 0.54

T-STAT 43.57 10.62 10.06 5.99 6.45 6.17RIV_S CORR 0.81 0.90 1.00 0.99 0.87 0.89 0.68

T-STAT 49.98 123.23 369.23 13.10 13.72 11.41RIV_L CORR 0.80 0.80 0.96 1.00 0.88 0.91 0.72

T-STAT 50.06 73.02 656.99 13.23 14.12 12.67AEGV_N CORR 0.78 0.75 0.91 0.93 1.00 0.94 0.68

T-STAT 37.53 48.95 171.45 204.14 66.78 11.76AEGV_S CORR 0.78 0.68 0.90 0.97 0.91 1.00 0.80

T-STAT 49.06 40.37 170.83 450.80 135.61 20.37AEGV_L CORR 0.70 0.56 0.79 0.89 0.78 0.95 1.00

T-STAT 46.25 27.08 82.11 149.70 65.60 315.60

38

The estimation sample consists of 35,473 firm-years. MVE is the market capitalization (CSHO × PRCC_F) at the end of year zero. ASSET is the total assets (AT) at the end of year zero. SALE is sales during the year zero. BM is the book to market ratio, defined as the book value of common equity (CEQ) over the market capitalization. EP is lead earnings to price ratio defined as I/B/E/S analysts’ consensus (median) forecast of one-year ahead earnings per share over stock price at the end of year zero. βLEV_FIRM is the estimated firm-level beta. See section 3.2 for detailed definitions. r is the cost of equity capital. See section 3.2 for detailed definitions. ROE1, ROE2, ROE3, ROE4, and ROE5 are expected return on equity, defined as earnings per share over the beginning balance of book value per share, for year one through five, respectively. RONEC 2, RONEC 3, RONEC 4, and RONEC 5 are expected return on new equity capital for years two through five. P is the share price at the I/B/E/S database release date. RIV_N, RIV_S, and RIV_L are the estimates of a firm’s share price at time zero based on RIV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. AEGV_N, AEGV_S, and AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. See section 3.1 and 3.2 for detailed definitions. Regarding the correlations shown in Panel C, Pearson product moment (Spearman rank order) correlations are shown above (below) diagonal. Correlations are calculated as the means of annual the correlations and t-statistics based on the temporal standard errors of the annual coefficients.

39

Table Two – Evaluation of Unsigned Valuation Errors (Estimation Sample) Panel A – Descriptive Statistics for Unsigned Valuation Errors

MEAN STD MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX|RIV_N-P|÷P 0.529 0.408 0.000 0.014 0.070 0.138 0.312 0.5280.712 0.847 0.920 1.337 32.822|RIV_S-P|÷P 0.470 0.497 0.000 0.011 0.053 0.105 0.238 0.4230.603 0.771 0.950 1.870 32.398|RIV_L-P|÷P 0.507 0.881 0.000 0.008 0.041 0.079 0.200 0.3960.607 0.860 1.149 2.944 81.628|AEGV_N-P|÷P 0.628 1.687 0.000 0.009 0.042 0.083 0.211 0.408 0.631 1.000 1.746 5.071 192.344|AEGV_S-P|÷P 0.662 1.991 0.000 0.007 0.035 0.072 0.184 0.376 0.633 1.143 1.902 5.719 152.269|AEGV_L-P|÷P 2.195 22.811 0.000 0.008 0.041 0.081 0.211 0.465 0.875 2.006 3.759 28.979 1993.076|BEST-P|÷P 0.203 0.246 0.000 0.001 0.008 0.016 0.046 0.124 0.288 0.497 0.650 0.928 14.776 Panel B – Comparison of Each Model to the Best Model

MEAN STD MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX(|RIV_N-P|-|BEST-P|)÷P 0.326 0.361 0.000 0.000 0.000 0.000 0.067 0.299 0.510 0.668 0.755 1.032 30.229(|RIV_S-P|-|BEST-P|)÷P 0.267 0.385 0.000 0.000 0.000 0.000 0.090 0.215 0.370 0.508 0.608 1.251 20.949(|RIV_L-P|-|BEST-P|)÷P 0.304 0.776 0.000 0.000 0.000 0.000 0.085 0.190 0.332 0.549 0.828 2.342 66.852(|AEGV_N-P|-|BEST-P|)÷P 0.425 1.587 0.000 0.000 0.000 0.000 0.055 0.209 0.394 0.722 1.346 4.494 177.569(|AEGV_S-P|-|BEST-P|)÷P 0.459 1.915 0.000 0.000 0.000 0.000 0.049 0.162 0.359 0.847 1.566 5.206 137.493(|AEGV_L-P|-|BEST-P|)÷P 1.992 22.799 0.000 0.000 0.000 0.000 0.000 0.242 0.647 1.796 3.525 28.714 1992.996 The estimation sample consists of 35,473 firm-years. P is the share price at the I/B/E/S database release date. RIV_N, RIV_S, and RIV_L are the estimates of a firm’s share price at time zero based on RIV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. AEGV_N, AEGV_S, and AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. BEST is the estimate of firm’s share price at time zero based on the model with the lowest unsigned valuation error for each observation. See section 3.1 and 3.2 for detailed definitions.

40

Table Three – Relative Model Rankings (Estimation Sample)

1 2 3 4 5 6RIV_N 19.10% 9.49% 7.53% 8.04% 16.12% 39.73%RIV_S 10.41% 21.92% 18.07% 19.87% 29.73% 0.01%RIV_L 10.50% 17.74% 31.08% 26.83% 9.98% 3.88%AEGV_N 17.06% 11.01% 25.28% 20.22% 8.37% 18.06%AEGV_S 14.41% 32.89% 10.24% 14.81% 26.91% 0.74%AEGV_L 28.53% 6.95% 7.81% 10.24% 8.89% 37.58%

Percentage of Obs. for which Model Ranks:

The estimation sample consists of 35,473 firm-years. Table Three shows the percentage of observations for which each valuation model ranks first, second, third, fourth, fifth, and sixth in terms of relative unsigned valuation errors. RIV_N, RIV_S, and RIV_L are the estimates of a firm’s share price at time zero based on RIV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. AEGV_N, AEGV_S, and AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. See section 3.1 and 3.2 for detailed definitions.

41

Table Four – Attributes of the Selected Model per the Original Algorithm (Holdout Sample)

Panel A – Percentage of Holdout-sample Observations for which a Particular Model is

Selected

RIV_N RIV_S RIV_L AEGV_N AEGV_S AEGV_L17.82% 1.73% 0.36% 0.39% 28.27% 51.42%

Panel B – Percentage of Holdout-sample Observations for which the Selected Model has

Rank N

1 2 3 4 5 635.80% 16.56% 9.23% 10.37% 13.72% 14.34%

The holdout sample consists of 23,696 firm-years with available data. RIV_N, RIV_S, and RIV_L are the estimates of a firm’s share price at time zero based on RIV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. AEGV_N, AEGV_S, and AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. See section 3.1 and 3.2 for detailed definitions.

42

Panel C – Descriptive Statistics for Unsigned Valuation Errors (Holdout Sample)

MEAN STD % < 0 % ≤ 0 MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX|SM-P|÷P 0.502 0.871 0.000 0.007 0.033 0.067 0.165 0.354 0.605 0.907 1.304 3.135 43.320(|SM-P|-|BEST-P|)÷P 0.307 0.813 0.00% 35.80% 0.000 0.000 0.000 0.000 0.000 0.104 0.355 0.718 1.138 2.867 42.045(|SM-P|-|RAN1-P|)÷P -0.276 6.827 50.89% 67.36% -629.766 -3.619 -0.744 -0.494 -0.233 -0.009 0.085 0.410 0.703 2.037 26.111(|SM-P|-|RAN2-P|)÷P -0.403 7.319 48.16% 70.70% -436.275 -4.551 -0.896 -0.544 -0.240 0.000 0.049 0.352 0.627 1.837 38.942(|SM-P|-|RAN3-P|)÷P -0.792 18.299 38.96% 76.86% -2188.144 -7.857 -1.323 -0.644 -0.186 0.000 0.000 0.240 0.456 1.303 14.447(|SM-P|-|AEGV_L-P|)÷P -1.490 21.917 32.56% 83.98% -2188.144 -24.351 -2.420 -1.073 -0.203 0.000 0.000 0.119 0.259 0.609 12.163

NA

The holdout sample consists of 23,696 firm-years with available data. SM is the estimate of a firm’s share price at time zero obtained assuming that for each observation we use the model selected by the algorithm described in section 4.3. BEST is the estimate of a firm’s share price at time zero based on the model with the lowest unsigned valuation error for each observation. RAN1 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a 1/6th probability of being selected. RAN2 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a probability of being selected that equals the percentage of observations for which that model is best per Table Three. RAN3 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a probability of being selected that equals the percentage of observations for which that model is selected by the algorithm that is based on the results of the ordered logistic regressions. AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after assuming a horizon of fifteen years. P is the share price at the I/B/E/S database release date. See section 4.3 for detailed definitions.

43

Table Five – Results of Factor Analysis (Estimation Sample)

INV_OPS LIFECYCLE NOA_TURN INV_OPS LIFECYCLE NOA_TURNLOGSALE 0.157 -0.693 0.011 0.0473 0.7092 0.0102AGE 0.072 -0.594 0.010 -0.0215 0.5977 0.0079VOL -0.111 0.616 0.001 -0.0140 -0.6260 0.0028RET -0.017 0.033 0.022 -0.0118 -0.0352 0.0224ROE 0.023 -0.071 0.008 0.0117 0.0738 0.0078LOSS -0.090 0.428 -0.010 -0.0220 -0.4368 -0.0096NOATO -0.001 0.023 0.831 -0.0060 -0.0204 0.8310∆NOATO 0.000 -0.006 0.831 -0.0094 0.0081 0.8305OPM 0.961 0.085 0.000 0.9621 0.0654 0.0099∆OPM 0.624 0.109 -0.001 0.6337 -0.0102 0.0056SGROW 0.010 0.137 0.004 0.0308 -0.1343 0.0047CASHFLOW 0.683 0.104 0.000 0.6905 0.0038 0.0071INVEST -0.767 -0.071 -0.001 -0.7682 -0.0497 -0.0082PAYER 0.091 -0.661 0.001 -0.0128 0.6670 -0.0004LEV -0.077 -0.302 0.001 -0.1234 0.2859 -0.0013∆LEV 0.054 0.044 -0.027 0.0609 -0.0354 -0.0261CASH -0.072 0.562 0.007 0.0166 -0.5665 0.0085∆CASH 0.013 -0.059 -0.003 0.0037 0.0605 -0.0035ACCRUALS 0.894 0.080 0.000 0.8956 0.0600 0.0087C-SCORE -0.009 0.049 0.006 -0.0009 -0.0497 0.0057REV 0.004 -0.019 0.003 0.0012 0.0199 0.0026LTG -0.024 0.367 0.000 0.0336 -0.3663 0.0011∆FOLLOW -0.016 0.128 0.008 0.0040 -0.1290 0.0080

EIGENVALUE 3.240 2.466 1.382

Raw Factor Pattern Factor Pattern after Varimax Rotation

This subset of the estimation sample consists of 23,799 firm-years with available data. LOGSALE is a natural log of contemporaneous sales. AGE is a year-to-date, duration in months since CRSP initiated coverage of the firm. VOL is a year-to-date, market-adjusted stock return volatility. RET is a year-to-date, market-adjusted, buy-hold stock return. ROE is the contemporaneous return on equity. LOSS equals one if contemporaneous net income is negative, zero otherwise. NOATO is contemporaneous net operating asset turnover. ΔNOATO is a change in net operating asset turnover. OPM is a contemporaneous operating profit margin. ΔOPM is a change in operating profit margin. SGROW is a contemporaneous sales growth. CASHFLOW is a contemporaneous cash flow to sales (cash flow equals the sum of GAAP cash flow from operations and research and development expenditures). INVEST is the contemporaneous investment to sales (investment equals the sum of research and development expenditures and capital expenditures). PAYER equals one if the firm paid dividends in the current year, zero otherwise. LEV is a Contemporaneous ratio of financial obligations to total assets. ∆LEV is a change in LEV. CASH is a contemporaneous ratio of financial assets to total assets. ∆CASH is a change in CASH. ACCRUALS is a contemporaneous ratio of working-capital accruals to sales. C-SCORE is a contemporaneous C-score per Penman and Zhang [2000]. REV is Ratio of most recent forecast revision to equity book value per share. LTG is the contemporaneous analysts' consensus long-term growth rate in earnings per share. ∆FOLLOW is a change in analysts' following during the last three months before the end of year zero.

44

Table Six – Results Obtained from Estimating Ordered Logistic Regressions of Ranks on Factor Scores (Estimation Sample)

Est. p-value Est. p-value Est. p-value Est. p-value Est. p-value Est. p-valueINV_OPS -0.024 0.185 -0.013 0.313 -0.010 0.415 -0.022 0.2770.027 0.074 0.027 0.157

ME_1 0.003 0.185 0.001 0.233 0.001 0.381 0.003 0.192 -0.004 0.101 -0.006 0.098ME_6 -0.006 0.185 0.000 0.381 -0.003 0.192 0.006 0.098

LIFECYCLE 0.692 0.000 0.533 0.000 0.229 0.000 -0.470 0.000 -0.292 0.000 -0.407 0.000ME_1 -0.090 0.000 -0.040 0.000 -0.021 0.000 0.055 0.000 0.038 0.000 0.089 0.000ME_6 0.170 0.000 0.007 0.000 -0.068 0.000 -0.091 0.000

NOA_TURN 0.130 0.004 0.056 0.019 0.008 0.562 -0.015 0.238 -0.025 0.075 -0.023 0.130ME_1 -0.017 0.009 -0.004 0.040 -0.001 0.526 0.002 0.187 0.003 0.059 0.005 0.097ME_6 0.032 0.009 0.000 0.526 -0.002 0.187 -0.005 0.097

INT_6 -0.253 0.000 -8.581 0.000 -3.418 0.000 -1.550 0.000 -4.880 0.000 -0.683 0.000INT_5 0.473 0.000 -0.646 0.000 -2.153 0.000 -1.031 0.000 -1.073 0.000 -0.320 0.000INT_4 0.837 0.000 0.246 0.000 -0.482 0.000 -0.017 0.199 -0.479 0.000 0.081 0.000INT_3 1.185 0.000 0.986 0.000 0.926 0.000 1.188 0.000 -0.0750.000 0.416 0.000INT_2 1.711 0.000 2.421 0.000 2.169 0.000 1.851 0.000 1.704 0.000 0.730 0.000Pseudo R-squared 0.035 0.023 0.004 0.016 0.007 0.013

NA

NA

NA

AEGV_L

NA

NA

NA

RIV_N RIV_S RIV_L AEGV_N AEGV_S

This subset of the estimation ample consists of 23,799 firm-years with available data. INV_OPS, LIFECYCLE, and EARN_QUAL are the factor scores from Factor 1, Factor 2, and Factor 3 shown in Table Five, respectively. ME_1 is the marginal effect on the probability that the related valuation model will rank first assuming a change of one standard deviation in the relevant factor and setting the remaining two factors equal to their mean values. ME_6 is the marginal effect on the probability that the related valuation model will rank sixth assuming a change of one standard deviation in the relevant factor and setting the remaining two factors equal to their mean values. INT_6, INT_5, INT_4, INT_3, and INT_2 are intercepts corresponding to the outcomes ranked sixth, fifth, fourth, third, and second, respectively.

45

Table Seven – Attributes of the Selected Model per the Modified Algorithm Using All Three Factors (Holdout Sample)

Panel A – Percentage of Holdout-sample Observations for which a Particular Model is

Selected

RIV_N RIV_S RIV_L AEGV_N AEGV_S AEGV_L12.15% 0.00% 0.00% 0.00% 39.49% 48.37%

Panel B – Percentage of Holdout-sample Observations for which the Selected Model has

Rank N

1 2 3 4 5 630.32% 18.17% 9.14% 11.24% 17.21% 13.93%

46

Panel C – Descriptive Statistics for Unsigned Valuation Errors

MEAN STD % < 0 % ≤ 0 MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX|SM-P|÷P 0.555 1.594 0.000 0.007 0.035 0.071 0.181 0.381 0.647 0.931 1.374 3.375 98.931(|SM-P|-|BEST-P|)÷P 0.359 1.558 0.00% 30.32% 0.000 0.000 0.000 0.000 0.000 0.144 0.400 0.755 1.182 3.029 98.638(|SM-P|-|RAN1-P|)÷P -0.223 6.917 48.28% 64.89% -629.766 -3.315 -0.673 -0.441 -0.200 0.000 0.108 0.423 0.726 2.062 98.050(|SM-P|-|RAN2-P|)÷P -0.351 7.426 46.32% 67.80% -436.275 -4.525 -0.813 -0.489 -0.203 0.000 0.079 0.379 0.659 1.922 98.289(|SM-P|-|RAN3-P|)÷P -0.740 14.224 34.33% 74.93% -1082.932 -7.631 -1.217 -0.550 -0.102 0.000 0.001 0.254 0.451 1.080 88.810(|SM-P|-|AEGV_L-P|)÷P -1.437 21.908 32.22% 80.59% -2188.144 -22.189 -2.218 -1.003 -0.143 0.000 0.000 0.156 0.280 0.620 5.182 The holdout sample consists of 23,696 firm-years with available data. RIV_N, RIV_S, and RIV_L are the estimates of a firm’s share price at time zero based on RIV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. AEGV_N, AEGV_S, and AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. See section 3.1 and 3.2 for detailed definitions. SM is the estimate of a firm’s share price at time zero obtained assuming that for each observation we use the model selected by the modified algorithm described in section 4.4. BEST is the estimate of a firm’s share price at time zero based on the model with the lowest unsigned valuation error for each observation. RAN1 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a 1/6th probability of being selected. RAN2 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a probability of being selected that equals the percentage of observations for which that model is best per Table Three. RAN3 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a probability of being selected that equals the percentage of observations for which that model is selected by the modified algorithm. AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after assuming a horizon of fifteen years. P is the share price at the I/B/E/S database release date.

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Table Eight – Attributes of the Selected Model per the Modified Algorithm Using only the Life Cycle Factor (Holdout Sample)

Panel A – Percentage of Holdout-sample Observations for which a Particular Model is

Selected

RIV_N RIV_S RIV_L AEGV_N AEGV_S AEGV_L12.12% 0.00% 0.00% 0.00% 39.50% 48.37%

Panel B – Percentage of Holdout-sample Observations for which the Selected Model has

Rank N

1 2 3 4 5 630.32% 18.13% 9.15% 11.24% 17.24% 13.92%

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Panel C – Descriptive Statistics for Unsigned Valuation Errors

MEAN STD % < 0 % ≤ 0 MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX|SM-P|÷P 0.554 1.589 0.000 0.007 0.035 0.071 0.181 0.381 0.647 0.931 1.373 3.375 98.931(|SM-P|-|BEST-P|)÷P 0.359 1.553 0.00% 30.32% 0.000 0.000 0.000 0.000 0.000 0.144 0.400 0.753 1.180 3.029 98.638(|SM-P|-|RAN1-P|)÷P -0.224 6.916 48.27% 64.87% -629.766 -3.323 -0.674 -0.441 -0.200 0.000 0.108 0.423 0.725 2.070 98.050(|SM-P|-|RAN2-P|)÷P -0.351 7.420 46.32% 67.78% -436.275 -4.525 -0.813 -0.490 -0.203 0.000 0.080 0.379 0.658 1.916 98.289(|SM-P|-|RAN3-P|)÷P -0.747 14.308 34.46% 74.99% -1082.932 -8.727 -1.203 -0.546 -0.105 0.000 0.000 0.253 0.451 1.079 98.601(|SM-P|-|AEGV_L-P|)÷P -1.438 21.907 32.22% 80.60% -2188.144 -22.357 -2.221 -1.003 -0.143 0.000 0.000 0.156 0.280 0.620 5.182 The holdout sample consists of 23,696 firm-years with available data. RIV_N, RIV_S, and RIV_L are the estimates of a firm’s share price at time zero based on RIV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. AEGV_N, AEGV_S, and AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after making a naïve assumption about the forecast horizon, assuming a horizon of five years, and assuming a horizon of fifteen years, respectively. See section 3.1 and 3.2 for detailed definitions. SM is the estimate of a firm’s share price at time zero obtained assuming that for each observation we use the model selected by the modified algorithm described in section 4.4. BEST is the estimate of a firm’s share price at time zero based on the model with the lowest unsigned valuation error for each observation. RAN1 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a 1/6th probability of being selected. RAN2 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a probability of being selected that equals the percentage of observations for which that model is best per Table Three. RAN3 is the estimate of a firm’s share price at time zero from the valuation model selected using the algorithm that for each observation, each valuation model has a probability of being selected that equals the percentage of observations for which that model is selected by the modified algorithm. AEGV_L are the estimates of a firm’s share price at time zero based on AEGV Model after assuming a horizon of fifteen years. P is the share price at the I/B/E/S database release date.

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Table A1 – Descriptive Statistics for Firm Attributes (Estimation Sample)

MEAN STD MIN P1 P5 P10 Q1 P50 Q3 P90 P95 P99 MAX NOBSLOGSALE 5.860 2.041 -6.908 0.637 2.707 3.436 4.542 5.811 7.183 8.501 9.260 10.659 12.960 35290AGE 191 190 11 13 19 27 56 127 255 439 642 873 1008 35471VOL 0.086 0.083 -0.051 -0.010 0.007 0.016 0.036 0.067 0.113 0.172 0.222 0.367 3.826 35443RET 0.164 0.735 -1.040 -0.833 -0.604 -0.459 -0.203 0.067 0.359 0.781 1.189 2.677 23.698 35449ROE -0.147 44.670 -8351.000 -1.755 -0.503 -0.209 0.041 0.130 0.202 0.306 0.420 1.103 431.531 35418LOSS 0.201 0.401 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 35470NOATO 5.001 77.796 -0.407 0.154 0.431 0.638 1.232 2.113 3.482 6.109 9.152 25.857 6595.471 32504∆NOATO -0.488 78.331 -6537.678 -16.615 -3.109 -1.451 -0.402 -0.016 0.240 0.834 1.681 8.388 6547.484 31622OPM -1.929 77.378 -8305.500 -8.700 -0.684 -0.144 0.020 0.060 0.112 0.181 0.239 0.494 285.986 35272∆OPM -0.573 70.256 -7865.466 -2.036 -0.259 -0.103 -0.024 0.001 0.023 0.104 0.281 3.042 4725.085 34992SGROW 0.286 3.175 -6.488 -0.537 -0.213 -0.101 0.015 0.113 0.260 0.544 0.874 2.659 385.485 35193CASHFLOW -0.524 41.705 -5428.000 -2.610 -0.128 0.002 0.0700.146 0.253 0.395 0.508 0.870 771.842 31516INVEST 1.592 54.804 -136.960 -0.276 -0.017 0.000 0.016 0.067 0.171 0.403 0.914 6.406 5500.846 35272PAYER 0.481 0.500 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 35473LEV 0.212 0.195 0.000 0.000 0.000 0.000 0.044 0.183 0.326 0.456 0.541 0.794 3.426 35473∆LEV 0.002 0.103 -3.126 -0.273 -0.119 -0.073 -0.028 0.000 0.023 0.087 0.146 0.334 1.953 35425CASH 0.217 0.231 -0.030 0.001 0.006 0.012 0.039 0.129 0.319 0.587 0.732 0.916 1.000 35473∆CASH -0.007 0.099 -0.978 -0.337 -0.166 -0.102 -0.034 -0.001 0.025 0.079 0.131 0.294 0.968 35425ACCRUALS -0.171 16.248 -1152.584 -1.425 -0.328 -0.184 -0.085 -0.035 0.004 0.052 0.097 0.270 1879.000 31516C-SCORE 0.529 11.582 -0.120 0.000 0.000 0.000 0.000 0.016 0.138 0.399 0.856 5.204 1088.263 35473REV -0.034 1.618 -159.136 -0.794 -0.205 -0.109 -0.038 -0.004 0.010 0.039 0.077 0.357 139.791 35473LTG 0.146 0.138 -0.927 0.000 0.000 0.000 0.070 0.130 0.200 0.300 0.350 0.700 1.000 35473∆FOLLOW 0.217 0.787 -0.933 -0.750 -0.500 -0.429 -0.167 0.000 0.333 1.000 1.500 3.167 16.000 28074 LOGSALE is a natural log of contemporaneous sales. AGE is the year-to-date duration in months since CRSP initiated coverage of the firm. VOL is the year-to-date market-adjusted stock return volatility. RET is the year-to-date market-adjusted, buy-hold stock return. ROE is the contemporaneous return on equity. LOSS equals one if contemporaneous net income is negative, zero otherwise. NOATO is contemporaneous net operating asset turnover. ΔNOATO is a change in net operating asset turnover. OPM is a contemporaneous operating profit margin. ΔOPM is a change in operating profit margin. SGROW is a contemporaneous sales

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growth. CASHFLOW is a contemporaneous cash flow to sales (cash flow equals the sum of GAAP cash flow from operations and research and development expenditures). INVEST is the contemporaneous investment to sales (investment equals the sum of research and development expenditures and capital expenditures). PAYER equals one if the firm paid dividends in the current year, zero otherwise. LEV is a Contemporaneous ratio of financial obligations to total assets. ∆LEV is a change in LEV. CASH is a contemporaneous ratio of financial assets to total assets. ∆CASH is a change in CASH. ACCRUALS is a contemporaneous ratio of working-capital accruals to sales. C-SCORE is a contemporaneous C-score per Penman and Zhang [2000]. REV is Ratio of most recent forecast revision to equity book value per share. LTG is the contemporaneous analysts' consensus long-term growth rate in earnings per share. ∆FOLLOW is a change in analysts' following during the last three months before the end of year zero.

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Table A2 – Results Obtained from Estimating Ordered Logistic Regressions of Ranks on Firm Attributes (Estimation Sample)

Est. p-value Est. p-value Est. p-value Est. p-value Est. p-value Est. p-valueLOGSALE 0.159 0.000 0.151 0.000 -0.037 0.000 0.011 0.167 -0.067 0.000 -0.126 0.000AGE 0.000 0.427 0.000 0.005 0.000 0.145 0.000 0.076 0.000 0.032 0.000 0.000VOL -2.991 0.000 -2.616 0.000 0.208 0.295 0.281 0.164 2.315 0.000 2.077 0.000RET 0.607 0.000 0.600 0.000 -0.053 0.005 0.092 0.000 -0.472 0.000 -0.566 0.000ROE 0.031 0.007 0.004 0.462 -0.002 0.702 -0.039 0.000 -0.0050.365 0.003 0.605LOSS -1.132 0.000 -0.711 0.000 -0.110 0.001 1.213 0.000 0.879 0.000 0.115 0.001NOATO 0.003 0.004 0.001 0.016 0.000 0.363 -0.001 0.004 -0.001 0.033 0.000 0.538∆NOATO -0.001 0.035 -0.001 0.102 0.000 0.719 0.001 0.021 0.000 0.084 0.000 0.895OPM -0.001 0.302 -0.002 0.111 0.000 0.863 -0.010 0.018 0.0020.080 0.002 0.137∆OPM 0.000 0.698 0.000 0.953 0.000 0.763 0.009 0.007 0.001 0.051 0.000 0.456SGROW 0.000 0.971 0.002 0.847 -0.009 0.316 -0.007 0.508 -0.022 0.043 0.011 0.312CASHFLOW 0.000 0.895 0.001 0.408 0.001 0.572 0.009 0.189 -0.002 0.202 -0.002 0.277INVEST 0.000 0.863 0.000 0.535 0.000 0.860 0.007 0.081 0.0000.917 -0.001 0.383PAYER 0.384 0.000 0.416 0.000 0.253 0.000 -0.003 0.920 -0.253 0.000 -0.464 0.000LEV -0.169 0.023 -0.118 0.099 0.377 0.000 -0.168 0.018 0.0380.600 0.126 0.083∆LEV 0.511 0.000 0.536 0.000 0.277 0.033 0.520 0.000 -0.597 0.000 -0.930 0.000CASH 0.092 0.235 0.800 0.000 -0.563 0.000 1.148 0.000 -0.3100.000 -0.778 0.000∆CASH -0.398 0.004 -0.676 0.000 0.199 0.135 -0.321 0.016 0.523 0.000 0.406 0.003ACCRUALS 0.000 0.939 0.003 0.437 0.001 0.792 0.016 0.117 -0.002 0.572 -0.005 0.209C-SCORE 0.000 0.853 0.001 0.311 0.000 0.622 0.002 0.193 -0.001 0.527 -0.001 0.264REV -0.007 0.381 -0.009 0.233 0.004 0.615 -0.003 0.673 0.0220.010 0.004 0.645LTG 0.836 0.000 0.030 0.763 -1.335 0.000 0.290 0.003 -2.300 0.000 0.941 0.000∆FOLLOW 0.021 0.208 0.002 0.893 -0.068 0.000 -0.008 0.612 -0.081 0.000 0.056 0.000INT_6 -1.187 0.000 -9.675 0.000 -3.046 0.000 -2.144 0.000 -4.486 0.000 0.024 0.705INT_5 -0.397 0.000 -1.617 0.000 -1.777 0.000 -1.593 0.000 -0.534 0.000 0.402 0.000INT_4 0.010 0.884 -0.662 0.000 -0.090 0.145 -0.533 0.000 0.115 0.069 0.820 0.000INT_3 0.399 0.000 0.144 0.022 1.336 0.000 0.698 0.000 0.562 0.000 1.168 0.000INT_2 0.983 0.000 1.658 0.000 2.590 0.000 1.371 0.000 2.451 0.000 1.496 0.000

Pseudo R-squared 0.073 0.053 0.01 0.033 0.04 0.031

AEGV_LRIV_N RIV_S RIV_L AEGV_N AEGV_S

This subset of the estimation sample consists of 23,799 firm-years with available data. LOGSALE is a natural log of contemporaneous sales. AGE is the year-to-date duration in months since CRSP initiated coverage of the firm. VOL is the year-to-date market-adjusted stock return volatility. RET is the year-to-date market-adjusted, buy-hold stock return. ROE is the contemporaneous return on equity. LOSS equals one if contemporaneous net income is negative, zero otherwise. NOATO is contemporaneous net operating asset turnover. ΔNOATO is a change in net operating asset turnover. OPM is a contemporaneous operating profit margin. ΔOPM is a change in operating profit margin. SGROW is a contemporaneous sales growth. CASHFLOW is a contemporaneous cash flow to sales (cash flow equals the sum of GAAP cash flow from operations and research and development expenditures). INVEST is the contemporaneous investment to sales (investment equals the sum of research and development expenditures and capital expenditures). PAYER equals one if the firm paid dividends in the current year, zero otherwise. LEV is a Contemporaneous ratio of financial obligations to total assets. ∆LEV is a change in LEV. CASH is a contemporaneous ratio of financial assets to total assets. ∆CASH is a change in CASH. ACCRUALS is a contemporaneous ratio of working-capital accruals to sales. C-SCORE is a contemporaneous C-score per Penman and Zhang [2000]. REV is Ratio of most recent forecast revision to equity book value per share. LTG is the contemporaneous analysts' consensus long-term growth rate in earnings per share. ∆FOLLOW is a change in

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analysts' following during the last three months before the end of year zero. INT_6, INT_5, INT_4, INT_3, and INT_2 are intercepts corresponding to the outcomes ranked sixth, fifth, fourth, third, and second, respectively.