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I wish to thank Ray Ball (the editor), Trevor Harris, Amir Ziv, an anonymous referee,
and especially James Ohlson for helpful comments and suggestions. I also thank seminar
participants at Baruch College, UC Berkeley, UCLA, Carnegie Mellon University, University of
Chicago, Columbia University, Cornell University, Duke University, Ohio State University, and
Yale University.
* Tel.: (510)-642-4789; fax: (510)-642-4700.
E-mail address: [email protected] (X. Zhang).
Journal of Accounting and Economics 29 (2000) 125}149
Conservative accounting and equityvaluation
Xiao-Jun Zhang*
Haas School of Business, University of California at Berkeley, Berkeley, CA 94720-1900, USA
Received 8 March 1999; received in revised form 12 June 2000
Abstract
This paper examines how conservative accounting a!ects the relation between
accounting data and "rm value. The analysis shows that conservative accountingcan be characterized equivalently in terms of book value, earnings, or book rate of
return. Furthermore, capitalized earnings generally provide a less biased estimate
of equity value than book value does. In addition, "rm growth a!ects the way
earnings and book value are combined in valuation. A weighted average of book value
and capitalized earnings, with the weight on earnings being an increasing and convex
function of growth, yields an asymptotically unbiased estimate of equity value. When
growth is positive, the weight on book value is negative. 2000 Elsevier Science B.V.
All rights reserved.
JEL classixcation: M41; G12
Keywords: Capital markets; Conservative accounting; Equity valuation; Book rate of
return; Residual income
0165-4101/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved.
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Because of accounting conservatism, intangibles such as unrealized bene"ts of R&D or future
growth opportunities are not recognized as assets on the balance sheet. Recognized assets may also
be assigned relatively low values based on rules such as lower of cost or market. In addition,
accountants follow guidelines such as &choosing a solution that is least likely to overstate assets and
income' when implementing accounting standards. For more discussions regarding the concept of
conservatism, see Statement of Financial Accounting Concepts No. 2.
P/B ratio is de"ned as market value of common equity (per share) divided by book value of
common equity (per share). P/E ratio is de"ned as price (adjusted for net dividends) divided by
earnings. Capitalized earnings is de"ned as R/(R!1)xR!d
R, where R denotes the discount factor,
xR
denotes earnings, and dR
denotes net dividends. P/Cap.E is de"ned as price divided by capitalized
earnings. Under unbiased accounting, P/B and P/Cap.E on average equal one.
Asymptotic analysis avoids the e!ect of idiosyncratic shocks and focuses on the general or on
average results. For example, the accounting policy of expensing R&D is generally considered to be
conservative, given that R&D on average generates future bene"ts. However, in cases when
particular R&D activity does not provide any future bene"t, expensing the cost is actually unbiased.
Asymptotic analysis can avoid such problems by focusing on ex ante long-term expected relations.
1. Introduction
This paper seeks to understand the general impact of conservative accounting
on the relation between accounting data and "rm value. The convention ofconservatism in#uences the establishment and implementation of accounting
standards, causing the average price-to-book (P/B) ratio to exceed one. Ab-
sence of on average equivalence between book value and market value leads to
questions regarding how accounting data relate to "rm value.
This study addresses two issues. First, I examine how conservatism in#uences
not only the average price-to-book (P/B) ratio but also the average price-to-
earnings (P/E) ratio. In this context, I also compare deviations of the average P/B
ratio and the average price-to-capitalized earnings (P/Cap.E) ratio from theirbenchmark of one. Second, I analyze how linear combinations of book value and
capitalized earnings yield unbiased estimates of"rm value. The emphasis is on the
properties of the weight coe$cients when the two weights sum to one.
A "rm is modeled in a multi-period setting with the going concern assumption
maintained throughout. Like Feltham and Ohlson (1995), I examine the asymp-
totic relations among book value, earnings and "rm value to capture the general
impact of conservatism. The study builds on basic accounting and economic
principles, including the clean surplus relation, the assumption that the presentvalue of expected future cash #ows determines market value, and a measure of
asymptotic growth. I measure "rm growth based on the changes in "rm value,
and examine the e!ects of long-term growth on the properties of accounting
data and their relation with "rm value.
The analysis provides a unifying framework regarding the in#uence of ac-
counting conservatism on earnings, book value and book rate of return. I
show that under rather general conditions the following characterizations of
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The term &terminal value' is used to refer to either the value of equity or the value of goodwill at
the end of the forecast horizon. For detailed discussions, see Copeland et al. (1994), Penman (1998a),
and Ohlson and Zhang (1999).
conservative accounting are equivalent: (i) book value is on average less than
market value; (ii) earnings are on average less than (equal to) economic income
when long-term growth exceeds (equals) zero; and (iii) book rate of return is on
average greater than the discount rate. Feltham and Ohlson (1995) derive (ii)and (iii) based on the assumption of linear information dynamics. This paper
generalizes their insights by showing that, under rather general conditions,
properties (ii) and (iii) hold as long as book value is on average less than market
value. In addition, these properties not only are implied by, but also imply,
conservative accounting.
Based on the analysis of how conservatism a!ects accounting data, I show
that the average P/E ratio equals (exceeds) the capitalization factor when growth
equals (exceeds) zero. By comparing the asymptotic deviations of P/B andP/Cap.E from their benchmark of one, I further show that capitalized earnings
generally provide a less biased estimate of"rm value than book value does.
A key result demonstrates that, under conservative accounting, "rm growth
plays an important role in combining earnings and book value in valuation.
A weighted average of book value and capitalized earnings, with the weights
being independent of the extent of conservatism in accounting, yields an asymp-
totically unbiased estimate of"rm value. The weight on capitalized earnings is
an increasing and convex function of growth. When long-term growth is posit-ive, the weight on capitalized earnings is greater than one and the weight on
book value is negative.
The above result provides a benchmark for research that relates earnings and
book value to "rm value under conservative accounting. In an empirical study,
Penman (1998b) examines how earnings and book value can be integrated in
equity valuation. He expresses market value as a weighted average of book value
and capitalized earnings, and analyzes the properties of the weight coe$cients.
He "nds that, in "ve out of the 12 sample groups, the median coe$cients on
book value are negative. These negative coe$cients on book value are viewed as
being di$cult to interpret (Burgstahler, 1998). My analysis demonstrates that it
is possible for negative coe$cients on book value to be attributed to conserva-
tism in accounting. This result also bears on market-based accounting research
that involves regressing market value on book value and earnings. Several
papers, including Amir and Lev (1996) and Francis and Schipper (1999), docu-
ment that in certain years the regression coe$cients on book value are negative.
My analysis helps interpret such results.
This study also relates to research on terminal value estimation, which arisesin equity valuation based on "nite forests. Frankel and Lee (1998), Penman and
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This approach serves two purposes. First, it allows the paper to incorporate Miller andModigliani's (1961) basic concept regarding debt by assuming that the "rm's borrowing (and
lending) activities, whether incremental or on average, yield zero net present value. Second, it
emphasizes that conservatism a!ects mainly the measurement of operating activities. Since"nancing
activities involve assets for which there are relatively perfect markets, one can conceptualize
that book values and market values of "nancial assets approximately coincide. In contrast,
operating assets are typically not individually traded. Accounting conventions such as conserva-
tism generally lead to divergence between the book value and the market value of operating
assets.
Sougiannis (1998) and Lee et al. (1999) estimate terminal year's goodwill by
assuming that residual earnings beyond the forecast horizon follow a random
walk. This paper shows that conservative accounting can cause residual earn-
ings to be positive on average and to grow at a rate greater than zero. In light ofthis result, assuming that residual earnings follow a random walk would under-
estimate terminal values.
The rest of the paper is organized as follows. Section 2 speci"es the basic
model. Section 3 examines how conservatism in#uences earnings, book value
and book rate of return. These properties of conservative accounting are then
used in Section 4 to demonstrate that capitalized earnings provide an asymp-
totically less biased estimate of"rm value than book value does. Section 4 also
analyzes the weighted average model and establishes a mapping between theasymptotic weights and the growth rate. Financing activities are added to the
"rm's operating activities in Section 5. Section 6 discusses some empirical
implications. Section 7 concludes the paper with a brief discussion of contribu-
tions, limitations and extensions.
2. Assumptions and concepts
I model a "rm in a multi-period setting and assume it is a going concern.
Initially I also assume that the "rm is entirely equity-"nanced, with no borrow-
ing or lending activities. This simpli"cation enables the paper to focus on
the measurement of the "rm's operating activities. I relax this assumption in
Section 5.
Given this all-equity setup, the "rm does not build up "nancial assets or
liabilities. Its operating cash #ows precisely o!set "nancial cash #ows to the
owner. I apply standard valuation concepts and assume that the present value ofexpected cash #ows determines the current market value of the "rm:
]
R#
ER[c
R>]
R#2 (1)
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Implicit in expression (1) is the assumption that the term structure of interest rates is #at. This
assumption can be relaxed by assuming that the relation holds asymptotically, and all conclusions
still hold. The analysis in this paper can also be extended to encompass risk aversion and
non-homogeneous beliefs. See Feltham and Ohlson (1999).
where R'1 speci"es the non-stochastic discount factor and cR
denotes net cash
#ows during period t.
On each date t (t"1,2,2) the "rm discloses accounting data pertaining to its
operating activities. These data are random prior to their disclosure, and theprobabilistic structure governing their stochastic behavior is exogenous. The
following notation is used to denote the accounting and market measures of"rm
activities:
oaR
book value of the "rm's net operating assets at time t;
oxR
accounting operating earnings during period t;
oeR
economic operating income during period t, de"ned as O]*0 for all *0 and lim
O
ER[&oa
R>O]
ER[O](R. (4)
Regularity condition (3) requires the market value of the "rm to be positive
and the "rm's asymptotic growth rate to be non-negative. The asymptotic
growth rate (g), which is measured by the change in "rm value, is restricted to be
less than R!1 to ensure that the equity value calculated using Eq. (1) is
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Although I assume the existence of g, it can be derived from more basic assumptions such as
Markovian evolution of transactions/events and linear accounting rules (See, for example, Ohlson
and Zhang, 1998). This growth rate depends on the "rm's retention of capital as well as its
implementation of positive net present value projects.
The growth rate is assumed to be less than R!1 to ensure convergence in the present value
calculation using Eq. (1). Strictly speaking, when g'R!1, the transversality condition used to
derive Eq. (1) would be violated and Eq. (1) would no longer hold.
"nite. Note that condition (3) allows for the possibility of bankruptcy, which
is incorporated in the expectation of future "rm values. I also assume that the
growth rate is non-negative to avoid negative (or zero) asymptotic market
values. It will become apparent that the analysis also applies to the case wheng is negative, i.e., when the "rm is assumed to be declining and eventually going
out of business.
Regularity condition (4) restricts accounting measurements. It requires that
the accounting not be so conservative that book values are on average negative.
Negative book values cause problems in interpreting ratios such as book rate of
return. Condition (4) also requires that the accounting constructs not be too
aggressive. Policies violating the second part of condition (4) have to be highly
aggressive in the sense that, regardless of the actual growth rate of the business,the accounting policy keeps the book value growing at a higher rate and
eventually drives the ratio between the expected book value and the expected
market value to in"nity.
Assumptions speci"ed in this section form the foundation of the analysis. In
the following section, I provide a unifying framework regarding how conserva-
tism in#uences book value, earnings, and book rate of return. These properties
of conservatism will then be used in Section 4 to analyze the link between
accounting data and "rm value.
3. Impact of conservatism on accounting data
Following Feltham and Ohlson (1995), conservative and unbiased accounting
are de"ned in terms of how book value di!ers, on average, from market value.
De,nition. An accounting policy is conservative if
lim
O
ER[&oa
R>O]/E
R[O](1 regardless of the date t information;
An accounting policy is unbiased if
lim
O
ER[&oa
R>O]/E
R[O]"1 regardless of the date t information.
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See, for example, Davidson et al. (1982, pp. 581}588).
The above de"nition of conservative accounting focuses on the systematic
bias in book value. Such bias is introduced by the principle of conservatism
which a!ects the setting and implementation of accounting standards. Conser-
vative accounting standards include, for example, not recognizing the bene"ts ofR&D and rapid depreciation of"xed assets. In addition, accounting rules such
as lower of cost or market require early recognition of unrealized losses, causing
earnings to re#ect &bad news' more quickly than &good news'. This aspect of
conservative accounting is emphasized in Basu (1997).
The following observation regarding unbiased accounting follows immediate-
ly from its de"nition.
Observation 1. Given unbiased accounting, i.e.,
lim
O
ER[&oa
R>O]/E
R[O]"1, (5)
then
lim
O
ER[&ox
R>O]/E
R[&oe
R>O]"1; (6)
and
lim
O
ER[&ox
R>O]/E
R[&oa
R>O\]"R!1. (7)
Eq. (5) compares the stock measure generated by accounting, &oaR>O
, with the
market stock measure, O
. By d e"nition, when accounting is unbiased,
book value on average equals market value. Eq. (6) compares the two #ow
measures. It shows that accounting earnings on average coincide with eco-
nomic income. Eq. (7) examines the relation between earnings and book
value and indicates that the ratio of the two on average equals the discount
rate R!1.
When accounting is conservative, two issues emerge. The "rst issue is how
these ratios change. Textbook examples illustrate that although book value is
less than market value under conservative accounting, earnings can be greater
than, equal to, or less than economic income depending on growth. Feltham
and Ohlson (1995) generalize these results to an uncertainty setting and also
relate earnings to book value by showing that book rate of return asymp-
totically exceeds the discount rate. Their proof, however, is based on the
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Strictly speaking, the claim is inaccurate in the sense that lim(ER[&ox
R>O/&oa
R>O\]) generally
does not equal to lim(ER[&ox R>O]/E
R[&oa
R>O\]). We can roughly check the di!erence by assuming
that&ox
R>Oand&oa
R>O\have independent log normal distributions, i.e., Log(
&ox
R>O)&N(
V,V
) and
Log(&oa
R>O)&N(
?,?
). It is easy to show that Log(ER[&ox
R>O/oa
R>O\])'E
R[Log(&ox
R>O/oa
R>O\)]
"V!
?and Log(E
R[&ox
R>O]/E
R[oa
R>O\])"
V!
?!(
?!
V)/2. Therefore, assuming
?*
V, lim(E
R[&ox
R>O]/E
R[&oa
R>O\])'R!1 implies lim(E
R[&ox
R>O/&oa
R>O\])'R!1.
suggests that when measurement error remains unchanged over time (due to
reasons such as consistent application of accounting standards and repeated
"rm operation), earnings will on average equal the true economic income
regardless of whether the accounting is unbiased or not.Statement (9) also shows that when growth is positive, the above canceling-
error argument no longer holds. Even though to some extent R
and R\
still
cancel in the earnings calculation, the reduction in current earnings (R) becomes
such a dominating factor that accounting earnings are on average less than
economic income. Implications of this partial canceling-error phenomenon are
explored in Section 4.
Because conservative accounting in general reduces book value, we can
conclude that book rate of return exceeds R!1 when there is no growth.However, when growth is positive, the conclusion is not obvious. In a book rate
of return evaluation, conservatism decreases not only the denominator (begin-
ning book value) but also the numerator (earnings). Nonetheless, we would
expect that the numerator e!ect will be dominated by the denominator e!ect
because reduction in book value is the cumulative impact of prior conservative
practices. Statement (10) formally demonstrates that book rate of return exceeds
R!1 even when growth is positive.
Proposition 1 derives the general impact of conservatism on earnings, bookvalue and book rate of return. It shows that characterizations of conservative
accounting in terms of book value, earnings, and book rate of return are
equivalent in that any one of them implies the other two. Therefore, under
rather general conditions, (9) and (10) not only are properties of conservative
accounting, but also imply that the book value is on average less than
the market value. These characteristics of conservatism are used in the
following section to examine the relation between accounting data and
"rm value.
4. Conservative accounting and valuation
Before studying the asymptotic relations among earnings, book value and "rm
value under conservative accounting, I "rst establish a benchmark regarding
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The relation between (ER[&oa
R>O], E
R[&ox
R>O], E
R[cJ
R>O]) and E
R[O] is emphasized because
simple methods such as the earnings-multiple approach are often used by investors as bench-marks,
or starting points, to achieve precise valuations. Even in more complicated schemes, valuation using
&bottom line' numbers still plays an important role. As discussed in Copeland (1994), Penman
(1998a), and Ohlson and Zhang (1999), equity valuation based on "nite forecasts involves estimating
terminal value, which is often a critical part of the analysis. By focusing on the shareholders' equity
statement, I include the most widely used terminal value estimation methods, such as the P/B, P/E
and free cash #ow method.
the link between &bottom line' numbers and "rm value under unbiased ac-
counting.
Observation 2. Given unbiased accounting, i.e.,
lim
O
ER[&oa
R>O]/E
R[O]"1,
then
lim
O
ER[&ox
R>O!c
R>O]/E
R[O]"1, where ,R/(R!1);
and
lim
O
ER[w(&ox
R>O!c
R>O)#(1!w)&oa
R>O]/E
R[O]"1 for any w.
Observation 2 states that when accounting is unbiased, both the balance sheet
approach (which uses book value as the estimator of market value) and the
income statement approach (which uses capitalized earnings as the estimator of
market value) are unbiased. These two approaches become perfect substitutes
asymptotically, and any convex combination of them also yields an unbiased
estimate of"rm value.
4.1. Balance sheet approach vs. income statement approach under conservative
accounting
When accounting is conservative, by de"nition, the balance sheet approach is
subject to asymptotic downward bias. However, the properties of the income
statement approach are more complicated because of growth. The following
corollary states the result.
Corollary 1. Assume regularity conditions (3) and (4) hold. Given conservative
accounting, we have
lim
O
ER[&ox
R>O!c
R>O]/E
R[O]"1 when g"0,
(1 when g'0.
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Corollary 1 states that, with zero growth, capitalized earnings asymptotically
equal "rm value. This result follows immediately from statement (9) of Proposi-
tion 1: zero-growth implies that "rm's earnings on average equal its economic
income; therefore, capitalized earnings provide a good indicator of"rm value.Corollary 1 also shows that the income statement approach is no longer
unbiased when growth is positive. In fact, under such circumstances, capitalized
earnings are asymptotically less than "rm value. The reason is intuitive: with
positive growth, errors due to past and present conservatism in accounting only
partially cancel each other.
Even though the strict canceling-error argument no longer holds when
growth is positive, the fact that measurement errors are still partially o!setting
in the income statement suggests that capitalized earnings on average providea less biased estimate of"rm value than book value does. Formally, one would
hypothesize that
"ER[O!(&ox
R>O!c
R>O)]"/"E
R[O!&oa
R>O]"(1 as PR. (12)
Recall that when accounting is conservative, both terms inside the absolute signs
are positive as PR; therefore, Inequality (12) is equivalent to
ER[&oa
R>O]/E
R[&ox
R>O!c
R>O](1 as PR, (13)
which states that capitalized earnings are on average greater than book value.
Lemma 1. Assume regularity conditions (3) and(4). Given conservative accounting,
Inequality (13) holds. Hence,
limO
"ER[O!(
&
ox R>O!cR>O )]"/"ER[O!
&
oa R>O]"(1.
Lemma 1 allows us to compare the balance sheet approach with the income
statement approach under conservative accounting. It shows that capitalized
earnings provide an asymptotically less biased estimate of market value than
book value does.
4.2. Convex combination of book value and capitalized earnings
Because capitalized earnings asymptotically di!er from book value when
accounting is conservative, one can express "rm value as a convex combination
of these two accounting measures. The result of Lemma 1 suggests that, in such
a scheme, more weight would be placed on capitalized earnings due to conserva-
tism in accounting.
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Proposition 2. Assume regularity conditions (3) and (4) hold. Suppose there exists
a weight (w) such that
limO
ER[(1!w)&
oa R>O#w(&
ox R>O!cR>O)]/ER[O]"1. (14)
Then conservative accounting implies
w"(R!1)(1#g)
R!(1#g)where w'1 ("1) when g'0 ("0).
In contrast, when accounting is unbiased, (14) holds for all w.
Proposition 2 shows that, under conservative accounting, "rm growth
plays an important role in combining earnings and book value in valuation.
In the weighted average model stated in (14), the weight on capitalized earnings
(w) is an increasing and convex function of the growth rate (g). When g is
positive, the weight on capitalized earnings is not only positive, but also greater
than one.
To understand the close relation between the weight coe$cient and the
growth rate, note that, ER[(1!w)
&
oa R>O#w(&
ox R>O!cR>O)],ER[
&
oa R>O#&ox?R>O
], where residual earnings (ox?R) is de"ned as ox
R!(R!1)oa
R\, and
,w/(R!1). One can therefore transform the weighted average formula of (14)
into the following format:
lim
O
ER[&oa
R>O#&ox?
R>O]/E
R[O]"1. (15)
The underpinning of Eq. (15) is the following well-known formula which ex-
presses market value in terms of book value plus the present value of expectedfuture residual earnings:
O]/RO. (16)
The next corollary shows how conservative accounting on average a!ects the
asymptotic dynamics of residual earnings.
Corollary 2. Assume regularity conditions (3) and (4) hold.
;nder conservative accounting, lim
O
ER[&ox?
R>O>]
ER[&ox?
R>O]"1#g. (17)
;nder unbiased accounting, lim
O
ER[&ox ?
R>O>]
ER[&ox ?
R>O](1#g when exists.
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Penman (1998a) hypothesizes a similar result regarding residual earnings dynamics. O'Hanlon
(1994) also shows that asymptotic residual earnings are positive with certain type of conservative
accounting policy. Unlike O'Hanlon (1994) which is based on speci"c assumptions regarding
accounting policies and residual earnings dynamics, this study relies on more basic accounting and
economics principles and emphasizes the generality of the result.
Corollary 2 shows that conservative accounting causes the residual earnings
to be on average positive and to grow at a rate greater than zero. In contrast,
when accounting is unbiased, residual earnings would either grow/decay at an
asymptotic rate that is less than g, or they would satisfy the following condition:
'0 such that ER[&ox?
R>O]"0 for all *,
in which case limO
ER[&ox?
R>O>]/E
R[&ox?
R>O] is not de"ned.
Substituting Eq. (17) into (16), it is easy to show that Eq. (15) holds. From this
analysis, one sees clearly how the weight coe$cient (w) links to growth (g)
through the capitalization factor ,w/(R!1)"(1#g)/(R!1!g).
Another interesting aspect about the weighted average model is that the
weight on book value, 1!w"!Rg(R!(1#g))\
, is zero (negative) whengrowth is zero (positive). Such a non-positive coe$cient may seem surprising,
but a close examination reveals the logic behind the result. To interpret the
non-positive coe$cient on book value, one can transform the weighted average
formula of (14) into the following equivalent expression:
lim
O
ER[&ox
R>O!(1#g)\g&oa
R>O]/E
R[O]"1. (18)
Expression (18) illustrates the weighted average formula from a &buying earn-
ings' perspective. The underpinning of this expression can be obtained by
substituting (2) into (1):
O]. As shown in
the proof of Proposition 2, wV and w? can be explicitly solved as functions ofs and
N@:
wV"
(N@!s)(R!1)(1#g)
(N@!1)(R!1!g)
and w?"s!w
V. (20)
It is easy to see from (20) that
wV
'0, w?
(0 when s(NC
;
wV*0, w
?*0 when
NC)s)
N@;
wV(0, w
?'0 when s'
N@,
where NC, lim
OER[O]/E
R[&oa
R>O!c
R>O] is the asymptotic P/Cap.E ratio.
Hence if one places no restriction on the sum of the two weights, the coe$cient
on book value can be positive.
Although s"1 is only one of many possible alternatives, comparing the result
of Proposition 2 with expressions in (20) reveals a unique characteristic of thestrict weighted average model.
Corollary 3. The weights wV
and w?
are independent of N@
if(and only if) s"1.
The asymptotic price-to-book ratio (N@
) is a function of growth, the discount
rate and the degree of conservatism in accounting policies. Corollary 3 shows
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that if (and only if ) s"1, the relative weights (wV
, w?
) are functions only of
growth and the discount rate, and they are immune to the degree of conserva-
tism.
The immunity of weights to the degree of conservatism is another importantaspect of the strict weighted average model. When growth is positive, a more
conservative accounting policy yields a lower book value and lower earnings.
However, because measurement errors due to conservatism cancel to some
extent in the income statement, book value and capitalized earnings are a!ected
di!erently. When the weights are chosen in such a fashion to incorporate growth
&correctly', the two e!ects cancel in the strict weighted average model.
5. E4ects of leverage
To this point the paper has assumed that the "rm's operation is entirely equity
"nanced. In this section I assume that the "rm also engages in debt "nancing
activities, in which case operating cash #ows may di!er from cash #ows to/from
the owner. The following notations are used:
bvR
book value of equity at time t;
faR
net "nancial assets at time t;PR
market value of equity at time t;
xR
accounting earnings during period t;
iR
net interest income during period t;
dR
net dividend during period t.
I assume that the "rm's book value equals the total of"nancial and operating
assets at all times:
bvR"fa
R#oa
R.
Financing activities are assumed to involve assets that have relatively perfect
markets such that the following net interest relation is satis"ed:
iR"(R!1)fa
R\. (21)
Accounting for "nancial assets is assumed to satisfy the following "nancial
assets relation:
faR"fa
R\#i
R!(d
R!c
R). (22)
Unlike the measurement of "nancing activities, assumed to be unbiased,
accounting for operating activities is assumed to be conservative. Book value of
equity and earnings are simple combinations of two fundamentally di!erent
kinds of measures. How the resulting accounting data relate to market value of
equity is the subject of this section.
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Observation 3. Dexne ,(1#g)(R!(1#g))\ and w,(R!1). If
ER[&ox
R>O>]
ER[&
oa R>O]
"oxR/oa
R\
and
ER[&oa
R>O>]
ER[&oa
R>O]"oa
R/oa
R\for all *0, (23)
then
w( oxR!c
R)#(1!w)oa
R#fa
R"P
R, (24)
and
w(xR!d
R)#(1!w)bv
R"P
R. (25)
Expression (24) separates "nancing activities and operating activities in valu-
ation. As discussed in Section 4, valuing operating assets involves extrapolation
of residual operating earnings based on the growth rate of operating assets:
w(oxR!c
R)#(1!w)oa
Requals "rm value when the "rm's operating activities
are on a steady-state trajectory. Contrary to this key role played by the growth
rate of operating assets, growth of"nancial assets does not matter in valuation.
Eq. (24) holds regardless of whether the "rm's "nancing activities are in a steady
state or not. Because of assumptions (21) and (22), all value-relevant information
regarding "nancial assets is summarized in the balance sheet data, faR.
To understand the aggregation that takes place in (25), observe that assump-
tion (21) implies that "nancial assets generate zero residual earnings. This result
implies that residual earnings are entirely attributable to operating activities, i.e.,
x?R"ox?
R. Hence, the two sets of accounting measures can be aggregated despite
the fact that they are quite di!erent in terms of bias.
Condition (23) holds asymptotically, which leads to the following proposition.
Proposition 3. Assume that the regularity conditions (3) and (4) hold. Dexne
leverage as RO,!E
R[&fa
R>O]/E
R[&bv
R>O]. Assume further that leverage con-
verges to R
as PR. Given conservative accounting, ifR'0 then
lim
O
ER[PI
R>O]/E
R[&bv
R>O]' lim
O
ER[O]/E
R[&oa
R>O]'1;
limO
ER[PIR>O#d
IR>O]/ER[xR>O]
' lim
OER[O#c
R>O]/E
R[&ox
R>O]'R/(R!1) when g'0,
" limO
ER[O#c
R>O]/E
R[&ox
R>O]"R/(R!1) when g"0,
lim
O
ER[x
R>O]/E
R[&bv
R>O\]' lim
O
ER[&ox
R>O]/E
R[&oa
R>O\]'R!1.
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Francis and Schipper (1999) run cross-sectional regressions of market value on earnings and
book value. Although they do not place any restriction on the sum of the coe$cients and they allow
for an intercept, negative coe$cients on book values appear in several regression results. Amir and
Lev (1996) document signi"cant negative coe$cients on book value after controlling for growth.
And
lim
O
ER[(1!w)&bv
R>O#w(x
R>O!dI
R>O)/E
R[PI
R>O]"1,
where
w"(R!1)(1#g)
R!(1#g).
Note that PR
and bvR
denote the market and book value of equity, whereas
O>]!E
R[c
R>O>] imply
lim
O
ER[O>]
ER[O]"1#g;
lim
O
ER[&oe
R>O>]
ER[O>]"
R!1
1#g;
lim
O
ER[c
R>O>]
ER[O>]"
R!1!g
1#g,
all of which are "nite. We now get
@N"
@N#0
1
1#g# limO
ER[&ox
R>O>]
ER[&oe
R>O>]!
@NR!1
1#g !(1!
@N)
R!1!g
1#g .Hence
lim
O
ER[&ox
R>O>]
ER[&oe
R>O>]"1!
g
R!1(1!
@N).
De"ne
VC, lim
O
ER[&ox
R>O]
ER[&oe
R>O]
.
Thus
VC"1!
g
R!1(1!
@N). (A.1)
Since @N(1, we get
VC
(1 when g'0,
"1 when g"0.
Step A.2.: (9) N (8). Proof is by contradiction. Supposing
@N,lim
O(E
R[&oa
R>O]/E
R[O])*1, we can repeat the proof in Step A.1 and
get
VC"1!
g
R!1(1!
@N).
144 X. Zhang/Journal of Accounting and Economics 29 (2000) 125}149
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Therefore,
(a) If@N"1, then
VC"1 regardless of g;
(b) If@N'1, then
VC'1 when g'0,
"1 when g"0.
Both cases contradict (9). Therefore, we have
@N,
ER[&oa
R>O]
ER[O](1.
Based on results of Steps A.1 and A.2 we conclude that (8) 0 (9).
Part B: (8) 0 (10)
Step B.1.: (8)N (10).
We know from (A.1) that
VC"1!
g
R!1(1!
@N),
which implies (and is implied by)
VC!
@N"1!
g
R!1(1!@N).Since (1) and (3) assume 1#g(R, and (8) requires
@N(1, we conclude that
VC
!@N
'0, i.e., lim
O
ER[&ox
R>O]
ER[&oe R>O]' lim
O
ER[&oa
R>O\]
ER[O\]
.
Multiplying both sides by
lim
O
ER[&oe
R>O]
ER[O\]
and lim
O
ER[O\]
ER[&oa
R>O\]
,
we get
lim
O
ER[&ox R>O]ER[&oa
R>O\]' lim
O
ER[&oe R>O]ER[O\]"R!1.
Step B.2.: (10)N (8). Proof is by contradiction. Supposing
@N, lim
O
ER[&oa
R>O]
ER[O]*1,
X. Zhang/Journal of Accounting and Economics 29 (2000) 125}149 145
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we can then repeat the proof in Step B.1 and get
VC!
@N"
1!
g
R!1
(1!
@N).
If@N*1, then
VC!
@N)0, which implies
lim
O
ER[&ox
R>O]
ER[&oa
R>O\]) lim
O
ER[&oe
R>O]
ER[&oa
R>O\]"R!1.
This contradicts (10). Therefore, (10)N (8).
Based on results of Steps B.1 and B.2, we conclude that (8) 0 (10).
Parts A and B complete the proof of Proposition 1.
Proof of Corollary 1. Follows directly from Proposition 2.
Proof ofemma 1. De"ne ROARR>O,E
R[&ox
R>O]/E
R[&oa
R>O\]. Then
ER[&ox
R>O!c
R>O]/E
R[&oa
R>O]
"R
R!1ROA
RR>OER[&oa
R>O\]/E
R[&oa
R>O]!E
R[c
R>O]/E
R[&oa
R>O]. (A.2)
Clean surplus relation (2) implies that
ER[&oa
R>O]"E
R[&oa
R>O\]#E
R[&ox
R>O]!E
R[c
R>O]
"ER[&oa
R>O\](1#ROA
RR>O)!E
R[c
R>O]. (A.3)
Substituting (A.3) into (A.2) and simplifying the expression we get
ER[&ox
R>O!c
R>O]/E
R[&oa
R>O]!1"
ER[&oa
R>O\]
ER[&
oa R>O]
ROARR>O
R!1
!1
. (A.4)
Given conservative accounting, we know from Proposition 1
lim
O
ER[&ox
R>O]
ER[&oa
R>O\]'R!1.
It is easy to show that, for any time t realization, there exists'0 such that for
all *, the right-hand side of (A.4) is greater than 0.
Therefore, we get
lim
O
ER[&oa
R>O]
ER[&ox
R>O!c
R>O](1.
Proof of Proposition 2. Given conservative accounting, Lemma 1 implies that
lim
O
ER[&ox
R>O!c
R>O!&oa
R>O]'0.
146 X. Zhang/Journal of Accounting and Economics 29 (2000) 125}149
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Therefore, for any s3R, there exists '0 such that for all * we can
"nd w?R>O
and wVR>O
which satisfy ER[O]"w
?R>OER[&oa
R>O]#
wVR>O
(ER[&ox
R>O]!E
R[c
R>O]) and w
?R>O#w
VR>O"s.
This implies1"w
?R>OER[&oa
R>O]/E
R[O]#w
VR>O(E
R[&ox
R>O]!E
R[c
R>O])/E
R[O].
Taking limits on both sides and solving for w?
and wV
, we get
wV"
(N@!s)(R!1)(1#g)
(N@!1)(R!1!g)
,
w?"s!w
V. (A.5)
Substituting s"1 into (A.5) we get
wV"
(R!1)(1#g)
R!(1#g)and w
?"!
Rg
R!(1#g).
Proof of Corollary 2. De"ne goodwill gwRR>O,E
R[O]!E
R[&oa
R>O].
If any one of the three statements ((8)}(10)) holds, then from Proposition 2 we
know
lim
O
ER[&ox?
R>O]
ER[O!&oa
R>O]"
R!(1#g)
1#g,
i.e.,
lim
O
ER[&ox?
R>O]
gwRR>O
"R!(1#g)
1#g
. (A.6)
By de"nition, gwRR>O,E
R[&ox ?
R>O>]/R#gw
RR>O>/R. Hence
gwRR>O
gwRR>O>
"1
R1#ER[&ox?
R>O>]
gwRR>O>
.Taking limits on both sides and using result (A.6) we get
lim
O
gwRR>O
gwRR>O>
"1
1#g.
Therefore,
lim
O
ER[&ox?
R>O>]
ER[&ox?
R>O]" lim
O
ER[&ox ?
R>O>]
gwRR>O>
)
gwRR>O>
gwRR>O
)
gwRR>O
ER[&ox ?
R>O]"1#g.
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