Zhang 2000

7/30/2019 Zhang 2000

1/25

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.

PII: S 0 1 6 5 - 4 1 0 1 ( 0 0 ) 0 0 0 1 6 - 1

7/30/2019 Zhang 2000

2/25

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

126 X. Zhang/Journal of Accounting and Economics 29 (2000) 125}149

7/30/2019 Zhang 2000

3/25

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

X. Zhang/Journal of Accounting and Economics 29 (2000) 125}149 127

7/30/2019 Zhang 2000

4/25

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)


7/30/2019 Zhang 2000

5/25

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


7/30/2019 Zhang 2000

6/25

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.


7/30/2019 Zhang 2000

7/25

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


7/30/2019 Zhang 2000

8/25

7/30/2019 Zhang 2000

9/25

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


7/30/2019 Zhang 2000

10/25

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.


7/30/2019 Zhang 2000

11/25

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.


7/30/2019 Zhang 2000

12/25

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.


7/30/2019 Zhang 2000

13/25

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


7/30/2019 Zhang 2000

15/25

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.


7/30/2019 Zhang 2000

16/25

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.


7/30/2019 Zhang 2000

17/25

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).


7/30/2019 Zhang 2000

21/25

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,


7/30/2019 Zhang 2000

22/25

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.


7/30/2019 Zhang 2000

23/25

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.


7/30/2019 Zhang 2000

24/25

7/30/2019 Zhang 2000

25/25

O' Hanlon, J., 1994. Earnings based valuation models with conservative accounting. Unpublished

working paper. Lancaster University.

Ohlson, J., Zhang, X., 1998. Accrual accounting and equity valuation. Journal of Accounting

Research 36, 85}111.

Ohlson, J., Zhang, X., 1999. On the theory of forecast horizon in equity valuation. Journal of

Accounting Research 37, 437}449.

Penman, S., 1998a. A synthesis of equity valuation techniques and the terminal value calculation for

the dividend discount model. Review of Accounting Studies 2, 303}323.

Penman, S., 1998b. Combining earnings and book value in equity valuation. Contemporary

Accounting Research 15, 291}324.

Penman, S., Sougiannis, T., 1998. A comparison of dividend, cash #ow, and earnings approaches to

equity valuation. Contemporary Accounting Research 15, 343}383.


Zhang 2000

Documents

Transcript of Zhang 2000