## Optimal impairment rules

### $

### Robert F. Go¨x

a### , Alfred Wagenhofer

b, a*University of Fribourg, Bd. de Pe´rolles 90, CH-1700 Fribourg, Switzerland*
b

*University of Graz, Universitaetsstrasse 15, A-8010 Graz, Austria*

### a r t i c l e

### i n f o

*Article history:*

Received 28 August 2008 Received in revised form 16 April 2009

Accepted 20 April 2009
Available online 6 May 2009
*JEL Classification:*

G32
M41
M44
*Keywords:*
Conservatism
Impairment
Debt contracting
Asset measurement

### a b s t r a c t

We study the optimal accounting policy of a ﬁnancially constrained ﬁrm that pledges assets to raise debt capital for ﬁnancing a risky project. The accounting system provides information about the value of the collateral. Absent accounting regulation, the optimal accounting system is conditionally conservative: it recognizes an impairment loss if the asset value is below a certain threshold, but never reports unrealized gains. We describe the optimal impairment rule and the optimal precision of the accounting information, and we provide comparative static results that lead to testable predictions on the determinants of impairment rules.

&2009 Elsevier B.V. All rights reserved.

1. Introduction

Conservatism is a primary characteristic of accounting systems worldwide. It introduces a downward bias in the value of
net assets reported in ﬁnancial statements. However, the decision usefulness of biased accounting information has recently
been under scrutiny by the IASB and FASB, who argue that unbiased or neutral accounting information is more useful for
decisions-making.1 _{As a consequence, the two standard setters tend to favor fair value measurement over more}

conservative measurement approaches such as the measurement at amortized cost less impairment.

This paper contributes to a better understanding of the economic roles of conservative accounting. In a setting in which a ﬁrm needs to pledge assets in order to raise outside capital for ﬁnancing a risky investment project we examine the following question: If the ﬁrm can design and commit to use an accounting system for valuing its existing assets, would it select a neutral or a biased accounting system and how would it value the assets? In our model, a demand for accounting information arises endogenously because the ﬁrm beneﬁts from providing information about the value of the collateral to the lender. We ﬁnd that the optimal accounting system is conditionally conservative, that is, it recognizes impairment losses but no unrealized gains in the asset value. We describe the optimal impairment rule and the optimal precision of the accounting information, and we provide testable predictions for the determinants of cross-industry differences in accounting covenants and cross-country differences in impairment rules.

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journal homepage:www.elsevier.com/locate/jae

## Journal of Accounting and Economics

0165-4101/$ - see front matter&2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jacceco.2009.04.004

$

Helpful comments by Birgit Beinsen, Joel Demski, Ralf Ewert, Frank Gigler (the referee), Christian Hofmann, Wayne Landsman, Ross Watts (the editor), Stefan Wielenberg, and participants at the Annual Conference of the Accounting Section of VHB joint with IAAER and at workshops at the University of Paderborn and University of Vienna are gratefully acknowledged.

_{Corresponding author. Tel.: +43 316 380 3500; fax: +43 316 380 9565.}

*E-mail addresses:*robert.goex@unifr.ch (R.F. Go¨x),alfred.wagenhofer@uni-graz.at (A. Wagenhofer).
1

The model consists of a ﬁrm facing a risky investment opportunity that needs to approach a lender to provide capital for ﬁnancing the project. The lending market is perfectly competitive and potential lenders hold rational expectations. The expected Net Present Value of the project is positive only if the ﬁrm’s management exerts effort. Since effort is unobservable and costly, an incentive problem arises. We assume that the expected return of the project is not sufﬁcient to guarantee high effort and a positive return to lenders, so that the ﬁrm must pledge existing assets from earlier investment projects to raise the required amount of debt. If the expected value of the ﬁrm’s assets is not sufﬁciently high to ensure ﬁnancing, there arises an endogenous demand for an accounting system that reports additional information about the asset value. We assume that the ﬁrm can design the accounting system and commit to use it for ﬁnancial reporting purposes. The objective is to maximize the expected proﬁt of the ﬁrm, which is equivalent to maximizing the probability of realizing the investment project.

We examine accounting systems that report a book value of the assets. The book value is based on signals about the
asset value generated from an underlying information system. Our main result is that the optimal accounting system
adjusts the original book value of the assets only if the asset value falls below a certain threshold. This accounting policy is
consistent with the notion of conditional conservatism and impairment rules that are required by leading accounting
standards. The lender will ﬁnance the project only if the assets are*not*reported as being impaired.

The optimal accounting policy in our model stands in sharp contrast with the intuitive idea that investors would want to
report high asset values to lenders in order to obtain the required funds. The impairment rule reverses this strategy by
requiring that the ﬁrm reports only very low asset values (impairment), so that the*absence*of an impairment indicates to
the lender that the assets are sufﬁciently valuable to meet its ﬁnancing condition. Since simple reversal implies a
conditional expected asset value for no impairment, which strictly exceeds the level required for ﬁnancing the project, the
ﬁrm can adjust the threshold for impairment downwards and, thus, increase the probability to obtain ﬁnancing. This result
is robust because it neither depends on the precision of the signal nor on the cost of the accounting system.

We further examine the choice of the precision of the underlying information system and ﬁnd that the ﬁrm wants to implement an imprecise information system under a broad set of circumstances. Moreover, we introduce a costly earnings manipulation opportunity by allowing the ﬁrm to bias the signal provided by the information system. This opportunity requires a stricter impairment rule and more conservatism. We provide comparative analyses for the threshold level and for the probability of reporting an impairment for the relevant economic parameters. Our results help to explain differences in accounting covenants for debt contracting across ﬁrms and industries, and they also contribute to a better understanding of differences in the impairment rules under different accounting standards, such as IFRS and US GAAP, with respect to the triggers for impairment. In line with the arguments provided byBall et al. (2008), our study focuses on debt ﬁnancing as the main source of demand for impairment rules. However, we also extend our analysis to equity ﬁnancing and show that the optimal accounting system exhibits similar characteristics.

There has been much interest in understanding the potential beneﬁts of conservative accounting in the literature
recently.Watts (2003)surveys explanations of conservatism and names contracting, litigation, tax reasons, and political
cost as the main drivers of accounting conservatism. He argues that the economic role of conservatism in contracting is to
mitigate moral hazard by the management, for example, by providing early signals of poorly performing investment
projects and by maintaining a minimum level of assets to back debt. Similarly, Ball and Shivakumar (2005)stress the
governance role of conservatism to increase management incentives in order to limit economic losses.Zhang (2008)ﬁnds
that more conservative accounting is more likely to violate debt covenants (*ex post*view) and to lower interest rates (*ex ante*

view). The present paper contributes to the debt contracting explanation, but focuses on the *ex ante*use of accounting
information to help raising debt capital to ﬁnance an investment project.

Formally, our model is a disclosure model in which the ﬁrm commits to a disclosure strategy in an adverse selection
setting.2_{It is related to the work by}_{Guay and Verrecchia (2007)}_{, who study disclosure of a ﬁrm’s private information in the}

context of a risk averse capital market. In their model a ﬁrm obtains private information with some probability and commits to a disclosure strategy. Similar to our paper, they ﬁnd that the ﬁrm prefers to commit to disclosing unfavorable information. Unlike our results, however, the committed disclosure complements the voluntary disclosure of favorable information, so that essentially full disclosure is induced. The beneﬁt of full disclosure stems from a lower discount in the market price. In our paper, disclosure of high asset values would destroy the optimality of the accounting system, so that the proposed impairment policy emerges in equilibrium. Moreover, we also consider the precision of the accounting system as well as earnings management.

Demski et al. (2008)study asset revaluation regulation in an investment setting, where the ﬁrm sells the asset in the
market. In their model, the ﬁrm has private information about the asset value, so an adverse selection problem arises,
which interacts with the optimal choice of the level of investment.3 _{Demski et al. (2008)} _{show that depending on}

exogenous costs, revaluation policies that resemble historical cost with impairment can be optimal. Unlike our paper,

Demski et al. (2008)restrict their attention to lower-tail revaluation policies and some of their results are off-equilibrium,
for example, due to its effect on*ex ante*investment the revaluation policy is tailored in a manner that the ﬁrm never reports
an impairment loss.

2

Verrecchia (2001)surveys the disclosure literature.

Other papers examine the role of conservatism in accounting systems for investment settings with different decisions.

Gigler et al. (2009)consider debt-ﬁnanced investment in a two-period setting, in which the accounting system reports information that is useful for deciding on whether or not to abandon the project. They ﬁnd that conservatism in the accounting report decreases the efﬁciency of debt contracting because it increases the cost of falsely liquidating the asset (type I error), and this cost is larger than the gain from the reduction in the type II error.Caskey and Hughes (2008)extend the analysis ofGigler et al. (2009)by introducing stochastic abandonment and continuation values and by allowing for different post-contract decisions by the manager. They ﬁnd that impairment accounting can improve the debt contract in that it avoids inefﬁcient project selection.Li (2009)considers renegotiation of debt covenants and ﬁnds that conservative accounting is welfare-enhancing if the cost of renegotiation is low.Smith (2007)uses a setting with staged investments and abandonment to examine the conservatism of accounting systems. In his model, a ﬁrm undertakes a ﬁrst investment project and must sell it to investors. A more conservative accounting system makes the sale potentially more attractive to investors, but also increases the opportunity cost of abandoning the project. He ﬁnds that conservative accounting is preferable if the second-stage investment is more important; otherwise, aggressive accounting is preferable. In our paper we do not consider potential abandonment of the investment project.

Further papers related to our study includeLin (2006), who shows that conservative accounting in the form of higher depreciation in the ﬁrst period is beneﬁcial in that it provides information about the project type.Chen et al. (2007)study the interaction of conservative accounting and earnings management. They ﬁnd that conservative accounting reduces management’s incentives to manage earnings and that this beneﬁt can outweigh the loss in information content due to the bias.

The paper proceeds as follows: In Section 2, we set up the basic model and describe the investment project, the outside ﬁnancing needs, and the incentive problem the ﬁrm faces. Section 3 introduces the demand for accounting information and speciﬁes the information systems we study. In Section 4, we derive the main results on the characteristics of the optimal accounting system. Section 5 provides some extensions of the analysis, and Section 6 concludes.

2. The basic model

This section introduces a simple model of the investment in a risky project that is subject to moral hazard. The model is based onHolmstro¨m and Tirole (1997)andTirole (2001, 2006). While they focus on ﬁnancing issues, we use this economic setting to study the properties of an accounting system.Fig. 1depicts the sequence of events that are explained in the following.

*2.1. The investment project*

Prior to*t*_{¼}1, the ﬁrm has the opportunity to invest in a risky project that requires an investment of*I4*0, which is
common knowledge.4_{The project pays off at the end of the period (}_{t}

¼2). For simplicity, we consider only two states of
nature, success (*S*) or failure (*F*): the project is successful with probability*p*and yields a cash ﬂow of*X*40; it fails with
probability (1*p*) and yields zero payoff. Without loss of generality we assume zero discounting. The expected Net Present
Value of the project is*NPV*¼*pXI*.

The probability of success depends on the unobservable effort of the ﬁrm’s manager. There are two possible effort levels,
high (*H*) and low (*L*). If the manager exerts high effort, the probability of success is*pH*, and for low effort it is *pL*. Let

*pHpL*

### Dp4

0, that is, high effort shifts the cash ﬂow distribution to the right in the sense of ﬁrst-order stochasticdominance. If the manager chooses low effort, he incurs a private beneﬁt ofv40 (e.g., value of leisure) whereas there is no such beneﬁt when he works hard; alternatively, the manager incurs a private disutilityvfor high effort and none for low effort. We assume that the project is proﬁtable only if high effort is exerted,

pHXI40 and pLXIo0. (1)

Firm acquires assets and uses them in

normal operations.

Investment opportunity requires outside

financing.

Firm designs accounting system.

Accounting system reports

information about the value of assets pledged.

Firm invests in project if contract is agreed upon.

Payoff from project realizes.

Contractual consequences

obtain. Firm

approaches lender and proposes contract that

specifies payments and pledged assets.

Fig. 1. Sequence of events.

The manager is the ﬁrm’s current owner and, thus, initially there is no conﬂict of interest between the manager and the owner of the ﬁrm. The manager is risk neutral and protected by limited liability.

*2.2. Pledging of assets*

Besides the new investment project, the ﬁrm owns other assets that it uses for its operations. To focus on the ﬁnancing
of the new project, we ignore any potential synergies with future projects and assume that the other operations just earn
the normal rate of return, which is normalized to zero. That is, if the existing projects lose value over the period, the loss is
exactly balanced by a reinvestment of the cash they earn over the same period.5_{We assume that the ﬁrm does not own any}

cash or cash equivalents that could be used to ﬁnance the new project.6

Thus, in order to carry out the investment, the ﬁrm needs outside ﬁnancing. We study debt ﬁnancing, but brieﬂy
consider equity ﬁnancing in a later section. The ﬁrm approaches a lender to obtain debt in the required amount of
investment*I* to ﬁnance the project. Potential lenders are risk neutral and the capital market is perfectly competitive.
Therefore, in equilibrium lenders expect to earn the market rate of return, which is normalized to zero. The lending
contract speciﬁes payments*dj*at*t*¼2 from the ﬁrm to the lender in the two states*j*¼*S*,*F*. Due to the lack of other sources
of cash, the payments speciﬁed in the debt contract must be recovered by the project’s cash ﬂows. Accordingly, the ﬁrm can
pay out*dS*r*X*in case of success and*dF*¼0 in case of failure.7

In addition, the ﬁrm can pledge assets in the amount of*A*Z0 from its assets in place.8The disadvantage of pledging is

that the net value of the pledged assets to the lender is typically lower than to the ﬁrm. Main reasons are different
preferences, information asymmetries between borrower and lender, speciﬁcity of the assets for the borrower’s business, or
the existence of liquidation costs. Therefore, liquidating assets is costly and results in a deadweight loss that is borne by the
ﬁrm as the residual claimant. To capture these differences in values, we assume that the lender values the assets with a
value *V*(*A*), where *V*(*A*)o*A,* the asset value from the ﬁrm’s perspective. To simplify the analysis, we

assume that the liquidating value of the pledged assets can be expressed as a constant percentage

### g

A[0, 1] of the assetvalue*A*,9_{i.e.,}

VðAÞ ¼

### g

A (2)A low value of

### g

indicates assets with a relatively low liquidation value, such as ﬁrm-speciﬁc factory equipment and machinery;### g

is relatively high for assets such as land, buildings, and ﬁnancial instruments. The higher### g

, the lower is the welfare loss in case of an eventual liquidation of the collateral deﬁned in the debt contract.*2.3. The optimal debt contract*

If the manager exerts high effort, the lender will provide the required amount of debt (*I*) if the following participation
constraint holds:

pHdSþ ð1pHÞVðAÞ I0. (3)

The project would not be ﬁnanced if the lender assumes*ex ante*that the manager exerts low effort. In that case the
project would earn a negative NPV and the lender’s participation constraint would command higher payments under low
than under high effort.

Since the debt contract affects the manager’s incentives to exert high effort*ex post*, the contract must be designed to be
incentive compatible. Incentive compatibility requires that the expected net proﬁt of the manager for high effort is greater
or equal to the expected net proﬁt for low effort, i.e.,

pHðXdSÞ ð1pHÞApLðXdSÞ ð1pLÞAþv

which is equivalent to

dSXþA v

### D

p. (4)This condition puts an upper bound on the payments to the lender,*dS*.10

5

For simplicity, we assume that the ﬁrm cannot divert the cash ﬂows for investment to pay back its debt obligations. 6

This assumption is not restrictive. As long as outside ﬁnancing is required for realizing the new investment project, the analysis would be similar for the net debt required for ﬁnancing the project.

7_{A value of}_{dF}_{o}_{0, i.e., the lender provides additional debt in case of failure, is feasible but clearly not part of an optimal debt contract as it can always}
replicated by a contract that requires*dF*¼0.

8

The pledging of assets complements the ‘‘pledgeable income’’ (Tirole, 2001) and is available to lenders in case of default. We do not allow the lender to gain access to other assets or income outside that what was contracted upon. For example, the ﬁrm may establish a new legal entity for the investment project.

9

Tirole (2006, p. 170), uses a similar assumption. A more general functional form of*V*() would not signiﬁcantly alter the analysis.
10

The payment*dS*can also be used to express the contract in terms of a nominal interest rate*r*, which is*dS*¼(1+*r*)*I*, so that the lender’s participation
constraint isr ½ðxv=DpþAÞ VðAÞ=pH=ð1pHÞ½ðxv=DpþAÞ VðAÞ

The following program determines the optimal debt contract. The ﬁrm maximizes its expected proﬁt over*dS*and*A*

subject to the lender’s participation constraint, the investor’s incentive compatibility condition: max

ds;A pHðXdsÞ ð1pHÞA, (5)

s:t:pHdsþ ð1pHÞ

### g

AI0, (6)XdsþAv=

### D

p0. (7)At the maximum, the expected utility must be positive since otherwise the manager would not invest (and earn zero return).

If the project is sufﬁciently proﬁtable or the moral hazard problem is not very strong, the pledging of assets is not
necessary to obtain debt ﬁnancing. This solution is efﬁcient, and there is no deadweight loss from outside ﬁnancing. To see
why, assume that *A*¼0. The lender’s break-even constraint (3) implies a maximum debt of *I*r*pHdS*, and incentive

compatibility requires that

dSX v

### D

p.Combining both conditions implies that the lender will only ﬁnance the project without pledging any assets if the required debt level satisﬁes

IpH X v

### D

p

. (8)

Condition (8) is met as long as the project’s NPV is greater than the expected agency cost,*pHv*/

### Dp

. If (8) does not hold, thepledging of assets is necessary to obtain ﬁnancing.

Proposition 1. *If I*4*pH*(*X*v/

### Dp

),*the optimal debt contract is uniquely defined by*

A¼A^I_{p}pHðXv=

### D

pÞHþ ð1pHÞ

### g

, (9)

dS¼X v

### D

pþA. (10)Since both,*dS*and*A*, are greater than zero if pledging is necessary for realizing the project, the two constraints (6) and (7)
must be binding. Solving the constraints for*A*and*ds*yields the expressions in the proposition.

The maximum expected proﬁt of the ﬁrm is equal to the project’s NPV less the expected efﬁciency loss caused by an eventual liquidation of assets in the unfavorable state,

### P

_{¼}pHXI

|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ}

NPV

ð1pHÞð1

### g

ÞA|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

Efficiency loss

. (11)

The efﬁciency loss stems from the moral hazard problem of the ﬁrm that ﬁnances the project with debt. A second type of efﬁciency loss arises if projects with a strictly positive expected NPV are not realized. This problem could arise if the ﬁrm’s expected utility under the optimal contract would be negative or, if it does not own sufﬁcient assets for pledging, i.e., ifAoA.

3. The accounting system

The result in Proposition 1 shows that both*A*and*V*(*A*) at the end of the project cycle (*t*_{¼}2) are important determinants
to the debt contract.11_{The lender needs to estimate both values, and not only the liquidating value of the pledged assets}
*V*(*A*), because the incentive compatibility of the debt contract depends on both*A*and*V*(*A*). It follows that information about
the value of the assets in place can be an important element in debt contracting.

To introduce uncertainty with respect to the value of the pledged assets into the model, we assume that the value of the
assets is a random variable_{A}˜with a strictly positive probability density function*f*() for all*A*. More speciﬁcally, we deﬁne
the asset value as

˜

A¼

### m

þ˜þ### x

˜, (12)where

### m

is the expected*ex ante*asset value and˜and

_{x}

˜_{are two independent noise terms with zero expectation. We do not}

assume speciﬁc forms of distribution functions for the two noise terms except that they have continuous and strictly
positive densities,*f*(

### e

) and*f*(

### x

) over their respective support. Given this extended model structure, the analysis of Section 2,including Proposition 1, still applies if*A*is replaced by the expected value of the asset pledged as collateral in the debt
contract.

In what follows, we consider a generic class of accounting systems that report information about the value of*A*(and

*V*(*A*)) in the ﬁnancial statements. The ﬁrm designs an accounting system and commits*ex ante*to its use, that is, before it
approaches the lender. To ensure that the ﬁrm uses this accounting system, it must be part of the debt contract. Otherwise,
the lender has no means to enforce it. The setup and use of the accounting system incurs a nonnegative cost of*k*Z0. Since

this cost does not drive the main results, it can be taken to be zero for the major part of the analysis. The accounting system
includes an information system and a reporting policy. The information system generates an informative signal on the
value of the assets in place and resolves some of the uncertainty over_{A}˜. The information system provides a signal˜y2Y,
where

˜y¼

### m

þ### x

˜. (13)The expected value of the signal equals the unconditional expected asset value,E½˜y ¼

### m

; and the expected value of the asset value conditional on the signal equals the value of the observed signal, that is,E½_{A}˜_{j}_{y}_{ ¼}_{E}_{½}_{y}_{þ}

### ¼

˜ y. (14)The reporting policy determines how the asset value for a given signal is measured in the ﬁrm’s ﬁnancial statements. It
contains two elements: First, the reporting policy deﬁnes the sets of signals that are used and those that are ignored in the
measurement of the assets. We denote with*D*D*Y*the set of signals that are used and with*N*D*Y*the set of signals that are

ignored, where

D[N¼Y and D\N¼+.

Second, whenever a signal is used in the measurement of the assets, the asset value is reported truthfully with its best estimate, given the available information. That is, the ﬁrm reports the conditional expected asset value,B¼BðyÞ ¼E½A˜jy ¼

yas deﬁned in (14). If the signal is ignored, the ﬁrm continues to report the original book value*B*¼*B*0. In other words, the

accounting system is designed as a reporting ‘‘technology’’ in that it is not affected by decisions of management or the auditor but reports book values according to a predetermined rule. We consider earnings management later.

4. Results

*4.1. Optimal accounting system with sufficient assets in place*

Accounting information can only have value if the contract makes non-trivial use of the reported book value of the
assets and the signal*y*, respectively. Based on the information contained in the accounting report, the lender updates the
value of the assets as follows:

E½A˜j ¼B y ifB¼BðyÞ

aB0
E½A˜y2N_{} ifB¼B0

(

. (15)

If *B*a*B0* then the lender knows that the book value contains information, and given the assumptions about the
information system the conditional expected asset value equals the reported book value. If*B*_{¼}*B*0then the lender cannot

infer the value of the signal from the book value, but knows that *y*A*N*, so that the conditional expected value equals
E½A˜jy2N.12

The lender accepts the contract and provides the requested amount of capital*I*if the conditional expected asset value is
sufﬁciently high to ensure that the participation constraint holds, i.e.,

E_{A}˜_{j}BA^,

whereA^ is the amount of assets required as a collateral for convincing the lender to fund the project as deﬁned in (9). Indeed, the ﬁrm will pledge an amount of assets equal toA^ so that this condition is satisﬁed with equality, provided the value of assets in place is at least as great asA^. The accounting system reports more precisely the value of the assets at hand. Therefore, it could also serve to reduce the amount of asset to be pledged in the ﬁrst place. However, the next result shows that there is no beneﬁt of introducing an accounting system as long as the ﬁrm owns sufﬁcient assets for pledging.13

Proposition 2. *If*A^

### m

,*the firm prefers to install no*(

*or a completely uninformative*)

*accounting system*.

The proposition follows directly from the fact that the lender provides ﬁnancing with probability 1 if the unconditional expected value of the assets pledged,

### m

, is greater than or equal to the minimum required asset valueA^. Any informative12

If*B0*¼*B*(*y*) for some*y*A_{D}_{this case is indistinguishable from a report resulting from some}* _{y}*A

_{N}_{. However, the event}

_{y}_{¼}

_{B0}_{has zero probability in}

_{Y}_{,}so it does not affect the analysis.

13

accounting system cannot do better than that, but it potentially incurs two costs: (i) The accounting system can be costly
(if*k*40); and (ii) the probability of investing reduces to the probability of those signals that induce a book valueBA^, so
that the ﬁrm foregoes investment opportunities with positive NPV. If the cost of the information system is positive, the ﬁrm
strictly prefers to not install it.

Proposition 2 implies that accounting information has a (weakly) negative value in this setting. As long as there are sufﬁcient assets available for pledging, the ﬁrm always beneﬁts more from providing additional assets as collateral than from installing an accounting system. A beneﬁt from an accounting system, therefore, requires that the ﬁrm does not own sufﬁcient assets for pledging, i.e.,A^4

### m

.*4.2. Optimal accounting system with insufficient assets in place*

If the expected amount of assets available for pledging is less than what would be required by the lender, the lender does not provide ﬁnancing and the ﬁrm foregoes the opportunity of investing in a positive NPV project. In this case the accounting system provides more information about the value of the assets and identiﬁes situations in which their value is sufﬁcient for the lender. Identifying the properties of the optimal accounting system for this scenario is the central issue of our paper.

At ﬁrst sight, it appears intuitive to assume that the ﬁrm would ﬁnd it useful to report all asset valuesBðyÞ A^y^

because doing so convinces the lender to fund the project. Indeed, this reporting policy is one equilibrium strategy if the ﬁrm is unable to commit to the accounting system.

Lemma 1. *If the firm cannot commit to the accounting system*,*but it can voluntarily and truthfully report the asset value based*
*on the signal*,*then there exist an infinite number of equilibria that are characterized by reporting sets D*0 _{with}_{½}_{y}_{^}_{;}_{y}_{} _{D}0_{}_{Y}_{.}

The proof is in the Appendix A. All these equilibria reportBðyÞ A^. They differ only in the reporting of*y*for values where

BðyÞoA^. Moreover, all of these reporting equilibria yield equivalent outcomes and an*ex ante*probability of investment equal

to 1Fðy^Þo1.

Since the ﬁrm can commit to the accounting system, it can do strictly better than reporting all favorable value signals derived from its information system directly to the lender. Proposition 3 describes the optimal accounting policy.

Proposition 3. *If*A^4

_{m}

,*it is optimal to install an accounting system, provided its cost k is not excessive. The optimal accounting*

*system reports*

B¼ BðyÞ ¼E½A˜jy ¼y if yoy

N

B0 otherwise

(

, (16)

*where the threshold value is*yN_{o}_{y}_{^}_{.}

The proof is in the Appendix A.

The proposition states that the optimal accounting system is distinctly different from the equilibrium strategy discussed
earlier. It reports only the lowest possible values of the assets up to a threshold value of*B*(*yN*_{). This threshold values is}

strictly less than the value of assets required for ﬁnancing,A^. Thus, the set of signals that are not reported equalsN¼ ½yN_{;}_{y}_{}_{}_{.}

The intuition behind this result is as follows: for a given investment project, the ﬁrm’s objective is to maximize the
probability of obtaining the funds from the lender. If the accounting system reports only high asset values, it cannot obtain
ﬁnancing if it reports the original book value*B*¼*B0*because the lender will rationally interpret an unadjusted book value
as bad news. However, the ﬁrm can achieve the same result if it reverses its reporting policy and reports only values of the
assets for whichBðyÞoA^. Then it will not obtain ﬁnancing for these low values, but the project will be funded if the ﬁrm
reports*B*¼*B0*because the unadjusted book value is now interpreted as good news. The lender infers that the revised asset
value equalsEA˜y2 ½y^;y4A^ asy^¼A^.

Precisely becauseE_{A}˜_{y}_{2 ½}_{y}_{^}_{;}_{y}_{}_{}is strictly greater thanA^, the ﬁrm can extend the set of*y*for which it reports*B0*until the
conditional expectation of the asset value is just equal toA^. The proof in the Appendix A shows that this is best achieved by
including the values of*y*closest toy^ in the set*N*because higher values lower the conditional expected value less than
including lower values with the same probability mass.

The threshold signal that is included,*yN*_{, is implicitly deﬁned by}

E_{A}˜ _{y}_{2 ½}_{y}N_{;}_{y}_{}

¼A^. This reporting policy maximizes the
probability that ﬁnancing is obtained, which equals*F*(*N*)¼1*F*(*yN*_{).}14_{The accounting system provides a strictly positive}

expected proﬁt to the ﬁrm. However, the ﬁrm will only use it if its cost*k*is less than that proﬁt. Otherwise, it does not
invest in the accounting system and obtains no funding.

A special feature of the optimal accounting system in Proposition 3 is that it aligns the*ex ante*interests of the ﬁrm and
the lenders. While lenders are indifferent with regard to ﬁnancing the project if they consider the expected proﬁt, as long as

14

they have an interest in doing business (and this is why they exist), this accounting system maximizes the probability of investment and, hence, also the probability of lending. Consequently, lenders do not want a different accounting system because that would reduce the probability of doing business with the ﬁrm. Similarly, a regulator that maximizes social welfare would prescribe exactly the same accounting system.

*4.3. Characteristics of the optimal accounting system*

The optimal accounting system singles out the set of asset values with the lowest value and uses all information
provided by the information system for reporting the asset values within this set. However, the ﬁrm does not adjust the
book value for high asset values outside this lower value set. The optimal reporting policy is consistent with accounting
systems that are based on historical cost: It reports the amortized cost of assets and recognizes impairment losses, but it
never reports an upward revaluation of existing assets. Moreover, whenever the cost of the assets does not exceed their
expected value, that is, *B0*r

### m

, reporting the book value*B*¼

*B0*implies that the expected asset value is larger, that is,

E½_{A}˜_{j ¼}_{B}_{0} A^4

_{m}

, which is also in line with a measurement at cost in that the reporting of an unimpaired book value indicates
a lower bound of the expected value of the assets.
Proposition 3 does not impose any constraints on the value of*B0*, so that the interpretation of the reporting policy
depends on the initial book value.15_{Formally, the optimal accounting system is consistent with any measurement that}

reports the same book value for all signals*y*A*N*. However, none of these reporting policies is consistent with an accounting

system that reports the fair value or the value in use of the assets because the ﬁrm*always*observes the realized signal*y*

and, hence, knows the (conditional) expected value*B*(*y*), but instead commits to report*B*0whenever*y*A*N*.

The optimal accounting system undermines the ﬁrm’s*ex ante*interest in reporting favorable asset values for convincing
the lender to fund the project. Indeed, if the ﬁrm were to report high asset values, or if this policy were anticipated by the
lender, it would destroy the reporting equilibrium and result in a decrease of total welfare.16

An important property of the optimal accounting system is that the information content of the book value depends on
whether an impairment is recognized (*B*¼*B*(*y*)) or not (*B*¼*B0*).

Corollary 1. *The book value of an impaired asset provides more precise information about the asset value than the book value of*
*an unimpaired asset*.

A book value of*B*a*B0*is more precise because it carries only the uncertainty about the asset value resulting from

*B*¼

*B0*contains also some uncertainty resulting from

### x

˜, depending on the size of the nondisclosure set*N*. The larger

*N*is, the less precise is the information contained in

*B*0about the asset valueA˜.

More speciﬁcally, the optimal accounting system generates conditional conservatism because bad news (low *y*) are
recognized as an impairment whereas good news (high*y*) are not immediately recognized. The more likely it is that an
impairment is reported (i.e., the higher *yN* _{is), the more conditionally conservative is the accounting system. More}

conditional conservatism is also associated with a lower probability of funding the project.

This feature of the optimal accounting system is consistent with empirical evidence on conditional conservatism ﬁnding stronger earnings response coefﬁcients for unfavorable information than for favorable information. It is also in line with the predictions of other theoretical models, such asSmith (2007)andGigler et al. (2009).

The following result records several results on the optimal strictness of the impairment rule with respect to economic parameters of the model.

Corollary 2. *Ceteris paribus*,*the optimal accounting system includes a stricter impairment rule*(*yN _{increases}*

_{)}

_{if:}(i) *the investment project is less profitable*(*NPV*¼*pHXI decreases*);
(ii) *less assets are available for pledging*;

(iii) *the loss from liquidating pledged assets is higher*,*e.g. if assets become more firm specific*(

### g

*decreases*); (iv)

*the agency problem is more severe*(v/

### D

*p increases*).

The statements in the corollary follow directly from

EA˜ y2 ½yN_{;}_{y}_{}

¼A^¼I_{p}pHðXv=

### D

pÞHþ ð1pHÞ

### g

. (17)

The cost of the accounting system does not affect its characteristics. A higher cost only reduces the probability that the accounting system is implemented and, as a consequence, the probability that the project is undertaken.

Corollary 2 suggests that the optimal impairment rules become stricter and the accounting system becomes more conditionally conservative if the economic conditions are more unfavorable, in general. These results can be used to

15

It may be that*yN*_{4}_{B0}_{, then the optimal accounting system would, to a limited extent, require an upward revaluation above cost.}
16

generate testable hypotheses. They can provide insights into variations in debt covenants that make adjustments to GAAP.17

The model predicts that ﬁrms with high growth options have lower values of assets in place and use stricter impairment rules for debt covenants. A similar prediction holds for ﬁrms with high proportions of intangibles, ﬁrm-speciﬁc or project-speciﬁc assets, which are harder to sell.18

Impairment rules create conditional conservatism in ﬁnancial reporting. Of course, given rational expectations, the
lender is never fooled and correctly revises the conditional expected asset value according to the information content
carried by the book value*B*. In that sense, the accounting system is neither conservative nor aggressive. However, if one
interprets the reported book values ‘‘literally’’ as best estimates of the asset value and compares these values with the
information available to the ﬁrms, the optimal accounting system also exhibits unconditional conservatism in that it
understates the expected asset value if no impairment is recognized (as long ifB0oE_{½}A˜y2N).19_{While a higher threshold}
*yN* _{increases conditional conservatism, as more unfavorable news are recognized, its effect on the unconditional}

conservatism is*ex ante*ambiguous. Assume thatB0¼E½A˜, then an increase in*yN*leads*ex post*to a greater value difference

betweenE½_{A}˜_{j}y2Nand*B0*but at the same time the probability that*y*A*N*decreases. Accordingly, the expected difference

between the book value and the best estimate of the asset value and thereby the expected degree of unconditional
conservatism can increase or decrease in*yN*

.

The analysis assumes a single ﬁrm with a given investment project and offers thereby insights into the design of
covenants in debt contracts. To gain insights into characteristics of accounting standards, we would need to extend the
analysis to an*ex ante*perspective. This problem can be addressed within our model structure by assuming that the ﬁrm
implements the accounting system before it learns the proﬁtability of the investment project (see againFig. 1). One simple
representation of this scenario would assume that all projects have the same cash ﬂow structure and differ only in their
required amounts of investment *I*, which are drawn from a commonly known distribution. Then the ﬁrm selects an
accounting system that maximizes its expected proﬁt over this range. Since linearity prevails, the properties of the optimal
accounting system are not affected by this change. The uncertainty of*I*only affects the boundary condition of A^ and,
consequently, the conditions for the value of the accounting system and the strictness of the impairment rule. In a similar
manner, the analysis extends to a continuum of ﬁrms with several investment projects of the same type. Thus, while we do
not explicitly model accounting regulation, our results are also relevant for the design of accounting standards.

Interpreting our results in these terms offers an explanation for differences in accounting standards across countries
because some of the characteristics vary systematically by country; particularly, they differ with respect to the conditions
under which an impairment loss must be recognized. For example, both IAS 36 (IASB, 2004) and SFAS 144 (FASB, 2001)
require an indicator for impairment before an impairment test is made. SFAS 144 further includes a trigger test that
requires recognition of an impairment loss only if the carrying amount exceeds the expected undiscounted cash ﬂow
resulting from the asset.20_{IAS 36 deﬁnes the lower value as the value in use, whereas SFAS 144 deﬁnes it as the fair value,}

which is usually less than the value in use. The Fourth European Directive states in Article 35 (European Union, 1978) that a ﬁxed asset should be written down to a lower value that is attributed to the asset if the value reduction is expected to be permanent. These examples of accounting rules suggest that the value of the assets can be signiﬁcantly less than the carrying amount before an impairment loss is reported. Our analysis provides a simple theory for understanding these different triggers and their effect on investment behavior.

*4.4. Optimal precision of the information system*

An important parameter of the accounting system is the precision of the underlying information system that produces
the value signals ˜y. In this section, we allow the ﬁrm to choose the precision and assume that the precision choice is
observable. As shown in Proposition 3, the optimal threshold value*yN*_{does not depend on the precision of the information.}

However, the precision has an effect on the expected proﬁt of the ﬁrm because it affects the probability of obtaining
ﬁnancing,*F*(*N*). Recall that

˜

A¼

### m

þ˜þ### x

˜and˜y¼

### m

þ### x

˜, so the signal*y*resolves the uncertainty in the asset value

_{A}˜that stems from

_{x}

˜_{. An information system is more}

precise if the conditional expected value ofA˜ after observing*y*is less uncertain. Since the total uncertainty ofA˜ does not
depend on the information system, a higher precision of the conditional distribution is equivalent with a decrease in

_{x}

˜_{. We formalize the precision by considering two arbitrary distribution functions,}

_{F1}_{(}

_{}

_{) and}

*F2*(), over

*y*A

*Y*. These functions have the same expected value

*E*(

*y*|

*F1*)

_{¼}

*E*(

*y*|

*F2*)

_{¼}

### m

, and*F2*is a mean-preserving spread of

*F*1, that is,

*F*2(

*y*)4

*F*1(

*y*) for

*y*o

### m

and*F*2(

*y*)o

*F*1(

*y*) for

*y4*

### m

. It follows that*F*2() is more precise than

*F*1(). We also allow for

the more precise information system to be more costly, i.e., the cost*k*may increase in the precision.

17

See, e.g.,Beatty et al. (2008)andSunder et al. (2009). 18

This observation is consistent withSmith (2007). 19

Proposition 4. *Let F2*()*be more precise than F1*().*Then the following holds*:

(i) *If yN*

r

### m

*then the firm unambiguously prefers the less precise information system*,

*F*1().

(ii) *If yN*_{4}

_{m}

_{then the firm prefers the more precise information system}_{,}

_{F}2(),*as long as the incremental cost of precision is*
*sufficiently low.*

To prove this result, notice that the ﬁrm’s objective is to maximize the probability of obtaining ﬁnancing, that is

*F*(*N*)¼1*F*(*yN*_{). Consider ﬁrst the case of}_{y}N

r

### m

, which is shown inFig. 2. Then*F1*(

*yN*)o

*F2*(

*yN*) because

*F2*(

_{}) is a

mean-preserving spread of*F1*(), and since*F1*is (weakly) less costly, the ﬁrm strictly prefers*F1*over*F2*, that is, it prefers the
information system that provides less precise information about_{A}˜_{given}_{y}_{. If}_{y}N_{4}

_{m}

, then*F1*(*yN*)4*F2*(*yN*) and the ﬁrm prefers
the more precise information system as long as its incremental cost is not larger than the beneﬁt of the expected
incremental proﬁt due to the higher probability that the project is ﬁnanced.

Corollary 2 implies that the optimal accounting system becomes more conditionally conservative (the threshold value

*yN*_{increases) if the economic conditions are more unfavorable, in general. We know from Proposition 4 that if the economic}

conditions are unfavorable then the optimal accounting system also provides more precise information. Thus, the precision of accounting information is positively correlated with conditional conservatism and negative correlated with the economic conditions.

5. Extensions

*5.1. Reporting manipulation*

So far, we have assumed that the manager cannot manipulate the report from the accounting system. However, empirical evidence suggests that impairment rules allow for discretion so that there is a potential for earnings management. For example,Ramanna and Watts (2008)observe that many companies do not impair goodwill even if it is likely that its value has decreased.

To examine the effect of earnings management on the characteristics of the optimal accounting system, we assume that
the manager can manipulate the signal *y* that is reported by the information system, but he cannot manipulate the
reporting policy. One reason may be that the reporting policy is easier to audit given the evidence of the (potentially
manipulated) signal. If a signal*y*is observed by the manager privately, he can pretend it was the signal*m*,

mðyÞ ¼yþb. (18)

Adding a bias*b*to the original signal*y*causes a personal cost of*b*2_{/2 for the manager. Consider a reporting policy with an}

impairment rule at threshold value*yN*_{. If the manager observes}_{y}_{Z}_{y}N_{then there is no beneﬁt from manipulation, as the}

project gets ﬁnanced anyway. If*y*o*yN*there is an incentive to report an*m*(*y*)Z*yN*. Since manipulation is costly, the manager

will not report a signal*m*4*yN*_{but exactly}_{m}_{(}_{y}_{)}

¼*yN*_{because the book value is unaffected. Moreover, he will over report}

only for values of*y*in the neighborhood of*yN*_{, which results in a cost of (}_{y}N

*y*)2_{/2. According to (11), the investment project}

yields an expected proﬁt of

### P

¼NPV ð1pHÞð1### g

ÞA^µ *y*

*B*(*y*)… _{B}_{(}_{y}N_{)} *B*_{0}

Reported book values contingent on signal *y*
*F*_{2}(*y*)

ˆ

*y*ˆ

*F*1(*y*)

*y* _{y}N

Fig. 2.Precision of the optimal accounting system. This ﬁgure shows the probability distribution of a more precise information system (*F2*(*y*)) and a less
precise information system (*F1*(*y*)) over the support of signalsy2 ½y;y, where*F2*() is a mean-preserving spread of*F1*() with meanm.^yindicates the
required value of assets to obtain ﬁnancing and*yN*_{the equilibrium threshold value.}_{B}_{(}

given that the project is ﬁnanced; otherwise,

### P

¼0. The boundary value of*y*for which the manager is indifferent between manipulating or not is given by

yb¼yNpﬃﬃﬃﬃﬃﬃﬃﬃﬃ2

### P

:Since rational lenders understand the manager’s incentives for manipulation they will take the manager’s optimal reporting strategy into account in making their lending decision. The following result obtains.

Proposition 5. *The optimal accounting system under potential manipulation is the same as described in Proposition 3*,*except*
*that the threshold value is yNb*_{,}_{where}_{y}Nb_{}_{y}N_{þ}pﬃﬃﬃﬃﬃﬃﬃﬃ_{2}

_{P}

_{.}

_{More specifically}_{,}

_{y}Nb

_{¼}

_{y}N

_{þ}pﬃﬃﬃﬃﬃﬃﬃﬃ

_{2}

_{P}

*0*

_{if f}_{(}

_{y}_{)}

_{Z}

_{0}

_{for y}_{Z}

_{y}N_{.}

The proof is in the Appendix A.

Intuitively, the original accounting system without manipulation can be replicated by increasing the threshold value*yN*

to*yNb*by the maximum bias the manager will choose, which equalspﬃﬃﬃﬃﬃﬃﬃﬃ2

### P

and is independent of*y*. The proposition provides a sufﬁcient condition that this policy is indeed optimal.

Lenders holding rational expectations anticipate the optimal bias, and in equilibrium they are not fooled by the
manipulation. They perfectly back it out by revising the expected value of the assets conditional on observing*B*¼*B0*to

E½_{A}˜_{j}_{B}_{0}_{ ¼}_{E}_{½}_{A}˜_{j}_{y}_{2}_{N}_{}, which is strictly less thanE_{A}˜_{j}_{y}_{2 ½}_{y}Nb_{;}_{y}_{}_{}_{, the conditional expected value if no manipulation were}

possible. Notice that the manager’s optimal strategy is to manipulate regardless of the fact that he bears the expected cost of manipulation. This cost adds to the other costs of the optimal accounting system. The proof of the proposition shows that the ﬁrm might even reduce the probability of ﬁnancing in order to reduce the cost of manipulation. As a result, the potential for manipulation makes it more likely that the accounting system is too costly to be used; hence, the ﬁrm is more likely to forego the investment project. Indeed, the ﬁrm would want to take measures for excluding or constraining the manager’s reporting manipulation. For example, it could implement alternative mechanisms that encourage truthful reporting or it could increase the level of auditing or regulatory scrutiny.

Proposition 5 is consistent with the empirical results inBharath et al. (2006), who ﬁnd that accounting quality, proxied
by discretionary accounting choices, induces more stringent debt contracts. Other empirical studies ﬁnd that the
opportunity for earnings management increases conservatism (see, e.g.,Chen et al., 2007;LaFond and Watts, 2008). Even
though Proposition 5 shows that the accounting system deﬁnes a higher threshold value for impairment and thereby
formally induces more conditional conservatism into the ﬁrm’s accounting policy, there will be no empirically observable
change in the reported book values. The reason for this conclusion is found in the manager’s earnings management
strategy: He reports book values satisfying *B*(*yN*_{)}

r*B*(*y*)o*B*(*yNb*) with zero probability because for all the values in this

interval, the manager manipulates the signal and reports*B*¼*B0*.21

*5.2. Equity financing*

In this section, we brieﬂy discuss the effect on the optimal accounting system if the ﬁrm raises equity rather than debt capital. Suppose the ﬁrm issues new shares and let

### a

A[0,1] be the percentage in the ﬁrm held by new investors afterissuance. Investors are risk neutral and require the market rate of return. New investors will provide the necessary amount*I*

of equity capital if their expected proﬁt share is greater than the invested amount of capital,

### a

½pHðXþAÞ þ ð1pHÞA I0or

A

_{a}

IpHX. (19)
Higher values of the assets increase the likelihood that condition (19) holds and so ensure equity ﬁnancing of the project.
The main difference between condition (19) and the lender’s participation constraint in (3) is that the equity investors
value the asset at its value for the ﬁrm (*A*), whereas lenders value them at their liquidating value*V*(*A*)o*A*. Therefore, equity

ﬁnancing does not induce an efﬁciency loss.

An incentive problem arises because the manager, who is the current owner, shares the investment returns with the new investors but bears the full cost of effort. Incentive compatibility requires that the manager’s expected utility (1

### a

)[*pHX*+

*A*] for high effort is greater than for low effort,

ð1

### a

Þ½pHXþA ð1### a

Þ½pLXþA þvor

### a

### a

¼1_{D}

_{p X}v . (20)

This condition provides an upper bound for the share the new investors can hold in the ﬁrm. The bound

### a

is higher – and the condition is less restrictive – if the agency problem becomes less severe or if the cash ﬂow of a successful project is higher.Equity ﬁnancing is only feasible if both conditions in (19) and (20) hold. The next lemma shows that these conditions are met if there are sufﬁcient assets in place, so that equity ﬁnancing is preferred over debt ﬁnancing. Equity ﬁnancing is preferable because it is costless whereas debt ﬁnancing is costly if the pledging of assets is required.

Lemma 2. *If*

AA^E¼XpHv

### D

p NPV### D

p Xv , (21)*the firm can raise equity capital. Equity financing is*(*weakly*)*preferable to debt financing.*

The boundA^Ein condition (21) obtains from equating the two conditions (19) and (20). If condition (21) does not hold, the ﬁrm can either look for debt ﬁnancing or implement an informative accounting system and then again consider equity and debt ﬁnancing.

The accounting system is used for exactly the same purpose as it is used for debt ﬁnancing, namely to convince the investor to provide equity ﬁnancing. Thus, the characteristics of the optimal accounting system derived for debt ﬁnancing carry over to equity ﬁnancing with the one exception that the required amount of assets held by the ﬁrm generally differs from that of the assets that are pledged. The necessary asset value in debt ﬁnancing is less than that for equity ﬁnancing, i.e.,A^oA^E, if

pHv=

### D

pNPV pHþ ð1pHÞ### g

oXpHv

### D

p NPV### D

p Xv .This condition is equivalent to

### g

4### a

pH1pH

. (22)

Debt ﬁnancing is a viable option if

### g

is high, that is, if the liquidating value*V*(

*A*) is not much less than the value in use of the pledged asset, which implies that the efﬁciency loss from debt ﬁnancing is small.

In their empirical studyLaFond and Roychowdhury (2008)hypothesize that lower managerial ownership (higher

### a

) makes the agency problem more severe and increases the demand for conservative accounting. This hypothesis is consistent with the predictions of this model: Managerial ownership (1### a

) is endogenous in this model; a more severe agency problem (higherv) induces a lower external ownership (### a

) and, consequently, a higher asset value is required to raise equity capital. This mechanism again induces a stricter impairment rule and more conditional conservatism, that is, a higher reporting threshold*yN*

_{.}

The question whether equity or debt ﬁnancing is preferable for insufﬁcient assets depends on several parameters, particularly on the proﬁtability of the investment, the severity of the agency problem, and the precision and the cost of the accounting system. As a tendency, debt ﬁnancing is more preferable for relatively low values of existing assets, so that debt ﬁnancing should be positively associated with accounting rules favouring conditional conservatism. Thus, our results provide a possible explanation for differences in accounting conservatism across countries, which complements the impact of legal systems and other institutional features.22

6. Conclusions

This paper provides an economic rationale for why conditionally conservative asset measurement can be optimal. In
particular, it shows why it is desirable that unfavorable information is recognized by an impairment of the book value of
assets whereas favorable information is not recognized. We ﬁnd that, absent any accounting regulation, a ﬁrm that seeks to
ﬁnance a risky project with outside capital will optimally design a conditionally conservative accounting system. This
system is also desired by lenders, because they would not want the ﬁrm to fully report its private information about asset
values. Although counter-intuitive at ﬁrst glance because reporting low asset values can be mistaken to impede ﬁnancing, a
conservative accounting system is indeed optimal because it increases the conditional expected value of the assets once no
impairment is recognized and it maximizes the*ex ante*probability of obtaining ﬁnancing.

The optimal accounting system is consistent with a measurement at cost and impairment. We describe its characteristics and provide comparative static analyses for the optimal impairment rule. In particular, we show that less favorable economic conditions lead to stricter impairment rules and, thus, to a higher degree of conditional conservatism. If the ﬁrm can also select the precision of the information that it obtains about the asset values, we ﬁnd that the less favorable conditions tend to induce ﬁrms to implement a more precise information system. The opportunity for earnings management further increases the strictness of the impairment rule. The optimality of a conditionally conservative

accounting system is robust with respect to debt or equity ﬁnancing. These results can be used to derive testable predictions about debt covenants and accounting standards.

We provide an economics-based argument for impairment rules and conditional conservative accounting within the context of raising outside ﬁnancing, but the model is silent on other potential uses of accounting information. It is based on several simplifying assumptions and captures only few, albeit important, facets of the real world. However, we believe that the main results are robust and carry over to more general situations.

Appendix A

Proof of Lemma 1. The equilibrium accounting system deﬁnes the set*D*0_{of signals that are reported as}_{½}_{y}_{^}_{;}_{y}_{} _{D}0_{}_{Y}_{. The}

set½y^;yincludes all*y*for whichBðyÞ A^, that is,D¼ fy y y^g, wherey^A^. Let*N*¼*YD*. Given*D*0_{the lender agrees to the}

contract ifBðyÞ A^and does not ifBðyÞoA^. If the ﬁrm reports*B0*then the lender infers thatE_{½}A˜jB¼B0 ¼E½A˜jy2N0NoA^

and does not agree to the contract either. The ﬁrm maximizes its reporting policy given the rational responses of the lender.
Therefore, it reports*B*(*y*) ifBðyÞ A^ because investing results in a strictly positive proﬁt, and it is indifferent between
reporting*B*(*y*) forBðyÞoA^ and*B0*because its proﬁt is zero. There exist an inﬁnite number of sets*D*0with_{½}y^;y D0_{}_{Y}_{for}

which these strategies form a rational expectations equilibrium. &

Proof of Proposition 3. The ﬁrm’s objective is to maximize the probability of obtaining ﬁnancing. This maximizes the expected proﬁt it can achieve because investing generates a strictly positive NPV whereas not investing generates zero proﬁt. The proof proceeds by showing that this probability is strictly the highest probability that the ﬁrm can obtain by any accounting system.

The proposition deﬁnes the set of signals that are reported,*D*¼{*y*|*B*¼*B*(*y*)} as a lower interval [*y*,*yN*_{) and the set of}

signals for which the original book value is reported as an upper interval,N¼ fyjB¼B0g ¼ ½yN;y. The threshold value*yN*is

implicitly deﬁned so that

E½A˜jy2N ¼A^.

Since*N*includes ally2 ½A^;y, it must be thatyN_{o}_{y}_{^}_{}_{A}_{^}_{. Therefore, if the ﬁrm reports}_{B}_{¼}_{B}

0, the lender will ﬁnance the

project. If the ﬁrm reportsB¼BðyÞoA^it will not ﬁnance the project. Thus, the probability of ﬁnancing isFðy2N¼ ½yN_{;}_{y}_{}

Þ. It is strictly greater thanFðy2 ½y^;yÞthat would be achieved by reporting all signals that indicateE½A˜jy A^.

Assume to the contrary that this accounting system is not optimal. Deﬁne another accounting system that reports*B0*for
an arbitrary set*Y1*¼[*y1*,*y2*]C*D*, where*y1*o*y2*. In order to obtain ﬁnancing for this alternative accounting system, the set*N*

must be adjusted to another set*N1*for which

E½A˜jy2Y1[N1 A^.

Otherwise, the probability of ﬁnancing would be zero. Assume therefore thatN1¼ ½y3;y, where*y*34*yN*such that

E½_{A}˜_{j}y2Y1[N1 ¼

FðY1ÞEðY1Þ þFðN1ÞEðN1Þ FðY1Þ þFðN1Þ ¼

^

A¼EðNÞ.

where*F*(*Z*) stands short for*F*(*y*A*Z*) and*E*(*Z*) forE_{½}A˜jy2Zfor any set*Z*D*Y*. Notice that it must be thaty_{3}oy_{^}because if some
subsetY0_{ ½}_{y}_{^}_{;}_{y}_{}_{}_{were removed from}_{N1}_{it would not be possible to guarantee}_{E}_{½}_{A}_{˜}_{j}_{y}_{2}_{Y}

1[N1 A^for any*Y1*as*Y1*C*D*.

To show the preferability of the accounting system described in the proposition requires proving that*F*(*N*)4*F*(*Y1*)+*F*(*N1*).
Denote*Y3*¼[*yN*_{,}_{y3}_{); then}_{N}

¼*Y3*[*N1*. Inserting this expression into the equation above results in

FðY1ÞEðY1Þ þFðN1ÞEðN1Þ FðY1Þ þFðN1Þ ¼

FðY3ÞEðY3Þ þFðN1ÞEðN1Þ FðY3Þ þFðN1Þ

.

Since*E*(*Y1*)o*E*(*Y3*), it remains to show that*F*(*Y1*)o*F*(*Y3*). Straight-forward calculation results in

FðY3ÞFðN1Þ½EðN1Þ EðY3Þ ¼FðY1ÞFðY3Þ½EðY3Þ EðY1Þ þFðY1ÞFðN1Þ½EðN1Þ EðY1Þ.

Assume to the contrary that*F*(*Y1*)4*F*(*Y3*). Then

FðY1ÞFðY3Þ½EðY3Þ EðY1Þ þFðY1ÞFðN1Þ½EðN1Þ EðY1Þ4FðY3ÞFðY3Þ½EðY3Þ EðY1Þ þFðY3ÞFðN1Þ½EðN1Þ EðY1Þ.

Inserting into the above equation yields

FðN1Þ½EðY1Þ EðY3Þ4FðY3Þ½EðY3Þ EðY1Þ

or

FðN1Þ4FðY3Þ,

which is a contradiction as all*F*()40. Therefore, it must be that*F*(*Y1*)o*F*(*Y3*) and*F*(*N*)4*F*(*Y1*)+*F*(*N1*). Since this holds for any

As shown in (11), the expected proﬁt of always investing is

### P

¼NPV ð1pHÞð1### g

ÞA^.The optimal accounting system results in an expected proﬁt of*F*(*N*)

### P

. Since the accounting system incurs cost*k*, this cost must be

*k*o

*F*(

*N*)

### P

in order to install the accounting system. Otherwise the ﬁrm would not invest. &Proof of Proposition 5. Consider the optimal accounting system described in (16),

B¼ BðyÞ ¼E½

˜

Ajy ifyoyN B0 otherwise

(

.

Deﬁne yNb_{¼}_{y}N_{þ}pﬃﬃﬃﬃﬃﬃﬃﬃ_{2}

_{P}

_{, then the manager manipulates all signals}

_{y}_{A}

_{[}

_{y}N_{,}

_{y}Nb_{) and reports}

_{m}_{(}

_{y}_{)}

¼*yNb*_{, but does not}

manipulate the other signals as it does not change the lender’s decision or is too costly. The reporting policy then provides book values

B¼

BðyÞ ¼E½_{A}˜_{j}y ifyoyN

BðmÞ ¼B0 ifyNyoyNb

B0 ifyNby

8 > <

> :

,

which are exactly the same as in the original accounting system in Proposition 3 without manipulation. Since this accounting system replicates the outcome produced by the optimal accounting in Proposition 3, the ﬁrm cannot do better by implementing another accounting system, if costs are ignored. Since manipulation is costly, the expected cost of manipulation is

1 2

Z yNb

yN ð

yNb

yÞ2fðyÞdy.

While the interval in which the manager manipulates the signal always has the same length pﬃﬃﬃﬃﬃﬃﬃﬃ2

### P

, the expected cost depends on the probability mass in the interval [*yN*

_{,}

_{y}Nb_{). Any accounting system that adjusts}

_{y}Nb_{by an amount of less than}

ﬃﬃﬃﬃﬃﬃﬃﬃ

2

### P

p

cannot be optimal as it would implyE½_{A}˜_{j}BðmÞ ¼B0oA^ and, hence, forego ﬁnancing. This provesyNb_{}yN_{þ}pﬃﬃﬃﬃﬃﬃﬃﬃ2

### P

.Since the probability functions are not constrained it is possible that a small shift of this interval to [*yN*+

### d

,*yNb*+

### d

), where### d

40, decreases the expected cost by an amount that is greater than*F*([

*yN*

_{,}

_{y}N_{+}

_{d}

_{])}

_{P}

_{, the expected loss in proﬁt if the}

threshold value is increased to*yN*+

### d

. A sufﬁcient condition that this is not the case isf0_{ð}_{y}_{Þ }_{0} _{for}_{y}_{}_{y}Nb_{,}

because a shift of the interval to the right increases the expected cost so that it can never be beneﬁcial. &

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