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Optimal impairment rules

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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.

Contents lists available atScienceDirect

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

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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

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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 arenotreported 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 theabsenceof 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 postview) and to lower interest rates (ex ante

view). The present paper contributes to the debt contracting explanation, but focuses on the ex anteuse 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.2It is related to the work byGuay 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 onex anteinvestment the revaluation policy is tailored in a manner that the firm never reports an impairment loss.

2

Verrecchia (2001)surveys the disclosure literature.

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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 tot¼1, the firm has the opportunity to invest in a risky project that requires an investment ofI40, which is common knowledge.4The 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 probabilitypand yields a cash flow ofX40; it fails with probability (1p) and yields zero payoff. Without loss of generality we assume zero discounting. The expected Net Present Value of the project isNPV¼pXI.

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 ispH, 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 stochastic

dominance. 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.

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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.5We 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 investmentI 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 paymentsdjatt¼2 from the firm to the lender in the two statesj¼S,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 outdSrXin case of success anddF¼0 in case of failure.7

In addition, the firm can pledge assets in the amount ofAZ0 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)oA, 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 asset

valueA,9i.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 assumesex antethat 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 effortex 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.

7A value ofdFo0, 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 requiresdF¼0.

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 ofV() would not significantly alter the analysis. 10

The paymentdScan also be used to express the contract in terms of a nominal interest rater, which isdS¼(1+r)I, so that the lender’s participation constraint isr ½ðxv=DpþAÞ VðAÞ=pH=ð1pHÞ½ðxv=DpþAÞ VðAÞ

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The following program determines the optimal debt contract. The firm maximizes its expected profit overdSandA

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 A¼0. The lender’s break-even constraint (3) implies a maximum debt of IrpHdS, 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, the

pledging of assets is necessary to obtain financing.

Proposition 1. If I4pH(Xv/

Dp

),the optimal debt contract is uniquely defined by

A¼A^IppHðXv=

D

Hþ ð1pHÞ

g

, (9)

dS¼X v

D

pþA. (10)

Since both,dSandA, are greater than zero if pledging is necessary for realizing the project, the two constraints (6) and (7) must be binding. Solving the constraints forAanddsyields 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

¼pHXI

|fflfflfflffl{zfflfflfflffl}

NPV

ð1pHÞð1

g

ÞA

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Efficiency loss

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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 bothAandV(A) at the end of the project cycle (t¼2) are important determinants to the debt contract.11The 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 bothAandV(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 variableA˜with a strictly positive probability density functionf() for allA. More specifically, we define the asset value as

˜

m

þ˜

þ

x

˜, (12)

where

m

is the expectedex anteasset 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

) andf(

x

) over their respective support. Given this extended model structure, the analysis of Section 2,

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including Proposition 1, still applies ifAis 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 ofA(and

V(A)) in the financial statements. The firm designs an accounting system and commitsex anteto 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 ofkZ0. 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 overA˜. 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,

A˜jy ¼E½yþ

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 withDDYthe set of signals that are used and withNDYthe 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 valueB¼B0. 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 signaly, 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 BaB0 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. IfB¼B0then the lender cannot

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

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

EA˜jBA^,

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. IfA^

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 informative

12

IfB0¼B(y) for someyADthis case is indistinguishable from a report resulting from someyAN. However, the eventy¼B0has zero probability inY, so it does not affect the analysis.

13

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accounting system cannot do better than that, but it potentially incurs two costs: (i) The accounting system can be costly (ifk40); 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 D0 with½y^;y D0Y.

The proof is in the Appendix A. All these equilibria reportBðyÞ A^. They differ only in the reporting ofyfor values where

BðyÞoA^. Moreover, all of these reporting equilibria yield equivalent outcomes and anex anteprobability 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. IfA^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 isyNoy^.

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 ofB(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 valueB¼B0because 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 reportsB¼B0because 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 becauseEA˜y2 ½y^;yis strictly greater thanA^, the firm can extend the set ofyfor which it reportsB0until 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 ofyclosest toy^ in the setNbecause 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

EA˜ y2 ½yN;y

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

expected profit to the firm. However, the firm will only use it if its costkis 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 theex anteinterests 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

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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, B0r

m

, reporting the book valueB¼B0implies that the expected asset value is larger, that is,

A˜j ¼B0 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 ofB0, so that the interpretation of the reporting policy depends on the initial book value.15Formally, the optimal accounting system is consistent with any measurement that

reports the same book value for all signalsyAN. 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 firmalwaysobserves the realized signaly

and, hence, knows the (conditional) expected valueB(y), but instead commits to reportB0wheneveryAN.

The optimal accounting system undermines the firm’sex anteinterest 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 (B¼B(y)) or not (B¼B0).

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 ofBaB0is more precise because it carries only the uncertainty about the asset value resulting from

˜, whereas a book value ofB¼B0contains also some uncertainty resulting from

x

˜, depending on the size of the nondisclosure setN. The largerNis, the less precise is the information contained inB0about 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 (highy) 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(yNincreases)if:

(i) the investment project is less profitable(NPV¼pHXI 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^¼IppHðXv=

D

Hþ 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 thatyN4B0, then the optimal accounting system would, to a limited extent, require an upward revaluation above cost. 16

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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 valueB. 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 ifB0oE½A˜y2N).19While a higher threshold yN increases conditional conservatism, as more unfavorable news are recognized, its effect on the unconditional

conservatism isex anteambiguous. Assume thatB0¼E½A˜, then an increase inyNleadsex postto a greater value difference

betweenE½A˜jy2NandB0but at the same time the probability thatyANdecreases. 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 inyN

.

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 anex anteperspective. 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 ofIonly 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.20IAS 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 valueyNdoes 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

˜

m

þ˜

þ

x

˜

and˜y¼

m

þ

x

˜, so the signalyresolves the uncertainty in the asset valueA˜that stems from

x

˜. An information system is more

precise if the conditional expected value ofA˜ after observingyis 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

˜and a corresponding increase in

x

˜. We formalize the precision by considering two arbitrary distribution functions,F1() and F2(), overyAY. These functions have the same expected valueE(y|F1)¼E(y|F2)¼

m

, andF2is a mean-preserving spread of F1, that is,F2(y)4F1(y) foryo

m

andF2(y)oF1(y) fory4

m

. It follows thatF2() is more precise thanF1(). We also allow for

the more precise information system to be more costly, i.e., the costkmay increase in the precision.

17

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

This observation is consistent withSmith (2007). 19

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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,F1().

(ii) If yN4

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)¼1F(yN). Consider first the case ofyN

r

m

, which is shown inFig. 2. ThenF1(yN)oF2(yN) becauseF2() is a

mean-preserving spread ofF1(), and sinceF1is (weakly) less costly, the firm strictly prefersF1overF2, that is, it prefers the information system that provides less precise information aboutA˜giveny. IfyN4

m

, thenF1(yN)4F2(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

yNincreases) 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 signalyis observed by the manager privately, he can pretend it was the signalm,

mðyÞ ¼yþb. (18)

Adding a biasbto the original signalycauses a personal cost ofb2/2 for the manager. Consider a reporting policy with an

impairment rule at threshold valueyN. If the manager observesyZyNthen there is no benefit from manipulation, as the

project gets financed anyway. IfyoyNthere is an incentive to report anm(y)ZyN. Since manipulation is costly, the manager

will not report a signalm4yNbut exactlym(y)

¼yNbecause the book value is unaffected. Moreover, he will over report

only for values ofyin the neighborhood ofyN, which results in a cost of (yN

y)2/2. According to (11), the investment project

yields an expected profit of

P

¼NPV ð1pHÞð1

g

ÞA^

µ y

B(y)… B(yN) B0

Reported book values contingent on signal y F2(y)

ˆ

yˆ

F1(y)

y yN

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, whereF2() is a mean-preserving spread ofF1() with meanm.^yindicates the required value of assets to obtain financing andyNthe equilibrium threshold value.B(

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given that the project is financed; otherwise,

P

¼0. The boundary value ofyfor 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,whereyNbyNþpffiffiffiffiffiffiffiffi2

P

.More specifically,yNb¼yNþpffiffiffiffiffiffiffiffi2

P

if f0(y)Z0for yZyN.

The proof is in the Appendix A.

Intuitively, the original accounting system without manipulation can be replicated by increasing the threshold valueyN

toyNbby the maximum bias the manager will choose, which equalspffiffiffiffiffiffiffiffi2

P

and is independent ofy. 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 observingB¼B0to

A˜jB0 ¼E½A˜jy2N, which is strictly less thanEA˜jy2 ½yNb;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)

rB(y)oB(yNb) with zero probability because for all the values in this

interval, the manager manipulates the signal and reportsB¼B0.21

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 after

issuance. Investors are risk neutral and require the market rate of return. New investors will provide the necessary amountI

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

a

½pHðXþAÞ þ ð1pHÞA I0

or

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 valueV(A)oA. 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 þv

or

a

a

¼1

D

p Xv . (20)

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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

pH

1pH

. (22)

Debt financing is a viable option if

g

is high, that is, if the liquidating valueV(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 thresholdyN.

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 theex anteprobability 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

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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 setD0of signals that are reported as½y^;y D0Y. The

set½y^;yincludes allyfor whichBðyÞ A^, that is,D¼ fy y y^g, wherey^A^. LetN¼YD. GivenD0the lender agrees to the

contract ifBðyÞ A^and does not ifBðyÞoA^. If the firm reportsB0then the lender infers thatE½A˜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 reportsB(y) ifBðyÞ A^ because investing results in a strictly positive profit, and it is indifferent between reportingB(y) forBðyÞoA^ andB0because its profit is zero. There exist an infinite number of setsD0with½y^;y D0Yfor

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,D¼{y|B¼B(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 valueyNis

implicitly defined so that

E½A˜jy2N ¼A^.

SinceNincludes ally2 ½A^;y, it must be thatyNoy^A^. Therefore, if the firm reportsB¼B

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 reportsB0for an arbitrary setY1¼[y1,y2]CD, wherey1oy2. In order to obtain financing for this alternative accounting system, the setN

must be adjusted to another setN1for which

E½A˜jy2Y1[N1 A^.

Otherwise, the probability of financing would be zero. Assume therefore thatN1¼ ½y3;y, wherey34yNsuch that

A˜jy2Y1[N1 ¼

FðY1ÞEðY1Þ þFðN1ÞEðN1Þ FðY1Þ þFðN1Þ ¼

^

A¼EðNÞ.

whereF(Z) stands short forF(yAZ) andE(Z) forE½A˜jy2Zfor any setZDY. Notice that it must be thaty3oy^because if some subsetY0 ½y^;ywere removed fromN1it would not be possible to guaranteeE½A˜jy2Y

1[N1 A^for anyY1asY1CD.

To show the preferability of the accounting system described in the proposition requires proving thatF(N)4F(Y1)+F(N1). DenoteY3¼[yN,y3); thenN

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

FðY1ÞEðY1Þ þFðN1ÞEðN1Þ FðY1Þ þFðN1Þ ¼

FðY3ÞEðY3Þ þFðN1ÞEðN1Þ FðY3Þ þFðN1Þ

.

SinceE(Y1)oE(Y3), it remains to show thatF(Y1)oF(Y3). Straight-forward calculation results in

FðY3ÞFðN1Þ½EðN1Þ EðY3Þ ¼FðY1ÞFðY3Þ½EðY3Þ EðY1Þ þFðY1ÞFðN1Þ½EðN1Þ EðY1Þ.

Assume to the contrary thatF(Y1)4F(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 allF()40. Therefore, it must be thatF(Y1)oF(Y3) andF(N)4F(Y1)+F(N1). Since this holds for any

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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 ofF(N)

P

. Since the accounting system incurs costk, this cost must bekoF(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 yNb¼yNþpffiffiffiffiffiffiffiffi2

P

, then the manager manipulates all signals yA[yN,yNb) and reports m(y)

¼yNb, 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

BðyÞ ¼E½A˜jy 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,yNb). Any accounting system that adjustsyNbby an amount of less than

ffiffiffiffiffiffiffiffi

2

P

p

cannot be optimal as it would implyE½A˜jBðmÞ ¼B0oA^ and, hence, forego financing. This provesyNbyNþpffiffiffiffiffiffiffiffi2

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,yN+

d

])

P

, the expected loss in profit if the

threshold value is increased toyN+

d

. A sufficient condition that this is not the case is

f0ðyÞ 0 foryyNb,

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

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