Accrual Accounting, Informational Sufficiency, and Equity Valuation

transcript

DOI: 10.1111/j.1475-679X.2011.00428.xJournal of Accounting Research

Vol. 50 No. 1 March 2012Printed in U.S.A.

Accrual Accounting, InformationalSufficiency, and Equity Valuation

A L E X A N D E R N E Z L O B I N ∗

Received 5 January 2010; accepted 2 August 2011

ABSTRACT

This paper studies accrual accounting and equity valuation in the context ofa firm that makes repeated and overlapping investments in productive capac-ity. The analysis identifies a particular accrual accounting (depreciation) rulethat is termed replacement cost accounting because the book value of exist-ing capacity assets is set equal to the value that such assets would have if acompetitive market were to exist for used assets. It is shown that replacementcost accounting aggregates past investment decisions of the firm without aloss of value-relevant information. The intrinsic value of the firm can then beexpressed as a function of current accounting data and certain parametersof the firm’s operating environment. Further, it is shown that replacementcost accounting is essentially the only accounting rule with this informationalsufficiency property.

∗Stern School of Business, New York University. This paper is a chapter from my disserta-tion that I completed at the Stanford University Graduate School of Business. I would like tothank my academic advisor, Stefan Reichelstein, for his continuous support, guidance, andencouragement. I am also grateful to Bill Beaver, Anne Beyer, Ilan Guttman, Jack Hughes,Maureen McNichols, Jim Ohlson, Joe Piotroski, Madhav Rajan, Haresh Sapra (the editor),Dan Taylor, and an anonymous referee for their comments and suggestions. This paper ben-efited from comments from seminar participants at Carnegie-Mellon University, ColumbiaUniversity, Dartmouth College, INSEAD, New York University, University of California, Berke-ley, University of California, Los Angeles, and University of Southern California. Last, but notleast, I wish to acknowledge the advice and patient support of Nikolai Nezlobin and PolinaNezlobin.

Copyright C© University of Chicago on behalf of the Accounting Research Center, 2011

234 A. NEZLOBIN

1. Introduction

This paper examines the informativeness of alternative accrual account-ing rules for the purpose of firm valuation. I model a firm that undertakesa sequence of investments in productive capacity. The firm’s accountingsystem aggregates the resulting investment history into current financialstatements. It is shown that, for certain accounting rules, this aggregationprocess does not entail a loss of value-relevant information, and investorswill be able to assess the firm’s value correctly by observing only the aggre-gate accounting data. Such accounting rules will be referred to as infor-mationally sufficient. For informationally sufficient rules, I derive valuationequations that express the firm’s intrinsic value in terms of the accountingnumbers observed by investors and certain parameters of the firm’s operat-ing environment. For all other accounting rules, I show that it is generallyimpossible to solve the valuation problem precisely based on current ac-counting information. I demonstrate that there may exist infinitely manyvaluations consistent with the firm’s financial statements.

Earlier theoretical literature on accounting-based valuation has beenlargely silent on the relative advantages of alternative accrual accountingrules in providing information useful to investors. I seek to address two limi-tations inherent in earlier valuation studies such as Ohlson [1995], Felthamand Ohlson [1995], and Ohlson and Juettner-Nauroth [2005]. First, inthese papers, the firm’s underlying transactions, as well as the accountingrules employed, are not modeled beyond their most basic properties, likethe clean surplus relation.1 Therefore, assumptions on the behavior of thetime-series of accounting numbers (e.g., linear information dynamics) areinherently joint constraints on the economic environment of the firm andthe accounting rules in use.2 My analysis seeks to disentangle the economicand reporting factors that affect the time-series of accounting numbers.Second, none of these models explicitly articulates limitations on the infor-mation available to outsiders. Without information asymmetry, it is difficult,

1 This criticism applies to earlier studies in varying degrees. More elaborate models of thefirm’s transactions and accounting rules can be found in Ohlson and Zhang [1998], Zhang[2000], and Zhang [2000]. On the other hand, in his discussion of the model in Ohlson andJuettner-Nauroth [2005], Penman [2005] poses the question: “Where’s the accounting?”

2 Several empirical studies challenged the linear information dynamics assumption ofOhlson [1995]. See, for instance, Myers [1999] and Dechow, Hutton, and Sloan [1999]. Dis-cussing this issue, Kothari [2001, p. 181] points out the following direction for future research:

While an autoregressive process in residual income as a parsimonious description iseconomically intuitive, there is nothing in economic theory to suggest that all firms’residual earnings will follow an autoregressive process at all stages in their life cycle.A more fruitful empirical avenue would be to understand the determinants of theautoregressive process or deviations from that process as a function of firm, industry,macroeconomic, or international institutional characteristics.

ACCRUAL ACCOUNTING AND EQUITY VALUATION 235

if not impossible, to demonstrate informational advantages of particular ac-counting rules over others, including cash accounting.

This paper models the activities of a firm as a sequence of capital invest-ments in productive capacity. The firm uses its capacity to deliver goodsand services that generate revenues. The accounting rules may reflectthe productivity pattern of the firm’s assets and provide aggregate infor-mation on the investments undertaken. Depreciation is the only accrualin my model. This focus was chosen for two reasons. First, depreciationis arguably the largest accrual in many industries. Second, earlier litera-ture on performance measurement has shown that insights obtained inconnection with capital investments and depreciation do carry over toother accrual accounting items (see, for instance, Dutta and Reichelstein[2005]).

A central point of departure for my analysis is that investors must relyonly on limited publicly available information for valuation purposes. Theequity value in my model is determined by future conditions in the firm’sproduct market and the history of investment decisions of the firm. I as-sume that the projections of future growth in the product market cannotbe incorporated into the accounting numbers. In practice, future growthopportunities may not be verifiable to the accounting system before theyare realized. Consequently, the requirement for an informationally suffi-cient rule is that it must provide enough information to value the companyunder alternative assumptions about future growth opportunities. This re-quirement is broadly consistent with the perspective taken by standardsetters in Statement of Financial Accounting Concepts No. 1 (FASB[1978]):

Financial accounting is not designed to measure directly the value ofa business enterprise, but the information it provides may be helpfulto those who wish to estimate its value . . . Although financial reportingshould provide basic information to aid them [investors, creditors, andothers], they do their own evaluating, estimating, predicting, assessing,confirming, changing, or rejecting.

Since I assume that growth opportunities cannot be incorporated into theaccounting data, fair value accounting, which would result in book valuesequal to market values at all times, is rendered infeasible in the context ofthe present model.

The main goal of depreciation rules in my model is to aggregate infor-mation about past investments of the firm such that current financial state-ments are sufficient for valuation. Clearly, if investors could observe all thepast transactions of the firm, it would be impossible to differentiate be-tween alternative accounting rules, because there would be no incrementalinformation in financial statements irrespective of the accrual rule in use. I,therefore, restrict attention to parsimonious valuation models, that is, theones that express the firm’s value as a function of only the current account-ing data and future parameters of the firm’s economic environment. De-preciation schedules for which there exists a parsimonious valuation modelare referred to as informationally sufficient.

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My framework of a firm undertaking a sequence of overlapping capacityinvestments builds on Rogerson [2008, 2011]. Investments in this modelgenerate capacity over multiple periods and thus represent a joint cost ofthose periods. In the context of overlapping investments, it becomes possi-ble to identify the marginal cost of providing an additional unit of capacityfor one period, holding capacity levels in other periods fixed. It can beshown that this marginal cost is equal to the rental price of a unit of ca-pacity in a hypothetical perfectly competitive rental market. The overalleconomic cost of production in a certain period is equal to the product ofcapacity utilized in that period and the marginal cost of capacity. I referto the difference between revenues and the economic cost of capacity aseconomic income. Rogerson [2008] demonstrates that there exists a par-ticular depreciation schedule, the Relative Replacement Cost, or the RRCrule, under which the historical cost of capacity, defined as the sum of depre-ciation expense and imputed interest on the book value of assets, is equalto the economic cost in every period. Therefore, the RRC rule equates eco-nomic and residual income, where the latter is traditionally defined as thedifference between revenues and the historical cost of capacity.

While in the formal model of Rogerson [2008] the firm does not own anyassets in the initial period, I allow for the firm to own a certain endowmentof assets at the valuation date. I define the replacement cost of assets in placeto be the present value of future economic costs associated with the assetsthat the firm already owns. It may be equivalently defined as the presentvalue of rental payments that the firm would have to incur in a perfectlycompetitive market to replicate the stream of capacity that the assets alreadyin place will generate in future periods. The first result of my paper showsthat the firm’s value is equal to the replacement cost of assets in place plusthe present value of its future economic incomes. I further show that theresidual income valuation formula implies that, under the RRC rule, thebook value of assets is precisely equal to the replacement cost of assets inplace. Therefore, I refer to the RRC rule as replacement cost accounting.

The second and third results of my paper provide valuation equationsfor replacement cost accounting. First, I consider the case when the cost ofnew capital assets is constant over time and demand for the firm’s productincreases proportionately at all price levels. Under these assumptions, resid-ual income calculated with the replacement cost rule will grow at the samerate as demand for the firm’s product. Therefore, investors can project fu-ture residual income by applying the growth rate in the product market toresidual income of the latest period. To make such projections, investorsdo not need to know the specific functional form of revenue functions. Ishow that the firm’s value is equal to the sum of the book value of assets andthe capitalized residual income, where the capitalization factor depends onfuture growth in the product market. As a special case of this formula, Idemonstrate that, if the product market is stationary, then the value of thefirm is equal to the forward earnings capitalized at the required rate ofreturn. This valuation formula is often referred to as the permanent earn-ings model, and it is known to hold for fair value accounting, under which

the book value of assets is equal to the present value of future cash flows.In contrast, in my paper, the permanent earnings model obtains under re-placement cost accounting, which generally results in book values being lessthan the present value of future cash flows. By the first result of my paper,the difference between the firm’s value and the replacement cost of assetsin place is equal to the present value of future economic profits. When theproduct market is stationary, the present value of future economic profitsis constant over time and, therefore, so are the “errors” in book values ofassets under replacement cost accounting. The permanent earnings modelthen follows from the well-known “Canceling Errors” theorem.

Next, I turn to the setting where the cost of new assets changes over time.In response to changes in the marginal cost of capacity, the firm will adjustthe price of its product. Therefore, future economic profits will depend notonly on changes in demand but also on changes in the cost of new capitalassets. Assuming that demand increases proportionately at all real price lev-els, I show that, given replacement cost accounting, the firm’s value is equalto the book value of assets plus capitalized residual income. The capitaliza-tion factor can be calculated by investors if they observe the growth rates ofthe cost of new assets and of the product market size.

The last result of this paper characterizes the set of informationally suffi-cient accounting rules. First, I show that the replacement cost rule remainsinformationally sufficient if it is modified to allow for direct expensing (orwrite-up) of a constant share of all investments. However, cash accounting,under which investments are fully expensed, is informationally insufficient.I provide a valuation formula for replacement cost accounting with directexpensing. The coefficients on accounting variables in this formula dependon the parameters of the firm’s economic environment as well as on theshare of investments that is expensed. I formulate the uniqueness resultof my paper in a setting where assets start to generate capacity in the pe-riod when they are acquired, that is, assets do not require installation orconstruction time. Then, I show that replacement cost accounting with di-rect expensing is unique in its informational sufficiency. To prove this, Iinvoke a basic result from aggregation theory. Applications of this theoryhave a long tradition in accounting (e.g., Lev [1968] and Ijiri [1967]; seealso Arya et al. [2000] for a more recent reference). My uniqueness resultrests on the argument that, in order for financial statements to permit val-uation for a broad range of possible output market growth projections, aninformationally sufficient rule must preserve information about both theeconomic cost of the current period and the replacement cost of assets inplace. This information has to be preserved through accounting book valueand depreciation. This requirement implies that book values must always beproportional to the replacement cost of assets in place, that is, the rule inuse has to conform to replacement cost accounting with direct expensing.3

3 In a setting where assets require installation or construction time, I show that there may beadditional modifications to the replacement cost rule that preserve informational sufficiency.

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The notion of proper accrual accounting and the economic significanceof “good” accounting have been explored in different strands of the ac-counting literature. Beaver and Dukes [1974] identify depreciation sched-ules for various types of assets, such that accounting rates of return equalthe economic rates of return. Rajan, Reichelstein, and Soliman [2007] char-acterize the biases in accounting rates of return that result from applyingdepreciation schedules that do not match the productivity profile of assets.Rajan and Reichelstein [2009] investigate how the reported historical costof capacity depends on the underlying depreciation rules. In the manage-rial literature, certain accounting rules were shown to provide goal congru-ent incentives for a manager to whom investment decisions have been del-egated, but whose decisions are subject to potential horizon biases.4 Thispaper provides a new perspective on the usefulness of alternative accrualaccounting rules in the context of a forward-looking equity valuation prob-lem, absent any incentive or contracting issues.

The remainder of this paper is organized as follows. The next section il-lustrates the concept of informational sufficiency by means of an example.In section 3, the economic and reporting model of the firm is presentedand an expression for the firm’s value is derived. In section 4, I derive eq-uity valuation models for replacement cost accounting. Section 5 providesthe general concept of informational sufficiency and identifies accountingrules that have this sufficiency property. Concluding remarks are providedin the last section of the paper.

2. An Illustrative Example

This section presents an example that illustrates the concept of informa-tional sufficiency. The role of accrual accounting is to aggregate informa-tion about past transactions of the firm in a manner that preserves infor-mation essential to investors. The example shows that, in this regard, someaccounting rules are more effective than others. In particular, I considera simplified model in which a firm employs a single type of capital assetswhose productivity is constant over their useful life. First, it will be shownthat straight-line accounting is informationally insufficient for such assets,meaning that the valuation problem is not solvable if investors observe onlythe latest period financial statements based on straight-line depreciation.In contrast, an alternative depreciation policy, the annuity rule, aggregatespast investment data without a loss of value-relevant information, and en-ables investors to value the company based solely on current aggregate ac-counting data.

In particular, if assets come online with a lag of one period, then replacement cost account-ing accelerated by one period is informationally sufficient. This result shows that there maybe a mismatch between the useful and depreciable life of assets, and the current financialstatements may still provide enough information for valuation purposes.

4 See, for example, Rogerson [1997, 2008], Reichelstein [1997, 2000].

Consider a firm that invests in capacity, produces a single type of product,and then sells this product to an outside market. In the simplified model,capacity is generated by capital assets with a useful life of four periods. Eachasset is idle in the acquisition period and then provides capacity to produceone unit of output in each of the following four periods. This productivitypattern will be referred to as one-hoss shay productivity.5 The price of capitalassets is constant over time and is normalized to unity so that an investmentof I t−4 dollars in period t − 4 generates I t−4 units of capacity in periods t −3, t − 2, t − 1, t. The total capacity available in period t is therefore givenby

Kt = It−1 + It−2 + It−3 + It−4.

To make the valuation problem particularly simple, assume that the firmfaces a kinked demand curve of the following nature in its product market.Any quantity of output up to some maximum level, qmax , can be sold at aprice p that is high enough to cover the costs of production. Beyond qmax ,the product price declines so rapidly that the firm never finds it optimalto supply more than qmax . Further, assume that demand is stationary overtime. Under these assumptions, it is optimal for the firm to generate capac-ity of exactly qmax in every period and sell all its output at price p. Revenuesin period t + τ are then equal to

Rt+τ = p qmax = p Kt+τ .

The value of the firm at date t is defined as the present value of its futurecash flows:

Vt =∞∑

(Rt+τ − It+τ ) γ τ ,

where future investments are assumed to be chosen optimally, and γ = 11+r

is the appropriate discount factor. On the optimal path, K t+τ must be equalto qmax for any τ , so in each period the firm will exactly replace the invest-ment that goes offline in that period:

It+τ = It+τ−4,

for τ ≥ 1. Hence, starting with period t + 1, the optimal investment policycycles through investments I t−3, I t−2, I t−1, It , I t−3, and so forth. Under thispolicy, the firm’s value is completely determined by the history of its latestfour investments:

5 This term originates from the poem “The Deacon’s Masterpiece, or, the Wonderful One-Hoss Shay: A Logical Story,” written by Oliver W. Holmes in 1858, in which a shay is describedthat does not require repairs for 100 years and then falls apart “all at once, and nothing first.”The term is widely used in the economic literature on regulation.

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Vt = (p qmax − It−3)γ + (p qmax − It−2)γ 2 + · · ·

p qmax − (γ + γ 5 + · · ·)It−3 − · · · − (γ 4 + γ 8 + · · ·)It

p qmax − γ

1 − γ 4It−3 − γ 2

1 − γ 4It−2 − γ 3

1 − γ 4It−1 − γ 4

1 − γ 4It .

2.1 STRAIGHT-LINE DEPRECIATION

Assume that the company prepares financial statements in accordancewith the straight-line depreciation rule. Thus, assets are capitalized in theacquisition period and then depreciated evenly over the next four periods.At the end of period t, the accounting system reports the following infor-mation: revenues, Rt ,

Rt = p (It−1 + It−2 + It−3 + It−4),

depreciation, Dt ,

Dt = 14

It−1 + 14

It−2 + 14

It−3 + 14

It−4,

book value at date t, BVt ,

BVt = It + 34

It−1 + 24

It−2 + 14

It−3,

and cash flows to investments, It . Given clean surplus accounting, investorscan also infer the beginning of period book value of assets, BV t−1, as

BVt−1 = BVt + Dt − It .

Define the firm’s state at date t as the history of its past five investments,

θt = (It , It−1, It−2, It−3, It−4).

Note that all accounting numbers above are linear combinations of the lat-est five investments, and therefore are completely determined by the statevector θt .

The main idea of the example is to demonstrate that, depending on thechoice of accounting rules, current period financial statements may be suf-ficient or insufficient for valuation purposes. While straight-line deprecia-tion may seem to be a natural method to account for assets whose produc-tivity is constant over time, it turns out that this rule entails a loss of value-relevant information. To demonstrate this claim, it suffices to describe twohypothetical firms, operating in the same market, and generating identicalfinancial statements, yet, because of different investment histories, their in-trinsic values differ. Observing only the financial statements, investors willnot be able to figure out the underlying investment history, and, conse-quently, they will not be able to value the two firms correctly. Specifically,assume that qmax = 40 and consider the following two histories at date t:

θ(1)t = (10, 10, 10, 10, 10)

θ(2)t = (10, 15, 0, 15, 10).

Firm 1 is in its steady state and invests $10 in every period. Firm 2 cyclesthrough investments 15, 0, 15, 10, also implementing capacity of 40 in ev-ery period. At date t, these two histories lead to exactly the same financialstatements. Indeed, for both firms, we have

Rt = 40p , Dt = 10, BVt = 25, It = 10.

On the other hand, applying equation (1) to these histories, one can com-pute the valuations of the two firms at date t:

V (1)t = 1

rp qmax − 10

1 − γ 4− 10

1 − γ 4

V (2)t = 1

rp qmax − 15

1 − γ 4− 0 − 15

1 − γ 4− 10

1 − γ 4.

Since γ 2 < (γ + γ 3)/2, the value of firm 1 is greater than that of firm 2:

V (1)t > V (2)

However, observing only the latest financial statements, it is impossible toinfer which of the two firms generated them. Hence, the valuation problemis not solvable in this case, and straight-line depreciation is informationallyinsufficient for assets corresponding to the one-hoss shay pattern.

To further illustrate this insufficiency result, consider the application ofthe residual income valuation model to firms 1 and 2. Residual income inperiod t, RIt , is defined as the difference between revenues and the aggre-gate historical cost of capacity, Ht , the latter being the sum of depreciationand an imputed charge on the beginning of period book value:

RIt = Rt − Ht ≡ Rt − Dt − r BV t−1.

It is well known that, regardless of the accounting rules, value can be ex-pressed by means of the residual income valuation formula:6

Vt = BV t +∞∑

γ τ RIt+τ .

In our example, book values of both firms are equal to 25 at date t − 1.Given straight-line depreciation, residual income of firm 1 is constant overtime and always equal to

RI (1)t+τ = 40p − 10 − 25r.

6 See, for instance, Penman [2010].

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One can check that the second firm’s residual income in period t is alsoequal to 40p − 10 − 25r , but, from period t + 1 onwards, it will iteratethrough the following four values:

40p − 10 − 25r, 40p − 10 − 30r, 40p − 10 − 20r, 40p − 10 − 25r, . . .

Therefore, the future residual income processes of the two firms are dif-ferent, although they operate in equivalent environments and report thesame accounting numbers at date t. For this reason, the current financialstatements do not convey enough information to predict future residualearnings, even though investors may correctly anticipate future conditionsin the product market.

2.2 ANNUITY DEPRECIATION

I now consider an alternative accounting policy—the annuity depreci-ation rule. Assets are again capitalized at historical cost in the acquisitionperiod and then fully depreciated over their useful life, yet the depreciationcharges now compound at the rate of 1 + r :

dτ+1 = (1 + r )dτ , (2)

where dτ is the depreciation charge per one dollar of assets in period τ .Since assets are fully depreciated, the depreciation charges must satisfy therelation

d1 + d2 + d3 + d4 = 1. (3)

Equations (2) and (3) imply that

dτ = r γ 5−τ

1 − γ 4, (4)

for 1 ≤ τ ≤ 4. One can also check that the book value at date t of one dollarof assets acquired in period t − τ is given by

bvτ = 1 − d1 − · · · − dτ = dτ+1 + · · · + d4 = 1 − γ 4−τ

1 − γ 4,

for 0 ≤ τ < 4. Therefore, the aggregate book value and depreciation at datet are

BV t = It + 1 − γ 3

1 − γ 4It−1 + 1 − γ 2

1 − γ 4It−2 + 1 − γ

1 − γ 4It−3, (5)

Dt = r γ 4

1 − γ 4It−1 + r γ 3

1 − γ 4It−2 + r γ 2

1 − γ 4It−3 + r γ

1 − γ 4It−4. (6)

It should also be noted that, for any investment history, the aggregate his-torical cost of capacity in a given period is proportional to capacity utilizedin that period:

Ht+τ = Dt+τ + r BVt+τ−1

= r1 − γ 4

(It+τ−1 + · · · + It+τ−4)

= r1 − γ 4

Kt+τ . (7)

In later sections, this observation will be shown to be a special case ofa broader replacement cost accounting property for assets conforming tothe one-hoss shay pattern. To demonstrate this link, let me also describe theannuity rule by invoking the concept of a hypothetical perfectly competitiverental market for capital assets. If a perfect rental market were to exist, thenthe rental price of a unit of capacity, c, would be such that a rental firmwould exactly break even over time. If the rental firm invests one dollarin period t and then rents out the resulting capacity in the following fourperiods, its net present value is

−1 + γ c + γ 2c + γ 3c + γ 4c .

Since the rental market is assumed to be perfectly competitive, the expres-sion above must be equal to zero. Therefore, the competitive rental priceof a unit of capacity is given by:

c = 1γ + γ 2 + γ 3 + γ 4

= r1 − γ 4

Under replacement cost accounting, book value at date t of an asset pur-chased in period t − τ is defined as the fair value of that asset in the hypo-thetical rental market at date t. For τ < 4, a unit investment in period t −τ adds one unit of capacity in periods t + 1 , . . . , t + 4 − τ . To replace thisstream of capacity, the firm would need to incur the following cost:

γ c + · · · + γ 4−τ c = 1 − γ 4−τ

1 − γ 4= bvτ .

Therefore, the annuity rule corresponds to replacement cost accounting ifa perfect rental market exists. Also, combining equations (7) and (8), oneobtains

Ht+τ = cKt+τ . (9)

Thus, the annuity depreciation rule also implies that the aggregate histor-ical cost of capacity is equal to the replacement cost of capacity utilized inthe current period.

To show that the annuity rule is informationally sufficient, one can applythe residual income valuation approach. Property (9) implies that, on theoptimal path of future capacity levels, residual income is given by

RIt+τ = Rt+τ − Ht+τ = p Kt+τ − cKt+τ = (p − c)qmax, (10)

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for τ ≥ 0. Therefore, the firm’s value can be computed as follows:

Vt = BVt +∞∑

γ τ RIt+τ = BVt + 1r

RIt . (11)

Beginning-of-period book value can be recovered from the financial state-ments at date t in the following way:

BVt−1 = BVt + Dt − It .

Substituting this expression into (11), one obtains:7

Vt = BVt + 1r

(Rt − Dt − r BVt−1)

Rt − 1r γ

Dt + It .

OBSERVATION 1. Given annuity depreciation,

Vt = 1r

Rt − 1r γ

Dt + It .

Since the firm’s value can now be expressed in terms of the current ac-counting data, we have demonstrated the informational sufficiency of theannuity depreciation rule. Central to this result is the fact that, under theannuity rule, residual income is constant over time and equals

(p − c)qmax

for any optimized investment history. It will be shown below that (p −c)qmax can be viewed as the economic profit of the firm in all periods. Sinceit was assumed that both the capital and the product markets are station-ary, the fact that optimized economic profits are also stationary should notcome as a surprise. In this simplified environment, I showed that, underannuity depreciation, the current residual income provides sufficient in-formation for predicting future residual incomes. This contrasts with thefindings in the straight-line scenario, where current accounting data wereinsufficient for predicting future residual earnings.

To conclude the discussion of this example, it is instructive to comparereplacement cost accounting with fair value accounting. When the productprice is high enough to cover the marginal costs of production, specificallywhen p is greater than c, book values under the annuity rule will be lessthan the present values of future cash flows, since residual earnings will bealways positive by equation (10). Therefore, the annuity rule is more conser-vative than fair value accounting in the sense that accounting book valuesare always below market values. It is also interesting to note that, when the

7 In this example, the book value of assets drops out from the valuation formula becausethe product market is assumed to be stationary. The general model allows for growth in theproduct market, and then the coefficient on the book value in the valuation formula will bedifferent from zero.

product market is stationary, the difference between market values and ac-counting book values under replacement cost accounting is constant overtime. Therefore, by the “Canceling Errors” theorem, accounting earningsunder the annuity rule are equal to the economic, or permanent, earnings.In particular, it can be verified that, under the annuity rule,

Vt = 1r

(Rt+1 − Dt+1),

that is, the permanent earnings model holds.8

3. Model Description

3.1 TRANSACTIONS

Consider a firm that invests in a single type of long-lived assets and pro-duces a single output good. I will assume that the cash cost of a unit ofinvestment, pt , changes geometrically over time:9

p t = p 0εt .

Values of ε less than one may represent technological progress, while valuesgreater than one may model inflation. Let period t be the interval betweendates t − 1 and t, and let It be the investment in that period measured inphysical units. Then, the investment cash outflow in period t is

CFI t = It p t .

A unit of asset purchased in period t generates capacity to produce xτ unitsof output in period t + τ for 0 ≤ τ ≤ T . I allow for the possibility thatassets require construction or installation during which time they are notproductive, x0 = · · · = xL−1 = 0. After the initial lag of L periods, assets startto generate capacity, and, from that time, their productivity weakly declinesover their useful life, xL ≥ xL+1 ≥ · · · ≥ xT > 0. The total capacity availablein period t, Kt , is given by

Kt = It x0 + It−1x1 + · · · + It−T xT .

The vector x ≡ (x0, . . . , xT ) will be referred to as the asset’s productivityprofile and the vector θt ≡ (CFI t , . . . , CFI t−T ) will be referred to as the

8 It is well known that, if the firm makes constant investments, then earnings are the sameirrespective of accounting rules. In the example considered in this section, I assume that thefirm generates constant capacity in every period. This assumption can be satisfied not only bythe constant investment trajectory, but also by infinitely many others, such as the trajectory θ(2)

described earlier. Under the annuity rule, the permanent earnings model holds for any suchinvestment trajectory. It can be easily verified that the annuity rule is the only depreciationrule with this property.

9 I present the model as one of certainty about the conditions that the firm experiences inthe future in its capital asset and product markets. Section 6 discusses several ways to introduceuncertainty into the model.

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relevant investment history at date t. If all xτ for τ ≥ L are equal, I willcall the productivity profile the one-hoss shay pattern.

In period t, the firm faces an inverse demand curve, Pt(qt), that definesprice as a function of the quantity of product sold, qt . Let Rt(qt) denotethe corresponding revenue function, which is assumed to be concave anddifferentiable in qt :

Rt (qt ) = Pt (qt )qt .

In the interest of parsimony, I assume that the firm is all-equity financed,there are no operating expenses other than depreciation, and all free cashflows are disbursed to shareholders immediately. Investors are interestedin valuing the company under the assumption that managers have perfectinformation and always act in the best interests of the firm’s owners.

For notational convenience, let T be the valuation date.10 An investor’svaluation of the firm at date T is the present value of its future cash flowsunder the optimal investment and production policies:

VT = max{qT+τ ,IT+τ },τ=1,...

∞∑τ=1

γ τ (RT+τ (qT+τ ) − p T+τ IT+τ ) (12)

subject to

IT+τ ≥ 0, (13)

qT+τ ≤ KT+τ , (14)

where γ = 11+r , r is the firm’s cost of capital. I assume that, at date T , the

firm has a history of at least L investments that were also made optimally.This assumption is made to ensure that the firm will have assets in produc-tive use in period T + 1. I further assume that the rate of inflation, ε − 1,does not exceed the firm’s cost of capital, r , so it is never optimal for thefirm to purchase assets more than L periods ahead of their first use.

Following Arrow [1964] and Rogerson [2008], I will first ignore con-straint (13) and characterize the optimal investments in the relaxed op-timization problem (12, 14), I ∗

t . Then, I will state Assumption 1, which willbe sufficient to ensure that constraint (13) is satisfied for I ∗

t . This will im-ply that, under Assumption 1, investments I ∗

t also represent the optimalinvestment policy in the original problem (12)–(14).

Define the marginal cost of capacity in period t, ct , as the incremental costto the firm, in period t dollars, of generating one additional unit of capacityin that period, holding fixed capacity levels in all other periods. To gener-ate one unit of capacity in period t, the firm needs to invest pt−L/xL dollarsin period t − L. However, this investment will generate capacity not only in

10 I do not exclude the possibilities that the firm starts operations before or after period 0,so there is no loss of generality in letting T be the valuation date.

period t but also in the following T − L periods. One can construct a seriesof adjustments to investments I t−L+1 and onwards that will ensure that ca-pacity levels beyond period t are unaffected. The marginal cost of capacityrepresents the present value (in period t dollars) of the initial investmentin period t − L and all these adjustments to future investments.

Since the productivity profile is invariant over time and the price of assetschanges geometrically, one might expect the marginal cost of capacity tochange geometrically as well. Rogerson [2008] showed that the marginalcost is given by11

c t = p t

x0 + x1εγ + · · · + xT εT γ T. (15)

As one might expect, when the cost of investments is constant over time,the marginal cost of capacity is also constant and given by12

c = px0 + x1γ + · · · + xT γ T

. (16)

The intuition behind equation (15) can be demonstrated by consider-ing again the notion of a hypothetical rental market for capital assets andassuming that this market is perfectly competitive. As in the numerical ex-ample in section 2, ct can be shown to be equal to the rental price per unitof capacity at which a competitive rental firm exactly breaks even over time.Indeed, assume that the rental firm invests pt dollars in period t and rentsout the resulting capacity in periods t through t + T . Then, present valueof its cash flows is

−p t + x0c t + x1c t+1γ + · · · + xT ct+T γ T

= −p t + c t (x0 + x1εγ + · · · + xT εT γ T ).

The value of ct in expression (15) equates the present value of the rentalfirm’s revenues with the costs to generate those revenues. Since the rentalbusiness is competitive, the producing firm is indifferent between investingin assets at unit price pt and renting capacity at unit price ct . Hence, by in-ternalizing a cost of ct per unit of capacity, the producing firm will generatethe first-best investment policy in acquiring new capacity. On the optimalpath, capacity levels K ∗

t must be chosen so that the marginal revenue froman additional unit of capacity is equal to the marginal cost of that unit, ct :

R ′t

(K ∗

) = c t . (17)

11 Arrow [1964] provided a general formula for the marginal cost of capacity in a settingwhere the cost of investment follows any arbitrary process. Application of this general formulais difficult because it involves an infinite series of recursively defined functions. Arrow [1964]further showed that a simple expression for the marginal cost exists if one assumes that assetshave infinite useful lives and their productivity declines geometrically. Rogerson [2008] de-rived formula (15), which holds for any productivity profile, but relies on the assumption thatthe cost of investment follows a geometric process.

12 When the cost of new investments is constant over time, I will drop subscripts on p and c.

248 A. NEZLOBIN

With an implicit reference to the notion of a hypothetical rental market, Iwill call ctKt the replacement cost of capacity in period t. Since ct is the marginalcost of capacity, it is also natural to label ctKt the economic cost, and thedifference between revenues and ctKt—the economic profit of the firm:

πt = Rt (Kt ) − c t Kt .

Let I ∗t be the minimum (possibly negative) investments required to gen-

erate capacity levels K ∗t defined by (17). Since asset productivity is weakly

declining after the initial lag of L periods, investments I ∗t will be nonnega-

tive if capacity levels K ∗t weakly increase over time. The following assump-

tion is sufficient to ensure that investments I ∗t are nonnegative and, there-

fore, solve the original problem, (12)–(14).

ASSUMPTION 1 (No-Excess Capacity Condition). Marginal revenues ad-justed for growth in the cost of investments are nondecreasing at any capacity level:

R ′t (q )εt

≤ R ′t+1(q )εt+1

for any q, t.

Under Assumption 1, the firm will invest so as to generate capacity levelsK ∗

t in every period. Once the capacity is in place, it will be fully utilized,since marginal revenues are greater than zero at K ∗

t . I summarize the dis-cussion above in Proposition 0 that is due to Rogerson [2008].

PROPOSITION 0. Under Assumption 1, the optimal capacity levels are character-ized by

R ′t

(K ∗

) = c t

for t ≥ T. The firm will operate at full capacity in every period, qt = K ∗t .

In the formal model of Rogerson [2008], the firm enters the initial pe-riod with no existing capital. In contrast, I allow for the firm to have acertain endowment of capacity assets at the valuation date. As was demon-strated in the numerical example, two firms operating in the same eco-nomic environment and implementing equal capacity in a given and allfuture periods, can have different valuations. This difference reflects thatthe composition of assets in place differs for the two firms at a given date.Clearly, if two firms utilize the same amount of capacity in period t, butfirm 1’s capacity is newer than that of firm 2, then firm 1 will have a greaterintrinsic value. Intuitively, assets in place can be essentially viewed as cost-savings in future periods. To quantify the benefits of owning a certain en-dowment of assets, I will define the replacement cost of assets in place at date tas follows.

Consider a dollar of investment made in period t − τ , where 0 ≤ τ ≤ T .This investment is still productive at date t and will generate the followingstream of capacity in periods t + 1 , . . . t + T − τ :

1p t−τ

xτ+1, . . . ,1

p t−τ

The value of this stream of capacity in a hypothetical rental market is givenby

vτ = γ c t+11

p t−τ

xτ+1 + γ 2c t+21

p t−τ

xτ+2 + · · · + γ T−τ c t+T1

p t−τ

I will call vτ the replacement cost factor of a dollar investment of age τ . Substi-tuting for ct+i , this replacement cost factor can be simplified to

vτ = xτ+1ετ+1γ + · · · + xT εT γ T−τ

x0 + x1εγ + · · · + xT εT γ T. (18)

The aggregate replacement cost of assets in place at date t is the sum of replace-ment costs of assets still productive at that date:

RCt = v0CFI t + · · · + vT CFI t−T = v · θt ,

where v = (v0, . . . , vT ). The replacement cost of assets in place representsthe benefit to the firm of owning a particular endowment of assets if thefirm were to sell the capacity generated by these assets at zero economicprofits. Alternatively, it can be viewed as the present value of rental pay-ments that the firm would have to make in a perfectly competitive rentalmarket to replicate the capacity that its assets in place at date t will gener-ate in the future. The following proposition expresses the firm’s value as afunction of its past cash investments and future optimized capacity levels.13

PROPOSITION 1. Under Assumption 1, the value of the firm at date T is equalto the sum of the replacement cost of assets in place and the present value of futureoptimized economic profits:

VT = RCT +∞∑

γ τ(RT+τ

(K ∗

) − cT+τ K ∗T+τ

). (19)

Proposition 1 separates the firm value into two components. The firstcomponent, the replacement cost of assets in place, is a function of therelevant investment history and is independent of the future market con-ditions. In contrast, the second component, the present value of futureeconomic profits, is determined only by future market conditions and isindependent of the firm’s investment history. This separation holds as longas Assumption 1 is satisfied and the firm is on the optimal capacity path.

13 Related results were established in the economics and finance literature; see, for exam-ple, Thomadakis [1976] and Lindenberg and Ross [1981]. However, in these studies, capitalassets are usually assumed to be infinitely lived and their productivity is assumed to declinegeometrically over time. For the purposes of this paper, it is important to state the result inProposition 1 for general productivity patterns.

250 A. NEZLOBIN

To illustrate the economic meaning of the replacement cost of assets inplace, assume that assets become productive immediately, L = 0, and con-sider a firm that does not own any assets at date T . Let V 0

T denote themarket value of this firm, and let VT be the market value of a firm thatowns some assets at date T and faces equivalent product market conditions.Clearly, V 0

T ≤ VT , since the firm that already owns certain assets will nothave to invest as much as the firm that is just about to start its operations.Proposition 1 then implies that RCT = VT − V 0

T ; that is, the valuation of thefirm with assets will exceed the valuation of the firm without assets by pre-cisely the replacement cost of assets in place of the former firm. Therefore,the replacement cost of assets in place is equal to the economic benefit tothe firm of owning those assets.14

3.2 FINANCIAL REPORTING

The accounting system aggregates the information about past transac-tions of the firm into the financial statements. Depreciation is the onlyaccrual considered in the model. Therefore, investors learn the followingfour numbers from the income statement and the balance sheet at datet: revenues for the latest period, Rt ; depreciation, Dt ; net income, Inct =Rt − Dt ; and book value of assets at date t, BVt . Investors further learnthe latest investment cash outflow, CFIt , from the statement of cash flows.I will assume that depreciation is computed according to a fixed scheduled = (d0, . . . , dT ). This schedule can be tailored to the anticipated physicaldecay of assets in the sense that d can depend on the productivity profile x .Total depreciation in period t then becomes

Dt = d0CFI t + · · · + dT CFI t−T = d · θt .

Let bvτ denote the share of investment CFIt that remains capitalized atthe end of period t + τ , and let bv = (bv0, . . . , bvT ). I will restrict attentionto depreciation rules satisfying the usual “tidiness” requirement that

bvτ = 1 −τ∑

andT∑

di = 1.

Hence, bvT = 0. Thus, the book value of each investment changes onlydue to depreciation charges related to that investment, and assets are fullydepreciated over their useful life. In this notation, the aggregate book valueat date t is equal to

BVt = bv0CFI t + · · · + bvT CFI t−T = bv · θt .

14 I am thankful to the anonymous referee for suggesting this interpretation.

To summarize, investors observe the following information set at date t:15

It = {Rt , Dt , BVt , CFI t }.It will be convenient to define the aggregate historical cost of capacity in

period t, Ht , as the sum of depreciation expense and a cost of capital chargeon the beginning book value of assets:

Ht = Dt + r BVt−1.

The historical cost is a function of the state vector at date t, since both Dt

and BV t−1 are determined by that vector:

Ht = d0CFI t + (d1 + r bv0)CFI t−1 + · · · + (dT + r bvT−1)CFI t−T .

Let z0 = d0 and zτ = dτ + rbvτ−1 for 0 < τ ≤ T and let z = (z0, . . . , zT ) bethe vector of historical cost charges. Then,

Ht = z · θt .

Prior literature has established that there exists a one-to-one mappingbetween depreciation charges and historical cost charges (see e.g., Roger-son [1997] and Reichelstein [1997]). In particular, it can be shown that thedepreciation vector will satisfy the clean surplus condition if and only if thecorresponding z -vector satisfies

T∑τ=0

zτ γτ = 1. (20)

Following Rogerson [2008], I now define replacement cost accounting, orthe replacement cost rule, in terms of its corresponding z -vector and thencheck that condition (20) is satisfied. Let z∗

τ be equal to the replacementcost of capacity provided in period t by a dollar investment made in periodt − τ :

z∗τ = 1

p t−τ

c t xτ = xτ ετ

x0 + x1εγ + · · · + xT εT γ T.

Intuitively, this rule allocates historical cost charges to a particular period,τ , in proportion to the capacity that the asset generates in that period,adjusted for growth or decline in the cost of new investments. Condition(20) is satisfied for the vector z∗:

T∑τ=0

z∗τ γ

τ = 1(x0 + x1εγ + · · · + xT εT γ T )

T∑τ=0

xτ ετ γ τ = 1.

15 Net income is contained in It , but it has no incremental information content beyond rev-enues and depreciation. Also, I will routinely assume that BV t−1 is in It , since BV t−1 = BVt +Dt − CFIt .

252 A. NEZLOBIN

Given this rule, the aggregate historical cost in period t is indeed equalto the replacement cost of capacity for any investment history:

H ∗t = z∗ · θt =

T∑τ=0

1p t−τ

c t xτ p t−τ It−τ = c t Kt . (21)

Define residual income in period t to be the difference between revenuesand the historical cost of capacity:

RIt = Rt − Dt − r BVt−1. (22)

Let d∗, bv∗ denote the depreciation and book value schedules correspond-ing to z∗ and let D∗

t , BV ∗t be the aggregate depreciation and book values

under this rule. It can be verified that if assets have one-hoss shay produc-tivity and become productive in the period following their acquisition, thenthe replacement cost rule corresponds to the annuity method.16

By (21), residual income under the replacement cost rule is equal to thefirm’s economic profits:

RI ∗t = Rt − H ∗

t = Rt (Kt ) − c t Kt (23)

for any t. This is precisely the property of the z∗-rule identified in Rogerson[2008].

Proposition 1 implies that replacement cost accounting possesses an-other useful property. Note that, on the optimal investment path, thepresent value of future optimized economic profits will be equal to thepresent value of future residual incomes calculated under replacement costaccounting:

∞∑τ=1

γ τ(RT+τ

(K ∗

) =∞∑

γ τ RI ∗T+τ .

By the residual income valuation formula, the firm’s value at date T can beexpressed as

VT = BV ∗T +

∞∑τ=1

γ τ RI ∗T+τ .

Therefore, Proposition 1 implies that, on the optimal investment path,book values corresponding to the z∗-rule are equal to the replacement costof assets in place, that is,

BV ∗T = RCT ,

hence the term replacement cost accounting.

16 If assets require more than one period for installation or construction, then this rule callsfor writing up the assets in periods preceding their productive use. These write-ups reflect theapproaching generation of economic benefits by the assets.

COROLLARY 1. Given the z∗-rule, the book value of assets is equal to the replace-ment cost of those assets:

BV ∗T = RCT

bv∗ = v.

To illustrate this result, assume again that L = 0 and consider two firmsthat face equivalent market conditions—one without assets at date T andthe other owning some endowment of assets at that date. By Proposition 1,the difference in market values between the two firms is equal to the re-placement cost of the second firm’s assets in place. Corollary 1 then impliesthat the difference between the market values of the two firms is equal tothe book value of assets of the second firm calculated under the z∗-rule. Inother words, given replacement cost accounting, the book value of assetsprecisely quantifies the benefits of owning those assets.

In much of the discussion, the replacement cost rule was described witha reference to the rental market for capital assets. It is important to notethat under the no-excess capacity condition (Assumption 1), the existenceof such a market is not required. When the product market expands, thefirm seeks to increase its capacity in every period. Under this assumption,the firm will never find it desirable to rent out its capacity. Therefore, theabsence of a rental market does not affect the firm’s first-best investmentpolicy nor does it affect the firm’s valuation. Replacement cost accountingmay then be viewed as a particular depreciation rule corresponding to his-torical cost accounting.

4. Equity Valuation and Replacement Cost Accounting

4.1 STATIONARY COST OF INVESTMENTS

In valuing the company, investors seek to estimate (i) the replacementcost of assets in place and (ii) the present value of future optimized eco-nomic profits. Given replacement cost accounting, the former componentof the firm’s value is reported in the financial statements as the book valueof assets. Estimating future economic profits naturally requires an assess-ment of how the product market conditions evolve over time. Moreover,even when the product market is stationary, changes in the cost of newinvestments will lead to changes in the marginal cost of capacity, optimalproduct prices, and optimal capacity levels. Therefore, future economicprofits are affected by changes in both input and output markets. In thissubsection, I will consider the case when the cost of new investments is con-stant over time, ε = 1. The following assumption will be imposed on theevolution of the firm’s inverse demand functions.17

17 A similar assumption is invoked in Nezlobin, Rajan, and Reichelstein [2011].

254 A. NEZLOBIN

ASSUMPTION 2. Market demand evolves such that

Pt+1 ((1 + μt+1) q ) = Pt (q ) (24)

for all q, t, where μt+1 ≥ 0 is the product market growth rate between period t andperiod t + 1.

Note that, while Assumption 2 constrains the evolution of demand curvesfrom one period to the next, it does not impose restrictions on their struc-tural form. In particular, this assumption can be satisfied by the constantelasticity demand curves with a stationary elasticity parameter or the lineardemand curves with the same intercept term.

Assumption 2 states that, for all price points, demand increases at therate μt+1 from period t to period t + 1. Let μT+1 = (μT+1, μT+2, . . .) de-note the sequence of market growth rates after the valuation date. The keyproperty of the proportionate growth parametrization is that the future op-timal prices and capacity levels can be expressed as simple functions of thecurrent optimal capacity and price. To see this, recall that the optimal ca-pacity level in period t, K ∗

t , maximizes the economic income in that period:

πt = Pt (Kt )Kt − cKt .

The economic profit in period t + 1 can be expressed as:

πt+1 = Pt+1(Kt+1)Kt+1 − cKt+1

= (1 + μt+1)[

1 + μt+1

1 + μt+1− c

1 + μt+1

It is readily seen that the economic profit in period t + 1 is maximized atK ∗

t+1 = (1 + μt+1)K ∗t . Therefore, the optimal prices are constant over time,

Pt+1(K ∗

) = Pt(K ∗

and optimized economic profits, π∗t , grow at the same rate as the product

market:

π∗t+1 = (1 + μt+1)π∗

To summarize, under the proportional growth assumption, optimal prod-uct prices are constant and optimal capacity levels, revenues, and economicprofits grow at the same rate as the demand for the firm’s product. It shouldbe noted that, to make such projections, investors do not need to know thestructural form of the inverse demand curves.

From now on, I assume that, at date T , the firm has at least one periodof revenue data. This assumption implies that the firm must have made itsfirst capacity investment not later than in period T − L. Given replacementcost accounting, investors know that the economic income in the periodimmediately preceding the valuation date is equal to the residual incomein that period:

π∗T = RI ∗

To project future economic incomes, it suffices to apply the product marketgrowth rates to residual income in period T :

π∗T+τ = RI ∗

T+τ = RI ∗T

τ∏i=1

(1 + μT+i ).

Note that Assumption 2 requires that the product market be expand-ing over time (μt ≥ 0), so the optimal capacity levels are nondecreasing.Therefore, Assumption 2 supersedes Assumption 1 and Proposition 1 canbe applied.

PROPOSITION 2. Assume that the cost of new investments is constant over timeand inverse demand curves satisfy Assumption 2. Then, the firm’s value is given by

VT = BV ∗T + αT RI ∗

T , (25)

where αT = ∑∞τ=1(γ τ

∏τi=1(1 + μT+i)).

Proposition 2 shows that, under replacement cost accounting, the firmcan be valued using the residual income valuation formula, where currentperiod residual income is used to project future period residual incomes.This implies that replacement cost accounting aggregates the firm’s trans-action history into financial statements at no loss of value-relevant informa-tion. The numerical example in section 2 demonstrated that not every ac-counting rule has that property. In particular, if straight-line depreciation isapplied to assets with one-hoss shay productivity, then it may be impossibleto project future residual incomes based on aggregate accounting informa-tion and product market growth rates.

An interesting special case of formula (25) obtains if one assumes thatdemand is stationary, μT+τ = 0 for all τ . Then, the firm’s residual incomeunder replacement cost accounting is constant over time:

RI ∗T+τ = RI ∗

Therefore, the firm’s value can be expressed as

VT = BV ∗T + 1

rRI ∗

T = BV ∗T + 1

rRI ∗

T+1 = 1r

I nc∗T+1,

which is known as the permanent earnings model.

COROLLARY 2. If the product market is stationary, then

VT = 1r

I nc∗T+1.

It is well known that the permanent earnings model arises under fairvalue accounting, that is, when the book value of assets is set equal to thepresent value of the firm’s cash flows in every period. Note, however, thatthis is not the case in the context of Corollary 2, since, by Proposition 2,

BV ∗T ≤ VT .

256 A. NEZLOBIN

Proposition 1 states that the difference between the intrinsic value of thefirm and the replacement cost of its assets is equal to the present valueof future economic incomes. Absent growth in the product market, thisdifference is constant over time:

VT − BV ∗T = VT+1 − BV ∗

Therefore, by the Canceling Errors Theorem, earnings under replacementcost accounting will be equal to the permanent earnings.

4.2 TIME-DEPENDENT INVESTMENT COST

In this subsection, I revert to the assumption that the cost of new invest-ments changes geometrically over time:

p t = p 0εt .

Under this assumption, the optimal product price, characterized by

P ′t

(K ∗

) = c t , (26)

will generally also be time-dependent. Therefore, to project economic in-come in a future period, investors need to account for both the change inthe optimal product price and the change in demand at the new price level.Assumption 2 alone is insufficient to formulate an answer to this predictionproblem, since the change in the optimal product price generally dependson the functional form of inverse demand curves.

A convenient assumption that makes the prediction problem analyticallytractable is that demand for the firm’s product increases proportionately atall real price levels. Formally, I state this assumption as follows:18

ASSUMPTION 3. Market demand evolves such that

Pt+1((1 + μt+1)q ) = εPt (q ) (27)

for all q, t, where μt+1 ≥ 0 is the demand growth rate calculated at any given realproduct price from period t to period t + 1.

Similarly to Assumption 2, Assumption 3 does not restrict the functionalform of inverse demand functions. In particular, linear inverse demandfunctions,

Pt (Kt ) = at − btqt ,

satisfy Assumption 3 if the parameters at and bt evolve such that at+1 = εat

and bt+1 ≤ εbt . Also, this assumption can be satisfied if the demand curvesare of constant elasticity form:

qt = mt Pt (qt )−η,

18 I thank the referee for suggesting this assumption, which is more general than the oneemployed in earlier versions of the paper.

where η > 1 is the price elasticity of demand and mt is the market size.Given the constant elasticity curves, demand will expand at real price levelsand the nonnegativity condition on μt in Assumption 3 will be satisfied aslong as mt+1 ≥ εηmt for all t.

In the proof of Proposition 3, I show that Assumption 3 implies that thereal product price is constant over time; that is, the nominal product pricechanges at the same rate as the cost of investment:

Pt+1(K ∗

) = εPt(K ∗

Since the real product price is constant, the optimal capacity levels will growat rates {μt}:

K ∗t+1 = (1 + μt+1)K ∗

Therefore, economic income will evolve according to the following process:

π∗t+1 = Pt+1

(K ∗

)K ∗

t+1 − c t+1K ∗t+1 = ε(1 + μt+1)π∗

t . (28)

Given replacement cost accounting, economic income is equal to resid-ual income for any history of investments, and the following propositionholds.

PROPOSITION 3. Under Assumption 3, the firm’s value is given by

T , (29)

αT =∞∑

(γ τ ετ

τ∏i=1

(1 + μT+i )

For certain values of parameters, valuation formula (29) may again re-duce to the permanent earnings model. For example, if ε < 1 and μt isconstant and satisfies

1 + μt = 1ε,

then the effects of market expansion and decreasing nominal prices onresidual income will exactly offset each other. The optimized residual in-come will then be constant over time, and, therefore, it will be the casethat

VT = I nc∗T+1

and earnings calculated under the replacement cost rule will be equal tothe permanent earnings.

It may appear that equation (28) and Proposition 3 imply that the higherthe growth rate of the cost of investment, ε, the faster the firm’s economicincome will grow and, therefore, the higher the firm’s value will be. Recall,however, that the growth rates μt were defined in Assumption 3 to mea-sure the growth in demand at given real product prices. Therefore, for a

258 A. NEZLOBIN

given sequence of inverse demand curves, Pt(·), higher values of ε will cor-respond to lower values of μt . To illustrate this point, consider again thescenario with inverse demand curves of constant elasticity form:

qt = mt Pt (qt )−η , (30)

where η > 1 is the price elasticity of demand and mt is the market size.Assume that, holding the nominal product prices fixed, the market grows atrates μ0

t (that is, μ0t are the growth rates corresponding to Assumption 2):

mt = (1 + μ0

)mt−1,

for all t. Then, for a given value of ε, the following equation holds:

εPt−1 (q ) = εm1/η

t−1q −1/η = ε(1 + μ0

)−1/ηm1/η

t q −1/η = Pt

((1 + μ0

From the equation above and the definition of μt in Assumption 3, it fol-lows that

1 + μt =(1 + μ0

and, therefore, μt is decreasing in ε. Holding the product price in periodt − 1 fixed, larger values of ε result in higher nominal product prices inperiod t, and, correspondingly lower quantities of the product demandedin that period. Therefore, the growth in demand measured at fixed realproduct prices will be lower for larger values of ε.

To conclude this section, I formulate a corollary that restates the valu-ation formula in Proposition 3 in terms of product market growth ratesmeasured at fixed nominal prices.

COROLLARY 3. If revenue functions are of constant elasticity form (30) and de-mand measured at fixed nominal prices grows at rates satisfying

1 + μ0t ≥ εη,

then the firm’s value is given by

αT =∞∑

(γ τ ε−τ(η−1)

τ∏i=1

(1 + μ0

))(31)

η = RT

RI ∗T

Corollary 3 uses a well-known result from the theory of monopoly pricingthat states that the optimal product price satisfies

Pt(K ∗

) − c t

Pt (K ∗t )

= 1η. (32)

The quantity in the left-hand side of (32) is known as the Lerner indexof monopoly power and the quantity in the right-hand side is the inverseof the price elasticity of demand. Given replacement cost accounting, theLerner index can be calculated from the financial statements as the ratioof residual income to sales:

RI ∗t

(K ∗

) − c t)

K ∗t

Pt (K ∗t ) K ∗

The assumption that demand elasticity is constant over time is equivalentto assuming that the ratio of residual income to sales is constant underreplacement cost accounting.19

Assuming that the firm uses replacement cost accounting and that in-vestors can project product market growth rates at fixed nominal prices,Corollary 3 suggests that investors can employ the following valuation pro-cedure in the constant elasticity scenario. First, they can calculate the priceelasticity of demand as the ratio of revenues to residual income. Using thisinformation, investors can calculate the capitalization factor, αT , accordingto equation (31), and apply this factor to the last realized residual incometo calculate the present value of future economic profits. Finally, the valueof the firm can be calculated as the sum of the replacement cost of its assetsand the present value of future economic profits. Note that, while investorsneed to know the future growth rates of the product market, they do notneed to observe the absolute size of the market, mt , at the valuation date.Neither do they have to know the price elasticity of demand, since it canbe inferred from the financial statements. Finally, it suffices to observe onlythe aggregate revenues, and not the capacity and the product price sepa-rately.

5. Informational Sufficiency of Depreciation Rules

5.1 INFORMATIONAL ASSUMPTIONS

While standard-setters generally agree that providing information usefulto equity investors is one of the most important objectives of financial re-porting, theoretical literature on firm valuation offers little guidance on therelative advantages of alternative accounting rules. Most accounting valua-tion models do not explicitly impose limits on the information available toinvestors, and the valuation problem is usually solved on the basis of perfectinformation. If investors have access to the whole transaction history of thefirm and can observe all the parameters of the firm’s economic environ-ment, then financial statements cannot carry any additional informationirrespective of the accrual rules in use.

19 Rajan and Reichelstein [2009] study how the accounting profit margin, defined as the ra-tio of residual income to sales, deviates from the Lerner index under alternative depreciationrules.

260 A. NEZLOBIN

In this section, I introduce into the model informational asymmetry be-tween investors and the firm’s manager. My goal is to characterize the setof accrual rules that facilitate parsimonious equity valuation models. I call avaluation model parsimonious if it takes as inputs only the current account-ing variables and does not require knowledge of the whole investment his-tory of the firm. I further assume that investors have limited knowledge ofthe firm’s operating environment: while investors can project the growthin the firm’s product market, they do not know the inverse demand curves.Therefore, I restrict attention to valuation models that do not depend onthe functional form of inverse demand curves, but may take as input thegrowth in demand for the firm’s product.

Propositions 2 and 3 showed how this problem of equity valuation basedon limited information can be solved if replacement cost accounting isused. I refer to the accounting rules that aggregate past investments at noloss of value relevant information as informationally sufficient. In contrast,the numerical example in section 2 demonstrated that, when the straight-line depreciation rule is applied to investments with one-hoss shay produc-tivity, it is impossible to value the firm observing only the current financialstatements, even when it is known that the product market is stationary.In other words, in such a scenario, there may exist several candidate solu-tions for the firm’s value consistent with the latest financial statement, andit may be impossible to identify the correct candidate without access to thewhole history of the firm’s investments. This example demonstrates that thestraight-line depreciation rule is informationally insufficient for assets withone-hoss shay productivity.

I now specify in more detail information available to different parties.I assume that, at date t, the manager has perfect information about pasttransactions of the firm and knows the inverse demand functions up toperiod t + L. This information allows the manager to implement the pathof the first-best investments. The accounting system tracks all transactionsof the firm and takes into consideration the asset’s productivity profile, x .This knowledge can be used in choosing an appropriate depreciation rulefor the firm’s assets. Investors face the problem of determining VT on thebasis of the information provided in the latest financial statements,

IT = {RT , DT , BVT , CFI T } .

Investors are not knowledgeable about the exact shape of demand curves,but they assume that demand will grow proportionately at all price levels,according to the vector of growth rates μt+1 = (μt+1, μt+2, . . .). I allow forthe possibility of investors observing the productivity profile of assets andthe depreciation rule in use. To avoid unnecessary notational burden, Irestrict attention to the case of constant cost of investment.

While investors know aggregate revenues for the latest period, I assumethat they do not observe separately the price and quantity of the prod-uct sold. In particular, the informational structure described precludes in-vestors from inferring capacity costs from revenues only. For example, if

investors knew the product price, PT (KT ), and the cost of investment,p, they would be able to infer the marginal cost of capacity using equa-tion (16) and, then, calculate the aggregate economic cost by dividingrevenues by the product price and multiplying the resulting capacity by c.Alternatively, if they knew the exact shape of the demand curve, they couldinfer c as the marginal revenue at the optimal point. Since my interest isin modeling the accounting system as the primary vehicle of informationtransfer to investors, both of these inferences are rendered infeasible.20

Finally, I assume that future growth rate projections, μt+1, cannot beincorporated into the financial statements at date t, either because theseprojections are not considered verifiable to the accounting system, or be-cause these projections may be investor-specific and the accounting systemneeds to accommodate all heterogeneous investors. A consequence of thisassumption is that fair value accounting, under which BVt is equal to Vt , isrendered infeasible, because Vt inherently depends on μt+1.

The discussion above motivates the following definition.

DEFINITION 1. A depreciation rule, d, is said to be informationally sufficient ifthere exists a function V̂T (IT ,μT+1)such that

VT (θT , PT (·),μT+1) = V̂t (IT ,μT+1)

for any inverse market demand function PT (·), any investment history θT , and anygrowth rate projections μT+1.

Accounting rules are called informationally sufficient if there exists avalue estimate that uses as inputs only the information available to investorsat date T , and that captures the value correctly for any projection of growthin the product market. If the accounting rules used for financial reportingdo not meet this criterion, then the valuation problem is generally not solv-able on the basis of limited information.

5.2 A UNIQUENESS RESULT

Proposition 2 has established that replacement cost accounting is infor-mationally sufficient. It turns out that there is a degree of freedom associ-ated with replacement cost accounting that does not impede its informa-tional properties. This degree of freedom corresponds to the possibility ofpartial direct expensing (or write-ups) in the acquisition period. If investorsknow that a constant share of all investments is expensed in the acquisitionperiod and replacement cost accounting is applied to the capitalized share,then they can undo the effects of such expensing by adjusting the coeffi-cients on accounting variables in the valuation formula.

20 For the argument in this section, it is sufficient to assume that investors do not know atleast one of the following: the price of the product, PT (KT ); the cost of investment, p; or theproductivity profile, x . If investors know all of these parameters, they can infer the efficiencyof the firm’s technology and they can calculate the economic income from revenues only.

262 A. NEZLOBIN

Formally, consider the class of depreciation rules for which

bv = λbv∗,

where λ > 0 is some constant. Depreciation schedules of this class havethe property that, at the time of asset acquisition, a share (1 − λ) of theinvestment is directly expensed (or writtenup). Then, the amount initiallycapitalized is depreciated in proportion to the replacement cost rule. Theparameter λ can be viewed as a degree of unconditional conservatism ofthe accounting system in use.21 Values of λ less than one define rules moreconservative than the replacement cost rule, while values greater than onedefine more liberal rules. The one-dimensional family of replacement costaccounting rules corresponding to λ > 0 will be called replacement cost ac-counting with partial expensing . It is readily verified that all accounting rulesin this class are informationally sufficient.

Under replacement cost accounting with partial expensing, book valuesare proportional to the replacement cost of assets in place for any historyof investments:

BVt = λBV ∗t . (33)

It can also be verified that the aggregate depreciation expense satisfies

Dt = λD∗t + (1 − λ)CFI t .

Therefore, observing {BVT , DT , CFIT}, investors can infer BV ∗T and D∗

T . Thefollowing corollary to Proposition 2 provides an exact valuation formula forreplacement cost accounting with partial expensing.

COROLLARY 4. Replacement cost accounting with partial expensing, that is,bv = λbv∗for some λ > 0, is informationally sufficient. The valuation formulais given by

VT = 1λ

BVT + αT

(RT − 1

λDT + 1 − λ

λCFI T − r

λBVT−1

), (34)

where αT is as given in Proposition 2.

When λ = 1, the expression in (34) reduces to the one in Proposition2, equation (25). There are three observations worth discussing in connec-tion with equation (34). First, note that, when λ = 1, the coefficients onrevenues and depreciation are not equal, and, hence, net income has tobe disaggregated into its cash flow and accrual components. Also, the in-vestment cash flows, CFIT , now enter the valuation formula in a nontrivialmanner. This was not the case in Proposition 2, where net income couldbe used as the aggregate measure and investment cash flows were required

21 In the empirical part of their paper, Rajan, Reichelstein, and Soliman [2007] operational-ize conservative accounting by the proportion of new investments that are expensed directly,for example, R&D and advertising, relative to total investments.

only to infer the beginning of period book value. The difference is that,while the rule with partial expensing preserves value-relevant information,it is imperfect from a measurement perspective in the sense that account-ing data, other than revenues, are not equal to the economic fundamentals.To obtain an unbiased valuation, these imperfections need to be adjustedby changing the coefficients on the accounting data.

The second observation with respect to (34) is that the coefficients in thevaluation equation depend on both the market growth rates, μT+1, and thedegree of conservatism, λ. This observation again stresses the importanceof separately considering the two determinants of accounting numbers—the economic environment of the firm and the accounting rules employed.A correct valuation function, if one exists, can only be constructed by takinginto account both of these determinants.

Finally, the preceding corollary requires that λ > 0. When λ = 0, cashaccounting is obtained under which investors observe only revenues andthe latest investment. Such a data set is informationally insufficient, sinceit is easy to construct two histories with the same latest investments and thesame capacity levels in period T , but with different replacement costs ofassets in place at date T . Formally, this result will follow from Proposition 4.

The following proposition shows that, if assets can be put to productiveuse immediately, then replacement cost accounting with partial expensingis the only informationally sufficient rule.

PROPOSITION 4. Assume that assets become productive immediately, L = 0.Then, any informationally sufficient accounting rule corresponds to replacement costaccounting with partial expensing for some λ > 0.

The main insight behind this finding can be explained in terms of thetools developed in aggregation theory (see, for instance, Ijiri [1968]). Thebasic algebraic problem addressed in this literature is as follows: investorsare interested in computing some dot-product of the investment historyvector, θT · y , yet they observe only some other dot-products of this history,θT · w1, . . . ,θT · wn. When does a function exist that maps the observeddata into precise estimates of θT · y for any vector θT ? Clearly, if y is in thelinear subspace, L, generated by vectors w1, . . . , wn, such a function exists.Indeed, if

y = λ1w1 + · · · + λnwn,

θT · y = λ1(θT · w1) + · · · + λn(θT · wn).

Moreover, the condition given above is also necessary. To see this, notethat, if y does not belong to L, then it can be decomposed into two parts,y (L), which belongs to L, and y (⊥L), which belongs to the orthogonal com-plement of L. Then, if we add y (⊥L) to any history θT , the investmenthistory θt + y (⊥L) will produce exactly the same observed data (because

264 A. NEZLOBIN

y (⊥L) · w i = 0 for any i), but will correspond to a different value of the tar-get function (y · y (⊥L) = 0).

To apply this result from aggregation theory to the economic model de-veloped in this paper, suppose the firm uses an informationally sufficientaccounting system defined by vectors d, bv. Then there exists a functionV̂T (BVT , DT , RT , CFI T ,μT+1) such that

V̂T (BVT , DT , RT , CFI T ,μT+1) = VT ,

for any history of investments, any μT+1, and any underlying environment.By Proposition 2, this means that

V̂T (BVT , DT , RT , CFI T ,μT+1) = bv∗ · θT + αT (RT − z∗ · θT ). (35)

Observing revenues, investors can estimate the present value of future rev-enues. However, since investors do not know the shape of demand curvesand the price of the product, they cannot use information in revenues toinfer the replacement cost of capacity of the replacement cost of assets inplace. Therefore, investors effectively face the problem of estimating

VT − αT RT = (bv∗ − αT z∗) · θt , (36)

while observing

BVT = bv · θT , DT = d · θT , CFI T = i · θT ,

where i = (1, 0, . . . , 0) . The result from aggregation theory discussed sug-gests that this problem is solvable if and only if the vector in parentheses,that is bv∗ − αT z∗, belongs to the linear subspace spanned by bv, d, i. Sincewe seek a system that provides sufficient information for any growth rateprojections, μT+1, the latter condition has to hold for any αT . The proof ofProposition 4 shows that this requirement is equivalent to bv being parallelto bv∗.22

The arguments underlying Proposition 4 were also instrumental in thenumerical example in section 2 concerning the straight-line depreciationrule for assets with one-hoss shay productivity. The two investment historiesin that example were different along dimension v = bv∗, but had the sameprojections onto x , bv, d, i. As a result, the two firms implemented equal ca-pacity in every period, had the same financial statements at date t, but haddifferent replacement costs of assets in place, and, consequently, differentvaluations at that date.

22 Vectors bv∗ and z∗ can be viewed as the value-relevant dimensions of the investmenthistory. If two firms face equivalent market conditions, and the difference between their in-vestment histories is orthogonal to both bv∗ and z∗, then the two firms will have the samevaluation. In other words, the difference between their investment histories is value irrele-vant. The depreciation expense is value relevant as long as it conveys information about eitherθ · bv∗ and θ · z∗, or, mathematically, as long as d is not orthogonal to both bv∗ and z∗. Al-ternative notions of value-relevant information are discussed in Holthausen and Watts [2001]and Barth, Beaver, and Landsman [2001].

If assets require some construction or installation time, that is, if L > 0,then investments made up to date T commit the firm to certain levels ofcapacity in periods T + 1 , . . . , T + L. Therefore, the manager needs toplan the optimal capacity path L periods in the future. Investors observethe product market growth rates and can, under the assumption that themanager has made optimal investments up to date T , infer certain addi-tional information about the investment history θT . For example, investorsknow that, since the firm is on the optimal investment path,

1 + μT+1 = K ∗T+1

K ∗T

= θT · x a

θT · x,

where x a = (x1, x2, . . . , xT , 0) . This additional information effectively re-duces the informational demands on the financial statements. To illustratethis point, I show that, if assets start to generate capacity in the period fol-lowing their acquisition, that is, if L = 1, then there is an additional possiblemodification to replacement cost accounting that preserves informationalsufficiency.

Consider the following accounting rule that accrues depreciationcharges one period ahead of the replacement cost schedule:

d0 = d∗1 , d1 = d∗

2 , . . . , dT−1 = d∗T , dT = 0. (37)

Since assets are assumed to be idle in the period when they are acquired(x0 = 0), the depreciation charge in the acquisition period under replace-ment cost accounting, d∗

0, is zero. Therefore, the accounting rule definedin (37) satisfies the clean surplus condition:

T∑τ=0

dτ =T∑

d∗τ = 1.

I will refer to this rule as replacement cost accounting accelerated by oneperiod. Under this rule, the aggregate depreciation expense in any periodis equal to the aggregate depreciation expense under replacement cost ac-counting in the following period:

Dt = D∗t+1.

It is also easy to verify that book values calculated under this rule are equalto

BVt−1 = BV ∗t − CFI t ,

for any t and any history of investments. Therefore, the information con-tained in {BVT , DT , CFIT} is sufficient to calculate the economic cost inperiod T + 1:

H ∗T+1 = D∗

T+1 + r BV ∗T

= DT + r (BVT−1 + CFI T ) = (1 + r ) DT + r BVT .

266 A. NEZLOBIN

The economic income in period T + 1 is then given by

πT+1 = (1 + μT+1) RT − (1 + r ) DT − r BVT .

The following corollary to Proposition 2 provides a valuation formula forreplacement cost accounting accelerated by one period.

COROLLARY 5. Assume that assets become productive one period after their ac-quisition. Replacement cost accounting accelerated by one period, that is, bva =(bv∗

1, bv∗2, . . . , bv∗

T , 0), is informationally sufficient. The valuation formula isgiven by

VT = BVT + DT + αT

1 + μT+1((1 + μT+1)RT − (1 + r )DT − r BVT ),(38)

where αT is as given in Proposition 2.

It can be shown that, when L is equal to one, any linear combination ofreplacement cost accounting with direct expensing and replacement costaccounting accelerated by one period is informationally sufficient. Specifi-cally, a depreciation rule is informationally sufficient if the correspondingasset valuation rule takes the form

bv = λ1bv∗ + λ2bva,

for some constants λ1, λ2.Clearly, when L is greater than one, replacement cost accounting acceler-

ated by one period is still informationally sufficient with the same valuationformula (38). More generally, I conjecture that replacement cost account-ing is informationally sufficient if it is accelerated by not more than L pe-riods. In other words, there can be a mismatch between the time intervalsover which an asset is depreciated and over which it is used, but the formermust precede the latter.

6. Concluding Remarks

The model presented in this paper demonstrates that certain accrual ac-counting rules aggregate past transaction history of the firm at no loss ofvalue relevant information. For such accrual rules, I provide equity valua-tion models that express the firm’s value as a function of current account-ing data and the anticipated growth in the firm’s product market. For allother accrual rules, I show that the valuation problem is generally not solv-able on the basis of current financial statements, and, to assess the firm’svalue precisely, investors need to have access to the history of the underly-ing transactions of the firm.

This paper has clearly relied on a number of restrictive assumptions, butI conjecture that my results are robust to relaxing some of them. For ex-ample, Propositions 2 and 3 remain essentially unaffected if future growthof the product market is uncertain. Since assets become productive with alag of L periods, the manager needs to foresee growth in the market only

L periods ahead in order to implement the first-best capacity path. If thiscondition is satisfied and the product market is expanding, then results inPropositions 2 and 3 are completely unchanged. The results are also robustto the introduction of additive random shocks to revenues.

While financial statements prepared with informationally insufficient ac-counting rules do not convey enough information to value the firm pre-cisely, they may reduce uncertainty that investors face about the firm’svalue. In a Bayesian framework where investors have prior beliefs aboutinvestment histories, one may view an accounting rule as more desirableif it is characterized by a smaller residual variance of the firm’s value afterconditioning on the accounting data. I conjecture that depreciation rulescloser to replacement cost accounting result in financial statements that aremore informative about the firm’s value.

The inferences in my analysis were made possible by explicitly modelingthe firm’s transactions as well as its financial reporting system. To that end,it was convenient to focus on a model of sequential investments in produc-tive capacity and examine depreciation as the relevant accrual accountingconcept. Future research may address the question of informational suffi-ciency in the context of other accruals such as revenue recognition or thevaluation of inventory. If a major goal of accounting information is to facil-itate parsimonious equity valuation, then the results in this paper suggestthat informational sufficiency is a natural criterion. Extending the modelto other accounting items may provide a framework for examining how thepricing of accruals depends on the underlying accounting rules.

APPENDIX

PROOF OF PROPOSITION 1. First, I show that, if all investments are chosenoptimally, then

R ′t

(K ∗

) = c t .

Let x, It , CFIt , Kt , K0t denote the following infinite sequences:

x = (x0, x1, . . . , xT , 0, . . .),

It = (It , It+1, . . .),

CFIt = (CFI t , CFI t+1, . . .),

Kt = (Kt , Kt+1, . . .),

K0t = (

K 0t−1,t , . . . , K 0

t−1,t+T−1, 0, . . .),

K 0t−1,t+τ = It−1xτ+1 + · · · + It+τ−T xT

is the capacity that will be generated in period t + τ by assets purchasedbefore date t − 1 for τ ≥ 0. Observe that

268 A. NEZLOBIN

Kt = K0t + It ∗ x, (A1)

where ∗ denotes the convolution operator.23

By the definition of the one-sided Z-transform, the present value (in pe-riod t dollars) of investment cash outflows {CFIt , CFI t+1 , . . .} is equal to thevalue of the Z-transform of this sequence calculated at 1 + r :

Z {CFIt }(1 + r ) = CFI t + γ CFI t+1 + γ 2CFI t+2 + · · ·Since, the cost of investment declines geometrically, this is equal to

Z {CFIt }(1 + r ) = p t Z {It }(ε−1(1 + r )).

Applying the Z-transform to (A1) and using the convolution property ofthe Z-transform, one obtains

Z {Kt } = Z{K0

} + Z {It }Z {x}.Now dividing both sides by Z {x}

p tand calculating all Z-transforms at ε−1(1 +

r) yields

p t Z {Kt }(ε−1(1 + r ))Z {x}(ε−1(1 + r ))

= p t Z{K0

}(ε−1(1 + r ))

Z {x}(ε−1(1 + r ))+ Z {CFIt }(1 + r ).

It can be immediately verified that equation (15) is equivalent top t

Z {x}(ε−1(1 + r ))= c t .

Therefore, we have

Z {CFIt }(1 + r ) = c t Z {Kt }(ε−1(1 + r )) − c t Z{K0

}(ε−1(1 + r )).

Expanding the Z-transforms yields∞∑

γ τ CFI t+τ =∞∑

γ τ c t+τ Kt+τ −T−1∑τ=0

γ τ c t+τ K 0t−1,t+τ . (A2)

When the firm chooses investment CFIt , the firm’s target function is∞∑

γ τ (R(qt+τ ) − CFI t+τ ).

Using equation (A2), this target function can be rewritten asT−1∑τ=0

γ τ c t+τ K 0t−1,t+τ +

∞∑τ=0

γ τ (R(qt+τ ) − c t+τ Kt+τ ).

If one ignores the nonnegativity constraint on investments, the optimal pol-icy is characterized by

R ′t+τ

(K ∗

) = c t+τ

23 The properties of convolution and the Z-transform that are used in this proof are dis-cussed, for instance, in Proakis and Manolakis [2006].

for τ ≥ L. Under Assumption 1, so defined capacity levels K ∗t are nonde-

creasing, and since the manager is assumed to act always (both before andafter date t) in the best interest of investors, capacity levels K ∗

t will be im-plemented in all periods.

At date T , the firm value is given by

VT =∞∑

γ τ(R

(K ∗

) − CFI ∗T+τ

where CFI ∗T+τ = pT+τ I ∗

T+τ . Applying (A2) for t = T + 1, one obtains

VT =T∑

γ τ c t+τ K 0t,t+τ +

∞∑τ=1

γ τ(R

(K ∗

It remains to show that the first term on the right-hand side is equal tothe replacement cost of assets in place. First, let us expand capacity levelsK 0

t,t+τ :

T∑τ=1

γ τ c t+τ K 0t,t+τ =

T∑τ=1

(γ τ c t+τ

T−τ∑i=0

CFI t−i

p t−ixi+τ

The coefficient on CFI t−i is equal to

T−i∑τ=1

γ τ c t+τ

p t−i= c t−i

p t−i

T−i∑τ=1

γ τ εi+τ xi+τ =

T−i∑τ=1

γ τ εi+τ xi+τ

T∑τ=1

γ τ ετ xτ

= vi .

PROOF OF PROPOSITION 2. Let π t(Kt) denote the economic profit of thefirm in period t if it operates at capacity Kt in that period. By definition,

πt (Kt ) = Rt (Kt ) − cKt .

Under Assumption 2, the economic profit in period t + 1 can be rewrittenas

πt+1(Kt+1) = Pt+1(Kt+1)Kt+1 − cKt+1

= (1 + μt+1)[

1 + μt+1

1 + μt+1− c

1 + μt+1

= (1 + μt+1)πt

1 + μt+1

Therefore, the optimal capacity levels will satisfy

K ∗t+1

1 + μt+1= K ∗,

and the optimized economic profits will evolve so that

π∗t+1 = (1 + μt+1)π∗

270 A. NEZLOBIN

Given replacement cost accounting, residual income is equal to the opti-mized economic profits in every period. Therefore, for any τ ≥ 1,

RI ∗T+τ = RI ∗

τ∏i=1

(1 + μT+i ).

To conclude the proof, it remains to apply the residual income valuationformula:

VT = BV ∗T +

∞∑τ=1

γ τ RI ∗T+τ = BV ∗

T + RI ∗T

∞∑τ=1

(γ τ

τ∏i=1

(1 + μT+i )

PROOF OF PROPOSITION 3. I will first show that, under Assumption 3,

π∗t+1 = (1 + μt+1)επ∗

Note thatπt+1(Kt+1) = Pt+1(Kt+1)Kt+1 − c t+1Kt+1

= εPt

1 + μt+1

)(1 + μt+1)Kt+1

1 + μt+1− εc t

(1 + μt+1)Kt+1

1 + μt+1

= ε(1 + μt+1)πt

1 + μt+1

Therefore,

K ∗t+1 = (1 + μt+1)K ∗

Pt+1(K ∗

) = εPt(K ∗

π∗t+1 = ε(1 + μt+1)π∗

The claim of the Proposition then follows from the valuation formula inProposition 1, since, under replacement cost accounting, BV ∗

T = RCT andRI ∗

t = π∗t for any t.

PROOF OF PROPOSITION 4. Assume that an accounting rule, d, is infor-mationally sufficient and let L be the linear subspace generated by vectors{bv, d, i}, where i = (1, 0, 0, . . .). I will first show that vectors bv∗ and z∗

must belong to L.Assume the contrary is true. Then, for all but maybe one αT > 1

r , it willbe the case that

y = bv∗ − αT z∗ /∈ L.

Fix some αT > 0 that satisfies the condition above. Let yL be the linearprojection of y onto L, and y⊥L = y − yL. Consider a trajectory of constantinvestments

θ(1)T = (CFI 0, . . . , CFI 0),

and let

θ(2)T = θ

(1)T + εy⊥L,

where ε is sufficiently small so that all components of θ(2)T are nonnegative.

A firm with investment history θ(2)T will report the same depreciation in

period T as a firm with history θ(1)T , since y⊥L is orthogonal to d:

D(2)T = d · θ

(2)T = d · (

θ(1)T + εy⊥L) = D(1)

Similarly, it can be shown that the firms will report the same book valueat date T and the same latest investment, CFIT . Let K (1)

T and K (2)T be the

capacity levels these two investment histories generate in period T . Assumethat the firms face revenue functions such that

R(1)′T

(K (1)

) = R(2)′T

(K (2)

) = c ,

and R(1)T (KT

(1)) = R(2)T (KT

(2)). Then, these firms will also report equal rev-enues and their financial statements at date T will be identical.

Assume that, starting in period T + 1, product markets for both firmsgrow at a constant rate μ such that αT = 1+μ

r −μ. Then, the firms’ equity values

will be equal to

V (1)T = αT R(1)

T + y · θ(1)T ,

V (2)T = αT R(2)

T + y · θ(2)T .

Note that V (1)T = V (2)

T , since R(1)T = R(2)

y · θ(2)T = y · (

θ(1)T + εy⊥L) = y · θ

(1)T + ε‖y⊥L‖2 = y · θ

(1)T .

Therefore, it is possible to find two firms with identical financial statementsand different valuations. This contradicts the informational sufficiency ofdepreciation schedule d, and, therefore, bv∗ and z∗ must belong to L.

I will now show that bv has to be proportional to bv∗. First, note that, bythe clean surplus relation and the definition of z∗,

z∗ = r bv∗ + (1 + r )d∗ − r i.

Vector i has only one nonzero component—the one in the first position.Therefore, it must be that bv∗(T) and d∗(T) are in the linear hull of bv(T)and d(T), where, by bv∗(T), d∗(T), bv(T), and d(T), I denoted the last Tcomponents of the respective vectors. Let

bv∗(T) = λ1bv(T) + λ2d(T) (A3)

d∗(T) = λ3bv(T) + λ4d(T). (A4)

For the last component of these vectors, we have the following expression:

d∗T = λ3bvT + λ4dT .

272 A. NEZLOBIN

By the clean surplus relation, bvT = 0. Therefore, since d∗T = 0, dT = 0. On

the other hand,

bv∗T = λ1bvT + λ2dT .

Both bv∗T and bvT are equal to zero, but dT = 0, therefore, λ2 = 0. I have

shown that the last T components of bv are proportional to bv∗. It remainsto check that bv∗

0 = λ1bv0. Again invoking the clean surplus relation, dT =bvT−1 and d∗

T = bv∗T−1. Hence,

λ4 = d∗T

dT= bv∗

bvT−1= λ1.

Now, writing condition (A4) for the second to last components of respectivevectors, one obtains

d∗T−1 = λ3bvT−1 + λ1dT−1.

From the proportionality of the last T components of bv∗ and bv, it followsthat

d∗T−1 = bv∗

T−2 − bv∗T−1 = λ1(bvT−2 − bvT−1) = λ1dT−1.

Therefore, λ3 = 0. I have shown that d∗(T) = λ1d(T). To conclude, observethat

bv∗0 = bv∗

1 + d∗1 = λ1(bv1 + d1) = λ1bv0.

REFERENCES

ARROW, K. “Optimal Capital Policy, Cost of Capital and Myopic Decision Rules.” Annals of theInstitute of Statistical Mathematics 1–2 (1964): 21–30.

ARYA, A.; J. C. FELLINGHAM; J. C. GLOVER; D. A. SCHROEDER; AND G. STRANG. “Inferring Trans-actions from Financial Statements.” Contemporary Accounting Research 17 (2000): 365–85.

BARTH, M. E.; W. H. BEAVER; AND W. R. LANDSMAN. “The Relevance of the Value-RelevanceLiterature for Financial Accounting Standard Setting: Another View.” Journal of Accountingand Economics 31 (2001): 77–104.

BEAVER, W., AND R. DUKES. “δ-Depreciation Methods: Some Analytical Results.” Journal of Ac-counting Research 9 (1974): 391–419.

DECHOW, P.; A. HUTTON; AND R. SLOAN. “An Empirical Assessment of the Residual IncomeValuation Model.” Journal of Accounting & Economics 26 (1999): 1–34.

DUTTA, S., AND S. REICHELSTEIN. “Accrual Accounting for Performance Evaluation.” Review ofAccounting Studies 10 (2005): 527–52.

FELTHAM, G., AND J. OHLSON. “Valuation and Clean Surplus Accounting for Operating andFinancial Activities.” Contemporary Accounting Research 11 (1995): 689–731.

FINANCIAL ACCOUNTING STANDARDS BOARD (FASB). “Objectives of Financial Reporting by Busi-ness Enterprises.” Statement of Financial Accounting Concepts No. 1 (1978), Norwalk, CT: FASB.

HOLTHAUSEN, R. W., AND R. L. WATTS. “The Relevance of the Value-Relevance Literature forFinancial Accounting Standard Setting.” Journal of Accounting and Economics 31 (2001): 3–75.

IJIRI, Y., The Foundations of Accounting Measurement: A Mathematical, Economic, and BehavioralInquiry. Englewood Cliffs, New Jersey: Prentice-Hall, 1967.

IJIRI, Y. “The Linear Aggregation Coefficient as the Dual of the Linear Correlation Coeffi-cient.” Econometrica 36 (1968): 252–59.

KOTHARI, S. “Capital Markets Research in Accounting.” Journal of Accounting & Economics 31(2001): 105–231.

LEV, B. “The Aggregation Problem in Financial Statements: An Informational Approach.” Jour-nal of Accounting Research 6 (1968): 247–61.

LINDENBERG, E. B., AND S. A. ROSS. “Tobin’s q Ratio and Industrial Organization.” Journal ofBusiness 54 (1981): 1–32.

MYERS, J. “Implementing Residual Income Valuation with Linear Information Dynamics.” TheAccounting Review 74 (1999): 1–28.

NEZLOBIN, A.; M. RAJAN; AND S. REICHELSTEIN. “Dynamics of Rate of Return Regulation.” Man-agement Science (2011): Forthcoming.

OHLSON, J. “Earnings, Book Values, and Dividends in Equity Valuation.” Contemporary Account-ing Research (1995): 661–87.

OHLSON, J., AND B. JUETTNER-NAUROTH. “Expected EPS and EPS Growth as Determinants ofValue.” Review of Accounting Studies 10 (2005): 349–65.

OHLSON, J., AND X. ZHANG. “Accrual Accounting and Equity Valuation.” Journal of AccountingResearch 36 (1998): 85–111.

PENMAN, S. H. “Discussion of ‘On Accounting-Based Valuation Formulae’ and ‘Expected EPSand EPS Growth as Determinants of Value’.” Review of Accounting Studies 10 (2005): 376–78.

PENMAN, S. H. Financial Statement Analysis and Security Valuation, 4th edition. New York, NY:McGraw-Hill/Irwin, 2010.

PROAKIS, J., AND D. MANOLAKIS, Digital Signal Processing . Upper Saddle River, NJ: Prentice Hall,2006.

RAJAN, M., AND S. REICHELSTEIN. “Depreciation Rules and the Relation Between Marginal andHistorical Cost.” Journal of Accounting Research 47 (2009): 823–67.

RAJAN, M.; S. REICHELSTEIN; AND M. SOLIMAN. “Conservatism, Growth, and Return on Invest-ment.” Review of Accounting Studies 12 (2007): 325–70.

REICHELSTEIN, S. “Investment Decisions and Managerial Performance Evaluation.” Review ofAccounting Studies 2 (1997): 157–80.

REICHELSTEIN, S. “Providing Managerial Incentives: Cash Flows Versus Accrual Accounting.”Journal of Accounting Research 38 (2000): 243–69.

ROGERSON, W. “Inter-Temporal Cost Allocation and Managerial Investment Incentives: A The-ory Explaining the Use of Economic Value Added as a Performance Measure.” Journal ofPolitical Economy 105 (1997): 770–95.

ROGERSON, W. “Inter-Temporal Cost Allocation and Investment Decisions.” Journal of PoliticalEconomy 105 (2008): 770–95.

ROGERSON, W. “On the Relationship Between Historic, Forward-Looking Cost, and Long RunMarginal Cost.” Review of Network Economics (2011): Forthcoming.

THOMADAKIS, S. “A Model of Market Power, Valuation, and the Firm’s Returns.” The Bell Journalof Economics 7 (1976): 150–62.

ZHANG, G. “Accounting Information, Capital Investment Decisions, and Equity Valuation: The-ory and Implications.” Journal of Accounting Research 38 (2000): 271–95.

ZHANG, X. “Conservative Accounting and Equity Valuation.” Journal of Accounting and Economics29 (2000): 125–49.

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