Keywords: profitability; accounting rate of return; economic rate of return; growth rate. Data availability: The data in the paper are publicly available from the ...
Accounting Returns Revisited: Evidence of their Usefulness in Estimating Economic Returns
Morris G. Danielson Assistant Professor of Finance & Eric Press Associate Professor of Accounting The Fox School of Business Temple University Philadelphia, PA 19122 February 2002
We acknowledge the helpful comments of Rich Aronson, Tom Dowdell, Peter Gillett, Ken Kopecky, Jayanthi Krishnan, David Mest, Kevin Sachs, and workshop participants at Lehigh, Marquette, Rutgers, and Temple Universities.
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Accounting Returns Revisited: Evidence of their Usefulness in Estimating Economic Returns
ABSTRACT: Shareholders, managers, and researchers rely on estimates of a firm's past investment returns when evaluating its performance, and estimating its value. In this paper, we revisit the question of whether accounting rates of return (ARR)—computed from financial statement data—are appropriate surrogates for a firm's realized internal rate of return (IRR). Although some authors argue accounting numbers are useless for this purpose, many managers and researchers employ accounting returns as a measure of firm profitability. We derive a model showing that a firm's ARR and historical growth rate define the range in which its IRR on past investments is likely to fall. Using panel data, we find ARR is close to IRR for a large number of firms. In addition, we identify conditions under which a firm's ARR is likely to be a misleading proxy for IRR. Given the theoretical relation we derive between ARR and IRR and the empirical linkage we observe, previous research results demonstrating the value relevance of accounting information are predictable. Keywords: profitability; accounting rate of return; economic rate of return; growth rate. Data availability: The data in the paper are publicly available from the sources listed in the text.
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I. INTRODUCTION In theory, information about the return a firm earns on its past investments enables shareholders to assess the performance of the firm's managers, guides the firm's future capital-allocation decisions, and helps investors predict the future investment returns and cash flows that support the firm's stock price.1 Although the internal rate of return (IRR) on past investment projects is more relevant for these purposes, the accounting rate of return (ARR)—being more accessible—is often used instead.2 However, such use of accounting information is appropriate only if ARR is a reasonable proxy for IRR. The goal of this paper is to identify conditions under which a firm's accounting return is a suitable proxy for its past internal rate of return, and circumstances when it is not. The role of ARR as a return measure is viewed at least two different ways in the literatures of accounting, economics, and finance. A number of studies—including Fisher and McGowan (1983), Fisher (1988), Salamon (1985, 1988), and Brief and Lawson (1991)—question whether investors can use accounting numbers to estimate IRRs. In the strongest critique, Fisher and McGowan (page 90) state, “there is no way in which one can look at accounting rates of return and infer anything about relative economic profitability.” Other papers, however, claim that accounting information is value relevant. Long and Ravenscraft (1984, p. 497) attempt to refute Fisher and McGowan (1983) by arguing their conclusions are supported by unrealistic examples. In addition, research by Ball and Brown (1968), Beaver (1968, 1998), Beaver et al. (1979, 1980, and 1987), Kormendi and Lipe (1987), Collins and Kothari (1989), Ohlson (1995), and Feltham and Ohlson (1995) establishes the connection between firms’ accounting information and market values. Their findings imply that a firm's ARR is informative about the economic returns, or IRR, the firm earned from its past investments. 3
1
The return on a firm's future investments plays a key role in many valuation models, including Miller and Modigliani (1961), Gordon (1962), Leibowitz and Kogelman (1990), and Danielson and Dowdell (2001). 2 We measure ARR as return on assets. The IRR of a project (or firm) is the discount rate that sets the present value of future cash flows equal to the investment in the project (or firm). 3 Peasnell (1996) provides a review of the ARR-IRR debate in the accounting and economics literature.
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In this paper, we derive a model linking ARR and IRR, using assumptions equivalent—in most respects—to those in Fisher and McGowan (1983). When applied in an unconstrained setting, the model supports Fisher and McGowan's main conclusion: ARR is often a misleading proxy for IRR. However, we extend the analysis from Fisher and McGowan by showing how the implications of generally accepted accounting procedures (GAAP) constrain the relation between ARR and IRR. With this constraint in place, accounting numbers are informative about past returns, and analysts can use our model to estimate a firm's past IRR from its ARR. The model we derive predicts that adherence to GAAP causes a firm's IRR to fall within a range between its historical growth rate and ARR. For example, if a firm's ARR is 10 percent and its growth rate is 20 percent, the firm's IRR is between 10 and 20 percent. While the model allows ARR and IRR to differ, we argue that conditions creating larger differences are often observable. For example, the difference between ARR and IRR can be larger for firms that invest heavily in intangible assets, such as research and development. These investments distort ARR because their costs are expensed immediately, even though they may produce cash flows across multiple periods. Our findings have implications for both security analysis and academic research. For security analysis and performance evaluation, analysts can use our model to estimate the direction and magnitude of the difference between ARR and IRR. For research purposes, our model identifies firms for which accounting information should be more value relevant (i.e., ARR and IRR are close), and firms for which the relevance of accounting information is lower (ARR and IRR are not close). We develop the model in Section II, and compare it to previous models of the ARR-IRR relation. Section III describes the constraints accounting rules place on the relation between ARR and IRR, and reevaluates several of the conclusions of Fisher and McGowan (1983). Rather than arguing that their examples are unrealistic, we show how problems interpreting ARR become less severe when GAAP constrains the relation between ARR and IRR. Thus we respond to the challenge posed by Fisher (1984, p. 512) that “those who wish to go on using accounting rates of return as measures of economic profit . . . must show rather than presume that the problems”—identified in Fisher and McGowan (1983)—“do not arise for real
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firms.” Section IV describes estimation errors that can arise when the model's underlying assumptions are not met, and evaluates the model's implications using empirical data. Section V shows how to use our model to interpret ARRs of individual firms, and Section VI concludes. II. THE ARR-IRR RELATION We begin by deriving a model that links a firm's ARR to the realized IRR from investments made in prior periods. We define a firm's accounting income (AI), its accounting rate of return (ARR), its economic book value (EBV), its economic income (EI), and its internal rate of return (IRR) as follows. AI is earnings before interest but after taxes, calculated from a firm's income statement. ARR is AI divided by total assets (accounting book value, ABV) at the beginning of the period. Thus, ARR is return on assets. Since IRR is usually calculated in terms of asset (rather than equity) investment, we measure accounting returns on this basis as well. 4 ARR =
AI ABV
(1)
We define a firm's economic income from operations, EI, as its AI, plus the accounting depreciation of its assets, less the economic depreciation of these assets. As defined in Hotelling (1925), and Fisher and McGowan (1983), an economic depreciation schedule allocates (expenses) the investment in a project over its useful life based upon the timing of the project's cash inflows. We define the EBV of a firm's assets in place as the total investment in those assets, less the accumulated economic depreciation. If a firm were to depreciate its assets in accordance with economic depreciation schedules, the decline in the firm's asset book value over time would track the decline in the future productive value of its assets. Thus, EBV is simply the ABV a firm would report if it depreciated its assets using the theoretically correct economic depreciation schedule.5
4
We add back interest expense to net income so that we consider the earnings produced by assets before any distribution to ownership groups (i.e., earnings before interest and dividends). 5 At the date of an investment, EBV is equal to the total dollar investment. Then, EBV is reduced each year by the amount of the economic depreciation. EBV is not directly related to the market value of the firm. A firm's total market value is the present value of cash flows from assets in place plus the net present value of cash flows from growth opportunities, using the risk adjusted cost of capital as the discount rate. At any point in time, EBV is the present value of the remaining cash flows from a firm’s assets in place calculated using the IRR as the discount rate (see Hotelling, 1925). Depending on the cash flow stream of a project, the economic depreciation during a given year can be
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Using these definitions, we define a firm's IRR each year in Equation (2): EI during the year divided by the investment in the firm (economic book value, EBV) at the beginning of the period.6 IRR =
EI EBV
(2)
Our definition of economic income differs slightly from the definition of economic income found in other applications. Economic income is defined in many texts as the difference between a firm's accounting income and its cost of capital, in dollars. Here, we attempt to measure the IRR of a firm. The IRR of a project (or firm) is generally stated before the subtraction of capital (opportunity) costs, and must be compared to the cost of capital to evaluate the profitability of the project (or firm). To align with this definition of IRR, we define economic income as a gross income measure, before capital costs. 7 We recognize that our measure of economic income is simplified, focusing only on differences between accounting and economic earnings arising from depreciation differences, including the accounting treatment of intangible investments. In practice, a firm's accounting and economic income can differ for reasons other than differences between accounting and economic depreciation.8 However, previous studies—including Fisher and McGowan (1983), Salamon (1985), and Brief and Lawson (1991)—suggest that depreciation differences alone are sufficient to make accounting returns poor proxies for economic returns. One of our objectives is to reconsider whether that conclusion is warranted.
positive (reflecting a return of the principal invested in the project) or negative (reflecting an additional investment in the project). 6 When the economic depreciation schedule is used to determine the EI and EBV of a project each year, the ratio EI/EBV will be constant over time and will be equal to the project's IRR. 7 Other authors define economic income more broadly. For example, Beaver (1998, Chapter 4) defines economic i ncome as the return from past investments, plus the change in the market value of future growth opportunities. In our application, we separate the return on a firm's past investments from the expected return on its future investments. The economic income (and IRR) that a firm earns from past investment can give some insight into the IRR the firm might earn on future investments. Therefore, our measure of economic income and IRR are measures of past performance an investor can use when evaluating the future prospects of the firm. Rather than being a function of market value changes, our measures of economic income (IRR) can help an investor estimate the market value of a firm. 8 For example, any unrealized appreciation in the value of a firm's assets will increase its economic income, but will not increase its accounting income in most cases. Similarly, there are some expenses—such as off-balance sheet hedging, joint venture activity, or employee-stock options—that can reduce a firm's economic income, but are not afforded full accounting recognition. Such differences add additional layers of complexity that are beyond the scope of this paper.
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To link IRR with ARR, we expand the right-hand side (RHS) of Equation (2).9 First, we add and subtract AI from the numerator. IRR =
AI EI − AI + EBV EBV
(3)
Next, we multiply the first term on the RHS of Equation (3) by 1 (ABV/ABV), and use Equation (1) to simplify the resulting expression. IRR =
AI ABV EI − AI ABV EI − AI = ARR + + ABV EBV EBV EBV EBV
(4)
Then, we add and subtract EBV from the numerator of the bracketed term on the RHS of Equation (4). ABV − EBV EI − AI ABV + EBV − EBV EI − AI IRR = ARR = ARR 1 + + + EBV EBV EBV EBV
(5)
Equation (5) can apply to an individual project, or to a firm comprising a collection of projects. In this paper, we apply Equation (5) to the firm as a whole. To simplify Equation (5) further, we note that a firm’s asset base will have changed over time as it invested in new projects. We make five assumptions about the composition of a firm's new investments. 1. Each unit of new investment can contain both tangible and intangible items, but the relative percentages of each type of asset remains constant over time. 2. Each unit of new investment produces cash flows over a period of N years. 3. The firm depreciates each unit of new investment using the same accounting depreciation schedule. 4. We assume the assets a firm acquires all have the same economic depreciation schedule, and wear out at the same rate over a period of N years. The assets the firm invests in each year will not necessarily earn the same economic rate of return, though. For example, a firm could invest in assets with an IRR of 20 percent in one year, and then as competition increases within the firm's industry, the IRR on new investments could decrease to 10 percent. 5. Finally, our model allows the amount the firm invests in new assets to change over time. For simplicity, we assume that a firm's annual investment growth rate g has been constant over time (g can be greater than, less than, or equal to zero). This assumption implies that the firm's capital expenditures, depreciation expense, and book values (both accounting and economic) have all increased at the constant annual rate, g. We consider the implications of the constant growth assumption in Section IV and in an appendix. 9
We use static rather than dynamic analysis, given the properties of financial statements. Measurement of accounting income derives from the economic concept of stocks and flows (Hicks, 1946). Because balance sheet account levels (stocks) and income statement items (flows) are reported only at the end of periods, instantaneous measures are unobservable except at the end of each period. Thus, static analysis is an appropriate tool.
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These assumptions are identical in most respects to those used by Fisher and McGowan (1983), differing mainly in matters of interpretation. For example, we allow a firm's assets to include both tangible and intangible items. In addition, we require only the economic depreciation schedule—not the economic returns (i.e., the IRRs)—to remain constant over time. These assumptions do not conflict with the Fisher and McGowan model, but were not explicitly stated in their study. We also derive our model using a slightly different set of variables than Fisher and McGowan. In their model, information about the time pattern of a project's cash flows is stated as an integral function. In our model, the variable EBV captures this information. This small but important difference allows us to use the implications of accounting rules to constrain the possible relations between a firm's ARR and IRR. We note the growth rate in our model (and in Fisher and McGowan) is a historical growth rate, not a projected growth rate. We are interested in the past growth rate because a firm's depreciation expense relates to investments made in prior years, and the historical growth rate helps define the composition of the firm's assets in place. The assets of a firm with a higher growth rate are more heavily weighted toward investments made in recent years, whereas the assets of a firm with a lower growth rate are more heavily weighted toward investments from earlier years. We define depreciation expense as comprising the reduction in the book value of any tangible or intangible asset, and thus treat the expensing of R&D, advertising, and investments in training as “immediate” depreciation charges. We assume these amounts are included in beginning book values as cash, or another liquid asset. When these costs are incurred, the accounting expense reduces both accounting income and accounting book value.10 We define the variable DA, n as a firm's accounting depreciation, and the variable DE, n as the firm's economic depreciation, during year n. Because a firm's accounting and economic depreciation schedules can differ, the accounting and economic incomes produced by its assets during any year can differ as well. We assume the difference between a firm's accounting and the economic income is caused solely by the
10
In theory, a firm can use either internal cash flows or new capital to fund R&D. We assume that any new capital is raised at the beginning of a period, and the proceeds are included in the beginning asset balances.
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difference between the firm's accounting and economic depreciation. Therefore, we write the difference between the two income amounts in year n as Equation (6).11
EI n − AI n = DA, n − DE , n
(6)
The accounting depreciation in year n can also be written as a function of beginning and ending accounting book values, and capital expenditures during year n, CEn:12
DA , n = ABVn − ( ABVn +1 − CE n )
(7)
As we assume the firm's investment base increases at a constant growth rate g each year, the ending accounting book value can be written as a function of the beginning accounting book value: ABVn+1 = ABVn (1 + g). Thus, we can write the accounting depreciation in year n as Equation (8).
DA , n = CE n − gABVn
(8)
Using similar reasoning, the economic depreciation in year n is Equation (9).
DE , n = CE n − gEBVn
(9)
The capital expenditure amount, CEn, will be the same on both an accounting and economic basis, as will the growth rate, g. Thus, we can substitute Equations (8) and (9) into Equation (6) to write the difference between the accounting and economic income in year n as Equation (10).
EI n − AI n = g ( EBVn − ABVn )
(10)
Suppressing the subscript n (for notational simplicity) and substituting Equation (10) into Equation (5) yields Equation (11).
ABV − EBV g (EBV − ABV ) IRR = ARR 1 + + EBV EBV
(11)
After combining terms, we can rewrite Equation (11) as Equation (12). 11
The IRR of a project (or firm) is a function of the actual tax liabilities of the firm. These tax liabilities are a function of the firm's accounting depreciation schedules. Thus, tax expense is the same for both AI and EI, and ARR and IRR. For simplicity, we ignore the complexities created by differences between a firm's depreciation policies for f inancial reporting and tax purposes. 12 To derive Equation (7), we assume that beginning ABV, plus net income, minus dividends is equal to ending ABV. Next, we assume that net income (NI) is equal to cash flows (CF) minus depreciation expense: NI = CF - D. Finally, we assume that cash flows are either reinvested in the firm (capital expenditures) or paid as dividends (DIV): DIV =
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ABV IRR = ARR + (g − ARR )1 − EBV
(12)
Equation (12) shows that a firm's IRR is a function of its ARR, its asset-growth rate g, and the ratio of its accounting book value to its economic book value. Fisher and McGowan (1983), Steele (1986), and Brief and Lawson (1991) use some or all of these variables to map accounting returns into economic returns. The difference between ARR and IRR in the Fisher and McGowan model depends on a firm's growth rate. By writing a firm's IRR as a function of both the growth rate and the ratio ABV/EBV at the beginning of the estimation year, our model is more descriptive than Fisher and McGowan’s. In Steele (1986) and Brief and Lawson (1991), the difference between ARR and IRR depends on the firm’s growth rate and on the ratio of the firm’s accounting book value to its economic book value at both the beginning and the end of an estimation period. Because IRR is a function of the ABV/EBV ratio at only one point in time, our model is easier to operationalize than the models in Steele and Brief and Lawson.13 Equation (12) identifies two cases when the difference between a firm's IRR and ARR is zero. In the first, the firm's asset-growth rate equals its ARR. In the second, the firm's ABV/EBV ratio is equal to one (which implies that a firm's accounting and economic depreciation schedules are equal). These conditions for the equivalence of ARR and IRR are not new, being noted in Fisher and McGowan (1983) and Fisher (1988). However, Fisher (1988, p. 256) describes these conditions as “remarkable, but not of great general interest.” Similarly, Fisher and McGowan (1983, page 83) conclude, “only by accident will accounting rates of return be in one-to-one correspondence with economic rates of return”.14
CF - CE. (If CE exceeds CF, DIV measures new capital raised by the firm.) The cash flow terms in our expressions for NI and DIV cancel out, leaving us with Equation (7). 13 Although our model appears similar to one developed by Beaver (1998, chapter 3), the two differ in a fundamental way. Beaver uses his framework to analyze the relations among a firm's future ARR, IRR, and stock returns. As a result, Beaver must distinguish between the case of certain outcomes (Chapter 3), and uncertain outcomes (Chapter 4). Our goal is more modest, in that we focus only on the relation between the ARR and IRR a firm has earned in the past. Therefore, we do not have to deal with the case of uncertain future outcomes. 14 The economic value-added literature identifies many other factors that create differences between ARR and IRR. Weaver (2001) notes there are 164 potential adjustments to convert accounting-based return measures to their economic counterparts. Weaver also notes the process of estimating economic value-added requires a very detailed
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However, we believe both of these conditions could be empirically descriptive for actual firms. For example, a firm’s asset-growth rate will equal its ARR if it retains (reinvests) its entire cash flow.15 Thus, our empirical findings in Section IV, showing that ARR is often close to g, are not surprising. We also argue that adherence to GAAP rules will cause ABV/EBV to be close to one for many firms. This topic is discussed in the next section. III. ACCOUNTING-POLICY CONSTRAINTS AND THE SIZE OF ABV/EBV Equation (12) implies the difference between a firm's IRR and ARR depends on the difference between g and ARR, and on the size of the ratio ABV/EBV. For this section, we assume the assumption of constant growth is met, and focus on determining the likely size of ABV/EBV. Complicating matters, the variable EBV depends on the value of a firm's IRR, introducing circularity into the model. However, we do not attempt to remove the circularity because accounting rules allow us to place constraints on ABV/EBV, and make it possible to estimate the likely size of the ratio for many firms. We describe these constraints, and their implications, in this section. A. Why ABV is typically less than EBV Firms cannot arbitrarily adopt procedures for expensing investment costs, but instead must comply with GAAP. We argue that for most firms, the expensing of investment costs in accordance with GAAP will cause ABV/EBV to be less than 1. For tangible assets, the inherent conservatism of GAAP encourages a firm to record accounting depreciation at a rate faster than the decline in the economic value of its assets. 16 For most intangible assets, firms must depreciate (expense) the investment immediately. Research and development outlays are ex-
knowledge of a company’s operations, making this calculation better suited to company insiders. Our approach is less detailed and can be applied by external analysts using publicly available data. 15 To illustrate this point, assume that an all-equity firm has beginning assets of $100, and income after taxes of $15. Further, assume that the firm’s depreciation expense is some value z. The firm’s ARR (as defined in this paper) will be 15 percent. Now, assume that the firm reinvests its entire cash flow (15 + z) in new (or replacement) assets. The firm’s ending assets will be $115 (= 100 - z + 15 + z). Thus, the asset-growth rate is 15 percent. We thank Rich Aronson for this insight. 16 In most cases, firms will depreciate assets over a period of time that is less than the time over which the assets will generate cash flows. While this idea is embedded in the principle of conservatism, SFAS No. 121 provides formal rules. If an asset has no remaining economic value—that is, the firm will realize no additional future cash flows —the asset is impaired, and SFAS No. 121 requires the remaining cost to be written off.
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pensed as incurred (Financial Accounting Standard No. 2, 1974), as are investments in advertising and employee training [i.e., investments that in the first case create brand-name capital (Klein and Leffler, 1981), and in the second, human capital]. Other intangible assets are omitted from the Balance Sheet if they are internally generated, or if acquired in a merger treated as a pooling transaction (Accounting Principles Board Opinion No. 16, 1970). The application of accounting rules thus typically causes firms to record accounting depreciation faster than the underlying assets are depreciating on an economic basis, and will cause ABV/EBV to be less than one. Nonetheless, it is possible for ABV to be greater than EBV for some firms, in certain years. For example, ABV/EBV could be greater than one in the years leading up to a significant write-down of a firm's assets. This could occur if a firm does not write-down the value of its assets quickly enough, perhaps because the firm overestimates the probability of a successful turnaround. In extreme cases, this problem could be the result of accounting irregularities; Rite-Aid, Cendant, Sunbeam, and Enron are examples of recent vintage. Given these reasons, it is possible for the ABV/EBV ratio of an individual firm to be larger than one in a given year. However, if internal and external auditing functions are reasonably effective, this situation should not be widespread, nor persist over time. We also note that ABV/EBV can be greater than one because there are some costs that reduce the economic value of a firm, but do not reduce the accounting book value of the firm. An example is the cost of incentive-stock options. To allow our model to handle firms that incur such costs, we define both ABV and EBV as measures of undepreciated investment balances before the recognition of the costs of incentive-stock options (or other costs that do not receive formal recognition in a firm's financial statements). When ABV and EBV are so defined, accounting rules still push ABV/EBV toward a value less than one. Therefore, our model can be used to analyze the ARR-IRR relation for firms that incur unrecognized costs (from an accounting standpoint). However, we must be careful to recognize that here, our IRR estimate is before the recognition of some potentially significant expenses.
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B. Analytical Estimates of ABV/EBV Equation (12) implies that a firm's IRR will fall in a range bounded by its ARR and asset-growth rate if ABV/EBV is greater than zero, but less than or equal to one. However, if the difference between a firm's accounting rate of return and its growth rate is large, the range in which the firm's IRR can fall also is large. To narrow the range, we must be able to place additional restrictions on ABV/EBV. In particular, is this ratio closer to one, or closer to zero? Tables 1 and 2 use simulations to illustrate how ABV/EBV is related to a firm's growth rate, its accounting depreciation schedules, and the timing of the cash inflows from projects. In both tables, the firm's IRR is 15 percent. In Table 1, each project generates a level cash flow, and is depreciated on a straight-line basis for accounting purposes. With this cash-flow pattern, accounting depreciation will exceed economic depreciation in the early years of a project's life.17 In Panel A, the average project has a five-year life; in Panel B, the average project has a ten-year life; and in Panel C, the average project has a fifteen-year life. —Insert Table 1 here— In all three panels, ARR decreases as the growth rate increases. As the growth rate increases, the firm's portfolio of assets becomes more heavily weighted with projects in the earlier years of their useful lives. Since accounting depreciation exceeds economic depreciation during these years, the accounting rate of return decreases as a firm's growth rate increases (holding IRR constant). This effect is more pronounced for firms with longer-lived projects. However, our main concern in Table 1 is with the size of the ratio ABV/EBV. In all three panels of Table 1, the ratio ABV/EBV is always greater than 0.75. In Table 2, the three projects all have a ten-year life and an IRR of 15 percent, but have different cash-flow patterns. 18 In Table 2, Panel A, cash inflows decrease over the project’s life, from $25 (per $100 investment) in year 1 to $11.5 (per $100 investment) in year 10. With this cash-flow pattern, the economic 17
The economic depreciation in this case is similar to the amortization of the principal associated with a schedule of constant mortgage payments. In the early years of a mortgage (project), the bulk of the level payment applies to interest (economic return on investment). Only a small portion of the level cash flow is applied to principal (depreciation) during these years. In the later years of a mortgage (project), the bulk of the level payment applies to principal (depreciation). If a project offers level cash flows and if accounting depreciation is recorded on a straight-line basis, accounting depreciation will exceed the economic depreciation during the early years of a project's life.
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depreciation is equal to 10 percent of the project's cost in each of the 10 years of the project’s life. If straight-line depreciation is also used for accounting purposes, the ARR is 15 percent at all growth rates, and the ratio ABV/EBV is always 1.0. —Insert Table 2 here— If, instead, the firm uses an accelerated accounting-depreciation method—double-declining balance, or immediate depreciation (consistent with GAAP treatment of R&D)—the ABV/EBV ratio is less than one. When the double-declining balance method is used, ARR remains within a relatively small range around 15 percent (12.6 percent to 18.5 percent), and ABV/EBV is 0.811 or higher (for the growth rates listed). When all investments are written off immediately (in year 1), the range of the potential ARRs becomes much larger, from -14.9 percent to +82.5 percent, and the range of ABV/EBV is from 0.182 to 0.334. Although a range of potential ARRs from -14.9 percent to +82.5 percent is very large, it is important to keep in mind that the difference between g and ARR is 82.5 percentage points when ARR is 82.5 percent in this example, and is 44.9 percentage points when ARR is -14.9 percent. The empirical results in Section IV suggest that so large a difference between g and ARR is uncommon.19 Panels B and C in Table 2 tell similar stories, but with slightly different cash-flow patterns. In Panel B, the projects produce level cash flows over a ten-year life, but in Panel C, the bulk of the cash flows are received in the last year of the projects' lives. Moving from Panel A to C, the economic depreciation occurs later in the projects' lives, while the accounting-depreciation schedules remain fixed across the three panels. As a result, the difference between the accounting- and economic-depreciation schedules is larger in Panels B and C than in Panel A. Because of these larger differences, the range of ARRs is also larger, and the values of ABV/EBV are smaller in Panels B and C than in A. In the extreme case—Table 2, Panel C with immediate deprecia-
18
Switching to five -year or fifteen-year projects changes Table 2 in ways consistent with the patterns described in Table 1. That is, as the project length increases, the range of ARR values increases, while the values of ABV/EBV decrease slightly. 19 In the empirical procedures of Section IV, the absolute value of the difference between g and ARR is larger than 40 percentage points for less than one percent of the sample firms.
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tion—the projects are depreciated in year 1, but the bulk of the cash flows occur in year 10. Even in this extreme case, ABV/EBV is 0.20 or higher when the firm's growth rate exceeds 20 percent. Tables 1 and 2 imply that ABV/EBV will typically be closer to one than to zero. For the ratio to drop below 0.5, the accounting and economic depreciation schedules must differ significantly. For example, the ratio drops below 0.5 if the firm depreciates all assets immediately for accounting purposes, even though the assets produce cash flows over a ten-year period. The ratio also drops below 0.5 for a lowgrowth firm if the accounting schedule is weighted toward the early years (double-declining balance), and the cash flows are weighted toward the later years (Table 2, Panel C). In all other cases, ABV/EBV is greater than 0.5. The examples in Tables 1 and 2 assume that all of the firm's assets depreciate over time. However, most firms have some assets that do not depreciate. For example, firms do not record accounting depreciation against current asset balances, or for land. Only when the market value of such assets falls below book value (i.e., the asset's cost) is a firm required to reduce the asset’s book value. That is, the firm will only record accounting depreciation if the asset depreciates on an economic basis. Thus, ABV/EBV is equal to one for a portion of a firm's assets. Since most firms have some assets that do not depreciate, the results in Tables 1 and 2 do not directly apply. Assume that 40 percent of a firm's assets do not depreciate (i.e., 40 percent of the firm's assets are current assets), and that 60 percent of the firm's assets depreciate. If ABV/EBV is 0.50 for the portion of the firm's assets that depreciates, ABV/EBV for the firm's entire asset base is 0.70 [= (40 + 30)/(40 + 60)]. A fortiori, because firms have assets that do not depreciate, it is more likely that ABV/EBV is closer to one than to zero. C. Fisher and McGowan (1983), Revisited Fisher and McGowan (1983) conclude that ARR provides little insight about the underlying IRR from a firm's investments, even in the unrealistically favorable case of constant growth. Below, we recount why Fisher and McGowan claim ARR is a misleading proxy for IRR, and explain why the implications of accounting principles allow an analyst or researcher to interpret accounting information despite each of the
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shortcomings. We challenge the conclusion that ARR provides little information about IRR in the constant growth case. First, Fisher and McGowan (1983, page 84) note that a firm's ARR and IRR fall on the same side of its historical growth rate, but argue that little else can be said about the relation between a firm's ARR and IRR. For example, if a firm's ARR is 20 percent, and its historical asset-growth rate is 10 percent, their analysis suggests the only inference to draw about the firm's IRR is that the return is greater than 10 percent. Their conclusion is consonant with the implication of our unconstrained model (i.e., when ABV/EBV can be greater than or less than one).20 However, we argue that the implications of accounting rules constrain ABV/EBV to be less than, but close to, one. If so, the IRR of the firm in the above example falls between 10 percent and 20 percent, but will be closer to 20 percent. Though this first-cut estimate is imprecise, it demonstrates that an analyst can garner information about a firm's IRR from its ARR. In addition, ABV/EBV depends in part on some observable variables. The mix of a firm's tangible and intangible investments can be estimated from a firm's financial statements, as can the firm's average accounting-depreciation schedule. The future cashflow pattern can be inferred from knowledge of the average product lifecycle in the firm's industry.21 Therefore, analysts can draw inferences about which firms will have higher, and lower, ABV/EBV ratios. A second shortcoming—per Fisher and McGowan (pages 89-91)—is that IRR and ARR may rank firms' profitability in different orders. This result is also an implication of both our constrained (i.e., ABV/EBV less than 1) and unconstrained model. For example, consider two firms, both with historical asset-growth rates of 20 percent. Firm A has an ARR of 8 percent, and an IRR of 11 percent. Firm B has an ARR of 6 percent and an IRR of 13 percent. With these assumptions, ARR and IRR rank the two firms in the opposite order. Therefore, an analyst focusing on only ARR will reach an incorrect conclusion about the relative profitability of the two firms.
20
The Fisher and McGowan model does include one implicit constraint: ABV/EBV cannot be negative. This constraint is also implied when we refer to our “unconstrained” model. 21 For example, the future cash flows for a pharmaceutical product will depend on the remaining patent life, the number of competing products, and the rate of innovation in that product category.
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Our model implies an analyst is not defenseless against this type of ranking conflict. For the assumed relations between ARR, IRR, and g to hold, ABV/EBV must be 0.75 and 0.50 for Firms A and B, respectively (see Equation 12). Therefore, the firms must differ in some fundamental ways, and these differences are likely to be observable. For example, Firm B may rely on R&D investments much more than Firm A. If so, an analyst would estimate a lower ABV/EBV ratio for Firm B, and Equation (12) would show that ARR understates IRR by a larger amount for Firm B than for Firm A. The appropriate conclusion is not that analysts should avoid making ARR comparisons across firms. Instead, analysts must be careful to make ARR comparisons across similar firms (e.g., firms within the same industry). Thus, we conclude that an analyst can draw inferences about the size of a firm's IRR from its ARR, if the firm satisfies the constant growth assumption embedded in our model. However, as pointed out by Fisher and McGowan (1983), this case is probably unrealistically favorable toward use of ARR. In the next section, we explore the implications of our model when firm-growth rates are not constant. IV. THE PROBLEM OF NON-CONSTANT GROWTH If a firm's growth rate is not constant, it is possible for our model to estimate the firm's IRR with error. Nonetheless, there may be a range of non-constant growth patterns for which the error is small enough that our model can provide useful IRR estimates. Guided by the results of simulations, we identify cross-sections of firms in which the IRR estimation error from Equation (12) is likely to be small. Then, we test our conjecture that the ratio ABV/EBV is less than one for most firms. A. Growth Variability and the IRR-Estimation Error The growth rate of any individual firm is unlikely to be perfectly constant. Therefore, the constant growth assumption underlying Equation (12) is not satisfied for most firms. In the extreme cases, the constant growth assumption will not be met because of a recent year of unusually high (or low) growth in response to an unusual event, such as an acquisition, a divestiture, or a large write-off of assets. Since these events create a severe violation of the constant growth assumption, our model cannot apply to firms in the years immediately following this type of unusual event.
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Even for other firms, growth rates may not be constant, and Equation (12) will estimate the firms' IRR with error. This raises two questions. First, how frequently, and how severely, is the constant growth assumption violated across samples of firms that have not experienced unusual events in recent years? Second, how large are the IRR-estimation errors that various non-constant growth patterns create? We use three cross-sectional samples to address the first question. Then, using the results of simulations reported in the appendix, we describe the IRR-estimation errors that are likely to arise when applying Equation (12) to these samples of firms. We construct our samples using data from Year 2000 Compustat Research Insight. To mitigate the impact of transient economic effects, we sample three years: 1999, 1994, and 1989. We include all active and research firms in each year that meet the following conditions: 1. The firm is not in the financial services industry (SIC codes 6000-6799). 2. Total assets are at least $50 million six years prior (t - 6) to the sample year, t. 3. Sales are at least $10 million in year t. 4. Assets, sales, and depreciation and amortization expense are positive in year t. 5. The firm has a positive total asset balance in each year from t-6 to t.
We use two additional filters when constructing our samples. First, we exclude firms for which data are not available to calculate all individual year asset-growth rates from year t-6 to t. Second, we excluded firms that had unusually high or low one-year asset growth rates in any year from t-6 to t. We define an unusually high growth rate as positive 50 percent, and an unusually low growth rate as negative 25 percent. These growth rates are the likely consequence of an acquisition or a divestiture. In any event, growth of this magnitude is unlikely to be sustainable for most firms over a five-year period.22 —Insert Table 3 here—
22
After growing at a 50 percent rate per year over five years, a firm will have increased its scale by 650 percent. After decreasing by 25 percent per year for five years, a firm would be less than a quarter of its former size. The firms that we exclude based upon this criteria have, on average, much larger growth-rate standard deviations than the firms within our samples. In all three samples, the mean growth-rate standard deviations of the excluded firms are significantly larger than those firms within our samples in two -tailed difference in means tests.
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We measure ARR as income before extraordinary items and discontinued operations during the sample year, plus interest expense (net of taxes), divided by beginning-of-the-year total assets. We measure the growth rate g as the geometric annual change in total assets over a five-year period ending one year before the sample year. That is, the estimate of g for each firm in the 1999 sample is the geometric annual growth in year-end total assets from 1993 to 1998. When the firm's growth rate is constant, the asset-growth rate will equal the year-to-year growth in new investments, which is the growth rate defined in our model. Although the asset-growth rate can differ from the investment-growth rate when the firm's growth is not constant, we measure asset growth because investment growth is difficult to measure. A firm’s total investment during a year consists of capital expenditures, investments in working capital, and investments in intangible assets. Firms do not disclose the amount of incremental working capital investments (the change in working capital is the difference between incremental investments and recoveries of investments made in prior periods). In addition, an important component of the new investments in non-R&D intangible assets—advertising, employee training, and marketing costs—is buried within a firm's income statement, and cannot be easily quantified. We illustrate the effects of non-constant investment-growth rates on asset-growth rates in the appendix. For each firm in our three samples, we calculate the standard deviation of the asset-growth rate over the five-year estimation period. We then sort the samples into quintiles based upon the size of this standard deviation. Table 4 presents information about growth volatility, growth levels (g), and accounting returns (ARR) for the three overall samples, and for quintiles in each sample. —Insert Table 4 here— Table 4 includes two measures of growth volatility. The first is the standard deviation of individual firms' asset-growth rates over the preceding five years. The second is the difference between the mean asset-growth rate, calculated over three- and five-year estimation periods. If these growth rates differ, then the annual growth rates over the past three years differ from the annual growth rates four and five years ago. This second measure captures the trend in the firm's growth rate.
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Equation (12) implies that the difference between ARR and IRR will be small when ARR is close to g, an idea that is also an implication of the model in Fisher and McGowan (1983). However, Fisher and McGowan (1983, page 83) argue that ARR is unlikely to equal the asset-growth rate. Table 4 lists the percentage of each sample (and quintile) with asset-growth rates within 2.5, 5.0, and 10.0 percentage points of ARR. This information quantifies empirically how often growth rates are close to ARR. It is important to keep in mind that a 10 percentage point difference between ARR and g may not create a large difference between ARR and IRR; if a firm's ABV/EBV ratio is 0.75, then the implied difference between ARR and IRR will be only 2.5 percentage points. In the appendix, we report results of simulations showing the estimation error created using Equation (12) to estimate the IRR of a firm with non-constant growth. The estimation error will generally increase as the standard deviation of the asset-growth rate increases, and as the difference between the threeand five-year growth rates increases. However, the differences between IRR and ARR are generally less than 3 percentage points when the standard deviation of the asset-growth rate is less than 15 percent, and when the difference between the three- and five-year growth rates is less than 5 percent. Whereas a 3 percentage-point difference between ARR and IRR can be meaningful, a difference of this size still defines a tight range in which a firm's IRR likely falls. Therefore, our model provides a good first-cut estimate of IRR for firms with growth-rate variability of this magnitude. With these broad cut-off points in mind, we interpret the results in Table 4 as follows. For most firms in Q1 through Q4, the asset-growth rate standard deviation is 10 percent or less (and it is always less than 15 percent). In addition, the mean difference between the three- and five-year growth rates is less than 5 percentage points. This evidence suggests that these firms do not violate the constant growth assumption by much, and that Equation (12) will estimate the IRR of these firms with an acceptably small error term. The difference between ARR and g is also small for most of these firms. For 80 to 90 percent of the firms in Q1 and Q2 and for 60 to 80 percent of the firms in Q3 and Q4, this difference is less than 10 percentage points. Combined, this evidence suggests that Equation (12) estimates the
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IRR of these firms with a small error, and the difference between ARR and IRR is probably small for most of the firms in four of the five quintiles. For firms in Q5, the measures of growth variability are higher, and fewer firms have ARR values close to g. However, the mean asset-growth standard deviation is less than 20 percent, and the difference between ARR and g is still less than 10 percentage points for well over 60 percent of the firms. For firms in this quintile, Equation (12) estimates IRR with less precision. Therefore, it becomes more important for analysts to obtain a good estimate of the useful life of a firm's assets, and its ABV/EBV ratio. Even without perfect estimates of these variables, however, Equation (12) provides an analyst with insight about the direction and magnitude of the difference between a firm's ARR and IRR. The results in Table 4, combined with the simulation results in the appendix, suggest that nonconstant growth can reduce the applicability of Equation (12) as a tool for estimating the IRR in some firms. However, the results also show that Equation (12) produced meaningful IRR estimates for over 80 percent of firms that had not experienced recent and large shocks from acquisitions, divestitures, or asset write-offs. B. The Size of ABV/EBV: Empirical Analysis To implement our model, an analyst must be able to estimate a firm's ABV/EBV ratio. In Section III, we predict that ABV/EBV will be less than one (but closer to one than zero) for most firms. If ABV/EBV is less than one, then a firm's historical asset-growth rate and its ARR define a range in which the firm's IRR will fall. In this section, we describe a test of this proposition, using our three samples. In this test, we estimate the IRR of each firm in the three samples using Equation (12) and five estimates of the average ABV/EBV ratio across firms. Since IRR and ABV/EBV cannot be observed, and since the literature does not provide us with a model for estimating IRR from observable variables, we cannot directly test the accuracy of our estimates. 23 23
The existing literature does provide other methods for estimating IRR using unobservable variables. For example, Salamon (1982, 1985, 1988) and Gordon and Hamer (1988) advocate estimating IRR as a function of the firm's cash recovery rate (CRR). However, this method requires an estimate of an unobservable variable, the time profile of the firm's assets. Griner and Stark (1991) and Hubbard and Jensen (1991) argue that CRR estimates can be systematically biased when the cash flow profile (or the average asset useful life) is incorrectly specified. Comparing IRR estimates from Equation (12) with IRR estimates based on a firm's cash recovery rate remains a topic for future research.
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However, we can still evaluate the reasonableness of our estimated distributions of IRR across firms. We start with the proposition that markets allocate resources to firms on an efficient basis. If correct, firms with high IRR-growth opportunities are able to fund more growth than firms with low IRRgrowth opportunities, and firms with higher growth rates should have higher estimated IRRs. Although it is mathematically possible for IRR to be negatively related to growth, this result would imply that firms with agency problems severe enough to allow managers to over-invest in negative net present value projects dominate the population of high growth firms. Given an active market for corporate control, we believe such a scenario could not long persist. Thus, we will interpret values of ABV/EBV that produce a negative relation between IRR and growth as implausible. Using this criterion, we identify IRR distributions that appear to be most consonant with economic theory. Our conjecture is that these distributions will be created using ABV/EBV values that are less than one.24 —Insert Table 5 here— In Table 5, we present the results of this test. This table lists the mean and the standard deviation of our IRR estimates for all firms in the three samples, and for those firms with high growth rates (i.e., above 10% and above 20%). We also report results of two-tailed tests of the hypothesis that there is no difference between the mean IRR of high growth firms and the mean IRR across all firms. In all three samples, high growth firms have significantly higher IRR estimates than other firms, when the average ABV/EBV ratio is less than or equal to one. As the ABV/EBV ratio increases past one, the difference between the estimated IRRs of high growth firms and other firms quickly becomes smaller, and loses significance when the average ABV/EBV ratio is 1.125. As we increase ABV/EBV, the IRR estimates of high growth firms eventually become significantly less than the IRR estimates across all firms. The results in Table 5 imply that values of ABV/EBV less than or equal to one produce plausible distributions of IRR across firms. As ABV/EBV increases past one, the distributions become less tenable. 24
As discussed in Heckman (2001), one of the most important challenges when working with unobservable variables is estimating the distribution of the unobservables. In our model, the unobservable variable ABV/EBV defines the
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Thus, even if the constraints placed upon the ARR-IRR relation by accounting rules are not binding, the empirical evidence suggests that, at worst, ABV/EBV should be only slightly larger than one. If ABV/EBV can be slightly larger than one, Equation (12) still provides a well-defined range in which a firm's IRR will fall. If ARR is 10%, the growth rate is 18%, and ABV/EBV falls somewhere in a range between 0.75 and 1.25, Equation (12) indicates the firm's IRR is between 8% and 12%. Therefore, ARR is a suitable proxy for IRR as long as a firm's ABV/EBV ratio is close to one, even if ABV/EBV is slightly higher than one. As further evidence that ABV/EBV is close to one, we calculate the ratio of current assets to total assets at the beginning of each sample year. The mean (median) values range from a low of 42.8 percent (43.1 percent) in 1999, to a high of 45.4 percent (49.2 percent) in 1989. As depreciation differences affect only slightly more than half of the average firm's asset book value, it is likely that the average value of ABV/EBV across our three samples is close to one. V. ANALYSIS OF INDIVIDUAL FIRMS’ RETURNS Financial statements provide the most accessible information about the returns a firm earns on past investments. Ideally, this accounting-based information helps analysts detect changes in a firm's performance, and allows analysts to compare relative performance across firms. Previous studies cast doubt on the usefulness of accounting information for these purposes. Fisher and McGowan (p. 82) claim that, although accounting information may reveal that one firm has more dollars of profits than another, " . . . that information alone does not tell us which firm is more profitable in the sense of having the higher economic rate of return." Our model supports this conclusion in its literal sense. Equation (12) implies that ARR must be adjusted for a firm's growth rate and its ABV/EBV ratio to estimate a firm's IRR. Therefore, accounting information alone will not provide complete information about a firm’s economic returns.
weighting of the difference (g - ARR). We calculate the implied distribution of the unobservable variable IRR using a variety of weights, and compare the characteristics of these implied distributions.
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However, the analytical procedures and empirical evidence in this paper suggest that a firm's unadjusted ARR frequently does provide valuable insights about a firm's underlying IRR. If either ABV/EBV is close to one, or ARR is close to g, ARR will be a suitable proxy for IRR. In this section, we illustrate how ARR provides information about IRR, using three pairs of firms. Each pair belongs to the same four-digit SIC code, and we selected industries with recognizable products. Since firms in the same industry will invest in the same types of assets (similar useful lives; similar mix of tangible/intangible investment), ABV/EBV should be similar. As with the procedures in Section IV, we estimate a firm's ARR and historical growth rate using financial statement information. We then estimate each firm's IRR using Equation (12), and selected values of ABV/EBV. The ABV/EBV ratios we use (0.75 and 0.50) are less than, but close to one, consistent with the arguments we present in this paper. The ABV/EBV value that is most appropriate for any two-firm comparison will depend on the industry. For an industry that requires a large amount of intangible investment (e.g. SIC 3674, with Intel and Advanced Micro Devices), ABV/EBV is likely to be at the lower end of this range. For an industry that uses less intangible investment, (i.e., SIC 5331, with Wal-Mart and KMart), the most appropriate ABV/EBV ratio will be higher. We also identify the ABV/EBV value that would, if applied to each firm within a two-firm comparison, produce the same estimated IRR for each firm. For each two-firm comparison, we attempt to determine if their ranking by ARR is likely to match the ranking of the firms by IRR. —Insert Table 6 here— In Table 6, Panel A, we compare two firms that manufacture paints and other coatings, Benjamin Moore and PPG. These firms conform to the constant growth assumption embedded in Equation (12) reasonably well; in five of the six annual observations, the standard deviation of the asset-growth rate is less than 5 percent. In all three years, Benjamin Moore has the higher ARR. The difference between the ARR of the two firms ranges from a low of 5.1 percentage points in 1994 to a high of 11.0 percentage points in 1999. Is this evidence sufficient to conclude that Benjamin Moore has the higher IRR? Fisher and McGowan
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(1983, footnote 3 on page 82) would argue “no”. However, when we estimate the IRR of these two firms for 1999, 1994, and 1989 using Equation (12), we find the differences between the estimated IRRs to be quite similar to the difference between the actual ARRs. These results suggest that Benjamin Moore had the higher IRR during these years. Indeed, for the estimated IRR for PPG to be higher than that of Benjamin Moore, the average ABV/EBV ratio in this industry must be less than zero! However, ARR alone does not tell the complete story about the profitability of these two firms. For each firm in each test year, Equation (12) suggests that ARR slightly overstates the economic profitability of the firms' underlying assets. In Panel B, we look at two firms that develop and manufacture computer chips, Advanced Micro Devices and Intel. These two firms do not conform to the constant growth assumption as well as the two firms of Panel A. In 1989, the growth standard deviation of both Advanced Micro Devices and Intel was greater than 15 percent. By 1999, this standard deviation had declined to approximately 9.5 percent for both firms. This information suggests that IRR estimates using Equation (12) will not be precise, especially for 1989. Because of the higher growth standard deviations, it is not obvious that Intel realized a higher IRR than Advanced Micro Devices in all three years, despite displaying higher ARRs. In 1989, for example, the difference between the IRR estimates when ABV/EBV is 0.50 (which is plausible given the large R&D investments of these firms) is less than 4 percent. By 1999, however, the difference between the estimated IRRs is more pronounced, and Intel appears to have the higher IRR. For Advanced Micro Devices to have the higher IRR in 1999, the average ABV/EBV ratio in this industry must be negative. In Panel C, we analyze two retail firms, K-Mart and Wal-Mart. The low growth standard deviations—less than 7 percent in five of the six observations—suggest that these two firms conform well to the constant growth assumption of our model. In all test years, Wal-Mart has the higher ARR, and the higher IRR estimates. When ABV/EBV is 0.75, the smallest difference between the IRR estimates of the two firms is 9.6 percentage points, in 1999. For K-Mart to have a higher IRR in any year, the average ABV/EBV ratio in this industry must be larger than 1.5. This evidence supports the propositions that Wal-
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Mart has consistently earned a higher investment IRR than K-Mart, and that ARR consistently ranks the two firms in the correct order. Nonetheless, the information in Panel C shows that ARR by itself is an incomplete measure of the relative profitability of the two firms, and of changes in their profitability over time. In all three years, the difference between the two ARRs understates the difference between our estimated IRRs. In addition, the changes in ARR do not track the changes in our estimated IRRs with precision. For example, Wal-Mart's ARR is lower in 1994 than in 1999, but our estimate of IRR in 1994 is higher than the 1999 IRR estimate. Our IRR estimates also suggest that the decline in Wal-Mart's IRR over time was more gradual than the decline in its ARR. Similarly, ARR suggests that K-Mart was more profitable in 1999 than in 1994, but our IRR estimates suggest otherwise. Despite these problems, the overall results in Table 6 suggest that ARR can be informative about the underlying IRR on a firm's investments. Even though ARR may measure IRR with error, the conditions creating this problem can be readily observed and identified. ARR is demonstrably a useful first-cut measure of economic profitability for many firms. VI. SUMMARY AND CONCLUSIONS Previous researchers express doubts that accounting numbers are useful in measuring the economic returns on a firm's past investments. We challenge this conclusion, arguing it is possible to fathom relations that include an unobservable component. We derive a model that relates accounting numbers to economic returns, and show how to implement it. We conclude that, for many firms, ARR is a suitable proxy for the IRR from past investments. We also acknowledge it is possible for a firm's ARR and IRR to differ by a large amount. However, we identify two conditions that will create this problem—a high asset-growth rate or a large amount of intangible investment. Since these conditions can be observed, our model helps investors or researchers identify firms for which ARR is a less suitable proxy for IRR. In addition, our model can guide the additional analyses necessary to adjust ARR toward IRR in these cases.
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The results in our paper have implications for both investors and researchers. For investors, our results justify the attention many analysts give to firms’ accounting performance. For researchers, our results justify the use of accounting returns as a measure of performance. We argue that investors and researchers should attend to a firm's financial statements because a firm's ARR is often a suitable proxy for the IRR from the firm's past investments, and because ARR is a useful starting point for estimating IRR when the two return measures are not close. Thus, we believe information in a firm's financial statements helps shareholders assess the performance of a firm's managers, and helps investors predict the future investment returns and cash flows that support a firm’s current stock price. Our paper focuses only on differences between ARR and IRR that arise from depreciation, including those created by the accounting treatment of intangible assets. This focus is appropriate because previous studies conclude that such differences are sufficient to render accounting numbers meaningless as surrogates for a firm's IRR. In reality, ARR and IRR differ for other reasons as well. For example, there are some expenses— such as off-balance sheet hedging, joint venture activity, and employee-stock options—that reduce a firm's IRR, but do not reduce its ARR. For firms that incur material amounts of these types of expenses, our model can serve only as the starting point for estimating the true IRR on a firm's investments. Therefore, the IRR estimate we generate should be interpreted as a profitability measure before taking into account expenses that do not receive accounting recognition. Extending our model to handle these types of expenses in a more formal sense remains a topic for further research.
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Appendix: Implications of Non-Constant Growth Rates The model we derive assumes a firm's asset-growth rate is constant over time. However, the empirical data reported in Table 4 reveal that constant growth is not descriptive for a large number of firms. Thus, Equation (12) will estimate the IRR of some firms with error. This appendix illustrates the effect of non-constant growth on IRR estimates from Equation (12). Our goal is to identify ranges of growth variability in which IRR estimates from Equation (12) have small error terms. Since we use a firm's asset-growth rate to measure g, the examples here describe the relation between asset-growth rate variability, and the size of the IRR estimation error created by Equation (12). To generate these examples, we make the following assumptions. A1. We assume that a firm's assets have, on average, a six-year life. This assumption allows us to calculate five annual growth rates, as in our empirical procedures. A2. We assume firms use double-declining balance depreciation for accounting purposes. This assumption yields higher error terms than examples using straight-line depreciation, and we do not want to understate the errors that could be embedded within our model. Since most firms have asset portfolios that contain items depreciated using a range of schedules (straight-line, accelerated, and immediate), the use of straight-line depreciation in these examples could understate the rate at which firms depreciate investments. A3. We assume all investments yield level cash-flow streams, with an IRR of 15%. We distinguish between two growth rates. The investment-growth rate measures the growth in new assets from one year to the next. The asset-growth rate measures growth in asset book value, and depends on the annual investment-growth rates over time, and on the accounting depreciation schedule. If the investment-growth rate is constant, the asset-growth rate will be equal to the investment-growth rate, and will be constant. In the following examples, we illustrate how changes in the variability of investment-growth rates affect the variability of asset-growth rates, and IRR estimation error. We employ the following procedures to construct the examples. 1. We assume a pattern of investment growth rates over a twelve-year period prior to the year of interest. The growth-rate pattern over the last six years directly affects the items still in the firm's asset portfolio. The growth patterns we select for this six-year period vary within a range of positive 150 percent to negative 50 percent. The growth over this period drives the results we report. The results are much less sensitive to the investment-growth rate between seven and twelve years ago, but we must quantify this growth to calculate asset-growth rates over time. For simplicity, we assume the investment growth over this period was constant, at 10 percent per year. (Changing this assumption to positive 20 percent, or zero percent, does not have a large affect on the results.)
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2. With these assumed investment growth rates, we construct a depreciation schedule for a firm's investments over the twelve-year period. For each investment cohort, the asset balance is depreciated to zero over a six-year period using the double-declining balance method. From this schedule, we calculate beginning and ending asset balances, and the asset-growth rate, for each of the last twelve years. 3. We calculate the geometric mean asset-growth rate across the last five years, and the last three years. The standard deviation of the last five annual asset-growth rates measures the variability of the asset-growth rates. The difference between the three- and five-year mean growth rates provides insight about the trend of the asset-growth rates. If the three-year growth rate is larger than the five-year growth rate, then the firm's asset portfolio is more heavily weighted toward younger assets than the five-year growth rate suggests, and the five-year growth rate may slightly understate the growth rate that should be used in Equation (12). 4. We calculate ARR and ABV/EBV values that would obtain using each set of investment-growth rates for the last six years. We plug these values into Equation (12), along with the five-year assetgrowth rate, to estimate the IRR in each case. The difference between this IRR estimate and the true IRR of 15 percent is the estimation error created by our model. Table 7 provides the results of this procedure. We organize the results into four panels, based on the size of the asset-growth rate standard deviation. In each panel, we list the underlying investmentgrowth rates, the standard deviation of the five-year asset-growth rate, the mean five-year asset-growth rate, the mean three-year asset-growth rate, the difference between these growth rates, the actual ARR, the estimated IRR, and the difference between this IRR estimate and the true IRR of 15 percent. The results in Table 7, Panel A imply that when the standard deviation of the five-year assetgrowth rate is less than 5 percent, the IRR estimates from Equation (12) are all within 1.1 percentage points of the true IRR of 15 percent. As growth standard deviation increases, the precision of the IRR estimates declines. However, the IRR estimates in Panel B—with a standard deviation between 5 percent and 10 percent—are still within a range of 0.8 and 2.2 percentage points of 15 percent. In all cases in Panels A and B, the IRR estimate is closer to the true IRR of 15 percent than is actual ARR. In Panel C—with a standard deviation between 10 percent and 15 percent—the IRR estimates are all within 3.4 percentage points of 15 percent. Besides having a larger error, the IRR estimates in this panel overshoot 15 percent in cases C.1 and C.2, and the IRR estimate is further than ARR from 15 percent in one case, C.5. These results indicate that greater caution must be used when estimating the IRR of firms
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with so high a growth rate standard deviation. Nevertheless, the IRR estimates in Panel C are still closer to 15% than is ARR for the other four cases. In Panel D—with a standard deviation between 15 percent and 30 percent (the largest growth standard deviation in Table 4 is less than 30 percent)—the IRR estimates are imprecise. The only estimation error less than 3 percentage points is for case D.1, the IRR estimates overshoot 15 percent in two of the cases (D.1 and D.3), and Equation (12) adjusts ARR away from IRR in one case (D.2). Despite these problems, the estimated IRR is still closer to 15 percent than ARR for four of the five cases in Panel D. By evaluating other information—such as the difference between the three- and five-year growth rates—an analyst could identify at least some potential problems. When the IRR estimate overshoots 15 percent, the five-year growth rate is usually further from ARR than the three-year rate, suggesting the use of the five-year growth rate misstates the firm’s growth, creating too large of an adjustment to ARR. In the cases when Equation (12) adjusts ARR away from IRR, the two growth rates fall on different sides of ARR. When the adjustment of ARR to IRR is incomplete, the three-year growth rate is further away from ARR than the five-year rate. Thus, even in cases where Equation (12) estimates IRR with a large error, the model can still help analysts understand the likely relation between a firm's ARR and its underlying IRR. Finally, it is important to keep in mind that the differences reported in Table 7 depend on the underlying assumptions used to construct the examples. If we had used straight-line depreciation, rather than double-declining balance depreciation, the IRR estimation errors would have been smaller, but if we had used immediate depreciation, the differences would have been larger. Regardless of the exact assumptions we use, the key conclusion remains the same. The ability of Equation (12) to estimate underlying IRRs is not constant across all firms, and depends on characteristics of individual firms' growth volatility. For some firms, this volatility is large enough to violate the constant growth assumption embedded in the model. For a significant number of firms, however, it is not. 25
25
Brief and Lawson (1992) conclude that moderate violations of a constant growth assumption do not produce large enough errors to offset the computational benefits of its use. They examine the effect of non-constant asset growth on the accuracy of stock-price predictions, using in a model that links accounting returns to stock prices though a constant growth term.
Page 30
References Abarbanell, J. 1997. Fundamental analysis, future earnings, and stock prices. Journal of Accounting Research 35: 1-24. American Institute of Certified Public Accountants. 1970. Business combinations, APB Opinion No. 16. Ball, R., and P. Brown. 1968. An empirical evaluation of accounting numbers. Journal of Accounting Research 6: 159-178. Beaver, W. 1968. The information content of annual earnings announcements. Empirical Research in Accounting: Selected Studies. Supplement to the Journal of Accounting Research 6: 67-92. -------------. 1998. Financial Reporting: An Accounting Revolution. 3e. Upper Saddle River, NJ: Prentice Hall. -------------, R. Clarke, and W. Wright. 1979. The association between unsystematic security returns and the magnitude of the earnings forecast error. Journal of Accounting Research 17: 316-340. -------------, R. Lambert, and D. Morse. 1980. The information content of security prices. Journal of Accounting and Economics 2: 3-28. -------------, R. Lambert, and S. Ryan. 1987. The information content of security prices: A second look. Journal of Accounting and Economics 9: 139-157. Brief, R., and R. Lawson. 1991. Approximate error in using accounting rates of return to estimate economic returns. Journal of Business Finance and Accounting 18: 13-20. -----------------------------. 1992. The role of accounting rate of return in financial statement analysis. The Accounting Review 67: 411-426. Collins, D., and S. P. Kothari. 1989. An analysis of intertemporal and cross-sectional determinants of earnings response coefficients. Journal of Accounting and Economics 11: 143-182. Danielson, M., and T. Dowdell. 2001. The return-stages valuation model and the expectations within a firm's P/B and P/E ratios. Financial Management 30: 93-124. Fama, E., and K. French. 2000. Forecasting profitability and earnings. Journal of Business 73: 161-175. Feltham, G., and J. Ohlson. 1995. Valuation and clean surplus accounting for operating and financial activities. Contemporary Accounting Research 11: 689-731. Financial Accounting Standards Board. 1974. Statement of Financial Accounting Standards No. 2: Accounting for research and development costs. ----------------------------------------------. 1995. Statement of Financial Accounting Standards No. 121: Accounting for the impairment of long-lived assets and for long-lived assets to be disposed of. Fisher, F. 1988. Accounting data and the economic performance of firms. Journal of Accounting and Public Policy 7: 253-260.
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-----------. 1984. The misuse of accounting rates of return: Reply. American Economic Review 74: 509517. Fisher, F., and J. McGowan. 1983. On the misuse of accounting rates of return to infer monopoly profits. American Economic Review 73: 82-97. Gordon, L., and M. Hamer. 1988. Rates of return and cash flow profiles: An extension. The Accounting Review 63: 514-521. Gordon, M. 1962. The Investment, Financing, and Valuation of the Corporation. Homewood, IL: Richard D. Irwin. Griner, E., and A. Stark. 1991. On the properties of measurement error in cash-recovery-rate-based estimates of economic performance. Journal of Accounting and Public Policy 10: 207-223. Heckman J. 2001. Micro data, heterogeneity, and the evaluation of public policy: Nobel lecture. Journal of Political Economy 109: 673-748. Hicks, J.R. 1946. Value and Capital. 2e. Oxford: Chaundon Press. Hotelling, H. 1925. A general mathematical theory of depreciation. Journal of the American Statistical Association 20: 340-53. Hubbard, C., and R. Jensen. 1991. Lack of robustness and systematic bias in cash recovery rate methods of deriving internal (economic) rates of return for business firms. Journal of Accounting and Public Policy 10: 225-242. Klein, B., and K. Leffler. 1981. The role of market forces in assuring contractual performance. Journal of Political Economy 89: 615-641. Kormendi, R., and R. Lipe. 1987. Earnings innovations, earnings persistence, and stock returns. Journal of Business: 323-345. Leibowitz, M., and S. Kogelman. 1990. Inside the P/E ratio: The franchise factor. Financial Analysts Journal 46: 17-35. Long W. and D. Ravenscraft. 1984. The misuse of accounting rates of return: Comment. American Economic Review 74: 494-508. Miller, M., and F. Modigliani. 1961. Dividend policy, growth, and the valuation of shares. Journal of Business 34: 411-433. Ohlson, J. 1995. Earnings, book values, and dividends in equity valuation. Contemporary Accounting Research 11: 661-687. Peasnell, K. 1996. Using accounting data to measure the economic performance of firms. Journal of Accounting and Public Policy 15: 291-303. Salamon, G. 1982. Cash recovery rates and measures of firm profitability. The Accounting Review 57: 292-302. --------------. 1985. Accounting rates of return. American Economic Review 75: 495-504.
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--------------. 1988. On the validity of accounting rates of return in cross-sectional analysis: theory, evidence, and implications. Journal of Accounting and Public Policy 7: 267-292. Steele, A. 1986. A note on estimating the internal rate of return from published financial statements. Journal of Business Finance and Accounting 13: 1-13. Weaver, S. 2001. Measuring economic value added: A survey of the practices of EVA® proponents. Journal of Applied Finance 11: 50-60.
Page 33
Table 1 The Effect of Growth and Project Length on ARR and ABV/EBV This table shows how ARR and ABV/EBV, the ratio of the firm's accounting book value to its economic book value, are related to the firm's growth rate, and the length of the firm's average project. Within the table, all projects have an IRR of 15 percent, and an initial investment of $100 that is depreciated on a straight-line basis over the project's life. The first project will produce a level cash flow of $29.83 per year over a five-year life. The second project will produce a level cash flow of $19.93 per year over a ten-year life. The third project will produce a level cash flow of $17.10 per year over a fifteen-year life.
Growth Rate
0%
5%
10%
15%
20%
25%
30%
Panel A: Five-Year Life ARR 16.4%
15.9%
15.4%
15.0%
14.6%
14.3%
14.0%
ABV/EBV
0.915
0.920
0.924
0.927
0.932
0.935
0.938
Panel B: Ten-Year Life ARR 18.0%
16.8%
15.8%
15.0%
14.3%
13.8%
13.3%
ABV/EBV
0.831
0.846
0.859
0.871
0.882
0.891
0.900
Panel C: Fifteen-Year Life ARR 19.6%
17.6%
16.1%
15.0%
14.2%
13.6%
13.1%
ABV/EBV
0.795
0.819
0.841
0.859
0.873
0.887
0.767
Page 34
Table 2 Analysis of ARR and ABV/EBV for a 10-Year Project The relation of a Firm’s ARR and ABV/EBV, the ratio of accounting book value to economic book value, to its growth rate, and depreciation method Panel A. Early Cash Flows In Panel A, the project requires a $100 investment and will produce cash inflows in the following amounts over a 10-year period: $25 (year 1), 23.5, 22, 20.5, 19, 17.5, 16, 14.5, 13, and 11.5 (year 10). The IRR of this project is 15 percent. Three depreciation methods are illustrated: Straight-line, double-declining balance, and immediate depreciation, in which the entire $100 investment is written off in year 1. Growth Rate
0%
5%
10%
15%
20%
25%
30%
15.0%
15.0%
15.0%
15.0%
15.0%
15.0%
15.0%
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Straight-Line Depreciation ARR ABV/EBV
Double-declining Balance Depreciation ARR
18.5%
17.2%
16.0%
15.0%
14.1%
13.3%
12.6%
ABV/EBV
0.811
0.820
0.829
0.838
0.846
0.854
0.862
Immediate Depreciation ARR
82.5%
52.8%
31.2%
15.0%
2.6%
-7.1%
-14.9%
ABV/EBV
0.182
0.209
0.236
0.262
0.287
0.311
0.334
Panel B. Level Cash Flows In Panel B, the project requires a $100 investment and will produce a level cash inflow of $19.93 per year over a 10-year period. The IRR of this project is 15 percent and the depreciation treatment is identical to Panel A. Growth Rate
0%
5%
10%
15%
20%
25%
30%
Straight-Line Depreciation ARR
18.0%
16.8%
15.8%
15.0%
14.3%
13.8%
13.3%
ABV/EBV
0.831
0.846
0.859
0.871
0.882
0.891
0.900
Double-declining Balance Depreciation ARR
22.2%
19.4%
17.0%
15.0%
13.3%
11.9%
10.6%
ABV/EBV
0.675
0.694
0.712
0.730
0.746
0.761
0.775
Immediate Depreciation ARR
99.3%
61.6%
34.7%
15.0%
0.2%
-11.1%
-19.9%
ABV/EBV
0.151
0.177
0.203
0.228
0.253
0.277
0.300
Page 35
Table 2 (continued) The relation of a Firm’s ARR and ABV/EBV, the ratio of accounting book value to economic book value, to its growth rate, and depreciation method Panel C. Late Cash Flows In Panel C, the project requires a $100 investment and produces cash inflows in the following amounts over a 10-year period: $15 (year 1), 15, 15, 15, 15, 15, 15, 15, 15, and 115 (year 10). The IRR of this project is 15 percent, and the depreciation treatment is identical to Panel A.
Growth Rate
0%
5%
10%
15%
20%
25%
30%
Straight-Line Depreciation ARR
27.3%
21.9%
18.0%
15.0%
12.8%
11.1%
9.7%
ABV/EBV
0.550
0.590
.627
.662
.693
0.720
0.745
Double-declining Balance Depreciation ARR
33.6%
25.7%
19.6%
15.0%
11.5%
8.7%
6.6%
ABV/EBV
0.446
0.484
0.520
0.554
0.586
0.615
0.642
150.0%
86.1%
43.8%
15.0%
-5.2%
-19.6%
-30.3%
0.100
0.123
0.148
0.173
0.199
0.224
0.249
Immediate Depreciation ARR ABV/EBV
Page 36
Table 3 Sample Selection
1999
1994
1989
Total Firms
17,301
17,301
17,301
Financial services firms (SIC 6000-6799)
(3,680)
(3,680)
(3,680)
(10,271)
(11,044)
(11,477)
Firms with no assets in year t
(829)
(369)
(524)
Firms with sales less than $10 million in year t
(401)
(109)
(96)
Firms with no depreciation and amortization in year t
(35)
(12)
(6)
Firms with missing data between year t-6 and year t
(11)
(18)
(5)
(812)
(580)
(458)
1,262
1,489
1,055
Firms with total assets less than $50 million in year t-6
Firms with an unusually large one-year asset-growth rate between year t-6 and year t Sample Size
Page 37
Table 4 Analysis of Growth Variability This table presents statistics about the variability of g, and the relation between ARR and g, for the 1999 (Panel A), 1994 (Panel B), and 1989 (Panel C) samples. The table lists results for the entire samples, and for quintiles based upon the standard deviation of individual firm asset-growth rates over a five-year period preceding the test year. The standard deviation of an individual firm's asset-growth rate is the standard deviation of the last five annual asset-growth rates preceding the test year. The table lists the cross sectional mean and standard deviation of three variables: ARR, g (calculated over five years), and the absolute value of the difference between estimates of g over three- and five-year periods. ARR is earnings before extraordinary items and discontinued operations plus interest expense, net of tax, for the sample year, divided by total assets at the beginning of the sample year. The asset-growth rate g is the geometric annual growth rate in total assets over a five-year period (or three-year period) ending one-year prior to the sample year. We compare each quintile mean value for these three variables to the overall sample mean in a difference in means test. Quintile means that are significantly different than the overall mean are denoted by "a" (1% significance level) and "b" (5% significance level). Finally, the table lists the percentage of each sample, and each quintile, that has growth rates within plus or minus 2.5, 5.0, and 10.0 percentage points of ARR. All
Q1
Q2
Q3
Q4
Q5
1,262
253
253
253
253
250
σ of Annual Asset Growth Rates (5 Years) Minimum Maximum Mean
0.3% 29.1% 10.1%
0.3% 4.9% 3.4%
5.0% 7.8% 6.4%
7.8% 11.1% 9.4%
11.2% 14.9% 12.9%
14.9% 29.1% 18.6%
ARR Mean Standard Deviation
6.7% (0.08)
7.4% (0.07)
7.4% (0.07)
6.9% (0.10)
5.5% b (0.08)
6.2% (0.10)
g (Estimated Over 5-Years) Mean Standard Deviation
8.2% (0.08)
5.6% a (0.06)
7.3% (0.08)
9.0% (0.09)
8.7% (0.10)
10.3%a (0.08)
Absolute Value of Difference Between 3 and 5-Year Estimates of g Mean 3.5% Standard Deviation (0.03)
1.1% a (0.01)
2.2% a (0.01)
3.2% (0.02)
4.6% a (0.03)
6.4%a (0.04)
g - ARR < |2.5%| < |5.0%| < |10.0%|
33.6% 63.6% 91.3%
24.5% 46.2% 79.8%
22.9% 43.5% 68.8%
20.9% 34.0% 64.0%
18.4% 37.6% 64.0%
Panel A: 1999 Sample Number of Firms
24.1% 45.0% 73.5%
Page 38
Table 4 (continued) Analysis of Growth Variability
All
Q1
Q2
Q3
Q4
Q5
1,489
298
298
298
298
297
σ of Annual Asset Growth Rates (5 Years) Minimum Maximum Mean
0.7% 28.1% 9.1%
0.7% 4.6% 3.2%
4.6% 6.7% 5.6%
6.7% 9.5% 8.0%
9.5% 13.3% 11.1%
13.3% 28.1% 17.4%
ARR Mean Standard Deviation
6.7% (0.07)
7.2% (0.06)
6.8% (0.07)
6.5% (0.05)
6.7% (0.09)
6.2% (0.08)
g (Estimated Over 5-Years) Mean Standard Deviation
5.5% (0.07)
4.6% b (0.05)
4.8% (0.06)
4.2% b (0.07)
6.6% b (0.08)
7.5%a (0.07)
Absolute Value of Difference Between 3 and 5-Year Estimates of g Mean 3.2% Standard Deviation (0.03)
1.0% a (0.01)
1.9% a (0.01)
2.8% a (0.02)
3.9% a (0.02)
6.2%a (0.04)
g - ARR < |2.5%| < |5.0%| < |10.0%|
34.2% 58.7% 85.2%
27.9% 52.3% 81.9%
27.9% 50.0% 76.5%
22.8% 44.0% 75.8%
23.2% 44.1% 77.8%
Panel B: 1994 Sample Number of Firms
27.2% 49.8% 79.4%
Page 39
Table 4 (continued) Analysis of Growth Variability
All
Q1
Q2
Q3
Q4
Q5
1,055
211
211
211
211
211
σ of Annual Asset Growth Rates (5 Years) Minimum Maximum Mean
0.7% 29.9% 9.5%
0.7% 4.6% 3.1%
4.6% 7.4% 6.0%
7.4% 10.3% 8.8%
10.3% 14.0% 11.9%
14.1% 29.9% 17.7%
ARR Mean Standard Deviation
7.0% (0.07)
8.4% a (0.06)
7.0% (0.06)
7.3% (0.07)
6.0% (0.08)
6.1% (0.08)
g (Estimated Over 5-Years) Mean Standard Deviation
8.2% (0.08)
6.6% a (0.05)
7.5% (0.07)
7.9% (0.08)
9.3% (0.09)
9.6%a (0.08)
Absolute Value of Difference Between 3 and 5-Year Estimates of g Mean 3.2% Standard Deviation (0.03)
1.1% a (0.01)
2.1% a (0.01)
2.9% (0.02)
4.1% a (0.02)
5.6%a (0.04)
g - ARR < |2.5%| < |5.0%| < |10.0%|
39.3% 61.1% 91.5%
29.4% 61.1% 86.3%
24.2% 47.4% 79.1%
18.0% 36.0% 69.2%
19.0% 37.0% 66.8%
Panel B: 1989 Sample Number of Firms
26.0% 48.5% 78.6%
Page 40
Table 5 IRR Estimates for Low Growth Variability Firms This table summarizes estimated IRRs for the 1999 (Panel A), 1994 (Panel B), and 1989 (Panel C) samples. We use Equation (12) to estimate the IRR for each firm. Each estimate uses the firm's actual ARR (earnings before extraordinary items and discontinued operations plus interest expense, net of tax, for the sample year, divided by total assets at the beginning of the sample year) and asset-growth rate for the sample year. The asset-growth rate we use in this calculation is the geometric annual growth rate in total assets over a five-year period ending one-year prior to the sample year. We make five IRR estimates for each firm using the following assumed ABV/EBV values: ABV/EBV = 0.75, 0.875, 1.00, 1.125, and 1.25. The table reports the mean IRR estimate, and the standard deviation of the estimate, for all firms in these samples, for firms with growth rates greater than 10%, and for firms with asset-growth rates greater than 20%. We compare the mean IRR for the two high-growth subsamples to the overall sample mean in a difference in means test. IRR means that are significantly different than the overall mean are denoted by "a" (1% significance level) and "b" (5% significance level). # of Firms
0.75
0.875
1.00
1.125
1.25
7.0% (0.07)
6.9% (0.08)
6.7% (0.08)
6.5% (0.09)
6.3% (0.10)
Panel A: 1999 Sample All Firms Mean IRR Standard Deviation
1,262
g > 10% Mean IRR Standard Deviation
470
10.1% a (0.06)
9.0% a (0.07)
7.9% a (0.08)
6.7% (0.09)
5.6% (0.10)
g > 20% Mean IRR Standard Deviation
115
13.0% a (0.08)
11.0% a (0.09)
9.0% a (0.10)
7.1% (0.11)
5.1% (0.12)
6.4% (0.06)
6.5% (0.06)
6.7% (0.07)
6.8% (0.08)
6.9% (0.09)
Panel B: 1994 Sample All Firms Mean IRR Standard Deviation
1,242
g > 10% Mean IRR Standard Deviation
361
10.0% a (0.05)
9.2% a (0.06)
8.4% a (0.07)
7.7% (0.08)
6.9% (0.08)
g > 20% Mean IRR Standard Deviation
39
14.3% a (0.05)
12.6% a (0.05)
10.8% a (0.06)
9.1% (0.07)
7.4% (0.07)
Page 41
Table 5 (continued) IRR Estimates for Low Growth Variability Firms
# of Firms
0.75
0.875
1.00
1.125
1.25
7.3% (0.06)
7.1% (0.06)
7.0% (0.07)
6.8% (0.08)
6.7% (0.09)
Panel C: 1989 Sample All Firms Mean IRR Standard Deviation
1,055
g > 10% Mean IRR Standard Deviation
394
10.1% a (0.04)
9.2% a (0.05)
8.2% a (0.06)
7.2% (0.07)
6.3% (0.07)
g > 20% Mean IRR Standard Deviation
73
12.3% a (0.05)
10.3% a (0.06)
8.3% (0.07)
6.3% (0.08)
4.3%b (0.09)
Page 42
Table 6 Analysis of Individual Firm Profitability This table lists the asset-growth rate, the standard deviation of the asset-growth rate, the accounting rate of return, two estimates of IRR, the ratio of research and development expenses to beginning of the year total assts (R&D/Total Assets), and the ratio of special items to beginning of the year total assets (SI/Total Assets) for firms in each of three 4-digit SIC codes. The asset-growth rate is the geometric annual growth rate in total assets over a five-year period ending one-year prior to the sample year. That is, the asset-growth rate for the 1999 sample is the geometric growth in (year-end) total assets from 1993 to 1998. The standard deviation of the asset-growth rate is the standard deviation of the last five annual asset-growth rates preceding the test year. ARR is earnings before extraordinary items and discontinued operations plus interest expense (net of tax) for the sample year, divided by total assets at the beginning of the sample year. We use Equation (12) to estimate IRR for two assumed values of ABV/EBV (= 0.75; 0.50). The break-even ABV/EBV ratio sets the estimated IRR of the two firms in an industry equal.
1999
1994
1989
7.3% 4.5% 20.0% 16.8% 13.6% 3.4% -1.0%
5.9% 3.0% 15.3% 12.9% 10.6% 5.6% 0.0%
12.7% 3.3% 18.1% 16.8% 15.4% 4.5% 1.4%
5.5% 1.5% 9.0% 8.1% 7.2% 4.4% -1.4%
1.9% 6.7% 10.2% 8.1% 6.0% 3.9% 0.0%
7.4% 3.9% 10.9% 10.0% 9.1% 4.6% 0.0%
-0.20
-3.64
-2.79
Panel A: SIC 2851 Benjamin Moore Asset-growth Rate, g Growth Standard Deviation ARR IRR (ABV/EBV = 0.75) IRR (ABV/EBV = 0.50) R&D/Total Assets SI/Total Assets PPG Asset-growth Rate, g Growth Standard Deviation ARR IRR (ABV/EBV = 0.75) IRR (ABV/EBV = 0.50) R&D/Total Assets SI/Total Assets Break-even ABV/EBV
Page 43
Table 6 (continued) Analysis of Individual Firm Profitability 1999
1994
1989
17.1% 9.5% -0.3% 4.1% 8.4% 14.9% 9.3%
12.3% 13.2% 16.2% 15.2% 14.2% 14.5% -3.0%
16.1% 22.6% 5.4% 8.1% 10.8% 18.7% 0.0%
22.6% 9.6% 23.3% 23.1% 23.0% 11.1% -1.2%
26.2% 11.7% 20.7% 22.1% 23.5% 9.8% -3.8%
16.1% 15.8% 12.9% 13.7% 14.5% 10.3% -2.2%
-0.30
1.48
0.00
-4.1% 5.4% 5.8% 3.3% 0.8% 0.0% 0.0%
7.6% 10.0% 3.5% 4.5% 5.6% 0.0% 0.6%
8.2% 3.2% 4.8% 5.6% 6.5% 0.0% -5.3%
13.9% 6.9% 12.6% 12.9% 13.1% 0.0% 0.0%
33.0% 4.4% 12.1% 17.3% 22.5% 0.0% 0.0%
30.9% 6.5% 18.5% 21.6% 24.7% 0.0% 0.0%
1.61
1.51
2.52
Panel B: SIC 3674 Advanced Micro Devices Asset-growth Rate, g Growth Standard Deviation ARR IRR (ABV/EBV = 0.75) IRR (ABV/EBV = 0.50) R&D/Total Assets SI/Total Assets Intel Asset-growth Rate, g Growth Standard Deviation ARR IRR (ABV/EBV = 0.75) IRR (ABV/EBV = 0.50) R&D/Total Assets SI/Total Assets Break-even ABV/EBV Panel C: SIC 5331 K-Mart Asset-growth Rate, g Growth Standard Deviation ARR IRR (ABV/EBV = 0.75) IRR (ABV/EBV = 0.50) R&D/Total Assets SI/Total Assets Wal-Mart Asset-growth Rate, g Growth Standard Deviation ARR IRR (ABV/EBV = 0.75) IRR (ABV/EBV = 0.50) R&D/Total Assets SI/Total Assets Break-even ABV/EBV
Table 7 Analysis of IRR Estimates Using Non-constant Growth Patterns This table illustrates IRR estimation errors that obtain when using Equation (12) to estimate the IRR of firms with non-constant growth. All simulations assume that the firm’s investment-growth rate was a constant 10 percent per year from t = -11 to t = -6. The investment-growth rates during the last five years are listed in the table. Using these investment-growth rates, a six year useful life, and double-declining balance depreciation, we calculated the firm’s asset-growth rate and asset-growth rate standard deviation over the past five (and three) years. Assuming an investment IRR of 15 percent, we calculated the ARR the firm would report in the current year, and the firm’s ABV/EBV ratio (not listed in the table). Using ARR, ABV/EBV, and the five-year asset-growth rate, we estimated IRR using Equation (12). The difference between the IRR estimate and 15 percent is the IRR estimation error.
gt-1
Investment Growth Rates gt-2 gt-3 gt-4
gt-5
σ Asset Asset Asset Growth Growth Growth Growth Rate Rate (5 Yr) Rate (5 Yr) Rate (3 Yr) Difference
ARR
IRREst
IRRError
Panel A: σ Asset-growth rate (5 Yr) < 5% A.1
5%
15%
-10%
15%
-10%
2.7%
3.5%
3.2%
0.3%
19.4%
14.9%
-0.1%
A.2
10%
25%
15%
35%
30%
3.6%
21.0%
20.3%
0.7%
13.2%
15.2%
0.2%
A.3
25%
10%
25%
35%
25%
4.0%
21.0%
21.4%
-0.4%
12.8%
14.8%
-0.2%
A.4
-10%
10%
5%
0%
15%
4.2%
6.6%
4.7%
1.9%
20.0%
16.1%
1.1%
A.5
15%
15%
5%
-10%
-10%
4.5%
2.3%
4.3%
-2.0%
18.8%
14.2%
-0.8%
Panel B: 5% < σ Asset-growth rate (5 Yr) < 10% B.1
10%
30%
-20%
30%
-20%
6.0%
3.6%
4.1%
-0.5%
18.2%
14.2%
-0.8%
B.2
25%
10%
45%
35%
60%
6.4%
30.4%
29.3%
1.1%
11.2%
15.8%
0.8%
B.3
-10%
-10%
5%
15%
15%
7.1%
6.6%
2.8%
3.8%
21.5%
16.9%
1.9%
B.4
-20%
10%
20%
-10%
30%
7.6%
8.5%
6.4%
2.1%
20.1%
16.6%
1.6%
B.5
35%
35%
25%
10%
10%
8.9%
17.9%
23.5%
-5.6%
11.3%
12.8%
-2.2%
Page 45
Table 7 (continued) Analysis of IRR Estimates Using Non-constant Growth Patterns
gt-1
Investment Growth Rates gt-2 gt-3 gt-4
gt-5
σ Asset Asset Asset Growth Growth Growth Growth Rate Rate (5 Yr) Rate (5 Yr) Rate (3 Yr) Difference
ARR
IRREst
IRRError
Panel C: 10% < σ Asset-growth rate (5 Yr) < 15% C.1
10%
10%
35%
60%
60%
11.5%
31.4%
27.1%
4.3%
12.1%
16.9%
1.9%
C.2
20%
70%
0%
50%
100%
11.8%
38.4%
32.2%
6.2%
9.5%
16.0%
1.0%
C.3
35%
60%
10%
60%
10%
12.2%
27.0%
32.0%
-5.0%
8.9%
12.8%
-2.2%
C.4
-20%
-20%
10%
30%
30%
13.0%
10.1%
3.3%
6.8%
22.2%
18.4%
3.4%
C.5
75%
-25%
25%
75%
-25%
14.3%
12.6%
16.2%
-3.6%
13.5%
13.3%
-1.7%
Panel D: 15% < σ Asset-growth rate (5 Yr) < 30% D.1
100%
-50%
25%
100%
-50%
16.3%
2.6%
6.0%
-3.4%
17.5%
13.4%
-1.6%
D.2
-25%
25%
0%
45%
75%
16.3%
23.2%
13.5%
9.7%
17.0%
18.8%
3.8%
D.3
-25%
50%
75%
25%
125%
20.9%
42.3%
38.5%
3.8%
11.0%
18.7%
3.7%
D.4
50%
100%
0%
100%
0%
22.0%
34.1%
42.2%
-8.1%
6.2%
11.5%
-3.5%
D.5
125%
50%
-25%
125%
-25%
29.7%
23.3%
29.6%
-6.3%
7.2%
10.2%
-4.8%
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