Consider a Firm Structured and Accounted for As Follows
Residual income valuation: Are inflation adjustments necessary?
John O'Hanlon* and Ken Peasnell**
Management School
Lancaster University
Lancaster LA1 4YX
UK
July 17, 2003
* John O'Hanlon: email - J.O'; telephone – (44) 1524 593631
** Ken Peasnell: email - ; telephone – (44) 1524 593977
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Residual income valuation: Are inflation adjustments necessary?
Abstract
This paper explores the question of whether the residual income valuation relationship (RIVR) should be written in inflation-adjusted terms. This question is of particular interest in the light of Ritter and Warr's (2002) claim that the standard nominal historical cost RIVR undervalues firms because it fails to deal with inflation. We present three inflation-adjusted formulations of RIVR, each of which is based on an income measure from the inflation accounting literature, and one of which is a general case of Ritter and Warr's formulation. We show that none of these formulations is any more or less correct than the standard formulation of RIVR, and find no support for the view that it is necessary to write RIVR in inflation-adjusted terms. Finally we argue that, in a setting in which accounting numbers and forecasts thereof are normally presented in nominal historical cost terms, the inflation adjustment of RIVR is likely to bring unnecessary complication to the valuation process, with increased scope for error.
Keywords: Equity valuation, residual income, inflation
JEL classification: M4
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Residual income valuation: Are inflation adjustments necessary?
This paper explores the question of whether the residual income-based valuation relationship (RIVR) should be written in terms of inflation-adjusted residual incomes rather than in terms of nominal residual incomes. Until recently, the RIVR literature has been silent on the question of whether and how residual income forecasts should be adjusted to reflect expected inflation. The standard practice has been to work with nominal forecasts of historical cost residual income, discounted at the nominal cost of capital. However, a recent study by Ritter and Warr (2002) (RW) claims that this practice, as exemplified in the empirical work of Lee, Myers and Swaminathan (1999), can lead to under-valuation of firms. RW claim that for residual income models to produce accurate measures of true economic value 'they should use real required returns, adjust depreciation for the distorting effects of inflation, and make adjustments for leverage-induced capital gains' (RW, pp. 59-60). Bradley and Jarrell (2003) express a similar concern that accounting-based valuation models, including RIVR, fail to deal properly with the effect of inflation. In order to remedy the claimed shortcomings of the nominal historical cost-based RIVR, RW propose a corrected inflation-adjusted formulation. RW report that this gives value estimates that differ markedly from those given by the nominal historical cost-based RIVR. Given the central role of RIVR in the theory and practice of accounting-based valuation, the possibility that RIVR fails to deal properly with the effects of inflation merits detailed analysis. In this paper, we carry out such an analysis.
In our analysis, we present three formulations of RIVR, each of which is based on an inflation-adjusted income measure that has appeared in prior literature. The first formulation is based on current cost residual income. The second is based on real current cost residual income, being current cost residual income less a purchasing-power capital maintenance charge. These two residual income measures derive from income measures familiar from both the seminal work of Edwards and Bell (1961) and the now-defunct Statement of Financial Accounting Standards No. 33: Financial Reporting and Changing Prices (Financial Accounting Standards Board, 1979). The third formulation is based on real current cost residual income expressed in real terms as at the valuation date. This is a development of the residual income measures used in the first two inflation-adjusted formulations of RIVR, and forms the basis of the RW model. In presenting the three inflation-adjusted formulations of RIVR, we demonstrate that each is equivalent to the standard nominal historical cost-based RIVR and that, therefore, none of the three is any more correct or any less correct than the standard formulation of RIVR. Second, we demonstrate that, subject to a minor error, the RW model is a special case of the third of our inflation-adjusted formulations of RIVR, and is likewise no more or less correct than the standard nominal historical cost-based RIVR. We conclude that there is no logical basis for the view that RIVR needs to be written in terms of inflation-adjusted residual incomes rather than nominal historical cost-based residual incomes. In the light of this we argue that, in a setting in which accounting numbers and forecasts thereof are normally presented in nominal historical cost terms, complexity and potential for error are likely to be introduced into the valuation process by working with an inflation-adjusted formulation of RIVR.
The paper is organized as follows. Section 1 outlines the basis for residual income-based valuation in general, and outlines the criticisms levelled by RW at the nominal historical cost-based formulation of RIVR.Section 2 presents three inflation-adjusted formulations of RIVR, as outlined above, and shows that all are theoretically equivalent to the standard nominal historical cost-based formulation of RIVR. Section 3 shows that the RW model is a special case of one of the inflation-adjusted formulations of RIVR presented in Section 2. Section 4 illustrates the complexity and potential for error that can be introduced into the valuation process by recasting nominal historical cost accounting inputs in inflation-adjusted terms. Conclusions are presented in Section 5.
1. The residual income-based valuation relationship (RIVR)
This section outlines the basis for residual income-based valuation in general, and outlines the criticisms levelled by RW at the nominal historical cost-based formulation of RIVR.
RIVR has three foundations. The first of these is the present value relationship (PVR) which is the cornerstone of the theory of asset valuation:
, (PVR)
whereV denotes the intrinsic value of equity, C denotes dividends net of new equity contributions, denotes the one-period nominal cost of equity applicable to the equity capital as at time t+k-1, and denotes expectations at time t. Here and throughout our exposition, all transactions are assumed to occur at the end of the relevant period. A second foundation of RIVR is the assumption that forecasts of dividends, accounting income and book value comply with the clean surplus relationship (CSR). A general statement of CSR is as follows:
, (CSR)
whereB denotes the book value of equity and Y denotes accounting income. The third foundation is the definition of residual income, denoted RI, as income less a capital charge comprising the product of the cost of equity and the beginning-of-period book value of equity:
. (RI)
Provided that as ,[1] the combination of PVR, CSR and RI yields the well-known RIVR:[2]
. (RIVR)
As long as forecast accounting numbers conform to CSR, the estimate of equity value given by RIVR is equal to the estimate, , given by PVR. It is important to note that the theoretical equivalence between RIVR and PVR does not rely upon the valuation convention used in arriving at the forecasts of accounting incomes and book values.
We now present the nominal historical cost-based formulation of RIVR that is conventionally used in the valuation literature, and outline RW's criticism of this formulation. We represent the historical cost balance sheet of the firm as comprising real (non-monetary) depreciable assets measured at historical cost net of depreciation, net debt, and equity measured on a historical cost basis.[3]These three items are denoted , D and , respectively, where the superscript h indicates that the accounting number in question is measured on a historical cost basis. To avoid unnecessary complication, we assume throughout that debt is measured on the same basis under both historical cost and current cost accounting.[4] The historical cost book value of shareholders' equity at time t+s is the excess (or shortfall) of assets over debt:
. (1)
Historical cost income for time t+s, denoted is represented as comprising historical cost net income before depreciation, denoted HCIED, less historical cost depreciation, denoted :
. (2)
Historical cost residual income for time t+s, denoted , is as follows:
. (3)
Provided that forecasts of historical cost income, historical cost book value of equity and dividends articulate in accordance with the historical cost CSR given by (4),
, (4)
the value of equity can be written as follows:
(RIVR-H)
RIVR-H is the nominal historical cost-based formulation of RIVR, where is the estimate of the value of equity in terms of the historical cost book value of equity and forecasts of future historical cost residual incomes and is equal to the value estimate, , given by PVR.
Residual income-based valuation is conventionally based on RIVR-H. Empirical applications of RIVR-H are commonly based on the present value of analyst-based forecasts of nominal historical cost residual incomes for periods up to a horizon of two to five years, plus a terminal value term that includes the present value of all expected post-horizon residual incomes (Francis, Olsson and Oswald, 2000; Frankel and Lee, 1998; Lee et al., 1999). For example, an empirical valuation model used by Lee et al. (1999) and referred to by RW (p. 36) in motivating their analysis, is as follows:
, (5)
where is the (assumed constant) nominal cost of equity. Here, the terminal value term comprises the nominal historical cost residual income forecast for time t+3 capitalized as a constant perpetuity.
RW argue (pp. 36-38) that valuation models of the form of (5) have four shortcomings. First, the post-horizon residual incomes are capitalized as a flat perpetuity, which RW argue could be related to the erroneous application of a nominal discount rate to real flows. Second, such models fail to recognize a gain to equity-holders resulting from erosion in the purchasing power of monetary liabilities due to inflation ('debt capital gain'). Third, the use of the nominal required rate of return to calculate the residual income capital charge means that, in inflationary times, residual incomes are underestimated. Fourth, the depreciation expense embedded within residual income takes no account of increases in the current cost of depreciable assets. RW claim that the overall effect of these shortcomings will be to cause RIVR to undervalue firms. RW also present what they claim to be a corrected inflation-adjusted formulation of RIVR, which gives value estimates that differ markedly from those given by the standard formulation of RIVR. Bradley and Jarrell (2003) express concerns that are related to those of RW, and argue that the use of earnings or residual incomes in valuation models without inflation adjustment will result in under-valuation of firms.
RIVR has a central role in the theory and practice of accounting-based valuation. The claim that the standard nominal historical cost-based formulation of the relationship is deficient with regard to its treatment of inflation, and that this results in the under-valuation of firms, is a serious one. In the next section, we present a number of inflation-adjusted formulations of RIVR, and examine the claimed superiority of these formulations over the nominal historical cost-based formulation.
2. Residual income valuation using nominal and inflation-adjusted numbers
In this section, we formulate versions of RIVR based on three inflation-adjusted residual income measures: (i) current cost residual income, (ii) real current cost residual income, being current cost residual income less a purchasing-power capital maintenance charge, and (iii) real current cost residual income expressed in real terms as at the valuation date. Current cost residual income and real current cost residual income are derived from income measures which appear in the seminal work of Edwards and Bell (1961), and which were required to be disclosed under the now-defunct Statement of Financial Accounting Standards No. 33. The third measure, real current cost residual income as restated to real terms at the valuation date, is embedded within RW's inflation-adjusted formulation of RIVR. The three inflation-adjusted formulations of RIVR are presented in subsections 2.1, 2.2 and 2.3, respectively. For each inflation-adjusted formulation, we show analytically that the inflation adjustment has no effect on the residual income-based value estimate.
2.1 Residual income and RIVR on a current cost basis
The first inflation adjustment that we consider involves restating income and residual income to a current cost basis. We follow the tradition in the literature of assuming that current cost will normally be defined as the cost of replacing the firm's assets.[5] Nothing fundamental is involved in changing from historical cost to current cost. The current cost book value of shareholders' equity at time t+s is as follows:
, (6)
where is the cost at time t+s of replacing the non-monetary assets, based on the prices of those assets, and is the book value of equity at time t+s measured on a current cost basis. Current cost income for time t+s, denoted , is as follows:
(7)
where is the current cost depreciation charge, based on the replacement cost of the related assets, is the adjustment required to convert the historical cost depreciation charge to the current cost depreciation charge (i.e., ), and is the rate of increase in the current cost of the firm's assets in period t+s. The item , reflecting the periodic change in the current cost of the specific non-monetary assets, is sometimes referred to in the inflation accounting literature as a 'holding gain' (Scapens, 1981, p. 61) or as a 'realizable cost saving' (Edwards and Bell, 1961, p. 94). Current cost residual income for time t+s, denoted , is as follows:
(8)
Provided that forecasts of current cost income, current cost book value of equity and forecasts of dividends articulate in accordance with the current cost CSR given by (9),
, (9)
the value of equity can be written as follows:
(RIVR-C)
RIVR-C is the current cost-based formulation of RIVR, where, is the value estimate in terms of the current cost book value of equity and forecasts of future current cost residual incomes. is equal to the value estimates, and , given by PVR and RIVR-H, respectively. Properly applied, RIVR-H and RIVR-C must yield the same value estimate, since the accounting in each conforms to CSR.
2.2 Real current cost residual income
The second inflation adjustment that we consider involves the subtraction from current cost income of a capital maintenance adjustment, to give real current cost income. This income measure is described in Edwards and Bell (1961), and was required to be disclosed under Statement of Financial Accounting Standards No. 33. The subtraction from this income measure of a real capital charge gives real current cost residual income. Real current cost income, denoted , is calculated by deducting from current cost income the amount by which opening equity needs to increase over the period in order for its beginning-of-period purchasing power to be maintained. This capital maintenance adjustment is calculated by reference to the rate of change in the general price level.[6] Real current cost income can be represented as follows:
, (10)
where is the periodic rate of change in the general price level for period t+s. The capital maintenance adjustment in (10), , can be decomposed using (6) to give (i) a price level adjustment to the holding gain on assets and (ii) a gain resulting from the decline in the purchasing power of debt:[7],[8]
. (11)
The real current cost income measure in (10) breaches CSR.[9] It is therefore not immediately obvious that a valuation model that uses the associated residual income measure will yield the same value estimate as models based on clean surplus residual income measures. Nevertheless, a formulation of RIVR based on real current cost residual incomes can be shown to be theoretically equivalent to a version based on nominal current cost residual incomes. This is so because nominal residual income contains an inflation component within the capital charge. The removal of this inflation component when the capital charge is restated to real terms exactly cancels against the capital maintenance charge that is applied in arriving at real current cost income. Real current cost residual income and nominal current cost residual income are therefore identical, and so are the theoretical value estimates based on the two residual income measures.
To establish this equivalence, note that subtraction of a real capital charge from the income measure in (11) gives real current cost residual income as follows:
(12)
where is the time t+s-1 (beginning-of-period) current cost book value of equity restated to express its purchasing power in terms of time t+s (end-of-period) money, and is the period t+s real cost of equity capital defined as follows:
(13)
Substitution of (13) into (12) gives:
(14)
Since , the inflation adjustment to the residual income capital charge, , cancels exactly against the net of (i) the inflation adjustment to the holding gains on real assets, , and (ii) the gain from erosion of the purchasing power of debt capital . Expression (14) therefore collapses to