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'hanlon@lancaster.ac.uk; telephone – (44) 1524 593631
** Ken Peasnell: email - K.Peasnell@lancaster.ac.uk; telephone – (44) 1524 593977
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
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 costbased 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 costbased 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
2
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:




Et [Ct  s ] 

Vt   s
,

s 1
  (1  Re,t  k ) 
 k 1


(PVR)
where V denotes the intrinsic value of equity, C denotes dividends net of new equity
contributions, Re,t  k denotes the one-period nominal cost of equity applicable to the equity
capital as at time t+k-1, and Et [.] 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:
Bt  s  Bt  s 1  Ct  s  Yt  s ,
(CSR)
3
where B 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 t  s  Yt  s  Re,t  s Bt  s 1 .
(RI)
s
Provided that Et [ Bt  s ] /  (1  Re,t  k )  0 as s   ,1 the combination of PVR, CSR and RI
k 1
yields the well-known RIVR:2




Et [ RI t  s ] 

Vt  Bt   s
.

s 1
  (1  Re,t  k ) 
 k 1


(RIVR)
As long as forecast accounting numbers conform to CSR, the estimate of equity value given by
RIVR is equal to the estimate, Vt , 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 A h , D and B h , 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
4
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:
Bths  Ath s  Dt s .
(1)
Historical cost income for time t+s, denoted Y h is represented as comprising historical cost net
income before depreciation, denoted HCIED, less historical cost depreciation, denoted DEPN th s
:
Yt hs  HCIEDt  s  DEPN th s .
(2)
Historical cost residual income for time t+s, denoted RI th s , is as follows:
RI th s  Yt h s  Re,t  s Bth s 1 .
(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),
Bths  Bths1  Ct s  Yt hs ,
(4)
the value of equity can be written as follows:




h
E
[
RI
]
ts

Vt h  Bth    s t


s 1
  (1  Re,t  k ) 
 k 1

 Vt .

(RIVR-H)
RIVR-H is the nominal historical cost-based formulation of RIVR, where Vt h 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, Vt , given by PVR.
5
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:
Vt h  Bth 
Et [ RI th1 ] Et [ RI th2 ]
Et [ RI th3 ]
,


(1  Re )
(1  Re ) 2 (1  Re ) 2 Re
(5)
where Re 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
6
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
7
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:
Btcs  Atc s  Dt s ,
(6)
where Atc s is the cost at time t+s of replacing the non-monetary assets, based on the prices of
those assets, and Btc s is the book value of equity at time t+s measured on a current cost basis.
Current cost income for time t+s, denoted Yt c s , is as follows:
Yt c s  HCIEDt  s  DEPN tc s   t  s Atc s 1
 Yt h s  ADEPN t  s   t  s Atc s 1 ,
(7)
where DEPN tc s is the current cost depreciation charge, based on the replacement cost of the
related assets, ADEPN t  s is the adjustment required to convert the historical cost depreciation
charge to the current cost depreciation charge (i.e., DEPN tcs  DEPN ths  ADEPNt s ), and
 t  s is the rate of increase in the current cost of the firm's assets in period t+s. The item
 t  s Atc s 1 , reflecting the periodic change in the current cost of the specific non-monetary assets,
8
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 RI tc s , is as follows:
RI tc s  Yt c s  Re,t  s Btc s 1
 Yt h s  ADEPN t  s   t  s Atc s 1  Re,t  s Btc s 1 .
(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),
Btcs  Btcs1  Ct s  Ytcs ,
(9)
the value of equity can be written as follows:




c
Et [ RI t  s ] 
c
c

Vt  Bt   s

s 1
  (1  Re,t  k ) 
 k 1


(RIVR-C)
 Vt  Vt h .
RIVR-C is the current cost-based formulation of RIVR, where, Vtc is the value estimate in terms
of the current cost book value of equity and forecasts of future current cost residual incomes. Vtc
is equal to the value estimates, Vt and Vt h , 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
9
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 Y c ,real , 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:
Ytc,sreal  Yt hs  ADEPNt s   t s Atcs1  pt s Btcs1 ,
(10)
where pt  s is the periodic rate of change in the general price level for period t+s. The capital
maintenance adjustment in (10), pt  s Btc s 1 , 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
Ytc,sreal  Yt hs  ADEPNt s  pt s Dt s1  ( t s  pt s ) Atcs1 .
(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
10
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:
RI tc, real
 Yt c,sreal  re,t  s [ Btc s 1 (1  p t  s )]
s
 Yt h s  ADEPN t  s  p t  s Dt  s 1
 ( t  s 
p t  s ) Atc s 1
 re,t  s [ Btc s 1 (1 
(12)
p t  s )] .
where Btcs1 (1  pt s ) 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 re,t  s is
the period t+s real cost of equity capital defined as follows:
re,t  s  (1  Re,t  s ) /(1  pt  s )  1
(13)
 ( Re,t  s  pt  s ) /(1  pt  s ) .
Substitution of (13) into (12) gives:
RI tc,real
 Yt h s  ADEPN t  s  pt  s Dt  s 1
s
 ( t  s  pt  s ) Atc s 1  ( Re,t  s  pt  s ) Btc s 1 .
(14)
Since Btcs1  Atcs1  Dt s1 , the inflation adjustment to the residual income capital charge,
pt  s Btc s 1 , cancels exactly against the net of (i) the inflation adjustment to the holding gains on
real assets, pt  s Atc s 1 , and (ii) the gain from erosion of the purchasing power of debt capital
pt  s Dt  s 1 . Expression (14) therefore collapses to
RI tc,real
 Yt h s  ADEPN t  s   t  s Atc s 1  Re,t  s Btc s 1 .
s
11
(15)
Since the right-hand side of (15) contains exactly the same elements as the right-hand side of (8),
it follows that
RI tc,real
 RI tcs .
s
(16)
In other words, real current cost residual income is equal to nominal current cost residual
income. From (16) and RIVR-C, the value of equity can be written as follows:
Vt c ,real



c , real 
E
[
RI
]
ts

 Btc    s t


s 1
  (1  Re,t  k ) 
 k 1


(RIVR-CR)
 Vt  Vt h  Vt c .
RIVR-CR is the real current cost residual income-based formulation of RIVR, where, Vt c,real is
the value estimate in terms of the current cost book value of equity and forecasts of future real
current cost residual incomes. Vt c,real is equal to the value estimates, Vt , Vt h and Vtc given by
PVR, RIVR-H and RIVR-C, respectively.
2.3. Restatement of residual income to real terms as at the valuation date
The third inflation adjustment that we consider involves restating real current cost residual
income to real terms as at the valuation date, with an appropriate adjustment to the cost of capital
applied to the residual income forecasts. This elaboration is embedded within the RW model.
,t
Real residual income at time t+s stated in real terms at the valuation date t is denoted RI tc,real
s
and is defined as follows:
12
,t
RI tc,real
s

RI tc,real
s
s
,
(17)
 (1  pt k )
k 1
where p denotes the periodic rate of change in the general price level, as previously defined. The
discount factor applicable to forecasts of this item is as follows:
s
s
 (1  re,t  k ) 
k 1
 (1  Re,t  k )
k 1
s
,
(18)
 (1  pt  k )
k 1
where re denotes the real cost of equity, as previously defined. Substitution of (17) and (18) into
RIVR-CR enables the value of equity to be written as follows:
Vt c ,real,t



c , real,t 
E [ RI t  s ] 
 Btc    s t

s 1
  (1  re,t  k ) 
 k 1


(RIVR-CRT)
 Vt  Vt h  Vt c  Vt c ,real .
RIVR-CRT is a formulation of RIVR in terms of real current cost residual incomes stated in real
terms at the valuation date, t. Vtc,real,t , the value estimate within this formulation, is equal to the
value estimates, Vt , Vt h , Vtc and Vt c,real given by PVR, RIVR-H, RIVR-C and RIVR-CR,
respectively.
2.4 Summary
We have now presented inflation-adjusted formulations of RIVR based on current cost residual
income, real current cost residual income, and real current cost residual income stated in real
terms as at the valuation date. The formulations encompass inflation-adjusted income measures
13
which feature both in Edwards and Bell (1961) and in Statement of Financial Accounting
Standards No. 33, as well as the inflation-adjusted residual income measure which is embedded
within the RW model. We have shown that the inflation-adjusted formulations of RIVR are
theoretically equivalent to the standard nominal historical cost-based formulation of RIVR, and
are no more correct and no less correct than that formulation. Provided that CSR is maintained,
there is no theoretical difference between the value estimate obtained from the use of nominal
historical cost residual incomes and that obtained from the use of current cost residual incomes.
Because of the compensating effects of the capital maintenance charge and the inflation
adjustment to the residual income capital charge, the restatement of current cost residual incomes
to real terms should have no effect either. Also, deflation of the real current cost residual
incomes to real terms as at the valuation date, together with use of a real cost of equity rather
than a nominal cost of equity, should have no effect. If the underlying forecasts have been
prepared on a consistent and comparable basis, they should all lead to the same estimates of
intrinsic value. Any differences that could arise must therefore be due to forecasting
inconsistencies. After consideration of the RW model in Section 3, we present in Section 4 a
numerical example which illustrates the equivalences that have been demonstrated in this
section, and which illustrates forecasting inconsistencies that could cause differences between
RIVR-H value estimates and those from inflation-adjusted formulations of RIVR.
14
3. The Ritter-Warr (RW) model
In this section we demonstrate that, with the exception of an insignificant difference that appears
to result from an error in the derivation of the RW model, the RW model is simply a special case
of RIVR-CRT. Expansion of the residual income-related term in RIVR-CRT, by substitution of
(12) and (17) gives
Vt c ,real,t  Btc
 

 
c
h
c

r
B
Yt  s
ADEPN t  s
pt  s Dt  s 1
( t  s  pt  s ) At  s 1
e ,t  s t  s 1
E 
 s
 s

 s 1

s
 t s

(1  pt  k )  (1  pt  k )  (1  pt  k )
(1  pt  k )
(1  pt  k )  



 

k 1
k 1
k 1
k 1
  .
    k 1
s

s 1
 (1  re,t  k )


k 1






(19)
The RW model contains a number of simplifying assumptions. First, the current cost
depreciation adjustment is expected to remain constant in real terms10:
s
ADEPN t  s  ADEPN t  (1  pt k )  s .
k 1
(20)
Second, the rate of holding gains is expected to remain equal to the rate of inflation: 11
 t  s  pt  s  s .
(21)
The RW representation does not therefore allow for the re-distributive effects of inflation
allowed by the real holding gains term, ( t s  pt s ) Atcs1 , that appears in (12). Third, it is
,t
assumed that, for s > 3, the rate of growth in RI tc,real
, being real current cost residual income as
s
stated in real terms at the valuation date t, is expected to remain constant at the rate of g. Fourth,
15
the real cost of equity is assumed to be constant at the rate of re . Application of the RW
assumptions to our expression (19) gives
Vt c ,real,t  Btc
 Yh

p D
E t  t 1  ADEPN t  t 1 t  re Btc 
1  pt 1
1  pt 1

 
1  re




Yt h 2
pt  2 Dt 1
re Btc1 
Et  2
 ADEPN t  2


1  pt 1 
  (1  pt  k )
 (1  pt  k )
k 1
 k 1





Yt h3
pt 3 Dt  2
re Btc 2 
Et  3
 ADEPN t  3
 2

  (1  pt  k )
 (1  pt  k )  (1  pt  k ) 
k 1
k 1
 k 1

(22)
(1  re ) 2
(1  re ) 2 (re  g )
.
Without providing any formal analysis, RW claim the theoretically correct residual incomebased valuation model to be as follows (RW, pp. 38-39):
Vt c ,real,t  Btc
 Yh

E t  t 1  ADEPN t  p t 1 Dt  re Btc 
1  p t 1

 
1  re




Yt h 2
p t  2 Dt 1
re Btc1 
Et  2
 ADEPN t 


(1  p t 1 ) 1  p t 1 
  (1  p t  k )
 k 1





Yt h3
p t 3 Dt  2
re Btc 2 
Et  3
 ADEPN t  2
 2

  (1  p t  k )
 (1  pt  k )  (1  pt  k ) 
k 1
k 1
 k 1

(23)
(1  re ) 2
(1  re ) 2 (re  g )
16
.
Expression (23) differs from (22) only in that the terms reflecting the inflation-related gain on
2
3
k 1
k 1
debt are greater by pt21 Dt /(1  pt 1 ) , pt22 Dt 1 /  (1  pt k ) and pt23 Dt 2 /  (1  pt k ) ,
respectively. This discrepancy appears to be due to an error in the RW model, which wrongly
implies that the debt-related component of the capital maintenance charge in (11) and (19) is
pt  s Dt  s 1 (1  pt  s ) rather than pt  s Dt  s 1 , an overstatement of pt2s Dt s1 . To illustrate the
nature of this error, suppose a firm owes $1,000 throughout a year when inflation is 5%. The
amount required to maintain the purchasing power of the amount owing to the creditor is $50 (=
0.05 * $1,000), but RW measure the related adjustment to the real current cost income in (11) as
$52.5 (= 0.05 * $1000 * 1.05), an overstatement of $2.5 (=0.052 * $1000). The impact of this
additional squared inflation-rate term is likely to be empirically unimportant in most cases.
We have shown that the RW model is essentially a special case of the formulation of
RIVR in terms of real current cost residual income stated in real terms at the valuation date. We
have previously shown that this formulation is theoretically equivalent to the standard historical
cost residual income-based valuation model as given by RIVR-H. The RW model is therefore no
more correct and, subject to an insignificant error, no less correct than the standard historical cost
residual income-based valuation model. We may therefore conclude that we find no basis for
RW's claims that the standard historical cost residual income-based valuation model must be
adjusted to deal with inflation and that the RW inflation-adjusted residual income valuation
model is theoretically more correct than the historical cost-based version. If the underlying
forecasts have been prepared on a consistent and comparable basis, they should all lead to the
17
same estimates of intrinsic value. Any differences that could arise in empirical applications must
therefore be due to forecasting inconsistencies. We consider this issue in the next section.
4. Potential problems is working with inflation adjusted formulations of RIVR
In Section 2 we presented formulations of RIVR in terms of a number of inflation-adjusted
residual income measures, and in Section 3 we showed that the RW model is a special case of the
last of these formulations. We have shown that all of these inflation-adjusted formulations are
theoretically equivalent to the standard nominal historical cost-based formulation of RIVR. We
can find no justification for RW's claim that, for residual income models to produce accurate
measures of true economic value, it is necessary to '… use real required returns, adjust
depreciation for the distorting effects of inflation, and make adjustments for leverage-induced
capital gains' (RW, pp. 59-60). Having demonstrated that there is no need to work with an
inflation-adjusted formulation of RIVR, we now highlight the possibility that there might be
important practical disadvantages in working with such a formulation in a setting in which the
raw accounting inputs to RIVR are conventionally stated in nominal historical cost terms. We
note that International Accounting Standard No. 29 (International Accounting Standards
Committee, 1994) recommends that, in hyperinflationary economies, financial statements should
be stated in terms of the measuring unit current at the balance sheet date, and where this happens
there may be practical advantages in working with inflation-adjusted numbers.12 However, in a
setting in which accounting numbers and forecasts thereof are not generally prepared in inflationadjusted form, the unnecessary inflation adjustment of readily available nominal numbers is
likely to introduce undesirable added complexity to the valuation procedure. We make this point
18
using two approaches. First, we use a numerical illustration of the equivalence of the value
estimates given by PVR, RIVR-H, RIVR-C, RIVR-CR and RIVR-CRT. This example is a
simplified one, but it allows us to illustrate a number of forecasting inconsistencies that can arise
when implementing an inflation-adjusted formulation of RIVR on the basis of inputs initially
framed in nominal historical cost terms. Second, we explore analytically the relationship between
growth in nominal historical cost residual incomes and current cost residual incomes. We argue
that the potentially complicated nature of this relationship is likely to hinder the recasting of
nominal forecasts of growth in inflation-adjusted terms.
4.1 A numerical illustration
Table 1 sets out a numerical example which illustrates that, properly done, valuations based on
PVR, RIVR-H, RIVR-C, RIVR-CR and RIVR-CRT all yield identical results. We then go on to
illustrate potential forecasting inconsistencies that could give rise, even in this very simple
setting, to differences between RIVR-H value estimates and value estimates from inflationadjusted formulations of RIVR. To this end, we include within the example a number of
apparently short-lasting differences between the properties of historical cost- and current costbased accounting measures. In particular, the example includes the following features. First, both
dividends and historical cost residual incomes grow at the constant rate of 5% after year 2 in
perpetuity. As observed by Lundholm and O'Keefe (2001), it is not correct in general to assume
that both items have the same constant post-horizon rate of growth, but this example is
constructed such that growth in both items is equal after year 2. Second, current cost residual
incomes do not grow at the same rate as dividends and historical cost residual incomes in years
19
1, 2 and 3, but they do grow at the same constant rate of 5% after year 3 in perpetuity. As
illustrated by the example, it is not correct in general to assume that current cost residual incomes
and historical cost residual incomes grow at the same rate, but this example is constructed such
that growth in both items is equal after year 3. The effect of these growth patterns is that PVR
and RIVR-H include a terminal value term reflecting flows from year 2 onward, and RIVR-C,
RIVR-CR and RIVR-CRT include a terminal value term reflecting flows from year 3 onward.
Third, the nominal cost of equity is 7% in year 1 and 10% thereafter. Fourth, the general inflation
rate is 1% in year 1 and 3% thereafter. Fifth, the rate of current cost holding gains is not equal to
the rate of general inflation in year 1 or year 2, but is equal to this rate from year 3 onwards
(3%). Note that the general equivalence of the valuation approaches does not depend upon any of
these features.
Panel A gives the value estimates for PVR and RIVR-H, Panel B gives the estimate for
RIVR-C, and Panel C gives those for RIVR-CR and RIVR-CRT. The value estimate given by
each valuation method is equal to 192.5234. To aid understanding, we explain here the
calculations for PVR, RIVR-H, RIVR-C and RIVR-CRT. The calculation for PVR is as follows:
V0 

E0 [C1 ]
E0 [C 2 ]

(1  Re,1 ) (1  Re,1 )( Re, 2  G3h )
6.0000
10.0000

1.07
(1.07)(. 10  .05)
 5.6075  186.9159
 192.5234 ,
where G3h is the rate of post-horizon (i.e.: post-year 2) growth in both historical cost residual
incomes and dividends. The calculation for RIVR-H is as follows:
20
V0h

B0h
E0 [ RI 1h ]
E0 [ RI 2h ]


(1  Re,1 ) (1  Re,1 )( Re, 2  G 3h )
 60.0000 
15.0000  (.07 * 60.0000) 13.4500  (.10 * 69.0000)

1.07
(1.07)(.10  .05)
 60.0000  10.0935  122.4299
 192.5234.
The calculation for RIVR-C is as follows:
V0c  B0c 
E 0 [ RI 1c ]
E 0 [ RI 2c ]
E 0 [ RI 3c ]


(1  Re,1 ) (1  Re,1 )(1  Re, 2 ) (1  Re,1 )(1  Re, 2 )( Re,3  G c )
4
21.0000  (.07 * 80.0000) 16.0625  (.10 * 95.0000)

1.07
(1.07)(1.10)
15.5531  (.10 *101.0625)

(1.07)(1.10)(. 10  .05)
 80.0000 
 80.0000  14.3925  5.5756  92.5553
 192.5234 ,
where G 4c is the rate of post-horizon (i.e.: post-year 3) growth in current cost residual incomes.
The calculation for RIVR-CRT, written in the form of expression (19) which links directly with
the RW model, is as follows:
21
V0c ,real,0  B0c
 Y1h

ADEPN1
p1 D0 ( 1  p1 ) A0c
E0 



 re,1 B0c 
1  p1
1  p1
1  p1
1  p1


1  re,1


c

ADEPN 2
Y2h
p 2 D1
( 2  p 2 ) A1c re, 2 B1 
E0  2
 2
 2
 2


1  p1 
  (1  p k )  (1  p k )  (1  p k )
(
1

p
)

k
 k 1

k 1
k 1
k 1

2
 (1  re,k )
k 1




re,3 B2c 
ADEPN 3
Y3h
p 3 D2
( 3  p3 ) A2c
E0  3
 3
 3
 3
 2

  (1  p k )  (1  p k )  (1  p k )
 (1  p k )  (1  p k ) 
k 1
k 1
k 1
k 1
 k 1

2
.
 (1  re,k )( re,3  g 4 )
k 1
 80.0000
 8.0000

 .01 * 120.0000

15.0000 (12.0000  10.0000) (.01 * 40.0000)  120.0000




 (.059406 * 80.0000)
1.01
1.01
1.01
 1.01
1.059406
 5.2125

 .03  * 135.0000

(13.5000  10.9000) (.03 * 40.0000)  135.0000
13.4500
.067961 * 95.0000





(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03)
1.01

(1.059406)(1.067961)
(14.3063  11.4450) (.03 * 42.0000) (.03  .03) * 143.0625 .067961 * 101.0625
14.1225




2
(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03) 2
(1.01)(1.03) 2
(1.01)(1.03) 2

(1.059406)(1.067961)(. 067961  .019417)
 80.0000  14.3925  5.5756  92.5553
 192.5234 ,
where g 4 is the rate of post-horizon (i.e.: post-year 3) growth in real current cost residual
incomes as expressed in real terms at the valuation date.
22
This example shows that proper application of the various valuation approaches to
internally consistent inputs will yield identical value estimates. Our prime purpose in presenting
this example is to illustrate errors that could be made in applying an inflation-adjusted
formulation of RIVR, and which could result in erroneous differences between value estimates
obtained from the historical cost and inflation-adjusted formulations of RIVR. Examples of
possible errors are considered below.
Incorrect assumption that current cost residual income grows at the same rate as historical cost
residual income in year 3.
In the example, the growth rate in historical cost residual income is constant at 5% after year 2.
The growth rate in nominal and real current cost residual income is -17% in year 3 and is
constant at 5% after year 3. The growth rates in real current cost residual income as stated in real
terms at the valuation date are -19.4175% (=(1-0.17)/(1+0.03)-1) for year 3 and 1.9417%
(=(1+0.05)/(1+0.03)-1) after year 3. Consider the effect of an erroneous assumption that, after
year 2, nominal (and real) current cost residual incomes are expected to grow at the same rate as
historical cost residual incomes, and are therefore expected to grow at the constant rate of 5%
after year 2. (Consistent with this, real current cost residual incomes as stated in real terms at the
valuation date are now expected to grow at the rate of 1.9417% after year 2.) This erroneous
assumption gives a value estimate of 217.0561. (See the Appendix for detailed calculations). The
apparently innocuous assumption that the year 3 rate of growth in nominal historical cost
residual incomes can be applied to current cost residual incomes has given rise to a value
estimate that is about 12.7% higher than the correct value estimate of 192.5234 calculated above.
In subsection 4.2, we consider more formally the potentially complicated nature of the
23
relationship between the rates of growth in nominal historical cost residual incomes and current
cost residual incomes, and highlight further the danger of assuming that the rates of growth are
equal.
Inconsistency between the nominal and real rates for the cost of equity and the post-horizon
growth rate.
Another potential pitfall is to make the apparently innocuous assumption that the real cost of
equity and the real rate of growth in post-horizon flows can be approximated by subtracting the
inflation rate from the nominal cost of equity and the nominal growth rate, respectively. This
yields estimated real costs of equity of 6% in year 1 and 7% thereafter (instead of 5.9406% and
6.7961%), and an estimated real post-horizon growth rate in real current cost residual incomes of
2% (instead of 1.9417%). The detailed calculations given in the Appendix show that these
apparently innocuous approximations within RIVR-CRT yield an incorrect value estimate of
185.9588, which is about 3.4% less than the correct value estimate of 192.5234.
Inconsistency (lack of articulation) between current cost holding gains, current cost depreciation
and movements in the current cost book value of assets.
It should be noted that the proper application of a formulation of RIVR employing current cost
income should use forecasts that comply with the current cost version of CSR. This requires that
changes in successive current cost balance sheet values of equity should articulate with periodic
dividends and current cost incomes, inclusive of both holding gains and current cost
depreciation. Failure to obey this articulation will result in value estimates that erroneously differ
from those given by RIVR-H. The complicated nature of the required articulation, even within
our simplified example, illustrates how easy it would be to fall into such error.
24
In this subsection, by reference to a numerical example we have illustrated some of the
complications and errors that might result from attempts to refashion nominal historical costbased accounting data to a form appropriate for use within an inflation-adjusted formulation of
RIVR. In the next subsection, we use a formal model of the relationship between growth in
nominal historical cost residual incomes and current cost residual incomes to emphasise this
point.
4.2 A model of the relationship between growth in nominal historical cost residual incomes
and growth in current cost residual incomes
We now explore analytically, in a simplified setting, the relationship between growth in
historical cost residual incomes and growth in current cost residual incomes (real or nominal). 13
The setting is as follows. An all-equity firm is created at time t, with an initial contribution of
equity capital, which is invested in its entirety in operating assets (i.e: Bth  Ath ). It is expected
that the firm will generate a constant pre-depreciation accounting rate of return of  , that
reducing balance depreciation at the rate d (where d   ) will be applied, that the time t+s
nominal historical cost pre-tax income of At  s 1 (  d ) will be taxed at the constant rate of  ,
that the dividend payout ratio will remain constant at  , that all retained earnings will be
invested in operating assets, and that the nominal cost of equity will remain constant at Re .
Under these assumptions,
Et [ RI th s ]  Ath (1  G h ) s 1 ((  d )(1   )  Re ) ,
(24)
where G h  (  d )(1   )(1   ) is the constant rate of growth in both the historical cost book
value of equity and the historical cost residual income.
25
Now consider the situation if expectations are framed in terms of current cost residual
incomes, where the assumed constant rate of holding gains is  and where the depreciation rate
d is now applied to the current cost of assets. Current cost residual income, both nominal and
real, is expected to evolve as follows:
RI tc1  Ath ((  d )(1   )  Re   )
RI tc 2  Ath (1  G h )((  d )(1   )  Re   )  Ath (  d  Re )
RI tc3  Ath (1  G h ) 2 ((  d )(1   )  Re   )  Ath [(1  G h )  (1    d )](  d  Re )
RI tc 4  Ath (1  G h ) 3 ((  d )(1   )  Re   )
 Ath [(1  G h ) 2  (1  G h )(1    d )  (1    d ) 2 ](  d  Re )
etc.....
(25)
Generalising,
Et [ RI tc s ]  Ath (1  G h ) s 1 ((  d )(1   )  Re   )
 (1  G h ) s 1  (1    d ) s 1 
 Ath 
 (  d  Re ) .
h
(
1

G
)

(
1



d
)


(26)
As s increases, the first term on the right-hand side of (26) grows at the constant rate of G h , but
the second term grows at the more complex time-varying rate of:
(1  G h ) s 1  (1    d ) s 1
1 .
(1  G h ) s  2  (1    d ) s  2
(27)
For G h    d , the growth rate given by (27) asymptotes to G h , and the overall growth rate in
current cost residual income asymptotes to G h ; for G h    d , by L'Hospital's rule, the term in
square brackets in (26) reduces in the limit to ( s  1)(1  G h ) s  2 , and the growth rate in current
cost residual income asymptotes to G h ; for G h    d , the growth rate given by (27)
26
asymptotes to   d , and the overall growth rate in current cost residual income also asymptotes
to   d . These asymptotic growth rates have an intuitive interpretation. If growth in nominal
historical cost residual income is greater than or equal to the rate of current cost holding gains
(less depreciation), then the nominal historical cost growth rate dominates; in the event that
growth in nominal historical cost residual income is less than the rate of current cost holding
gains (less depreciation), then the holding gain term (net of depreciation) dominates. Note,
however, that these are rates to which current cost residual income growth will asymptote in the
long term, and it is not correct in general to use as the rate of growth in current cost residual
incomes the rate of growth in forecasts of historical cost residual incomes, or a simple transform
thereof. Table 2 illustrates the patterns of current cost residual income growth in the presence of
constant residual income growth for two different rates of current cost holding gains,  : 1%
(0.01) and 5% (0.05). Other parameters are as follows: A th = 100,  = 0.40, d = 0.20,  = 0.25,
 = 0.60, and the nominal cost of equity, Re , is 0.10. The parameters  , d, ,  and  give
rise to a rate of growth in historical cost residual income of 6% (0.06):
G h  (  d )(1   )(1   )
 (0.40  0.20)(1  0.25)(1  0.60)
 0.06 .
For  = 0.01, the growth rate in current cost residual income reaches 0.036 after 5 years, 0.053
after 10 years, and 0.058 after 15 years. In contrast, where  takes a value of 0.05, which is
much closer to that of G h , the growth in current cost residual income takes a lot longer to
approximate to the rate of growth in historical cost residual income. Under this circumstance, the
27
growth rate in current cost residual income is -0.031 after 5 years, 0.018 after 10 years, and 0.044
after 15 years.
As can be seen from this simplified illustration, the theoretical link between growth in
historical cost residual income and growth in its current cost counterpart can be complex. If the
valuer of a company has access to a readily available estimate of growth in historical cost
residual incomes, the unnecessary and potentially complex task of correctly recasting this
estimate to a current cost basis is likely to create significant scope for error.
4.3 Summary
In previous sections, we have demonstrated that there is no need to work with an inflationadjusted formulation of RIVR. In this section, using both a numerical example and a model of
the relationship between growth in historical cost and current cost residual incomes, we have
illustrated the potentially complicated nature of adjustments that have to be made in transforming
the nominal historical cost-based inputs to RIVR into a form that could be used in inflationadjusted formulations of RIVR. On the basis of our analysis, we conclude that inflation
adjustment of RIVR is not only unnecessary, but is a likely source of complexity and error.
5. Conclusion
Inflation can have marked effects on both the appearance and the reality of corporate
profitability. An issue that has exercised many accountants and economists over the years is
whether these effects are of such a nature and magnitude as to require the wholesale
transformation of traditional accounting data before reliable inferences can be drawn. Until
28
recently, the literature on accounting-based valuation models has been silent on the questions of
whether and how such models should be written in inflation-adjusted form. However, a recent
study by Ritter and Warr (2002) claims that failure to deal with inflation within RIVR can lead to
under-valuation of firms, a point that has been echoed in Bradley and Jarrell (2003). Our paper
explores the issue of whether it is indeed necessary to adjust RIVR in order to deal properly with
inflation. We find that formulations of the RIVR based on a number of inflation-adjusted income
measures from prior literature, including a formulation of which the Ritter and Warr (2002)
model is a special case, are each no more correct and no less correct than the standard
formulation based on nominal historical cost residual incomes. We have therefore established
that inflation adjustment of RIVR, as advocated by Ritter and Warr (2002) is unnecessary. Any
theoretical superiority claimed for the inflation-adjusted residual income valuation model over
the standard nominal historical cost formulation, due for example to inclusion of gains from debt
in periods of inflation, is therefore illusory.
Any case for using inflation-adjusted forecasts must rest on the practical ground that such
forecasts are easier to obtain or more reliable or both. Only in economies experiencing high
inflation are inflation-adjusted income forecasts likely to be readily available. As we have
illustrated, in other settings the inflation adjustment of readily available forecasts of nominal
accounting numbers may well introduce unnecessary and undesirable complexity, and scope for
error.
We emphasise that we do not argue that the effects of inflation should be ignored in
framing forecasts for use in RIVR by, for example, assuming zero growth for flows that are
expected to grow in line with inflation. The estimation of post-horizon residual income growth
29
should take account not only of matters related to competitive advantage, but also of the impact
of inflation on expectations regarding accounting numbers, under whatever convention those
numbers are constructed. However, we emphasis that, as long as forecast flows are represented in
a consistent manner, a nominal historical cost formulation of RIVR is as correct as an inflationadjusted formulation.
30
Appendix
Calculations relating to numerical example in Section 4
Incorrect assumption that current cost residual income grows at the same rate as historical cost
residual income in year 3.
Consider the effect of an incorrect assumption that current cost residual incomes (and real
current cost residual incomes) grow at the constant rate of 5% after year 2. This gives an
incorrect value estimate of 217.0561, about 12.7% greater than the correct estimate of 192.5234:
V0c  80.0000
21.0000  (.07 * 80.0000) 16.0625  (.10 * 95.0000)

1.07
(1.07)(.10  .05)
 80.0000  14.3925  122.6636

 217.0561.
The erroneous assumption that nominal (and real) current cost residual incomes grow at the
constant rate of 5% after year 2 is equivalent to assuming that real current cost residual incomes
as stated in real terms at the valuation date grow at the rate of 1.9417% after year 2. This
representation, based on expression (19), gives the same value estimate:
V0c ,real,0  80.0000
 8.0000

 .01 * 120.0000

15.0000 (12.0000  10.0000) (.01 * 40.0000)  120.0000




 (.059406 * 80.0000)
1
.
01
1
.
01
1
.
01
1
.
01

1.059406
 5.2125

 .03  * 135.0000

(13.5000  10.9000) (.03 * 40.0000)  135.0000
13.4500
.067961 * 95.0000





(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03)
1.01

(1.059406)(. 067961  .019417)
 80.0000  14.3925  122.6636
 217.0561
31
Inconsistency between the nominal and real rates for the cost of equity and the post-horizon
growth rate.
The assumption that the real cost of equity and the real rate of growth in post-horizon flows can
be approximated by subtracting the inflation rate from the nominal cost of equity and the
nominal growth rate, respectively, yields estimated real costs of equity of 6% in year 1 and 7%
thereafter (instead of 5.9406% and 6.7961%), and an estimated real post-horizon growth rate in
real current cost residual incomes of 2% (instead of 1.9417%). Use of these apparently
innocuous approximations within RIVR-CRT yields an incorrect value estimate of 185.9588,
which is about 3.4% less than the correct value estimate of 192.5234. The calculation is as
follows, based on expression (19):
V0c ,real,0  80.0000
 8

 .01 * 120.0000

15.0000 (12.0000  10.0000) (.01 * 40.0000)  120




 (.06 * 80.0000)
1
.
01
1
.
01
1
.
01
1
.
01

1.06
 5.2125

 .03  * 135.0000

(13.5000  10.9000) (.03 * 40.0000)  135.0000
13.4500
.07 * 95.0000





(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03)
1.01

(1.06)(1.07)
(14.3063  11.4450) (.03 * 42.0000) (.03  .03) * 143.0625 .07 * 101.0625
14.1225




2
(1.01)(1.03)
(1.01)(1.03)
(1.01)(1.03) 2
(1.01)(1.03) 2
(1.01)(1.03) 2

(1.06)(1.07)(. 07  .02)
 80.0000  14.3396  5.3928  86.2264  185.9588 .
32
Notes
1
This condition is exactly met if the firm is expected to be wound up at some finite date in the
future, since after that date book value is zero.
2
Our exposition allows the cost of equity to vary over time. In practical applications, it is
commonly assumed that the cost of equity is constant, i.e., its term structure is flat.
3
We assume the firm's balance sheets contain only depreciable plant and equipment financed by
debt and equity, and this assumption is also made by RW. The assumption is less restrictive than
appearances might suggest. Assets such as inventories and real estate, which are not normally
subject to depreciation, can be thought of as special forms of depreciable asset. For such assets,
the depreciation rate can be taken to be zero until the asset is sold or otherwise disposed of, at
which time the depreciation rate is 100%. Likewise, a typical firm will have significant monetary
liabilities other than interest-bearing debt (notably amounts owed to suppliers); it will also have
monetary assets, such as cash balances and trade credit extended to customers. The concept of
debt must therefore be expanded to include monetary working capital items, with the rate of
interest on these items being set equal to zero.
4
This treatment is in line with that required by all of the inflation accounting standards that have
appeared in Anglo-Saxon countries, including Statement of Financial Accounting Standards No.
33. An alternative way of motivating this treatment is to assume that all debt is issued on a
floating-rate basis.
5
This assumption is also made by RW. It should be noted however that all the arguments made
in this paper would apply equally to other ways of measuring changes in the value of a firm's
assets, such as by reference to realizable proceeds.
33
6
Further details on the real current cost income measure are provided in Edwards and Bell
(1961, chapter 8), Scapens (1981, chapter 6) and Edwards, Kay and Mayer (1987, chapter 5).
The now defunct Statement of Financial Accounting Standards No. 33 (Financial Accounting
Standards Board, 1979) required real current cost income to be presented as supplementary
information within U.S. financial statements.
7
The terms on the right-hand side of expression (11) correspond to items in the illustration
appearing in Appendix A of Schedule A of Statement of Financial Accounting Standards No. 33.
The first two terms taken together correspond to the 'loss from continuing operations adjusted for
changes in specific prices'. (In the illustration given in the standard, current cost income is
negative.) The third term corresponds to the 'gain from decline in purchasing power of net
amounts owed'. The fourth term corresponds to the 'increase in specific prices (current cost) of
inventories and property, plant, and equipment held during the year' less the 'effect of increase in
general price level' in respect of those items.
8
Of course, any anticipated decline in the purchasing power of money will be reflected in
interest charged on debt, and deducted as an expense in arriving at Yt h s .
9
This is so because the last term, pt  s Btc s 1 , is not reflected in the change in book value between
time t+s-1 and time t+s.
10
The assumption that the extra depreciation charge is expected to increase at the general rate of
inflation is a restrictive one. A set of circumstances under which it would hold is as follows.
First, the firm is in steady state with a seasoned stock of identical depreciable assets of varying
vintage such that the annual outlay on replacement is equal to the current cost depreciation. (The
34
method used to depreciate the assets is of no consequence.) Second, the rate of change in the
current replacement cost of the assets remains equal to the (constant) rate of general inflation. If
inflation were time-varying and current cost depreciation increased in step with inflation, its
historical cost counterpart would not do so, and neither would the extra depreciation charge (the
difference between the two).
11
This is consistent with RW's assumption that the extra depreciation charge (ADEPN) rises in
line with general inflation.
12
However, even in a hyperinflationary economy, RIVR calculations based on nominal historical
cost residual incomes should, if done properly, yield the same value estimates as those obtained
using inflation-adjusted residual incomes.
13
Recall from Section 2 that real current cost residual income, RI c,real , is equal to nominal
current cost residual income RI c .
References
Bradley, M. and G. Jarrell. (2003). "Inflation and the constant growth valuation model: A
clarification." Working Paper, Duke University and University of Rochester.
Edwards, E. and P. Bell. (1961). The Theory and Measurement of Business Income, University of
California Press.
Edwards, J., Kay, J. and C. Mayer. (1987). The Economic Analysis of Accounting Profitability,
Oxford University Press.
Financial Accounting Standards Board. (1979). Statement of Financial Accounting Standards
No. 33: Financial Reporting and Changing Prices.
Francis, J., P. Olsson and D. Oswald. (2000). "Comparing the accuracy and explainability of
dividend, free cash flow, and abnormal earnings equity value estimates." Journal of
Accounting Research 38, 45-70.
Frankel, R. and C. Lee. (1998). "Accounting valuation, market expectation and cross-sectional
stock returns." Journal of Accounting and Economics 25, 283-319.
International Accounting Standards Committee. (1994). International Accounting Standard No.
29: Financial Reporting in Hyperinflationary Economies.
Lee, C., J. Myers and B. Swaminathan. (1999). "What is the intrinsic value of the Dow?" Journal
of Finance 54, 1693-1741.
Lundholm, R. and T. O'Keefe. (2001). "Reconciling Value Estimates from the Discounted Cash
Flow Model and the Residual Income Model." Contemporary Accounting Research, 18,
331-335.
36
Ritter, J., and R. Warr. (2002). "The decline of inflation and the bull market of 1982-1999."
Journal of Financial and Quantitative Analysis 37, 29-61.
Scapens, R. (1981). Accounting in an Inflationary Environment, 2nd edition, Macmillan.
37
Table 1 Numerical example of the equivalence of formulations of RIVR based on historical cost and inflation-adjusted residual income measures
This example illustrates the consistency between valuation approaches based on dividends, historical cost residual incomes, and various inflation adjusted
residual income measures. In this example, both dividends and historical cost residual incomes grow at the constant rate of 5% after year 2. Current cost residual
incomes grow at this rate after year 3, but not in year 3. The cost of equity and inflation vary from year 1 to year 2, but each remains constant thereafter. These
features are incorporated to facilitate exposition, and the general equivalence of the approaches does not depend upon them.
Panel A: Value estimates based on dividends and historical cost (HC) residual incomes (PVR and RIVR-H)
Year 0
Income statement
HC income (excluding depreciation) (1)
HC depreciation (10%) (2)
HC net income
Dividend
HC retained earnings
Balance sheet
HC Assets
Debt
HC Equity
Year 1
Year 2
Year 3
Year 4
-
25.0000
-10.0000
15.0000
-6.0000
9.0000
24.3500
-10.9000
13.4500
-10.0000
3.4500
25.5675
-11.4450
14.1225
-10.5000
3.6225
26.8459
-12.0173
14.8286
-11.0250
3.8036
100.0000
40.0000
60.0000
109.0000
40.0000
69.0000
114.4500
42.0000
72.4500
120.1725
44.1000
76.0725
126.1811
46.3050
79.8761
Cost of equity (3)
-
0.07
0.10
0.10
0.10
Growth in dividends
Present value at year 0 of dividends
-
5.6075
0.6667
186.9159
(Terminal value) (6)
0.0500
0.0500
HC capital charge (4)
HC residual income
Growth in HC residual incomes (5)
Present value at year 0 of HC residual
incomes
-
4.2000
10.8000
10.0935
6.9000
6.5500
-0.3935
122.4299
(Terminal value) (6)
7.2450
6.8775
0.0500
38
Year 0
Equity
TOTAL
192.5234
(PVR)
7.6072
7.2214
0.0500
60.0000
192.5234
(RIVR-H)
39
Table 1 Numerical example of the equivalence of formulations of RIVR based on historical cost and inflation-adjusted residual income measures
Panel B: Value estimate based on current cost (CC) residual incomes (RIVR-C)
Year 0
Income statement
HC income (excluding depreciation) (1)
CC depreciation (1) (2)
CC holding gain (8)
CC net income
Dividend
CC retained earnings
Balance Sheet
CC Assets
Debt
CC Equity
Year 1
Year 2
Year 3
Year 4
-
25.0000
-12.0000
8.000
21.0000
-6.0000
15.0000
24.3500
-13.5000
5.2125
16.0625
-10.0000
6.0625
25.5675
-14.3063
4.2919
15.5531
-10.5000
5.0531
26.8459
-15.0216
4.5065
16.3308
-11.0250
5.3058
120.0000
40.0000
80.0000
135.0000
40.0000
95.0000
143.0625
42.0000
101.0625
150.2156
44.1000
106.1156
157.7264
46.3050
111.4214
-
Cost of equity (3)
-
.07
.10
.10
.10
CC capital charge (4)
CC residual income
Growth in CC residual incomes (7)
Present value at year 0 of CC residual
incomes
-
5.6000
15.4000
14.3925
9.5000
6.5625
-0.5739
5.5756
10.1062
5.4469
-0.1700
92.5553
(Terminal value) (6)
10.6116
5.7192
0.0500
40
Year 0
Equity
TOTAL
80.0000
192.5234
(RIVR-C)
Table 1 Numerical example of the equivalence of formulations of RIVR based on historical cost and inflation-adjusted residual income measures
Panel C: Value estimates for real current cost (CC) residual incomes (RIVR-CR and RIVR-CRT)
Year 0
Income Statement
HC income (excluding depreciation) (1)
CC depreciation (1) (2)
CC holding gain (8)
Capital maintenance charge: Assets (9)
Capital maintenance charge: Debt (9)
Real CC net income
Dividend
Real CC retained earnings (11)
Balance Sheet
CC Assets
Debt
CC Equity
-
120.0000
40.0000
80.0000
Year 1
Year 2
Year 3
Year 4
25.0000
-12.0000
8.000
-1.2000
0.4000
20.2000
-6.0000
14.2000
24.3500
-13.5000
5.2125
-4.0500
1.2000
13.2125
-10.0000
3.2125
25.5675
-14.3063
4.2919
-4.2919
1.2600
12.5212
-10.5000
2.0212
26.8459
-15.0216
4.5065
-4.5065
1.3230
13.1473
-11.0250
2.1223
135.0000
40.0000
95.0000
143.0625
42.0000
101.0625
150.2156
44.1000
106.1156
157.7264
46.3050
111.4214
Nominal cost of equity (3)
General Inflation rate
Real cost of equity (3)
-
.07
.01
.059406
.10
.03
.067961
.10
.03
.067961
.10
.03
.067961
Base for real CC capital charge (4)
Real CC capital charge (4)
Real CC residual income
Growth in real CC residual income
Present value of real CC residual incomes
-
80.8000
4.8000
15.4000
14.3925
97.8500
6.6500
6.5625
-0.5739
5.5756
109.2991
7.4281
5.7192
0.0500
Real CC residual income in real terms as at
year 0 (10)
Growth in real CC residual income in real
terms as at year 0
Present value of real CC residual incomes in
real terms as at year 0
-
15.2475
6.3083
104.0944
7.0743
5.4469
-0.1700
92.5553
(Terminal value) (6)
5.0834
-0.586273
-0.194175
0.019417
5.5756
92.5553
(Terminal value) (6)
-
14.3925
41
Year 0
equity
TOTAL
80.0000
192.5234
(RIVR-CR)
80.0000
192.5234
(RIVR-CRT)
5.1821
Notes to Table 1:
1. HC denotes historical cost. CC denotes current cost.
2. Depreciation is at the rate of 10% on a reducing balance basis, applied to the opening net book value of assets.
3. The nominal cost of equity is allowed to vary between years 1 and 2, and remains constant at 10% from year 2 onwards. The real costs of equity for years 1
and 2, based on the inflation rates for years 1 and 2 of 1% and 3%, are calculated as follows: ((1.07)/(1.01)) - 1 = 0.059406; ((1.10)/(1.03)) - 1 = 0.067961.
4. The historical cost capital charge is calculated by applying the cost of equity for the year to the opening historical cost book value of equity for the year. For
example, the capital charge for year 1 is 0.07 * 60.0000 = 4.2000. The current cost capital charge is calculated by applying the cost of equity for the year to
the opening current cost book value of equity for the year. For example, the capital charge for year 1 is 0.07 * 80.0000 = 5.6000. The real current cost
capital charge is calculated by applying the real cost of equity for the year to the opening current cost book value of equity as augmented at the rate of
inflation for the year. For example, the capital charge for year 1 is 0.059406*(80.0000 * 1.01) = 0.059406* 80.8000 = 4.8000.
5. It is not true in general that the rate of growth in historical cost residual incomes is equal to that in dividends. This example is constructed such that the two
items grow at the same rate after year 2.
6. All terminal value calculations reflect expected growth at a constant rate in perpetuity.
The terminal value of dividends is calculated as follows: [10.0000/(0.10-0.05)]/1.07 = 186.9159.
The terminal value of nominal historical cost residual incomes is calculated as follows: [6.5500/(0.10-0.05)]/1.07 = 122.4299.
The terminal value of current cost residual incomes is calculated as follows: [5.4469/(0.10-0.05)]/[(1.07)(1.10)] = 92.5553. The same calculation applies for
the terminal value of real current cost residual incomes, since nominal and real current cost residual incomes are identical to each other.
The terminal value of real residual incomes as stated in real terms at the valuation date is calculated as follows:
[5.0834/(.067961-0.019417)]/[(1.059406)(1.067961)] = 92.5553.
7. It is not true in general that the rate of growth in current cost residual incomes is equal to that of historical cost residual incomes. This example is
constructed such that the two items grow at the same rate after year 3.
8. In years 1 and 2, the current cost holding gains do not accrue at the rate of general inflation. From year 3 onwards, they accrue at the general rate of inflation
(=3%). The year 3 holding gain is 3% of 143.0625 = 4.2919.
9. The capital maintenance charges are calculated by applying the general rate of inflation for the period to the beginning-of-period balances of the items in
question.
10. The real CC residual income in real terms as at year 0 is calculated by dividing by the cumulative effect of inflation. For example, the figure for year 2 is
calculated as follows: 6.5625/(1.01)(1.03) = 6.3083.
11. Note that, because of the capital maintenance charge, real current cost retained earnings is not equal to the periodic change in the book value of equity.
42
Table 2 Illustration of patterns of growth in nominal historical cost residual income and current cost residual income
Year
t+1
t+2
t+3
t+4
t+5
t+6
t+7
t+8
t+9
t+10
t+11
t+12
t+13
t+14
t+15
Nominal historical cost
residual income
Level
Growth
5.000
5.300
.060
5.618
.060
5.955
.060
6.312
.060
6.691
.060
7.093
.060
7.518
.060
7.969
.060
8.447
.060
8.954
.060
9.491
.060
10.061
.060
10.665
.060
11.305
.060
Current cost residual income
(holding gain rate =1%)
Level
Growth
6.000
6.070
.012
6.199
.021
6.381
.029
6.610
.036
6.881
.041
7.193
.045
7.543
.049
7.929
.051
8.351
.053
8.809
.055
9.302
.056
9.832
.057
10.398
.058
11.003
.058
43
Current cost residual income
(holding gain rate =5%)
Level
Growth
10.000
9.350
-0.065
8.849
-0.054
8.476
-0.042
8.217
-0.031
8.058
-0.019
7.987
-0.009
7.994
0.001
8.073
0.010
8.217
0.018
8.421
0.025
8.680
0.031
8.991
0.036
9.353
0.040
9.763
0.044
Note to Table 2:
This table illustrates the behaviour of growth in historical cost residual incomes ( RI h ) and in current cost residual incomes ( RI c ), in
accordance with the following generating processes described in the text:
Et [ RI th s ]  Ath (1  G h ) s 1 ((  d )(1   )  Re )
(24)
 (1  G h ) s 1  (1    d ) s 1 
Et [ RI tc s ]  Ath (1  G h ) s 1 ((  d )(1   )  Re   )  Ath 
(26)
 (  d  Re ) .
h
 (1  G )  (1    d ) 
As described in the text, the setting is as follows. An all-equity firm is created at time t, with an initial contribution of equity capital of
Ath (=100), which is invested in its entirety in operating assets. The firm generates a constant pre-depreciation accounting rate of
return of  (=0.40), reducing balance depreciation is charged at the rate d (=0.20), historical cost pre-tax income is taxed at the rate
of  (=0.25), the dividend payout ratio is  (=0.60), all retained earnings are invested in operating assets, and the nominal cost of
equity is constant at Re (=0.10). Two rates of current cost holding gains,  , are considered: 0.01 (1%) and 0.05 (5%). The rate of
growth in historical cost residual incomes, G h , is as follows:
G h  (  d )(1   )(1   )
 (0.40  0.20)(1  0.25)(1  0.60)
 0.06 .
44