Review of Accounting Studies, 9, 419–441, 2004
2004 Kluwer Academic Publishers. Manufactured in The Netherlands.
Accounting-Based Valuation with Changing Interest
Rates
DAN GODE
dgode@stern.nyu.edu
Stern School of Business, New York University, 44 W 4th Street, Suite KMC10-86, New York, NY 10012,
USA
JAMES OHLSON
johlson@stern.nyu.edu
W.P. Carey School of Business, Arizona State University, P.O. Box 873606, Tempe, AZ 85287, USA
Abstract. This paper generalizes Ohlson’s [Contemporary Accounting Research Vol. 11 No. 2. 661–687
(1995)] equity valuation framework to allow for stochastic interest rates. Much of this analysis initially
deals with the specialized setting in which earnings suffice for cum-dividend value. In such a case, the
beginning-of-period (lagged) rate determines the capitalization factor, not the current rate. The underlying
earnings dynamic modifies the traditional random walk model via an additional term, namely current
earnings multiplied by the percentage change in interest rates. The general model retains these basic
aspects of the earnings-sufficiency setting. Empirical implications bear on the returns-to-earnings regression: The earnings-response coefficient decreases as the beginning-of-period rate increases.
Keywords: stochastic interest rates, valuation, Ohlson model, random walk model of earnings, permanent
earnings, earnings response coefficient
JEL Classification: M41, G12
Accounting-based valuation models build upon the dividend discount model. Instead of focusing on the expected dividend sequence, they use the Modigliani–Miller
dividend policy irrelevancy assumption to focus on the creation of wealth as captured by accounting data. The existing literature models the discount-factor as fixed
and equivalent to a fixed interest rate, which precludes analysis of how rates and
their changes influence valuation coefficients associated with accounting data. In a
more realistic setting, notions of ‘‘earnings multiple’’, in particular, must be reconsidered based on first principles. Similarly, the underlying information-dynamics
require modification due to stochastic rates. To fill this void in the literature,1 the
analysis here introduces stochastic interest rates in a class of accounting-based valuation models.
Following Ohlson (1995), two benchmarks span our valuation models: ‘‘mark-tomarket accounting,’’ or the ‘‘balance-sheet approach,’’ where book value suffices for
valuation; and ‘‘earnings-sufficiency accounting,’’ or the ‘‘income-statement
approach,’’ where earnings suffice for valuation. A weighted average of the two
benchmarks, combined with information besides current accounting data, generalizes the model.
Extending the mark-to-market model to stochastic rates is straightforward
because this model equates book value to market value and thus no coefficient
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GODE AND OHLSON
depends on interest rates. The simplicity of the mark-to-market model still provides
a powerful insight: An accounting-based valuation model need not specify anything
about interest rates if the accounting numbers incorporate market prices, which, in
turn, incorporate interest rates. As a specific example, the securities owned by an
investment fund always reflect interest rates if the fund uses mark-to-market
accounting. Though an analyst trying to value securities in an investment fund might
well consider interest rates, an accountant applying the mark-to-market model to the
fund simply sets book values to security prices without any reference to interest rates.
In contrast to the mark-to-market model, where the book value multiple is 1, in
the earnings-sufficiency model [named ‘‘permanent-earnings model’’ in Ohlson
(1995)], an earnings multiple must depend on interest rates. Ohlson (1995) shows
that with constant interest rates, the multiple for current earnings is R/r, where r is
the risk-free rate and R”1+r. Much of this paper considers the multiple-specification
issue: How does one generalize the multiple R/r when rates are stochastic?
Extending Ohlson’s (1995) earnings-sufficiency model to stochastic interest rates
leads to certain subtleties. Replacing r by rt, the spot interest rate prevailing at time t,
does not work. Instead, we show that r is replaced by rt1 , the spot interest rate
prevailing at the beginning of the earnings measurement period. This result holds
regardless of the stochastic process of interest rates. The use of rt)1 instead of rt in the
multiplier rests on a simple yet useful insight: the expected earnings rate for period
(t)1,t) is rt)1, the rate at the beginning of the period, not rt, the rate at the end of the
period.
Three main findings about the earnings-sufficiency model are as follows:
1. Valuation function: Proposition 1 shows that if earnings xt for the period (t)1, t)
suffice for the cum dividend value Pt + dt, then the Modigliani–Miller dividend
policy irrelevancy condition developed by Yee (Forthcoming) implies that
Pt þ dt ¼ ðRt1 =rt1 Þxt . Moreover, Proposition 2 shows that if the forthcoming
expected earnings are capitalized by the current rate, then the current earnings
must be capitalized by the beginning-of-period rate. Proposition 2 highlights the
idea of capitalization consistency: the capitalization multiple for expected or
realized earnings for a period must be based on the rate prevailing at the
beginning of the corresponding earnings measurement period.
2. Residual earnings dynamic: Proposition 3 shows that with earnings sufficiency the
persistence of residual earnings equals rt =rt1 . With constant rates, residual
earnings have a persistence of 1, i.e., they are ‘‘permanent.’’ With stochastic rates,
residual earnings are no longer permanent because the persistence parameter
(rt =rt1 ) oscillates around 1 rather than equal 1. In this sense, earnings sufficiency
no longer corresponds to the concept of ‘‘permanent earnings.’’
3. Earnings dynamic: Corollary 1 shows that the traditional random walk of
earnings requires an additional term. Specifically, it includes a term, which equals
the percentage change in interest rates multiplied by current earnings. This term
captures the idea that the expected earnings improve if interest rates increase,
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
421
keeping current earnings and dividends constant. The usual random walk model
under constant interest rates disallows the possibility.
Proposition 4 derives the implied information dynamic for the weighted-average
valuation model with other information to highlight a new aspect of earnings
forecasting. In the expression for forecasted earnings, the relative weight of current
earnings vis-à-vis book value is inversely related to the current rate (Corollary 2).
Analysis of this more general model provides two additional testable hypotheses:
1. Levels regression: In a regression of price on book value, capitalized current
earnings, and capitalized expected earnings, the first two coefficients decrease in
the current rate, while the third coefficient increases in the current rate. That is, as
the current rate increases, the information content of current accounting data
decreases relative to information content of capitalized expected earnings. (The
beginning-of-period rate does not affect the relative information content.)
2. Returns regression: In a returns-on-earnings regression, the coefficient of unexpected earnings (earnings response coefficient or ERC), decreases in the beginning-of-period rates. In contrast, a comparative static analysis of Ohlson’s (1995)
model fails to tell us whether the current rate or the beginning-of-period rate
should affect ERC. Further, even though comparative statics suggest that the
ERC should decrease as the rate increases, it is not a foregone result when rates
are stochastic.
Three aspects of the model make it general and versatile. First, the model does not
assume a specific stochastic process of interest rates. Second, realizations of
accounting data and other information can depend on historical and expected
interest rates. Third, unexpected changes in interest rates can correlate with unexpected earnings and growth in expected earnings.
The paper proceeds as follows. Section 1 introduces notations and core assumptions while Sections 2–4 analyze increasingly complex settings. Section 2 examines
the mark-to-market model. Section 3 analyzes the earnings-sufficiency model. Section 4 investigates the weighted average model with other information. Section 5
summarizes the findings.
1.
Notations and Basic Assumptions
At date t, the ‘‘preceding’’ period refers to the period from date t)1 to date t, and the
‘‘forthcoming’’ period refers to the period from date t to date t+1.
xt ¼ earnings for the period t)1 to t, i.e., the preceding period
dt ¼ dividends, net of capital contributions, date t
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GODE AND OHLSON
Pt ¼ ex dividend market price of equity, date t
bt ¼ book value, date t
gt ¼ Pt)bt ¼ goodwill, date t
rt ¼ risk-free interest rate for the period t to t+1 (at date t, rt is the current rate and
rt)1 is the beginning-of-period rate)
Rt ¼ 1 þ r t ;
xat ¼ xt rt1 bt1 is the residual earnings for the preceding period
We assume the following throughout the paper and introduce additional
assumptions later.
1. Risk neutrality and no arbitrage:2
Pt ¼
d tþ1 Þ
Et ðPe tþ1 þ e
Rt
(RNNA):
Note that Rt is observed only at date t; it is random from the perspective of prior
dates.
2. A clean surplus relation:
btþ1 ¼ bt þ xtþ1 dtþ1
(CSR):
The above assumptions imply the following goodwill equation, which we use later:
gt ¼
2.
Et ð~gtþ1 þ ~xatþ1 Þ
Rt
(GE).
The Mark-to-Market Model: The Balance-Sheet Approach
We start with a simple but important benchmark—the mark-to-market model,
where the balance sheet (i.e., the book value, bt) provides sufficient value-relevant
information. The three pertinent points are the following:
1. Price depends only on book value, and goodwill is zero. Pt ¼ bt.
2. Setting goodwill to zero in the goodwill equation (GE) yields Et ½~
xatþ1 ¼ 0 and
xtþ1 ¼ rt bt ¼ rt Pt , so that forthcoming expected earnings depend on the
Et ½~
current rate.
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
423
3. Applying mark-to-market accounting does not require one to model the stochastic process of interest rates because prices already impound such information.
For example, marking an investment fund to market simply requires setting the
book value of each security to its market value. It does not require one to know
the stochastic process of interest rates that the market is using to price the
securities.
3.
The Earnings-Sufficiency Model: The Income-Statement Approach
We now turn to the second benchmark. The income statement provides sufficient
value-relevant information. Though earnings and cum dividend price both represent
the same underlying information, one views earnings as a ‘‘sufficient statistic’’
without specifying the accounting rules.3
As a reference point, consider earnings sufficiency in a constant rate setting
(Ohlson, 1995; Ryan, 1988):
Pt ¼
R
xt d t ;
r
where R/r is the price-earnings multiple.
Alternatively, because RNNA and constant interest rates also imply that
xtþ1 =r, one can express earnings sufficiency in terms of expected earnings
Pt ¼ Et ½e
rather than current earnings.
The subsequent subsections specify how the P/E multiples depend on interest rates
in a stochastic rate setting.
3.1.
The Valuation Function under Stochastic Discount Rates
One may be tempted to incorporate stochastic rates by simply replacing r with rt in
the valuation function so that Pt ¼ ðRt =rt Þ xt dt . The analysis below shows, however, that under reasonable conditions, the multiple must be based on the beginningof-period rate rt)1, not the current rate rt. The valuation function is therefore as
follows:
Pt ¼
Rt1
xt dt :
rt1
Before deriving the valuation function formally, we illustrate the intuition using a
savings account. In a savings account, the price Pt is the account balance, dt is the
withdrawal, and earnings are the interest earned for the period (t)1,t), i.e.,
xt ¼ rt1 Pt1 . RNNA implies Pt ¼ Rt1 Pt1 dt . Substituting Pt1 ¼ xt =rt1 , we get
Pt ¼ ðRt1 =rt1 Þ xt dt . The earnings rate for the period (t)1,t) is the rate prevailing
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GODE AND OHLSON
at t)1, not t, so the multiple for earnings for the period (t)1, t) depends on the rate at
t)1, not t.
A savings account is a special case of certainty. Uncertainty requires a more
sophisticated analysis because under uncertainty xt 6¼ rt1 Pt1 and Pt þd t 6¼ Rt1 Pt1 .
We therefore can no longer derive the valuation function by substituting
Pt1 ¼ xt =rt1 in RNNA. We show below, however, that we can still derive
Pt ¼ ðRt1 =rt1 Þ xt dt by using a Modigliani–Miller (MM) dividend policy irrelevancy condition.
We adapt an MM condition in Yee (forthcoming) to allow for stochastic rates.
Specifically, a firm’s value at date t should be the same if the firm had lowered its
dividends by $z at date t)1, invested $z in a zero NPV investment such as a Treasury
bill, and raised date t dividends by Rt)1z. The interest earned on the investment
would have increased earnings for the period (t)1,t) by rt)1z. Yee’s MM condition
therefore corresponds to a requirement that P(xt,dt;,rt,rt)1,…) ¼ P(xt+rt)1z,
dt+Rt)1z; rt,rt)1,…) for any value of z. We next show that Yee’s MM condition
implies that the multiplier equals Rt)1/rt)1 as in the savings account.
Proposition 1 Assume that (i) Pt + dt ¼ f(rt, rt)1, …)xt, where f(.) depends only on the
history of interest rates,4 and (ii) Pt satisfies a Modigliani–Miller condition P(xt,dt;
rt,rt)1,…) ¼ P(xt + rt)1z, dt + Rt)1z; rt,rt)1,…), where z is any number. Then f(.) =
Rt)1/rt)1.
Proof: See Appendix I.
Two aspects of Proposition 1 are noteworthy. First, Proposition 1 needs no
assumptions about the stochastic process of interest rates because applying the MM
condition at date t)1 only requires rt)1, a rate that is known at the beginning of the
period (t)1, t). If applying the MM condition at date t)1 required earnings rate(s)
beyond the period (t)1, t), such as earnings rate rt for the period (t, t + 1), the MM
condition would need a stochastic process of interest rates. Second, Proposition 1
does not explicitly assume RNNA. The MM condition, however, implies that
RNNA holds at the margin with respect to a change in dividends.
We can extend the intuition behind the capitalization of current earnings to
capitalization of forthcoming earnings by adding RNNA to Proposition 1. RNNA
implies Pt ¼ Et ½P~ tþ1 þ d~tþ1 Rt . Substituting from Proposition 1, we get
Pt ¼ Et ½ðRt =rt Þ ~xtþ1 . Because rt is known at date t, Et ½ðRt =rt Þ ~xtþ1 ¼ ðRt =rt ÞEt ½~xtþ1 =Rt .
Therefore, Pt ¼ Et ½~xtþ1 =rt . In fact, as Proposition 2 shows, assuming a capitalization
multiple for expected earnings provides another way of deriving the multiple for
current earnings.
Proposition 2. Assume (i) risk neutrality and no arbitrage (RNNA); (ii)
Pt ¼ Et ½~xtþ1 =rt ; (iii) Pt+dt = f(rt, rt)1, …)xt, where f(.) depends only on the history of
interest rates. Then f(.) = Rt)1/rt)1.
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
425
Proof: See Appendix I.
Propositions 1 and 2 are two ways of representing earnings sufficiency. One relates
current price to current earnings, Pt ¼ ðRt1 =rt1 Þ xt dt , and the other relates current price to expected earnings, Pt ¼ Et ½~xtþ1 =rt . Both relations use the rate at the
beginning of the respective earnings measurement periods in the earnings multiples,
i.e., there is ‘‘capitalization consistency.’’
We now show how earnings sufficiency relies on earnings smoothing.
3.2.
‘‘Smoothing’’—The Core Concept Underlying Earnings Sufficiency
We show that the earnings-sufficiency model yields valid and intuitive accounting
measurements even for an investment fund, a setting that seems suited only to markto-market accounting. The discussion shows that earnings sufficiency relies on
‘‘earnings smoothing,’’ a concept underlying accounting for defined benefit pension
plans under GAAP.
Consider two accounting alternatives for an investment fund that holds one share
of the same equity security at all times and passes on dividends received to its
investors. RNNA implies the following equation:
Pt þ dt ¼ Rt1 Pt1 þ ~et ;
where Et1 ½~et ¼ 0.
Mark-to-market Accounting: Set bt ¼ Pt . CSR yields xt ¼ rt1 Pt1 þ ~et .
Earnings-sufficiency Accounting: Set xt ¼ rt1 Pt1 þ ðrt1 =Rt1 Þ ~et . Compute
book value by applying CSR recursively with the initial condition of b0 ¼ 0. It is easy
to verify that Ptþ1 þ dtþ1 ¼ ðRt =rt Þxtþ1 .
Mark-to-market accounting includes the entire value shock ~et in current earnings
while earnings-sufficiency accounting includes only a fraction rt1 =Rt1 of the value
shock ~et in current earnings. GAAP for pension assets approximates earnings-sufficiency accounting. GAAP recognizes two components of earnings from pension
assets: (i) expected return on plan assets, rt1 Pt1 ; and (ii) part of unexpected gains
and losses ~et on plan assets.
The equation for xt under earnings sufficiency shows how such earnings can depend on both beginning-of-period and current rates. xt depends on rt1 because of
the component rt1 Pt1 . xt can depend on rt if ~et correlates with rt because Pt depends
on rt . Because earnings, xt , measure change over the period (t)1, t), it is intuitive that
xt relates to both rt and rt1 .
We now derive the earnings dynamics underlying earnings sufficiency.
3.3.
Residual Earnings Dynamic: Interest-Rate-Dependent Persistence
With earnings sufficiency and constant interest rates, Ohlson (1995) shows that
residual earnings follow a strict random walk, i.e., they have a persistence of 1:
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GODE AND OHLSON
~xatþ1 ¼ xat þ ~etþ1 ;
where Et ½~etþ1 ¼ 0:
Stochastic interest rates raise the following questions: Is the persistence still 1?
Does it oscillate around 1? Does the persistence depend on the entire history of
interest rates or does it depend on a smaller subset of past interest rates? Proposition
3 provides answers.
Proposition 3. Assume RNNA, clean surplus (CSR), and the earnings dynamic
~xatþ1 ¼ xt xat þ ~etþ1 , where xt can depend only on the history of interest rates. Then,
Pt ¼
Rt1
xt dt
rt1
implies
xt ¼
rt
:
rt1
Proof: See Appendix I.
Proposition 3 generalizes Ohlson’s (1995) to stochastic interest rates. The residual
earnings persistence, xt, decreases in the beginning-of-period rate and increases in
the current rate, but it is otherwise independent of the history of interest rates.
Further, if the distribution of interest rates meets reasonable regularity conditions,
the median of xt is 1, which is its value with constant interest rates.
The intuition behind xt ¼ rt =rt1 follows from the capitalization consistency referred to in Proposition 2, i.e., current earnings are capitalized by the lagged rate and
forthcoming earnings are capitalized by the current rate. Proposition 3 holds for all
stochastic processes of interest rates because forthcoming earnings are capitalized by
rt, which is known at date t. If the forthcoming earnings had been capitalized by rt+1,
then one would have needed to specify the stochastic process of interest rates and its
covariance with earnings to compute Et(xt+1/rt+1).
The converse of Proposition 3 holds; xt ¼ rt = rt1 , RNNA, and CSR imply
earnings sufficiency as defined. If we assume xt ¼ 1 along with RNNA and CSR,
then earnings sufficiency cannot hold; i.e., a strict random walk (permanent earnings) and earnings sufficiency are mutually inconsistent when interest rates are stochastic. In fact, as Appendix II notes, xt ¼ 1 implies that Pt must also depend on bt
in addition to xt.
Observation 1 Proposition 3 implies that goodwill follows a random walk.
Et ½gtþ1 ¼ gt .
Proof: See Appendix I.
Thus, with stochastic rates, goodwill, not earnings or residual earnings, follow a
random walk. The result holds because, under earnings sufficiency, goodwill equals
capitalized residual earnings; and Proposition 3 shows that capitalized residual
earnings follow a random walk.
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
3.4.
427
Earnings Dynamic: Modifying Random Walk for Changing Interest Rates
Assuming constant interest rates, Ohlson (1995) derives the following earnings
dynamic:
Et ½~xtþ1 ¼ xt þrD bt ;
where D bt bt bt1 .
The first term represents the standard random walk of earnings and equals expected earnings when new net investment (Dbt) is zero. The second term represents
incremental earnings from new investment. Because incremental earnings equal
current rate times new net investment, rt replaces r when interest rates are stochastic.
The corollary below reveals that replacing r with rt is not enough; stochastic interest
rates introduce an additional term in the random walk.
Corollary 1
Et ½xtþ1 ¼ xt þ rt D bt þ% Drt xt ;
where %D rt ðrt rt1 Þ= rt1 .
Proof: See Appendix I.
Because xt/rt)1 equals cum-dividend value at time t, the third term represents
incremental earnings from existing investment due to an increase in earnings yield
from rt)1 to rt. The third term, which prior research has not recognized, shows that
the percentage change in interest rates, not just the level of current interest rate,
affects earnings forecasts; an up-tick in interest rates leads to higher earnings forecasts, and vice versa.
4.
A Weighted-Average of the Two Models with ‘‘Other’’ Information
So far, we have extended the mark-to-market and earnings-sufficiency benchmarks
to stochastic rates. We now extend Ohlson’s (1995) general model, which is a
weighted average of the two benchmarks with other value relevant information besides accounting data. We examine the general model because analysts use both
balance sheets and income statements for valuation, and complement their forecasts
with other information.
Other information makes the model realistic by picking up expectations of future
interest rates, which are generally not picked up by accounting numbers under the
current transaction-based GAAP. For example, suppose a rise in the current rate
raises the stock price because the market expects future income to rise with an
increase in interest rates. Other information can immediately pick up the effect of
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GODE AND OHLSON
current rates on value while accounting data picks it up as future transactions occur.5
If the model did not permit other information and if accounting statements were to
suffice for valuation, then current book value and earnings would have to rise to
reflect higher expected future income, which is not permissible under GAAP. Thus,
introduction of other information allows @ Pt =@ rt 6¼ 0 when xt, bt, and dt do not
respond to changes in forthcoming rates.
Our analysis of the general model has the following implications: (i) In a regression
of returns on unexpected earnings and unexpected other information, the coefficient
for unexpected earnings (ERC) decreases in the beginning-of-period rate. (ii) When
expected earnings represent other information, a rise in interest rates raises the
weights on capitalized expected earnings vis-à-vis current book value and capitalized
current earnings.
4.1.
The Weighted Average Valuation Function with Other Information
Ohlson’s (1995) model derives the valuation function as
R
xt dt þ ð1 k Þ bt þ b tt ;
Pt ¼ k
r
where k 2 ½0; 1 given
e
xatþ1 ¼ x xat þ tt þ e
e1;tþ1 ;
e
c tt þ e
e2;tþ1 ;
ttþ1 ¼
where Et ½~ej;tþs ¼ 0 for all s ‡ 1 and j ¼ 1, 2. The variable tt thus represents all other
information influencing residual earnings forecasts. Given the RNNA condition, the
parameters satisfy
x¼
kR
;
kþr
c¼ R
kþr
:
rb
Allowing x and c to depend on interest rates while keeping k and b constant generalizes the model. We therefore consider the following valuation function:6
Rt1
xt dt þ ð1 k Þ bt þ b tt :
Pt ¼ k
rt1
Proposition 4 Given risk neutrality and no arbitrage (RNNA), clean surplus (CSR),
and the following information dynamics:
e
e1;tþ1 ;
xatþ1 ¼ xt xat þ tt þ e
e
ttþ1 ¼
ct tt þ e
e2;tþ1 ;
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
429
where Et ½~ej;tþs ¼ 0 for all s ‡ 1 and j = 1, 2, and the two persistence parameters xt
and ct depend at most on k, b, and the history of interest rates,
Rt1
x
¼
Pt k
dt þ ð1 k Þ bt þb tt implies
t
rt1
rt þ1 rt
rt þ k
xt ¼
:
;
c t ¼ Rt k
rt þ k rt1
rt b
Proof: See Appendix I.7
The remaining subsections draw out the implications of the solution with no
additional assumptions or modifications. We also discuss the critical role of
other information, and state some empirical hypotheses that follow from the
model.
The generalized model requires no specification of the stochastic process of
interest rates because k and b are constants, and the multiple for current
earnings is Rt1 =rt1 , which allows us to evaluate Et1 ½Pe t þ e
d t (the numerator of
RNNA) without knowing the distribution of rt at date t)1. This model can be
extended without specifying the stochastic process of interest rates as long as the
‘‘structure’’ of the valuation function at date t is known at date t)1. Appendix
II details such a generalization in which k and b depend on the beginning-ofperiod rate, i.e., k is replaced by kt)1, and b is replaced by bt)1 in the valuation
function.
4.2.
Residual Earnings Dynamic: Robustness of Ohlson (1995)
We now discuss the implications of the functional form of xt and ct derived in Proposition 4. As in the earnings-sufficiency setting, xt depends on the beginning-of-period
rate as well as the current rate. In contrast to xt, ct depends only on the current rate.
Because mt represents generic other information, there are no reasons why ct should
include a capitalization component based on the beginning-of-period rate.
The earnings-sufficiency setting suggests that xt should increase in rt and decrease
in rt)1. Differentiating xt with respect to rt and rt)1 yields the anticipated conclusion:
For k > 0, xt increases in rt (@ xt =@ rt > 0) and decreases in rt)1 (@ xt =@ rt1 < 0).
Although the sensitivity of xt to interest rates may be expected, its precise functional
form is not obvious. The first term, ððrt þ1Þ=ðrt þ k ÞÞk, is the expression for x when
interest rates are not stochastic, as in Ohlson (1995), while the second term, (rt/rt)1),
reflects the ‘‘correction’’ due to the changing interest rates.
Differentiating xt with respect to k yields @ xt =@ k > 0 for any rt, rt)1 > 0. That is,
residual earnings are more persistent when earnings have more weight in the valuation function. We elaborate on this consistency between the relative importance of
current earnings in forecasting and in valuation in the Section 4.3.
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GODE AND OHLSON
4.3.
Earnings Dynamic: Consistency Between Valuation and Forecasting
Ohlson (1995) residual earnings dynamic implies the following earnings dynamic:
Et ½~xtþ1 ¼ xðxt þrD bt Þ þ ð1 xÞr bt :
Three aspects of the expression are noteworthy. First, expected forthcoming
earnings are a weighted average of the expected forthcoming earnings under the
two benchmark models. Second, the residual earnings persistence parameter (x)
determines the weight assigned to the first component. Third, the higher the weight
placed on the earnings-sufficiency component in the valuation function (k), the
higher is the weight placed on the earnings sufficiency component in the earnings
dynamics (x). That is, there is a consistency between the weights in the valuation
function and the weights in the earnings dynamics. These three aspects generalize
to the setting with stochastic interest rates. But there is an important qualification:
As we
move from constant rates to stochastic rates, we replace x not by xt, but by
ðrt1 rt Þxt as the weight placed on the earnings-sufficiency component.
Corollary 2
Et ½~xtþ1 ¼ ht ðxt þ rt D bt þ %D rt xt Þ þ ð1 ht Þ rt bt þ tt ;
where
ht ¼
1 þ rt
k:
k þ rt
Proof: See Appendix I.
The weight ht has several interesting properties. First, ht ¼ ðrt1 rt Þxt , which
equals xt when rates are constant. Second, ht increases as rt decreases; i.e., the
earnings-sufficiency component is relatively more important vis-à-vis the mark-tomarket component when the current rate is low. In contrast, beginning-of-period
rates do not affect ht. Third, ht increases in k; i.e., as in Ohlson (1995), the weights
used in the earnings dynamics and the valuation function are consistent.
4.4.
The Role of Other Information: The Effect of Current Rates on Value
We now provide an example that shows how current rates can affect value via other
mtþ1 ) satinformation, mt. Suppose unexpected other information (~e2;tþ1 ¼ m~tþ1 Et ½~
isfies the following equation,: ~e2;tþ1 ¼ qð~rtþ1 Et ½~rtþ1 Þ þ ~utþ1 ; where, Et ½~utþ1 ¼ 0,
and q 6¼ 0 is some fixed constant that reflects the firm’s type of business, accounting
rules, etc. The specification satisfies Et ½~e2;tþ1 ¼ 0 as required. The specification also
allows the unexpected other information (~e2;tþ1 ) to correlate with unexpected changes
in interest rates (~rtþ1 Et ½~rtþ1 ).
Recursive substitution then yields mt as an explicit function of rt, rt)1, Et1 ½~rt ,
Et2 ½~rt1 … and ut , ut1 ,… This example underscores even though mt depends
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
431
explicitly on rt, the evaluation of Et ½~
mtþ1 is independent of the stochastic process that
determines the distribution of ~rtþ1 given any date t information.
If mt depends on rt, then the accounting rules that produce xt and bt do not
fully reflect the value implications of a change in interest rates for the period
(t)1,t). Only with the passage of time will the accounting data reflect the prior
history of interest rates. Yet this scenario also requires that the earnings forecasts
depend on the current (and past) rates via mt. In analytical terms, Et ½~xatþ1 depends
on mt, which in turn depends on the history of interest rates. Introducing mt thus
allows considerable generality in how interest rates can influence value and
earnings forecasts.
4.5.
Two Key Implications: A Returns Model and an Expected Earnings Model
We now generalize two key implications of Ohlson (1995). First, the surprise in
residual earnings ~e1;t and the surprise in other information ~e2;t explain the unexpected
returns. As Appendix I shows,
~e1;t
~e2;t
P~ t þ d~t
Rt1 ¼ ð1 þ a1;t1 Þ
þ a2;t1
;
Pt1
Pt1
Pt1
where [Because of RNNA, the expected value of the left-hand side equals zero.]
k
;
rt1
¼ b:
a1;t1 ¼
a2;t1
The price-normalized response coefficient, a1,t)1, is known at the beginning of the
return interval (t)1,t). As 1+a1,t)1 is the ERC for unexpected earnings, an empirical
hypothesis follows: the ERC should be larger when beginning-of-period interest rate
is lower because for a fixed k, a1;t1 increases as rt)1 decreases.
Second, following Ohlson (2001), we can substitute expected next-period earnings
for tt . As Appendix I shows,
Pt ¼ w1;t bt þ w2;t
where
P
j
Rt1
xt dt
rt1
þ w3;t
Et ½~xtþ1 ;
rt
wj;t ¼ 1:
r
w1;t ¼ ð1 k b rt þb xt rt1 Þ ¼ ð1 kÞ 1 b rt t
;
rt þ k
r þ1
w2;t ¼ ðk b xt rt1 Þ ¼ k 1 b rt t
;
rt þk
w3;t ¼ b rt :
432
GODE AND OHLSON
Expressing value as a function of bt, xt, and Et ½~xtþ1 has an intuitive interpretation.
Value derives from a weighted average of three benchmark models: (i) mark-to[ðRt1 rt1 Þ xt dt ], and (iii) capitalized expected
market [bt], (ii) earnings-sufficiency
earnings [Et ½~xtþ1 rt ]. Further note that components (ii) and (iii) reinforce the key
idea of ‘‘capitalization consistency.’’ The multiple for current earnings is based on
the beginning-of-period rate, while the multiple for expected forthcoming earnings
depends on the current rate.
The weights associated with book value and capitalized earnings decrease as the
interest rate rises, while the weight associatedwith capitalized expected
forthcoming
w
w
r
earnings
increases
(with
k
>
0,
@
@
@ rt 0, but
0
and
@
1;t
t
2;t
@ w3;t @ rt > 0). We therefore predict that as interest rates rise, current accounting
data has less information content than capitalized expected earnings.
5.
Summary and Conclusions
This paper relates value to accounting data and other information when interest rates
are stochastic. The paper starts with two benchmark models—mark-to-market
accounting, where the balance sheet suffices for valuation and earnings-sufficiency
accounting, where the income statement suffices for valuation—and then examines a
weighted average of the two models with other information.8
Stochastic rates play no role in mark-to-market accounting because the book
value is simply set to the price, which impounds expectations of interest rates. In
contrast, stochastic rates play a key role in earnings-sufficiency accounting because
the multiplier for earnings depends on interest rates. Our dynamic analysis shows
how the current rate rt and the beginning-of-period rate rt)1 affect valuation and
earnings forecasting in earnings-sufficiency accounting.
Propositions 1 and 2 show capitalization consistency: the multiple for current
earnings is based on the beginning-of-period rate (1+1/rt)1), while the multiple for
expected forthcoming earnings is based on the current rate (1/rt). Propositions 1 and
2 use Yee’s Modigliani–Miller dividend policy irrelevancy conditions, and rely on
the intuition that the earnings rate for a given period is the rate prevailing at the
beginning of the earnings measurement period rather than the rate prevailing at the
end of the earnings measurement period.
Proposition 3 shows that the under earnings-sufficiency accounting, residual
earnings do not follow a random walk, i.e., their persistence is not 1. Instead, Proposition 3 shows that capitalized residual earnings follow a random walk, and therefore
the persistence of residual earnings is rt/rt)1. Corollary 1 shows that the traditional
random walk of expected earnings now requires an new term reflecting changes in
interest rates because the beginning-of-period rate is needed to scale current earnings,
while the current rate is needed to forecast forthcoming earnings. Thus, the percentage
change in interest rate, not just the level of interest rates, affects earnings forecasts; an
up tick in interest rates leads to higher earnings forecasts, and vice versa.
Proposition 4 and Corollary 2 make the model versatile by introducing other
information about expected future events, expectations that are not captured by the
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
433
transaction-based accounting system. Allowing expected earnings to proxy for other
information yields another important empirical prediction: an increase in interest
rates reduces the information content of current accounting data relative to that of
capitalized expected earnings.
A returns specification shows that unexpected returns correlate with unexpected
earnings and revisions in earnings expectations, consistent with traditional empirical
specifications. The returns specification also provides a new empirical prediction:
The earnings response coefficient is independent on the current rate; instead, it
depends inversely on the beginning-of-period rate.
Overall, we generalize Ohlson’s (1995) model to stochastic interest rates. Allowing
stochastic interest rates clarifies key dynamic aspects of the accounting-based valuation framework that are obscured by constant interest rates in Ohlson (1995).
Specifically, the beginning-of-period rate, not the current rate, relates prices to
contemporaneous earnings, while the current rate relates earnings forecasts to current earnings and price. Our analysis formalizes, sharpens, and augments empirical
hypotheses about how prices relate to accounting numbers.
Acknowledgements
We acknowledge the helpful comments of an anonymous referee, Suresh
Govindaraj, Stefan Reichelstein (editor), Merav Rom, Stephen Ryan, Ken Skogsvik,
and seminar participants at the Chinese University of Hong Kong, the National
University of Singapore, Rutgers University, Stanford University, the Stern School,
the Stockholm School of Economics, and the Wharton School.
Appendix I: Proofs
Proof of Proposition 1: Because the one-period interest rate at time t)1 is rt)1,
withholding $z of dividends at date t)1 increases earnings for the period (t)1, t) by
rt)1 z and increases dividends at date t by (1+rt)1)z. Withholding dividends should
not affect date t value if dividend policy is irrelevant. That is,
Pt ¼ f ð:Þxt dt ¼ f ð:Þ½xt þ rt1 z ½dt þ ð1 þ rt1 Þz:
This implies f(.)rt)1z ) (1+rt)1)z ¼ 0 for all z. Thus, f(.) = (1+rt)1)/rt)1.
Remark: The proof works only because f(.) is assumed not to depend on earnings or
dividends. If f(.) were dependent on earnings, then Pt + dt would be non-linear in xt.
Proof of Proposition 2: Define f(.)= f(rt, rt)1,…); RNNA combined with assumption
(ii) implies
Rt1 Pt1 ¼ Et1 ½Pt þ dt ¼ Et1 ½F ð:Þxt :
434
GODE AND OHLSON
Due to assumption (i), Rt)1Pt)1 ¼ Rt)1Et)1[xt]/rt)1, so that (Rt)1/rt)1)Et)1[xt] =
Et)1[f(.)xt].
Because there are no particular restrictions on the stochastic process related to rt
given the history of interest rates, this equation must be satisfied for all feasible
stochastic processes of interest rates. In particular, consider the case when rt happens
to be known at date t)1 for some history of interest rates. It then follows that
Et1 ½f ð:Þxt ½ ¼ f ð:ÞEt1 ½xt ; so that ðRt1 =rt1 ÞEt1 ½xt ¼ f ð:ÞEt1 ½xt :
Hence, f(.)= Rt)1/rt)1 as asserted.
Proof of Proposition 3: Earnings sufficiency implies
Pt ¼
Rt1
xt dt :
rt1
Substituting for Pt and Pt+1 in RNNA and recognizing that rt is known at date t
yields,
Rt1
E x
xt dt ¼ t tþ1 :
rt1
rt
Rearranging terms, we get,
Et xtþ1 ¼
rt
rt1
xt þ rt ½xt dt :
CSR implies xt dt ¼ bt bt1 . Substituting and rearranging terms we get,
a
Et ½xtþ1 ¼
rt a
x :
rt1 t
j
Proof of Observation 1: Given earnings sufficiency, we can restate the expression for
Pt as
P t ¼ bt þ
xat
:
rt1
That is,
gt ¼
xat
:
rt1
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
435
Substituting in Proposition 3, we get
Et ½gtþ1 ¼ gt :
j
Proof of Corollary 1: From the proof of Proposition 3, we have
Et xtþ1 ¼
rt
xt þ rt ½xt dt :
rt1
Substituting xt dt ¼ bt bt1 from CSR, we get,
Et ½xtþ1 ¼
rt
rt1
xt þ rt ðbt bt1 Þ:
Adding and subtracting xt, we get,
Et ½xtþ1 ¼ xt þ
rt
xt xt þ rt ðbt bt1 Þ;
rt1
rt
1xt þ rt ðbt bt1 Þ;
Et ½xtþ1 ¼ xt þ ½
rt1
Et ½xtþ1 ¼ xt þ rt Dbt þ %Drt xt :
j
Modigliani–Miller Restrictions on the Weighted Average Valuation Function:
Let Pt ¼ a1t xt þ a2t dt þ ð1 kÞbt þ btt .
Dividend policy irrelevancy implies the following conditions:
1. @Pt =@dt ¼ 1, and where @xt =@dt ¼ 0,@bt =@dt ¼ 1, and @tt =@dt ¼ 0 (from
Ohlson, 1995).
2. @Pt =@dt1 ¼ ð1 þ rt1 Þ, and where @xt =@dt1 ¼ rt1 ,@bt =@dt1 ¼ ð1 þ rt1 Þ
(from Yee, forthcoming), and @tt =@dt1 ¼ 0 (from Ohlson, 1995).
These two conditions yield the following:
a2t ð1 kÞ ¼ 1;
a1t rt1 ð1 kÞð1 þ rt1 Þ ¼ ð1 þ rt1 Þ:
Solving the above equation, we get the following:
a2t ¼ k;
kð1 þ rt1 Þ
:
a1t ¼
rt1
436
GODE AND OHLSON
Proof of Proposition 4:
gt ¼ Pt bt ¼ k ðRt1 xt dt bt Þ þ b tt :
rt1
Substituting for bt from the clean surplus relation, bt + dt ¼ xt + bt)1, and using
the definition of residual earnings,
gt ¼ k
xat
þ b tt :
rt1
Using the goodwill equation (GE), we get
a
xtþ1
xat
a
þ Rt b t t ¼ Et k
þ b ttþ1 þ xtþ1 :
Rt k
rt1
rt
Because k and b are fixed and ttþ1 ¼ ct tt þ e2;tþ1 , we get
a
Et ½xtþ1 ¼
xa
rt þ1
rt
rt k t þ
b ð Rt c t Þ t t :
rt1 rt þ k
rt þ k
This implies
xt ¼
rt rt þ1
k
rt1 rt þ k
and
rt
b ð Rt c t Þ ¼ 1 :
rt þ k
Thus,
c t ¼ Rt rt þ k
:
rt b
Proof of Corollary 2: From Proposition 4,
a
Et ½xtþ1 ¼
1 þ rt
k a
x þ tt :
rt
rt þ k rt1 t
Substituting for residual earnings, we get
Et ½xtþ1 ¼
1 þ rt
ðx rt1 bt1 Þ
rt k t
þ r t bt þ t t :
rt1
rt þ k
j
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
437
Define
ht ¼
1 þ rt
k:
k þ rt
Thus,
Et ½xtþ1 ¼ ht
rt
rt1
ðxt rt1 bt1 Þ þ rt bt
þ ð1 ht Þ rt bt þ tt ;
which can be restated as follows:
Et ½xtþ1 ¼ ht ðxt þ rt D bt þ %D rt xt Þ þ ð1 ht Þ rt bt þ tt :
Derivation of the Coefficients in the Returns Specification:
Pt1 ¼ bt1 þ
P t ¼ bt þ
k a
x þb mt1 ;
rt2 t1
k a
x þb mt ;
rt1 t
Pt þ dt ¼ bt þ dt þ
k a
x þb mt :
rt1 t
From CSR,
bt þ dt ¼ bt1 þ xt :
Pt þ dt ¼ bt1 þ xt þ
k
rt1
xat þb mt
¼ bt1 þ rt1 bt1 þ xt rt1 bt1 þ
k a
x þb mt
rt1 t
¼ Rt1 bt1 þ
k þ rt1 a
xt þb mt
rt1
¼ Rt1 bt1 þ
k þ rt1
ðxt1 xat1 þ mt1 þ e1;t Þ þ b ðct1 mt1 þ e2;t Þ:
rt1
Substituting for xt)1 and ct)1, we get
j
438
GODE AND OHLSON
k þ rt1
k þ rt1
Rt1 k a
x þ
Þ mt1
¼ Rt1 bt1 þ
þ bðRt1 rt1
rt1 b
rt2 t1
k þ rt1
þ
e1;t þb e2;t
rt1
k þ rt1
Rt1 k a
x þb Rt1 mt1 þ
¼ Rt1 bt1 þ
e1;t þb e2;t
rt1
rt2 t1
k þ rt1
¼ Rt1 Pt1 þ
e1;t þb e2;t :
rt1
Pt þ d t
Rt1 ¼ ð1 þ a1;t1 Þ e1;t = Pt1 þ a2;t1 = Pt1 ;
Pt1
k
a1;t1 ¼
;
rt1
a2;t1 ¼ b:
Derivation of the Triple Weighted Average: We can express mt in terms of expected
earnings as follows:
a
Et ½xtþ1 ¼ xt xat þ mt
mt ¼ Et ½xatþ1 xt xat
¼ Et ½xtþ1 rt bt xt ðxt rt1 bt1 Þ
¼ Et ½xtþ1 rt bt xt ðxt rt1 ðbt xt þ dt ÞÞ
¼ Et ½xtþ1 rt bt xt ðRt1 xt rt1 ðbt þ dt ÞÞ
¼ Et ½xtþ1 rt bt xt ðRt1 xt rt1 ðbt þ dt ÞÞ:
Substituting for mt in the valuation function, we get the following:
Rt1
xt dt þ b Et ½xtþ1 rt bt xt ½Rt1 xt rt1 ðbt þ dt Þ
Pt ¼ ð1 kÞ bt þk
rt1
Rt1
E ½x xt dt þ b rt t tþ1 :
¼ ð1 k b rt þb xt rt1 Þ bt þðk b xt rt1 Þ
rt1
rt
From Proposition 4, we know
xt ¼
rt þ1
k
rt
:
rt þ k rt1
Thus,
xt rt1 ¼
rt þ1
rt k;
rt þ k
which depends only on rt.
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
439
We therefore can express price as follows:
Pt ¼ w1;t bt þ w2;t
X
Rt1
xt dt
rt1
þ w3;t
Et ½xtþ1 ;
rt
wj;t ¼ 1;
j
w1;t ¼ ð1 k b rt þb xt rt1 Þ;
w2;t ¼ ðk b xt rt1 Þ;
w3;t ¼ b rt :
Appendix II: The Weighted Average Model with Variable but Known Weights
We now examine a setting in which the weights can vary over time, but are known
at the beginning of a period. For brevity, we consider the setting without other
information. The concepts illustrated in this appendix will remain unchanged if we
introduce other information. Thus, price is expressed as follows:
Rt1
xt dt þ ð1 kt1 Þ bt :
Pt ¼ kt1
rt1
If we rearrange the terms in the above equation and apply CSR, it follows that
a
xtþ1
xa
þ xatþ1 :
Rt kt1 t ¼ Et kt
rt1
rt
Inserting the last equation into the goodwill equation (GE) yields
a
xtþ1
xat
a
x
¼ Et k t
þ tþ1 :
Rt kt1
rt1
rt
Because kt and rt are known at time t, the RHS equals
kt
Et ½xatþ1 þ Et ½xatþ1 ;
rt
and we obtain the following:9
a
Et ½xtþ1 ¼
rt þ1 kt1 a
x :
rt
rt þ kt rt1 t
The residual earnings persistence parameter therefore is represented by
440
GODE AND OHLSON
xt ¼
rt þ1 kt1
rt
:
rt þ kt rt1
In this expression, we can think of xt as being the endogenous result of realization of
interest rates and kt’s, where the kt’s follow some exogenous stochastic process
(though, as noted, the weights are determined at the beginning of a period). More
important, as shown below, we can rearrange the terms to state kt in terms xt, kt)1,
rt, and rt)1; i.e., we can think of xt as being exogenous.
kt ¼
rt kt1
rt :
Rt
xt rt1
If we substitute recursively, it follows that kt is some function of the history of
interest rates and the history of xt; i.e., we can write kt+1 = f(rt, rt)1, …, xt, xt)1,
…), where the xt are determined by some exogenous process.
We can now ask the following question: What happens if xt is simply a constant,
such as xt ¼ 1? That is, what happens if residual earnings follow a random walk and
interest rates can change over time? The answer is clear: Pt will generally depend on
book value as well as capitalized earnings (adjusted for dividends). This is because, in
contrast to the setting in which rt is constant, kt need not be 1 when xt ¼ 1; i.e., the
weight on book value (1)kt) can be non-zero even when xt ¼ 1. We therefore cannot
view earnings as a sufficient statistic when residual earnings follow a random walk, as
book value still generally enters the valuation function.
Notes
1. Beaver (1999), p. 37, questions the assumption of constant discount rates in empirical studies.
Although Feltham and Ohlson (1999) extend the residual income valuation to stochastic rates, the do
not consider the closed-form valuation functions, PE multiples, and supporting information dynamics
develop here.
2. Three aspects of our model are noteworthy:
One can construct an economy such that the equilibrium expected return on an asset is the risk-free
rate, and yet the rate changes stochastically over time. The construction is easy with unconstrained,
event-contingent preferences.
For risk aversion, one can replace the expectation operator E with E*, which reflects risk-adjusted
probabilities. [Huang and Litzenberger (1988)].
P
d tþs ). As is well
Ohlson (1995) and others assume constant interest rates and PVED (Pt ¼ Rs Es ½e
known, RNNA implies PVED when rates are constant.
3. Section 3.2 illustrates the intuition and an application of the earnings-sufficiency model. Ohlson and
Zhang (1998) show how to account for transactions in an earnings-sufficiency model.
4. The multiple f(.) can depend on expected interest rates as long as such expectations are based solely on
the history of interest rates. Allowing the multiple to depend on accounting or non-accounting
information other than interest rates is inconsistent with the notion that earnings suffice for valuation.
5. Nissim and Penman (2000) empirically study how interest rates influence subsequent accounting data.
6. Appendix I shows how this function is motivated by the Modigliani-Miller dividend policy irrelevancy
conditions.
ACCOUNTING-BASED VALUATION WITH CHANGING INTEREST RATES
441
7. If one additionally assumes that Pt ¼ a1 þ a2 /t1 xt þ a3 dt þ a4 bt þ a5 mt , then one can show that the
dynamics imply the weighted-average valuation function, i.e., that the converse of Proposition 4 is true
under this assumption.
8. It is unclear how stochastic interest rates affect valuation under conservative accounting as in Zhang
(2000) or under non-linearities due to bankruptcies [Barth, Beaver, and Landsman (1998)].
9. If kt, instead of kt)1, is the weight in the pricing equation, i.e., the weight is determined at the end of
the period rather than at the beginning, then we would need to know the covariance of kt + 1 and
xatþ1 rt :
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