A GENERAL MODEL OF EARNINGS,
DIVIDENDS AND RETURNS
by
G. Hobbes*, G. Partington** and M. Stevenson**
* Macquarie University
** University of Technology, Sydney
First Draft January 1994
Second Draft February 1996
Third Draft January 1997
1
Abstract
In this paper we develop a general linear model for the relation between earnings, dividends and
returns. Arising from this is a model for realised returns and an equity valuation model. We show
how two classes of earnings, dividend and return models that have appeared in the literature arise
as special cases of our general model. The differences between these models are a function of the
restrictions imposed on the general model. Depending on the restrictions imposed the model can
also generate results consistent with either Miller and Modigliani’s (1961) dividend irrelevance
proposition, or their dividend signalling proposition. We note that the single equation models
typically used in earnings return work cannot adequately distinguish between alternative models.
The general model developed here suggests model testing should be conducted using a system of
simultaneous equations with constraints on the coefficients. Possible practical uses of the model
include estimating terminal values and equity discount rates.
A General Model of Earnings, Dividends and Returns
Introduction
In recent years there has been a growing interest in theoretical models of the relationship between
earnings, dividends and returns. For example, Ohlson (1991) develops a theoretical model where
the level of earnings determines returns, and Ohlson and Shroff (1992) develop a model where
earnings levels, or earnings changes, can be used to explain returns. Hobbes, Partington and
Stevenson (1993,1996) provide a model where earnings levels and changes, and dividend levels
and changes, together with the firm’s dividend policy parameters explain returns. This latter
model has been extended by Goya and Beg (1995) to include cash flow. While in a continuous
time framework, Chiang, Davidson and Okunev (1996) derived a model similar to the Hobbes,
Partington and Stevenson model.
This theoretical work was preceded by over two decades of primarily empirical work stimulated
by the seminal paper of Ball and Brown (1968). However, the theoretical modeling has had an
influence on some of the later empirical work. For example, Easton and Harris (1991) utilise an
Ohlson style of model and find empirical support for using both levels and changes in earnings as
explanators of returns.
In recent empirical work there has been an increasing interest in the interaction between dividend
policy and earnings in explaining returns. Both Mande (1994) and Kallipan (1994) find that
earnings response coefficients are affected by dividend policy variables. These results are
consistent with the theoretical model of Hobbes, Partington and Stevenson (1996).
Empirics and theory can therefore be called upon to support more than one model. The question
naturally arises - How should we choose between them? How do we reach agreement about
which model best helps explain observed phenomena and best assists empirical researchers in
designing experiments? We might be able to make a choice based on the existing empirical
evidence, or perhaps specially designed tests might be necessary. However, the purpose of this
paper is to argue that such effort is not required. Rather than viewing the Ohlson style of model,
or the Hobbes, Partington, Stevenson style of model as competing alternatives, we demonstrate
that they are special cases of a more general model.
2
These special cases arise as a result of coefficient restrictions placed on the elements of the
general model. The circumstances under which the model is to be applied determines the
particular set of coefficient restrictions that are appropriate, and therefore which model is to be
preferred. An important conclusion that flows from our analysis is that the single equation
earnings return regressions typically used in empirical work cannot adequately distinguish
between the possible theoretical models. Furthermore, the use of such single equation regressions
may easily lead the researcher to erroneous conclusions.
The paper is organised as follows. First we detail the elements of the general model under
uncertainty, and then provide a general solution to the model. We next examine the impact of
different coefficient restrictions that lead to specific forms of the model. We start by considering
the simple certainty case and then extend the analysis to alternative coefficient restrictions under
uncertainty.
The General Model
Generating processes for earnings and dividends
We assume that dividends are generated by a linear process. The process is such that this period’s
dividend is a function of the last period’s dividend and current earnings. The dividend process is
defined by equation (1).1 From an empirical perspective, this equation is consistent with the
Lintner (1956) model of dividends, which has substantial empirical support (Fama and Babiak
(1968), Laub (1972), Shevlin (1982)). From a theoretical perspective, equation (1) is consistent
with the general dividend function used by Ohlson (1991), and the specific use of the Lintner
model in Hobbes et.al. (1996).
Dt  a1  a2Yt  a3 Dt 1  et
(1)
Here Dt is the dividend announced and paid, Yt is the earnings announced, et is the management
determined shock to dividends all at time  t . The expected value of the shock term is zero i.e.
Et 1 (et )  0. The coefficients a1, a2 and a3 are the dividend policy parameters for the firm. The
magnitude of these parameters are subject to substantial managerial discretion.2
Equation (2) describes the earnings generating process as a general function of lagged earnings
and lagged dividends. This is consistent with the form of the earnings equation used by Ohlson
(1991) which he justifies in terms of a reinvestment model for current earnings. Equation (2) is
Yt  b1  b2Yt 1  b3 Dt 1   t
(2)
where t is the shock to earnings at time  t , and Et 1 ( t )  0 . The coefficients b1, b2 and b3 are
the parameters for the firm’s earnings process.
Equation (2) provides quite a flexible specification for the earnings process. For example, if the
coefficients b1 and b3 are set to zero and b2 is set to one, we are left with the random walk model
for earnings. The random walk model has considerable empirical support and much popular use
1
Equation 1 and subsequent equations are firm specific, but for economy of notation the subscript i representing the i’th firm has
been omitted.
2
This discretion is, of course, not without limit and some values for these parameters are more plausible than others. For
example, we would expect a2 to generally be less than 1.
3
in empirical studies. In particular it has been extensively used in papers that study earnings
announcement effects3.
The return equation (3) is simply an identity that expresses realised returns as price change plus
dividends scaled by beginning of period price as follows
Rt 
Pt  Pt 1 Dt

Pt 1
Pt 1
(3)
where Rt is the realised return for time  t , and Pt is the price at time t.
Expectation processes and price
In forming the expectation equations we assume rational expectations formed at t-1. Thus, we
assume that investors form their expectation for dividends and earnings on the basis of the
generating processes introduced above. We also assume that they substitute into the generating
process the values for dividends and earnings observed at t-1. The resulting expectations process
for dividends is given by equation (4) and the expectation process for earnings is given by
equation (5).
Et 1 ( Dt )  a1  a2 Et 1 (Yt )  a3 Dt 1
(4)
Et 1 (Yt )  b1  b2Yt 1  b3 Dt 1
(5)
The price equation is given by equation (6) which is the standard discounted dividend model,
with the price being given ex-dividend. We assume equilibrium prices, thus at the date of price
formation expected and required rate of returns are equal.

E (D )
Pt 1   t 1 tii1
(6)
i  0 (1  r )
The solution for the general model
To find the solution for the model, we use the earnings expectations process in equation (5) to
substitute for the expected earnings in the dividend equation (4) and then use this expected
dividend equation to substitute for expected dividends in the price equation (6). After some
manipulation (see Appendix 1), the resulting solutions for price and returns are given by equation
(7) and (8) respectively.
Pt 

(1  b2 )(a1  a2 b1 )  a2 b1 b2

r(1  b2 )(1  a3  a2 b3 )  a2 b2 b3 
(1  b2 )(1  r  b2 )( a3  a2 b3 )  a2 b2 b3   a1  a2 b1   (1  r )a2 b1 b2 

(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
a2 b2 (1  r )b3 (a1  a2 b1 )  b1b2 (1  r  a3 )


(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 

3
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3
D 
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  t
See for example Brown, Finn and Hancock (1977).
4

Rt =
(1  r )a2 b2
Y
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  t
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 

(7)
Dt

Pt 1
(1  r )a2 b2
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
Yt
D
 t
Pt 1 Pt 1
(8)
The current price is shown to be a function of levels in current dividends and current earnings.
Current realised returns are a function of changes in current dividends and current earnings, plus
the dividend yield. The changes in dividends and earnings scaled by price determine the capital
gain component of returns. Note also that by using equation (1) and/or equation (2) to substitute
for dividends Dt and/or earnings Yt , it is possible to express price as a function of several
additional variables including last period’s dividends, the shock to current dividends, last
period’s earnings and the shock to earnings. Similar substitutions can be made in the return
equation.
There is, therefore, plenty of scope in single equation empirical studies to find a range of
statistically significant variables which “explain” realised returns.4 Such variables may be
selection of those identified here, or proxies. Additionally, in cross-sectional analysis, the
coefficients from the dividend and earnings generating processes, or their proxies, can be used to
“explain” returns.
Given the general nature of our model, the statistical significance of a particular variable may be
consistent with several alternative explanations. Thus, we suggest that some care should be
exercised in interpreting the results of single equation studies. We further suggest that
consideration be given in empirical work to estimating simultaneous equation systems with
coefficient restrictions as appropriate.
Specific Cases of the General Model
In this section we consider some of the possible restrictions on the general model.
The Certainty Case
The certainty case is a highly restrictive case. The parameter restrictions are such that only one
price model can result from substitution in the general model. Under the assumption of certainty
the generating process and the expectations process are identical, since by definition there is no
error in expectation. One consequence of this is that in equilibrium all assets earn the risk free
rate of return rf.
The earnings process and earnings expectation simply depend on the scale of firm’s investment.
Given an initial investment I0 then the earnings realised in one period’s time are given by:
4
Using a less general model Hobbes, Partington and Stevenson (1996) demonstrate that similar comments apply to abnormal
returns.
5
Y 1 = rf I 0
If the firm pays out 100% of earnings as dividends then there will be no change in earnings over
time. They will remain constant at Y1. If less than 100% of earnings are distributed as a dividend
then the profit retained in any period t is given as t = Yt - Dt. This is reinvested at the rate rf.
Thus the earnings as t=2 can be written as
Y 2 = rf ( I 0 ) + rf (Y 1 - D1)
= Y 1 + rf (Y 1 - D1)
and, in general, the earnings can be written as:
Yt  Yt 1  r f Yt 1  Dt 1 


 1  r f Yt  1  r f Dt 1
(9)
In the certainty case, therefore, the earnings process in equation (2) is subject to the restrictions
that b1  0, b2  1  rf , b3  rf . The coefficient on the error term et is no longer one, but is
restricted to zero since under certainty there is no error.
In equation (9) growth in earnings arises from profit retention and we make no explicit allowance
for the possibility of additional contributions of capital by owners. One way to handle such a
possibility, which we adopt, is to follow the approach of Ohlson (1991). This involves treating
injections of cash by owners as negative dividends. In this case Dt is measured net of capital
contributions. Then total additional funds for investment are correctly measured as Yt - Dt5.
The only restriction on the coefficients of the dividend equation arises from the restriction that
dividends paid cannot exceed cash available. Given the results of Miller and Modigliani (1961)
that, under certainty, dividend policy is irrelevant to value, we would expect that the choice of
parameters for the dividend policy equation would have no effect on value. In Appendix 2 we
show that by substitution of the coefficient restrictions into the general model that this is indeed
the case. The result is Equation (10), which gives the price in the certainty case. In this case, the
magnitude of the current dividend is only relevant in determining the ex-dividend price.
Pt 
1  r  Y  D
f
rf
t
t
(10)
The realised returns equation is given by
Rt  rf
(11)
The above result for the price equation is the same as that derived by Ohlson (1991). It is also
equivalent to the result obtained by applying Miller and Modigliani’s (1961) valuation model to
a no growth case under certainty.
5
Alternatively, we could add an additional variable and funds for investment could be measured as Yt - Dt + St. Where St
represents additional capital contributions from owners. For notational simplicity, however, we adopt the Ohlson (1991)
approach.
6
Alternatives Under Uncertainty
The uncertainty case allows consideration of a richer set of alternatives than the certainty case.
Unlike the certainty case, the nature of the parameter restrictions provide alternatives that lead to
differing results in the model for price and returns. How we characterise uncertainty also has
important implications for the resulting model. For example, in the first case we consider below,
the characterisation of uncertainty is such that we obtain an identical result to the certainty case.
General Model for Dividends and Reinvestment Model for Earnings
Under this heading we consider two alternatives. First we consider a characterisation of
uncertainty which implies constant realised returns and then we consider stochastic realised
returns.
Return on Investment is Constant
Perhaps the simplest characterisation for uncertainty is to allow for shocks to dividends, while
holding the return on investment constant. The shocks to dividends are transmitted to the level of
reinvestment, and this in turn generates some uncertainty in earnings forecasts more than one
step ahead.
The generating process for dividends is as in the general model equation (1) as follows:
Dt  a1  a2Yt  a3 Dt 1  et
The expectation of dividends formed at t-1 is as previously given by equation (4).
Et 1 ( Dt )  a1  a2 Et 1 (Yt )  a3 Dt 1
The reinvestment process for earnings is as in the certainty case equation (9), except that we
replace rf with ROI. Where ROI is the return on investment, defined as Yt/It.. By definition, if
ROI is constant, there is no difference between the return on old or new investment.6 The
resulting reinvestment equation for earnings is given below as equation (12).
Yt  ROI [ It 1  Yt 1  Dt 1 ]
 Yt 1  ROI Yt 1  Dt 1 
(12)
The expectation of earnings formed at t-1 is given by the right hand side of equation (12). There
is no uncertainty in the one period ahead earnings forecast7 since it is based entirely on
observables and a constant.
6Thus,
7.
ROI could also be defined as Yt / It
However, there is uncertainty regarding earnings forecasts more than one period ahead. For example, the earnings forecast two
periods ahead is given by:
Et  2(Yt )  Et  2(Yt 1)  ROI  Et  2(Yt 1)  Et  2( Dt 1)
The expectation for Yt-1 can be formed without error, but the expectation of dividends Dt -1 formed at t-2 is subject to error.
7
By multiplying out the brackets in equation (12) we can see that the coefficient restrictions
imposed on the general model are b1 = 0, b2 = 1 + ROI and b3 = -ROI. As we demonstrate in
Appendix 3, the result of substituting into the general model is the following price equation:
Pt 
a1 (1  r )(r  ROI )

r[r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )

(r  ROI )( a3  a2 ROI )  a2 ROI (1  ROI )
Dt 
(r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )

a2 (1  r )(1  ROI )
Yt
(r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
(13)
We make the argument below that r and ROI are equal to rf in equilibrium. Given this equality
we can simplify equation (13) and the result is identical to the certainty case.
Given a constant return on investment in a competitive market, it seems reasonable to assume
that competition results in the following equilibrium between required return, expected return
and ROI.
r  E  Rt   ROI
Under the characterisation of uncertainty used in this section, the primary uncertainty is in the
magnitude of the cash dividends. As is clearly explained at the textbook level by Brealey and
Myers (1984), if the expected return on investment is equal to the investors’ required rate of
return, then the magnitude of the dividend and reinvestment decision has no effect on investors
wealth, it merely changes the form of their wealth.8 A dollar increase in dividends is exactly
offset by a dollar fall in share price and vice-versa.
Since the return on investment ROI was defined to be constant and dividend policy has no effect
on investors wealth, realised returns on equity must also be constant. In other words, investors’
realised returns are risk free. Therefore the equilibrium required rate of return is rf. Thus:
Rt  E Rt   ROI  rf
This equality between realised and expected returns leads to a result which is the same as the
certainty case:
Pt 
1  r  Y  D
f
rf
t
t
This result is consistent with that of Ohlson (1991) since his analysis implicitly assumes the
equality of realised returns and equilibrium expected returns. It is an interesting result that
although there is uncertainty in dividend forecasts and earnings forecasts, more than one period
ahead, this uncertainty has no effect on value.
8
Note that this point is distinct from M & M’s (1961) argument on dividend irrelevance. In that paper investment is held
constant. Here we discuss the condition where dividend changes are irrelevant to value, despite these dividend changes being
coupled to changes in investment.
8
Return on Investment is Stochastic
In this section the return on investment is assumed to have a mean equal to investors’ required
return, but investment returns are also assumed to have a stochastic component.9 The stochastic
component of ROI has a zero expectation. The shocks to investment returns feed through into
shocks to earnings. As a result uncertainty in future earnings now arises from both uncertainty in
ROI, and uncertainty about the level of reinvestment which arises from uncertainty about future
dividends.
The dividend generating processes is as in the general model, and identical to that used in the
constant ROI case above. As in the constant ROI case, the expectation process for dividends is as
given by equation (4). The earnings process differs from the constant ROI case, by the addition of
a subscript t to the ROI variable. This indicates that the realisation of ROI on re-investment may
vary in a stochastic fashion over time.
That is
Yt  Yt 1  ROI t Yt 1  Dt 1 ,
(14)
where
ROIt = ROI + t,
t is a random shock with zero mean, and the following restrictions on covariances of t are
assumed:
E(t+k , t+i) = 0
E(t+k , et+i) = 0
E(t+k , t+i) = 0
with 0 < i < k, for all t.
These restrictions eliminate the possibility of using the shocks to ROI to form improved forecasts
of future ROI, future earnings, or future dividends.
Substituting for ROIt in equation (14), expanding and re-arranging gives:
Yt  Yt 1  ROI (Yt 1  Dt 1 )  t (Yt 1  Dt 1 )
(15)
The random shock in ROI is reflected in earnings as a random shock scaled by the level of reinvestment. Thus equation (15) can be rewritten as:
Yt  Yt 1  ROI (Yt 1  Dt 1 )   t
Given E(t)=0 and the covariance restrictions on t , E(t)=0. The expectation of earnings is
therefore given by:
(12)
Et 1 (Yt )  Yt 1  ROI Yt 1  Dt 1 
The resulting coefficient restrictions are therefore, b1  0, b2  1  ROI, b3   ROI .
Substituting in the general model gives:
9
It does not matter whether changes in ROI are assumed to only affect new investment or to affect all investment, the resulting
price model is the same. For simplicity in the derivation the results are based on the former assumption.
9
Pt 
a1 (1  r ) r  ROI 
...
r[r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )]
...
(r  ROI )( a3  a2 ROI )  a2 ROI (1  ROI )
Dt ...
r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )

a2 (1  r )(1  ROI )
Yt
(r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
The form of the model is unchanged from the constant ROI case. This is because the expectation
for ROIt is:
E( ROI t)  ROI
The expectation of ROI is a constant, and it hardly seems feasible that in equilibrium this
constant is different from the required return, r. Substituting
ROI = r into the above price equation gives:
Pt 
(1  r )
Yt  Dt
r
As explained above, given equality between the expected return on investment and the required
return, then dividend policy per se is irrelevant to value and, as such, does not have any
information content. Consequently dividends do not appear in the final price model, except to
determine the ex-dividend price.
There is one change from the earlier models in this case. The discount rate is no longer rf.. This is
because there is now period to period uncertainty in the level of return realised by investors. As
we show in Appendix 4, the realised return is now given by:
Rt  r 
(1  r )  t
r Pt 1
We could continue and assume that stochastic changes in investment returns affect both reinvestment and existing investment, but the resulting price model would be the same as that
immediately above. The bottom line is that as long as we make assumptions that imply equality
between the expected return on investment and the required return, we will get essentially the
same pricing model. This is because, what we have is essentially an investment model of
valuation, in which all investments are expected to be zero NPV, and in which dividend policy is
irrelevant to value. Indeed, the model equates to Miller and Modigliani’s (1961) no growth case.
Lintner Model for Dividends and Random Walk for Earnings
Lintner model and simple random walk
In this model the generating process for earnings is not explicitly defined, in contrast to the
reinvestment model above. Rather we define the output of the earnings process as being well
approximated by a random walk. The earnings process has many degrees of freedom and
processes with many degrees of freedom often have the appearance of a random walk. Also, there
is an extensive empirical literature which suggests that annual earnings are well approximated by
a random walk, Ball and Watts (1972), Whittred (1978). Thus the earnings process is given by:
10
Yt  Yt 1   t
The corresponding expectation process for earnings is:
Et 1 (Yt )  Yt 1
The dividend process follows the Lintner (1956) model which, as noted earlier, is well supported
in the empirical literature. That is:
Dt  a  c Yt  1  c Dt 1  et
Where, is the target payout ratio (dividends as a proportion of profits) and c is the speed of
adjustment coefficient which governs how fast the firm adjusts the actual payout to the target
payout.
When we earlier considered the certainty case we followed the approach of Ohlson (1991) and
defined Dt as dividends net of capital contributions by owners. This is clearly contrary to the
spirit of the Lintner model which is intended only to model the dividend payment, and not capital
contributions. In order to have consistent notation throughout the paper, therefore, we must here
impose the restriction that there be no capital contributions by owners from t onwards.
The expectation process for dividends is:
Dt  a  c E Yt   1  c Dt 1
Using the solution for the general model with the coefficient restrictions
a1  a , a 2  c , a 3  1  c, b1  0, b2  1, b3  0 , we show in Appendix 5 that
the price and realised returns equations are given by:
or
Pt 
 1  c
a r  1 c  r  1

Yt 
D
 r  c t
r  r  c r  r  c 
Rt 
c  r  1 Yt  1  c Dt Dt


r r  c Pt 1  r  c Pt 1 Pt 1
1  r  et
c  r  1  t
Rt  r 

r r  c Pt 1  r  c Pt 1
2
Lintner Model for Dividends and Random Walk with Drift for Earnings
In this case the only change from the equations in the preceding case is the addition of a drift
term b to the earnings equation. Thus earnings are given by
Yt  b  Yt 1   t
Using the solution for the general model with the following coefficient restrictions
a1  a , a2  c , a3  (1  c) , b1  b , b2  1 , b3  0
We show in Appendix 6 that the price and realised return equations are given by
11
Pt 
Rt 
a  cb c (1  r )
(1  c)

Yt 
Dt
rc
r (r  c)
(r  c)
D
c (1  r ) Yt (1  c) Dt

 t ,
r (r  c) Pt 1 (r  c) Pt 1 Pt 1
or
Rt  r 
c (1  r ) 2  t
(1  r ) et

r(r  c) Pt 1 (r  c) Pt 1
Lintner Model for Dividends and Random Walk with Growth for Earnings
The output from the earnings process is now assumed to be well approximated by compound
growth in earnings which follows a random walk. This is given by
Yt   1  g Yt 1   t
With the coefficient restrictions a1  a, a2  c , a3  1 c, b1  0, b2  1 g, b3  0, in Appendix 7
we derive the price and realised returns equations to be
or
Pt 
 1  c
a r  1 c  1  g   r  1

Yt 
D
 r  c t
r  r  c
 r  g   r  c
Rt 
c  1  g   r  1 Yt  1  c Dt Dt


 r  g  r  c Pt 1  r  c Pt 1 Pt 1
c  r  1
 t  1  r  et
Rt  r 

 r  g r  c Pt 1  r  c Pt 1
2
These three models which use the Lintner equation are consistent with the models of Hobbes,
Partington and Stevenson (1996). However, the model with drift represents an extension to their
work. Notice also the different possibilities for expressing these models. As we show above the
return equations can be expressed as a function of dividend and earnings changes and current
dividends, or as a function of the equilibrium expected return for the firm and shocks to earnings
and dividends. Also note that by using the Lintner model to substitute for the dividend term,Dt,
the level of earnings, lagged dividends and the shock to dividends can be introduced into the
returns model which contains changes in earnings and dividends.
Summary and Conclusion
As can be seen from the foregoing the general model can be useful in analysing a variety of
special cases. The models are derived in a time series, but can be used to explain cross sectional
variation in price and return, in response to changes in dividends and earnings.10 Table 1
summarises the particular special cases examined above.
10This,
of course, presumes that you can determine which of the special cases should be used as the explanatory model.
12
Table 1: Models of equity returns
State of nature
Certain
Uncertain
Main assumptions
Return on investment is equal to the risk
free rate
Distribution of future dividends is
uncertain, return on investment is
constant
Distribution of future dividends is
uncertain, return on investment is
stochastic with an expected value equal
to r
Lintner model for dividends,
random walk for earnings
Lintner model for dividends,
random walk with drift for earnings
Lintner model for dividends,
random walk with growth for earnings
Equity valuation model,
and reference
1  rf
Pt 
Yt  Dt ,
rf


Ohlson (1991)
1  rf
Pt 
Yt  Dt ,
rf


Ohlson (1991)
1  r  Y  D ,
Pt 
t
t
r
This paper, page 11.
Pt  f  r , c,  , Yt , Dt  ,
Hobbes, Partington and
Stevenson (1996)
Pt  f  r, c,  , b, Yt , Dt  ,
This paper, page 12.
Pt  f r , c,  , g , Yt , Dt  ,
Hobbes, Partington and
Stevenson (1996)
The list in Table 1 is not exhaustive of the possibilities. The general model encompasses models
which comply with the following:
1. Price can be expressed as the discounted dividend model.
2. Expected earnings can be expressed as a linear function of one period lagged earnings and/or
one period lagged dividends in a stochastic, or non-stochastic framework.
3. Expected dividends can be expressed as a linear function of current earnings, and/or one
period lagged dividends in a stochastic, or non-stochastic framework..
The general model is consistent with the intuition that current earnings and dividends are relevant
to value when they can be used by investors to form improved forecasts of future cash flows
from the business. This will generally be through an improved forecast of earnings, or dividends,
or return on investment. Note, however that some information variables affect price, but they do
not affect returns. For example, in the random walk with drift model, earnings drift is a variable
in the price equation but is not in the corresponding return equation.11
Value changes can be of two types, scale effects and wealth changes. Scale effects are simply a
consequence of changes in the level of re-investment, while wealth changes arise from shocks in
the value relevant variables. A question of particular importance in analysing wealth effects, is
whether investors are able to determine if future returns on the firm’s investment will differ from
the required rate. This is, of course, Miller and Modigliani’s (1961) criteria for identifying
growth stocks. If the restrictions on the model imply that the expected return on investment is
11Once
the earnings drift has been incorporated into the price at t=0, it no longer affects either expected return, or shocks to
realised returns. However, if the expected magnitude of the earnings drift changed, it would enter the return equation as a
shock to value and hence return.
13
equal to the required rate, a rather simple valuation model emerges. This model can be expressed
in terms of Ohlson’s (1991) earnings based valuation model, or equivalently Miller and
Modigliani’s no growth valuation model.
The general model allows cases consistent with Miller Modigliani’s (1961) dividend irrelevance
proposition, such as their no growth model. However, it goes a step further by allowing cases
where their dividend signaling proposition can be incorporated into the valuation model.12 Thus,
depending on the particular restrictions imposed on the general model dividends may be relevant
to value or irrelevant.
In the Hobbes, Partington, Stevenson (1996) model dividends are relevant to value because of
dividend signaling. Dividend signaling is implicit in the use of the Lintner (1956) model, which
directly conditions expectations of future dividends and implicitly indicates sustainable cash
flow. In the case of the Ohlson (1991) model dividend policy is not relevant to value. This result
is driven by the re-investment process, and assumptions about the return on investment. The
nature of the model is such that there is uncertainty about future dividends, which are relatively
unconstrained. There is also uncertainty about future earnings, which stems primarily from
uncertainty over the level of re-investment. However, it turns out that neither of these sources of
uncertainty affects the wealth of investors, given that the expected return on investment is equal
to the required return on investment.
In the Ohlson (1991) model, therefore, uncertainty over dividends and future earnings has no
substantive impact on value. This demonstrates the interesting result; that it is possible to create a
characterisation of uncertainty that is irrelevant to valuation. Under such a characterisation all
discounting naturally takes place at the risk free rate. Changes in current dividend payments
merely create changes in value that are a consequence of scale effects, as discussed above.
The general model and special cases of the general model arise from a system of equations. The
common practice in empirical work of estimating a single equation return model is unlikely to
adequately distinguish between alternative models. For example, is the significance of earnings
in an earnings/return equation evidence in support of the Ohlson (1991) model, or the Hobbes,
Partington, Stevenson (1996) model? It is consistent with both, it is also consistent with earnings
proxying for some omitted, but correlated, causal variable. Another problem, as demonstrated in
the section on earnings/re-investment models, is that the same reduced form equation for returns
may result from different processes. It is also possible to get different single equation models
from the same processes. This is shown in the section involving the Lintner model.
It seems better to test the system of equations that form the model. This would involve
addressing the question, are the estimated coefficients consistent between the earnings and
dividends equations, and the return equation? This naturally leads to estimation of the models as
simultaneous equation systems with consistency constraints on the coefficients. This is not a
trivial task. Not only is there a requirement for a substantial body of data, but the constraints on
the coefficients are likely to be non-linear.
The general model is not only of theoretical interest but may also have some practical use. It
provides a basis for solutions to the discounted dividend pricing model that do not require
explicit forecasts of future values of dividends, only currently observable values for dividends
12
Miller and Modigliani (1961) invoke the argument that the price changes observed in response to dividend changes are a
consequence of dividends conveying information about future cash flow. However, they do not incorporate dividend signaling
into their valuation model.
14
and earnings. The cost of this is that assumptions must be made about the explicit processes for
dividends and earnings. The particular parameterisation assumed must be consistent with the
linear form of the general model. Given an acceptable parameterisation, the general model may
be a possible alternative to the standard dividend growth model, in providing horizon values
beyond the forecasting horizon of explicit dividend forecasts. Conversely, given today’s price
and assumptions about the dividend and earnings processes it is possible to back out an implied
discount rate. This may be an additional tool for those faced with the vexing question of how to
estimate a company’s cost of equity.
A final point to note is that the general model provides a model for realised returns, as distinct
from expected, or required returns. Although, as Hobbes, Partington and Stevenson (1996) show,
it is possible to derive an expression for expected returns by taking expectations of specific forms
of the realised returns model. A general model of realised returns may turn out to be useful given
that much empirical work is based on realised returns. This is in accord with Fisher Black’s
(1993) suggestion that empirical investigations of asset pricing models, being based on realised
returns, rather than expected returns, likely tell us more about the variance of returns than they do
about asset pricing models. Black also suggests that more is to be learned about asset pricing
models from theoretical as opposed to empirical investigation. Perhaps more theory about
realised returns will help bridge the gap.
15
References
Ball, R., and P. Brown, 1968, An empirical evaluation of accounting income numbers, Journal
of Accounting Research, 6, 159-178.
Ball, R. and R. Watts, 1972, Some time series properties of accounting income, Journal of
Finance, 27, 663-682.
Black, F., 1993, Estimating expected return, Financial Analysts Journal, September- October,
36-38.
Brealey, R., and S. Myers, 1984, Priciples of Corporate Finance, 2nd ed., McGraw-Hill, New
York.
Brown, P., F.J. Finn, and P. Hancock, 1977, Dividend changes, earnings reports and share prices:
Some Australian findings, Australian Journal of Management, 127-147.
Chiang, R., Davidson, I. And J. Okunev, 1996, Some further theoretical and empirical
implications regarding the relationship between earnings, dividends and stock prices, Journal of
Banking and Finance.
Easton, P.D., and T. Harris, 1991, Earnings as an explanatory variable for returns, Journal of
Accounting Research, 29, 19-36.
Fama, E.F. and H. Babiak, 1968, Dividend policy: An empirical analysis, Journal of the
American Statistical Association, 63, 1132-1161.
Goyal, M. and R. A. Beg, 1995, A theoretical model of earnings, dividends, returns and cash
flows, European Accounting Association Conference, Conference paper, May 1995.
Hobbes, G., G. Partington and M. Stevenson, 1993, The earnings, dividends and return
relationship, Accounting Association of Australia and New Zealand, Conference paper.
Hobbes, G., G. Partington and M. Stevenson, 1996, Earnings, dividends and returns: A
theoretical model, Research in Finance, Supplement 2, 221-244.
Kallapur, S., 1994, Dividend payout ratios as determinants of earnings response coefficients: A
test of the free cash flow theory, Journal of Accounting and Economics, 17, 359-375.
Laub, M.P., 1972, Some aspects of the aggregation problem in the dividend-earning relationship,
Journal of the American Statistical Association, 67, 552-559.
Lintner, J., 1956, Distribution of incomes of corporations among dividends, retained earnings,
and taxes, American Economic Review, 46, 97-113.
Mande, V., 1994, Earnings response coefficients and dividend policy parameters, Accounting
and Business Research, 24(94), 148-156.
Miller, M. and F. Modigliani, 1961, Dividend policy, growth and the valuation of shares,
Journal of Business, 34, 411-433.
16
Ohlson, J., 1991, The theory of value and earnings and an introduction to the Ball-Brown
analysis, Contemporary Accounting Research, 8(1),
1-19.
Ohlson, J. A. and P. K. Shroff, 1992, Changes versus levels in earnings as explanatory variables
for returns: Some theoretical considerations, Journal of Accounting Research, 30, 210-225.
Shevlin, T., 1982, Australian corporate dividend policy: Empirical evidence, Accounting and
Finance, 22(1), 1-22.
Whittred, G., 1978, The time series behaviour of corporate earnings, Australian Journal of
Management, 195-202.
17
APPENDIX 1 GENERAL SOLUTION FOR PRICE AND RETURNS
General process for dividends:
Dt = a1 + a2Yt + a3Dt-1 + et
Expectation process for dividends:
Et-1(Dt) = a1 + a2Et-1(Yt) + a3Dt-1
as Et-1(et) = 0
General process for earnings:
Yt = b1 + b2Yt-1 + b3Dt-1 + t
Expectation process for earnings:
Et-1(Yt) = b1 + b2Yt-1 + b3Dt-1
as Et-1(t) = 0
Realised returns:
Rt 
Pt  Pt 1 Dt

Pt 1
Pt 1
Now
Dt
=
a1 + a2[b1 + b2Yt-1 + b3Dt-1 + t] + a3Dt-1 + et
=
a1 + a2b1 + a2b2Yt-1 + a2b3Dt-1 + a3Dt-1 + et + a2t
 Dt =
a1 + a2b1 + a2b2Yt-1 + (a3 + a2b3)Dt-1 + et + a2t
Yt =
or
 Dt 
  =
 
 Yt 
and
J
~t

b1
+ b2Yt-1
+
 a1  a2 b1   a3  a2 b3

 

 
 b1   b3
F
~

GJ
~ ~ t 1
t
b3Dt-1 +
a2 b2   Dt 1  1 a2   et 

 
 
 
 

b2   Yt 1   0 1    t 

MK
~
~ t
.....(A1.1)
 a3  a2 b3 a2 b2 
 Dt 
 et 
 a1  a2 b1 
1 a2 




 




where J    , F  


 , K~   
, G
, M
~
~
~t
~
b2 
0 1 
 b1

 Yt 
 t 
 b3
Equation (A1.1) demonstrates that current dividends and earnings follow a first-order
autoregressive process.
Solving A1.1 by recursion:
18
J1
=
F G J  M K
=
F G F G J  M K  M K
~
J2
~
J3
~

~
~
~1
~
~ ~o
~1

~
~ 2
F G F G J  G M K  M K
=
F G J  M K
=
F  G F  G F  G2 J  G M K  M K  M K
2
~
~ ~
~
~
~
~ ~2
~

~
~o
~
~
~
~1
~
~ 2
~ 3
~ ~
2
~
~o
~
2
~
~1
~
~ 2

~
~ 3
=
F G F G F G J  G M K  G M K  M K
=
F G J  M K
=
F  G F  G F  G 2 F  G3 J  G 2 M K  G M K  M K  M K
~
~
~ ~
~
~

~
~
~
~
~ ~3
~
3
~
~o
~
~
~1
~
~
~ 2
~
~ 3
~ 4
~ ~
~
~
~
~o
~
~
~1
o
~
~
~ 2
~
~ 3

~
~ 4
=
G F  G F  G F  G F  G J  G M K  G M K  G M K  G3 M K
J
=
Go F  G F  Gt 1 F  Gt J  Go M K  G M K
J
=
~t

~
~ ~o
=
~
J4
~
o
~
~
2
~
~ ~
~
3
~
~
~ ~
~t
~
~
~
~
~
t 1
4
~
~o
~o
~
~
~
~
~ t
2
~ 4
~
~
~
~
~ 3
~ t 1
~
~
~ 2
~
~
~1
 Gt 1 M K
~
~
~1
t 1
 Gi F Gt J   Gi M K
i0
~
~
~
~o
~
~
i0
~ t i
When the focal point is time equal to 0, then
k 1
J
=
~k
k 1
 Gi F G k J   G k M K
i0
~
~
~
~o
i0
~ k i
~
~
When the focal point is time equal to t, then
k 1
J
~ t k
=
G
i0
k 1
i
F G J   Gi M K
k
~
~
~
~t
~
~
i0
~ t  k i
Taking expectations,
 
=
 
=
Et J
~ t k
Et J
~ t k
k 1
i0
Pt
k 1
i
=
G
i 0
F  G k J   G i M Et ( K
~
~
k 1
=
Now
G
i
~
~
~
i0
F G k J
~
~
~t
~
~
~
~

t
k 1
1
k
~
~ t k
k
(1  r )
~
)
as
t
   I  G  F G J
N E J 
E D 



I G
~ t  k i

Et K
~ t  k i
0
k
~

k 1
~
~
~t
t
~ t k
k
(1  r )
where N = (1 0)
~
19

=
=
Pt
=
where N  (0 1) ,
~
Now
 
I G
~
 
I G
~
~
~
~
~
~
~
~ ~
~t
(1  r )
k 1
1
 N ( I  G) F

Gk
1
~ ~
~
 N ( I  G)  ~ k F  

k
~
~ ~
~
(1  r )
k 1
k 1 (1  r )

Gk
 N ~ k J
~
~t
k 1 (1  r )
k
G
1
1


 N ( I  G)
I
G F  

~
~
~
~
r
(1  r )  (1  r ) ~  ~
1
G 
1

~
 N
I
G J
~ (1  r )  ~
 (1  r ) ~  ~ t
 a3  a2 b3 a2 b2 
 Dt 
 a1  a2 b1 
1 a 2 
, M  
, G  
, J   
F  



 


~
~
b2 
 0 1  ~  Yt 
 b1  ~  b3
~
~
~
~
1
~
=
1  a3  a2 b3  a2 b2 





b
1

b

3
2 
=
a2 b2
 (1  b2 )

1



[(1  a3  a2 b3 )(1  b2 )  a2 b2 b3 ]  b3
(1  a3  a2 b3 )
=
a2 b2
 (1  b2 )

 D

D1
1




 b3 (1  a3  a2 b3 ) 


D1
 D1

1
1
1


G =
 I~ 
~
 (1  r ) 
=
~
N ( I  G) 1 F

~

N ( I  G) 1 ( I  G k ) F  N G k J
where D1  (1  a3  a2 b3 )(1  b2 )  a2 b2 b3
 1  r  a3  a2 b3  a2 b2 


(1  r )
(1  r ) 



 b3
(1  r  b2 



(1  r )
(1  r ) 
1
a2 b2
 (1  r  b2



(
1

r
)
(
1

r
)
1


=


1  r  a3  a2 b3 1  r  b2   a2 b2 b3  b3
(1  r  a3  a2 b3 ) 


(1  r ) 2
 (1  r )

(1  r )
 (1  r  b2 a2 b2



D
D
2
2
(1  r  a3  a2 b3 )(1  r  b2 )  a2 b2 b3



 where D 2 
(1  r )
1  r  a3  a2 b3 
 b3


D2
 D2

20
 
I G
~
 (1  b2 ) a2 b2 
 D
 a a b
D1
1

  1 2 1 



 b3 1  a3  a2 b3   b1 


D1
 D1

 (1  b2 )( a1  a2 b1 )  a2 b1 b2 


D1




 b3 ( a1  a2 b1 )  b1 (1  a3  a2 b3 ) 


D1


1
=
F
~
~
=
N ( I  G ) 1 F
~
~
~
~
r
=
(1  b2 )( a1  a2 b1 )  a2 b1 b2
r[(1  a3  a2 b3 )(1  b2 )  a2 b2 b3 ]
=
(1  b2 )( a1  a2b1 )  a2b1b2
rD1
G
1
G
1
1


G =
 I~ 
~
(1  r )  (1  r ) 
=
~
.....(A1.2)


1
I
G F

(1  r )  ~ (1  r ) ~  ~
 (1  r  b2 )( a3  a2 b3 )  a2 b2 b3  a1  a2 b1   (1  r )a2 b1 b2 


D2 (1  r )


=


(1  r )( a1  a2 b1 )b3  b1 b2 (1  r  a3 )




D2 (1  r )


~
G
1
1


N ( I  G)
I
G F

~ ~
~
(1  r )  ~ (1  r ) ~  ~
 [(1  r  b2 )( a3  a2 b3 )  a2 b2 b3 ][a1  a2 b1 ]  (1  r )a2 b1 b2 


D2 (1  r )
 (1  b2 ) a2 b2  

=



D1 
 D1
(1  r )( a1  a2 b1 )b3  b1 b2 (1  r  a3 )




D2 (1  r )


1

~
(1  b2 ) [(1  r  b2 )( a3  a2 b3 )  a2 b2 b3 ][a1  a2 b1 ]  (1  r )a2 b1 b2 


D1 
D2 (1  r )


a2 b2  (1  r )b3 ( a1  a2 b1 )  b1 b2 (1  r  a3 ) 


D1 
D2 (1  r )

21


N ( I  G) F
~
~
~
 N ( I  G) 1
~
r
~
~
~
1
G
1


I
G F

~
~
(1  r )  (1  r )  ~
~
(1  b2 )(a1  a2 b1 )  a2 b1 b2 
r(1  b2 )(1  a3  a 2 b3 )  a 2 b2 b3 


(1  b2 )(1  r  b2 )( a3  a 2 b3 )  a 2 b2 b3  a1  a 2 b1   (1  r )a 2 b1 b2 
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
a2 b2 (1  r )b3 ( a1  a2 b1 )  b1b2 (1  r  a3 )
.....(A1.3)
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
1
G
1


N
I
G J

~ (1  r ) ~
 (1  r ) ~  ~ t
~

(1  r  b2 )(a3  a2 b3 )  a2 b2 b3

 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
 Pt 

N ( I  G) 1 F
~
~
r
~
  Dt 
(1  r )a2 b2
 
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3   Yt 
G
1
1
NG 
1
1



~ ~
 N ( I  G)
I

G
F

I
G J



~ ~
~
(1  r )  ~ (1  r ) ~  ~ (1  r )  ~ (1  r ) ~  ~ t
1
~
(1  b2 )( a1  a2 b1 )  a2 b1b2

r(1  b2 )(1  a3  a2 b3 )  a2 b2 b3 

(1  b2 )(1  r  b2 )( a3  a2 b3 )  a2 b2 b3   a1  a2 b1   (1  r )a2 b1 b2 

(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
a2 b2 (1  r )b3 (a1  a2 b1 )  b1b2 (1  r  a3 )


(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3
D 
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  t
(1  r )a2 b2

Y
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  t

......(A1.4)
22
Recall
Rt
However, Pt - Pt-1
P  Pt 1
and t
Pt 1
Dt
Pt 1
Also
=
Pt  Pt 1 Dt

Pt 1
Pt 1
=
1
NG 
1

~ ~
I
G
J J
(1  r )  ~ (1  r ) ~  ~ t ~ t 1
=
1
NG 
1
  J~ t
~ ~
I
G
(1  r )  ~ (1  r ) ~  Pt 1


NJ
=
~ ~t
Pt 1
 Realised returns can be written as
or
Rt
=
1
NG 
1
  J~ t N~ J~ t
~ ~
I
G

(1  r )  ~ (1  r ) ~  Pt 1
Pt 1
Rt
=
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 

Dt

Pt 1
(1  r )a2 b2
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
Yt
D
 t
Pt 1 Pt 1
.....(A1.5)
23
APPENDIX 2 - PRICE AND RETURNS FOR THE CERTAINTY CASE
For the certainty case the earnings process is subject to the following restrictions:
b1  0, b2  1  rf , b3  rf
Let
N ( I  G) 1 F
~
~
~
~
r
1
G
1


 N ( I  G)
I
G F  A  B  C

~
~ ~
~
~
(1  r )  (1  r )  ~
1
~
From (A1.3) in APPENDIX 1 it follows that:
A 
(  rf )( a1 )
(1  b2 )( a1  a2 b1 )  a2 b1 b2

r f (1  b2 )(1  a3  a2 b3 )  a2 b2 b3  r f (  rf )(1  a3  r f a2 )  a2 (1  r f )(  rf


B
=
=
C
=
=
A - B - C
a1
rf (1  a3  a2 )


(1  b2 ) (1  rf  b2 )(a3  a2 b3 )  a2 b2 b3 a1  a2 b1   (1  rf )a2 b1 b2
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  rf


 b2 )(1  rf  a3  a2 b3 )  a2 b2 b3

 r a )  a (1  r )(  r )  a (1  r )( r )

( rf ) a2 (1  rf )( rf )a1
(r )(1  a
3
f
=

2
f
2
f
2
f
f
f
 a1
(1  a3  a2 )


a2 b2 (1  rf )( a1  a2 b1 )b3  b1 b2 (1  r f  a3 )
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  rf


a2 (1  rf ) (1  rf )a1 (  rf )
(  rf ) (1  a3  a2 )(  a2 )(1  rf )
2
=


 b2 )(1  r f  a3  a2 b2 b3 )
(1  rf )a1
rf (1  a3  a2 )
rf a1
(1  rf )a1
a1


 0
rf (1  a3  a2 ) rf (1  a3  a2 ) rf (1  a3  a2 )
1
NG 
1

~ ~
I
G J
(1  r )  ~ (1  r ) ~  ~ t
  Dt 
 
(1  rf  b2 )(1  rf  a3  a2 b3 )  a2 b2 b3   Yt 
(1  rf )(1  rf )a2   Dt 
  
 a2 (1  rf )(rf )   Yt 

(1  rf  b2 )(a3  a2 b3 )  a2 b2 b3
= 
 (1  rf  b2 )(1  rf  a3  a2 b3 )  a2 b2 b3
=
 a2 (1  rf )(rf )

  a2 (1  rf )(rf )
(1  rf )a2 b2
24
=
 Dt 
 
 
 Yt 

(1  rf ) 
 1

rf 

Now
N ( I  G) 1 F
Pt
=
Pt
=
~
~
~
~
r
 Dt 
(1  rf )
rf
Yt
G
1
1
NG
1
1


~ ~ 
 N ( I  G)
I

G
F

I
G J



~ ~
~
(1  r )  ~ (1  r ) ~  ~ (1  r )  ~ 1  r ~  ~ t
1
~
when b1  0, b2  1  rf , b3  rf
Realised returns can be calculated as follows:
Rt
Pt  Pt 1 Dt

Pt 1
Pt 1
=
Now Pt - Pt-1 =
and

NJ
~ ~t
Rt
=
=
=
(1  rf )Yt 1  rf Dt 1
=
Pt-1Rt =
=
=
 (1  rf )

rf 
Yt 1  Dt 1 
 rf

rf Pt1
=
rf
=

Rt
Dt
1
J

  J~ t N
1
N GI 
G
 ~ ~t
~ ~ ~
Pt 1
 (1  rf ~  Pt 1
 Dt 
(1  rf   Dt 
1 
  1 (1 0) 
 1
 
 
Pt 1 
rf   Yt  Pt 1
 Yt 
(1  rf )
Yt  Yt 1   Dt 1
rf
(1  rf )
(rf )Yt 1  Dt 1   Dt 1
rf
(1  rf )Yt 1  (1  rf ) Dt 1  Dt 1
=

1
NG 

1
~ ~
G  J ,
I 
~t
(1  rf )  ~ (1  rf ) ~ 
25
APPENDIX 3 - PRICE WHEN RETURN ON INVESTMENT IS CONSTANT
For the case when return on investment is constant the earnings process is subject to the
following restrictions:
b1 = 0,
Let
N ( I  G) 1 F
~
~
~
r
~
b2 = 1 + ROI,
b3 = -ROI
1
G
1


 N ( I  G)
I
G F  A  B  C

~
~ ~
~
~
(1  r )  (1  r )  ~
1
~
(1  b2 )( a1  a2 b1 )  a2 b1 b2
 ROI ( a1 )

r(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  r(  ROI )(1  a3  ROIa2 )  a2 (1  ROI )(  ROI )
A

B

(1  b 2 )(1  r  b2 )(a3  a2 b3 )  a2 b2 b3  a1  a2 b1   (1  r )a2 b1b2 
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
(  ROI )(1  r  1  ROI )(a3  a2 ROI )  a2 (1  ROI )(  ROI )a1
( ROI )(1  a3  a2 ROI )  a2 (1  ROI )( ROI ) 1  r  1  ROI )(1  r  a3  a2 ROI )  a2 (1  ROI )( ROI )

C

a1
r1  a3  a2 
a1 (r  ROI )(a3  a2 ROI )  a2 ROI (1  ROI )
1  a3  a2  r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
a2 b2 (1  r )b3 (a1  a2 b1 )  b1b2 (1  r  a3 )
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
a2 (1  ROI )a1 (1  r )(  ROI )
( ROI )(1  a3  a2 ROI )  a2 (1  ROI )( ROI ) (1  r  1  ROI )(1  r  a3  a2 ROI )  a2 (1  ROI )( ROI )

a1a2 (1  ROI )(1  r )
1  a3  a2  (r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
a1 r  ROI )(a3  a2 ROI )  a2 ROI (1  ROI )
a1

r1  a3  a2  1  a3  a2  (r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
a1 a2 (1  ROI )(1  r )

1  a3  a2 (r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
a1 (1  r )(r  ROI )

r(r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )
A BC 
Also
26
1
NG 
1

~ ~
I
G J
(1  r )  ~ (1  r ) ~  ~ t
 (1  r  b2 )( a 3  a 2 b3 )  a 2 b2 b3
  Dt 
(1  r )a 2 b2

 
 (1  r  b2 )(1  r  a 3  a 2 b3 (1  r  b2 )(1  r  a 3  a 2 b3 )  a 2 b2 b3  Yt 
 (r  ROI )(a3  a2 ROI )  a2 ROI (1  ROI )
  Dt 
a2 (1  r )(1  ROI )

 
 (r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI ) (r  ROI )(1  r  a3  a2 ROI )  a2 ROI (1  ROI )  Yt 
Now
Pt 
N ( I  G) 1 F
~
~
~
~
r
G
1
1
NG 
1
1



~ ~
 N ( I  G)
I

G
F

I
G J



~ ~
~
(1  r )  ~ (1  r ) ~  ~ (1  r )  ~ (1  r ) ~  ~ t
1
~
a (1  r )(r  ROI )
1
P 

t r[r  ROI )(1  r  a  a ROI )  a ROI (1  ROI )]
3
2
2
(r  ROI )( a  a ROI )  a ROI (1  ROI )
3
2
2
D 
(r  ROI )(1  r  a  a ROI )  a ROI (1  ROI ) t
3
2
2
a (1  r )(1  ROI )
2

Y
(r  ROI )(1  r  a  a ROI )  a ROI (1  ROI ) t
3
2
2

When r = ROI = rf ,
Pt   Dt 
(1  rf )
rf
Yt
27
APPENDIX 4 REALISED RETURN WHEN RETURN ON INVESTMENT IS
STOCHASTIC
Recall from APPENDIX 1 that current earnings and dividends follow a first-order autoregressive
process previously defined by equation (A1.1) as:
J  F  GJ
~t
~ ~ t 1
~
MK
~
.....(A4.1)
~ t
 a3  a2 b3 a2 b2 
 Dt 
 et 
 a1  a2 b1 
1 a 2 
, M  
, G  
, K   
where J    , F  



 


~
~t
0 1  ~ t  t 
 Yt  ~  b1  ~  b3
b3 
It follows that
J IJ
~t
or
 F  (G  I ) J
MK
 J  F  (G  I ) J
MK
~ ~ t 1
~t
~
~
~
~
~
~
~ t 1
~ t 1
~
~
~ t
.....(A4.2)
~ t
Recall equation (A1.5 ) from APPENDIX 1 which gives
1
J
NG 
1
  J~ t N
~ ~t
~ ~
Rt 
I

G

(1  r )  ~ (1  r ) ~  Pt 1
Pt 1
Substituting (A4.1) and (A4.2) into the above results in
1

G 
G 
1  N
~ ~
~


Rt 
I

F

N
F


Pt 1  (1  r )  ~ (1  r )  ~ ~ ~ 


1

G 
G 
1  N
 ~ ~  I  ~  (G  I )  N G  J  

~ ~  ~ t 1
Pt 1  (1  r )  ~ (1  r )  ~ ~


1

G 
G 
1  N
 ~ ~ I  ~  M  N M K

Pt 1  (1  r )  ~ (1  r )  ~ ~ ~  ~ t


.....(A4.3)
Using equation (A1.2) in APPENDIX 1 we can write
1
NG 
1

~ ~
I
G
(1  r )  ~ (1  r ) ~ 
 (1  r  b2 )(a3  a 2 b3 )  a 2 b2 b3

(1  r )a 2 b2


 (1  r  b2 )(1  r  a3  a 2 b3 )  a 2 b2 b3 (1  r  b2 )(1  r  a3  a 2 b3 )  a 2 b2 b3 
28
and
1
NG 
1

~ ~
I
G F N F
(1  r )  ~ (1  r ) ~  ~ ~ ~

(1  r  b2 )(a3  a2 b3 )  a2 b2 b3  a1  a2 b1   (1  r)a2 b1b2  (a
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3
1
 a2 b1 )
 c0 ,
1
NG 
1

~ ~
I
G (G  I )  N G
~ ~
(1  r )  ~ (1  r ) ~  ~ ~
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3   a3  a2 b3  1  (1  r )a2 b2 b3

 ( a3  a2 b3 )

(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3






(1  r  b2 )(a3  a2 b3 )  a2 b2 b3 a2 b2  (1  r )a2 b2 (b2  1)  a b


2 2
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3


 c1
c2  , and
1
NG 
1

~ ~
I

G
M N M
(1  r )  ~ (1  r ) ~  ~ ~ ~
 (1  r  b2 )( a3  a2 b3 )  a2 b2 b3
(1  r  b2 )(a3  a2 b3 )  a2 b2 b3 a2  (1  r )a2 b2  a 

1
2
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3
 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3

 c3 c4 
(A4.3) can be rewritten as
 1 


 et 
 Pt 1 
P 


t 1
Dt 1 


Rt   co c1 c2 
  c3 c 4 

 Pt 1 
 t 


 Pt 1 
Y 
 t 1 
 Pt 1 
c Z
~ z ~ t 1
c K
~K ~ t
.....(A4.4)
29
Taking conditional expectations of equation (A4.4) gives
Et 1 ( Rt )  c Z
as E t-1 ( K )  0
~z ~ t
~ t
Under the assumption that there is no information content in the shock to ROI, earnings and
dividends, which allows for improved predictions of future ROI, we can write
E( ROIt )  ROI  r
Using the restrictions that b1  0, b2  1  r and b3  r, it follows that
Et 1 ( Rt )  c z Z t 1
1

 Pt 1 


 Dt 1

 (co c1 c2 )
Pt 1 


Y

 t 1

Pt 1 

1

 Pt 1 


 Dt 1

 [0  r (1  r )]
Pt 1 


Y

 t 1

Pt 1 


1
r  (1  r )
(1  r )Yt 1  rDt 1  
Yt 1  Dt 1 


Pt 1
Pt 1  r


r
. Pt 1  r
Pt 1
 Equation (A4.4) becomes
Rt  Et 1 ( Rt )  c K
~K ~ t
 et 
P 
t 1

  c3 c 4  

 t 
 Pt 1 
30
i. e. Rt  r  0


 Rt  r 
 t 


(1  r )   Pt 1 
for b1  0, b2  1  r and b 3   r,
r    t 


 Pt 1 
(1  r )  t
r Pt 1
(1  r )  t
r Pt 1
31
APPENDIX 5 LINTNER MODEL FOR DIVIDENDS AND RANDOM WALK
FOR EARNINGS
Lintner Model for Dividends:
General Generating Process
for Dividends:
Dt  a  cYt  (1  c) Dt 1  et
Dt  a1  a2Yt  a3 Dt 1  et
Equating the above two equations results in
a1  a , a2  c
, a3  (1  c)
Random Walk for Earnings:
General Generating Process
for Earnings:
Yt  Yt 1   t
Yt  b1  b2 Yt 1  b3 Dt 1   t
Equating the above two equations results in
b1  0 , b2  1 , b3  0
Recall equation (A1.4),
Pt 


(1  b2 )( a1  a2 b1 )  a2 b1b2

r(1  b2 )(1  a3  a2 b3 )  a2 b2 b3 
(1  b2 )(1  r  b2 )(a3  a2 b3 )  a2 b2 b3  a1  a2 b1   (1  r )a2 b1b2 
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 

a2 b2 (1  r )b3 (a1  a2 b1 )  b1b2 (1  r  a3 )
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
(1  r  b2 )(a3  a2 b3 )  a2 b2 b3
D 
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  t
(1  r )a2 b2

Y
(1  r  b2 )(1  r  a2  a2 b3 )  a2 b2 b3  t


.....(A5.1)
After substitution for a1, a2, a3, b1, b2 and b3 the constant term is undefined.
Consider the limit of each of the three terms comprising the constant as b2 approaches
1 with the above restrictions applying to the other parameters.
32
(1  b2 )( a1  a2 b1 )  a2 b1 b2
a(1  b2 )

r(1  b2 )(1  a3  a2 b2 b3 )  a2 b2 b3  rc(1  b2 )
i.e.

lim
a(1  b2 ) 0
 (undefined)
b2  1 rc(1  b2 ) 0
By L’Hopital’s Rule:
lim
lim
a(1  b2 )
a
a


b2  1 rc(1  b2 ) b2  1  rc rc
(1  b2 )(1  r  b2 )(a3  a2 b3 )  a2 b2 b3   a1  a2 b1   (1  r )a2 b1b2 
(1  b2 )(1  a3  a2 b3 )  a2 b1b2 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 

a(1  b2 )(1  r  b2 )(1  c)
c(1  b2 )(1  r  b2 )(r  c)
lim
a(1  b2 )(1  r  b2 )(1  c) 0

b2  1 c(1  b2 )(1  r  b2 )(r  c) 0
(undefined)
By L’Hopital’s Rule:
lim
a(1  c)(1  r  2b2  rb2  b22 )
a(1  c)( 2  r  2b2 )

2
b2  1 c(r  c)(1  r  2b2  rb2  b2 ) b2  1 c(r  c)( 2  r  2b2 )
lim

a(1  c)
c( r  c )
a2 b2 (1  r )b3 ( a1  a2 b1 )  b1 b2 (1  r  a3 )
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 

b22 b1
(1  b2 )(1  r  b2 )
lim
b22 b1
b2  1 (1  b2 )(1  r  b2 )

0
(undefined)
0

2b2 b1
0
b2  1 2  r  2b2
By L’Hopital’s Rule:
lim
b22 b1
b2  1 (1  b2 )(1  r  b2 )
 Constant 
lim
a a(1  c) a(1  r )


rc c(r  c) r(r  c)
33
After substitution for the parameters in the coefficients of the Dt and Yt terms we have
Pt 
a(1  r ) c (1  r )
(1  c)

Yt 
Dt
r ( r  c) r ( r  c)
( r  c)
34
Further, recall equation (A1.5),
Rt 
(1  r  b2 )(a3  a2 b3 )  a2 b2 b3
Dt

(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  Pt 1
Yt
D
(1  r )a2 b2
 t
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3  Pt 1 Pt 1
.....(A5.2)

and equation (A4.4) where
Rt 

1 (1  r  b2 )( a3  a2 b3 )  a2 b2 b3   a1  a 2 b1   (1  r )a2 b1 b2
 ( a1  a 2 b1 )  

Pt 1 
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3

(1  r  b2 )( a3  a2 b3 )  a2 b2 b3   a3  a2 b3  1  (1  r )a2 b2 b3
D

 ( a3  a2 b3 ) t 1  
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3

 Pt 1
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3 a2 b2  (1  r )a2 b2 (b2  1)
Y

 a2 b2  t 1  
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3

 Pt 1
 (1  r  b2 )(a3  a2 b3 )  a2 b2 b3
 e
 
 1 t  
 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3
 Pt 1
(1  r  b2 )( a3  a2 b3 )  a2 b2 b3 a2  (1  r )a2 b2
 

 a2  t
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3

 Pt 1
,
or
 (1  r  b2 )( a3  a2 b3 )  a2 b2 b3
 e
Rt  r  
 1 t  
 (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3
 Pt 1
 a2 (1  r  b2 )( a3  a2 b3 )  a2 b2 b3   (1  r )a2 b2
 

 a2  t
(1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3

 Pt 1
.....(A5.3)
After substitution for the parameters a1, a2, a3, b1, b2 and b3 in
(A5.1), (A5.2) and (A5.3) we have:
Rt 
D
c (1  r ) Yt (1  c) Dt

 t , and
r (r  c) Pt 1 (r  c) Pt 1 Pt 1
Rt  r 
c (1  r ) 2  t
(1  r ) et

r (r  c) Pt 1 (r  c) Pt 1
35
APPENDIX 6 LINTNER MODEL FOR DIVIDENDS AND RANDOM WALK
WITH DRIFT FOR EARNINGS
Lintner Model for Dividends :
General Generating Process
for Dividends
:
Dt  a  cYt  (1  c) Dt 1  et
Dt  a1  a2Yt  a3 Dt 1  et
Equating the above two equations results in
a1  a , a2  c , a3  (1  c)
Yt  b  Yt 1   t
Yt  b1  b2 Yt 1  b3 Dt 1   t
Random Walk with Drift for Earnings:
General Generating Process for Earnings:
Equating the above two equations results in
b1  b , b2  1 , b3  0
After substitution in (A5.1), (A5.2) and (A5.3) the constant term in (A5.1) is
undefined.
We now consider the limit of each of the three terms comprising the constant as b2
approaches 1.
i. e.
(1  b2 )( a1  a2 b1 )  a2 b1 b2
(1  b2 )( a  cb)  cb)

r(1  b2 )(1  a3  a2 b3 )  a2 b2 b3 
rc(1  b2 )
lim
(1  b2 )( a  cb)  cb cb

b2  1
rc(1  b2 )
o
(undefined)
By L’Hopital’s Rule:
lim
lim
(1  b2 )(a  cb)
a  cb a  cb


b2  1
b2  1
rc(1  b2 )
rc
rc
(1  b2 )(1  r  b2 )( a3  a2 b3 )  a2 b2 b3   a1  a2 b1   (1  r )a2 b1 b2 
(1  b2 )(1  a3  a2 b3 )  a2 b1b2  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
(1  b2 )(1  r  b2 )(1  c)(a  cb)  (1  r )cbb2 

c(1  b2 )  cbb2 (1  r  b2 )(r  c)
lim
(1  b2 )(1  r  b2 )(1  c)(a  cb)  (1  r )cbb2 
0

0
b2  1
cbr(r  c)
c(1  b2 )  cbb2 (1  r  b2 )(r  c)
36
a2 b2 (1  r )b3 (a1  a2 b1 )  b1 b2 (1  r  a3 )
(1  b2 )(1  a3  a2 b3 )  a2 b2 b3  (1  r  b2 )(1  r  a3  a2 b3 )  a2 b2 b3 
cb(r  c)b22

c(r  c)(1  r  b2 )(1  b2 )
lim
cb(r  c)b22
cb(r  c)

( undefined)
b2  1 c(r  c)(1  r  b2 )(1  b2 )
0
By L’Hopital’s Rule:
lim
2bb2
2b

b2  1 ( 2  r  2b2 )
r
 Constant 
a  cb 2b

rc
r

a  cb  2cb
rc

a  cb
rc
After substitution for the parameters in the coefficients of the Dt and Yt terms we
have:
Pt 
a  cb c (1  r )
(1  c)

Yt 
Dt
rc
r (r  c)
(r  c)
Further,
Rt 
D
c (1  r ) Yt (1  c) Dt

 t , and
r (r  c) Pt 1 (r  c) Pt 1 Pt 1
Rt  r 
c (1  r ) 2  t
(1  r ) et

r(r  c) Pt 1 (r  c) Pt 1
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APPENDIX 7 LINTNER MODEL FOR DIVIDENDS AND RANDOM WALK
WITH GROWTH FOR EARNINGS
Lintner Models for Dividends:
General Generating Process for Dividends:
Dt  a  cYt  (1  c) Dt 1  et
Dt  a1  a2Yt  a3 Dt 1  et
Equating the above two equations results in
a1  a , a2  c , a3  (1  c)
Yt  (1  g)Yt 1   t
Yt  b1  b2 Yt 1  b3 Dt 1   t
Random Walk with Growth for Earnings:
General Generating Process for Earnings:
Equating the above two equations results in
b1  0 , b2  1  g , b3  0
After substitution in (A5.1), (A5.2) and (A5.3) we have
Pt 
a(1  r ) c (1  r )(1  g)
(1  c)

Yt 
Dt
r (r  c)
(r  g)(r  c)
(r  c)
Rt 
D
(1  c) Dt c (1  r )(1  g) Yt

 t
(r  c) Pt 1
(r  g)(r  c) Pt 1 Pt 1
r
(1  r ) et
c (1  r ) 2  t

(r  c) Pt 1 (r  g)(r  c) Pt 1
38