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Some further theoretical and empirical implications regarding the relationship between earnings, dividends and stock prices
Journalof ELSEVIER
Journal of Banking & Finance 21 (1997) 17-35
BANKING & FINANCE
Some further theoretical and empirical implications regarding the relationship between earnings, dividends and stock prices Raymond Chiang a, Ian Davidson
b, John Okunev
c,,
a Chinese University ofHong Kong, Shatin, New Territories, Hong Kong b University of Warwick, Coventry, CV4 7 AL, UK c Lend Lease Investment Management, Sydney 2000, Australia
Received 10 November 1994; accepted 11 March 1996
Abstract In this paper earnings, dividends and stock prices are modelled within a plausible economic framework. The first stage in the analysis involves characterization of the dynamic behavior of earnings, evidence was found for mean reverting behavior in the long term, and weaker evidence for mean reversion in the short term. The relationship between dividends and earnings is then examined using a modified form of the Lintner model. The empirical results suggest the modified formulation performs as effectively as the original Lintner approach. Using these findings, we then develop the functional form of the corresponding share price relationship. As a consequence of using a generalized model for earnings we are able to examine theoretically, the effect of different earnings processes on share price behavior. The empirical results imply that changes in earnings per share and earnings per share are important in explaining returns. JEL classification: GI2 Keywords: Earnings; Dividends; Share pricing
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R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35
1. Introduction
There has been considerable debate recently whether stock prices have a strong predictable component, and consequently, if this is true it casts some doubt on the validity of the efficient market hypothesis. In examining this issue researchers have adopted one of two approaches, these being modelling the behavioral characteristics of share prices without regard to the underlying macroeconomic variables, and secondly identifying fundamental variables which explain movements in share prices or returns. Studies in the former category are typically identified by DeBondt and Thaler (1985), Thaler (1987), Fama and French (1988b), Poterba and Summers (1988), Campbell (1991), McQueen and Thorley (1991). These types of studies put forward the view that share prices revert back towards some fundamental value and therefore stock price movements are not random but move in such a way as to move back to fundamentals in the long term. However Kim et al. (1991) and McQueen (1992) have criticized these conclusions and suggest the conclusion of mean reversion may be overstated. The second approach is to identify variables which predict share price movements or returns. Studies by Summers (1986), Korrnendi and Lipe (1987), Fama and French (1988a), Campbell and Shiller (1988b), West (1988), Campbell and Kyle (1993) and Chiang et al. (1995) typically fall into this category. For example, Fama and French (1988a) propose dividends per share as being significant in explaining returns. Similarly, Campbell and Shiller (1988b) propose dividends per share, dividend growth and long term earnings per share as being significant in explaining returns. Each of these studies find strong predictability of returns in longer term investment horizons. There is also debate whether earnings or dividends should be used as an explanatory variable. Kormendi and Lipe (1987) and Campbell and Shiller (1988b) propose that earnings have significant explanatory power in predicting share returns. In this paper we adopt the second approach. We model share prices as being dependent upon a stochastic earnings and dividend process. Our first stage is to model earnings as a generalized mean reverting process, the proposed model capturing various combinations of possible earnings dynamics, a special case being that earnings follows a random walk. Our next step is to examine the relationship between dividend changes and earnings, and we go on to incorporate into our analysis a modified form of the Lintner (1956) hypothesis which states that a firm's target payout ratio is a constant proportion of current earnings, but that adjustment in any particular year is only partial. We then develop the functional form of the share price under the modified Lintner model, assuming that the share price is the discounted value of future expected dividends. Specific scenarios of theoretical share price movements based upon different earnings processes are examined. Of the papers cited earlier only Campbell and Kyle (1993) adopt a similar approach. In their paper they model detrended dividends as a combination of an
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 19
additive Brownian motion and Ornstein Uhlenbeck process. J They proceed by determining the functional form of the share price, but include in their model a variable which identifies the action of noise traders. Our analysis differs from theirs in many respects, in that we do not detrend the earnings or dividend processes. We model the earnings process and the relationship between earnings and dividends, whereas Campbell and Kyle do not model earnings. Furthermore, our dividend dynamics are more general than the process assumed by Campbell and Kyle (1993). In their model dividends are assumed to have a mean of zero. We also discount the future cashflows at the risk adjusted rate where as Campbell and Kyle discount by the risk free rate. We do not directly address the issue of noise traders in our analysis. To foreshadow some results we find that earnings can be represented reasonably well by the Ornstein Uhlenbeck process and find significant evidence of mean reversion towards a exponentially growing long term mean in nominal terms. Furthermore, we find that the modified form of the Lintner model performs as effectively as the Lintner model. The empirical results imply that changes in earnings per share and earnings per share are important in explaining returns. The progression of the paper follows. Section 2 contains the description of the generalized earnings process and the modified Lintner formulation, with empirical estimates of the parameters. In Section 3 we develop the corresponding share price equation based upon the assumption of the Lintner model and estimate the parameters. Section 4 outlines a possible approach for modelling the intrinsic value of the firm. The conclusions are set out in Section 5.
2. Modelling the earnings process Previous empirical research has found that earnings can be reasonably described by a random walk with drift (see Ball and Brown, 1968; Ball et al., 1976; Foster, 1977). However, in the light of recent evidence that share prices and dividends may be mean reverting, and given that there is generally a strong relationship between earnings and dividends, one might expect earnings to exhibit mean reversion also. Studies by DeBondt and Thaler (1985), Fama and French (1988a,b), Poterba and Summers (1988), Campbell and Shiller (1988b), Campbell (1991), McQueen and Thorley (1991) and Campbell and Kyle (1993) have provided evidence that returns have a strong predictable component in long term investment horizons, implying that there is mean reversion of stock prices towards to some fundamental value rather than stock prices being a random walk. Furthermore, Lintner (1956), Shiller (1981) and Fama and French (1988a) have
i Campbell and Kyle (1993) work in terms of detrended variables. That is, actual dividends are divided by the continuouslycompoundedgrowth of dividends.
R. Chiang et al./ Journal of Banking & Finance 21 (1997) 17-35
20
found evidence of mean reversion in the dividends series over the long term. However, the pattern of mean reversion in share prices and dividends is weak in the more recent time frame, but this may be due to the inability of current statistical tests to distinguish between a random walk and a weakly mean reverting process. Studies by Lothian and Taylor (1992) and A b a u f and Jorion (1990) have shown that current statistical tests have low power in distinguishing between a random walk and weak mean reversion in the short term. Given that there is evidence of mean reversion in the dividend series, earnings are modelled as a process reverting towards a time varying mean. W e suggest that the instantaneous change in earnings may be described by 2 dE(t)=
13e( a-~eekt-E(t))dt+o'ldZl(t )
(1)
where dE(t)
= = /~e = (c~//3e)e kt =
E(t)
the instantaneous change in earnings, earnings at time t, the speed of adjustment, the long term mean which grows or decays exponentially at rate
e kt, ~rl2 dzl(t)
= the variance of d E ( t ) per unit time, = a Wiener process with mean zero and unit variance.
The formulation described by Eq. (1) allows for numerous combinations of dynamic processes for earnings. For example, k > 0 and /3e > 0 reflects mean reversion towards a time varying mean which grows at a rate of e ~t. If on the other hand, k = 0 and J~e > 0, then earnings reverts to a stable long term m e a n oL//j~ e. The formulation also permits the possibility of a random walk when /3e = 0. It is apparent that Eq. (1) offers considerably more flexibility in modelling earnings than the previous models employed in accounting based empirical studies. However, care must be taken in using a continuous time process to model earnings and dividends, given that earnings and dividends are announced quarterly, even though shares are traded continuously. It is on the basis that share price represents the discounted value of expected future dividends, and that changes in share prices reflect continuously occurring changes in the market's expectations of future earnings or dividends, that it is appropriate to use the continuous time framework. 3
2 The earnings process of Eq. (1) is implicitly dependent upon dividend policy. Clearly if all earnings are paid out as dividends then this would affect the growth of earnings in the future. Thus if 3' = 1, all earnings are paid out as dividends. The growth of earnings in this situation will clearly be less than when 3' = 0. We therefore put forward the notion that the growth of earnings implicitly reflects dividend policy. We are thankful to one of the referees for bringing this point to our attention. 3 Other researchers have also modelled dividends in the continuous time framework, see Merton (1973), Campbell and Kyle (1993).
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 21
Using standard techniques the solution to Eq. (1) is given by
E(t) - ~(k + fle)
[e k t - e -/3~t ] q-
e-fetfotO'lefeSdzl(s ) .
(2)
In order to operationalize Eq. (2), we need to determine the discrete time version of changes in earnings of Eq. (2). 4 This is given by the following expression:
AE(I)
t~
--ekt(ekat--e-[3eZlt) (k-t-fie)
q-(e - f e A t - 1 ) E ( t )
+ e -fie(t+ At)ftt+ Ato'le f'Sd Z l ( S ) .
(3)
From Eq. (3) we see that changes in earnings are dependent upon a nonlinear function of time and it is therefore appropriate to use nonlinear methods to estimate the parameters a, /3e and k. Using the specification of Eq. (3), the parameters were estimated by nonlinear maximum likelihood techniques, using annual observations of the Standard & Poors Composite Price Index, with corresponding earnings and dividend price index series for the period 1871-1986. The Standard & Poors Composite Stock Price index (S & P) are January stock prices and have been extended back to 1871 using the data in Cowles (1939). The dividend and earnings series are total dividends and earnings accruing to the index for each year. 5 The parameters are estimated for the three periods 1871-1986, 1925-1986 and 1960-1986. These periods were selected as a number of studies of mean reversion have focussed on the first two sample periods. However, the majority of the studies in the accounting area have focussed on the more recent time frame from the 1960s to the present, and it is of interest to see whether the results of the more recent time frame are consistent with those of the longer term. As previously mentioned, studying the time series behavior of earnings in the more recent time frame may give quite different results to those in the long tenn. 6 Table 1 shows summary results of the nonlinear maximum likelihood estimation of the parameters c~, /3e and k. If one examines the longest term results it is apparent that earnings revert significantly towards a time dependent mean. The extent of mean reversion is approximately 32% per year and the growth of the
4 This is obtained by substituting t + At into Eq. (2) and then subtracting E(t) from E(t + ,:It). This gives the correct discrete time approximation to Eq. (2). 5 This data set has been used extensively by numerous researchers (see Campbell and Shiller, 1988a,b; West, 1988; and is similar to Shiller (1981), Mankiw et al. (1985). For further information regarding the data set see Campbell and Shiller (1988a). 6 We refer to the period 1960-1986 as the short term. This is similar to the terminology adopted in examining the behavior of real exchange rates in more recent times.
R. Chiang et al. /Journal of Banking & Finance 21 (1997) 17-35 22 Table 1 Modelling earnings using nonlinear maximum likelihood estimation a Variable
Period
o~
/3
k
Coefficient DW R2
Nominal
1871-1986
0.0119 (2.39) 0.0115 (2.02) 0.0147 (1.00)
0.324 (3.28) 0.375 (2.54) 0.371 (1.51)
0.0528 (13.76) 0.054 (11.80) 0.052 (5.00)
1.85
0.17
1.82
0.17
1.83
0.12
1925-1986 1960-1986
a Figures in parentheses are asymptotic t statistics. DW = Durbin-Watson statistic. long term mean is 5.28% per year. 7 This means that the change in earnings in the next period will move towards the time dependent mean at a rate of 32% of [ ( ~ / / 3 ) e k ' - E(t)]. For example, if earnings are above the mean, then ( a / / 3 ) e k' < E(t). The change in earnings will be negative, hence earnings will move back towards the mean, but will only move back at rate of 32% of this difference. This is referred to as partial adjustment model. Examination of the other two subperiods reveals similar estimates of all parameters. Even for the 1960-1986 period, the parameter estimates are essentially the same. However the t statistic for the speed of adjustment coefficient is not significant in the short term, suggesting that earnings follows a random walk with a drift term. Overall, the results suggest that even for the most recent sample period earnings are 'mean reverting' towards a time dependent mean. From this point on we proceed on the basis that earnings can be represented as a mean reverting process and attempt to model the relationship between earnings and dividends. 2.1. Dividends and earnings relationship Possibly the best known relationship between dividend changes and earnings is the model proposed by Lintner (1956). In this classic study Lintner suggested that a firm's dividend policy through time behaves as though the managers have in mind a target dividend, which is a fixed proportion of current earnings, but that they adjust to this target from last year's dividend to the target level gradually according to a speed of adjustment parameter. Ignoring the constant which Lintner included in his adjustment equation (this constant is generally more significant on aggregate data than at the individual form level) gives the following discrete time equation: AO(t) =/3(yE(t
+ 1) - O ( t ) ) + e ( t )
(4)
7 The average long term growth of earnings was 6.2% for the period 1871-1986. The estimation from the nonlinear regression understates this figure by 0.9%.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 23
where ADO) = D ( t + 1 ) - D(t), D(t) = the level of dividends at time t, E(t + 1) = the level of earnings at time t + 1, = the dividend speed of adjustment coefficient, /3 = the payout ratio, 3' = the residual term. e(t) We adopt a similar approach to Lintner, but instead define the formulation in continuous time. The process for the instantaneous change in dividends is taken to be 8 d D ( t ) =/3( 3/E(t) - D( t ) ) d t + o-2dz2(t) (5) where the terms are defined in a similar manner to those of Eq. (1). Eq. (5) is solved under the assumption that earnings are mean reverting (or, as a special case, follow a random walk). We determine the appropriate discrete time approximation to Eq. (5) and show that the ensuing discrete time regression equation is consistent with the model proposed by Lintner in Eq. (4).Solving Eq. (5) yields the dividend process:
Y/3 E( t) - ~3//3(ekt-e-t~') e-~t Ctcr2e~Sdz2( O ( t ) = ( /3 -- /3e ) ( /3 + k ) ( /3 - /3e ) + 3//3e - ~ t ( "~ _--~-e) fore t3~O'ldz, ( s).
(6)
Eq. (6) implies that dividends at time t are a function of (i) the current earnings level, (ii) a time dependent term which reflects the growth of earnings, and also (iii) the difference between the accumulation of stochastic shocks to the earnings and dividend processes. The corresponding discrete time approximation to Eq. (6) is given by
AD(t)
3//3
A E ( t ) + (e -t3 - 1 ) O ( t )
([3--/3e)
(/3-/3~)
E(t)
( [~ -- /3e )
- aY/3ek'(et-e-~) ( /3 q'- k ) ( / 3 - ~e) T/3e-/3 ( t + l )
3//3(e -¢ - 1)
+e
#(t+ l) f t + l c r 2 e ~ S d z 2 ( s)
rt+ 1 fls ] e crld Zl(S )
J,
(7)
at
Eq. (7) represents a generalized form of the Lintner proposition and is 8 Eq. (5) represents the change in dividends from t to t + dt. However, we show at a later stage that the discrete time approximation of Eq. (5) is consistent with the Lintner model. The reason we have adopted the alternative form of Eq. (5) is that it is consistent with the earnings formulation, and also it is easier to solve the dividend process when earnings and dividends are observed at the same point in time. In the Lintner model earnings and dividends are observed at different points in time.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 24 Table 2 Modelling nominal changes in dividends using AD(t)= 3'0 + 3,1 AE(t)+ 3"2E(t)+ 3,3D(t)+ e(t)
Period
Variable
1871-1986 1871-1986 1925-1986 1925-1986 1960-1986 1960-1986
Constant
AE(t)
0.0126 (1.33) b 0.012 (1.32) 0.020 (1.13) 0.019 (1.13) 0.016 (0.36) 0.011 (0.29)
0.088 (7.81)
E(t + 1)
E(t)
D(t)
fl a
3"
R2
0.092 (8.25)
-0.145 (-5.88) -0.14 ( - 7.40) -0.144 (-4.44) -0.14 (-5.67) -0.11 ( - 3.44) -0.10 (-4.00)
0.157
0.63
0.76
0.14
0.64
0.76
0.155
0.63
0.75
0.14
0.63
0.75
0.116
0.66
0.77
0.10
0.73
0.78
0.09 (10.8) 0.086 (6.06)
0.091 (6.33) 0.088 (8.41)
0.069 (4.46)
0.077 (5.07) 0.073 (6.69)
Beta was calculated using the following formula fl = -log(1 b Figures in parentheses are t statistics. a
+
3,3).
applicable whether earnings are mean reverting or follow a random walk. Furthermore we can be modify Eq. (7) by removing the growth term of earnings. This is accomplished by substituting for e kt in terms of AE(t), E(t) and the residual term from Eq. (3). This produces the following regression equation: 9
AD( t) = YlAE( t) + TeE(t) +
Y3O(t) + e( t).
(8)
Table 2 contains summary results of the estimation of Eq. (8), and also the Lintner reduced form Eq. (4) for the three periods. It is immediately apparent that the two formulations generate very similar results. Furthermore it is evident that the three variables AE(t), E(t) and D(t) generally capture at least 75% of the variation in dividend changes and supports the view that both earnings and dividends are mean reverting in the long and short term. It is also interesting to note that the speed of adjustment of the dividend process appears to be quite stable across the differing time periods and is of the order of 11% to 15% per year. The implied payout ratios from the two different formulations are also included in Table 2. The results reveal that the payout ratio is of the order of 60-69% and is reasonably stable across the sample time periods, and across the two different models. 10 The validity of the t tests can be called into question if the residuals are heteroskedastic or serially dependent, so we conducted the Breusch and Pagan test
9 Eq. (8) is consistent with the Lintner formulation of Eq. (4). This is obtained by adding and subtracting 3"E(t) to the term enclosed in mean reversion portion. This produces D(t + At)-- D(t) = fl(yE(t + 1 ) - 3,E(t)+ 3,E(t)- D(t))+ e(t). This reduces to D(t + At)-- D(t) = fl(y(E(t + 1 ) E(t))+ 3"E(t)- D(t))+ e(t), which is essentially the same as Eq. (8). 10 The payout figure is consistent with those obtained by Lintner (1956).
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35
25
for heteroskedasticity, and the t test and L a n g r a n g e Multiplier test for serial correlation. No evidence of heteroskedasticity was found, b u t on testing up to 23 lags there was slight evidence of serial correlation which occurred at lag1 for all sample periods. ~1 The evidence was only minor, and we therefore conclude that changes in dividends can be reasonably described b y Eq. (8), and that the process is homoskedastic and essentially serially i n d e p e n d e n t for the time frames considered in this analysis. 12 Proceeding on the basis that changes in dividends can be described as a m e a n reverting process of the form outlined above the corresponding share price formulation will n o w be developed.
3. Modelling stock prices The share price is modelled as the discounted value of future expected dividends. For the sake of simplicity it is assumed that the long term cost of equity is constant and therefore the discounted value of expected future dividends is given by
P ( t) = E( f t ~ e - i ( S - ° D ( s ) d s )
(9)
where P ( t ) is the current share price at time t, i is the risk adjusted discount rate (cost of capital) applied to r a n d o m rate of dividends D(s) received at time s, and E is the expectation operator at time t. U n d e r the a s s u m p t i o n that dividends can be described by the process of Eq. (5) and substituting for D(s), the solution of Eq. (9) is g i v e n b y 13
P(t) =
y a f l e k' (t~+k)(i-k)(i+fle) Tfl ot e - ¢3t (fl+k)(i+fl)(i+fl,)
O( t ) + - (i+fle) "
(10)
11 Using GLS to correct for serial correlation did not alter the results significantly. ~2Alternatively we modelled changes in dividends towards long term sustainable earnings as suggested by Marsh and Merton (1987). Where sustainable earnings is captured by the deterministic time trend of the earnings process. The empirical results from these regressions did not offer any improved explanatory power (R 2 = 51%) over the Lintner model. These results are available from the authors on request. 13 Eq. (10) contains the payout ratio y and on initial inspection it appears that the share price can be increased by increasing the payout ratio, which is contrary to the Modigliani and Miller (1961) (MM) concept of dividend policy irrelevancy. However this is not the case. This is because as the payout ratio increases the growth of earnings correspondingly decreases which offsets any increase in the payout ratio. For example it can be shown from Eq. (10) that there exists a relationship between the payout ratio and the growth rate of earnings which is consistent with dividend policy irrelevancy. In the case where no dividends are paid the share price is given by P(t) = aekt/[(i -- k)(i + fie)]+ E(t)/(i + fie)This is obtained by assuming that the share price is the discounted value of expected future earnings.
R. Chiang et aL / Journal of Banking & Finance 21 (1997) 17-35 26
Eq. (10) is a generalized expression for the share price which is flexible enough to incorporate assumptions such as mean reversion on the part of earnings and dividends. This expression is similar to that derived by Campbell and Kyle (1993). Dividing Eq. (10) through by e kt yields a similar expression to Eq. (2.8) in Campbell and Kyle if one removes the effects of noise traders. However Eq. (10) is different in that dividends are discounted by the long term risk adjusted cost of equity whereas in the Campbell and Kyle model they discount at the riskless rate of interest, and also contains different parameters reflecting the earnings process. It is also consistent with other specifications presented in the literature - for example, the Lang (1991) share price model (in his formulation the share price is the discounted value of future expected earnings where earnings are a random walk), can be generated by substituting /3~ = 0 into Eq. (10) to give the share price when earnings follow a random walk with exponential growth: 14 ya/3e k' P(t)
=
(~+k)(i-k)i
-
yo~e -~' ([3+k)(i+~)i
Note that substituting for e kt from Eq. (2) with form of the share price:
D( t) i
+ - -
/~e =
0 yields the following
Y~ [ (i-k) ~ P(t) = ( ~ + kY~ ) ( i - k ) E( t) + i(~+k) + (/3+k)(i+/3)
i(~+k)
+ -e- ~tfotO.2et3Sdz2(s) i
(i
k)
(11)
1] 1
o',dZl(S)
ye-~tfotO.le~sdZl(S). i
(12)
It is evident from Eq. (12) that the Lang specification is essentially the same in structure with regards to the earnings variable and the constant term, but it omits the other two terms. The negative exponential term will probably have little effect in the long term: however the accumulation of stochastic shocks may not be small. The difference between the two formulations is due to the difference in modelling the expected growth in earnings. Lang formulates his model with constant expected growth, whereas we have modelled expected growth as an exponential term, and also incorporated a modified form of the Lintner model to reflect dividend policy. Our specification receives empirical support from the empirical results in Table 1, which provides evidence that k is significantly different from zero.
14 I f / 3 e = k = 0, that is d E ( t ) = a d t + o-dz, then the share price is given by P(t) = E ( t ) / i + a / i 2. This formulation is essentially the same as derived by Lang (1991).
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35
27
Continuing on the basis that earnings are not a random walk and adopting Eq. (10) the discrete time change in the share price is given by 15 3"a/3ek'(e k - 1) ziP(t)
=
(~+k)(i-k)(i+¢l,)
AD(t) + -
-
(i+~)
y / 3 a e - / 3 t ( e - g - 1) -
(13)
(~+k)(i+~)(i+13~)"
However one can develop an alternative formulation of Eq. (13) by incorporating changes in earnings and lagged earnings. This is accomplished by removing a e ~' from Eq. (13) by using Eq. (3). A p ( t ) = T, a E ( t )
+ 3'2 E ( , ) + 3'3 a D ( t )
+ y4 e - e ' + e ( t )
(14)
where
,//3(k +/3e) (e k - 1) 3'1 =
( ~ + k)(i-
k)(i + ~e)(e k- e-~e) '
+ t3e)(e k 3,2=-
1)
( ~ + k)( i_ k)( i + ~e)(ek_e_t~,)
,
1 3'3 = ( i + 3"13a(e -~ - 1) 3"4 = -- ( [3 + k ) ( i + [3 ) ( i + [3e ) "
The formulation of Eq. (14) is similar in structure to that proposed by Ohlson (1991). 16 From Eq. (14) it is evident that both changes in earnings and dividends and lagged earnings levels are relevant in explaining stock price changes. J7 The majority of empirical studies do not focus on share price changes, but rather focus upon returns. Eq. (14) can easily be modified to reflect returns by
~5It is evident that E(t), D(t) and P(t) are nonstationary as the mean and variance are time dependent. However, the variance of changes in E(t), D(t) and P(t) are time independent and can be regarded as trend stationary. This would indicate E(t), D(t) and P(t) are I(1). Campbell and Kyle (1993) have shown that real detrended D(t) and P(t) are cointegrated. 16The Ohlson (1991) model can he described as P( t ) = el)l E( t ) + 42 B V( t ) + 4)3D( t ). Where B V( t ) is the book value at time t. Therefore changes in the share price can be expressed as P(t + 1)- P(t) = 41(E(t + 1)- E(t))+ 42(BV(t + 1)- BV(t))+ 43(D(t + 1)- D(t)). Incorporating the clean surplus relationship modifies the above to Ap(t) = 41 AE(t) + 42( E(t + 1) - D(t + 1)) + 43 AD(t). Note this expression is slightly different to that proposed in Eq. (15). 17 It can be shown that e(t) is homoskedastic and serially independent, and therefore estimation is efficiently estimated using ordinary least squares.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 28
adding dividends paid at (t + 1) and dividing through by P(t). is The appropriate return equation becomes 19 ,aE(t)
R(t+ l ) = y l - P(t)
E(t)
AD(t)
- { - ~ 2 P - - - - ~ -'~'-~3 P ( / ' ~ [ - ~ 4 P ~
+E(t)
e -t3t
D(t+ 1) --~- P(t)
(15)
where R ( t + 1) = -
AP(t) + D(t + 1) P(t)
e(t) e(t)
Table 3 presents parameter estimates of Eq. (15) without the e -t3' term. This term was found to offer no explanatory power. We have also included regression equations which relate to the earnings process (that is, changes in earnings and earnings per share), and the dividend process (dividends per share). For the dividend process we did not include changes in dividends per share, as this term offered little explanatory power. The reason we have included these additional regressions is to see whether earnings dominates dividends in explaining returns. It is evident that for the period 1871-1986, both changes in earnings and lagged earnings are significant in explaining returns. Whereas for the period 1925-1986 no variable is significant in explaining returns. This result may be due to multicollinearity. When comparing the earnings and dividend regressions it appears that both earnings and dividend processes offer similar explanatory power. When considering all variables the R 2 value of 11% is marginally greater than the explanatory power obtained by Fama and French (1988a) (R 2 = 7%) for the period 1927-1986.
18 Eq. (15) is essentially the same as that derived by Chiang et al. (1995), the only term which differs is the time dependent term and the change in dividends per share term, otherwise the regression equation is identical. The expression is also similar to the formulation of Campbell and Shiller (1988b). They model real returns as being dependent upon dividends per share, dividend growth and long term earnings per share. It is apparent that the two are essentially the same if the change in earnings per share variable is not significant. The approaches adopted in each of the two papers mentioned above were different to the one proposed here. In the Chiang et al. (1995) paper they model mean reversion of asset prices towards the fundamental value, and in order to operationalize their approach they assume that the fundamental value is a function of current level of earnings or dividends. In the Campbell and Shiller (1988b) analysis they model retums as being dependent upon long term earnings per share, dividends per share and changes in dividends. Neither paper models the earnings or dividends processes and subsequently does not develop the functional form of asset prices to earnings or dividends. 19 For empirical estimation we inserted a constant term Y0, though from Eq. (14) this is not required. We desired to test whether the constant term was significantly different from zero.
R. Chiang et al./Journal of Banking & Finance 21 (1997) 17-35 29 Table 3 Modelling returns using the following formulation:
aE(t) Period
Coefficient To
1871-1986
-0.07 ( - 1.00) a - 0.04 (-0.75) -0.146 (-0.63) -0.12 ( - 1.19) - 0.053 ( - 0.68) - 0.12 ( - 1.22) 0.44 (-2.86) 0.023 (0.48) * - 0.07 ( - 1.03) ~
1925-1986
1960-1986
E(t) +
TI
,aDO)
O(t +1)
Y2 3.08 (3.08) 2.76 (3.32)
0.86 (0.40) 1.99 (1.26)
-4.68 (-1.57) - 5.87 ( - 2.49)
1.31 (1.66) 1.68 (2.64)
0.68 (0.46) 2.07 (2.28)
-7.71 (-2.25) 1.36 (2.26)
3'3
"Y4
R2
- 1.50 ( - 0.61)
1.24 (0.61)
0.11 0.105
1.96 (0.48)
2.96 (2.05) 3.73 (1.38)
0.036 0.11 0.093
- 19.78 (-0.88)
5.03 (2.56) 30.30 (3.12)
0.097 0.46 0.43
4.32 (2.51)
0.34
a Figures in parentheses are t statistics. * Corrected for serial correlation.
The results from 1960-1986 fare reasonably well when considered in the light of previous empirical studies for this period. For example, Easton and Harris (1991) obtain R 2 values of the order of 7.7% (this is the average for the entire sample period 1968-1986). However their regressions only incorporated changes in earnings per share and earnings per share. When considering only these two variables our data resulted in an R 2 of 23% (not corrected for serial correlation), these results being consistent with those obtained by Ali and Zarowin (1992) (R 2 = 19%). Closer inspection of Table 4, however, reveals the puzzling nature of the coefficients, in that the change in earnings and earnings levels appear to have a negative influence on returns. One would expect that as earnings increased this should translate to increases in share prices and hence positive returns. Therefore changes in earnings and earnings levels should have positive coefficients. Tests for heteroskedasticity and serial correlation revealed little evidence of it for the two longer sample periods. However, there was considerable evidence of both in the short term results, and these influences may contribute to the negative signs of the earnings variables. Adjustments for serial correlation and heteroskedasticity produced the same signs in the coefficients for the sample period 1960-1986. However, we could not completely remove the effects of serial correlation. The
R. Chiang et al./ Journal of Banking & Finance 21 (1997) 17-35 3o Table 4 Summary of studies cited for returns Period Author
Nominal independent variables
R2
1969-1985 1969-1986 1927-1986 1871-1986
All and Zarowin ( 1 9 9 2 ) Eastonand Harris (1991) Fama and French (1988a) Chianget al. (1995)
JE(t)/P(t), E(t + 1)/P(t) AE(t)/P(t), E(t + l)/P(t) D(t + 1)/P(t) AE(t)/P(t), E(t)/P(t) and D(t + 1)/P(t)
0.19 0.08 0.07 0.11
Period
Author
Real independent variables
R2
1927-1986 Fama and French (1988a) D(t + 1)/P(t) 0.07 1900-1986 Campbelland Shiller (1988b) E(t + 1)l / P ( t ) a, D(t + 1)/P(t) and /~D(t) 0.076 1871-1986 Chianget al. (1995) AE(t)/P(t), E(t)/P(t) and D(t+ 1)/P(t) 0.05 E(t + 1)~ is the average of the previous 30 year earnings. earnings regression produced similar explanatory power to the regression containing all variables. The dividend model produced marginally less explanatory power. The parameter estimates reported for the earnings and dividend regression were corrected for serial correlation and heteroskedasticity. The results in the longer term ( 1 8 7 1 - 1 9 8 6 ) are consistent with dividend policy irrelevance. For the other two sample periods the earnings process offers at least as much explanatory power as dividends and appears to be consistent with dividend policy irrelevance. Table 4 depicts a summary of the dependent variables and time frames used in the studies cited above.
4. Modelling the fundamental value Alternatively it is possible to model the fundamental value by using Eq. (10). That is the deterministic portion of the fundamental value may be defined as by 20
ElF(t)]---
T a r e kt (f+k)(i_k)(i+fle)
D(t) + ~( i + ~. g e
(16)
This formulation relates the share price to the discounted value of expected future dividends and could be reasonably regarded as reflecting the fundamental value of the stock. In examining the parameters of Eq. (16) a , f , fie and k can be estimated from the dividends and earnings processes, and E ( t ) and D ( t ) are known at time t. The only variable which is not known is the cost of capital i. This discount rate may not be the same as the rate applied by the market to the stock index. The discount rate will be identical if the observed market price is the
20 We deleted the negative exponential term from Eq. (10) as this term had very little effect on the empirical results that follow.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 31
same as the fundamental value. However the difficulty of estimating the appropriate discount rate when the fundamental value and the observed stock price differ can be overcome by assuming that on average the percentage mispricing between the fundamental value and the market price is zero ( P ( t ) - F ( t ) ) / F ( t ) . Using an iterative procedure we estimated the discount rate until the average mispricing equalled zero. The implied cost of equity capital was 7.1%. This estimate appears to be low as the average return on the S &P inclusive of dividends for the period 1871-1986 is approximately 10.2%. We now proceed to model mean reversion of the stock price towards the fundamental value. One possible approach may be described by the following process:
P(t)
= [V(t)] Z'eL°<'>-¢~2+z'~z++~'-z+
(17)
where
~(t)
is a mean reverting process governed by d a ( t ) = - A a ( t ) d t + 6d z(t), is the speed of adjustment coefficient, is the standard deviation of d a(t), dz(t) is a Wiener process with mean zero and unit variance, is a parameter determining the non-linear nature of mean reversion, t3+ o- and p are parameters of the fundamental value process that will be defined below. A 6
The attractive features of the process described by Eq. (17) are that P(t) is always non-negative and that the variance of changes in the stock price are a function of the level of stock price. Furthermore, c~(t) determines the extent of mean reversion towards the fundamental value. We also allow for the possibility that the relation between the stock price and the fundamental value may be non-linear. This is reflected by the /37 term. If /3/ is not significantly different from one this yields a linear relation between P(t) and F(t). To completely capture the process of Eq. (17) we need to specify some type of diffusion process of the fundamental value. We make the assumption that the fundamental value may be described by a Geometric Brownian motion and is given by dF(t) - -/xdt+ F(,)
o-dw(t)
(18)
where
dF(t)/F(t) tx o
dw(t) P
= = an = = =
the return of the fundamental value, the instantaneous drift term and is given by Eq. (16), as this is expectation at time t, the standard deviation per unit time of the return, a Wiener process with mean zero and unit variance, and the correlation coefficient between dw(t) and d z(t).
R. Chianget al./Journal of Banking& Finance21 (1997)17-35
32
Applying Ito's lemma to Eq. (17) yields the instantaneous return of the stock price relative to the fundamental value. This is given by the following: dP(t) P--~
[ --
J~fl[~ --
Alog P ( t ) ]dt+6dz(t)+Ctfo'dw(t). F ( t ) ps]
(19)
In adopting the process described by Eq. (19) we note that the instantaneous return in the stock price is dependent upon how far the current stock price is from the fundamental value. The greater the deviation from the fundamental value the greater the economic forces will drive the stock price towards to the fundamental value. Economic forces are constantly affecting the fundamental value which is reflected in our stochastic modeling of this rate. The speed of reversion is not only dependent upon the magnitude of the difference between log P(t) and log F(t), but also on the speed of adjustment coefficient, A. The larger the value of the speed of adjustment the stronger the restoring force back to equilibrium. If the speed of adjustment coefficient is small, then mean reversion towards fundamental value may be quite slow, even though deviations from fundamental value may be large. In using discrete time data it is necessary to transform the continuous time process of Eq. (17) into a discrete time formulation. This is accomplished by substituting t + At for t into Eq. (17) then subtracting P(t). The discrete time change in the stock price is given by log
P(t + at) P(t) = (e - A a t - 1)
+/3/1o8
2A
F(t + m) F(t) +(e Aat--1)[l°gP(t)--Jgfl°gF(t)]
+ e- x(, +,~t)f/+ ateXS~5d z(S).
(20)
We note from Eq. (20) that the change in the stock price is not only a function of the change in the fundamental value, but also is a function of the difference between log P(t) and log F(t) which has a speed of adjustment coefficient of (e -hat- 1). This means that the change in the stock price is not only mean reverting towards the fundamental value, but it is also dependent upon the change in the fundamental value. Note that for mean reversion to exist we must have A > 0. Eq. (20) is efficiently estimated by ordinary least squares as the residual term is homoskedastic. Table 5 presents summary results of the regression Eq. (20). It is evident that ~ix is not significantly different from 1, and hence supports the hypothesis of a
R. Chiang et al./ Journal of Banking & Finance 21 (1997) 17-35 33 Table 5 Returns as a function of the fundamental value P(t+I) log
P(t)
F(t+l)
Y°+yll°g~
- +y21°gP(t)+ y31ogF(t)+e(t)
Coefficient Variable
3'0
Yl
Y2
"/3
h
R2
0.047 (0.97) a
0.93 (1.28)
-- 0.09 ( - 2.16)
0.073 (2.50)
0.10
0.06
a Figures in parentheses are t statistics.
linear relationship between the P(t) and F(t). The speed of adjustment coefficient A is significant at the 5% level and illustrates that mean reversion towards the fundamental value occurs at a rate of 10% per year. This means that deviations from the fundamental value can exist for prolonged periods. The significance of the t tests can called into question if there exists significant evidence of heteroskedasticity or serial correlation in the residuals. An Arch test and Breusch and Pagan test were used to test for the presence of heteroskedasticity. Neither test detected any significant evidence of heteroskedasticity. A Lagrange Multiplier test revealed little evidence of serial correlation, and we can reasonably conclude that the residuals are homoskedastic and serially independent. This is accordance with the model put forward in Eq. (20).
5. Conclusion
In this paper we have endeavored to model earnings, dividends and stock prices within a viable economic framework. Our starting point was to model the earnings process and to proceed to investigate it empirically, the main finding being that there existed significant long term mean reversion with some weaker evidence of short term mean reversion. The parameter estimates of the earnings process appear to be reasonably stable within the three periods considered, justifying our conclusion that earnings are mean reverting towards a time varying mean. These results are consistent with the results obtained by other researchers when modelling returns in the long term. We then examined the relationship between dividends and earnings and incorporated a modified form of the Lintner model. The empirical results revealed that the modified formulation performed as effectively as the original Lintner approach. Furthermore, the payout ratio of approximately 64%, and appears to be reasonably stable across the three sample periods. Proceeding on the basis the that the share price is the discounted value of future expected dividends, and that dividends are generated by the modified Lintner model, we developed the functional form of the corresponding share price
R. Chiang et al./Journal of Banking & Finance 21 (1997) 17-35
34 equation. W e e x a m i n e d different scenarios for the earnings process and found that our results were consistent with L a n g (1991) w h e n earnings are a r a n d o m walk. Estimation of our formulation suggested that changes in earnings per share and earnings per share levels are important in explaining returns, w h i c h supports d i v i d e n d policy irrelevance. The results in the longer term sample periods are similar to those of other empirical studies, h o w e v e r the results in the short term present i m p r o v e d explanatory p o w e r to other studies considering this period.
Acknowledgements W e w o u l d like to thank the two a n o n y m o u s referees for their constructive c o m m e n t s , and w o u l d also like to thank Stuart H o d g e s and the seminar participants at Strathclyde University, Lancaster University, U n i v e r s i t y o f W a r w i c k , U n i v e r s i t y of T e c h n o l o g y and the E u r o p e a n F i n a n c e C o n f e r e n c e for their valuable c o m m e n t s . W e are also grateful to John C a m p b e l l for p r o v i d i n g the data.
References Abanf, N. and P. Jorion, 1990, Purchasing power parity in the long term, Journal of Finance 45, 157-174. Ali, A. and Zarowin, 1992, The role of earnings levels in annual earnings-returns studies, Journal of Accounting Research 30, no. 2, 286-296. Ball, R. and P. Brown, 1968, An empirical evaluation of accounting income numbers, Journal of Accounting Research 6, 159-178. Ball, R., B. Lev and R. Watts, 1976, Income variation and balance sheet compositions, Journal of Accounting Research, 1-9. Campbell, J.Y. and J., Shiller, 1988a, The dividend-price ratio and expectations of future dividends and discount factors, Review of Financial Studies 1, 195-228. Campbell, J.Y. and R.J. Shiller, 1988b, Stock prices, earnings and expected dividends, Journal of Finance 43, 661-676. Campbell, J.Y., 1991, A variance decomposition of stock returns, The Economic Journal 101, 157-179. Campbell, J.Y. and A.S. Kyle, 1993, Smart money, noise trading and stock price behavior, Review of Economic Studies 60, 1-34. Chiang, R., P. Liu and J. Okunev, 1995, Modelling mean reversion of asset prices towards their fundamental value, Journal of Banking and Finance 19, 1327-1340. Cowles, A., 1939, Common stock indexes, 2rid ed. (Principia Press, Bloomington, IN). DeBondt, W.F. and R. Thaler, 1985, Does the stock market overreact?, Journal of Finance 40, 793-805. Easton, P. and T. Harris, 1991, Earnings as an explanatory variable for returns, Journal of Accounting Research, 19-36. Fama, E.F. and K.R. French, 1988a, Dividend yields and expected stock returns, Journal of Financial Economics 22, 3-25. Fama, E.F. and K.R. French, 1988b, Permanent and transitory components of stock prices, Journal of political economy 96, 246-273.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 35 Foster, G., 1977, Quarterly accounting data: Time series properties and predictive ability, Accounting Review, 1 21. Kim, M., C. Nelson and R. Startz, 1991, Mean reversion in stock prices: A reappraisal of the empirical evidence, Review of Economics Studies 58, 515-528. Kormendi, R~ and R. Lipe, 1987, Earnings innovations, earnings, persistence, and stock returns, Journal of Business 60, 323-345. Lang, M., 1991, Time-varying stock price response to earninsg induced by uncertainty about the time series process of earnings, Journal of Accounting Research 29, 229-257. Lintner, J., 1956, Distribution of incomes of corporations among dividends, retained earnings, and taxes, American Economic Review 61, 97-113. Lothian, J. and M. Taylor, 1992, Real exchange rate behaviour: The recent float from a perspective of the past two centuries, Working paper (Fordham University, New York). Mankiw, N., D. Romer and M. Shapiro, 1985, An unbiased reexamination of stock market volatility, Journal of Finance 40, 677-687. Marsh, T.A. and R.C. Merton, 1987, Dividend behavior for aggregate stock markets, Journal of Business 60, 1-40. McQueen, G. and S. Thorley, 1991, Are stock returns predictable? A test using Markov chains, Journal of Finance 46, 239-163. McQueen, G., 1992, Long term mean reverting stock prices revisited, Journal of Financial and Quantitative Analysis 27, 1-18. Merton, R., 1973, The theory of rational option pricing, Bell Journal of Management 4, 141-183. Modigliani, F. and M. Miller, 1961, Dividend policy, growth and the valuation of shares, Journal of Business 34, 411-433. Ohlson, J., 1991, Earnings, book value, and dividends in security valuation, Working paper (Columbia University, New York). Poterba, J.M. and L.H. Summers, 1988, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics 22, 27-60. Shiller, R.J., 1981, Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review 71,421-436. Summers, L.H., 1986, Does the stock market rationally reflect fundamental values?, Journal of Finance 41,591-601. Thaler (1987): Author, please complete. West, K.D., 1988, Bubbles, fads and stock price volatility tests: A partial view, Journal of Finance 43, 619-639.
Journal of Banking & Finance 21 (1997) 17-35
BANKING & FINANCE
Some further theoretical and empirical implications regarding the relationship between earnings, dividends and stock prices Raymond Chiang a, Ian Davidson
b, John Okunev
c,,
a Chinese University ofHong Kong, Shatin, New Territories, Hong Kong b University of Warwick, Coventry, CV4 7 AL, UK c Lend Lease Investment Management, Sydney 2000, Australia
Received 10 November 1994; accepted 11 March 1996
Abstract In this paper earnings, dividends and stock prices are modelled within a plausible economic framework. The first stage in the analysis involves characterization of the dynamic behavior of earnings, evidence was found for mean reverting behavior in the long term, and weaker evidence for mean reversion in the short term. The relationship between dividends and earnings is then examined using a modified form of the Lintner model. The empirical results suggest the modified formulation performs as effectively as the original Lintner approach. Using these findings, we then develop the functional form of the corresponding share price relationship. As a consequence of using a generalized model for earnings we are able to examine theoretically, the effect of different earnings processes on share price behavior. The empirical results imply that changes in earnings per share and earnings per share are important in explaining returns. JEL classification: GI2 Keywords: Earnings; Dividends; Share pricing
* Fax: (+61) 2-92321850. 0378-4266/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S037 8-4266(96)00029-5
18
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35
1. Introduction
There has been considerable debate recently whether stock prices have a strong predictable component, and consequently, if this is true it casts some doubt on the validity of the efficient market hypothesis. In examining this issue researchers have adopted one of two approaches, these being modelling the behavioral characteristics of share prices without regard to the underlying macroeconomic variables, and secondly identifying fundamental variables which explain movements in share prices or returns. Studies in the former category are typically identified by DeBondt and Thaler (1985), Thaler (1987), Fama and French (1988b), Poterba and Summers (1988), Campbell (1991), McQueen and Thorley (1991). These types of studies put forward the view that share prices revert back towards some fundamental value and therefore stock price movements are not random but move in such a way as to move back to fundamentals in the long term. However Kim et al. (1991) and McQueen (1992) have criticized these conclusions and suggest the conclusion of mean reversion may be overstated. The second approach is to identify variables which predict share price movements or returns. Studies by Summers (1986), Korrnendi and Lipe (1987), Fama and French (1988a), Campbell and Shiller (1988b), West (1988), Campbell and Kyle (1993) and Chiang et al. (1995) typically fall into this category. For example, Fama and French (1988a) propose dividends per share as being significant in explaining returns. Similarly, Campbell and Shiller (1988b) propose dividends per share, dividend growth and long term earnings per share as being significant in explaining returns. Each of these studies find strong predictability of returns in longer term investment horizons. There is also debate whether earnings or dividends should be used as an explanatory variable. Kormendi and Lipe (1987) and Campbell and Shiller (1988b) propose that earnings have significant explanatory power in predicting share returns. In this paper we adopt the second approach. We model share prices as being dependent upon a stochastic earnings and dividend process. Our first stage is to model earnings as a generalized mean reverting process, the proposed model capturing various combinations of possible earnings dynamics, a special case being that earnings follows a random walk. Our next step is to examine the relationship between dividend changes and earnings, and we go on to incorporate into our analysis a modified form of the Lintner (1956) hypothesis which states that a firm's target payout ratio is a constant proportion of current earnings, but that adjustment in any particular year is only partial. We then develop the functional form of the share price under the modified Lintner model, assuming that the share price is the discounted value of future expected dividends. Specific scenarios of theoretical share price movements based upon different earnings processes are examined. Of the papers cited earlier only Campbell and Kyle (1993) adopt a similar approach. In their paper they model detrended dividends as a combination of an
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 19
additive Brownian motion and Ornstein Uhlenbeck process. J They proceed by determining the functional form of the share price, but include in their model a variable which identifies the action of noise traders. Our analysis differs from theirs in many respects, in that we do not detrend the earnings or dividend processes. We model the earnings process and the relationship between earnings and dividends, whereas Campbell and Kyle do not model earnings. Furthermore, our dividend dynamics are more general than the process assumed by Campbell and Kyle (1993). In their model dividends are assumed to have a mean of zero. We also discount the future cashflows at the risk adjusted rate where as Campbell and Kyle discount by the risk free rate. We do not directly address the issue of noise traders in our analysis. To foreshadow some results we find that earnings can be represented reasonably well by the Ornstein Uhlenbeck process and find significant evidence of mean reversion towards a exponentially growing long term mean in nominal terms. Furthermore, we find that the modified form of the Lintner model performs as effectively as the Lintner model. The empirical results imply that changes in earnings per share and earnings per share are important in explaining returns. The progression of the paper follows. Section 2 contains the description of the generalized earnings process and the modified Lintner formulation, with empirical estimates of the parameters. In Section 3 we develop the corresponding share price equation based upon the assumption of the Lintner model and estimate the parameters. Section 4 outlines a possible approach for modelling the intrinsic value of the firm. The conclusions are set out in Section 5.
2. Modelling the earnings process Previous empirical research has found that earnings can be reasonably described by a random walk with drift (see Ball and Brown, 1968; Ball et al., 1976; Foster, 1977). However, in the light of recent evidence that share prices and dividends may be mean reverting, and given that there is generally a strong relationship between earnings and dividends, one might expect earnings to exhibit mean reversion also. Studies by DeBondt and Thaler (1985), Fama and French (1988a,b), Poterba and Summers (1988), Campbell and Shiller (1988b), Campbell (1991), McQueen and Thorley (1991) and Campbell and Kyle (1993) have provided evidence that returns have a strong predictable component in long term investment horizons, implying that there is mean reversion of stock prices towards to some fundamental value rather than stock prices being a random walk. Furthermore, Lintner (1956), Shiller (1981) and Fama and French (1988a) have
i Campbell and Kyle (1993) work in terms of detrended variables. That is, actual dividends are divided by the continuouslycompoundedgrowth of dividends.
R. Chiang et al./ Journal of Banking & Finance 21 (1997) 17-35
20
found evidence of mean reversion in the dividends series over the long term. However, the pattern of mean reversion in share prices and dividends is weak in the more recent time frame, but this may be due to the inability of current statistical tests to distinguish between a random walk and a weakly mean reverting process. Studies by Lothian and Taylor (1992) and A b a u f and Jorion (1990) have shown that current statistical tests have low power in distinguishing between a random walk and weak mean reversion in the short term. Given that there is evidence of mean reversion in the dividend series, earnings are modelled as a process reverting towards a time varying mean. W e suggest that the instantaneous change in earnings may be described by 2 dE(t)=
13e( a-~eekt-E(t))dt+o'ldZl(t )
(1)
where dE(t)
= = /~e = (c~//3e)e kt =
E(t)
the instantaneous change in earnings, earnings at time t, the speed of adjustment, the long term mean which grows or decays exponentially at rate
e kt, ~rl2 dzl(t)
= the variance of d E ( t ) per unit time, = a Wiener process with mean zero and unit variance.
The formulation described by Eq. (1) allows for numerous combinations of dynamic processes for earnings. For example, k > 0 and /3e > 0 reflects mean reversion towards a time varying mean which grows at a rate of e ~t. If on the other hand, k = 0 and J~e > 0, then earnings reverts to a stable long term m e a n oL//j~ e. The formulation also permits the possibility of a random walk when /3e = 0. It is apparent that Eq. (1) offers considerably more flexibility in modelling earnings than the previous models employed in accounting based empirical studies. However, care must be taken in using a continuous time process to model earnings and dividends, given that earnings and dividends are announced quarterly, even though shares are traded continuously. It is on the basis that share price represents the discounted value of expected future dividends, and that changes in share prices reflect continuously occurring changes in the market's expectations of future earnings or dividends, that it is appropriate to use the continuous time framework. 3
2 The earnings process of Eq. (1) is implicitly dependent upon dividend policy. Clearly if all earnings are paid out as dividends then this would affect the growth of earnings in the future. Thus if 3' = 1, all earnings are paid out as dividends. The growth of earnings in this situation will clearly be less than when 3' = 0. We therefore put forward the notion that the growth of earnings implicitly reflects dividend policy. We are thankful to one of the referees for bringing this point to our attention. 3 Other researchers have also modelled dividends in the continuous time framework, see Merton (1973), Campbell and Kyle (1993).
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 21
Using standard techniques the solution to Eq. (1) is given by
E(t) - ~(k + fle)
[e k t - e -/3~t ] q-
e-fetfotO'lefeSdzl(s ) .
(2)
In order to operationalize Eq. (2), we need to determine the discrete time version of changes in earnings of Eq. (2). 4 This is given by the following expression:
AE(I)
t~
--ekt(ekat--e-[3eZlt) (k-t-fie)
q-(e - f e A t - 1 ) E ( t )
+ e -fie(t+ At)ftt+ Ato'le f'Sd Z l ( S ) .
(3)
From Eq. (3) we see that changes in earnings are dependent upon a nonlinear function of time and it is therefore appropriate to use nonlinear methods to estimate the parameters a, /3e and k. Using the specification of Eq. (3), the parameters were estimated by nonlinear maximum likelihood techniques, using annual observations of the Standard & Poors Composite Price Index, with corresponding earnings and dividend price index series for the period 1871-1986. The Standard & Poors Composite Stock Price index (S & P) are January stock prices and have been extended back to 1871 using the data in Cowles (1939). The dividend and earnings series are total dividends and earnings accruing to the index for each year. 5 The parameters are estimated for the three periods 1871-1986, 1925-1986 and 1960-1986. These periods were selected as a number of studies of mean reversion have focussed on the first two sample periods. However, the majority of the studies in the accounting area have focussed on the more recent time frame from the 1960s to the present, and it is of interest to see whether the results of the more recent time frame are consistent with those of the longer term. As previously mentioned, studying the time series behavior of earnings in the more recent time frame may give quite different results to those in the long tenn. 6 Table 1 shows summary results of the nonlinear maximum likelihood estimation of the parameters c~, /3e and k. If one examines the longest term results it is apparent that earnings revert significantly towards a time dependent mean. The extent of mean reversion is approximately 32% per year and the growth of the
4 This is obtained by substituting t + At into Eq. (2) and then subtracting E(t) from E(t + ,:It). This gives the correct discrete time approximation to Eq. (2). 5 This data set has been used extensively by numerous researchers (see Campbell and Shiller, 1988a,b; West, 1988; and is similar to Shiller (1981), Mankiw et al. (1985). For further information regarding the data set see Campbell and Shiller (1988a). 6 We refer to the period 1960-1986 as the short term. This is similar to the terminology adopted in examining the behavior of real exchange rates in more recent times.
R. Chiang et al. /Journal of Banking & Finance 21 (1997) 17-35 22 Table 1 Modelling earnings using nonlinear maximum likelihood estimation a Variable
Period
o~
/3
k
Coefficient DW R2
Nominal
1871-1986
0.0119 (2.39) 0.0115 (2.02) 0.0147 (1.00)
0.324 (3.28) 0.375 (2.54) 0.371 (1.51)
0.0528 (13.76) 0.054 (11.80) 0.052 (5.00)
1.85
0.17
1.82
0.17
1.83
0.12
1925-1986 1960-1986
a Figures in parentheses are asymptotic t statistics. DW = Durbin-Watson statistic. long term mean is 5.28% per year. 7 This means that the change in earnings in the next period will move towards the time dependent mean at a rate of 32% of [ ( ~ / / 3 ) e k ' - E(t)]. For example, if earnings are above the mean, then ( a / / 3 ) e k' < E(t). The change in earnings will be negative, hence earnings will move back towards the mean, but will only move back at rate of 32% of this difference. This is referred to as partial adjustment model. Examination of the other two subperiods reveals similar estimates of all parameters. Even for the 1960-1986 period, the parameter estimates are essentially the same. However the t statistic for the speed of adjustment coefficient is not significant in the short term, suggesting that earnings follows a random walk with a drift term. Overall, the results suggest that even for the most recent sample period earnings are 'mean reverting' towards a time dependent mean. From this point on we proceed on the basis that earnings can be represented as a mean reverting process and attempt to model the relationship between earnings and dividends. 2.1. Dividends and earnings relationship Possibly the best known relationship between dividend changes and earnings is the model proposed by Lintner (1956). In this classic study Lintner suggested that a firm's dividend policy through time behaves as though the managers have in mind a target dividend, which is a fixed proportion of current earnings, but that they adjust to this target from last year's dividend to the target level gradually according to a speed of adjustment parameter. Ignoring the constant which Lintner included in his adjustment equation (this constant is generally more significant on aggregate data than at the individual form level) gives the following discrete time equation: AO(t) =/3(yE(t
+ 1) - O ( t ) ) + e ( t )
(4)
7 The average long term growth of earnings was 6.2% for the period 1871-1986. The estimation from the nonlinear regression understates this figure by 0.9%.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 23
where ADO) = D ( t + 1 ) - D(t), D(t) = the level of dividends at time t, E(t + 1) = the level of earnings at time t + 1, = the dividend speed of adjustment coefficient, /3 = the payout ratio, 3' = the residual term. e(t) We adopt a similar approach to Lintner, but instead define the formulation in continuous time. The process for the instantaneous change in dividends is taken to be 8 d D ( t ) =/3( 3/E(t) - D( t ) ) d t + o-2dz2(t) (5) where the terms are defined in a similar manner to those of Eq. (1). Eq. (5) is solved under the assumption that earnings are mean reverting (or, as a special case, follow a random walk). We determine the appropriate discrete time approximation to Eq. (5) and show that the ensuing discrete time regression equation is consistent with the model proposed by Lintner in Eq. (4).Solving Eq. (5) yields the dividend process:
Y/3 E( t) - ~3//3(ekt-e-t~') e-~t Ctcr2e~Sdz2( O ( t ) = ( /3 -- /3e ) ( /3 + k ) ( /3 - /3e ) + 3//3e - ~ t ( "~ _--~-e) fore t3~O'ldz, ( s).
(6)
Eq. (6) implies that dividends at time t are a function of (i) the current earnings level, (ii) a time dependent term which reflects the growth of earnings, and also (iii) the difference between the accumulation of stochastic shocks to the earnings and dividend processes. The corresponding discrete time approximation to Eq. (6) is given by
AD(t)
3//3
A E ( t ) + (e -t3 - 1 ) O ( t )
([3--/3e)
(/3-/3~)
E(t)
( [~ -- /3e )
- aY/3ek'(et-e-~) ( /3 q'- k ) ( / 3 - ~e) T/3e-/3 ( t + l )
3//3(e -¢ - 1)
+e
#(t+ l) f t + l c r 2 e ~ S d z 2 ( s)
rt+ 1 fls ] e crld Zl(S )
J,
(7)
at
Eq. (7) represents a generalized form of the Lintner proposition and is 8 Eq. (5) represents the change in dividends from t to t + dt. However, we show at a later stage that the discrete time approximation of Eq. (5) is consistent with the Lintner model. The reason we have adopted the alternative form of Eq. (5) is that it is consistent with the earnings formulation, and also it is easier to solve the dividend process when earnings and dividends are observed at the same point in time. In the Lintner model earnings and dividends are observed at different points in time.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 24 Table 2 Modelling nominal changes in dividends using AD(t)= 3'0 + 3,1 AE(t)+ 3"2E(t)+ 3,3D(t)+ e(t)
Period
Variable
1871-1986 1871-1986 1925-1986 1925-1986 1960-1986 1960-1986
Constant
AE(t)
0.0126 (1.33) b 0.012 (1.32) 0.020 (1.13) 0.019 (1.13) 0.016 (0.36) 0.011 (0.29)
0.088 (7.81)
E(t + 1)
E(t)
D(t)
fl a
3"
R2
0.092 (8.25)
-0.145 (-5.88) -0.14 ( - 7.40) -0.144 (-4.44) -0.14 (-5.67) -0.11 ( - 3.44) -0.10 (-4.00)
0.157
0.63
0.76
0.14
0.64
0.76
0.155
0.63
0.75
0.14
0.63
0.75
0.116
0.66
0.77
0.10
0.73
0.78
0.09 (10.8) 0.086 (6.06)
0.091 (6.33) 0.088 (8.41)
0.069 (4.46)
0.077 (5.07) 0.073 (6.69)
Beta was calculated using the following formula fl = -log(1 b Figures in parentheses are t statistics. a
+
3,3).
applicable whether earnings are mean reverting or follow a random walk. Furthermore we can be modify Eq. (7) by removing the growth term of earnings. This is accomplished by substituting for e kt in terms of AE(t), E(t) and the residual term from Eq. (3). This produces the following regression equation: 9
AD( t) = YlAE( t) + TeE(t) +
Y3O(t) + e( t).
(8)
Table 2 contains summary results of the estimation of Eq. (8), and also the Lintner reduced form Eq. (4) for the three periods. It is immediately apparent that the two formulations generate very similar results. Furthermore it is evident that the three variables AE(t), E(t) and D(t) generally capture at least 75% of the variation in dividend changes and supports the view that both earnings and dividends are mean reverting in the long and short term. It is also interesting to note that the speed of adjustment of the dividend process appears to be quite stable across the differing time periods and is of the order of 11% to 15% per year. The implied payout ratios from the two different formulations are also included in Table 2. The results reveal that the payout ratio is of the order of 60-69% and is reasonably stable across the sample time periods, and across the two different models. 10 The validity of the t tests can be called into question if the residuals are heteroskedastic or serially dependent, so we conducted the Breusch and Pagan test
9 Eq. (8) is consistent with the Lintner formulation of Eq. (4). This is obtained by adding and subtracting 3"E(t) to the term enclosed in mean reversion portion. This produces D(t + At)-- D(t) = fl(yE(t + 1 ) - 3,E(t)+ 3,E(t)- D(t))+ e(t). This reduces to D(t + At)-- D(t) = fl(y(E(t + 1 ) E(t))+ 3"E(t)- D(t))+ e(t), which is essentially the same as Eq. (8). 10 The payout figure is consistent with those obtained by Lintner (1956).
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35
25
for heteroskedasticity, and the t test and L a n g r a n g e Multiplier test for serial correlation. No evidence of heteroskedasticity was found, b u t on testing up to 23 lags there was slight evidence of serial correlation which occurred at lag1 for all sample periods. ~1 The evidence was only minor, and we therefore conclude that changes in dividends can be reasonably described b y Eq. (8), and that the process is homoskedastic and essentially serially i n d e p e n d e n t for the time frames considered in this analysis. 12 Proceeding on the basis that changes in dividends can be described as a m e a n reverting process of the form outlined above the corresponding share price formulation will n o w be developed.
3. Modelling stock prices The share price is modelled as the discounted value of future expected dividends. For the sake of simplicity it is assumed that the long term cost of equity is constant and therefore the discounted value of expected future dividends is given by
P ( t) = E( f t ~ e - i ( S - ° D ( s ) d s )
(9)
where P ( t ) is the current share price at time t, i is the risk adjusted discount rate (cost of capital) applied to r a n d o m rate of dividends D(s) received at time s, and E is the expectation operator at time t. U n d e r the a s s u m p t i o n that dividends can be described by the process of Eq. (5) and substituting for D(s), the solution of Eq. (9) is g i v e n b y 13
P(t) =
y a f l e k' (t~+k)(i-k)(i+fle) Tfl ot e - ¢3t (fl+k)(i+fl)(i+fl,)
O( t ) + - (i+fle) "
(10)
11 Using GLS to correct for serial correlation did not alter the results significantly. ~2Alternatively we modelled changes in dividends towards long term sustainable earnings as suggested by Marsh and Merton (1987). Where sustainable earnings is captured by the deterministic time trend of the earnings process. The empirical results from these regressions did not offer any improved explanatory power (R 2 = 51%) over the Lintner model. These results are available from the authors on request. 13 Eq. (10) contains the payout ratio y and on initial inspection it appears that the share price can be increased by increasing the payout ratio, which is contrary to the Modigliani and Miller (1961) (MM) concept of dividend policy irrelevancy. However this is not the case. This is because as the payout ratio increases the growth of earnings correspondingly decreases which offsets any increase in the payout ratio. For example it can be shown from Eq. (10) that there exists a relationship between the payout ratio and the growth rate of earnings which is consistent with dividend policy irrelevancy. In the case where no dividends are paid the share price is given by P(t) = aekt/[(i -- k)(i + fie)]+ E(t)/(i + fie)This is obtained by assuming that the share price is the discounted value of expected future earnings.
R. Chiang et aL / Journal of Banking & Finance 21 (1997) 17-35 26
Eq. (10) is a generalized expression for the share price which is flexible enough to incorporate assumptions such as mean reversion on the part of earnings and dividends. This expression is similar to that derived by Campbell and Kyle (1993). Dividing Eq. (10) through by e kt yields a similar expression to Eq. (2.8) in Campbell and Kyle if one removes the effects of noise traders. However Eq. (10) is different in that dividends are discounted by the long term risk adjusted cost of equity whereas in the Campbell and Kyle model they discount at the riskless rate of interest, and also contains different parameters reflecting the earnings process. It is also consistent with other specifications presented in the literature - for example, the Lang (1991) share price model (in his formulation the share price is the discounted value of future expected earnings where earnings are a random walk), can be generated by substituting /3~ = 0 into Eq. (10) to give the share price when earnings follow a random walk with exponential growth: 14 ya/3e k' P(t)
=
(~+k)(i-k)i
-
yo~e -~' ([3+k)(i+~)i
Note that substituting for e kt from Eq. (2) with form of the share price:
D( t) i
+ - -
/~e =
0 yields the following
Y~ [ (i-k) ~ P(t) = ( ~ + kY~ ) ( i - k ) E( t) + i(~+k) + (/3+k)(i+/3)
i(~+k)
+ -e- ~tfotO.2et3Sdz2(s) i
(i
k)
(11)
1] 1
o',dZl(S)
ye-~tfotO.le~sdZl(S). i
(12)
It is evident from Eq. (12) that the Lang specification is essentially the same in structure with regards to the earnings variable and the constant term, but it omits the other two terms. The negative exponential term will probably have little effect in the long term: however the accumulation of stochastic shocks may not be small. The difference between the two formulations is due to the difference in modelling the expected growth in earnings. Lang formulates his model with constant expected growth, whereas we have modelled expected growth as an exponential term, and also incorporated a modified form of the Lintner model to reflect dividend policy. Our specification receives empirical support from the empirical results in Table 1, which provides evidence that k is significantly different from zero.
14 I f / 3 e = k = 0, that is d E ( t ) = a d t + o-dz, then the share price is given by P(t) = E ( t ) / i + a / i 2. This formulation is essentially the same as derived by Lang (1991).
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35
27
Continuing on the basis that earnings are not a random walk and adopting Eq. (10) the discrete time change in the share price is given by 15 3"a/3ek'(e k - 1) ziP(t)
=
(~+k)(i-k)(i+¢l,)
AD(t) + -
-
(i+~)
y / 3 a e - / 3 t ( e - g - 1) -
(13)
(~+k)(i+~)(i+13~)"
However one can develop an alternative formulation of Eq. (13) by incorporating changes in earnings and lagged earnings. This is accomplished by removing a e ~' from Eq. (13) by using Eq. (3). A p ( t ) = T, a E ( t )
+ 3'2 E ( , ) + 3'3 a D ( t )
+ y4 e - e ' + e ( t )
(14)
where
,//3(k +/3e) (e k - 1) 3'1 =
( ~ + k)(i-
k)(i + ~e)(e k- e-~e) '
+ t3e)(e k 3,2=-
1)
( ~ + k)( i_ k)( i + ~e)(ek_e_t~,)
,
1 3'3 = ( i + 3"13a(e -~ - 1) 3"4 = -- ( [3 + k ) ( i + [3 ) ( i + [3e ) "
The formulation of Eq. (14) is similar in structure to that proposed by Ohlson (1991). 16 From Eq. (14) it is evident that both changes in earnings and dividends and lagged earnings levels are relevant in explaining stock price changes. J7 The majority of empirical studies do not focus on share price changes, but rather focus upon returns. Eq. (14) can easily be modified to reflect returns by
~5It is evident that E(t), D(t) and P(t) are nonstationary as the mean and variance are time dependent. However, the variance of changes in E(t), D(t) and P(t) are time independent and can be regarded as trend stationary. This would indicate E(t), D(t) and P(t) are I(1). Campbell and Kyle (1993) have shown that real detrended D(t) and P(t) are cointegrated. 16The Ohlson (1991) model can he described as P( t ) = el)l E( t ) + 42 B V( t ) + 4)3D( t ). Where B V( t ) is the book value at time t. Therefore changes in the share price can be expressed as P(t + 1)- P(t) = 41(E(t + 1)- E(t))+ 42(BV(t + 1)- BV(t))+ 43(D(t + 1)- D(t)). Incorporating the clean surplus relationship modifies the above to Ap(t) = 41 AE(t) + 42( E(t + 1) - D(t + 1)) + 43 AD(t). Note this expression is slightly different to that proposed in Eq. (15). 17 It can be shown that e(t) is homoskedastic and serially independent, and therefore estimation is efficiently estimated using ordinary least squares.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 28
adding dividends paid at (t + 1) and dividing through by P(t). is The appropriate return equation becomes 19 ,aE(t)
R(t+ l ) = y l - P(t)
E(t)
AD(t)
- { - ~ 2 P - - - - ~ -'~'-~3 P ( / ' ~ [ - ~ 4 P ~
+E(t)
e -t3t
D(t+ 1) --~- P(t)
(15)
where R ( t + 1) = -
AP(t) + D(t + 1) P(t)
e(t) e(t)
Table 3 presents parameter estimates of Eq. (15) without the e -t3' term. This term was found to offer no explanatory power. We have also included regression equations which relate to the earnings process (that is, changes in earnings and earnings per share), and the dividend process (dividends per share). For the dividend process we did not include changes in dividends per share, as this term offered little explanatory power. The reason we have included these additional regressions is to see whether earnings dominates dividends in explaining returns. It is evident that for the period 1871-1986, both changes in earnings and lagged earnings are significant in explaining returns. Whereas for the period 1925-1986 no variable is significant in explaining returns. This result may be due to multicollinearity. When comparing the earnings and dividend regressions it appears that both earnings and dividend processes offer similar explanatory power. When considering all variables the R 2 value of 11% is marginally greater than the explanatory power obtained by Fama and French (1988a) (R 2 = 7%) for the period 1927-1986.
18 Eq. (15) is essentially the same as that derived by Chiang et al. (1995), the only term which differs is the time dependent term and the change in dividends per share term, otherwise the regression equation is identical. The expression is also similar to the formulation of Campbell and Shiller (1988b). They model real returns as being dependent upon dividends per share, dividend growth and long term earnings per share. It is apparent that the two are essentially the same if the change in earnings per share variable is not significant. The approaches adopted in each of the two papers mentioned above were different to the one proposed here. In the Chiang et al. (1995) paper they model mean reversion of asset prices towards the fundamental value, and in order to operationalize their approach they assume that the fundamental value is a function of current level of earnings or dividends. In the Campbell and Shiller (1988b) analysis they model retums as being dependent upon long term earnings per share, dividends per share and changes in dividends. Neither paper models the earnings or dividends processes and subsequently does not develop the functional form of asset prices to earnings or dividends. 19 For empirical estimation we inserted a constant term Y0, though from Eq. (14) this is not required. We desired to test whether the constant term was significantly different from zero.
R. Chiang et al./Journal of Banking & Finance 21 (1997) 17-35 29 Table 3 Modelling returns using the following formulation:
aE(t) Period
Coefficient To
1871-1986
-0.07 ( - 1.00) a - 0.04 (-0.75) -0.146 (-0.63) -0.12 ( - 1.19) - 0.053 ( - 0.68) - 0.12 ( - 1.22) 0.44 (-2.86) 0.023 (0.48) * - 0.07 ( - 1.03) ~
1925-1986
1960-1986
E(t) +
TI
,aDO)
O(t +1)
Y2 3.08 (3.08) 2.76 (3.32)
0.86 (0.40) 1.99 (1.26)
-4.68 (-1.57) - 5.87 ( - 2.49)
1.31 (1.66) 1.68 (2.64)
0.68 (0.46) 2.07 (2.28)
-7.71 (-2.25) 1.36 (2.26)
3'3
"Y4
R2
- 1.50 ( - 0.61)
1.24 (0.61)
0.11 0.105
1.96 (0.48)
2.96 (2.05) 3.73 (1.38)
0.036 0.11 0.093
- 19.78 (-0.88)
5.03 (2.56) 30.30 (3.12)
0.097 0.46 0.43
4.32 (2.51)
0.34
a Figures in parentheses are t statistics. * Corrected for serial correlation.
The results from 1960-1986 fare reasonably well when considered in the light of previous empirical studies for this period. For example, Easton and Harris (1991) obtain R 2 values of the order of 7.7% (this is the average for the entire sample period 1968-1986). However their regressions only incorporated changes in earnings per share and earnings per share. When considering only these two variables our data resulted in an R 2 of 23% (not corrected for serial correlation), these results being consistent with those obtained by Ali and Zarowin (1992) (R 2 = 19%). Closer inspection of Table 4, however, reveals the puzzling nature of the coefficients, in that the change in earnings and earnings levels appear to have a negative influence on returns. One would expect that as earnings increased this should translate to increases in share prices and hence positive returns. Therefore changes in earnings and earnings levels should have positive coefficients. Tests for heteroskedasticity and serial correlation revealed little evidence of it for the two longer sample periods. However, there was considerable evidence of both in the short term results, and these influences may contribute to the negative signs of the earnings variables. Adjustments for serial correlation and heteroskedasticity produced the same signs in the coefficients for the sample period 1960-1986. However, we could not completely remove the effects of serial correlation. The
R. Chiang et al./ Journal of Banking & Finance 21 (1997) 17-35 3o Table 4 Summary of studies cited for returns Period Author
Nominal independent variables
R2
1969-1985 1969-1986 1927-1986 1871-1986
All and Zarowin ( 1 9 9 2 ) Eastonand Harris (1991) Fama and French (1988a) Chianget al. (1995)
JE(t)/P(t), E(t + 1)/P(t) AE(t)/P(t), E(t + l)/P(t) D(t + 1)/P(t) AE(t)/P(t), E(t)/P(t) and D(t + 1)/P(t)
0.19 0.08 0.07 0.11
Period
Author
Real independent variables
R2
1927-1986 Fama and French (1988a) D(t + 1)/P(t) 0.07 1900-1986 Campbelland Shiller (1988b) E(t + 1)l / P ( t ) a, D(t + 1)/P(t) and /~D(t) 0.076 1871-1986 Chianget al. (1995) AE(t)/P(t), E(t)/P(t) and D(t+ 1)/P(t) 0.05 E(t + 1)~ is the average of the previous 30 year earnings. earnings regression produced similar explanatory power to the regression containing all variables. The dividend model produced marginally less explanatory power. The parameter estimates reported for the earnings and dividend regression were corrected for serial correlation and heteroskedasticity. The results in the longer term ( 1 8 7 1 - 1 9 8 6 ) are consistent with dividend policy irrelevance. For the other two sample periods the earnings process offers at least as much explanatory power as dividends and appears to be consistent with dividend policy irrelevance. Table 4 depicts a summary of the dependent variables and time frames used in the studies cited above.
4. Modelling the fundamental value Alternatively it is possible to model the fundamental value by using Eq. (10). That is the deterministic portion of the fundamental value may be defined as by 20
ElF(t)]---
T a r e kt (f+k)(i_k)(i+fle)
D(t) + ~( i + ~. g e
(16)
This formulation relates the share price to the discounted value of expected future dividends and could be reasonably regarded as reflecting the fundamental value of the stock. In examining the parameters of Eq. (16) a , f , fie and k can be estimated from the dividends and earnings processes, and E ( t ) and D ( t ) are known at time t. The only variable which is not known is the cost of capital i. This discount rate may not be the same as the rate applied by the market to the stock index. The discount rate will be identical if the observed market price is the
20 We deleted the negative exponential term from Eq. (10) as this term had very little effect on the empirical results that follow.
R. Chiang et al. / Journal of Banking & Finance 21 (1997) 17-35 31
same as the fundamental value. However the difficulty of estimating the appropriate discount rate when the fundamental value and the observed stock price differ can be overcome by assuming that on average the percentage mispricing between the fundamental value and the market price is zero ( P ( t ) - F ( t ) ) / F ( t ) . Using an iterative procedure we estimated the discount rate until the average mispricing equalled zero. The implied cost of equity capital was 7.1%. This estimate appears to be low as the average return on the S &P inclusive of dividends for the period 1871-1986 is approximately 10.2%. We now proceed to model mean reversion of the stock price towards the fundamental value. One possible approach may be described by the following process:
P(t)
= [V(t)] Z'eL°<'>-¢~2+z'~z++~'-z+
(17)
where
~(t)
is a mean reverting process governed by d a ( t ) = - A a ( t ) d t + 6d z(t), is the speed of adjustment coefficient, is the standard deviation of d a(t), dz(t) is a Wiener process with mean zero and unit variance, is a parameter determining the non-linear nature of mean reversion, t3+ o- and p are parameters of the fundamental value process that will be defined below. A 6
The attractive features of the process described by Eq. (17) are that P(t) is always non-negative and that the variance of changes in the stock price are a function of the level of stock price. Furthermore, c~(t) determines the extent of mean reversion towards the fundamental value. We also allow for the possibility that the relation between the stock price and the fundamental value may be non-linear. This is reflected by the /37 term. If /3/ is not significantly different from one this yields a linear relation between P(t) and F(t). To completely capture the process of Eq. (17) we need to specify some type of diffusion process of the fundamental value. We make the assumption that the fundamental value may be described by a Geometric Brownian motion and is given by dF(t) - -/xdt+ F(,)
o-dw(t)
(18)
where
dF(t)/F(t) tx o
dw(t) P
= = an = = =
the return of the fundamental value, the instantaneous drift term and is given by Eq. (16), as this is expectation at time t, the standard deviation per unit time of the return, a Wiener process with mean zero and unit variance, and the correlation coefficient between dw(t) and d z(t).
R. Chianget al./Journal of Banking& Finance21 (1997)17-35
32
Applying Ito's lemma to Eq. (17) yields the instantaneous return of the stock price relative to the fundamental value. This is given by the following: dP(t) P--~
[ --
J~fl[~ --
Alog P ( t ) ]dt+6dz(t)+Ctfo'dw(t). F ( t ) ps]
(19)
In adopting the process described by Eq. (19) we note that the instantaneous return in the stock price is dependent upon how far the current stock price is from the fundamental value. The greater the deviation from the fundamental value the greater the economic forces will drive the stock price towards to the fundamental value. Economic forces are constantly affecting the fundamental value which is reflected in our stochastic modeling of this rate. The speed of reversion is not only dependent upon the magnitude of the difference between log P(t) and log F(t), but also on the speed of adjustment coefficient, A. The larger the value of the speed of adjustment the stronger the restoring force back to equilibrium. If the speed of adjustment coefficient is small, then mean reversion towards fundamental value may be quite slow, even though deviations from fundamental value may be large. In using discrete time data it is necessary to transform the continuous time process of Eq. (17) into a discrete time formulation. This is accomplished by substituting t + At for t into Eq. (17) then subtracting P(t). The discrete time change in the stock price is given by log
P(t + at) P(t) = (e - A a t - 1)
+/3/1o8
2A
F(t + m) F(t) +(e Aat--1)[l°gP(t)--Jgfl°gF(t)]
+ e- x(, +,~t)f/+ ateXS~5d z(S).
(20)
We note from Eq. (20) that the change in the stock price is not only a function of the change in the fundamental value, but also is a function of the difference between log P(t) and log F(t) which has a speed of adjustment coefficient of (e -hat- 1). This means that the change in the stock price is not only mean reverting towards the fundamental value, but it is also dependent upon the change in the fundamental value. Note that for mean reversion to exist we must have A > 0. Eq. (20) is efficiently estimated by ordinary least squares as the residual term is homoskedastic. Table 5 presents summary results of the regression Eq. (20). It is evident that ~ix is not significantly different from 1, and hence supports the hypothesis of a
R. Chiang et al./ Journal of Banking & Finance 21 (1997) 17-35 33 Table 5 Returns as a function of the fundamental value P(t+I) log
P(t)
F(t+l)
Y°+yll°g~
- +y21°gP(t)+ y31ogF(t)+e(t)
Coefficient Variable
3'0
Yl
Y2
"/3
h
R2
0.047 (0.97) a
0.93 (1.28)
-- 0.09 ( - 2.16)
0.073 (2.50)
0.10
0.06
a Figures in parentheses are t statistics.
linear relationship between the P(t) and F(t). The speed of adjustment coefficient A is significant at the 5% level and illustrates that mean reversion towards the fundamental value occurs at a rate of 10% per year. This means that deviations from the fundamental value can exist for prolonged periods. The significance of the t tests can called into question if there exists significant evidence of heteroskedasticity or serial correlation in the residuals. An Arch test and Breusch and Pagan test were used to test for the presence of heteroskedasticity. Neither test detected any significant evidence of heteroskedasticity. A Lagrange Multiplier test revealed little evidence of serial correlation, and we can reasonably conclude that the residuals are homoskedastic and serially independent. This is accordance with the model put forward in Eq. (20).
5. Conclusion
In this paper we have endeavored to model earnings, dividends and stock prices within a viable economic framework. Our starting point was to model the earnings process and to proceed to investigate it empirically, the main finding being that there existed significant long term mean reversion with some weaker evidence of short term mean reversion. The parameter estimates of the earnings process appear to be reasonably stable within the three periods considered, justifying our conclusion that earnings are mean reverting towards a time varying mean. These results are consistent with the results obtained by other researchers when modelling returns in the long term. We then examined the relationship between dividends and earnings and incorporated a modified form of the Lintner model. The empirical results revealed that the modified formulation performed as effectively as the original Lintner approach. Furthermore, the payout ratio of approximately 64%, and appears to be reasonably stable across the three sample periods. Proceeding on the basis the that the share price is the discounted value of future expected dividends, and that dividends are generated by the modified Lintner model, we developed the functional form of the corresponding share price
R. Chiang et al./Journal of Banking & Finance 21 (1997) 17-35
34 equation. W e e x a m i n e d different scenarios for the earnings process and found that our results were consistent with L a n g (1991) w h e n earnings are a r a n d o m walk. Estimation of our formulation suggested that changes in earnings per share and earnings per share levels are important in explaining returns, w h i c h supports d i v i d e n d policy irrelevance. The results in the longer term sample periods are similar to those of other empirical studies, h o w e v e r the results in the short term present i m p r o v e d explanatory p o w e r to other studies considering this period.
Acknowledgements W e w o u l d like to thank the two a n o n y m o u s referees for their constructive c o m m e n t s , and w o u l d also like to thank Stuart H o d g e s and the seminar participants at Strathclyde University, Lancaster University, U n i v e r s i t y o f W a r w i c k , U n i v e r s i t y of T e c h n o l o g y and the E u r o p e a n F i n a n c e C o n f e r e n c e for their valuable c o m m e n t s . W e are also grateful to John C a m p b e l l for p r o v i d i n g the data.
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