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Source: http://www.doksinet A GENERAL MODEL OF EARNINGS, DIVIDENDS AND RETURNS by G. Hobbes*, G. Partington* and M. Stevenson* * Macquarie University * University of Technology, Sydney First Draft January 1994 Second Draft February 1996 Third Draft January 1997 1 Source: http://www.doksinet Abstract In this paper we develop a general linear model for the relation between earnings, dividends and returns. Arising from this is a model for realised returns and an equity valuation model We show how two classes of earnings, dividend and return models that have appeared in the literature arise as special cases of our general model. The differences between these models are a function of the restrictions imposed on the general model. Depending on the restrictions imposed the model can also generate results consistent with either Miller and Modigliani’s (1961) dividend irrelevance proposition, or their dividend signalling proposition. We note that the single equation models typically

used in earnings return work cannot adequately distinguish between alternative models. The general model developed here suggests model testing should be conducted using a system of simultaneous equations with constraints on the coefficients. Possible practical uses of the model include estimating terminal values and equity discount rates. A General Model of Earnings, Dividends and Returns Introduction In recent years there has been a growing interest in theoretical models of the relationship between earnings, dividends and returns. For example, Ohlson (1991) develops a theoretical model where the level of earnings determines returns, and Ohlson and Shroff (1992) develop a model where earnings levels, or earnings changes, can be used to explain returns. Hobbes, Partington and Stevenson (1993,1996) provide a model where earnings levels and changes, and dividend levels and changes, together with the firm’s dividend policy parameters explain returns. This latter model has been extended

by Goya and Beg (1995) to include cash flow. While in a continuous time framework, Chiang, Davidson and Okunev (1996) derived a model similar to the Hobbes, Partington and Stevenson model. This theoretical work was preceded by over two decades of primarily empirical work stimulated by the seminal paper of Ball and Brown (1968). However, the theoretical modeling has had an influence on some of the later empirical work. For example, Easton and Harris (1991) utilise an Ohlson style of model and find empirical support for using both levels and changes in earnings as explanators of returns. In recent empirical work there has been an increasing interest in the interaction between dividend policy and earnings in explaining returns. Both Mande (1994) and Kallipan (1994) find that earnings response coefficients are affected by dividend policy variables. These results are consistent with the theoretical model of Hobbes, Partington and Stevenson (1996). Empirics and theory can therefore be called

upon to support more than one model. The question naturally arises - How should we choose between them? How do we reach agreement about which model best helps explain observed phenomena and best assists empirical researchers in designing experiments? We might be able to make a choice based on the existing empirical evidence, or perhaps specially designed tests might be necessary. However, the purpose of this paper is to argue that such effort is not required. Rather than viewing the Ohlson style of model, or the Hobbes, Partington, Stevenson style of model as competing alternatives, we demonstrate that they are special cases of a more general model. 2 Source: http://www.doksinet These special cases arise as a result of coefficient restrictions placed on the elements of the general model. The circumstances under which the model is to be applied determines the particular set of coefficient restrictions that are appropriate, and therefore which model is to be preferred. An important

conclusion that flows from our analysis is that the single equation earnings return regressions typically used in empirical work cannot adequately distinguish between the possible theoretical models. Furthermore, the use of such single equation regressions may easily lead the researcher to erroneous conclusions. The paper is organised as follows. First we detail the elements of the general model under uncertainty, and then provide a general solution to the model. We next examine the impact of different coefficient restrictions that lead to specific forms of the model. We start by considering the simple certainty case and then extend the analysis to alternative coefficient restrictions under uncertainty. The General Model Generating processes for earnings and dividends We assume that dividends are generated by a linear process. The process is such that this period’s dividend is a function of the last period’s dividend and current earnings. The dividend process is defined by

equation (1). 1 From an empirical perspective, this equation is consistent with the Lintner (1956) model of dividends, which has substantial empirical support (Fama and Babiak (1968), Laub (1972), Shevlin (1982)). From a theoretical perspective, equation (1) is consistent with the general dividend function used by Ohlson (1991), and the specific use of the Lintner model in Hobbes et.al (1996) Dt = a1 + a2Yt + a3 Dt −1 + et (1) Here Dt is the dividend announced and paid, Yt is the earnings announced, et is the management determined shock to dividends all at time = t . The expected value of the shock term is zero ie Et −1 ( et ) = 0 . The coefficients a1, a2 and a3 are the dividend policy parameters for the firm The magnitude of these parameters are subject to substantial managerial discretion. 2 Equation (2) describes the earnings generating process as a general function of lagged earnings and lagged dividends. This is consistent with the form of the earnings equation used by

Ohlson (1991) which he justifies in terms of a reinvestment model for current earnings. Equation (2) is Yt = b1 + b2Yt −1 + b3 Dt −1 + ε t (2) where ε t is the shock to earnings at time = t , and Et −1 (ε t ) = 0 . The coefficients b1, b2 and b3 are the parameters for the firm’s earnings process. Equation (2) provides quite a flexible specification for the earnings process. For example, if the coefficients b 1 and b 3 are set to zero and b 2 is set to one, we are left with the random walk model for earnings. The random walk model has considerable empirical support and much popular use 1 Equation 1 and subsequent equations are firm specific, but for economy of notation the subscript i representing the i’th firm has been omitted. 2 This discretion is, of course, not without limit and some values for these parameters are more plausible than others. For example, we would expect a2 to generally be less than 1. 3 Source: http://www.doksinet in empirical studies. In

particular it has been extensively used in papers that study earnings announcement effects 3. The return equation (3) is simply an identity that expresses realised returns as price change plus dividends scaled by beginning of period price as follows Rt = Pt − Pt −1 Dt + Pt −1 Pt −1 (3) where Rt is the realised return for time = t , and P t is the price at time t. Expectation processes and price In forming the expectation equations we assume rational expectations formed at t-1. Thus, we assume that investors form their expectation for dividends and earnings on the basis of the generating processes introduced above. We also assume that they substitute into the generating process the values for dividends and earnings observed at t-1. The resulting expectations process for dividends is given by equation (4) and the expectation process for earnings is given by equation (5). Et −1 ( Dt ) = a1 + a2 Et −1 (Yt ) + a3 Dt −1 (4) Et −1 (Yt ) = b1 + b2Yt −1 + b3 Dt −1 (5)

The price equation is given by equation (6) which is the standard discounted dividend model, with the price being given ex-dividend. We assume equilibrium prices, thus at the date of price formation expected and required rate of returns are equal. ∞ E (D ) Pt −1 = ∑ t −1 ti++i1 (6) i = 0 (1 + r ) The solution for the general model To find the solution for the model, we use the earnings expectations process in equation (5) to substitute for the expected earnings in the dividend equation (4) and then use this expected dividend equation to substitute for expected dividends in the price equation (6). After some manipulation (see Appendix 1), the resulting solutions for price and returns are given by equation (7) and (8) respectively. Pt = − (1 − b2 )( a1 + a 2 b1 ) + a 2 b1 b2 − r[(1 − b2 )(1 − a3 − a 2 b3 ) − a 2 b2 b3 ] (1 − b2 ){[(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a1 + a2 b1 ] + (1 + r )a2 b1 b2 } − [(1 − b2 )(1 − a3 − a2 b3 ) −

a2 b2 b3 ] [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )} − + [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] + 3 (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 D + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] t See for example Brown, Finn and Hancock (1977). 4 Source: http://www.doksinet + Rt = (1 + r )a2 b2 Y [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] t (7) (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] ∆Dt + Pt −1 (1 + r )a2 b2 + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] D ∆Yt + t Pt −1 Pt −1 (8) The current price is shown to be a function of levels in current dividends and current earnings. Current realised returns are a function of changes in current dividends and current earnings, plus the dividend yield. The changes in

dividends and earnings scaled by price determine the capital gain component of returns. Note also that by using equation (1) and/or equation (2) to substitute for dividends D t and/or earnings Y t , it is possible to express price as a function of several additional variables including last period’s dividends, the shock to current dividends, last period’s earnings and the shock to earnings. Similar substitutions can be made in the return equation. There is, therefore, plenty of scope in single equation empirical studies to find a range of statistically significant variables which “explain” realised returns. 4 Such variables may be selection of those identified here, or proxies. Additionally, in cross-sectional analysis, the coefficients from the dividend and earnings generating processes, or their proxies, can be used to “explain” returns. Given the general nature of our model, the statistical significance of a particular variable may be consistent with several alternative

explanations. Thus, we suggest that some care should be exercised in interpreting the results of single equation studies. We further suggest that consideration be given in empirical work to estimating simultaneous equation systems with coefficient restrictions as appropriate. Specific Cases of the General Model In this section we consider some of the possible restrictions on the general model. The Certainty Case The certainty case is a highly restrictive case. The parameter restrictions are such that only one price model can result from substitution in the general model. Under the assumption of certainty the generating process and the expectations process are identical, since by definition there is no error in expectation. One consequence of this is that in equilibrium all assets earn the risk free rate of return r f . The earnings process and earnings expectation simply depend on the scale of firm’s investment. Given an initial investment I 0 then the earnings realised in one

period’s time are given by: 4 Using a less general model Hobbes, Partington and Stevenson (1996) demonstrate that similar comments apply to abnormal returns. 5 Source: http://www.doksinet Y 1 = rf I 0 If the firm pays out 100% of earnings as dividends then there will be no change in earnings over time. They will remain constant at Y 1 If less than 100% of earnings are distributed as a dividend then the profit retained in any period t is given as π t = Y t - D t . This is reinvested at the rate r f Thus the earnings as t=2 can be written as Y 2 = rf ( I 0 ) + rf (Y 1 - D1) = Y 1 + rf (Y 1 - D1) and, in general, the earnings can be written as: Yt = Yt −1 + r f (Yt −1 − Dt −1 ) ( ) = 1 + r f Yt − 1 − r f Dt −1 (9) In the certainty case, therefore, the earnings process in equation (2) is subject to the restrictions that b1 = 0, b2 = 1 + rf , b3 = − rf . The coefficient on the error term e t is no longer one, but is restricted to zero since under certainty

there is no error. In equation (9) growth in earnings arises from profit retention and we make no explicit allowance for the possibility of additional contributions of capital by owners. One way to handle such a possibility, which we adopt, is to follow the approach of Ohlson (1991). This involves treating injections of cash by owners as negative dividends. In this case D t is measured net of capital contributions. Then total additional funds for investment are correctly measured as Y t D t 5 The only restriction on the coefficients of the dividend equation arises from the restriction that dividends paid cannot exceed cash available. Given the results of Miller and Modigliani (1961) that, under certainty, dividend policy is irrelevant to value, we would expect that the choice of parameters for the dividend policy equation would have no effect on value. In Appendix 2 we show that by substitution of the coefficient restrictions into the general model that this is indeed the case. The

result is Equation (10), which gives the price in the certainty case In this case, the magnitude of the current dividend is only relevant in determining the ex-dividend price. Pt = (1 + r ) Y − D f rf t t (10) The realised returns equation is given by Rt = rf (11) The above result for the price equation is the same as that derived by Ohlson (1991). It is also equivalent to the result obtained by applying Miller and Modigliani’s (1961) valuation model to a no growth case under certainty. 5 Alternatively, we could add an additional variable and funds for investment could be measured as Yt - Dt + St. Where St represents additional capital contributions from owners. For notational simplicity, however, we adopt the Ohlson (1991) approach. 6 Source: http://www.doksinet Alternatives Under Uncertainty The uncertainty case allows consideration of a richer set of alternatives than the certainty case. Unlike the certainty case, the nature of the parameter restrictions provide

alternatives that lead to differing results in the model for price and returns. How we characterise uncertainty also has important implications for the resulting model. For example, in the first case we consider below, the characterisation of uncertainty is such that we obtain an identical result to the certainty case. General Model for Dividends and Reinvestment Model for Earnings Under this heading we consider two alternatives. First we consider a characterisation of uncertainty which implies constant realised returns and then we consider stochastic realised returns. Return on Investment is Constant Perhaps the simplest characterisation for uncertainty is to allow for shocks to dividends, while holding the return on investment constant. The shocks to dividends are transmitted to the level of reinvestment, and this in turn generates some uncertainty in earnings forecasts more than one step ahead. The generating process for dividends is as in the general model equation (1) as follows:

Dt = a1 + a2Yt + a3 Dt −1 + et The expectation of dividends formed at t-1 is as previously given by equation (4). Et −1 ( Dt ) = a1 + a2 Et −1 (Yt ) + a3 Dt −1 The reinvestment process for earnings is as in the certainty case equation (9), except that we replace r f with ROI. Where ROI is the return on investment, defined as Y t /I t By definition, if ROI is constant, there is no difference between the return on old or new investment. 6 The resulting reinvestment equation for earnings is given below as equation (12). Yt = ROI [ It −1 + Yt −1 − Dt −1 ] = Yt −1 + ROI (Yt −1 − Dt −1 ) (12) The expectation of earnings formed at t-1 is given by the right hand side of equation (12). There is no uncertainty in the one period ahead earnings forecast 7 since it is based entirely on observables and a constant. Thus, ROI could also be defined as ∆Yt / ∆It 6 7 . However, there is uncertainty regarding earnings forecasts more than one period ahead For example, the

earnings forecast two periods ahead is given by: Et − 2(Yt ) = Et − 2(Yt −1 ) + ROI ( Et − 2(Yt −1 ) − Et − 2( Dt −1 ) ) The expectation for Yt-1 can be formed without error, but the expectation of dividends Dt -1 formed at t-2 is subject to error. 7 Source: http://www.doksinet By multiplying out the brackets in equation (12) we can see that the coefficient restrictions imposed on the general model are b 1 = 0, b 2 = 1 + ROI and b 3 = -ROI. As we demonstrate in Appendix 3, the result of substituting into the general model is the following price equation: Pt = a1 (1 + r )(r − ROI ) + r[r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI ) + (r − ROI )( a3 − a2 ROI ) − a2 ROI (1 + ROI ) Dt + (r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI ) + a2 (1 + r )(1 + ROI ) Yt (r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI ) (13) We make the argument below that r and ROI are equal to r f in equilibrium. Given this equality we can simplify

equation (13) and the result is identical to the certainty case. Given a constant return on investment in a competitive market, it seems reasonable to assume that competition results in the following equilibrium between required return, expected return and ROI. r = E ( Rt ) = ROI Under the characterisation of uncertainty used in this section, the primary uncertainty is in the magnitude of the cash dividends. As is clearly explained at the textbook level by Brealey and Myers (1984), if the expected return on investment is equal to the investors’ required rate of return, then the magnitude of the dividend and reinvestment decision has no effect on investors wealth, it merely changes the form of their wealth. 8 A dollar increase in dividends is exactly offset by a dollar fall in share price and vice-versa. Since the return on investment ROI was defined to be constant and dividend policy has no effect on investors wealth, realised returns on equity must also be constant. In other words,

investors’ realised returns are risk free. Therefore the equilibrium required rate of return is r f Thus: Rt = E( Rt ) = ROI = r f This equality between realised and expected returns leads to a result which is the same as the certainty case: Pt = (1 + r ) Y − D f rf t t This result is consistent with that of Ohlson (1991) since his analysis implicitly assumes the equality of realised returns and equilibrium expected returns. It is an interesting result that although there is uncertainty in dividend forecasts and earnings forecasts, more than one period ahead, this uncertainty has no effect on value. 8 Note that this point is distinct from M & M’s (1961) argument on dividend irrelevance. In that paper investment is held constant. Here we discuss the condition where dividend changes are irrelevant to value, despite these dividend changes being coupled to changes in investment. 8 Source: http://www.doksinet Return on Investment is Stochastic In this section the

return on investment is assumed to have a mean equal to investors’ required return, but investment returns are also assumed to have a stochastic component. 9 The stochastic component of ROI has a zero expectation. The shocks to investment returns feed through into shocks to earnings. As a result uncertainty in future earnings now arises from both uncertainty in ROI, and uncertainty about the level of reinvestment which arises from uncertainty about future dividends. The dividend generating processes is as in the general model, and identical to that used in the constant ROI case above. As in the constant ROI case, the expectation process for dividends is as given by equation (4). The earnings process differs from the constant ROI case, by the addition of a subscript t to the ROI variable. This indicates that the realisation of ROI on re-investment may vary in a stochastic fashion over time. That is Yt = Yt −1 + ROI t (Yt −1 − Dt −1 ), (14) where ROI t = ROI + η t , η t is

a random shock with zero mean, and the following restrictions on covariances of η t are assumed: E(η t+k , η t+i ) = 0 E(η t+k , e t+i ) = 0 E(η t+k , ε t+i ) = 0 with 0 < i < k, for all t. These restrictions eliminate the possibility of using the shocks to ROI to form improved forecasts of future ROI, future earnings, or future dividends. Substituting for ROI t in equation (14), expanding and re-arranging gives: Yt = Yt −1 + ROI (Yt −1 − Dt −1 ) + η t (Yt −1 − Dt −1 ) (15) The random shock in ROI is reflected in earnings as a random shock scaled by the level of reinvestment. Thus equation (15) can be rewritten as: Yt = Yt −1 + ROI (Yt −1 − Dt −1 ) + ε t Given E(η t )=0 and the covariance restrictions on η t , E(ε t )=0. The expectation of earnings is therefore given by: (12) Et −1 (Yt ) = Yt −1 + ROI (Yt −1 − Dt −1 ) The resulting coefficient restrictions are therefore, b1 = 0, b2 = 1 + ROI , b3 = − ROI . Substituting in the

general model gives: 9 It does not matter whether changes in ROI are assumed to only affect new investment or to affect all investment, the resulting price model is the same. For simplicity in the derivation the results are based on the former assumption 9 Source: http://www.doksinet Pt = a1 (1 + r )( r − ROI ) +. r[r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )] .+ (r − ROI )( a3 − a2 ROI ) − a2 ROI (1 + ROI ) Dt +. r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI ) + a2 (1 + r )(1 + ROI ) Yt (r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI ) The form of the model is unchanged from the constant ROI case. This is because the expectation for ROI t is: E( ROI t) = ROI The expectation of ROI is a constant, and it hardly seems feasible that in equilibrium this constant is different from the required return, r. Substituting ROI = r into the above price equation gives: Pt = (1 + r ) Yt − Dt r As explained above, given equality between the expected

return on investment and the required return, then dividend policy per se is irrelevant to value and, as such, does not have any information content. Consequently dividends do not appear in the final price model, except to determine the ex-dividend price. There is one change from the earlier models in this case. The discount rate is no longer r f This is because there is now period to period uncertainty in the level of return realised by investors. As we show in Appendix 4, the realised return is now given by: Rt = r + (1 + r ) ε t r Pt −1 We could continue and assume that stochastic changes in investment returns affect both reinvestment and existing investment, but the resulting price model would be the same as that immediately above. The bottom line is that as long as we make assumptions that imply equality between the expected return on investment and the required return, we will get essentially the same pricing model. This is because, what we have is essentially an

investment model of valuation, in which all investments are expected to be zero NPV, and in which dividend policy is irrelevant to value. Indeed, the model equates to Miller and Modigliani’s (1961) no growth case Lintner Model for Dividends and Random Walk for Earnings Lintner model and simple random walk In this model the generating process for earnings is not explicitly defined, in contrast to the reinvestment model above. Rather we define the output of the earnings process as being well approximated by a random walk. The earnings process has many degrees of freedom and processes with many degrees of freedom often have the appearance of a random walk. Also, there is an extensive empirical literature which suggests that annual earnings are well approximated by a random walk, Ball and Watts (1972), Whittred (1978). Thus the earnings process is given by: 10 Source: http://www.doksinet Yt = Yt −1 + ε t The corresponding expectation process for earnings is: Et −1 (Yt ) = Yt

−1 The dividend process follows the Lintner (1956) model which, as noted earlier, is well supported in the empirical literature. That is: Dt = a + cτ Yt + (1 − c) Dt −1 + et Where, τ is the target payout ratio (dividends as a proportion of profits) and c is the speed of adjustment coefficient which governs how fast the firm adjusts the actual payout to the target payout. When we earlier considered the certainty case we followed the approach of Ohlson (1991) and defined D t as dividends net of capital contributions by owners. This is clearly contrary to the spirit of the Lintner model which is intended only to model the dividend payment, and not capital contributions. In order to have consistent notation throughout the paper, therefore, we must here impose the restriction that there be no capital contributions by owners from t onwards. The expectation process for dividends is: Dt = a + cτ E( Yt ) + (1 − c) Dt −1 Using the solution for the general model with the coefficient

restrictions a1 = a , a 2 = cτ , a 3 = 1 − c, b1 = 0, b2 = 1, b3 = 0 , we show in Appendix 5 that the price and realised returns equations are given by: or Pt = ( 1 − c) a( r + 1) cτ ( r + 1) Yt + D + ( r + c) t r ( r + c) r ( r + c) Rt = cτ ( r + 1) ∆Yt ( 1 − c) ∆Dt Dt + + r ( r + c) Pt −1 ( r + c) Pt −1 Pt −1 ( 1 + r ) et cτ ( r + 1) ε t + Rt = r + r ( r + c) Pt −1 ( r + c) Pt −1 2 Lintner Model for Dividends and Random Walk with Drift for Earnings In this case the only change from the equations in the preceding case is the addition of a drift term b to the earnings equation. Thus earnings are given by Yt = b + Yt −1 + ε t Using the solution for the general model with the following coefficient restrictions a1 = a , a2 = cτ , a3 = (1 − c) , b1 = b , b2 = 1 , b3 = 0 We show in Appendix 6 that the price and realised return equations are given by 11 Source: http://www.doksinet Pt = Rt = (1 − c) a − cτb cτ (1 + r ) + Yt + Dt (r + c) rc r

(r + c) Dt cτ (1 + r ) ∆Yt (1 − c) ∆Dt + + , r (r + c) Pt −1 (r + c) Pt −1 Pt −1 or Rt = r + (1 + r ) et cτ (1 + r ) 2 ε t + r (r + c) Pt −1 (r + c) Pt −1 Lintner Model for Dividends and Random Walk with Growth for Earnings The output from the earnings process is now assumed to be well approximated by compound growth in earnings which follows a random walk. This is given by Yt = ( 1 + g ) Yt −1 + ε t With the coefficient restrictions a1 = a , a2 = cτ , a3 = 1 − c, b1 = 0, b2 = 1 + g , b3 = 0 , in Appendix 7 we derive the price and realised returns equations to be or Pt = ( 1 − c) a( r + 1) cτ ( 1 + g ) ( r + 1) + D Yt + ( r + c) t r ( r + c) ( r − g ) ( r + c) Rt = cτ ( 1 + g ) ( r + 1) ∆Yt ( 1 − c) ∆Dt Dt + + ( r − g )( r + c) Pt −1 ( r + c) Pt −1 Pt −1 cτ ( r + 1) ε t ( 1 + r ) et Rt = r + + ( r − g )( r + c) Pt −1 ( r + c) Pt −1 2 These three models which use the Lintner equation are consistent with the models of

Hobbes, Partington and Stevenson (1996). However, the model with drift represents an extension to their work. Notice also the different possibilities for expressing these models As we show above the return equations can be expressed as a function of dividend and earnings changes and current dividends, or as a function of the equilibrium expected return for the firm and shocks to earnings and dividends. Also note that by using the Lintner model to substitute for the dividend term,D t , the level of earnings, lagged dividends and the shock to dividends can be introduced into the returns model which contains changes in earnings and dividends. Summary and Conclusion As can be seen from the foregoing the general model can be useful in analysing a variety of special cases. The models are derived in a time series, but can be used to explain cross sectional variation in price and return, in response to changes in dividends and earnings. 10 Table 1 summarises the particular special cases

examined above. 10 This, of course, presumes that you can determine which of the special cases should be used as the explanatory model. 12 Source: http://www.doksinet Table 1: Models of equity returns State of nature Main assumptions Certain Return on investment is equal to the risk free rate Uncertain Distribution of future dividends is uncertain, return on investment is constant Distribution of future dividends is uncertain, return on investment is stochastic with an expected value equal to r Lintner model for dividends, random walk for earnings Lintner model for dividends, random walk with drift for earnings Lintner model for dividends, random walk with growth for earnings Equity valuation model, and reference 1 + rf Pt = Yt − Dt , rf Ohlson (1991) 1 + rf Pt = Yt − Dt , rf Ohlson (1991) (1 + r ) Y − D , Pt = t t r This paper, page 11. ( ) ( ) Pt = f (r , c, τ , Yt , Dt ) , Hobbes, Partington and Stevenson (1996) Pt = f ( r , c, τ , b, Yt , Dt ) , This paper,

page 12. Pt = f (r , c, τ , g , Yt , Dt ) , Hobbes, Partington and Stevenson (1996) The list in Table 1 is not exhaustive of the possibilities. The general model encompasses models which comply with the following: 1. Price can be expressed as the discounted dividend model 2. Expected earnings can be expressed as a linear function of one period lagged earnings and/or one period lagged dividends in a stochastic, or non-stochastic framework. 3. Expected dividends can be expressed as a linear function of current earnings, and/or one period lagged dividends in a stochastic, or non-stochastic framework. The general model is consistent with the intuition that current earnings and dividends are relevant to value when they can be used by investors to form improved forecasts of future cash flows from the business. This will generally be through an improved forecast of earnings, or dividends, or return on investment. Note, however that some information variables affect price, but they do not

affect returns. For example, in the random walk with drift model, earnings drift is a variable in the price equation but is not in the corresponding return equation. 11 Value changes can be of two types, scale effects and wealth changes. Scale effects are simply a consequence of changes in the level of re-investment, while wealth changes arise from shocks in the value relevant variables. A question of particular importance in analysing wealth effects, is whether investors are able to determine if future returns on the firm’s investment will differ from the required rate. This is, of course, Miller and Modigliani’s (1961) criteria for identifying growth stocks. If the restrictions on the model imply that the expected return on investment is 11 Once the earnings drift has been incorporated into the price at t=0, it no longer affects either expected return, or shocks to realised returns. However, if the expected magnitude of the earnings drift changed, it would enter the return

equation as a shock to value and hence return. 13 Source: http://www.doksinet equal to the required rate, a rather simple valuation model emerges. This model can be expressed in terms of Ohlson’s (1991) earnings based valuation model, or equivalently Miller and Modigliani’s no growth valuation model. The general model allows cases consistent with Miller Modigliani’s (1961) dividend irrelevance proposition, such as their no growth model. However, it goes a step further by allowing cases where their dividend signaling proposition can be incorporated into the valuation model. 12 Thus, depending on the particular restrictions imposed on the general model dividends may be relevant to value or irrelevant. In the Hobbes, Partington, Stevenson (1996) model dividends are relevant to value because of dividend signaling. Dividend signaling is implicit in the use of the Lintner (1956) model, which directly conditions expectations of future dividends and implicitly indicates sustainable

cash flow. In the case of the Ohlson (1991) model dividend policy is not relevant to value This result is driven by the re-investment process, and assumptions about the return on investment. The nature of the model is such that there is uncertainty about future dividends, which are relatively unconstrained. There is also uncertainty about future earnings, which stems primarily from uncertainty over the level of re-investment. However, it turns out that neither of these sources of uncertainty affects the wealth of investors, given that the expected return on investment is equal to the required return on investment. In the Ohlson (1991) model, therefore, uncertainty over dividends and future earnings has no substantive impact on value. This demonstrates the interesting result; that it is possible to create a characterisation of uncertainty that is irrelevant to valuation. Under such a characterisation all discounting naturally takes place at the risk free rate. Changes in current

dividend payments merely create changes in value that are a consequence of scale effects, as discussed above. The general model and special cases of the general model arise from a system of equations. The common practice in empirical work of estimating a single equation return model is unlikely to adequately distinguish between alternative models. For example, is the significance of earnings in an earnings/return equation evidence in support of the Ohlson (1991) model, or the Hobbes, Partington, Stevenson (1996) model? It is consistent with both, it is also consistent with earnings proxying for some omitted, but correlated, causal variable. Another problem, as demonstrated in the section on earnings/re-investment models, is that the same reduced form equation for returns may result from different processes. It is also possible to get different single equation models from the same processes. This is shown in the section involving the Lintner model It seems better to test the system of

equations that form the model. This would involve addressing the question, are the estimated coefficients consistent between the earnings and dividends equations, and the return equation? This naturally leads to estimation of the models as simultaneous equation systems with consistency constraints on the coefficients. This is not a trivial task. Not only is there a requirement for a substantial body of data, but the constraints on the coefficients are likely to be non-linear. The general model is not only of theoretical interest but may also have some practical use. It provides a basis for solutions to the discounted dividend pricing model that do not require explicit forecasts of future values of dividends, only currently observable values for dividends and earnings. The cost of this is that assumptions must be made about the explicit processes for 12 Miller and Modigliani (1961) invoke the argument that the price changes observed in response to dividend changes are a consequence of

dividends conveying information about future cash flow. However, they do not incorporate dividend signaling into their valuation model. 14 Source: http://www.doksinet dividends and earnings. The particular parameterisation assumed must be consistent with the linear form of the general model. Given an acceptable parameterisation, the general model may be a possible alternative to the standard dividend growth model, in providing horizon values beyond the forecasting horizon of explicit dividend forecasts. Conversely, given today’s price and assumptions about the dividend and earnings processes it is possible to back out an implied discount rate. This may be an additional tool for those faced with the vexing question of how to estimate a company’s cost of equity. A final point to note is that the general model provides a model for realised returns, as distinct from expected, or required returns. Although, as Hobbes, Partington and Stevenson (1996) show, it is possible to derive an

expression for expected returns by taking expectations of specific forms of the realised returns model. A general model of realised returns may turn out to be useful given that much empirical work is based on realised returns. This is in accord with Fisher Black’s (1993) suggestion that empirical investigations of asset pricing models, being based on realised returns, rather than expected returns, likely tell us more about the variance of returns than they do about asset pricing models. Black also suggests that more is to be learned about asset pricing models from theoretical as opposed to empirical investigation. Perhaps more theory about realised returns will help bridge the gap. 15 Source: http://www.doksinet References Ball, R., and P Brown, 1968, An empirical evaluation of accounting income numbers, Journal of Accounting Research, 6, 159-178. Ball, R. and R Watts, 1972, Some time series properties of accounting income, Journal of Finance, 27, 663-682. Black, F., 1993,

Estimating expected return, Financial Analysts Journal, September- October, 36-38. Brealey, R., and S Myers, 1984, Priciples of Corporate Finance, 2nd ed, McGraw-Hill, New York. Brown, P., FJ Finn, and P Hancock, 1977, Dividend changes, earnings reports and share prices: Some Australian findings, Australian Journal of Management, 127-147. Chiang, R., Davidson, I And J Okunev, 1996, Some further theoretical and empirical implications regarding the relationship between earnings, dividends and stock prices, Journal of Banking and Finance. Easton, P.D, and T Harris, 1991, Earnings as an explanatory variable for returns, Journal of Accounting Research, 29, 19-36. Fama, E.F and H Babiak, 1968, Dividend policy: An empirical analysis, Journal of the American Statistical Association, 63, 1132-1161. Goyal, M. and R A Beg, 1995, A theoretical model of earnings, dividends, returns and cash flows, European Accounting Association Conference, Conference paper, May 1995. Hobbes, G., G Partington and M

Stevenson, 1993, The earnings, dividends and return relationship, Accounting Association of Australia and New Zealand, Conference paper. Hobbes, G., G Partington and M Stevenson, 1996, Earnings, dividends and returns: A theoretical model, Research in Finance, Supplement 2, 221-244. Kallapur, S., 1994, Dividend payout ratios as determinants of earnings response coefficients: A test of the free cash flow theory, Journal of Accounting and Economics, 17, 359-375. Laub, M.P, 1972, Some aspects of the aggregation problem in the dividend-earning relationship, Journal of the American Statistical Association, 67, 552-559. Lintner, J., 1956, Distribution of incomes of corporations among dividends, retained earnings, and taxes, American Economic Review, 46, 97-113. Mande, V., 1994, Earnings response coefficients and dividend policy parameters, Accounting and Business Research, 24(94), 148-156. Miller, M. and F Modigliani, 1961, Dividend policy, growth and the valuation of shares, Journal of

Business, 34, 411-433. 16 Source: http://www.doksinet Ohlson, J., 1991, The theory of value and earnings and an introduction to the Ball-Brown analysis, Contemporary Accounting Research, 8(1), 1-19. Ohlson, J. A and P K Shroff, 1992, Changes versus levels in earnings as explanatory variables for returns: Some theoretical considerations, Journal of Accounting Research, 30, 210-225. Shevlin, T., 1982, Australian corporate dividend policy: Empirical evidence, Accounting and Finance, 22(1), 1-22. Whittred, G., 1978, The time series behaviour of corporate earnings, Australian Journal of Management, 195-202. APPENDIX 1 GENERAL SOLUTION FOR PRICE AND RETURNS 17 Source: http://www.doksinet General process for dividends: D t = a 1 + a 2 Y t + a 3 D t-1 + e t Expectation process for dividends: E t-1 (D t ) = a 1 + a 2 E t-1 (Y t ) + a 3 D t-1 as E t-1 (e t ) = 0 General process for earnings: Y t = b 1 + b 2 Y t-1 + b 3 D t-1 + ε t Expectation process for earnings: E t-1 (Y t ) = b 1 +

b 2 Y t-1 + b 3 D t-1 as E t-1 (ε t ) = 0 Realised returns: Rt = Pt − Pt −1 Dt + Pt −1 Pt −1 Now Dt = a 1 + a 2 [b 1 + b 2 Y t-1 + b 3 D t-1 + ε t ] + a 3 D t-1 + e t = a 1 + a 2 b 1 + a 2 b 2 Y t-1 + a 2 b 3 D t-1 + a 3 D t-1 + e t + a 2 ε t ∴ Dt = a 1 + a 2 b 1 + a 2 b 2 Y t-1 + (a 3 + a 2 b 3 )D t-1 + e t + a 2 ε t Yt = or  Dt    =    Yt  and J ~t = b1 + b 2 Y t-1 +  a1 + a2 b1   a3 + a2 b3  +      b1   b3 + F ~ GJ ~ ~ t −1 εt b 3 D t-1 + a2 b2   Dt −1  1 a2   et    +        b2   Yt −1   0 1   ε t  + MK ~ ~ t .(A11)  a3 + a2 b3 a2 b2   Dt   et   a1 + a2 b1  1 a2            where J =   , F =  = =  , K~ =   , G , M ~ ~ ~t ~ b2  0 1    b1  Yt  ε t   b3 Equation (A1.1)

demonstrates that current dividends and earnings follow a first-order autoregressive process. Solving A1.1 by recursion: = J1 F+ G J + M K ~ ~ ~ ~o ~ ~1 18 Source: http://www.doksinet ] F+ G F+ G J + M K + M K = F+ G F+ G J + G M K + M K = F+ G J + M K = F+ G F+ G F+ G2 J + G M K + M K + M K = F+ G F+ G F+ G J + G M K + G M K + M K = F+ G J + M K = F + G F + G F + G2 F + G3 J + G2 M K + G M K + M K + M K = G F + G F + G F + G F + G J + G M K + G M K + G M K + G3 M K J = G o F + G F ++ G t −1 F + G t J + G o M K + G M K J = ~ J3 ~ J4 ~ ~t ∴ [ = J2 ~ ~ ~ ~ ~ ~o ~ ~ ~ ~ ~ ~ ~ ~2 ~ [ ~ ~o ~ ~ ~ 2 ~ ~ ~ ~ ~ ~ ~ ~ ~3 ~ [ ~ ~ ~o ~ ~ ~1 ~ ~ 2 ] ~ ~ 3 ~ ~ ~1 ~ ~ ~ 2 ~ ~ 3 ~ 4 ~ ~ ~ ~ ~ ~ ~o ~ ~ ~ 2 ~ ~ 2 2 ~ o ~ ~ 3 ~ ~ ~1 ~ 3 2 ~ ~ ~ ~ ~ ~o 3 ~ ~ ~ ~ 4 ~ ~ ~ ~ ~ t −1 ~t ~1 2 ~ ~o ~o ~1 o ~ ~ ~ ~ ~ t ~ ~ ~ 2 ~ ~ 3 ] ~ ~ 4 2 ~ 4 ~ ~ ~ ~ ~ 3

~ t −1 ~ ~ ~ 2 ~ ~ ~1 ++ G t −1 M K ~ ~ ~1 t −1 ∑ Gi F+ Gt J + ∑ Gi M K i=0 ~ ~ ~ ~o ~ ~ i=0 ~ t −i When the focal point is time equal to 0, then k −1 = J ~k k −1 ∑ Gi F+ G k J + ∑ G k M K i=0 ~ ~ ~ ~o ~ k −i ~ ~ i=0 When the focal point is time equal to t, then k −1 J ~ t +k = k −1 ∑ Gi F+ G k J + ∑ Gi M K i=0 ~ ~ ~ ~t ~ ~ i=0 ~ t + k −i Taking expectations, ( ) = ( ) = Et J ~ t+k Et J ~ t +k Pt ∑ G i F + G k J + ∑ G i M Et ( K i=0 ~ ~ k −1 = Now k −1 k −1 = ∑G i=0 i ~ ~ ~t ~ ~ i=0 F+ G k J ~ ~ ~ ~ ~ ∞ t k =1 −1 k ~ ~ t +k k (1 + r ) ~ ) as t ( ) (I − G ) F+ G J N E (J ) E (D ) =∑ ∑ I− G ~ t + k −i ( Et K ~ t + k −i )=0 k ~ ∞ k =1 ~ ~ ~t t ~ t +k k (1 + r ) where N = (1 0) ~ 19 Source: http://www.doksinet ∞ = = Pt = where N = (0 1) , ~ Now ( ) I− G ~ ( I − G) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t (1 + r ) k

=1 −1 ∞ N ( I − G) F ∞ Gk −1 ~ ~ ~ − N ( I − G) ∑ ~ k F +  ∑ k ~ ~ ~ ~ (1 + r ) k =1 k =1 (1 + r ) ∞ Gk + N∑ ~ k J ~ ~t k =1 (1 + r ) k G −1 1   − N ( I − G) I− G F +   ~ ~ ~ ~ ~ (1 + r )  (1 + r )  ~ r −1 G  1  ~ + N I− G J ~ (1 + r )  ~  (1 + r ) ~  ~ t  a3 + a2 b3 a2 b2   Dt   a1 + a2 b1  1 a 2  , M =  , G =  , J =   F =         ~ ~ b2   0 1  ~  Yt   b1  ~  b3 ~ ~ ~ ~ −1 ~ = 1 − a3 − a2 b3 − a2 b2      1 − b2   − b3 = a2 b2  (1 − b2 )  1    [(1 − a3 − a2 b3 )(1 − b2 ) − a2 b2 b3 ]  b3 (1 − a3 − a2 b3 ) = a2 b2  (1 − b2 )   D  D1 1      b3 (1 − a3 − a2 b3 )    D1  D1  −1 −1 1   G =  I~ −  (1 + r ) ~  = ~ N ( I − G) −1 F ∴ ~ ∑ N ( I

− G ) −1 ( I − G k ) F + N G k J where D1 = (1 − a3 − a2 b3 )(1 − b2 ) − a2 b2 b3  1 + r − a3 − a2 b3 − a2 b2    (1 + r ) (1 + r )     − b3 (1 + r − b2     (1 + r ) (1 + r )  −1 a2 b2  (1 + r − b2   ( + r)  (1 + r ) 1  1  =   (1 + r − a3 − a2 b3 )(1 + r − b2 ) − a2 b2 b3  b3 (1 + r − a3 − a2 b3 )    (1 + r ) 2  (1 + r )  (1 + r )   (1 + r − b2 a2 b2   D2 D2 (1 + r − a3 − a2 b3 )(1 + r − b2 ) − a2 b2 b3    where D 2 =  (1 + r ) 1 + r − a3 − a2 b3   b3   D2   D2 20 Source: http://www.doksinet ( ) I− G ~  (1 − b2 ) a2 b2   D  a +a b D1 1    1 2 1      b3 1 − a3 − a2 b3   b1    D1  D1   (1 − b2 )( a1 + a2 b1 ) + a2 b1 b2    D1      b3 ( a1 + a2 b1 ) + b1 (1 − a3 − a2 b3 )    D1  

−1 = F ~ ~ = N ( I − G) −1 F ~ ~ ~ ~ r = (1 − b2 )( a1 + a2 b1 ) + a2 b1 b2 r[(1 − a3 − a2 b3 )(1 − b2 ) − a2 b2 b3 ] = (1 − b2 )(a1 + a2b1 ) + a2b1b2 rD1 −1 G 1   G =  I~ − (1 + r )  (1 + r ) ~  ~ = .(A12) −1 G   1 I− G F  ~ ~ (1 + r )  (1 + r )  ~  [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ][ a1 + a2 b1 ] + (1 + r )a2 b1 b2    D2 (1 + r )   =   (1 + r )( a1 + a2 b1 )b3 + b1 b2 (1 + r − a3 )     D2 (1 + r )   ~ G −1 1   N ( I − G) I− G F  ~ ~ ~ (1 + r )  ~ (1 + r ) ~  ~  [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ][ a1 + a2 b1 ] + (1 + r )a2 b1 b2    D2 (1 + r )  (1 − b2 ) a2 b2    =    D1   D1 (1 + r )( a1 + a2 b1 )b3 + b1 b2 (1 + r − a3 )     D2 (1 + r )   −1 ~ 21 Source: http://www.doksinet = (1 − b2 ) [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ][a1 + a2 b1

] + (1 + r )a2 b1 b2   + D1  D2 (1 + r )  + ∴ = a2 b2  (1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )    D1  D2 (1 + r )  N ( I − G) F ~ ~ ~ r r{(1 − b2 )(1 − a3 − a 2 b3 ) − a 2 b2 b3 } ~ − (1 − b2 ){[(1 + r − b2 )( a3 + a 2 b3 ) + a 2 b2 b3 ][ a1 + a 2 b1 ] + (1 + r )a 2 b1 b2 } {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 }{(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 } a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1b2 (1 + r − a3 )} .(A13) {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 }{(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 } −1 G N ~ {(1 − b2 )(a1 + a2 b1 ) + a2 b1 b2 } − −1 G 1   − N ( I − G) I− G F  ~ ~ ~ (1 + r )  ~ (1 + r ) ~  ~ −1 ~ 1   I− G J  ~ ~ (1 + r )  (1 + r )  ~ t ~  (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 =  [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] ∴ Pt = = N ( I − G ) −1 F ~

~ r ~ − N ( I − G ) −1 ~ ~ ~  (1 + r )a2 b2  [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] G −1  Dt       Yt  −1 NG  1 1    ~ ~ + − I G F I− G J    ~ ~ ~ ~ ~ (1 + r )  (1 + r )  (1 + r )  (1 + r )  ~ t ~ (1 − b2 )( a1 + a2 b1 ) + a2 b1b2 − r[(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] − (1 − b2 ){[(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a1 + a2 b1 ] + (1 + r )a2 b1 b2 } − [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )} − + [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 D + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] t (1 + r )a2 b2 Y + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] t + .(A14) 22

Source: http://www.doksinet Recall Rt However, P t - P t-1 and Pt − Pt −1 Pt −1 Dt Pt −1 Also = Pt − Pt −1 Dt + Pt −1 Pt −1 = −1 NG  1  ~ ~ I− G J −J (1 + r )  ~ (1 + r ) ~  ~ t ~ t −1 = −1 NG  1  ∆ J~ t ~ ~ − I G (1 + r )  ~ (1 + r ) ~  Pt −1 ( ) NJ = ~ ~t Pt −1 ∴ Realised returns can be written as or Rt = Rt = −1 NG  1  ∆ J~ t N~ J~ t ~ ~ + I− G (1 + r )  ~ (1 + r ) ~  Pt −1 Pt −1 (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ∆Dt + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] Pt −1 ∆Yt D (1 + r )a2 b2 + + t [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] Pt −1 Pt −1 .(A15) 23 Source: http://www.doksinet APPENDIX 2 - PRICE AND RETURNS FOR THE CERTAINTY CASE For the certainty case the earnings process is subject to the following restrictions: b1 = 0, b2 = 1 + r f , b3 = − r f Let N ( I − G) −1 F ~ ~ ~ ~ r − N ( I

− G) −1 ~ ~ ~ −1 G 1   I− G F = A − B − C  ~ ~ (1 + r )  (1 + r )  ~ ~ From (A1.3) in APPENDIX 1 it follows that: A = ( − r f )( a1 ) (1 − b2 )( a1 + a2 b1 ) + a2 b1 b2 = r f {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } r f ( − r f )(1 − a3 + r f a2 ) − a2 (1 + r f )( − r f { = B = = C = = ∴A - B - C a1 r f (1 − a3 − a2 ) {[ ] (1 − b2 ) (1 − r f − b2 )( a3 + a2 b3 ) + a2 b2 b3 [ a1 + a2 b1 ] + (1 + r f )a2 b1 b2 {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } {(1 + rf { } − b2 )(1 + r f − a3 − a2 b3 ) − a2 b2 b3 } + r a ) − a (1 + r )( − r )} {− a (1 + r )( − r )} } ( − rf ) a2 (1 + rf )( − rf )a1 {(−r )(1 − a 3 f = } 2 f 2 f 2 f f f − a1 (1 − a3 − a2 ) { } a2 b2 (1 + r f )( a1 + a2 b1 )b3 + b1 b2 (1 + r f − a3 ) {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } {(1 + rf { } a2 (1 + rf ) (1 + rf )a1 ( − rf ) ( − rf ) (1 − a3 − a2 )( − a2

)(1 + rf ) 2 = = } − b2 )(1 + r f − a3 − a2 b2 b3 ) (1 + rf )a1 rf (1 − a3 − a2 ) (1 + rf )a1 r f a1 a1 + − = 0 r f (1 − a3 − a2 ) r f (1 − a3 − a2 ) r f (1 − a3 − a2 ) −1 NG  1  ~ ~ I− G J (1 + r )  ~ (1 + r ) ~  ~ t  (1 + rf − b2 )( a3 + a2 b3 ) + a2 b2 b3 =   (1 + rf − b2 )(1 + rf − a3 − a2 b3 ) − a2 b2 b3 =  a2 (1 + r f )( − r f )   − a2 (1 + r f )( − r f )   (1 + rf − b2 )(1 + rf − a3 − a2 b3 ) − a2 b2 b3  (1 + rf )(1 + r f )a2   − a2 (1 + rf )( − rf )  (1 + rf )a2 b2  Dt       Yt   Dt       Yt  24 Source: http://www.doksinet =  Dt       Yt   (1 + r f )   −1  r f   Now N ( I − G) −1 F Pt = ∴P t = ~ ~ ~ ~ r − Dt + G −1 −1 NG 1 1   ~ ~  + − N ( I − G) − I G F I− G  J    ~ ~ ~ (1 + r )  ~

(1 + r ) ~  ~ (1 + r )  ~ 1 + r ~  ~ t (1 + rf ) rf −1 Yt ~ when b1 = 0, b2 = 1 + rf , b3 = − rf Realised returns can be calculated as follows: Rt Pt − Pt −1 Dt + Pt −1 Pt −1 = Now P t - P t-1 = and ∴ NJ ~ ~t Rt = = = ∴ P t-1 R t = = Rt Dt −1   ∆ J~ t N~ J~ t 1 N GI − G + ~ ~ ~ Pt −1  (1 + rf ~  Pt −1  Dt  (1 + r f   ∆Dt  1   + 1 (1 0)   −1     Pt −1  r f   ∆Yt  Pt −1  Yt  (1 + rf ) (Yt − Yt −1 ) + Dt −1 rf (1 + rf ) (rf )(Yt −1 − Dt −1 ) + Dt −1 rf = (1 + rf )Yt −1 − (1 + rf ) Dt −1 + Dt −1 = (1 + rf )Yt −1 − rf Dt −1 =  (1 + rf )  rf  Yt −1 − Dt −1   rf  rf Pt −1 = rf = ∴ −1 NG   1 ~ ~ G ∆ J , I − ~t (1 + r f )  ~ (1 + r f ) ~  25 Source: http://www.doksinet APPENDIX 3 - PRICE WHEN RETURN ON INVESTMENT IS CONSTANT For

the case when return on investment is constant the earnings process is subject to the following restrictions: b 1 = 0, Let N ( I − G) −1 F ~ ~ ~ r ~ − N ( I − G) −1 ~ ~ ~ b 2 = 1 + ROI, b 3 = -ROI −1 G 1   I− G F = A − B − C  ~ ~ (1 + r )  (1 + r )  ~ ~ (1 − b2 )( a1 + a2 b1 ) + a2 b1 b2 − ROI ( a1 ) = r{(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } r{( − ROI )(1 − a3 + ROIa2 ) − a2 (1 + ROI )( − ROI )} A= = B= = (1 − b 2 ){[(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a1 + a2 b1 ] + (1 + r )a2 b1b2 } {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } {(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 } ( − ROI ){[(1 + r − 1 − ROI )( a3 − a2 ROI ) + a2 (1 + ROI )( − ROI )]a1} {(− ROI )(1 − a3 + a2 ROI ) − a2 (1 + ROI )(− ROI )} {1 + r − 1 − ROI )(1 + r − a3 + a2 ROI ) − a2 (1 + ROI )(− ROI )} = C= = a1 r{1 − a3 − a2 } a1 {(r − ROI )( a3 − a2 ROI ) − a2 ROI (1 + ROI )} {1

− a3 − a2 } {r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )} a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )} {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } {(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 } a2 (1 + ROI )a1 (1 + r )( − ROI ) ( )( 1 ) ( 1 )( − ROI − a + a ROI − a + ROI − ROI )} {(1 + r − 1 − ROI )(1 + r − a3 + a2 ROI ) − a2 (1 + ROI )( − ROI )} { 3 2 2 = a1a2 (1 + ROI )(1 + r ) {1 − a3 − a2 } {(r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )} a1 {r − ROI )( a3 − a2 ROI ) − a2 ROI (1 + ROI )} a1 − r{1 − a3 − a2 } {1 − a3 − a2 } {(r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )} a1 a2 (1 + ROI )(1 + r ) − {1 − a3 − a2 }{(r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )} a1 (1 + r )(r − ROI ) = r{(r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )} ∴A− B−C = Also 26 Source: http://www.doksinet −1 NG  1  ~ ~ I− G J (1 + r )  ~ (1 + r

) ~  ~ t  (1 + r − b2 )( a 3 + a 2 b3 ) + a 2 b2 b3   Dt  (1 + r )a 2 b2 =    (1 + r − b2 )(1 + r − a 3 − a 2 b3 (1 + r − b2 )(1 + r − a 3 − a 2 b3 ) − a 2 b2 b3  Yt   (r − ROI )( a3 − a2 ROI ) − a2 ROI (1 + ROI )   Dt  a2 (1 + r )(1 + ROI ) =    (r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI ) (r − ROI )(1 + r − a3 + a2 ROI ) + a2 ROI (1 + ROI )  Yt  Now Pt = N ( I − G ) −1 F ~ ~ ~ ~ r G −1 −1 NG  1 1    ~ ~ + − N ( I − G) − I G F I− G J    ~ ~ ~ (1 + r )  ~ (1 + r ) ~  ~ (1 + r )  ~ (1 + r ) ~  ~ t −1 ~ a (1 + r )(r − ROI ) 1 ∴P = + t r[r − ROI )(1 + r − a + a ROI ) + a ROI (1 + ROI )] 3 2 2 (r − ROI )( a − a ROI ) − a ROI (1 + ROI ) 3 2 2 D + (r − ROI )(1 + r − a + a ROI ) + a ROI (1 + ROI ) t 3 2 2 a (1 + r )(1 + ROI ) 2 + Y (r − ROI )(1 + r − a + a ROI ) + a ROI

(1 + ROI ) t 3 2 2 + When r = ROI = rf , Pt = − Dt + (1 + rf ) rf Yt 27 Source: http://www.doksinet APPENDIX 4 REALISED RETURN WHEN RETURN ON INVESTMENT IS STOCHASTIC Recall from APPENDIX 1 that current earnings and dividends follow a first-order autoregressive process previously defined by equation (A1.1) as: J = F + GJ ~t ~ ~ t −1 ~ +MK ~ .(A41) ~ t  a3 + a2 b3 a2 b2   Dt   et   a1 + a2 b1  1 a 2  , M =  , G =  , K =   where J =   , F =         ~ ~t 0 1  ~ t ε t   Yt  ~  b1  ~  b3 b3  It follows that J −IJ ~t or ~ ~ t −1 = F + (G − I ) J ~ ~ ~ ~ t −1 ∆ J = F + (G − I ) J ~t ~ ~ ~ ~ t −1 +MK ~ ~ t +MK ~ .(A42) ~ t Recall equation (A1.5 ) from APPENDIX 1 which gives −1 NG  1  ∆ J~ t N~ J~ t ~ ~ + Rt = I− G (1 + r )  ~ (1 + r ) ~  Pt −1 Pt −1 Substituting (A4.1) and (A42) into the above

results in −1  G   1  N~ G ~ ~  Rt = I −  F+ N F +  Pt −1  (1 + r )  ~ (1 + r )  ~ ~ ~    −1  G   1  N~ G ~ ~  J + − + + − I ( ) G I N G   ~ ~  ~ t −1 Pt −1  (1 + r )  ~ (1 + r )  ~ ~   −1  G   1  N~ G ~ ~  K + + − I M N M   Pt −1  (1 + r )  ~ (1 + r )  ~ ~ ~  ~ t   .(A43) Using equation (A1.2) in APPENDIX 1 we can write −1 NG  1  ~ ~ I− G (1 + r )  ~ (1 + r ) ~   (1 + r − b2 )( a 3 + a 2 b3 ) + a 2 b2 b3  (1 + r )a 2 b2 =   (1 + r − b2 )(1 + r − a 3 − a 2 b3 ) − a 2 b2 b3 (1 + r − b2 )(1 + r − a 3 − a 2 b3 ) − a 2 b2 b3  28 Source: http://www.doksinet and −1 NG  1  ~ ~ − I G F+ N F (1 + r )  ~ (1 + r ) ~  ~ ~ ~ = [(1 + r − b2 )(a3 + a2 b3 ) + a2 b2 b3 ] [a1 + a2 b1 ] + (1 + r )a2 b1b2 + (a (1 + r − b2 )(1 + r − a3

− a2 b3 ) − a2 b2 b3 1 + a2 b1 ) = c0 , −1 NG  1  ~ ~ I− G (G − I ) + N G ~ ~ (1 + r )  ~ (1 + r ) ~  ~ ~  [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a3 + a2 b3 − 1] + (1 + r )a2 b2 b3  + ( a3 + a2 b3 )  (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3   =    [(1 + r − b2 )(a3 + a2 b3 ) + a2 b2 b3 ]a2 b2 + (1 + r )a2 b2 (b2 − 1) + a b   2 2 (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3   = [c1 c2 ] , and −1 NG  1  ~ ~ I− G M+ N M (1 + r )  ~ (1 + r ) ~  ~ ~ ~  (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 [(1 + r − b2 )(a3 + a2 b3 ) + a2 b2 b3 ]a2 + (1 + r )a2 b2 + a  = +1 2 (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  = [ c3 c4 ] (A4.3) can be rewritten as  1     et   Pt −1  P    t −1 Dt −1    Rt = [ co c1 c2 ] + [ c3 c 4 ] 

 Pt −1   εt     Pt −1  Y   t −1   Pt −1  =c Z ~ z ~ t −1 +c K ~K ~ t .(A44) 29 Source: http://www.doksinet Taking conditional expectations of equation (A4.4) gives Et −1 ( Rt ) = c Z as E t-1 ( K ) = 0 ~z ~ t ~ t Under the assumption that there is no information content in the shock to ROI, earnings and dividends, which allows for improved predictions of future ROI, we can write E( ROI t ) = ROI = r Using the restrictions that b1 = 0, b2 = 1 + r and b3 = − r, it follows that Et −1 ( Rt ) = c z Z t −1  1  Pt −1      Dt −1 = (co c1 c2 ) Pt −1     Y   t −1 P  t −1   1  Pt −1      Dt −1 = [0 − r (1 + r )] Pt −1     Y   t −1 P  t −1  = 1 r  (1 + r ) (1 + r )Yt −1 − rDt −1 ] = Yt −1 − Dt −1  [ Pt −1 Pt −1  r  = r . Pt −1 = r Pt −1 ∴

Equation (A4.4) becomes Rt − Et −1 ( Rt ) = c K ~K ~ t  et  P  t −1  = [ c3 c 4 ]    εt   Pt −1  30 Source: http://www.doksinet i.e Rt − r = 0  = ∴ Rt = r +  εt    (1 + r )   Pt −1  for b1 = 0, b2 = 1 + r and b 3 = − r, r   ε t     Pt −1  (1 + r ) ε t r Pt −1 (1 + r ) ε t r Pt −1 31 Source: http://www.doksinet APPENDIX 5 LINTNER MODEL FOR DIVIDENDS AND RANDOM WALK FOR EARNINGS Lintner Model for Dividends: General Generating Process for Dividends: Dt = a + cτYt + (1 − c) Dt −1 + et Dt = a1 + a2 Yt + a3 Dt −1 + et Equating the above two equations results in a1 = a , a2 = cτ , a3 = (1 − c) Random Walk for Earnings: General Generating Process for Earnings: Yt = Yt −1 + ε t Yt = b1 + b2 Yt −1 + b3 Dt −1 + ε t Equating the above two equations results in b1 = 0 , b2 = 1 , b3 = 0 Recall equation (A1.4), Pt = − − (1

− b2 )( a1 + a2 b1 ) + a2 b1b2 − r[(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] (1 − b2 ){[(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a1 + a2 b1 ] + (1 + r )a2 b1b2 } {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } {(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 } − a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )} {(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 } {(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 } (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 D + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] t (1 + r )a2 b2 Y + [(1 + r − b2 )(1 + r − a2 + a2 b3 ) − a2 b2 b3 ] t + + .(A51) After substitution for a 1 , a 2 , a 3 , b 1 , b 2 and b 3 the constant term is undefined. Consider the limit of each of the three terms comprising the constant as b 2 approaches 1 with the above restrictions applying to the other parameters. a(1 − b2 ) (1 − b2 )( a1 + a2 b1 ) + a2 b1 b2 = r[(1 − b2 )(1 − a3 − a2 b2 b3 )

− a2 b2 b3 ] rc(1 − b2 ) i.e ∴ lim a(1 − b2 ) 0 = (undefined) b2 1 rc(1 − b2 ) 0 32 Source: http://www.doksinet By L’Hopital’s Rule: lim lim a(1 − b2 ) −a a = = b2 1 rc(1 − b2 ) b2 1 − rc rc (1 − b2 ){[(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a1 + a2 b1 ] + (1 + r )a2 b1b2 } [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b1b2 ][(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] = a(1 − b2 )(1 + r − b2 )(1 − c) c(1 − b2 )(1 + r − b2 )(r + c) a(1 − b2 )(1 + r − b2 )(1 − c) 0 = b2 1 c(1 − b2 )(1 + r − b2 )(r + c) 0 lim (undefined) By L’Hopital’s Rule: lim lim a(1 − c)(1 + r − 2b2 − rb2 + b22 ) a(1 − c)( −2 − r + 2b2 ) = 2 b2 1 c(r + c)(1 + r − 2b2 − rb2 + b2 ) b2 1 c(r + c)( −2 − r + 2b2 ) = a(1 − c) c( r + c ) a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )} [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] = τb22 b1 (1 − b2 )(1

+ r − b2 ) lim τb22 b1 0 = (undefined) b2 1 (1 − b2 )(1 + r − b2 ) 0 By L’Hopital’s Rule: lim lim τb22 b1 2τb2 b1 = =0 b2 1 (1 − b2 )(1 + r − b2 ) b2 1 −2 − r + 2b2 ∴ Constant = a a(1 − c) a(1 + r ) − = rc c(r + c) r (r + c) After substitution for the parameters in the coefficients of the D t and Y t terms we have 33 Source: http://www.doksinet Pt = (1 − c) a(1 + r ) cτ (1 + r ) + Yt + Dt (r + c) r (r + c) r (r + c) 34 Source: http://www.doksinet Further, recall equation (A1.5), Rt = (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ∆Dt + [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] Pt −1 D ∆Yt (1 + r )a2 b2 + t [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] Pt −1 Pt −1 .(A52) + and equation (A4.4) where Rt =  1 [(1 + r − b2 )( a3 + a 2 b3 ) + a 2 b2 b3 ] [ a1 + a 2 b1 ] + (1 + r )a2 b1 b2 + ( a1 + a2 b1 ) +   Pt −1  (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  [(1 + r

− b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a3 + a2 b3 − 1] + (1 + r )a2 b2 b3 D + + ( a3 + a2 b3 ) t −1 +  (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3   Pt −1 [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ]a2 b2 + (1 + r )a2 b2 (b2 − 1) Y + + a2 b2  t −1 +  (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  Pt −1   (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3  e + + 1 t +   (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  Pt −1 [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ]a2 + (1 + r )a2 b2  ε + + a2  t (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  Pt −1  , or  (1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3  e Rt = r +  + 1 t +   (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  Pt −1  ε  a2 [(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] + (1 + r )a2 b2 + + a2  t (1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3  Pt −1

 .(A53) After substitution for the parameters a 1 , a 2 , a 3 , b 1 , b 2 and b 3 in (A5.1), (A52) and (A53) we have: Rt = D cτ (1 + r ) ∆Yt (1 − c) ∆Dt + + t , and r (r + c) Pt −1 (r + c) Pt −1 Pt −1 Rt = r + cτ (1 + r ) 2 ε t (1 + r ) et + r (r + c) Pt −1 (r + c) Pt −1 35 Source: http://www.doksinet APPENDIX 6 LINTNER MODEL FOR DIVIDENDS AND RANDOM WALK WITH DRIFT FOR EARNINGS Lintner Model for Dividends : General Generating Process for Dividends : Dt = a + cτYt + (1 − c) Dt −1 + et Dt = a1 + a2 Yt + a3 Dt −1 + et Equating the above two equations results in a1 = a , a2 = cτ , a3 = (1 − c) Yt = b + Yt −1 + ε t Yt = b1 + b2 Yt −1 + b3 Dt −1 + ε t Random Walk with Drift for Earnings: General Generating Process for Earnings: Equating the above two equations results in b1 = b , b2 = 1 , b3 = 0 After substitution in (A5.1), (A52) and (A53) the constant term in (A51) is undefined. We now consider the limit of each of the three terms

comprising the constant as b 2 approaches 1. i. e (1 − b2 )( a1 + a2 b1 ) + a2 b1 b2 (1 − b2 )( a + cτb) + cτb) = r[(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] rc(1 − b2 ) lim (1 − b2 )( a + cτb) + cτb cτb = b2 1 rc(1 − b2 ) o (undefined) By L’Hopital’s Rule: lim (1 − b2 )( a + cτb) a + cτb a + cτb = = b2 1 b2 1 rc(1 − b2 ) rc rc lim (1 − b2 ){[(1 + r − b2 )( a3 + a2 b3 ) + a2 b2 b3 ] [ a1 + a2 b1 ] + (1 + r )a2 b1 b2 } [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b1b2 ] [(1 + r − b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] (1 − b2 ){(1 + r − b2 )(1 − c)( a + cτb) + (1 + r )cτbb2 } = [c(1 − b2 ) − cτbb2 ] [(1 + r − b2 )(r + c)] lim (1 − b2 ){(1 + r − b2 )(1 − c)( a + cτb) + (1 + r )cτbb2 } 0 = =0 b2 1 − cτbr (r + c) [c(1 − b2 ) − cτbb2 ] [(1 + r − b2 )(r + c)] 36 Source: http://www.doksinet a2 b2 {(1 + r )b3 ( a1 + a2 b1 ) + b1 b2 (1 + r − a3 )} [(1 − b2 )(1 − a3 − a2 b3 ) − a2 b2 b3 ] [(1 + r −

b2 )(1 + r − a3 − a2 b3 ) − a2 b2 b3 ] cτb(r + c)b22 = c(r + c)(1 + r − b2 )(1 − b2 ) cτb(r + c)b22 cτb(r + c) ( undefined) = b2 1 c(r + c)(1 + r − b2 )(1 − b2 ) 0 lim By L’Hopital’s Rule: lim 2τbb2 −2τb = b2 1 ( −2 − r + 2b2 ) r ∴ Constant = a + cτb 2τb − rc r = a + cτb − 2cτb rc = a − cτb rc After substitution for the parameters in the coefficients of the D t and Y t terms we have: Pt = (1 − c) a − cτb cτ (1 + r ) + Yt + Dt (r + c) rc r (r + c) Further, Rt = D cτ (1 + r ) ∆Yt (1 − c) ∆Dt + + t , and r (r + c) Pt −1 (r + c) Pt −1 Pt −1 Rt = r + (1 + r ) et cτ (1 + r ) 2 ε t + r (r + c) Pt −1 (r + c) Pt −1 37 Source: http://www.doksinet APPENDIX 7 LINTNER MODEL FOR DIVIDENDS AND RANDOM WALK WITH GROWTH FOR EARNINGS Lintner Models for Dividends: Dt = a + cτYt + (1 − c) Dt −1 + et General Generating Process for Dividends: Dt = a1 + a2 Yt + a3 Dt −1 + et Equating the above two equations results

in a1 = a , a2 = cτ , a3 = (1 − c) Yt = (1 + g)Yt −1 + ε t Yt = b1 + b2 Yt −1 + b3 Dt −1 + ε t Random Walk with Growth for Earnings: General Generating Process for Earnings: Equating the above two equations results in b1 = 0 , b2 = 1 + g , b3 = 0 After substitution in (A5.1), (A52) and (A53) we have Pt = a(1 + r ) cτ (1 + r )(1 + g) (1 − c) + Yt + Dt r (r + c) (r − g)(r + c) (r + c) Rt = D (1 − c) ∆Dt cτ (1 + r )(1 + g) ∆Yt + + t (r + c) Pt −1 (r − g)(r + c) Pt −1 Pt −1 cτ (1 + r ) 2 ε t (1 + r ) et =r+ + (r + c) Pt −1 (r − g)(r + c) Pt −1 38