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Dividend policy and tax structure
Economics Letters North-Holland
DIVIDEND
POLICY
Kian-Guan
269
31 (1989) 269-272
AND TAX STRUCTURE
LIM
National
University of Singapore,
Received Accepted
27 January 1989 8 May 1989
Singapore,
MII,
Republic of Singapore
We show that a certain class of progressive tax structure induces an optimal dividend The theoretical model is consistent with empirically observed dividend policies.
policy which is autoregressive
in nature.
1. Introduction A classic paper by Lintner (1956) provided the empirical observation that dividend depends in part on the firm’s current earnings and in part on the dividend for the previous year. Fama and Babiak (1968) confirmed this observation in a study of 392 companies in the United States between 1946 and 1964. They found that the frequency of a dividend increase depended on both the number of occasions on which earnings had risen and on how recently these had risen. The evidence suggested the following empirical model for the determination of dividend policy: D ItI =XE,+,+(l-X)D,,
O
where D,+ I and El+, are the dividend and earning in period t + 1. However, there has been no theoretical model which is consistent with this empirical evidence. Much of the literature veered towards the issue of the relevance or irrelevance of dividends raised in the seminal paper by Miller and Modigliani (1961). When the tax effects are considered, most papers, except Miller and Scholes (1978) etc., indicated that dividends are less desirable to investors. This arises from the preferential tax treatment of capital gains over dividend income. If dividend policy is irrelevant, which means that it has no effect on security valuation, then it is of no consequence to analyse the empirical evidence we observed. The observed policy would be just a matter of folklore. But if dividend policy is relevant, then it is important to explain why de facto dividend policy followed the particular autoregressive structure on current and past earnings. Brennan (1970) developed a two-period model which showed that high dividends fetched an added premium to the required rate of return. Litzenberger and Ramaswamy (1979) also provided an analysis and empirical results which supported the premium hypothesis. In this paper we discuss a simple multi-period equilibrium model where the representative shareholder chooses an optimal dividend policy. We show that a certain class of progressive tax structure induces an optimal dividend policy identical with the empirical observation. This tax structure is typical of the existing tax structure. The relationship between the tax structure and the implied dividend policy is presented in the model. We present the model in the next section, and also analyse the implications. Section 3 concludes. 01651765/89/$3.50
0 1989, Elsevier Science Publishers
B.V. (North-Holland)
K.-G. Lim / Dividendpolicy
270
and tax structure
2. The model There is a representative firm faced with an investment opportunity set with a stochastic return kt in period t, for each t. The firm issues a share to a representative shareholder who maximizes the after-tax present value of the firm,
subject
to the stochastic
equation
of motion
of firm value
Vt+1 =(v,4,)(1+&)
with V. given. Here 6 is the discount rate on the after-tax dividend income. This is lower the higher the cost of funds in the investment. The function r( .) is concave, strictly increasing, and twice continuously differentiable, and maps the pre-tax dividends into the after-tax dividends. It characterizes the tax structure in force. _eo( .) is the shareholder’s expectation conditional on market information available at t = 0, and R, is the real rate of return on the investment project the firm undertakes. This value maximization program is commonly used elsewhere in the context of price determination, for example, Shiller (1981). In this model we abstract from the issue of tax income re-distribution by the tax authority. The shareholder is solely concerned about his after-tax dividend income. The effect of capital gains tax is not explicitly captured in the model, but is included in the discount. The equation of motion supposes that the firm finances its investment totally from retained earnings. ’ The model is essentially a discounted dynamic programming problem, and similar structures occur in the consumption models of Lucas (1978) and Prescott and Mehra (1980). Under sufficient regularity conditions, 2 an optimal solution to the above problem exists, and the optimality condition is
where r’( 0,) is the after-tax marginal income to the investor when dividend is increased in period t. This optimality equation can be loosely interpreted as equating the marginal rate of substitution of after-tax current for future income to the project’s return. The sequence of dividend choices which are optimally chosen should then satisfy the equation. Next we show how the optimal dividend policy is found given the tax structure. Suppose the income tax schedule is progressive. In other words, it is an increasing convex function of dividend income. Then the after-tax income function is increasing and concave. In particular, if this function r( .) is of the form log D,“, where k is any finite constant, then we show that an optimal dividend policy exists in the multi-period problem. Without loss of generality we shall use the natural logarithm throughout; the linear transform to other bases is quite straightforward. We state and prove two propositions in the rest of this section. First, we want to relate the multi-period model to ’ We can allow for outside borrowing
in the following way. Consider such a borrowing, B, to fetch a return &II. After paying and identically the cost, the increment to y+, through such added debt is j,+t. Assume this increment is independently distributed. Then all the analyses in this paper can be trivially extended. * We may assume 8, is independently and identically distributed. Furthermore, lim, _ + m S,V, = 0.
271
K. -G. Lim / Dividend policy and tax structure
the standard irrelevance result of Miller and Modigliani. This will indicate the plausibility of our approach. Second, we show the main proposition of this paper, that the autoregressive dividend policy which has been empirically documented can be explained theoretically. In Proposition 1 below, we revert to the no-tax situation. Proposition 1. The value of the firm is independent funds is equal to the expected rate of return. Proof
When there is no tax, the optimality c,[S(l
+ &)]
of the firm’s dividend policy, provided
condition
the cost of
becomes
= 1.
The cost of funds can be written as 6-l i.i.d. If this condition is satisfied, then the Since the optimality condition is satisfied the value of the firm is independent of its
1 which is e(i). The subscript t is dropped because i?, is variables { 0,) do not appear in the optimality condition. independent of the dividend policy, we can conclude that dividend policy. Q.E.D.
This result is not surprising apropos of Miller and Modigliani. However, when there is taxation on dividend, the existing literature is mainly divided into two categories: One school argues that dividends are too costly because shareholders can always realize higher capital gains in lieu of dividends, the other contends that even with tax, dividend policy is still irrelevant in some sense. Brennan and Litzenberger and Ramaswamy wrote papers addressing the differential impact of that in dividends, implying its higher cost to the firm. Black and Scholes (1974) explained equilibrium tax clienteles formed made it irrelevant to consider whether adjusting dividend up or down would improve any firm’s value. Indeed switching either way may diminish the value since the firm has come to attract its clientele in equilibrium. Miller and Scholes also suggested some institutional reasons why issuing dividends may not be relatively more costly to the investors. However, in all, there is little if any discussion about why empirically a distinct policy has been observed. 3 In the next proposition we show that the autoregressive dividend policy which has been empirically observed by Lintner, Fama and Babiak, and others, is an optimal policy in our model. Proposition 2. If the progressive tax is such that after-tax of the firm is maximized when the dividend follows D f+l = Xl?,,,
dividend income is log D, V t, then the value
+ (1 -X)0,,
where h = 1 - S. Proof Substitute D-’ for 7’(D) in the optimality equation. The optimal dividend policy function is then D, = AK V t. Put this in the optimality equation to obtain X = 1 - 8. Now the firm’s earning E ItI can be expressed as V,, 1 - (y - 0,). Therefore, D 1+1= Put v = A-‘0,
w+1
+h(v-D,).
in the last equation,
’ Of course, the autoregressive over all firms.
=AE,+,
nature
hence.
of the dividend
Q.E.D. payouts
could be interpreted
as merely
a statistical
aggregation
result
272
K.-G. Lim / Diuidendpolicy
and tax structure
In equilibrium the cost of funds can be derived from the result 8 = 1 - X. More important, the optimal dividend payout X is seen to increase when the cost of funds increases. Intuitively this happens when the firm finds it expensive to retain earnings for future investments. When this is so, the amount of dividends would depend largely on current earnings. On the other hand, when it is cheap to use retained earnings, the dividends issued would more likely follow closely that of the past dividend.
3. Conclusion We show in this paper that a simple multi-period equilibrium model can be constructed in which an optimal dividend policy exists. Moreover, under a progressive tax scheme which is quite typical of existing regimes, the optimal dividend policy is exactly that which is empirically observed in the literature. The empirically observed autoregressive nature of dividends does not have to be treated as a folklore; we provide a rational explanation. Although our model does not deal with capital gains tax and external borrowings explicitly, the multi-period nature of the model is well suited to the question at hand, which essentially begs an understanding of the time series property of dividends. Existing equilibrium models, though richer in details, nevertheless do not have the multi-period framework. Further extensions of our model along the details explored by the other papers seem to be useful, although the mathematical difficulties of dynamic programming may eventually prove the task to be intractable analytically.
References Black, F. and M.S. Scholes, 1974, The effects of dividend yield and dividend policy on common stock prices and returns, Journal of Financial Economics, December l-22. Brennan, M.J., 1970, Taxes, market valuation and corporate financial policy, National Tax Journal, December, 417-427. Fama, E.F. and H. Babiak, 1968, Dividend policy: An empirical analysis, Journal of American Statistical Association, December, 1132-1161. Lintner, J., 1956, Distributions of incomes of corporations among dividends, retained earnings, and taxes, American Economic Review, May, 97-113. Litzenberger, R. and K. Ramaswamy, 1979, The effects of personal taxes and dividends on capital asset prices: Theory and empirical evidence, Journal of Financial Economics, 163-195. Lucas, R., Jr., 1978, Asset prices in an exchange economy, Econometrica. Miller, M.H. and F. Modigliani, 1961, Dividend policy, growth and the valuation of shares, Journal of Business, October, 411-433. Miller, M.H. and MS. Scholes, 1978, Dividends and taxes, Journal of Financial Economics, December, 333-364. Prescott, E.C. and R. Mehra, 1980, Recursive competitive equilibrium: The case of homogeneous households, Econometrica, Vol. 48,1365-1380. Shiller, R., 1981, Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review 7. 421-436.
DIVIDEND
POLICY
Kian-Guan
269
31 (1989) 269-272
AND TAX STRUCTURE
LIM
National
University of Singapore,
Received Accepted
27 January 1989 8 May 1989
Singapore,
MII,
Republic of Singapore
We show that a certain class of progressive tax structure induces an optimal dividend The theoretical model is consistent with empirically observed dividend policies.
policy which is autoregressive
in nature.
1. Introduction A classic paper by Lintner (1956) provided the empirical observation that dividend depends in part on the firm’s current earnings and in part on the dividend for the previous year. Fama and Babiak (1968) confirmed this observation in a study of 392 companies in the United States between 1946 and 1964. They found that the frequency of a dividend increase depended on both the number of occasions on which earnings had risen and on how recently these had risen. The evidence suggested the following empirical model for the determination of dividend policy: D ItI =XE,+,+(l-X)D,,
O
where D,+ I and El+, are the dividend and earning in period t + 1. However, there has been no theoretical model which is consistent with this empirical evidence. Much of the literature veered towards the issue of the relevance or irrelevance of dividends raised in the seminal paper by Miller and Modigliani (1961). When the tax effects are considered, most papers, except Miller and Scholes (1978) etc., indicated that dividends are less desirable to investors. This arises from the preferential tax treatment of capital gains over dividend income. If dividend policy is irrelevant, which means that it has no effect on security valuation, then it is of no consequence to analyse the empirical evidence we observed. The observed policy would be just a matter of folklore. But if dividend policy is relevant, then it is important to explain why de facto dividend policy followed the particular autoregressive structure on current and past earnings. Brennan (1970) developed a two-period model which showed that high dividends fetched an added premium to the required rate of return. Litzenberger and Ramaswamy (1979) also provided an analysis and empirical results which supported the premium hypothesis. In this paper we discuss a simple multi-period equilibrium model where the representative shareholder chooses an optimal dividend policy. We show that a certain class of progressive tax structure induces an optimal dividend policy identical with the empirical observation. This tax structure is typical of the existing tax structure. The relationship between the tax structure and the implied dividend policy is presented in the model. We present the model in the next section, and also analyse the implications. Section 3 concludes. 01651765/89/$3.50
0 1989, Elsevier Science Publishers
B.V. (North-Holland)
K.-G. Lim / Dividendpolicy
270
and tax structure
2. The model There is a representative firm faced with an investment opportunity set with a stochastic return kt in period t, for each t. The firm issues a share to a representative shareholder who maximizes the after-tax present value of the firm,
subject
to the stochastic
equation
of motion
of firm value
Vt+1 =(v,4,)(1+&)
with V. given. Here 6 is the discount rate on the after-tax dividend income. This is lower the higher the cost of funds in the investment. The function r( .) is concave, strictly increasing, and twice continuously differentiable, and maps the pre-tax dividends into the after-tax dividends. It characterizes the tax structure in force. _eo( .) is the shareholder’s expectation conditional on market information available at t = 0, and R, is the real rate of return on the investment project the firm undertakes. This value maximization program is commonly used elsewhere in the context of price determination, for example, Shiller (1981). In this model we abstract from the issue of tax income re-distribution by the tax authority. The shareholder is solely concerned about his after-tax dividend income. The effect of capital gains tax is not explicitly captured in the model, but is included in the discount. The equation of motion supposes that the firm finances its investment totally from retained earnings. ’ The model is essentially a discounted dynamic programming problem, and similar structures occur in the consumption models of Lucas (1978) and Prescott and Mehra (1980). Under sufficient regularity conditions, 2 an optimal solution to the above problem exists, and the optimality condition is
where r’( 0,) is the after-tax marginal income to the investor when dividend is increased in period t. This optimality equation can be loosely interpreted as equating the marginal rate of substitution of after-tax current for future income to the project’s return. The sequence of dividend choices which are optimally chosen should then satisfy the equation. Next we show how the optimal dividend policy is found given the tax structure. Suppose the income tax schedule is progressive. In other words, it is an increasing convex function of dividend income. Then the after-tax income function is increasing and concave. In particular, if this function r( .) is of the form log D,“, where k is any finite constant, then we show that an optimal dividend policy exists in the multi-period problem. Without loss of generality we shall use the natural logarithm throughout; the linear transform to other bases is quite straightforward. We state and prove two propositions in the rest of this section. First, we want to relate the multi-period model to ’ We can allow for outside borrowing
in the following way. Consider such a borrowing, B, to fetch a return &II. After paying and identically the cost, the increment to y+, through such added debt is j,+t. Assume this increment is independently distributed. Then all the analyses in this paper can be trivially extended. * We may assume 8, is independently and identically distributed. Furthermore, lim, _ + m S,V, = 0.
271
K. -G. Lim / Dividend policy and tax structure
the standard irrelevance result of Miller and Modigliani. This will indicate the plausibility of our approach. Second, we show the main proposition of this paper, that the autoregressive dividend policy which has been empirically documented can be explained theoretically. In Proposition 1 below, we revert to the no-tax situation. Proposition 1. The value of the firm is independent funds is equal to the expected rate of return. Proof
When there is no tax, the optimality c,[S(l
+ &)]
of the firm’s dividend policy, provided
condition
the cost of
becomes
= 1.
The cost of funds can be written as 6-l i.i.d. If this condition is satisfied, then the Since the optimality condition is satisfied the value of the firm is independent of its
1 which is e(i). The subscript t is dropped because i?, is variables { 0,) do not appear in the optimality condition. independent of the dividend policy, we can conclude that dividend policy. Q.E.D.
This result is not surprising apropos of Miller and Modigliani. However, when there is taxation on dividend, the existing literature is mainly divided into two categories: One school argues that dividends are too costly because shareholders can always realize higher capital gains in lieu of dividends, the other contends that even with tax, dividend policy is still irrelevant in some sense. Brennan and Litzenberger and Ramaswamy wrote papers addressing the differential impact of that in dividends, implying its higher cost to the firm. Black and Scholes (1974) explained equilibrium tax clienteles formed made it irrelevant to consider whether adjusting dividend up or down would improve any firm’s value. Indeed switching either way may diminish the value since the firm has come to attract its clientele in equilibrium. Miller and Scholes also suggested some institutional reasons why issuing dividends may not be relatively more costly to the investors. However, in all, there is little if any discussion about why empirically a distinct policy has been observed. 3 In the next proposition we show that the autoregressive dividend policy which has been empirically observed by Lintner, Fama and Babiak, and others, is an optimal policy in our model. Proposition 2. If the progressive tax is such that after-tax of the firm is maximized when the dividend follows D f+l = Xl?,,,
dividend income is log D, V t, then the value
+ (1 -X)0,,
where h = 1 - S. Proof Substitute D-’ for 7’(D) in the optimality equation. The optimal dividend policy function is then D, = AK V t. Put this in the optimality equation to obtain X = 1 - 8. Now the firm’s earning E ItI can be expressed as V,, 1 - (y - 0,). Therefore, D 1+1= Put v = A-‘0,
w+1
+h(v-D,).
in the last equation,
’ Of course, the autoregressive over all firms.
=AE,+,
nature
hence.
of the dividend
Q.E.D. payouts
could be interpreted
as merely
a statistical
aggregation
result
272
K.-G. Lim / Diuidendpolicy
and tax structure
In equilibrium the cost of funds can be derived from the result 8 = 1 - X. More important, the optimal dividend payout X is seen to increase when the cost of funds increases. Intuitively this happens when the firm finds it expensive to retain earnings for future investments. When this is so, the amount of dividends would depend largely on current earnings. On the other hand, when it is cheap to use retained earnings, the dividends issued would more likely follow closely that of the past dividend.
3. Conclusion We show in this paper that a simple multi-period equilibrium model can be constructed in which an optimal dividend policy exists. Moreover, under a progressive tax scheme which is quite typical of existing regimes, the optimal dividend policy is exactly that which is empirically observed in the literature. The empirically observed autoregressive nature of dividends does not have to be treated as a folklore; we provide a rational explanation. Although our model does not deal with capital gains tax and external borrowings explicitly, the multi-period nature of the model is well suited to the question at hand, which essentially begs an understanding of the time series property of dividends. Existing equilibrium models, though richer in details, nevertheless do not have the multi-period framework. Further extensions of our model along the details explored by the other papers seem to be useful, although the mathematical difficulties of dynamic programming may eventually prove the task to be intractable analytically.
References Black, F. and M.S. Scholes, 1974, The effects of dividend yield and dividend policy on common stock prices and returns, Journal of Financial Economics, December l-22. Brennan, M.J., 1970, Taxes, market valuation and corporate financial policy, National Tax Journal, December, 417-427. Fama, E.F. and H. Babiak, 1968, Dividend policy: An empirical analysis, Journal of American Statistical Association, December, 1132-1161. Lintner, J., 1956, Distributions of incomes of corporations among dividends, retained earnings, and taxes, American Economic Review, May, 97-113. Litzenberger, R. and K. Ramaswamy, 1979, The effects of personal taxes and dividends on capital asset prices: Theory and empirical evidence, Journal of Financial Economics, 163-195. Lucas, R., Jr., 1978, Asset prices in an exchange economy, Econometrica. Miller, M.H. and F. Modigliani, 1961, Dividend policy, growth and the valuation of shares, Journal of Business, October, 411-433. Miller, M.H. and MS. Scholes, 1978, Dividends and taxes, Journal of Financial Economics, December, 333-364. Prescott, E.C. and R. Mehra, 1980, Recursive competitive equilibrium: The case of homogeneous households, Econometrica, Vol. 48,1365-1380. Shiller, R., 1981, Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review 7. 421-436.