نرخ رشد پایدار، نرخ رشد بهینه، و نسبت پرداخت بهینه: رویکرد بهینه سازی مشترک

This study investigates the investment decision and dividend policy jointly from a non-steady state to a steady state. We extend Higgins, 1977, Higgins, 1981 and Higgins, 2008 sustainable growth rate model and develop a dynamic model which jointly optimizes the growth rate and payout ratio. We optimize the firm value to obtain the optimal growth rate in terms of a logistic equation and find that the steady state growth rate can be used as the benchmark for the mean-reverting process of the optimal growth rate. We also investigate the specification error of the mean and variance of dividend per share when introducing the stochastic growth rate. Empirical results support the mean-reverting process of the growth rate and the importance of covariance between the profitability and the growth rate in determining dividend payouts. The intertemporal behavior of the covariance may shed some light on the fact of disappearing dividends over decades.

The relationship between the optimal dividend policy and the growth rate has been analyzed at length by Gordon, 1962, Lintner, 1964, Lerner and Carleton, 1966, Miller and Modigliani, 1961 and Miller and Modigliani, 1966, and others. In addition, Higgins, 1977, Higgins, 1981 and Higgins, 2008 derives a sustainable growth rate assuming that a firm can use retained earnings and issue new debt to finance the growth opportunities of the firm. Furthermore, a growing body of empirical literature focuses on the relationship between the optimal dividend payout policy and the growth rate. For example, Rozeff (1982) showed that the optimal dividend payout is related to the fraction of insider holdings, the growth of the firm, and the firm’s beta coefficient. Grullon et al., 2002, DeAngelo et al., 2006 and DeAngelo and DeAngelo, 2006 suggest that increases in dividends convey information about changes in a firm’s life cycle from a higher growth phase to a lower growth phase. Benartzi et al., 1997 and Grullon et al., 2002 show that dividend changes are related to the change in the growth rate and the change in the rate of return on assets. Fama and French (2001) use a free cash flow hypothesis to find that firms tend to pay dividends when they experience high profitability and low growth rates. Finally, DeAngelo and DeAngelo, 2006, Blau and Fuller, 2008 and Lee et al., 2011 reexamine issues of dividend policy based upon the dividend flexibility hypothesis. Although pioneering in their efforts, these authors focus their analyses at the equilibrium point, but not on the time path that leads to the equilibrium. More importantly, these authors do not jointly consider the investment decision and finance decision at the same time. This study attempts to provide a way of jointly finding the optimal investment and payout decisions from a non-steady state to a steady state. We start by extending Higgins, 1977, Higgins, 1981 and Higgins, 2008 sustainable growth rate model by allowing new equity to be issued. Under this framework, the new assets of a firm can be financed by new debt, external equity, and internal equity through retained earnings. We also assume that the growth rate varies over time and profitability follows a stochastic process. A dynamic model is then developed expressing a firm’s risk-adjusted stock price. Under this dynamic model, a firm’s investment decision and financing decision will interact with each other and the time path of the optimal growth rate is traced and analyzed. By maximizing the current stock price, we derive closed forms of the optimal growth rate and the optimal payout ratio. We find that the optimal growth rate is a logistic equation with characteristics of a mean-reverting process. We show that a specification error of the expected dividend per share may exist when a stochastic growth rate is introduced. The model shows that the covariance between the rate of return on equity and the total asset growth, which represents the interaction between the profitability risk and the growth risk, has a negative impact on dividend payout policy. Ignoring the covariance term may therefore result in a misspecification error in determining the dividend payout policy. Such a negative impact in the covariance between the profitability risk and the growth risk may provide an alternative explanation for the declining number of firms paying cash dividends documented by Fama and French, 2001 and DeAngelo et al., 2004.1 We further develop three testable hypotheses and determine their implications based on our theoretical model. Hypothesis 1 investigates the nature of the mean-reverting process of the optimal growth rate of a firm. Hypothesis 2 examines whether the covariance between the profitability risk and the growth risk can be one of the key factors in determining a firm’s dividend payout ratio. Hypothesis 3 examines whether the covariance term affects a firm’s decision to stop or initiate a dividend payout. To tackle our research questions, we empirically test the hypotheses using 31,255 firm-years of data during the period from 1969 to 2011. We use a partial adjustment model to test the mean-reverting process of the growth rate as indicated in our theoretical model while acknowledging that the model has been used passively in testing the adjustment process of dividend policy (see Fama and Babiak, 1968, Lee et al., 1987 and Andres et al., 2009) and capital structure (see Flannery and Rangan, 2006, Lemmon et al., 2008 and Huang and Ritter, 2009). In addition, we use a structural change model and a probit model to examine the determinants and decisions of dividend policy. Empirical results support the mean-reverting process in total asset growth and the importance of the interaction between the profitability risk and the growth risk in determining the dividend payout ratio. We also find that a positive covariance term can increase the probability of stopping a dividend payout while a negative covariance term cannot assure an increase in the probability of initiating a dividend payout. These results may, to some extent, explain the puzzle of disappearing dividends over past decades (see Fama and French, 2001). The present study deviates from earlier studies in two major aspects. First, this study presents a fully dynamic model for determining the optimal growth rate and the optimal dividend policy under stochastic conditions. Second, the focus is on tracing the time optimal path of the relevant decision variables and exploring their inter-temporal dependencies. The entire analysis in this paper is under the joint consideration of the investment and finance decisions, and is carried out in the stochastic control theory framework. The implications induced from our theoretical model are confirmed by the empirical work. We believe that this study is the first of its kind to provide a theoretical basis for combining the investment decision and the finance decision under uncertainties. The paper is organized as follows. In Section 2, we develop a theoretical model jointly considering the investment decision and the finance decision. The optimal growth rate is obtained and analyzed. In Section 3, we present the optimal dividend payouts under conditions of a static growth rate and a dynamic growth rate. In Section 4, we develop three hypotheses and discuss the empirical results of optimal growth rate and dividend payout policy. The conclusion is in Section 5.

In this paper, our concern has been to sharpen the focus on corporate decisions about optimal growth and dividend payout policy. The interdependence of corporate decisions on these two important policy variables is explicitly recognized in deriving the solution to the problem of maximizing the present value of a firm. In more detail, we examine the time path of the optimal growth rate, starting from the stochastic initial conditions, and approaching its steady state value. We find that the optimal growth rate follows a mean-reverting process. Important insights are developed into the effects of the degree of market perfection, rate of return on equity, and initial growth rate on the optimal growth rate. Furthermore, we derive the optimal payout policy under most genera1 conditions with regard to the market imperfection, nonstationarity, and the stochasticity of the growth rate and the profitability rate. In particular, the error introduced in the optimal payout ratio by the misspecification of the growth rate as deterministic is identified and related to its initial stochastic conditions. The model we develop in this paper explicitly brings out the influence of covariance between risks of non-stationary profitability and growth rate on the optimal dividend policy. Empirical work confirms the mean-reverting process of the optimal growth rate and the importance of the interaction between the profitability risk and the growth risk. The partial adjustment regressions show that, on average, firms will partially adjust their growth rates and take five years to make the growth rate reach the optimal target rate. The fixed-effects model with a structural change not only confirms the flexibility and the free cash flow hypotheses proposed by Lee et al. (2011), but also shows that the covariance between the profitability and the growth rate is one of the determinants of the dividend payout. In addition, the probit model shows that the interaction between the profitability risk and the growth rate can affect a firm’s decisions on not only stopping dividend payments but on not initiating dividend payments. Such an irreversible effect along with the intertemporal behavior of the covariance term may shed some light on the puzzle of disappearing dividends over the last 40 years. As mentioned earlier, important work in determining the optimal equilibrium values of corporate decision variables has been done by Gordon, 1962, Lintner, 1964, Miller and Modigliani, 1961, Miller and Modigliani, 1966, DeAngelo et al., 1996, Fama and French, 2001, DeAngelo and DeAngelo, 2006 and Lee et al., 2011. The need, however, for further research in tracing the optimal time path under uncertainty leading to these equilibrium values and the relevant stability conditions remains, and the present paper is a modest step forward in that direction.