Dividend growth, stock valuation, and long-run risk

Abstract

In this paper, we integrate the long-run concept of risk into the stock valuation process. We use the intertemporal consumption capital asset pricing model to demonstrate that a stock’s long-run dividend growth is negatively related to its current dividend-price ratio and positively related to its long-run covariance between dividends and consumption. Then, we show that the equilibrium price of a stock is determined by its current dividend, long-run dividend growth, and long-run risk. In all, our work suggests that risk cumulated over many periods represents an important parameter in assessing the theoretical value of a firm.

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Notes

  1. 1.

    See Rubinstein (1976).

  2. 2.

    See Bakshi and Chen (2005, p. 111 and p. 112).

  3. 3.

    See also Parker and Julliard (2005), Hansen et al. (2008), Malloy et al. (2009), Beeler and Campbell (2009), and Bansal and Kiku (2011).

  4. 4.

    The operators E t , VAR t , and COV t refer respectively to mathematical expectations, variance, and covariance, where index t implies that we consider the available information at time t (t = 0, 1, 2, …, ∞). The tilde (~) indicates a random variable.

  5. 5.

    See Rubinstein (1976).

  6. 6.

    The premium (\( U\prime \)) is a derivative of a function.

  7. 7.

    See, for example, Huang and Litzenberger (1988, page 202).

  8. 8.

    If x and y are bivariate normally distributed: \( COV(y,\;f(x)) = E(f\prime (x))\;COV(y,\;x) \). See Huang and Litzenberger (p. 101).

  9. 9.

    Rozeff (1982), Eades (1982), Baskin (1989), Gillet et al. (2008), Carter and Schmidt (2008), and many others, present similar results regarding the dividend-risk relationship.

  10. 10.

    See, for example, Esteve and Prats (2010), for a more complete definition of cointegration or our Eqs. 8 and 9.

  11. 11.

    In this paper, the indice m replace i, when refering to market portfolio.

  12. 12.

    See, in particular, Bansal et al. (2002, p. 5 and p. 6).

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Acknowledgements

I would like to thank Guy Charest, from Université du Québec à Montréal, for helpful comments. I also thank John Y. Campbell, from Harvard University, for helpful suggestions and references.

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Correspondence to Claude Bergeron.

Appendices

Appendix A

In this appendix, we assume that the dividend growth rate on a stock may be stated as a linear function of the K-periods average of future consumption rates, plus a disturbance term.

From Eq. 21 we have:

$$ \sum\limits_{{t = 0}}^{{T - 1}} {E[{{\tilde{g}}_{{i,\;t + 1}}}]} = \sum\limits_{{t = 0}}^{{T - 1}} {\left( { - 1 + \frac{{{\lambda_{{1t}}}}}{{(1 + \theta_i^{{ - 1}})}} + \gamma \frac{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma - 1}}}]}}{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma }}}]}}COV[{{\tilde{g}}_{{t + 1}}},\;{{\tilde{g}}_{{i,\;t + 1}}}]} \right)} . $$
(A1)

Introducing Eq. 31 in A1 gives:

$$ \begin{array}{*{20}{c}} {\sum\limits_{{t = 0}}^{{T - 1}} {E[{{\tilde{g}}_{{i,t + 1}}}] = } } \hfill \\ {\sum\limits_{{t{ } = { }0}}^{{T - 1}} {\left( { - 1 + \frac{{{\lambda_{{1t}}}}}{{(1 + \theta_i^{{ - 1}})}} + \gamma \frac{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma - 1}}}]}}{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma }}}]}}COV[{{\tilde{g}}_{{t + 1}}},\;{a_i} + {b_i}{{\tilde{x}}_t} + {{\tilde{\varepsilon }}_{{i,\;t + 1}}}]} \right)} .} \hfill \\ \end{array} $$
(A2)

The properties of the covariance show that:

$$ \sum\limits_{{t = 0}}^{{T - 1}} {E[{{\tilde{g}}_{{i,\;t + 1}}}]} = \sum\limits_{{t = 0}}^{{T - 1}} {\left( { - 1 + \frac{{{\lambda_{{1t}}}}}{{(1 + \theta_i^{{ - 1}})}} + \gamma \frac{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma - 1}}}]}}{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma }}}]}}COV[{{\tilde{g}}_{{t + 1}}},\;{{\tilde{x}}_t}]\;{b_i}} \right)} $$
(A3)

or

$$ \sum\limits_{{t = 0}}^{{T - 1}} {E[{{\tilde{g}}_{{i,\;t + 1}}}]} = - T + {(1 + \theta_i^{{ - 1}})^{{ - 1}}}\sum\limits_{{t = 0}}^{{T - 1}} {{\lambda_{{1t}}} + {b_i}\sum\limits_{{t = 0}}^{{T - 1}} {\lambda_{{2t}}^{\prime }} } . $$
(A4)

where \( \lambda_{{2t}}^{\prime } \equiv COV[{\tilde{g}_{{t + 1}}},\;{\tilde{x}_t}]\gamma E[{(1 + {\tilde{g}_{{t + 1}}})^{{ - \gamma - 1}}}]/E[{(1 + {\tilde{g}_{{t + 1}}})^{{ - \gamma }}}] \). Multiplying by T −1 in each side of Eq. A4 yields:

$$ {\bar{g}_i} = - 1 + {\lambda_1}{(1 + {D_{{i0}}}/{P_{{i0}}})^{{ - 1}}} + \lambda_2^{\prime }{b_i} $$
(A5)

where \( \lambda_2^{\prime } \equiv \sum\limits_{{t = 0}}^{{T - 1}} {\lambda_{{2t}}^{\prime }} /T \). Thus:

$$ {P_{{i0}}} = \frac{{1 + {{\bar{g}}_i} - \lambda_2^{\prime }{b_i}}}{{{\lambda_1} + \lambda_2^{\prime }{b_i} - {{\bar{g}}_i} - 1}}{D_{{i0}}}. $$
(A6)

Appendix B

We can also facilitate our estimation of parameters \( \lambda_{{2t}}^{\prime } \), if we assume the existence of a riskless asset. In fact, for the market portfolio, Eq. 31 shows that:

$$ {\tilde{g}_{{m,\;t + 1}}} = {a_m} + {b_m}{\tilde{x}_t} + {\tilde{\varepsilon }_{{m,\;t + 1}}}, $$
(B1)

in which \( E[{\tilde{\varepsilon }_{{m,\;t + 1}}}] = 0 \), \( E[{\tilde{g}_{{t + k}}}{\tilde{\varepsilon }_{{m,\;t + 1}}}] = 0 \), and:

$$ {b_m} = \frac{{COV[{{\tilde{x}}_t},\;{{\tilde{g}}_{{m,\;t + 1}}}]}}{{{\sigma^2}[{x_t}]}}. $$

For the market portfolio, Eqs. 8 and 20 also indicate that:

$$ E[1 + {\tilde{g}_{{m,\;t + 1}}}] = \frac{{{\lambda_{{1t}}}}}{{(1 + {d_{{mt}}})}} - \gamma \frac{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma - 1}}}]}}{{E[{{(1 + {{\tilde{g}}_{{t + 1}}})}^{{ - \gamma }}}]}}COV[({\tilde{g}_{{t + 1}}},\;{\tilde{g}_{{m,\;t + 1}}}]. $$
(B2)

Introducing Eq. B1 in B2 gives, after simplifications:

$$ E[{\tilde{g}_{{m,\;t + 1}}}] = - 1 + {\lambda_{{1t}}}{(1 + {d_{{mt}}})^{{ - 1}}} + \lambda_{{2t}}^{\prime }{b_m}. $$
(B3)

Thus:

$$ \lambda_{{2t}}^{\prime } = [1 + E[{\tilde{g}_{{m,\;t + 1}}}] - (1 + {r_{{f,\;t + 1}}}){(1 + {d_{{mt}}})^{{ - 1}}}]/{b_m}. $$
(B4)

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Bergeron, C. Dividend growth, stock valuation, and long-run risk. J Econ Finan 37, 547–559 (2013). https://doi.org/10.1007/s12197-011-9196-5

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Keywords

  • Valuation Model
  • Dividends
  • Long-Run Risk
  • Intertemporal Model
  • CCAPM

JEL Classification

  • D91
  • G12