Stochastic dividend discount model: covariance of random stock prices

Abstract

The price of common stocks, defined as the sum of all future discounted dividends, is at the heart of both the dividend discount models (DDM) and the stochastic DDM (SDDM). Gordon and Shapiro (Manag Sci 3:102–110 1956) assume a deterministic and constant dividends’ growth rate, whereas Hurley and Johnson (Financ Anal J 4:50–54 1994, J Portf Manag 25(1)27–31 1998) and Yao (J Portf Manag 23(4)99–103 1997) introduce randomness by letting the growth rate be a finite-state random variable and random dividends behave in a Markovian fashion. In this second case expected stock price is determined, but what if higher-order moments are needed? In order to address a number of financial topics, the present contribution presents an explicit formula for the covariance between (possibly) correlated stock prices.

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Correspondence to Arianna Agosto.

Appendices

Appendix: A

The analytic expression for

$$\mathbb E\left[ \tilde P_{A0}\tilde P_{B0}\right] =\sum\limits_{j = 1}^{+\infty}\sum\limits_{p = 1}^{+\infty } \frac{\mathbb E\left[ \tilde d_{Aj}\tilde d_{Bp}\right]}{(1 + k_{A})^{j}(1 + k_{B})^{p}} $$

is obtained recalling (12). It results that

$$ \mathbb E\left[ \tilde P_{A0}\tilde P_{B0}\right] = d_{A0}d_{B0}\sum\limits_{p = 1}^{+\infty} \left( \underset{\text{\#1}}{\underbrace{\sum\limits_{j = 1}^{p}\frac{(1 + G_{A})^{j}(1 + \bar g_{B})^{p}}{(1 + k_{A})^{j}(1 + k_{B})^{p}}}} + \underset{\text{\#2}}{\underbrace{\sum\limits_{j = p + 1}^{+\infty}\frac{(1 + \bar g_{A})^{j}(1 + G_{B})^{p}}{(1 + k_{A})^{j}(1 + k_{B})^{p}}}}\right). $$
(19)

For ease of notation, let, for i = A,B,

$$\gamma_{\bar g_{i}} = \frac{1 + \bar g_{i}}{1 + k_{i}} \hspace{1cm} \text{and} \hspace{1cm} \gamma_{G_{i}} = \frac{1 + G_{i}}{1 + k_{i}}. $$

Sum #1 becomes

$$\gamma_{\bar g_{B}}^{p}\sum\limits_{j = 1}^{p}\gamma_{G_{A}}^{j} =\frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\left( 1 - \gamma_{G_{A}}^{p}\right)\gamma_{\bar g_{B}}^{p}, $$

while sum #2, that converges to a positive value as soon as \(\bar g_{A} < k_{A}\), reduces to

$$\gamma_{G_{B}}^{p}\sum\limits_{j = p + 1}^{+\infty}\gamma_{\bar g_{A}}^{j} =\frac{\gamma_{\bar g_{A}}}{ 1-\gamma_{\bar g_{A}}}\left( \gamma_{G_{B}}\gamma_{\bar g_{A}}\right)^{p}. $$

Observing that

$$\gamma_{G_{B}}\gamma_{\bar g_{A}} =\frac{\left( 1 + \bar g_{A}\right)\left( 1 + \bar g_{B}\right) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}{\left( 1 + k_{A}\right)\left( 1 + k_{B}\right)} =\gamma_{G_{A}}\gamma_{\bar g_{B}}, $$

sum (19) results being

$$\begin{array}{@{}rcl@{}} \mathbb E \left[ \tilde P_{A0}\tilde P_{B0}\right] & =& d_{A0}d_{B0}\sum\limits_{p = 1}^{+\infty}\left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\gamma_{\bar g_{B}}^{p} - \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}} \left( \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)^{p} + \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}}\left( \gamma_{G_{B}}\gamma_{\bar g_{A}}\right)^{p}\right)\\ & =& d_{A0}d_{B0}\left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\sum\limits_{p = 1}^{+\infty }\gamma_{\bar g_{B}}^{p} +\left( \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}} - \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\right)\sum\limits_{p = 1}^{+\infty }\left( \gamma_{G_{B}}\gamma_{\bar g_{A}}\right)^{p}\right). \end{array} $$

Now, \({\sum }_{p = 1}^{+\infty }\gamma _{\bar g_{B}}^{p}\) converges to

$$\frac{\gamma_{\bar g_{B}}}{1 - \gamma_{\bar g_{B}}} = \frac{1 + \bar g_{B}}{k_{B} - \bar g_{B}} > 0 $$

if \(\bar g_{B} < k_{B}\), whereas \({\sum }_{p = 1}^{+\infty }\left (\gamma _{G_{B}}\gamma _{\bar g_{A}}\right )^{p}\) converges to

$$\frac{\gamma_{G_{A}}\gamma_{\bar g_{B}}}{1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}} =\frac{\left( 1 + \bar g_{A}\right)\left( 1 + \bar g_{B}\right) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]} {\left( 1 + k_{A}\right) \left( 1 + k_{B}\right) - \left( 1 + \bar g_{A}\right)\left( 1 + \bar g_{B}\right) - \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]} $$

if

$$\vert\left( 1 + \bar g_{A}\right)\left( 1 + \bar g_{B}\right)+ \text{Cov}\left[ \tilde g_{A},\tilde g_{B}\right]\vert < \left( 1 + k_{A}\right)\left( 1 + k_{B}\right). $$

Moreover, observe that

$$\bar P_{m0} = \frac{d_{m0}\gamma_{\bar g_{m}}}{1 - \gamma_{\bar g_{m}}},\hspace{1cm} m = A,B. $$

Covariance between \(\tilde P_{A0}\) and \(\tilde P_{B0}\) is, then,

$$\begin{array}{@{}rcl@{}} && \text{Cov}\left[ \tilde P_{A0}\tilde P_{B0}\right]\\ & =& d_{A0}d_{B0}\left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\frac{\gamma_{\bar g_{B}}}{1 - \gamma_{\bar g_{B}}} +\left( \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}} - \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\right)\frac{\gamma_{G_{A}}\gamma_{\bar g_{B}}}{1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}} - \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}}\frac{\gamma_{\bar g_{B}}}{1 - \gamma_{\bar g_{B}}}\right)\\ & =& d_{A0}d_{B0}\left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}} - \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}}\right) \left( \frac{\gamma_{\bar g_{B}}}{1 - \gamma_{\bar g_{B}}} -\frac{\gamma_{G_{A}}\gamma_{\bar g_{B}}}{1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}}\right)\\ & =& d_{A0}d_{B0} \left( \frac{\gamma_{G_{A}} - \gamma_{\bar g_{A}}}{\left( 1 - \gamma_{G_{A}}\right)\left( 1 - \gamma_{\bar g_{A}}\right)}\right) \left( \frac{\gamma_{\bar g_{B}}\left( 1 - \gamma_{G_{A}}\right)}{\left( 1 - \gamma_{\bar g_{B}}\right)\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)}\right)\\ & =& \left( \frac{d_{A0}\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}}\right) \left( \frac{d_{B0}\gamma_{\bar g_{B}}}{1 - \gamma_{\bar g_{B}}}\right) \left( \frac{\gamma_{G_{A}} - \gamma_{\bar g_{A}}}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)}\right)= \frac{\bar P_{A0}\bar P_{B0}\left( \gamma_{G_{A}} - \gamma_{\bar g_{A}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)}. \end{array} $$
(20)

Now, recall the definition of \(\gamma _{G_{A}}\), \(\gamma _{\bar g_{A}}\), \(\gamma _{\bar g_{B}}\), and GA. Then,

$$\gamma_{G_{A}} - \gamma_{\bar g_{A}} =\frac{\text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}{(1 + k_{A})(1 + \bar g_{B})}, $$

and

$$\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right) =\frac{\left( 1 + \bar g_{A}\right)\left( (1 + k_{A})(1 + k_{B}) - (1 + \bar g_{A})(1 + \bar g_{B}) - \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)}{(1 + k_{A})^{2}(1 + k_{B})} $$

Substituting these expressions in (20),

$$\begin{array}{@{}rcl@{}} && \text{Cov}\left[ \tilde P_{A0}\tilde P_{B0}\right]\\ & =& \bar P_{A0}\bar P_{B0}\times \frac{\text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}{(1 + k_{A})(1 + \bar g_{B})} \times\frac{(1 + k_{A})^{2}(1 + k_{B})}{(1 + \bar g_{A})\left( (1 + k_{A})(1 + k_{B}) - (1 + \bar g_{A})(1 + \bar g_{B}) - \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)}\\ & =& \bar P_{A0}\bar P_{B0}\times \frac{(1 + k_{A})(1 + k_{B})}{(1 + \bar g_{A})(1 + \bar g_{B})} \times\frac{\text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}{(1 + k_{A})(1 + k_{B}) - (1 + \bar g_{A})(1 + \bar g_{B}) - \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}. \end{array} $$

This proves the claimed formula.

Appendix: B

The analytic expression for

$$\mathbb E\left[ \tilde P_{At}\tilde P_{Bt}\right] =\sum\limits_{j = t + 1}^{+\infty}\sum\limits_{p = t + 1}^{+\infty } \frac{\mathbb E\left[ \tilde d_{Aj}\tilde d_{Bp}\right]}{(1 + k_{A})^{j - t}(1 + k_{B})^{p - t}} $$

is obtained analogously to that of \(\mathbb E\left [ \tilde P_{A0}\tilde P_{B0}\right ]\) by means of the substitutions u := jt and v := pt, and recalling that

$$\mathbb E\left[ \tilde d_{A,u + t}\tilde d_{B,v + t}\right] =\left\{\begin{array}{ll} d_{A0}d_{B0}\left( 1 + G_{A}\right)^{u + t}\left( 1 + \bar g_{B}\right)^{v + t} u \leq v \\ d_{A0}d_{B0}\left( 1 + G_{B}\right)^{v + t}\left( 1 + \bar g_{A}\right)^{u + t} u >v. \end{array}\right. $$

Therefore

$$ \mathbb E\left[ \tilde P_{At}\tilde P_{Bt}\right] = d_{A0}d_{B0}\sum\limits_{v = 1}^{+\infty} \left( \underset{\text{\#1}}{\underbrace{\sum\limits_{u = 1}^{v}\frac{(1 + G_{A})^{u + t}(1 + \bar g_{B})^{v + t}}{(1 + k_{A})^{u}(1 + k_{B})^{v}}}} + \underset{\text{\#2}}{\underbrace{\sum\limits_{u = v + 1}^{+\infty}\frac{(1 + \bar g_{A})^{u + t}(1 + G_{B})^{v + t}}{(1 + k_{A})^{u}(1 + k_{B})^{v}}}}\right). $$
(21)

Under the assumptions of Proposition 1, sum #1 becomes

$$(1 + G_{A})^{t}(1 + \bar g_{B})^{t}\gamma_{\bar g_{B}}^{v}\sum\limits_{u = 1}^{v}\gamma_{G_{A}}^{u} =(1 + G_{A})^{t}(1 + \bar g_{B})^{t}\frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\left( 1 - \gamma_{G_{A}}^{v}\right)\gamma_{\bar g_{B}}^{v}, $$

while sum #2 reduces to

$$(1 + G_{B})^{t}(1 + \bar g_{A})^{t}\gamma_{G_{B}}^{v}\sum\limits_{u = v + 1}^{+\infty}\gamma_{\bar g_{A}}^{u} =(1 + G_{B})^{t}(1 + \bar g_{A})^{t}\frac{\gamma_{\bar g_{A}}}{ 1-\gamma_{\bar g_{A}}}\left( \gamma_{G_{B}}\gamma_{\bar g_{A}}\right)^{v}. $$

Recalling that

$$(1 + G_{A})(1 + \bar g_{B}) =(1 + G_{B})(1 + \bar g_{A}) =(1 + \bar g_{A})(1 + \bar g_{B}) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right], $$

and that \(\gamma _{G_{B}}\gamma _{\bar g_{A}} = \gamma _{G_{A}}\gamma _{\bar g_{B}}\), sum (21) results being

$$\begin{array}{@{}rcl@{}} &&\mathbb E \left[ \tilde P_{At}\tilde P_{Bt}\right]\\ & = &f(t) \sum\limits_{v = 1}^{+\infty}\left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\gamma_{\bar g_{B}}^{v} - \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}} \left( \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)^{v} + \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}}\left( \gamma_{G_{B}}\gamma_{\bar g_{A}}\right)^{v}\right)\\ & =& f(t) \left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\sum\limits_{v = 1}^{+\infty }\gamma_{\bar g_{B}}^{v} +\left( \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}} - \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\right){\sum}_{v = 1}^{+\infty }\left( \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)^{v}\right)\\ & =& f(t) \left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\frac{\gamma_{\bar g_{B}}}{1 - \gamma_{\bar g_{B}}} +\left( \frac{\gamma_{\bar g_{A}}}{1 - \gamma_{\bar g_{A}}} - \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\right)\frac{\gamma_{G_{A}}\gamma_{\bar g_{B}}}{1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}}\right), \end{array} $$

where

$$f(t) = d_{A0}d_{B0}\left( (1 + \bar g_{A})(1 + \bar g_{B}) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)^{t}. $$

Observing that

$$f(t) = \bar P_{A0}\bar P_{B0}\left( (1 + \bar g_{A})(1 + \bar g_{B}) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)^{t} \frac{1 - \gamma_{\bar g_{A}}}{\gamma_{\bar g_{A}}}\frac{1 - \gamma_{\bar g_{B}}}{\gamma_{\bar g_{B}}}, $$

one can write

$$\mathbb E \left[ \tilde P_{At}\tilde P_{Bt}\right] = g(t) \left( \frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\frac{1 - \gamma_{\bar g_{A}}}{\gamma_{\bar g_{A}}} + \left( 1 - \frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\right)}\right) \frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{B}}\right)}{1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}}\right), $$

where

$$g(t) = \bar P_{A0}\bar P_{B0}\left( (1 + \bar g_{A})(1 + \bar g_{B}) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)^{t}. $$

Trivial algebra proves that

$$\frac{\gamma_{G_{A}}}{1 - \gamma_{G_{A}}}\frac{1 - \gamma_{\bar g_{A}}}{\gamma_{\bar g_{A}}} +\left( 1 - \frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\right)}\right) \frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{B}}\right)}{1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}} =\frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\gamma_{\bar g_{B}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)}, $$

so that

$$\mathbb E \left[ \tilde P_{At}\tilde P_{Bt}\right] =\bar P_{A0}\bar P_{B0}\frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\gamma_{\bar g_{B}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)} \left( (1 + \bar g_{A})(1 + \bar g_{B}) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)^{t}. $$

Moreover, for m = A,B we have

$$\bar P_{mt} =\mathbb E\left[\sum\limits_{j = t + 1}^{+\infty}\frac{\tilde d_{mj}}{(1 + k_{m})^{j - t}}\right] =\mathbb E\left[\sum\limits_{u = 1}^{+ \infty}\frac{\tilde d_{m,u + t}}{\left( 1 + k_{m}\right)^{u}}\right] =\bar P_{m0}\left( 1 + \bar g_{m}\right)^{t}, $$

and covariance between \(\tilde P_{At}\) and \(\tilde P_{Bt}\) is

$$\begin{array}{@{}rcl@{}} && \text{Cov}\left[ \tilde P_{At}\tilde P_{Bt}\right]\\ & =& \bar P_{A0}\bar P_{B0}\frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\gamma_{\bar g_{B}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)}\left( (1 + \bar g_{A})(1 + \bar g_{B}) + \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]\right)^{t} -\bar P_{A0}\bar P_{B0}\left( 1 + \bar g_{A}\right)^{t}\left( 1 + \bar g_{B}\right)^{t}\\ & =& \bar P_{A0}\bar P_{B0}\left( 1 + \bar g_{A}\right)^{t}\left( 1 + \bar g_{B}\right)^{t} \left( \frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\gamma_{\bar g_{B}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)} \left( 1 + \frac{\text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}{(1 + \bar g_{A})(1 + \bar g_{B})}\right)^{t} -1\right). \end{array} $$

Recalling the definitions of \(\gamma _{G_{A}}\), \(\gamma _{\bar g_{A}}\), \(\gamma _{\bar g_{B}}\), and GA,

$$\frac{\gamma_{G_{A}}\left( 1 - \gamma_{\bar g_{A}}\gamma_{\bar g_{B}}\right)}{\gamma_{\bar g_{A}}\left( 1 - \gamma_{G_{A}}\gamma_{\bar g_{B}}\right)} =\left( 1 + \frac{\text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}{(1 + \bar g_{A})(1 + \bar g_{B})}\right) \frac{(1 + k_{A})(1 + k_{B}) - (1 + \bar g_{A})(1 + \bar g_{B})}{(1 + k_{A})(1 + k_{B}) - (1 + \bar g_{A})(1 + \bar g_{B}) - \text{Cov}\left[\tilde g_{A},\tilde g_{B}\right]}, $$

and the claimed formula holds.

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Agosto, A., Mainini, A. & Moretto, E. Stochastic dividend discount model: covariance of random stock prices. J Econ Finan 43, 552–568 (2019). https://doi.org/10.1007/s12197-018-9455-9

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Keywords

  • Stock valuation
  • Dividend discount model
  • Markov chain
  • Financial risk management

JEL Classification

  • G11
  • G12
  • G32