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Another look at the dividend-price relationship in the accounting valuation framework

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Abstract

We examine the association between dividends and price through the lens of the Ohlson (Contemp Account Res 11:661–687, 1995, Contemp Account Res 18:107–120, 2001) accounting valuation framework. Employing the price-level model and coefficient definitions detailed in Ohlson (Contemp Account Res 18:107–120, 2001), we show that in a properly specified pricing model a positive dividend coefficient does not violate Miller and Modigliani (J Bus 34:411–433, 1961), imply market irrationality, nor signify an informational role for dividends. Using a simple illustration, we explain the intuition behind the positive dividend coefficient. We also demonstrate that in the price-level model Ohlson’s “other information” variable νt, is better specified as the expected change in earnings—not just the earnings forecast. Our results hold through a number of robustness tests.

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Notes

  1. Hereafter, MM.

  2. The terms “dividends,” “net dividend,” and “net distributions to shareholders” are often used interchangeably in the literature when discussing the accounting valuation framework, and we do the same throughout this paper. When we refer to individual components of net dividends (i.e., cash dividends, share repurchases, share issues), we use those specific terms.

  3. Residual income is accounting earnings less a charge for the cost of capital. The terms “residual income” and “abnormal earnings” are used interchangeably in accounting literature. We use the term “residual income” for the remainder of the paper.

  4. Boonlert-U-Thai et al. (2020) argue that it may be possible for dividends to convey information if there are no other credible information sources present. However, with analysts’ forecasts and/or other reliable information sources available in almost all market environments, there is no reason to believe that dividends provide anything other than a wealth effect in normal settings.

  5. All model development variable definitions are summarized in Appendix A.1.

  6. Ohlson (1995, 2001) defines νt as other information not included in current residual income that affects forecasts of future residual income. We interpret this as information about the firm’s ability to innovate in a competitive environment and create additional value worth paying for.

  7. We thank an anonymous referee who suggested we highlight this point.

  8. Both θ1 and θ3 are positive by definition. In the case of θ3, \( \left| {\beta _{2} \varphi } \right| < {\raise0.7ex\hbox{${\beta _{3} }$} \!\mathord{\left/ {\vphantom {{\beta _{3} } r}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$r$}} \).

  9. As a check, we also ran our tests using Fama–MacBeth (1973) regressions with a Newey-West adjustment for serial correlation. Results were nearly identical.

  10. Deflation narrows the measurement range but does not change distribution(s) of regression variables.

  11. No deflation; a returns specification; deflation by shares, book value, lagged price and market value of equity.

  12. Multiplicative and additive omitted variable effects; scale-varying parameter estimates; survivorship; and heteroscedasticity.

  13. All regression model variable and coefficient definitions are summarized in Appendix A.2.

  14. This variable, which they call “net capital contributions,” is measured as repurchases of both common and preferred stock minus sales of both common and preferred stock.

  15. Coulton et al. (2014), Skinner (2008), and Skinner and Soltes (2011) are notable exceptions.

  16. We follow Fama and French (2001) in calling firms “payers” if they pay cash dividends and “non-payers” if they do not. For all descriptors shown in Table 2 the mean differences between payers to common only and payers to common and/or preferred were insignificantly different from zero.

  17. We do not report estimates for θ0 to conserve space and because regression intercepts are irrelevant to our research questions.

  18. US GAAP defines comprehensive income as follows: “The change in equity (net assets) of a business entity during a period from transactions and other events and circumstances from non-owner sources. It includes all changes in equity during a period except those resulting from investments by owners and distributions to owners. Comprehensive income comprises both of the following:

    a. All components of net income.

    b. All components of other comprehensive income.” FASB ASC Glossary.

    Other comprehensive income is defined as follows: “Revenues, expenses, gains, and losses that under generally accepted accounting principles (GAAP) are included in comprehensive income but excluded from net income.” FASB ASC Glossary.

    Net income is defined as follows: “A measure of financial performance resulting from the aggregation of revenues, expenses, gains, and losses that are not items of other comprehensive income. A variety of other terms such as net earnings or earnings may be used to describe net income.” FASB ASC Glossary.

  19. Two examples are unrealized gains (losses) on available-for-sale debt investments and foreign currency translation gains (losses). Both are consequences of market value changes and/or macroeconomic events over which management exerts little control.

  20. FASB Accounting Standards Update 2011–05 (2011, page 1).

  21. We thank an anonymous referee for suggesting that we test this possibility.

  22. We are grateful to an anonymous referee who brought this point to our attention.

  23. Such an assumption or requirement would be irrational because components are measured with different signs.

  24. There is no upper limit on R, but we characterize this range as representing realistic values.

  25. Beginning in 2005 the SEC transitioned to a three-tier reporting regime: 90 days for non-accelerated filers (public float less than $75 million); 75 days for accelerated filers (public float from $75 million to less than $700 million); 60 days for large accelerated filers (public float of $700 million or more). We apply the shortest window (60 days) to missing earnings report dates because the average float is considerably higher than $700 million in every year of our sample.

  26. Our Eq. (4) is the same as Ohlson’s expression of price as a function of current book value and abnormal earnings plus next period expected abnormal earnings (Ohlson 2001, page 113, Eq. (7)).

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Authors and Affiliations

  1. Wright State University, Dayton, OH, USA

    Kathryn E. Easterday

  2. University of WA—Bothell, Bothell, WA, USA

    Pradyot K. Sen

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  1. Kathryn E. Easterday
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Correspondence to Kathryn E. Easterday.

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Appendices

Appendix A.1: Model development variable definitions

\(P_{t}\)

 = 

Price at the end of period t

\(b_{t}\)

 = 

Book value at the end of period t

\(R\)

 = 

\(1 + r =\) one plus the discount rate, r

\(\tilde{\varepsilon }_{1,t + 1}\), \(\tilde{\varepsilon }_{2,t + 1}\)

 = 

Unpredictable residuals with means equal to zero

\(\omega , \gamma\)

 = 

Parameters assumed to be known by the market but unknown to researchers, and must satisfy the following conditions for unbiased accounting: \(0 \le \omega \le 1\); \(0 \le \gamma \le 1\); and \(\omega = \gamma \ne 1\)

\(x_{t}\)

 = 

Accounting earnings during period t

\(\tilde{x}_{t + 1}\)

 = 

Expected accounting earnings for period t + 1

\(x_{t}^{a}\)

 = 

\(x_{t + 1} - rb_{t}\) = residual earnings during period t

\(\nu_{t}\)

 = 

Other value-relevant information not captured in accounting numbers at the end of period t

\(\tilde{\nu }_{t}\)

 = 

\(\gamma \nu_{t} + \tilde{\varepsilon }_{2,t + 1}\)

\(\tilde{x}_{t + 1}^{a}\)

 = 

\(\tilde{x}_{t + 1} - rb_{t} =\) expected residual income for period t + 1 \(= \omega x_{t}^{a} + \nu_{t} + \tilde{\varepsilon }_{1,t + 1}\)

\(d_{t}\)

 = 

Dividends in year t

\(\beta_{1}\)

 = 

\({\raise0.7ex\hbox{${R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)}$} \!\mathord{\left/ {\vphantom {{R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}\)

\(\beta_{2}\)

 = 

\({\raise0.7ex\hbox{${ - r\omega \gamma }$} \!\mathord{\left/ {\vphantom {{ - r\omega \gamma } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}\)

\(\beta_{3}\)

 = 

\({\raise0.7ex\hbox{${Rr}$} \!\mathord{\left/ {\vphantom {{Rr} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}\)

\(\varphi\)

 = 

\({\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R r}}\right.\kern-0pt} \!\lower0.7ex\hbox{$r$}}\)

Appendix A.2: Regression variable and coefficient definitions

\({MVE}_{t}\)

 = 

Total market value of equity ($MM), computed as the closing price per share times number of common shares outstanding at the end of year t

\({SEQ}_{t}\)

 = 

Total shareholders’ equity ($MM) at the end of year t

\({CASH}_{t}\)

 = 

Cash dividends ($MM) to common and/or preferred shareholders declared in year t

\({EARN}_{t}\)

 = 

Earnings in year t, measured as net income ($MM)

\({EXPGROWTH}_{t}\)

 = 

Expected earnings growth ($MM): forecasted earnings for year t + 1 minus earnings in year t

\({AT}_{t}\)

 = 

Firm size in total assets at the end of year t ($MM)

\({NETDIV}_{t}\)

 = 

Net dividend to shareholders in year t, measured as cash dividends declared in year t plus share purchases in year t minus share sales in year t

\({PURCH}_{t}\)

 = 

Repurchases of shares from stockholders ($MM) during year t

\({SALES}_{t}\)

 = 

Sales of shares ($MM) during year t, measured as a negative number

\({CI}_{t}\)

 = 

comprehensive income

\({PFGROWTH}_{t}\)

 = 

A “perfect forecast” of comprehensive income growth, measured as actual OCI in year t + 1 plus forecasted earnings for year t + 1 minus comprehensive income in year t

\(\theta_{0}\)

 = 

Intercept

\(\theta_{1}\)

 = 

\(\beta_{1}\)

\(\theta_{2}\)

 = 

\(- \beta_{2}\)

\(\theta_{3}\)

 = 

\(\beta_{2} \varphi + {\raise0.7ex\hbox{${\beta_{3} }$} \!\mathord{\left/ {\vphantom {{\beta_{3} } r}}\right.\kern-0pt} \!\lower0.7ex\hbox{$r$}}\)

\(\theta_{4}\)

 = 

\({\raise0.7ex\hbox{${\beta_{3} }$} \!\mathord{\left/ {\vphantom {{\beta_{3} } r}}\right.\kern-0pt} \!\lower0.7ex\hbox{$r$}}\)

Appendix A.3: Sample selection process

Number of observations in initial Compustat dataset

169,570

 

Missing or ineligible variable values

(106,987)

 

Number of Compustat observations

 

62,583

 Number of observations in initial I/B/E/S dataset

3,618,629

 

Missing or ineligible variable values

(25,587)

 

Filter out all but the latest forecast for each firm (see discussion below)

(3,503,913)

 

 Number of I/B/E/S observations

 

89,129

Compustat observations with no I/B/E/S match

(18,912)

 

Windsorization losses

(6,377)

 

 Number of observations in the final dataset

 

37,294

The initial sample consists of US firms trading ordinary common shares on NYSE, AMEX, or NASDAQ, with fiscal year end closing price of at least $1 whose data appear in Compustat and I/B/E/S. Except in cases of missing I/B/E/S earnings report dates, we follow Hand and Landsman (2005) in deleting observations with missing values for our regression variables, along with observations having incorrectly signed values for firm equity, cash dividends, share repurchases, or share sales.

For each firm, we use the last forecast for year t earnings that occurs in year t-1. The observation is deleted if that forecast in year t–1 does not also occur after the announcement of year t-2 earnings. Figure 1 shows the timeline of earnings announcements and forecasts.

We treat missing I/B/E/S earnings report dates in a manner similar to Hand and Landsman (2005), who simulate report dates by adding 90 days (the SEC’s filing deadline during their time period of study) to the fiscal year-end of the company in question. We set our missing report date to 60 days after the fiscal year end.Footnote 25 We delete any firm with an actual report date that is more than 100 days after the report year’s fiscal end, based on evidence that accounting information suffers when firms are in financial distress (Hayn 1995; Elliott and Hanna 1996; Joseph and Lipka 2006).

Observations for which there are no Compustat—I/B/E/S matches are deleted. Regression variables are Windsorized at 1% top and bottom (unless the minimum value is already zero). The final dataset consists of 37,294 firm-year observations for the years 2005—2021, inclusive.

Appendix B

Start with Eq. (4)Footnote 26:

$$ P_{t} = b_{t} + \left[ {{\raise0.7ex\hbox{$\omega $} \!\mathord{\left/ {\vphantom {\omega {\left( {R - \omega } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)}$}} - {\raise0.7ex\hbox{${R\omega }$} \!\mathord{\left/ {\vphantom {{R\omega } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right]x_{t}^{a} + \left[ {{\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right]\tilde{x}_{t + 1}^{a} $$
(4)

where \({\raise0.7ex\hbox{$\omega $} \!\mathord{\left/ {\vphantom {\omega {\left( {R - \omega } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)}$}} \ge 0\) and \({\raise0.7ex\hbox{${R\omega }$} \!\mathord{\left/ {\vphantom {{R\omega } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} \ge 0\) and \({\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} > 0\).

Define residual income in period t and expected residual income for period t + 1:

$$ x_{t}^{a} = x_{t} - \left( {R - 1} \right)b_{t - 1} $$
(A.1)
$$ \tilde{x}_{t + 1}^{a} = \tilde{x}_{t + 1} - \left( {R - 1} \right)b_{t} $$
(A.2)

then use CSR to substitute for bt-1 in A.1, and expand:

$$ x_{t}^{a} = Rx_{t} - Rb_{t} - Rd_{t} + b_{t} + d_{t} $$
(A.3)

Substituting A.2 and A.3 into Eq. (4), then factoring and collecting terms we obtain:

$$ P_{t} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} \times \left[ {R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)b_{t} - r\omega \gamma \left( {\varphi x_{t} - d_{t} } \right) + Rx_{t + 1} } \right] $$
(A.4)

Define

$$ \beta_{1} = {\raise0.7ex\hbox{${R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)}$} \!\mathord{\left/ {\vphantom {{R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} $$
$$ \beta_{2} = {\raise0.7ex\hbox{${ - r\omega \gamma }$} \!\mathord{\left/ {\vphantom {{ - r\omega \gamma } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} $$
$$ \beta_{3} = {\raise0.7ex\hbox{${Rr}$} \!\mathord{\left/ {\vphantom {{Rr} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} $$

then substitute to obtain:

$$ P_{t} = \beta_{1} b_{t} + \beta_{2} \left( {\varphi x_{t} - d_{t}}\right) + \beta_{3} \tilde{x}_{t + 1} $$
(A.5)

Appendix C

Start with Ohlson (1995) equation expressing price as a weighted average of earnings, book value, and other information:

$$ P_{t} = k\left( {\varphi x_{t} - d_{t} } \right) + \left( {1 - k} \right)b_{t} + \left[ {{\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right]\nu_{t} $$
(B.1)

where

$$ k = {\raise0.7ex\hbox{${r\omega }$} \!\mathord{\left/ {\vphantom {{r\omega } {\left( {R - \omega } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)}$}} $$
$$ \varphi = {\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R r}}\right.\kern-0pt} \!\lower0.7ex\hbox{$r$}} = {\raise0.7ex\hbox{${\left( {r + 1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {r + 1} \right)} r}}\right.\kern-0pt} \!\lower0.7ex\hbox{$r$}} $$
$$ \nu_{t} = \tilde{x}_{t + 1}^{a} - \omega x_{t}^{a} $$
$$ \tilde{x}_{t + 1}^{a} = \tilde{x}_{t + 1} - rb_{t} $$

Substitute the definitions of k, φ, R, and νt:

$$ P_{t} = \left[ {{\raise0.7ex\hbox{${\left( {r\omega x_{t} + \omega x_{t} - r\omega d_{t} + rb_{t} + b_{t} - \omega b_{t} - r\omega b_{t} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {r\omega x_{t} + \omega x_{t} - r\omega d_{t} + rb_{t} + b_{t} - \omega b_{t} - r\omega b_{t} } \right)} {\left( {R - \omega } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)}$}}} \right] + \left[ {{\raise0.7ex\hbox{${\left( {r\tilde{x}_{t + 1} - r^{2} b_{t} - r\omega x_{t} + r^{2} \omega b_{t - 1} + \tilde{x}_{t + 1} - rb_{t} - \omega x_{t} + r\omega b_{t - 1} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {r\tilde{x}_{t + 1} - r^{2} b_{t} - r\omega x_{t} + r^{2} \omega b_{t - 1} + \tilde{x}_{t + 1} - rb_{t} - \omega x_{t} + r\omega b_{t - 1} } \right)} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right] $$
(B.2)

Multiply the first term by \({\raise0.7ex\hbox{${R - \gamma }$} \!\mathord{\left/ {\vphantom {{R - \gamma } {R - \gamma }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R - \gamma }$}} = {\raise0.7ex\hbox{${\left( {r + 1 - \gamma } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {r + 1 - \gamma } \right)} {\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \gamma } \right)}$}}\), then cancel and collect terms:

$$ P_{t} = {\raise0.7ex\hbox{${\begin{array}{*{20}c} {(r^{2} \omega x_{t} + r\omega x_{t} - r\omega \gamma x_{t} - \omega \gamma x_{t} - r^{2} \omega d_{t} - r\omega d_{t} + r\omega \gamma d_{t} + rb_{t} - 2r\omega b_{t} + r\omega b_{t - 1} - r^{2} \omega b_{t} } \\ { + r^{2} \omega b_{t - 1} + b_{t} - \omega b_{t} - r\gamma b_{t} - \gamma b_{t} + \omega \gamma b_{t} + r\omega \gamma b_{t} + r\tilde{x}_{t + 1} + r\omega \tilde{x}_{t + 1} )} \\ \end{array} }$} \!\mathord{\left/ {\vphantom {{\begin{array}{*{20}c} {(r^{2} \omega x_{t} + r\omega x_{t} - r\omega \gamma x_{t} - \omega \gamma x_{t} - r^{2} \omega d_{t} - r\omega d_{t} + r\omega \gamma d_{t} + rb_{t} - 2r\omega b_{t} + r\omega b_{t - 1} - r^{2} \omega b_{t} } \\ { + r^{2} \omega b_{t - 1} + b_{t} - \omega b_{t} - r\gamma b_{t} - \gamma b_{t} + \omega \gamma b_{t} + r\omega \gamma b_{t} + r\tilde{x}_{t + 1} + r\omega \tilde{x}_{t + 1} )} \\ \end{array} } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} $$
(B.3)

Apply CSR by replacing (\(- r\omega b_{t} + r\omega b_{t - 1} = - r\omega x_{t} + r\omega d_{t}\)) and (\(- r^{2} \omega b_{t} + r^{2} \omega b_{t - 1} = - r^{2} \omega x_{t} + r^{2} \omega d_{t}\)). Then cancel terms, factor, rearrange, and multiply the last term by r/r:

$$ \left[ {{\raise0.7ex\hbox{${R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)}$} \!\mathord{\left/ {\vphantom {{R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right]b_{t} - \left[ {{\raise0.7ex\hbox{${r\omega \gamma }$} \!\mathord{\left/ {\vphantom {{r\omega \gamma } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right]\left( {\varphi x_{t} - d_{t} } \right) + \left[ {{\raise0.7ex\hbox{${Rr}$} \!\mathord{\left/ {\vphantom {{Rr} {r\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}}} \right]\tilde{x}_{t + 1} $$
(B.4)

where

$$ {\raise0.7ex\hbox{${R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)}$} \!\mathord{\left/ {\vphantom {{R\left( {1 - \omega } \right)\left( {1 - \gamma } \right)} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} = \beta_{1} $$
$$ {\raise0.7ex\hbox{${ - r\omega \gamma }$} \!\mathord{\left/ {\vphantom {{ - r\omega \gamma } {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} = \beta_{2} $$
$$ {\raise0.7ex\hbox{${Rr}$} \!\mathord{\left/ {\vphantom {{Rr} {\left( {R - \omega } \right)\left( {R - \gamma } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {R - \omega } \right)\left( {R - \gamma } \right)}$}} = \beta_{3} $$

resulting in

$$ P_{t} = \beta_{1} b_{t} + \beta_{2} \left( {\varphi x_{t} - d_{t} } \right) + \left( {{\raise0.7ex\hbox{${\beta_{3} }$} \!\mathord{\left/ {\vphantom {{\beta_{3} } r}}\right.\kern-0pt} \!\lower0.7ex\hbox{$r$}}} \right)\tilde{x}_{t + 1} $$
(B.5)

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Easterday, K.E., Sen, P.K. Another look at the dividend-price relationship in the accounting valuation framework. Rev Quant Finan Acc 61, 879–925 (2023). https://doi.org/10.1007/s11156-023-01167-y

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