Using accounting information for consumption planning and equity valuation

Abstract

This article develops a consumption-based valuation model that treats earnings and cash flow as complementary information sources. The model integrates three ideas that do not appear in traditional valuation models: (i) earnings provide information about future shocks to cash flow; (ii) earnings contain indiscernible transient accruals; and (iii) investors use cash flow and earnings to make allocation and consumption decisions and set price. Accordingly, the quality of earnings affects production and consumption as well as price. Among other implications, the model reveals that a valuation coefficient is not just a capitalization factor; it is the product of a capitalization factor and a structural factor reflecting earnings quality and accounting bias.

Appendix

1.1 Inferring the value of the trailing accrual shock

Investors infer the realization of ${\varepsilon_{t}}$ by inverting Eq. (1): ${\varepsilon_{t}=c_{t}-\gamma c_{t-1}-\theta_{t}}$. On the other hand, investors do not observe and cannot infer the realizations of the contemporaneous accrual shock ${\delta_{t}}$. The best they can do at date t is to infer the trailing accrual shock from public information ${\{\delta_{t-2}, c_{t-1}, x_{t-1}, \theta_{t}\}}$ by inverting Eq. (2): ²¹

$$ \delta_{t-1} =\delta_{t-2} +\left({x_{t-1} -c_{t-1}} \right)\lambda^{-1}-\theta_t +\theta_{t-1}. $$

This means that at any date t ≥ 1, investors know the values of ${\{\varepsilon_{t}, \delta_{t-1}\}}$ by inference. Thus, the public information set at each date t ≥ 1 is effectively ${\left\{ {\left\{{c_\tau } \right\}_{\tau =0}^t, \left\{{x_\tau } \right\}_{\tau =0}^t, \left\{{\theta_\tau } \right\}_{\tau =1}^t, \left\{{\varepsilon_\tau } \right\}_{\tau =1}^t, \left\{ {\delta_\tau } \right\}_{\tau =0}^{t-1}} \right\}}$.

1.2 Proofs of lemmas and propositions

1.2.1 Setup for remainder of proofs

The rest of the proofs build on the following setup. Each investor i solves the program given in assumption A3 for her optimal portfolio-consumption plan. When the variables of information set ${\Upomega_{\tau}}$ are normally distributed and P _t is linear in these variables, this dynamic optimization problem can be solved using standard methods (He & Wang, 1995), which I only sketch here. The optimization problem is equivalent to

$$ J\left({W_\tau^i } \right)=\mathop {\max }\limits_{z_\tau^i, q_\tau^i } \left\{{-e^{-\rho z_\tau^i }+\beta E\left[ {J\left({W_{\tau +1}^i } \right)\left| {\Upomega_\tau^i } \right.} \right]} \right\} $$

subject to two conditions:

$$ \begin{array}{l} \hbox{budget: }W_{\tau +1}^i =\left({W_\tau^i -z_{_\tau }^i } \right)R+\left({P_{\tau +1} +c_{\tau +1} -RP_\tau } \right)q_\tau^i \quad \forall \tau \geq t \\ \hbox{transversality: }\mathop {\hbox{lim}}\limits_{\tau \to \infty } E\left[ {\beta^\tau e^{-\rho z_\tau^i }\left| {\Upomega_t^i } \right.} \right]=0\hbox{ and }\mathop {\hbox{lim}}\limits_{\tau \to \infty } E\left[ {\frac{W_{t+\tau }^i }{R^\tau }\left| {\Upomega_t^i } \right.} \right]=0. \\ \end{array} $$

Market clearance,

$$ \sum\limits_{i\hbox{=}1}^M {q_\tau^i =1} \quad \forall \tau \geq t, $$

determines equilibrium market prices at each date. The second transversality condition prevents unlimited borrowing in perpetuity. In addition, I require (as is standard practice in solving these models) that the solution does not involve “doubling down” strategies that may entail going infinitely into debt, which is unrealistic (Harrison & Kreps, 1979). To solve for equilibrium prices and consumption paths:

• (a) Guess a trial solution ${J(W_{\tau}^{i})=-e^{-\alpha W_{\tau}^{i}+\varsigma}}$ for all ${\tau\geq t}$, where ${\{\alpha, \varsigma\}}$ are undetermined constants.

• (b) Assuming this trial function, derive the two first order conditions for ${\{z_{\tau}^{i},q_{\tau}^{i}\}}$ for each date ${\tau \geq t}$. Solving these two first order conditions for all ${\tau \geq t}$ yields ${\{z_{\tau}^{i,*}, q_{\tau}^{i,*}\}}$ in terms of the undetermined coefficients ${\{\alpha, \varsigma\}}$.

• (c) Require ${J({W_\tau^i })=-e^{-\rho z_\tau^{i,\ast } }+\beta E[ {J({({W_\tau^i -z_{_\tau }^{i,\ast }})R+\left({P_{\tau +1} +c_{\tau +1} -RP_\tau } \right)q_\tau^{i,\ast }})\left| {\Upomega_\tau^i } \right.}]}$ to obtain two equations. These equations imply that ${\alpha =\rho \phi^{-1}}$ and

  $$ \varsigma =\ln \phi +\left({\frac{1}{R-1}} \right)\left\{{\ln \left({R\beta } \right)-\rho \phi^{-1}\left({E\left[ {q_\tau^i \Re_{\tau +1} \left| {\Upomega_\tau^i } \right.} \right]-\frac{\rho \phi^{-1}}{2}\hbox{var}\left[ {q_\tau^i \Re_{\tau +1} \left| {\Upomega_\tau^i } \right.} \right]} \right)} \right\}, $$

  where ${\phi \equiv \frac{R}{R-1}}$ and ${\Re_{\tau +1} \equiv P_{\tau +1} +c_{\tau +1} -RP_\tau}$.

• (d) Plug these equilibrium values of ${\{\alpha, \varsigma\}}$ into the expressions for ${\{z_{\tau}^{i,*},q_{\tau}^{i,*}\}}$ determined in Step b to obtain the expressions for the equilibrium portfolio-consumption plans stated in Lemma 1.

• (e) Impose ${\sum\nolimits_{i=1}^{M}q_{\tau}^{i}=1}$ to obtain the fundamental pricing equation

  $$ E\left[ {P_{\tau +1} +c_{\tau +1} -RP_\tau \left| {\Upomega_\tau^i } \right.} \right]=\frac{\rho }{\phi M}\Upsigma_P \quad \forall \tau \geq 0, $$

  (A1)

where ${\Upsigma_P \equiv \hbox{var}\left[ {\Re_{\tau +1} \left| {\Upomega_\tau^i } \right.} \right]}$. While not used in this article, the stochastic discount factor in this economy is ${\mu_{t+1}^i =\beta \exp \left({-\rho \left({z_{i+1}^{i\ast } -z_i^{i\ast }} \right)} \right)}$, where ${z_i^{i\ast}}$ is the equilibrium consumption of any investor i.

Proof of Lemma 1

Step d of “Setup for remainder of proofs” states the procedure for deriving ${q_t^{i\ast}}$ and ${z_t^{i\ast}}$. Note that ${q_t^{i\ast } =1/M}$ simply because each of the M identical investors owns an identical fraction of the firm. When homogeneous information, ${\Upomega_{t}^{i}=\Upomega_{t}}$ for all i and Eq. (A1) implies Eq. (3) with ${\Upsigma_{P}}$ as defined following Eq. (A1). □

Proof of Proposition 1

Lemma 1 implies that ${\Upsigma_P \equiv \hbox{var}\left[ {\Re_{t+1} \left| {\Upomega_t } \right.} \right]}$, ${CE\Re =\frac{\phi^{-1}\rho }{2M^2}\Upsigma_P}$, ${E\left[ {q_t^{i\ast } \Re_{t+1} \left| {\Upomega_t} \right.} \right]=\frac{\phi^{-1}\rho }{M^2}\Upsigma_P}$, and ${z_{t+1}^{i\ast } -z_t^{i\ast } =\phi^{-1}\left\{{q_t^{i\ast } \Re_{t+1} +\hbox{I}-CE\Re } \right\}}$, which implies ${\hbox{var}\left[ {z_{t+1}^{i\ast } -z_t^{i\ast } \left| {\Upomega_t } \right.} \right]=({\frac{1}{\phi M}})^2\Sigma_P}$. Since ${\frac{\partial \Upsigma_P }{\partial Q_j } < 0}$ for ${j\in \left\{{1,2} \right\}}$ by Corollary 2 (proved independently below), parts (a) and (b) follow. Parts (c) and (d) are proved by relating ${E\left[ {z_{t+s}^{i\ast } \left| {\Upomega_t} \right.} \right]}$ and ${E\left[ {\Pi_{t+s}^{i\ast } \left| {\Upomega_t } \right.} \right]}$ to ${\Upsigma_P}$. The formulas in Lemma 1 imply ${W_{t+s}^{i\ast } =W_t^{i\ast} +\hbox{I}\,s+\sum\nolimits_{u=1}^s {\left\{{q_{t+u-1}^{i\ast } \Re_{t+u} -CE\Re } \right\}}}$,

$$ E\left[ {z_{t+s}^{i\ast} \left| {\Upomega_t } \right.} \right]=\phi^{-1}\left\{{W_t^{i\ast} -\left({\frac{1}{R-1}-s} \right)\hbox{I}+\left({\frac{\rho \phi^{-1}}{2M^2}} \right)\left({\frac{1}{R-1}+s} \right)\Upsigma_P } \right\}, $$

and

$$ E\left[ {\Pi_{t+s}^{i\ast } \left| {\Upomega_t } \right.} \right]=W_t^{i\ast } -s{\rm I}+\left\{{\phi +\frac{1}{2}\left({s-2} \right)} \right\}\frac{\rho \phi^{-1}}{M^2}\Upsigma_P -\frac{1}{M}\left({\frac{\gamma^s}{R-\gamma }} \right)E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]. $$

The latter two formulas are valid for all ${t\geq 1}$ and ${s\geq 0}$. Plugging in ${W_t^{i\ast } =B_t^{i\ast } +\left({P_t +c_t } \right)q_{t-1}}$ and taking the partial derivatives holding ${B_t^{i\ast}}$ and ${E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]}$ fixed yields parts (c) and (d). Finally, the discussion leading up to Eq. (A1) and the formulas in Lemma 1 imply that the value of investor i’s equilibrium utility function (a.k.a. the “derived utility”) ${J\left({W_\tau^{i\ast }} \right)=E[ {-\sum\nolimits_{\tau =t}^\infty {\beta^{\tau -t}e^{-\rho z_\tau^{i\ast }}} \left| {\Upomega_t } \right.}]}$ equals

$$ \begin{array}{ll} J\left({W_\tau^{i\ast }} \right)&=-e^{-\rho \phi^{-1}W_\tau^{i\ast } +\ln \phi +\left({\frac{1}{R-1}} \right)\left\{{\ln \left({R\beta } \right)-\left({E\left[ {q_\tau^i \Re_{\tau +1} \left| {\Upomega_\tau^i } \right.} \right]-\frac{\rho \phi^{-1}}{2}\hbox{var}\left[ {q_\tau^i \Re_{\tau +1} \left| {\Upomega_\tau^i } \right.} \right]} \right)} \right\}} \\ &=-\phi e^{-\rho \phi^{-1}\left\{{W_\tau^{i\ast } -\left({\frac{{\rm I}-CE\Re }{R-1}} \right)I} \right\}} \\ &=-\phi e^{-\rho z_\tau^{i\ast}}.\\ \end{array} $$

Hence,

$$ \frac{\partial}{\partial Q_j}_{\left|{\mathop{}\limits_{ E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]}^ {B_t^{i\ast }}}\right.} E\left[ {-\sum\limits_{\tau =t}^\infty {\beta^{\tau -t}e^{-\rho z_\tau^{i\ast }}} \left| {\Upomega_t } \right.} \right]=-\phi \frac{\partial }{\partial Q_j}_{\left|{\mathop{}\limits_{ E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]} ^ {B_t^{i\ast }}}\right.} e^{-\rho z_\tau^{i\ast }}=+\phi \rho e^{-\rho z_\tau^{i\ast }}\frac{\partial }{\partial Q_j }_{\left|{\mathop{}\limits_{ E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]} ^ {B_t^{i\ast }}}\right.} z_\tau^{i\ast } > 0, $$

which proves part (e). I thank Peter Christensen for pointing out an error in an earlier statement of part (e). □

Proof of Proposition 2

Equation (1) implies ${\sum\nolimits_{\tau =1}^\infty {\frac{E\left[ {c_{t+\tau } \left| {\Upomega_t } \right.} \right]}{R^\tau }} =\frac{E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]}{R-\gamma}}$, which means evaluating Eq. (8) requires evaluating ${E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]}$. Plugging ${s_t \equiv \frac{1}{\lambda }\left\{{x_t -c_t +\lambda \theta_t +\lambda \delta_{t-1}} \right\}}$ into Eq. (7) and rearranging yields

$$ \begin{array}{ll} E\left[ {c_{t+1} \left| {\Upomega_t } \right.} \right]&=\gamma c_t +\frac{Q_P }{\lambda }\left\{{x_t -c_t +\lambda \theta_t +\lambda \delta_{t-1}} \right\} \\ &=\gamma c_t +\frac{Q_P }{\lambda }\left\{{\left({\frac{R-\gamma }{R-1}} \right)\hat{x}_t -c_t } \right\} \\ &=\left({R-\gamma } \right)\left\{{\frac{\gamma c_t }{R-\gamma }+\frac{Q_P }{R\lambda }\left[ {\phi \hat{x}_t -\frac{R\,c_t }{R-\gamma}} \right]} \right\},\\ \end{array} $$

where ${\phi}$ and ${\hat{x}_{t}}$ are as defined in the Proposition. Rearranging this further yields

$$ \frac{E\left[c_{t+1}|\Upomega_{t}\right]}{R-\gamma}= \left(1-\frac{Q_{P}}{R\lambda}\right)\frac{\gamma c_{t}} {R-\gamma}+\frac{Q_{P}}{R\lambda}\left[\phi \hat{x}_{t}-c_{t}\right]. $$

(A2)

Plugging Eq. (A2) into Eq. (8) yields Eq. (10). Next, we evaluate ${\Upsigma_{P}}$ starting from Eq. (9). To this end, recalling that ${Q_{P}\equiv \frac{\sigma_{\theta}^{2}}{\sigma_{\theta}^{2}+\sigma_{\delta}^{2}}}$, ${Q_{c}\equiv \frac{\sigma_{\theta}^{2}}{\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}}}$, and ${\sigma_{c}^{2}\equiv \sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}}$ and rearranging Eq. (9) yields

$$ \begin{aligned} \Upsigma_{P}&=\left(\frac{R}{R-\gamma}\right)^{2} \left\{\left(1-Q_{P}\right)\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}\right\}+ \left(\frac{Q_{P}}{R-\gamma}\right)^{2} \left\{\sigma_{\theta}^{2}+\sigma_{\delta}^{2}\right\}\\ &=\left(\frac{R}{R-\gamma}\right)^{2}\left\{\left[\left(1-Q_{P}\right) \sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}\right]+ \left(\frac{1}{R}\right)^{2}Q_{P}\sigma_{\theta}^{2}\right\}\\ &=\left(\frac{R}{R-\gamma}\right)^{2}\left\{\left[\left(1-Q_{P}\right) Q_{C}+\left(1-Q_{C}\right)\right]+\left(\frac{1}{R}\right)^{2} Q_{P}Q_{C}\right\}\sigma_{c}^{2}. \end{aligned} $$

Finally, observing that ${(1-Q_{P})Q_{C}+(1-Q_{C})=1-Q_{P}Q_{C}}$ proves the formula for ${\Upsigma_{P}}$. □

Proof of Corollary 1

${\frac{\partial P_t}{\partial V_t^c }=0}$ implies ${P_{t}=V_{t}^{x}-\frac{\rho}{RM}\Upsigma_{P}=\phi \hat{x}_{t}-c_{t}-\frac{\rho}{RM}\Upsigma_{P}}$. Hence, ${\frac{\partial P_{t}}{\partial V_{t}^{x}}=1}$. The second part of the Proposition is proved in the main text preceding the Proposition. □

Proof of Corollary 2

Follows from taking the partial derivative of Eq. (11). □