Accounting rules, equity valuation, and growth options

Abstract

In a model with irreversible capacity investments, we show that financial statements prepared under replacement cost accounting provide investors with sufficient information for equity valuation purposes. Under alternative accounting rules, including historical cost and value in use accounting, investors will generally not be able to value precisely a firm’s growth options and therefore its equity. For these accounting rules, we describe the range of valuations that is consistent with the firm’s financial statements. We further show that replacement cost accounting preserves all value-relevant information if the firm’s investments are reversible. However, the directional relation between the value of the firm’s equity and the replacement cost of its assets is different from that in the setting with irreversible investments.

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Fig. 1

Notes

  1. 1.

    Most commonly cited valuation models, such as Ohlson (1995), Feltham and Ohlson (1995), and Ohlson and Juettner-Nauroth (2005), express the value of a firm as a function of accounting variables, assuming an exogenous process for firm’s residual earnings. These papers do not specify the accounting rules that need to be applied in different economic environments to generate a residual earnings process conforming to the assumed specification. For example, one of the main questions raised by Penman (2005) in his discussion of Ohlson and Juettner-Nauroth (2005) was “Where’s the accounting?” Several papers explicitly model accounting rules but treat both investment and cash flow processes as exogenous (e.g., Feltham and Ohlson 1996; Ohlson and Zhang 1998; and Zhang 2000b).

  2. 2.

    We take it as given that the firm’s shareholders seek to estimate the present value of the firm’s cash flows, which is consistent with much of the neoclassical and real options literature. The demand for accounting information in our model then stems from this exogenously assumed goal of shareholders. The focus of our paper is similar to that of Hughes et al. (2004), who provide a more in-depth discussion of the different perspectives on demand for accounting disclosures.

  3. 3.

    In our model, the firm is all equity financed and there are no taxes and no accruals other than the ones related to capital assets (depreciation and revaluations). Therefore the operating cash flow is essentially equal to the firm’s earnings before interest, taxes, depreciation and amortization (EBITDA).

  4. 4.

    Similar decompositions have been obtained in other settings; see, for instance, Lindenberg and Ross (1981), Berk et al. (1999), Abel and Eberly (2011), and Kogan and Papanikolaou (2014).

  5. 5.

    Accordingly, the firm’s equity value is bounded by the present value of cash flows from existing assets and by the value of the firm had it been at the investment threshold.

  6. 6.

    Here, we consider a conditionally conservative system based on replacement cost accounting. In practice, the standards for asset impairments are different under U.S. GAAP and IFRS, and the write-down amounts are determined by different quantities, including the asset’s value in use and its fair value. We discuss an application of our model to a setting where asset write-downs are recognized according to IAS 36 in Appendix B.

  7. 7.

    This result also holds when the only source of uncertainty in the model is demand in the firm’s output market. In this special case, there is no difference between replacement cost and historical cost disclosures. Therefore our result suggests that the direction of the relationship between the firm’s equity value and its historical cost book value of assets also depends on investment reversibility.

  8. 8.

    See, for instance, Rajan and Reichelstein (2009), Nezlobin (2012), McNichols et al. (2014), Nezlobin et al. (2012, 2016).

  9. 9.

    For example, the specification above obtains if one assumes (i) the standard Cobb-Douglas production technology, \(q_{t}={K_{t}^{s}}\) with 0 < s, where q t is the number of units of the output product the firm can make at time t, and (ii) constant elasticity demand curves:

    $$P_{t}\left( q_{t}\right)=X_{t}\cdot q_{t}^{-\frac{1}{\eta}}, $$

    where η > 1 is the price elasticity of demand. Then, the total cash flow is given by:

    $$CF_{t}=P_{t}\left( q_{t}\right)\cdot q_{t}=X_{t}K_{t}^{s\left( -\frac{1}{\eta}+1\right)}. $$

    To ensure that the optimal production volume is always finite, we impose the requirement that \(0<s\left (-\frac {1}{\eta }+1\right )<1\) . This requirement is satisfied if the firm’s production technology exhibits decreasing returns to scale (s < 1), or if returns to scale are constant or increasing (s ≥ 1) but η is sufficiently small (i.e., demand is sufficiently inelastic). It is also straightforward to extend our results to a setting where the firm’s production function requires a second input, labor, that can be purchased instantaneously after the firm observes the current demand.

  10. 10.

    With this notation, I t is the total quantity of capital goods installed from the firm’s inception up to time t.

  11. 11.

    Generally, such situations may arise when prices in both product and capital goods markets are affected by common macroeconomic factors.

  12. 12.

    Without loss of generality, we assume that the firm starts it operations at date 0 without any assets in place, that is, I 0− = 0.

  13. 13.

    To implement such an extension, one would have to convert the processes that govern the firm’s cash flows from the physical to the risk-neutral measure. Under the risk-neutral measure, the results would be equivalent to the ones presented in our paper. For papers that model the pricing kernel as a geometric Brownian motion, see Berk et al. (2004) and Li (2011).

  14. 14.

    In contrast, Nezlobin (2012) assumed that investors observe only the current financial statements and do not have access to the firm’s investment history.

  15. 15.

    The amount of gross investment at time t is given by

    $$GI_{t}{\equiv{\int}_{0}^{t}}P_{\tau}\cdot dI_{\tau}. $$

    Since the whole history of investments, {P τ d I τ } τt , is in \(\mathcal {I}_{t}\) , investors observe the full path of the gross investment function.

  16. 16.

    It will be shown that, when the firm invests, the NPV of its investment is strictly positive (otherwise, it would be optimal to postpone the investment). Under replacement cost accounting, assets are capitalized at their acquisition cost, which is less than the present value of cash flows that they are expected to generate.

  17. 17.

    In contrast, in the model of Dixit and Pindyck (1994, Chapter 11), observing the firm’s current operating cash flow and the history of investments would be sufficient to solve for the state variable.

  18. 18.

    See Rajan and Reichelstein (2009), Dutta and Reichelstein (2010), Nezlobin (2012), McNichols et al. (2014), Nezlobin et al. (2016).

  19. 19.

    Recall that 0 < α < 1, so \(\frac {\alpha \left (\alpha -1\right )}{2}<0\) . Therefore \(\bar {r}\) increases in \({\sigma _{K}^{2}}\) , and the present value of cash flows from assets in place decreases in \({\sigma _{K}^{2}}\) .

  20. 20.

    This definition can be consistent with fixed depreciation schedules only if the physical depreciation of assets is nonstochastic (σ K = 0). Since in our model investors observe all cash flows and all parameters of the economic environment of the firm, historical cost accounting in conjunction with fixed depreciation schedules will not generate information incrementally useful to investors if σ K > 0. In fact, we show below that, even if shocks to asset productivity are immediately incorporated in the book value of assets, the range of valuations consistent with the firm’s fundamentals is the same under historical cost accounting as under cash accounting.

  21. 21.

    To verify inequality (9 ), substitute the definition of \(B_{t}^{cc}\) from Eq. 8 into the left-hand side:

    $$GI_{t}-\sup_{\tau\le t}\left( GI_{\tau}-P_{\tau}K_{\tau}\right)^{+}\le P_{t}K_{t}, $$

    or, equivalently,

    $$GI_{t}-P_{t}K_{t}\le\sup_{\tau\le t}\left( GI_{\tau}-P_{\tau}K_{\tau}\right)^{+}. $$

    The inequality above must hold by the definition of supremum. To verify the second statement, note that G I t is monotonically increasing. Therefore \(dB_{t}^{cc}<0\) implies that the supremum in the right-hand side of Eq. 8 is attained at t:

    $$\sup_{\tau\le t}\left( GI_{\tau}-P_{\tau}K_{\tau}\right)^{+}=GI_{t}-P_{t}K_{t}. $$

    The equation above is equivalent to \(B_{t}^{cc}=P_{t}K_{t}\) .

  22. 22.

    To simplify notation, we will drop the subscript t when it is not needed and use subscripts on V to denote partial derivatives, for example, V P X in the expression below denotes the cross-derivative of V with respect to P and X.

  23. 23.

    To ensure that a solution for the firm’s equity value exists and is always finite, we impose the following two constraints on the parameters of the model: \(\lambda >\frac {\alpha }{1-\alpha }\) and \(\bar {r}>0\) . In addition, to guarantee uniqueness, we restrict attention to value functions that approach zero when X t → 0. The latter assumption excludes, for example, solutions with deterministic “bubbles” growing at the discount rate.

  24. 24.

    It is straightforward to verify that, when the price of new assets is constant over time and the physical depreciation is nonstochastic, the expression for the optimal investment barrier in Eq. 14 reduces to the one of Dixit and Pindyck (1994, p. 376).

  25. 25.

    This is consistent with the intuitive argument of Edwards and Bell (1961), who emphasize that one of the advantages of the “current operating profit,” that is, net income under replacement cost accounting, is that it evaluates firm as a going concern and can be used for predictive purposes (e.g., pp. 99–100).

  26. 26.

    Recall that investors observe P t d I t as this is the investment cash outflow.

  27. 27.

    When the firm can adjust its capacity both upward and downward, the optimal investment policy is to set the marginal revenue equal to the (properly defined) user cost of capital at all times. It can then be verified that, on the optimal investment path, the controlled process, K t , follows a geometric Brownian motion, and the control process, I t , has an unbounded variation. However, the standard approach in the singular stochastic control theory limits the set of controls to processes of bounded variation (e.g., Fleming and Soner, 2006, Chapter 8). Therefore, we will first solve the model in discrete time; once we obtain a solution, we will, for the purposes of comparison, describe its limit as the duration of the time period approaches zero.

  28. 28.

    Since we allow for stochastic depreciation, the firm does not know the exact capacity of its assets in period t + 1 before g K,t+1 is realized.

  29. 29.

    We show in the proof of Proposition 2 that the constants C 1and C 2are givenby:

    $$C_{2}\equiv\left( \frac{\alpha\beta E\left[g_{X}g_{K}^{\alpha}\right]}{1-\beta\left( 1-\delta\right)E\left[g_{K}g_{P}\right]}\right)^{\frac{1}{1-\alpha}} $$

    and

    $$C_{1}\equiv\frac{\left( 1-\alpha\right)\left( 1-\beta\left( 1-\delta\right)E\left[g_{K}g_{P}\right]\right)}{\alpha\left( 1-\beta E\left[g_{X}^{\frac{1}{1-\alpha}}g_{P}^{-\frac{\alpha}{1-\alpha}}\right]\right)}. $$

    Wewill discuss below the limiting values of these constants as the length of the timeperiod approaches zero.

  30. 30.

    However, observing cash flows alone is not sufficient for equity valuation purposes in the scenario with reversible investments. According to Eq. 19 , the firm’s equity value depends on \(P_{t}\hat {K}_{t+1}\) , but investors cannot solve for this quantity if they observe only \(X_{t}K_{t}^{\alpha }\) and P t I t .

  31. 31.

    These disclosures were used by, for instance, Lindenberg and Ross (1981) to construct their Tobin’s q estimates.

  32. 32.

    When d t → 0, there is no difference between the cum-dividend and ex-dividend valuations of the firm since the instantaneous cash flows approach zero. The proof of expressions (21 ) and (22 ) is available from authors upon request.

  33. 33.

    We will write z K,t todenote the value of process z K at time t.

  34. 34.

    IFRS 13 explicitly states that value in use is not fair value (paragraph 6).

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Acknowledgements

We thank Tim Baldenius, Mary Barth, Jeremy Bertomeu, Davide Cianciaruso (discussant), Sunil Dutta, Jonathan Glover (discussant), Moritz Hiemann, Alastair Lawrence, Charles Lee, Guoyu Lin, Dmitry Makarov (discussant), Jim Ohlson, Stephen Penman, Tarun Ramadorai, Stefan Reichelstein (editor), Igor Vaysman, two anonymous reviewers, seminar participants at Columbia University, the University of Vienna, and the participants of the 4th International Moscow Finance Conference, the 12th Workshop on Accounting and Economics in Tilburg, 2015 Berkeley-Stanford Joint Workshop, and 2016 Review of Accounting Studies conference for their helpful comments and suggestions.

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Correspondence to Alexander Nezlobin.

Appendices

Appendix A

Proof Proof of Observation 1

The present value of cash flows from existing assets is equal to:

$$ B_{t}^{viu}=E_{t}\left[{\int}_{t}^{\infty}e^{-r(s-t)}X_{s}K_{s}^{\alpha}ds\right] $$
(23)

with the state variables following

$$dK_{t}=-\delta K_{t}dt+\sigma_{K}K_{t}dz_{K}, $$
$$dX_{t}=\mu_{X}X_{t}dt+\sigma_{X}X_{t}dz_{X}. $$

From the above SDEs, we can solve for X t and K t :Footnote 33

$$\begin{array}{@{}rcl@{}} K_{t} &=&K_{0}e^{-\left( \delta+\frac{1}{2}{\sigma_{K}^{2}}\right)t+\sigma_{K}z_{K,t}},\\ X_{t} &=&X_{0}e^{\left( \mu_{X}-\frac{1}{2}{\sigma_{X}^{2}}\right)t+\sigma_{X}z_{X,t}}. \end{array} $$

Then the ratio of \(X_{s}K_{s}^{\alpha }/X_{t}K_{t}^{\alpha }\) is

$$\begin{array}{@{}rcl@{}} \frac{X_{s}K_{s}^{\alpha}}{X_{t}K_{t}^{\alpha}} &=& \exp\left( -\left( -\mu_{X}+\alpha\delta+\frac{\alpha}{2}{\sigma_{K}^{2}}+\frac{1}{2}{\sigma_{X}^{2}}\right)(s-t)\right.\\ && \left. +~\alpha\sigma_{K}(z_{K,s}-z_{K,t})+\sigma_{X}(z_{X,s}-z_{X,t})\!\vphantom{\frac{1}{2}}\right)\!. \end{array} $$
(24)

Consider the followingprocess:

$$\widetilde{z}_{t}\equiv\alpha\sigma_{K}z_{K,t}+\sigma_{X}z_{X,t}\sim\mathcal{N}\left( 0,\left( \alpha^{2}{\sigma_{K}^{2}}+{\sigma_{X}^{2}}+2\alpha\rho_{XK}\sigma_{K}\sigma_{X}\right)t\right), $$

and rewrite Eq. 24 as

$$\frac{X_{s}K_{s}^{\alpha}}{X_{t}K_{t}^{\alpha}}=\exp\left( -\left( -\mu_{X}+\alpha\delta+\frac{\alpha}{2}{\sigma_{K}^{2}}+\frac{1}{2}{\sigma_{X}^{2}}\right)(s-t)+\widetilde{z}_{s}-\widetilde{z}_{t}\right). $$

Thenthe value of cash flows from assets in place can be expressed as:

$$\begin{array}{@{}rcl@{}} B_{t}^{viu} & =&E_{t}\left[{\int}_{t}^{\infty}e^{-r(s-t)}X_{s}K_{s}^{\alpha}ds\right]=X_{t}K_{t}^{\alpha}{\int}_{t}^{\infty}e^{-r(s-t)}E_{t}\left[\frac{X_{s}K_{s}^{\alpha}}{X_{t}K_{t}^{\alpha}}\right]ds\\ & =&X_{t}K_{t}^{\alpha}{\int}_{t}^{\infty}e^{-\left( r-\mu_{X}+\alpha\delta+\frac{\alpha}{2}{\sigma_{K}^{2}}+\frac{1}{2}{\sigma_{X}^{2}}\right)(s-t)}E_{t}\left[e^{\widetilde{z}_{s}-\widetilde{z}_{t}}\right]ds. \end{array} $$

Since \(\widetilde {z}_{s}-\widetilde {z}_{t}\) is normallydistributed,

$$E_{t}\left[e^{\widetilde{z}_{s}-\widetilde{z}_{t}}\right]=e^{\frac{1}{2}\left( \alpha^{2}{\sigma_{K}^{2}}+{\sigma_{X}^{2}}+2\alpha\rho_{XK}\sigma_{K}\sigma_{X}\right)\left( s-t\right)}. $$

Therefore,

$$\begin{array}{@{}rcl@{}} B_{t}^{viu} & =&X_{t}K_{t}^{\alpha}{\int}_{t}^{\infty}e^{-\left( r-\mu_{X}+\alpha\delta+\frac{\alpha}{2}{\sigma_{K}^{2}}+\frac{1}{2}{\sigma_{X}^{2}}\right)(s-t)}E_{t}\left[e^{\widetilde{z}_{s}-\widetilde{z}_{t}}\right]ds\\ & =&X_{t}K_{t}^{\alpha}{\int}_{t}^{\infty}e^{-\bar{r}(s-t)}ds=\frac{XK^{\alpha}}{\bar{r}}. \end{array} $$

Proof Proof of Proposition 1

Recall that the Hamilton-Jacobi-Bellman equation for the firm’s value is:

$$ \min\left( rV-\mathcal{L}V,P-V_{K}\right)=0, $$
(25)

where

$$\begin{array}{@{}rcl@{}} \mathcal{L}V&\equiv& XK^{\alpha}-\delta KV_{K}+\mu_{X}XV_{X}+\mu_{P}PV_{P}+\frac{1}{2}{\sigma_{K}^{2}}K^{2}V_{KK}\\ && +~\frac{1}{2}{\sigma_{P}^{2}}P^{2}V_{PP}+\frac{1}{2}{\sigma_{X}^{2}}X^{2}V_{XX} + \rho_{KP}\sigma_{K}\sigma_{P}KPV_{KP}\\ && +~\rho_{KX}\sigma_{K}\sigma_{X}KXV_{KX}+\rho_{XP}\sigma_{X}\sigma_{P}PXV_{XP}. \end{array} $$

We will first construct a solution to the variational inequality (25 ) that is continuously differentiableeverywhere and is C 2 in the inaction region (Note that the PDE in the investment region, the second part of thevariational inequality in Eq. 25 , does not depend on second derivatives of V.) The boundarybetween investment and inaction regions will be determined jointly with the solution. It will thenfollow from a standard verification argument that the value function so constructed is indeed equalto the present value of optimized cash flows, and the investment policy characterized by theboundary is indeed optimal.

We guess the solution of the variational inequality (25 ) to be of the following form,

$$ V(X,P,K)=XK^{\alpha}f(X,P,K), $$
(26)

where f(X,P,K)is to be determined. Substituting (26 ) into (25 ) results in the following variational inequality for f(X,P,K)

$$ \min\left( \overline{r}f-\mathcal{L}^{f}f,\frac{P}{XK^{\alpha-1}}-Kf_{K}-\alpha f\right)=0, $$
(27)

where

$$\begin{array}{@{}rcl@{}} \mathcal{L}^{f}f &=& 1+\bar{\mu}_{K}Kf_{K}+\bar{\mu}_{X}Xf_{X}+\bar{\mu}_{P}Pf_{P}+\frac{1}{2}{\sigma_{K}^{2}}K^{2}f_{KK}\\ && +~\frac{1}{2}{\sigma_{P}^{2}}P^{2}f_{PP} +\frac{1}{2}{\sigma_{X}^{2}}X^{2}f_{XX}+\rho_{KP}\sigma_{K}\sigma_{P}KPf_{KP}\\ && +~\rho_{KX}\sigma_{K}\sigma_{X}KXf_{KX}+\rho_{XP}\sigma_{X}\sigma_{P}PXf_{XP}. \end{array} $$

The new coefficients \(\bar {\mu }_{P}\) ,\(\bar {\mu }_{K}\) ,\(\bar {\mu }_{X}\) aredefined as

$$ \begin{array}{l} \bar{\mu}_{P}=\mu_{P}+\alpha\rho_{KP}\sigma_{K}\sigma_{P}+\rho_{XP}\sigma_{X}\sigma_{P},\\ \bar{\mu}_{K}=-\delta+{\alpha\sigma_{K}^{2}}+\rho_{KX}\sigma_{X}\sigma_{K},\\ \bar{\mu}_{X}=\mu_{X}+\alpha\rho_{KX}\sigma_{K}\sigma_{X}+{\sigma_{X}^{2}}. \end{array} $$
(28)

First, we will solve the part of the variational inequality that is responsible for the no-investmentregion (\(\overline {r}f-\mathcal {L}^{f}f=0\) ).In the no-investment region, we have:

$$\begin{array}{@{}rcl@{}} \overline{r}f&=&1+\bar{\mu}_{K}Kf_{K}+\bar{\mu}_{X}Xf_{X}+\bar{\mu}_{P}Pf_{P}+\frac{1}{2}{\sigma_{K}^{2}}K^{2}f_{KK}\\ && +~\frac{1}{2}{\sigma_{P}^{2}}P^{2}f_{PP} +\frac{1}{2}{\sigma_{X}^{2}}X^{2}f_{XX}+\rho_{KP}\sigma_{K}\sigma_{P}KPf_{KP}\\ && +~\rho_{KX}\sigma_{K}\sigma_{X}KXf_{KX}+\rho_{XP}\sigma_{X}\sigma_{P}PXf_{XP}. \end{array} $$
(29)

We look for a candidate solution of the following form:

$$ f(X,P,K)=G(X,P,K)+\frac{1}{\overline{r}}. $$
(30)

Then G(X,P,K)satisfies the following homogeneous PDE:

$$\begin{array}{@{}rcl@{}} \overline{r}G&=&\bar{\mu}_{K}KG_{K}+\bar{\mu}_{X}XG_{X}+\bar{\mu}_{P}PG_{P}+\frac{1}{2}{\sigma_{K}^{2}}K^{2}G_{KK}\\ && +\frac{1}{2}{\sigma_{P}^{2}}P^{2}G_{PP} +\frac{1}{2}{\sigma_{X}^{2}}X^{2}G_{XX}+\rho_{KP}\sigma_{K}\sigma_{P}KPG_{KP}\\ && +\rho_{KX}\sigma_{K}\sigma_{X}KXG_{KX}+\rho_{XP}\sigma_{X}\sigma_{P}PXG_{XP}. \end{array} $$
(31)

Next, we implement the change of variables

$$ p=\log P,\text{} k=\log K,\text{} x=\log X, $$
(32)

thus leading to

$$\begin{array}{@{}rcl@{}} \overline{r}G &=& \left( \bar{\mu}_{K}\,-\,\frac{1}{2}{\sigma_{K}^{2}}\right)G_{k}\,+\,\left( \bar{\mu}_{X}\,-\,\frac{1}{2}{\sigma_{X}^{2}}\right)G_{x}\,+\,\left( \bar{\mu}_{P}\,-\,\frac{1}{2}{\sigma_{P}^{2}}\right)G_{p} \,+\,\frac{1}{2}{\sigma_{K}^{2}}G_{kk}\\ && +~\frac{1}{2}{\sigma_{P}^{2}}G_{pp} \,+\,\frac{1}{2}{\sigma_{X}^{2}}G_{xx}\,+\,\rho_{KP}\sigma_{K}\sigma_{P}G_{kp}\,+\,\rho_{KX}\sigma_{K}\sigma_{X}G_{kx}\,+\,\rho_{XP}\sigma_{X}\sigma_{P}G_{xp}.\\ \end{array} $$
(33)

Let wk(α − 1) + xp, andnote that w = ln ω.Then, we have:

$$\begin{array}{@{}rcl@{}} G_{p}&=&-G_{w},\\ G_{k}&=&\left( \alpha-1\right)G_{w},\\ G_{x}&=&G_{w},\\ G_{pp}&=&G_{xx}=G_{ww},\\ G_{kk}&=&\left( \alpha-1\right)^{2}G_{ww},\\ G_{px}&=&-G_{ww},\\ G_{kx}&=&\left( \alpha-1\right)G_{ww},\\ G_{kp}&=&-\left( \alpha-1\right)G_{ww}. \end{array} $$

Now implementing the change of variables w = k(α − 1) + xp,we obtain a second-order ordinary differential equation for G(w):

$$ \frac{1}{2}\bar{\sigma}^{2}G^{\prime\prime}+\bar{\mu}G^{\prime}-\bar{r}G=0, $$
(34)

where \(\bar {\mu }\) and \(\bar {\sigma }^{2}\) are defined as

$$\begin{array}{@{}rcl@{}} \bar{\mu} &=& \bar{\mu}_{X}-\bar{\mu}_{P}+\left( \alpha-1\right)\bar{\mu}_{K}+\frac{1}{2}\left( {\sigma_{P}^{2}}-{\sigma_{X}^{2}}+\left( 1-\alpha\right){\sigma_{K}^{2}}\right),\\ \bar{\sigma}^{2} &=& {\sigma_{X}^{2}}+{\sigma_{P}^{2}}+\left( \alpha-1\right)^{2}{\sigma_{K}^{2}}+2\left( \alpha-1\right)\rho_{KX}\sigma_{K}\sigma_{X}\\ && -~2\rho_{PX}\sigma_{P}\sigma_{X}-2\left( \alpha-1\right)\rho_{PK}\sigma_{P}\sigma_{K}. \end{array} $$
(35)

By substituting Eq. 28 into Eq. 35 , it is straightforward to verify that the definition of\(\bar {\mu }\) here isconsistent with the one in Eq. 11.

Our conjectured optimal investment policy will be characterized by a threshold value w , such that the firmdoes not invest if w < w .Therefore the firm’s value must be always finite for w < w , and, in particular, it mustapproach zero as w →− (The firm’svalue must approach zero as X t goes to zero, since the current and expected future cash flows are proportional to X t .) Hence we choose the following solution of the ODE (34 )

$$ G(w)=\frac{A}{\bar{r}}\exp\left( \lambda\left( w-w^{\ast}\right)\right), $$
(36)

where

$$ \lambda=-\frac{\bar{\mu}}{\bar{\sigma}^{2}}+\sqrt{\left( \frac{\bar{\mu}}{\bar{\sigma}^{2}}\right)^{2}+\frac{2\bar{r}}{\bar{\sigma}^{2}}}, $$
(37)

is the positive root of the quadraticequation

$$\lambda^{2}+\frac{2\bar{\mu}}{\bar{\sigma}^{2}}\lambda-\frac{2\bar{r}}{\bar{\sigma}^{2}}=0. $$

The positive rootis chosen since \(\underset {w\rightarrow -\infty }{\lim }G(w)=0\) .The constant of integration A and the no-investment boundary w can befound from the boundary conditions.

The boundary conditions at w = w are the standard value-matching and smooth-pasting conditions that come from the second part ofthe variational inequality (27 ) (e.g., Dixit and Pindyck 1994, p.364). Implementing the samecandidate solution (30 ) and the same changes of variables as in the no-investment region, thesecond part of the variational inequality becomes:

$$ \left( \alpha-1\right)G^{\prime}+\alpha G=e^{-w}-\frac{\alpha}{\bar{r}}. $$
(38)

The first boundary condition (value-matching) states that the equality above has to be satisfiedat w .The second (smooth-pasting) boundary condition can be obtained by differentiating (38 ) withrespect to w:

$$-e^{-w}=\left( \alpha-1\right)G^{\prime\prime}+\alpha G^{\prime}. $$

The equality abovehas to be satisfied at w .

To summarize, we have:

$$ \left( \alpha-1\right)G^{\prime}\left( w^{*}\right)+\alpha G\left( w^{*}\right)=e^{-w^{*}}-\frac{\alpha}{\bar{r}}, $$
(39)
$$ -e^{-w^{*}}=\left( \alpha-1\right)G^{\prime\prime}\left( w^{*}\right)+\alpha G^{\prime}\left( w^{*}\right). $$
(40)

Expressing \(e^{-w^{*}}\) from the first condition and substituting it into the second, we obtain:

$$ \left( \alpha-1\right)G^{\prime\prime}\left( w^{*}\right)+\left( 2\alpha-1\right)G^{\prime}\left( w^{*}\right)+\alpha G\left( w^{*}\right)+\frac{\alpha}{\bar{r}}=0. $$
(41)

Now we substitute the expression for G(⋅)from Eq. 36 to find A:

$$A=\frac{\alpha}{\left( \lambda+1\right)\left( \lambda-\alpha\lambda-\alpha\right)}. $$

Finally, substituting (36 ) and the expression for A above into Eq. 39 , we obtain:

$$\begin{array}{@{}rcl@{}} e^{-w^{*}} & =&\frac{\alpha}{\bar{r}}-\left( 1-\alpha\right)\frac{A}{\bar{r}}\lambda+\alpha\frac{A}{\bar{r}}\\ & =&\frac{\alpha\lambda}{\bar{r}\left( 1+\lambda\right)}. \end{array} $$

It then follows that

$$\omega^{*}=e^{w^{*}}=\frac{\bar{r}\left( 1+\lambda\right)}{\alpha\lambda}. $$

We have now found both constants that were still to be identified in equation (36 ), A and w . The optimal investmentprocess \(I_{t}^{*}\) is such that w serves as a reflecting barrierfor w t . The candidate solutionis C 1 everywhere. Note furtherthat the process w t willnever cross the threshold w ,and our solution is C 2 for w t < w . Since oursolution is C 2 in the interior of its domain, it is indeed the value function for problem (25 )(e.g., Strulovici and Szydlowski 2015). Note further that since, by construction,\(dI_{t}^{*}\geq 0\) , thecandidate control we have identified is monotonic and therefore has bounded variation.It is therefore indeed the optimal control (e.g., Strulovici and Szydlowski 2015). □

Proof Proof of Proposition 2

We need to solve the following Bellman equation:

$$\begin{array}{@{}rcl@{}} V(\hat{K}_{t},X_{t},P_{t},g_{K,t}) &=& X_{t}g_{K,t}^{\alpha}\hat{K}_{t}^{\alpha}+P_{t}\left( 1-\delta\right)\hat{K}_{t}g_{K,t}\\ && +~\max\limits_{\hat{K}_{t+1}}\{ \beta\cdot E[V(\hat{K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1})]-P_{t}\cdot\hat{K}_{t+1}\}.\\ \end{array} $$
(42)

The first-order condition for \(\hat {K}_{t+1}\) is:

$$\beta\cdot E\left[V_{\hat{K}_{t+1}}\left( \hat{K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1}\right)\right]=P_{t}. $$

Calculating\(V_{\hat {K}}\) from Eq. 42 and substituting into the equation above, weobtain:

$$\beta\cdot E\left[\alpha X_{t+1}g_{K,t+1}^{\alpha}\hat{K}_{t+1}^{\alpha-1}+\left( 1-\delta\right)P_{t+1}g_{K,t+1}\right]=P_{t}. $$

Therefore,

$$ \alpha\beta E\left[g_{X}g_{K}^{\alpha}\right]X_{t}\hat{K}_{t+1}^{\alpha-1}+\left( 1-\delta\right)\beta E\left[g_{P}g_{K}\right]P_{t}=P_{t}. $$
(43)

Let aE[g P g K ] and \(b\equiv E\left [g_{X}g_{K}^{\alpha }\right ]\) .Then, from Eq. 43 , we can find the optimal value of\(\hat {K}_{t+1}\) :

$$\hat{K}_{t+1}=\left( \frac{\alpha\beta b}{1-\left( 1-\delta\right)\beta a}\right)^{\frac{1}{1-\alpha}}\left( \frac{X_{t}}{P_{t}}\right)^{\frac{1}{1-\alpha}}. $$

Therefore,

$$C_{2}=\left( \frac{\alpha\beta b}{1-\left( 1-\delta\right)\beta a}\right)^{\frac{1}{1-\alpha}}. $$

We will now look for a solution for the firm’s equity value of the following form:

$$ V\left( \hat{K}_{t},X_{t},P_{t},g_{K,t}\right)=X_{t}g_{K,t}^{\alpha}\hat{K}_{t}^{\alpha}+P_{t}\left( 1-\delta\right)\hat{K}_{t}g_{K,t}+C_{3}X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}, $$
(44)

where the constant C 3 is to be determined. Given the candidate solution in Eq. 44 , we can calculate\(E_{t}\left [V\left (\hat {K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1}\right )\right ]\) as:

$$\begin{array}{@{}rcl@{}} E_{t}[V(\hat{K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1})] \!&=&\! C_{2}^{\alpha}\left( \frac{X_{t}}{P_{t}}\right)^{\frac{\alpha}{1-\alpha}}X_{t}b\,+\,C_{2}P_{t}\left( \frac{X_{t}}{P_{t}}\right)^{\frac{1}{1-\alpha}}P_{t}(1\,-\,\delta)a\\ && +~C_{3}X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}E\left[g_{X}^{\frac{1}{1-\alpha}}g_{P}^{-\frac{\alpha}{1-\alpha}}\right]. \end{array} $$

Let \(c\equiv E\left [g_{X}^{\frac {1}{1-\alpha }}g_{P}^{-\frac {\alpha }{1-\alpha }}\right ]\) .Then the expression above can be simplifiedto:

$$E_{t}\left[V\left( \hat{K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1}\right)\right]=X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}\left\{ bC_{2}^{\alpha}+C_{2}\left( 1-\delta\right)a+C_{3}c\right\} . $$

Now substituting the candidate solution (44 ) and the expression for\(E_{t}\left [V\left (\hat {K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1}\right )\right ]\) above into the Bellman Eq. 42 and simplifying, weget:

$$C_{3}=\beta\left\{ bC_{2}^{\alpha}+C_{2}\left( 1-\delta\right)a+C_{3}c\right\} -C_{2}. $$

From this we can find C 3:

$$C_{3}=\frac{C_{2}}{1-\beta c}\left( \beta bC_{2}^{\alpha-1}+\left( 1-\delta\right)\beta a-1\right). $$

Nownote that the candidate solution in Eq. 44 can be written as:

$$\begin{array}{@{}rcl@{}} V\left( \hat{K}_{t},X_{t},P_{t},g_{K,t}\right) & =&X_{t}g_{K,t}^{\alpha}\hat{K}_{t}^{\alpha}+P_{t}\left( 1-\delta\right)\hat{K}_{t}g_{K,t}+C_{3}X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}\\ & =&CF_{t}+P_{t}\left( 1-\delta\right)K_{t}+P_{t}I_{t}-P_{t}I_{t}+C_{3}X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}\\ & =&CF_{t}-P_{t}I_{t}+P_{t}\hat{K}_{t+1}+\frac{C_{3}}{C_{2}}C_{2}\left( \frac{X_{t}}{P_{t}}\right)^{\frac{1}{1-\alpha}}P_{t}\\ & =&CF_{t}-P_{t}I_{t}+P_{t}\hat{K}_{t+1}\left( 1+C_{1}\right), \end{array} $$

where \(C_{1}\equiv \frac {C_{3}}{C_{2}}\) . It remainsto simplify C 1as follows:

$$\begin{array}{@{}rcl@{}} C_{1} & =&\frac{1}{1-\beta c}\left( \beta bC_{2}^{\alpha-1}+\left( 1-\delta\right)\beta a-1\right)\\ & =&\frac{1}{1-\beta c}\left( \frac{1-\left( 1-\delta\right)\beta a}{\alpha}+\left( 1-\delta\right)\beta a-1\right)\\ & =&\frac{\left( 1-\alpha\right)\left( 1-\left( 1-\delta\right)\beta a\right)}{\alpha\left( 1-\beta c\right)}. \end{array} $$

Appendix B

In this section, we consider an application of our model to valuation of a firm that i) uses the historical cost model of IAS 16 to calculate the book value of its property, plant, and equipment and ii) recognizes write-downs as prescribed by IAS 36. Under IAS 36, an impairment loss has to be recognized if the carrying amount of an asset is greater than the maximum of its fair value less costs of disposal and value in use. The definition of value in use in IAS 36 is consistent with that used in our paper: it is equal to the present value of future cash flows expected to be derived from an asset. Different approaches to fair value measurement are defined in IFRS 13. According to this standard, the fair value of an asset should be measured as the price of an identical asset in an orderly transaction (if such price is observable) or using an appropriate valuation technique, such as the cost approach or the income approach.Footnote 34 Since in the setting with irreversible investments there is no secondary market for used capital goods, we will assume that the firm uses the cost approach to fair value measurement. We further assume that the cost of disposal is immaterial.

Under IAS 36, when an impairment is recognized, the firm has to disclose whether the new book value of the asset reflects its fair value or value in use. In addition, past impairments are reversed if the recoverable amount increases, but such reversals cannot lead to a carrying amount greater than what it would have been had not impairment loss been recognized before. In this setting, we obtain the following bounds on the firm’s equity value.

Corollary 3

Assume the firm uses the cost approach to fair value measurement and recognizes impairments according to IAS 36. Then the tightest bounds on the firm’s equity value that hold almost surely conditional on \(\mathcal {I}_{t}\) are:

If P t d I t > 0,

$$ V_{t}=\frac{CF_{t}}{\bar{r}}\left( 1+A\right); $$
(45)

if a revaluation to replacement cost is recognized \(\left (B_{t}=B_{t}^{rc}\right )\) ,

$$ V_{t}=\frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{CF_{t}}{B_{t}\cdot\omega^{*}}\right]^{\lambda}\right); $$
(46)

if a revaluation to value in use is recognized,

$$ \frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{\bar{r}}{\omega^{*}}\right]^{\lambda}\right)\le V_{t}\le\frac{CF_{t}}{\bar{r}}\left( 1+A\right); $$
(47)

if \(B_{t}>\frac {CF_{t}}{\bar {r}}\) ,

$$ \frac{CF_{t}}{\bar{r}}\le V_{t}\le\frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{CF_{t}}{B_{t}\cdot\omega^{*}}\right]^{\lambda}\right); $$
(48)

otherwise

$$ \frac{CF_{t}}{\bar{r}}\le V_{t}\le\frac{CF_{t}}{\bar{r}}\left( 1+A\right). $$
(49)

We now briefly discuss the five cases described in the corollary above. First, if new investment is observed, P t d I t > 0, then the firm is at the investment threshold and its equity can be valued accordingly. If the firm recognizes a write-down (or reverses a past write-down) to fair value, then investors know that the current book value of assets reflects their replacement cost, and the firm’s equity can be valued according to Eq. 46. If a write-down to value in use is recognized, then the new book value of assets is equal to

$$B_{t}=B_{t}^{viu}=\frac{CF_{t}}{\bar{r}} $$

and it has to be that \(B_{t}\geq B_{t}^{rc}\) (Otherwise a write-down to fair value would have been recognized.) It follows that

$$\begin{array}{@{}rcl@{}} V_{t}&=&\frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{CF_{t}}{B_{t}^{rc}\cdot\omega^{*}}\right]^{\lambda}\right)\geq\frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{CF_{t}}{B_{t}^{viu}\cdot\omega^{*}}\right]^{\lambda}\right)\\ &=&\frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{\bar{r}}{\omega^{*}}\right]^{\lambda}\right), \end{array} $$

and inequality (47 ) obtains.

Now assume that investors observe a book value of assets that is greater than the value in use:

$$B_{t}>\frac{CF_{t}}{\bar{r}}. $$

Then it has to be that

$$B_{t}\le B_{t}^{rc} $$

since otherwise a write-down would have been recognized. Therefore, in this case, investors know that

$$\frac{CF_{t}}{\bar{r}}\le V_{t}\le\frac{CF_{t}}{\bar{r}}\left( 1+A\cdot\left[\frac{CF_{t}}{B_{t}\cdot\omega^{*}}\right]^{\lambda}\right). $$

Lastly, when \(B_{t}<B_{t}^{viu}\) , the firm’s replacement cost of assets can be arbitrarily close to zero or \(\frac {CF_{t}}{\omega ^{*}}\) , and the tightest inequality that holds almost surely is given by Eq. 49.

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Livdan, D., Nezlobin, A. Accounting rules, equity valuation, and growth options. Rev Account Stud 22, 1122–1155 (2017). https://doi.org/10.1007/s11142-017-9402-6

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Keywords

  • Equity valuation
  • Real options
  • Irreversible investment
  • Accounting rules

JEL Classification

  • E22
  • G31
  • M41