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.

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 :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 w ≡ k(α − 1) + x − p, 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) + x − p,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 a ≡ E[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.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.