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.

1 Introduction

The FASB Conceptual Framework for Financial Reporting (FASB 2010) states that “the objective of general purpose financial reporting is to provide financial information about the reporting entity that is useful to existing and potential investors, lenders, and other creditors in making decisions about providing resources to the entity.” To achieve this objective, it is important to understand the informational needs of different groups of financial statement users, including the current and potential future shareholders of the firm. However, the theoretical literature on accounting-based equity valuation provides little guidance on the relative desirability of alternative accounting rules (e.g., Ohlson 1995; Feltham and Ohlson 1995).¹ In a model without uncertainty, Nezlobin (2012) shows that replacement cost accounting is essentially the only depreciation policy under which there exists a mapping between a firm’s current accounting data and its equity value. The goal of this study is to characterize how the resolution of different types of uncertainty in the firm’s economic environment should get reflected in its financial statements to provide information useful to equity investors.

We apply the real options framework to the problem of accounting-based valuation under uncertainty. In the model, a representative firm makes repeated investments in capital goods and uses the resulting capital stock for production. Initially, we assume that the firm’s investments are irreversible – once purchased and put in place (or constructed), the firm’s capital goods cannot be sold. In practice, investments can be fully or partially irreversible because they are firm-specific (e.g., highly specialized equipment or R&D), industry-specific, or because the market for used capital goods is affected by the “lemons” problem (e.g., Dixit and Pindyck 1994, p.8). Pindyck (1988) characterizes the firm’s optimal investment policy under the assumption that demand for its product is stochastic. We extend the model of Pindyck (1988) and Dixit and Pindyck (1994, Chapter 11) to allow for (i) a stochastic rate of physical depreciation of the firm’s assets in place and (ii) a stochastic price of new capital goods. All three stochastic processes that determine the firm’s economic environment – demand for the firm’s output, the physical productivity of its assets in place, and the price of new capital goods – are allowed to be mutually correlated. In the second part of the paper, we solve the model under the assumption that investments are fully reversible and juxtapose these results to the ones obtained in the setting with irreversibility.

Equity investors in our model know the constant parameters of the firm’s economic environment, for example, the mean and variance of the growth rate of the output market size. However, they do not observe the realizations of the underlying stochastic processes, for example, the actual size of the output market. In valuing the firm’s equity, investors rely on the firm’s financial statements to gather information about its assets in place and cash flows. Our objectives are twofold. First, we identify accounting rules with the property that the firm’s equity value can be precisely determined based on its accounting information.² Second, for commonly considered accounting rules without this property, we seek to characterize the range of valuations consistent with the firm’s financial statements.

We start by showing that, in the setting with irreversible investments, the firm’s optimal investment policy is of a barrier control type. Specifically, the firm invests only when the ratio of its operating cash flow to the replacement cost of its capital stock exceeds a certain threshold. We refer to this ratio as the cash return on economic assets or, simply, cash return on assets when there is no room for confusion.³ The firm’s investment increases both the numerator and the denominator of this ratio. Under the standard assumption that the marginal revenue with respect to capital is decreasing in capital stock, the overall ratio falls with investment. The firm chooses its investment so as to return the ratio of operating cash flows to the replacement cost of assets to a level just below the optimal investment threshold. Consistent with much of the real options literature, the NPV of the marginal project at the investment threshold is strictly greater than zero: the firm optimally takes into account the fact that the investment can be postponed if market conditions improve but cannot be undone if they deteriorate. The NPV of the marginal project is thus equal to the opportunity cost of the option to wait.

The model allows us to derive an analytical expression for the firm’s equity value. Specifically, the firm’s value is equal to the sum of two components: the present value of cash flows to be generated by existing assets and the value of the firm’s capacity expansion (growth) options.⁴ The value of cash flows from existing assets can be calculated by multiplying the current operating cash flow by a capitalization factor that reflects the firm’s cost of capital and other parameters of the firm’s economic environment, such as the expected growth in demand for firm’s output and the drift and variance of the asset productivity process. The value of the capacity expansion options is proportional to the present value of cash flows from existing assets, with the coefficient of proportionality being determined by the distance between the current cash return on assets ratio and the optimal investment threshold.

It follows that, if the book value of assets in the firm’s financial statements reveals their replacement cost, investors will have sufficient information to value the firm precisely at each point in time. Note that we obtain this result in the setting with irreversible investment, that is, even though the firm cannot sell its current assets, knowing their replacement cost is crucial for investors in valuing the firm’s future growth options. In contrast, we show later that in the absence of optionality, that is, when the firm’s investments are fully reversible, precise valuation may still be possible even when the replacement cost of assets is not directly reported to investors. The assumption of investment irreversibility is solely responsible for the unique importance of replacement cost disclosures.

We refer to an accounting system that sets the book value of assets equal to their replacement cost as replacement cost accounting. The replacement cost of assets can be viewed as the amount that the firm would have to pay at the current price of new capital goods to replicate the productive capacity of its current capital stock. Alternatively, the replacement cost can be calculated for each vintage of capital goods as the depreciated historical cost of that vintage (reflecting all shocks to the productivity of assets incurred up to the current date) multiplied by the ratio of the price of new assets today to the price of new assets at the time of purchase. The notion of replacement cost accounting used in this study has a long history in accounting thought: for example, it is closely related to the concept of “current operating profit” in Edwards and Bell (1961). It also corresponds exactly to the cost approach to fair value measurement defined in IFRS 13 and FASB Codification Topic 820 as “a valuation technique that reflects the amount that would be required currently to replace the service capacity of an asset.”

Next we explore the informational properties of alternative accounting rules. We start by assuming that the firm prepares financial statements on a cash basis, that is, investors observe only the cash flows from operating and investment activities. While cash flow information alone is not sufficient for valuation purposes in all states of the world, investors can (i) value the firm precisely when new investment is observed and (ii) always put upper and lower bounds on the firm’s equity value. In the investment region, where the firm is expanding its capacity, the replacement cost of assets in place does not provide incremental value-relevant information relative to the firm’s operating cash flow. The mere fact that the firm is exercising its expansion options tells investors that the cash return-on-assets ratio is at the investment threshold. This information is then sufficient to value the firm conditional on the value of the operating cash flow. In the inaction region (i.e., when cash return on assets is below the investment threshold), the firm’s expansion options cannot be valued precisely without knowing the replacement cost of assets in place. However, their value can be bounded from below by zero and from above by the value of growth options assuming the firm had been at the investment threshold.⁵

These results demonstrate that, while financial statements prepared on a cash basis provide useful information to investors, they are not sufficient to accurately value the firm’s growth options in the inaction region. We further show that the bounds on the firm’s value obtained under cash accounting cannot be improved if financial statements are prepared using value in use accounting (where the book value of assets at each date is set equal to the present value of cash flows that they are expected to generate) or historical cost accounting (assets are carried at historical cost with depreciation reflecting the physical shocks to productivity). The reason for this is that the value of the firm’s capacity expansion options depends on the current price of new capital goods, and the changes in the price of new capital goods do not get reflected in the financial statements under historical cost or value in use accounting if the firm is in the inaction region.

Next, we model a conditionally conservative accounting system under which the book value of assets is written down immediately if it exceeds their replacement cost but is not written up if the replacement cost is higher.⁶ We show that financial statements prepared under such rules allow investors to improve (relative to cash basis accounting) only the upper bound on the value of the firm’s growth options and therefore on its equity value. If accounting is conservative, then the book value of assets always understates their replacement cost. Therefore the reported cash return on assets, that is, the ratio of operating cash flow to the book value of assets, can serve as an upper bound for the cash return on economic assets. The value of capacity expansion options is monotonically increasing in the cash return on economic assets, so this value can be more precisely bounded from above if accounting is conditionally conservative.

We also consider a scenario where the firm’s investments are fully reversible: the firm’s capital goods can be sold at the price of new capital goods with equivalent productive capacity. In this setting, the value of the firm’s equity is also shown to depend on the current replacement cost of the firm’s capital stock. However, the direction of the relation between the firm’s equity value and the replacement cost of its assets switches to the opposite: with reversible investments, the replacement cost of assets is positively associated with the equity value.⁷ When the firm cannot revise its capital stock downward, a high replacement cost of assets indicates that the value of growth options is low (new assets are too expensive or the firm has too much capacity already). On the other hand, if investments are reversible, then the firm would keep assets with a high replacement cost only if the present value of cash flows to be generated by those assets is even higher. Therefore the relation between the replacement cost of assets and the firm’s equity value critically depends on whether the firm can sell its used capital goods. These results caution against judging the value relevance of an accounting amount by the magnitude of its coefficient in a market value regression without explicitly controlling for liquidity of the assets in question.

Interestingly, in the setting with reversible investment, the firm can be valued precisely even if the financial statements are prepared under value in use accounting. When the firm can freely adjust its capital stock in either direction, the replacement cost of assets and value in use are proportional to each other at all times on the optimal investment path and thus carry the same information. In contrast, in the setting with irreversible investment, the ratio of the assets’ value in use to their replacement cost is stochastic in the inaction region.

This study is related to several strands of research in finance and accounting. First, our finding that investment is determined by the operating cash flow and replacement cost of assets in place is consistent with much of the theoretical and empirical literature on investment (e.g., Hayashi 1982; Lewellen and Badrinath 1997; Abel and Eberly 2011). We extend this literature by allowing for stochastic productivity of assets in the setting with irreversible investment. Our finding that replacement cost accounting provides useful information to investors is further consistent with the empirical evidence of Gordon (2001). Our paper also contributes to the growing accounting literature on real options. Several papers examined the problems of incentive provision and product pricing with optionality (e.g., Arya and Glover 2001; Pfeiffer and Schneider 2007; Caskey and Hughes 2012; Johnson et al. 2013; Reichelstein and Rohlfing-Bastian 2015; Baldenius et al. 2016). To the best of our knowledge, the only other theoretical paper that studies accounting-based equity valuation with real options is by Zhang (2000a). Unlike us, Zhang (2000a) studies a European rather American real option related to a single investment or abandonment decision, that is, the firm’s decision cannot be postponed or advanced in his model.

Replacement cost accounting has also emerged as a natural benchmark in studies employing the vintage capital framework of Rogerson (2008).⁸ In Rogerson (2008), the economic depreciation of assets is deterministic but age-dependent. As a consequence, the vintage composition of the firm’s assets in place becomes relevant for valuation. As shown by Nezlobin (2012), financial statements prepared under replacement cost accounting provide information useful to investors who do not observe the full investment path of the firm by properly reflecting the vintage composition of its assets in place. In contrast, the main focus of our paper is on the accounting treatment of uncertainty regarding the economic productivity of assets and their acquisition price. Accordingly, the role of replacement cost accounting in our model is to provide information about past shocks to productivity and capital good prices.

The rest of the paper is organized as follows. We describe the firm’s transactions in Section 2 , and the information that is available to investors under alternative accounting rules in Section 3. The firm’s optimal investment policy and its equity value are characterized in Section 4. In Section 5 , we consider a scenario with reversible investments. Section 6 concludes.

2 Firm transactions

Our model builds on the continuous-time capacity choice model of Dixit and Pindyck (1994, Chapter 11). Consider a firm that makes repeated investments in capital assets and uses the productive capacity of those assets to generate its output. Let K _t denote the firm’s physical capital stock at time t. Assume that the firm’s operating cash flow at time t is given by

$$ CF_{t}=X_{t}K_{t}^{\alpha}, $$

(1)

where 0 < α < 1 is the elasticity of the operating cash flow to capital, and X _t is a shift parameter that reflects the current demand for the firm’s output. The parameter α is less than one, possibly reflecting diminishing physical returns to production or a downward-sloping demand curve for the firm’s product.⁹ We assume that the shift parameter X _t follows a geometric Brownian motion

$$ \frac{dX_{t}}{X_{t}}=\mu_{X}dt+\sigma_{X}dz_{X}, $$

(2)

where μ _X is the drift in demand for the firm’s product, σ _X is the instantaneous variance of the demand process, and X ₀ is greater than zero.

The firm can purchase capital goods instantaneously and frictionlessly at time t at a price of P _t per unit. The price of new capital goods follows a geometric Brownian motion with constant drift μ _P and constant instantaneous variance ${\sigma _{P}^{2}}$ , so

$$ \frac{dP_{t}}{P_{t}}=\mu_{P}dt+\sigma_{P}dz_{P}, $$

(3)

where P ₀ > 0. The drift in the price of new assets, μ _P , can reflect expected inflation (if μ _P > 0) or the notion that new assets become more efficient over time due to technology improvements (if μ _P < 0).

Let d I _t denote the physical quantity of capital goods that the firm purchases at time t; the corresponding cash outflow at time t (cash flow from investing activities) is then given by d I _t ⋅ P _t .¹⁰ We assume that the firm cannot adjust its capital stock downwards; that is, the investments in capital goods are irreversible (d I _t ≥ 0). The firm’s capital stock evolves according to the following equation:

$$ dK_{t}=-\delta K_{t}dt+\sigma_{K}K_{t}dz_{K}+dI_{t}, $$

(4)

The first two terms in the right-hand side of Eq. 4 model the economic depreciation of the firm’s assets in place at time t. The physical rate of depreciation is stochastic with mean δ and instantaneous variance ${\sigma _{K}^{2}}$ . The last term in the right-hand side of Eq. 4 reflects the installation of the newly purchased capital goods.

To make our problem as general as possible, we allow for the Brownian motions d z _X , d z _P , and d z _K to be mutually correlated:

$$\rho_{XP}=\frac{dz_{X}dz_{P}}{dt}, $$

$$\rho_{KP}=\frac{dz_{K}dz_{P}}{dt}, $$

$$\rho_{XK}=\frac{dz_{X}dz_{K}}{dt}. $$

Therefore our analysis allows for situations where the prices of inputs move together with the price of output (e.g., costs of constructing and maintaining drilling rigs can be correlated with the oil price).¹¹ The model also accommodates technology shocks that affect, to varying degrees, both the productivity of the firm’s assets in place as well as the price of new ones. Lastly, it is also conceivable that shocks to demand in the output market are correlated with the productivity of the firm’s assets: for example, disruptions in supply (caused by unexpectedly low productivity of assets) may damp the current and future demand for the firm’s product.

At time t, the firm observes its current capital stock, K _t , the price of new capital goods, P _t , and the current demand for its product, X _t and decides whether to purchase another d I _t units of new capital goods. The firm’s investment policy, I _t , is therefore determined by the three state variables: K _t , X _t , and P _t . The firm chooses its investments to maximize the present value of its cash flows, that is, it solves the following dynamic program:

$$ \underset{\left\{ I_{t}\right\}_{0}^{\infty}}{\max}E_{0}\left[{\int}_{0}^{\infty}e^{-rt}CF_{t}dt-{\int}_{0}^{\infty}e^{-rt}P_{t}dI_{t}\right] $$

(5)

$$s.t.\text{} dK_{t}=-\delta K_{t}dt+\sigma_{K}K_{t}dz_{K}+dI_{t}, $$

$$\frac{dX_{t}}{X_{t}}=\mu_{X}dt+\sigma_{X}dz_{X}, $$

$$\frac{dP_{t}}{P_{t}}=\mu_{P}dt+\sigma_{P}dz_{P}, $$

$$I_{0-}=0,\,\,dI_{t}\geq0\,\,\forall t\geq0,\text{} $$

where r is the firm’s discount rate, assumed to be constant.¹² Our analysis can be extended to a setting in which the firm’s cash flows are discounted by a stochastic discount factor that evolves as a geometric Brownian motion.¹³

Let V _t (X _t ,P _t ,K _t ) denote the firm’s value at time t assuming that it has followed the optimal investment policy, $I_{t}^{*}\left (X_{t},P_{t},K_{t}\right )$ , in the past. To summarize, the model discussed in this section generalizes the one of Dixit and Pindyck (1994, Chapter 11) to allow for stochastic shocks to the productivity of assets (d z _K ) and to the price of new assets (d z _P ). Accordingly, the state variable in our model is three dimensional – (X _t ,P _t ,K _t ). While the firm makes investment decisions based on all information available up to the current date, we assume that the information set of investors who are valuing the firm is imperfect. Specifically, while the investors know the main parameters of the firm’s economic environment, such as the elasticity of cash flow to capital (α) and the drift, variance and correlation parameters of all stochastic processes (μ _X , ${\sigma _{X}^{2}}$ , μ _P , ${\sigma _{P}^{2}}$ , δ, ${\sigma _{K}^{2}}$ , ρ _{X P} , ρ _{K P} , ρ _{X K} ), they do not directly observe the current state variable (X _t ,P _t ,K _t ). To estimate the value of the firm, they must rely on the information in the firm’s financial statements.

3 Investors’ information set

The information set of investors at time t includes the current and all past financial statements of the firm.¹⁴ Formally, we write the investors’ information set as:

$$\mathcal{I}_{t}=\left[\left\{ B_{\tau}\right\}_{\tau\le t},\left\{ CF_{\tau}\right\}_{\tau\le t},\left\{ P_{\tau}\cdot dI_{\tau}\right\}_{\tau\le t}\right]. $$

Here, B _τ denotes the book value of assets at time τ, C F _τ is the firm’s operating cash flow at time τ, and P _τ ⋅ d I _τ is the investment cash outflow.¹⁵ Note that the information in past income statements is subsumed by $\mathcal {I}_{t}$ : the firm’s net income from time τ − d t to τ is given by:

$$CF_{\tau}dt-P_{\tau}dI_{\tau}-B_{\tau-dt}+B_{\tau}. $$

Since $\mathcal {I}_{t}$ includes the whole history of book values and operating and investment cash flows, the path of the firm’s net income is also in $\mathcal {I}_{t}$ .

We now turn to characterizing the book value of assets under alternative accounting rules. Under cash accounting, the book value of assets is always set to zero, $B{}_{\tau }^{cash}=0$ ; that is, the investors only observe the firm’s cash flows. In our model, the net cash flows, C F _t − P _t d I _t , are disbursed to shareholders immediately, so the firm’s cash balance is zero at all times.

Next consider replacement cost accounting where

$$B_{t}^{rc}\equiv P_{t}\cdot K_{t} $$

for all t. Under this rule, at the time of acquisition, new assets are recorded at their cost, P _t ⋅ d I _t .¹⁶ At each date after initial recognition, the book value of assets reflects past shocks to both the productivity of assets in place (d z _K ) and to the price of new assets (d z _P ). In particular, if the price of new capital goods has changed since the time of last investment, all of the firm’s assets in place must be revalued accordingly. The book value of assets in place under replacement cost accounting reflects the amount that the firm would have to pay today for new capital goods to replicate the current capacity of its assets purchased in the past. This definition corresponds exactly to the cost approach to fair value measurement as defined in IFRS 13 and FASB Codification Topic 820.

Note that the information set of investors is imperfect even if replacement cost accounting is used to calculate the book value of assets. To see this, assume that the firm has not been investing for some time. Then, investors effectively observe the evolution of the firm’s operating cash flows, $CF_{t}=X_{t}\cdot K_{t}^{\alpha }$ , and the book value of assets, $B_{t}^{rc}=P_{t}\cdot K_{t}$ . Even though the complete paths of these two processes are revealed to investors over time, it is still impossible for them to solve for the underlying state variable, (X _t ,P _t ,K _t ), because it has more dimensions than their information set.¹⁷

Several recent studies have employed the vintage capital framework of Rogerson’s (2008) to examine a broad set of questions in managerial performance evaluation, financial statement analysis, and equity valuation.¹⁸ These studies have shown that a related concept of replacement cost accounting has a number of desirable features from the perspective of internal and external users of the firm’s financial information. There are two major differences between the economic setting of this paper and the model of Rogerson (2008). First, whereas Rogerson (2008) studies assets with age-dependent but deterministic economic efficiency, the efficiency of capital goods in our model is vintage-independent but stochastic. As a consequence, the goal of replacement cost accounting in the current setting is not to convey information about the vintage composition of the firm’s capital stock (as such information is value-irrelevant) but to reflect past shocks to asset productivity and capital acquisition prices. In practice, an ideal implementation of replacement cost accounting would arguably need to encompass both of these dimensions of economic information.

The second important difference between Rogerson’s (2008) model and ours is that we allow for nontrivial effects of irreversibility on the firm’s investment policy. Rogerson (2008) assumes that demand curves shift out over time, which ensures that the irreversibility constraint never binds. In our setting, the market conditions can improve or worsen stochastically over time, thus leading to a binding irreversibility constraint. It is well known that the problem of dynamic capacity expansion is generally not analytically tractable when the economic depreciation of assets is age-dependent (e.g., Dixit and Pindyck 1994).

Another accounting system considered in this paper is value in use accounting, where the book value of assets at each point in time, $B_{t}^{viu}$ , is set equal to the present value of cash flows that these assets are expected to generate in the future. To calculate the present value of cash flow attributable to the firm’s assets at time t, we assume that I _{t + τ} = 0 for τ > 0, that is, the firm will not make additional investments after date t. The following observation provides an expression for value in use.

Observation 1

The present value of cash flows from the firm’s assets in place at time t is given by

$$ B_{t}^{viu}=\frac{X_{t}K_{t}^{\alpha}}{\bar{r}}, $$

(6)

where

$$ \bar{r}\equiv r-\mu_{X}+\alpha\delta-\alpha\rho_{KX}\sigma_{K}\sigma_{X}-\frac{\alpha\left( \alpha-1\right)}{2}{\sigma_{K}^{2}}. $$

(7)

The present value of cash flows from existing assets can be calculated by capitalizing the current operating cash flow, $X_{t}K_{t}^{\alpha }$ , with an adjusted interest rate $\bar {r}$ . This rate has an intuitive interpretation. Assume first that there is no physical depreciation of capital goods: $\delta ={\sigma _{K}^{2}}=0$ . Then the rate $\bar {r}$ reduces to the one corresponding to the Gordon growth model, $\bar {r}=r-\mu _{X}$ . The third term in Eq. 7 reflects the effect of the expected physical depreciation of assets (δ), taking into account the concavity of the firm’s cash flows in the capital stock (α). When depreciation is stochastic (${\sigma _{K}^{2}} >0$ ), the concavity of cash flows in the capital stock leads to a Jensen’s inequality effect: the expected value of future cash flows declines in ${\sigma _{K}^{2}}$ . Accordingly, the last term in Eq. 7 serves as the concavity adjustment to $\bar {r}$ .¹⁹ Lastly, a positive correlation between shocks to X _t and K _t has a positive effect on the expected value of $X_{t}K_{t}^{\alpha }$ (holding the mean and variance of shocks fixed); this effect is captured by the penultimate term in Eq. 7. Note that, if depreciation is nonstochastic (σ _K = 0), then the last two terms in Eq. 7 are zero, and uncertainty about future demand does not affect the capitalization factor in the calculation of value in use.

The concept of value in use will prove instrumental in interpreting the expression for the firm’s value that we derive below. However, it is clear from Eq. 6 that, in the irreversible investment setting, financial statements prepared under value in use accounting do not provide information useful to investors over and above the firm’s operating cash flow. Since we assume that investors know all parameters of the firm’s economic environment (except for the realizations of stochastic processes), they can calculate $\bar {r}$ and infer $B_{t}^{viu}$ as

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

Therefore the bounds that investors can derive on the firm’s equity value will be precisely the same under value in use accounting as under cash accounting.

We model historical cost accounting as a system under which assets are capitalized at cost when they are acquired, and, at each date thereafter, their book value reflects their current productive capacity. Therefore we assume that shocks to asset productivity are timely reflected in the book value of assets, while the shocks to the price of new assets are not.²⁰ Formally, the book value of assets under historical cost accounting evolves according to the following process:

$$\begin{array}{@{}rcl@{}} dB_{t}^{hc} & =&-\delta B_{t}^{hc}dt+\sigma_{K}B_{t}^{hc}dz_{K}+P_{t}dI_{t},\\ B_{0-}^{hc} & =&0. \end{array} $$

The last accounting system that we consider in this paper is a variant of replacement cost accounting combined with asymmetric recognition of gains and losses. Specifically assume that all capital goods are initially recognized at their acquisition cost, P _t d I _t . Then, at each date, the firm compares the current carrying value of its assets to their total replacement cost: if the former amount exceeds the latter, then all assets are written down to their current replacement cost. However, if the opposite relation holds, then no write-up is recognized. Under such system, the amount of accumulated depreciation (and revaluations) is strictly increasing over time and, at time t, is equal to:

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

where G I _τ is the gross investment up to time τ and (x)⁺ = max {x,0}. Therefore under this rule, the book value of assets at time t, $B_{t}^{cc}$ , is given by:

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

(8)

We will refer to the system described above as conditionally conservative accounting. For our future discussion, it is important to observe that

$$ B_{t}^{cc}\le P_{t}K_{t} $$

(9)

for all t, and $dB_{t}^{cc}<0$ only when $B_{t}^{cc}=P_{t}K_{t}$ .²¹

4 Firm value

We now turn to characterizing the firm’s optimal investment policy and the firm’s equity value. Let ω _t denote the ratio of the firm’s operating cash flow to the replacement cost of its assets in place:

$$\omega_{t}\equiv\frac{CF_{t}}{P_{t}K_{t}}=\frac{X_{t}K_{t}^{\alpha-1}}{P_{t}}. $$

We will refer to ω _t as the cash return on (economic) assets. Note that, when d I _t = 0 (the firm is not investing), the three processes, X _t , K _t , and P _t , evolve as geometric Brownian motions, and therefore so does ω _t . Cash return on assets increases in demand for the firm’s output, X _t , and decreases in the price of new capital goods, P _t . Furthermore, since α < 1, ω _t decreases in the firm’s capital stock. In particular, the firm’s investment, d I _t > 0, has a negative effect on the cash return-on-assets ratio, due to diminishing returns to capital inherent in the firm’s production function.

We show in the proof of Proposition 1 that the firm’s optimal investment policy is fully characterized by a certain threshold value of the cash return-on-assets ratio, ω ^∗ , that serves as a reflecting barrier for the process ω _t . When cash return on assets is below this barrier, ω _t < ω ^∗ , the firm does not invest. When cash return on assets reaches the barrier ω ^∗ , the firm makes a sequence of investments that is just sufficient to prevent ω _t from crossing the threshold; the process ω _t is thus reflected from the barrier as illustrated in Fig. 1 below. Intuitively, the firm expands when the product market is sufficiently profitable; as the firm increases its capacity, the marginal, as well as average, profitability of its sales falls and the firm returns to the inaction (no-investment) region. After that point, new investments will be made only when the output market expands sufficiently more to push ω _t to the barrier again. Therefore after the initial investment at date 0, the cash return-on-assets ratio will follow a geometric Brownian motion, reflected at ω ^∗ .

Fig. 1

Optimal investment policy

We will now write down the Hamilton-Jacobi-Bellman equation that determines the firm’s value function, V(X,P,K).²² First, assume that the firm is not investing, d I _t = 0. Then, by Ito’s lemma, we can write E[C F d t + d V] (i.e., the instantaneous dividend plus the expected change in the value of future cash flows) in the inaction region as:

$$\mathcal{L}V\cdot dt, $$

where

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

On the other hand, since V is the present value of expected cash flows, it has to satisfy:

$$E\left[CFdt+dV\right]=rV\cdot dt. $$

Therefore, in the inaction region, we have:

$$rV=\mathcal{L}V. $$

If the firm’s investment is positive, d I _t > 0, then $rV>\mathcal {L}V$ , since the right-hand side does not include the (positive) NPV of the marginal investment project.

At times when the firm invests, the amount of investment is chosen so that the unit price of new capital goods is equal to the marginal benefit of investment,

$$P=V_{K}. $$

The firm is in the inaction region when the cost of new capital goods is greater than the marginal increase in value that a new investment would generate, P − V _K > 0. To summarize the observations above, the firm’s value function is determined by the following variational inequality:

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

(10)

In the investment region, the first term is positive, and the second term is equal to zero; in the inaction region, $rV-\mathcal {L}V=0<P-V_{K}$ .

Our first proposition provides an expression for the firm’s equity value at each date. In formulating this proposition, the following notation will be convenient. First, let $\bar {\mu }$ be equal to:

$$\begin{array}{@{}rcl@{}} \bar{\mu} &\equiv& \mu_{X}-\mu_{P}-\left( \alpha-1\right)\delta+\frac{1}{2}{\sigma_{X}^{2}}+\frac{1}{2}{\sigma_{P}^{2}}+\left( \alpha-1\right)\left( \alpha-\frac{1}{2}\right){\sigma_{K}^{2}}\\ && +~ (\alpha-1)\rho_{KX}\sigma_{K}\sigma_{X}-\rho_{PX}\sigma_{P}\sigma_{X}-\alpha\rho_{KP}\sigma_{K}\sigma_{P}. \end{array} $$

(11)

Note that, in the absence of uncertainty, $\bar {\mu }$ would be equal to μ _X − μ _P −(α − 1)δ, that is, it would be equal to the growth rate of the cash return on assets ratio, ω. Next, let $\bar {\sigma }^{2}$ be the following measure of aggregate uncertainty of the firm’s economic environment:

$$\begin{array}{@{}rcl@{}} \bar{\sigma}^{2} &\equiv& \sigma_{X}^{2} +{\sigma_{P}^{2}}+(\alpha-1)^{2}{\sigma_{K}^{2}}+2(\alpha-1)\rho_{KX}\sigma_{K}\sigma_{X}\\ && -~2\rho_{PX}\sigma_{P}\sigma_{X} -2(\alpha-1)\rho_{PK}\sigma_{P}\sigma_{K}. \end{array} $$

(12)

Lastly, let λ and A be

$$\lambda\equiv-\frac{\bar{\mu}}{\bar{\sigma}^{2}}+\sqrt{\left( \frac{\bar{\mu}}{\bar{\sigma}^{2}}\right)^{2}+\frac{2\bar{r}}{\bar{\sigma}^{2}}}>0, $$

and

$$A\equiv\frac{\alpha}{\left( \lambda+1\right)\left( \lambda-\alpha\lambda-\alpha\right)}, $$

where $\bar {r}$ is given by Eq. 7. Note that $\bar {\mu }$ , $\bar {\sigma }^{2}$ , λ, and A depend only on the parameters of the model that are known to investors and do not depend on the realizations of stochastic processes, {X _t ,P _t ,K _t }.²³

Proposition 1

The firm’s equity value at time t is equal to

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

(13)

where ω ^∗ is the optimal threshold for investment, given by:

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

(14)

The firm makes its first investment so that

$$I_{0}^{*}=\left( \frac{X_{0}}{P_{0}}\omega^{*}\right)^{\frac{1}{1-\alpha}}, $$

and then invests only when its cash return-on-assets ratio, ω _t , is equalto ω ^∗ .

The equity valuation formula in Eq. 13 has an intuitive interpretation. Recall that the expected value of cash flows from assets in place (value in use) is given by:

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

Proposition 1 shows that the firm’s equity value exceeds the value of cash flows from assets in place by the expectation of payoffs from capacity expansion options. The value of these growth options is given by:

$$\frac{CF_{t}}{\bar{r}}\cdot A\cdot\left[\frac{CF_{t}}{B_{t}^{rc}\cdot\omega^{*}}\right]^{\lambda}. $$

The quantity above is proportional to the present value of cash flows from assets in place times the cash return on assets ratio raised to power λ. Growth options become more valuable as cash return on assets approaches the optimal exercise threshold, ω ^∗ .²⁴ Note that term in square brackets in the expression above is equal to $\frac {\omega _{t}}{\omega ^{*}}$ and thus measures how far the current cash return-on-assets ratio is from the investment threshold. Given replacement cost accounting, investors can accurately value both the cash flows from assets in place and the capacity expansion options.²⁵

Recall that the price of new capital goods, P _t , as well as the parameters of the {P _t } process do not enter the expression for the present value of cash flows from existing assets. The parameters of the {P _t } process do, however, affect the constants A, λ, and ω ^∗ and therefore the valuation of capacity expansion options. Importantly, the value of capacity expansion options also depends on P _t : the higher the price of new capital goods today, the less likely it is that the firm will exercise its growth options soon. Therefore the total value of the firm’s equity depends on all three components of the state variable, (X _t ,P _t ,K _t ). Proposition 1, however, shows that investors do not need to observe the individual components of the state variable to be able to value the firm’s cash flows as long as they observe the two aggregate variables, C F _t and $B_{t}^{rc}$ .

While we do not explicitly model agency problems in this paper, it is interesting to note that, given replacement cost accounting, investors can always verify that the firm is indeed following the optimal investment policy: positive investment should happen if and only if

$$\frac{CF_{t}}{B_{t}^{rc}}=\omega^{*}, $$

and investments should be chosen such that ω ^∗ becomes a reflecting barrier for the cash return-on-investment process. Specifically, on the optimal investment path, the cash return-on-assets ratio must never exceed ω ^∗ , and the firm must not invest when ω < ω ^∗ . This finding is consistent with the recent results in the managerial performance evaluation literature that show that replacement cost accounting can be used to achieve goal congruence in settings with overlapping investments (e.g., Rogerson 2008 and Nezlobin et al. 2015).

We now turn to characterizing the bounds on the firm equity value that investors can calculate under accounting systems that do not convey enough information to value the firm precisely. First, note that, immediately after (or very soon after) observing a positive investment, the cash return on economic assets is close to ω ^∗ . Therefore, even without directly observing the replacement cost of assets in place, investors know that when P _t d I _t > 0,²⁶

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

(15)

On the other hand, if P _t d I _t = 0, it has to be that ω _t < ω ^∗ . The firm’s value must then be bounded from below by the present value of expected cash flows from assets in place and from above by the value of the firm if it were at the investment threshold:

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

(16)

It is straightforward to see that, if investors only observe the firm’s cash flows, then the bounds in Eq. 16 are the tightest possible based on the investors’ information set, $\mathcal {I}_{t}$ . Indeed, if P _t is very large, then the value of growth options approaches zero and $V_{t}\rightarrow \frac {CF_{t}}{\bar {r}}$ . On the other hand, the price of new assets can be arbitrarily close to

$$\frac{CF_{t}}{K_{t}\cdot\omega^{*}}, $$

in which case V _t is going to be very close to $\frac {CF_{t}}{\bar {r}}\left (1+A\right )$ . Given cash accounting, the changes in P _t since the firm’s last investment do not get reflected in $\mathcal {I}_{t}$ . Therefore, any inequality on V _t that is tighter than Eq. 16 will be violated with positive probability.

Since, according to Observation 1, the present value of cash flows from the firm’s assets in place is collinear with C F _t , the bounds in Eq. 16 also cannot be improved under value in use accounting. Similarly, if the firm uses historical cost accounting, then changes in P _t when ω _t < ω ^∗ do not affect any of the observable variables in $\mathcal {I}_{t}$ . Therefore we obtain the following corollary.

Corollary 1

Given historical cost accounting, value in use accounting, or cash accounting, 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,

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

if P _t d I _t = 0,

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

Given conditionally conservative accounting, investors know that the book value of assets understates their replacement cost, and therefore

$$\omega_{t}\le\frac{CF_{t}}{B_{t}^{cc}}. $$

Therefore, the following upper bound on the firm’s value must always hold:

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

If the firm is in the inaction region, the lower bound on the firm’s value from inequality (16 ) cannot be improved: as the price of new capital goods rises, the firm’s value can get arbitrarily close to $\frac {CF_{t}}{\bar {r}}$ , and the price increases in the capital goods market do not get reflected in $\mathcal {I}_{t}$ under conditionally conservative accounting. However, if investors observe a write-down, $dB_{t}^{cc}<0$ , then it has to be that $B_{t}^{cc}=P_{t}K_{t}$ , and the firm can be valued precisely. These observations are summarized in the following corollary.

Corollary 2

Given conditionally conservative accounting, 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,

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

if $dB_{t}^{cc}<0$ ,

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

if P _t d I _t = 0 and$dB_{t}^{cc}\geq 0$ ,

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

While the actual accounting standards (such as U.S. GAAP and IFRS) are generally different from the stylized accounting rules modeled here, they share certain common features. For example, under IAS 36, firms are required to recognize an impairment if the carrying amount of an asset exceeds the higher of its fair value (which, for an illiquid asset, can be measured as its replacement cost) and its value in use. In Appendix B , we provide bounds on the equity value for firms that account for their property, plant, and equipment using the cost model of IAS 16 and recognize write-downs according to IAS 36.

5 Reversible investments

In this section, we assume that the firm’s investments are reversible, that is, the firm can sell its used capital goods at a price of new capital goods with equivalent productive capacity. For technical reasons, we will now consider the optimal investment and equity valuation problems in discrete time.²⁷ Consistent with previous sections, we allow for stochastic physical depreciation, stochastic demand and stochastic price of new capital goods.

Let period t be the interval of time between dates t − 1 and t. We assume that the cash flow from operations, $CF_{t}=X_{t}K_{t}^{\alpha }$ , and the investment cash outflow, P _t I _t , are realized at the end of period t (i.e., just before date t). For notational convenience, we will focus on the cum-dividend value of the firm defined as:

$$V_{t}=CF_{t}-P_{t}I_{t}^{*}+\sum\limits_{\tau=1}^{\infty}\beta^{\tau}\cdot E_{t}\left[CF_{t+\tau}-P_{t+\tau}I_{t+\tau}^{*}\right], $$

where $I_{t+\tau }^{*}$ denotes the optimal investment policy and $\beta =\frac {1}{1+r}$ is the firm’s discount factor.

We assume the following evolution of stochastic processes, X _t , P _t , K _t :

$$X_{t}=g_{X,t}\cdot X_{t-1}, $$

$$P_{t}=g_{P,t}\cdot P_{t-1}, $$

$$K_{t}=g_{K,t}\cdot\left[\left( 1-\delta\right)\cdot K_{t-1}+I_{t-1}\right], $$

where g _X,t , g _P,t , g _K,t are random variables realized in period t before the operating cash flow of that period is observed and the new investment I _t is chosen. Let 𝜃 _t ≡ (g _X,t ,g _P,t ,g _K,t ) denote the three-dimensional innovation in the state variable (X _t ,P _t ,K _t ). We assume that 𝜃 _t are i.i.d. over time, but the components of each 𝜃 _t can be mutually correlated. Since the distribution of 𝜃 _t is time-invariant, we will drop the time subscript when we refer to expectations of new innovations, for example, we will write E[g _X ] for E _t−1[g _X,t ] and $E\left [g_{X}g_{K}^{\alpha }\right ]$ for $E_{t-1}\left [g_{X,t}g_{K,t}^{\alpha }\right ]$ .

Let $\hat {K}_{t}\equiv \left (1-\delta \right )\cdot K_{t-1}+I_{t-1}$ , then we have:

$$K_{t}=g_{K,t}\cdot\hat{K}_{t}. $$

Note further that the firm’s investment in period t can be expressed as:

$$ I_{t}=\hat{K}_{t+1}-\left( 1-\delta\right)\hat{K}_{t}g_{K,t}. $$

(17)

Intuitively, $\hat {K}_{t+1}$ is the firm’s expectation (at the end of period t) of the productive capacity of its assets in period t + 1.²⁸ Note that $\hat {K}_{t+1}$ includes the productive capacity of the assets just purchased, I _t . The total replacement cost of assets in place at date t can therefore be written as:

$$RC_{t}=P_{t}\cdot\hat{K}_{t+1}. $$

In formulating th ent value of the firm’s optimized cash flows, it will be convenient to write V _t as $V_{t}\left (\hat {K}_{t},X_{t},P_{t},g_{K,t}\right )$ . Then the Bellman equation takes the following form:

$$\begin{array}{@{}rcl@{}} V_{t}(\hat{K}_{t},X_{t},P_{t},g_{K,t})\,=\,CF_{t}\,+\,\max\limits_{I_{t}}\{ \beta\!\cdot\! E_{t}[V_{t+1}(\hat{K}_{t+1},X_{t+1},P_{t+1},g_{K,t+1})]\,-\,P_{t}\!\cdot\! I_{t}\}. \end{array} $$

Now applying (17 ), we can simplify the equation above as:

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

(18)

To solve the equation above, we will impose two regularity conditions. First, to ensure that the firm’s valuation is always finite, we will assume that X _t does not grow too quickly relative to P _t :

$$1+r>E\left[g_{X}^{\frac{1}{1-\alpha}}g_{P}^{-\frac{\alpha}{1-\alpha}}\right]. $$

Second, if the price of new assets is expected to grow very fast, then the firm could make infinite profits by buying capital goods today and selling them in future periods. To avoid such behavior, we assume that

$$1+r>\left( 1-\delta\right)E\left[g_{K}g_{P}\right]. $$

The following proposition provides an equity valuation formula for a firm that makes reversible investments.

Proposition 2

The firm’s cum-dividend equity value at time t is equal to

$$ V_{t}=CF_{t}-P_{t}I_{t}+\left( 1+C_{1}\right)\cdot RC_{t}, $$

(19)

and the firm’s optimal investment policy is characterized by:

$$ \hat{K}_{t+1}=C_{2}\cdot\left( \frac{X_{t}}{P_{t}}\right)^{\frac{1}{1-\alpha}}, $$

(20)

where C ₁ and C ₂ are two non-negative constants that depend only on the parameters of the stochastic processes X _t ,P _t ,K _t and not on their realizations.²⁹

The valuation equation in Eq. 19 states that the cum-dividend value of the firm at date t is equal to the current cash flow (C F _t − P _t I _t ) plus another term that is proportional to the replacement cost of assets in place; in other words, the ex-dividend value of the firm equity is equal to (1 + C ₁) ⋅ R C _t . In the investment literature, it is common to decompose the value of the firm’s equity into two components: the replacement cost of its assets in place and the discounted sum of future economic profits (e.g., Thomadakis 1976; Lindenberg and Ross 1981; Fisher and McGowan 1983; Salinger 1984; and Abel and Eberly 2011). The constant C ₁ in our model reflects the ratio of the present value of future economic profits to the replacement cost of assets in place. Note that C ₁ does not depend on the current state variable (X _t ,P _t ,K _t ). All the value-relevant information about the underlying state is summarized in the replacement cost of the firm’s assets in place, R C _t .

It is interesting to compare our firm valuation equations in Propositions 1 and 2. It turns out that replacement cost of assets in place is an important variable for equity valuation in both scenarios. However, as discussed above, the replacement cost of assets in place alone is a sufficient statistic for the firm’s equity value when investments are reversible. In contrast, according to Proposition 1, to value a firm with irreversible investments in the inaction region, one needs to observe both the replacement cost of its assets and its operating cash flow.

To understand this difference between the two results, recall that the underlying state in both scenarios is three-dimensional, that is, it is determined by variables X _t , P _t , K _t . The process {K _t } is controlled: it is affected by the firm’s investments. When the firm can adjust its capital stock in both directions (investments are reversible), the firm ensures that the marginal expected benefit of investment is precisely equal to the marginal cost of new assets, P _t , at each date. Therefore, the three processes, X _t , P _t , and the expected forward capacity ($\hat {K}_{t+1}$ ) are always linked by Eq. 20. This effectively eliminates one dimension from the underlying state variable, and, relative to the scenario with irreversible investments, it is sufficient for investors to observe one financial variable less to be able to value the firm precisely.

In particular, it is interesting to note that the firm’s equity can also be valued precisely even in the absence of replacement cost information if financial statements are prepared under value in use accounting. It is straightforward to verify that value in use in the current setting is proportional to the expected next period’s operating cash flow, E _t [C F _t+1] ,which in turn is proportional to

$$E_{t}\left[CF_{t+1}\right]\sim X_{t}\hat{K}_{t+1}^{\alpha}. $$

Given the expression for $\hat {K}_{t+1}$ in Eq. 20 , it follows that:

$$E_{t}\left[CF_{t+1}\right]\sim X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}\sim P_{t}\hat{K}_{t+1}=RC_{t}. $$

Thus, in contrast to our results in the setting with irreversible investment, replacement cost accounting is informationally equivalent to value in use accounting. The importance of replacement cost accounting in our model arises from the investor’s need to value the firm’s growth options.³⁰

When the firm’s investments are irreversible and the firm is in the inaction region, the three components of the state variable, (X _t ,P _t ,K _t ), move independently: the firm cannot adjust its capacity downward and the output market is not profitable enough to justify additional investments, so d I _t is zero. Then investors need to have more information to value the firm. Accordingly, the valuation equation in Proposition 1 depends on both C F _t and P _t K _t . However, when the firm reaches its investment boundary, investors know that an additional constraint on the components of the state variable is binding, ω _t = ω ^∗ , and the firm’s equity can be valued based on just one financial variable as Corollary 1 demonstrates.

Propositions 1 and 2 also show that the replacement cost of assets in place affects the value of the firm’s equity differently in the two scenarios. First, note that the value of a firm with reversible investments always (weakly) exceeds the replacement cost of its assets, V _t ≥ R C _t . This has to be the case at all times because otherwise the firm could simply sell all of its assets today and generate more value than by participating in the output market in future periods. The same inequality, however, does not necessarily hold for a firm with irreversible investments. In particular, note that the expression for V _t in Proposition 1 approaches zero as X _t → 0; so for small values of X _t , the firm’s value will be less than the replacement cost of its assets.

Second, Proposition 1 states that, controlling for the current operating cash flow, the value of the firm with irreversible investments decreases in R C _t . In contrast, in the scenario with reversible investments, V _t strictly increases in R C _t . To understand this difference, consider what happens after a positive shock to the price of new assets, P _t . In both cases, the value of the firm’s future cash flows falls, since the cost of its inputs has risen. In the scenario with irreversible investments, the replacement cost of the firm’s assets, P _t K _t , rises since the firm cannot change K _t in response to the change in P _t . Therefore, an increase in P _t (an event unfavorable to the firm) has an effect of increasing the replacement cost of its assets in place, R C _t . In the scenario with reversible investments, the firm can sell some of its assets at the new price, P _t , and choose a new level of capital stock. In fact, according to Eq. 20 , the firm’s replacement cost of assets at date t is:

$$P_{t}\hat{K}_{t+1}=C_{2}\cdot X_{t}^{\frac{1}{1-\alpha}}P_{t}^{-\frac{\alpha}{1-\alpha}}. $$

The quantity above decreases in P _t . Therefore, in the scenario with reversible investments, an increase in the price of new capital goods (which is, again, an event unfavorable to the firm) leads to a decrease in the replacement cost of assets after taking into account the firm’s optimal capacity adjustment. To summarize, our results indicate that, while the replacement cost of assets in place is an important variable for equity valuation in both scenarios, the exact way this variable enters the equity value function crucially depends on the reversibility of investments.

In 1980s, under SFAS No. 33, large firms were required to disclose the current (replacement) cost of their assets such as plant, property, equipment, and inventories.³¹ Several papers failed to find incremental value of SFAS 33 disclosures over the historical cost earnings (e.g., Beaver and Landsman 1983; Beaver and Ryan 1985). However, Revsine (1973) suggested that positive holding gains under replacement cost accounting can be good news for some firms and bad news for others, depending on how well each firm is able to react to price changes in its input markets. Hopwood and Schaefer (1989) provide empirical support for Revsine’s argument. Consistent with the intuition described in this literature, our model formally shows that the direction of the relation between equity value and replacement cost of assets depends critically on the firm’s ability to adjust its capital stock downwards. Our results also help explain the small or negative coefficients on the book value of assets in equity valuation regressions (e.g., Dechow et al. 1999; Hao et al. 2011).

To conclude this section, we describe the limits of the coefficients C ₁ and C ₂ as the length of the time period approaches zero. It can be verified that as d t → 0, the firm’s value becomes:³²

$$V_{t}=(1+\hat{C}_{1})\cdot RC_{t} $$

where

$$ \hat{C}_{1}=\frac{\left( 1-\alpha\right)}{\alpha}\cdot\frac{r+\delta-\mu_{P}-\frac{1}{2}{\sigma_{K}^{2}}-\rho_{PK}\sigma_{P}\sigma_{K}}{r-\frac{\left( \mu_{X}-\alpha\mu_{P}\right)}{1-\alpha}-\frac{\alpha}{2\left( 1-\alpha\right)^{2}}\left( {\sigma_{X}^{2}}+{\sigma_{P}^{2}}-2\rho_{XP}\sigma_{X}\sigma_{P}\right)}, $$

(21)

and the firm’s optimal investment policy is such that

$$K_{t}=\hat{C}_{2}\cdot\left( \frac{X_{t}}{P_{t}}\right)^{\frac{1}{1-\alpha}}, $$

where

$$ \hat{C}_{2}=\left( \frac{\alpha}{r+\delta-\mu_{P}-\frac{1}{2}{\sigma_{K}^{2}}-\rho_{PK}\sigma_{P}\sigma_{K}}\right)^{\frac{1}{1-\alpha}}. $$

(22)

In the expressions above, the constants μ _X ,μ _P ,σ _X ,σ _P ,σ _K ,ρ _{P K} ,ρ _{X P} are defined consistently with the specification in Section 2.

The expressions in Eqs. 21 and 22 can be intuitively interpreted in certain special cases. Assume, for example, that depreciation is nonstochastic, σ _K = 0, and there is no drift in the price of new assets, μ _P = 0. Then $\hat {C}_{2}$ becomes:

$$\hat{C}_{2}=\left( \frac{\alpha}{r+\delta}\right)^{\frac{1}{1-\alpha}}. $$

The denominator of the ratio in brackets, r + δ, is the standard user cost of capital (e.g., Jorgensen 1963; Abel and Eberly 2011). The remaining terms in the denominator of $\hat {C}_{2}$ in Eq. 22 adjust the user cost of capital for uncertainty in the prices of new capital goods and the productive capacity of assets in place.

Now assume that there is no uncertainty about future values of X _t ,P _t ,K _t and μ _P = μ _X = 0 and consider the expression for the firm’s equity value:

$$V_{t}=\left( 1+\frac{\left( 1-\alpha\right)\left( r+\delta\right)}{\alpha r}\right)P_{t}K_{t}. $$

Intuitively, the firm’s equity value is equal to the replacement cost of its assets, P _t K _t , plus the present value of future economic profits:

$$\frac{\left( 1-\alpha\right)\left( r+\delta\right)}{\alpha r}P_{t}K_{t}. $$

To see that the expression above is indeed equal to the present value of future economic profits, recall that (r + δ) P _t K _t is the user cost of capital employed, $\frac {1-\alpha }{\alpha }$ is the optimal mark-up in the firm’s output market, and $\frac {1}{r}$ is the capitalization factor for a stream of payments equal in expectation. In the special case considered here, there is no drift in the price of new assets or demand for the firm’s output, μ _P = μ _X = 0, so the stream of future economic profits effectively becomes an annuity.

6 Conclusion

Our paper studies the problem of equity valuation based on accounting information in a setting where the firm makes investments in capital goods, facing uncertainty regarding future conditions in its output and capital goods markets. We initially assume that the firm’s investments are irreversible and the productivity of its assets is stochastic. Our main result shows that, if the firm’s financial statements are prepared using replacement cost accounting, then outside investors will have sufficient information to value the firm’s equity. The demand for replacement cost disclosures comes from investors’ need to value the firm’s growth options. In a setting with reversible investments, we show that replacement cost accounting also provides information useful for equity valuation. However, the relation between the firm’s equity value and the replacement cost of its assets depends on the firm’s ability to sell its used capital goods.

Our results help understand the informational needs of equity investors. However, in setting accounting standards, the regulators are concerned with interests of a broader group of financial statement users, including the firm’s lenders and other creditors. Characterizing the disclosure preferences of broader groups of financial statement users in a setting with real options is an interesting direction for future research. Furthermore, while this paper focuses only on the growth options faced by the firm, in practice, firms face a richer set of decisions with option-like nature. For instance, in the presence of variable costs, firms need to consider the possibility of idling parts of their capacity (operating options) or shutting down their operation completely (abandonment option). We leave extending accounting-based valuation models to such situations to future research. Lastly, throughout the paper, we have assumed that the firm’s manager acts in the best interests of equity holders, thus exercising all real options optimally. In practice, the inherent opacity of real options is likely to create a host of agency problems. Another promising direction for future study is to investigate the problem of equity valuation based on accounting information in the presence of information asymmetry and incentive issues.

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.