2.1. Entry and Exit Strategies under Uncertainty: A Conceptual Framework

There are three different states in which an ethanol refinery can be and five multiple switching options. The refinery states are idle, active, or mothballed. The switching options are idle to active, active to mothballed, mothballed to active, mothballed to exit (abandoned), and active to exit. The switching options this study evaluates include the following: (1) idle to active, (2) active to mothballed, (3) mothballed to active, and (4) active to exit.

In an idle state, a firm does not pay fixed or capital costs because the plant has not yet been built. An idle firm will invest when the demand conditions for output become sufficiently favorable, and an active firm will abandon the plant if demand becomes sufficiently adverse (Dixit and Pindyck, Reference Dixit and Pindyck1994). In this study, we assume that the firm incurs a lump-sum fixed cost K to invest in an energy beet ethanol refinery. If the refinery decides to shut down (exit), there is a lump-sum exit/abandonment cost l to close it. A refinery can recoup part of the investment cost K upon exit, reflecting that part of K is not fully sunk. Once the investment is made, the refinery generates a given flow of ethanol production until the plant is shut down. A refinery firm incurs a constant operating cost of ω to produce each unit of output. The ethanol refinery will generate a flow of operating profit equal to P − ω per period, where P is the firm's ethanol gross margin (dollars per gallon). The ethanol gross margin is defined as the sum of ethanol and beet pulp prices minus beet price. The nonbeet operating cost (ω) is the sum of all operating costs less beet feedstock cost (dollars per gallon). Two possibilities exist: the ethanol refinery may be shut down if P < ω and later reactivated if P > ω. This makes the ethanol refinery have an infinite sequence of instantaneous operating options, each of which is exercised if P > ω and can be valued accordingly (Dixit and Pindyck, Reference Dixit and Pindyck1994). Instead of abandoning, an ethanol refinery may choose to suspend operations and mothball, then later reactivate. In a mothballed state, a firm incurs an ongoing maintenance cost, m, to maintain existing capital. Mothballing requires a sunk cost per unit of expected output, E_m . In addition, the refinery can be reactivated in the future at an additional sunk cost, r. Finally, we assume m < ω for the mothballing option to be feasible.

The key variables to the investment decision in this particular investment analysis are ethanol gross margin and investment costs. The decisions to invest and exit are made assuming that the gross margin of ethanol is exogenous and stochastic and that it evolves according to geometric Brownian motion (GBM), described in equation (1):

(1) $$\begin{equation} dP = \mu Pdt + \sigma Pdz, \end{equation}$$

where P is the gross margin of the refinery, μ is an instantaneous drift in the gross margins of ethanol, σ is the standard deviation or volatility of a given random increment to the gross margins of ethanol, and dz is an increment to the standard Wiener processes for the uncertain variable and is uncorrelated across time t and $dz = \sigma \sqrt {\Delta t} $ , E[dz ²] = dt.

2.1.1. The Decision to Enter

The conceptual framework and the mathematical derivations of entry and exit decisions in this section are adapted from Dixit and Pindyck (Reference Dixit and Pindyck1994) and Schmit, Luo, and Tauer (Reference Schmit, Luo and Tauer2009) and Schmit, Luo, and Conrad (Reference Schmit, Luo and Conrad2011). We assume that the value of a project is a function of the exogenous state variable (i.e., the gross margin of ethanol). Let V ₀(P)be the discounted expected value of an idle project or the option to invest. Similarly, V ₁(p) is the value for the active project. The solution consists of these functions and the rules for optimally switching between idle and active states. If the investors “sold” the refinery instead of deciding to invest, they would earn V ₀(P). Equilibrium in the asset market requires that

(2) $$\begin{equation} \delta {V_0}(P) = \frac{{{E_t}[d{V_0}(p)]}}{{dt}}, \end{equation}$$

where P is the gross margin of ethanol, δ is the discount rate, δV ₀(P) represents the normal return if the option to invest is sold, and $\frac{{{E_t}[d{V_0}(p)]}}{{dt}}$ is the expected capital gain from holding the option to invest. Equation (2) defines the entry trigger gross margin.

From Ito's lemma, for a function V = V ₀(P),

(3) $$\begin{equation} dV = [V(P) + \mu P{V'_0}(P) + 0.5{P^2}{V''_0}(P){\sigma ^2}]dt + \sigma P{V'_0}(P)dZ, \end{equation}$$

where μ and σ are respectively the drift and standard deviation parameters of P; ${V_t} = {{\partial V} / {\partial t}} = 0$ , given the infinite time horizon, ${V'_0} = {{\partial V} / {\partial P}}$ , ${V''_0} = {{{\partial ^2}V} / {\partial {P^2}}},$ and E[dZ] = 0. Simplifying equation(3) and substituting into equation(2) gives us the asset equilibrium condition represented by the differential equation:

(4) $$\begin{equation} \delta {V_0} = \mu P{V'_0} + (0.5{\sigma ^2}){P^2}{V''_0}. \end{equation}$$

The general solution for an idled firm for this homogenous second-order ordinary differential equation gives us:

(5) $$\begin{equation} {V_0}(P) = {A_0}{P^{ - \alpha }} + {B_0}{P^\beta }, \end{equation}$$

where A ₀ and B ₀ are unknown constants to be determined, and α and β are the root mean of the quadratic equation parameters that capture and incorporate the uncertainty modeled by GBM into the model:

$$\begin{equation*} \beta = \frac{{(1 - m) + {{[{{(1 - m)}^2} + 4g]}^{1/2}}}}{2} > 0 \end{equation*}$$

$$\begin{equation*}- \alpha = \frac{{(1 - m) - {{[{{(1 - m)}^2} + 4g]}^{1/2}}}}{2} < 0 \end{equation*}$$

where m = 2μσ⁻² and g = 2δσ⁻², and for the convergence condition, g > m implies that δ > μ; δ is the discount rate. A ₀ P ^−α and B ₀ P ^β capture the option value of switching to another state if margin P changes. An idle project has no value if P approaches zero; if A ₀ P ^−αapproaches zero, the first term in equation (5) vanishes. Because –α < 0 and β > 1, this requires A ₀ = 0 and simplifies equation(5), which gives the following solution to the differential equation:

(6) $$\begin{equation} {V_o}(P) = {B_0}{P^\beta }, \end{equation}$$

where V_o (P) is the value function of an idle investment and can be interpreted as the value of the option to enter.

2.1.2. The Decision to Mothball

If the gross margin of ethanol falls below ω, instead of permanently abandoning the refinery, the refinery owner has the option to put the refinery into a state of temporary suspension, allowing it to be reactivated in the future at a sunk cost that is much less than the cost of building a new refinery.

Consider a refinery that is operating and earning instantaneous net revenue (p – ω). Let V ₁(p) denote the value function of the active refinery. Equilibrium conditions require that

(7) $$\begin{equation} \delta {V_1}(P) = \frac{{{E_t}[d{V_1}]}}{{dt}} + (P - \omega ) \end{equation}$$

where V ₁(p) is the value function of an active refinery, and ω is the operating cost. The left-hand side is the normal return if the refinery is sold and proceeds invested at discount rate δ. The right-hand side of equation (7) is the net revenue flow plus the expected capital gain. The derivation of the value of the active firm is done in a similar way to the idle firm, except that an active firm receives the flow of operating profit (P − ω)dt, in addition to the expected capital gain. The general solution for an active refinery can be expressed as follows:

(8) $$\begin{equation} {V_1}(p) = {A_1}{P^{ - \alpha }} + {B_1}{P^\beta } + \left(\frac{P}{{\delta - \mu }} - \frac{\omega }{\delta }\right), \end{equation}$$

where A ₁ and B ₁ are unknown constants to be determined, A ₁ P ^−α + B ₁ P ^β captures the option value of the mothballed refinery, and P(δ − μ)⁻¹ − ωδ⁻¹ is the present value of the net cash flow. The value of the mothballing option goes to zero as P gets very large, implying that B ₁ = 0. Therefore, equation (8) becomes

(9) $$\begin{equation} {V_1}(p) = {A_1}{P^{ - \alpha }} + \left(\frac{P}{{\delta - \mu }} - \frac{\omega }{\delta }\right), \end{equation}$$

where A ₁ P ^−α represents the value of the option to mothball and P(δ − μ)⁻¹ − ωδ⁻¹ captures the expected present value of continuing production forever. The value of the mothballing option is derived from future possibilities of reactivation or abandonment. Note that an economically meaningful solution for V ₀ must be nonnegative. Therefore, we must haveB ₁ ⩾ 0 in equation (8). Similarly, an active firm has an expected value of P(δ − μ)⁻¹ − ωδ⁻¹ from the feasible strategy of never shutting down. Also, in equation (9), we must haveA ₁ ⩾ 0 (Dixit, Reference Dixit1989).

2.1.3. The Decision to Reactivate and Exit

Mothballing requires a sunk cost, E_m . In a mothballed state, the refinery requires maintenance at a cost of m. Let V_m (p) denote the value function of the mothballed refinery with the option of reactivating or exiting. Equilibrium in the asset market requires that

(10) $$\begin{equation} \delta {V_m}(P) = \frac{{{E_t}[d{V_m}]}}{{dt}} - m, \end{equation}$$

where the left-hand side is the normal return if the firm sells the mothballed refinery and invests the earnings at the discount rate of δ, and the right-hand side is the expected capital gain of the mothballed refinery less ongoing maintenance costs. The resulting value function can be expressed as follows:

(11) $$\begin{equation} {V_m}(p) = {A_m}{P^{ - \alpha }} + {B_m}{P^\beta } - m{\delta ^{ - 1}}, \end{equation}$$

where A_m and B_m are constants to be determined, A_mP ^−α is the value of the option to reactivate the mothballed refinery, B_mP ^βis the value of the option to exit, and mδ⁻¹ is the capitalized maintenance cost assuming that the refinery remains in the mothballed state forever. Mothballing only makes sense if the maintenance cost (m) is less than the cost of actual operation and if the reactivation cost is less than the cost of reinvesting. The option value to exit is positive only if the gross margin decreases, and the option value to reactivate is positive only if the gross margin increases.

2.1.4. Deriving Trigger Gross Margin

For our representative ethanol refinery, four switching options are considered: from idle to active, active to mothballed, mothballed to active, and active to exit (abandon). At each switching option point, the value-matching and smooth-pasting conditions must hold. Each of these options will be exercised at a specific gross margin value. Smooth-pasting conditions require tangency of the value functions at the respective trigger gross margins.

For the active investment option, the value-matching and smooth-pasting conditions are, respectively,

(12) $$\begin{equation} {V_o}({P_h}) = {V_1}({P_h}) - K \quad {\rm{and}}\quad {V^\prime_o}({P_h}) = {V^\prime_1}({P_h}), \end{equation}$$

where P_h is the real options entry trigger gross margin. For mothballing, the value-matching and smooth-pasting conditions are, respectively,

(13) $$\begin{equation} {V_1}({P_m}) = {V_m}({P_m}) - {E_m}\quad {\rm{and}}\quad {V^\prime_1}({P_m}) = {V^\prime_m}({P_m}), \end{equation}$$

where V_m (P) is the value function of the mothballed refinery, and P_m is the real options mothball trigger gross margin. For reactivation, the value-matching and smooth-pasting conditions are, respectively,

(14) $$\begin{equation} {V_m}({P_r}) = {V_1}({P_r}) - r\quad {\rm{and}}\quad {V'_m}({P_r}) = {V'_1}({P_r}), \end{equation}$$

where P_r is the real options reactivation trigger gross margin. At P_r , the value of the option to reactivate must equal the value of the active project minus the sunk cost of reactivation. Finally, for the exit option, the value-matching and smooth-pasting conditions are, respectively,

(15) $$\begin{equation} {V_m}({P_l}) = {V_o}({P_l}) - l\quad {\rm{and}}\quad {V'_m}({P_l}) = {V'_o}({P_l}), \end{equation}$$

where P_l is the real options exit trigger gross margin (dollars per gallon). At P_l , the value of the option to exit must equal the value of exiting less any sunk costs of exit.

Substituting the values of equations (6), (9), and (11) into the corresponding value-matching equations (12) through (15), and the derivative of the value functions with respect to P into the smooth-pasting equations (12) through (15), results in a nonlinear system of eight equations with eight unknowns in equations (16) to (23). The first four equations constitute the value-matching conditions:

(16) $$\begin{equation} {B_1}P_h^\beta - P_h^\beta (\delta - \mu ) + \omega {\delta ^{ - 1}} - {A_1}{P^{ - \alpha }} = k, \end{equation}$$\\

(17) $$\begin{equation} {P_m}{(\delta - \mu )^{ - 1}} - \omega {\delta ^{ - 1}} + {A_1}P_m^\alpha = {A_m}P_m^\alpha + {B_m}P_m^\beta - m{\delta ^{ - 1}} - {E_m}, \end{equation}$$\\

(18) $$\begin{equation} {A_m}P_r^\alpha + {B_m}P_r^\beta - m{\delta ^{ - 1}} = {P_r}{(\delta - \mu )^{ - 1}} - \omega {\delta ^{ - 1}} + {A_1}P_r^\alpha - r, \end{equation}$$\\

(19) $$\begin{equation} {A_m}P_l^\alpha + P_l^\beta {(\delta - \mu )^{ - 1}} - \omega {\delta ^{ - 1}} = {B_1}P_l^\beta - l. \end{equation}$$

The second corresponding four equations are derived from the smooth-pasting conditions:

(20) $$\begin{equation} \beta {B_1}P_h^{\beta - 1} = - {P_h}{(\delta - \mu )^{ - 2}} + \alpha {A_1}P_h^{\alpha - 1}, \end{equation}$$\\

(21) $$\begin{equation} - {P_m}{(\delta - \mu )^{ - 2}} + \omega {\delta ^{ - 2}} + \alpha {A_1}P_m^{\alpha - 1} = \alpha {A_m}P_m^{\alpha - 1} + \beta {B_m}P_m^{\beta - 1}, \end{equation}$$\\

(22) $$\begin{equation} \alpha {A_m}P_r^{\alpha - 1} + \beta {B_m}P_r^{\beta - 1} - m{\delta ^{ - 2}} = - {P_r}{(\delta - \mu )^{ - 2}} + \alpha {A_1}P_r^{\alpha - 1}, \end{equation}$$\\

(23) $$\begin{equation} \alpha {A_m}P_l^{\alpha - 1} + \beta {B_m}P_l^{\beta - 1} = \beta {B_1}P_l^{\beta - 1}. \end{equation}$$

The previous eight equations (16–23) determine the four unknown trigger gross margins (P_h, P_m, P_r , and P_l ) and the four unknown scalar coefficients associated with the option value of switching states (A ₁, B ₁, A_m , and B_m ). The equations are nonlinear in the trigger gross margins and solved numerically.Footnote ¹ The trigger gross margins are endogenous and satisfy the condition 0 < P_l < P_h < ∞, and the constant coefficients A ₁ and B ₁ are nonnegative. Moreover, P_r < P_h because the cost of reactivation is less than that of investing from scratch. An idle firm will find it optimal to remain idle as long as ethanol gross margins (P) remain below P_h and will invest as soon as P moves above P_h . Moreover, an active firm will remain active as long as P remains above P_l but will abandon if P falls to P_l . In the range of gross margins between the thresholds, P_l and P_h , the optimal policy is to continue with the status quo, whether it be active operation or waiting. The discount rate (δ) must be greater than the drift rate (μ), otherwise it would never be optimal to invest because the growth rate would outpace the discount rate (Dixit and Pindyck, Reference Dixit and Pindyck1994).

3.1. Operating and Investment Costs

Operating costs comprise energy beet feedstock, processing including chemicals (denaturants, yeast and others), electricity, natural gas, stillage powder, makeup water and wastewater disposal, and labor, administration, and maintenance, among others. Net operating cost is computed after subtracting the value of coproduct sales and stillage powder from the total operating cost. Our cost breakdown shows that feedstock costs represent approximately 72% of the operating cost, implying that any change in the price of energy beets eventually affects operating costs. Hence, energy beet prices affect entry and exit prices. Table 1 contains baseline operating costs for our representative ethanol plant.

Table 1. Annual Operating Costs, 20 Million Gallon per Year Ethanol Plant

Note: Processing cost represents net processing cost after crediting $0.131/gal. equivalent for the use of stillage powder generated in the plant.

Sources: Wamisho, Ripplinger, and De Laporte (Reference Wamisho, Ripplinger and De Laporte2015), Maung and Gustafson (Reference Maung and Gustafson2011), and Green Vision Group (2014) project documents. Equipment prices were either obtained from the literature or provided by manufacturers through formal quotes.

Capital investment costs (K) comprise engineering and construction costs (e.g., equipment and tank packages, piping and mechanical, steel and enclosures, electrical and instrumentation, engineering, site development, rail, piler and other storage equipment, startup inventories, and other contingencies). Capital costs are calculated ($3.18/gal.) as the present value of the investment costs. Details on capital cost are presented in Table 2. Our investment cost assumes 50% of loan financing of the total investment with a discount rate of 8.25%. To satisfy the smooth-pasting conditions of the equalities, we assume that once the investment is made, an ethanol refinery produces a unit of output per period until abandoned.

Table 2. Investment Costs, 20 Million Gallon per Year Ethanol Plant

Sources: Wamisho, Ripplinger, and De Laporte (Reference Wamisho, Ripplinger and De Laporte2015), Maung and Gustafson (Reference Maung and Gustafson2011), and Green Vision Group (2014) project documents. Equipment prices were either obtained from the literature or provided by manufacturers through formal quotes.

For our baseline analysis, the value of each sunk cost is calculated as a percentage of the initial capital investment (Schmidt, Luo, and Tauer, Reference Schmit, Luo and Tauer2009). We assume the liquidation/exit cost (l), mothballing (E_m ) and reactivation (r) are valued as −25%, 5%, and 10% of initial capital investment, respectively, and the maintenance cost (m) in the mothballed state is valued at 2.5% of K. The baseline sunk costs per gallon of ethanol are as follows: l = $0.795, E_m = $0.159, r =$0.318, and m = $0.0795. We assume that mothballing and reactivation costs comprise expenditures on wages and other costs paid in the process of mothballing and reactivation. Preparation costs to sell the plant and clean up the site upon shutting down the refinery permanently are assumed to be negligible and not included in the cost. All costs are assumed to be constant in real terms.

The gross margin of beet ethanol is calculated as the sum of ethanol and beet pulp prices minus the beet price. To compute the gross margin, we convert the sugar beet and beet pulp prices into a dollars per gallon of ethanol equivalent using a conversion rate of 26.5 gallons of ethanol and 0.05 ton of beet pulp produced per ton of energy beets (Maung and Gustafson, Reference Maung and Gustafson2011 ; Wamisho, Ripplinger, and De Laporte, Reference Wamisho, Ripplinger and De Laporte2015). Whereas the stillage powder is credited back into the operating cost, ethanol prices are computed by averaging monthly ethanol prices from January 2000 to December 2014 from the Nebraska Energy Office (2015), representing rack (wholesale) prices (free on board [FOB] price, Omaha, Nebraska). The data for sugar beet prices are obtained from the Farm Financial Database (FINBIN), representing the average value of sugar beets for North Dakota and Minnesota form 2000 to 2014 (FINBIN, 2016). The beet pulp price is from monthly prices representing the Minneapolis market (USDA, 2016).

3.2. Stochastic Gross Margin Parameters

In this section, we estimate the gross margin stochastic parameters. The random variable considered in this study is the ethanol gross margin.

Our data show that ethanol prices were in the range of $1 to $2 per gallon between 2000 and 2004. After 2005, the ethanol price increased steadily, with the exception of a downward blip in 2009 and 2010. The monthly ethanol price peaked in July 2006 at nearly $3.58 per gallon and in April 2014 at $3.15 per gallon. In order to investigate the impact of increasing price volatility starting in late 2006 (see Figure 1), we split the sample into two periods: the full sample period from January 2000 to December 2014 and the subsample period from August 2006 to December 2014. The drift and volatility parameters for these subsample periods are calculated.

Figure 1. Historical Monthly Ethanol Rack Price at Omaha, Nebraska

The gross margins are deflated into real terms using the urban consumer price index (U.S. Department of Labor, 2010). An augmented Dickey-Fuller (ADF) test is applied to determine if the evolution of gross margins follows a random walk or has a unit root. The ADF unit root test results (Table 3) show that we cannot reject the null hypothesis that log of gross margin (ln P) exhibits a unit root at the 99% and 95% confidence level.

Table 3. Augmented Dickey-Fuller Test for Unit Root

Notes: Full sample period represents data from January 2000 to December 2014, and subsample period is from August 2006 to December 2014. CV represents critical value.

Choosing the appropriate stochastic process for modeling commodity prices should also be based on theoretical consideration in addition to the statistical test results (Dixit and Pindyck, Reference Dixit and Pindyck1994). In addition, Schmit, Luo, and Tauer (Reference Schmit, Luo and Tauer2009), Schmit, Luo, and Conrad (Reference Schmit, Luo and Conrad2011), and Gonzalez, Karali, and Wetzstein et al. (Reference Gonzalez, Karali and Wetzstein2012) found no evidence of mean reversion in U.S. ethanol prices. These authors argue that ethanol price is better modeled using GBM than mean reversion. As such, Postali and Pichcetti (Reference Postali and Pichcetti2006) show that crude oil prices reasonably follow GBM and are strongly correlated with ethanol prices. In fact, Dixit and Pindyck (Reference Dixit and Pindyck1994) argue that commodity prices are shown to exhibit GBM and typically require many years of data to determine with any degree of confidence whether a variable certainly follows mean reversion. Given the ADF test result and having shorter sample data periods, the stochastic variables used in our real option analysis are assumed to evolve according to GBM. Hence, we assume that the gross margin of ethanol evolves exogenously over time through GBM as in equation (1), which is the continuous-time formulation of the random walk. As a result, ln P follows a Brownian motion with drift parameter $\mu = \beta - \frac{1}{2}{\sigma ^2}$ and volatility σ. The total derivative of ln P is the following:

(24) $$\begin{equation} d\ln P = (\beta - \frac{1}{2}{\sigma ^2})dt + \sigma dz, \end{equation}$$

where P is the gross margin and β contains the parameters to be estimated. GBM requires d ln P to have a unit root, and our resulting series are nonstationary. To remedy this problem, the drift and volatility parameters are computed as the sample mean and standard deviation of the first difference of the natural log of the annual gross margin series. Note that in some months the gross margins are less than zero. To avoid negative values, we vertically scaled up the gross margins by $0.14/gal. for all data points before we ran the regression.Footnote ² We recovered stochastic parameters displayed in Table 4 using ordinary least squares by regressing Δ ln P on an intercept term and lagged values Δ ln P_{t – 1}.

Table 4. Regression Results and Parameters of Gross Margins

Notes: Standard errors in parentheses. Plus sign indicates significance at the 0.10 level. Asterisks (*, **, and ***) indicate significance at the 0.05, 0.01, and 0.001 level, respectively.

Because the investment analysis is based on annual production, the baseline solutions of real options trigger gross margins are calculated after converting monthly parameters to annual equivalents assuming that both the drift rate and variance of the increments of a Brownian motion are linear in time. For the full data period model, January 2000 to December 2014, the annual drift and volatility parameters are −0.015 and 0.483, respectively, and for the subsample period of August 2006 to December 2014, these parameters are −0.037 and 0.277, respectively (Table 4). The gross margin volatility parameters evolve directly from the root-mean-square error estimation results. Although correlation among all cost and revenue variables certainly determines investment behavior relative to entry and exit decisions, the lack of stochastic data specifically on beet ethanol investment cost and other operating costs do not allow us to capture correlated uncertainty in empirical specification. Therefore, we assume that the magnitude of other operating, investment, mothballing, and reactivation costs are constant, not stochastic.