A real options approach to environmental R&D project evaluation

Abstract

With the immense evolution of technology and consumers’ maturing anticipations, novel products’ development becomes the primary way that companies adapt to survive in the growing international contention. Creation or innovation can allow the organization to stand one step forward of the competition and to conduct the competition in the direction it selects. We examined the evaluation system of real options approach to a biomimicry R&D project together with conventional R&D and environmental R&D projects. This model considers uncertainties of investment cost and cash flow of the new product in each project. We find that investment on environmental and biomimicry technologies will achieve promising expectations in Asian area when certain environmental demands increase.

Appendices

Appendix 1: Proof of the modeling valuation

When the investment of the R&D project has been successfully accomplished, the value of the project depends only on the generated net cash flows. Assume that the product loses its uniqueness at time, where $T = {\raise0.7ex\hbox{$Y$} \!\mathord{\left/ {\vphantom {Y {\Delta t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\Delta t}$}}$ and ∆t is the step size; also assume that the residuary value of the project, represented by the possible cash flows created after the singularity disappears, is a multiple M of the cash flows at time, and r stands for the risk-free rate, the value V must satisfy the following Bellman equation (dynamic programming equation):

$$V = C{\text{d}}t + e^{{ - r{\text{d}}t}} \left[ {V + E\left( {{\text{d}}V} \right)} \right]$$

(9)

The first term shows the obtained cash flow during, and the second term represents the discounted value of the project after the period of. Then, use Ito’s Lemma to divide:

$${\text{d}}V = V_{t} {\text{d}}t + V_{C} {\text{d}}C + \frac{1}{2}V_{\text{CC}} ({\text{d}}C)^{2}$$

(10)

Put Eqs. (4) and (10) into Eq. (9):

$$V = Cd + e^{{ - r{\text{d}}t}} \left\{ {V + E\left[ {V_{t} {\text{d}}t + V_{C} \left( {\left( {\alpha - \eta } \right)C{\text{d}}t + \sigma C{\text{d}}z} \right) + \frac{1}{2}V_{\text{CC}} ( {\left({\alpha - \eta } \right)^{2} C^{2} \left( {{\text{d}}t} \right)^{2} + \sigma^{2} C^{ 2} \left( {{\text{d}}z} \right)^{2} + 2\left( {\alpha - \eta } \right)\sigma^{2} C^{2} {\text{d}}t{\text{d}}z})} \right]} \right\}$$

(11)

E(dw) = 0 for dz is subject to Wiener distribution; E(dw)² = dt because of Ito’s Lemma. Let:

$$\left[ {e^{{ - r{\text{d}}t}} \left( {\frac{1}{2}\sigma^{2} C^{2} V_{\text{CC}} + \left( {\alpha - \eta } \right)CV_{C} + V_{t} } \right) + C} \right]{\text{d}}t - \left( {1 - e^{{ - r{\text{d}}t}} } \right)V = 0$$

(12)

Dividing Eq. (12) by the partial differential equation (PDE) of the project value is:

With boundary condition:

$$V(C,T) = M \times C$$

(13)

Therefore, the solution of the PDE is:

$$V\left( {C,t} \right) = \frac{C}{{r - \left( {\alpha - \eta } \right)}}\left\{ {1 - e^{{ - \left[ {r - \left( {\alpha - \eta } \right)} \right]\left( {T - t} \right)}} } \right\} + MCe^{{ - \left[ {r - \left( {\alpha - \eta } \right)} \right]\left( {T - t} \right)}}$$

(14)

Using Ito’s Lemma and putting Eq. (3) to Eq. (14), the true stochastic process for the value of the successful R&D project can be represented by:

$$\frac{{{\text{d}}V}}{V} = \left( {r + \eta } \right){\text{d}}t + \sigma {\text{d}}z$$

(15)

Before the accomplishment of the investment, the value of R&D project depends on the expected total investing cost K and the possible cash flow after completion. Also, it must satisfy the following Bellman equation:

$$F\left( {C,K,t} \right) = - I{\text{d}}t + e^{{ - (r + \lambda ){\text{d}}t}} \left[ {F\left( {C,K,t} \right) + E\left( {{\text{d}}F\left( {C,K,t} \right)} \right)} \right]$$

(16)

The first term stands for the cost to invest during at the time point of, and the second term means the discounted value of the project after the period of λ donates for the Possion probability of failure. According to Schwartz and Moon (2000), it can be directly put into the discounted rate. Then, use Ito’s Lemma to divide:

$${\text{d}}F = F_{t} {\text{d}}t + F_{C} {\text{d}}C + F_{K} {\text{d}}K + \frac{1}{2}F_{\text{CC}} ({\text{d}}C)^{2} + \frac{1}{2}F_{\text{KK}} ({\text{d}}K)^{2} + F_{\text{CK}} {\text{d}}C{\text{d}}K$$

(17)

Put Eqs. (2), (3), and (10) into Eq. (17). Afterwards, by letting and removing the terms of dt with high order, the PDE of F(C, K, T) can be shown to be:

$${\text{Max}}_{I} \left[ {\frac{1}{2}\sigma^{2} C^{2} F_{\text{CC}} + \frac{1}{2}\rho^{2} (IK)F_{\text{KK}} + \sigma \rho \varphi C(IK)^{{\frac{1}{2}}} F_{\text{CK}} + (a - \eta )CF_{C} - IF_{K} + F_{t} - (r + \lambda )F - 1} \right] = 0$$

(18)

And the boundary condition for this PDE is:

$$F\left( {C,0,\tau } \right) = V\left( {C,\tau } \right)$$

(19)

Appendix 2: Progress of the LSM solution

The LSM method applied for valuing options is based on Monte Carlo simulation and least squares regression. The decision for project abandonment is evaluated at discrete points in time, instead of continuously. This seems to be a more reasonable assumption on R&D projects’ analysis (Schwartz 2004).

In the simulations, the discrete approximation to remaining investment cost process and cash flow process in risk-neutral version is:

$$K(t + \Delta t) = K(t) - I\Delta t + \rho \left[ {IK} \right]^{1/2} \Delta t^{1/2} \varepsilon_{x}$$

(20)

$$C\left( {t + \Delta t} \right) = C\left( t \right)\exp \left[ {\left( {\alpha - \frac{1}{2}\sigma^{2} - \eta } \right)\Delta t + \sigma \left( {\Delta t} \right)^{1/2} \varepsilon_{z} } \right]$$

(21)

where φ donates for the correlation between cost and cash flow uncertainty. In these simulations, ε _x and ε _c can be calculated as ε _x = u ₁ and $\varepsilon_{Z} = \phi u_{1} + u_{2} \left( {1 - \phi^{2} } \right)^{{\frac{1}{2}}}$, where u ₁ and u ₂ are independent random drawings from a standardized normal distribution. These two parameters can be obtained with the pertinent function of the MATLAB™.

If Y is the time to the expiration of the uniqueness and ∆t is the step size, $T = {\raise0.7ex\hbox{$Y$} \!\mathord{\left/ {\vphantom {Y {\Delta t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\Delta t}$}}$ is the number of periods per path in the simulation. Equations (20) and (21) are used to generate G paths of T periods each of costs to completion and cash flows. Each path is then described by each parameter (K, C). During the duration of R&D project, the cash flows represent the occurred value if the investment is terminated.

If the project has not been abandoned before, at the final observation date of a given period (time, the last stage of operational period), the value of the R&D project is given by the boundary condition:

$$W\left( {g,j} \right) = \left\{ {\begin{array}{*{20}l} { M \cdot C\left( {g,T} \right), \quad K\left( {g,T} \right) = 0 } \hfill \\ {0,\quad \qquad \qquad K\left( {g,T} \right) > 0} \hfill \\ \end{array} } \right.$$

(22)

where K(g, T) = 0 means the R&D project is finished; K(g, T) > 0 means the project is still ongoing.

At any period, if investment has been terminated in those paths, the value of the project will be computed recursively by:

$$W\left( {g,j} \right) = e^{ - r\Delta t } \Delta W\left( {g,j + 1} \right) + C\left( {g,j} \right)\Delta t$$

(23)

If investment has not been finished in those paths, the conditional expected value of duration will be estimated by regression. The dependent variable is the discounted value of R&D project at j + 1 period defined as e ^−rΔt W(g, j + 1), and the independent variable is the awaited cash flow of the new product at period. The fitted value $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{W} \left( {g,f} \right)$ can be calculated by polynomial regression. By comparing the conditional expected value of the new product $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{W} \left( {g,j} \right)$ with the investment expenditure, we can obtain the following expression:

$$W\left( {g,j} \right) = \left\{ {\begin{array}{*{20}l} { 0, \quad \qquad \qquad \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{W} \left( {g,j} \right) < I(g,j) } \hfill \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{W} \left( {g,j} \right) - I\left( {g,j} \right),\quad \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{W} \left( {g,j} \right) \ge I(g,j)} \hfill \\ \end{array} } \right.$$

(24)

The recursion continues by rolling back in time and repeating the procedure until the decisions at each possible exercise time for all paths have been determined.

The computation starts from time zero, proceeding along each path until the final observation date of a given period or the first stopping time. Then by discounting the resultant cash flows to time zero, and taking the average value over all the G paths, we can eventually work out the value of the R&D project with abandonment option (RV). More detail information and discussion on this methodology can be found in the study of Schwartz (2004).

Appendix 3: Tables that describe inputs and outputs of each firm

See Tables 6 and 7.

Table 6 Parameters of each firm

Table 7 Project values of each firm for five scenarios

Appendix 4: Tables of sensitivity analysis for each firm

See Tables 8, 9, 10 and 11.

Table 8 Sensitivity analysis of success rate for each firm (Scenario ENV2)

Table 9 Sensitivity analysis of cash flow for each firm (Scenario ENV2)

Table 10 Sensitivity analysis of success rate for each firm (Scenario BIO2)

Table 11 Sensitivity analysis of cost for each firm (Scenario BIO2)