^{1}

^{*}

^{2}

^{*}

We discuss the valuation of investment project in a firm applying a real option method with abandonment and reset investment proportion. We take the depreciation value of the facilities and the research and development (R & D) fee into consideration. Our contribution is to derive a pricing model of two-stage optimal decisions allowing abandonment and reset investment proposition. Different from the net present value (NPV) or discount cash flow (DCF), the real option method can efficiently catch the uncertainty in the market, and it can help managers to make the optimal policy for the project. We can improve our method for a multi-stage decision model or a continuous decision model in the further researches.

We discuss the valuation of investment project in a firm applying a real option model with abandonment and reset investment proportion. We take the depreciation value of the facilities and the research and development (R & D) fee into consideration. Because of the uncertainty of the real world, the policy-making of a firm should be dynamic. We focus on deriving a pricing model of two-stage optimal decisions allowing abandonment and reset investment proposition in this study.

The traditional method to evaluate a project is through the net present value (NPV) or discount cash flow (DCF). If the net present value of the project is positive after considering the future cash flow of each period, we would accept the project. Otherwise we will decline it. Robichek and Horne [

Because of the maturation of appropriate valuation models in barrier options and reset options recently, we could apply the concepts of barrier options and reset options to evaluate a project and derive the valuation of two-stage decision model, which can extend or reduce the scope of a project in the policy-making point. We could further study which models are more compatible with the real world—multi-stage decisions models or continuous decisions models.

The rest of this paper is organized as follows. Section 2 is the literature review. Section 3 is the methodology about considering only abandonment, or both abandonment and reset investment proportion. Finally, Section 4 discusses the conclusions and recommendations of further researches.

Robichek and Horne [

Since Gray and Whaley [

We focus on applying the concepts of reset options and barrier options to the valuation of a project in a firm. We derive the valuation of two-stage decision model, and we put more emphasis on the research and development (R & D) fee into consideration. We could derive a general valuation model in the future researches.

This paper discusses the two-stage decisions of a project in a firm using the valuation of real options with abandonment and reset conditions. The valuation is a series of initial value problems because of discrete policy- makings in multi-stage time. According to different criterions at different policy-making points, it is a series of free boundary problems. Therefore, the valuation of the decision problem can be viewed as a series of combinations of initial value problems and free boundary value problems. We adapt an approach to solve a partial differential equation (PDE)—Green’s function, which is useful to solve the initial value problem and boundary value problem, to discuss the valuation of a project in a firm.

We adapt the fundamental assumptions in Kulatilaka and Perotti [

1) We assume that the demand of the goods is linear in prices and increases with a random variable

where Q is the total supply in the market, and the random variable

2) If no initial investment is made, the firm will produce only when the market is profitable with a unit cost of

quality Q reaches the balance

Next, we define A to be a part of the project asset which can be sold in the market, such as the real estate, facilities or factory buildings. Suppose the average depreciation rate of facilities is

Furthermore, we suppose that the products have a maximum demand in the real market, and the facilities of the project have finite ability to produce the products. We consider the present maximum balance benefit is H, and it increases with time by the natural growth rate

When the maximum benefit of production

efit of production

cient to pay the increasing research and development (R & D) fee

where

where

In addition, other firms would buy the old facilities of the project only if the benefit is greater than the cost they pay. On the contrary, if the benefit which the facilities would create is lesser than the cost of the facilities, there will be no firm buying the facilities. The project will be worthless. Therefore, the balance point will satisfy the following equation:

where

where

When we continue to produce the products, we can obtain more benefit. However, there is a maximum balance benefit at each policy-making point after considering the demand of population and the ability of the facilities. It means that the benefit is finite. Therefore, the threshold

where

where

From above explanation, the benefit function for all situations at the maturity date T is as follows:

where

According to assumptions in the Subsection 3.1, the valuation of the project is transformed as a contingent claim that depends on the underlying random valuable

The underlying

Then, the Equation (8) can be transformed to the Heat Equation (9) by the variable transformation (10).

Therefore, the maturity date T will become_{1} will be_{0} will be

where

We adapt the most useful method—Green’s function to solve the heat Equation (9). The translated valuation

where Green’s function

We combine the result

where

Therefore, we can graph Equation (13), the translated valuation of the project in each interval and each criterion of the threshold, as

After using the integral representation with Green’s function, the present translated valuation of the project can be presented as follows:

Finally, we can obtain the present valuation of the project as Equation (18).

The demand is changing with time in the real world, so the investment of the project should adjust accordingly to cope with the situation in the market. For example, when the forecasted demand increases, we add the investment proportion of the project at the policy-making point. Then we set the increasing proportion for the demand of population to be

The policy-making depends on the situation of the transformed random variable x at time

where

The translated valuation of the project after reducing investment at the policy-making point is as equation (20).

The criterion

That is,

On the contrary, when

where

The translated valuation of the project after adding investment at the policy-making point is as Equation (23).

The criterion

That is,

That is, when random variable

Furthermore, when the operating benefit of the project is too bad after reducing the investment of the project at the policy-making point, we will even consider to close the project and to sell the facilities to the market with its depreciation value. Hence, the criterion of the threshold

On the other hand, if the operating benefit of the project is very good after adding investment, even over the upper bound of production of the factory, the critical value

Comparing with Equation (13), we can rearrange above equations as Equation (28).

where

sent as Equations (12), (20), and (23), respectively.

Equation (28) will be the initial condition of PDE in the next interval

Then, the valuation of the project at present time to maturity

We discuss the valuation of the project in a firm applying real option with abandonment and reset investment proportion. We take the depreciation value of the facilities, the research and development (R & D) fee into consideration in this study. The traditional method to evaluate a project is by the net present value (NPV) or discount cash flow (DCF). Many scholars reported that above methods were hard to catch the uncertainty of the market in time. It would undervalue or miss the opportunity value of the investments, so the managers cannot make the optimal policy to the project immediately during a short time when the transient investment opportunity appears.

Our suggestion is to evaluate the project in a firm with the valuation of real options with abandonment and reset investment proportion. The advantage is that we can adjust the investment proportion according to the situation of the market, even abandon the project to reduce our loss. When we reduce the investment proportion to some level in the project, we would rather stop the project and sell the facilities to cover our loss. We could improve our method to derive a multi-stage decision model or a continuous decision model in the further researches.