The Model

In this paper, we apply the rational real-option approach to analyzing investment decision under uncertainty for all-equity firm. In this static framework, uncertainty of the economy is from a complete probability space

(

Ω, ,F Ρ

)

. Using linear setting as our valuation benchmark (Berk, Green, and Naik, 1999, and Shleifer and Vishny, 2003), we develop two assets model to investigate investment decision problems. In contrast to previous literature that is limited to discuss only the value of new capital stocks, we argue that both the value of new capital and the value of existing capital have apparent effects on the expansion decision. In this section, we build our basic two assets model and briefly introduce the interaction between existing assets and investment.

According to Berk, Green, and Naik (1999), we assume that assets in place and new investment create the value of the firm in this framework. Moreover, investment is irreversible, so that it cannot be used for any other purpose. Managers can postpone the expansion options until new information about the valuation of existing and new capital is revealed. Hence, investment decision can hinge on the valuation of both

assets. We further assume that the all-equity firm only has one investment opportunity, but the optimal investment scale can be distinct among firms. In addition, we assume that the irreversible investment option is infinite-lived.

Moreover, we presume that productivity of existing and new capital stocks are different but can affect each other. This is the so-called learning-by-doing effect. The simplest case of learning-by-doing is when learning occurs as a side effect of the production of new capital. Given and , which represent the present value of future cash flows per unit of existing and new capital, respectively, after investment the valuation per unit of capital can be shown as :

In equation(1), G represents the valuation per unit of existing assets, and H stands for the valuation per unit of newly investing capital. Suppose that the valuation of each asset has two components. The first factor is the present value of the future cash flows generated by their original operation, and ; the second factor is the potential extra benefits created by new investment waiting to implement. We assert that assets in place benefit from new investment and the synergy from new investment is conditional on the valuation of existing assets. Therefore, the implicit value of each asset is dependent. In brief, if the learning-by-doing effect is under consideration, the valuations of existing and new capital stocks are related and cannot evaluate separately.

Gt H_t

If the capital stocks of existing and new assets are K₁ andK, respectively, the value of the firm is given by

(

,

) (

1

)

^{( )}

(

1

)

,

V G H ⁼ K ⁺K ^⎡⎣ λ α+ G^{+ − − +}λ α αβ ^H⎤⎦ (2) whereλ=K1/

(

K1+K

)

. λ refers to called book ratio and is applied to capture the

relative importance of existing and new capital stocks. We further assume the learning-by-doing effect is distinct among new and existing capital stock. In such setting, it is easy to identify what kind of driving force, improvement on productivity of existing capital stocks or improvement on productivity of new capital stocks, is behind investment decision. In our model, α, and β are parameters describing the improvement on productivity from expansion for existing and new capital stocks, in which α is shared by both assets but β is only beneficial to new capital stocks. In addition, α is observable to all outside investors but β is only observable to inside managers. From equation(2), we assert that given an investment option the productivity of these two capital stocks will change in a predictable way if both α and β are observable. For simplicity, we do not discuss the heterogeneous investor problem in this model and assume that all investors have the same opinion about these changes.

Thus the information parameters α and β are constant for all investors but can vary among firms to investigate heterogeneous productivity.

The source of investment uncertainty in our framework is the future cash flows generated by these two assets. Prior to investment, we assume the present value of these cash flows evolve as follows:

/ _{G G} ,

dG G=μ dt+σ dW_G (3)

/ _{H H H}.

dH H =μ dt+σ dW (4)

μi and σ_i are, respectively, the drift and volatility of the growth rate of , . is the standard Brownian motion on

i i=G H,

Wi

(

Ω, ,F Ρ

)

. Besides, W_G and W are two _H dependent standard Brownian motions with constant correlationρ . Furthermore, by settingρ<1, our model captures the feature that changes in the value of existing asset

can be the results of economic shocks other than those driving new investment.

When growth options are under consideration, the synergy created by the new project can be expressed as:

where HK is the cost of investment and it is time-varying to verify the importance of timing to investment. Once the firm undertakes new investment, it is irreversible in that the project cannot be abandoned. However, we need two additional assumptions, α> and0 β >1, to make sureI_G = ∂ ∂ > andI G 0 I_H = ∂ ∂ > . In other word, we I H 0 need the value of the firm and the value of growth options can increase with the valuation of existing and new capital stocks. Equation(5) shows that the more improvement on productivity,α andβ , the larger synergy that new project can create for this company. If the synergy created by new investment is less than zero, the firm will not undertake any investment as it need internal funds to finance new projects.

This criterion is not valid, however, especially when investment is irreversible and faces uncertainty. The following proposition shows the optimal timing of investment and the corresponding value of this growth options when investment is irreversible.

Proposition 1: Suppose that the true value of the synergy parameter isβ β= ^*. The optimal investment strategy of a firm is to expand when the relative valuation ratio,R=G H/ , is at or above this level

(

* *

1 1 .

R ^{= η β}_−η ⁻

)

⁽⁶⁾

Moreover, the corresponding value of this growth options is

( )

where η denotes the positive root of the following familiar quadratic equation

(

^{2 2}

) ^{( ) ( ) ( )}

1 2 1

2 σ^G − ρσ σ^{G H} +σ η η^H − + μ^G −μ η^H + μ^H −r =0, (8) in which η <1.¹⁵

As shown in Proposition 1, a firm’s optimal investment policy is governed by a constant thresholdR . The value-maximizing expansion policy is to expand when the ^* relative valuation ratio reaches this cutoff level. This implies that only when the existing capital stocks have higher profitability or there is no idle capacity problem, then new capital is valuable. Our investment decision model differs from the previous studies in which assets in place do not affect the firm’s investment decisions, such as Berk, Green, and Naik (1999). However, our work is close to Cooper (2006) that the optimal timing of expansion dose depends on the profitability of the firm’s existing assets. He suggests that investment is triggered only when the productivity is high enough relative to the stocks of existing capital, so that the benefits of adjusting the capital stock cover the costs by doing so. Prior to investment, the value of the growth options will depend on the timing of expansion and contain uncertainty. In the following sections, we will discuss the implications of this optimal investment strategy.

3.5 The Optimal Investment Strategy

15 We choose η<1 as possible solution because it is reasonable to assume that the value of growth option is increasing function of R but the increasing speed is declining with R because the value of

在文檔中 Productivity, Corporate Investment, and Liquidity (頁 79-83)