The Effects of Suboptimal Investment Rules on Value

The Effects of Suboptimal Investment Rules on Value

George Y. Wang

Department of International Business National Kaohsiung University of Applied Science

Kaohsiung, Taiwan Email: [email protected] Abstract—This study evaluates optimal investment rules under

various stochastic processes and investigates the cost of adopting suboptimal investment rules to firm value. It is found that with consideration of the effects of preemptive competition or mean reversion, optimal investment trigger is lower than that under a geometric Brownian motion (GBM) process. Therefore, acknowledging option to wait, under the underlying GBM assumption, unnecessarily delays corporate investment. The study also proposes a loss function to measure the costs of adopting suboptimal investment rules. The main finding is that the best investment rule is the optimal investment rule itself. Any deviation from the optimal investment rule is suboptimal and may lead to a substantial loss in value.

Keywords- optimal investment rules, preemptive competition, mean reversion, geometric Brownian motion

I. INTRODUCTION

The valuation of irreversible investments is a crucial topic in financial economics. Over the last decades, real options theory has become the dominant focus of capital investment theory. Dissatisfaction with traditional discounted cash flow (DCF) approach and inspiration from the option concepts have generated considerable interest in applying options pricing theory to capital investment decisions.

The basic insight of the options thinking in capital budgeting is first mentioned by Miller and Modigliani (1961), who propose the “investment opportunities” approach to valuation. 1 _{Their followers show that growth opportunities} can contribute a substantial portion to the value of firm. Another line of research leading to real options is the study of “irreversibility” in early environmental economics. These studies, including Weisbrod (1964), Arrow and Fisher (1974), and Henry (1974a and 1974b), basically analyze optimal investment behaviors in governmental decisions in irreversible and unrecoverable environments. The most influential impact on real options is inspired by the seminal financial option pricing models of Black and Scholes (1973) and Merton (1973). Since then, numerous studies have applied the option pricing framework to analyze investment problems on capital assets.

1 _{Although both Kester (1984) and Trigeorgis (1996) have} mentioned that Myers (1977) was the first to initiate the idea of “growth options”, yet Myers and Turnbull (1977) admits that the growth opportunities approach to valuation is inspired by Miller and Modigliani (1961).

Myers (1977) and Myers and Turnbull (1977) further elaborate on this idea of “opportunities” and suggest that a firm in an uncertain world should be considered as a portfolio of tangible assets and intangible assets. The tangible assets are regarded as physical assets with productive capacity and the intangible assets are as options to grow in future. Consequently, the market value of firm is the present value of capital assets plus the sum of values of all possible growth options.2 _{Kester (1984) proposes that investment opportunities} are conceptually analogous to ordinary financial call options on securities, also termed options on real assets. It is argued that investment projects with negative NPVs can still be valuable as long as management can arbitrarily postpone investments and wait for favorable future conditions. This argument means that discretionary managerial actions to the changing conditions may increase the chance of eventually realizing the upside potential without raising the possibility of incurring downside loss. Therefore, Kester suggests that project risk has a significant and positive impact on the value of growth options as long as flexibility is available to management.

The theory on irreversible investments and real options is further advanced by two of the most influential books, by Dixit and Pindyck (1994) and Trigeorgis (1996), both of which include all of their earlier works with a different focus. Dixit and Pindyck (1994), from a perspective of financial economic behavior, emphasize how optimal investment rules are determined under the assumptions of irreversibility, uncertainty, and corporate ability to delay investments. In general, the optimal decision rules associated with entry and exit decisions must be the point at which options values are maximized. Trigeorgis (1996), from a prospective of capital budgeting, primarily deals with the valuation of various types of real options embedded within capital investments. He postulates that an appropriate analysis of project appraisal should consider not only the present values of cash flows derived from the dynamic DCF analysis but also the values of embedded real options.

Several recent studies emphasize on early ignition of investment under various circumstances. For example, Chi and Fan (1997) examine the bias of potential optimal investment rule under uncertain development time and completion cost. Their main finding is that the new investment rule is smaller than the potential one due to the dissipating effect of project value, leading to preference of short-term over long-term

2 _{See Myers and Turnbull (1977), p. 332.}

projects. Boyle and Guthrie (2003) analyze the dynamic relationship between investment and financing considerations, and argue that the threat of future funding shortfalls lowers the value of the firm’s timing options and encourages acceleration of investment beyond the original optimal level. Malchow-Moller and Thorsen (2005) explore repeated real options implied within an investment project. The main finding is that the value of option to wait is smaller than in the case of a single option.

On the ground of literature on investment under uncertainty, this paper argues that optimal investment rule, in the presence of option to wait, would be at the point that project value exceeds investment trigger. Early exercise or over delay investment is suboptimal and thus may incur substantial loss in option value. The rest of the paper is organized as follows: Section 2 presents the derivation of investment triggers under various stochastic processes, including geometric Brownian motion, mixed diffusion-jump, and mean-reverting process. Then, a numerical analysis on investment triggers is conducted for a near-zero-NPV project in Section 3, since option value is less crucial in investment decision-making for a significant positive NPV or a deep negative NPV project. Section 4 proposes a method to estimate the cost of adopting suboptimal investment rules. Section 5 gives concluding remarks.

II. THE VALUE OF OPTION TO WAIT

Most classic papers on real options study optimal investment timing to pay an investment cost in return for an irreversible project whose value is a major source of uncertainty, evolving as a geometric Brownian motion (GBM). In this section, we first review a general investment timing problem discussed in both McDonald and Siegel (1986) and Pindyck (1991), assuming that the project value follows a GBM. Then, we describe the same problem under a mixed diffusion-jump process and a mean-reverting process.

A. GBM Model

Modern investment theory centers on searching for optimal investment trigger, V* , such that the value of the investment opportunity, F(V) , is maximized. We assume that the project value, V, follows a GBM as follows:

dV_=αVdt_+σVdz (1)

where α , σand dz denote drift rate, volatility, and an increment of a standard Wiener process, respectively.

For a project whose value follows a GBM, literature has shown that the solution of F(V) is as follows (see McDonald and Siegel, 1986; Dixit, 1989; Pindyck, 1991):

1 ( ; ) b F V V∗ ₌AV ₍₂₎ where A=

(

)

μ is opportunity cost of investment.

At the maximum of option values, the optimal investment trigger equals the sum of investment cost and the value of investment opportunity, which is also called the value-matching condition. By substituting Equation (2) into the value-matching condition, V* is solved as follows:

1 1 1 GBM b V I b ∗ _{= ⎜}⎛ ⎞_⎟ − ⎝ ⎠ (5)

where VGBM∗ denotes the optimal GBM trigger. B. Mixed Difusion-Jump Model

In this subsection, we extend the preceding analysis to the case in which project value (or cash flows) follows a mixed diffusion-jump (MX) process. This process is often specifically used to describe the situation in that the value of an investment opportunity can become worthless as potential competitors enter the market as first-movers.3 _{In other words,} the preemptive competitive effect may lead to the project value appropriated by the competitors, which thus can be characterized by a mixed diffusion-jump process.4 A mixed diffusion-jump process is formalized as follows:

dV=αVdt+σVdz Vdq− (6) where dq is the increment of a Poisson process with a mean arrival rate of λ and is expressed by

with a probability of 0 with a probability of 1-dt dq dt φ λ λ ⎧ =⎨ ⎩ (7)

where _φ(0≤ ≤_φ 1) stands for the constant percentage of loss in V should the jump event, i.e. competitive arrivals, occur. Meanwhile, dq is assumed to be independent of dz, i.e., E(dqdz)=0.

With the same boundary conditions as in the GBM model, both McDonald and Siegel (1986) and Dixit and Pindyck (1994, Ch. 5) have verified that if _φ=1, the solutions of F V( )

and VMX∗ are exactly the same as Equation (2) and (5),

respectively, except that b1 is replaced with b2. If φ≠1, the

value of the investment opportunity is still the form of

( ) b

F V =AV . However, the solution needs to be found numerically together with the boundary conditions.

2 2 2 2 2 1 ( ) ( ) 1 2( ) 2 2 r r r b μ α μ α λ σ σ σ − − − − + ⎛ ⎞ ⎛ ⎞ =⎜_⎝ − ⎟_⎠+ ⎜_⎝ − ⎟_⎠ + (8)

The parameter b is similar to the parameter b1 in the

functional form except that the jump intensity λgets added into the interest rate in the constant term. It is easy to see that b is equal to b1 if λ=0 and greater than b1 if λ>0. Since both

b1 and b2 are inversely correlated with optimal investment

trigger, for the same set of parameter values the relationship of

3 _{Trigeorgis (1990) deals with the preemptive competitive} effect by treating the competitors’ actions as dividends which are the proportions of the project value appropriated by the competitors. His analysis is limited by the assumption that the erosion effect can be completely anticipated and quantified by the firm, which appears to be less realistic.

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