The Optimal Investment Strategy

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

(

) ^{( ) ( ) ( )}

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

This section investigates the optimal investment activity of the firm derived in Proposition 1. From equation(6), we find that the firm’s investment decision involves two sources of uncertainty: the information set about improvement on productivity, and the dynamics of future cash flows. In this section, we discuss the impact of these two characteristics on optimal investment.

First, from our closed-form solution in equation(6), we find that only the unknown productivity parameterβ is critical to the timing of expansion. Our intuition is that because α is observable and shared by both assets, it cannot reveal any useful information to the dynamics of relative valuation ratioR. Hence, only the unrevealed information has impact on the optimal timing of investment. In addition, because the relative valuation ratio is non-negative, the constant investment threshold should be positive. From equation(6), we can verify that∂R^* ∂ > . That is the firm that creates β 0 large learning-by-doing effect through investment is not eager to chase profitable investment by setting a strict threshold. Our explanation is that if the improvement on productivity is large, the firm will hold the growth options to maximize the value of waiting to invest. Becauseβ is not observable to the outside investors, managers will hold the growth options until existing capital has higher valuation. In brief, waiting becomes more valuable to managers because this growth options can make existing assets more valuable.

Next, we discuss how the dynamics about cash flows affect the investment threshold. Figure 3 shows some comparative static to discuss the effects of cash flows dynamics in our framework. First, we present a number of key model parameters used in our analysis. The mean and volatility of cash flows from new projects are 5% and 21%, respectively, from Ang and Liu (2004). The volatility of cash flows from existing

capital stock is 29% to match the standard deviation of the annual earnings growth of U.S. corporate earnings in the period 1929 to 2001 as reported by Longstaff and Piazzesi (2004). The drift of existing capital stock is set to 12%. This implies that the average of equity return is 8.5%, consistent with the equity premium data from Campbell, Lo, and MacKinlay (1997). The appropriate discount rate is equal to 8% to keep firms holding the options. The investment ratio1− is equal to 15% from Abel λ and Eberly (2001). The correlation between existing and new capital stocks is set to 0.1.

The improvement on productivity of new capital stocksβ is 1.3, which is consistent with the estimated reported by Hennessy (2004). Finally, because α is irrelevant to the investment threshold, we set it equal to one.

0.27 0.28 0.29 0.30 0.

Panel A: The volatilit

31 0.19 0.20 0.21 0.22 0.23

Panel B: The volatility of cash flow from new assets

0.5

y of cash flow from existing assets

Figure 3: The effect of cash flows’ volatility on the investment threshold.

This figure shows the comparative static of investment threshold. Two driving forces are discussed here including the volatility of cash flows from existing assets (Panel A) and the volatility of cash flows from new assets (Panel B). Input parameter values are set from previous research as described in the article.

Figure 3 presents the comparative static of the investment threshold. We demonstrate that cash flows uncertainty would time investment because of irreversibility. When a firm faces a higher uncertainty about investment, proxy byσ_H, it would prefer to hold this growth options and wait to invest. This finding is consistent with the previous research that a higher level of uncertainty will increase the critical investment trigger level (Sarkar, 2000). Greater uncertainty increases the incentive to keep the growth options in order to obtain more information about future prices and market conditions. Most importantly, we find that uncertainty about profitability from existing assets also times investment. Because of learning-by-doing, the valuation of existing assets also has impact on the synergy of expansion. When the profitability of existing capital stocks contains more uncertainty, managers will set a stricter investment threshold to expand latter.

Next, Figure 4 shows the impact of the cash flows volatility on the value of growth options. We find that the higher uncertainty about profitability from existing or new capital stocks reduces the value of growth options. This finding is opposite to the real options literature that a higher level of uncertainty increases options value (McDonald and Siegel, 1986). However, according to Boyle and Guthrie (2003), if capital market has frictions such that a firm’s investment decision is subject to its internal funds, then greater cash flows volatility reduces the value of the expansion options because the firm has to choose the suboptimal investment timing. Consistent with Boyle and Guthrie (2003), we argue that because of learning-by-doing and the assumption of all-equity firm, the value of growth options depends on the valuation of existing and new capital stocks. Uncertainty about profitability reduces the value of a firm’s investment opportunity and makes its market value go down. Thus, waiting is

still worth when investment is irreversible, but gains from delaying expansion decrease as profitability become more uncertain.

0.1 0.2 0.3 0.4 0.

Panel A: The relative valuation ratio R=G/H

0.000

Panel B: The relative valuation ratio R=G/H

0.000

Figure 4: The effect of cash flows’ volatility on the value of growth options.

This figure shows the comparative static of the value of growth options. Two driving forces are discussed here including the volatility of cash flows from existing assets (Panel A) and the volatility of cash flows from new assets (Panel B). Input parameter values are set from previous research as described in the article. Total amount of capital stocks, K₁+ , is one. K