# Investment timing for dynamic business expansion.

We investigate the timing of business expansion. With an indefinite
sequence of growth opportunities that have constant returns to scale,
current investment neither displaces nor impairs future returns. In a
dynamic setting with expansion restricted to a fraction of firm size,
the endogenously determined cost of capital uniformly exceeds the value
maximizing return threshold for expansion. Taking this into account, a
manager accelerates investment to facilitate larger and more valuable
future investments when earnings stochastically improve. This result is
the opposite of deferral that the investment literature recommends due
to irreversibility. This means that the managerial application of the
cost of capital as an expansion hurdle rate is improperly conservative.

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We investigate the managerial trade-off between the investment delaying force of irreversibility and the accelerating force of follow-on options when managerial choice endogenously determines corporate growth. If the firm has an infinite sequence of growth opportunities with constant returns to scale, current investment neither displaces nor impairs future investment returns. These expansion features weaken the managerial incentive to defer irreversible investment.

Furthermore, when expansion is restricted to a fraction of firm size, we demonstrate that the endogenously determined cost of capital, expected return on business value, uniformly exceeds the value maximizing expansion return threshold. In other words, a manager accepts marginal investments that return less than the cost of capital to facilitate larger and more valuable future investments when earnings stochastically improve. In this instance, the force of increasing the follow-on options dominates the force of irreversibility for expansion timing. We confirm that the managerial application of the cost of capital as an expansion hurdle rate is improperly conservative. A value maximizing manager expands sooner than a manager who uses the cost of capital as an expansion hurdle rate.

There are four features of our business expansion model that, when combined, lead to new results. First, we relate the process for operating profit with the process for capital investment so that growth in both is endogenous. Rather than exogenous profit growth driving capital growth, value maximizing managerial decisions jointly determine profit and capital growth. Profit that grows only with capital growth removes a managerial attraction to waiting. Second, we presume indefinite expansion opportunities with constant returns to scale. Current investment neither displaces nor impairs future investment returns, which weakens the managerial incentive for investment delay. Third, the manager has a dynamic option to grow that includes the ability to suspend growth upon inadequate profitability. Current investment does not impair future investment returns because the manager can suspend growth upon inadequate profitability. This ability to suspend unprofitable growth reduces the incentive to delay current investment. Fourth, we apply an expansion proportionality constraint so that small firms make small investments and large firms make large investments. If a manager expands business prematurely when profit is possibly inferior, current investment increases the size and dollar value of future expansion upon stochastically improved profitability. Downside earnings risk is no greater as the manager can defer growth once started.

It is important to compare value maximizing models with the cost of capital analysis since the cost of capital is ingrained in the corporate practice of business expansion evaluation. As managers recognize that the cost of capital is not an appropriate benchmark for business investment decisions in dynamic settings in the presence of various real options, they will look to the financial literature for guidance. The most likely resource is Dixit and Pindyck (1994). Managers will discover that the predominant modeling feature is the investment deferral option. Managers exercise their "call option" on the value of an underlying investment when downside value protection no longer exceeds foregone profit from investment delay. This delay indicates that the investment return boundary exceeds the cost of capital. Because the cost of capital is relatively easy to measure, when following the existing literature, managers will add a positive factor to the cost of capital to reflect the deferral option when calculating the expansion boundary for their businesses. We illustrate that this conjectured managerial heuristic for business expansion is inappropriately conservative.

McDonald and Siegel (1986) and Dixit and Pindyck (1994) illustrate that the possibility of disappointing earnings leads a manager to exercise the "start option" on a solitary irreversible investment only when the investment is "in the money" and net present value (NPV) exceeds a strictly positive threshold. Because current investment precludes future investment, current investment creates an opportunity loss that encourages delay. The return boundary for starting the investment exceeds the cost of capital and the manager defers a timeable investment. There are a number of economic factors that influence a manager's decision to defer an irreversible investment: 1) partial reversibility (Kandel and Pearson, 2002), 2) investment learning (Grenadier and Weiss, 1997; Bergemann and Hege, 1998), 3) strategic investment (Cottrell and Sick, 2002), 4) capital stock adjustment costs (Abel and Eberly, 1994; Carlson, Fisher, and Giammarino, 2004), and 5) agency costs (Bergemann and Hege, 1998; Christensen, Feltham, and Wu, 2002). (1) These factors modify the extent of managerial delay but do not induce investment earlier than recommended by Modigliani and Miller's (1958, 1963) conventional cost of capital analysis. Our model justifies such early investment.

In a piece described by Brealey, Myers, and Allen (2006) that predates the modern real options literature, Roberts and Weitzman (1981) and Weitzman, Whitney, and Rabin (1981) indicate that a manager might accept a negative NPV "now-or-never" investment if it has a follow-on investment with positive NPV contingent on initial investment success. Of course, if the manager could defer the investment, then the delaying force of irreversibility might dominate the accelerating force of follow-on options.

Irreversibility delays, while follow-on options accelerate investment timing. In some circumstances investment can be induced at rates below the conventional cost of capital. By exaggerating the size or profitability of follow-on options, we could manufacture stylized problems where the value attraction of follow-on options dominates the downside earnings risk of irreversible investment so that the manager does not delay but alternatively accepts a negative NPV initial investment. Therefore, in the current paper, we impose some economic realism on the timing decision by studying what is arguably the most important investment for practicing managers: expansion that has the same profitability as an existing business.

McDonald and Siegel (1986) and Dixit and Pindyck (1994) demonstrate that while volatility increases both downside risk and upside earning potential, because downside risk avoidance dominates managerial decision making for one-time investments, volatility increases profit thresholds for investment. There is a literature suggesting that dynamic models of sequential growth can have distinct characteristics from solitary one-time investments. In Dixit and Pindyck (1994), Abel (1983), Pindyck (1988), Caballero (1991), and Aguerrevere (2003), volatility can encourage sequential investment despite irreversibility. Optimal investment thresholds can decrease rather than increase with profit volatility. The reason for this difference is that upside earnings potential is more important to managers for a sequence of investments when compared to a solitary investment. With upside earnings surprises, managers make supplementary investments. Even if earnings disappoint, eventually, once earnings improve, managers also make supplementary investments. Volatility increases upside earnings potential that managers pursue with lower investment profit thresholds.

Despite the possibility of distinct characteristics, managers also delay investment in the above sequential growth literature. In Pindyck (1988), current investment impairs future investment returns due to diminishing productivity that encourages investment delay beyond that for a solitary investment. On the other hand, since current investment does not displace future investment, the imperative for investment delay is less for an indefinite sequence of future investment options. These two incremental timing effects exactly offset one another. A manager who faces a sequence of investments delays current investment exactly like it was a solitary investment with no further expansion opportunities. Abel and Eberly (1994) investigate a capital adjustment cost model where small investments have modest costs, but large investments are increasing costly. Their model still results in investment delay despite the indefinite growth options it contains. The incentive to delay in this model is driven by the managerial desire to avoid costly capital adjustments should profitability reverse in either direction. Last, in Dixit and Pindyck (1994), Abel (1983), Pindyck (1988), Caballero (1991), and Aguerrevere (2003), capital investment over time is a managerial reaction to an exogenously specified profit process. Managers await improvement in the profit process over which they have no control. Spontaneous profit improvement without immediate investment costs encourages delay.

The paper is organized as follows. Section I models business expansion. Section II derives the value-maximizing expansion boundary and compares it to the cost of capital. Section III formulates conclusions in a summary.

I. Dynamic Business Expansion

A. Expansion

Because growth is neither spontaneous nor unplanned, a corporate manager invests to produce profit growth. In free cash flow valuation, asset value depends on predicted future free cash flow, part of which is the required expenditure to generate profit growth. We presume this expenditure depends on existing asset replacement costs, which we represent with, [B.sub.t] > 0, capital stock. Monitoring capital is relevant because of this dependence. A manager controls capital stock by undertaking irreversible investment. The instantaneous investment rate is g [greater than or equal to] 0 if the manager expands the business and zero otherwise (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Equation (1) implicitly assumes that there is no cost for switching between growth and no-growth regimes.

Because capital stock adjustment costs are zero for an investment up to gB dt and infinitely great thereafter, and because production technology has constant returns to scale, there are no benefits or costs, proportionately, to large investments. While the manager might prefer a larger investment, the limit on investment spending is gB dt. Total yearly investment is a random amount between 0 and gB (uncompounded). If the manager neglects expansion for a period of time, this neglect cannot subsequently be reversed with more intense future expansion. Regardless of the past, the limit on investment spending over dt is gB dt.

Kogan (2001, 2004) also limits the rate of investment between zero and a positive constant, which is a special case of the standard convex adjustment cost specification and restricts a manager's instantaneous capital adjustment between zero and g. Our study extends this literature by endogenizing the cost of capital and demonstrating that a value maximizing manager accelerates business expansion when compared to the standard NPV criteria.

Operating profit, a cash-flow-based measure, is generated from a constant returns to scale technology with stochastic productivity parameter [Y.sub.t], the return on capital (ROC), with dynamics [dY.sub.t]/[Y.sub.t] = [sigma] dz, where dz is a standard Gauss-Weiner process. (3) Because operating profit, [X.sub.t], is ROC times capital, [X.sub.t] = [Y.sub.t] [B.sub.t], the process for operating profit is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Because operating profit, X, is strictly positive, the manager neither abandons nor suspends existing operations. The business generates instantaneous operating profit, X dt, whether the manager invests for expansion or not. Strictly positive operating income is characteristic of large industrial and commercial firms with existing operations, but not characteristic of small (often) development-oriented firms. While creating commercially viable products, start-up firms bear not only development risk, but also negative operating income before they create markets and earn revenues. We study the expansion of firms with existing operations rather than new venture investments. (4)

B. Investment Return

The process for ROC, [dY.sub.t][dY.sub.t]/[Y.sub.t] = [sigma] dz, is dependent upon neither the manager's expansion decision nor the proportionality constraint on investment and profit growth, g. ROC is a martingale. ROC for a small firm is expected future ROC, regardless of the growth factor, g, when the small firm, perhaps, becomes large, E[[??].sub.t]] = [Y.sub.0].

Investment return for either the existing business or for an expansion investment is ROC and not ROC plus a growth factor as there is no ROC growth like there is for spontaneous profit growth investments. (5) Any investment on its own, either the existing business or an expansion investment, before ensuing growth investments, which also have the same character, generates a perpetual flow of nongrowing expected profit Y per dollar of capital. Without additional capital, profit does not increase. The IRR satisfies Y/IRR - 1 = 0, therefore, IRR = Y.

In a two-period model, Abel, Dixit, Eberly, and Pindyck (1996) illustrate that if capital's cost is expected to increase, then because future investment is more costly than current investment, the incentive to invest currently increases. The results in our model do not depend on this cost of capital effect. Whether or not capital is costly is dependent upon the profit that capital generates, ROC. We expect no change in capital's cost, E[[??].sub.t]] = [Y.sub.0]. Constrained capital growth neither encourages nor discourages current investment. The playing field is level for our comparison of the dynamic expansion boundary and the cost of capital.

C. Assumptions

Investments are equity financed and we do not investigate financing costs, taxes, or other frictions except to the extent that they are encompassed within the investment constraint, g. We avoid extensive investigation of frictions in order to maintain simplicity, a simplicity that leads to both a closed-form solution for the value maximizing return expansion boundary and analytic results. Despite simplicity, the model is not stylized and represents, rather accurately, the expansion decisions faced by practicing business managers. Adding more detailed frictions requires that return on investment, the value maximizing return expansion boundary, and the cost of capital be calculated numerically. There is considerable real options literature regarding how various frictions might influence business investments. The next section investigates the manager's dynamic value maximizing investment strategy.

II. Cost of Capital and the Value Maximizing Expansion Boundary

A. Market-to-Capital Ratio

Our paper contributes to a recent growing literature on risk measurement in dynamic models of managerial choice (e.g., Berk, Green, and Naik, 1999; Goldstein, Ju, and Leland, 2001; Carlson, Fisher, and Giammarino, 2004; Zhang, 2005; Cooper, 2006). Berk, Green, and Naik (1999) investigate corporate growth options. Because these opportunities are of heterogeneous risk, their model is not suited to expansion analysis. Goldstein, Ju, and Leland (2001) study optimal bankruptcy in a dynamic setting, but they do not consider capital growth. Since real spontaneous earnings growth is unrealistic, we presume, alternatively, that earnings growth depends upon capital growth. Zhang (2005) and Cooper (2006) use real options models with dynamic risk adjustment for idle physical production capacity in economic downturns to explain the relationship between equity returns and the book-to-market ratio. They do not consider the association between the value maximizing profit boundary and the cost of capital. Carlson, Fisher, and Giammarino (2004, 2006) explore the impact of corporate operating leverage and growth opportunities on asset returns. Their model provides a rational explanation for both size and book-to-market effects in the cross-section of asset returns. However, it offers neither closed-form solutions nor analytic results in a general context nor a comparison of optimal investment policy with the cost of capital.

We use the valuation methodology of Goldstein, Ju, and Leland (2001) to find the value of a business, V(X, B), that has investment and profit growth restricted to a fraction, g, of capital, B. Goldstein, Ju, and Leland model profit growth, but not the capital expansion required to produce this growth. In Appendix A, we formally relate profit growth with capital growth through the manager's expansion decision.

The risk-adjusted process, X', for operating profit is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [theta] [greater than or equal to] 0 is the coefficient of constant relative risk aversion for a representative investor; [[sigma].sub.xc] is the covariance of the log of operating profit, X, with the log of aggregate consumption, c = log(C); and aggregate consumption follows a geometric Brownian motion. We presume positive covariance risk, [[sigma].sub.xc] [greater than or equal to] 0.

Let V(X, B) be the value of the business. In Appendix A, we confirm that the form of this value function is

V(X, B) = B [pi](Y), (4)

where ROC = Y = X/B. Since the value to capital ratio, [phi](Y), depends only on Y, the manager expands the business depending on Y and not its separate components (X and B). The manager expands when ROC exceeds a boundary that we denote as, [xi]. In Appendix A, we demonstrate that for this arbitrary expansion boundary, [xi], the value to capital ratio is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [r.sup.*] [equivalent to] r + [theta][[sigma].sub.xc] and we define the constants [alpha] [greater than or equal to] 1, [lambda] [less than or equal to] 0 in Equation (A6).

On the "no-growth" branch of Equation (5), the first term is the value of operating profit if the manager never expands the business. This amount, which does not depend on the parameter g, is the discounted value of expected future profit at the risk-adjusted rate [r.sup.*]. The second term (positive) and third term (negative) are, respectively, the value of expected future incremental profit generated and the cost of expansion investments incurred when, at some time in the future, ROC exceeds the expansion boundary, [xi], and the manager expands the business. The positive value of future incremental profit becomes more positive and the negative value for the cost of future expansion investment becomes more negative with Y because both incremental profit and expansion investment become more likely.

On the "growth" branch of Equation (5), the first term is the value of operating profit if the manager permanently expands the business. Because the manager grows the business, this term, for a growing perpetuity, recognizes the growth parameter, g. The second term (negative) is the value of expected future profit foregone when ROC declines below the expansion boundary and the manager defers investment until ROC once more exceeds the expansion boundary, [xi]. This term becomes less negative with Y as the possibility of foregone profit decreases with greater profitability. The third term (negative) is the cost of expansion expenditures, which recognizes that the manager avoids these costs when they defer investment for ROC below the expansion boundary. This term becomes increasingly negative with Y as the likelihood of continued growth investment increases with profitability.

B. Value-Maximizing Expansion Boundary

To find the value maximizing expansion boundary, [[xi].sup.*], we take the derivative of the value function, [pi](Y) with respect to [xi] on either branch of Equation (5), set the result to zero, evaluate at Y = [xi], and solve for [[xi].sup.*]

[[xi].sup.*] = [r.sup.*] x [[[r.sup.*]-r-g]] x [[alpha]/[alpha] -1]] x [[lambda]/[lambda] -1]]. (6)

The first two terms on the right-hand side of Equation (6) equal the manager's ROC boundary for a hypothetical "now-or-never" permanent expansion decision. The value of the permanent growth firm, [[X.sub.0]/[r.sup.*]-g]] - gB/r-g, exceeds the value of the permanent no-growth firm, [[X.sub.0]/[r.sup.*]], when Y = ROC [greater than or equal to] [r.sup.*] x ([r.sup.*] - g/r-g]).

The first three terms on the right-hand side of Equation (6), [r.sup.*] x [[r.sup.*] - g]/r-g]] x [[alpha]/[alpha]-1]], equal the manager's expansion boundary for a hypothetical firm that has a perpetual option to start permanent growth (see Appendix B). Since [[alpha]/[alpha]-1] [greater than or equal to] 1, the manager defers permanent growth as compared to the now-or-never manager. Investment irreversibility induces delay. A manager who awaits permanent growth does not have a follow-on option because growth, once started, is permanent and, therefore, is effectively a solitary investment (with lump- sum cost [[g.sup.*]B/r-g]). In this setting, results in McDonald and Siegel (1986) and Dixit and Pindyck (1994) apply and it is not surprising that the manager defers growth.

The first two terms times the fourth term on the right-hand side of Equation (6), [r.sup.*] x [[r.sup.*]-g/r-g]] x [[lambda]/[lambda]-1], is the threshold return boundary for a hypothetical growing firm that has a perpetual option to permanently stop growth (see Appendix B). Since [[[lambda]/[lambda]-1]] [less than or equal to] 1, even though there is no cost to accepting permanent no growth, the manager of the growing firm defers no growth as compared to the now-or-never manager. In fact, not only does the manager not incur a cost in accepting no growth, but they also avoid the irreversible investment cost of growth. Permanence of the decision dominates foregone irreversible investment and leads to no-growth deferral.

The product of the final two terms on the right-hand side of Equation (6), [[[alpha]/[alpha]-1]] x [[[lambda]/[lambda]- 1]], represents the dynamic combination of a growing manager's inclination to defer permanent no growth and a no-growth manager's inclination to defer permanent growth. Of course, the juxtaposition of these two options in a dynamic setting makes neither decision permanent, reducing the force of both inclinations. Further, in the dynamic setting, because the no-growth decision for a growing firm is immediately reversible without cost and each subsequent decision is also reversible, decision permanence is not a motivating factor for no-growth delay (or equivalently temporally accelerated investment for a no-growth manager awaiting growth). Rather, a no-growth firm with an indefinite stop/start growth option accelerates growth as compared to the now-or-never manager, that is, [[xi].sup.*] [less than or equal to] [r.sup.*] x [[r.sup.*-g/r-g]], for two reasons.

First, the now-or-never manager who opts for permanent growth accepts significant growth leverage. We establish in Appendix C that a sufficient condition for the now-or-never expansion boundary, [r.sup.*] x ([r.sup.*] - g/r-g]) to exceed the dynamic expansion boundary, [[xi].sup.*], is positive covariance risk, [theta][[sigma].sub.xc] > 0. In this case, the term ([[r.sup.*]-g/r-g]) exceeds one and represents the impact of growth leverage, unavoidable fixed costs of growth investment, on the now-or-never manager's permanent growth decision. Because the dynamic manager avoids the burden of growth leverage at will, the dynamic expansion boundary, [[xi].sup.*], is less than the now-or-never permanent expansion boundary, [r.sup.*] x ([r.sup.*] - g/r-g])). The dynamic manager's ability to avoid growth leverage makes the manager less reluctant to begin growth. The growth-leverage-induced deferral motive remains, however, because suspended growth reduces, but does not eliminate, growth leverage. Growth leverage exists whenever the firm grows.

Second, the principal motive for accelerated investment in our model is follow-on options when indefinite future growth investments are proportional to existing capital. In Appendix C, we demonstrate that for positive maximal growth, g > 0, even if the covariance risk is zero, [theta][sigma].sub.x,c] = 0, so that growth leverage risk is irrelevant, the dynamic expansion boundary, [[xi].sup.*], is strictly less than the now-or-never permanent expansion boundary (which in this instance is the riskless interest rate, r). The dynamic manager temporally accelerates growth as compared to the now-or-never manager as not only does current investment not impair future returns, but it also increases the dollar value of future growth options. Current investment has a size attraction that promotes growth.

With reasonable parameter values, the dynamic expansion boundary, [[xi].sup.*], is significantly less than the now-or-never permanent expansion boundary, [r.sup.*] x ([r.sup.*] - g/r-g]). For example, with parameter values that we use in Figure 1, the now-or-never permanent expansion boundary is 54%, whereas the dynamic expansion boundary is [[xi].sup.*] = 11.82%. The now-or-never permanent expansion boundary is very large. Not only do growth fixed costs impose risk on shareholders, but these costs grow unavoidably over time.

The now-or-never permanent growth manager, the growing manager with an option to permanently stop growth, and the nongrowing manager with an option to permanently grow are all stylized characters not meant to represent real business managers or business situations but are helpful in understanding manager's motives with dynamic stop/start growth options. Our central interest is comparing value maximizing managers that take advantage of stop/start growth options with cost of capital managers and their investment mistakes. We derive the cost of capital and make this comparison in Sections D and E that follow.

C. Net Value Creation

In Appendix D, we demonstrate that the market-to-capital ratio equals one when ROC is at the expansion boundary, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As the value to expenditure ratio, [pi](Y), is increasing in ROC = Y, if Y [greater than or equal to] [[xi].sup.*], the manager expands the business at the maximum rate g and the firm creates positive net value because value exceeds capital V(X, B) [greater than or equal to] B and [pi](Y) [greater than or equal to] 1 when Y [greater than or equal to] [[xi].sup.*]. If the manager invests at any return less than the expansion boundary, [[xi].sup.*], net value creation is negative.

Marginal and average market to capital ratios both equal one at the expansion threshold, [[xi].sup.*]. Therefore, the value maximizing manager follows the traditional investing rule that calls for expansion when the market-to-book ratio exceeds one. A market-to-book ratio in excess of one reflects the value of waiting to invest. In our model, the market-to-book is always one when the firm switches from growth to no growth and vice versa. This result is distinct from McDonald and Siegel (1986) type models where current investment displaces future investment and waiting protects against downside losses. The market-to-book ratio strictly exceeds one at the investment profit threshold with positive net value creation upon starting an irreversible solitary investment.

D. The Cost of Capital and Expected Return

Modigliani and Miller (1958) define the cost of capital in two ways: 1) the expected return on overall business value and 2) the return threshold for net value creation with incremental investment. The first definition is the average cost of capital and the second is the marginal cost of capital. Modigliani and Miller (1958, 1963) favored the second definition due to risk measurement limitations that preceded the capital asset pricing model (CAPM) and other modem asset pricing models. However, with risk measurement advances, the first definition has become the underpinning of the common place cost of capital measurement with the weighted average cost of capital (WACC). In a static environment, the average and marginal costs of capital equal one another, and therefore, the distinction is irrelevant. However, as we demonstrate in this paper, in a dynamic setting, the average and the marginal costs of capital diverge. Therefore, business persons risk decision errors when using the average cost of capital for expansion benchmarking. Expected return on business market value is a growth-levered cost of capital. As we show in this subsection, a primary determinant of this expected return is growth leverage. It is also an average cost of capital as it represents a risk average of the firm's existing business and future expansion options. When the firm has no growth opportunities, g = 0, the cost of capital unlevered by growth investments, the growth unlevered cost of capital, is [r.sup.*] representing the risk of in place operations. The results in Subsection C above illustrate that the value maximizing return threshold, [[xi].sup.*], satisfies Modigliani and Miller's (1958, 1963) second cost of capital definition. Because our economic environment is dynamic, the manager can start and suspend growth at any future time, it is a dynamic cost of capital. It is also the marginal cost of capital since it is the return for a marginal expansion investment made by a value maximizing manager.

We measure the average cost of capital as operating profit, X, less expansion expenditures (if incurred) plus expected capital gain from changes in operating profit, all divided by business value. We denote [omega](Y) as the average cost of capital, the instantaneous rate of return on business value. Using Ito's lemma, expected return is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [[pi].sub.G](Y) and [[pi].sub.NG](Y) are the market to capital ratios, [V(X,B)/B = [pi](Y)], in the growth and no-growth states, respectively, as given in Equation (5).

[FIGURE 1 OMITTED]

Because a call option on an asset has greater risk than the asset itself and the average cost of capital represents a risk average of the existing business and future expansion options, the average cost of capital exceeds the growth unlevered cost of capital, [omega](Y) [greater than or equal to] [r.sup.*]. In Appendix E, we confirm this result and the assertions that we make in the following paragraph. Figure 1 depicts these results for a numerical example.

Unlike Modigliani and Miller (1958), expected return, [omega](Y), changes with profitability, Y, in a dynamic environment. This dependence represents the changing prospects of incurring growth investments and leverage. Of course, Modigliani and Miller (1958) do not consider a dynamic economic environment, and therefore, they do not identify these altering prospects. As the return on capital, Y, increases from zero, growth leverage increases initially and then decreases. Leverage risk increases, as ROC increases from zero, where the business is in the no-growth state due to the increasing likelihood that ROC will exceed the expansion boundary, [[xi].sup.*], and the manager invests for growth-incurring growth expenditures. As the return on capital, Y, approaches its lower bound of zero, the likelihood of an increase back to the expansion boundary, [[xi].sup.*], is remote. With no likelihood of incurring capital expenditures for growth, the risk cost of growth leverage disappears and expected return, [omega](Y), reflects only the risk cost of in place operations. As ROC falls to zero from the right, expected return approaches the growth unlevered cost of capital, [lim.sub.Y[right arrow]0+] [omega](Y) = r + [theta][[sigma].sub.x,c]. Expected return, [omega](Y), increases as ROC increases through the expansion boundary, [[xi].sup.*], before it reaches its maximum. Further, [omega](Y) has an inflection point at the expansion boundary, Y = [[xi].sup.*]. The change in expected return, [d[omega](Y)/dY], is greatest at this point. Both of these observations indicate that when ROC is at the expansion boundary, Y = [[xi].sup.*], the fraction of future time that the business will be in the growth state is modest. ROC must be greater than the expansion boundary, Y > [[xi].sup.*], before the likelihood of remaining in the growth state is significant. Only then, does expected return reach its maximum (at approximately Y = 20% in Figure 1). At this return, above the expansion boundary, the likelihood of regularly incurring expansion expenditures is great enough that the risk cost of leverage is at a maximum. The ROC that maximizes expected return, [omega](Y), is greater than the expansion boundary, [[xi].sup.*]. Expected return, [omega](Y), eventually decreases as profitability increases and the business is better able to cover increasing capital expenditures required for continued profit growth. As ROC increases without limit, [lim.sub.Y[right arrow][infinity]] [omega](Y) = r + [theta][[sigma].sub.x,c].

E. Comparing the Expansion Boundary and the Cost of Capital

In Appendix E, we demonstrate that the growth unlevered cost of capital, [r.sup.*] = r + [theta][[sigma].sub.x,c], exceeds the value maximizing ROC benchmark for growth, [[xi].sup.*]. Consequently, expected return (the average cost of capital), [omega](Y), exceeds the growth boundary (the marginal cost of capital), [omega](Y) [greater than or equal to] [[xi].sup.*]. This result establishes that the value maximizing expansion boundary is uniformly less than the expected return, the average cost of capital. Consequently, practical application of the average cost of capital for business expansion analysis is overly conservative. A value maximizing manager undertakes an expansion today that has a marginal return less than the overall business expected return because this investment enhances the capacity to capture greater future value with large investments when business earnings improve. Marginal investments today create the facilities and competence that a small firm needs to capture greater value with large and more profitable future investments.

In Figure 1, the average cost of capital exceeds the expansion boundary by approximately 20 basis points when ROC = 0% and by over 100 basis points when ROC [approximately equal to] 22%. When the manager expands the business (ROC [greater than or equal to] 11.8 %) but ROC is less that the right most boundary in Figure 1 (ROC = 70%), the average cost of capital exceeds the expansion boundary by at least 60 basis points. These differences illustrate that those managers using the average cost of capital as a benchmark for growth analysis risk grave decision errors.

Other parameterizations of our model lead to more dramatic differences between the maximum expected return and the dynamic expansion boundary. Panels A and B of Figure 2 illustrate that profit volatility, [sigma], and maximum growth, g, increase this difference. For instance, if operating profit volatility is [sigma] = 0.4 and maximum growth is g = 0.049, with other parameters the same as those in Figure 1, then the difference between the maximum expected return and the dynamic expansion boundary is about 270 basis points. However, rather than concentrate on the magnitude of this difference, we believe that the essential lesson proffered from our research is that the dynamic expansion boundary is uniformly less than the average cost of capital. A manager who uses the average cost of capital as an investment threshold under invests and is unable to fully take advantage of valuable market opportunities when profitability increases.

Two forces in our model work together to accelerate investment: 1) current investment neither impairs nor displaces future returns, and 2) current investment increases the size and dollar value of future investments. We leave to future research an investigation of which of these forces is more significant. However, together these two forces offset the investment delaying force of growth leverage that exists whenever the manager exercises the growth option and expands business. We can measure this delaying force with the return differential between the average cost of capital and the growth unlevered cost of capital, [omega](Y) - [r.sup.*]. In Figure 1, when the average cost of capital is at its maximum, [omega](Y) - [r.sup.*] | [sub.y=0.22] [congruent to] 12.9 - 12 = 90 basis points. We measure the accelerating force of indefinite sequential proportional growth options with the return differential between the average cost of capital and the marginal cost of capital, [omega](Y) - [[xi].sup.*]. In Figure 1, when the average cost of capital is at its maximum, [omega](Y) - [[xi].sup.*][|.sub.y=0.22] [approximately equal to] 12.9 - 11.8 = 110 basis points. Finally, we measure the net accelerating force of indefinite proportional growth options as the return differential between the growth unlevered cost of capital and the marginal cost of capital, [r.sup.*] - [[xi].sup.*] = 12 - 11.8 = 20 basis points. The fact that this difference is not substantial suggests that one can approximate the marginal cost of capital with the growth unlevered cost of capital. A business person might calculate the growth unlevered cost of capital in a number of ways: 1) directly by estimating the risk premium [theta][[sigma].sub.x,c], 2) by unlevering the average cost of capital, or 3) in an appropriate asset pricing model by zeroing the estimated growth risk premium. We leave to future applied research how best to estimate/approximate the marginal cost of capital.

[FIGURE 2 OMITTED]

Proportional growth must be combined with indefinite rather than finite growth opportunities to accelerate investment. Current investment does not increase the size of future growth investments because the assumed facts predetermine growth investment size. Only if proportional growth investments are indefinite with constant returns to scale does current investment neither impair nor displace future investment and also increases the size of future investments at any future date for a fixed ROC growth threshold and for any ROC path between now and that future date.

In the following four subsections and in the panels of Figure 2, we present a comparative static numerical analysis of the value maximizing expansion boundary, [[xi].sup.*], the growth unlevered cost of capital, [r.sup.*] = r [theta][[sigma].sub.x,c], and the maximum expected return, [[omega].sup.*] [equivalent to] [max.sub.0<y<[infinity]][omega](Y).

F. Cost of Capital and the Expansion Boundary versus Volatility

The expansion boundary, [[xi].sup.*], decreases with volatility, [sigma] (see Panel A of Figure 2). The proportionality of growth option value with firm size enhances the appeal of an investment's upside earnings potential to a manager. This appeal reduces the dynamic expansion boundary, [[xi].sup.*], below the growth unlevered cost of capital, [r.sup.*]. This appeal is greater for greater earnings volatility, [sigma], and therefore, the dynamic expansion boundary, [[xi].sup.*], decreases with volatility. Note, in the left-most section of Panel A of Figure 2, that when earnings volatility, [sigma], approaches zero, the dynamic expansion boundary approaches the growth unlevered cost of capital, [r.sup.*].

The comparative static result above is opposite of the investment deferral option for a solitary investment in Dixit and Pindyck (1994). Downside earnings risk for a one-time irreversible investment is an essential feature of Dixit and Pindyck's study of investment under uncertainty and the investment deferral option. With greater earnings volatility, this downside risk is greater and the profit boundary to start a solitary investment increases because the investment becomes less attractive to the manager.

When profit volatility approaches zero, [sigma] [right arrow] 0, the maximum expected return, [[omega].sup.*], exceeds the expansion boundary, [[xi].sup.*], by almost 90 basis points. If profit volatility is greater, [sigma] [right arrow] 0.40, then the maximum expected return, [[omega].sup.*], exceeds the expansion boundary, [[xi].sup.*], by about 150 basis points. This divergence between the maximum expected return, [[omega].sup.*], and the expansion boundary, [[xi].sup.*], illustrates that the maximum expected return, [[omega].sup.*] (and by extension, expected return, [omega](Y), itself) is insensitive to volatility as compared to the growth boundary, [[xi].sup.*]. The maximum expected return, [[omega].sup.*], decreases, though modestly, with volatility, [sigma], because the expansion option's value increases with volatility, which in turn increases business value, decreasing expected return.

G. Cost of Capital and the Expansion Boundary versus Maximum Growth

In Panel B of Figure 2, maximum expected return, [[omega].sup.*], increases with maximal growth, g, due to increasing growth leverage. Alternatively, the expansion boundary, [[xi].sup.*], decreases modestly with maximal growth, g. Greater growth, g, enhances the appeal of upside earnings potential to managers, who, therefore, reduce the expansion boundary, [[xi].sup.*]. However, the optimal growth boundary, [[xi].sup.*], is relatively insensitive to the rate of growth, g, as the benefit of expansion, profit growth, g, matches rather closely the burden of expansion, capital growth, g. Maximal growth, g, has a more pronounced impact on expected return, [[omega].sup.*], than it does on the expansion boundary, [[xi].sup.*]. Maximum growth, g, is a primary determinant of expected return, but not the value maximizing expansion boundary. Furthermore, the difference between the growth unlevered cost of capital and the maximum expected return ([r.sup.*] and [[omega].sup.*]) increases substantially as the maximum growth rate increases.

H. Cost of Capital and the Expansion Boundary versus Covariance Risk

In Panel C of Figure 2, covariance, [theta][[sigma].sub.x.c], increases both the expected return, [[omega].sup.*], and the expansion boundary, [[xi].sup.*]. Covariance risk makes operating profit less attractive to managers, who, therefore, increase the expansion boundary, [[xi].sup.*].

Note that when covariance risk is zero, [theta][[sigma].sub.x,c] = 0, both the maximum expected return and the growth unlevered cost of capital equal the riskless rate, [[omega].sup.*] = r and [r.sup.*] = r. However, the expansion boundary, [[xi].sup.*], is less than the riskless interest rate ([[xi].sup.*] < r, when [[theta][[sigma].sub.x,c] = 0). The difference between the riskless interest rate and the expansion boundary, r - [[xi].sup.*], is about 150 basis points for zero covariance risk, [theta][[sigma].sub.x,c] = 0 in Panel C of Figure 2.

Measure the impact of growing growth options on managerial incentive to expand with the difference between the maximum expected return and the expansion boundary, [[omega].sup.*] - [[xi].sup.*]. Panel C of Figure 2 illustrates that this incentive is invariant to covariance risk, [theta][[sigma].sub.x,c]. Note that even though the expansion boundary approaches the unlevered cost of capital as covariance risk becomes large, [[xi].sup.*] [right arrow] [r.sup.*], when [theta][[sigma].sub.x,c] [right arrow] 0.1 in Panel C of Figure 2, the difference between the maximum average cost of capital and the expansion boundary, [[omega].sup.*] - [[xi].sup.*], is always about 150 basis points. Greater covariance increases the discounting of future growth investments reducing the incentive to make expansion investments today. However, at the same time, increasing covariance increases expected return. Thus, covariance risk has little impact on the difference between the maximum expected return and the expansion boundary, [[omega].sup.*] - [[xi].sup.*].

I. The Cost of Capital and the Expansion Boundary versus Riskless Interest Rates

In Panel D of Figure 2, as one would expect, interest rates increase the maximum expected return, [[omega].sup.*]. In addition, as greater interest discourages investment, greater interest increases the expansion boundary, [[xi].sup.*]. The riskless rate of interest, r, has little influence on the difference between the expansion boundary and the maximum expected return, [[omega].sup.*] - [[xi].sup.*]. This difference is approximately 120 basis points in Panel D of Figure 2.

J. Primary Determinants of the Cost of Capital and the Expansion Boundary

The panels of Figure 2 illustrate that the parameters of business value that determine the growth unlevered cost of capital, r + [theta][[sigma].sub.x,c], influence the expected return and the expansion boundary in a like manner. In Panels C and D, the maximum average cost of capital, [[omega].sup.*], and the expansion boundary, [[xi].sup.*], both increase parallel with the riskless rate, r, and covariance risk, [theta][sigma].sub.x.c.

In Panel B of Figure 2, growth, g, has a modest negative effect on the growth boundary, [[xi].sup.*], whereas it has a positive impact on expected return, [[omega].sup.*]. Growth, g, has little impact on the expansion boundary, [[xi].sup.*], since the benefit of expansion (profit growth) closely matches the cost of expansion (capital investment). Alternatively, a manager's discretionary expansion expenditures impose a leverage risk on financial asset holders. Expected return reflects this risk and the expected return, [[omega].sup.*], increases with maximal growth, g.

In Panel A of Figure 2, volatility, [sigma], has a modest negative effect on the expected return through the expansion option's value. Volatility, [sigma], has a negative impact on the expansion boundary, [[xi].sup.*]. Volatility has little impact on expected return as investors anticipate the influence of volatility on the manager's expansion decision. As in many asset pricing models, volatility is largely unpriced in equilibrium because portfolio diversification diminishes its influence. The proportionality of growth option value with firm size increases the appeal of an investment's upside earnings potential to a manager as compared to a sequence of same-sized growth investments. Greater earnings volatility further enhances this appeal, which induces the manager to reduce the expansion boundary, [[xi].sup.*].

These observations indicate that the primary determinants of expected return are: 1) riskless interest rates, 2) covariance risk, and 3) growth leverage. The primary determinants of the optimal expansion boundary are: 1) riskless interest rates, 2) covariance risk, and 3) profit volatility. The primary determinants of the dynamic expansion boundary and expected return differ. These differences are important for framing empirical tests of our dynamic business expansion model.

K. Implications for Capital Budgeting

A common view in the managerial practice of corporate finance, which most corporate finance text books represent, is that an investment that expands existing operating profit, but otherwise is of the same character as in place operations, is of the same risk as in place operations. This commonly held view is erroneous. Business persons can use WACC to measure the average cost of capital that they can then use as the marginal cost of capital for expansion investments. However, as we demonstrate in this paper, in a dynamic setting, the average and the marginal costs of capital diverge. Therefore, business persons risk decision errors when using the average cost of capital for expansion benchmarking. This theoretical result is consistent with the empirical findings of Bernardo, Chowdhry, and Goyal (2007).

Our results indicate that even if expansion investments are identically scaled copies of existing operations, they are of greater risk. Expected return for the existing business, a hypothetically nongrowing business, is the growth unlevered cost of capital, r + [theta][[sigma].sub.x,c]. Conversely, because expected return, [omega](Y), which reflects the risk of the existing business plus growth opportunities, exceeds the growth unlevered cost of capital, [omega](Y) [greater than or equal to] r + [theta][[sigma].subx,c], growth investments are riskier than the existing business. Since growth is not spontaneous, the manager must invest to create profit growth. This investment creates growth leverage risk that does not exist for in place operations. Expected return for a firm with expansion options exceeds that of a no-growth firm.

Even with optional expansion, financial asset holders are not immune to growth leverage. The numeric example of Figure 1 reflects a rather modest fall in expected return, [omega](Y), with increasing profitability, Y, for large profitability. As ROC increases from approximately 20% (above the expansion boundary and near the maximum expected return) to 70% (well above the expansion boundary), the risk premium falls only (approximately) from 12.8% - 5% = 7.8% to 12.6% 5% = 7.6%. As profit growth requires capital growth (increasing fixed costs), leverage "risk cost" is insensitive to profitability in this range. ROC must be very great to cover increasing fixed costs and eliminate growth risk costs. Possibly this decrease in expected return is modest because while 70% seems a high return, it is only approximately three standard deviations above the value maximizing expansion boundary in Figure 1. Further, since the logarithm of ROC follows an arithmetic Brownian motion, despite the fact that ROC is very high (70%), ROC falls below the expansion boundary at some time in the future with probability one (Cox and Miller, 1965).

III. Conclusion

Our main result is that the cost of capital as it is typically calculated with the weighted average cost of capital (the expected return on business value) is inadequate for expansion decisions when a firm has indefinite future investments proportional to capital. McDonald and Siegel's (1986) and Dixit and Pindyck's (1994) option to defer investment due to irreversibility is dominated by another option; the option to invest currently to increase the value of future growth investments when profits stochastically improve. This suggests that managers should grow the business even at levels below the current cost of capital. Such growth is justified as it allows the firm to fully take advantage of favorable investment opportunities when profitability improves.

Appendix A

Business Value

With a constant riskless interest rate, r, the business value function, V(X,B), satisfies the differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Use the branches for dB/B from Equation (1), apply Ito's lemma to dV, and with the risk-adjusted process for operating profit in Equation (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We conjecture (and verify) that the value function V(X,B) is of the form

V(X, B) = B[pi](Y), (A2)

where [pi] (Y) is a function of Y = ROC. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A3)

Substitute Equation (A3) into Equation (A1) and after dividing both sides by B

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A4)

For, 0 [less than or equal to] g < r, the solutions to these ordinary differential equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A6)

Determine the parameters [C.sub.1] and [C.sub.2] with value matching and smooth pasting conditions at Y = [xi] (see Dixit and Pindyck, 1994, for a discussion of these conditions). Solve these two equations (not given) to determine the value to expenditure ratio, [pi], in Equation (5).

Our problem has three unknowns: 1) [C.sub.1], which establishes the value of the option to start growth; 2) [C.sub.2], which establishes the value of the option to stop growth; and 3) [[xi].sup.*], the value maximizing expansion return boundary. Value matching and smooth pasting determine only two of these three unknowns. Consequently, smooth pasting and value matching do not ensure value maximization. In using smooth pasting and value matching to determine [C.sub.1] and [C.sub.2], we ensure that the value function has no kinks, and, therefore, no arbitrage. Shackleton and Sodal (2005) demonstrate that smooth pasting is equivalent to return equalization between the investment option and its levered payoff, which prevents arbitrage at the investment threshold. The optimization that we describe in Section II.B is the third condition required to determine the third unknown, the value-maximizing expansion boundary, [[xi].sup.*].

Appendix B

In this appendix, we determine the value of a no-growth firm that awaits permanent growth (at the rate g) and a growing firm that has the option to permanently stop growth. In each case, we determine the critical return threshold that separates growth from no growth.

For both problems, since the decision to grow or stop growth is permanent, each problem has two unknowns: 1) a constant that establishes the value of the option to either start or stop growth and 2) the return boundary for either starting or stopping growth. Because there are two unknowns, value matching and smooth pasting not only determine both unknowns but also ensure value maximization.

Using the same methodology as in Appendix A, the value of the no-growth business awaiting permanent growth is given on the upper branch of the equation below, whereas the value of the permanently growing business is given on the lower branch.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Parameter C establishes the value of the option to permanently grow. Use smooth pasting and value matching to find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the value maximizing return threshold for permanent growth, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, the value of the growing business waiting to permanently stop growth is given on the lower branch of the equation below. The value of the permanently nongrowing business is given on the upper branch. Parameter C establishes the value of the option to permanently stop growth.

Use smooth pasting and value matching to find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the value maximizing return threshold for permanent growth, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Appendix C

In this appendix, we use the definitions of [alpha] and [lambda] in Equation (A6) to verify that a sufficient condition for the final term in Equation (6), [alpha][lambda]/(1-[alpha])(l-[lambda]), to be less than one, is positive covariance risk, [theta][[sigma].sub.x,c][greater than or equal to] 0. For g = 0, [alpha][lambda]/(l-[alpha])(l- [lambda])[less than or equal to] 1, with equality achieved when the covariance risk is zero.

Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So, [alpha][lambda]/(l-[alpha])(l-[lambda]) is one when there is no covariance risk and growth, g, is zero, and strictly less than one for positive covariance risk and any maximum growth rate, g.

Appendix D

In this appendix, we prove V(X,B)/B|Y=[[xi].sup.*] = [pi]([[xi].sup.*]) = 1. Substitute the expression for [[xi].sup.*] on the right-hand side of Equation (6) into either branch of Equation (5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This expression equals one if

r([lambda] - [alpha]) + r([[alpha].sup.2] - [[lambda].sup.2]) + [alpha]g(1 - [alpha]) + [[alpha].sup.2][lambda][theta][[sigma].sub.x,c] - [alpha][[lambda].sup.2][theta][[sigma].sub.x,c] = O.

Substitute the definitions for [alpha] and [lambda] to verify this equation.

Appendix E

This appendix illustrates that the optimal expansion boundary, [[xi].sup.*], is always below expected return, [omega](Y). We presume positive covariance risk, [[sigma].sub.xc] [greater than or equal to] O. Simplify Equation (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (E1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We demonstrate the following steps:

1. The growth unlevered cost of capital is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

a. Expected return equals the growth unlevered cost of capital for Y = 0, [omega](0) = [r.sup.*].

b. Expected return approaches [r.sup.*] as Y increases without bound [lim.sub.Y[right arrow][infinity]][omega](Y) = [r.sup.*].

c. On the no-growth branch, [omega](Y) [greater than or equal to] [r.sup.*], for 0 [less than or equal to] Y [less than or equal to] [xi] because expected return, Equation (E1), is an increasing function of Y.

d. On the growth branch of Equation (El), [omega](Y) [greater than or equal to] [r.sup.*] because [omega](Y) has only one extreme point, which is a maximum for Y [greater than or equal to] [xi].

2. The growth unlevered cost of capital exceeds the expansion boundary [[xi].sup.*] [less than or equal to] [r.sup.*].

Therefore, expected return exceeds the expansion boundary, [omega](Y) [greater than or equal to][[xi].sup.*] for Y [greater than or equal to] 0. la. Expected return equals the unlevered cost of capital at Y = 0. On the no-growth branch, expected return is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (E2)

At Y = O, expected return simplifies to [omega](O) = [r.sup.*]. lb. Expected return approaches [r.sup.*] as Y increases without bound, [lim.sub.Y[right arrow][infinity]][omega](Y) = [r.sup.*]. On the growth branch, expected return is

[omega](Y) = Y-g + 1/2 [[partial derivative].sup.2][[pi].sub.G]/[partial derivative][Y.sup.2][[sigma].sup.2][Y.sup.2] + g[[pi].sub.G]/[[pi].sub.G]. (E3)

The denominator of (E3) tends to infinity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E4)

Therefore, the limit of expected return is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E5)

The first term of (E5) tends to [r.sup.*] - g, the second term tends to zero, and the third term is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E6)

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E7)

1c. On the no-growth branch, expected return, Equation (E1), increases with Y. Differentiate the no-growth branch of Equation (E1) with respect to Y

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E8)

This derivative is positive because

[[sigma].sup.2]([alpha]-1)/2(1/[r.sup.*] + [gamma]/[alpha][Y.sup.[alpha]-1])[greater than or equal to] 1/[alpha](1 + 1/2 [gamma]([alpha]-l)[[sigma].sup.2][Y.sup.[alpha]-l]), (E9)

or

[alpha]([alpha]-1)[greater than or equal to] [2r.sup.*]/[[sigma].sup.2]. (E10)

Substitute for [alpha] and simplify to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (E11)

1d. On the growth branch, expected return, Equation (E1), has only one extreme point, which is a maximum for Y > [xi].

Differentiate the growth branch of Equation (E1) with respect to Y

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E12)

The numerator of Equation (E12) has at most two roots for Y > 0. Note that the smooth pasting condition implies [partial derivative][omega]/[partial derivative]Y|Y=[xi] > 0 on either branch of Equation (E 1). It is easily verified that the first derivative of the growth branch, Equation (E12), is negative for small ROC, [lim.sub.Y[right arrow]0]([partial derivative][omega]/[partial derivative]Y) < 0. (6) Therefore, Equation (E12) has one root for Y [less than or equal to] [xi]. Hence, Equation (E12) has, at most, one root for Y > [xi]. In other words, expected return, Equation (E1), has, at most, one extreme value for Y > [xi]. Combined with parts 1a, 1b, and 1c of this proof, this result indicates that [omega](Y) > [r.sup.*].

To review, because expected return, [omega](Y), has, at most, one extreme point for Y > [xi], and [omega](Y) is an increasing function at Y = [xi] above [r.sup.*] at Y = [xi] and approaches [r.sup.*] as Y tends to infinity, the extreme point is a maximum. Expected return, [omega](Y), therefore, exceeds [r.sup.*] for any Y [greater than or equal to] [xi].

2. The expansion boundary never exceeds the growth unlevered cost of capital, [[xi].sup.*] [less than or equal to] [r.sup.*]. A rearrangement of Equation (6) reveals that [[xi].sup.*] [less than or equal to] [r.sup.*] when

S [equivalent to] [theta][[sigma].sub.x,c][alpha][lambda] - (r - g)(1 - [alpha] - [lambda]) [greater than or equal to] 0. (E13)

Consider S as a function of the parameter g. We demonstrate that S(g) [greater than or equal to] 0 for 0 [less than or equal to] g [less than or equal to] r. Substitute the definitions for [alpha] and [lambda] and verify that S(g)[|.sub.g=o] = 0 and S(g)[|.sub.g=r] = 0. The first-order condition for the maximum value of the function S with respect to g is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E14)

Denote the right-hand side of Equation (E 14) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E15)

The second derivative of h(g) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (E16)

which is clearly positive for 0 [less than or equal to] g [less than or equal to] r. Therefore, the function h(g) is convex for 0 [less than or equal to] g [less than or equal to] r. Further, it is easy to verify that h(g)]g=0 > 0 and h(g)]g=r < r. Consequently, Equation (E14) has exactly one solution for 0 [less than or equal to] g [less than or equal to] r, which is the only extreme point of the function S(g). The last step of the proof requires that we establish that S is positive rather than negative at this extreme point.

To verify that this extreme point is a maximum not a minimum, take the derivative of S with respect to g and evaluate it at g = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (E17)

Simplification illustrates that as [partial derivative]S/[partial derivative]g[|.sub.g=0] [greater than or equal to] 0 if r(r + [theta][[sigma].sub.x,c]) [greater than or equal to] 0, which clearly holds.

Because the function S(g) is increasing at g = 0 and has only one extreme value for 0 [less than or equal to] g [less than or equal to] r, this extreme value must be a maximum. Therefore, S(g) is nonnegative, S(g) [greater than or equal to] 0 for 0 [less than or equal to] g [less than or equal to] r.

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We thank an anonymous reviewer and the Editor who were instrumental in forcing clarity of the exposition and economic intuition. A number of colleagues assisted with valuable comments including Robert McDonald, Robert Pindyck, Rob Grauer, Gordon Sick, Daniel Smith, Chris Robinson, Ross Valkanov, and Vijay Jog. We presented a version of this paper entitled "Corporate Performance and Dynamic Business Expansion" at the 2004 Northern Finance Association Conference in St. Johns, Newfoundland. Of course, the authors retain responsibility for errors.

Andrey D. Pavlov, George W. Blazenko is a Professor of Finance at Simon Fraser University in Burnaby, British Columbia.

Andrey D. Pavlov is a visiting Associate Professor of Real Estate at the University of Pennsylvania, Philadelphia, PA and an Associate Professor of Finance at Simon Fraser University in Burnaby, British Columbia, Canada.

(1) See, with the related literature review, Cottrell and Sick (2002). A manager undertakes strategic investments that preempt competitive entry earlier than absent this competition. However, no one has shown that this investment is sooner in a real options setting as compared to standard cost of capital analysis in the same strategic environment. Cottrell and Sick compare the attractions of follower to first mover strategies.

(2) We could add development risk, randomness to capital expenditures, without changing the fundamental results in this paper. Further, we could add development risk in two ways. First, we could add development risk as a common random increment to the upper and lower branches of Equation (1) correlated with both operating profit, X, and aggregate consumption (see the upcoming section). This modeling might represent, for example, construction costs that vary randomly throughout the economy regardless of whether or not a particular firm grows its business. Second, if a firm bears development risk only when it grows (e.g., R&D risk), we could add a random increment only to the growth branch of Equation (1).

(3) The essence of the constant returns to scale assumption in our model is that investment returns between small and large firms do not differ simply because of size measured by capital, B. In practice, smaller firms are more likely to be development oriented with the potential for exceptionally great returns, whereas larger firms are more likely to be survivors with great ex post return on capital. Otherwise, we expect that there is no difference between return on capital for small versus large firms. Of course, this is an empirical question.

(4) We thank the reviewer for a clearer exposition of our model presented here as compared to earlier drafts and the insight that indefinite proportional expansion opportunities are vital to the results.

(5) The static environment is helpful to illustrate the point. If permanent profit growth at the rate g requires capital growth at the rate g, then IRR satisfies (X - [g.sup.*]B)/(IRR - g) - B = 0, and IRR = ROC regardless of the growth factor, g. For comparison purposes, for spontaneous profit growth, which is not the nature of the investment we study, the IRR satisfies X/(IRR - g) - B = 0, and IRR = ROC + g.

(6) The third term of the numerator determines the sign of the derivative as Y approaches zero from the right. Substituting [lambda] into the third term confirms that this term is negative.

**********

We investigate the managerial trade-off between the investment delaying force of irreversibility and the accelerating force of follow-on options when managerial choice endogenously determines corporate growth. If the firm has an infinite sequence of growth opportunities with constant returns to scale, current investment neither displaces nor impairs future investment returns. These expansion features weaken the managerial incentive to defer irreversible investment.

Furthermore, when expansion is restricted to a fraction of firm size, we demonstrate that the endogenously determined cost of capital, expected return on business value, uniformly exceeds the value maximizing expansion return threshold. In other words, a manager accepts marginal investments that return less than the cost of capital to facilitate larger and more valuable future investments when earnings stochastically improve. In this instance, the force of increasing the follow-on options dominates the force of irreversibility for expansion timing. We confirm that the managerial application of the cost of capital as an expansion hurdle rate is improperly conservative. A value maximizing manager expands sooner than a manager who uses the cost of capital as an expansion hurdle rate.

There are four features of our business expansion model that, when combined, lead to new results. First, we relate the process for operating profit with the process for capital investment so that growth in both is endogenous. Rather than exogenous profit growth driving capital growth, value maximizing managerial decisions jointly determine profit and capital growth. Profit that grows only with capital growth removes a managerial attraction to waiting. Second, we presume indefinite expansion opportunities with constant returns to scale. Current investment neither displaces nor impairs future investment returns, which weakens the managerial incentive for investment delay. Third, the manager has a dynamic option to grow that includes the ability to suspend growth upon inadequate profitability. Current investment does not impair future investment returns because the manager can suspend growth upon inadequate profitability. This ability to suspend unprofitable growth reduces the incentive to delay current investment. Fourth, we apply an expansion proportionality constraint so that small firms make small investments and large firms make large investments. If a manager expands business prematurely when profit is possibly inferior, current investment increases the size and dollar value of future expansion upon stochastically improved profitability. Downside earnings risk is no greater as the manager can defer growth once started.

It is important to compare value maximizing models with the cost of capital analysis since the cost of capital is ingrained in the corporate practice of business expansion evaluation. As managers recognize that the cost of capital is not an appropriate benchmark for business investment decisions in dynamic settings in the presence of various real options, they will look to the financial literature for guidance. The most likely resource is Dixit and Pindyck (1994). Managers will discover that the predominant modeling feature is the investment deferral option. Managers exercise their "call option" on the value of an underlying investment when downside value protection no longer exceeds foregone profit from investment delay. This delay indicates that the investment return boundary exceeds the cost of capital. Because the cost of capital is relatively easy to measure, when following the existing literature, managers will add a positive factor to the cost of capital to reflect the deferral option when calculating the expansion boundary for their businesses. We illustrate that this conjectured managerial heuristic for business expansion is inappropriately conservative.

McDonald and Siegel (1986) and Dixit and Pindyck (1994) illustrate that the possibility of disappointing earnings leads a manager to exercise the "start option" on a solitary irreversible investment only when the investment is "in the money" and net present value (NPV) exceeds a strictly positive threshold. Because current investment precludes future investment, current investment creates an opportunity loss that encourages delay. The return boundary for starting the investment exceeds the cost of capital and the manager defers a timeable investment. There are a number of economic factors that influence a manager's decision to defer an irreversible investment: 1) partial reversibility (Kandel and Pearson, 2002), 2) investment learning (Grenadier and Weiss, 1997; Bergemann and Hege, 1998), 3) strategic investment (Cottrell and Sick, 2002), 4) capital stock adjustment costs (Abel and Eberly, 1994; Carlson, Fisher, and Giammarino, 2004), and 5) agency costs (Bergemann and Hege, 1998; Christensen, Feltham, and Wu, 2002). (1) These factors modify the extent of managerial delay but do not induce investment earlier than recommended by Modigliani and Miller's (1958, 1963) conventional cost of capital analysis. Our model justifies such early investment.

In a piece described by Brealey, Myers, and Allen (2006) that predates the modern real options literature, Roberts and Weitzman (1981) and Weitzman, Whitney, and Rabin (1981) indicate that a manager might accept a negative NPV "now-or-never" investment if it has a follow-on investment with positive NPV contingent on initial investment success. Of course, if the manager could defer the investment, then the delaying force of irreversibility might dominate the accelerating force of follow-on options.

Irreversibility delays, while follow-on options accelerate investment timing. In some circumstances investment can be induced at rates below the conventional cost of capital. By exaggerating the size or profitability of follow-on options, we could manufacture stylized problems where the value attraction of follow-on options dominates the downside earnings risk of irreversible investment so that the manager does not delay but alternatively accepts a negative NPV initial investment. Therefore, in the current paper, we impose some economic realism on the timing decision by studying what is arguably the most important investment for practicing managers: expansion that has the same profitability as an existing business.

McDonald and Siegel (1986) and Dixit and Pindyck (1994) demonstrate that while volatility increases both downside risk and upside earning potential, because downside risk avoidance dominates managerial decision making for one-time investments, volatility increases profit thresholds for investment. There is a literature suggesting that dynamic models of sequential growth can have distinct characteristics from solitary one-time investments. In Dixit and Pindyck (1994), Abel (1983), Pindyck (1988), Caballero (1991), and Aguerrevere (2003), volatility can encourage sequential investment despite irreversibility. Optimal investment thresholds can decrease rather than increase with profit volatility. The reason for this difference is that upside earnings potential is more important to managers for a sequence of investments when compared to a solitary investment. With upside earnings surprises, managers make supplementary investments. Even if earnings disappoint, eventually, once earnings improve, managers also make supplementary investments. Volatility increases upside earnings potential that managers pursue with lower investment profit thresholds.

Despite the possibility of distinct characteristics, managers also delay investment in the above sequential growth literature. In Pindyck (1988), current investment impairs future investment returns due to diminishing productivity that encourages investment delay beyond that for a solitary investment. On the other hand, since current investment does not displace future investment, the imperative for investment delay is less for an indefinite sequence of future investment options. These two incremental timing effects exactly offset one another. A manager who faces a sequence of investments delays current investment exactly like it was a solitary investment with no further expansion opportunities. Abel and Eberly (1994) investigate a capital adjustment cost model where small investments have modest costs, but large investments are increasing costly. Their model still results in investment delay despite the indefinite growth options it contains. The incentive to delay in this model is driven by the managerial desire to avoid costly capital adjustments should profitability reverse in either direction. Last, in Dixit and Pindyck (1994), Abel (1983), Pindyck (1988), Caballero (1991), and Aguerrevere (2003), capital investment over time is a managerial reaction to an exogenously specified profit process. Managers await improvement in the profit process over which they have no control. Spontaneous profit improvement without immediate investment costs encourages delay.

The paper is organized as follows. Section I models business expansion. Section II derives the value-maximizing expansion boundary and compares it to the cost of capital. Section III formulates conclusions in a summary.

I. Dynamic Business Expansion

A. Expansion

Because growth is neither spontaneous nor unplanned, a corporate manager invests to produce profit growth. In free cash flow valuation, asset value depends on predicted future free cash flow, part of which is the required expenditure to generate profit growth. We presume this expenditure depends on existing asset replacement costs, which we represent with, [B.sub.t] > 0, capital stock. Monitoring capital is relevant because of this dependence. A manager controls capital stock by undertaking irreversible investment. The instantaneous investment rate is g [greater than or equal to] 0 if the manager expands the business and zero otherwise (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Equation (1) implicitly assumes that there is no cost for switching between growth and no-growth regimes.

Because capital stock adjustment costs are zero for an investment up to gB dt and infinitely great thereafter, and because production technology has constant returns to scale, there are no benefits or costs, proportionately, to large investments. While the manager might prefer a larger investment, the limit on investment spending is gB dt. Total yearly investment is a random amount between 0 and gB (uncompounded). If the manager neglects expansion for a period of time, this neglect cannot subsequently be reversed with more intense future expansion. Regardless of the past, the limit on investment spending over dt is gB dt.

Kogan (2001, 2004) also limits the rate of investment between zero and a positive constant, which is a special case of the standard convex adjustment cost specification and restricts a manager's instantaneous capital adjustment between zero and g. Our study extends this literature by endogenizing the cost of capital and demonstrating that a value maximizing manager accelerates business expansion when compared to the standard NPV criteria.

Operating profit, a cash-flow-based measure, is generated from a constant returns to scale technology with stochastic productivity parameter [Y.sub.t], the return on capital (ROC), with dynamics [dY.sub.t]/[Y.sub.t] = [sigma] dz, where dz is a standard Gauss-Weiner process. (3) Because operating profit, [X.sub.t], is ROC times capital, [X.sub.t] = [Y.sub.t] [B.sub.t], the process for operating profit is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Because operating profit, X, is strictly positive, the manager neither abandons nor suspends existing operations. The business generates instantaneous operating profit, X dt, whether the manager invests for expansion or not. Strictly positive operating income is characteristic of large industrial and commercial firms with existing operations, but not characteristic of small (often) development-oriented firms. While creating commercially viable products, start-up firms bear not only development risk, but also negative operating income before they create markets and earn revenues. We study the expansion of firms with existing operations rather than new venture investments. (4)

B. Investment Return

The process for ROC, [dY.sub.t][dY.sub.t]/[Y.sub.t] = [sigma] dz, is dependent upon neither the manager's expansion decision nor the proportionality constraint on investment and profit growth, g. ROC is a martingale. ROC for a small firm is expected future ROC, regardless of the growth factor, g, when the small firm, perhaps, becomes large, E[[??].sub.t]] = [Y.sub.0].

Investment return for either the existing business or for an expansion investment is ROC and not ROC plus a growth factor as there is no ROC growth like there is for spontaneous profit growth investments. (5) Any investment on its own, either the existing business or an expansion investment, before ensuing growth investments, which also have the same character, generates a perpetual flow of nongrowing expected profit Y per dollar of capital. Without additional capital, profit does not increase. The IRR satisfies Y/IRR - 1 = 0, therefore, IRR = Y.

In a two-period model, Abel, Dixit, Eberly, and Pindyck (1996) illustrate that if capital's cost is expected to increase, then because future investment is more costly than current investment, the incentive to invest currently increases. The results in our model do not depend on this cost of capital effect. Whether or not capital is costly is dependent upon the profit that capital generates, ROC. We expect no change in capital's cost, E[[??].sub.t]] = [Y.sub.0]. Constrained capital growth neither encourages nor discourages current investment. The playing field is level for our comparison of the dynamic expansion boundary and the cost of capital.

C. Assumptions

Investments are equity financed and we do not investigate financing costs, taxes, or other frictions except to the extent that they are encompassed within the investment constraint, g. We avoid extensive investigation of frictions in order to maintain simplicity, a simplicity that leads to both a closed-form solution for the value maximizing return expansion boundary and analytic results. Despite simplicity, the model is not stylized and represents, rather accurately, the expansion decisions faced by practicing business managers. Adding more detailed frictions requires that return on investment, the value maximizing return expansion boundary, and the cost of capital be calculated numerically. There is considerable real options literature regarding how various frictions might influence business investments. The next section investigates the manager's dynamic value maximizing investment strategy.

II. Cost of Capital and the Value Maximizing Expansion Boundary

A. Market-to-Capital Ratio

Our paper contributes to a recent growing literature on risk measurement in dynamic models of managerial choice (e.g., Berk, Green, and Naik, 1999; Goldstein, Ju, and Leland, 2001; Carlson, Fisher, and Giammarino, 2004; Zhang, 2005; Cooper, 2006). Berk, Green, and Naik (1999) investigate corporate growth options. Because these opportunities are of heterogeneous risk, their model is not suited to expansion analysis. Goldstein, Ju, and Leland (2001) study optimal bankruptcy in a dynamic setting, but they do not consider capital growth. Since real spontaneous earnings growth is unrealistic, we presume, alternatively, that earnings growth depends upon capital growth. Zhang (2005) and Cooper (2006) use real options models with dynamic risk adjustment for idle physical production capacity in economic downturns to explain the relationship between equity returns and the book-to-market ratio. They do not consider the association between the value maximizing profit boundary and the cost of capital. Carlson, Fisher, and Giammarino (2004, 2006) explore the impact of corporate operating leverage and growth opportunities on asset returns. Their model provides a rational explanation for both size and book-to-market effects in the cross-section of asset returns. However, it offers neither closed-form solutions nor analytic results in a general context nor a comparison of optimal investment policy with the cost of capital.

We use the valuation methodology of Goldstein, Ju, and Leland (2001) to find the value of a business, V(X, B), that has investment and profit growth restricted to a fraction, g, of capital, B. Goldstein, Ju, and Leland model profit growth, but not the capital expansion required to produce this growth. In Appendix A, we formally relate profit growth with capital growth through the manager's expansion decision.

The risk-adjusted process, X', for operating profit is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [theta] [greater than or equal to] 0 is the coefficient of constant relative risk aversion for a representative investor; [[sigma].sub.xc] is the covariance of the log of operating profit, X, with the log of aggregate consumption, c = log(C); and aggregate consumption follows a geometric Brownian motion. We presume positive covariance risk, [[sigma].sub.xc] [greater than or equal to] 0.

Let V(X, B) be the value of the business. In Appendix A, we confirm that the form of this value function is

V(X, B) = B [pi](Y), (4)

where ROC = Y = X/B. Since the value to capital ratio, [phi](Y), depends only on Y, the manager expands the business depending on Y and not its separate components (X and B). The manager expands when ROC exceeds a boundary that we denote as, [xi]. In Appendix A, we demonstrate that for this arbitrary expansion boundary, [xi], the value to capital ratio is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [r.sup.*] [equivalent to] r + [theta][[sigma].sub.xc] and we define the constants [alpha] [greater than or equal to] 1, [lambda] [less than or equal to] 0 in Equation (A6).

On the "no-growth" branch of Equation (5), the first term is the value of operating profit if the manager never expands the business. This amount, which does not depend on the parameter g, is the discounted value of expected future profit at the risk-adjusted rate [r.sup.*]. The second term (positive) and third term (negative) are, respectively, the value of expected future incremental profit generated and the cost of expansion investments incurred when, at some time in the future, ROC exceeds the expansion boundary, [xi], and the manager expands the business. The positive value of future incremental profit becomes more positive and the negative value for the cost of future expansion investment becomes more negative with Y because both incremental profit and expansion investment become more likely.

On the "growth" branch of Equation (5), the first term is the value of operating profit if the manager permanently expands the business. Because the manager grows the business, this term, for a growing perpetuity, recognizes the growth parameter, g. The second term (negative) is the value of expected future profit foregone when ROC declines below the expansion boundary and the manager defers investment until ROC once more exceeds the expansion boundary, [xi]. This term becomes less negative with Y as the possibility of foregone profit decreases with greater profitability. The third term (negative) is the cost of expansion expenditures, which recognizes that the manager avoids these costs when they defer investment for ROC below the expansion boundary. This term becomes increasingly negative with Y as the likelihood of continued growth investment increases with profitability.

B. Value-Maximizing Expansion Boundary

To find the value maximizing expansion boundary, [[xi].sup.*], we take the derivative of the value function, [pi](Y) with respect to [xi] on either branch of Equation (5), set the result to zero, evaluate at Y = [xi], and solve for [[xi].sup.*]

[[xi].sup.*] = [r.sup.*] x [[[r.sup.*]-r-g]] x [[alpha]/[alpha] -1]] x [[lambda]/[lambda] -1]]. (6)

The first two terms on the right-hand side of Equation (6) equal the manager's ROC boundary for a hypothetical "now-or-never" permanent expansion decision. The value of the permanent growth firm, [[X.sub.0]/[r.sup.*]-g]] - gB/r-g, exceeds the value of the permanent no-growth firm, [[X.sub.0]/[r.sup.*]], when Y = ROC [greater than or equal to] [r.sup.*] x ([r.sup.*] - g/r-g]).

The first three terms on the right-hand side of Equation (6), [r.sup.*] x [[r.sup.*] - g]/r-g]] x [[alpha]/[alpha]-1]], equal the manager's expansion boundary for a hypothetical firm that has a perpetual option to start permanent growth (see Appendix B). Since [[alpha]/[alpha]-1] [greater than or equal to] 1, the manager defers permanent growth as compared to the now-or-never manager. Investment irreversibility induces delay. A manager who awaits permanent growth does not have a follow-on option because growth, once started, is permanent and, therefore, is effectively a solitary investment (with lump- sum cost [[g.sup.*]B/r-g]). In this setting, results in McDonald and Siegel (1986) and Dixit and Pindyck (1994) apply and it is not surprising that the manager defers growth.

The first two terms times the fourth term on the right-hand side of Equation (6), [r.sup.*] x [[r.sup.*]-g/r-g]] x [[lambda]/[lambda]-1], is the threshold return boundary for a hypothetical growing firm that has a perpetual option to permanently stop growth (see Appendix B). Since [[[lambda]/[lambda]-1]] [less than or equal to] 1, even though there is no cost to accepting permanent no growth, the manager of the growing firm defers no growth as compared to the now-or-never manager. In fact, not only does the manager not incur a cost in accepting no growth, but they also avoid the irreversible investment cost of growth. Permanence of the decision dominates foregone irreversible investment and leads to no-growth deferral.

The product of the final two terms on the right-hand side of Equation (6), [[[alpha]/[alpha]-1]] x [[[lambda]/[lambda]- 1]], represents the dynamic combination of a growing manager's inclination to defer permanent no growth and a no-growth manager's inclination to defer permanent growth. Of course, the juxtaposition of these two options in a dynamic setting makes neither decision permanent, reducing the force of both inclinations. Further, in the dynamic setting, because the no-growth decision for a growing firm is immediately reversible without cost and each subsequent decision is also reversible, decision permanence is not a motivating factor for no-growth delay (or equivalently temporally accelerated investment for a no-growth manager awaiting growth). Rather, a no-growth firm with an indefinite stop/start growth option accelerates growth as compared to the now-or-never manager, that is, [[xi].sup.*] [less than or equal to] [r.sup.*] x [[r.sup.*-g/r-g]], for two reasons.

First, the now-or-never manager who opts for permanent growth accepts significant growth leverage. We establish in Appendix C that a sufficient condition for the now-or-never expansion boundary, [r.sup.*] x ([r.sup.*] - g/r-g]) to exceed the dynamic expansion boundary, [[xi].sup.*], is positive covariance risk, [theta][[sigma].sub.xc] > 0. In this case, the term ([[r.sup.*]-g/r-g]) exceeds one and represents the impact of growth leverage, unavoidable fixed costs of growth investment, on the now-or-never manager's permanent growth decision. Because the dynamic manager avoids the burden of growth leverage at will, the dynamic expansion boundary, [[xi].sup.*], is less than the now-or-never permanent expansion boundary, [r.sup.*] x ([r.sup.*] - g/r-g])). The dynamic manager's ability to avoid growth leverage makes the manager less reluctant to begin growth. The growth-leverage-induced deferral motive remains, however, because suspended growth reduces, but does not eliminate, growth leverage. Growth leverage exists whenever the firm grows.

Second, the principal motive for accelerated investment in our model is follow-on options when indefinite future growth investments are proportional to existing capital. In Appendix C, we demonstrate that for positive maximal growth, g > 0, even if the covariance risk is zero, [theta][sigma].sub.x,c] = 0, so that growth leverage risk is irrelevant, the dynamic expansion boundary, [[xi].sup.*], is strictly less than the now-or-never permanent expansion boundary (which in this instance is the riskless interest rate, r). The dynamic manager temporally accelerates growth as compared to the now-or-never manager as not only does current investment not impair future returns, but it also increases the dollar value of future growth options. Current investment has a size attraction that promotes growth.

With reasonable parameter values, the dynamic expansion boundary, [[xi].sup.*], is significantly less than the now-or-never permanent expansion boundary, [r.sup.*] x ([r.sup.*] - g/r-g]). For example, with parameter values that we use in Figure 1, the now-or-never permanent expansion boundary is 54%, whereas the dynamic expansion boundary is [[xi].sup.*] = 11.82%. The now-or-never permanent expansion boundary is very large. Not only do growth fixed costs impose risk on shareholders, but these costs grow unavoidably over time.

The now-or-never permanent growth manager, the growing manager with an option to permanently stop growth, and the nongrowing manager with an option to permanently grow are all stylized characters not meant to represent real business managers or business situations but are helpful in understanding manager's motives with dynamic stop/start growth options. Our central interest is comparing value maximizing managers that take advantage of stop/start growth options with cost of capital managers and their investment mistakes. We derive the cost of capital and make this comparison in Sections D and E that follow.

C. Net Value Creation

In Appendix D, we demonstrate that the market-to-capital ratio equals one when ROC is at the expansion boundary, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As the value to expenditure ratio, [pi](Y), is increasing in ROC = Y, if Y [greater than or equal to] [[xi].sup.*], the manager expands the business at the maximum rate g and the firm creates positive net value because value exceeds capital V(X, B) [greater than or equal to] B and [pi](Y) [greater than or equal to] 1 when Y [greater than or equal to] [[xi].sup.*]. If the manager invests at any return less than the expansion boundary, [[xi].sup.*], net value creation is negative.

Marginal and average market to capital ratios both equal one at the expansion threshold, [[xi].sup.*]. Therefore, the value maximizing manager follows the traditional investing rule that calls for expansion when the market-to-book ratio exceeds one. A market-to-book ratio in excess of one reflects the value of waiting to invest. In our model, the market-to-book is always one when the firm switches from growth to no growth and vice versa. This result is distinct from McDonald and Siegel (1986) type models where current investment displaces future investment and waiting protects against downside losses. The market-to-book ratio strictly exceeds one at the investment profit threshold with positive net value creation upon starting an irreversible solitary investment.

D. The Cost of Capital and Expected Return

Modigliani and Miller (1958) define the cost of capital in two ways: 1) the expected return on overall business value and 2) the return threshold for net value creation with incremental investment. The first definition is the average cost of capital and the second is the marginal cost of capital. Modigliani and Miller (1958, 1963) favored the second definition due to risk measurement limitations that preceded the capital asset pricing model (CAPM) and other modem asset pricing models. However, with risk measurement advances, the first definition has become the underpinning of the common place cost of capital measurement with the weighted average cost of capital (WACC). In a static environment, the average and marginal costs of capital equal one another, and therefore, the distinction is irrelevant. However, as we demonstrate in this paper, in a dynamic setting, the average and the marginal costs of capital diverge. Therefore, business persons risk decision errors when using the average cost of capital for expansion benchmarking. Expected return on business market value is a growth-levered cost of capital. As we show in this subsection, a primary determinant of this expected return is growth leverage. It is also an average cost of capital as it represents a risk average of the firm's existing business and future expansion options. When the firm has no growth opportunities, g = 0, the cost of capital unlevered by growth investments, the growth unlevered cost of capital, is [r.sup.*] representing the risk of in place operations. The results in Subsection C above illustrate that the value maximizing return threshold, [[xi].sup.*], satisfies Modigliani and Miller's (1958, 1963) second cost of capital definition. Because our economic environment is dynamic, the manager can start and suspend growth at any future time, it is a dynamic cost of capital. It is also the marginal cost of capital since it is the return for a marginal expansion investment made by a value maximizing manager.

We measure the average cost of capital as operating profit, X, less expansion expenditures (if incurred) plus expected capital gain from changes in operating profit, all divided by business value. We denote [omega](Y) as the average cost of capital, the instantaneous rate of return on business value. Using Ito's lemma, expected return is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [[pi].sub.G](Y) and [[pi].sub.NG](Y) are the market to capital ratios, [V(X,B)/B = [pi](Y)], in the growth and no-growth states, respectively, as given in Equation (5).

[FIGURE 1 OMITTED]

Because a call option on an asset has greater risk than the asset itself and the average cost of capital represents a risk average of the existing business and future expansion options, the average cost of capital exceeds the growth unlevered cost of capital, [omega](Y) [greater than or equal to] [r.sup.*]. In Appendix E, we confirm this result and the assertions that we make in the following paragraph. Figure 1 depicts these results for a numerical example.

Unlike Modigliani and Miller (1958), expected return, [omega](Y), changes with profitability, Y, in a dynamic environment. This dependence represents the changing prospects of incurring growth investments and leverage. Of course, Modigliani and Miller (1958) do not consider a dynamic economic environment, and therefore, they do not identify these altering prospects. As the return on capital, Y, increases from zero, growth leverage increases initially and then decreases. Leverage risk increases, as ROC increases from zero, where the business is in the no-growth state due to the increasing likelihood that ROC will exceed the expansion boundary, [[xi].sup.*], and the manager invests for growth-incurring growth expenditures. As the return on capital, Y, approaches its lower bound of zero, the likelihood of an increase back to the expansion boundary, [[xi].sup.*], is remote. With no likelihood of incurring capital expenditures for growth, the risk cost of growth leverage disappears and expected return, [omega](Y), reflects only the risk cost of in place operations. As ROC falls to zero from the right, expected return approaches the growth unlevered cost of capital, [lim.sub.Y[right arrow]0+] [omega](Y) = r + [theta][[sigma].sub.x,c]. Expected return, [omega](Y), increases as ROC increases through the expansion boundary, [[xi].sup.*], before it reaches its maximum. Further, [omega](Y) has an inflection point at the expansion boundary, Y = [[xi].sup.*]. The change in expected return, [d[omega](Y)/dY], is greatest at this point. Both of these observations indicate that when ROC is at the expansion boundary, Y = [[xi].sup.*], the fraction of future time that the business will be in the growth state is modest. ROC must be greater than the expansion boundary, Y > [[xi].sup.*], before the likelihood of remaining in the growth state is significant. Only then, does expected return reach its maximum (at approximately Y = 20% in Figure 1). At this return, above the expansion boundary, the likelihood of regularly incurring expansion expenditures is great enough that the risk cost of leverage is at a maximum. The ROC that maximizes expected return, [omega](Y), is greater than the expansion boundary, [[xi].sup.*]. Expected return, [omega](Y), eventually decreases as profitability increases and the business is better able to cover increasing capital expenditures required for continued profit growth. As ROC increases without limit, [lim.sub.Y[right arrow][infinity]] [omega](Y) = r + [theta][[sigma].sub.x,c].

E. Comparing the Expansion Boundary and the Cost of Capital

In Appendix E, we demonstrate that the growth unlevered cost of capital, [r.sup.*] = r + [theta][[sigma].sub.x,c], exceeds the value maximizing ROC benchmark for growth, [[xi].sup.*]. Consequently, expected return (the average cost of capital), [omega](Y), exceeds the growth boundary (the marginal cost of capital), [omega](Y) [greater than or equal to] [[xi].sup.*]. This result establishes that the value maximizing expansion boundary is uniformly less than the expected return, the average cost of capital. Consequently, practical application of the average cost of capital for business expansion analysis is overly conservative. A value maximizing manager undertakes an expansion today that has a marginal return less than the overall business expected return because this investment enhances the capacity to capture greater future value with large investments when business earnings improve. Marginal investments today create the facilities and competence that a small firm needs to capture greater value with large and more profitable future investments.

In Figure 1, the average cost of capital exceeds the expansion boundary by approximately 20 basis points when ROC = 0% and by over 100 basis points when ROC [approximately equal to] 22%. When the manager expands the business (ROC [greater than or equal to] 11.8 %) but ROC is less that the right most boundary in Figure 1 (ROC = 70%), the average cost of capital exceeds the expansion boundary by at least 60 basis points. These differences illustrate that those managers using the average cost of capital as a benchmark for growth analysis risk grave decision errors.

Other parameterizations of our model lead to more dramatic differences between the maximum expected return and the dynamic expansion boundary. Panels A and B of Figure 2 illustrate that profit volatility, [sigma], and maximum growth, g, increase this difference. For instance, if operating profit volatility is [sigma] = 0.4 and maximum growth is g = 0.049, with other parameters the same as those in Figure 1, then the difference between the maximum expected return and the dynamic expansion boundary is about 270 basis points. However, rather than concentrate on the magnitude of this difference, we believe that the essential lesson proffered from our research is that the dynamic expansion boundary is uniformly less than the average cost of capital. A manager who uses the average cost of capital as an investment threshold under invests and is unable to fully take advantage of valuable market opportunities when profitability increases.

Two forces in our model work together to accelerate investment: 1) current investment neither impairs nor displaces future returns, and 2) current investment increases the size and dollar value of future investments. We leave to future research an investigation of which of these forces is more significant. However, together these two forces offset the investment delaying force of growth leverage that exists whenever the manager exercises the growth option and expands business. We can measure this delaying force with the return differential between the average cost of capital and the growth unlevered cost of capital, [omega](Y) - [r.sup.*]. In Figure 1, when the average cost of capital is at its maximum, [omega](Y) - [r.sup.*] | [sub.y=0.22] [congruent to] 12.9 - 12 = 90 basis points. We measure the accelerating force of indefinite sequential proportional growth options with the return differential between the average cost of capital and the marginal cost of capital, [omega](Y) - [[xi].sup.*]. In Figure 1, when the average cost of capital is at its maximum, [omega](Y) - [[xi].sup.*][|.sub.y=0.22] [approximately equal to] 12.9 - 11.8 = 110 basis points. Finally, we measure the net accelerating force of indefinite proportional growth options as the return differential between the growth unlevered cost of capital and the marginal cost of capital, [r.sup.*] - [[xi].sup.*] = 12 - 11.8 = 20 basis points. The fact that this difference is not substantial suggests that one can approximate the marginal cost of capital with the growth unlevered cost of capital. A business person might calculate the growth unlevered cost of capital in a number of ways: 1) directly by estimating the risk premium [theta][[sigma].sub.x,c], 2) by unlevering the average cost of capital, or 3) in an appropriate asset pricing model by zeroing the estimated growth risk premium. We leave to future applied research how best to estimate/approximate the marginal cost of capital.

[FIGURE 2 OMITTED]

Proportional growth must be combined with indefinite rather than finite growth opportunities to accelerate investment. Current investment does not increase the size of future growth investments because the assumed facts predetermine growth investment size. Only if proportional growth investments are indefinite with constant returns to scale does current investment neither impair nor displace future investment and also increases the size of future investments at any future date for a fixed ROC growth threshold and for any ROC path between now and that future date.

In the following four subsections and in the panels of Figure 2, we present a comparative static numerical analysis of the value maximizing expansion boundary, [[xi].sup.*], the growth unlevered cost of capital, [r.sup.*] = r [theta][[sigma].sub.x,c], and the maximum expected return, [[omega].sup.*] [equivalent to] [max.sub.0<y<[infinity]][omega](Y).

F. Cost of Capital and the Expansion Boundary versus Volatility

The expansion boundary, [[xi].sup.*], decreases with volatility, [sigma] (see Panel A of Figure 2). The proportionality of growth option value with firm size enhances the appeal of an investment's upside earnings potential to a manager. This appeal reduces the dynamic expansion boundary, [[xi].sup.*], below the growth unlevered cost of capital, [r.sup.*]. This appeal is greater for greater earnings volatility, [sigma], and therefore, the dynamic expansion boundary, [[xi].sup.*], decreases with volatility. Note, in the left-most section of Panel A of Figure 2, that when earnings volatility, [sigma], approaches zero, the dynamic expansion boundary approaches the growth unlevered cost of capital, [r.sup.*].

The comparative static result above is opposite of the investment deferral option for a solitary investment in Dixit and Pindyck (1994). Downside earnings risk for a one-time irreversible investment is an essential feature of Dixit and Pindyck's study of investment under uncertainty and the investment deferral option. With greater earnings volatility, this downside risk is greater and the profit boundary to start a solitary investment increases because the investment becomes less attractive to the manager.

When profit volatility approaches zero, [sigma] [right arrow] 0, the maximum expected return, [[omega].sup.*], exceeds the expansion boundary, [[xi].sup.*], by almost 90 basis points. If profit volatility is greater, [sigma] [right arrow] 0.40, then the maximum expected return, [[omega].sup.*], exceeds the expansion boundary, [[xi].sup.*], by about 150 basis points. This divergence between the maximum expected return, [[omega].sup.*], and the expansion boundary, [[xi].sup.*], illustrates that the maximum expected return, [[omega].sup.*] (and by extension, expected return, [omega](Y), itself) is insensitive to volatility as compared to the growth boundary, [[xi].sup.*]. The maximum expected return, [[omega].sup.*], decreases, though modestly, with volatility, [sigma], because the expansion option's value increases with volatility, which in turn increases business value, decreasing expected return.

G. Cost of Capital and the Expansion Boundary versus Maximum Growth

In Panel B of Figure 2, maximum expected return, [[omega].sup.*], increases with maximal growth, g, due to increasing growth leverage. Alternatively, the expansion boundary, [[xi].sup.*], decreases modestly with maximal growth, g. Greater growth, g, enhances the appeal of upside earnings potential to managers, who, therefore, reduce the expansion boundary, [[xi].sup.*]. However, the optimal growth boundary, [[xi].sup.*], is relatively insensitive to the rate of growth, g, as the benefit of expansion, profit growth, g, matches rather closely the burden of expansion, capital growth, g. Maximal growth, g, has a more pronounced impact on expected return, [[omega].sup.*], than it does on the expansion boundary, [[xi].sup.*]. Maximum growth, g, is a primary determinant of expected return, but not the value maximizing expansion boundary. Furthermore, the difference between the growth unlevered cost of capital and the maximum expected return ([r.sup.*] and [[omega].sup.*]) increases substantially as the maximum growth rate increases.

H. Cost of Capital and the Expansion Boundary versus Covariance Risk

In Panel C of Figure 2, covariance, [theta][[sigma].sub.x.c], increases both the expected return, [[omega].sup.*], and the expansion boundary, [[xi].sup.*]. Covariance risk makes operating profit less attractive to managers, who, therefore, increase the expansion boundary, [[xi].sup.*].

Note that when covariance risk is zero, [theta][[sigma].sub.x,c] = 0, both the maximum expected return and the growth unlevered cost of capital equal the riskless rate, [[omega].sup.*] = r and [r.sup.*] = r. However, the expansion boundary, [[xi].sup.*], is less than the riskless interest rate ([[xi].sup.*] < r, when [[theta][[sigma].sub.x,c] = 0). The difference between the riskless interest rate and the expansion boundary, r - [[xi].sup.*], is about 150 basis points for zero covariance risk, [theta][[sigma].sub.x,c] = 0 in Panel C of Figure 2.

Measure the impact of growing growth options on managerial incentive to expand with the difference between the maximum expected return and the expansion boundary, [[omega].sup.*] - [[xi].sup.*]. Panel C of Figure 2 illustrates that this incentive is invariant to covariance risk, [theta][[sigma].sub.x,c]. Note that even though the expansion boundary approaches the unlevered cost of capital as covariance risk becomes large, [[xi].sup.*] [right arrow] [r.sup.*], when [theta][[sigma].sub.x,c] [right arrow] 0.1 in Panel C of Figure 2, the difference between the maximum average cost of capital and the expansion boundary, [[omega].sup.*] - [[xi].sup.*], is always about 150 basis points. Greater covariance increases the discounting of future growth investments reducing the incentive to make expansion investments today. However, at the same time, increasing covariance increases expected return. Thus, covariance risk has little impact on the difference between the maximum expected return and the expansion boundary, [[omega].sup.*] - [[xi].sup.*].

I. The Cost of Capital and the Expansion Boundary versus Riskless Interest Rates

In Panel D of Figure 2, as one would expect, interest rates increase the maximum expected return, [[omega].sup.*]. In addition, as greater interest discourages investment, greater interest increases the expansion boundary, [[xi].sup.*]. The riskless rate of interest, r, has little influence on the difference between the expansion boundary and the maximum expected return, [[omega].sup.*] - [[xi].sup.*]. This difference is approximately 120 basis points in Panel D of Figure 2.

J. Primary Determinants of the Cost of Capital and the Expansion Boundary

The panels of Figure 2 illustrate that the parameters of business value that determine the growth unlevered cost of capital, r + [theta][[sigma].sub.x,c], influence the expected return and the expansion boundary in a like manner. In Panels C and D, the maximum average cost of capital, [[omega].sup.*], and the expansion boundary, [[xi].sup.*], both increase parallel with the riskless rate, r, and covariance risk, [theta][sigma].sub.x.c.

In Panel B of Figure 2, growth, g, has a modest negative effect on the growth boundary, [[xi].sup.*], whereas it has a positive impact on expected return, [[omega].sup.*]. Growth, g, has little impact on the expansion boundary, [[xi].sup.*], since the benefit of expansion (profit growth) closely matches the cost of expansion (capital investment). Alternatively, a manager's discretionary expansion expenditures impose a leverage risk on financial asset holders. Expected return reflects this risk and the expected return, [[omega].sup.*], increases with maximal growth, g.

In Panel A of Figure 2, volatility, [sigma], has a modest negative effect on the expected return through the expansion option's value. Volatility, [sigma], has a negative impact on the expansion boundary, [[xi].sup.*]. Volatility has little impact on expected return as investors anticipate the influence of volatility on the manager's expansion decision. As in many asset pricing models, volatility is largely unpriced in equilibrium because portfolio diversification diminishes its influence. The proportionality of growth option value with firm size increases the appeal of an investment's upside earnings potential to a manager as compared to a sequence of same-sized growth investments. Greater earnings volatility further enhances this appeal, which induces the manager to reduce the expansion boundary, [[xi].sup.*].

These observations indicate that the primary determinants of expected return are: 1) riskless interest rates, 2) covariance risk, and 3) growth leverage. The primary determinants of the optimal expansion boundary are: 1) riskless interest rates, 2) covariance risk, and 3) profit volatility. The primary determinants of the dynamic expansion boundary and expected return differ. These differences are important for framing empirical tests of our dynamic business expansion model.

K. Implications for Capital Budgeting

A common view in the managerial practice of corporate finance, which most corporate finance text books represent, is that an investment that expands existing operating profit, but otherwise is of the same character as in place operations, is of the same risk as in place operations. This commonly held view is erroneous. Business persons can use WACC to measure the average cost of capital that they can then use as the marginal cost of capital for expansion investments. However, as we demonstrate in this paper, in a dynamic setting, the average and the marginal costs of capital diverge. Therefore, business persons risk decision errors when using the average cost of capital for expansion benchmarking. This theoretical result is consistent with the empirical findings of Bernardo, Chowdhry, and Goyal (2007).

Our results indicate that even if expansion investments are identically scaled copies of existing operations, they are of greater risk. Expected return for the existing business, a hypothetically nongrowing business, is the growth unlevered cost of capital, r + [theta][[sigma].sub.x,c]. Conversely, because expected return, [omega](Y), which reflects the risk of the existing business plus growth opportunities, exceeds the growth unlevered cost of capital, [omega](Y) [greater than or equal to] r + [theta][[sigma].subx,c], growth investments are riskier than the existing business. Since growth is not spontaneous, the manager must invest to create profit growth. This investment creates growth leverage risk that does not exist for in place operations. Expected return for a firm with expansion options exceeds that of a no-growth firm.

Even with optional expansion, financial asset holders are not immune to growth leverage. The numeric example of Figure 1 reflects a rather modest fall in expected return, [omega](Y), with increasing profitability, Y, for large profitability. As ROC increases from approximately 20% (above the expansion boundary and near the maximum expected return) to 70% (well above the expansion boundary), the risk premium falls only (approximately) from 12.8% - 5% = 7.8% to 12.6% 5% = 7.6%. As profit growth requires capital growth (increasing fixed costs), leverage "risk cost" is insensitive to profitability in this range. ROC must be very great to cover increasing fixed costs and eliminate growth risk costs. Possibly this decrease in expected return is modest because while 70% seems a high return, it is only approximately three standard deviations above the value maximizing expansion boundary in Figure 1. Further, since the logarithm of ROC follows an arithmetic Brownian motion, despite the fact that ROC is very high (70%), ROC falls below the expansion boundary at some time in the future with probability one (Cox and Miller, 1965).

III. Conclusion

Our main result is that the cost of capital as it is typically calculated with the weighted average cost of capital (the expected return on business value) is inadequate for expansion decisions when a firm has indefinite future investments proportional to capital. McDonald and Siegel's (1986) and Dixit and Pindyck's (1994) option to defer investment due to irreversibility is dominated by another option; the option to invest currently to increase the value of future growth investments when profits stochastically improve. This suggests that managers should grow the business even at levels below the current cost of capital. Such growth is justified as it allows the firm to fully take advantage of favorable investment opportunities when profitability improves.

Appendix A

Business Value

With a constant riskless interest rate, r, the business value function, V(X,B), satisfies the differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Use the branches for dB/B from Equation (1), apply Ito's lemma to dV, and with the risk-adjusted process for operating profit in Equation (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We conjecture (and verify) that the value function V(X,B) is of the form

V(X, B) = B[pi](Y), (A2)

where [pi] (Y) is a function of Y = ROC. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A3)

Substitute Equation (A3) into Equation (A1) and after dividing both sides by B

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A4)

For, 0 [less than or equal to] g < r, the solutions to these ordinary differential equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A6)

Determine the parameters [C.sub.1] and [C.sub.2] with value matching and smooth pasting conditions at Y = [xi] (see Dixit and Pindyck, 1994, for a discussion of these conditions). Solve these two equations (not given) to determine the value to expenditure ratio, [pi], in Equation (5).

Our problem has three unknowns: 1) [C.sub.1], which establishes the value of the option to start growth; 2) [C.sub.2], which establishes the value of the option to stop growth; and 3) [[xi].sup.*], the value maximizing expansion return boundary. Value matching and smooth pasting determine only two of these three unknowns. Consequently, smooth pasting and value matching do not ensure value maximization. In using smooth pasting and value matching to determine [C.sub.1] and [C.sub.2], we ensure that the value function has no kinks, and, therefore, no arbitrage. Shackleton and Sodal (2005) demonstrate that smooth pasting is equivalent to return equalization between the investment option and its levered payoff, which prevents arbitrage at the investment threshold. The optimization that we describe in Section II.B is the third condition required to determine the third unknown, the value-maximizing expansion boundary, [[xi].sup.*].

Appendix B

In this appendix, we determine the value of a no-growth firm that awaits permanent growth (at the rate g) and a growing firm that has the option to permanently stop growth. In each case, we determine the critical return threshold that separates growth from no growth.

For both problems, since the decision to grow or stop growth is permanent, each problem has two unknowns: 1) a constant that establishes the value of the option to either start or stop growth and 2) the return boundary for either starting or stopping growth. Because there are two unknowns, value matching and smooth pasting not only determine both unknowns but also ensure value maximization.

Using the same methodology as in Appendix A, the value of the no-growth business awaiting permanent growth is given on the upper branch of the equation below, whereas the value of the permanently growing business is given on the lower branch.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Parameter C establishes the value of the option to permanently grow. Use smooth pasting and value matching to find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the value maximizing return threshold for permanent growth, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, the value of the growing business waiting to permanently stop growth is given on the lower branch of the equation below. The value of the permanently nongrowing business is given on the upper branch. Parameter C establishes the value of the option to permanently stop growth.

Use smooth pasting and value matching to find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the value maximizing return threshold for permanent growth, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Appendix C

In this appendix, we use the definitions of [alpha] and [lambda] in Equation (A6) to verify that a sufficient condition for the final term in Equation (6), [alpha][lambda]/(1-[alpha])(l-[lambda]), to be less than one, is positive covariance risk, [theta][[sigma].sub.x,c][greater than or equal to] 0. For g = 0, [alpha][lambda]/(l-[alpha])(l- [lambda])[less than or equal to] 1, with equality achieved when the covariance risk is zero.

Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So, [alpha][lambda]/(l-[alpha])(l-[lambda]) is one when there is no covariance risk and growth, g, is zero, and strictly less than one for positive covariance risk and any maximum growth rate, g.

Appendix D

In this appendix, we prove V(X,B)/B|Y=[[xi].sup.*] = [pi]([[xi].sup.*]) = 1. Substitute the expression for [[xi].sup.*] on the right-hand side of Equation (6) into either branch of Equation (5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This expression equals one if

r([lambda] - [alpha]) + r([[alpha].sup.2] - [[lambda].sup.2]) + [alpha]g(1 - [alpha]) + [[alpha].sup.2][lambda][theta][[sigma].sub.x,c] - [alpha][[lambda].sup.2][theta][[sigma].sub.x,c] = O.

Substitute the definitions for [alpha] and [lambda] to verify this equation.

Appendix E

This appendix illustrates that the optimal expansion boundary, [[xi].sup.*], is always below expected return, [omega](Y). We presume positive covariance risk, [[sigma].sub.xc] [greater than or equal to] O. Simplify Equation (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (E1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We demonstrate the following steps:

1. The growth unlevered cost of capital is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

a. Expected return equals the growth unlevered cost of capital for Y = 0, [omega](0) = [r.sup.*].

b. Expected return approaches [r.sup.*] as Y increases without bound [lim.sub.Y[right arrow][infinity]][omega](Y) = [r.sup.*].

c. On the no-growth branch, [omega](Y) [greater than or equal to] [r.sup.*], for 0 [less than or equal to] Y [less than or equal to] [xi] because expected return, Equation (E1), is an increasing function of Y.

d. On the growth branch of Equation (El), [omega](Y) [greater than or equal to] [r.sup.*] because [omega](Y) has only one extreme point, which is a maximum for Y [greater than or equal to] [xi].

2. The growth unlevered cost of capital exceeds the expansion boundary [[xi].sup.*] [less than or equal to] [r.sup.*].

Therefore, expected return exceeds the expansion boundary, [omega](Y) [greater than or equal to][[xi].sup.*] for Y [greater than or equal to] 0. la. Expected return equals the unlevered cost of capital at Y = 0. On the no-growth branch, expected return is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (E2)

At Y = O, expected return simplifies to [omega](O) = [r.sup.*]. lb. Expected return approaches [r.sup.*] as Y increases without bound, [lim.sub.Y[right arrow][infinity]][omega](Y) = [r.sup.*]. On the growth branch, expected return is

[omega](Y) = Y-g + 1/2 [[partial derivative].sup.2][[pi].sub.G]/[partial derivative][Y.sup.2][[sigma].sup.2][Y.sup.2] + g[[pi].sub.G]/[[pi].sub.G]. (E3)

The denominator of (E3) tends to infinity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E4)

Therefore, the limit of expected return is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E5)

The first term of (E5) tends to [r.sup.*] - g, the second term tends to zero, and the third term is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E6)

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E7)

1c. On the no-growth branch, expected return, Equation (E1), increases with Y. Differentiate the no-growth branch of Equation (E1) with respect to Y

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E8)

This derivative is positive because

[[sigma].sup.2]([alpha]-1)/2(1/[r.sup.*] + [gamma]/[alpha][Y.sup.[alpha]-1])[greater than or equal to] 1/[alpha](1 + 1/2 [gamma]([alpha]-l)[[sigma].sup.2][Y.sup.[alpha]-l]), (E9)

or

[alpha]([alpha]-1)[greater than or equal to] [2r.sup.*]/[[sigma].sup.2]. (E10)

Substitute for [alpha] and simplify to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (E11)

1d. On the growth branch, expected return, Equation (E1), has only one extreme point, which is a maximum for Y > [xi].

Differentiate the growth branch of Equation (E1) with respect to Y

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E12)

The numerator of Equation (E12) has at most two roots for Y > 0. Note that the smooth pasting condition implies [partial derivative][omega]/[partial derivative]Y|Y=[xi] > 0 on either branch of Equation (E 1). It is easily verified that the first derivative of the growth branch, Equation (E12), is negative for small ROC, [lim.sub.Y[right arrow]0]([partial derivative][omega]/[partial derivative]Y) < 0. (6) Therefore, Equation (E12) has one root for Y [less than or equal to] [xi]. Hence, Equation (E12) has, at most, one root for Y > [xi]. In other words, expected return, Equation (E1), has, at most, one extreme value for Y > [xi]. Combined with parts 1a, 1b, and 1c of this proof, this result indicates that [omega](Y) > [r.sup.*].

To review, because expected return, [omega](Y), has, at most, one extreme point for Y > [xi], and [omega](Y) is an increasing function at Y = [xi] above [r.sup.*] at Y = [xi] and approaches [r.sup.*] as Y tends to infinity, the extreme point is a maximum. Expected return, [omega](Y), therefore, exceeds [r.sup.*] for any Y [greater than or equal to] [xi].

2. The expansion boundary never exceeds the growth unlevered cost of capital, [[xi].sup.*] [less than or equal to] [r.sup.*]. A rearrangement of Equation (6) reveals that [[xi].sup.*] [less than or equal to] [r.sup.*] when

S [equivalent to] [theta][[sigma].sub.x,c][alpha][lambda] - (r - g)(1 - [alpha] - [lambda]) [greater than or equal to] 0. (E13)

Consider S as a function of the parameter g. We demonstrate that S(g) [greater than or equal to] 0 for 0 [less than or equal to] g [less than or equal to] r. Substitute the definitions for [alpha] and [lambda] and verify that S(g)[|.sub.g=o] = 0 and S(g)[|.sub.g=r] = 0. The first-order condition for the maximum value of the function S with respect to g is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E14)

Denote the right-hand side of Equation (E 14) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E15)

The second derivative of h(g) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (E16)

which is clearly positive for 0 [less than or equal to] g [less than or equal to] r. Therefore, the function h(g) is convex for 0 [less than or equal to] g [less than or equal to] r. Further, it is easy to verify that h(g)]g=0 > 0 and h(g)]g=r < r. Consequently, Equation (E14) has exactly one solution for 0 [less than or equal to] g [less than or equal to] r, which is the only extreme point of the function S(g). The last step of the proof requires that we establish that S is positive rather than negative at this extreme point.

To verify that this extreme point is a maximum not a minimum, take the derivative of S with respect to g and evaluate it at g = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (E17)

Simplification illustrates that as [partial derivative]S/[partial derivative]g[|.sub.g=0] [greater than or equal to] 0 if r(r + [theta][[sigma].sub.x,c]) [greater than or equal to] 0, which clearly holds.

Because the function S(g) is increasing at g = 0 and has only one extreme value for 0 [less than or equal to] g [less than or equal to] r, this extreme value must be a maximum. Therefore, S(g) is nonnegative, S(g) [greater than or equal to] 0 for 0 [less than or equal to] g [less than or equal to] r.

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We thank an anonymous reviewer and the Editor who were instrumental in forcing clarity of the exposition and economic intuition. A number of colleagues assisted with valuable comments including Robert McDonald, Robert Pindyck, Rob Grauer, Gordon Sick, Daniel Smith, Chris Robinson, Ross Valkanov, and Vijay Jog. We presented a version of this paper entitled "Corporate Performance and Dynamic Business Expansion" at the 2004 Northern Finance Association Conference in St. Johns, Newfoundland. Of course, the authors retain responsibility for errors.

Andrey D. Pavlov, George W. Blazenko is a Professor of Finance at Simon Fraser University in Burnaby, British Columbia.

Andrey D. Pavlov is a visiting Associate Professor of Real Estate at the University of Pennsylvania, Philadelphia, PA and an Associate Professor of Finance at Simon Fraser University in Burnaby, British Columbia, Canada.

(1) See, with the related literature review, Cottrell and Sick (2002). A manager undertakes strategic investments that preempt competitive entry earlier than absent this competition. However, no one has shown that this investment is sooner in a real options setting as compared to standard cost of capital analysis in the same strategic environment. Cottrell and Sick compare the attractions of follower to first mover strategies.

(2) We could add development risk, randomness to capital expenditures, without changing the fundamental results in this paper. Further, we could add development risk in two ways. First, we could add development risk as a common random increment to the upper and lower branches of Equation (1) correlated with both operating profit, X, and aggregate consumption (see the upcoming section). This modeling might represent, for example, construction costs that vary randomly throughout the economy regardless of whether or not a particular firm grows its business. Second, if a firm bears development risk only when it grows (e.g., R&D risk), we could add a random increment only to the growth branch of Equation (1).

(3) The essence of the constant returns to scale assumption in our model is that investment returns between small and large firms do not differ simply because of size measured by capital, B. In practice, smaller firms are more likely to be development oriented with the potential for exceptionally great returns, whereas larger firms are more likely to be survivors with great ex post return on capital. Otherwise, we expect that there is no difference between return on capital for small versus large firms. Of course, this is an empirical question.

(4) We thank the reviewer for a clearer exposition of our model presented here as compared to earlier drafts and the insight that indefinite proportional expansion opportunities are vital to the results.

(5) The static environment is helpful to illustrate the point. If permanent profit growth at the rate g requires capital growth at the rate g, then IRR satisfies (X - [g.sup.*]B)/(IRR - g) - B = 0, and IRR = ROC regardless of the growth factor, g. For comparison purposes, for spontaneous profit growth, which is not the nature of the investment we study, the IRR satisfies X/(IRR - g) - B = 0, and IRR = ROC + g.

(6) The third term of the numerator determines the sign of the derivative as Y approaches zero from the right. Substituting [lambda] into the third term confirms that this term is negative.

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Author: | Blazenko, George W.; Pavlov, Andrey D. |
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Publication: | Financial Management |

Date: | Dec 22, 2009 |

Words: | 10900 |

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