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Optimal Investment Timing When External Financing Is Costly Stefan Hirth ∗ Marliese Uhrig-Homburg ∗ Chair of Financial Engineering and Derivatives a Universit¨t Karlsruhe (TH) First version: August 2004 Current version: December 2004 ∗ a Chair of Financial Engineering and Derivatives, Universit¨t Karlsruhe (TH), 76128 Karlsruhe, Ger- many. Corresponding Author: Stefan Hirth, Phone +49 (721) 608-8185, Fax +49 (721) 608-8190, Email stefan.hirth@fbv.uni-karlsruhe.de. Optimal Investment Timing When External Financing Is Costly Abstract We examine how a ﬁrm’s investment timing decision is inﬂuenced by costly external ﬁnancing. If both internal funds and the ﬁrm’s risk-free debt capacity are not suﬃcient to permit investment, the ﬁrm may issue risky debt or equity. Then immediate investment is possible, but due to information asymmetries there are ﬁnancing costs that decrease the value of the investment option. Although all outside investors suﬀer equal information asymmetries, risky debt is cheaper than equity, because the debtholders’ undervaluation is relevant only in default states. We compare investment timing when external ﬁnancing is costly to the bench- mark case of a ﬁrm facing only a ﬁnancing quantity constraint. While it is known that a pure-quantity constrained ﬁrm might accelerate investment in order to avoid future quantity constraints, we ﬁnd that ﬁnancing costs can induce both voluntary delay and acceleration of investment. This can be explained by the ﬁrm’s objective to avoid current or future ﬁnancing costs, respectively. We show that if risky debt is available, investment acceleration and delay are mainly ampliﬁed by the presence of ﬁnancing costs, so we conﬁrm and extend the qualitative structure of the pure-quantity constrained ﬁrm’s optimal investment timing. However, if the ﬁrm is restricted to equity ﬁnancing, it will make use of external ﬁnancing only for very favorable investment projects. In states in which no equity ﬁnancing is needed, possible future equity ﬁnancing costs induce signiﬁcant acceleration of investment. JEL Classiﬁcation: G31, G32 Keywords: Investment timing ﬂexibility, costly external ﬁnancing, debt vs. equity ﬁnanc- ing, option pricing 1 Introduction Since the seminal article of Modigliani and Miller (1958) it is a well-known fact in corpo- rate ﬁnance theory that investment and ﬁnancing decisions can be made independently on frictionless markets. In consequence, there is a wide range of literature analyzing either the investment or the ﬁnancing decision of the ﬁrm. Sophisticated models have been devel- oped, and it is possible to examine corporate decision making in a modern continuous-time dynamic framework. A well-known article dealing with the ﬁrm’s investment decision is the work of McDonald and Siegel (1986), which explains the value of waiting to invest. The authors examine an investment project that, once started, pays a continuous cash-ﬂow stream. While waiting enables the ﬁrm to gather more information about the project value, the foregone cash- ﬂows, which are proportional to the project value, represent opportunity costs of waiting. It is optimal to trade oﬀ beneﬁts and costs of waiting, and to exercise the investment option as soon as a critical project value is exceeded. Since there are no ﬁnancing frictions, the investment decision can be analyzed without considering the source of funding for the investment project. More recent literature deals with the fact that investment and ﬁnancing decisions cannot be separated in the presence of ﬁnancing frictions. Among the work that extends real options literature in order to include ﬁnancing frictions is a recent article by Boyle and Guthrie (2003): In their world, the ﬁrm owns an investment option as introduced by McDonald and Siegel (1986), but the option can be exercised only if the ﬁrm’s cash balance is above a critical level. In consequence, there are states in which the constrained ﬁrm cannot invest although it would be optimal in the unconstrained case, and there are also states in which the constrained ﬁrm invests in order to avoid a future funding shortfall, although an unconstrained ﬁrm would prefer to wait. Both of these suboptimal policies lower today’s value of the investment option compared to McDonald and Siegel (1986). However, in the world of Boyle and Guthrie (2003) there are no ﬁnancing costs – where we deﬁne the term ﬁnancing costs as the amount that has to be paid beyond the fair compensation for the risk transferred to external investors. If investment is possible at all, then it is possible without extra costs – even if the ﬁrm relies on a tremendous amount of external ﬁnancing. The loss in today’s option value, compared to an unconstrained ﬁrm, results only of the fact that there are states in which the constrained ﬁrm cannot invest due to ﬁnancing quantity constraints. As Lyandres (2003) points out, both ﬁnancing quantity constraints and ﬁnancing costs have to be taken into account in order to correctly analyze the eﬀects of ﬁnancial constraints on investment timing. In this work we extend Boyle and Guthrie (2003) to allow for ﬁnancing costs. Upon in- vestment the ﬁrm suﬀers a loss in value, compared to the situation of a ﬁrm facing no ﬁnancing costs. We explain this loss in value by information asymmetries, which are such that outside investors undervalue the ﬁrm’s investment project. Our model’s assumptions imply a ﬁnancing policy that is consistent with the pecking order derived by Myers and 1 Majluf (1984): The ﬁrm uses the safest securities ﬁrst, since they give rise to the lowest ﬁ- nancing costs. If possible, internal funds and risk-free debt are used exclusively. Additional funding is raised using risky debt rather than new equity. Taking into account this ﬁnancing policy and the respective ﬁnancing costs, we solve for both today’s ﬁrm value and the investment threshold for optimal investment timing. We ﬁnd that ﬁnancing costs can induce both voluntary delay and acceleration of investment: While for high current ﬁnancing costs the ﬁrm waits longer than a ﬁrm facing only ﬁnanc- ing quantity constraints, there are also states in which the ﬁrm invests earlier in order to avoid future ﬁnancing costs. There are other lines of research also considering both the investment decision and ﬁnan- cial market frictions: Mauer and Triantis (1994) determine optimal investment, operating, and ﬁnancing policies given that there are switching costs for policy changes. They ﬁnd that debt ﬁnancing has a negligible inﬂuence on investment timing. However, they do not focus on the source of funding for the initial investment amount, but instead they consider tax shield eﬀects of debt ﬁnancing after investment. Therefore there are no direct eﬀects of ﬁnancial constraints on the initial investment decision. Mello and Parsons (2000) justify market frictions very similar to Boyle and Guthrie (2003) and the approach followed in this paper by a lower value of the ﬁrm’s assets to outside investors. However, they do not consider the initial investment decision at all, but they examine a ﬁrm that has already invested. Their focus is on the value that is created by hedging in ﬁnancially constrained ﬁrms, which is pioneered by Froot, Scharfstein, and Stein (1993) and is examined for our modelling framework by Boyle and Guthrie (2004). The research that is most related to Boyle and Guthrie (2003) and the approach followed in this paper is done by Lyandres (2003). He also analyzes the impact of ﬁnancial constraints on investment timing, pointing out that constraints can take the shape of both insuﬃcient funds and ﬁnancing costs. However, he uses a discrete-time 3-date model similar to Froot, Scharfstein, and Stein (1993), and therefore he fails to explain the full range of voluntary delay and acceleration of investment. While he ﬁnds that increasing cash balance always lowers the investment threshold, Boyle and Guthrie (2003) show that above a critical level of cash balance the ﬁrm makes use of its timing option and waits longer, i.e. the threshold is increasing again. We conﬁrm this result even in the presence of ﬁnancing costs. This work is organized as follows: Section 2 presents the framework of the model. In Section 3 we oﬀer several alternatives how the ﬁrm can mitigate its ﬁnancing constraint, and we derive ﬁnancing costs. Then we compare these alternatives and determine the optimal ﬁnancing policy given that immediate investment is optimal, but a certain amount of additional funding is needed. After that we turn to the investment decision in Section 4, given the ﬁnancing policies and costs derived above. We compare the investment threshold function, as well as today’s value of the ﬁrm, to both an unconstrained ﬁrm and a quantity constrained ﬁrm with costless ﬁnancing. Finally, Section 5 summarizes insights compared to the two benchmark cases and concludes. 2 2 Framework Consider a ﬁnancially constrained value-maximizing ﬁrm at time t consisting of • physical assets worth G, • a cash balance of Xt , and • the rights to an investment project worth Vt . The physical assets create an income stream of νGdt + φGdZt where ν and φ are constant parameters and Zt is a Wiener process. While G remains constant over time, the income stream augments or reduces the ﬁrm’s cash balance Xt . As long as the investment project is not launched, the cash balance is invested in riskless securities yielding an interest rate of r, so it evolves according to dXt = rXt dt + νGdt + φGdZt . (1) The focus of the work is on the ﬁrm’s perpetual rights to a project whose value Vt follows the geometric Brownian motion dVt = µVt dt + σVt dWt (2) where µ and σ are constant parameters, and the correlation of the Wiener processes Zt and Wt is given by ρ. In the following, we will leave out time t subscripts unless they are needed for clariﬁcation. At any point in time, the ﬁrm can decide to exercise its rights by paying an investment amount I, launching the project, and receiving the project value. The net payoﬀ upon exercise of this investment option is V − I. Alternatively, the ﬁrm can delay investment and retain the rights, or sell both the existing assets and the project rights to outside investors, which causes liquidation costs discussed in more detail below. There is a value of waiting to invest, since the ﬁrm can avoid sunk costs in case the project does not develop favorably. However, delay is not always optimal as the project provides a cash- ﬂow stream of δ, which is the opportunity cost of waiting. Prior to investment, the value of this investment option is F . In order to analyze the implications of diﬀerent sources of ﬁnancing, we assume that the ﬁrm has limited access to external ﬁnancing. Consequently, the state of the ﬁrm’s cash balance has an impact on the ﬁrm’s investment decision, and the value of the ﬁrm’s investment option will be below that of an unconstrained1 ﬁrm, which we denote by F u (V ). 1 The value of the unconstrained ﬁrm’s investment option as derived by McDonald and Siegel (1986) will be given in Section 4.2.1. 3 If the ﬁrm had no access at all to external ﬁnancing, it could exercise its investment option only if it had enough cash to do so, i.e. investment would be possible only if X ≥ I. (3) In Section 3 we will discuss various possibilities to mitigate this hard cash-ﬂow constraint in investment states. We will allow access to external ﬁnancing, but there will be ﬁnancing costs, i.e. the cost of capital is above the ”fair” price in a frictionless world. However, in states in which the ﬁrm postpones investment, we will assume throughout this paper that both the existing assets G and the project rights F are accepted as a collateral for risk-free debt at no cost but the ”fair” price. Therefore the ﬁrm has access to a short-term risk-free credit up to the amount of G+F u (γV ), and the ﬁrm can continue operation as long as X ≥ −[G + F u (γV )]. (4) Thus, we allow the cash balance to become negative while according to (1) the ﬁrm has to pay the risk-free rate on X. In Section 3 we will explain why the project value is treated diﬀerently in investment states. Notice that the value of the project rights to ﬁnancially unconstrained outside investors is only F u (γV ) instead of F u (V ): Although they accept the project rights as a collateral, the outside investors account for the fact that they would receive only a project value of γV upon investment of I. This discount reﬂects that they might not have the knowledge required to extract full project value, but only a fraction γ ∈ [0, 1). This is the central assumption of our model, which makes ﬁnancing constraints relevant for valuation. For γ = 1 the ﬁrm could sell the investment option to outside investors at the unconstrained value F u (V ), and therefore the state of the ﬁrm’s cash balance would have no eﬀect on the valuation of the investment option. As soon as (4) is not satisﬁed any longer, liquidation is enforced: The risk-free debtholders receive both the existing assets and the project rights, which they can sell to outside investors at exactly the value of their loan, |X|, and the owners of the ﬁrm receive nothing. The ﬁrm may also liquidate voluntarily at any time before. Again, the existing assets and the project rights are sold to outside investors. The owners of the ﬁrm receive an amount of X + G + F u (γV ), where in case of a negative cash balance the risk-free debtholders receive the value of their loan, |X|. This assumption ensures that the liquidation value of the investment option, F u (γV ), is always a lower bound for the actual option value. While the project rights F are intangible and investment requires special knowledge in- corporated in the existing assets, both G and V are physical assets after investment has taken place, and their value is fully transferable to outside investors. So after investment, the ﬁrm’s owner is free to sell the assets at the full value of G + V . 4 3 Financing Policy 3.1 Funding Capacity As long as the ﬁrm is restricted to internal funding, it can exercise its investment option only if X ≥ I. From now on we consider states in which immediate investment is optimal, however the cash balance is not suﬃcient for investment. Firstly, the existing assets G can serve as a collateral for risk-free debt, just as introduced for states in which investment is postponed. As long as X + G ≥ I, the ﬁrm can therefore invest using internal funds and risk-free debt exclusively, and there are no ﬁnancing costs. Moreover, in states in which X + G < I, the ﬁrm has the possibility to raise the addi- tional funding by issuing claims written on the project value that will be extracted after investment. We deﬁne the amount missing as ∆X = max{I − (X + G), 0}, (5) which is zero in states in which internal funds and risk-free debt are suﬃcient to permit investment. Due to information asymmetries, the ﬁrm is not able to credibly communicate the true project value to outside investors, who consequently value the project to be worth only αV with α ∈ [0, 1). We will see later on how the amount αV can actually be raised. Overall, investment using both internal and external funding is possible only if X + G + αV ≥ I. (6) However, the ﬁrm cannot get any more risk-free debt than G, since the ﬁrm’s extraction of project value during the investment procedure is not considered to be risk-free. One might think of α as the fraction of the project value that external investors expect the ﬁrm to extract, while both lower and higher fractions are possible. Instead of using risk-free debt, the ﬁrm has to oﬀer risky claims written on the project value to debt and equity markets. However, whenever ∆X > 0 the undervaluation of the project will cause ﬁnancing costs, which will be derived in the remainder of this section. Notice that investment, i.e. exercise of the option, occurs right after raising the external funding. This restriction signiﬁcantly reduces both the complexity of the problem and the set of alternatives available to the owner of the ﬁrm. It might be in the interest of the owner of the ﬁrm to raise new funds but still wait with the exercise of the option. However, external investors do not trust the owner’s capabilities. If he does not precommit to immediate investment they will not provide funding, since they fear that their contribution sinks into the depths of the ﬁrm and a part of their value is lost. After investment, the cash balance will be oﬀset to zero, and the total distributable value consists of the existing assets and the investment project. As we explained at the closing 5 of Section 2, there is no more use in keeping the ﬁrm together, since the existing assets have already helped to extract full project value V . We will show that it is optimal to issue claims on the project value only after the full risk- free debt capacity of G has been used. In that case, the existing assets as a whole will be transferred to the risk-free debtholders after investment, and the remaining stakeholders own claims to the investment project exclusively. If the total amount of funding is less than G, then the only external funding will be risk-free debt of G or less, which the ﬁrm can easily repay by selling the existing assets. 3.2 Equity Issuance We ﬁrst discuss how the ﬁrm can raise the amount ∆X using new equity exclusively. The ﬁrm oﬀers a share of the project to outside investors, who contribute an amount ∆X to the cash balance. Since they value the project to be worth αV instead of V , equity issuance is possible only for ∆X ∈ [0, αV ], since the outside investors will not provide more capital than the total project value they assume to be distributable after the investment. The new equityholders’ share ω is determined such that ∆X ω= . αV Notice that the actual project value after equity issuance and exercise of the option is V , which means that the new equityholders demand a higher share in the project than they deserve. They receive a value of ωV = ∆X/α for their contribution of ∆X = ωαV. We deﬁne the diﬀerence as equity issuance costs 1−α EIC(∆X) = ∆X = ω(1 − α)V. (7) α The second expression in (7) allows an alternative interpretation of equity issuance: The ﬁrm decides to sell a fraction ω of the project to outside investors, who value the project to be worth only αV . Therefore the remaining value (1 − α)V of the fraction sold is lost. We see that equity issuance can be interpreted as the sale of a fraction ω of an undervalued project, and the costs are simply the same fraction ω of the maximum possible costs (1 − α)V . Notice that EIC(∆X) is linear in ∆X. For the maximum equity ﬁnancing amount of αV we have EIC(αV ) = (1 − α)V , which can be interpreted as a sale of the project V as a whole to outside investors, who value it to be worth only αV . 6 Although it seems intuitive that the ﬁrm should use its risk-free debt capacity as much as possible, one might think that it could be optimal to use only a part of the risk-free debt capacity of G in order to get lower equity issuance costs. However, we prove now that this actually never is the case. Let Drf and E denote the amounts raised using risk-free debt and equity, respectively, where Drf + E = I − X. The new equityholders assume a total value of G + αV − Drf to be available for both new and old equityholders after investment. Equity issuance is possible only for E ≤ G + αV − Drf , i.e. Drf + E = I − X ∈ [0, G + αV ]. (8) The new equityholders demand a share of E I − X − Dr f ω= = . (9) G + αV − Drf G + αV − Drf Since the new equityholders contribute E = ω(G + αV − Drf ) but receive a value of ω(G + V − Drf ), equity issuance costs amount to EIC(Drf ) = ω(1 − α)V with ω as in (9). These are at the same time equal to the total ﬁnancing costs, since risk-free debt causes no costs. The ﬁrm’s optimization problem is equivalent to the choice of Drf in order to minimize ω. For I − X = G + αV , new equityholders demand the whole equity regardless of the choice of Drf , and ﬁnancing costs are (1 − α)V . For I −X < G+αV , which holds according to (8), we ﬁnd that an increase of Drf has two adverse eﬀects on ω: First, the nominator of (9), which represents the capital contribution of new equityholders, decreases. Second, the denominator of (9), which represents the total value that new equityholders assume to be available for both new and old equityholders after investment, decreases as well, which leads to an increase in ω. However, the ﬁrst eﬀect is more important, and therefore the new equityholders’ share and at the same time total ﬁnancing costs are decreasing in Drf , so we have a minimum of total ﬁnancing costs for Drf = G. 3.3 Risky Debt Issuance Similar to the equity issuance case, the ﬁrm again has used its full risk-free debt capacity of Drf = G, but now the missing amount ∆X is raised using risky debt issuance ex- clusively. The risky debtholders’ claim consists of a zero bond with ﬁxed maturity date T and a notional amount N (∆X, T ) as determined below. After investment, the risk- free debtholders own the rights to the existing assets G, and we can describe the risky debtholders’ position using the framework of Merton (1974): They take the remaining assets, namely the investment project, as a collateral and write a call option to equity- holders which entitles the latter to buy back the project at date T by paying the notional 7 amount of N . Thus the position of the risky debtholders consists of a long position in the underlying project and a short position in the call option with date t value of C(Vt , N, T ). The latter can be calculated using the well-known Black/Scholes formula. From now on we will leave out time t indices and only mention time T . Similar to the equity issuance case, ﬁnancing costs arise due to asymmetric information: The risky debtholders value their collateral to be worth αV instead of V . They determine the notional amount of debt N such that the amount of cash they pay out equals the value of the position they think to hold, that is ! ∆X = αV − C(αV, N, T ). (10) Since the value of a call option can never be negative, debt ﬁnancing is only possible for a cash shortage of ∆X ∈ [0, αV ]. The limiting case of ∆X = αV is obtained when debtholders choose N = ∞, which results in a worthless call option. Financing costs arise since the notional amount N is chosen higher than the debtholders actually deserve. Notice that the actual value that the risky debtholders receive is V − C(V, N, T ), while they give ∆X to the ﬁrm. We deﬁne the diﬀerence as debt issuance costs DIC(∆X, V ) = V − C(V, N, T ) − ∆X. (11) It is easy to show that the debt issuance costs are never negative, i.e. the position V − C(V, N, T ) that the risky debtholders receive is always greater than or equal to their contribution ∆X. More precisely, the debtholders’ proﬁt today, DIC(∆X, V ), is equal to the value of a portfolio which has payoﬀs greater than or equal to zero at maturity. To see this, substitute (10) into (11) yielding the portfolio DIC(∆X, V ) = V − C(V, N, T ) − (αV − C(αV, N, T )) . (12) Using (12) we can interpret the current value of debt issuance costs as the current value of a portfolio consisting of a long position in what the debtholders actually receive at issuance time and a short position in what the debtholders think that they receive. At maturity, for VT ≤ N there is default and the debtholders expect to get αVT , but they actually get the higher payoﬀ VT . Similarly, in the case N < VT < N/α the debtholders get their full loan of N instead of αVT – there actually is no default although the debtholders expected one. Only if N/α ≤ VT , there is no default and the debtholders expected none, so they get what they expected, and they make no extra proﬁt. Overall, the portfolio yields: payoﬀ in T VT ≤ N N < VT < N/α N/α ≤ VT min{VT , N } − min{αVT , N } (1 − α)VT N − αVT =0 >0 >0 8 Notice that although the current values of the positions in (11) and (12) are equal, this new portfolio’s payoﬀs at maturity are in general not equal to the payoﬀs of the original DIC position given by (11). The latter pays the debtholders’ actual proﬁt at maturity, which is min{VT , N } − ∆Xer(T −t) and can clearly become negative for small VT . 3.4 Comparison So far we have introduced equity issuance and risky debt issuance as mutually exclusive sources of funding. Now we will compare the ﬁnancing costs of these two types of funding, given that they are actually used mutually exclusive. In Section 3.5 we will then analyze ﬁnancing costs for a mix of equity and risky debt ﬁnancing. We showed above that the maximum amount of funding that can be raised using equity issuance or risky debt issuance equals the project value αV that outsiders see. We will see below that this maximum amount stays unchanged for a mix of funding as well. First we discuss the extreme cases: Obviously EIC(0) = DIC(0, V ) = 0 has to hold. For the maximum amount of funding, ∆X = αV , we also have equal costs, since the risky debtholders choose N = ∞, which implies C(V, ∞, T ) = 0 and DIC(αV, V ) = (1 − α)V = EIC(αV ). This is the maximum possible cost, and the situation can be interpreted as a project sale, as pointed out in the discussion of (7). To summarize, we have: ∆X = 0 0 < ∆X < αV ∆X = αV 1−α EIC(∆X) 0 α ∆X (1 − α)V DIC(∆X, V ) 0 V − C(V, N, T ) − ∆X (1 − α)V For any 0 < ∆X < αV we use (10) to rewrite the EIC as2 EIC(∆X) = EIC(α[V − C(V, N/α, T )]) = (1 − α)[V − C(V, N/α, T )] and from (12) we have EIC(∆X) − DIC(∆X, V ) = C(V, N, T ) − C(V, N/α, T ) > 0, i.e. for any situation in between the two extreme cases discussed above, we have that risky debt issuance is cheaper than equity issuance. Figure 1 shows shapes of EIC and DIC. If the ﬁrm has to decide between these two alternative means of ﬁnancing, it will be better oﬀ choosing risky debt ﬁnancing. In the following section we will show that this is true even if the ﬁrm may choose a ﬁnancing mix of risky debt and new equity, and we give an economic explanation for this pecking order. 2 We use the fact that C(·) is homogenous of degree one, i.e. C(αV, N, T ) = αC(V, N/α, T ). 9 Issuance Costs (1 − α)V 20 17.5 15 12.5 EIC(∆X) 10 7.5 DIC(∆X, V ) 5 2.5 ∆X 10 20 30 40 50 60 70 80 Figure 1: Comparision of risky debt and equity issuance costs. The risky debt issuance costs DIC(∆X, V ) and the equity issuance costs EIC(∆X) are plotted for diﬀerent amounts of funding ∆X. Parameter values are given in Table 1. While the equity issuance costs are linear in the amount of funding, the risky debt issuance costs are negligible for small amounts of funding, but their slope is increasing for every additional unit of funding. For the maximum amount of funding, ∆X = αV = 80, both costs reach the same maximum value of (1 − α)V = 20. 10 3.5 Mixed Financing Having compared mutually exclusive equity and risky debt ﬁnancing, we now consider raising ∆X using a ﬁnancing mix of risky debt D and new equity E, i.e. ∆X = D + E. In this case, the risky debtholders still take the project as a collateral for their contri- bution of D, and they have absolute priority in case of default. Their position therefore remains unchanged by the new equityholders’ contribution of E. The debt issuance costs DIC(D, V ) also remain unchanged as deﬁned in (11). The new equityholders contribute E, and they are oﬀered a share in the call option instead of a share in the project itself. They value the call option to be worth C(αV, N, T ) = αV − D where the equality holds according to (10). Notice that the new equityholders and the risky debtholders have exactly the same perception of project and option values. Equity issuance is possible only for E ∈ [0, αV − D] which means that we have ∆X = D + E ∈ [0, αV ], and thus the total funding capacity is not inﬂuenced by mixed ﬁnancing, as we claimed in Section 3.1. The new equityholders’ share ω is determined such that E E ω= = . (13) C(αV, N, T ) αV − D However, the actual option value is C(V, N, T ), which means that the new equityholders receive a value of ωC(V, N, T ) for their contribution of E. Therefore the new equity issuance costs amount to EIC(D, E, V ) = ωC(V, N, T ) − E. (14) We now turn to compare these new equity issuance costs (with mixed ﬁnancing) to the pure equity issuance costs. Firstly, for D = 0 we have N = 0 and therefore C(V, N, T ) = V . With E = ∆X and ω = ∆X , (14) reduces to the original EIC deﬁned by (7), αV 1−α EIC(∆X) = ω(1 − α)V = ∆X. α Below (7) we gave an interpretation of equity issuance as a project sale: The maximum possible costs are given by the diﬀerence between the true and the external valuation of the project, (1−α)V , and the actual costs are proportional to the fraction ω of the project that is sold. We rewrite (14) to EIC(D, E, V ) = ω[C(V, N, T ) − C(αV, N, T )] (15) 11 and give a similar interpretation: Understanding equity issuance as the sale of an un- dervalued call option to outside investors, we see that the maximum possible costs of C(V, N, T ) − C(αV, N, T ) apply to the extent ω that the old equityholders make use of new equity. Now we focus on the total issuance costs. First we rewrite (12) yielding (1 − α)V = DIC(D, V ) + C(V, N, T ) − C(αV, N, T ). (16) That is, the maximum possible total costs (1 − α)V can be decomposed into the debt issuance costs for a given debt amount of D and the maximum possible equity issuance costs when selling the call option received from the debtholders to outside investors. Notice that these maximum costs are independent of the debt/equity distribution, since after all the whole project is sold to outside investors in any case. But what are the total issuance costs if the ﬁrm does not issue the maximum possible amount of equity? Substituting (15) into (16) and solving for the total issuance costs T IC(D, E, V ) = EIC(D, E, V ) + DIC(D, V ), we get T IC(D, E, V ) = ω(1 − α)V + (1 − ω)DIC(D, V ). (17) As shown in Figure 2, the T IC are therefore composed as a convex combination of the DIC for the debt level D and the maximum possible cost of (1 − α)V , where the weight is the extent ω that new equity is used. We obviously have the result that the ﬁrm will never choose any equity ﬁnancing, even if a mix of risky debt and equity ﬁnancing is permitted. Formally, due to the convexity of the DIC function, any point on the DIC curve is below the corresponding convex combination of any other DIC point to its left, i.e. for a smaller amount of debt ﬁnancing, and the maximum point (αV, (1 − α)V ). The economic intuition behind this result is as follows: When issuing equity, the ﬁrm basically sells undervalued assets to outside investors. Therefore given a certain fraction that is sold to outside investors, the part of the asset value that outside investors do not see due to information asymmetries is lost. If we allow mixing, the ﬁrm sells the call option instead of the assets themselves. Still, the value that is lost is linear in the amount of equity ﬁnancing. However this is diﬀerent when issuing risky debt: The outside investors’ undervaluation is only relevant in states in which they expect default. For states in which they expect the full notional amount to be payed back, it does not matter how much they think that the project value is above their notional amount, as long as it is above the notional amount at all. Such states are more likely for small amounts of ﬁnancing. The higher the amount raised by risky debt ﬁnancing, the higher is the notional amount and thereby the probability of states in which outside investors expect default. Therefore the debt issuance costs approach the equity issuance costs when debt ﬁnancing is converging to its maximum amount of αV . 12 Issuance Costs (1 − α)V 20 17.5 15 12.5 EIC(∆X) 10 7.5 DIC(D, V ) 5 2.5 D or ∆X 10 20 30 40 50 60 70 80 Figure 2: Total issuance costs. The total issuance costs T IC(D, E, V ) are plotted for diﬀerent debt levels D (increased in steps of 10 monetary units) and changing equity levels E. In addition, the risky debt issuance costs DIC(∆X, V ) (lower bold line) and the equity issuance costs EIC(∆X) (upper bold line), are plotted for diﬀerent amounts of funding ∆X. Parameter values are given in Table 1. While the risky debt issuance costs and the equity issuance costs are the same as in Figure 1, the respective total issuance costs for a given debt level D result as a convex combination of the DIC for that debt level (on the lower bold line) and the maximum possible cost of (1 − α)V = 80, where the weights correspond to the extent that equity ﬁnancing is used. 13 To summarize, we have the following pecking order, which is consistent with the results derived in the modelling framework of Myers and Majluf (1984): The ﬁrm will use internal funds or risk-free debt (which is the safest external funding source) exclusively if possible. If it needs more funding than available using these two sources, i.e. I > X + G, it will switch to the safest security apart from risk-free debt, which is risky debt. Only for the part of funding that cannot be ﬁnanced using debt the ﬁrm will issue new equity, which is the most expensive source of funding. In our framework equity ﬁnancing is never used, since the ﬁrm has no access to equity ﬁnancing after using the maximum amount of debt ﬁnancing. Introducing an exogenous capacity restriction of less than αV for risky debt would lead to an optimal ﬁnancing policy of using risky debt ﬁnancing up to that amount and raising any additional funding using new equity, thus overall using a mix of funding. 4 Investment Policy 4.1 Problem So far we discussed the ﬁnancing policy of a constrained ﬁrm in states in which immediate investment is optimal, and we derived the optimal choice of external ﬁnancing given that internal funds were not suﬃcient to permit investment in that state. Now we turn to answer the question in which states immediate investment actually is optimal. We start by visualizing investment conditions for the whole state space, where each state is fully determined by the two state variables, the cash balance X and the project value V . Figure 3 shows that these conditions are closely connected to the funding capacity intro- duced in Section 3.1. While in Area E internal funds are suﬃcient to permit investment, the ﬁrm can invest using risk-free debt in Area D, but still there are no ﬁnancing costs. Even in Area C investment is possible, but the outsiders’ valuation of the project causes ﬁnancing costs as derived in Section 3. In Area B the ﬁrm can survive, but immediate investment is not possible. Finally, as soon as Area A is reached, immediate liquidation takes place. In the following we will derive investment thresholds for the constrained ﬁrm, which are dependent on the ﬁrm’s cash balance. It will be useful to refer to the border between Areas B and C, which is the condition X + G + αV ≥ I as given by (6), in terms of the minimum investment threshold that the constrained ﬁrm is allowed to choose: ˆ I − (X + G) ∆X V cm (X) = = . (18) α α We ﬁrst assume that the ﬁrm uses risky debt ﬁnancing in Area C. Then the free boundary ˆ above which immediate investment is optimal is denoted by V d (X). It is chosen by the ﬁrm in order to maximize the investment option value, subject to the ﬁnancing quantity constraint that ˆ ˆ V d (X) ≥ V cm (X). (19) 14 300 250 200 project value V 150 X+G+Fu(γV) ≤ 0 X+G+αV ≥ I X+G ≥ I X≥I 100 A B C D E 50 ˆ V cm (X) = ∆X/α 0 −250 −200 −150 −100 −50 0 50 100 150 cash balance X Figure 3: State Space and Investment Conditions. In the two-dimensional state space (cash balance X and project value V ) we identify ﬁve areas characterized by diﬀerent investment conditions. Parameter values are given in Table 1. Area A is bounded by the ﬁrm’s solvency constraint. Areas B and C are ˆ separated by V cm (X), which is the minimum project value required for investment. The border between Areas C and D is the ∆X = 0 line, which means that in Area C there are costs of risky debt and equity ﬁnancing, while there are no such costs in Area D. As soon as the ﬁrm’s cash balance X exceeds the investment amount I, we are in Area E where internal funds are suﬃcient to permit investment. 15 ˆ For all states in which it is optimal to postpone investment, i.e. for all V < V d (X), one d can show using standard replication arguments that the option value F (X, V ) satisﬁes 1 2 2 d 1 σ V FV V + ρσφV GFXV + φ2 G2 FXX + (r − δ)V FV + r(X + G)FX = rF d . d d d d (20) 2 2 If immediate investment is optimal, we have ¯ ˆ F d (X, V ) = F d (X, V ) ∀V ≥ V d (X), (21) where the pure exercise value of V − I is reduced to ¯ F d (X, V ) = V − I − DIC(∆X, V ) (22) ˆ due to the risky debt issuance costs. Determination of the free boundary V d (X) is equiv- alent to an optimal stopping problem, since the decision to invest can also be seen as a decision to stop waiting. However we emphasize that we do not derive an explicit stop- ping time, since a state is fully characterized by (X, V ) in our model and does not depend on time. But still, continuity of V implies that a higher project value will be reached later, given a speciﬁc (lower) current project value. So we can interpret higher and lower investment thresholds as policies inducing later and earlier investment, respectively. The other boundary conditions that determine (20) are as follows: For a worthless project, we also have a worthless option, therefore F d (X, 0) = 0 ∀X. (23) As soon as the cash balance becomes so negative that it cannot be covered by the ﬁrm’s risk-free debt capacity of G + F u (γV ), the ﬁrm suﬀers an enforced liquidation, and we have F d (X, V ) = F u (γV ) ∀X < −[G + F u (γV )]. (24) On the other hand, a ﬁrm with extensive cash holdings has an investment option whose value approaches that of an unconstrained ﬁrm: lim F d (X, V ) = F u (V ). (25) X→∞ An analogous problem arises when the ﬁrm uses new equity instead of risky debt.3 In this case, the exercise value is given by ¯ ˆ F e (X, V ) = F e (X, V ) = V − I − EIC(∆X) ∀V ≥ V e (X), (26) ˆ with V e (X) being the respective free boundary. The ﬁnancing constraint (19) still has ˆ to be satisﬁed for V e (X). In states in which waiting is optimal, (20) holds just as for F d (X, V ), as well as the boundary conditions (23) to (25). Before we analyze the investment policy of a ﬁrm facing ﬁnancing costs, we will introduce two benchmark cases. 3 Although we found in Section 3 that risky debt is preferred to equity ﬁnancing, we also examine the situation for a ﬁrm using only risk-free debt and new equity, but no risky debt, e.g. since the ﬁrm has no access to risky debt markets. 16 4.2 Benchmark Cases 4.2.1 The Unconstrained Firm We saw in (25) that the situation of an unconstrained ﬁrm can be interpreted as the limit case of a constrained ﬁrm with a cash balance X being high enough that the probability of suﬀering ﬁnancing restrictions or even a forced liquidation can be neglected. Then, the investment option value does not depend on the cash balance X, and (20) simpliﬁes to 1 2 2 u ˆ u σ V FV V + (r − δ)V FV = rF u ∀V < V u . (27) 2 for the unconstrained option value F u (V ). We have the free boundary condition ¯ F u (V ) = F u (V ) = V − I ˆ ∀V ≥ V u , (28) combined with the usual value matching and smooth pasting conditions at the free bound- ary, and the worthless project boundary condition (23) still holds, i.e. F u (0) = 0. The ﬁrm then faces a simple trade-oﬀ between the value of waiting due to increasing information about the project value and the costs of waiting due to foregone cash ﬂows. McDonald and Siegel (1986), and later Dixit and Pindyck (1994) in a simpliﬁed version, derived a solution for this option value. Since the rights to the project are perpetual and the ﬁrm’s cash balance X does not aﬀect valuation, the only remaining state variable is ˆ the project value V . The ﬁrm chooses the critical project value V u above which immediate investment is optimal, i.e. the investment threshold, in order to maximize today’s value F u (V ). The solution to the problem is an investment threshold of ˆ βI Vu = , β−1 which results in an investment option value of 1−β β u I V ˆ F (V ) = ∀V < V u , β−1 β with 2 1 r−δ 2r 1 r−δ β= − + + − . 2 σ2 σ2 2 σ2 Figure 4 depicts the typical shape of the unconstrained ﬁrm’s investment option value. The distance between the option value and the exercise value, i.e. F u (V ) − (V − I), (29) is the time value of the investment option. In can also be used to get a diﬀerent interpre- tation for the exercise value of a ﬁrm facing ﬁnancing costs. 17 150 F u (V ) V −I 100 50 0 ˆ Vu −50 −100 0 50 100 150 200 250 project value V Figure 4: The Unconstrained Firm’s Investment Option. The value of the unconstrained ﬁrm’s investment option F u (V ) and the payoﬀ V − I given immediate exercise are plotted for diﬀerent project values V . Parameter values are ˆ I = 100, σ = 0.2, δ = 0.03, and r = 0.03. For V ≥ V u = 222 immediate exercise is u optimal, and the option value F (V ) equals the exercise value V − I. Otherwise the value of waiting to invest dominates the opportunity costs of waiting, and the option value exceeds the exercise value. 18 We rewrite this exercise value given in (22) to ¯ F d (X, V ) = F u (V ) − DIC(∆X, V ) − [F u (V ) − (V − I)]. That is, there are two reasons why the exercise value of a ﬁrm facing ﬁnancing costs is below the unconstrained ﬁrm’s option value: First, it is reduced by ﬁnancing costs (DIC in the risky debt issuance case). The second component (in square brackets) is the time value that an unconstrained ﬁrm would enjoy for the given project value, and that is lost by following the early exercise policy of the constrained ﬁrm. Figure 5 discusses all possible cases within the state space that was introduced in Figure 3. In addition to the well-known X = 0 line below which there are ﬁnancing costs, we ˆ plot the unconstrained threshold V u that we have just derived, as well as the investment ˆ threshold of a constrained ﬁrm using risky debt ﬁnancing V d (X), as it will result in the ˆ numerical solution below. The area above V d , where immediate investment is optimal for the risky debt ﬁnancing ﬁrm, is divided in four quadrants by the two straight lines: ˆ For X < 0 and V ≥ V u there are ﬁnancing costs DIC(∆X, V ), but no time value is lost since an unconstrained ﬁrm would also invest immediately. For X ≥ 0 and V < V u ˆ there are no ﬁnancing costs upon investment, but the constrained ﬁrm loses the time value F u (V ) − (V − I) due to its accelerated investment. If both X < 0 and V < V u ,ˆ ¯ d u we conclude that F (X, V ) < F (V ) both due to ﬁnancing costs and loss in time value, ¯ while in the remaining quadrant we have F d (X, V ) = F u (V ) = V − I, and there is no loss at all due to ﬁnancing constraints in these states. 4.2.2 The Pure-Quantity Constrained Firm As a second benchmark case we consider a ﬁrm facing no ﬁnancing costs, i.e. in Area C it can use a fraction α of the project value at no cost as a collateral for raising external ﬁnancing. But still it can invest only if the condition X + G + αV ≥ I as given by (6) is satisﬁed, therefore it faces a pure ﬁnancing quantity constraint. This case was ﬁrst analyzed by Boyle and Guthrie (2003). One interpretation could be that outside investors know for sure that the ﬁrm can extract a value of exactly αV upon investment, and therefore are willing to issue risk-free debt against αV , which causes no ﬁnancing costs. This case is in between the ﬁrst benchmark case of an unconstrained ﬁrm and the situation of a ﬁrm facing ﬁnancing costs considered in this paper. We denote its investment option value by F c (X, V ). We still have to respect ˆ the ﬁnancing constraint (19) for its threshold V c (X). In states in which waiting is optimal, d (20) still holds just as for F (X, V ), as well as the boundary conditions (23) to (25). But since we have no ﬁnancing costs, ¯ F c (X, V ) = F c (V ) = V − I ˆ ∀V ≥ V c (X), (30) so the exercise value is the same as for an unconstrained ﬁrm. In states in which waiting is optimal, however, the pure-quantity constrained ﬁrm’s option value is below that of an unconstrained ﬁrm: The option’s time value is lower for the pure-quantity constrained 19 350 300 V − I − DIC < V − I V −I 250 ˆ Vu project value V 200 ˆ Vd 150 V − I < F u (V ) 100 V − I − DIC < F u (V ) 50 0 −300 −200 −100 0 100 200 cash balance X Figure 5: Exercise Value, Time Value, and Financing Costs. ˆ The investment threshold of a constrained ﬁrm using risky debt ﬁnancing (V d (X), thick) is plotted for diﬀerent values of cash balance X. Additionally, the X = 0 line and the ˆ investment threshold of the unconstrained case, V u , are shown. Parameter values are given in Table 1. Given that immediate exercise is optimal for the risky debt ﬁnancing ﬁrm, ˆ i.e. V > V d (X), we distinguish four diﬀerent regions in the state space, separated by the two straight lines: The exercise value of the constrained ﬁrm is below the unconstrained ˆ value due to ﬁnancing costs for X < 0, and due to foregone value of waiting for V < V u . While both of these apply if both the project value and the cash balance are low, the exercise value is V − I for both types of ﬁrms if there are no ﬁnancing costs and if even the unconstrained ﬁrm wants to invest immediately. 20 ﬁrm, since it follows exercise policies that are suboptimal compared to an unconstrained ﬁrm. 4.3 Results Our goal now is to analyze investment policies in the costly ﬁnancing cases and compare them to the two benchmark cases. However, the interdependence of the two state variables X and V given by the boundary conditions precludes an analytical solution for all cases but the benchmark case of an unconstrained ﬁrm. Therefore we use a numerical method in order to solve (20). The procedure we use yields an iterative solution based on ﬁnite diﬀerence methods, and except for changes to account for ﬁnancing costs according to (22) and (26) instead of (30), it follows the procedure described in Boyle and Guthrie (2003). The parameter values that we use are given in Table 1. Table 1: Parameter Values used in Numerical Examples. Values are identical to those in Boyle and Guthrie (2003) (except for the risky debt maturity, which is not used there). Parameter Value Project investment cost ($) I = 100 Project value volatility σ = 0.2 Project cash-ﬂow rate δ = 0.03 Riskless interest rate r = 0.03 Project value – ﬁrm cash ﬂow correlation ρ = 0.5 Cash ﬂow volatility φ = 0.6 Market value of existing assets ($) G = 100 Market friction α = γ = 0.8 Risky debt maturity (years) T =5 The resulting investment thresholds are shown in Figure 6. Compared to the visualization of the state space introduced in Figure 3, we only took up the boundaries of Area C, ˆ namely V cm (X), below which investment is not possible, and ∆X = 0, above which there ˆ are no ﬁnancing costs. The threshold of the unconstrained ﬁrm, V u , is a horizontal line, since it does not depend on the cash balance X. In the following we analyze the thresholds of the three ﬁnancially constrained cases, which have by far more interesting shapes. We point out that they are signiﬁcantly diﬀerent from what might be an intuitive guess, namely that the investment threshold should be monotonically decreasing in the ﬁrm’s initial cash balance, i.e. that less constrained ﬁrms always invest earlier. This is even the result of some previous work, e.g. Lyandres (2003). Since the latter uses a discrete-time 3-date model, he does not fully capture the dynamic properties and the value of waiting that a less constrained ﬁrm enjoys by postponing investment. 21 350 300 ˆ V ul 250 ˆ Ve ˆ Vu project value V 200 ˆ Vd 150 ˆ Vc 100 ˆ V cm 50 0 −300 −200 −100 0 100 200 cash balance X Figure 6: The Constrained Firm’s Investment Threshold. The value of the constrained ﬁrm’s investment threshold is plotted for diﬀerent values of cash balance X. We distinguish the pure-quantity constrained ﬁrm with costless ﬁnancing ˆ ˆ (V c (X), dotted) and the costly ﬁnancing cases using risky debt (V d (X), dark) or new equity (V ˆ ˆ e (X), light), respectively. Additionally, the boundaries of Area C, V cm and ˆ ∆X = 0, and the investment threshold of the unconstrained case, V u , are shown, as well ˆ ul as the threshold after liquidation, V . Parameter values are given in Table 1. All the three constrained cases have a minimum threshold. For higher cash balance, the value of waiting dominates the risk of future ﬁnancing constraints and raises the investment ˆ threshold. For lower cash balance, the exogenous lower bound V cm becomes binding in the pure-quantity constrained case, or current ﬁnancing costs become prohibitively high and raise the investment threshold. 22 First, we discuss the threshold of the pure-quantity constrained ﬁrm with costless ﬁnanc- ˆ ing, V c (X). Compared to an unconstrained ﬁrm, we recognize two areas where the con- ˆ strained ﬁrm has to invest suboptimally: Above V u and for lower cash balance than at the ˆ u ˆ c intersection of V and V (X), the constrained ﬁrm is forced to delay investment, although an unconstrained ﬁrm would invest immediately. For higher cash balance than at that in- ˆ ˆ tersection and between the thresholds V c (X) and V u , we observe that the pure-quantity constrained ﬁrm invests earlier, i.e. for smaller project values, than an unconstrained ﬁrm. As Boyle and Guthrie (2003) point out, this voluntary acceleration occurs in order to avoid future funding shortfalls. For low cash balance, the constrained ﬁrm invests as soon as ˆ ˆ possible, i.e. V c (X) = V cm (X). Although this is an endogenous decision, notice that we have a boundary solution: The ﬁrm invests as soon as the exogenous constraint V cm (X) ˆ allows to do so. Above a certain level of cash balance we observe that the value of waiting becomes more important than the risk of future funding shortfalls, and therefore V c (X) ˆ starts rising in X. As we expected, for very large X the ﬁnancing constraint is unlikely to become important anymore, and therefore the constrained ﬁrm’s threshold approaches that of an unconstrained ﬁrm. ˆ Second, we compare the risky debt threshold V d (X) to that of the pure-quantity con- strained ﬁrm, V ˆ ˆ ˆ c (X). We similarly start at the intersection of V d (X) and V c (X). For lower cash balance than at that intersection there is voluntary investment delay: The risky debt ﬁnancing ﬁrm does not invest although the pure-quantity constrained ﬁrm does, and although investment is feasible, i.e. the ﬁnancing constraint (19) is satisﬁed. Investment is postponed in order to avoid prohibitively high ﬁnancing costs. Notice that this is an endogenous decision, which is in contrast to the forced delay that we observed ˆ above. Graphically, this can be shown by the fact that the threshold V d (X) leaves the ˆ cm exogenous minimum threshold V (X) while it is still falling in X, and that especially at the new critical level of cash balance where the value of waiting becomes more important ˆ than the risk of future ﬁnancing costs and V d (X) starts rising from its minimum point, it is signiﬁcantly above V ˆ cm (X). The policy for cash balance levels above the intersection of ˆ ˆ V d (X) and V c (X) can be described as accelerated investment: Even though the risky debt ﬁnancing ﬁrm faces costs, immediate investment takes place in order to avoid even higher ﬁnancing costs in the future. In contrast, the pure-quantity constrained ﬁrm does not invest immediately, although it has more favorable investment conditions in these states. ˆ ˆ Just as in the comparison of V u and V c (X), it may be surprising at ﬁrst glance that the ﬁrm facing stronger constraints invests earlier. Finally, for very large X the ﬁnancing constraint is again unlikely to become important anymore, and therefore the risky debt ﬁnancing ﬁrm’s threshold approaches that of the pure-quantity constrained ﬁrm, and in the end that of an unconstrained ﬁrm. For the third case of an equity ﬁnancing ﬁrm, the shape of the optimal investment thresh- old looks quite diﬀerent: The only obvious resemblance to the cases discussed above is that ˆ the threshold still approaches the minimum possible value of V cm (X) for very low values ˆ u of cash balance and the unconstrained threshold V for suﬃciently high cash balance. For low levels of cash balance we observe that the threshold is ﬁrst close to a project value ˆ of about V ul = 278, then it is smoothly decreasing. At X = 0 it jumps down far below 23 ˆ the unconstrained threshold V u and afterwards smoothly approaches this threshold for increasing cash balance. ˆ The threshold V ul that we have just introduced refers to the investment decision of an unconstrained outside investor holding the constrained ﬁrm’s investment option after liq- uidation. It can be shown4 to be ˆ ˆ V ul = V u /γ. (31) ˆ Since equity ﬁnancing costs are substantial, the equity ﬁnancing ﬁrm’s threshold V e (X) ˆ approaches V ul when the cash balance becomes negative and ﬁnancing costs come into ef- fect. This means that for low cash balance, the equity ﬁnancing ﬁrm follows an investment policy very similar to that of an unconstrained outside investor who owns the constrained ˆ ﬁrm’s investment option after liquidation: Postpone investment for V < V ul , and on the ˆ ul other hand invest even for small cash balance as soon as V ≥ V . This behavior can be explained by the fact that for low cash balance, equity issuance costs are so prohibitively high that the ﬁrm owns a value that is close to the liquidation value, and therefore also adopts the policy of an investor owning the liquidation value. In terms of project sale, as introduced in the discussion of (7), one could regard the project as virtually sold to outside investors, who would then exactly face the same situation as after liquidation, as long as the friction in case of liquidation γ equals the ﬁnancing friction α. ˆ Now we explain the jump of V e (X) at X = 0, i.e. we explain why there is a certain range of project values ˆ ˆ V e (0 + ǫ) < V < V e (0 − ǫ) (with ǫ being a small positive number) in which the ﬁrm’s policy switches between waiting and immediate investment when, for a given V , the X = 0 line is crossed. For that purpose, we examine the exercise value of the equity ﬁnancing ﬁrm’s investment option as given by (26). The partial derivative of that value with respect to the cash balance is ∂(V − I − EIC(∆X)) 1−α = · 1X<0 , ∂X α where the equity issuance costs EIC(∆X) are deﬁned as in (7), and 1 is an indicator function. While a marginal unit of cash raises the exercise value by 1−α for negative X, α it has no eﬀect for positive cash balance, so there is a kink in the marginal value of cash. This kink in the exercise value boundary condition induces a jump in the investment ˆ threshold V e (X). Notice that the usual value matching and smooth pasting of the option value and the exercise value at the free boundary are still satisﬁed separately for positive and negative cash balance. That is, for negative cash balance the option value merges into a plane that has a slope of 1−α in X direction, while for positive cash balance the corresponding plane α is parallel to the X axis. 4 For derivation we use the liquidation value F u (γV, I) = γF u (V, I/γ) of the constrained ﬁrm’s investment option. Here we explicitly mention the exercise price as an argument, and we use the fact that F u (V, I) is homogenous of degree one. 24 ˆ ˆ An economic explanation for the policy switch in the range V e (0 + ǫ) < V < V e (0 − ǫ) is that for negative cash balance, each additional unit of cash balance raises the option value signiﬁcantly by reducing ﬁnancing costs, and therefore the value of waiting dominates the threat of future ﬁnancing costs, should the cash balance further decrease. The ﬁrm therefore requires a relatively high project value to justify immediate investment for X < 0. For increasing cash balance, however, as soon as X = 0 is reached, each additional unit of cash balance does not raise the exercise value, but it only makes future ﬁnancing costs less likely. In this situation, the value of waiting consists solely of the chance of better project values, and no longer of a possible reduction in ﬁnancing costs. Should the cash balance decrease again by one marginal unit, however, the cost of waiting is unalteredly signiﬁcant. Therefore there is a strong incentive for the ﬁrm to invest immediately as soon as X = 0 is reached, even for relatively low project values. ˆ However, for suﬃciently low project values V < V e (0 + ǫ), immediate investment is unfavorable even for X > 0, and the ﬁrm decides to postpone investment on both sides of X = 0. Although there are no present ﬁnancing costs for positive cash balance, but rather the threat of future ﬁnancing costs, the ﬁrm is better oﬀ waiting for a better ˆ project value. For suﬃciently high project values V > V e (0 − ǫ), respectively, immediate investment is favorable on both sides of X = 0. In these states, the current project value and its foregone cash-ﬂows dominate possible future savings that could be achieved by ˆ ˆ waiting for increasing cash balance. Only for V e (0 + ǫ) < V < V e (0 − ǫ), we observe the policy switch described above at X = 0. In all other cases discussed so far but the equity issuance case, there is no kink in the exercise value. This is obvious for the unconstrained and the pure-quantity constrained case, since the exercise value does not at all depend on X. For the risky debt ﬁnancing ﬁrm, it can be seen in Figure 1 that the risky debt issuance costs smoothly approach the abscissa for ∆X → 0, and therefore there is also no kink in the exercise value. Therefore we do not observe jumps in the investment threshold for all these cases. Overall, we ﬁnd that equity issuance is used only for very favorable project values. Besides an actual relevance as a source of funding, its more remarkable eﬀect is that it signiﬁcantly drives down the investment threshold of a ﬁrm without access to debt markets even for the area of positive cash balance, where there are no ﬁnancing costs at all, but the ﬁrm anticipates the ﬁnancing costs in case of funding shortfalls in the future. Figure 7 shows option values as a function of the cash balance X for selected project values of V = 100, V = 180, and V = 230. For each project value, we show the option value of a pure-quantity constrained ﬁrm with costless ﬁnancing, F c (X, V ), and of ﬁrms using risky debt or equity ﬁnancing, F d (X, V ) or F e (X, V ), respectively. Notice that for each project value, all option values have common boundary values: For very low cash balance it is the liquidation value F u (γV ), and if the cash balance is high enough that future ﬁnancing constraints are unlikely to become important, it is the unconstrained option value F u (V ). First, consider a project value of V = 100. As we see in Figure 6, it is optimal to wait regardless of the cash balance, since this project value is far below the thresholds of all 25 140 120 100 V = 230 option value Fc,d,e(X,V) 80 V = 180 60 40 V = 100 20 0 −300 −200 −100 0 100 200 cash balance X Figure 7: The Constrained Firm’s Option Value. The constrained ﬁrm’s option value is plotted for selected project values of V = 100, V = 180, and V = 230, and diﬀerent values of cash balance X. We distinguish the pure-quantity constrained ﬁrm with costless ﬁnancing (F c (X, V ), dotted) and the costly ﬁnancing cases using risky debt (F d (X, V ), dark) or new equity (F e (X, V ), light), respectively. Parameter values are given in Table 1. For each project value, the boundary value for low cash balance is the liquidation value F u (γV ). Then the option value rises to the exercise value. In cash balance areas where immediate exercise is optimal, the option value remains constant at the exercise value for the pure-quantity constrained case, while it still rises for the costly ﬁnancing cases due to decreasing ﬁnancing costs. For even higher cash balance, the option value rises to the unconstrained option value F u (V ). 26 cases. The value of the investment option is aﬀected mainly by the low project value and by the doubt if investment ever becomes favorable and feasible at all, and not very much by the question if there are ﬁnancing costs given that investment once has become favorable and feasible. Therefore, the costly ﬁnancing option values are hardly below the pure-quantity constrained option value. In the numerical calculation, they can even become equal. For V = 180 we see in Figure 6 that all the constrained thresholds are intersected, therefore for each case there is a certain X value above which immediate investment is optimal and the option value equals the exercise value. While the option value remains constant for the pure-quantity constrained case, it still rises for the other cases due to decreasing ﬁnancing costs. For even higher X values there is a point above which the ﬁrm is better oﬀ waiting. We see that the option value again rises from the exercise value to the unconstrained value. ˆ V = 230 is a value above the unconstrained threshold V u . Therefore in the pure-quantity constrained case, the option value jumps up from the liquidation value to the exercise value (actually it rises through Area B, which though is very narrow at V = 230). For the costly ﬁnancing cases and low cash balance, waiting is still optimal due to ﬁnancing costs. The option value rises to the exercise value with increasing cash balance as costs decrease. For V = 230, risky debt ﬁnancing is used within nearly the whole range where it is possible at all, and we can see well that the shape of the option value curve resembles much the inverse of the debt issuance cost curve (Figure 1). 5 Conclusion We analyzed the investment timing decision of a ﬁnancially constrained ﬁrm with costly external ﬁnancing. To do so, we ﬁrst deﬁned equity and risky debt issuance costs. They were justiﬁed by the fact that although outside investors are able to correctly assess the value of the ﬁrm’s existing assets, they undervalue the new investment project due to information asymmetries. Securities that are issued against the project value therefore transfer value from the old equityholders to risky debtholders and new equityholders. Our cost deﬁnitions imply that equity issuance costs are proportional to the size of the issue. Equity issuance can be interpreted as the sale of a fraction of the project at an undervalued price to outside investors. In contrast, risky debt is fairly cheap for small to medium size issues. Only for issue sizes close to the full project value that outside investors see, the issuance costs approach those of equity. This overall advantage of debt against equity can be explained by the fact that the outside investors’ undervaluation is relevant only in future states in which there is default in the outside investors’ perspective. States in which the outside investors expect no default, which are more likely for smaller issues, have no contribution to the ﬁnancing costs. The conclusion that the ﬁrm prefers risky debt to equity holds even if we allow mixed ﬁnancing. In this case we show that the total costs are a convex combination of the 27 respective risky debt issuance costs and the maximum possible equity issuance costs. Overall, the modelling is consistent with the pecking order put forward by Myers and Majluf (1984): The ﬁrm prefers internal funds and risk-free debt to risky debt, and uses new equity only as a last resort. Then we analyzed the investment threshold of a constrained ﬁrm facing costly external ﬁnancing. Benchmark cases were an unconstrained ﬁrm and a pure-quantity constrained ﬁrm that can also use a part of the project value as a collateral for costless ﬁnancing, as introduced by Boyle and Guthrie (2003). We got an interior solution for the investment threshold, which is in contrast to Boyle and Guthrie (2003). They have a boundary solution, and for low cash balance, the ﬁrm invests as soon as it is feasible. We derive an interior minimum point where ﬁnancing costs induce the most accelerated investment. For lower cash balance, the ﬁrm waits longer due to higher ﬁnancing costs, and for higher cash balance, it waits longer since the risk of a costly funding shortfall is suﬃciently low, and the ﬁrm can enjoy the value of waiting and observing the evolution of the project value. Compared to a pure-quantity constrained ﬁrm, we observed both voluntary delay and acceleration of investment, while an intuitive guess might be that more constrained ﬁrms always invest later. This can be explained by the ﬁrm’s objective to avoid current or fu- ture ﬁnancing costs, respectively. The result is similar to Boyle and Guthrie (2003), who compare the pure-quantity constrained ﬁrm to an unconstrained ﬁrm: They ﬁnd forced delay and voluntary acceleration of investment, which they explain by a binding current quantity constraint and by the ﬁrm’s objective to avoid future quantity constraints, re- spectively. However, the results are not consistent with Lyandres (2003), who postulates that the investment threshold is monotonically decreasing in the ﬁrm’s initial cash bal- ance. Since he uses a discrete-time 3-date model, he does not fully capture the dynamic properties and the value of waiting. We found that ﬁrms restricted to equity ﬁnancing hardly use costly ﬁnancing unless the project value is so favorable that even an external investor who receives only the liquidation value would also invest immediately. Firms with access to risky debt make by far more use of their ﬁnancing capacities. To conclude, we showed that ﬁnancing constraints, in the form of both quantity constraints and ﬁnancing costs, have a signiﬁcant impact on investment policy, and we extended the pure-quantity constrained ﬁrm’s case given Boyle and Guthrie (2003) for the presence of ﬁnancing costs. We found that either form of ﬁnancing constraint can lead to both voluntary delay and acceleration of investment. The importance of suﬃcient internal funds gives rise to the question how it can be ensured that these are available when necessary. For example, Froot, Scharfstein, and Stein (1993) take into account the correlation between the ﬁrm’s cash ﬂows and the value of investment opportunities and show how hedging can add value to the ﬁrm. In general, endogenizing the dynamics of the cash balance gives rise to a diﬀerent strand of literature. 28 References Boyle, Glenn W., and Graeme A. Guthrie, 2003, Investment, Uncertainty, and Liquidity, Journal of Finance 58, 2143–66. Boyle, Glenn W., and Graeme A. Guthrie, 2004, Hedging the Value of Waiting, Working Paper, http://ssrn.com/abstract=427900. Clementi, Gian Luca, and Hugo Hopenhayn, 2002, A Theory of Financing Constraints and Firm Dynamics, NYU Stern 2004 Working Paper No. 04-25. Dixit, Avinash, and Robert Pindyck, 1994, Investment under Uncertainty. (Princeton University Press Princeton, NJ, USA). Froot, Kenneth A., David S. 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