Optimal Investment Timing When External Financing Is Costly Stefan Hirth ∗ Ma by yhb15493

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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
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