The E ects of Risky Debt on Investment under Uncertaintyhompi.sogang.ac.kr/peteryou/peter_risky...
Transcript of The E ects of Risky Debt on Investment under Uncertaintyhompi.sogang.ac.kr/peteryou/peter_risky...
The Effects of Risky Debt on Investment underUncertainty
Seung Dong You∗
December 18, 2012
Abstract
This paper investigates investment and disinvestment decisions when an in-vestor finances debt to fund the lump-sum cost at the time of investment. Thestudy examines investment timing decisions in a frictionless financial market. Byextending the model presented in Dixit (1989), this paper argues that, as riskydebt increases, an investor’s trigger price for investment decreases while the trig-ger price for disinvestment increases. Such an investor hastens both investmentand disinvestment decisions with risky debt. This paper focuses on stand-alonefinancing rather than expansion financing, as in Lyandres and Zhdanov (2010).
JEL classification: G30; G31Keywords: Investment; Leverage; Real Option; Hysteresis
∗I thank two anonymous referees, Lorenzo Garlappi, Chang-Soo Kim (Editor), Jaehyon Lee (AKFAsdiscussant) and participants at the 2011 AKFAs Conference for helpful discussions and suggestions. Atravel support from the KAFA is is gratefully acknowledged. All errors are mine.
1
1 Introduction
Under uncertainty, an investor’s investment decisions are characterized by “hysteresis”;
with uncertainty an investor’s trigger price for investment increases while the trigger
price for disinvestment decreases because uncertainty makes waiting valuable (Dixit,
1992). Investment and disinvestment can be defined as entry into and exit from in-
vestment, respectively (Dixit, 1989). In order to invest in an irreversible project, an
investor can pay a lump-sum cost, which may be capital-intensive. To fund the cost,
such an investor often finances debt in the capital market. For example, Berger and
Udell (1998) show that the debt-to-equity ratio for US small businesses is on average
1.01.1 Esty (2004) reports that project companies carry on average a 70% debt-to-total-
capitalization ratio.2
The main objective of the paper is to examine the effects of debt (newly issued at
the time of an investment) on the timing of investment and disinvestment decisions for
leveraged investors. We restrict our attention to investment decisions with stand-alone
financing rather than investment decisions with expansion financing, as in Lyandres and
Zhdanov (2010). With no assets-in-place to consider, the (equity) investor posited in
the paper acquires newly issued debt to finance the cost of an investment. We assume
that our investor issues a new debt instrument with a fixed coupon per unit of time at
the time of investment; our goal is not to determine an optimal capital structure, but
rather to analyze leveraged decisions for investment and disinvestment. Our assumption
of exogenous debt is consistent with assumptions made in previous corporate finance
studies.3 As a result, we can highlight the effects of risky debt on investment timing
decisions.4
1The figure is for nonfarm, nonfinancial, and nonreal-estate small businesses; debt is provided byfinancial institutions such as commercial banks and finance companies, nonfinancial institutions andgovernments, and individuals, including credit cards.
2This figure represents the mean debt-to-total-capitalization ratio according to the book valuesof 1,050 project companies that were on the Thompson Financial Securities Data Project FinanceDatabase (TFSD) and were financed between 1990 and 2001. For the purpose of a single project,project companies, as legally independent entities, finance nonrecourse debt.
3See Geske (1979), Mella-Barral and Perraudin (1997), Mauer and Sarkar (2005), Sarkar (2007) andLyandres and Zhdanov (2010).
4A large body of literature in corporate finance addresses interactions between an investment deci-sion and a financing decision. A recent development with respect to real options has been successfullyincorporating two traditional agency problems: Jensen and Meckling’s (1976) overinvestment and My-ers’s (1977) underinvestment.
2
We work backward in order to solve a leveraged investor’s problem. The critical
level of the price that triggers disinvestment decisions increases with risky debt (Mauer
and Sarkar, 2005; and Jou, 2001). The value of waiting for disinvestment increases
with leverage, as such an investor becomes exposed to financial risks. We assume that
a disinvestment decision for a (leveraged) investor is a strategic default decision such
that, when the value of the investment reaches the value of the debt, the investor
defaults by ceasing to execute a predetermined coupon payment. Notice that an equity
value maximizer’s disinvestment decision differs from a value maximizer’s disinvestment
decision, which is to abandon investment.
On the other hand, the critical price level that triggers investment decisions decreases
with risky debt (Lyandres and Zhdanov, 2010). The value of a call option related to the
risk of uncertainty is reduced with leverage; the value of an investment that includes
an uncertainty premium is distributed between an investor and a lender. The value of
equity decreases with an increase in leverage and the reduced equity effect dominates the
financial risk effect. Waiting produces value for both an investor and a lender because
the prices of the underlying asset provide information about the likelihood of default. A
leveraged investor nonetheless takes no account of the value of waiting that is available
to a lender.
We make two specific contributions to the literature. First, we show that, under
uncertainty, the zone between the investment trigger and the disinvestment trigger for
an investor shrinks with risky debt. Risky debt encroaches on Dixit’s (1989) zone
between the investment trigger and the disinvestment trigger.5 Second, we prove that,
as the amount of risky debt that an investor takes out increases, the value of waiting to
invest decreases while the value of waiting to disinvest increases. By creating the simplest
possible model, we provide closed-form expressions for the values of an investment option
and a disinvestment (or default) option that are available to an investor. Consequently,
we stress the effects of risky debt on leveraged decisions with regard to investment and
disinvestment.
We make a key assumption in our analysis that a leveraged investor maximizes the
value of an equity investment. Our model differs from that of Mauer and Ott (2000) and
5At the personal level, consumers will be encouraged to purchase automobiles, mobile homes, andother goods with non-recourse installment financing, and also they are more likely to default on thelarge outstanding balance of a loan.
3
Jou (2001), who posit an investor who maximizes the total value of an investment. Like
Myers (1977), Mauer and Ott (2000) consider an investor who exercises a growth option
with equity alone. Like investors posited by Jou (2001) and Mauer and Sarkar (2005),
however, our investor exercises such an option with a mix of debt and equity. Our
investor nonetheless should be distinguished from Jou’s (2001) investor, who maximizes
the project value, which is determined by the tax advantage of debt, or from Mauer’s
and Sarkar’s (2005) equity value maximizer, who takes advantage of an interest tax
shield. Abstracting from the tax issue, we study a leveraged investment problem in the
absence of a mechanism through which leverage changes the value of an investment. In
the paper, an investment decision is determined by risky debt, even though the total
value of an investment is invariant with respect to the debt.
In the spirit of Geske’s (1979) compound option, we analyze a call option on equity.
In the case of a compound option, uncertainty risks and financial risks interact; the
uncertainty risks are related to fluctuations in the price of an underlying asset, while
the financial risks originate from the financial status of an investor. Unlike Geske (1979)
who provides valuation for a European call option on stocks, however, we approach an
investment problem within an optimal stopping framework.
Our results are similar to those reported in previous papers that have studied over-
investment within a real option framework. With tax and bankruptcy costs, however,
Mauer and Sarkar (2005) estimate the size of agency cost with leverage; for empirical
evidence of a relationship between corporate tax and financial structure, see Jung and
Kim (2008) and Ko and Yoon (2011). Yagi et al. (2008) investigate the timing of in-
vestment with convertible debt. Lyandres and Zhdanov (2010) propose an option to
invest with assets-in-place. To isolate an underinvestment problem for existing capi-
tal structure, Lyandres and Zhdanov (2010) propose cases that involve neither wealth
expropriation nor taxes. Previous papers have moved towards covering realistic but
complicated structures, which should be analyzed numerically. Nonetheless, our ap-
proach is distinguishable from previous approaches, because for stand-alone financing
we have no need to consider existing shareholders and we can innocuously abstract
from sources of market friction such as taxes and bankruptcy cost.6 In doing so, the
6When disinvesting, our (equity) investor transfers a project “costlessly” to a lender. The mainimplications of the model remain invariant with the exit cost. Dixit (1989) shows that hysteresis exists,even though the sunk cost of exit is zero.
4
paper proposes a more realistic and simple structure, which increases our understand-
ing of leveraged investments; unlike previous studies, this paper employs closed-form
expressions for leveraged investments, with which we investigate a leveraged investor’s
strategies systematically.
The structure of the paper is as follows. Section 2 provides the analytic values of an
irreversible investment with no consideration of debt. In order to consider the effects of
leverage on investment decisions, section 3 provides the values of equity and debt with
fixed debt services and derives the value of an disinvestment (or default) option. Section
4 presents a simple model of a leveraged investment. The optimal development trigger
and the value of an investment option are presented. Section 5 concludes.
2 The Value of Investment
Our model embeds risky debt in the classical model of an irreversible investment pro-
posed by Dixit (1989). Nonetheless, our model differs from Dixit’s (1989) model in two
respects. First, a fraction of the sunk cost k is funded with risky debt b, as given. An
investor spends the net investment k− b ≥ 0 at the time an investment occurs. Second,
the investor disinvests by defaulting on the risky debt but has no reentry option, because
the investment is under the lender’s control.
Before considering leverage, we derive the value of an investment with no debt (or
leverage) in this section. By investing k, the project produces a single commodity at the
rate of one unit per unit of time. The commodity has the exogenous price P (t), which
evolves according to the following process:
dP
P= µdt+ σdz, (1)
where the constant parameters of µ and σ are growth rate and volatility, respectively,
and dz is the increment of a standard Wiener process ; if the project produces more than
one commodity, P (t) represents the value of the cash flows from those commodities. The
discount rate for the cash flows from the risky asset is constant and equals to ρ, with
ρ > µ. The marginal cost of production w is constant. In order to study how price
changes affect the value of an investment, we assume that it is optimal not to abandon
that investment over an infinitesimal period dt. The value of an investment with an
5
abandonment option is
V1(P ) = eρdtV1(P (dt)) + E
∫ dt
0
(P (τ)− ω)eρτdτ, (2)
where P (dt) is the price at time t+ dt. Over dt, the investor receives P and pays ω for
one unit of commodity. Based on Ito’s lemma, we expand the right-hand side of (2) in
a Taylor series. By using (1) for the change in P and ignoring the terms of order higher
than linear in dt, we derive the inhomogeneous second-order differential equation for the
value of the investment V1(P ):
σ2
2P 2V ′′1 (P ) + µPV ′1(P )− ρV1(P ) = ω − P, P > PA. (3)
The general solution to (3) is known to be V1(P ) = A0(P ) + A1Pα + A2P
β, where
a particular solution A0(P ) and constants A1 and A2 are to be determined, and α =
[(1−m)−(1−m)2 +4r1/2]/2 < 0, and β = [(1−m)+(1−m)2 +4r1/2]/2 > 1, where
m ≡ 2µ/σ2 and r ≡ 2ρ/σ2. The parameters α and β are the roots of the characteristic
equation of σ2
2ξ(ξ − 1) + µξ − ρ = 0.
The value of investment in (3) requires initial, value-matching, and smooth-pasting
conditions:
limP→∞ V1(P ) =P
ρ− µ− ω
ρ, (4a)
V1(PA) = 0, (4b)
V ′1(PA) = 0. (4c)
The investor will neither disinvest nor abandon a project if P is high enough in (4a).
Note that we include no cost of abandonment in (4b). In (4c), holding the project and
abandoning the project are indifferent at the time of abandonment, which is known
as the smooth-pasting condition. Note that abandonment of the project should be
distinguished from an equity investor’s default decision as discussed in the next section.
Substituting (3) into (4) yields
V1 =P
ρ− µ− ω
ρ− 1
α− 1
ω
ρ(P
PA)α, (5a)
where the trigger level of an abandonment is
PA =α
α− 1
ρ− µρ
ω, (5b)
6
which is same without both a reentry option and the abandonment cost in Dixit (1989).7
3 The Values of a Leveraged Investment
This section shows that, when risky debt is involved, a leveraged equity investor moves
the disinvestment decision forward and the value of waiting for the disinvestment in-
creases with debt. We will solve the investor’s problem by backward induction; we
assume that capital markets have no friction and no information asymmetry. To pin
down the values of both equity and debt, moreover, an investor pays a continuous coupon
c per unit of time immediately following an investment.8
3.1 The Value of the Equity
We derive the values of the equity and the debt to extend the model in Dixit (1989) to
an investor with risky debt outstanding at the time of investment. In order to study
how price changes affect the equity value, we assume that a leveraged investor optimally
does not disinvest or default over an infinitesimal period dt,
V1S(P ) = eρdtV1S(P (dt)) + E
∫ dt
0
(P (τ)− ω − c)eρτdτ, (6)
where P (dt) is the price at time t + dt. Over dt, the leveraged investor receives P and
pays ω + c. With the same method that yielded (3), we derive the differential equation
for the value function to obtain the value of the equity V1S(P ):
σ2
2P 2V ′′1S(P ) + µPV ′1S(P )− ρV1S(P ) = ω + c− P, P > PD, (7)
where PD is the trigger level of the disinvestment: default for a leveraged investor. The
general solution of (7) is V1S(P ) = A0S(P ) +A3Pα +A4P
β, where a particular solution
A0S(P ) and constants A3 and A4 are to be determined. The value of the equity in (7)
7For simplicity, we abstract from interactions between an entry option and an exit option and noinvestment lags as in Bar-Ilan and Strange (1996) are considered. Also notice that with no abandonmentoption, the value of which is the second term of the right-hand side of (5a), the value of the durableinvestment that the investor cannot abandon is V NA1 ≡ P
ρ−µ −ωρ .
8Prior to exercising of an investment option, an investor negotiates a loan contract with a lender;this assumption is consistent with that in Mella-Barral and Perraudin (1997), Mauer and Ott (2000),Mauer and Sarkar (2005), Sarkar (2007) and Lyandres and Zhdanov (2010).
7
requires initial, value-matching and smooth-pasting conditions:
limP→∞ V1S(P ) =P
ρ− µ− ω + c
ρ, (8a)
V1S(PD) = 0, (8b)
V ′1S(PD) = 0. (8c)
The equity investor has no incentive to default if P is high enough in (8a). When
defaulting, the leveraged investor incurs no default-related cost in (8b). The investor
makes a strategic default decision in (8c).
Substituting (7) into (8) gives
V1S =P
ρ− µ− ω + c
ρ− (
1
α− 1
ω + c
ρ)(P
PD)α, (9a)
where the level of the default trigger is
PD =α
α− 1
ρ− µρ
(ω + c), (9b)
which represents Dixit’s (1989) result with both no default cost and no reentry option.
Jou (2001) also has the same trigger level of disinvestment in (9b). Equation (9b) yields:
Proposition 1. The trigger level of a default (or disinvestment) for a leveraged invest-
ment increases with risky debt.
The proof:∂PD∂c
> 0.
The intuition behind proposition 1 is that the value function of the equity decreases
as the value of the debt increases. With the decreased value function, the investor has
an incentive to default at higher prices. Proposition 1 also holds under no uncertainty,
because the Marshallian trigger for default is PMD = ω + c; note the hysteresis under
uncertainty in (9b) which we can see 0 < αα−1
< 1, as α is negative.
The equity value represents the value of a call option on the cash flows from an
investment (Merton, 1974); the equity value on the left-hand side of (9a) is replicated
by the portfolio involving a long position in an investment, a short position in the riskless
debt and a long position in a put option on default, respectively.9 The value of a default
option that a leveraged investor longs is
9This holds only when the claims to the investment, the debt, and the put option are tradable in africtionless market such as the paper assumes; we greatly appreciate a referee’s insightful comments.
8
Ω(c) ≡ (1
1− αω + c
ρ)(P
PD)α. (10)
Proposition 2. The value of waiting to default (or disinvestment) for a leveraged in-
vestment increases with risky debt.
See appendix A.1 for the derivation.
By taking the financial risk, our investor obtains the positive value of an option of
waiting to default, which increases the value of the equity. The greater leverage exposes
a leveraged investor to higher levels of financial risk, and makes the default option more
valuable.10
An investor nonetheless makes a strategic default decision according to the equity
value, which includes the positive value of waiting in (10). The effect of the reduced
value of the equity dominates the effect of the increased financial risk, even though
higher levels of financial risk caused by an increase in risky debt increase the equity
value. When P > PD, note that ∂V1S(P )∂c
is negative as ∂V1S(P )∂c
= −1ρ− 1
ρ(α−1)( PPD
)α +
( αα−1
ω+cρ
)( PPD
)αP−1D
∂PD∂c
= −1ρ
+ 1ρ( PPD
)α = −1ρ(1 − ( P
PD)α), where the first term is the
effect of the reduced equity value and the second term is the effect of the increased
financial risks.11 Due to a default option, moreover, the marginal value of waiting,∂V1S∂P
= 1ρ−µ(1 − ( P
PD)α−1), decreases as leverage increases;
∂V 21S
∂P∂c= ρ
α(ω+c)( PPD
)α−1 < 0.
Note that when P ≥ PD an increase in risky debt leads to a decrease in the marginal
value of the equity. A leveraged investor has a weaker incentive for waiting to default
because she shares the marginal value of waiting with a debt provider.
In closing this section, note that the value of a default option decreases as the value
of cash flows from a project increases: ∂Ω(c)∂P
< 0. As the value of cash flows from the
project increases, the equity investor becomes less exposed to risk; the leveraged investor
is less likely to default.
10With safe debt, the equity value in (9a) converts to V NA1S ≡ Pρ−µ −
ω+cρ for a leveraged investor
who cannot default on the debt and the value of the debt for a lender who bears no risk is constant atV NA1D ≡ c
ρ over time.11This seems to be strange because a leveraged investor’s marginal value function at P = PD is
irrelevant to financial structure; ∂V1S(PD)∂c =
∂V NA1S
∂c , which equals the marginal value of equity with nodefault option (under no uncertainty) with respect to the coupon. For the investor, however, holdingthe project and defaulting on the loan are indifferent at the trigger.
9
3.2 The Value of the Debt
In order to study how price changes affect the value of debt, we can define the value of
the debt as:
V1D(P ) = eρdtV1D(P (dt)) + E
∫ dt
0
ceρτdτ. (11)
Following the same steps that yielded equation (7) yields
σ2
2P 2V ′′1D(P ) + µPV ′1D(P )− ρV1D(P ) = −c, P > PD (12)
The general solution of (12) is V1D(P ) = A0D(P )+A5Pα+A6P
β, where a particular
solution A0D(P ) and constants A5 and A6 are to be determined. The value of the debt
in (12) requires initial and boundary conditions:
limP→∞ V1D(P ) =c
ρ, (13a)
V1D(PD) = V1(PD). (13b)
If the price is high enough in (13a), the value of risky debt equals the value of
safe debt. When an investor disinvests, the default property is transferred costlessly
to the lender in (13b). Notice that the lender has an abandonment option, but the
abandonment option will be exercised after default, which is determined by the leveraged
investor in (9b).
Substituting (12) into (13) gives the value of the debt as
V1D(P ) =c
ρ+ (
1
α− 1
ω + c
ρ)(P
PD)α − 1
α− 1
ω
ρ(P
PA)α, (14)
where PA < PD in (5b) and (9b), respectively. In equation (14), the lender evaluates
the investor’s incentives and incorporates them in the debt valuation (Harris and Raviv,
1991).
Waiting for information is beneficial to the lender; the value of the debt increases
with the output price. As the output price increases, both the default option and the
abandonment option decrease in value; as the price increases, the equity investor is less
likely to default on the loan and the lender is less likely to abandon the project. While
the lender shorts a default option, the lender longs an abandonment option in (14). The
former default effect dominates the latter abandonment effect. To see this, take the
derivative of (14) with respect to P ,
∂V1D(P )
∂P=Pα−1
ρ− µ(P 1−α
D − P 1−αA ) > 0. (15)
10
In closing this section, note that the value of an investment in (5a) is
V1 = V1S + V1D, (16)
in (9a) and (14), respectively. Equation (16) is independent of the tax rate or bankruptcy
cost, which can be important for firms with debt financing, as shown in Jung and Kim
(2008) and Ko and Yoon (2011). Our approach diverges from that of Mauer and Ott
(2000) and Mauer and Sarkar (2005), but is close to that of Modigliani and Miller (1958)
and Mella-Barral and Perraudin (1997). We contribute to the literature by considering
a lender’s abandonment in equation (16).
4 The Value of the Leveraged Investment Option
This section analyzes the effects of newly issued debt on the timing of an investment.
With risky debt, a leveraged investor moves the investment decision forward and the
value of waiting for the investment decreases.
In order to study how price changes affect the value of an investment opportunity,
we assume that it is optimal not to invest over an infinitesimal period dt.
V0(P ) = e−ρdtV0(P (dt)). (17)
An investor with no assets-in-place holds only an investment option and has no cash
inflow before exercising such an option in (17). Our modeling approach differs from
that of Lyandres and Zhdanov (2010) and Sarkar (2007) because V0 is independent of
assets-in-place with no preexisting debt. As a result, we examine an investment for a
new project with debt outstanding rather than expansion.
Using the same method that yielded (3), we derive the differential equation for an
investment option V0(P )
σ2
2P 2V ′′0 (P ) + µPV ′0(P )− ρV0(P ) = 0, P < PS, (18)
where PS is the trigger level of the leveraged investment.
The general solution for the homogeneous second-order differential equation in (18)
is V0(P ) = A7Pα + A8P
β, where A7 and A8 are constants to be determined.
11
The value of an investment opportunity must satisfy the following boundary condi-
tions12:
limP→0 V0(P ) = 0, (19a)
V0(PS) = V1S(PS)− (k − b), (19b)
V ′0(PS) = V ′1S(PS), (19c)
where
b = V1D(PS) (19d)
in (14). An investor will not exercise an investment option if P is low enough in (19a).
At the time of an investment, the value of an option equals the value of a leveraged
investment in (19b); the equity investor incurs no loss or receives no gain. In addition,
when investing, the investor takes out a new loan. The investor makes an optimal
decision in (19c) and the lender underwrites the loan at fair market value in (19d).
Nonetheless, the pre-determined coupon must be weakly less than a maximum coupon
c, which is determined by ∂V1D(PS :c)∂c
= 0.13
Substituting (18) into (19) gives the value of an investment option,
V0(P ) = (PSρ− µ
− 1
ρ(ω + c+ ρ(k − b))− (
1
α− 1
ω + c
ρ)(PSPD
)α)(P
PS)β
= (PSρ− µ
− ω
ρ− 1
α− 1
ω
ρ(PSPA
)α − k)(P
PS)β, (20a)
where the trigger level of a leveraged investment is
PS =
ββ−1
ρ−µρ
(ω + ρk) + 1α
ββ−1
(PSPA
)αPA
1 + 1β−1
( PSPD
)α−1. (20b)
For the proof of (20b), see appendix A.2.
Proposition 3. A unique PS in (20b) exists where P > PD.
12If an investor maximizes the value of an investment, the value-matching condition in (19b) isV0(P ) = V1(P ) − k, and the smooth-pasting condition in (19c) is V ′0(P ) = V ′1(P ), where P is theinvestment trigger. With no tax advantage, the value maximizer has no incentive to move the investmentdecision even with leverage. For details, see Jou (2001).
13We know that c is determined by ∂V1D(PS :c)∂c = 1
ρ (1− ( PSPD(c) )α) = 0. Furthermore, the second-order
condition is ∂2V1D(P :c)∂c2 = −∂
2Ω(P :c)∂c2 < 0.
12
For the proof, see appendix A.3.1.
Proposition 4. The trigger level of a leveraged investment decreases with risky debt.
The proof:∂PS∂c
< 0. (21)
See appendix A.3.2 for the derivation.
Proposition 4 is distinguished from proposition 2 in Jou (2001), who suggests an
investment timing decision for a value maximizer. Due to the tax advantage of debt, an
investor who maximizes the project value is likely to delay a timing decision with risky
debt; the project value is positively determined by the debt. However, our investor,
who maximizes the equity value, enjoys no tax advantage; in equation (16), the value of
an investment is independent of the tax. Therefore, proposition 4 is also distinguished
from Mauer’s and Sarkar’s (2005) argument that an equityholder hastens an investment
decision in order to “enjoy earlier receipt of interest tax shields.” It is surprising that
an equity value maximizer brings an investment decision forward with risky debt with
no tax advantage.
Proposition 5. The value of waiting to make a leveraged investment in (20a) decreases
with risky debt.
For the proof, from (20a):
∂V0(P )
∂c=Pα−1S
ρ− µ[P 1−αD − P 1−α
A ](P
PS)β∂PS∂c
< 0, (22)
with (20b) and proposition 4. For details, see Appendix A.4.
The reduced value of the equity lowers the value of a call option; the value of a
compound option on the equity decreases as the value of an underlying asset decreases.
With an American option, proposition 5 confirms Geske’s (1979) argument that the
value of a compound option on equity decreases with risky debt.
The effect of the financial risks is dominated by the effect of the uncertainty risks.
The reduced value of waiting with the change in equity value dominates the increased
value of waiting with leverage in the long run, when an investor reacts to a change
in c. Nonetheless, in the short run, when an investor does not react to a change in
c, the value of waiting increases because the leveraged investor assumes the financial
13
risk. For details, see appendix A.5. The marginal benefit of waiting decreases with
leverage, as the equity value maximizer takes no account of the lender’s value of waiting;∂π∂c
= α−1ρ−µ( P
PD)α(P )−1 ∂PD
∂c< 0, where π in (29). Our approach is distinguishable from
that of Lyandres and Zhdanov (2010) because, by analyzing a unique structure of stand-
alone financing, we stress the effects of debt on the zone between the investment trigger
and the disinvestment trigger. In addition, we propose analytic expressions for both the
investment option and the disinvestment option.
With no leverage and no abandonment option, the value of an investment option
is V NL0 (P ) = ( PH
ρ−µ −ωρ− k)( P
PH)β, where PH = β
β−1ρ−µρ
(ω + ρk) as a special case in
Dixit (1989). Compared with PH , PS in (20b) is reduced due to both a leverage effect
and an abandonment effect; because 1β−1
( PSPD
)α−1 > 0 in the denominator in (20b) and1α
ββ−1
(PSPA
)αP−1A < 0 in the numerator in (20b), PS is lower than PH .
With leverage and no abandonment option, the value of an investment option for an
equity value maximizer is V NA0 (P ) = (PHS
ρ−µ −ωρ− k)( P
PHS)β, where PHS =
ββ−1
ρ−µρ
(ω+ρk)
1+ 1β−1
(PHSPD
)α−1,
where PD is in (9b). We can derive the closed-form trigger of PHS as the default trigger
is determined without a reentry (investment) trigger.
With no uncertainty, moreover, the Marshallian trigger for an investment is PMH =
ω + ρk. Comparing PH and PMH , we notice investment hysteresis under uncertainty. In
addition, comparing PH and PHS, leverage speeds up the investment trigger. Notice
that PHS(1 + 1β−1
(PHSPD
)α−1) = ββ−1
ρ−µρ
(ω + ρk); we can prove PHS < PH with the logic
employed in Appendix A.3.
The key assumption on the basis of which derive our results is that the leveraged
investor maximizes the value of the equity, excluding the value of risky debt. Taking
no account of the debt holder’s information, the investor will, in some states of the
world, pass up future opportunities that make the investment more valuable. One
caveat pertaining to our model is that, for comparative statics, a numerical analysis is
required.14
14Unlike models employed in previous studies, our model does not need numerical examples to illus-trate a leveraged investor’s strategies because it models analytic investment strategies. For comparativestatics, a numerical example is available upon request; the example makes it possible to analyze theeffects of variation in σ, µ, k and r on strategies of investment, disinvestment, and abandonment.
14
5 Conclusion
We have created a model of leveraged investment in an uncertain investment environ-
ment. In particular, we have explored the effects of risky debt on timing decisions for
investment and disinvestment for a leveraged investor; with risky debt, a leveraged eq-
uity investor moves both investment decisions and disinvestment decisions forward. In
addition, risky debt lowers the value of the investment option, while increasing the value
of the disinvestment (or default) option.
Nevertheless, to obtain empirical evidence, it is critical to control for interactions
between existing leverage and the investment decision; unlike in our analysis, previous
studies, such as Mauer and Ott (2000) and Lyandres and Zhdanov (2010), stress the
roles of assets-in-place before an investment is made. In addition, a key assumption
of our model is that a leveraged investor maximizes the value of an equity investment.
Moreover, we rule out a decision to invest on the part of the value maximizer in Jou
(2001). Therefore, referencing general managers of firms, who are likely to consider
the total value of an investment, would be inappropriate for testing our theory. We
suggest seeking empirical evidence from industries in which equity investors with no
assets-in-place play a key role in timing decisions.
15
A Proofs
A.1 Proposition 3
By taking the derivative of equation (10) with respect to c, we have
∂Ω(c)
∂c=
1
1− α1
ρ(P
PD)α − α
1− αω + c
ρ(P
PD)αP−1
D
∂PD∂c
=1
ρ(P
PD)α > 0,
with equation (9b).
A.2 Equation (20b)
PS =
1ρ(ω + ρk)− b+ c
ρ+ ( 1
α−1ω+cρ
)( PSPD
)α
β−1β
1ρ−µ + α
β( 1α−1
ω+cρ
)( PSPD
)α(PS)−1
=
1ρ(ω + ρk) + 1
α−1ωρ(PSPA
)α
β−1β
1ρ−µ + α
β( 1α−1
ω+cρ
)( PSPD
)α(PS)−1
=
ββ−1
ρ−µρ
(ω + ρk) + β 1α−1
1β−1
ρ−µρω(PS
PA)α
1 + α 1α−1
1β−1
ρ−µρ
(ω + c)( PSPD
)αPS−1
=
ββ−1
ρ−µρ
(ω + ρk) + 1α
ββ−1
(PSPA
)αPA
1 + 1β−1
( PSPD
)α−1. (23)
A.3 Propositions 3 and 4
A.3.1 The proofs of proposition 3
From equation (20b), we define f(P ) ≡ P− ββ−1
ρ−µρ
(ω+ρk) and g(P ) ≡ − 1β−1
( PPD
)αPD+1α
ββ−1
( PPA
)αPA = 1β−1
(−P−α+1D + β
αP−α+1A )Pα. By taking the derivative of g(P ), we
obtain g′(P ) > 0, and g′′(P ) < 0 and also we know limP→∞ g(P ) = 0−. By showing
f(P )− g(P ) ≤ 0 at the lower bound P = PD, we can complete the proof. First,
16
f(PD) = PD −β
β − 1
ρ− µρ
(ω + ρk) ≤ g(PD) = − 1
β − 1PD +
β
α
1
β − 1(PDPA
)αPA ⇔
β
β − 1PD −
β
β − 1
ρ− µρ
(ω + ρk) ≤ β
α
1
β − 1(PDPA
)αPA ⇔
α
α− 1(ω + c)− (ω + ρk) ≤ 1
α
ρ
ρ− µ(PDPA
)αPA ⇔
ω
α− 1+
αc
α− 1− ρk ≤ 1
α
ρ
ρ− µ(PDPA
)αPA =ω
α− 1(PDPA
)α. (24)
From equation (19d), we know, second, that the maximum loan amount is weakly
less than the sunk cost.
b = V1D(PD) =c
ρ+ (
1
α− 1
ω + c
ρ)− 1
α− 1
ω
ρ(PDPA
)α ≤ k ⇔
c+1
α− 1(ω + c)− ρk ≤ ω
α− 1(PDPA
)α ⇔
ω
α− 1+
αc
α− 1− ρk ≤ ω
α− 1(PDPA
)α. (25)
According to both equations (24) and (25), we know f(P ) < g(P ) where P > PD.
Note that the maximum coupon cm is determined by ωα−1
+ αcm
α−1−ρk− ω
α−1(PD(cm)
PA)α = 0
A.3.2 The proofs of proposition 4
While ∂f(P )∂c
= 0, ∂g(P )∂c
= −1−αβ−1
PαP−αD∂PD∂c
< 0 with ∂PD∂c
> 0. Along with the proofs of
existence and uniqueness of a solution, ∂PS∂c
< 0.
17
A.4 Equation (22).
∂V0(P )
∂c= (
1
ρ− µ− α
α− 1
ω
ρ(PSPA
)αP−1S )(
P
PS)β∂PS∂c
−β(PSρ− µ
− ω
ρ− 1
α− 1
ω
ρ(PSPA
)α − k)(P
PS)βP−1
S
∂PS∂c
= [1
ρ− µ(1− (
PSPA
)α−1) + −βρ− µ
+β
PS(1
ρ(ω + ρk) +
1
α− 1
ω
ρ(PSPA
)α)]( PPS
)β∂PS∂c
= [1
ρ− µ(1− (
PSPA
)α−1)
+ −βρ− µ
+β − 1
ρ− µ+
α
α− 1
ω + c
ρ(PSPD
)α(PS)−1)]( PPS
)β∂PS∂c
, in (23)
= [1
ρ− µ(1− (
PSPA
)α−1)− 1
ρ− µ1− (
PSPD
)α−1]( PPS
)β∂PS∂c
=Pα−1S
ρ− µ[P 1−αD − P 1−α
A ]︸ ︷︷ ︸+
(P
PS)β∂PS∂c
< 0. (26)
A.5 Uncertainty Risks vs. Financial Risks.
From the right-hand side of the value-matching condition in (19b), we can see that a
benefit of a leveraged investment is
Π(P ) ≡ P
ρ− µ− ω
ρ− k − (
1
α− 1
ω + c
ρ)(P
PD)α + (
1
α− 1
ω + c
ρ)(PSPD
)α − 1
α− 1
ω
ρ(PSPA
)α,
(27)
with both (14) and (19d). At P = PS, the cost of waiting V0(P ) equals the benefit of a
leveraged investment Π(PS) = V1(PS)− k in (27); we can define the continuation region
R ≡ P | Π(P ) < V0(P ).To show the conflicting effects, we adopt the proof strategy employed in Geske (1979).
In the short run, when an investor does not react to an increase in c, the value of waiting
increases with leverage, because the leveraged investor assumes the financial risk. This
can be demonstrated by ∂Π(P )∂c
= 1ρ( PPD
)α − 1ρ( PSPD
)α = 1ρP−αD [(P )α − (PS)α] > 0, when
P < PS. In the long run, when an investor reacts to an increase in c, nonetheless, the
benefit of a leveraged investment decreases, because the value of the debt increases;
∂Π(P )
∂c=
1
ρP−αD [(P )α − (PS)α] +
1
ρ− µ[P 1−αD − P 1−α
A ]Pα−1S
∂PS∂c
< 0, P > PS − P ,(28)
18
where P (> 0) is constant.15 The reduced value of the equity dominates the change in
the value of the equity; due to the financial risks, the value of the leveraged investment
increases with leverage.
On the right-hand side of the smooth-pasting condition in (19c), the marginal benefit
of a leveraged investment is
π(P ) ≡ ∂Π(P )
∂P= V ′1S(P ) =
1
ρ− µ(1− (
P
PD)α−1), (29)
where 0 < ( PPD
)α−1 < 1. At P = PS, the marginal cost V ′0(P ) equals the marginal
benefit π(P ). The value of a default option that an investor holds in (10) decreases as
the output price increases; the investor is less likely to default.
References
Bar-Ilan, A. and W. C. Strange (1996, June). Investment lags. American Economic
Review 86 (3), 610–22.
Berger, A. N. and G. F. Udell (1998). The economics of small business finance: The
roles of private equity and debt markets in the financial growth cycle. Journal of
Banking & Finance 22 (68), 613 – 673.
Dixit, A. (1992, Winter). Investment and hysteresis. Journal of Economic Perspec-
tives 6 (1), 107–32.
Dixit, A. K. (1989, June). Entry and exit decisions under uncertainty. Journal of
Political Economy 97 (3), 620–38.
15P is determined by the following condition: 1ρP−αD [(P )α − (PS)α] + 1
ρ−µ [P 1−αD −
P 1−αA ]Pα−1
S∂PS∂c = 0. We can find P (> 0) as follows. When P > PS , ∂Π(P )
∂c =1
ρP−αD [(P )α − (PS)α]︸ ︷︷ ︸
−
+1
ρ− µ[P 1−αD − P 1−α
A ]Pα−1S
∂PS∂c︸ ︷︷ ︸
−
< 0. When P = PS , ∂Π(P )∂c =
1
ρ− µ[P 1−αD − P 1−α
A ]Pα−1S
∂PS∂c︸ ︷︷ ︸
−
< 0. Nonetheless, when P < PS − P , ∂Π(P )∂c =
1
ρP−αD [(P )α − (PS)α]︸ ︷︷ ︸
+
+1
ρ− µ[P 1−αD − P 1−α
A ]Pα−1S
∂PS∂c︸ ︷︷ ︸
−
> 0 because, from the first term of ∂Π(P )∂c ,
∂[ 1ρP−αD ((P )α−(PS)α)]
∂P = αρP−αD Pα−1 < 0.
19
Esty, B. C. (2004). Why study large projects? an introduction to research on project
finance. European Financial Management 10 (2), 213–224.
Geske, R. (1979, March). The valuation of compound options. Journal of Financial
Economics 7 (1), 63–81.
Harris, M. and A. Raviv (1991). The theory of capital structure. The Journal of
Finance 46 (1), pp. 297–355.
Jensen, M. C. and W. H. Meckling (1976, October). Theory of the firm: Managerial
behavior, agency costs and ownership structure. Journal of Financial Economics 3 (4),
305–360.
Jou, J.-B. (2001). Entry, financing, and bankruptcy decisions: The limited liability
effect. The Quarterly Review of Economics and Finance 41 (1), 69–88.
Jung, K. and B. Kim (2008). Corporate cash holdings and tax-induced debt financing.
Asia-Pacific Journal of Financial Studies 37, 983–1023.
Ko, J. K. and S.-S. Yoon (2011). Tax benefits of debt and debt financing in korea.
Asia-Pacific Journal of Financial Studies 40, 824–855.
Lyandres, E. and A. Zhdanov (2010). Accelerated investment effect of risky debt. Journal
of Banking & Finance 34 (11), 2587 – 2599.
Mauer, D. C. and S. Ott (2000). Agency costs, underinvestment, and optimal capital
structure: The effect of growth options to expand. In M. Brennan and L. Trigeorgis
(Eds.), Project Flexibility, Agency, and Competition. Oxford University Press, pp.
151–180. Oxford University Press.
Mauer, D. C. and S. Sarkar (2005, June). Real options, agency conflicts, and optimal
capital structure. Journal of Banking & Finance 29 (6), 1405–1428.
Mella-Barral, P. and W. Perraudin (1997, June). Strategic debt service. Journal of
Finance 52 (2), 531–56.
Merton, R. C. (1974, May). On the pricing of corporate debt: The risk structure of
interest rates. Journal of Finance 29 (2), 449–70.
20
Modigliani, F. and M. H. Miller (1958). The cost of capital, corporation finance and the
theory of investment. The American Economic Review 48 (3), pp. 261–297.
Myers, S. C. (1977). Determinants of corporate borrowing. Journal of Financial Eco-
nomics 5 (2), 147 – 175.
Sarkar, S. (2007). Expansion financing and capital structure. SSRN eLibrary .
Yagi, K., R. Takashima, H. Takamori, and K. Sawaki (2008, August). Timing of con-
vertible debt financing and investment. CARF F-Series CARF-F-131, Center for
Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
21