The Non-Neutrality of Debt in Investment Timing: A New NPV Rule

The Non-Neutrality of Debt in Investment Timing:
A New NPV Rule
Tarun Sabarwal1
Department of Economics, BRB 1.116
The University of Texas at Austin
1 University Station C3100
Austin TX 78712-0301, USA
sabarwal@eco.utexas.edu
This Version: April 8, 2005
JEL Numbers: G00, D92, E50
Keywords: Debt, Default, Limited Liability, Investment, NPV, Option Value
1The author is deeply grateful to Bob Anderson for his advice and encouragement. He thanks Joydeep
Bhattacharya, Svetlana Boyarchenko, Gabriele Camera, Helios Herrera, Warren Hrung, Dwight Jaffee, Sergei
Levendorski˘i, Preston McAfee, Chris Shannon, Max Stinchcombe, and referees for helpful comments.

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The Non-Neutrality of Debt in Investment Timing:
A New NPV Rule
Abstract
Limited liability debt financing of irreversible investments can affect investment timing
through an entrepreneur’s option value, even after compensating a lender for expected de-
fault losses. This non-neutrality of debt arises from an entrepreneur’s unique investment
opportunity, and it is shown in a standard model of irreversible investment that is enhanced
in a straightforward manner to include the equilibrium effect of a competitive lending sec-
tor. The analysis is partial, in that it takes as exogenously given an entrepreneur’s use of
debt. Intuitively, limited liability lowers downside risk for the entrepreneur by truncating the
lower tail of risks, thereby lowering the investment threshold. Compensating the lender for
expected default losses reduces project profitability to the entrepreneur, thereby increasing
the investment threshold. The net effect is negative, because lower downside risk has an
additional impact on the option value of delaying investment. The standard NPV rule in
real options theory implicitly assumes debt to be neutral. With non-neutrality of debt, an
investment threshold is higher than investment cost, but lower than the standard NPV rule.
Comparisons with other standard investment thresholds show similar relationships.

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1
Introduction
The Modigliani-Miller theorem (1958) guarantees that the mix of debt and internal financing
does not affect the overall value of a firm, or as is the case in this paper, the overall value
of an investment project. It implies that a financing decision cannot be used to increase the
overall value of a project, and therefore, it obviates theoretical arguments about any role
of debt financing in an investment decision.2 In practice, debt financing appears to play a
significant role in investment decisions; for example, aggregate debt growth rates are well-
known to be pro-cyclical for both households and firms, there is a literature on the role of
debt financing on real estate construction decisions, there is another literature on the effects
of financing on firm decisions, and there is a further literature on the role of debt financing
on household financial decision-making.
Exceptions to the Modigliani-Miller theorem, including explanations for its observed vio-
lations, usually rely on some form of market imperfection, including tax distortions, agency
costs, market power, other deadweight losses, and so on.
This paper presents another exception to the Modigliani-Miller theorem. Using ideas
regarding value of waiting that have been formalized in real options theory, it is shown that
if an investment option resides with an entrepreneur, (as opposed to a lender,) then even
after compensating a lender for expected default losses, limited liability debt financing affects
optimal investment timing by affecting an entrepreneur’s value of waiting.3 In particular,
with such debt financing, the optimal investment threshold is lower than that consistent
2The well-known result by Merton (1977) shows the validity of the Modigliani-Miller theorem even with
a positive probability of bankruptcy, an application of this theorem to Arrow-Debreu economies is given in
Stiglitz (1969), and another application to open market operations is given in Wallace (1981).
3For a development of real options theory, see the seminal papers by McDonald and Siegel (1986), Pindyck
(1988), Dixit (1989), and others. More recent work is presented in the collection by Brennan and Trige-
orgis (2000). For generalizations to non-Gaussian processes, see Boyarchenko and Levendorsk˘i (2000), and
Boyarchenko (2003).
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with the standard theory of irreversible investment under uncertainty, but higher than cost
of investment. Therefore, whereas the classical NPV rule of investment (invest at the first
time when present value exceeds cost of investment) does not consider the value of waiting for
new information, and gives a threshold that is too low, the standard NPV rule of irreversible
investment under uncertainty (invest at the first time when present value exceeds a multiple
of investment cost) does not consider the effect of limited liability on the value of waiting,
and gives a threshold that is too high.
Consider an entrepreneur who finances an investment cost I with debt D ≤ I using a
simple, limited liability debt contract.4 The lender has a first claim on project revenues
up to a fixed coupon C, determined by the lender’s zero profit condition. If revenue is
below C, the entrepreneur’s liability is limited to available revenue, and she turns over
the entire revenue to the lender. In this case, there are several effects of debt financing
on the optimal investment threshold. To the extent that debt reduces an entrepreneur’s
share of investment cost, the investment threshold is lower, and to the extent that limited
liability lowers downside risk of the project, the investment threshold is lower. But default
is not costless, because the lender anticipates its probability, and sets coupon in part to
offset expected losses. To the extent a higher coupon decreases project profitability for
4The analysis here does not explicitly consider a reason for debt financing, but takes the decision about
debt as given, assumes that the entrepreneur can finance the remainder from either an endowment, or
equivalently, from existing internal funds, and focuses on the effect of debt on the optimal investment
threshold. In particular, a priori, there is no guarantee that optimal debt is not zero. The debt contract
assumed here is a reduced-form of the simple debt contract that is optimal in the costly state verification
(CSV) model of Townsend (1979). In a more recent paper, Krasa and Villamil (2000) show that among
other things, the CSV model can be thought of as a reduced-form of a more fundamental model with limited
committment and with costly enforcement as a decision variable, and in such a model, simple debt remains an
optimal contract. Qualitative effects of alternative limited liability debt contracts are the same, as described
below.
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the entrepreneur, the investment threshold is higher. Moreover, a lower default probability
decreases coupon, increases project profitability for the entrepreneur, and therefore, lowers
the investment threshold, and similarly, a higher default probability raises coupon, reduces
project profitability, and raises the threshold. What might be the net effect of such debt
financing on the optimal investment threshold? Intuitively, one might expect that after
compensating the lender for expected default losses, its net effect is zero. In contrast, the
analysis here shows that the net effect is not zero; in fact, it is negative.
The non-neutrality of debt can be viewed in terms of a skewed evaluation of a mean-
preserving spread in an entrepreneur’s option value. Recall that option value derives in part
from an ability to avoid downside risk; that is, in case of adverse realizations, an option
does not have to be exercised, and therefore, other things equal, if downside risk of a project
is lower, (its option value is lower, and) its investment threshold is lower. With limited
liability, some of the downside risk is transferred to the lender, and from an entrepreneur’s
viewpoint, the lower tail of risks is truncated. This has two effects — one on project value,
and the other on option value. The mean-preserving aspect arises from the lender’s zero
profit condition, and it implies an equal and opposite impact on total project value. The
skewed evaluation arises from an additional impact of the truncation of the lower tail of risks
on the entrepreneur’s option value. Limited liability reduces downside risk, and this has an
additional negative impact on an entrepreneur’s option value, thus lowering the investment
threshold.
The formal analysis in this paper is based on a straightforward extension of a simple
version of a well-established model (Pindyck (1988), and McDonald and Siegel (1986)) of
irreversible investment to include a competitive lending sector, and it yields a closed-form
solution for the optimal investment threshold.5 A unique feature of this analysis is the intro-
5As mentioned in Pindyck (1991), examples of irreversible investments include investments that are
specific to a firm or industry, (for example, a large, industry-specific production unit,) investments the resale
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duction of a lending sector, and consequently, an endogenous determination of the coupon
or interest rate on debt based on the equilibrium feedback from the lending sector. An
appealing feature of this analysis is that it provides an equilibrium solution that is unique,
analytically tractable, and intuitive. The equilibrium solution allows a natural decompo-
sition of the optimal threshold into two parts, one that can be compared directly to the
standard model of irreversible investment under uncertainty, and another that arises from
the introduction of debt financing, thereby facilitating a better understanding of the impact
of debt financing on optimal investment thresholds.
The paper proceeds as follows. Section 2 specifies the model, defines an equilibrium, and
shows the existence of a unique equilibrium. Section 3 discusses the new NPV rule, and
relates it to some established notions in the theory of investment under uncertainty.
2
Specification of the Model
The model in this paper adapts a simple version of a well-established model (Pindyck (1988),
and McDonald and Siegel (1986)) of irreversible investment to include a competitive lending
sector. To facilitate comparison to the standard model, notation from Dixit and Pindyck
(1994) is used. Suppose, as usual, that an entrepreneur is considering an irreversible invest-
ment in a project with a fixed scale, infinite life, and no marginal cost. If inverse demand for
project output is P = Y ·D(Q), where P is price of the firm’s output, Q is quantity of output,
and Y a stochastic shift variable, then fixing Q = 1 allows P to be the stochastic variable,
and P follows a geometric brownian motion with drift, given by dP = αP dt + σP dW . Let
µ be the discount rate for future revenues, and let δ = µ − α > 0.
The cost of investment is I. The investment is irreversible, in the sense that the invest-
of which involve informational asymmetries, and investments that cannot be divested because of government
or institutional restrictions.
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ment cost is sunk, once it is incurred, but this can be relaxed with a disinvestment cost, as
explained below. For convenience, there is no depreciation or time-to-build, and once the
investment cost is incurred, the project yields a unit flow of output forever. The entrepreneur
finances this project with the following limited liability debt contract. If an entrepreneur
desires debt D ≤ I,6 then in exchange for D, he agrees to give the lender a first claim on
project revenues up to a fixed coupon C, determined by the lender. If revenue is below
C, the entrepreneur’s liability is limited to available revenue, and he turns over the entire
revenue to the lender. There is no benefit from selling the project, because the investment
is irreversible, and therefore, the lender does not foreclose the project when revenue is below
coupon, and the project continues to produce output forever.
The lender knows the project’s price process, and output is observable. The lender has
access to funds at the risk-free rate r. For example, we may think of a lender as a bank
with access to consumer deposits. The lender cannot directly invest in the project; it is only
the entrepreneur that has an opportunity to invest.7 The lending sector is competitive, so
that in equilibrium, the coupon set by the lender offsets the expected loss to the lender from
6As mentioned above, this analysis takes the debt decision as given, and focuses on the effect of debt on
the optimal investment threshold. An entrepreneur might decide to utilize debt for several reasons: perhaps
investment cost is large and cannot be financed by savings or retained earnings, or perhaps there is a first-
mover advantage and quick entry into a project might yield some monopoly rents, or perhaps there is a tax
advantage for using debt, or perhaps debt is relatively cheaper in times of expansionary monetary policy.
See also, for example, Lambrecht (2001), Mello and Parsons (1992), and Mauer and Triantis (1994). In
particular, this paper does not consider the problem of optimal capital structure, for example, in the spirit of
Jensen and Meckling (1976), Myers (1977), (including extensions to real options by Mauer and Ott (2000),
and by Nachman (2003),) Brander and Lewis (1986), Brander and Lewis (1988), Leland (1994), and Leland
and Toft (1996), and others.
7For example, an entrepreneur might have a monopoly right, such as a patent, or she might have some
specialized knowledge to produce a particular output, or a bank might be prohibited by law from equity
investment in a project.
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future default, and the net value of the project to the lender is zero.8
The entrepreneur’s problem is mathematically the same as in a simple version of Pindyck
(1988).9 The entrepreneur’s profit flow is given by πE(C, P ) = max(P − C, 0). For fixed C,
using Ito’s lemma, the value of this project to the entrepreneur, denoted V E(C, P ), satisfies


 B1Pβ1 + B2Pβ2 + P − C if P ≥ C, and
V E(C, P ) =
δ
r

 K1Pβ1 + K2Pβ2
if P < C.
The terms β1 > 1 and β2 < 0 are solutions to the quadratic equation 1σ2β(β − 1) + (r −
2
δ)β − r = 0, and the terms B1P β1 and K2P β2 are related to speculative bubbles, which are
ruled out by assumption, implying that B1 = 0 and K2 = 0. Standard value matching and
smooth pasting conditions (at P = C) give
C1−β2
β
β
C1−β1
β
β
B
1
1 − 1
2
2 − 1
2 =
−
> 0, and K
−
> 0.
β
1 =
1 − β2
r
δ
β1 − β2
r
δ
Notice that B2 is a function of C. When convenient, the dependence of B2 on C is denoted
B2(C).
For P ≥ C, the value of the project to the entrepreneur is the expected present value of
capitalized revenues P minus the capitalized value of the sure coupon flow C adjusted for
δ
r
his benefit from future default when revenue is below coupon, given by B2P β2.
The entrepreneur has an option to delay investment, if he so chooses. For fixed debt D and
coupon C, the entrepreneur’s problem is to determine the revenue threshold (or equivalently,
the demand threshold) that maximizes the value of this project to him, and he invests at the
8The model is described for the case of a risk-neutral entrepreneur and lender. Alternatively, using a
standard change-of-measure, the results here are true with the same degree of risk-aversion for both an
entrepreneur and a lender.
9This formulation is well-known, and is presented in Pindyck (1991), and in Dixit and Pindyck (1994),
chapter 6, as a model in which a firm can costlessly suspend and resume operations. This paper presents a
different application of that model, and therefore, details are presented here both for completeness, and to
aid appropriate interpretation of that model.
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first time when revenue (or demand) crosses this threshold. For fixed C and D, the value of
his option to invest in this project, denoted F (P ), is given by F (P ) = A1P β1 + A2P β2. The
no-arbitrage, smooth-pasting, and absorbing barrier at zero boundary conditions are
(1) F (P ∗) = V E(C, P ∗) − (I − D),
(2) FP (P ∗) = V E(C, P ∗),
and
(3) F (0) = 0.
P
Consequently, the optimal threshold P satisfies
P
C
(β1 − β2)B2P β2 + (β1 − 1)
− β
+ I − D = 0.
δ
1
r
Notice that, for fixed C and D, the entrepreneur’s problem is mathematically the same as
the one in which there is a fixed operating cost C, cost of investment I − D, and operations
that can be costlessly suspended and resumed. In that interpretation, B2P β2 is the expected
present value to the entrepreneur from costlessly suspending operations when revenue falls
below cost.
In the formulation in this paper, the project continues to operate even when revenue
is below coupon, but the entrepreneur’s profit is zero, because revenue is turned over to
the lender. Therefore, B2P β2 is the expected present value to the entrepreneur from not
having to pay coupon when revenue falls below coupon; in other words, it is the value to
the entrepreneur from defaulting when revenue falls below coupon. Default correspondingly
reduces the value of the project to the lender. In this paper, the term B2P β2 denotes the
default value of a project. In equilibrium, default is not costless, because the lender sets
coupon to offset the default value of a project, as shown below.
Applying a well-known result by Pindyck (1988), for fixed C and I − D, the entrepreneur
will invest only when P > C, and the equation for his optimal threshold has a unique solution
for P , where P > C.
The lender’s problem is formulated as follows. The lender’s profit flow is given by
πL(C, P ) = min(C, P ). For fixed C, the value of this project to the lender evolves with
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P , and using the same technique as above, satisfies


 BLP β1 + BLP β2 + C if P ≥ C, and
1
2
V L(C, P ) =
r

 KLP β1 + KLP β2 + P if P < C.
1
2
δ
As above, the terms β1 > 1 and β2 < 0 are solutions to the quadratic equation 1σ2β(β −
2
1) + (r − δ)β − r = 0, and the terms BLP β1 and KLP β2 are associated with speculative
1
2
bubbles, which are ruled out by assumption, implying that BL = 0 and KL = 0.10 The value
1
2
matching and smooth pasting conditions with respect to P imply (at P = C),
BLCβ2 = KLCβ1 + C − C , and
2
1
δ
r
β2BLCβ2−1 = β
Cβ1 + 1,
2
1K L
1
δ
from which it follows that
C1−β2
β
β
C1−β1
β
β
BL =
1 − 1 − 1
= −B
=
2 − 1 − 2
= −K
2
β
2, and K L
1
1.
1 − β2
δ
r
β1 − β2
δ
r
For P ≥ C, the value of the project to the lender is the capitalized value of the sure
coupon flow C adjusted for the expected loss in present value from default BLP β2. In other
r
2
words, if the lender sets coupon C, and price is P , the expected present value of this project
to the lender is C − B
r
2P β2 . Because the entrepreneur invests only when P ≥ C , the zero
profit condition for the lender is
C
0 = V L − D =
− D − B
r
2P β2 .
Thus, for fixed D and P , the lender chooses coupon to cover both the debt extended, and
its expected losses from future default.11
10When P is very large, default probability is very low, implying that the value of the project to the lender
is close to C , and therefore BL
, so
r
1 = 0. When P is close to zero, the value of the project is close to P
δ
KL
2 = 0.
11Irreversibility can be replaced by a constant cost of disinvestment equal to K1Cβ1, which equals
C
β1 − β1−1 , and this makes the lender strictly prefer to remain in the project when the entrepreneur
β1−β2
r
δ
8

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