Discussion Papers In Economics
And BusinessCorporate sustainability, investment, and capital structureMichi NISHIHARA
Discussion Paper 22-05

November 2022

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANCorporate sustainability, investment, and capital structure∗
Michi NISHIHARA†

Abstract
This study develops a real options model in which a ﬁrm invests in either a sustainable projector an unsustainable project. The sustainable project requires a high investment cost and yields
cash ﬂows perpetually, whereas the unsustainable project requires a low investment cost and yields
cash ﬂows until a random maturity. The random termination of cash ﬂows reﬂects the project’s
environmental, social, and governance (ESG) risk. In the model, the optimal investment choice andtiming are analytically derived, and the eﬀects of key parameters on the choice are also examined.
Higher ESG risk, growth rate, and volatility, and lower discount rate encourage sustainable investing
mainly through their impacts on the net present value (NPV) and timing option value. The lesssustainable ﬁrm chooses higher leverage to enjoy a greater beneﬁt of debt ﬁnancing. Therefore,
access to debt ﬁnancing and a higher corporate tax rate (tax shield) discourage sustainable investing.
JEL Classiﬁcations Code: G13; G31; G32.
Keywords: sustainability; ESG; real options; capital structure.

∗

This version was written on October 31, 2022. This work was supported by the JSPS KAKENHI (Grant numbers

JP20K01769, JP21H00730). The author thanks Stein-Erik Fleten, Norio Hibiki, Haejun Jeon, and Motoh Tsujimura for
helpful comments. The author also thanks the participants at FMA 2021 in Kyoto (online), EURO 2022 in Espoo, and
ORSJ 2022 fall conference in Niigata for helpful feedback.
†
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka
560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277

1

1Introduction

Over the last decade, ﬁrms have identiﬁed the importance of the sustainability of their operations
in capital budgeting decisions. International organizations and governments have also become
more active in resolving sustainability issues (e.g., the 17 sustainable development goals by theUnited Nations). In these circumstances, an increasing number of academic researchers have also
investigated a variety of aspects of sustainability issues. For instance, several studies have applied
operations research methodologies, such as multi-criteria decision-making methods, to the assessment of sustainability (for a survey, see Thies, Kieckhafer, Spengler, and Sodhi (2019)). In theﬁnance area, many papers have investigated ESG investment (e.g., Hong and Kacperczyk (2009),
Bolton and Kacperczyk (2021), and Pedersen, Fitzgibbons, and Pomorski (2021)). Many presentations at the 32nd European Conference on Operational Research also discussed the eﬀectivenessof ESG investment (e.g., Grassetti and Stocco (2022), D’Ecclesia, Stefanelli, and Morelli (2022),
Nicolosi, Cerqueti, Ciciretti, and Dalo (2022), and Castellano (2022)). 1 Most of the existing literature addresses the relationships among ESG performance, ﬁrm characteristics, ﬁrm risk, and ﬁrm
value (for a survey, see Gillan, Koch, and Starks (2021)).
Despite the growing body of research, few papers have attempted to address sustainability
issues in the framework of the capital budgeting decision theory. To this point, no theory has
explained the economic mechanisms of market, ﬁrm, and institutional characteristics’ eﬀects on
corporate investing in sustainable projects. This paper aims to ﬁll that research gap and apply
the real options modeling approach to reveal the conditions that stimulate corporate investing insustainable projects.2
For this purpose, the present study uses the following model. Consider a ﬁrm that has an option
to invest in one of two projects (i.e., sustainable and unsustainable projects). There is a tradeoﬀ
between ESG risk and investment cost. The sustainable project requires a high investment costand yields cash ﬂows perpetually, while the unsustainable project requires a low investment costand yields cash ﬂows until a random maturity, which follows a Poisson arrival process and reﬂects
the project’s ESG risk. After evaluating the tradeoﬀ, the ﬁrm optimizes its investment choice andtiming to maximize ﬁnancial value.3 In addition to investigating an unlevered model, this paper
examines a levered model in which the ﬁrm also optimizes debt issuance (i.e., capital structure) at
the time of investment.

1More than ﬁfty presentations at the 32nd European Conference on Operational Research were related to sustainability

issues (not limited to ESG investment) according to the program.
2
The model is based on real options theory (e.g., Dixit and Pindyck (1994)). The real options method is now widely
accepted as a better method than the NPV method speciﬁcally in capital budgeting decisions involving uncertainty andirreversibility (for a review, see Kozlova (2017) and Trigeorgis and Tsekrekos (2018)).
3This paper focuses on an ESG-aware ﬁrm that recognizes ESG risk and maximizes ﬁnancial value. On the other
hand, some papers consider that ﬁrms or investors take nonﬁnancial beneﬁts from sustainable projects into account (e.g.,Pedersen, Fitzgibbons, and Pomorski (2021) and Oehmke and Opp (2022)). Such ﬁrms are regarded as ESG-motivatedﬁrms. The diﬀerences are explained in Section 2.3.

2

Certain practical examples can validate the model. Imagine a hypothetical mining ﬁrm thatplans to exploit natural resources (say, gold). Generally, mining activities cause many environmental problems, but it is possible to mitigate environmental damage at a cost. In this case, theﬁrm chooses one of the two means of exploitation (i.e., traditional exploitation with environmental
damage or costly but eco-friendly exploitation), with the knowledge that environmentally destructive mining might be prohibited in the future. The ﬁrm makes its investment choice and timing
decisions by observing the international market price of gold, which determines the project’s cash
ﬂows. Another example is an apparel ﬁrm that plans to invest in a factory in a developing country.
The ﬁrm chooses one of two factory types (i.e., a factory with or without suﬃcient labor rights),
and learns that the labor-exploiting factory has a risk of disaster and closure as a reaction to public criticism (e.g., the Dhaka garment factory ﬁre in Bangladesh). The ﬁrm makes its investment
choice and timing decisions based on the ESG risk and market demand for its apparel products.
This paper analytically solves the model and derives the explicit condition (hereafter, the sustainability condition) under which the ﬁrm chooses the sustainable project. The eﬀects of key
parameters on the sustainability condition are also examined analytically and numerically. These
parameters can impact the ﬁrm’s investment choice through three channels: (i) the NPV of project
cash ﬂows, (ii) the value of the option to defer investment, and (iii) optimal capital structure.
Higher ESG risk, unsustainable investment cost, and lower sustainable investment cost lead theﬁrm to invest in the sustainable project through the NPV channel. Lower discount rate and higher
cash ﬂow growth rate increase the value of cash ﬂows in the far future and, through the NPV
and option channels, encourage sustainable investing. Higher cash ﬂow volatility encourages sustainable investing through the option channel. In fact, higher volatility increases the option valueof waiting for a better economic state and investing in the high-cost and high-proﬁt project (i.e.,the sustainable project) rather than in the low-cost and low-proﬁt project (i.e., the unsustainableproject).
Debt ﬁnancing diﬃculties and a lower corporate tax rate do not have the NPV and option
channels but nonetheless encourage sustainable investing through the capital structure channel.
The unsustainable project increases its value by debt ﬁnancing more than the sustainable project
because the unsustainable ﬁrm cares less about bankruptcy costs, increases leverage, and enjoys thehigher leverage multiplier eﬀect. Consequently, diﬃculty in debt ﬁnancing and a lower corporate
tax rate suppress debt ﬁnancing and lead the ﬁrm to invest in the sustainable project, although
this capital structure channel is relatively weak.
These results have signiﬁcant implications for sustainability issues. Not only cost structures(e.g., technological innovations and subsidies) and degree of ESG risk (e.g., regulations and publicity), but also market conditions (e.g., discount rate, growth rate, and volatility) and the government
policies (e.g., interest rate and tax rate) aﬀect corporate decisions to invest in sustainable projects.
Typically, large and listed ﬁrms in developed countries are active in resolving sustainability issues. This trend can be rationally explained by these ﬁrms’ and countries’ characteristics of lower
discount rate and higher ESG risk (or publicity). That is, these ﬁrms have a greater economic3

rationale to invest sustainably than small and unlisted ﬁrms in developing countries do.
So far, I have focused on the choice between sustainable and unsustainable projects, but in termsof investment timing, sustainable project investment timing is later (due to its high investment cost)than unsustainable project investment timing is. By consequence, parameter changes (e.g., market
and policy changes) that promote sustainable investing do not necessarily accelerate corporate
investment. Although parameter changes could lead ﬁrms to invest in more sustainable projects,
they delay investment timing. Thus, the eﬀects of parameters on investment timing become nonmonotonic.
Last, I will explain technically related literature. In the real options modeling literature on
sustainability issues, many papers have studied investment timing and valuation of renewable energy
projects (for a review, see Kozlova (2017)). For instance, Boomsma, Meade, and Fleten (2012)
analyze investment timing and capacity decisions on renewable energy projects under two support
schemes and reveal those support schemes’ diﬀering impacts on the decisions. As in this paper’s
use of a Poisson jump to model ESG risk, Boomsma and Linnerud (2015) and Adkins and Paxson(2016) model renewable policy uncertainty using Poisson jumps and show the eﬀects of the policy
uncertainty on renewable investment timing and valuation. However, these previous studies do
not examine the process of making a decision between sustainable and unsustainable projects that
accounts for the tradeoﬀ between ESG risk and investment cost, nor do their models incorporate
debt ﬁnancing. Therefore, unlike this paper, they do not clarify the conditions that promotesustainable investment and those conditions’ relation to capital structure.
Apart from sustainability issues, many papers examine real options models of investment anddebt ﬁnancing. Seminal studies by Mauer and Sarkar (2005), Sundaresan and Wang (2007), andSundaresan, Wang, and Yang (2015) show the eﬀects of debt ﬁnancing on investment timing, agency
cost, and ﬁrm value by integrating McDonald and Siegel (1986)’s investment timing model with
Leland (1994)’s capital structure model. In this context, Shibata and Nishihara (2012) and Shibata
and Nishihara (2018) show the eﬀects of debt borrowing constraints on investment timing, capital
structure, and ﬁrm value. However, these previous studies do not consider the process of selecting
one project from among several alternatives. By incorporating the choice between sustainable
and unsustainable projects in their levered setup, this paper unveils the relationships of between
corporate sustainability and capital structure.
The remainder of this paper is organized as follows. Section 2 introduces the model setup.
Section 3 shows the solutions in the unlevered and levered models. In particular, the sustainability
conditions are explicitly derived in both models. Section 4 numerically explores the eﬀects of the
key parameters on investment choice, timing, and capital structure. Section 5 concludes the paper.

42Model setup

2.1

Unlevered model

Consider a ﬁrm that has an option to invest in either an unsustainable project by paying investment
cost I1 or a sustainable project by paying investment cost I2 , where 0 < I1 < I2 holds. Throughout
the paper, subscripts 1 and 2 stand for the unsustainable and sustainable projects. The ﬁrm mustchoose only one of the available projects for reasons such as operational and ﬁnancial constraints.
After investing in either project, the ﬁrm receives continuous streams of earnings before interestand taxes (EBIT) X(t), which follows a geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion deﬁned in a ﬁltered probability space (Ω, F, P, {Ft })and µ, σ(> 0) and x(> 0) are constants. Apply the corporate tax rate τ ∈ (0, 1) to X(t). EBIT
X(t) lasts perpetually for the sustainable project, but for the unsustainable project, X(t) matures
at a random time, which follows an exponential distribution with rate λ(> 0). The maturity means
the time of the ﬁrm’ exit from the market. For simplicity, the model assumes the independence
between X(t) and the exponential distribution.
Parameter λ stands for ESG risk of the unsustainable project. For instance, high-pollution
technologies and products made with exploited labor could be legally banned on the grounds of
ESG concerns. Even if such projects are not legally banned, ESG concerns could nonetheless force
them to exit the market by attracting signiﬁcant attention in the era of social networks. 4 To reduce
the risks of such future penalties, ﬁrms would have to pay higher costs (e.g., by implementing
low-pollution technologies and making products with expensive labor). To capture this situation
as simply as possible, the model assumes the unsustainable project to have ESG risk λ and low
investment cost I1 and the sustainable project to have no ESG risk and high investment cost I2 .
By assessing the tradeoﬀ of ESG risk and investment cost, the ﬁrm optimizes investment choice
and timing.5
4

For simplicity, the model assumes that ESG risk will arise only after the unsustainable project initiation. Thismeans that the ﬁrm’s poor ESG activity attracts public attention and causes regulatory measures or backlash. In somesituations, unsustainable investment options may also entail ESG risk even before investment. In fact, the ﬁrst draftof this paper studied such a model. In this case, the ﬁrm acts as if it faces alternative investment opportunities with
diﬀerent discount rates. Then, the choice is dynamically inconsistent, and there is no explicit sustainability condition
like (3) and (24). I computed the ﬁrm’s policy and value under the assumption that the ﬁrm chooses a higher-valued
project at time 0 and found that the results are qualitatively the same as those of this paper.
5
In practice, the lifetime of sustainable projects may be ﬁnite, but it is expected to be longer than that of unsustainableprojects. Firms may have options to invest in (i.e., switch to) sustainable projects after unsustainable projects are forced
to terminate. However, the remaining option values will be lower than the values of ﬁrms that choose sustainableprojects from the beginning, due to switching costs, reputational damage, and competitors’ preemption, etc. Although
the diﬀerence between the sustainable and unsustainable projects (i.e., the impacts of ESG risk λ) is smaller in thesedetailed setups, this paper’s main results will remain qualitatively unchanged.

5The assumption of the unsustainable project’s characteristics is based on empirical evidence
that ﬁrms with poor ESG practices have high credit, legal, and downside risks (see Table 4 in
Gillan, Koch, and Starks (2021)). For instance, Jagannathan, Ravikumar, and Sammon (2018)
show that ESG risks are identiﬁed as rare, large, and nondiversiﬁable downside risks, and Seltzer,
Starks, and Zhu (2022) show that ﬁrms with poor environmental performance have high credit risk
particularly in states with stringent environmental regulations. On the other hand, there have been
mixed empirical ﬁndings about the eﬀects of ESG performance on ﬁrm (project) value (see Table 5in Gillan, Koch, and Starks (2021)). Based on the ambiguous ﬁndings, the model assumes the same
cash ﬂow process for both projects. This simplifying assumption renders the model one-dimensionaland facilitates an analytical solution.

2.2

Levered model

As in the standard models of dynamic investment and ﬁnancing based on tradeoﬀ theory (e.g.,
Shibata and Nishihara (2012) and Sundaresan, Wang, and Yang (2015)), the ﬁrm issues consol
debt at investment time. The ﬁrm optimizes debt level (i.e., the coupon of debt) in addition toinvestment choice and timing. With debt, shareholders are better oﬀ declaring default than continuing operation and coupon payments when EBIT X(t) falls to a certain threshold. In bankruptcy,
shareholders receive nothing, and debt holders takes over the unlevered ﬁrm value, 6 where a fraction α ∈ (0, 1) of the ﬁrm value is lost to the deadweight costs of bankruptcy (e.g., ﬁling andattorney fees). If the ﬁrm invests in the unsustainable project, the ﬁrm could potentially exit fromthe market before default.

2.3

Firm types

In the baseline model, the ﬁrm is fully aware of ESG risk and optimizes its investment and ﬁnancing
policy to maximize ﬁnancial value. That is, the ﬁrm can choose either a sustainable project or anunsustainable project by taking into account the tradeoﬀ between ESG risk and investment cost.
For comparison, this paper considers ESG-unaware and ESG-motivated ﬁrms.

The ESG-unaware ﬁrm mistakenly assumes no ESG risk, i.e., λ = 0. The ESG-unaware ﬁrm optimizes
its investment and ﬁnancing policy to maximize ﬁnancial value under the faulty assumption. Because of the assumption λ = 0, the ESG-unaware ﬁrm prefers the unsustainable project with lower
investment costs I1 . By contract, the ESG-motivated ﬁrm gains extra value ∆I = I2 − I1 in addition to ﬁnancial value gained by investing in the sustainable project. The extra value can beinterpreted as nonﬁnancial beneﬁts or utility from the improvement of ESG performance (cf. Pedersen, Fitzgibbons, and Pomorski (2021)). The ESG-motivated ﬁrm prefers the sustainable project
because the extra value ∆I fully compensates for the project’s high investment cost I2 . For simplicity, this paper assumes that equity and debt holders have the same information and preferences
6

This paper does not distinguish between liquidation and reorganization bankruptcy. The unlevered value can beinterpreted as the liquidation or operating-concern value.

6about ESG risk. It could be an interesting issue for future study to investigate the agency conﬂicts
on ESG risk by incorporating asymmetric information and preferences between equity and debt
holders.
This paper’s classiﬁcation of ﬁrms into one of three categories (i.e., the ESG-aware, ESGunaware, and ESG-motivated ﬁrms) is based on those introduced by Pedersen, Fitzgibbons, andPomorski (2021). I focus on the ESG-aware ﬁrm as the baseline case for the following two reasons.
First, the ESG-aware ﬁrm’s problem is the most complicated and interesting because it addresses
the tradeoﬀ between ESG risk and investment cost. In fact, because the other two types of ﬁrm
face no such tradeoﬀ, their choice of project is clear-cut. Second, several studies (e.g., Bellon (2020)
and Cai, Zhu, Zhang, Li, and Xie (2020)) indicate that ESG-aware ﬁrms seem prevalent in reality.

3

Model solutions

3.1

Unlevered model solutions

The expected NPV of cash ﬂows starting from X(0) = x at time 0 is calculated as
∫ ∞
∫ s
−λs
E[
λe
e−rt (1 − τ )X(t)dtds] = aU
1x
0

(1)

0

for the unsustainable project andE[∫ ∞e−rt (1 − τ )X(t)dt] = aU
2x0Ufor the sustainable project, where aU
1 = (1 − τ )/(r − µ + λ) and a2 = (1 − τ )/(r − µ). ThroughoutUthe paper, superscript U represents the unlevered model. Multiplier aU
1 is lower than a2 becauseof random termination of cash ﬂows. By the strong Markov property of X(t), the expected NPVof cash ﬂows starting from X(t) = t at time t becomes aU
1 X(t) for the unsustainable project andaU
2 X(t) for the sustainable project. The unlevered ﬁrm (option) value at time 0 is expressed as
V U (x) = sup E[e−rT max {aUi X(T ) − Ii }],
i=1,2

T ∈T

(2)

where T denotes the set of Ft -stopping times. That is, the ﬁrm chooses its investment time T anda higher-valued project using information available until time T .
Optimal stopping problem (2) is technically the same as the one solved by Décamps, Mariotti,
and Villeneuve (2006). By using their results (see also Nishihara and Ohyama (2008)), I can show√2
the following proposition, where β = 0.5 − µ/σ 2 + (µ/σ 2 − 0.5) + 2r/σ 2 (> 1) is the positive
characteristic root. For the proof, see Appendix A.
Proposition 1 If sustainability condition
(

r−µ+λ
r−µ

β
) β−1

7

≥

I2
,
I1

(3)

holds, the ﬁrm chooses the sustainable project. The value function and investment threshold are
given by( )β (aU xU − I ) x
22 2xU
V U (x) =2
 aU x − I
2

2

xU
2 =

βI2
.
(β − 1)aU
2(x < xU
2)(4)(x ≥ xU
2 ),

and
(5)

Otherwise, the ﬁrm chooses either a sustainable project or an unsustainable project. The valuefunction and investment threshold are given by

)β
(
x

U U


(x < xU
(a
x
−
I
)
1

11 )
U
 1 11
x

11

 U
U
a1 x − I1(x ∈ [xU
11 , x12 ])V U (x) =


U
U
U U
U
U
U

(aU

1 x12 − I1 )∆ (x) + (a2 x21 − I2 )Σ (x) (x ∈ (x12 , x21 )) aU x − I(x ≥ xU
2
2
21 ),

(6)

U
where ∆U (x) and ΣU (x) denote the state prices for X(t) ﬁrst reaching xU
12 from above and x21 frombelow, respectively (for the details, see (29) and (30) in Appendix A), and
xU
11 =

βI1
,
(β − 1)aU
1(7)

U
where xU
12 and x21 are determined by the smooth pasting conditions (31) and (32) in Appendix A.

Remark 1 The ESG-unaware ﬁrm invests in the unsustainable project at threshold
xU
3 =

βI1
,
(β − 1)aU
2(8)

whereas the ESG-motivated ﬁrm invests in the sustainable project at threshold xU
3 . Inequalities
UUU
xU
3 < x2 and x3 < x11 hold, which means that the ESG-unaware and ESG-motivated ﬁrms invest

earlier than the ESG-aware ﬁrm does. Because of the suboptimality of xU
3 , the ﬁnancial values of
the ESG-unaware and ESG-motivated ﬁrms are lower than that of the ESG-aware ﬁrm.
Sustainability condition (3) is equivalent to the condition that value function (4) dominates the
payoﬀ of the unsustainable project aU
1 x − I1 for all x ≥ 0. That is, under sustainability condition(3), the ﬁrm has no reason to invest in the unsustainable project for any X(t). Value function (4)and investment threshold (5) are equal to those of the case in which only the sustainable projectis available. In other words, under sustainability condition (3), the unsustainable project does not
aﬀect the ﬁrm at all.
If sustainability condition (3) does not hold, the solution is more complicated. The ﬁrm’s
project choice depends on the initial state variable X(0) = x. For x < xU
11 , the ﬁrm waits forU
the unsustainable investment, for x ∈ [xU
11 , x12 ], the ﬁrm immediately invests in the unsustainableproject, and for x ≥ xU
21 , the ﬁrm immediately invests in the sustainable project. A notableUregion is (xU
12 , x21 ), where the ﬁrm waits to invest. The ﬁrm will undertake either an unsustainableUproject when X(t) falls to xU
11 or a sustainable project when X(t) increases to x12 . That is, for8Ux ∈ (xU
12 , x21 ), the ﬁrm postpones its project choice. Then, the value function includes the state

prices for the unsustainable and sustainable investments. The ﬁrst term represents the value ofthe option of investing in the unsustainable project, while the second term represents the value ofthe option of investing in the unsustainable project. An interesting insight from this area is that adecrease in EBIT can trigger the unsustainable investment, while an increase in EBIT can trigger
the sustainable investment.
U
Although the intermediate region (xU
12 , x21 ) is interesting, most of the real options literatureassumes that the initial state variable is low enough to exclude immediate investment. FollowingUthis standard, Section 4 will assume baseline parameter values satisfying x < xU
11 and x < x2 . In

this case, the ﬁrm chooses the unsustainable project if sustainability condition (3) is not satisﬁed.
Thus, sustainability condition (3) determines whether the ﬁrm chooses the sustainable project. The
next proposition shows the comparative statics of sustainability condition (3) and helps answer thefollowing questions: What institutional and market environments encourage sustainable investing?
What types of ﬁrms are more eager to invest in sustainable projects? For the proof, see Appendix
B.
Proposition 2 Sustainability condition (3) is more likely to hold for higher I1 , λ, σ, and µ, as well
as for lower I2 and r.
Obviously, higher I1 , λ, and lower I2 increase the advantage of the sustainable NPV overU
the unsustainable NPV (i.e., aU
2 /a1 ). Higher volatility σ increases the option value of postponing

investment, and through the option eﬀect, it increases the advantage of the high-cost and high-proﬁt
project (i.e., the sustainable project) over the low-cost and low-proﬁt project (i.e., the unsustainableproject). The impact of σ coincides with those of Décamps, Mariotti, and Villeneuve (2006)
and Nishihara and Ohyama (2008). Higher growth rate µ and lower discount rate r increase
the advantage of the sustainable project through the NPV and option eﬀects. With higher µ andlower r, EBIT will remain at a signiﬁcant level for a longer time, and hence, the advantage of theUsustainable NPV over the unsustainable NPV (i.e., aU
2 /a1 ) increases. Thus, the ﬁrm tends to avoid

short-termism (i.e., the unsustainable project). Higher µ and lower r, like higher σ, also increase
the option value of postponing investment and hence increase the advantage of the sustainableproject.
Proposition 2 generates many insights. First, ﬁrm characteristics can aﬀect a ﬁrm’s choice
between the alternative projects. For example, large and listed ﬁrms are more likely to face theinstitutional and social pressures of ESG concerns, which implies higher λ than that of small andprivate ﬁrms. Large and listed ﬁrms also tend to be more diversiﬁed, less ﬁnancially constrained,
and less operationally constrained; hence they use lower discount rates than small and private ﬁrms
do (e.g., Harrington (1989), Krugger, Landier, and Thesmar (2015), Jagannathan, Matsa, Meier,and Tarhan (2016), Décaire (2020), and Nishihara (2021)). Because of higher λ and lower r, large
and listed ﬁrms are more likely to choose sustainable projects than small and private ﬁrms are. In
terms of economic rationality, this result explains observed trends that large and listed ﬁrms are9more engaged in sustainability.
Proposition 2 also entails policy implications. What governmental policies will encourage sustainable investing? Of course, subsidies for sustainable projects decrease I2 , and penalties forunsustainable projects increase I1 . The government’s proactive stances on regulations and information disclosures regarding ESG risks cause ﬁrms to perceive higher λ. These policies are
frequently implemented to promote sustainability in the real world. Notably, Proposition 2 alsosuggests that monetary policies, such as decreasing interest rates and quantitative easing, encourage sustainable investments by decreasing r. I believe that one of the reasons that ﬁrms have paid
great attention to sustainability over the last decade lies in the historically low interest rates that
followed the Great Recession.
From an international perspective, developed countries frequently blame developing countries
for poor ESG practices when there are externalities (e.g., climate change caused by greenhousegas emissions). However, interest rates vary among countries and tend to be higher in developing
countries. All else being equal, developing countries with higher interest rates valuate unsustainableprojects higher (i.e., short-termism) than developed countries do. To maintain the levels of ESG
practices established by developed countries, developing countries have to take more proactive ESG
policies (i.e., subsidies, penalties, and regulations) than developed countries do. In other words, thepromotion of sustainability is more costly in developing countries than it is in developed countries,
which causes conﬂict between developed and developing countries over ESG issues.

3.2
3.2.1

Levered model solution
Default decisions

The problem is solved backward. First, consider the unsustainable ﬁrm that issued debt withcoupon C at time 0. Following the standard literature (e.g., Leland (1994) and Goldstein, Ju, andLeland (2001)), shareholders do not take into account debt in place and optimize default time T D
for their own interests. The equity value of the ﬁrm is the solution to the optimal stopping problem∫ ∞∫ min{s,T D }−λs
E1 (x; C) = sup E[
λee−rt (X(t) − C)dtds](9)
T D ∈T
∫ ∞

0
−λs

0

∫ s

∫ ∞

−rt

−λs

∫ s

(X(t) − C)dtds] + sup E[
λee−rt (C − X(t))dtds]
T D ∈TTD
TD()(1 − τ )C(1 − τ )C−(r+λ)T D
U
D
+
sup
E[e
−
a
X(T
)
]
(10)
= aU
x
−
1
1
r+λ
r+λ
T D ∈T) γ1 (()(1 − τ )C(1 − τ )C
x
U D
= aU
x
−
+
−
a
x
(C)
,
(11)
1
1 1
r+λ
r+λ
xD
1 (C)

= E[

λe

0

e

0

where the strong Markov property of X(t) is used in (10), γ1 denotes the negative characteristicrootµγ1 = 0.5 − 2 −
σ

√

(µ
σ

− 0.5
2

)2
+

2(r + λ)(< 0),σ2(12)

and default threshold xD
1 (C) is derived asxD
1 (C) =

γ1 (r − µ + λ)C.
γ1 − 1
r+λ

10

(13)

More intuitively, one can also derive equity value (11) and default threshold (13) as the solution todiﬀerential equationµx∂E1 (x; C)∂ 2 E1 (x; C)+ 0.5σ 2 x2
+ λ(−E1 (x; C)) + x − C = rE1 (x; C)
∂x
∂2x

(14)

with the value matching condition E1 (xD (C); C) = 0 and smooth pasting condition ∂E1 (xD (C); C)/∂x =0, and boundary condition limx→∞ E1 (x; C)/x < ∞ (cf. Chapter 5.5.B of Dixit and Pindyck(1994)).7 Note that in (14), term λ(−E1 (x; C)) represents that E1 (x; C) becomes 0 at probability
λdt in inﬁnitesimal time interval dt. The solution to (14) is exactly the same as (11) and (13).
Similarly, the debt and ﬁrm values are derived as follows:)γ 1 (
(
)
C
x
C
U D
−
−
(1
−
α)a
x
(C)
,
D1 (x; C) =
1 1
r+λ
r+λ
xD
1 (C)
(
) γ1 (
)
τC
x
τC
U
U D
F1 (x; C) = a1 x +−+ αa1 x1 (C) .
r+λ
r+λ
xD
1 (C)

(15)
(16)

The equity, debt, and ﬁrm values (11), (15), and (16) resemble the standard valuations without
random maturity (e.g., Leland (1994), Goldstein, Ju, and Leland (2001), Shibata and Nishihara(2012), and Sundaresan, Wang, and Yang (2015)). Indeed, (11), (15), and (16) are equal to thestandard valuations with discount rate r + λ. That is, as in Chapter 6.4.A of Dixit and Pindyck(1994), one can regard the ﬁrm as inﬁnitely lived but augment the discount rate by adding thetermination probability parameter λ. As in the standard literature, equations (11), (15), and (16)can be interpreted as follows. The ﬁrst two terms in (11) are the equity value of operating theﬁrm with no default, while the last term represents the value of the default option. Debt value(15) is equal to the value of receiving coupons with no bankruptcy minus the loss associated withbankruptcy. Firm value (16) is composed of the unlevered ﬁrm value, tax beneﬁts of debt, andbankruptcy costs.c
Now, consider optimal capital structure. As described in the standard literature (e.g., Leland(1994) and Goldstein, Ju, and Leland (2001)), the ﬁrm chooses coupon C to maximize the ﬁrm value
F1 (x; C) (which is equal to the ex-ante equity value). By the ﬁrst-order condition ∂F1 (x; C)/∂C =0, the optimal coupon is derived as
C1 (x) =r + λ γ1 − 1 x.
r − µ + λ γ 1 h1

(17)

By (13), (16), and (17), the optimal ﬁrm value is derived as
F1 (x; C1 (x)) = ϕ1 aU
1 x,

(18)

where h1 and ϕ1 are constants deﬁned by[(α )]− γ11h1 = 1 − γ 1 1 − α +(> 1),ττ(> 1).
ϕ1 = 1 +(1 − τ )h1

7

I present this solution method because this method may be more prevalent in the ﬁnance area.

11

(19)
(20)

The leverage and credit spreads are also derived explicitly as
D1 (x; C(x))γ 1 − 1 1 − ξ1=,F1 (x, C(x))γ1 ϕ1 h1 (1 − τ )C1 (x)rξ1 + λ
CS1 =
−r =
,
D1 (x; C1 (x))
1 − ξ1
LV1 =

(21)
(22)

where ξ1 is a constant deﬁned byξ1 =(1 − (1 − α)(1 − τ )γ1 )(h1 )γ1.
γ1 − 1

These solutions are the same as those of the inﬁnitely-lived ﬁrm with augmented discount rater + λ in previous studies (e.g., Nishihara and Shibata (2010), Shibata and Nishihara (2012), andSundaresan, Wang, and Yang (2015)). The diﬀerence between the unlevered ﬁrm value (1) andlevered ﬁrm value (18) is the multiplier ϕ1 (> 1), which measures the eﬀect of optimal leverage on
ﬁrm value. That is, the ﬁrm obtains higher value by choosing the optimal capital structure basedon the tradeoﬀ between the tax beneﬁts of debt and bankruptcy costs.
The equity, debt, ﬁrm values, coupon, leverage, and credit spreads of the sustainable ﬁrm areequal to the above values with λ = 0 and are also derived in existing literature (e.g., Nishihara andShibata (2010), Shibata and Nishihara (2012), and Sundaresan, Wang, and Yang (2015)). Hence,
I will omit the equations and let subscript 2 represent the corresponding value of the sustainableﬁrm. For instance, C2 (x) and ϕ2 represent optimal coupon (17) and leverage eﬀect (20) with λ = 0,
respectively, for the sustainable ﬁrm.

3.2.2

Investment and ﬁnancing decisions

Now, consider the investment and ﬁnancing decisions. By the strong Markov property of X(t), theexpected NPV of cash ﬂows starting from X(t) = t at time t becomes aL
1 X(t) for the unsustainable
L
U
L
U
project and aL
2 X(t) for the sustainable project, where a1 = ϕ1 a1 and a2 = ϕ2 a2 . The ﬁrm(option) value at time 0 is expressed as
V L (x) = sup E[e−rT max {aLi X(T ) − Ii }],
i=1,2

T ∈T

(23)

where the ﬁrm chooses its investment time T and a higher-valued and leveraged project. The
diﬀerences between problems (2) and (23) are leverage multipliers ϕ1 and ϕ2 . Optimal stopping
problem (23) is technically the same as the one solved by Décamps, Mariotti, and Villeneuve(2006). The following proposition is shown in the same manner as in Proposition 1. For the proof,
see Appendix C.
Proposition 3 If sustainability condition() β(r − µ + λ)ϕ2 β−1I2≥ ,(r − µ)ϕ1I1(24)holds, the ﬁrm chooses the sustainable project. The value function and investment threshold are
given by( )β (aL xL − I ) x
22 2xLV L (x) =2
 aL x − I
2

2

12

(x < xL2)(x ≥ xL
2 ),

(25)

and
xL
2 =

βI2
= xU
2 /ϕ2 .
(β − 1)aL
2(26)

Otherwise, the ﬁrm chooses either a sustainable project or an unsustainable project. The valuefunction and investment threshold are given by

)β
(
x

L L


 (a1 x11 − I1 ) xL


11

 L
L
a
x
−
I
1
1
V (x) =


L L
L
L

(a1 x12 − I1 )∆L (x) + (aL
2 x21 − I2 )Σ (x) aL x − I22(x < xL
11 )L(x ∈ [xL
11 , x12 ])(27)

L(x ∈ (xL
12 , x21 ))(x ≥ xL
21 ),L
where ∆L (x) and ΣL (x) denote the state prices for X(t) ﬁrst reaching xL
12 from above and x21 frombelow, respectively (for the details, see (33) and (34) in Appendix C), and
xL
11 =

βI1
= xU
11 /ϕ1 ,(β − 1)aL
1

(28)

L
where xL
12 and x21 are determined by the smooth pasting conditions (35) and (36) in Appendix C.

Remark 2 The ESG-unaware and ESG-motivated ﬁrms invest in the unsustainable and sustainU
L
L
L
L
able projects, respectively at threshold xL
3 = x3 /ϕ2 . Inequalities x3 < x2 and x3 < x11 hold, whichmeans that the ESG-unaware and ESG-motivated ﬁrms invest earlier than the ESG-aware ﬁrm does.
Because of the suboptimality of xU
3 , the ﬁnancial values of the ESG-unaware and ESG-motivatedﬁrms are lower than that of the ESG-aware ﬁrm.
Proposition 3 can be interpreted as in Proposition 1. Indeed, sustainability condition (24) isequivalent to the condition that value function (25) dominates the unsustainable project’s payoﬀaL
1 x − I1 for all x ≥ 0. Value function (25) and investment threshold (26) are equal to those of
the levered model in which only the sustainable project is available. If sustainability condition(24) does not hold, the ﬁrm’s project choice depends on the initial state variable X(0) = x. The
L
regions x < xL
11 and x ≥ x21 are the unsustainable and sustainable investment regions, respectively,
L
whereas the other two regions are the waiting regions. A notable region is (xL
12 , x21 ), where theLﬁrm waits to invest until X(t) reaches either low threshold xL
11 or high threshold x12 .

As explained after Proposition 1, Section 4 will assume a suﬃciently small initial state variable
L
X(0) = x satisfying x < xL
11 and x < x2 . Then, under sustainability condition (24), the ﬁrm
Lwill invest in the sustainable project and issue debt with coupon C2 (xL
2 ) at threshold x2 and go
LLbankrupt at threshold xD
2 (C2 (x2 )) = x2 /h2 . If sustainability condition (24) does not hold, theLﬁrm will invest in the unsustainable project and issue debt with coupon C1 (xL
11 ) at threshold x11
LL
and either go bankrupt at threshold xD
1 (C1 (x11 )) = x11 /h1 or be forced to exit before bankruptcy.

Obviously, sustainability condition (24) is more likely to hold for higher I1 and lower I2 . I cannot
mathematically show the impacts of λ, σ, µ, and r on condition (24) because their impacts on theratio of the leverage eﬀect ϕ2 /ϕ1 are complicated. However, Section 4 will show numerically that
their impacts remain unchanged from Proposition 2. The main reason is that the impact throughthe leverage eﬀect is indirect and weaker than the impacts through the NPV and option eﬀects. The

13

question remains as to whether access to debt ﬁnancing encourages sustainable investing. Notably,
the next proposition answers this question in the negative. For the proof, see Appendix D.
Proposition 4 The leverage eﬀect ϕ1 increases in λ, which implies ϕ1 > ϕ2 . Hence, sustainability
condition (24) is less likely to hold than sustainability condition (3).
The positive derivative ∂ϕ1 /∂λ > 0 means that the leverage eﬀect is larger for the less sustainable ﬁrm. The tradeoﬀ between the tax beneﬁts of debt and bankruptcy costs explains thiseconomic mechanism as follows. The unsustainable ﬁrm is forced to exit regardless of its debt
level at a random maturity, and hence, bankruptcy costs are not relevant in the forced exit case.
That is, bankruptcy costs matter only when the ﬁrm defaults earlier than the forced exit. Thus,
with a less sustainable project (i.e., higher λ), the ﬁrm perceives lower bankruptcy costs, increases
leverage LV1 , and enjoys larger leverage multiplier ϕ1 .8 For instance, if the ﬁrm knows that it will
be terminated in one year, it only needs to care about one year of operation. Accordingly, theunsustainable ﬁrm chooses higher leverage to increase its value compared to the sustainable ﬁrm,
which cares about long-term operation. That is, one could interpret the unsustainable ﬁrm’s debt
policy as short-termism. Thus, access to debt ﬁnancing decreases the advantage of the sustainableproject over the unsustainable project through the leverage eﬀect and hence discourages sustainable
investing.
The result also implies that unrestricted lending can lead ﬁrms to invest in unsustainableprojects. How can lenders prevent this negative impact of lending on ESG issues? In practice,some lenders refuse to ﬁnance unsustainable projects (i.e., negative screening for ESG issues). Naturally, such debt ﬁnancing restrictions on unsustainable projects encourage sustainable investments
as follows.
Remark 3 Suppose that the ﬁrm is able to use debt ﬁnancing only for the sustainable project. In
this case, problem (23) is replaced withL
V L (x) = sup E[e−rT max{aU
1 X(T ) − I1 , a2 X(T ) − I2 }],T

which can be solved in the same way as problem (23). Sustainability condition (24) is replaced with() β(r − µ + λ)ϕ2 β−1
I2
≥ ,
r−µ
I1
which is more likely to hold than condition (3). That is, the ﬁrm is more likely to choose thesustainable project in the negative screening model than in the unlevered model.

4

Numerical illustrations

4.1

Impacts of ESG riskThis section conducts numerical analysis, including comparative statics with respect to key parameters: ESG risk λ, discount rate r, growth rate µ, volatility σ, and corporate tax rate τ . The
8

Although I cannot analytically show dLV1 /dλ > 0, but Section 4 veriﬁes dLV1 /dλ > 0 in numerical examples.

14

Table 1: Baseline parameter values.
r

µ

σ

τ

λ

α

I1

I2

x

0.05

0.01

0.2

0.15

0.1

0.4

10

100

0.5

baseline parameter values are set as in Table 1.9 The values of r, µ, σ, τ, and α are standard indynamic corporate ﬁnance literature and reﬂect a typical S&P ﬁrm (e.g., Morellec (2001), Arnold(2014), Shibata and Nishihara (2018), and Nishihara and Shibata (2021)). Following Boomsma and
Linnerud (2015) and Adkins and Paxson (2016), who assume renewable energy policy uncertainty
as a Poisson jump process with hazard rate 0.1, ESG risk is set at λ = 0.1. This means that theunsustainable project’s lifetime is, on average, 1/λ = 10 years. For instance, regulations for greenhouse gas emissions are greatly aﬀected by international agreements, such as the Kyoto Protocol
in 1997 and the Paris agreement in 2015, and the time span is about 10 to 20 years. Blowing up
due to ESG concerns in social networks might occur more frequently, although it depends on the
publicity of the ﬁrm or project. This subsection shows the results for λ = 0 to 0.2. The baseline
case also assumes x = X(0) = 0.5, I1 = 10, and I2 = 100 to make the sustainable and unsustainableproject values comparable. These values are not essential because they can be normalized. Notethat the initial state variable x = X(0) = 0.5 is low enough to exclude immediate investment for
all examples in this section.
For the baseline parameter values, sustainability conditions (3) and (24) are satisﬁed. The
unlevered ﬁrm’s value and investment threshold are V U (x) = 0.44 and xU
2 = 10.237, whereas thoseof the levered ﬁrm are V L (x) = 0.482 and xL
2 = 9.745, where the leverage eﬀect ϕ2 = 1.051. In
the levered case, the coupon, default threshold, leverage, and credit spreads are equal to C2 (xL2) =LL
6.072, xD
2 (C2 (x2 )) = x2 /h2 = 2.791, LV2 = 0.485, and CS2 = 0.0075. Note that LV2 and CS2 areevaluated at the time of investment.
Figure 1 plots investment thresholds xL , xU , default threshold xD , coupon C, ﬁrm values
V L (x), V U (x), leverage LV , and credit spreads CS for ESG risk λ = 0 to 0.2, where the blacksolid and blue dashed lines represent the levered and unlevered cases, respectively. Sustainability
conditions (3) and (24) hold for λ ≥ 0.076 and λ ≥ 0.078, respectively. The two conditions arealmost equal because the leverage eﬀect on the choice is very weak (i.e., ϕ2 /ϕ1 ≈ 1). As shownin Proposition 2, higher λ decreases the NPV of the unsustainable project with no debt and thus
induces the ﬁrm to choose the sustainable project. This holds true in the levered model because
the impact of r on the NPV is much stronger the impact of λ on ϕ2 /ϕ1 .
Figure 1 shows the jumps in the graphs other than V L (x) and V U (x) at the switching points
between the unsustainable and sustainable projects. For instance, sustainable investment thresholds
ULU
xL
2 and x2 are much higher than unsustainable investment thresholds x11 and x11 even when the9

I computed the results for a wide range of parameter values and presented the qualitatively robust results in thissection.

15

two project values are comparable. In fact, the ﬁrm postpones the sustainable project until thestate process X(t) reaches quite a high threshold because of its high-cost and high-proﬁt structure.
That is, sustainable investment timing is later than unsustainable investment timing when the
choice between the two projects is comparable.
On the other hand, as explained in Remarks 1 and 2, ESG-unaware and ESG-motivated ﬁrms
invest in the unsustainable and sustainable projects, respectively, at xU
3 = 1.024 in the unleveredULcase and xL
3 = 0.975 in the levered case. These thresholds agree with x and x for λ = 0 in the

top-left panel of Figure 1. Compared to these values, the baseline (ESG-aware) ﬁrm’s investment
threshold is much higher especially in the case of sustainable investing. These results entail thefollowing policy implication. The government’s proactive stances on sustainability issues may beable to increase λ and I2 /I1 and cause ESG-aware ﬁrms to switch from unsustainable projects tosustainable projects. In terms of investment timing, however, these sustainable investments willtake place much later than unsustainable investments with comparable project values.
Now, I examine each panel of Figure 1 closely. All values are constant in the sustainable project
region because the sustainable project does not depend on λ. In the unsustainable project region,
U
D
L
U
xL
11 , x11 , x1 , C1 , LV1 , and CS1 increase in λ, whereas V1 (x) and V1 (x) decrease in λ. The delay ininvestment and decrease in ﬁrm value are straightforward because the unsustainable NPV decreases
with higher λ. What is more notable is the increase in LV1 in the bottom-left panel of Figure 1.
The reason has been already mentioned below Proposition 4. Higher λ decreases the probabilityof bankruptcy prior to the random maturity and hence decreases the expected bankruptcy costs.
Accordingly, the ﬁrm adopts a more aggressive capital structure as λ increases. The increase in
CS1 in the bottom-right panel is due partially to the increase in LV1 , but the main driver is theincrease of the forced exit probability (i.e., the second term of the numerator in (22)). These results
predict a positive correlation between ESG risk and leverage (or credit spreads). In keeping withthe prediction, many papers empirically show that bad ESG practices increase credit risk (see Table
4 in Gillan, Koch, and Starks (2021) and Seltzer, Starks, and Zhu (2022)). Although I could notﬁnd any empirical paper on the relationship between ESG and leverage, it will be an interesting
issue for future empirical studies.

4.2

Impacts of discount rate

Apart from ESG risk λ, the discount rate r greatly aﬀects the ﬁrm’s choice between the sustainable
and unsustainable projects. Intuitively, with high r, the ﬁrm focuses on cash ﬂows in the near
future and puts less weight on long-term sustainability of cash ﬂows (i.e., short-termism). The ﬁrm
with short-termism tends to prefer the unsustainable project. More precisely, Proposition 2 show
the impact of r on the project choice thorough the NPV and option eﬀects in the unlevered model.
The impact of r remains unchanged in the levered model because the impact of r on the leverage
eﬀect (i.e., ϕ2 /ϕ1 ) is relatively weak. Figure 2 plots xL , xU , xD , C, V L (x), V U (x), LV , and CS for
discount rate r = 0.02 to 0.08, where the black solid and blue dashed lines represent the levered
and unlevered cases, respectively. Sustainability conditions (3) and (24) hold for r ≤ 0.0576 and16

r ≤ 0.0567, respectively, where the graphs other than V L (x) and V U (x) jump at the switching
points.
As explained below Proposition 4, low discount rate r is associated with large and listed ﬁrms,
and low interest rate policies. In such situations, ﬁrms face fewer ﬁnancial constraints and can easily
use debt ﬁnancing. Although unrestricted access to debt ﬁnancing hinders sustainable investing (seeProposition 4), the impact through the leverage channel is much weaker than the impacts throughthe NPV and option channels. Overall, large and listed ﬁrms are likely to choose sustainableprojects, and low interest rate policies help sustainable investment trends.
In each panel of Figure 2, the impacts of r are greater in the sustainable project region thanin the unsustainable project region. The main reason is that the unsustainable project’s cash
ﬂows may end in the short term and are less sensitive to the level of r. In the top-left panel, xL
and xU jump downward at the switching points from the sustainable project to the unsustainable
L
U
U
L
U
project, although xL
within each project region) increase in r.
11 , x2 , x11 , and x2 (i.e., x and x

It is well known in real options theory that higher r reduces the NPV of long-term cash ﬂows anddelays investment (e.g., see Proposition 2 in Nishihara (2021) for a formal proof). This standard
result does not hold due to the switch from the sustainable project to the unsustainable project.
For instance, investment occurs earlier for r = 0.06 than for r = 0.04 because the ﬁrm invests in
diﬀerent projects for r = 0.04 and 0.06. This implies that low interest policies do not necessarily
accelerate corporate investment.
In the center-right panel of Figure 2, V L (x) and V U (x) greatly decrease in r mainly because
the project values are exponentially discounted by rate r. In the bottom-left panel, LV increases inr. This result is due to the eﬀect of λ on the unsustainable ﬁrm’s LV (cf. Figure 1 in Section 4.1).
Indeed, with higher r, the ﬁrm puts less weight on bankruptcy costs in the future and increases
leverage to receive the tax beneﬁts. That is, the ﬁrm with short-termism adopts a riskier capital
structure. This also increases CS in the bottom-right panel.

4.3

Impacts of cash ﬂow growth rate

This subsection examines the comparative statics with respect to growth rate µ. Figure 3 plots
xL , xU , xD , C, V L (x), V U (x), LV , and CS for growth rate µ = 0 to 0.04, where the black solid andblue dashed lines represent the levered and unlevered cases, respectively. Sustainability conditions(3) and (24) hold for µ ≥ 0.0045 and µ ≥ 0.0051, respectively, where the graphs other than V L (x)and V U (x) jump at the switching points. All else being equal, higher µ encourages sustainable
investing. This results primarily from the NPV and option eﬀects explained below Proposition
2, that is, higher µ, like r, increases the NPV of long-term cash ﬂows and the option value of
postponing investment. Note that the impact of µ on the leverage eﬀect (i.e., ϕ2 /ϕ1 ) is much
weaker than the impact of µ on the NPV and option value. Empirical studies show mixed evidenceof whether good ESG practices help ﬁrm performance (e.g., Table 5 in Gillan, Koch, and Starks(2021)). This paper reminds us of the importance of endogeneity for the research question. Indeed,
the model indicates that ﬁrms that have high growth rates are more likely to invest in sustainable17

projects.
In the top-left panel of Figure 3, xL and xU increase in µ in the unsustainable project region
but decrease in µ in the sustainable project region. The non-monotonic result is explained by thetradeoﬀ between the NPV and option eﬀects as follows. Higher µ increases the NPV of cash ﬂows,
which plays a role in accelerating investment. On the other hand, higher µ also increases the option
value of waiting for a better economic state, which plays a role in delaying investment. The NPV
eﬀect (i.e., the investment acceleration eﬀect) dominates the option eﬀect (i.e., the investment delay
eﬀect) for the sustainable project with perpetual cash ﬂows, and vice versa for the unsustainableproject with short-term cash ﬂows.
The bottom-left panel of Figure 3 also shows a notable result. Although LV1 and LV2 (i.e.,
LV within each project region) increase in µ, LV is not monotonic due to the switch between theunsustainable and sustainable projects. Most of the structural models (e.g., Leland (1994)) basedon the tradeoﬀ theory generate a positive relation between LV and µ, although empirical studies(e.g., Titman and Wessels (1988) and Frank and Goyal (2015)) show a negative relation. This
paper shows the possibility that a ﬁrm with higher µ can invest in the sustainable project withlower LV than a ﬁrm with lower µ. For instance, the panel shows that LV for µ = 0 to 0.0051 ishigher than LV for µ = 0.0051 to 0.04. That is, the model can generate a negative relation between
LV and µ by incorporating an investment choice between sustainable and unsustainable projects.

4.4

Impacts of cash ﬂow volatility

In the real option analysis, it is important to study the comparative statics with respect to σ
because σ aﬀects the results thorough the option channel rather than the NPV channel. As shownin Proposition 2, higher σ increases the value of the option to invest in the high-proﬁt and highcost project (i.e., the sustainable project) rather than the low-proﬁt and low-cost project (i.e.,
the unsustainable project) in the unlevered model. This result holds true in the levered model
because the impact of σ on the leverage eﬀect (i.e., ϕ2 /ϕ1 ) is relatively weak. Figure 4 plots
xL , xU , xD , C, V L (x), V U (x), LV , and CS for volatility σ = 0.1 to 0.5, where the black solid andblue dashed lines represent the levered and unlevered cases, respectively. Sustainability conditions(3) and (24) hold for σ ≥ 0.147 and σ ≥ 0.152, respectively, where the graphs other than V L (x)and V U (x) jump at the switching points.
Although the jumps are characteristic of a model with project choice, most of the other results
are straightforward and consistent with the standard results in real options theory. For instance,
xL , xU , V L (x), and V U (x) increase in σ in the top-left and center-right panels . This is consistent
with the well-known result that higher σ increases the option value of waiting for a better economic
state (e.g., Dixit and Pindyck (1994)). In the bottom-left panel, LV decreases in σ, which isconsistent with the well-known result that higher σ increases bankruptcy risk and hence decreases
leverage (e.g., Leland (1994)). Although xD is not monotonic in the upper-right panel, it is notnotable. In fact, the gap between xL and xD increases in σ, which is consistent with the standard
result.

18

The bottom-right panel is most interesting in Figure 4. In the panel, CS1 and CS2 (i.e., CS
within each project region) increase in σ. This is because adjustment of capital structure (i.e., adecrease in LV with higher σ) does not fully oﬀset the increase in bankruptcy risk with higher
σ. Although this result is consistent with the standard result in the capital structure literature(e.g., Leland (1994)), the switch between the unsustainable and sustainable projects changes themonotonic result. Note that CS1 is much higher due to the forced exit risk (i.e., the second term of
the numerator in (22)) than CS2 . Lower σ can cause the ﬁrm to choose the unsustainable projectand hence increase CS. In fact, in the bottom-right panel, CS is higher for σ = 0.1 to 0.152 thanfor σ = 0.152 to 0.5. This counter-intuitive result may warn against the apparent argument thata less volatile economy encourages stable development and hence improves sustainability. Indeed,
the model indicates the opposite possibility, that is, lower σ discourages sustainable investments
and increases corporate bankruptcies.

4.5

Impacts of corporate tax rate

It is also interesting to study the eﬀects of corporate tax rate τ on the results in terms of policy
implications. Obviously, tax reductions only for sustainable projects, like subsidies, reduce investment cost I2 and encourage sustainable investing. The question is as to whether lower taxationof ﬁrms encourages sustainable investing. Sustainability condition (3) does not depend on τ inthe unlevered model, while sustainability condition (24) depends on τ through the leverage eﬀect
ϕ2 /ϕ1 . In other words, τ aﬀects the choice between the sustainable and unsustainable projects only
through the capital structure channel. As seen in the previous subsections, ϕ2 /ϕ1 does not change
greatly with parameters. Even though τ signiﬁcantly aﬀects the capital structure in tradeoﬀ theory,
ϕ2 /ϕ1 changes from 0.993 to 0.935 with variation of τ = 0.05 to 0.55. Because of the small impactof τ on ϕ2 /ϕ1 , sustainability condition (24) holds for τ = 0.05 to 0.55. To see the weak impactsof τ on the sustainability choice, I reset I2 = 140, where the other parameter values remain as in
Table 1. Note that the sustainable and unsustainable project values are almost equal for I2 = 140.
Figure 5 plots xL , xU , xD , C, V L (x), V U (x), LV , and CS for corporate tax rate τ = 0.05 to 0.55,
where the black solid and blue dashed lines represent the levered and unlevered cases, respectively.
Sustainability condition (3) holds true regardless of τ in the unlevered model, while sustainability
condition (24) hold for τ ≤ 0.307, where the graphs other than V L (x) jump at the switching points.
This result implies that lower τ encourages sustainable investing. The reason is explained as follows.
According to tradeoﬀ theory, lower τ reduces the tax advantages of debt and hence reduce LV (seethe bottom-left panel of Figure 5). That is, the ﬁrm is less leveraged and approximates the unlevered
ﬁrm as τ decreases. As explained in Proposition 4, the unleverd ﬁrm is more likely to choose thesustainable project than the highly levered ﬁrm is. Thus, lower τ encourages sustainable investingby decreasing LV . As a policy implication, tax reductions could lead to safer capital structure andencourage sustainable investing, although the eﬀects on sustainability through capital structure areindirect and weaker than the NPV and option eﬀects. Note that this result is based on permanently
lower levels of corporate tax rates. Tax reductions for a short period can discourage sustainable

19

investing because the ﬁrm focuses on short-term cash ﬂows rather than long-term cash ﬂows.
The center-right panel of Figure 5 shows that V L (x) and V U (x) decrease in τ . The results
are straightforward because higher τ decreases the NPV of cash ﬂows. More interestingly, thegap between V L (x) and V U (x) increases in τ . In fact, V U (x) decreases almost linearly, but theslope of V L (x) moderates, especially after the switching point τ = 0.307. That is, the levered ﬁrm
mitigates the negative impact of higher τ by choosing the unsustainable project and higher LV (seethe bottom-left panel) with greater tax advantages. In the top-left panel, xL jumps downward atL
the switching point from the sustainable project to the unsustainable project, whereas xL
11 , x2 (i.e.,

xL within each project region), and xU = xU
2 increase in τ . The monotonic increase within eachregion is explained by the straightforward eﬀect that higher τ decreases the NPV of cash ﬂows anddelays investment. However, in the levered model, the monotonic result does not hold due to theswitch from the sustainable project to the unsustainable project. In fact, xL is higher for τ = 0.05
to 0.307 than for τ = 0.307 to 0.55. This implies that low tax rate policies may improve corporate
sustainability but delay corporate investment.

5

Conclusion

This paper investigates a ﬁrm’s choice of project sustainability, investment timing, and capital
structure in the real options model. In the model, the ﬁrm chooses between sustainable andunsustainable projects, where the sustainable project with high investment cost yields perpetualcash ﬂows, but the unsustainable project with low investment cost has risk of cash ﬂow termination(i.e., ESG risk). This paper analytically derives the unlevered and levered model solutions including
the sustainability conditions and shows the eﬀects of key parameters on the results. The main
results are summarized below.
With higher ESG risk, lower discount rate, and higher growth rate, the ﬁrm is more likely
to invest in the sustainable project mainly because these factors increase the advantage of thesustainable NPV over the unsustainable NPV. Higher volatility encourages sustainable investing
mainly through the option eﬀect. The ﬁrm with a less sustainable project can increase its value by
adopting more leverage because bankruptcy costs matter less. This means that the leverage eﬀect islarger for the less sustainable ﬁrm. Through the leverage eﬀect, access to debt ﬁnancing decreases
the advantage of the sustainable project over the unsustainable project and hence discourages
sustainable investing. A higher corporate tax rate can also discourage sustainable investing throughthe leverage eﬀect, but the leverage eﬀect is weaker than the NPV and option eﬀects.
These results clarify the market conditions and ﬁrm characteristics that promote sustainable
investing and entail many policy implications. However, it must be noted that parameter shifts(e.g., market and policy shifts) that encourage sustainable investing do not necessarily accelerate investment because sustainable investment timing is later than unsustainable investment timing, due
to higher investment cost. For instance, contrary to straightforward expectations, lower discountrate and higher growth rate can delay corporate investment through switch from the unsustainable

20

investment to the sustainable investment.

A

Proof of Proposition 1

It is easy to show that the value function (4) with investment threshold (5) is the solution to the
problem with only the sustainable project, i.e., supT ∈T E[e−rT (aU
2 X(T ) − I2 )].
Deﬁne f (x) = (4) − (aU
1 x − I1 ). Below, I will show that f (x) ≥ 0 holds for all x ≥ 0 if and only
1/(β−1)
if condition (3) holds. The solution to f ′ (x∗ ) = 0 becomes x∗ = xU
∈ (0, xU
2 (a1 /a2 )2 ). Notethat f (x) is convex because of β > 1. Therefore, by the ﬁrst-order condition, I havemin f (x) = f (x∗ )x≥0β1( U ) β−1
( U ) β−1
a1
a1
U U
− a1 x2
+ I1
aU
aU
2
2
β
( U ) β−1
a1
= I1 − I2
,
aU
2

U
= (aU
2 x2 − I2 )which is nonnegative if and only if condition (3) holds.
Under condition (3), the value function (4) dominates the payoﬀ of the unsustainable project,and hence, the ﬁrm has no possibility of investing in the unsustainable project. Then, problem isreduced to supT ∈T E[e−rT (aU
2 X(T ) − I2 )], and the solution is (4).
Next, assume that condition (3) does not hold. In this case, the ﬁrm has a possibility of investingU
in the unsustainable project because of f (x∗ ) < 0. On the other hand, aU
2 x − I2 > a1 x − I1 holds

for x → ∞. Then, I can conjecture the two stopping regions, namely the unsustainable investment
U
U
U
U
region [xU
11 , x12 ] and sustainable investment region [x21 , ∞), where x12 < x21 . This conjecture

is veriﬁed in Décamps, Mariotti, and Villeneuve (2006). By the value matching conditions at
UUthresholds xU
11 , x12 , and x21 , the value function is expressed as (6), and the state prices are deﬁned

by
γ2
β γ2
xβ (xU− (xU
21 )21 ) x,U
UU
γ2(x12 )β (x21 )γ2 − (x21 )β (xU12 )γ2(xU )β xγ2 − xβ (xU
12 )ΣU (x) = U β12 U γ,
β U γ2(x12 ) (x21 ) 2 − (xU
21 ) (x12 )∆U (x) =where γ2 represents the negative characteristic root γ2 = 0.5−µ/σ 2 −

√

(29)
(30)
2

(µ/σ 2 − 0.5) + 2r/σ 2 . Notethat ∆(x) and Σ(x) stand for the present values of $1 contingent on the unsustainable investment
and sustainable investment, respectively. Equation (7) follows from the smooth pasting condition
U
U
at xU
11 . Thresholds x12 and x21 are determined by the smooth pasting conditions:
U
U ′ U
U U
U ′ U
U
(aU
1 x12 − I1 )(∆ ) (x12 ) + (a2 x21 − I2 )(Σ ) (x12 ) = a1 ,

(31)

U
U ′ U
U U
U ′ U
U
(aU
1 x12 − I1 )(∆ ) (x21 ) + (a2 x21 − I2 )(Σ ) (x21 ) = a2 .

(32)

21

B

Proof of Proposition 2

Clearly, higher I1 , λ, and lower I2 expand region (3). Higher σ increases β/(β − 1) because of∂β/∂σ < 0.10 Then, higher σ expands the region (3). Higher µ and lower r increase (r−µ+λ)/(r−µ)
as far as r > µ is satisﬁed. Higher µ and lower r also increases β/(β − 1) because of ∂β/∂µ < 0and ∂β/∂r > 0. Then, higher µ and lower r expands the region (3).
Note that investment thresholds (5) and (7) are β/(β − 1) times the zero-NPV thresholds, andhigher β/(β − 1) means the higher value of waiting. Thus, I can state that σ aﬀects sustainability
condition (3) through the option eﬀect, whereas µ and r aﬀects (3) through the NPV and option
eﬀects.

C

Proof of Proposition 3

ULL
I can trace the proof in Appendix A by replacing aU
1 and a2 with a1 and a2 respectively, and henceLthe proof is omitted. Sustainability condition (24) follows from aL
2 /a1 = (r − µ + λ)ϕ2 /((r − µ)ϕ1 ).

The state prices ∆L (x) and ΣL (x) are deﬁned by∆L (x) =

β γ2
γ2
− (xL
xβ (xL21 ) x21 ),β L γ2 − (xL )β (xL )γ2(xL12 ) (x21 )
21
12

(33)

ΣL (x) =

β γ2
γ2
(xL
− xβ (xL
12 ) x12 ).
β L γ2 − (xL )β (xL )γ2(xL12 ) (x21 )
21
12

(34)

L
Thresholds xL
12 and x21 are determined by the smooth pasting conditions:

D

L
L ′ L
L L
L ′ L
L
(aL
1 x12 − I1 )(∆ ) (x12 ) + (a2 x21 − I2 )(Σ ) (x12 ) = a1 ,

(35)

L
L ′ L
L L
L ′ L
L
(aL
1 x12 − I1 )(∆ ) (x21 ) + (a2 x21 − I2 )(Σ ) (x21 ) = a2 .

(36)

Proof of Proposition 4

Clearly, ∂ϕ1 /∂h1 < 0 holds in (20), and ∂γ1 /λ < 0 holds in (12). Then, it is suﬃcient to show that∂h1 /∂γ1 > 0. Simplify the notation by deﬁning the function h1 (γ1 ) = (1 − Aγ1 )1/γ1 for γ1 < 0,where A is a positive constant. By calculating the derivative, I have()
Aγ1
h1 (γ1 )′ln(1 − Aγ1 ) +.
h1 (γ1 ) =γ12
1 − Aγ1
Deﬁneg(γ1 ) = ln(1 − Aγ1 ) +Aγ1
1 − Aγ1and calculate the derivativeg ′ (γ1 ) =A2 γ 1< 0 (γ1 < 0).
(1 − Aγ1 )2Then, g(γ1 ) > g(0) = 0 holds for all γ1 < 0, which implies h′1 (γ1 ) > 0 in (37).
10

The sensitivities of β to σ, µ, and r are well known (e.g., see Dixit and Pindyck (1994)).

22

(37)

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3

10
xL

8

x

2.5

xD

U

2

6
1.5
4

1

2

0.5

0

0
0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

4

6

V L (x)C

5

V U(x)

3
4
2

3
2

1
1
0

0
0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

0.1
LV

0.56

CS

0.08

0.54

0.06

0.52

0.04

0.5

0.02
0
0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

Figure 1: Comparative statics with respect to sustainability parameter λ.

26

0.2

3

xL

10

x

xD

U

2.5
8
2
6
1.5
4
0.02

0.04

0.06

1
0.02

0.08

0.04

r

0.06

0.08

r
20
V L (x)C

6

V U(x)

15
5
10

4

5

3
2
0.02

0.04

0.06

0
0.02

0.08

0.04

r

0.06

0.08

0.06

0.08

r
0.12

0.6

CS

LV

0.1
0.55

0.08
0.06

0.5

0.04
0.45
0.02
0.4
0.02

0.04

0.06

0
0.02

0.08

r

0.04

r

Figure 2: Comparative statics with respect to discount rate r.

27

3

10

xD

9

2.5

8
xL

7

2

xU

6
5

1.5

4
1
0

0.01

0.02

0.03

0.04

0

0.01

0.02

0.03

0.04

30
V L (x)

C

20

25

15

V U(x)

20
15

10

10
5

5

0

0
0

0.01

0.02

0.03

0.04

0

0.6

0.01

0.02

0.03

0.04

0.12
LV

CS

0.1

0.55

0.08
0.06

0.5

0.04
0.02

0.45

0
0

0.01

0.02

0.03

0.04

0

0.01

0.02

Figure 3: Comparative statics with respect to growth rate µ.

28

0.03

0.04

3
20

x

L

x

U

xD

2.5

15
2
10
1.5

5
0
0.1

0.2

0.3

0.4

1
0.1

0.5

15

0.2

0.3

0.4

0.5

0.3

0.4

0.5

4
V L (x)C

3

V U(x)

10
2
5
1
0
0.1

0.2

0.3

0.4

0
0.1

0.5

0.2

0.8
0.1

LV

0.7

CS

0.08
0.6
0.06

0.5

0.04

0.4

0.02

0.3
0.2
0.1

0.2

0.3

0.4

0
0.1

0.5

0.2

0.3

Figure 4: Comparative statics with respect to volatility σ.

29

0.4

0.5

30

6
xL

25

xD

xU

5

20
4
15
3
10
2

5
0

1
0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.45

14
C

V L(x)

0.4

12

V U (x)

0.35

10

0.3
8
0.25
6

0.2

4

0.15

2

0.1
0.1

0.2

0.3

0.4

0.5

0.1

0.9

0.2

0.3

0.4

0.5

0.3

0.4

0.5

0.14
LV

0.8

CS

0.12

0.7

0.1

0.6

0.08

0.5

0.06

0.4

0.04

0.3

0.02

0.2

0
0.1

0.2

0.3

0.4

0.5

0.1

0.2

Figure 5: Comparative statics with respect to corporate tax rate τ . I2 = 140.

30