日本オペレーションズ・リサーチ学会
2021年 秋季研究発表会

2-E-7

Corporate investment, ﬁnancing, and exit decisions withan earnings-based borrowing constraint
01406054

大阪大学

*西原 理

NISHIHARA Michi01208084

東京都立大学芝田

SHIBATA Takashi

隆志

William Paterson University

1.

Introduction

ZHANG Chuanqian
based on EBIT. This type of EBC is the most preva-

Economics literature has traditionally investigatedlent among various types.

the eﬀects of liquidation value-based borrowing con-We assume partial investment reversibility, where astraint (LBC) on corporate investment and ﬁnancing

fraction k ∈ (0, 1) of capital expenditure I remains

decisions. Under LBC, a ﬁrm faces a debt issuance ca-

as liquidation value. The ﬁrm choose an exit type

pacity based on the liquidation value of the ﬁrm’s spe-among sellout, liquidation bankruptcy, and reorgani-ciﬁc assets. However, recent empirical studies, suchzation bankruptcy when EBIT X(t) deteriorates. On

as [1], have shown the prevalence of earnings-based
borrowing constraint (EBC) over LBC. Under EBC,

sellout, debt holders are repaid the principal of debt,

debt capacity is based on operating earnings ratherliquidation bankruptcy, debt holders receive liquida-than asset values. This paper shows how diﬀerently

tion value, while shareholders receive nothing. On

from LBC, EBC aﬀects corporate investment, ﬁnanc-reorganization bankruptcy, debt holder receive going-ing, and exit decisions.

concern value, while shareholders receive nothing.

2.

3.

Model Setup

and shareholders receive positive residual value. On

Model Solutions

The model builds on the standard setup of invest-

The model is solved backward. First, we derive thement with optimal capital structure based on tradeoﬀ

going-concern value of the ﬁrm for debt holders. Attheory. Consider a ﬁrm that has an option to invest inbankruptcy threshold xd (C), the going-concern value

a new project by incurring capital expenditure I(> 0).

is derived as

At the investment time, the ﬁrm can issue consol debt
with coupon C. The project generates continuous
streams of earnings before interest and taxes (EBIT)
X(t), which follows( dx (C)G(x (C)) = (1 − τ )(1 − α)r−µ( d)γ ())
lx (C)x+kI −(xd (C) ≥ xl ),xlr−µdX(0) = x,where xl = γ(r − µ)kI/(γ − 1) is the liquidation

where B(t) denotes the standard Brownian motion,

threshold (optimized by former debt holders), anddX(t) = µX(t)dt + σX(t)dB(t) (t > 0),and µ, σ(> 0) and x(> 0) are constants. For con-α ∈ (0, 1) denotes
bankruptcy cost. Notation γ =√2(µ/σ 2 − 0.5) + 2r/σ 2 is the negativevergence, we assume that r > µ, where a positive

0.5 − µ/σ 2 −constant r denotes the risk-free interest rate.

Forcharacteristic root. Debt holders choose reorganiza-X(t) − C, the corporate tax rate τ ∈ (0, 1) is applied.

tion bankruptcy if G(xd (C)) is higher than liquidation

The ﬁrm optimizes both investment time T i andbankruptcy value (1 − α)kI. Otherwise, they choose

coupon C to maximize the investment option value.

liquidation bankruptcy. With lower k and higher Cand xd (C), the ﬁrm is more likely to go into reorgani-i

At investment time T , EBC
Dd (X(T i ), C) ≤ ϕE X(T i )d(1)ization bankruptcy, which is consistent with empiricalevidence.

is enforced, where D (X(T ), C) and ϕE denote theNext, we consider shareholders’ exit choice between

risky debt value at time T i and the tightness param-sellout and default. Shareholders choose sellout foreter of EBC. EBC (1) means that the cap of debt ispositive residual value, i.e., C ≤ C s = r(1 − τ )kI.

In this region, at investment threshold xi , the equity
value is derived as((iidiﬀerences between EBC and LBC. In the LBC model,)γxCx− +E s (xi , C) = (1 − τ )r−µrxs (C)())τCxs (C)kI −−(xi ≥ xs (C)),(1 − τ )rr−µ
where x (C) = γ(r − µ)(kI − τ C/(1 − τ )r)/(γ − 1)
sis the sellout threshold (optimized by shareholders).
dWe also examine a model with LBC to highlight theiwe constrain the risky ﬁrm by Dd (xi , C) ≤ ϕL (1 −α)kI, where ϕL denotes the tightness parameter ofLBC. This LBC model is essentially the same as in[2].
We numerically show the following results.

Al-though very tight EBC (i.e., very low ϕE ) induces theThe debt value is the riskless value D (x , C) = C/r.

ﬁrm to resort to riskless debt, plausible levels of EBC

Shareholders choose default for negative residual

lead the ﬁrm to use risky debt ﬁnancing with realis-svalue, i.e., C > C . In this region, the equity value istic levels of leverage. The ﬁrm can increase the cap

derived asof debt under EBC by delaying investment, although((iixCxE d (xi , C) = (1 − τ )− +dr−µrx (C)())
Cxd (C)−(xi ≥ xd (C)),rr−µ)γinvestment timing is not related to the cap of debt
under LBC. Then, unlike with LBC, the ﬁrm with
EBC delays investment to utilize more debt ﬁnancing. Investment reversibility k does not largely aﬀect

where x (C) = γ(r − µ)C/(γ − 1)r is the default

the ﬁrm with EBC, although it greatly aﬀects the ﬁrm

threshold (optimized by shareholders).

with LBC. The diﬀerence implies that the ﬁrm withdvalue is(The debt

low investment reversibility, which leads to reorgani-)γ

xiC+
Dd (xi , C) =r
xd (C))(Cd.
max{(1 − α)kI, G(x (C))} −r

zation bankruptcy rather than liquidation bankruptcy

Finally, we consider the ﬁrm’s investment and ﬁ-uidation value. Higher volatility σ increases the cap

nancing decisions. For riskless debt ﬁnancing, i.e.,C ∈ [0, C s ], no ﬁnancial constraints is imposed. Theof debt under EBC by delaying investment, althoughﬁrm optimally chooses C s that maximizes the tax ben-in case of ﬁnancial distress, prefers EBC to LBC. This
also leads to a positive relation between prevalence ofEBC and reorganization bankruptcy through low liq-

the cap of debt under LBC does not depend on investment timing. Then, contrary to the results in uncon-eﬁts of debt. At time 0, the riskless ﬁrm value becomes( x )β ( (1 − τ )x τ C sV s (x) = sup+− I+
ir−µr
xi ≥x x()γ ())
xi(1 − τ )xs (C s ) τ C s−(1 − τ )kI −,xs (C s )r−µr

strained and LBC models, under EBC, the ﬁrm with

where xi is the investment threshold (optimized bying constraints. Quarterly Journal of Economics,

ex-ante shareholders). Notation β = 0.5 − µ/σ 2 +√2(µ/σ 2 − 0.5) + 2r/σ 2 is the positive characteristic

136:229–291, 2021.

root.

higher volatility increases leverage and choose reorganization bankruptcy in case of ﬁnancial distress.

References[1] C. Lian and Y. Ma. Anatomy of corporate borrow-[2] T. Shibata and M. Nishihara. Investment timing,reversibility, and ﬁnancing constraints. Journal ofs

For risky debt ﬁnancing, i.e., C > C , EBC (1) isimposed. The risky ﬁrm value becomes( x )βV d (x) =sup(E d (xi , C) + Dd (xi , C) − I)ixi ≥x,C>C s x
subject to Dd (xi , C) ≤ ϕE xi .
The ﬁrm optimally chooses between riskless and risky
debt ﬁnancing, and hence, the initial ﬁrm value isV (x) = max{V s (x), V d (x)}.

Corporate Finance, 48:771–796, 2018.