Online Appendix to Financing Asset Sales and
Business Cycles
Marc Arnold∗
Dirk Hackbarth†
Tatjana Xenia Puhan‡
March 11, 2016
∗
University of St. Gallen, Rosenbergstrasse 52, 9000 St. Gallen, Switzerland. Telephone: +41–71–224–7076.
Email: marc.arnold@unisg.ch.
†
Boston University Questrom School of Business, 595 Commonwealth Avenue, Boston, MA 02215, United States.
Telephone: +1–617–358–4206. Email: dhackbar@bu.edu.
‡
Swiss Life Asset Managers and University of Mannheim, General-Guisan-Quai 40, 8002 Zurich, Switzerland.
Telephone: +41–76–602–7715. Email: tatjana.puhan@swisslife.ch.
Appendix A. Derivations
Appendix A.1. The stochastic discount factor, risk-free rates, and market prices
of risk
We assume that the representative agent has the continuous-time analog of Epstein-Zin-Weil preferences of stochastic differential utility type (e.g., Duffie and Epstein 1992a, Duffie and Epstein
1992b). The utility index Ut over a consumption process Cs solves
"Z
P
Ut = E
t
∞
#
1−δ
ρ Cs1−δ − ((1 − γ) Us ) 1−γ
ds |Ft ,
1−δ
1 − δ ((1 − γ) U ) 1−γ
−
1
s
(A.1)
in which ρ is the rate of time preference, γ the coefficient of relative risk aversion for a timeless
gamble, and Ψ :=
1
δ
the elasticity of intertemporal substitution for deterministic consumption paths.
Incorporating the separability of time and state preferences and assuming that Ψ > 1, i.e., that
the representative agent has a preference for early resolution of uncertainty and require expected
returns that increase in the uncertainty about future consumption, are necessary to capture the
impact of aggregate risk on corporate security values.
According to Bhamra, Kuehn, and Strebulaev (2010) and Chen (2010), solving the Bellman
equation associated with the consumption problem of the representative agent implies that the
stochastic discount factor mt follows the dynamics
dmt
= −ri dt − ηi dWtC + (eκi − 1) dMt ,
mt
(A.2)
in which Mt is the compensated process associated with the Markov chain, ri are the regimedependent risk-free interest rates, and ηi denote the risk prices for systematic Brownian shocks
affecting aggregate output. The market prices of consumption risk ηi increase with the agents’
risk aversion and consumption volatility. The parameters κi denote the relative jump sizes of the
discount factor when the Markov chain leaves state i, i.e., they are the market prices of the discount
factor jump risk.
1
Risk-free rates, and the market prices of consumption and jump risk are defined as
ri = r̄i + λi
γ − δ − γ−1
−1
γ−δ
w
−1 − w −1 ,
γ−1
ηi = γσiC ,
(A.3)
(A.4)
κi = (δ − γ) log
hj
hi
,
(A.5)
with i, j = G, B, i 6= j. The parameters hG , hB solve the following non-linear system of equations
(e.g. Bhamra, Kuehn, and Strebulaev 2010):
0=ρ
1 − γ 2
1 − γ 1−γ
hiδ−γ + (1 − γ) θi − 12 γ (1 − γ) σiC − ρ
hi + λi hj1−γ − h1−γ
i
1−δ
1−δ
(A.6)
The risk-free rates ri contain the interest rate if the economy stayed in state i forever, r̄i , plus
a second term that incorporates a possible switch in the state. The no-jump part of the interest
rates, r̄i , are
2
1
r̄i = ρ + δθi − γ (1 + δ) σiC ,
2
(A.7)
w := eκB = e−κG
(A.8)
and
measures the size of the jump in the real-state price density when the economy shifts from bad
states to good states (see for example Proposition 1 in Bhamra, Kuehn, and Strebulaev 2010).
Appendix A.2. Derivation of the values of corporate securities after investment
The valuation of corporate debt. Our valuation of corporate debt of a firm that consists of only
invested assets in a two state setting follows (Hackbarth, Miao, and Morellec 2006), and (Arnold,
Wagner, and Westermann 2013). We illustrate the case in which the default boundary in good
states is lower than the one in bad states, i.e., D̂G < D̂B . If a firm defaults, debtholders receive
a fraction Λi αi of the unleveraged after tax asset value (1 − τ )Xyi . A debt investor requires an
instantaneous return equal to the risk-free rate ri . The instantaneous debt return corresponds to
the realized rate of return plus the coupon proceeds from debt. Hence, an application of Ito’s
lemma with possible state switches shows that debt satisfies the following system of ODEs.
2
For 0 ≤ X ≤ D̂G :


 dˆG (X) = αG ΛG (1 − τ )XyG
(A.9)

 dˆB (X) = αB ΛB (1 − τ )XyB .
For D̂G < X ≤ D̂B :


 rG dˆG (X) = c + µ̃G X dˆ0 (X) + 1 σ̃ 2 X 2 dˆ00 (X) + λ̃G αB ΛB (1 − τ )XyB − dˆG (X)
G
G
2 G


(A.10)
dˆB (X) = αB ΛB (1 − τ )XyB .
For X > D̂B :


 rG dˆG (X) = c + µ̃G X dˆ0 (X) + 1 σ̃ 2 X 2 dˆ00 (X) + λ̃G dˆB (X) − dˆG (X)
G
G
2 G

1
0
2
2
 rB dˆB (X) = c + µ̃B X dˆ (X) + σ̃ X dˆ00 (X) + λ̃B dˆG (X) − dˆB (X) .
B
B
2 B
(A.11)
The boundary conditions read
dˆi (X)
< ∞,
X→∞
X
lim
i = G, B,
lim dˆG (X) = lim dˆG (X),
X&D̂B
(A.12)
(A.13)
X%D̂B
lim dˆ0G (X) = lim dˆ0G (X),
(A.14)
lim dˆG (X) = αG ΛG (1 − τ )DG yG ,
(A.15)
X&D̂B
X%D̂B
X&D̂G
and
lim dˆB (X) = αB ΛB (1 − τ )DB yB .
(A.16)
X&D̂G
Condition (A.12) captures the no-bubbles condition. The other boundary conditions are the valuematching conditions (A.13), (A.15), and (A.16), and the smooth-pasting condition at the higher
default threshold D̂B for the debt function in the good state dˆG (·), Eq. (A.14). The functional form
3
of the solution is



αi Λi (1 − τ )Xyi
X ≤ D̂i
i = G, B



G
G
dˆi (X) =
Ĉ1 X β1 + Ĉ2 X β2 + C3 X + C4 D̂G < X ≤ D̂B , i = G




 Âi1 X γ1 + Âi2 X γ2 + Ai5
X > D̂B ,
i = G, B,
(A.17)
in which ÂG1 , ÂG2 , ÂB1 , ÂB2 , AG5 , AB5 , Ĉ1 , Ĉ2 , C3 , C4 , γ1 , γ2 , β1G , and β2G are real-valued parameters
to be determined.
First, we consider the region X > D̂B . We start with the standard approach by plugging
the functional form dˆi (X) = Âi1 X γ1 + Âi2 X γ2 + Ai5 into both equations of (A.11). Comparing
coefficients and solving the resulting two-dimensional system of equations for Ai5 , we obtain
Ai5 =
c (rj + λ̃i + λ̃j )
ri rj + rj λ̃i + ri λ̃j
=
c
,
rip
and find that ÂGk is a multiple of ÂBk , k = 1, 2, with the factor lk :=
1 2
2 σ̃G γk (γk
(A.18)
1
(r
λ̃G G
+ λ̃G − µ̃G γk −
− 1)), i.e., ÂBk = lk ÂGk . Using these results when comparing coefficients again, it can
be shown that γ1 and γ2 are the negative roots of the quadratic equation
2
2
(µ̃B γ + 12 σ̃B
γ(γ − 1) − λ̃B − rB )(µ̃G γ + 21 σ̃G
γ(γ − 1) − λ̃G − rG ) = λ̃B λ̃G .
(A.19)
To satisfy the no-bubbles condition for debt in Eq. (A.12), we take the negative roots.
G
Next, we consider the region D̂G ≤ X ≤ D̂B . Plugging the functional form dG (X) = Ĉ1 X β1 +
G
Ĉ2 X β2 + C3 X + C4 into the first equation of (A.10), we find by comparison of coefficients that
G
β1,2
=
C3 =
C4 =
1 µ̃G
− 2 ±
2 σ̃G
s
1 µ̃G
− 2
2 σ̃G
λ̃G αB ΛB (1 − τ )yB
rG + λ̃G − µ̃G
c
.
rG + λ̃G
2
+
2(rG + λ̃G )
2
σ̃G
(A.20)
We then plug the functional form (A.17) into conditions (A.13)–(A.16), and obtain a four-dimensional
4
linear system in the remaining four unknown parameters ÂG1 , ÂG2 , Ĉ1 , and Ĉ2 :
βG
βG
γ1
γ2
ÂG1 D̂B
+ ÂG2 D̂B
+ AG5 = Ĉ1 D̂G1 + Ĉ2 D̂B2 + C3 D̂B + C4
γ2
γ1
+ ÂG2 γ2 D̂B
ÂG1 γ1 D̂B
βG
βG
= Ĉ1 β1G D̂B1 + Ĉ2 β2G D̂B2 + C3 D̂B
βG
βG
(A.21)
αG ΛG (1 − τ )D̂G yG = Ĉ1 D̂B1 + Ĉ2 D̂B2 + C3 D̂B + C4
γ2
γ1
+ AB5 = αB ΛB (1 − τ )D̂B yB .
+ l2 ÂG2 D̂B
l1 ÂG1 D̂B
We define the matrices

γ1
D̂B


γ1 D̂γ1

B
M̂ := 

 0

γ1
l1 D̂B
βG
βG
γ2
D̂B
−D̂B1
γ2
γ2 D̂B
βG
−β1G D̂B1
0
D̂B1
D̂B2
γ2
l2 D̂B
0
0
−D̂B2

βG 
−β2G D̂B2 

βG
βG





(A.22)
and

C3 D̂B + C4 − AG5







C3 D̂B


b̂ := 
,


αG ΛG (1 − τ )D̂G yG − C3 D̂B − C4 


αB ΛB (1 − τ )D̂B yB − AB5
(A.23)
T
such that M̂ ÂG1 ÂG2 Ĉ1 Ĉ2 = b̂. The solution for the unknown parameters is
T
ÂG1 ÂG2 Ĉ1 Ĉ2
= M̂ −1 b̂.
(A.24)
The value of the tax shield is calculated by using the formula for the value of debt, in which
c is replaced by τ c, and α is equal to zero. Similarly, we obtain the value of bankruptcy costs by
simply replacing c with zero, and α with 1 − α.
Default policy. The value of equity equals the firm value minus the value of debt. The firm value
is given by the value of assets in place plus the value of the growth option and the tax shield minus
default costs. After debt has been issued, firms choose the ex post default policy that maximizes
the value of equity. Formally, the default policy is determined by equating the first derivative of
5
the equity value to zero at the corresponding default boundary level:


 ê0 (D̂∗ ) = 0
G
G
(A.25)

 ê0 (D̂∗ ) = 0.
B
B
We solve this problem numerically.
For a firm that receives scaled earnings after investment, the value of corporate securities is
solved similarly by replacing X with the scaled level of earnings. For example, if the firm exercises
the option in the good state, and finances the exercise cost by issuing equity, the scaled earnings
∗ and D̂ ∗ are then expressed in terms of the
correspond to (sG + 1)X. The default boundaries D̂G
B
scaled earnings levels.
Appendix A.3. Derivation of the value of the growth option
The case with XG < XB :
We present the derivation of the value of the growth option for a firm that finances the option
exercise by issuing equity in good states and selling assets in bad states. The value of the growth
option for a firm with an alternative financing strategy can be derived similarly. For each state
i, the option is exercised immediately whenever X ≥ Xi (option exercise region); otherwise, it
is optimal to wait (option continuation region). This structure results in the following system of
ODEs for the option’s value function.
For 0 ≤ X < XG :


 rG GG (X) = µ̃G XG0 (X) + 1 σ̃ 2 X 2 G00 (X) + λ̃G (GB (X) − GG (X))
G
G
2 G
(A.26)

 rB GB (X) = µ̃B XG0 (X) + 1 σ̃ 2 X 2 G00 (X) + λ̃B (GG (X) − GB (X)) .
B
B
2 B
For XG ≤ X < XB :



GG (X) = (1 − τ )sG XyG − KG (1 + ΥG )

 rB GB (X) = µ̃B XG0 (X) + 1 σ̃ 2 X 2 G00 (X) + λ̃B ((1 − τ )sG XyG − KB (1 + ΥB ) − GB (X)) .
B
B
2 B
(A.27)
6
For X ≥ XB :


 GG (X) = (1 − τ )sG XyG − KG (1 + ΥG )
(A.28)

 GB (X) = (1 − τ )sB XyB − KB /ΛB .
When X is in the option continuation region, which corresponds to system (A.26) and the second
equation of (A.27), the required rate of return ri (left-hand side) must be equal to the realized rate
of return (right-hand side). We calculate the realized rate of return by using Ito’s lemma for state
switches. In this region, the last term captures the possible jump in the value of the growth option
due to a state switch. It can be expressed as the instantaneous probability of a shift in the state,
λ̃G or λ̃B , times the corresponding change in the value of the option. The first equation of (A.27)
and the system (A.28) describe the payoff of the option at exercise. The process is in the option
exercise region in these cases. The boundary conditions are
lim Gi (X) = 0,
X&0
i = G, B,
(A.29)
lim GB (X) = lim GB (X),
(A.30)
lim G0B (X) = lim G0B (X),
(A.31)
lim GB (X) = (1 − τ )sB XB yB − KB /ΛB ,
(A.32)
lim GG (X) = (1 − τ )sG XG yG − KG (1 + ΥG ).
(A.33)
X&XG
X&XG
X%XG
X%XG
X%XB
and
X%XG
Condition (A.29) ensures that the option value goes to zero as earnings approach zero. Conditions
(A.30) and (A.31) are the value-matching and smooth-pasting conditions of the value function in
bad states at the exercise boundary in good states. The other conditions (A.32)–(A.33) are the
value-matching conditions at the exercise boundaries in a good state and a bad state, respectively.
7
The functional form of the solution is
Gi (X) =



Āi3 X γ3 + Āi4 X γ4
0 ≤ X < XG ,
i = G, B





 C̄1 X β1B + C̄2 X β2B + C̄3 X + C̄4 X ≤ X < XB , i = B
G


(1 − τ )sB XyB − KB /ΛB





 (1 − τ )s Xy − K (1 + Υ )
G
G
G
G
X ≥ XB
i=B
X ≥ XG
i = G,
(A.34)
in which ĀG3 , ĀG4 , ĀB1 , ĀB2 , C̄1 , C̄2 , C̄3 , C̄4 , γ3 , γ4 , β1B , and β2B are real-valued parameters that need
to be determined.
First, we consider the region 0 ≤ X < XG , and plug the functional form Gi (X) = Āi3 X γ3 +
Āi4 X γ4 into both equations of (A.26). A comparison of coefficients implies that ĀGk is a multiple of
ĀBk , k = 3, 4, with the multiple factor ¯lk :=
1
2 γ (γ −1)),
(r + λ̃G − µ̃G γk − 12 σ̃G
k k
λ̃G G
i.e., ĀBk = ¯lk ĀGk .
Using this result when comparing coefficients, we find that γ3 and γ4 are the positive roots of the
quadratic equation
1 2
1 2
(µ̃B γ + σ̃B
γ(γ − 1) − λ̃B − rB )(µ̃G γ + σ̃G
γ(γ − 1) − λ̃G − rG ) = λ̃B λ̃G .
2
2
(A.35)
Boundary condition (A.29) implies to take the positive roots.
Next, we consider the region XG ≤ X < XB . Plugging the functional form GB (X) = C̄1 X β1 +
C̄2 X β2 + C̄3 X + C̄4 into the second equation of (A.27), we find by comparison of coefficients that
B
β1,2
=
1 µ̃B
− 2 ±
2 σ̃B
s
1 µ̃B
− 2
2 σ̃B
(1 − τ )sG yG
,
rB − µ̃B + λ̃B
KB /ΛB
= −λ̃B
.
rB + λ̃B
C̄3 = λ̃B
C̄4
2
+
2(rB + λ̃B )
,
2
σ̃B
(A.36)
The remaining unknown parameters are ĀG3 , ĀG4 , C̄1 , and C̄2 . Plugging the functional form (A.34)
8
into conditions (A.30)–(A.33) yields
βB
βB
γ4
γ3
C̄1 XG1 + C̄2 XG2 + C̄3 XG + C̄4 = ¯l3 ĀG3 XG
,
+ ¯l4 ĀG4 XG
βB
βB
γ3
γ4
C̄1 β1B XG1 + C̄2 β2B XG2 + C̄3 XG = ¯l3 ĀG3 γ3 XG
+ ¯l4 γ4 ĀG4 XG
,
βB
βB
C̄1 XB1 + C̄2 XB2 + C̄3 XB + C̄4 = (1 − τ )sB yB XB − KB /ΛB ,
(A.37)
(A.38)
(A.39)
and
γ3
γ4
ĀG3 XG
+ ĀG4 XG
= (1 − τ )sG yG XG − KG (1 + ΥG ).
(A.40)
This four-dimensional system is linear in its four unknowns ĀG3 , ĀG4 , C̄1 and C̄2 . We define the
matrices

¯l3 X γ3
G
βB
−XG1
¯l4 X γ4
G


¯l3 γ3 X γ3 ¯l4 γ4 X γ4

G
G
M̄ := 

0
 0

γ3
γ4
XG
XG
βB
−β1B XG1
βB
XB1
βB
−XG2


βB 
−β2B XG2 

βB
XB2
0
0
,



(A.41)
and

C̄3 XG + C̄4







C̄
X
3
G


b̄ := 
,


−C̄3 XB − C̄4 + (1 − τ )sB yB XB − KB /ΛB 


(1 − τ )sG yG XG − KG (1 + ΥG )
(A.42)
T
such that M̄ ĀG3 ĀG4 C̄1 C̄2
= b̄. The solution to the remaining four unknowns is
T
ĀG3 ĀG4 C̄1 C̄2
= M̄ −1 b̄.
(A.43)
The unleveraged value of the growth option. The unleveraged value of the growth option can be
9
calculated by additionally imposing the smooth-pasting boundary conditions at option exercise:
0
lim
Gunlev
(X) = (1 − τ )sB yB
B
lim
Gunlev
(X) = (1 − τ )sG yG .
G
unlev
X%XB
(A.44)
and
unlev
X%XG
0
(A.45)
The solution method is analog to the one for the leveraged growth option value up to and including
Eq. (A.36). The system of equations (A.37)–(A.40) needs to be augmented by the two equations
corresponding to the additional smooth-pasting boundary conditions:
β1B −1
β2B −1
unlev
unlev
C̄1unlev β1B XB
+ C̄2unlev β2B XB
+ C̄3 = (1 − τ )sB yB
(A.46)
and
γ −1
γ −1
unlev 3
unlev
unlev 4
Āunlev
γ
X
+
Ā
γ
X
= (1 − τ )sG yG .
3
4
G3
G
G4
G
(A.47)
unlev
unlev , C̄ unlev X unlev , and
The full system is six-dimensional with the six unknowns Āunlev
2
G3 , ĀG4 , C̄1
G
unlev , linear in the first four unknowns and nonlinear in the last two unknowns. We solve it
XB
numerically.
The case with XG ≥ XB :
The solution of the case XG ≥ XB can be obtained immediately by renaming states in the
solution of the presented case for XG < XB .
Appendix A.4. Firms with invested assets and an expansion option
We first present a proof for the value of corporate debt in the case in which DG < DB , D̂G < D̂B ,
and XB > XG . The argument of the proof is adapted from (Arnold, Wagner, and Westermann
2013).
10
Proof of Proposition 2. A debt investor requires an instantaneous return equal to the riskfree rate ri . The application of Ito’s lemma with state switches shows that debt must satisfy the
following system of ODEs.
For 0 ≤ X ≤ DG :


 dG (X) = αG ΛG (1 − τ )XyG + Gunlev (X)
G


dB (X) = αB ΛB (1 − τ )XyB + Gunlev
(X) .
B
(A.48)


2 X 2 d00 (X)

rG dG (X) = c + µ̃G Xd0G (X) + 21 σ̃G

G


+λ̃G αB ΥB (1 − τ )XyB + Gunlev
(X) − dG (X)
B





dB (X) = αB ΛB (1 − τ )XyB + Gunlev
(X) .
B
(A.49)
For DG < X ≤ DB :
For DB < X < XG :


 rG dG (X) = c + µ̃G Xd0 (X) + 1 σ̃ 2 X 2 d00 (X) + λ̃G (dB (X) − dG (X))
G
G
2 G
(A.50)

 rB dB (X) = c + µ̃B Xd0 (X) + 1 σ̃ 2 X 2 d00 (X) + λ̃B (dG (X) − dB (X)) .
B
B
2 B
For XG ≤ X < XB :



dG (X) = dˆG ((sG + 1)X)

2 X 2 d00 (X) + λ̃
ˆ
 rB dB (X) = c + µ̃B Xd0B (X) + 1 σ̃B
d
((s
+
1)X)
−
d
(X)
.
B
G
G
B
B
2
For X ≥ XB :


 dG (X) = dˆG ((sG + 1)X)

 dB (X) = dˆB (sB + 1 − KB /ΛB )X .
(1−τ )X yB
(A.51)
(A.52)
t̄
In system (A.48), the firm is in the default region in both good and bad states. In this region,
debtholders receive αi Λi (1 − τ )Xyi + Gunlev
(X) at default. The firm is in the continuation
i
region in good states, and in the default region in bad states in system (A.49). For the continuation
region in good states, the left-hand side of the first equation is the instantaneous rate of return
required by investors for holding corporate debt. The right-hand side is the realized rate of return,
computed by Ito’s lemma as the expected change in the value of debt plus the coupon payment
11
c. The last term expresses the possible jump in the value of debt in case of a state switch that
triggers immediate default. Eqs. (A.50) describe the case in which the firm is in the continuation
region in both good and bad states. The next system, (A.51), treats the case in which the firm is
in the exercise region in good states and in the continuation region in bad states. After exercising
the option, the firm owns total assets in place with value (1 − τ )Xyi + (1 − τ )sī Xyi , reflecting the
notion that the exercise cost of the growth option can be financed by issuing equity in good states.
The value of debt must then be equal to the value of debt of a firm with only invested assets, i.e.,
dG (X) = dˆG ((sG + 1) X), which is the first equation in (A.51). We obtain the second equation in
this case by using the same approach as in (A.50). The last term captures the notion that a switch
from bad states to good states triggers immediate exercise of the expansion option with equity
financing. Finally, (A.52) describes the case in which the firm is in the exercise region in both good
and bad states. In good states, the earnings of the firm are scaled by sG + 1. In bad states, the
exercise cost KB is financed by selling
the firm are scaled by (sB + 1 −
KB /ΛB
(1−τ )Xt̄ yB
of the assets in place, such that the earnings of
KB /ΛB
(1−τ )Xt̄ yB ).
The system is subject to the following boundary conditions.
lim dG (X) = lim dG (X) ,
(A.53)
lim d0G (X) = lim d0G (X) ,
(A.54)
(D
)
,
lim dG (X) = αG ΛG (1 − τ )DG yG + Gunlev
G
G
(A.55)
lim dB (X) = αB ΛB (1 − τ )DB yB + Gunlev
(DB ) ,
B
(A.56)
lim dB (X) = lim dB (X) ,
(A.57)
lim d0B (X) = lim d0B (X) ,
(A.58)
lim dG (X) = dˆG ((sG + 1)XG ) ,
(A.59)
X&DB
X&DB
X%DB
X%DB
X&DG
X&DB
X&XG
X&XG
X%XG
X%XG
X%XG
and
ˆ
lim dB (XB ) = dB (sB + 1 −
X%XB
KB /ΛB
)XB .
(1 − τ )Xt̄ yB
(A.60)
Eqs. (A.53) and (A.54) are the value-matching and smooth-pasting conditions for the debt value
12
in the good state at the default boundary of the bad state. Eqs. (A.57) and (A.58) reflect the
corresponding conditions for the debt value in the bad state at the option exercise boundary of the
good state. Eqs. (A.55) and (A.56) show the value-matching conditions at the default thresholds,
and Eqs. (A.59) and (A.60) are the value-matching conditions at the option exercise boundaries.
The default thresholds and option exercise boundaries are chosen by equityholders. Hence, we do
not need the corresponding smooth-pasting conditions for debt.
To solve this system, we start with the functional form of the solution, in which
AG1 , AG2 , AB1 , AB2 , C1 , C2 , C3 , C4 , C5 , C6 , B1 , B2 , B4 , β1G , β2G , β1B , β2B , γ1 , γ2 , γ3 , and γ4 are real-valued
parameters to be determined.
We first consider the region DB < X ≤ XG . Plugging the functional form di (X) = Ai1 X γ1 +
Ai2 X γ2 + Ai3 X γ3 + Ai4 X γ4 + Ai5 into both equations of (A.50) and comparing coefficients, we
obtain
Ai5 =
c(rj + λ̃i + λ̃j )
ri rj + rj λ̃i + ri λ̃j
=
c
.
rip
(A.61)
As in Appendix A.2, AGk is a multiple of ABk , k = 1, . . . , 4, with the multiple factor lk :=
1
(r
λ̃G G
2 γ (γ − 1)), i.e., A
+ λ̃G − µ̃G γk − 12 σ̃G
k k
Bk = lk AGk . Using this relation and comparing
coefficients, it can be shown that γ1 , γ2 , γ3 , and γ4 correspond to the roots of the quadratic equation
2
2
γ(γ − 1) − λ̃B − rB )(µ̃G γ + 21 σ̃G
γ(γ − 1) − λ̃G − rG ) = λ̃B λ̃G .
(µ̃B γ + 21 σ̃B
(A.62)
According to Guo (2001), this quadratic equation always has two negative and two positive distinct
real roots. The value of debt in both states is subject to boundary conditions from below (default)
and above (exercise of expansion option). To satisfy these boundary conditions, we use four terms
with the corresponding factors Aik as well as the exponents γk , which requires the usage of all four
roots of Eq. (A.62). We do not incorporate the no-bubbles condition again because it is already
implemented in the value function dˆi of a firm with only invested assets. The unknown parameters
for this region are AGk , k = 1, . . . , 4.
G
Next, we examine the region DG ≤ X ≤ DB . Plugging the functional form dG (X) = C1 X β1 +
G
C2 X β2 + C3 X + C4 + C5 X γ3 + C6 X γ4 into the second equation of (A.49), we find by comparison
13
of coefficients that
G
β1,2
1 µ̃G
= − 2 ±
2 σ̃G
s
1 µ̃G
− 2
2 σ̃G
2
+
2(rG + λ̃G )
,
2
σ̃G
(A.63)
αB ΛB (1 − τ )yB
,
rG + λ̃G − µ̃G
c
C4 =
,
rG + λ̃G
¯l3
,
C5 = αB ΛB Āunlev
l3 G3
(A.64)
C3 = λ̃G
(A.65)
(A.66)
and
C6 = αB ΛB
¯l4
Āunlev .
l4 G4
(A.67)
The unknown parameters left in this region are C1 and C2 .
B
B
Finally, we consider the region XG < X ≤ XB . Plugging the functional form B1 X β1 +B2 X β2 +
Z (X) + λ̃B rP
i
c
(rB +λ̃B )
+
c
rB +λ̃B
into the second equation of (A.51) and comparing coefficients, we
find that
Z(X) = λ̃B B5 X γ1 + λ̃B B6 X γ2 .
(A.68)
(A.69)
The parameters B5 and B6 are given by
B5 =
(sB + 1)γ1 ÂG1
,
2 γ (γ − 1) + λ̃
rB − µ̃B γ1 − 12 σ̃B
1
1
B
(A.70)
B6 =
(sB + 1)γ2 ÂG2
.
2 γ (γ − 1) + λ̃
rB − µ̃B γ2 − 12 σ̃B
2
2
B
(A.71)
and
The unknown parameters in this region are B1 and B2 .
To obtain the unknown parameters AG1 , AG2 , AG3 , AG4 , C1 , C2 , B1 , and B2 , we plug the func-
14
tional form into the system of boundary conditions (A.53)–(A.60):
4
X
βG
βG
γk
AGk DB
+ AG5 = C1 DB1 + C2 DB2 + C3 X + C4 + C5 X γ3 + C6 X γ4
k=1
4
X
γk
AGk γk DB
βG
βG
= C1 β1G DB1 + C2 β2G DB2 + C3 X + C5 γ3 X γ3 + C6 γ4 X γ4
k=1
βG
βG
γ3
γ4
αG ΛG (1 + τ )DG yG + Gunlev
(D
)
= C1 DG1 + C2 DG2 + C3 DG + C4 + C5 DG
+ C6 D G
G
G
4
X
k=1
4
X
γk
lk AGk DB
+ AB5 = αB ΛB (1 + τ )DB yB + Gunlev
(DB )
B
βB
βB
γk
lk AGk XG
+ AB5 = B1 XG1 + B2 XG2 + Z(XG ) + B4
(A.72)
k=1
4
X
γk
lk AGk γk XG
βB
βB
= B1 β1B XG1 + B2 β2B XG2 + XG Z 0 (XG )
k=1
4
X
γk
+ AG5 = dˆG ((sG + 1)XG )
AGk XG
k=1
βB
B1 XB1
+
βB
B2 XB2
+ Z(XB ) + B4
ˆ
= dB (sB + 1 −
KB /ΛB
)XB .
(1 − τ )Xt̄ yB
Using matrix notation, we can write

γ1
DB


 γ Dγ1
 1 B


 0


 l1 Dγ1

B
M := 

γ
 l1 XG1


γ1
l1 γ1 XG


 X γ1

G

0
γ2
DB
γ2
γ2 DB
γ3
DB
γ3
γ3 DB
βG
−DB1
γ4
DB
β1G
βG
−DB2
β2G

0
γ4
γ4 DB
−β1G DB
−β2G DB
0
βG
DG2
0
0
0
0
0
βG
DG1
γ2
l2 DB
γ3
l3 DB
γ4
l4 DB
0
0
0
γ2
l2 XG
γ3
l3 XG
γ4
l4 XG
0
0
−XG1
γ2
l2 γ2 XG
γ3
l3 γ3 XG
γ4
l4 γ4 XG
0
0
−β1B XG1
γ2
XG
γ3
XG
γ4
XG
0
0
0
0
0
0
0
0
XB1
βB
βB
βB



0




0



0


β2B 
−XG 

βB 
−β2B XG2 



0


β2B
XB
(A.73)
15
and

γ1
C5 D B

γ2
C6 DB
−AG5 + C3 DB + C4 +
+




γ1
γ2


C3 DB + γ1 C5 DB + γ2 C6 DB






γ3
γ4
unlev
−C3 DG − C4 − C5 DG
− C6 DG + αG ΛG (1 − τ )DG yG + GG (DG ) 




unlev


−AB5 + αB ΛB (1 − τ )DB yB + GB (DB )


b := 
.


−AB5 + Z (XG ) + B4






0


XG Z (XG )






−AG5 + dˆG ((sG + 1)XG )




−Z (X ) + B4 + dˆB (sB + 1 − KB /ΛB )XB
B
(A.74)
(1−τ )Xt̄ yB
The solution to the remaining unknowns is
T
AG1 AG2 AG3 AG4 C1 C2 B1 B2
= M −1 b.
(A.75)
The case with DG < DB , D̂G < D̂B , and XG > XB :
Going through the same steps as in the previous case gives us

γ1
DB


γ Dγ1
 1 B


 0


 l1 Dγ1

B
M := 
 γ1
 XB


γ1
γ1 XB


 l1X γ1

B

0
γ2
DB
γ3
DB
γ4
DB
βG
−DB1
β1G
βG
−DB2
β2G

0
0
γ2
γ2 DB
γ3
γ3 DB
γ4
γ4 DB
−β1G DB
0
0
0
DG1
DG2
0
γ2
l2 DG
γ3
l3 DB
γ4
l4 DB
0
0
0
γ2
XB
γ3
XB
γ4
XB
0
0
−XB1
γ2
γ2 XB
γ3
γ3 XB
γ4
γ4 XB
0
0
−β1G XB1
γ2
l2XB
γ3
l3XB
γ4
l4XB
0
0
0
0
0
0
0
0
XG1
βG
16
−β2G DB
βG
0
βG
βG
βG



0




0



0


β2G 
−XB 

βG 
−β2G XB2 



0


G
β
XG2
(A.76)
and

γ1
C5 D B

γ2
C6 D B
−AG5 + C3 DB + C4 +
+




γ1
γ2


C3 DB + γ1 C5 DB + γ2 C6 DB






γ3
γ4
unlev
−C3 DG − C4 − C5 DG
− C6 DG + αG ΛG (1 − τ )DG yG + GG (DG ) 




unlev


−AB5 + αB ΛB (1 − τ )DB yB + GB (DB )


b := 
.


−AG5 + Z (XB ) + B4






0


XB Z (XB )




K
/Λ
B
B


−AB5 + dˆB (sB + 1 − (1−τ )X yB )XB


t̄


−Z (XG ) + B4 + dˆG ((sG + 1)XG )
(A.77)
The solution to the unknowns is again determined by
T
AG1 AG2 AG3 AG4 C1 C2 B1 B2
= M −1 b.
(A.78)
Appendix A.5. Bankruptcy costs
For the calculation of bankruptcy costs, the ODEs are given by the following system:
For 0 ≤ X ≤ DG :


 bG (X) = (1 − αG ΛG )(1 − τ )XyG + GG (X) − αG ΛG Gunlev (X)
G
(A.79)

 bB (X) = (1 − αB ΛB )(1 − τ )XyB + GB (X) − αB ΛB B unlev (X) .
B
For DG < X ≤ DB :


2 X 2 b00 (X)

rG bG (X) = µ̃G Xb0G (X) + 21 σ̃G

G


unlev (X) − b (X)
+
λ̃
(1
−
α
Λ
)(1
−
τ
)Xy
+
G
(X)
−
α
Λ
G
G
B
B
B
B
B
B
G
B





bB (X) = (1 − αB ΛB ) (1 − τ )XyB + GB (X) − αB ΛB Gunlev
(X) .
B
(A.80)
17
For DB < X < XG :


 rG dG (X) = c + µ̃G Xb0 (X) + 1 σ̃ 2 X 2 b00 (X) + λ̃G (bB (X) − bG (X))
G
G
2 G
(A.81)

 rB dB (X) = c + µ̃B Xd0 (X) + 1 σ̃ 2 X 2 b00 (X) + λ̃B (bG (X) − bB (X)) .
B
B
2 B
For XG ≤ X < XB :



bG (X) = dˆG ((sG + 1)X)

2 X 2 b00 (X) + λ̃
ˆG ((sG + 1)X) − bB (X) .
 rdB (X) = c + µ̃B Xb0B (X) + 1 σ̃B
d
B
B
2
For X ≥ XB :


 bG (X) = b̂G ((sG + 1)X)

 bB (X) = b̂B (sB + 1 − KB /ΛB )X .
(1−τ )X yB
(A.82)
(A.83)
t̄
The boundary conditions are as follows:
lim bG (X) = lim bG (X) ,
(A.84)
lim b0G (X) = lim b0G (X) ,
(A.85)
lim bG (X) = (1 − αG ΛB )(1 − τ )DG yG + GG (DG ) − αG ΛG Gunlev
(DG ) ,
G
(A.86)
lim bB (X) = (1 − αG ΛG )(1 − τ )DB yB + GB (DB ) − αB Gunlev
(DB ) ,
B
(A.87)
lim bB (X) = lim bB (X) ,
(A.88)
lim b0B (X) = lim b0B (X) ,
(A.89)
lim bG (X) = b̂G ((sG + 1)XG ) ,
(A.90)
X&DB
X&DB
X%DB
X%DB
X&DG
X&DB
X&XG
X&XG
X%XG
X%XG
X%XG
and
lim bB (XB ) = b̂B
X%XB
KB /ΛB
(sB + 1 −
)XB .
(1 − τ )Xt̄ yB
(A.91)
Eqs. (A.84) and (A.85) are the value-matching and smooth-pasting conditions for bankruptcy costs
18
in good states at the default boundary in bad states. Similarly, Eqs. (A.88) and (A.89) are the
corresponding conditions for bankruptcy costs in bad states at the option exercise boundary in good
states. Eqs. (A.86) and (A.87) are the value-matching conditions at the default thresholds. They
incorporate the fact that upon default, the value of the leveraged growth option switches to the
value of the unleveraged growth option. Eqs. (A.90) and (A.91) are the value-matching conditions
at the option exercise boundaries.
To solve for the unknown parameters, we plug the functional form



(1 − αi Λi )(1 − τ )Xyi − αi Λi Gunlev
(X) + Gi (X)

i




B
B


C1 X β1 + C2 X β2 + C5 X γ3 + C6 X γ4





ΛB yB (1−τ )


X + r +c λ̃
+λ̃G αB

rG −µ̃G +λ̃G

G
G

γ
γ
γ
1
2
3
bi (X) =
Ai1 X + Ai2 X + Ai3 X + Ai4 X γ4 + rcp

i



c
β1B + B X β2B + Z (X) + λ̃

+ r +c λ̃
B
X

1
2
B rP r +λ̃

(
B
B)
B
B

i



 b̂G ((sG + 1)X)






 b̂B (sB + 1 − KB /ΛB )X
(1−τ )X yB
t̄
X ≤ Di ,
i = G, B
DG < X ≤ DB , i = G
DB < X ≤ XG , i = G, B
XG < X ≤ XB , i = B
X > XG ,
i=G
X > XB ,
i=B
(A.92)
into the system of boundary conditions (A.84)-(A.91). The solution to the unknown parameters is
given by
T
AG1 AG2 AG3 AG4 C1 C2 B1 B2
in which
19
= M −1 b,
(A.93)

γ1
DB


 γ Dγ1
 1 B


 0


 l1 Dγ1

B
M := 

γ
 l1 XG1


γ1
l1 γ1 XG


 X γ1

G

0
γ2
DB
γ3
DB
γ4
DB
βG
−DB1
β1G

βG
−DB2
0
β2G
0
γ2
γ2 DB
γ3
γ3 DB
γ4
γ4 DB
−β1G DB
0
0
0
DG1
DG2
0
γ2
l2 DB
γ3
l3 DB
γ4
l4 DB
0
0
0
γ2
l2 XG
γ3
l3 XG
γ4
l4 XG
0
0
−XG1
γ2
l2 γ2 XG
γ3
l3 γ3 XG
γ4
l4 γ4 XG
0
0
−β1B XG1
γ2
XG
γ3
XG
γ4
XG
0
0
0
0
0
0
0
0
XB1
βG
−β2G DB
0
βG
βB
βB
βB



0




0



0

,
β2B 
−XG 

βB 
−β2B XG2 



0


β2B
XB
(A.94)

γ1
C5 DB

γ2
C6 D B
+
−AG5 + C3 DB + C4 +




γ1
γ2


C
D
+
γ
C
D
+
γ
C
D


3 B
1 5 B
2 6 B




γ3
γ4
−C3 DG − C4 − C5 DG − C6 DG + (1 − αG ΛG ) (1 − τ )DG yG − αG ΛG Gunlev
(DG ) + GG (DG )
G




unlev


−AB5 + (1 − αB ΛB ) (1 − τ )DB yB − αB ΛB GB (DB ) + GB (DB )


b := 
,


−AB5 + Z (XG ) + B4








XG Z 0 (XG )






−AG5 + dˆG ((sG + 1)XG )




−Z (X ) + B4 + dˆB (sB + 1 − KB /ΛB )XB
B
(1−τ )Xt̄ yB
(A.95)
C5 =
¯l3 unlev
Ālev
−
α
Λ
Ā
,
B
B
G3
G3
l3
(A.96)
¯l4 unlev
Ālev
−
α
Λ
Ā
.
B
B
G4
G4
l4
(A.97)
and
C6 =
The case in which DG < DB , D̂G < D̂B , and XG > XB can be solved analogously.
20
References
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Bhamra, H. S., L.-A. Kuehn, I. A. Strebulaev, 2010. The levered equity risk premium and credit
spreads: A unified framework. Review of Financial Studies 23(2), 645–703.
Chen, H., 2010. Macroeconomic conditions and the puzzles of credit spreads and capital structure.
Journal of Finance 65, 2171–2212.
Duffie, D., L. G. Epstein, 1992a. Asset pricing with stochastic differential utility. Review of Financial
Studies 5, 411–436.
, 1992b. Stochastic differential utility. Econometrica 60, 353–394.
Guo, X., 2001. An explicit solution to an optimal stopping problem with regime switching. Jounal
of Applied Probability 38, 464–481.
Hackbarth, D., J. Miao, E. Morellec, 2006. Capital structure, credit risk, and macroeconomic
conditions. Journal of Financial Economics 82, 519–550.
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