A Dynamic Model of Optimal Capital Structure andDebt Maturity with Stochastic Interest Rates
Nengjiu Ju and Hui Ou-Yang∗
This Version: December 15, 2003
∗Ju is with the Smith School of Business, University of Maryland, College Park, MD 20742. E-mail:[email protected]. Ou-Yang is with the Fuqua School of Business, Duke University, Durham, NC 27708.E-mail: [email protected]. We thank an anonymous referee, an Associate Editor, Gurdip Bakshi, and JayHuang for their valuable comments and suggestions. We also thank Henry Cao, Hua He, Hayne Leland,Mark Rubinstein, and Matt Spiegel for their advice on an earlier version where a static model was developed.
A Dynamic Model of Optimal Capital Structure and Debt Maturity
with Stochastic Interest Rates
Abstract
This paper develops a model in which an optimal capital structure and an optimaldebt maturity are jointly determined in a stochastic interest rate environment. Valu-ation formulas are derived in closed form and numerical solutions are used to obtaincomparative statics. The model yields leverage ratios and debt maturities that are con-sistent in spirit with empirical observations. It is demonstrated that a dynamic modelis crucial to obtain reasonable leverage ratios and that a stochastic interest rate is im-portant to study optimal debt maturity structure. It is also demonstrated that a modelof optimal capital structure with a constant interest rate cannot price risky bonds anddetermine optimal capital and debt structures simultaneously in a satisfactory manner.
1
1 Introduction
The problem of optimal capital structure has long been an intriguing one among researchers.
Brennan and Schwartz (1978) are perhaps the first to study this problem using the contingent-
claims analysis approach of Black and Scholes (1973), Merton (1974), and Black and Cox
(1976).1 In a more recent development, Leland (1994) introduces a model of optimal capital
structure based on a perpetuity. Leland and Toft (1996) extend Leland (1994) to examine
the effect of debt maturity on bond prices, credit spreads, and optimal leverage. The debt
maturity in Leland and Toft is assumed rather than optimally determined.2 While insightful,
these models assume that the risk-free interest rate is constant, thus ignoring the impact of
the stochastic nature of the interest rate on the firm’s optimal capital structure. Empirical
evidence has indicated that firms do take into account the slope of the default-free term
structure when they issue debt. See, for example, Barclay and Smith (1995), Guedes and
Opler (1996), Stohs and Mauer (1996), and Graham and Harvey (2001). In particular,
Graham and Harvey report that CFOs state that the slope of the term structure is an
important consideration in their refinancing decisions. These empirical evidences call for a
model that includes both optimal leverage and stochastic interest rates.
An active and growing body of work has studied the valuation of risky corporate bonds
and other derivative instruments in a stochastic interest rate environment. Kim, Ramaswamy,
and Sundaresan (1993) calculate various corporate bond prices in a series of numerical ex-
amples. Longstaff and Schwartz (1995) derive closed-form valuation expressions for fixed
and floating rate debt and find that the correlation between the underlying asset return and
interest rate has a significant effect on credit spreads.3 These models, however, assume that
the Modigliani-Miller (1958) theory holds, that is, the value of a firm is independent of its
1See also Kim (1978) for a mean-variance analysis of optimal capital structure.2See also Kane, Marcus and McDonald (1985), Fischer, Heinkel and Zechner (1989), Mello and Parsons
(1992), Leland (1998), Fan and Sundaresan (2000), Goldstein, Ju and Leland (2001), Miao (2002), andMorellec (2003).
3Duffie and Singleton (1999), Jarrow and Turnbull (1995), and many others specify the default outcomesand value credit risk by no arbitrage.
2
capital structure. This implies that the firm does not derive tax benefits from issuing bonds.
In this paper, we develop a dynamic model of optimal capital structure with stochastic
interest rates. At time zero the firm issues a T -year coupon bond. If the firm has not gone
bankrupt in T years, the firm issues a new T -year coupon bond at time T . The optimal value
of the new bond will depend on the firm value and interest rate at time T . If at the end
of the second T -year period the firm is still solvent, it issues another T -year coupon bond
optimally. This process goes on indefinitely as long as the firm is solvent. The optimally
levered firm value at time zero takes into account the tax benefits associated with all future
leverages. To obtain the optimally levered firm value, we need to compute explicitly the total
tax benefit, the total bankruptcy cost, and the total transactions cost of all future issues of
debt. Generally, this is a difficult problem because the issuance of future debt depends on the
fact that the firm has not gone bankrupt. We employ a scaling property and a fixed-point
argument to obtain the present values of the total tax benefit, the total bankruptcy cost,
and the total transactions cost explicitly. Valuation formulas are derived in closed form and
numerical solutions are used to obtain comparative statics.
In our model, the trade-off between tax shields and bankruptcy costs associated with
debts yields an optimal capital structure. Firms may issue new debt as firm values change
over time. The trade-off between the gains of dynamically adjusting the debt level and
the transactions costs of doing so yields an optimal debt maturity structure. The optimal
capital structure and the optimal maturity structure are interdependent. The interest rate
is modelled as a three-factor Vasicek (1977) mean-reverting process. Modelling the interest
rate as a mean-reverting process allows us to examine separately the impact of the long-run
mean and the initial value of the interest rate.
Our numerical results indicate that the long-run mean of the interest rate is an important
determinant of the optimal capital structure. This is intuitive because the long-run mean
plays a key role in the determination of the tax shields and bankruptcy costs associated with
all future debt issues. The initial interest rate level is important in determining the price of
3
current outstanding risky bonds, especially those with short and moderate maturities. The
reason is that it takes time for the interest rate to revert to its mean level.
Our results indicate that the maturity of a bond is also an important determinant in
capital structure considerations. For example, if a firm can issue debt more frequently due
to lower issuing costs, then it may issue less debt in the current period so as to reduce the
likelihood of bankruptcy because the firm has an option to issue more debt in the future. The
optimal leverage ratio of about 30% obtained from our dynamic model compares favorably
with the historical average for a typical large publicly traded firm in the U.S. On the other
hand, static models such as the Leland (1994) model typically predict a very high leverage
ratio (e.g., 80%). Furthermore, the optimal maturities generated from our dynamic model
are consistent in spirit with the empirical observations of Barclay and Smith (1995) and
Stohs and Mauer (1996).
In summary, our results suggest that a dynamic model of capital structure and debt
maturity with stochastic interest rates is important to price risky debt and determine optimal
capital and maturity structures appropriately.
The remainder of this paper is organized as follows. Section 2 introduces the model
and derives various closed-form valuation expressions. Section 3 presents numerical results.
Section 4 summarizes and concludes. More technical materials can provided in the two
appendices. Appendix A reviews the forward risk-neutral measure used to derive the valua-
tion formulas in Section 2 and Appendix B presents the proof of a scaling property used in
Subsection 2.4.
2 The Model
In this section we first set up the valuation problem and then derive the valuation formulas
in closed form.
4
2.1 The Setup
In this subsection we provide the main assumptions and notations used throughout the rest
of the paper.
Assumption 1 Financial markets are dynamically complete, and trading takes place contin-
uously. Therefore, there exists an equivalent martingale measure (Harrion and Kreps, 1979)
or a risk-neutral measure (Cox and Ross, 1976), Q, under which discounted price processes
are martingales.
Assumption 2 The total value of the firm’s unlevered asset, Vt, is described by a geometric
Brownian motion process given by (under the risk-neutral measure Q)
dVt
Vt
= (rt − δ) dt + σv dWQvt , (1)
where rt is the interest rate at time t, δ is a constant payout rate, σv denotes the constant
volatility of asset returns, and WQvt is a standard Wiener process defined on a complete prob-
ability space (Ω, P, F).
Assumption 3 The interest rate rt is modelled by three factors, that is, rt = y1t + y2t + y3t,
where each factor yit under Q follows a Vasicek (1977) process,
dyit = (αi − βi yit) dt + σi dWQit , i = 1, 2, 3. (2)
The coefficients αi, βi, and σi are constants, and WQ1t , WQ
2t , and WQ3t are independent standard
Wiener processes on the same probability space (Ω, P, F). The instant correlation between
dWQvt and dWQ
it is given by ρi dt.
Given this interest rate process, the price of a zero-coupon bond at time t with a maturity
of T years is given by4
Λ(yt, t; T ) ≡ Λ(y1t, y2t, y3t, t; T ) =3∏
i=1
eAi(t;T )−Bi(t;T )yit , (3)
4Note that for notational simplicity, we have used yt to denote a vector of three state variables (y1t, y2t, y3t).
5
where
Ai(t; T ) =
(σ2
i
2β2i
− αi
βi
)(T − t)−
(σ2
i
β2i
− αi
βi
)1− e−βi(T−t)
βi
+
(σ2
i
2β2i
)1− e−2βi(T−t)
2βi
, (4)
Bi(t; T ) =1− e−βi (T−t)
βi
. (5)
Assumption 4 Bankruptcy occurs when the firm’s asset value falls below a default boundary
VB(yt, t; P, T ), which is specified by
VB(yt, t; P, T ) = P Λ(yt, t; T )eγ(T−t), (6)
where P is the face value of the debt and γ is a constant parameter. If Vt > VB(yt, t; P, T ),
the firm is solvent and makes the contractual coupon payment to its bondholders. In the event
of bankruptcy, bondholders receives φVB(yt, t; P, T ) with φ ∈ [0, 1) and shareholders receive
nothing.
Note that VB(yt, t; P, T ) has the desired property that at the maturity date, it equals the
face value of the debt, implying that to avoid default the asset value has to be at least as
large as the debt’s face value. VB(yt, t; P, T ) has a stochastic part that depends on Λ(yt, t; T )
as well as a deterministic part that depends on eγ(T−t). When γ = 0, the default boundary
is the riskless price of the face value and it may become very small compared to the face
value of the debt for a large T and t << T . This implies that the firm would have to serve
the debt (paying the contractual coupon) even if the asset value becomes very low as long
as it is above a very low boundary. The shareholders may choose not to do so. Although it
is extremely difficult to determine an endogenous default boundary, especially in a dynamic
model like ours, the factor eγ(T−t) can ensure that the asset value will not become too low
before default is triggered.5 The exponent, γ, controls the level of the net worth before
5Indeed, Leland and Toft (1996) demonstrate that their endogenously determined default level may even beabove the face value of the debt in their debt-rollover model for very short maturities. For longer maturities,they find that the optimal default level is below the face value of the debt. Thus the firm may continue tooperate even if its net worth is negative (Vt < P ) if no bond covenants force the firm to default.
6
default is triggered. Let VB(y0, 0; P, T ) = PΛ(y0, 0; T )eγT = ϕP . We choose
ϕ = Λ(y0, 0; T )eγT (7)
to control the level and shape of the default boundary and fix γ accordingly. For example, if
ϕ = 0.9, the initial value of the boundary equals 0.9 of the face value of the debt.
Assumption 5 The firm rebalances its capital structure every T years. At time zero the
firm issues a T -year coupon bond. If the firm has not gone bankrupt in T years, it optimally
issues a new T -year coupon bond at time T . This process continues indefinitely as long as
the firm is solvent. Further assume that the firm incurs a transaction cost proportional to
the value of the debt issued.
The capital structures in Leland (1994) and Leland and Toft (1996) are static in the sense
that as the firm’s asset value evolves, the coupon rate and the face value of the debt remains
the same. This is clearly not optimal. For example, if the firm’s asset value increases, the
debt value should be increased to better exploit the tax shields. The new capital structure
strategy in Assumption 5 allows the firm to scale up (down) the debt value depending on
whether the firm’s asset value is above (below) the initial asset value when the firm rebalances
its capital structure. For example, if the firm’s asset value has increased, the firm’s optimal
debt capacity has also increased. Then it is optimal to scale up the debt value when the firm
issues new debt.
To examine the optimal maturity in a dynamic capital structure model, we need to
introduce a transaction cost associated with issuing and servicing the debt. The reason
is that without the cost of issuing and retiring debt, the firm would rebalance its capital
structure continuously.6 For tractability, we assume a transaction cost proportional to the
value of the debt issued.
6For similar reasons, Kane, Marcus, and McDonald (1985) and Fischer, Heinkel, and Zechner (1989) havealso included transactions costs in their models.
7
2.2 Preliminary Development
The stochastic nature of the interest rate and the default boundary complicate the pricing
of securities in this model tremendously. Nevertheless, the change of measure technique can
be used fruitfully to obtain the formulas in closed form. Before we present these formulas in
the next two subsections, we derive some preliminary results.
First, define the first passage time τ as τ = mint : Vt ≤ VB(yt, t; P, T ), which is the
first time at which the asset value Vt hits VB(yt, t; P, T ) in some state ω ∈ Ω under Q. Next,
define
Xt = log(Vt/VB(yt, t; P, T )). (8)
It is clear that τ is the first passage time that Xt, starting at X0 = log(V0/VB(y0, 0; P, T )) > 0,
reaches the origin. Ito’s lemma yields
dXt =
(3∑
i=1
σ2pi(t; T )/2− σ2
v/2 + γ − δ
)dt + σvdWQ
vt +3∑
i=1
σpi(t; T )dWQit , (9)
where
σpi(t; T ) = σiBi(t; T ). (10)
Note that although the drifts of dVt/Vt and dVB(yt, t; P, T )/VB(yt, t; P, T ) are stochastic, that
of dXt is deterministic.
In the next two subsections, we need the following three quantities:
F (t; T, X0) = EQ0
e−
∫ t
0r(u)du
Λ(y0, 0; t)1(τ < t)
, (11)
G(t; T, X0) = EQ0
e−
∫ T
0r(u)du
Λ(y0, 0; T )1(τ < t)
, (12)
H(t; T, X0) = EQ0
[e−σ2
vt/2+σvW Qvt1(τ > t)
], (13)
where 1(·) is the indicator function and where F (·), H(·) and G(·) all depend on T because
σpi(t; T ) in equation (10) does.
To obtain F (t; T,X0) in closed form, using the results in Appendix A,7 we can define the
7For simplicity of exposition, a one-factor interest process is assumed in Appendix A. The results can beeasily applied to our three-factor model.
8
following equivalent measure (to Q), Rt, by
dWRtvs = dWQ
vs +3∑
i=1
ρiσpi(s; t)ds, (14)
dWRtis = dWQ
is + σpi(s; t)ds, i = 1, 2, 3, (15)
where σpi(s; t) = σiBi(s; t). Under Rt, WRtvs and WRt
is , i = 1, 2, 3, are standard Wiener
processes with instant correlation ρids. Under the newly defined measure, Rt, Xs is given by
dXs =
(3∑
i=1
σ2pi(s; T )/2− σ2
v/2 + γ − δ − σv
3∑
i=1
ρiσpi(s; t)−3∑
i=1
σpi(s; T )σpi(s; t)
)ds
+σvdWRtvs +
3∑
i=1
σpi(s; T )dWRtis
=(−σ2(s; T )/2 + m(s; t, T )
)ds + σ(s; T )dWRt
s , (16)
where
σ(s; T ) =
√√√√3∑
i=1
σ2pi(s; T ) + σ2
v + 2σv
3∑
i=1
ρiσpi(s; T ), (17)
m(s; t, T ) = γ − δ +3∑
i=1
(σpi(s; T ) + ρiσv) (σpi(s; T )− σpi(s; t)) , (18)
and WRts is a new standard Wiener process under Rt.
Using the Girsanov theorem and the newly defined Rt, we have
F (t; T, X0) = ERt0 [1(τ < t)]. (19)
Hence, F (t; T, X0) is the distribution function (under Rt) of the first passage time τ .
Tedious derivation yields8
F (t; T, X0) = N
−X0 − µf (t; T )√
Σ(t; T )
+ e−
2µf (t;T )X0Σ(t;T ) N
−X0 + µf (t; T )√
Σ(t; T )
. (20)
Here N(·) is the standard normal distribution function and Σ, µf are given by
Σ(t; T ) =∫ t
0σ2(s; T )ds =
3∑
i=1
χ1i(t; T ) + 2σv
3∑
i=1
ρiχ2i(t; T ) + σ2vt, (21)
µf (t; T ) = −∫ t
0σ2(s; T )ds/2 +
∫ t
0m(s; t, T )ds = −Σ(t; T )/2 + (γ − δ)t +
3∑
i=1
χ1i(t; T )−3∑
i=1
χ3i(t; T ) + σv
3∑
i=1
ρiχ4i(t; T ), (22)
8Details for obtaining F (t;T, X0), G(t; T,X0), and H(t;T, X0) in closed form are available on request.
9
where the χki’s (k = 1, 2, 3, 4, i = 1, 2, 3) are given by
χ1i(t; T ) =∫ t
0σ2
pi(s; T )ds =σ2
i
β2i
(t + e−2βi(T−t)B2i(t)− 2e−βi(T−t)B1i(t)
), (23)
χ2i(t; T ) =∫ t
0σpi(s; T )ds =
σi
βi
(t− e−βi(T−t)B1i(t)
), (24)
χ3i(t; T ) =∫ t
0σpi(s; T )σpi(s; t)ds =
σ2i
β2i
(t + e−βi(T−t)B2i(t)
−(1 + e−βi(T−t))B1i(t)), (25)
χ4i(t; T ) =∫ t
0(σpi(s; T )− σpi(s; t))ds =
σi
βi
(1− e−βi(T−t)
)B1i(t), (26)
where B1i(t) = (1− e−βit)/βi and B2i(t) = (1− e−2βit)/(2βi).
Similar steps show that G(t; T, X0) is the distribution function of τ under RT and
G(t; T, X0) = ERT0 [1(τ < t)]
= N
−X0 − µg(t; T )√
Σ(t; T )
+ e−
2µg(t;T )X0Σ(t;T ) N
−X0 + µg(t; T )√
Σ(t; T )
, (27)
where
µg(t; T ) =∫ t
0
(−σ2(s; T )/2 + γ − δ
)ds = −Σ(t, T )/2 + (γ − δ)t. (28)
Finally, to obtain H(t; T,X0) in closed form, we define another equivalent measure (to
Q), R′t, such that W
R′tvs and W
R′tis defined by
dWR′tvs = dWQ
vs − σvds, (29)
dWR′tis = dWQ
is − ρiσvds, i = 1, 2, 3, (30)
are standard Wiener processes under R′t with the instant correlation between dW
R′tvs and
dWR′tis being given by ρids. It is easy to see that under R′
t,
dXs =(σ2(s; T )/2 + γ − δ
)ds + σ(s; T )dWR′t
s , (31)
where WR′ts is a new standard Wiener process under R′
t.
Using the Girsanov theorem and the newly defined R′t, we have
H(t; T, X0) = ER′t0 [1(τ > t)]. (32)
10
Thus, H(t; T, X0) is the complementary distribution function of τ under R′t and given by
H(t; T, X0) = 1− ER′t0 [1(τ < t)] =
N
X0 + µh(t; T )√
Σ(t; T )
− e−
2µh(t;T )X0Σ(t;T ) N
−X0 + µh(t; T )√
Σ(t; T )
, (33)
where
µh(t; T ) =∫ t
0
(σ2(s; T )/2 + γ − δ
)ds = Σ(t; T )/2 + (γ − δ)t. (34)
2.3 Valuation Formulas in Closed Form
Consider a bond that pays a coupon rate C, has a face value P , and matures at time T . The
payment rate d(s) to the bondholders at any time s is equal to (s ≤ T )
d(s) = C 1(s ≤ τ) + P δ(s− T ) 1(T ≤ τ) + φVB(ys, s; P, T ) δ(s− τ), (35)
where δ(·) is the Dirac delta function.
Using the results from the previous subsection, we have the debt value
D(T, X0; y0, P ) =∫ T
0EQ
0 [e−∫ s
0r(u)dud(s)]ds = C
∫ T
0EQ
0 [e−∫ s
0r(u)du1(s ≤ τ)]ds +
PEQ0 [e−
∫ T
0r(u)du1(T ≤ τ)] + φP
∫ T
0EQ
0 [e−∫ s
0r(u)duEQ
s [e−∫ T
sr(u)du]eγ(T−s)δ(s− τ)]ds
C∫ T
0Λ(y0, 0; s)ERs
0 [1(s ≤ τ)]ds + P Λ(y0, 0; T )ERT0 [1(T ≤ τ)] +
φP Λ(y0, 0; T )∫ T
0eγ(T−s)ERT
0 [δ(s− τ)]ds = C∫ T
0Λ(y0, 0; s)(1− F (s; T,X0))ds +
P Λ (y0, 0; T )(1−G(T ; T, X0)) + φP Λ(y0, 0; T )(G(T ; T, X0) + G(T ; T, X0)
), (36)
where
G(T ; T, X0) = γ∫ T
0eγ(T−s)G(s; T, X0)ds. (37)
Note that, to obtain the last two terms in (36), we have used both the relation: ERT0 [δ(s −
τ)]ds = g(s; T,X0)ds = dG(s; T, X0), where g(s; T, X0) is the density function of τ under
RT , and the integration by parts.
Let D(T, X0; y0, P ) = λP Λ(y0, 0; T ). That is, the debt is issued at λ times the price of
a riskless discount bond with the same face value. The debt is issued at (above or below)
11
par if λ is equal to (greater than or smaller than) 1/Λ(y0, 0; T ). We require that the initial
debt be priced at par by setting λ = 1/Λ(y0, 0; T ). We assume that all future debts are also
issued at λ times the riskless discount bond price with the face value of the debt issued.9
When the coupon of C is paid, θC, where θ is the tax rate, is deducted from corporate
taxes. Thus, the tax shield from the current issue of debt is θ times the first term in (36).
That is, the tax shield is given by
tb(T, X0) = θC∫ T
0Λ(y0, 0; s)(1− F (s; T, X0))ds. (38)
Given that D(T,X0; y0, P ) = λP Λ(y0, 0; T ) = λV0e−X0−γT , from (36) we have
tb(T,X0) = θ V0e−X0−γT
(λ− 1 + (1− φ)G(T ; T, X0)− φG(T ; T,X0)
). (39)
When bankruptcy occurs, φ fraction of the asset value is paid to bondholders and the
rest, (1 − φ) fraction, is lost to the bankruptcy process. Therefore, the bankruptcy cost is
easily seen to be
bc(T,X0) = (1− φ)V0e−X0−γT
(G(T ; T, X0) + G(T ; T,X0)
). (40)
The (proportional) transaction cost of issuing the debt is given by
tc(T, X0) = κD(T, X0; y0, P ) = κλV0e−X0−γT , (41)
where κ is the transaction cost per dollar issued.
From (39), (40), and (41), it is clear that, given both T and those parameters exogenous
to the model, the tax shield, bankruptcy cost, and transactions cost depend only on one
choice variable, X0, which in term depends on P since X0 = log(V0/λPΛ(y0, 0, T )).
2.4 Total Values in the Presence of Future Periods
Even though at any one point in time there is only one issue of debt outstanding, the total
levered firm value will reflect the benefits and costs of all future issues of debt. Therefore9This means that future debts may not be issued at par. This is not uncommon in models where future
debts are involved, for example, in the debt-rollover model of Leland and Toft (1996), the debt replacementmay be issued over/under par.
12
we need to find the total tax benefit, total bankruptcy cost, and total transaction cost from
all future periods. But the existence of all future periods depends on the firm not having
gone bankrupt in all previous periods. This is a multi-period conditional first passage time
problem. Generally, such problems are very difficult to solve. Fortunately, we are able to
transform this difficult problem into a simple one-period fixed-point problem as we show
now.
Consider the value of the tax shield at time T from the debt issued at time T . Similarly
to (39), the tax shield is given by
tb2(T, XT ) = θVT e−XT−γT(λ− 1 + (1− φ)G(T ; T, XT )− φG(T ; T, XT )
), (42)
where XT = log(VT /(P2Λ(yT , 0; T )eγT )) with P2 being the face value of the second debt.
Note that (42) has exactly the same functional form as (39) except for the scaling factor, VT
versus V0. Indeed, it is clear that the tax benefit of any debt issued in the future will have
the same functional form as (39) but scale with the firm’s asset value. But the scaling factor
of the firm’s asset value does not affect the optimal X. Hence, if the initial optimal capital
structure corresponds to an optimal value X∗ for X0, X∗ must also be the optimal value for
XT for the optimal capital structure at time T . In fact, X∗ will be the optimal value for all
future debt issues. Appendix B presents a detailed proof of this result.
Consequently, the total tax shield from the current and all future issues of debt also scales
with the asset value when the debt is issued. For example, if we let TB(T, X0) denote the
total tax benefit at time zero from the first issue of debt at time zero and all succeeding
issues of debt and TB2(T, X0) denote the total tax benefit at time T from the debt issued at
time T and all succeeding issues of debt, then TB2(T, X0) = TB(T, X0) ∗ VT /V0. However,
the total tax benefit TB(T, X0) at time zero is the tax benefit tb(T, X0) from the debt issued
at time zero, plus the present value of the total tax benefit TB2(T, X0) = TB(T,X0)∗VT /V0
at time T , conditional on no default occurred. Therefore, we have
TB(T,X0) = tb(T, X0) + EQ0
[VT
V0
TB(T, X0) 1(τ > T ) e−∫ T
0r(u)du
]= tb(T, X0) +
13
TB(T,X0) EQ0
[e∫ T
0(r(u)−δ−σ2
v/2)du+σvW QvT 1(τ > T ) e−
∫ T
0r(u)du
]
= tb(T, X0) + TB(T, X0) e−δT EQ0
[e−σ2
vT/2+σvW QvT 1(τ > T )
]
= tb(T, X0) + TB(T, X0)e−δT H(T ; T, X0), (43)
where
H(T ; T, X0) = EQ0 [e−σ2
vT/2+σvW QvT 1(τ > T )],
and its closed form formula is given in (33).
Equation (43) indicates that the total tax shield TB(T, X0) is a linear function of TB(T, X0)
itself. Thus, we have reduced the complex multi-period problem to a simple fixed-point prob-
lem whose solution is given by
TB(T, X0) =tb(T, X0)
1− e−δT H(T ; T, X0).
Similar arguments show that the total bankruptcy cost BC and transaction cost TC are,
respectively, given by
BC(T, X0) =bc(T, X0)
1− e−δT H(T ; T, X0)and TC(T,X0) =
tc(T, X0)
1− e−δT H(T ; T,X0). (44)
Notice that TB, BC, and TC do not depend directly on the interest rate or its three factors.
They only depend indirectly on the interest rate through the variable X0.
The total levered firm value consists of four terms: the firm’s unlevered asset value, plus
the value of tax shields, less the value of bankruptcy costs, less the transactions costs of
issuing debt:
TV (T, X0) = V0 + TB(T, X0)−BC(T, X0)− TC(T, X0)
= V0 +tb(T, X0)− bc(T, X0)− tc(T, X0)
1− e−δT H(T ; T, X0). (45)
The equity value E(T, X0; P ) is given by the total levered firm value less the debt value, i.e.
E(T, X0; P ) = TV (T, X0)− P , where P is the value of the debt because it is priced at par.
14
Note that given other parameters of the model, the total levered firm value TV (T, X0) is
a function of only T and X0.10 The firm chooses these two variables to maximize TV (T, X0).
This is a simple unconstrained bivariate maximization problem and a number of efficient
library routines are well suited for this problem.11
3 Numerical Results and Comparative Statics
In this section, we implement the dynamic optimal capital structure model developed in
the previous section. Although the valuation formulas are obtained in closed form, the joint
determination of optimal capital structure and debt maturity structure needs to be performed
numerically. In the numerical calculations, the base parameters are fixed as follows: the asset
return volatility σv = 0.2; the corporate tax rate θ = 0.35; the per dollar transaction cost of
issuing debt κ = 2%;12 the bankruptcy cost parameter φ = 0.5; the initial values of the three
interest rate factors y1(0) = 6%, y2(0) = 0, y3(0) = 0; the correlation coefficients between
the firm’s asset return and the three interest rate factors ρ1 = 0, ρ2 = 0, ρ3 = 0. The payout
rate δ is set so that initially the payout covers the net coupon rate and the dividend on
equity. That is, δ is the solution to δ = [(1− θ)C + δEE]/V0, where δE is the dividend yield
on equity.13 We set δE = 1.5% which is about the current dividend yield on equity. The
parameter values for the interest rate process are: α1 = 0.02, β1 = 0.3333, σ1 = 0.02; α2 = 0,
β2 = 0.5, σ2 = 0.03; α3 = 0, β3 = 0.1, σ3 = 0.01.14 Note that, because we have set the initial
values of the second and the third factors at zero, the first factor is the ‘major’ one whose
base parameter values resemble the empirical estimates from a one-factor Vasicek model.15
10Recall that X0 = log(V0/λPΛ(y0, 0, T )). Therefore, once the optimal T and X0 are obtained, the optimaldebt value P can be easily recovered.
11For example, ‘bconf’ from IMSL Math Library (2003) or ‘frprmn’ from Numerical Recipes (2003).12The transaction cost of issuing bonds is usually between 1% and 4%. Fischer, Heinkel and Zechner
(1989) have used transaction costs ranging from 1% to 10% while Kane, Marcus and McDonald (1985) haveconsidered 1% and 2% transaction costs.
13Since the optimal C and E are functions of δ, this is a nonlinear equation for δ.14To overcome the identification issue, α2 and α3 are exogenously fixed at 0. For more details, see Dai and
Singleton (2000) and Liu, Longstaff and Mandell (2003). We thank Jay Huang for bring the second referenceto our attention.
15See, for example, Aıt-Sahalia (1999). In particular, the long-run mean of the first factor is 6%.
15
The second and third factors add fluctuations around the value of the first factor. In our
calculations we find that the added variability from these two ‘minor’ factors has small effects
on a firm’s capital structure and maturity structure. The reason is that the variability of the
interest rate process is small relative to that of the firm’s asset return process. We consider
three different values (0.8, 0.9, 1.0) for the default boundary level and shape parameter ϕ.16
The numerical results are reported in Table 1 and Table 2.
3.1 Optimal Capital Structure
Table 1 reports the comparative statics with constant interest rates and Table 2 reports the
comparative statics with stochastic interest rates. Several features are notable.
Generally, the stochastic nature of the interest rate process, except for the long-run mean,
has a small impact on firms’ optimal leverage ratios. The reason is simple. While at any time
there is only one finite maturity debt outstanding, the optimal capital structure is based on
the total tax shield and default cost associated with the existing debt and all future debts.
It is the long-run mean rather than the current interest rate level that will have the greatest
impact on the values of these future debts. However, this is not to say that the current level
and the stochastic nature of the interest rate process are not important. It clearly affects the
value of the existing debt. For example, for the middle entry of Panel E in Table 1 with a
constant interest rate 4%, the bond price is 31.562, whereas a bond with the same maturity,
coupon, face value, (T , C, P ) = (5.901, 1.305, 31.562), and an initial interest level 4%, the
bond price with the stochastic interest rate is 30.090. If we keep the coupon and face value
the same, but change the debt maturity to 10 years, then the bond price with a 4% constant
interest rate will be 31.116 while that with the stochastic interest rate will be 28.495.
When the initial interest rate is at or close to its long-run mean, however, the price of the
same bond (same maturity, coupon rate, and face value) with the stochastic interest rate will
be close to that when the interest rate is a constant. For example, for the bond in the middle
16Leland and Toft (1996) consider a model with finite-maturity debt which is rolled over. Their endoge-nously determined bankruptcy level for five year maturity debt is about 0.9 of the face value. ϕ = 1.0corresponds to protected debts.
16
entry of Panel A in Table 1, where the initial interest rate is 6%, the bond price with the
stochastic interest rate is 36.771, which is close to the price of 36.623 with a constant interest
6%. If we change the maturity to 10 years, the prices, 36.321 and 36.092, are still close.
Clearly, if the current interest rate is close to its long-run mean, the stochastic nature of the
interest rate has small effects on the optimal capital structure or the bond price. However,
when the initial interest rate is far away from its long-run mean, the prices of bonds with
the same maturity, coupon and face value can be quite different whether the interest rate
is a constant or stochastic. In short, the capital structure is affected more by the long-run
mean of the interest rate process while the value of the existing debt is affected more by the
current interest rate level.
Second, the leverage ratios obtained from our model compare favorably with the historical
average of about 30% for a typical large publicly traded firm in the US, while most static
optimal capital structure models predict leverage ratios much higher than what is observed.
(See, e.g., Leland (1994).) The reason for the lower leverage ratio in our dynamic model
is that, in a static model, a firm cannot adjust its debt level in the future, and therefore
issues debt more aggressively. In contrast, a firm with an option to restructure in the future
issues debt less aggressively, for it can adjust its capital structure when the firm’s asset value
changes. Moreover, the cost of default for the firm includes not only the cost of bankruptcy,
but also the loss of the option value of adjusting the debt level in the future. Hence, default
is more costly in the dynamic setting, further decreasing the initial optimal leverage ratio.
Third, with a lower bankruptcy trigger level (smaller ϕ), a firm optimally levers more. The
reason is that with a lower bankruptcy level, the probability and expected cost of bankruptcy
are both lower. We can think of the level of ϕ as the strength of the bond covenants to force
bankruptcy. As the rights of bondholders to force default increase (higher ϕ), firms find it
optimal to use less leverage. However, the differences are very small. In our dynamic model,
the optimally chosen debt maturity is inversely related to ϕ. While a higher ϕ implies a
higher probability of default for a given maturity, a lower maturity decreases the default
17
probability. These two effects partially offset each other. This indicates that a different
default triggering boundary or mechanism will not have affected our results significantly.
Four, the comparison of the last two panels in Table 2 reveals that the correlation between
the return of a firm’s asset value and the factors of the stochastic interest rate process has
little impact on a firm’s capital structure decisions. The reason is that the covariances σvρiσi
between the asset return and the interest rate factors are much smaller than the variance
of asset value returns σ2v for typical values of σv.
17 Thus, σv has a first-order effect on the
pricing of securities.
Last, in Panels K-P of Table 2, we examine the comparative statics on the term structure
parameters α1, β1, and σ1. When α1 changes, the long-run mean of y1t also changes. The
corresponding long-run mean is 3% in Panel K and 9% in Panel L. Note that a firm optimally
issues more debt if the long-run mean of the interest rate process is higher, similar to the
effects of a higher initial interest rate level as Panels E and F indicate.18 Panels M and N
consider the effect of the speed of the mean-reverting parameter β1. Since α1/β1 represents
the long-run mean, when α1 is fixed, a higher β1 means a lower long-run mean. Similar to
the results for changing α1, the firm issues more debt when β1 is lower. Panels O and P
indicate that although the interest rate volatility increases four times from 0.01 in Panel O
to 0.04 in Panel P, the effect of it is quite small. The reason is that the return of the asset
value is much more volatile than the interest rate process and thus the added volatility in
the interest rate process has little impact on the first passage time to the default boundary.
Our calculations thus suggest that the long-run mean is more important than the typical
volatility of the stochastic interest rate process.
3.1.1 Further Comments on the Relation between Firm Value and Interest RateLevel
Notice that the comparison of Panels E and F in the two tables indicates that a firm optimally
levers more when the interest rate is higher. This result seems counterintuitive. We must note
17See, for example, (17).18Caution is required in interpreting this comparative static result. See the next subsection for a discussion.
18
that, however, the results in the tables are comparative statics, which means that, among
other inputs, the initial asset value V0 is maintained the same (at $100 in our calculations).
Given the same cash flow from a firm’s assets, the initial asset value of the firm will be
different for a different interest rate level. We have scaled the firm values so that the initial
values are always 100. Therefore, one should not conclude that for the same cash flow, a
firm optimally levers more at a higher interest rate level.
To illustrate our point, we consider a simple example. Assume that the cash flow of the
firm’s assets follows a geometric Brownian motion process:
dπt
πt
= µ dt + σπ dWt, (46)
where µ and σπ are constants and Wt is a standard Wiener process. Further assume that
the interest rate is a constant r and that the risk premium of the asset return is a constant
λ. The value of the firms’s assets is then given by
Vt =∫ ∞
0Et[πt+s]e
−(r+λ)sdt =πt
r + λ− µ. (47)
Note that Equation (47) is similar to the constant growth rate dividend discount model
where µ is the growth rate of the cash flow and λ the risk premium. Equation (47) clearly
demonstrates that, given the same cash flow process, the firm’s asset value is inversely related
to the interest rate.
To provide a concrete numerical example, we assume that π0 = 6, r = 0.06, λ = µ.
Hence, the firm’s asset value is given by V0 = 100. Panel A in Table 1 shows that with
φ = 0.9 and κ = 0.02, the firm optimally issues 36.623 in debt and the levered firm value is
118.661. Now suppose instead r = 0.04. The firm’s asset value is then given by V0 = 150.
Therefore, for the same cash flow, the debt and levered firm value in the middle entry of
Panel E should be multiplied by 1.5.19 That is, corresponding to V0 = 150, the optimal
debt and firm value are 47.343 and 169.485, respectively. If on the other hand, r = 0.08,
then V0 = 75 and the debt and levered firm value in the middle entry of Panel F should
19Recall that all of the results are obtained for V0 = 100.
19
be multiplied by 0.75. Therefore, the optimal debt and firm value are 30.069 and 92.056,
respectively, when r = 0.08 and V0 = 75. Consequently, we have obtained that the optimal
debt value and levered firm value are inversely related to the interest rate level, consistent
with the intuition.
However, even though the firm’s asset value, levered value and optimal debt value all
decrease as the interest rate increases, the optimal leverage ratio is higher. The reason is
that at a higher interest rate, the risk-neutral drift of the asset value process is higher and
the risk-neutral default probability becomes lower for the same debt value. Therefore, the
proportional decrease of the optimal debt value is less than that of the firm’s asset value and
levered value. As a consequence the firm optimally borrows less but the optimal leverage
ratio is higher when the interest rate is higher. Hence, care should be taken to interpret some
of the comparative statics.
Note that Vt as given in Equation (47) is proportional to πt, so Vt also follows a geometric
Brownian motion process. If r follows a stochastic process (e.g., a Vasicek process), the
resulting Vt will still be inversely related to the initial interest rate level but Vt will no
longer follow a geometric Brownian motion process. For tractability and following the capital
structure literature, we have assumed that Vt follows a geometric motion process. But the
intuition that the optimal debt value and firm value are inversely related to the interest rate
level should still hold.
3.2 Optimal Maturity Structure
The shareholders’ strategic consideration of dynamic adjustments of the debt level yields an
optimal maturity structure. The trade-off in this case is between the transactions costs of
issuing debt and the gains of adjusting debt level dynamically. On the one hand, the firm
should issue short maturity debt and therefore gives itself the opportunity to issue new debt
optimally, depending on the firm value when the old debt matures. On the other hand, if
new debt is issued too often, transactions costs will become too large. When the firm behaves
20
optimally, in addition to an optimal capital structure, an optimal maturity structure emerges.
The optimal maturities are reported in Column 3 in Table 1 and Table 2. The tables
show that the two most important parameters are the tax rate and the transactions costs
because the optimal maturity is the result of the trade-off between the transactions costs
and the gains of adjusting the debt level depending on future firm values.
Barclay and Smith (1995) find that during the period of 1974–1992, firms in their sample
have 51.7% of their debts due in more than three years. Because on average, the debt would
have existed for half of the lifetime at any point in time, the median maturity at issue appears
to be a little over six years. Stohs and Mauer (1996) find that the median time to maturity
is 3.38 years for their sample. Therefore the mean maturity of all debts at issue appears
to be between six and seven years, consistent with Barclay and Smith (1995). Most of the
optimal maturities in the two tables are between 4 and 6 years, comparing favorably with
the empirical values. While most of our theoretical values are smaller than 6 years, it is to
be noted that the optimal maturities in Panel B with a 20% effective tax rate are quite close
to 6 years. Even though 35% is the top corporate tax rate, the effective corporate tax rate
for most firms is likely to be lower due to investment credit, loss-carry forward, and personal
tax effects.
The comparison of the middle part of Panel A with those of Panel C and Panel D
indicates that the optimal debt maturity is inversely related to asset volatility. This is due
to the option-like effect. The higher the asset volatility, the greater the value of the option to
adjust the capital structure in the future. The gains of dynamically adjusting the debt level
are reduced if the firm is less volatile. Thus, the firm restructures its capital structure less
frequently. This appears to be consistent with the empirical evidence. For example, in the
sample of Barclay and Smith (1995), 36.6% of equally weighted debts mature in more than
five years, but 45.9% of the value-weighted debts mature in more than five years. Therefore
larger firms tend to have longer maturity debts. Because larger and more mature firms are
less volatile than smaller and less mature ones, it follows that less volatile firms have longer
21
maturity debts.
Panels A, E, and F indicate that the optimal debt maturity is inversely related to the
level of the interest rate. When the interest rate is low, the percentage growth in the firm’s
asset value, as given in Equation (1), is also low. When the growth in the asset value is
small, the potential benefit of adjusting the debt level is small. Therefore, the firm optimally
issues longer maturity debts when the term structure is upward sloping, i.e., when the initial
interest rate level is below the long-run mean. When the initial interest rate is above the
long-run mean with a downward sloping term structure, the firm correctly infers that the
future interest rate will likely be lower, so the firm optimally issues debts with shorter
maturity dates so that it can issue new debt sooner at a lower interest rate. Similarly, the
term structure is downward sloping in Panel K and upward sloping in Panel L. Consistent
with Panel E and Panel F, Panel K and Panel L demonstrate that the optimal maturity is
shorter in a downward sloping term structure environment. Notice that because the optimal
maturity does not depend on the initial asset value, the results presented in this subsection
are not subject to the comments of the last subsection.
4 Concluding Remarks
The existing models of the optimal capital structure consider neither stochastic interest
rates nor a maturity structure. This paper develops a model of optimal capital structure and
maturity structure with a three-factor Vasicek interest rate process. Valuation formulas are
obtained in closed form. A novel fixed-point argument is used to obtain the total tax shield,
default cost, and transaction cost for the dynamic model with potentially an infinite number
of debt issues.
The trade-off between the bankruptcy costs and the tax shields of debts yields an optimal
capital structure. The trade-off between the gains of adjusting the capital structure periodi-
cally and the costs of doing so yields an optimal maturity structure. Indeed, optimal capital
structure and optimal maturity structure are interdependent and must be determined jointly
22
and simultaneously.
The optimal maturity in our dynamic model appears to be reasonable compared to the
median maturities observed in empirical studies. The trade-off between the transaction costs
of issuing bonds and the gains of adjusting bonds dynamically may not be the only factors
determining an optimal maturity structure. Nevertheless, our model indicates that they could
be important factors. In addition, the term structure of the interest rate is an important
factor in the determination of an optimal maturity structure.
Our dynamic model indicates that the firm does not issue debt to take advantage of the
tax shield as aggressively as a static model implies. The reason is that, due to the possibility
of bankruptcy, the firm does not want to issue too much debt to risk losing the ability to
adjust its debt level in the future. This brings the leverage ratio in the dynamic model
much closer to the actual leverage ratios observed in practice than the high leverage ratios
predicted by most static models.
When the interest rate is assumed to be a constant, the level of the interest rate has
a significant impact on both the optimal coupon and optimal maturity. When the interest
rate is assumed to follow a mean-reverting stochastic process, however, both the long-run
mean as well as the current level of the interest rate process are required to price the risky
bond and determine the optimal capital structure of the firm. On the one hand, the current
interest rate level is crucial in the pricing of risky debts. On the other hand, the long-run
mean plays a key role in the determination of the tax shields and bankruptcy costs resulting
from the future debts. Therefore, a model of optimal capital structure with a constant
interest rate cannot simultaneously price risky corporate debts and determine the optimal
capital structure appropriately. A stochastic interest rate process is needed to account for
the evolution of the interest rate. While the long-run mean is shown to be important in
determining the optimal capital structure, numerical results indicate that the correlation
between the stochastic interest rate and the return of the firm’s asset has little impact.
For tractability, the default boundary VB(yt, t; P, T ) is exogenously specified. Extending
23
our model to allow for an endogenous default boundary, in the spirit of Leland (1994) and
Leland and Toft (1996), would be an important but challenging topic for future research. In
addition, Darrough and Stoughton (1986) develop a capital structure model in which moral
hazard and adverse selection problems exist and financing decisions affect investments.20
Incorporating these features into our model would be an another fruitful but difficult topic
for future research.
20Titman and Tsyplakov (2002) also allow a firm’s financing decisions to affect its investment choices.
24
A The Forward Risk-Neutral Measure
In this appendix, we use the Girsanov theorem to derive the T -forward risk-neutral measure
in a multi-dimensional setting. Without loss of generality, we assume a probability space Q
generated by two standard Wiener processes
WQt =
[WQ
1t
WQ2t
], (48)
with correlation matrix
ρ(t) =
[1 ρ(t)
ρ(t) 1
]. (49)
In the following, Q should be interpreted as the risk-neutral probability measure and rt the
riskless interest rate given by
drt = µ(r, t)dt + σ(r, t)dWQ2t . (50)
We leave other random variables generated by WQ1t and WQ
2t unspecified.
Suppose we want to compute the following expectation
h = EQ0 [e−
∫ T
0r(u)duZ(· · ·, T )], (51)
where · · · indicates that Z(· · ·, T ) may depend on the sample path in space Q from 0 to
T . Let Λ(r0, 0; T ) be the discount bond price at t = 0 with maturity T . Define
ξT =e−
∫ T
0r(u)du
Λ(r0, 0; T ). (52)
Then, we have
h = Λ(r0, 0; T )EQ[ξT Z(· · ·, T )]. (53)
It is clear that ξT is strictly positive and EQ[ξT ] = 1. Therefore it can be used as a Radon-
Nikodym derivative to define a new probability measure RT equivalent to the original measure
Q such that
ERT [1A] = EQ[ξT 1A] (54)
25
for any event A. Under the new forward risk-neutral measure RT ,
h = Λ(r0, 0; T )ERT [Z(· · ·, T )]. (55)
To find the Wiener processes under RT , define the likelihood ratio
ξt = EQt [ξT ] =
e−∫ t
0r(s)dsΛ(rt, t; T )
Λ(r0, 0; T ). (56)
Ito’s lemma implies that
d log ξt = −rdt +dΛ(rt, t; T )
Λ(rt, t; T )− 1
2
(dΛ(rt, t; T )
Λ(rt, t; T )
)2
= (57)
[−rΛ + Λt + u(r, t)Λr +
1
2σ2(r, t)Λrr
]dt/Λ +
σ(r, t)Λr
ΛdWQ
2 (t)− 1
2
(σ(r)Λr
Λ
)2
dt. (58)
The term inside the square bracket is the fundamental PDE satisfied by the discounted bond
price Λ, therefore (with another application of Ito’s lemma)
dξt = ξtσ(r, t)Λr
ΛdWQ
2 (t) = ξtβ(t)T dWQt , (59)
where T denotes transpose and
β(t) =
[0
σ(r,t)Λr
Λ
], dWQ
t =
[dWQ
1t
dWQ2t
]. (60)
Now the multi-dimensional Girsanov theorem implies that WRTt defined by21
WRTt =
[WRT
1t
WRT2t
]= WQ
t −∫ t
0ρ(s)β(s)ds (61)
are two standard Winner processes with the correlation matrix ρ(t). In differential form, we
have
dWRTt = dWQ
t − ρ(t)β(t)dt. (62)
21See, for example, Duffie (2001) for a review.
26
B Further Discussion of the Scaling Property
In this appendix, we provide a detailed discussion of the scaling property used in Subsection
2.4 to calculate the total firm value with potentially an infinite number of periods. A key
result is that the optimal solution X∗, remain the same for all the bonds issued by the firm.
Let Tn = (n − 1)T and τn be the first passage time when Vt = VB(yt, t; P, T ) in period
n. Note that from (39)-(41), the net benefit of debt (tax benefit of debt less the default cost
and the issuance cost) in the first period can be rewritten as22
NBT1(T, X0) ≡ tb(T, X0)− bc(T,X0)− tc(T, X0) = VT1 f(T, XT1), (63)
where f(T, XT1) is given by
f(T,XT1) = e−XT1−γT
[θ (λ− 1) + (θ − 1)(1− φ)
(G(T ; T, XT1) + G(T ; T, XT1)
)−
θ G(T ; T, XT1)− κλ]. (64)
Similarly, the net benefit of debt in period n is given by
NBTn = VTnf(T,XTn). (65)
Therefore, the total net benefit at T1 = 0 is given by
TNBT1(T, XT1 , XT2 , · · ·) =∞∑
n=1
EQ0
[e−
∫ Tn
0r(s)dsVTn f(T, XTn)Πn
i=11(τi > T )]
=
VT1
∞∑
n=1
e−δTnEQ0
[e−σ2
vTn/2+σvW QvTnf(T, XTn)Πn
i=11(τi > T )], (66)
where Πni=11(τi > T ) = 1(τ1 > T )1(τ2 > T ) · · · 1(τn > T ) is the product of indicator functions.
Notice that the relation VTn = VT1 exp[∫ Tn
0 (r(s)− δ − σ2v/2)ds + σvW
QvTn
]has been used to
obtain the second equality, and, as a result, the second equality no longer depends directly
on the stochastic interest rate.
In (66), XTn is the choice variable of the firm in period n and τn is the first passage time
of the process Xt given in (8), starting at XTn > 0, to reach the origin during period n.
22Note that T1 = 0. Thus VT1 = V0 and XT1 = X0.
27
To obtain (66) in closed form, consider the total net benefit at T2 = T , TNBT2 . It is easy
to see that TNBT2 is given by23
TNBT2(T, X ′T1
, X ′T2
, · · ·) = VT2
∞∑
m=1
e−δTmEQ0
[e−σ2
vTm/2+σvW QvTmf(T, X ′
Tm)Πm
i=11(τ ′i > T )], (67)
where X ′Tm
is the choice variable of the firm in period m and τ ′m is the first passage time of
the process Xt given in (8), starting at X ′Tm
> 0, to reach the origin during period m.
Note that, besides the factors VT1 and VT2 , (66) and (67) have the same functional form
of the choice variables. That is, if we let
TNBT1(T, XT1 , XT2 , · · ·) = VT1 U(T,XTn |i=1,···,∞), (68)
then we have
TNBT2(T, X ′T1
, X ′T2
, · · ·) = VT2 U(T,X ′Tn|i=1,···,∞). (69)
Therefore, if the optimal value of XT1 in (66), which is the choice variable at time zero,
is X∗, X∗ must also be the optimal value for X ′T1
in (67), which is the choice variable at
time T . Note that at time zero the firm’s objective is to maximize the total firm value
given by VT1 + TNBT1 = VT1 [1 + U(T, XTn|i=1,···,∞)]. Equivalently, the firm maximizes
U(T, XTn|i=1,···,∞), which is independent of the firm’s initial value VT1 . Likewise, the firm
maximizes U(T, X ′Tn|i=1,···,∞) at time T . Suppose that at time zero the optimal solution for
the first period is given by X∗0 . At time T , the optimal solution for the first period, which
is the second period viewed at time zero, must be the same as X∗0 or X
′∗0 = X∗
0 , because
U(T, XTn|i=1,···,∞) and U(T, X ′Tn|i=1,···,∞) have the same form. In other words, the optimal
solutions for X0 and XT are the same in the first two periods. Similarly, it can be seen that
the optimal value for XTn in all periods must be the same. We denote this solution by X∗.24
Now the optimal total net benefits at time zero and time T can be concisely expressed
as TNB0(T, X∗) and TNBT (T, X∗), respectively. It is easy to see that TNBT (T, X∗) =
23For notational purpose, we have renamed T as time zero. In other words, the first period starts at Twhen we compute the total net benefit of debts at T .
24A key point is that at times zero, T , 2T , · · ·, the firm faces an infinite horizon and the U functions takethe same form. Hence, the optimal solution for the first period viewed at different times must be the same.
28
VT /V0 TNB0(T, X∗). This relation is used next to obtain the optimal total net benefit in
closed form.
The optimal total net benefit TNB0(T, X∗) at time zero is the net benefit from the debt
issued at time zero, NB0(T, X0), which is given in (63), plus the present value of the optimal
total net benefit TNBT (T,X∗) = VT /V0 TNB0(T,X∗) at time T , conditional on no default
occurred. Therefore, we have
TNB0(T, X∗) = NB0(T, X0) + EQ0
[VT
V0
TNB0(T, X∗) 1(τ > T ) e−∫ T
0r(u)du
]. (70)
Solving for TNB0(T, X∗), we obtain
TNB0(T, X∗) =NB0(T,X∗)
1− e−δT H(T ; T,X∗), (71)
where H(T ; T, X∗) is given in (33). Note that V0 + TNB0(T, X∗) is (45) in the text and it
is the objective function of the firm.
29
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32
Table 1: Comparative Statics of the Dynamic Model with Constant Interest Rate
1 2 3 4 5 6 7 8Optimal Optimal Optimal Credit Optimal OptimalMaturity Coupon Debt Value Spread Leverage Firm Value
ϕ κ (years) (Dollars) (Dollars) (Basis Point) (Percent) (Dollars)Panel A: θ = 0.35, r0 = 0.06, σv = 0.20, φ = 0.5
0.800 0.015 3.738 2.383 38.941 11.923 32.357 120.3470.900 0.015 3.666 2.353 38.511 11.124 32.042 120.1881.000 0.015 3.599 2.321 38.030 10.308 31.687 120.0160.800 0.020 4.929 2.288 37.183 15.292 31.287 118.8450.900 0.020 4.845 2.250 36.623 14.278 30.864 118.6611.000 0.020 4.649 2.198 35.907 12.258 30.310 118.4640.800 0.025 6.108 2.207 35.760 17.262 30.388 117.6780.900 0.025 5.898 2.162 35.124 15.653 29.901 117.4691.000 0.025 5.724 2.117 34.477 14.176 29.404 117.254
Panel B: θ = 0.20, r0 = 0.06, σv = 0.20, φ = 0.50.800 0.015 5.949 1.935 31.800 8.637 29.228 108.8000.900 0.015 5.827 1.908 31.372 8.031 28.856 108.7191.000 0.015 5.717 1.878 30.922 7.438 28.464 108.6340.800 0.020 7.818 1.834 30.055 10.368 27.847 107.9300.900 0.020 7.599 1.799 29.522 9.495 27.377 107.8361.000 0.020 7.413 1.764 28.988 8.687 26.906 107.7410.800 0.025 9.799 1.772 28.965 11.944 27.006 107.2530.900 0.025 9.440 1.728 28.300 10.772 26.412 107.1461.000 0.025 9.148 1.687 27.667 9.738 25.847 107.041
Panel C: θ = 0.35, r0 = 0.06, σv = 0.15, φ = 0.50.800 0.020 6.064 2.815 46.058 11.113 37.572 122.5840.900 0.020 5.846 2.769 45.377 10.123 37.080 122.3771.000 0.020 5.657 2.719 44.635 9.152 36.541 122.151
Panel D: θ = 0.35, r0 = 0.06, σv = 0.25, φ = 0.50.800 0.020 4.167 1.876 30.335 18.279 26.205 115.7630.900 0.020 4.078 1.845 29.904 16.893 25.869 115.5971.000 0.020 4.000 1.813 29.453 15.571 25.517 115.426
(Continued on next page)
33
Table 1: (Continued)
1 2 3 4 5 6 7 8Optimal Optimal Optimal Credit Optimal OptimalMaturity Coupon Debt Value Spread Leverage Firm Value
ϕ κ (years) (Dollars) (Dollars) (Basis Point) (Percent) (Dollars)Panel E: θ = 0.35, r0 = 0.04, σv = 0.20, φ = 0.5
0.800 0.020 6.046 1.328 32.023 14.716 28.301 113.1520.900 0.020 5.901 1.305 31.562 13.511 27.933 112.9901.000 0.020 5.777 1.282 31.076 12.377 27.544 112.823
Panel F: θ = 0.35, r0 = 0.08, σv = 0.15, φ = 0.50.800 0.020 4.201 3.312 40.679 14.084 33.092 122.9280.900 0.020 4.088 3.259 40.092 12.979 32.664 122.7411.000 0.020 3.988 3.204 39.461 11.884 32.203 122.540
Panel G: θ = 0.35, r0 = 0.06, σv = 0.20, φ = 0.40.800 0.020 4.766 2.238 36.405 14.780 30.689 118.6250.900 0.020 4.655 2.205 35.925 13.751 30.329 118.4521.000 0.020 4.557 2.170 35.409 12.741 29.939 118.269
Panel H: θ = 0.35, r0 = 0.06, σv = 0.20, φ = 0.60.800 0.020 5.055 2.319 37.749 14.349 31.694 119.1060.900 0.020 4.901 2.278 37.169 12.990 31.259 118.9081.000 0.020 4.769 2.236 36.559 11.701 30.799 118.701
Column 1 represents the level and shape parameter ϕ in the default boundary. Column 2denotes the transaction cost of every dollar debt issued κ. r0 is the constant interest rate.The other parameters are given in the panels. Columns 3-8 represent the optimal maturity,optimal coupon rate, optimal debt value, credit spread, optimal leverage ratio, and optimallevered firm value, respectively.
34
Table 2: Comparative Statics of the Dynamic Model with Stochastic Interest Rate
1 2 3 4 5 6 7 8Optimal Optimal Optimal Credit Optimal OptimalMaturity Coupon Debt Value Spread Leverage Firm Value
ϕ κ (years) (Dollars) (Dollars) (Basis Point) (Percent) (Dollars)Panel A: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0
0.800 0.015 3.321 2.330 38.784 10.999 32.401 119.6990.900 0.015 3.274 2.304 38.379 10.317 32.102 119.5541.000 0.015 3.230 2.275 37.921 9.600 31.761 119.3960.800 0.020 4.211 2.191 36.521 13.321 30.956 117.9740.900 0.020 4.138 2.161 36.055 12.365 30.604 117.8121.000 0.020 4.071 2.128 35.548 11.400 30.218 117.6390.800 0.025 5.099 2.085 34.807 15.386 29.853 116.5980.900 0.025 4.993 2.050 34.279 14.130 29.444 116.4191.000 0.025 4.898 2.014 33.719 12.912 29.010 116.232
Panel B: θ = 0.20, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.00.800 0.015 5.106 1.833 30.994 7.787 28.633 108.2470.900 0.015 5.036 1.810 30.619 7.285 28.305 108.1761.000 0.015 4.971 1.786 30.218 6.779 27.953 108.1000.800 0.020 6.503 1.684 28.599 9.169 26.671 107.2320.900 0.020 6.392 1.658 28.174 8.483 26.293 107.1541.000 0.020 6.293 1.630 27.729 7.816 25.897 107.0730.800 0.025 7.924 1.572 26.805 10.404 25.185 106.4340.900 0.025 7.763 1.542 26.321 9.520 24.750 106.3491.000 0.025 7.621 1.512 25.834 8.693 24.311 106.264
Panel C: θ = 0.35, y1(0) = 0.06, σv = 0.15, φ = 0.5, ρ1 = 0.00.800 0.020 4.740 2.650 44.569 9.787 36.767 121.2210.900 0.020 4.649 2.617 44.039 9.052 36.381 121.0501.000 0.020 4.565 2.578 43.429 8.275 35.933 120.859
Panel D: θ = 0.35, y1(0) = 0.06, σv = 0.25, φ = 0.5, ρ1 = 0.00.800 0.020 3.769 1.825 30.156 16.852 26.186 115.1570.900 0.020 3.709 1.797 29.749 15.663 25.868 115.0051.000 0.020 3.656 1.768 29.320 14.500 25.530 114.847
(Continued on next page)
35
Table 2: (Continued)
1 2 3 4 5 6 7 8Optimal Optimal Optimal Credit Optimal OptimalMaturity Coupon Debt Value Spread Leverage Firm Value
ϕ κ (years) (Dollars) (Dollars) (Basis Point) (Percent) (Dollars)Panel E: θ = 0.35, y1(0) = 0.04, σv = 0.20, φ = 0.5, ρ1 = 0.0
0.800 0.020 5.813 1.650 32.592 17.179 28.247 115.3830.900 0.020 5.654 1.616 32.087 15.621 27.855 115.1911.000 0.020 5.520 1.582 31.561 14.175 27.446 114.995
Panel F: θ = 0.35, y1(0) = 0.08, σv = 0.15, φ = 0.5, ρ1 = 0.00.800 0.020 3.383 2.881 39.806 11.431 33.009 120.5930.900 0.020 3.337 2.848 39.342 10.708 32.664 120.4451.000 0.020 3.294 2.811 38.821 9.942 32.275 120.283
Panel G: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.4, ρ1 = 0.00.800 0.020 4.127 2.158 35.938 13.396 30.516 117.7680.900 0.020 4.063 2.129 35.506 12.555 30.188 117.6151.000 0.020 4.003 2.099 35.031 11.698 29.826 117.450
Panel H: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.6, ρ1 = 0.00.800 0.020 4.310 2.228 37.173 12.954 31.445 118.2130.900 0.020 4.226 2.194 36.666 11.854 31.062 118.0401.000 0.020 4.149 2.158 36.116 10.759 30.644 117.855
Panel I: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = −0.30.800 0.020 4.493 2.241 37.420 13.135 31.576 118.5070.900 0.020 4.406 2.209 36.927 12.182 31.204 118.3391.000 0.020 4.327 2.174 36.391 11.227 30.798 118.159
Panel J: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.30.800 0.020 3.990 2.145 35.694 13.391 30.379 117.4960.900 0.020 3.927 2.115 35.250 12.431 30.041 117.3381.000 0.020 3.869 2.083 34.763 11.458 29.669 117.169
(Continued on next page)
36
Table 2: (Continued)
1 2 3 4 5 6 7 8Optimal Optimal Optimal Credit Optimal OptimalMaturity Coupon Debt Value Spread Leverage Firm Value
ϕ κ (years) (Dollars) (Dollars) (Basis Point) (Percent) (Dollars)Panel K: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0, α1 = 0.01
0.800 0.020 3.503 1.733 36.223 9.394 31.747 114.0990.900 0.020 3.475 1.714 35.822 8.858 31.427 113.9831.000 0.020 3.447 1.693 35.375 8.288 31.070 113.858
Panel L: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0, α1 = 0.030.800 0.020 4.982 2.815 37.636 17.169 31.039 121.2550.900 0.020 4.819 2.755 37.015 15.572 30.581 121.0381.000 0.020 4.680 2.694 36.372 14.074 30.106 120.812
Panel M: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0, β1 = 0.166650.800 0.020 5.432 2.972 37.571 19.279 30.836 121.8400.900 0.020 5.192 2.891 36.868 17.183 30.320 121.5961.000 0.020 4.999 2.815 36.170 15.326 29.808 121.346
Panel N: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0, β1 = 0.66660.800 0.020 3.679 1.441 34.806 8.795 31.073 112.0140.900 0.020 3.648 1.426 34.430 8.285 30.766 111.9091.000 0.020 3.618 1.410 34.014 7.750 30.425 111.796
Panel O: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0, σ1 = 0.010.800 0.020 4.350 2.209 36.618 13.619 30.986 118.1770.900 0.020 4.268 2.178 36.143 12.621 30.627 118.0101.000 0.020 4.193 2.143 35.625 11.623 30.234 117.832
Panel P: θ = 0.35, y1(0) = 0.06, σv = 0.20, φ = 0.5, ρ1 = 0.0, σ1 = 0.040.800 0.020 3.799 2.131 36.201 12.388 30.877 117.2430.900 0.020 3.749 2.104 35.765 11.549 30.544 117.0951.000 0.020 3.702 2.074 35.284 10.683 30.174 116.935
Column 1 represents the level and shape parameter ϕ in the default boundary. Column 2denotes the transaction cost of every dollar debt issued κ. The initial values of the secondand third factor of the interest rate process are: y2(0) = 0 and y3(0) = 0. The correlationsbetween these two factors and the return of the firm asset value are: ρ2 = 0 and ρ3 = 0. Theparameter values of interest rate process are: α1 = 0.02, β1 = 0.3333, σ1 = 0.02; α2 = 0,β2 = 0.5, σ2 = 0.03; α3 = 0, β3 = 0.1, and σ3 = 0.01. The other parameters are given inthe panels. Columns 3-8 represent the optimal maturity, optimal coupon rate, optimal debtvalue, credit spread, optimal leverage ratio, and optimal levered firm value, respectively.
37