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:nju@rhsmith.umd.edu. Ou-Yang is with the Fuqua School of Business, Duke University, Durham, NC 27708.E-mail: huiou@duke.edu. 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

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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.

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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.

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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)

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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