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Debt Maturity and the Dynamics of Leverage∗

Thomas Dangl†

ISK Vienna and

Vienna University of Technology

Josef Zechner‡

Dept. of Finance

University of Vienna

February 2006

Abstract

This paper shows that long debt maturities destroy equityholders’ incentives to re-

duce leverage in response to poor firm performance. By contrast, a sufficiently short

debt maturity commits equityholders to implement such leverage reductions. However,

a short debt maturity also generates transactions costs associated with rolling over matur-

ing bonds. We show that this tradeoff between higher expected transactions costs against

the commitment to reduce leverage when the firm is doing poorly motivates an optimal

maturity-structure of corporate debt. Since firms with high costs of financial distress

benefit most from committing to leverage reductions, they have a stronger incentive to

issue short-term debt. The debt maturity required to commit to future leverage reductions

decreases with the volatility of the firm’s cash flows. We also find that the equityholders’

incentives to reduce debt is non-monotonic in the firm’s leverage. If the firm is pushed

to bankruptcy by a persistent series of low cash flows, then equityholders resume issuing

debt to refinance maturing bonds, even when debt maturities are short.

Keywords: debt maturity, optimal capital structure choice

JEL: G3, G32

∗A previous version of this paper was circulated under the title “Voluntary Debt Reductions”. We thank

Michael Brennan, Kent Daniel, Hayne Leland, Pierre Mella-Barral, Kristian Miltersen, participants of the semi-

nars at LondonBusiness School, Norwegian School of Economics and Business Administration, participants of

the 2004 Annual Summer UBC Finance Conference, 2004 Annual Meeting of the German Finance Association

in T¨ ubingen for their suggestions and comments.

†Institute of Strategic Capital Market Research, Coburgbastei 4/1, A-1010 Vienna, Austria, phone: +43-1-

51818-970,e-mail: thomas.dangl@iskwien.at

‡Department of Finance,University of Vienna,

Josef.Zechner@univie.ac.at,Tel:+43 - 1 - 4277 38071, Fax: +43 - 1 - 4277 38074

Br¨ unner Straße 72,A-1210 Vienna, e-mail:

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1Introduction

Significant progress has been made towards understanding firms’ dynamic financing deci-

sions. Majorcontributionstothisliteraturemodelafirm’sassetsorcashflowsasacontinuous-

timestochasticprocess and assumethat debt enjoyssomebenefit, such as a tax advantage, but

generates dead weight costs associated with excessively high leverage, such as bankruptcy

costs.1While these models have been quite successful in explaining firms’ optimal target

leverageratios and theirdecisionsto dynamically increase debt levelsin responseto increases

in their asset values or cash flows, they are much less successful in explaining leverage reduc-

tions. Many dynamic capital structure models imply that equityholders never find it optimal

to reduce dividends or issue equity to reduce debt.2In these models debt reductions only

occur following bankruptcy or if equityholders obtain partial debt forgiveness. Obviously

this implication is in contrast to empirical evidence showing that firms frequently reduce debt

even when bankruptcy has not yet occurred and no debt forgiveness has been negotiated.3

In this paper we develop a dynamic capital structure model where equityholders reduce

leverage without bankruptcy. We demonstrate how voluntary debt reductions are driven by

the firm’s debt maturity. Thus, we identify and analyze a largely unexplored aspect of debt

maturity, namely its effect on future capital structure dynamics. We specifically address

the following questions. How is debt maturity related to equityholders’ dynamic leverage

adjustments? How do firms optimally refinance expiring debt? What is the optimal debt

maturity structure given its implications for dynamic capital structure adjustments? Which

firms are most likely to issue short-term debt? We address these questions in a framework in

which firms issue a portfolio of bonds with different maturities. They are allowed to optimize

the average debt maturity, to adjust their capital structure at any point in time, and to optimize

the mix of debt and equity used to refinance maturing debt.

1See, for example, Fischer, Heinkel, and Zechner (1989), Leland (1994), Leland and Toft (1996), Goldstein,

Ju, and Leland (2001), Dangl and Zechner (2004), Strebulaev (2004).

2This is true for the already cited papers on the dynamic choice of corporate capital structure, as well as for

papersthat focusonthe renegotiationbetweendebtholdersandequityholders,such as AndersonandSundaresan

(1996), Mella-Barral and Perraudin (1997), Mella-Barral (1999), Hege and Mella-Barral (2000) or Hege and

Mella-Barral (2004).

3Surveying392 CFOs, Graham and Harvey(2001) report that 81% of firms in their sample use at least flexi-

ble targetleverageratios. Ifhighlylevered,firms tendto issue equityto maintaintheirtargetratios. Hovakimian,

Opler, and Titman (2001) report strong evidence that firms use (time varying) target leverage ratios. They find

the deviation from this target as the dominant economic factor in determining whether a firm retires debt.

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We find that equityholders never reduce leverage if the debt maturities are sufficiently

long. In thiscasereplacing maturingdebtwithequitywouldalwayslead toasufficientlylarge

wealth transfer to the remaining debtholders so that new debt is used to refinance maturing

debt. This result is in accordance with empirical evidence provided by Hovakimian, Opler,

and Titman (2001), who find that long debt maturities seem to be major impediments to debt

reductions.

By contrast, we find that sufficiently short debt maturities make it optimal for equity-

holders to replace maturing debt with equity when the firm’s profitability drops. To see this,

consider a firm which has issued short term debt and subsequently experiences a decrease in

its profitability. This firm can only issue new debt at unfavorable terms, i.e., at high credit

spreads. In addition the short maturity of the remaining debt reduces the debt overhang prob-

lem, i.e., the wealth transfer to other existing bondholders due to replacing debt with equity.

Thus, equityholders find it in their own best interest to refinance the short-term debt at least

partly with equity.

We also find that shorter debt maturities lead to more pronounced debt reductions since

this requires the firm to refund a larger fraction of its debt over any given period of time.

This implies that the firm will lower its debt level more quickly in response to a drop in its

profitability.

However, we find that the equityholders’ willingness to refund maturing debt with equity

is non-monotonic in its profitability. If the firm’s profitability drops sufficiently, then the

equityholders are no longer willing to reduce their dividends or inject new equity to refund

maturing debt, even for arbitrarily short debt maturities. Instead, they find it optimal to fully

utilize the possibilities to roll over maturing debt with new debt issues. This is so, since the

firm is already very close to bankruptcy so that a reduction of leverage largely benefits the

remaining bondholders. In such a situation equityholders are no longer willing to contribute

capital to reduce debt.

This is so since equityholders effectively own a put option on the risky bonds they have

issued and bankruptcy costs are borne by debtholders ex post. Voluntarily reducing debt

would dilute this put option without reducing bankruptcy costs for equityholders and thus,

voluntary debt reductions never occur.

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Hovakimian, Opler, and Titman (2001) present strong empirical support for this non-

monotonicity in voluntary debt reductions. Interestingly, existing literature such as Welch

(2004) has interpreted the fact that highly levered firms issue debt as evidence against the

trade off theory of capital structure choice, since it moves the leverage ratio away from the

optimal target ratio. Our analysis demonstrates that this behavior is in full accordance with a

dynamic tradeoff paradigm.

In our dynamic setting, debt maturity significantly influences the expected probability of

bankruptcy. This is so since short debt maturities lead to more rapid debt reductions when

the firm’s profitability starts to decrease. Investors take this into account when they price

the debt initially. This implies that firms’ debt capacity generally increases as they choose

shorter debt maturities. This result is also in contrast to existing literature which unanimously

predicts the opposite, i.e., that short-term debt reduces the firm’s debt capacity.

We find that total firm value is a non-monotonic function of debt maturity. As discussed

above, for long-term debt, firms never engage in debt reductions but still incur some transac-

tions costs when debt is rolled over. Thus, locally, total firm value is maximized for infinite-

maturity debt, since this saves transactions costs, but does not change the equityholders’

incentives to reduce leverage when profitability drops.4When further shortening the debt

maturity, however, firms start to engage in debt reductions when their profitability decreases,

thereby reducing the probability of financial distress. Beyond this critical debt maturity,

therefore, the firm’s debt capacity increases with shorter maturity and total firm value starts

to rise, until the transactions costs associated with refinancing maturing debt outweighs the

benefits due to faster debt reductions in financial distress. Thus, total firm value exhibits

another local maximum at the short end of debt maturity.5The location of this maximum

4Furthermore it is shown by Leland (1994), Leland and Toft (1996), Leland and Toft (1996) that the tax

advantage of debt is maximized when issuing infinite-maturity debt. Hence, when finite-maturity debt does not

imply more efficient downwards restructuring, it is clearly dominated by debt with infinite maturity.

5Alternativerationales forshort-termdebt are based on agencycosts originatingfrom the ‘asset substitution’

problem, first analyzed by (Jensen and Meckling 1976). Limited liability gives equity a put option. Hence, after

debt contracts are sold, equityholders have an incentive to increase the firm’s risk to make equity more valuable

(usually at the expense of debtholders). Short-term debt limits this incentive. This is so because short-term debt

must be renewed frequentlyand so debtholderscan easily react to a changein the riskiness of the firms assets by

making debt capital more expensive (see (Barnea, Haugen, and Senbet 1980) for a discussion of this argument).

Two papers that explicitly regard the opportunity of asset substitution are Leland (1998) and Ericsson (2000).

While Leland (1998) concludes that long-term debt maximizes firm value, Ericsson (2000) finds evidence that

potential asset substitution makes long-term debt inferior, however, in his examples optimal maturity ranges

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depends on characteristics of the firm’s cash flows such as its growth rate and its volatility, as

well as on the costs for rolling over debt and on bankruptcy costs.

If the costs of financial distress are large and the transactions costs for rolling over debt

are low, then firm value is maximized by choosing short debt maturities. If costs of financial

distress are low and costs for rolling over debt are high, then the indirect benefit for equity-

holders originating from debt reductions is negligiblecompared to the additional transactions

costs associated with short-term debt. In this case, it is better to issue debt with infinite

maturity.6

Inan influentialpaper, Leland(1994)first analyzed optimaldebtmaturityinacontinuous-

time tradeoff model. However, in Leland (1994) and Leland and Toft (1996) firms must roll

overmaturingdebtwithnewdebtissues, thereby keepingtheface valueoftheirdebtconstant.

Our model allows firms to optimally choose the mix of new debt and equity used to refund

maturing debt and can therefore be used to analyze the effect of debt maturity on downward

restructurings. Titman and Tsyplakov (2005) present a model where the firm issues debt

with finite maturity and chooses whether to refinance maturing entirely with new debt or

entirely with equity. The paper focuses on the interaction between capital structure dynamics

and investment decisions. Our model concentrates on the analysis of the maturity choice on

future capital structure adjustments. In contrast to Titman and Tsyplakov (2005) we provide

closed form solutions, solve for the optimal debt maturity and allow firms to use a mix of

debt and equity to refinance maturing debt.

Childs, Mauer, and Ott (2004) and Ju, Parrino, Poteshman, and Weisbach (2003) also

explore debt maturity. In these models firms can only change their debt levels after the en-

tire existing debt has matured. Also, at each point in time firms can only have one bond

outstanding with a given maturity. In our model firms are allowed to issue more debt or to

reduce debt at any point in time. As a result, our model is able to isolatethe pure commitment

effect of debt maturity on equityholders’ willingness to adjust debt levels downwards after

between 11 and 39 years. Fan, Titman, and Twite (2003) study debt maturity choice in 39 different countries.

Consistent with the agency-basedtheories mentioned above,they find evidence that the presence of information

intermediaries which are likely to reduce agency conflicts by facilitating information dissemination reduces the

fraction of short-term debt.

6This is in contrast to the finding of Ericsson (2000). In a model with a constant debt level, he finds that

optimal maturity increases with bankruptcy costs.

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a decrease in profitability. Furthermore, firms in our model issue a portfolio of bonds with

different maturities. Therefore, when one bond matures and is replaced with debt or equity,

this influences the value of the remaining bonds outstanding.

Also related to our paper is an interesting literature which explores the effect of renego-

tiations with debtholders. Several authors recognized that costly bankruptcy might prompt

debtholders to make concessions to equityholders, for example, in the form of debt service

holidays, or as debt write-down, see Mella-Barral and Perraudin (1997), Mella-Barral (1999),

or Anderson and Sundaresan (1996). Since we frequently observe firms reducing debt even

without first renegotiating with existing bondholders we focus on a setting in which renego-

tiations with bondholders are not possible. This may be due to coordination problems faced

by bondholders.

The remainder of the paper is structured as follows. Section 2 introduces the main build-

ing blocks of the model. The valuation of debt and equity claims and the optimal refinancing

of expiring debt are derived in Section 3. Section 4 analyzes optimal discrete capital struc-

ture adjustments and Section 5 provides numerical examples and comparative statics results.

Section 6 concludes.

2 The Model

Considera firm that has debt outstandingwithface valueBtand afixed coupon rate i. Coupon

payments are tax deductible so that there is a tax advantage of debt. See Table 1 for the no-

tation used throughout this paper. Following the modeling of finite maturity debt in Leland

(1994), Leland (1998), and Ericsson (2000), we assume that debt has no single explicit ma-

turity date but that a constant fraction m of the outstanding debt matures at any instant of

time.7Ignoring default and debt repurchase, the average maturity of a debt contract is then

1/m years.

The firm must repay maturing debt at par, and thus must make principal repayments of

mBtat each point in time. The retired portion of debt may be replaced by a new debt issue.

However, the bond indenture ensures that the new bond issue may not increase the total

7An issue of $1 is therefore a portfolio of contracts with different maturities in the range (0,∞), the relative

weight of contracts that expire in an interval [t,t +dt] is given by me−mtdt.

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Table 1: Notation

a firm’s instantaneous free cash flow after corporate tax

expected rate of change of ct

risk adjusted drift of the cash flow process

riskless rate of interest

instantaneous variance of the cash flow process

face value of debt

debt retirement rate

average debt maturity

debt roll over rate

value of equity

value of debt

total value of the firm

instantaneous coupon rate

firm’s inverse leverage ratio

personal tax rate on ordinary income

corporate tax rate

proportional bankruptcy costs

proportional transactions costs for rolling over debt

proportional transactions costs for issuing debt after recapitalization

proportional call premium

ct

µ

ˆ µ

r

c2

Bt

m

T = 1/m

δ

E

D

V

i

yt

τp

τc

g

ki

kr

λ

tσ2

initial face value of debt, so that the rate δtat which the firm may issue new debt must satisfy

0 ≤ δt≤ m. The new debt issue is associated with proportional transactions costs ki, has the

same priority as existing debt, and is amortized at the same constant rate m. This ensures

that the entire debt of the firm is homogeneous and no distinction between early issues and

later issues must be made. Although this modelling approach is a simplification it allows us

to analyze the implications of debt maturity in the realistic setting in which firms have more

than one debt issue outstanding and where the refinancing decision influences the value of

the remaining bonds.

As discussed above, covenants prohibit the firm from issuing debt that would increase

the total face value. The total amount of debt outstanding can therefore only be increased by

repurchasing all outstanding debt contracts and subsequently issuing new bonds with higher

face value. Again, proportional transactions costs krare associated with the new bond issue.

The coupon rate of the new issue is set to ensure that it can be sold at par.

The firm is not required to roll over the entire amount of maturing debt. For certain

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leverage ratios, the firm may find it optimal to replace only part of the retired debt with new

debt or it might entirely refrain from issuing new debt contracts. If the firm does not fully

replace retired debt then the face value of debt outstanding shrinks at a rate m−δtwhich in

turn may help the firm to avoid financial distress.

Debt covenants restrict the face value of debt issued in any given period to be less or

equal to the face value of the retired debt. Therefore, after a phase of debt reduction the firm

cannot return to the original debt level unless it eliminates the bond indenture by calling all

outstanding debt.

If the firm’s equityholders stop coupon payments and thereby trigger bankruptcy all con-

trol rights over the firm’s productive assets are handed over to debtholders who are then

allowed to relever the firm. Bankruptcy costs are assumed to be a certain fraction g of the

total value of the relevered firm.

We assume that the firm’s instantaneous free cash flow after corporate tax, ct, follows a

geometric Brownian motion given by

dct

ct

c0 = c(0),

= µdt+σdWt,

(1)

wheretheexpectedinstantaneousdriftand theinstantaneousvarianceofthecashflowprocess

are defined by ctµ and c2

tσ2respectively, and dWt is the increment to a standard Wiener

process.

In a given instant equityholders can decide to increase the amount of debt by a discrete

amount to a new face value B∗

t. Alternatively, equityholders may maintain the current debt

level and only decide on the rate δt∈ [0,m] at which new debt is issued to roll-over maturing

debt. If δt= m, then the firm issues new bonds with a face value exactly equal to the face

value of the bonds retired at time t.8The dynamics of the face value of debt are therefore

8Depending on the market value of debt, the proceeds may be considerably less than what is required by

repayment obligations even when m = δt. In this case the remaining amount is equity financed. Alternatively,

it may as well be the case that debt trades above par, then the net proceeds are paid out as a dividend to

equityholders.

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given by

dBt

Bt

=

B∗

Bt

−(m−δt)dt : firm replaces maturing debt at a rate δ ∈ [0,m] at time t

t

−1: debt is increased from Btto B∗

tat time t,

B0 = B(0).

(2)

We define ytas the inverse leverage ratio with respect to the unlevered firm value

yt=1

Bt

ct

r(1−τp)− ˆ µ,

(3)

where τpis the personal income tax rate and ˆ µ is the risk neutral drift rate of the free cash

flow ct.9Then the risk neutral dynamics of ytare

dyt

yt

=

Bt

B∗

(ˆ µ+(m−δt))dt+σdWt : maturing debt is replaced at a rate δtat time t

y0 = y(c0,B0) =1

B0

t

−1: debt is increased from Btto B∗

tat time t,

c0

r(1−τp)− ˆ µ.

(4)

(See Appendix A.1 for the derivationof Equation (4).) A discrete adjustment of the debt level

following a debt repurchase leads to an immediate jump in the inverse leverage ratio. When

the face value of debt is maintained at a constant level (i.e., δt= m), then the inverse leverage

ratio follows a geometric Brownian motion with the same drift rate and volatility as the cash

flow process ct. When only part of the maturing debt is rolled over (δt< m), then the drift

rate of the inverse leverage ratio is ˆ µ+(m−δt) > ˆ µ, i.e., due to the shrinking debt level, the

firm’s leverage ratio tends to fall, and thus, the inverse leverage ratio tends to rise.

Dynamic capital structure models with infinite maturity debt have utilized the fact that

equity value and debt value are homogeneous of degree one in the face value of debt, B. This

9In a world in which equity income is also taxed at the personal level, τp should be interpreted as the

differential tax on interest income over equity income. For a discussion of the effect of personal taxes on debt

dynamics, see Hennessy and Whited (2004).

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implies that all firm-relevant decisions can be made contingent on the leverage ratio y, hence

B serves as a scaling factor only. In the following, it is shown that this homogeneity can be

preserved also in the case of finite-maturity debt, even if debt reduction leads to a gradually

decreasing debt level. Therefore, all claims contingent on the cash flow ctare re-interpreted

as claims contingent on the two state variables, debt level Btand inverse leverage ratio yt.

This formulation is the key to obtain closed form solutions for the optimal roll-over schedule

δtand for the value of debt and equity of the firm.

3 Claim Valuation and Optimal Funding of Debt Repay-

ment

In this section we derive the valuation equations for the firm’s debt and equity as well as

propositions on the optimal refinancing mix for maturing debt. Consider a firm which has

debt outstanding with face value Bt. Contingent on the choice of δt, the firm’s debt level

changes at a rate −(m−δt) and, consequently, the drift rate of the inverse leverage ratio yt

is ˆ µ+(m−δt). The required instantaneous pricipal repayment is mBtdt, and the after-tax

coupon payment is i(1−τp)Btdt. Therefore the value of debt, D, must satisfy the partial

differential equation

1

2σ2y2∂2D

∂y2+(ˆ µ+(m−δt))y∂D

∂y+∂D

∂t+Bt(i(1−τp)+m) = r(1−τp)D.

(5)

Using the homogeneity with respect to the face value Bt, we can write D = Bt˜D(y). The

fact that debt amortizes at a rate m then leads to ∂D/∂t = −mBt˜D(y). Then the value of

debt per unit of face value,˜D(y), is not explicitly time dependent and satisfies the following

differential equation

1

2σ2y2∂2˜D

∂y2+(ˆ µ+(m−δt))y∂˜D

∂y+(i(1−τp)+m) = (r(1−τp)+m)˜D.

(6)

We next turn to the valuation of equity. Equityholders must provide a cash flow of mBt

to service expiring debt contracts. Furthermore, debt requires coupon payments of iBtwhich

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are tax deductible. The tax-adjusted outflow to debtholders is therefore (i(1−τc)+m)Bt.

At the same time equityholders issue new debt at a rate δtto (partly) replace maturing debt.

They receive the proceeds, i.e., the market value of the newly issued contracts, δtD(y,B),

and have to bear proportional transactions costs ki. The inflow from rolling over debt is

therefore δt(1−ki)D(y,B). Finally, equityholders receive the cash flow of the assets of the

firm, c = (r(1−τp)− ˆ µ)yBt.

Again using the homogeneity of the model with respect to the face value of debt we write

E = Bt˜E(y), where˜E is the equity value per face value of debt. The value of equity thus

satisfies the following differential equation

1

2σ2y2∂2˜E

∂y2+(ˆ µ+(m−δt))y∂˜E

−(i(1−τc)+m)+(1−ki)δtD(y)

+(r(1−τp)− ˆ µ)y =

(r(1−τp)+(m−δt))˜E.

∂y

(7)

We are now able to derive the equilibrium roll-over rate for maturing debt, δ. We hereby

assume that the firm cannot ex-ante commit to a roll-over rate. Suppose that a firm announces

a roll-over rate δ′and the market prices the bond issue accordingly. As long as the partial

derivative of equity value with respect to the roll-over rate is positive at δ′, the equityhold-

ers have an incentive to re-enter the market and issue more debt. Rational investors must

anticipate this incentive and price the new bonds, conjecturing a roll-over rate from which

equityholders have no incentive to deviate, given the price of the bonds.

Since it followsfrom the two Hamilton-Jacoby-Bellmanequations (6) and (7) that there is

no explicittimedependence, theoptimaldebt roll-overrate depends only on thecurrent lever-

age of the firm, i.e., δt= δ(y). The optimal roll-over schedule δ∗(y) is therefore determined

as the subgame-perfect MarkovianNash-equilibriumof thegamebetween equityholders(set-

tingtheroll overrate δ)and themarket (valuingequityand debt).10To derivetheequilibrium,

the following corollary will be useful.

10For a game theoretic analysis of a trading environmentin which buyers or sellers cannot commit to a single

trade, see DeMarzo and Bizer (1993). For a comprehensive discussion of differential games, see Dockner,

Jørgensen, Van Long, and Sorger (2000)

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Corollary 1. The partial derivative of equity with respect to the roll-over rate δ is given by

∂˜E

∂δ=K1−(r(1−τp)+m)K2

(r(1−τp)+(m−δ))2,

where K1and K2are given by

K1 =

1

2σ2y2∂2˜E

−(i(1−τc)+m)+(r(1−τp)− ˆ µ)y,

K2 = y∂˜E

∂y−(1−ki)˜D(y).

∂y2+(ˆ µ+m)y∂˜E

∂y

The partial derivative of debt with respect to the debt roll over rate δ is given by

∂˜ D

∂δ= −

y

r(1−τp)+m

∂˜D

∂y.

(See Appendix A.2 for the proof of Corollary 1.)

Corollary 1 implies that the sign of the partial derivative of equity with respect to the

roll-over rate depends on the value of debt per unit of face value,˜D(y). For sufficiently large

values of debt it is positive whereas it is negative for sufficiently low values. The partial

derivative is zero for a critical value˜DI. These results imply the following proposition.

Proposition 1. Equityholders are indifferent to changes in the debt roll over rate δ(y) if and

only if the value of debt per unit of face value satisfies

˜ D(y)=

1

1−ki

˜DI(y).

?

y∂˜E

∂y(y)−˜E(y)

?

=:

If and only if ˜D(y) >˜DI(y), the firm optimally rolls over debt at δ∗= m. If and only if

˜D(y) <˜DI(y) the firm optimally finances debt repayments entirely with equity, i.e., δ∗= 0.

(See Appendix A.3 for the proof of Proposition 1.)

This result is quite intuitive. Suppose the firm issues one unit of debt dB then it will

receive the proceeds of this issue (net transactions costs). In addition to the proceeds there

will be a change in equity value because the issue influences both B and y. Equityholders find

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it optimal to go ahead with this debt issue only if the sum of these effects is positive, i.e.,

0 < (1−ki)˜D(y)dB+dE

= (1−ki)˜D(y)dB+∂E

∂BdB+∂E

∂y

dy

dBdB

?

(8)

=

?

(1−ki)˜D(y)+˜E(y)−y∂˜E

∂y(y)

dB,

which is equivalent to the statement in Proposition 1.

On first inspection one may conclude from Proposition 1 that the optimal solution for

δ is characterized by a ’bang-bang’ solution, i.e., either full re-issuance of no re-issuance.

This first intuition is, however, not correct since the value of debt per unit of face value,

˜D(y) reflects the roll-over rate δ∗. In many situations it will not be optimal to fully roll-over

maturing debt, since this would imply a˜D(y) less than˜DI. At the same time it will not be

optimal to set the roll-over rate to zero, since this would imply a debt value larger than˜DI,

thus implying a positive partial derivativeof equity value with respect to the roll-over rate. In

all these cases there exists an interior equilibrium which implies that˜D =˜DI.

This situation represents a differential game between equityholders, who determine the

roll over rate δ∗and the market, who determines the value of debt and equity. For a given

value of˜D, the best response of equityholders is characterized by Proposition 1. The best

response of the capital market to a given roll-over rate δ is to price debt at the value given by

Equation(6). Therefore, theresponsecurveisastraightlinewithslope∂˜D/∂δ=−y∂˜D/∂y

Figure 1 illustrates the typical shape of the response functions δ(˜D) and˜D(δ) in the case of

1

r(1−τp)+m.

an interior equilibrium.

This interior equilibrium with 0 < δ∗< m is characterized by the following equilibrium

conditions on˜E,˜D, and δ∗.

Proposition 2. In an interior equilibrium for δ the value of equity, debt, and the roll over

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δ

˜D

˜D(δ)

˜D =˜DI

δ(˜D)

m

0

δ∗

Figure 1: The shape of the response functions δ(˜D) and˜D(δ) in the case of an interior equi-

librium. The equilibrium debt roll over rate is δ∗

rate must satisfy

˜E

=

K1

(r(1−τp)+m),

˜DI(y),

1

y∂˜D

∂y

˜D(y) =

0 < δ∗

=

?1

2σ2y2∂2˜D

∂y2+(ˆ µ+m)y∂˜D

∂y

(9)

+(i(1−τp)+m)−(r(1−τp)+m)˜D

?

< m.

Furthermore, the existence of an interior equilibrium requires

∂2˜E

∂y2

∂˜D

∂y

> 0,

> 0.

(See Appendix A.4 for the proof of Proposition 2.)

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The following Proposition gives the analytic solutions for debt and equity for all possible

roll-over rates. For δ = m and for δ = 0, analytic solutions are straightforward. However,

a closed-form solution can also be obtained for the case of an interior equilibrium since the

valuation equations for equity and debt do not explicitly depend on the equilibrium roll-over

rate, δ∗(see Proposition 2).

Proposition 3. In a region where the firm fully rolls over its debt, i.e., δ = m, the value of

equity and debt are given by

˜E(y) = E1yβm1+E2yβm2−i(1−τc)+m

r(1−τp)

i(1−τp)+m

(r(1−τp)+m)

D1yγ1

r(1−τp)− ˆ µγ1−1

+m(1−ki)

?

1

r(1−τp)

+

2σ2γ1(γ1−1)

D2yγ2

2σ2γ2(γ2−1)

+

r(1−τp)− ˆ µγ2−1

?

+y,

˜D(y) = D1yγ1+D2yγ2+i(1−τp)+m

r(1−τp)+m

In a region where the firm rolls over its debt at an interior optimum δ∗, the value of equity

and debt are given by

˜E(y) = E1yβ01+E2yβ02−i(1−τc)+m

r(1−τp)+m+y,

˜D(y) =

˜DI(y).

In a region where the firm funds repayment of retiring debt entirely with equity, i.e., where

δ = 0, the value of equity and debt are given by

˜E(y) = E1yβ01+E2yβ02−i(1−τc)+m

r(1−τp)+m+y,

˜D(y) = D1yβ01+D2yβ02+i(1−τp)+m

r(1−τp)+m.

The exponents β and γ are the characteristic roots of the homogeneous differential equations

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