Debt Maturity and the Dynamics of Leverage∗
ISK Vienna and
Vienna University of Technology
Dept. of Finance
University of Vienna
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-
‡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:
Significant progress has been made towards understanding firms’ dynamic financing deci-
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
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.
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
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
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.
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
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
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.
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
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
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.
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
T = 1/m
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
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
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
c0 = c(0),
are defined by ctµ and c2
tσ2respectively, and dWt is the increment to a standard Wiener
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
−(m−δt)dt : firm replaces maturing debt at a rate δ ∈ [0,m] at time t
−1: debt is increased from Btto B∗
tat time t,
B0 = B(0).
We define ytas the inverse leverage ratio with respect to the unlevered firm value
r(1−τp)− ˆ µ,
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
(ˆ µ+(m−δt))dt+σdWt : maturing debt is replaced at a rate δtat time t
y0 = y(c0,B0) =1
−1: debt is increased from Btto B∗
tat time t,
r(1−τp)− ˆ µ.
(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).
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.
3Claim Valuation and Optimal Funding of Debt Repay-
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
∂t+Bt(i(1−τp)+m) = r(1−τp)D.
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
∂y+(i(1−τp)+m) = (r(1−τp)+m)˜D.
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
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
+(r(1−τp)− ˆ µ)y =
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)
Corollary 1. The partial derivative of equity with respect to the roll-over rate δ is given by
where K1and K2are given by
−(i(1−τc)+m)+(r(1−τp)− ˆ µ)y,
K2 = y∂˜E
The partial derivative of debt with respect to the debt roll over rate δ is given by
(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
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
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
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
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
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
0 < δ∗
Furthermore, the existence of an interior equilibrium requires
(See Appendix A.4 for the proof of Proposition 2.)
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)− ˆ µγ1−1
r(1−τp)− ˆ µγ2−1
˜D(y) = D1yγ1+D2yγ2+i(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
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
˜D(y) = D1yβ01+D2yβ02+i(1−τp)+m
The exponents β and γ are the characteristic roots of the homogeneous differential equations
The constants E1,2and D1,2have to be determined separately for each of the regions by
proper boundary conditions (see below).
(See Appendix A.5 for the proof of Proposition 3.)
In equilibrium, the financing strategy of the firm and the corresponding valuation given
by Proposition 3 are in accordance with the optimality conditions stated by Proposition 1.
In addition to choosing the optimal roll-overrate of maturing debt, equityholders have the
possibilities to adjust the firm’s capital structure. They may declare bankruptcy, thereby ef-
fectively creating an all-equity financed firm which can be relevered optimally. Alternatively
they can repurchase all existing debt to eliminate the existing bond indenture, thereby creat-
ing the possibility to subsequently increase the total face value of debt. In the next chapter
we analyze these discrete financial restructurings.
4 Discrete Restructuring
For a given average debt maturity, equityholders will find it optimal to default on their con-
tracted obligations if the inverse leverage ratio reaches a lower threshold, y. Debtholders then
take over the unlevered assets and relever optimally. Alternatively, if the cash flow from pro-
ductive assets is high compared to the level of debt, equityholders have the incentive to call
the bonds and subsequently issue a larger amount of debt to generate higher tax shields. This
happens when y reaches an upper threshold y. Within these bounds, equityholders can either
keep the debt level constant by fully rolling over maturing debt, or reduce debt gradually by
choosinga roll-overrateless than m. So equityholdersspecify a rolloverscheduleδ(y) which
is consistent with the conditions of Propositions 1 and 2, i.e., they divide the range of feasible
inverse leverage ratios [y,y] into regions (i.e., sub-intervals) where δ = m, regions of interior
optimum (i.e., δ = δ∗from Proposition 2) and regions where there is no debt issuance, i.e.,
δ = 0. Anticipating these dynamic strategies, the initial firm owners choose a starting capital
structure, ´ y, and the average debt maturity m.
Therefore, a firm’s capital structure choice can be characterized by
(y, y, δ(y) ; ´ y,m),
with: y ≤
0 ≤ δ(y) ≤ m
where the critical restructuring thresholds y and y as well as the roll over schedule δ(y) are
set to maximize equity value. The initial capital structure ´ y and m is set to maximize firm
value. Due to homogeneity, it is always optimal to reestablish this initial capital structure and
maturity structure after each discrete capital structure adjustment.
The boundary conditions are the following.
Debt is assumed to be sold at par so that the coupon rate i is determined endogenously by
choose i such that
D(´ y,B) = B.
In the case of default, equity is worthless. If equityholders repurchase the entire debt at
y, thereby paying a call premium of λ times the face value, they receive an all-equity firm
which they immediately relever to achieve the inverse leverage ratio ´ y.11This leads to the
E(y,B) = 0,
Condition (11) is already used in (13).
As discussed above, debtholders take control over the productive assets of the firm after
11As introduced in Section 2 we differentiate between transactions costs kifor rolling over debt and transac-
tions costs krfor placing a discrete portion of debt in the case of a recapitalization of the firm.
a bankruptcy. They have to incur bankruptcy costs and transactions costs due to relevering
the firm. When debt is called by equityholders, debtholders receive the face value plus a
proportional call premium λ. This implies
D(y,B) = (1+λ)B.
The inverse leverage ratio y is a diffusion that can freely move inside the interval [y,y].
Thus, to ensure consistent expectation formation under the equivalent martingale measure,
both equity and debt must be continuous and smooth in the entire interval [y,y], independent
of the segmentation into sub-regions induced by the choice of δ(y).
The first order conditions of optimality at the upper and the lower reorganization thresh-
old follow from the ‘smooth pasting’ condition (see Dixit (1993) for a discussion of these
∂y(y,B) = 0,
´ y[E(´ y,B)+B(1−kr)].
Recognizing that the optimal values of y, ˜ y, y, and δ(y) are functions of ´ y and m, the
initial firm value V can be written as V(y,B; ´ y,m). Taking into account transaction costs the
´ y,m(V(y,B; ´ y,m)−krB),
with the first order conditions
∂m(y,B; ´ y,m) = 0,
∂y(´ y,B; ´ y,m)+∂V
∂´ y(´ y,B; ´ y,m)−1
´ y(V(´ y,B; ´ y,m)−krB) = 0.
Table 2: Base case parameters
riskless rate of interest r
personal tax rate τp
corporate tax rate τc
standard deviation σ
risk adjusted drift ˆ µ
transactions costs for rolling over debt ki
transactions costs after recapitalization kr
call premium λ
In this section we analyze how debt maturity affects equityholders’ debt roll-over rates and
firm value. The parameter values for the base-case numerical example are summarized in
Table 2. We start by exploring firms which have issued debt with relatively long maturi-
ties. Panels (a) and (b) of Figure 2 illustrate the optimal roll over rate δ/m and the partial
∂δ(y) for m = 0.08 (T = 12.5). As can be seen, the partial derivative of equity
with respect to δ is always positive, except for the small region near the upper restructuring
threshold y, where it is negative. Consequently, equityholders will not engage in voluntary
debt reductions when the firm’s cash flows decrease but instead the firm always fully rolls
over all debt by setting δ = m. Only immediately before calling the bonds to subsequently
issue more debt, i.e. in the region y ∈ [˜ y3,y] does it become optimal for equityholders to use
equity to repay maturing debt. The intuition for this latter result is straightforward. In this
leverage region, it is optimal to use retained earnings to finance principal repayments since it
would be inefficient to incur transactions costs for a new bond issue, knowing that the bond
will be called in the near future with high probability.
Panels(c) and (d)ofFigure2 showtheoptimalrolloverrate δ/m and thepartial derivative
∂δ(y) for the critical debt maturity of T = 10.584 (m = 0.0945). This is the lowest average
maturity for which there is no voluntary debt reduction, given the base case parameterization.
We see that the partial derivative of debt with respect to the roll-over rate,∂˜E
mfor m = 0.08 (T = 12.5)
∂δ(y) for m = 0.08 (T = 12.5)
mfor m = 0.0945 (T = 10.584)
∂δ(y) for m = 0.0945 (T = 10.584)
Figure 2: (a) and (b): The optimal roll over rate δ(y)/m for g = 0.4 and two different choices
of debt maturity which are large so that firms do not engage in debt reductions. (c) and (d):
The corresponding partial derivatives∂˜E
for δ. The maturity choice m = 0.0945 (T = 10.584) presents the critical value at which the
debt reductions in some range between y and ´ y.
∂δ(y) illustrate the optimality of the boundary solution
∂δ(y) touches the abscissa. A choice of m > 0.0945 (T < 10.584) would induce
zero between ´ y and y. That is, there is one point between ´ y and y at which equityholders are
indifferent between rolling over all debt and refraining from issuing debt to replace retired
debt. Thus, shortening the debt maturity from T = 12.5 to T = 10.584 considerably weakens
equityholders’ incentives to always fully roll-over maturing debt. We illustrate below that if
the average maturity is less than the critical value of T = 10.584 then there exists a region
where equityholders choose an interior roll-over rate, δ∗.
The example plotted in Figure 3 considers an even shorter debt maturity. Now m is set
to 0.15 which corresponds to an average debt maturity of T = 6.7. We find that there exists
a region [˜ y1, ˜ y2] between the initial inverse leverage ratio ´ y and the bankruptcy threshold y
where equityholders find it optimal to reduce δ below m to voluntarily reduce the debt level.
Figure 4 shows the partial derivative of equity with respect to δ. For y ∈ [˜ y1, ˜ y2], this deriva-
tive vanishes, thus, the choice how to fund debt repayments results in an interior equilibrium.
Intuitively, equityholders find the prices of new bonds too low, since they would reflect ex-
cessively high leverage and future costs of financial distress. It is in their own interest to
partly use equity to refund maturing debt, despite the fact that it implicitly benefits existing
bondholders. For all parameter values we have used, voluntary debt reduction was associated
with an interior choice of δ. I.e., we could not find a case where equityholders stopped issu-
ing debt completely and funded debt repayment exclusively with equity. However, we do not
have an analytic proof that this is a general result.
Interestingly, the firm’s willingness to use equity to repay debt is non-monotonic in the
inverse leverage ratio, y. When the firm approaches bankruptcy, i.e. for y < ˜ y1equityholders
terminate their effort to reduce debt. In this region they once again fully roll over debt and
exploit existing debtholders by re-issuing the expired debt. The intuition for this result is
the following. When pushed very close to bankruptcy, equityholders are no longer willing to
make equity investments in the firm. On the contrary, they would rather extract dividends and
keep the firm alive by issuing new debt to fund principal repayments, even if this can only be
done at unfavorable terms, i.e. at high credit spreads.
To summarize, there are four main insights that the above numerical analysis provides.
First, for sufficiently long maturities, we find that equityholders never use equity to repay
maturing debt except immediately before a discrete leverage increase. This result changes
if the average debt maturity is shortened sufficiently. In this case there exists a range of
leverage ratios strictly above the initial optimum for which equityholders find it optimal to
partly use equity to repay maturing debt. This is in accordance with the empirical findings of
Hovakimian, Opler, and Titman (2001) who report that long term debt is an impediment to
movements toward the target leverage ratio.
Second, at the initial leverage ratio ´ y the firm always holds its debt level constant and
fully rolls over maturing debt, δ = m. This follows directly from the optimality of the initial
leverage ratio. Since the initial issue of debt is associated with proportional transactions costs
kr, equityholders would not incur these costs if they would immediately find it optimal to
Figure 3: The optimal roll over rate δ/m for g = 0.4 and m = 0.15 (T = 6.7). This maturity
is sufficiently short to create an incentive for debt reductions in the range [˜ y1, ˜ y2]. There, the
roll over rate is optimally reduced to some interior value.
Figure 4: The partial derivative∂˜E
derivative vanishes, the firm is indifferent to the choice of δ. This is the requirement for an
∂δ(y) for g = 0.4 and m = 0.15 (T = 6.7). In [˜ y1, ˜ y2] the
reduce debt by repaying debt with equity.
Third, near the restructuring threshold y the firm entirely refrains from issuing debt. This
is so because approaching y is associated with the repurchase of all debt in order to reestab-
lish the optimal initial capital structure. Therefore, near this threshold, equityholders do not
find it optimal to incur issuing costs kifor contracts which will (with high probability) be
repurchased only after a short period. With ki→ 0 this region of δ = 0 vanishes.
Fourth, for g<100%, near the bankruptcy threshold y the firm fully rolls overall expiring
debt, δ = m. The only exception is g = 1, i.e., 100% bankruptcy costs. Thus, even with short
term debt outstanding, equityholders resume issuing debt if the leverage ratio is sufficiently
high. In this case the equityholders are not longer willing to invest in debt reductions to keep
their equity option alive. This latter result can also be derived analytically.
Proposition 4. If bankruptcy costs are less than 100% (i.e., g < 1), it is optimal to roll over
at the maximum rate δ = m in a neighborhood above the bankruptcy threshold y.
(See Appendix A.6 for the proof of Proposition 4.)
We find that bankruptcy costs represent the main determinant for the critical average
maturity below which equityholders find it optimal to engage in voluntary debt reductions.
The lower the bankruptcy costs the shorter the maturity required to give sufficient incentives
for voluntary debt reductions. Figure 5 plots the critical average maturity for different levels
of bankruptcy costs. While at g = 0.8 an average maturity of less than 30.2 years is sufficient
to induce voluntary debt reductions, at g = 0.1 it requires T < 2.46 years to create a region
of debt reductions.
We next consider the effect of debt maturity on firm value. To illustrate the potential
benefit of a short debt maturity more clearly, we consider our base case example for high
bankruptcy costs, i.e. for g = 0.8. Figure 6 displays the initial total-firm value divided by the
value of the unlevered assets as a function of the debt maturity, m.
The figure also displays the relative value of a reference firm (dotted line) which is as-
sumed to always fully roll-over maturing debt with new debt issues.12This reference firm
always chooses the longest possible maturity for its debt, as reported in Leland (1994) and
12Thisis modelledas in Leland(1994). Inaddition,we also allowthe firmto increaseits debtbyrepurchasing
all debt outstanding and to issue a higher amount of debt.
Figure 5: The critical value for the average maturity of debt below which voluntary debt
reduction exists as a function of bankruptcy costs.
Leland and Toft (1996). By contrast, if the firm can engage in debt reductions, the relation-
ship between total firm value and the maturity structure of debt is not longer monotonic.13
Thisis sobecausedebt withsufficientlyshortmaturityinduces moreefficient capital structure
adjustments by equityholderswhen the firm’s cash flows decrease, thereby lowering expected
As illustrated in Figure 6, the beneficial effect of shorter debt maturity on future capital
structure dynamics can outweigh the disadvantages due to higher transactions costs from
rolling over maturing debt. In the above example, overall firm value is maximized at a debt
maturity of ≈ 2 years.
The commitment effect of debt maturity also has a significant effect on the optimal initial
leverage ratio. In contrast to existing results in the finance literature we find that shorter debt
maturities lead to higher debt capacities.
13Empirical evidence for this nonmonotonicity is provided by Guedes and Opler (1996) who report that
investment grade firms seem to be indifferentbetween issuing debt at the long end of the maturity spectrum and
issuing debt at the short end of the spectrum.
Figure 6: The optimal initial total-firm value divided by the value of the unlevered assets
plotted against the retirement rate m for high bankruptcy costs. The dotted line shows the
corresponding firm value for a firm that has to keep the debt level constant and therefore rolls
over all expiring debt. The relation between the maturity structure of debt and firm value is
nonmonotonous. Firms with high bankruptcy costs prefer short-term debt.
This effect is illustrated by Figure 7, which plots the initial optimal leverage as a function
of m for the above example. Unlike firms which must roll over all maturing debt, firms
that choose the roll-over rate optimally actually increase their debt capacity as they shorten
their debt maturities. The optimal initial leverage increases from less than forty percent for
perpetual bonds to about eighty percent for an average debt maturity of approximately 1.5
5.1 Comparative Statics
In this section we explore the effect of various model parameters on firm value, optimal
debt maturity and dynamic capital structure policy. We first focus on the role of bankruptcy
costs. The key role of bankruptcy costs for the commitment to debt reductions was already
Figure 7: Optimal initial leverage ratios 1/´ y plotted over the retirement rate m. Without
allowing for downward restructuring, debt capacity decreases when moving from long to
short-term debt. For firms that explicitly consider debt reduction, debt capacity increases
once maturity is sufficiently short in order to commit to debt reductions to avoid financial
discussed above. Figure 8 plots the initial firm value as a multiple of the unlevered firm value
for different levels of bankruptcy costs. Several effects can be seen: (i) lower bankruptcy
costs require a shorter debt maturity in order to induce voluntary debt reductions, (ii) lower
bankruptcy costs imply that the local maximum of total firm value for finite debt maturities
is less pronounced, (iii) lowering bankruptcy costs moves the local maximum for finite debt
maturities towards shorter maturities, (iv) for lower bankruptcy costs it becomes relatively
more advantageous to issue console bonds.
When focusing on the local value maximum for finite maturity debt, the most surprising
effect is that higherbankruptcy costs implyhigherfirm values. Higherbankruptcy costsmake
it easier for equityholdersto credibly committo debt reductions. Theresultingdecrease in the
expected probability of bankruptcy more than offsets the effect of the increased costs given a
0.10.20.3 0.40.5 0.6
debt reduction, g = 0.8
g = 0.4
g = 0.2
g = 0.1
roll over, g = 0.8
g = 0.4
g = 0.2
g = 0.1
Figure 8: Total firm value (as a multiple of the unlevered firm value) plotted over the retire-
ment rate m for different levels of bankruptcy costs.
The costs associated with rolling over debt are another key determinant of firm value
when debt with finite maturity is issued. Figure 9 illustrates the effect on firm value for
different values of ki. When moving to lower values of kiwe observe that (i) firms with
shorter-term debt gain relatively more and (ii) the local maximum of total firm value for finite
debt maturity moves towards shorter maturities.
Figure 10 shows how changes in cash flow characteristics affect total firm value. Panel
10(a) plots total firm value as a function of debt maturity for several values of cash flow
volatility σ. Moving to higher volatility (i) results in lower firm value, (ii) requires shorter
debt maturity to induce debt reductions, (iii) moves the local maximum of total firm value
at the short end of debt maturity towards shorter maturity debt. High cash flow volatilities
imply a larger option value for equityholders and thus makes them more reluctant to default
on their debt obligations. At the point where they find it optimal to default, firm value and
therefore also bankruptcy costs are low. This makes the commitment effect of short-term
0.10.2 0.3 0.40.5 0.6
g = 0.2
Figure 9: Total firm value (as a multiple of the unlevered firm value) plotted over the retire-
ment rate m for different costs kiassociated with rolling over debt for g = 0.2.
debt relatively less advantageous and requires short debt maturities to induce voluntary debt
Panel 10(b) plots total firm value as a function of the debt maturity for several values
of the risk-adjusted cash flow growth rate ˆ µ. Moving to higher growth simply causes an
approximately parallel shift of the firm value towards higher levels.
Finally, Figure 11 plots the firm’s optimal initial leverage ratio, which we refer to as
the firm’s debt capacity, for different debt maturities and for different levels of bankruptcy
costs. It is again counterintuitive that high bankruptcy costs are associated with a higher
debt capacity. As discussed above, the higher bankruptcy costs effectively ensure that the
equityholders can commit to aggressive debt reductions when cash flows decrease. This
results in a reduced bankruptcy probability which more then offsets the higher bankruptcy
costs conditional on default.
g = 0.2
σ = 0.15%
σ = 0.20%
σ = 0.25%
(a) firm value for different values of σ
0.1 0.20.30.4 0.50.6
g = 0.2
ˆ µ = 0.1%
ˆ µ = 0.0%
ˆ µ = −0.1%
(b) firm value for different values of ˆ µ
Figure 10: Total firm value (as a multiple of the unlevered firm value) plotted over the retire-
ment rate m for different values of σ and ˆ µ.
0.10.2 0.30.4 0.50.6 0.7
g = 0.8
g = 0.4
g = 0.2
Figure 11: Optimal initial leverage ratios 1/´ y plotted over the retirement rate m. High
bankruptcy costs lead to high debt capacity if using short-term debt.
This paper has explored the effects of debt maturity on subsequent dynamic capital struc-
ture adjustment. We demonstrate that long debt maturities destroy equityholders’ incentives
to engage in future voluntary debt reductions. By contrast, short debt maturities serve as a
commitment to lower leverage in times when the firm’s profitability decreases. This posi-
tive effect of short debt maturities must be balanced against the increased transactions costs
associated with the higher frequency of rolling over maturing bonds. The resulting tradeoff
generates a new theory of optimal debt maturity.
We find, however, that the equityholders’ incentives to engage in debt reductions is non-
monotonic in its leverage. After intermediate deteriorations in the firm’s profitability, equi-
tyholders find it in their own best interest to repay maturing debt at least partly with equity.
In contrast, if the firm’s profitability drops so far that it is pushed close to bankruptcy, then
equityholders resume issuing new debt and gamble for resurrection.
Ex ante, the debt capacity of the firm increases if it uses debt with sufficiently short ma-
turity. We find that high costs of bankruptcy induce a stronger incentive to use short-term
debt since this reduces the expected probability of bankruptcy, for a given level of debt. Ad-
ditional comparative statics analyses reveal that increased cash flow risk reduces the optimal
debt maturity, whereas the growth rate of the cash flow process and the transactions costs of
rolling over debt have the opposite effect.
All our main results are in accordance with findings of existing empirical studies which
confirm that firms readjust their capital structure if they are highly levered and that firms
with a high portion of long-term debt are more reluctant to reduce debt in financial distress
compared to firms with a high portion of short-term debt. Other empirical predictions of our
theory, such as the effects of growth and firm risk on firms leverage adjustments in financial
distress remain to be tested.
A.1 Derivation of Equation 4
The inverse leverage ratio with respect to the unlevered firm value, yt, depends on two state
variables, the cash flow of the firm’s productive assets, ct, and the current face value of
debt, Bt. Thus one can write yt= y(ct,Bt). If the debt level is adjusted by repurchasing all
existing debt with face value Btand issuing new debt with face value B∗
t, the leverage ratio
immediately jumps to the new value, i.e., in this case we have
r(1−τp)− ˆ µ,
In the absence of a discrete adjustment, the inverse leverage ratio, yt, follows a diffusion
and its dynamics can be determined using a Taylor-series expansion and Itˆ o’s Lemma
Neglecting all terms that are o(dt) gives
r(1−τp)− ˆ µct(µdt+σdWt)
r(1−τp)− ˆ µ(−(m−δ)Btdt)
= yt((ˆ µ+(m−δ))dt+σdWt)
for 0 ≤ δ ≤ m.
A.2Proof of Corollary 1
From Equation (7) it follows that˜E can be written as
hence, the partial derivative of˜ E with respect to δ is given by the expression in Corollary 1.
The partial derivativeof˜D with respect to δ can be directly determined from Equation (6)
to be equal to −y∂˜D
A.3Proof of Proposition 1
Consider the expression for∂˜E
∂δfrom Corollary 1. Since δ≤m it follows that the denominator
in this expression is always strictly positive, the sign of the partial derivative equals the sign
of K1−(r(1−τp)+m)K2. From (7) it follows that
K1= (r(1−τp)+m−δ)˜E +δK2,
so we have
K1−(r(1−τp)+m)K2= (r(1−τp)+m−δ)(˜E −K2).
Since (r(1−τp)+m−δ) > 0 the sign of the partial derivative∂˜E
∂δequals the sign of
˜E −K2=˜E −(y∂˜E
Consequently, the firm is indifferent with respect to δ if and only if˜D satisfies
∂δ> 0 if and only if the value of˜D exceeds the value of the right-hand-side expression and
it is optimal to choose δ = m.∂˜E
∂δ< 0 if and only if the value of˜D is lower than the value of
the right-hand-side expression and it is optimal to choose δ = 0.
A.4Proof of Proposition 2
In a region of internal optimum for the roll-over rate 0 < δ∗< m we require˜D = 1/(1−
∂y−˜E]. From Proposition 1 we know that under this condition we have∂˜E
the value of equity determined by valuation equation (7) is independent of the particular
choice of δ. For simplicity, we substitute δ = 0 into (7) to receive the expression for˜E stated
∂δ= 0, thus,
in Proposition 2.
The equilibrium roll-over rate δ∗is then determined by solving Equation (6) for δ. Since
equityholders are indifferent with respect to the choice of δ the particular choice δ∗does not
change the valuation of equity.
The local response function of the value of debt to a re-issuance rate δ has the slope
∂˜ D/∂δ = −y ∂˜ D/∂y
the equilibrium is stable only if˜D(δ) is downward sloping, i.e., ∂˜D/∂δ < 0 which requires
∂˜ D/∂y > 0. The latter condition simply requires that the value of debt per unit of face value
r(1−τp)+m, (see equation (6)). From Figure 1 we can conclude that
increases as the inverse leverage ratio increases, i.e. leverage decreases.
In the case of an internal equilibrium we have ˜D(y) =˜DI(y), hence
(∂2˜E/∂y2) > 0. which concludes the proof.
A.5 Proof of Proposition 3
Inregionswhereδ=morδ=0, thevaluefunctionfor˜Dand˜E arethegeneralsolutionsofthe
second-order ordinary differential equations (6) and (7) which can be proved by substituting
the solution into the equation. In a region of an interior equilibrium 0 < δ∗< m we know
from Proposition 2 that˜D =˜DI. The value of equity must be the solution of the Hamilton-
Jacobi-Bellman equation (7) with δ∗from Proposition 2 substituted for δ. However, since we
know that in an internal equilibrium the value of equity is invariant with respect to the choice
of δ we solve (7) for δ = 0 and argue that this solution must hold for every 0 ≤ δ ≤ m, and in
particular for δ= δ∗. Substitution of this solution together with the equilibrium conditions of
Proposition 2 into (7) constitutes an alternative proof.
A.6Proof of Proposition 4
Suppose g < 1, then it follows from boundary condition (14) that˜D(y) > 0. However, from
boundary condition (12) and optimality condition (16) it follows that
y→y˜E(y) = lim
Therefore, in a neighborhood of y it is true that
According to Proposition 1 this implies that δ = m is the optimal strategy.
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