Debt Maturity and the Dynamics of Leverage

This paper shows that short debt maturities commit equityholders to leverage reductions when refinancing expiring debt in low-profitability states. However, shorter maturities lead to higher transactions costs since larger amounts of expiring debt need to be refinanced. We show that this tradeoff between higher expected transactions costs against the commitment to reduce leverage in low-profitability states, motivates an optimal maturity structure of corporate debt. Since firms with high costs of financial distress and risky cash flows benefit most from committing to leverage reductions, they have a stronger motive to issue short-term debt. Empirical evidence supports the model predictions.


Introduction
Significant progress has been made towards understanding firms' dynamic financing decisions. Major contributions to this literature model a firm's assets or cash flows as a stochastic process and assume that debt generates some benefit, such as a tax advantage, but also generates dead weight costs associated with excessively high leverage, such as bankruptcy costs. 1 While these models have been successful in explaining firms' optimal target leverage ratios and their decisions to dynamically increase debt levels in response to increases in their asset values or cash flows, they have been much less successful in explaining leverage reductions. In fact, these models generally imply that equityholders never find it optimal to use retained earnings or to issue equity to reduce leverage. This is due to a version of Myers (1977) underinvestment problem: Equityholders "underinvest"in leverage reductions and do not inject capital or reduce dividends to repurchase debt, since this would transfer wealth to the remaining bondholders. As shown by Admati, DeMarzo, Hellwig, and Pfleiderer (2018), equityholders not only lack any incentive to actively repurchase outstanding debt but frequently have incentives to increase debt even if this reduces total firm value. Thus, in these models debt reductions only occur following bankruptcy. 2 This implication is in contrast to empirical evidence that debt reductions frequently occur in the absence of bankruptcy or negotiated debt forgiveness. 3 In this paper we 1 See, for example, Fischer, Heinkel, and Zechner (1989), Leland (1994a), Leland and Toft (1996), Goldstein, Ju, and Leland (2001), Dangl and Zechner (2004), Strebulaev (2007). 2 Leland and Hackbarth (2019) extend the analysis in Admati, DeMarzo, Hellwig, and Pfleiderer (2018) and analyze multiple rounds of debt issues when existing debt is senior to new debt. They provide results how maturity affects incentives for subsequent debt issues. Other models consider debt renegotiations and derive partial debt forgiveness outside of bankruptcy (See, e.g. Anderson and Sundaresan (1996), Mella-Barral and Perraudin (1997), Mella-Barral (1999), Christensen, Flor, Lando, and Miltersen (2014)). Mao and Tserlukevich (2015) present a model where non-coordinated debt holders may accept repurchase offers below the market price when firms pay with existing safe assets or cash. Lehar (2018) considers multilateral bargaining and explicitly regards renegotiation breakdowns and subsequent inefficient liquidation. In contrast to these papers we focus on situations where coordination problems among bondholders prevent renegotiation solutions.
3 Surveying 392 CFOs, Graham and Harvey (2001) report that 81% of firms in their sample use target leverage ratios. If highly levered, firms tend to issue equity to maintain their target ratios. Hovakimian, Opler, and Titman (2001) find 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. DeAngelo, Gonçalves, and Stulz (2018) report that firms deleverage regularly, but that this deleveraging is typically small when compared to retained earnings or new equity issues. develop a dynamic capital structure model where leverage reductions not only occur as a consequence of bankruptcy or renegotiations with debtholders and show that such voluntary leverage reductions are driven by firms' finite debt maturities. The key insight is that equityholders of a firm with perpetual debt do not find it optimal to repurchase debt to reduce leverage, due to a version of the Myers (1977) underinvestment problem. However, a firm that must repay some of its debt due to its finite maturity may not want to issue new debt with the same face value, when its profitability is low.
Essentially, a firm with a higher fraction of maturing debt (shorter maturity), has a greater flexibility to manage its leverage in relatively bad states. 4 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 and which firms are most likely to issue short-term debt? We analyze these questions in a framework where equityholders are allowed to optimize the mix of debt and equity used to refinance maturing debt, but they cannot commit ex-ante to a particular refinancing mix. Firms can also increase the face value of debt at any point in time, but in order to do so, they first need to repurchase the existing debt. This can be interpreted as eliminating debt covenants, which prevent firms from diluting existing debtholders by issuing more debt.
We find that firms' equityholders may not wish to rollover maturing debt by issuing a new bond with the same face value. Instead, it may be optimal for them to issue a new bond with lower face value and at least partly refinance expiring debt with equity. This happens after a deterioration in the firm's profitability and debt maturity is sufficiently short. In this case new debt can only be issued at a high credit spread, since the price of the new bonds reflects the increased default probability and the resulting increase in expected costs of financial distress. Equityholders and may therefore wish to only partially rollover maturing debt.
If, by contrast, debt maturity is sufficiently long, then replacing maturing debt with equity always leads to a significant wealth transfer to the remaining bonds outstanding, since debt with a longer maturity is subject to more credit risk. This makes the rollover decision subject to a more severe debt overhang problem and makes the use of equity to refinance maturing debt suboptimal for equityholders. We find that, for sufficiently long debt maturities, firms always prefer to rollover debt at the maximum rate, i.e., to issue a new bond with a face value that corresponds to the face value of the maturing bonds. This result is in accordance with empirical evidence provided by Hovakimian, Opler, and Titman (2001) and Jungherr and Schott (2020b), who find that long debt maturities are major impediments to debt reductions.
We show that shorter debt maturities lead to more pronounced debt reductions since they require the firm to refund a larger fraction of its debt during any given period of time. Thus, ceteris paribus, the shorter the maturity, the faster the debt face value declines in response to deteriorating firm cash flows.
In accordance with Admati, DeMarzo, Hellwig, and Pfleiderer (2018) and DeMarzo and He (2020), firms never wish to actively repurchase non-expiring debt. Instead, debt reduction is determined by the ex-post decision how to repay maturing debt. Shortterm debt can therefore be interpreted as an ex-ante commitment to engage in debt reductions when the firm's profitability deteriorates.
Equityholders' incentives to refund maturing debt with equity are non-monotonic in the firm's profitability and thus in firm value. For values around the initial cash flow level it is optimal to fully rollover maturing debt by issuing new bonds with the same face value. If the firm's profitability drops sufficiently, then equityholders reduce the rollover rate, as explained above. However, if the firm's cash flows continue to deteriorate and the firm is pushed towards the default boundary, equityholders eventually find it optimal again to choose the maximum rollover rate. Since the firm is close to bankruptcy a reduction in leverage largely benefits the remaining bondholders, even if the maturity of the remaining debt is short. Thus, the resulting debt overhang problem implies that equityholders are no longer willing to contribute capital to reduce debt.
Our analysis is closely related to DeMarzo and He (2020) who analyze capital structure dynamics in the absence of any commitment or debt covenants. They show that in such a setting equityholders' incentives to dilute existing bondholders lead to rollover and debt issuing decisions that make the net present value of all future debt issues exactly zero. As a result relatively simple valuation expressions and a rich set of results are obtained. The focus of our paper is very different. Whereas in DeMarzo and He (2020) debt maturity is irrelevant and the value of the optimally levered firm equals that of a firm that never issues any debt, we focus precisely on the optimal maturity choice and the tradeoffs that motivate it. In doing so we allow for two realistic features, namely transactions costs associated with debt rollover and a covenant that protects existing bondholders from being diluted via leverage increasing new pari-passu issues. This generates an interior optimal debt maturity and also results in a positive net-benefit of leverage. Hovakimian, Opler, and Titman (2001) present strong empirical support for this non-monotonicity in voluntary debt reductions. Our own empirical results, presented in this paper, confirm these findings and also provide novel support for our theoretical results. 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 can be in full accordance with a dynamic tradeoff paradigm once new debt issues and rollover decisions are considered.
In our setting, debt maturity significantly influences the expected probability of bankruptcy since short 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 in contrast to the earlier literature (e.g. Leland (1994b), Leland and Toft (1996), and Leland (1998)) which predicts that short-term debt leads to early and inefficient default and therefore reduces debt capacity, as measured by the firm's initial target leverage. 5 5 This result can be understood as follows. In bad states of the world, issuing a bond with the same face value to refinance expiring debt leads to a funding gap, since the new bond must be sold below face value. This funding gap increases with the fraction of debt that must be rolled over. As a result, firms with short maturities default sooner i.e., at more profitable states, since equityholders would have to cover this higher funding gap. In contrast to these papers, we do not force firms to always Our analysis therefore generates a novel theory of optimal debt maturity where, for reasonable parameter values, total firm value is maximized at an interior debt maturity.
Firms hereby trade off an increased flexibility in future leverage reductions induced by short-term debt against the additional transactions costs incurred when refinancing expiring debt.
We show that firm value has two local maxima when plotted against its debt maturity. One local maximum is obtained when debt maturity goes to infinity. This is the case since firms with debt maturities beyond a critical threshold never engage in leverage reductions when rolling over debt. Increasing maturity beyond this critical threshold therefore no longer has any effect on leverage dynamics, but leads to a reduction in transactions costs since a smaller fraction of debt must be refinanced at any given period of time. Therefore, total firm value is locally maximized for inifinite-maturity debt. However, when shortening debt maturity below the critical threshold, firms start to engage in debt reductions when their profitability decreases, thereby reducing the probability of financial distress. Over this maturity range, shortening maturity increases debt capacity and total firm value starts to rise, until the increase in transactions costs associated with refinancing maturing debt outweigh the increased benefits due to faster debt reductions along unfavorable cash flow paths. Thus, total firm value exhibits another local maximum at an interior value of debt maturity. The exact location of this maximum depends on the parameters of the firm's cash flow process, such as its growth rate and its volatility, as well as on the transactions costs associated with rolling over debt and the magnitude of bankruptcy costs. For empirically reasonable model parameterizations we find that firm value is indeed maximized for interior debt maturities.
Infinite-maturity debt maximizes firm value globally only if the costs of financial distress and/or the tax advantage of debt are very low and/or transactions costs associated with rolling over debt are very high. In this case the benefit from increasing debt capacity and reducing the bankruptcy probability by committing to future leverage reductions via short-term debt is too low compared to the additional transactions.
Existing evidence as well as our own empirical results accord well with our theokeep the face value of debt constant and thus, we capture the effect of short debt maturity on future leverage reductions. This reverses the relation between debt capacity and debt maturity. retical predictions. When we double-sort firms by market leverage and the fraction of short-term debt into quintiles, we find that more highly levered firms subsequently delever more when a larger fraction of their debt is short-term. We also find evidence that the incentive to reduce leverage is non-monotonic, as predicted by the model. Furthermore, our regression results imply that firms with high cash flow volatilities and low bankruptcy costs prefer debt with shorter maturities, in line with our model results.
Debt maturity was first analyzed in a dynamic tradeoff setting by Leland (1994b), Leland and Toft (1996), and Leland (1998). Titman and Tsyplakov (2007) extend this literature by endogenizing investment decisions. These contributions develop important modelling strategies, allowing the analysis of debt maturity in a tractable continuoustime framework. They also provide insights on the interplay between leverage and debt maturity. However, they cannot explain interior optimal debt maturities since in all these models it would be optimal to issue perpetual debt. We extend the above literature in one crucial respect: we allow firms to choose how they refinance expiring debt whereas firms in the papers discussed above must always issue new bonds with a face value that equals the face value of the expiring debt. Thus, the total face value of debt remains constant. This paper differs from this literature, since it focusses on the role debt maturity to mitigate conflicts of interest between debtholders and equityholders that arise in the context of subsequent capital structure adjustments and find that this may generate an interior optimal debt maturity. 6 The growing interest in the interaction between debt maturity and leverage dynamics by corporate theorists has recently generated several papers which are related to our work. Benzoni, Garlappi, Goldstein, Hugonnier, and Ying (2020) focus on the main result of DeMarzo and He (2020), namely that in the absence of any commitment or bond covenants, agency problems imply that the present value of future tax benefits of 6 Childs, Mauer, and Ott (2005) and Ju, Parrino, Poteshman, and Weisbach (2005) also explore debt maturity. However, in these models firms only have a singly bond outstanding and they can only change their debt levels after the entire existing debt has matured. In our model firms are allowed to change the debt level at any point in time. As a result, we are able to isolate the commitment effect of debt maturity on equityholders' willingness to adjust debt levels downwards after a decrease in profitability. Furthermore, firms in our model have many bonds with different maturities outstanding, as is frequently the case in practice. At any point in time firms retire only a fraction of all outstanding bonds. Therefore, when some bonds mature and are refinanced with new debt or via equity, this influences the value of the remaining bonds outstanding. debt are perfectly offset by the present value of expected future bankruptcy costs. They show that this result may not be robust once one allows for fixed costs of debt issuance.
Such costs lead to a more conservative debt strategy, which in fact re-establishes a netbenefit of debt, even without commitment. Benzoni, Garlappi, Goldstein, Hugonnier, and Ying (2020) conclude that this result holds even in the limit, as fixed issuance costs go to zero. Our model allows for debt covenants, proportional costs of debt rollover, and fixed costs of discrete debt restructurings. This setting motivates a net-benefit of leverage and generates insights about the tradeoffs between debt maturity, rollover costs and the resulting leverage dynamics and about the way they are affected by firm characteristics.
Geelen (2016) studies a firm that issues non-callable debt with a single finite maturity. After having repaid the maturing debt, firms implement a new, total firm-value maximizing capital structure. This setting also captures some of the benefits of shortterm debt that we identify, since firms issue a lower amount of debt after having repaid outstanding debt in a bad state of the world. As in our model, the tradeoff between the flexibility of finite maturity debt, which reduces bankruptcy costs, on the one hand, and issuance costs on the other hand motivates an optimal debt maturity. However, in Geelen (2016) leverage adjustments cannot take place prior to the lumpy debt maturity and the model cannot account for the agency conflicts that lead to, what we refer to as underinvestment in leverage reduction which arises in the presence of multiple debt maturity issues. As we discuss in the conclusion, extending our setting, where maturity structure is perfectly granular, to allow some lumpiness of maturities, while still allowing for heterogeneous debt maturities, would be an interesting avenue for future research.
In a somewhat different vein, Chen, Xu, and Yang (2020) study debt maturity over the business cycle and model optimal maturity choice as a tradeoff between higher liquidity discounts associated with long-term debt and higher rollover costs associated with short-term debt. As in Geelen (2016), debt maturity is lumpy, i.e., perfectly correlated across all units of outstanding debt. The authors are able to link maturity choice to firms' systematic risk and to the business cycle. Our model does not generate predictions for the interaction between systematic risk and maturity choice, but is able to show that the disadvantage of long-term debt arises endogenously from agency conflicts.
Our paper is also related to debt maturity theories which are driven by informational asymmetries. In this literature short debt maturities signal positive inside information, as demonstrated by the seminal work by Diamond (1991Diamond ( , 1993 and Flannery (1986Flannery ( , 1994. Other authors, such as Calomiris and Kahn (1991) and Diamond and Rajan (2001) have emphasized the disciplinary role of short term debt. Debt maturity has also been linked to the debt overhang or underinvestment problem. While the original work by Myers (1977) concludes that short-term debt mitigates these problems, Diamond and He (2014) show that maturing short-term debt can lead to more severe debt overhang than non-maturing long-term debt. Different from these papers, we analyze a debt overhang problem that is not related to the asset side of the corporate balance sheet, but instead to the liability side. Long-term debt in our context leads equityholders to underinvest in leverage reductions. 7 Gomes, Jermann, and Schmid (2016) emphasize the role of long-term nominal corporate debt in the presence of unexpectedly low inflation and the resulting debt overhang problem in a general equilibrium model. They identify corporate long-term debt as an important channel that transmits inflationary shocks into the real economy.
There is also an interesting related literature on the interaction between debt maturity, rollover risk, and capital structure. Examples are Xiong (2012a,b), He andMilbradt (2014), Cheng and Milbradt (2012) and Chen, Cui, He, and Milbradt (2017). In a similar vein, He and Xiong (2012a) and Acharya, Gale, and Yorulmazer (2011) analyze debt maturity when short-term debt can lead to early and inefficient asset liquidation.
Some papers analyze debt maturity adjustments over time, such as Brunnermeier and Oehmke (2013), and He and Milbradt (2016), but in these papers the initial maturity is exogenous. Finally, our paper is also related to an emerging literature that analyzes rollover risk and the volatility of credit spreads and the optimal dispersion 7 Jungherr and Schott (2020a) analyze the effects of Myers' underinvestment problem and the incentive to dilute long-term debt via additional pari-passu issues on maturity in a dynamic framework. of debt maturities (see Choi, Hackbarth, and Zechner (2018) and Chaderina (2018)).
While credit spreads at future rollover dates are also stochastic in our model and therefore affect optimal maturity choices, we do not explicitly model the dispersion of debt maturities. 8 None of the contributions discussed above shares the focus of our paper, namely the effect of debt maturity on equityholders' future incentives to delever.
The remainder of the paper is structured as follows. Section 2 introduces the main building blocks of the model. The valuation of debt and equity claims and the optimal capital structure strategy is discussed in Section 3. Section 4 describes how the model is calibrated and Section 5 provides numerical analyses. Section 6 provides empirical evidence for the model predictions and Section 7 concludes.

The Model
We consider a firm with unlevered instantaneous free cash flow after corporate tax, c t , following a geometric Brownian motion where dW t is the increment to a standard Wiener process. Table 1 summarizes the notation used throughout this paper.
Let B t denote the firm's face value of debt outstanding at time t, with fixed coupon rate i. Coupon payments are deductible from the corporate tax base. In the spirit of Leland (1994b), Leland (1998), and Ericsson (2000), a constant fraction m of the outstanding debt matures at any instant of time. Ignoring default and debt repurchase, the average maturity of a debt contract is then 1/m years.
Retired debt may be replaced with a new debt issue, but the face value of the new bond may not exceed the face value of the retired debt, m B t . Thus, the bond indenture ensures that the rate δ t at which the firm issues new debt satisfies δ t ≤ m. The new Table 1: Notation a firm's instantaneous free cash flow after corporate tax c t expected rate of change of c t µ risk adjusted drift of the cash flow processμ riskless rate of interest r instantaneous variance of the cash flow process c 2 t σ 2 face value of debt B t debt retirement rate m average debt maturity T = 1/m debt rollover rate δ value of equity E value of debt D total value of the firm V instantaneous coupon rate i firm's inverse leverage ratio y t personal tax rate on ordinary income τ p corporate tax rate τ c proportional bankruptcy costs g proportional transactions costs for rolling over debt k i proportional transactions costs for issuing debt after recapitalization k r proportional call premium λ debt issue generates proportional transactions costs, k i , has the same priority as existing debt, and is amortized at the same constant rate m.
If the firm wishes to increase its face value of debt, it must first remove the existing debt covenants by repurchasing all outstanding debt. The subsequent new bond issue is also associated with proportional transactions costs, denoted by k r . The coupon rate of the new issue is set so that the bond can be sold at par.
The important difference to existing models with finite average maturity is that the firm is not forced to rollover the entire amount of maturing debt. While firms in Leland (1994b), Leland (1998), and Ericsson (2000) must always set the rollover rate δ t = m, we allow the firm to choose δ optimally. In some states of the world, the firm may find it optimal to replace only part of the retired debt with new debt or it might entirely abstain from issuing new debt contracts. If the firm does not fully replace retired debt then the face value of debt outstanding is reduced at a rate m − δ t which in turn may help the firm to avoid future financial distress.
If the firm's equityholders stop coupon and pricipal repayments and thereby trigger bankruptcy, all control rights over the firm's productive assets are transferred to debtholders who will then optimally relever the firm. As in Fischer, Heinkel, and Zechner (1989), this transfer of control rights is associated with bankruptcy costs, assumed to be a fraction g of the outstanding face value of the firm's debt. 9 Our model captures two central features of many countries' tax systems. First, we assume that interest payments are deductible from the corporate tax base while dividends are not. 10 Second, we allow for interest income to be taxed more heavily at the personal level than equity income. To capture this feature in a parsimonious way, we assume that equity income is not taxed at the personal level, whereas debt income is taxed at rate τ p . Therefore the appropriate discount rate for expected after-corporatetax income for equityholders under the risk-neutral probability measure is given by r(1 − τ p ), see Section 3. For a discussion of the calibration of the tax parameters and how they relate to the current US tax code we refer to Section 4.
Since the values of equity and debt will be shown to be lineare homogeneous in the face value of debt, we can redefine the state variable as the inverse leverage ratio with respect to the unlevered firm value, y t : where τ p is the personal income tax rate on debt andμ is the risk neutral drift rate of the cash flow c t .
Since firms can at any point in time either increase debt by a discrete amount from B t to B * t , or rollover maturing debt at rate δ t ≤ m, the dynamics of the face value of debt are given by −(m − δ t ) dt : firm replaces maturing debt at a rate δ t ∈ [0, m] at time t As a result, the risk neutral dynamics of y t are (μ + (m − δ t )) dt + σ dW t : maturing debt is replaced at a rate δ t at time t (For a derivation of the dynamics see Appendix A.1.) Thus, a discrete adjustment of the debt level following a debt repurchase leads to an immediate jump in the inverse leverage ratio. Alternatively, if the face value of debt is maintained at a constant level (i.e., δ t = m), then the inverse leverage ratio follows a geometric Brownian motion with the same drift rate and volatility as the original cash flow process for c t . When only part of the maturing debt is rolled over (δ t < m), then the drift rate of the inverse leverage ratio isμ + (m − δ t ) >μ, i.e., due to the shrinking debt level, the firm's leverage ratio tends to fall, and thus, the inverse leverage ratio, y, tends to rise.

Strategies
In this section we derive the valuation equations for the firm's debt and equity, given the state variables B and y, as well as propositions for the optimal refinancing mix for maturing debt. Consider first the value of equity, E t . Equityholders have a claim on the firm's after corporate tax cash flow, c t , reduced by the required payments to debtholders, i.e., after tax coupon payments, (1 − τ c )iB t plus principal repayment, mB t . The cash flow to equityholders also includes the proceeds from (partially) rolling over maturing where k i represents the proportional transactions costs associated with the new debt issue, δ t is the rate at which debt is rolled over and D t denotes its market value. When the resulting dividends to equityholders become negative, they can be interpreted as seasoned equity issues. 11 Equityholders default when it is optimal for them by stopping coupon and principal payments. At the resulting random default time, T D , the equity claim becomes worthless.
When the firm decides to increase the face value of debt at time T K , it must repurchase its outstanding debt at a price (1 + λ)B T K . Following such a call, equityholders hold a claim to the value of the firm, V K , endogenously determined later, which anticipates optimal relevering. For a given contingent strategy on rollover, recapitalization, and bankruptcy, the value of the firm's equity is coupons and amortization where 1 {.} is the indicator function, which equals 1 if the expression in braces is true and 0 otherwise.
Next we derive the valuation equation for the debt claim. Debtholders receive an after-tax coupon flow of i(1 − τ p )B t and a flow of principal repayment mB t . Depending on the firm's choice of the rollover rate δ t , debtholders buy the new issues at market 11 We do not consider equity issuance costs since this would considerably complicate the analysis but would not qualitatively alter the main drivers of our results. Short-term debt would still reduce agency conflicts in bad states, but the net-benefits are likely to be smaller. This is consistent with Chen, Xu, and Yang (2020), who model bondholders' liquidity preferences over the business cycle and find that in their model higher equity issuance costs lead to less short-term debt and higher default probabilities.
value D t , resulting in a cash flow of −δ t D t as well as a change of the firm's debt level at a rate −(m − δ t ). Debtholders receive this cash flow until either the firm defaults at random time T D or the entire debt is called at random time T K . At default, debtholders receive the value D D , equal to the value of the firm's relevered productive assets net of bankruptcy costs, which will be determined later. When debt is called, debtholders Anticipating the firm's future decisions regarding rollover, recapitalization, and bankruptcy, debtholders price the firm's debt as the expected present value of the total after tax cash flow under the risk neutral measure

Equityholders' Optimal Capital Structure Strategy
In this subsection we derive a Markov perfect Nash equilibrium capital structure strategy where investors price debt based on rational beliefs about the firm's debt rollover, default and recapitalization decisions and, given these prices, firms have no incentive to deviate from the conjectured capital structure strategy. There is no pre-commitment except that firms must repurchase the existing debt (i.e., eliminate a debt covenant) before they can issue new debt with a higher face value.
Anticipating that debt and equity values are homogenous of degree one in the face value of debt, B, we derive time-invariant dynamic equilibrium capital structure strategies which only depend on the firm's inverse leverage ratio, y. Specifically, equityholders' capital structure strategy, S, consists of the initial leverage,ý and debt maturity, m, a time-invariant rollover schedule δ(y), which determines the refinancing of expiring debt as a function of leverage as well as two time-invariant leverage thresholds, y and y.
As mentioned above, the specific choices of δ(y), y and y must be time-consistent (no pre-commitment). S can therefore be written as with 0 ≤ y ≤ý ≤ y, Default is triggered at random time T D at which y first hits the lower threshold y. In this case debtholders become the owners of the firm's productive assets, which they can re-lever optimally. If y first hits the upper threshold, y, at random time T K , then all debt is repurchased and the firm is subsequently re-levered optimally. 12 Thus, both at the default boundary and at the recapitalization boundary the firm becomes unlevered and its owners will therefore find it optimal to choose the initial leverageý and maturity m.
Next we present the values of equity and debt for a given strategy of type (7).
We hereby use the fact that the value of equity and debt are functions of the state variables B and y, and that valuations are homogeneous in B. Therefore, we write E(B, y) = BẼ(y) and D(B, y) = BD(y).
Proposition 1. For a given capital structure strategy (7), the value of equity and debt per unit of face value of debt,Ẽ(y) andD(y), must satisfy the following valuation 12 Equityholders' recapitalization and default strategies can be interpreted as indicator functions I y≥y and I y≤y , which map the state variable y into zero or one.

equations (Hamilton-Jacobi-Bellman)
Before determining the optimal equilibrium rollover rate δ, we discuss the system of equations stated in Proposition 1. Valuation equations for equity and debt, (8) and (11), are standard HJB equations, with the exception that the rollover rate δ(y) determines the dynamics of the state y cash flows to investors. They have analytically tractable solutions for all possible ranges of rollover rates, i.e., 0 ≤ δ(y) ≤ m.
Equation (9) reflects absolute priority which makes equity worthless in default, occurring at y. At the upper restructuring threshold y, all debt is repurchased at a premium over face value equal to (1 + λ) and the firm recapitalizes to the initial inverse leverage ratio ofý. The value of equity at y is therefore endogenously determined by the value of the total firm minus costs associated with the recapitalization, stated in Equation (10). Boundary condition (12) defines the value of debt in default as the value of the newly relevered firm (at inverse leverage ratioý) minus bankruptcy costs, g. If this does not result in a positive value, the firm is liquidated and debtholders get zero.
Finally, condition (13) states that at the upper restructuring threshold, debtholders get paid the face value plus a call premium λ.
In the setup explained above, the valuation of corporate debt depends on the capital structure strategy, conjectured by bond investors. At the same time, the firm's (i.e., equityholders') capital structure decisions depend on the valuation of the debt claims. This is so since equity value and debt value are interrelated, as can be seen from Equations (5) and (6). In particular, equity value in Equation (5)  Consequently, the optimal capital structure strategy and securities valuation must satisfy equilibrium conditions. I.e., investors price securities based on a conjectured capital structure strategy and firms must have no incentive to deviate from the conjectured strategy if securities are priced in this way. Employing the theory of stochastic control we therefore model a dynamic game and derive a Markov perfect Nash equilibrium in which debtholders price debt based on rational beliefs about equityholders' capital structure strategy and equityholders correctly anticipate the pricing of debt claims when making capital structure decisions. 14 The optimal equilibrium strategy is formally derived in Appendix A.3 and A.4. The key feature of the equilibrium strategy is that the optimal rollover depends on the value of debt relative to the value of equity (and its first derivative) which can be expressed as a critical thresholdD I (y) given in the following proposition.
Proposition 2. In any state y, the critical thresholdD(y) 13 In addition, the value V K that represents the value of the optimally relevered firm after restructuring debt, depends on the market value of debt at the optimal initial leverage ratio.

Proof: See Appendix A.3
According to Proposition 2, equityholders wish to fully rollover expiring debt and therefore set δ * (y) = m if debt value is above the critical thresholdD I (y). For debt values below this threshold, equityholders do not rollover debt at all by setting δ * (y) = 0.
Finally, when debt value matches the threshold valueD I (y), equityholders are indifferent between alternative values of δ(y).
The economic intuition behind this optimal decision rule can be seen by considering equityholders' costs and benefits from issuing an additional unit of debt. Issuing an additional unit of debt generates net proceeds of (1 − k i )D(y). On the other hand, the additional debt issue increases the firm's leverage (decreases y) and, thus, reduces the valuation of equity. This valuation effect is given by dE dB =Ẽ(y) − y ∂Ẽ(y) ∂(y) . Equating these costs and benefits and solving for the critical debt value yieldsD I (y) in Equation (14).
One might suspect that the optimal debt rollover is of a bang-bang type, i.e., that δ * is either at its minimum, δ * = 0, or at its maximum, δ * = 0 and that the states y at which debt value exactly equalsD I (y) serve as singular switching points where the optimal debt strategy jumps from δ * = 0 to δ * = m or vice versa. However, we demonstrate below that, for sufficiently short debt maturities, there exist extended regions of firm leverage, where an interior choice 0 < δ * < m is the equilibrium. In this region, the optimal rollover rate δ * (y) is set such that the resulting debt value exactly matches the critical threshold (14), which in turn makes equityholders indifferent between alternative rollover rates.
Interior optima arise in states y, where neither δ = 0 nor δ = m are feasible equilibrium solutions. This is the case if the choice δ = 0 results in a debt valuation D (y) >D I (y), which according to the decision rule (15) implies an optimal choice of δ = m or, alternatively, if full rollover, i.e., δ = m, implies a debt valueD(y) below the critical thresholdD I (y), and therefore a rollover choice δ = 0.
The following proposition summarizes valuation equations of equity and debt in the interior equilibrium as well as the equilibrium rollover rate together with conditions that ensure the stability of the equilibrium, i.e., conditions that ensure that a small perturbation of the conjectured rollover rate does not destroy the equilibrium.
Proposition 3. In an interior equilibrium 0 < δ * < m the value of equity and debt as well as the optimal rollover rate must satisfy the systems of equations In particular, equity valuation (16) is independent of the rollover rate, while δ * is strictly positive in this region. Furthermore, the existence of an interior equilibrium implies that equity is convex in y and that the value of debt per unit of face value increases with decreasing leverage (increasing y), We can now derive equity and debt valuation expressions for the three strategy regions. In the first one, δ(y) = m, i.e., equityholders fully rollover expiring debt, since proceeds from issuing new debt at market value are sufficiently high to make it optimal to choose the maximum rollover rate that the covenant allows. In the second one δ(y) = 0, proceeds from reissuing expiring debt at market value excessively dilutes the firm's equity and, thus, equityholders do not wish to issue any new debt. Finally, in the region with interior rollover we know from Proposition 2 that equityholders are indifferent between all feasible choices of δ(y). SubstitutingD(y) =D(y) I into the valuation equation (8)  Proposition 4. In a region where the firm fully rolls over its debt, i.e., δ = m, the value of equity and debt are given bỹ In a region where the firm rolls over its debt at an interior optimum δ * , the value of equity and debt are given bỹ In the region where the firm funds repayment of retiring debt entirely with equity, i.e., where δ = 0, the values of equity and debt are given bỹ The exponents β and γ are the characteristic roots of the homogeneous differential equations given by The constants E 1,2 and D 1,2 can be determined for each region by the proper boundary conditions that ensure that value functions are continuous across changes in the rollover regime.
(See Appendix A.5 for the proof of Proposition 4.)

Endogenous Bankruptcy and Optimal Discrete Recapitalization
While the choice of the rollover threshold δ(y) constitutes instantaneous stochastic control, the selection of the recapitalization threshold y as well as the choice of the bankruptcy threshold y are optimal stopping problems. 15 Proper boundary conditions (9), (10), (12), and (13) determine equity and debt values at these critical thresholds.
First order conditions of optimality at the upper and the lower reorganization threshold follow from inspecting boundary conditions (10) and (12) with respect to an equityvalue maximizing choice of the reorganization thresholds y and y. This leads to the following 'smooth pasting' or 'super contact' conditions (see Dixit, 1993, or Dumas, 1991, for a discussion of these optimality conditions), These conditions together with optimal rollover determine equityholders' optimal capital structure strategy contingent on all-equity firm owners' initial choice of leverage and debt maturity, S = {δ(y), y, y |ý, m}. As discussed above, discrete refinancing at y or y creates an unlevered firm, which all-equity owners will re-lever immediately.
A discussion of optimal debt rollover δ near bankruptcy closes this section. In Proposition 5 below, we prove that in states y which are sufficiently close to the bankruptcy state y, firms always find it optimal to fully rollover expiring debt, i.e., to set δ equal to m. In other words, when cash flows are sufficiently low, setting δ < m and thus reducing leverage would transfer wealth from equityholders to remaining debtholders. To see this, note that setting δ < m reduces the overall face value of debt outstanding and therefore increases the value of the remaining bonds in bankruptcy. For sufficiently bad states, this value transfer to remaining bondholders dominates any potential benefits to equityholders, since bankruptcy occurs in any case with very high probability. 16 Thus, in bad states the firm faces a classical debt overhang problem where equityholders do not engage in debt reducing activities. Instead, they wish to minimize their investment in debt reduction near the bankruptcy boundary, by choosing the highest feasible debt rollover rate. This is formally stated in the following proposition: Proposition 5. If loss-given-default is strictly less than 100%, it is optimal to rollover debt at the maximum rate δ = m for y in a neighborhood above the bankruptcy threshold y.
(See Appendix A.6 for the proof of Proposition 5.)

Optimal Initial Leverage
All-equity firms choose target leverageý and average debt maturity m to maximize total firm value, fully anticipating the effects on equityholders' optimal capital structure 16 This can be seen most intuitively if one considers the effects of a debt reduction for given default decisions. Remaining bondholders benefit from the debt reduction since they receive a higher value per unit of face value in default. Equityholders do not enjoy a similar benefit, since they receive zero in bankruptcy, independent of the shortfall of firm value in bankruptcy relative to the total face value of debt. strategy S E . We assume that i is set such that debt is initially issued at par, 17 The maximization problem of all-equity owners is therefore given by max y,m with the first order conditions The optimization problem of all-equity owners after recapitalization is identical to the problem of the initial firm owners who decide over initial leverage and maturity since capital structure strategies S preserve the model's linear homogeneity in the debt level B. Hence, the initial choice ofý, m, and i is identical to the choice made by all-equity owners after reorganization at y or y.

Calibration
For our numerical analysis our parameters are calibrated to capture important features of the US tax code, which are also shared by many other countries. The average marginal tax rate after interest expense over the last 20 years, i.e., from 1994 to 2013, is estimated to be 25.9%. Again, average simulated marginal tax rates in the sample period are lowest in 2010 (18.7%) and highest in 1995 (30.7%). The total-asset-weighted average marginal tax rate before interest expenses over the stated period is 33.1%. This is close to the asset-weighted mean value from COMPUSTAT MTR reported above. In the base case of our numerical analysis we therefore use a corporate tax rate of 30.6%. We also provide comparative statics with respect to our tax parameters to illustrate how sensitive our results are with respect to changing tax rates.
Recent empirical estimates of corporate bankruptcy costs have considerably changed the academic community's view of their magnitude. Early papers estimate bankruptcy costs investigating sets of defaulted firms and estimated these costs to be only a few 19 We thank John Graham for providing us with his comprehensive set of simulated marginal tax rates covering the period from 1980 to 2013 and for his advice on calibrating our model to the US tax code. Please see Graham (1996a) and Graham (1996b) for details on the applied simulation procedure. Graham and Mills (2008) use federal government tax return data and show that simulated marginal tax rates provided in the file are close approximations.  Ang, Chua, and McConnell (1982) (mean 7.5%, median 1.7%), Weiss (1990) (3.1%), Altman (1984) (6.0%), Andrade and Kaplan (1998) (10% to max. 23%).

Debt Maturity, Capital Structure Dynamics and Firm Value
We start by analyzing the effect of average debt maturity on the firm's optimal refinancing decision. If not otherwise mentioned, base-case parameters listed in Table 2 are used. First, we explore firms which have issued debt with long maturity. Panel (a) of Figure 1 illustrates the optimal rollover rate nomalized by the retirement rate, δ/m, over the inverse leverage ratio y for a bond with long maturity, T = 30 years, m = 0.03.
Since the state variable y is proportional to the firm's cash flow level, y serves as a proxy of the firm's profitability. We can see that equityholders optimally choose to rollover all expiring debt (δ/m = 1) over a large range of firm states, especially in bad states, i.e., low y. Only immediately before calling the bonds to subsequently issue more debt, i.e., in a region near 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.
Panel (b) of Figure 1 shows the optimal rollover rate δ/m for a bond with T = 10 years maturity, m = 0.1. In contrast to the 30 year bond, debt with shorter maturity is not fully rolled over if the firm's state deteriorates. There is a region of y below the initial inverse leverageý near the bankruptcy threshold y where rollover is an interior optimum as derived in Propositions 2 and 3. Intuitively, under full rollover, prices of new bonds would be too low, since they reflect high leverage and future costs of financial distress. It is in equityholders' own interest to partly use equity to refund maturing debt, despite the fact that it implicitly also benefits the remaining bondholders. In equilibrium, bondholders anticipate this and price bonds more attractively.
When increasing debt maturity above a critical value, this region of debt reduction vanishes. Under base-case parametrization, the critical maturity at which this happens is T = 23.86 years, m = 0.04192. This is the shortest maturity for which there is  Table 2.
Low and high levels of y correspond to states of low and high profitability of the firm. Equityholders will not engage in debt reduction in bad states of the firm. (b): A maturity of T = 10 years serves as a commitment that equityholders will not rollover all expiring debt in bad states, i.e., engage in debt reduction.
no debt reduction. Debt with maturity shorter than this critical value induces debt reduction, maturities longer than this critical value do not induce debt reductions in bad states.
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 near y, equityholders terminate their effort to reduce debt. In this region they once again fully rollover debt and dilute existing debtholders by re-issuing the expired debt. This is in accordance with Proposition 5 which proves that when loss-given-default is less than 100%, firms engage in full rollover when close to bankruptcy. Thus, when pushed very close to bankruptcy, equityholders are no longer willing to make additional voluntary equity investments in the firm. On the contrary, they would rather issue new debt at the maximum rate allowed by debt covenants, even if this can only be done at unfavorable prices, i.e., when high credit spreads are charged by investors. Figure 1 also reveals that there is an additional range in which the firm does not fully rollover maturing debt. This occurs when the inverse leverage ratio approachesȳ, i.e., the threshold, at which all outstanding debt is repurchased and replaced by a new, larger debt issue. The intuition for this is straightforward. In this range it would not be worthwhile to incur the transactions costs associated with debt rollover, since the firm anticipates that the repurchase of the entire outstanding debt is imminent. Thus, transactions costs associated with debt rollover are not expected to be "amortized" via the present value of the additional tax shields. In this region it is therefore optimal to repay maturing debt with retained earnings. 21 To summarize, there are four main insights that the above numerical analysis provides. First, for sufficiently long maturities, equityholders never use retained earnings or equity issues to repay maturing debt except immediately before a discrete leverage increase. This result changes if the average debt maturity is shortened. In this case there exists a range of leverage ratios strictly above the initial optimum for which equityholders find it optimal to partly use retained earnings of equity to repay maturing debt. This is in accordance with the empirical evidence presented in Section 6 and with the 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 ratioý 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.
Third, near the restructuring threshold y the firm entirely refrains from issuing debt.
In this range it would not be worthwhile to incur the transactions costs associated with debt rollover, since the firm anticipates that the repurchase of the entire outstanding debt is imminent.
Fourth, near the bankruptcy threshold y the firm fully rolls over all expiring debt, i.e. δ = m. Thus, even with short-term debt outstanding, equityholders resume issuing debt if the leverage ratio becomes sufficiently high. In this case the equityholders are no longer willing to invest in debt reductions to keep their equity option alive. This 21 This region where debt is not rolled over in states close to y has a negligible effect on total firm value, as we discuss below when we calibrate the model. This is so since firms spend only a small fraction of time in this region and because there are no significant agency problems associated with the rollover behavior of the firm in this region, since debt is essentially riskless. In contrast, the effect of debt reduction in bad states of the firm has a large effect on firm value, as we argue throughout the paper. We quantify the valuation effect of the former region below, when discussing optimal maturity choice and the associated tax benefits of debt.

latter result follows from Proposition 5. 22
Bankruptcy costs, corporate taxes and critical debt maturity: We find that bankruptcy costs as well as the magnitude of the tax shield of debt financing represent the main determinants for the critical average maturity that triggers voluntary debt reductions. The lower the bankruptcy costs the shorter the maturity required to provide incentives for voluntary debt reductions. Figure 2 plots the critical average maturity over bankruptcy costs for two different levels of corporate tax, τ c . The lower line represents our base case where the corporate tax rate is calibrated to the average marginal tax rate provided by COMPUSTAT MTR database (τ c = 30.6%, see Section 4).
The upper line shows critical debt maturities over bankruptcy costs when using the lower average effective corporate tax rate implied by the marginal-tax rate data provided by John Graham (τ c = 25.9%,). It is evident, that in the case of higher tax shields it requires shorter debt maturities to induce sufficient incentive for equityholders to engage in active debt reduction when the firm's cash flows deteriorate. This result is quite intuitive, since actively replacing retired debt with equity reduces the firms tax shields and, hence, providing larger tax shields reduces the incentive to substitute debt with equity.
With base-case parameterization, i.e., τ c = 30.6%, g = 34.39%, debt maturity below the critical maturity of 23.86 years induces debt reductions in bad times. With lower tax shields, using τ c = 25.9% the critical debt maturity at g = 34.39% is 33.4 years.
Bankruptcy costs as low as g = 25% require average maturities of 15.49 years and 21.53 years when using corporate tax rates of 30.6% and 25.8% respectively. Bankruptcy costs as high as g = 45% induce debt reduction for average maturities below 37.16 years and 53.07 years respectively. Thus, with lower bankruptcy costs, it needs shorter-term debt to induce debt reductions.
Debt maturity and firm value: We next consider the effect of debt maturity 22 In the online Appendix A.7. we derive a condition under which firms never have an incentive to increase the debt retirement rate m above the originally contracted level. This condition holds in all of our numerical analyses and is a manifestation of the leverage ratchet effect in Admati, DeMarzo, Hellwig, and Pfleiderer (2018).  Figure 2: Critical average debt maturity below which the commitment to debt reductions in bad times is credible as a function of bankruptcy costs. Critical maturities are plotted for the base case parameterization τ c = 30.6%, which is the average marginal tax rate estimated from COMPUSTAT MTR database, which employs the approach of Blouin, Core, and Guay (2010). Additionally, critical maturity for the average tax rate from John Graham's database, i.e., τ c = 25.9%, is also plotted. See Section 4 for more details.
on firm value and illustrate the potential benefit of a short debt maturity with basecase parameters. Results for different parameterizations are reported below. Figure 3 displays the initial tax advantage, i.e., the extent to which the value of the optimally levered firm exceeds the value of the unlevered productive assets as a function of the retirement rate of debt, m. The tax advantage of debt at the optimal initial leverage for the base-case firm plotted against the retirement rate m. The dotted line shows the corresponding tax advantage 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 non-monotonous. Firm value is n optimal endogenous average maturity of ≈ 4.26 years for its debt, as reported in Leland (1994b) and Leland and Toft (1996). By contrast, if the firm can engage in debt reductions, the relationship between total firm value and the maturity structure of debt is not monotonic. 24 This is so because debt with sufficiently short maturity induces more efficient capital structure adjustments by equityholders when the firm's cash flows decrease, thereby lowering probability of default and, hence, expected bankruptcy costs. This result is driven by the fact that the firm with the higher fraction of maturing debt (shorter maturity) essentially has greater flexibility of managing, i.e., reducing, its leverage in the relatively bad states. This flexibility 24 Empirical evidence for this nonmonotonicity is provided by Guedes and Opler (1996) who report that investment grade firms seem to be indifferent between issuing debt at the long end of the maturity spectrum and issuing debt at the short end of the spectrum.
increases the value of the firm since it can operate with higher debt ratios, thereby shielding its taxable income more effectively. 25 As illustrated in Figure 3, the beneficial effect of shorter debt maturity on future capital structure dynamics outweighs the disadvantage due to higher transactions costs from rolling over maturing debt. In the base case, overall firm value is maximized at a debt maturity of ≈ 4.26 years. 26,27 Debt capacity: 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. This effect is illustrated by Figure 4, which plots the initial optimal leverage as a function of m for the base-case firm. Unlike firms which must rollover all maturing debt, firms that choose the rollover rate optimally actually increase their debt capacity as they shorten their debt maturities. The optimal initial leverage increases from approximately 39% percent for perpetual bonds and reaches its maximum with 80% at an average debt maturity of approximately 1.5 years. For very short maturities, debt capacity decreases, due to transactions costs incurred when rolling over debt. At the firm-maximizing debt maturity of 4.26 years, the firm's debt capacity is approx. 65%.
Compared to Leland (1994b), Leland andToft (1996), or Leland (1998), our model generates higher initial target leverage ratios. However, over the lifetime of the firm, average leverage ratios are comparable. The relatively high initial leverage in our model is due to the fact that investors rationally anticipate low bankruptcy risk due to less than full debt rollover in bad states. This commitment effect of short debt maturities leads to higher initial but similar average leverage ratios due to subsequent leverage 25 We are grateful to an anonymous referee for providing this intuition. 26 We simulate 50,000 firms over 100 years and find that the optimally levered firm (optimal maturity ≈ 4.26 years) spends on average 8.9% of the time in states of debt reduction. Debt reduction decreases the risk neutral probability of default to 0.3% per year compared to a 1.3% default probability per year of a firm with an optimal amount of infinite horizon debt, which is the optimal maturity choice when debt must always be fully rolled over.
27 It is part of the optimal strategy that equityholders stop debt rollover close to the restructuring boundary y to save transactions costs. In contrast to the region where the firm does not fully rollover debt in bad states, the effects of stopping rollover in good states on firm value are small. In our calibrated base case a firm following the optimal rollover restructures at a leverage ratio 1/y = 35.16% while a firm that does not stop rollover near the restructuring boundary chooses to restructure at a leverage ratio of 34.59%. The effect on initial firm value is less than 1 basis point. : Optimal initial leverage ratios 1/ý 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 distress. Only at the very short end, transaction costs lead to a deterioration in debt capacity.
reductions. Lower initial leverage ratios would obtain if debt maturities were lumpy. In this case the leverage reducing effect of maturing debt would only take place at discrete points in time. 28

Comparative Statics
In this section we explore the effect of various model parameters on firm value, optimal debt maturity and dynamic capital structure policy. 28 As discussed in the introduction, Geelen (2016) and Chen, Xu, and Yang (2020) analyze the opposite polar case, in which there is only a single debt maturity outstanding. This indeed leads to lower initial target leverage and higher expected bankruptcy probabilities, compared to our setup. Analyzing the effect of different degrees of maturity granularity appears to be a fruitful avenue for future research.
Bankruptcy costs: We first focus on the role of bankruptcy costs. The key role of bankruptcy costs for the commitment to debt reductions was already discussed above. Figure 5 plots the tax advantage of debt, i.e., the extend to which the initial firm value exceeds 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 reduce the maximum attainable tax advantage of debt, (iii) lowering bankruptcy costs moves the optimal finite maturity towards shorter maturities, (iv) for very low bankruptcy costs it becomes relatively more advantageous to issue console bonds.
The most surprising effect is that higher bankruptcy costs imply higher firm values.
Higher bankruptcy costs make it easier for equityholders to credibly commit to debt reductions. The resulting decrease in the expected probability of bankruptcy more than offsets the effect of the increased costs given a default.
Transactions costs: The costs associated with rolling over debt are another key determinant of firm value when finite maturity debt is issued. Figure 6 illustrates the effect on firm value for different values of k i . When moving to lower values of k i we 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.
In our model, transactions costs k i are assumed to be proportional to the notional amount of debt that is rolled over. Thus, shorter bond maturity is-ceteris paribusassociated with higher costs per unit of time. 29 Cash flow characteristics: Figure 7 shows how changes in cash flow characteristics affect total firm value. Panel 7a plots the initial tax advantage as a function of the retirement rate for several values of cash flow volatility σ. Moving to higher volatilities (i) results in lower firm value, (ii) requires shorter debt maturity to induce debt reductions, (iii) moves the local maximum of total firm value towards shorter ma-29 It might be argued that it is actually the higher transactions cost burden for lower maturities which reduces bond prices and in turn incentivizes equityholders to engage in debt reductions, rather than short debt maturity by itself. To see that this is not the case we have solved a model which features constant transactions costs per unit of time, m k i = const. Thus, we assume that it is cheaper to rollover shorter term debt than it is to rollover long-term debt, so that the transactions costs per period are the same for different maturities. Under this alternative assumption, debt reduction still only occurs when debt maturity is below some threshold. m debt reduction Figure 5: Initial tax advantage plotted over the retirement rate m for different levels of bankruptcy costs. Lower bankruptcy costs require a shorter debt maturity in order to induce voluntary debt reductions, lower bankruptcy costs reduce the maximum attainable tax advantage of debt, lowering bankruptcy costs moves the optimal finite maturity towards shorter maturities. turity debt. High cash flow volatilities reduce the firm's debt capacity but increase the option value for equityholders and thus make them more reluctant to default on their debt obligations. This makes the commitment effect of short-term debt relatively less advantageous and requires shorter debt maturities to induce voluntary debt reductions.
Panel 7b displays the tax advantage of debt as a function of the retirement rate for several values of the risk-adjusted cash flow growth rateμ. Moving to higher growth (i) increases firm value and (ii) shifts optimal maturity towards long-term debt. This is so since the commitment to decrease leverage in response to decreasing cash flows is less valuable for firms with high expected cash flow growth rates.
Bankruptcy costs and debt capacity: Next, Figure 8 plots the firm's optimal initial leverage ratio, which we refer to as the firm's debt capacity, for different debt m k i = 0.4% Figure 6: Optimal initial tax advantage for different costs k i associated with rolling over debt plotted over the retirement rate m. With lower k i , the optimal maturity moves towards shorter-term debt.  Firm value and corporate tax rates: Finally, Figure 9 shows the tax advantage of debt plotted against the retirement rate m for the base-case firm with τ c = 30.6%.
As a comparison we plot the tax advantage when corporate taxes are estimated from John Graham's marginal tax rate data, τ c = 25.9%. Higher tax shields caused by higher corporate tax rates lead to (i) lower optimal debt maturity and (ii) a higher tax advantage at the optimal debt maturity. While it is intuitive that a higher corporate tax rate leads to higher tax advantage if debt is used optimally, the result that higher tax rates reduce optimal maturity is less obvious. As a direct consequence, higher tax rates make debt reduction less desirable, because reducing debt diminishes the associated tax shield. A secondary effect is that debt capacity increases with shorter debt, and higher m τ c = 30.6% (COMPUSTAT MTR) Figure 9: Tax advantage of debt for different levels auf corporate tax, τ c , plotted over the retirement rate m. High corporate tax rates lead to higher tax advantages and lower optimal maturity. debt capacity ex ante allows the firm to use debt more aggressively, which increases the debt tax shield. From Figure 9 we see that the latter effect dominates the direct effect and, over all, higher corporate tax rates reduce optimal debt maturity, from an optimal average maturity of 6.25 years for τ c = 25.9% to 4.26 years in the base case with an effective corporate tax rate of τ c = 30.6%.

Empirical Evidence
Our analysis generates several predictions, linking capital structure choices to various firm characteristics. Most importantly, the analysis implies a distinct relation between debt maturity and leverage dynamics, i.e., firms with shorter debt maturities exhibit more pronounced debt reductions when firm value drops. In addition the model implies relations between debt maturities and firm characteristics, such as cash flow volatilities, corporate tax rates, or bankruptcy costs. This section discusses empirical evidence on these model predictions. While it is not intended to represent a fully-fledged empirical test, it documents that even simple analyses reveal patterns which accord well with our theory.

Sample Construction
To relate our theoretical predictions to empirical evidence, we use annual Compustat data for the period from 04-1962 to 03-2017. As in Graham and Leary (2011), and DeAngelo, Gonçalves, and Stulz (2018), we exclude firms not incorporated in the US, utilities (SIC codes in the range 4900 to 4949), financials (SIC codes 6000 to 6999), and firm years where the book value of total assets is below USD 10 mio. We select only firm years for which either long-term debt or debt in short-term liabilities are available, (as in DeAngelo, Gonçalves, and Stulz, 2018). We exclude observations with missing total assets, cash, or market capitalization (i.e., share price at the end of fiscal year times common shares outstanding), with negative net-debt ratio and with negative cash. Finally, we truncate at the 97.5% EBIT volatility quantile to exclude implausibly high values. This results in a sample of 73640 firm years.
To measure the extent to which a firm uses short-term debt in its capital structure, we calculate the fraction of debt due within three years, normalized by total debt. 30 Market leverage is measured as long-term debt plus debt in short liabilities over total market value of the firm's assets (total liabilities plus market capitalization). We use the net debt ratio (i.e., book leverage minus cash ratio) to calculate changes in leverage over consecutive years.

Empirical Findings
In a first step we analyze whether firms' leverage changes are associated with the fraction of short-term debt. To do this we calculate transition probabilities in the spirit of 30 Our findings are qualitatively robust with respect to alternative definitions of short-term debt. For example, we have also used debt due within two years to total assets as a measure of short-term debt usage and found very similar result. Table 3: Number of observations in the buckets created by the double sort on short-term debt, i.e., debt due within three years, normalized by total debt, and market leverage, i.e., long-term debt plus debt in short liabilities over book value of total liabilities plus market capitalization of equity. DeAngelo, Gonçalves, and Stulz (2018), who show that firms with high market leverage tend to reduce their net debt ratio. We build on their study and apply a double sort.
First, with respect to market leverage, and second, with respect to short-term leverage, i.e., debt due within 3 years as a fraction of total debt. Along both dimensions we sort firms into quintiles. Breakpoints for short-term debt sorts are 7.40%, 15.89%, 26.64%, and 42.91% of debt due within three years. Breakpoints for market leverage sorts are 7.05%, 14.95%, 25.13%, and 41.34%. 31 Table 3 states the number of observations in each bucket. We see that buckets along the main diagonal are most densely populated, i.e., firms' market leverage and their fraction of short-term debt are positively correlated, with a correlation coefficient of 28.85%.
Our capital structure theory predicts that firms with higher levels of short-term debt have stronger incentives to reduce debt than firms with longer debt maturities.
Furthermore, our results also imply that the incentive to delever is non-monotonic in market leverage. Moving from low market leverage to high market leverage, the incentive to delever first increases, but for very high market leverage, the incentive to reduce net debt vanishes. Table 4 explores empirical evidence for these predictions. It reports the average 31 Comparing these breakpoints to our calibrated numerical examples, one needs to consider that the above empirical values are based on a leverage definition that only considers long-term debt plus debt in short liabilities. When focussing on firms' total liabilities, which include, for example, accounts receivables, pension obligations etc., one obtains substantially higher leverage ratios. In this case the firm at the 80 percentile exhibits a market leverage of 71.4%. changes in net debt ratios, stated in percent and measured over the year that follows the sort. 32 In each of the market-leverage buckets (columns of Table 4), firms with more short-term debt delever significantly more (or lever less) than firms with low levels of short-term debt.
For example, consider the column for market leverage 20%-40% which displays the change in the net debt ratio in the subsequent year for firms in this market leverage bucket. It shows that firms which are in the bottom quintile in terms of usage of short-term debt, i.e., row 0% to 20%, on average increase their net-debt ratio by approximately 1 %. By contrast, firms that are in the top quintile, i.e., row 80%-100%, decrease their net-debt ratio by approximately 1.9% in the subsequent year. In fact, as can be seen from the first row of Table 4, the average firm which relies almost entirely on long-term debt does not engage in leverage reductions, irrespective of its initial market leverage, consistent with our model. Moving from low to high fractions of short-term debt, firms increasingly engage in deleveraging. For example, row 5 shows that the average firm which almost exclusively uses short term debt and which is in a market leverage bucket between 60% and 80% reduces its market leverage in the subsequent year by more than 2%. These findings are also in accordance with Hovakimian, Opler, and Titman (2001), who also conclude that long-term debt hinders leverage reductions.
Thus, the first prediction of our theory seems to be empirically relevant and strongly supported by firm behavior.
We next turn to the non-monotonicity of leverage adjustments when conditioned on initial market leverage. Recall that in our model firms increase leverage when they reach a lower leverage threshold. Then, for increasing initial market leverage ratios, firms do not adjust the face value of debt over a range. Eventually they start reducing debt, by rolling over less than 100 percent of the expiring debt before they start gambling for resurrection by fully rolling over expiring debt again as they get close to their bankruptcy thresholds. This predicted non-monotonicity along increasing initial market leverage is also apparent in Table 4. The change in net debt is generally u-shaped when moving from low to high market leverage buckets. Consider row 3, for example. While firms in the market leverage bucket 60% to 80% reduce their leverage in the following period by aproximately 1.5%, firms in the highest market leverage bucket, i.e., 80%-100% reduce leverage by only less than 1%. Consistent with our model, firms with intermediate market leverage have the highest incentive to reduce their leverage. Table 4: Average changes in net debt in the year following the double quintile sort on short-term debt, i.e., debt due within three years, normalized by total debt, and market leverage, i.e., long-term debt plus debt in short liabilities over book value of total liabilities plus market capitalization of equity. Changes are measured as averages of (net debt ratio) t+1 − (net debt ratio) t in percent. Short-term debt is defined as debt expiring within three years over total assets. Row 1-5 exhibits the differences between changes in leverage of the average first-quintile firm and that of the average fifth-quintile firm. The p-value of the corresponding t-test is given in the last row. We believe that the results in Table 4 understate the true non-monotonicity of debt reductions, since our dataset does not fully capture the leverage effects of firm defaults. Compustat has no flag that indicates the reason why a firm drops out of the database. However, Table 5 shows that exits are more frequent for firms with high market leverage. For example, Table 5 reveals that almost 6% of those firms in the highest market leverage bucket exit the sample in the subsequent year, whereas this is the case for less than 3.4% of firms in the lowest market leverage bucket. Hence, we conclude that part of this increase is due to more frequent bankruptcies of firms in high leverage buckets. Since these firms will generally exhibit increasing market leverage until they default, the non-monotonicity of debt reductions is likely to be even more pronounced than what is illustrated in Table 4.
In addition to the main result that links debt maturity to firms' propensities to delever, our model also delivers comparative statics with respect to debt maturities and firms' corporate tax rates, their cash flow volatilities and their bankruptcy costs. To explore these predictions, we use a simple linear regression framework, with the fraction of short-term debt in a firm's capital structure over total liabilities as the dependent variable. Cash flow volatility is proxied by the standard deviation of firms' EBIT over total assets. We hereby require at least eight annual observations. As a proxy for the corporate tax rate we use past income taxes over EBIT averaged over the last eight years. We use this definition since a forward looking measure of marginal tax rates, as analyzed in Blouin, Core, and Guay (2010); Graham (1996a,b), would induce a severe endogeneity problem. This is so since leverage changes directly affect future effective corporate tax rates. Since bankruptcy costs are not directly observable at the firm level, we use estimates provided by Reindl, Stoughton, and Zechner (2017), aggregated for the Fama and French 17 industries.
Columns (1) to (3) of Table 6 show univariate regressions of the fraction of shortterm debt over total liabilities on our proxies for bankruptcy costs, cash flow volatility and corporate tax rates. Column (4) displays the multivariate regression results for the three explanatory variables. Column (5) also includes the additional controls suggested in Table 1 of Graham and Leary (2011).
Focussing first on columns (1) to (4), we can see that the coefficients of the modelbased explanatory variables all have the right sign in the univariate regressions as well as in the multivariate regression. Furthermore, RSZ MeanBC and ebitVola are both significant at the 1% level in all four regression specifications. Thus, higher bankruptcy costs are associated with longer debt maturities whereas higher cash flow volatilities are associated with shorter debt maturities, consistent with the model.
In column (5) we control for additional firm characteristics, such as size, leverage, asset tangibility, profitability and market-to-book ratio of equity. We see that the coefficient on ebitVola still remains positive at the 1% significance level. The negative relation between short-term debt usage and bankruptcy costs is also obtained with the extra control variables. However, this latter result is no longer statistically significant.
When interpreting this, one needs to consider that the bankruptcy cost proxies are necessarily noisy estimates of the firms' true bankruptcy costs, as they are aggregated at industry levels. For example, we know that firms within industries display a significant degree of heterogeneity, especially in terms of their size, which is usually highly right skewed. It is therefore likely that the bankruptcy costs of very large firms, expressed as a percentage of total assets are different from those of our proxies. We therefore also run the regression with the control variables when we truncate observations at the 95th size percentile. As can be seen in column (6), for this sample the coefficient on bankruptcy costs is indeed statistically significant and has the predicted sign. We also note that the coefficients of the additional control variables in columns (5)and (6) all have intuitive signs and are highly significant, except for market-to-book ratio of equity, which is insignificant.
Thus, although the empirical analysis provided in this subsection does not represent a full empirical test of our dynamic capital structure theory, it produces evidence that supports the model predictions. Firms with high ratios of short-term debt reduce leverage more aggressively than firms with more long-term debt and there is evidence for the predicted non-monotonicity in leverage reductions as a function of initial leverage.
Furthermore, firms' cash flow volatilities are positively related to their usage of shortterm debt whereas their bankruptcy costs are associated with longer debt maturities.
While these results support our model, some of the findings may also be consistent with other debt maturity theories. We therefore believe that a detailed and fully fledged empirical analysis of the documented strong link between debt maturity, delevering, and firm characteristics is a promising direction for future research.

Conclusions
This paper explores the effects of debt maturity on subsequent dynamic capital structure adjustments. We find that long debt maturities create agency costs in states where the  Reindl, Stoughton, and Zechner (2017), aggregated for the Fama and French 17 industries. Cash flow volatility, denoted by ebitVola, is proxied by the standard deviation of firms' EBIT over total assets. Finally, tax rates, denoted by pastIncomeTaxes, are proxied by past income taxes over EBIT, averaged over the past eight years. The additional control variables in column (5) and (6) are: assets, defined by the log of total book assets; leverage, defined by the sum of short-term and long-term debt over total book assets; tangible, defined by property plants and equipment over total book assets; profitability, defined as operating income over total book assets, market to book equity, defined by the ratio of market value of stocks outstanding over book value of equity.

Dependent variable:
fraction of short-term debt over total liabilities (1) (3)  firm's profitability is low, since equityholders have no incentive to reduce debt. Thus, equityholders of firms with long debt maturities "underinvest" in leverage reductions in low-profitability states. By contrast, short debt maturities imply a higher fraction of maturing debt, thereby creating more flexibility to reduce leverage in bad states without transferring value to debtholders and thereby mitigating the debt overhang problem. This value-enhancing effect of short debt maturities must be traded off against increased transactions costs associated with refinancing a larger fraction of debt in any given period. This tradeoff generates a new theory of optimal debt maturity.
We find that the equityholders' incentives to engage in debt reductions is nonmonotonic in the firm's leverage. For moderate drops in the firm's profitability, equityholders find it in their own best interest to repay maturing debt at least partly with equity, thereby mitigating the leverage increasing effect of decreasing cash flows. However, if the firm's cash flows continue to drop until it is pushed towards 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 maturity. 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 given debt level. Higher tax shields caused by a higher corporate tax rate also makes shorterterm debt more advantageous, since increased debt capacity associated with short-term debt allows for a better utilization of debt tax shields. Since long-term debt is stickier in downturns, it has a particularly adverse effect on the probability of bankruptcy for higher risk firms. Firms with higher cash flow risk therefore prefer shorter debt maturities.
In our numerical examples, the leverage-reducing effect of short debt maturities in bad states leads to relatively high initial leverage ratios and low bankruptcy probabilities. This is a direct consequence of the perfectly granular maturity structure that we assume in our model. Any lumpiness of firm' debt maturities would limit the leverage reducing effects that we identify to discrete points in time whereas in between maturity dates, leverage would still remain sticky. Thus, exploring the effects of lumpy debt maturities on initial leverage, leverage dynamics and bankruptcy probabilities appears to be a promising direction for future research.
Our main findings and comparative statics accord well with empirical studies and with our own empirical evidence presented in this paper. Most importantly, firms with high ratios of short-term debt reduce leverage more aggressively in downturns, there is evidence for the predicted non-monotonicity between leverage and leverage reductions, firms' cash flow volatilities are positively related to their use of short-term debt whereas bankruptcy costs are associated with longer debt maturities. Other empirical predictions of our theory, such as the effects of growth or transactions costs on maturity remain to be tested. Also, a fully-fledged empirical test how our theory of corporate debt maturity stands up against alternative explanations would be interesting.
value of debt, B t . Thus one can write y t = y(c t , B t ). If the debt level is adjusted by repurchasing all existing debt with face value B t and issuing new debt with face value B * t , the leverage ratio immediately jumps to the new value, i.e., in this case we have and therefore In the absence of a discrete adjustment, the inverse leverage ratio, y t , follows a diffusion and its dynamics can be determined using a Taylor-series expansion and Itô's Lemma dy t = ∂y ∂c dc + ∂y ∂B dB + 1 2 Neglecting all terms that are o(dt) gives

A.2 Proof of Proposition 1
For a given time-invariant, B-homogeneous capital structure strategy S of the form (7), we can apply the theorem of Feynman and Kač (see, e.g., Øksendal, 2003) and transfer valuation Equations (5) and (6)   We confirm in Proposition 1, 3, and 4 that a B-homogeneous capital structure strategy S (see Equation (7)) results in value functions that are linear-homogeneous in the debt level B. Thus, we substitute E(B, y) = BẼ(y) and D(B, y) = BD(y) into Equations (A.5) and (A.6). This yields Hamilton Jacobi Bellman Equations (8) and (11) for equity value and debt value per unit of face value of debt,Ẽ(y) andD(y), respectively.

A.3 Proof of Proposition 2
In this section we derive a Markov perfect Nash equilibrium capital structure strategy where investors price debt based on rational beliefs about the firm's debt rollover, default and recapitalization decisions and, given these prices, firms have no incentive to deviate from the conjectured capital structure strategy. There is no pre-commitment except that firms must repurchase the existing debt (i.e., eliminate a debt covenant) before they can issue new debt with a higher face value.
Proposition 1 states the system of valuation equations for a given strategy S. From there we proceed in two steps. First, we determine the optimal equilibrium rollover rate δ when bankruptcy and recapitalication thresholds y and y are given. In a second step we develop optimality conditions for the optimal choice of these thresholds. The choice of the initial leverageý and the maturity amortization m is done from the perspective of total firm owners. Optimality conditions are discussed in Sections 3.2 and 3.3.
In the first step we formulate the dynamic game in which investors price debt and equity based on rational beliefs about the firm's optimal capital structure. In the resulting Markov perfect Nash equilibrium, the conjectured optimal strategy results in market values of debt and equity that do not create incentives for the firm to deviate.
See, e.g., the chapter on optimal stochastic control in Björk (2004) or the chapter on stochastic differential games in Dockner, Jørgensen, Van Long, and Sorger (2000).
Let us define the differential operator Conjecturing that equityholders apply the optimal rollover δ * for given restructuring thresholds y and y, we determine the the resulting values of debt and equity must satisfy the following simultaneous system of equations. Thus, ifẼ andD derived under anticipation of the optimal rollover schedule δ * (y) satisfy verification (A.8), thenẼ andD are the value functions of equity and debt in equilibrium, see Björk (2004).
The first-order condition of optimal choice of δ in (A.7) is derived by differentiating the expression in the braces with respect to δ, which yields −y ∂Ẽ ∂y (y) +Ẽ(y) − (1 − k i )D(y). Hence, the derivative with respect to δ is positive if and only ifD(y) exceeds the critical thresholdD I (y) stated in Proposition 2. Consequently, δ(y) = m is the optimal choice of the rollover rate in this case. IfD(y) <D I (y), the derivative is negative and, thus, δ(y) = 0 is the optimal choice. Only ifD(y) =D(y) I , equityholders are indifferent in their choice of δ(y). This completes the proof of Proposition 2.

A.4 Proof of Proposition 3
While forD(y) <D I (y) andD(y) >D I (y) the choices δ(y) = 0 and δ(y) = m are stable equilibria (robust to perturbations in δ) we have to determine conditions under which an interior choice of δ is a stable Nash equilibrium in the sense that small perturbations in δ will not push the equilibrium to one of the boundary points. Figure  The marginal reaction of the market value of debt to perturbations in the rollover rate can be derived as follows. Consider a state y with 0 < δ * < m and increase debt rollover from the optimum equilibrium rollover δ * to some δ for a time span of length dt.
The resulting change in state y equals ∆ y = −(δ − δ * )y dt, according to the dynamics of y in (4). And consequently, the marginal impact on the debt value (through controlling y by perturbing δ) is ∆D = − ∂D ∂y (δ − δ * ) dt. (A.10) I.e., a positive perturbation in δ results in a marginal decrease inD if and only if ∂D ∂y > 0, which serves as a necessary condition for a stable interior equilibrium. One essential result of analyzing the interior equilibrium is that equityholders are indifferent with respect to the particular choice of δ, which allows for an analytical solution of the entire problem. Since equityholders are indifferent, equity value in a region with interior choice of δ * can be determined by simply setting δ = 0 in (8).
Value of debt follows directly fromD(y) =D I (y) and the particular value for δ * is determined by substitution ofD(y) =D I (y) into Equation (11) and solving for δ.

A.5 Proof of Proposition 4
In regions where δ * = m or δ * = 0, the value function forẼ andD are the general solutions of the second-order ordinary differential equations (8) and (11) after substituting δ * = m or δ * = 0. In a region of an interior equilibrium 0 < δ * < m we know from Proposition 3 thatD =D I . The value of equity must be the solution of the Hamilton-Jacobi-Bellman equation (8) with the optimal δ * from Proposition 3 substituted for δ. After substitution, the valuation equation for equity becomes independent of δ and exactly equal to the valuation equation for δ st to zero. Therefore, the solution corresponds to the value function in regions with δ = 0. This is so because in an internal equilibrium the value of equity is invariant with respect to the choice of δ.
The constants E 1,2 and D 1,2 are determined by boundary conditions (9), (10), (12), (13) which apply at y and y and by value-matching and smooth-pasting conditions at the interior boundary between the regions of full rollover, no rollover, and partial rollover, see, e.g., Dixit (1993). These conditions state that if such a boundary,ỹ, is transitory (i.e. the state variable y can freely move back and forth across the boundary in both directions), then the value functions of debt and equity must be continuous and smooth atỹ,

A.6 Proof of Proposition 5
Suppose loss-given-default is less than 100%. This is the case if g < 1 and optimal bankruptcy occurs at a level y such that the value of the remaining assets exceeds bankruptcy costs. Then it follows from boundary condition (12) thatD(y) > 0. However, for the value of equity and its partial derivative with respect to the inverse leverage ratio it follows from boundary condition (9) and optimality condition (18) that lim y→yẼ (y) = lim y→y ∂Ẽ(y) ∂y = 0.
Therefore, in a neighborhood of y it is true that D(y) > 1 1 − k i y ∂Ẽ ∂y (y) −Ẽ(y) .
According to Proposition 2, Equation (15) this implies that δ = m is the optimal strategy.
A.7 Online Appendix: Non-Optimality of Voluntary Debt Repayment In our model, firms can choose the financing mix used to pay for expiring debt, but the retirement rate, m, is fixed once the initial capital structure decision is made. In this Appendix we explore whether firms have ex-post an incentive to repay outstanding debt at a rate exceeding m. We show that in all states where firms rollover at least part of their debt this is not the case as long as they cannot retire outstanding debt at a price below the price at which new bonds can be issued. However, this cannot be the case in equilibrium, since it would not be rational for investors to pay a price for a newly issued bond above the price at which the firm can immediately repurchase it.
For the remaining states, i.e., the ones where the firm does not rollover any expiring debt, we derive an intuitive condition on the sensitivity of equity with respect to new debt, which eliminates the incentive to retire debt faster than contracted. In all our numerical analyses these conditions hold, i.e., firms never have an incentive to repay outstanding debt at a rate exceeding m. with A δ,m = 1 2 σ 2 y 2 ∂ 2 ∂y 2 + (μ + [m − δ])y ∂ ∂y .
Deriving the first order condition of optimal control from the valuation Equation For all parameter constellations explored in Section 5 we find that firms choose the initially optimalý such that there is no incentive to immediately repay debt at a rate exceeding m. Furthermore, we find that equity is convex for all y <ý, hence we conclude that firms never find it optimal to accelerate repaying debt at a rate which exceeds m. In particular, equityholders will sometimes use equity to repay debt at the contracted repayment rate m but never at a rate that exceeds m.