Debt Seniority and Sovereign Debt Crises

Is the seniority structure of sovereign debt neutral for a government's decision betweendefaulting and raising surpluses? In this paper, we address this question using a model ofdebt crises where a discretionary government endogenously chooses distortionary taxationand whether to apply an optimal haircut to bondholders. We show that when the size ofsenior tranches is small, a version of the Modigliani-Miller theorem holds: tranching justredistributes government revenues from junior to senior bondholders, while taxes andgovernment borrowing costs remain unchanged. However, as senior tranches becomesufficiently large, default costs on senior debt transpire into a stronger commitment to repaynot only the senior tranche, but also the junior one. We show that there is a lower thresholdfor senior bonds above which tranching can eliminate default on both junior and senior debt,and an upper threshold beyond which the government defaults also on senior debt.


Introduction
Tranching of public debt features prominently in recent debates on the European sovereign debt crisis, with particular attention devoted to proposals for jointly issued "Eurobonds", that would be senior relative to national sovereign debt (see e.g. Brunnermeier et al., 2016;Delpla and von Weizsäcker, 2010;European Commission, 2011). 1 More generally, governments in countries with weak economic and …scal fundamentals often pursue tranching by granting seniority to o¢ cial creditors or issuing a portion of their bonds under foreign jurisdiction. In such cases, seniority is often associated not only with priority in the repayment order, but with a stronger commitment to repayment. 2 While tranching and seniority are typically discussed in relation to the need to create a 'safe asset' in economies with risky debt, they also raise a number of important questions concerning …scal policy and debt sustainability. First and foremost, can issuing senior and junior debt a¤ect prospective (state-contingent) primary de…cits, by strengthening the government incentive to raise revenue through distortionary instruments, as opposed to undergoing costly default? The question is whether the revenue that the government generates to service its debt is independent of the seniority structure of public debt, that is, whether this form of '…nancial engineering'is neutral with respect to overall debt sustainability.
Second, if the seniority structure of government debt is not neutral for …scal policy, can a government rely on it to contain vulnerability to self-ful…lling debt crises? In other words, can managing the seniority structure be seen as an instrument to shield sovereign debt from arbitrary, highly destabilizing, market dynamics -without any need for the central bank to provide a monetary backstop to government debt, or for an external lender of last resort?
In this paper, we build a stylized framework to address these questions. In the model, default is costly and a discretionary government trades o¤ distortionary taxation against an optimal haircut to bondholders. Our key result is that the e¤ect of tranching, when associated with plausible, marginally higher default costs on senior bonds, is highly nonlinear in the size of the senior tranche. We show that the seniority structure is irrelevant (but for the relative price of bonds) when the senior tranche is either small or very large. In these cases, a version of the Modigliani and Miller (1958) theorem holds: issuing more or less senior debt at the margin does not a¤ect state-contingent primary surpluses, but only the distribution of cash ‡ows across bond types. However, intermediate levels of tranching may reduce the cost of …nancing and increase the incentive for the government to service not only senior, but also junior tranches. A key implication is that seniority can prevent sovereign default, addressing the instability created by self-ful…lling debt crises.
We provide analytical and economic insights on this result using a bare-bones model of debt crises. Households are risk neutral and may invest their endowment either in a safe asset or (risky) sovereign bonds. The government is benevolent in that it seeks to maximize household welfare, but discretionary in that it may not pre-commit to repaying its debt. To meet its …nancing needs, the government sells sovereign bonds at a market price. When debt repayments come due, the government optimally chooses the level of taxation, whether to default on its debt, and if it chooses to default, the extent of the haircut. In the model, household expectations about sovereign default a¤ect the government's borrowing costs and incentives to default, raising the possibility of multiple equilibria. In a 'bad equilibrium', high sovereign yields in anticipation of default risk raise the amount of distortionary taxation required to repay debt, therefore making default preferable. Vulnerability to this bad equilibrium hinges on the initial …nancing need of the government.
Our main results are as follows. First, we characterize a threshold below which issuing senior bonds simply leads the government to increase the haircut on junior bondholders in order to repay the senior tranche. Tranching is neutral for everything but the relative price of (or interest rates on) senior and junior debt-taxes, default incentives (on the junior tranche) and the overall cost of borrowing for the government remain unchanged. Tranching simply redistributes a given cash ‡ow among holders of di¤erent types of debt. Once the senior tranche size exceeds this threshold, however, default on the junior tranche in some state realizations may be complete.
Our second result concerns equilibria conditional on senior debt being large enough that the default on junior bonds is 100 percent -so that there are no residual cash ‡ows to redistribute from junior to senior bondholders: the government may only avoid default on senior bonds by raising tax revenues. We show that if, plausibly, there is a default cost di¤erential, the extra costs associated with reneging on senior liabilities create a discontinuity in the government's optimal …scal policy. Depending on the cost di¤erential, the government may choose to increase revenues, rather than defaulting on senior bonds. 3 We show that this discontinuity has far reaching consequences on debt pricing and sustainability. Namely, to the extent that higher total payments to bondholders reduce ex-ante borrowing costs, a higher valuation of sovereign bonds at issuance reduces the incentive for the government to default on all, junior and senior, bonds. A key result from our analysis is that, potentially, this e¤ect may eliminate bad equilibria with self-ful…lling default altogether. Depending on the cost of default on senior bonds, however, raising the senior tranche size to very high levels eventually makes default attractive again. Our …nal result is that there is a upper threshold above which senior debt issuance causes the government to default on both junior and senior debt -reinstating the validity of the Modigliani-Miller ine¤ectiveness result proven for low levels of tranching.
Our paper relates to a large literature on debt crises which includes Cole and Kehoe (2000), Lorenzoni and Werning (2013), and Nicolini et al. (2015) among many others. We build upon the framework in Calvo (1988). While we introduce …xed costs of default as in Corsetti and Dedola (2016), our focus is on tranching instead of monetary backstops. Similar to Hatchondo et al. (2017), we …nd that tranching may reduce the government's borrowing costs. However, our framework di¤ers from this literature in two fundamental ways. First, we allow for default on the senior tranche, which e¤ectively places an upper bound on the senior tranche size. Second, since the government optimally determines the haircut, we account for the possibility that tranching may reduce payments to junior bondholders.
Our paper is also closely related to the recent literature on Eurobonds. 4 Delpla and von Weizsäcker (2010) propose the creation of a senior tranche of jointly issued sovereign debt guaranteed by all euro area member states, while nationally issued debt becomes a junior tranche. To preserve market discipline, their proposal restricts the amount of senior bonds each country may issue. Muellbauer (2011) and German Council of Economic Experts (2012) argue that joint liability should be combined with conditionalities to prevent moral hazard while Philippon and Hellwig (2011) suggest restricting Eurobonds to only short term maturities.
In these studies, the main bene…t of Eurobonds is to pool risk and prevent liquidity runs on individual countries through mutualization, while tranching serves to alleviate moral hazard. In a di¤erent vein, Beck et al. (2011) and Brunnermeier et al. (2016 analyze Eurobonds without joint guarantees. Rather than risk sharing, these studies focus on the bene…ts of providing a safe asset for banks in default-risky countries through a combination of diversi…cation and tranching. We contribute to this literature by bringing to attention an additional reason for tranching: costly default. In our framework, there is neither scope for risk pooling, nor any demand for safe assets, that is, the irrelevance results from Modigliani and Miller (1958) hold in the absence of costly default. We show that introducing di¤erential default costs into this framework creates a non-linear relationship between tranching and government incentives to tax and default, potentially eliminating default risk from not only the senior tranches but also the junior.
We proceed as follows: Section 2 describes a bare-bones model of self-ful…lling debt crises. Section 3 introduces tranching. Section 4 considers default on senior debt. Section 5 provides a numerical illustration. Section 6 concludes.

A bare-bones model of sovereign debt crises
Consider a two-period endowment economy populated by a unit continuum of identical riskneutral households which derive utility from consuming in period 2 only. In the …rst period, households are endowed with a stock of …nancial wealth W 0 which they can invest in two assets: public debt B issued at an endogenous market price q b and a safe asset K sold at exogenously given price q with an in…nitely elastic supply. 5 Households' period 1 budget constraint can then be written as In period 2, households receive a random output realization: it may be "high"Y H and "low" Y L with probabilities (1 ) and respectively. Households also receive payo¤s from their assets, pay taxes, and consume, leading to the set of budget constraints where i 2 fL; Hg indexes the state of nature, C i is consumption, T i is the tax bill and z i (T i ) captures dead-weight losses associated with tax distortions. i 2 [0; 1] represents a potential haircut on public debt which leads to a …xed default cost on the economy In line with the literature on tax smoothing, we posit that z i (:) is strictly convex and that at a given level of tax revenues, dead-weight losses are larger and grow faster in taxation in the state with lower output, that is The government is benevolent -it seeks to maximize the utility of the representative household -but lacks the ability to commit ex-ante to a tax schedule. In period 1, the government rolls over its (exogenous) …nancing need B 0 by issuing discount bonds B = B 0 =q b . In period 2, the government observes the output realization and set tax and haircut rates under discretion, consistent with its budget constraint where 2 (0; 1) represents a budgetary cost of default proportional to defaulted payments and G is public spending. Since what matters in our analysis is the size of the primary surplus rather than individual budget components, we posit without loss of generality that G is a parameter invariant across states of nature.

Optimal default and taxation plans under discretion
The government's optimization problem can be written as subject to (4). Due to the boundary constraints on the haircut, there are two corner solutions, respectively with no default ( i = 0) and complete default ( i = 1), as well as an interior solution i =^ i 2 (0; 1). We …nd it convenient to begin with a description of the interior solution, where tax revenues are pinned down by a …rst order condition that trades-o¤ marginal tax distortions against (fractional) budgetary default costs. Combining (4) and (5) yields an expression for the (interior) haircut These expressions indicate that, in an interior solution, tax revenues are not a¤ected by a rise in the public debt bill B: Rather, additional debt simply increases the size of the haircut. In contrast, in states with lower (marginal) tax distortions as per (3), the government optimally collects more tax revenues and reduces the haircut. In other words, the government uses the haircut to smooth tax distortions across states of nature. The …xed costs associated with default create a discontinuity in the government's problem such that default is optimal only when it increases household welfare (i.e. consumption). This leads to the default condition When considering an interior solution, the default condition can be written in terms of a "minimum haircut" i , de…ned implicitly by the expression such that the interior solution is optimal for^ i i . Observe that i is increasing in the …xed default cost but independent of B. Moreover, the minimum haircut is higher in states with less tax distortions, that is L < H .
For^ i < i , optimal policy is characterized by a no-default corner outcome i = 0 with a tax bill Finally, the corner solution with complete default may be optimal when^ i > 1. In this case, the relevant default condition is given by combining (7) with i = 1 such that and taxes are set at Observe that in the corner solutions tax revenues become contingent on the debt bill B instead of tax distortions: facing constraints on its ability to adjust the haircut, the government sets taxes according to its debt repayment plans rather than marginal distortions. This also implies that tax revenues cease to be state-contingent. To summarize, the government's optimal policy plan is written compactly in Table 1.

Debt pricing
The price of public debt is pinned down by an interest parity condition, which equates (under risk neutrality) the expected returns on government bonds with those of safe assets Given the government's need to roll over its initial …nancing need B 0 , any decline in the price of its bonds q b due to expectations of default leads to a rise in market …nancing B = D=q b , raising the government's debt burden in period 2. We capture this in a market …nancing schedule

Rational expectations equilibrium and regularity conditions
Under rational expectations, households anticipate the optimal discretionary plan of the government conditional on its debt B. The market …nancing schedule (11), together with the government budget constraint (4), the expressions for interior and minimum haircuts respectively given by (6) and (8), and the government's optimal discretionary plan as described in Table 1 de…ne a rational expectations equilibrium.
To rule out equilibria on the right side of the debt-La¤er curve's peak (i.e. where the price of public debt q b is increasing in the government's …nancing need B 0 ), we impose the following regularity restrictions To keen our exposition as compact as possible, in the remainder of the paper we drop the subscript L. Therefore, the market …nancing schedule (11) becomes where refers to the haircut in state L.

Equilibria
As in our economy there may be two possible equilibrium types, one with and one without default, we adopt the following notation. Variables in a no-default equilibrium will be written with a superscript "N DE", whereas N DE = 0; the superscript "DE" will be use for the equilibrium with default, whereas the haircut DE > 0 will be strictly positive in state L: Proposition 1 characterizes these equilibria and shows that they exist in regions of …nancing need B 0 that line up monotonically, that is, as B 0 rises, we move from a region of N DE to

DE.
Most importantly, the proposition shows that, when there is a …xed default cost > 0, these two regions overlap leading to multiple equilibria.
and for B 0 B 0 , DE is self-con…rming and characterized by Finally, there are multiple equilibria when …nancing needs are within the region We provide intuition about the mechanism generating multiplicity with the use of a graphical example in Figure 1. In doing so, we represent the model as a system of three equations: (a) A minimum haircut given by (8) which is increasing in the …xed cost of default and independent of the government's period 2 debt burden B, (b) a non-linear optimal haircut schedule (B) described by (6) and Table 1, and (c) a market …nancing schedule B ( ; B 0 ) given by (12).
The market …nancing schedule is upward sloping as an increase in the anticipated haircut on sovereign bonds reduces their market price q b , driving the government to issue more bonds to meet its initial …nancing need B 0 . The optimal haircut schedule is also upward sloping in the interior region 2 ( ; 1) because the government's optimal policy plan prescribes a constant primary surplus T G at a level pinned down by marginal distortions as explained in Section 2.1. Figure 1 illustrates the equilibrium solutions at levels of …nancing need B 0 < B 0 0 < B 00 0whereby a higher …nancing need causes a shift to the right in the market …nancing schedule. Equilibrium occurs at the combinations of (B; ) where the market …nancing and optimal haircut schedules intersect. At B 0 , these schedules only intersect at (B; ) = B 0 q ; 0 such that there is a unique no default equilibrium denoted as N DE. In this equilibrium, the government has access to funds at the risk-free rate q 1 . In contrast, there are multiple equilibria at B 0 0 with N DE 0 pointing at an equilibrium with no default and DE 0 at an equilibrium with default in state L. The source of multiplicity here is the discontinuity of the optimal haircut in the region [0; L ], which stems from the …xed default cost > 0. Finally, at B 00 0 , there is a unique equilibrium DE 00 at the upper corner for haircuts = 1 such that B DE 00 = [q (1 )] 1 B 00 0 . 6 Figure 1: Equilibria across initial debt Note: The market …nancing, minimum haircut and optimal haircut schedules are respectively given by (11), (8) and Table 1. In the next section, we describe the changes in these equilibrium allocations as a result of interventions that alter the seniority structure of sovereign debt.

The e¤ect of tranching when senior debt is riskless
In this section, we reconsider the model equilibrium by allowing for a share 0 ! 1 of government bonds to have senior status. To begin with, we …nd it convenient to assume that the senior tranche is free of default risk and priced on par with risk-free assets at q s = q. This is equivalent to positing prohibitively high costs of defaulting on senior bonds. We will drop the assumption of non-defaultability in Section 4.
For a given share of senior debt, junior bonds are traded at the price q b re ‡ecting expected returns as per the debt pricing condition (10). The market …nancing schedule can then be written as where is the haircut on junior bonds. The government's period 2 budget constraint and default condition (for junior bonds) become, respectively in re ‡ection of full repayment of senior bonds.
In an interior solution, the discretionary government internalizes only the budgetary cost of a marginal increase in the haircut on junior bonds. This leaves the trade-o¤ between tax distortions and default costs same as in Section 2.1 with (5) determining the primary surplus T G independently of the senior tranche size !. A key implication is that, in the absence of a rise in the primary surplus, tranching leads to the re-allocation of repayments from junior to senior bondholders. To avoid default on the senior tranche, the government haircuts bonds in the junior tranche bŷ which is increasing in !.
Note also that tranching reduces the revenues gained from default at a given haircut. With a …xed default cost > 0, this increases the minimum haircut required to make (interior) default optimal. is implicitly de…ned by the expression (1 !) ! + (1 !) (1 (1 ) ) T G and positively related to ! while remaining independent of B.
For^ < , the optimal policy plan is free of default and remains identical to the case without tranching. For^ > 1 and the optimal policy plan involves complete default = 1 on junior bonds. In this case, tax revenues are given by the expression and naturally increasing in the share of senior bonds !, which must still be repaid in full. Finally, Table 2 summarizes the revised optimal policy plan under tranching, where the …rst two columns describe the government's optimal haircut schedule for junior bonds Observe that, on the one hand, tranching moves the optimal haircut towards complete default on junior bonds (i.e. the upper corner solution with = 1) since both the interior and minimum haircut expressions ^ ; increase in !. On the other hand, with su¢ ciently high !, the default condition in the upper corner given by (21) is not satis…ed such that tranching may eliminate default. 7 Elaborating on these two e¤ects, in the rest of the section we will analyze whether and under what conditions tranching sovereign debt may prevent debt crises. A full description of equilibria under tranching is provided in the Appendix B.

A Modigliani-Miller irrelevance result
We start by establishing that, as long as the optimal haircut lies in the interior region, tranching changes neither the equilibrium boundaries (in terms of government …nancing needs B 0 ) nor the government's debt bill B. This stems from our earlier …nding that the government optimally increases the haircut on junior bonds to fully repay the senior tranche. In anticipation of a lower expected return, investors discount junior bond prices q b which, under risk neutrality, exactly o¤sets the move to a risk-free price q in the senior tranche.
Proposition 2 Let ! denote the minimum senior tranche size that leads to a corner solution in the default equilibrium (DE) such that For all ! < !, we can write where B DE ; ; B 0 ; B 0 are respectively given by (15), (20), (13) and (14).

Proof. See Appendix A.2.
This …nding is naturally interpreted in light of Modigliani and Miller (1958)'s irrelevance result. Without a change in total payments to bondholders, tranching is irrelevant under risk neutrality. In other words, the price of bond, !q + (1 !) q b ; and the debt bill B are independent of ! such that tranching has no impact on government incentives to default. Constancy of the marginal (budgetary) default cost and hence the (interior) primary surplus T G are crucial for this result; were to increase (decrease) in , tranching would lead to a rise (fall) in the primary surplus and lower (higher) government borrowing costs and incentives to default.

Figure 2: Tranching at an interior equilibrium
Note: The market …nancing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market …nancing schedule re ‡ects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.
We use Figure 2 to provide further intuition. At senior tranche size !, the market …nancing and optimal haircut schedules intersect twice, leading to multiple equilibria. First, there is a no default equilibrium (N DE) with no haircut on public debt and bonds issued at the risk-free price B N DE = B 0 =q. Second, there is a default equilibrium (DE) with a haircut DE on junior bonds. In this equilibrium, the anticipation of default leads to a reduction in junior bond prices and a higher debt bill B DE .
Increasing the size of the senior tranche to ! 0 > ! has three distinct e¤ects. First, it pivots the market …nancing schedule B ( ; B 0 ; !) leftward. Provided that investors anticipate a default, senior bonds are sold at a higher price q > q b than junior bonds. Therefore, issuing a greater share of its debt in a senior tranche reduces the government's total debt bill B at a given haircut for junior bonds. Second, it increases the minimum haircut from to 0 . A rise in the share of senior bonds reduces the "default base" from which the government may extract resources through default. This necessitates a larger haircut to make default optimal given a …xed default cost . Third, it causes an upward shift in the optimal haircut schedule (B; !) within the interior region. Since the primary surplus remains constant in an interior solution, the government increases the haircut on junior bonds in order to fully repay a larger share of senior bonds.
N DE naturally remains unchanged in response to the rise in ! as there is no distinction between senior and junior tranches without default risk. In contrast, the default equilibrium moves from DE to DE 0 due to the …rst and third e¤ects. As explained above, the haircut on junior bonds rises from DE to a level DE0 that exactly o¤sets any movements in the government's debt bill B DE . Moreover, Proposition 2 indicates that DE rises by a greater amount than the minimum haircut . Therefore, equilibrium boundaries are not a¤ected and tranching is ine¤ective at eliminating debt crises.

Non-neutrality
The irrelevance result above is conditional on an interior solution DE < 1. We now show that our version of the Modigliani-Miller theorem fails when default rates on junior bonds are at the corner, that is, default is complete.
In Figure 2, an increase in the share of senior debt ! drives the optimal haircut towards the upper corner. Proposition 2 establishes that there is a threshold ! such that, for any share of the senior debt in excess of it, DE has a corner solution with DE = 1. We now show that, at the corner solution, the …ndings for the interior case are reversed: increasing ! reduces the government's borrowing costs and the total debt bill B, and eventually eliminates default. These …ndings are formally stated by Proposition 3 below. This proposition also shows that the minimum senior tranche share ! 2 (!; 1) required to eliminate default increases in the government's …nancing need B 0 .
Proposition 3 For ! > !, the default equilibrium has a corner solution characterized by There exists a minimum senior tranche ! above which there is no default, implicitly de…ned by the expression The key insight here is that the government may not respond to tranching by increasing the haircut on junior bonds when it is already at the upper corner. Instead, it is driven to increase tax revenues in order to fully repay the senior tranche. Since tranching leads to a rise in total payments to bondholders, the ine¤ectiveness result from Modigliani and Miller (1958) no longer holds.
There are two distinct channels through which tranching a¤ects the government's default decision. First, without a rise in the haircut, junior bond prices q b remain constant in response to an increase in !. As bonds in the senior tranche trade at a higher price q > q b , tranching reduces the government's average borrowing costs !q + (1 !) q b 1 and total debt bill B. Consequently, tax revenues T = G + B required to avoid default and the associated tax distortions decrease.
Second, with higher tax revenues needed to repay senior bonds during default, tax distortions z (G + (! + (1 !) ) B) under default increase with a rise in !. Together, these two channel decrease the potential reduction in tax distortions by defaulting on public debt and  Table 2. The market …nancing schedule re ‡ects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds. therefore the government's temptation to do so. Given the …xed cost of default, a su¢ ciently large senior tranche then prevents default.
Finally, we use Figure 3 to contrast the e¤ects of tranching in a corner solution with the interior case shown in Figure 2. The baseline scenario with multiple equilibria (N DE; DE) is identical in the two …gures but ! 00 in Figure 3 is su¢ ciently large to move the minimum haircut to the upper corner. Consequently, the optimal haircut schedule (B 0 ; !) loses its interior region and jumps directly from no default to a complete haircut at a boundary determined by (21). An increase in ! moves the default boundary to the right while the market …nancing schedule B ( ; B 0 ; !) pivots leftward as in Figure 2. When the senior tranche is large enough, the market …nancing and optimal haircut schedules no longer intersect above = 0, and the default equilibrium DE is eliminated.
Note that, as before, N DE is not a¤ected by tranching since seniority is irrelevant in the absence of default risk. Indeed, when tranching is successful in ruling out debt crises, the prospect of a complete haircut on junior bonds remains completely o¤-equilibrium and both junior and senior bonds are priced at risk free level q b = q.

Risky senior debt
In this section, we move away from the extreme assumption that the senior tranche is nondefaultable. We show that our main results generalize to the plausible case in which, even if the costs of defaulting on senior bonds are larger than for junior bonds, the government may (choose to) default also on senior bonds The government's budget constraint and market …nancing schedule now are: where s denotes the haircut on the senior tranche, and the prices of senior and junior bonds are respectively given by In an interior solution, the haircut on the senior tranche is determined by the expression with taxesT pinned down by the same …rst order condition (5) and self-con…rming for all ! > ! where ! is implicitly de…ned by the expression For su¢ ciently high s , the set ! 2 (! ; !) is non-empty.
Proof. See Appendix A.4 According to Proposition 4, government's funding costs and taxes are equivalent between SDE and the default equilibrium DE Once the government defaults on the senior tranche, the Modigliani-Miller irrelevance result described in Section 3.1 holds. The government has no incentive to raise tax revenues and average funding costs remain constant regardless of how revenues are allocated between the two tranches. Therefore, tranching is ine¤ective not only when the senior tranche size falls short of ! (and there is an interior haircut on the junior tranche) but also when it exceeds ! leading to default on the senior tranche.
The key implication is that tranching requires higher …xed default costs on the senior tranche to generate commitment to repay. The last part of Proposition 4 indicates that, for su¢ ciently high s , there exists an intermediate region of senior tranche size ! 2 (! ; !) where tranching eliminates default. Since ! > !, the junior tranche is haircut fully in this region. Provided s is high enough, however, the government chooses to increase tax revenues rather than defaulting on the senior tranche. The resulting decrease in the government's exante funding costs then eliminate default. Since ! is increasing in B 0 while ! decreases in it, greater government …nancing needs require higher levels of s for tranching to be e¤ective.

A numerical illustration
In this section, we provide a numerical example to demonstrate debt crises in the model economy and the circumstances under which tranching may eliminate these crises.
We adopt the following calibration for the numerical example. For tax distortions, we use the functional form z i (T i ) = i T 2 i such that tax revenues in the interior solutions are given byT i = [2 i (1 )] 1 . We set H to an arbitrarily small value in line with ruling out default in state H and calibrate L > H to attain a primary surplus ofT L G = 0:40 (where G = 0 without loss of generality). Expected output is normalized to unity such that the primary surplus can be interpreted as 40% of expected GDP. The …xed and budgetary costs of default are calibrated to = = 0:10 amounting to 10% of expected GDP each when B 0 = 1 (we consider di¤erent calibrations for s below). Given the two-period structure of our model, these …gures are in terms of present discounted values. Finally, q is set to 0:99 consistent with a risk-free interest rate of 1% and we also set = 0:10 in line with the regularity conditions in Section 2.3. Figure 4 depicts the equilibria without tranching (! = 0). Panel 1 shows the evolution of haircuts L (on the y-axis) against the government's initial …nancing needs B 0 (on the Due to the …xed default costs > 0, the haircut in state L rises discretely to a minimum haircut = 0:76 as the equilibrium switches from N DE to DE. In the range of B 0 displayed, the haircut remains in the interior region and therefore continues to rise with B 0 . Panel 2 shows that discrete changes in haircuts also cause jumps in the government's debt bill B. It also indicates that, under our calibration, sovereign spreads over the risk-free rate are around 10% in DE. Figure 5 plots the relevant threshold values for the senior tranche size ! (on the y-axis) across B 0 . In Panel 1, ! depicts the minimum tranche size required to take the haircut to the upper corner. For ! < ! L , tranching is completely ine¤ective as per the Modigliani-Miller ine¤ectiveness result and only serves to re-allocate payments from the junior to the senior tranche. The schedule ! shows the minimum senior tranche size required to prevent default, thus ruling out DE.
Observe that as B 0 rises, ! decreases while ! increases. Therefore, at high levels of B 0 , smaller tranche sizes may have an e¤ect while eliminating default require as very large senior tranche. Eliminating the bad equilibrium in multiplicity regions, however, only requires a senior tranche size of approximately 50%.

Figure 5: Tranching
Finally, Panel 2 plots the threshold ! above which there is default on the senior tranche at di¤erent levels of s , the additional …xed default cost for doing so. At s = 0, there is no intermediate region between ! and ! where tranching may have an e¤ect. We show that as s rises, however, ! shifts up and senior tranche sizes may get large to prevent default at increasingly higher values of B 0 .

Conclusion
We have analyzed the e¤ects of tranching on sovereign risk in a standard model of debt crises with risk neutral creditors, costly default and a government that faces a trade-o¤ between distortionary taxation and applying an (optimal) haircut to bondholders. Introducing tranching into the model yields two important insights. First, default costs on senior bonds may transpire into commitment to repay both the senior and junior tranches. Second, the e¤ect of tranching on resilience to debt crises is highly non-linear in the senior tranche size.
We found that tranching only redistributes government revenues from junior to senior bondholders unless default costs are higher for senior than for junior bonds, and the senior tranche size is su¢ ciently but not too large. We characterize a lower and an upper threshold for this tranche that de…nes the region over which the seniority structure is not neutral.
Remarkably, in this intermediate region, tranching reduces vulnerability to debt crises. This is because, without any remaining funds to redistribute from the junior tranche, the government is forced to raise tax revenues to pay the senior tranche. This results in an increase in total payments to bondholders, which in turn leads to a fall in government borrowing costs given default expectations. Borrowing costs may then fall enough to eliminate the default equilibrium altogether. We should stress in conclusion that our results can be applied more broadly. The mechanisms considered here are relevant for any situation in which there are …xed costs associated with default, and agents may take costly but non-contractible actions to increase recovery rates. Therefore, there may be a role for tranching corporate debt in sectors with signi…cant moral hazard. (1 ) which in terms of initial …nancing need becomes Using (3), a su¢ cient condition for N DE to be self-con…rming is Default equilibrium (DE) Now consider an equilibrium with default in state L such that 2 [ ; 1]. In an interior solution =^ 2 [ ; 1), we will havê In terms of initial …nancing needs, the equilibrium is then characterized aŝ There are three conditions for this interior equilibrium to exist: and in terms of …nancing needs, they become and the equilibrium condition iŝ With continuity between interior and corner equilibria, the boundary above with DE is self-con…rming is given by^ This is non-empty under the condition 1 1 (1 ) T G <T G 1 (1 ) ) > 0 which will be satis…ed when there is a strictly positive …xed cost of default > 0.

A.2 Proof of Proposition 2
The solution for the interior case of DE is provided in Appendix B. Using this solution, (32a) indicates that @B DE @! = 0 as long as ! remains at a level that does not lead to a corner solution DE = 1. Using (31), we can also show that To determine how ! a¤ects the minimum haircut, let we can show that This is negative under the condition which must be satis…ed since z 0 T = 1 by virtue of the interior FOC and z 00 (:) > 0. Moreover, using (28), we can also write Through the implicit function theorem, we can deduce that for all (!; ) within the unit circle. Using (29), we can then write Finally, to show that the boundaries B 0 ; B 0 are independent of !, let where (18) indicates that g (!; B) = 0 at a boundary. This yields the derivatives ©International Monetary Fund. Not for Redistribution In an interior solution, we know that Combining these with the derivatives above indicates and by the implicit function theorem we have that, on a boundary B 0 ; B 0 Since, B is a positive function of B 0 and independent of ! in an interior solution, we can further write @B 0 @! = @ B 0 @! = 0 In other words, ! has no e¤ect on the equilibrium boundaries when the haircut falls short of an upper corner DE = 1.

A.3 Proof of Proposition 3
B DE is given by (33) Since < 1, a su¢ cient condition for @f (!;D) @! < 0 and @f (!;D) which must be true since Therefore, using the implicit function theorem and f (! ; B 0 ) = 0, we can write To begin with, consider the interior equilibrium. Combining (24) with (25) yields where taxesT are pinned down by (5). The interior equilibrium is valid under the condition Otherwise, the equilibrium is at the corner SDE s = 1 with debt and taxes given by The default condition can be written as which in an interior equilibrium becomes To analyze the properties of ! let such that f (B 0 ; s ; !) = 0. We can then write the derivatives it is straightforward to show that we will have under the regularity condition 1 > . The derivatives of f (!; B 0 ) with respect to (!; B 0 ) will then also be positive under the condition which must be true since z 0 T = 1 , z 00 (:) > 0 Using the implicit function theorem, we can then write @ ! @ s > 0 @ ! @B 0 < 0 38 ©International Monetary Fund. Not for Redistribution Note also that using (30) and (22) , we can also show that f (B 0 ; s ; !) = s which indicates that we will have ! > ! for any s > 0. Finally, we conduct the same analysis for the corner solution, in which case the default condition is We can then write We begin with the interior case, where DE is characterized by the set of expressions which in terms of initial …nancing needs becomê Observe that tax revenues in the defaulting state are not a¤ected by !. Instead, the government increases the haircuts on junior bonds to fully pay the senior tranche and junior bond prices q DE b decline in anticipation of a higher haircut. Under risk neutrality, the decline in q DE b exactly o¤sets the higher prices for the senior tranche such that the government's total debt bill remains unchanged compared to the case without tranching In this case, the condition for DE to be self-con…rming remains identical to the case without tranching described in Proposition 1 When ! > !, on the other hand, DE is characterized by the (upper) corner case such that Since the government is unable to increase the haircut on junior bonds further, a rise in ! leads to a rise in tax revenues collected during default. Moreover, as the junior bond prices remain …xed at q DE+ b < q b in the corner solution, further increases in ! reduce the government's borrowing costs. Therefore, the debt bill B DE+ decreases in ! as per the expression Finally, DE is self-con…rming in the range of …nancing need B 0 B 0 where B 0 is implicitly by the expression z G + 1 1 (1 !) such that @B 0 @! > 0 8 ! 2 (!; 1)