1 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 fiscal fundamentals often pursue tranching by granting seniority to official 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 fiscal policy and debt sustainability. First and foremost, can issuing senior and junior debt affect prospective (state-contingent) primary deficits, 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 ‘financial engineering’is neutral with respect to overall debt sustainability.

Second, if the seniority structure of government debt is not neutral for fiscal policy, can a government rely on it to contain vulnerability to self-fulfilling 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 of distortionary taxation against an optimal haircut to bondholders. Our key result is that the effect 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 affect state-contingent primary surpluses, but only the distribution of cash flows across bond types. However, intermediate levels of tranching may reduce the cost of financing 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-fulfilling 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 financing 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 affect 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 financing 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 low among holders of different 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 flows 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 differential, the extra costs associated with reneging on senior liabilities create a discontinuity in the government’s optimal fiscal policy. Depending on the cost differential, 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 effect may eliminate bad equilibria with self-fulfilling 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 final 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 ineffectiveness 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 fixed 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 find that tranching may reduce the government’s borrowing costs. However, our framework differs from this literature in two fundamental ways. First, we allow for default on the senior tranche, which effectively 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 benefit of Eurobonds is to pool risk and prevent liquidity runs on individual countries through mutualization, while tranching serves to alleviate moral hazard. In a different vein, Beck et al. (2011) and Brunnermeier et al. (2016, 2017) analyze Eurobonds without joint guarantees. Rather than risk sharing, these studies focus on the benefits of providing a safe asset for banks in default-risky countries through a combination of diversification 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 differential 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-fulfilling debt crises. Section 3 introduces tranching. Section 4 considers default on senior debt. Section 5 provides a numerical illustration. Section 6 concludes.

2 A bare-bones model of sovereign debt crises

Consider a two-period endowment economy populated by a unit continuum of identical risk-neutral households which derive utility from consuming in period 2 only. In the first period, households are endowed with a stock of financial wealth W₀ 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 infinitely 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 − γ) and γ respectively. Households also receive payoffs from their assets, pay taxes, and consume, leading to the set of budget constraints

where i ∈ {L, H} 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. θ_i ∈ [0,1] represents a potential haircut on public debt which leads to a fixed 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) financing need B₀ by issuing discount bonds B = B₀/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 α ∈ (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.

2.1 Optimal default and taxation plans under discretion

The government’s optimization problem can be written as

$\underset{T_i>0,{\theta }_i\in [0,1]}{\max }{Y}_i-z_i\left(T_i\right)-{\text{Φ}}_i-T_i+K+(1-{\theta }_i)B$

subject to (4). Due to the boundary constraints on the haircut, there are two corner solutions, respectively with no default (θ_i = 0) and complete default (θ_i = 1), as well as an interior solution ${\theta }_i={\hat{\theta }}_i\in (0,1)$. We find it convenient to begin with a description of the interior solution, where tax revenues are pinned down by a first order condition

$\begin{array}{cc}{z}_i^{\prime }\left({\hat{T}}_i\right)=\frac{\alpha }{1-\alpha }& \left(5\right)\end{array}$

that trades-of marginal tax distortions against (fractional) budgetary default costs. Combining (4) and (5) yields an expression for the (interior) haircut

$\begin{array}{cc}{\hat{\theta }}_i=\frac{1}{1-\alpha }(1-\frac{{\hat{T}}_i-G}{B})& \left(6\right)\end{array}$

These expressions indicate that, in an interior solution, tax revenues are not affected 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 fixed costs Φ 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

$\begin{array}{cc}{z}_i(G+B)>z_i\left(T_i\right)+\mathrm{\Phi }+\alpha {\theta }_{i}B& \left(7\right)\end{array}$

When considering an interior solution, the default condition can be written in terms of a “minimum haircut” ${\underset{\_}{\theta }}_i$, defined implicitly by the expression

$\begin{array}{cc}{z}_i\left({\hat{T}}_i\right)+\mathrm{\Phi }=z_i({\hat{T}}_i+\frac{{\underset{\_}{\theta }}_i(1-a)}{1-{\underset{\_}{\theta }}_i(1-a)}({\hat{T}}_i-G))-\frac{a{\underset{\_}{\theta }}_i}{1-{\underset{\_}{\theta }}_i(1-a)}({\hat{T}}_i-G)& \left(8\right)\end{array}$

such that the interior solution is optimal for ${\hat{\theta }}_i\ge {\underset{\_}{\theta }}_i$. Observe that ${\underset{\_}{\theta }}_i$ is increasing in the fixed default cost Φ but independent of B. Moreover, the minimum haircut is higher in states with less tax distortions, that is ${\underset{\_}{\theta }}_L<{\underset{\_}{\theta }}_H$.

For ${\hat{\theta }}_i<{\underset{\_}{\theta }}_i$, optimal policy is characterized by a no-default corner outcome θ_i = 0 with a tax bill

$T_i=G+B$

that is consistent with complete repayment of debt.

Finally, the corner solution with complete default may be optimal when ${\hat{\theta }}_i>1$. In this case, the relevant default condition is given by combining (7) with θ_i = 1 such that

$\begin{array}{cc}{z}_i(G+B)>z_i(G+\alpha B)+\mathrm{\Phi }+\alpha B& \left(9\right)\end{array}$

and taxes are set at

$T_i=G+\alpha B$

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.

Table 1:

Government’s optimal discretionary plan

Table 1:

Government’s optimal discretionary plan

$\text{I}\text{f}{\hat{\theta }}_i>1,z_i(G+B)>z_i(G+\alpha B)+\mathrm{\Phi }+\alpha B$:	θi = 1	Ti = G + αB
$\text{I}\text{f}{\hat{\theta }}_i\in [{\underset{\_}{\theta }}_i,1]$:	${\theta }_i={\hat{\theta }}_i$	$T_i={\hat{T}}_i=z_i^{\prime -1}\left(\frac{\alpha }{1-\alpha }\right)$
Otherwise :	θi = 0	Ti = G + B

View Table

Table 1:

Government’s optimal discretionary plan

$\text{I}\text{f}{\hat{\theta }}_i>1,z_i(G+B)>z_i(G+\alpha B)+\mathrm{\Phi }+\alpha B$:	θi = 1	Ti = G + αB
$\text{I}\text{f}{\hat{\theta }}_i\in [{\underset{\_}{\theta }}_i,1]$:	${\theta }_i={\hat{\theta }}_i$	$T_i={\hat{T}}_i=z_i^{\prime -1}\left(\frac{\alpha }{1-\alpha }\right)$
Otherwise :	θi = 0	Ti = G + B

2.4 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 “NDE”, whereas θ^NDE = 0; the superscript “DE” will be use for the equilibrium with default, whereas the haircut θ^DE > 0 will be strictly positive in state L. Proposition 1 characterizes these equilibria and shows that they exist in regions of financing need B₀ that line up monotonically, that is, as B₀ rises, we move from a region of NDE to DE. Most importantly, the proposition shows that, when there is a fixed default cost Φ > 0, these two regions overlap leading to multiple equilibria.

Proposition 1 Let

$\begin{array}{cc}{\overline{B}}_0\equiv \frac{q(\hat{T}-G)}{1-(1-\alpha )\underset{\_}{\theta }}& \left(13\right)\end{array}$

$\begin{array}{cc}{\underset{\_}{B}}_0\equiv (1-\gamma \underset{\_}{\theta })\frac{q(\hat{T}-G)}{1-(1-\alpha )\underset{\_}{\theta }}& \left(14\right)\end{array}$

For $B_0<{\overline{B}}_0$, NDE is self-confirming such that

$\begin{array}{ccc}{q}_b^{NDE}& =& q\hfill \\ B^{NDE}& =& B_0/q\hfill \\ {\theta }^{NDE}& =& 0\hfill \\ T^{NDE}& =& G+B^{NDE}\hfill \end{array}$

and for $B_0\ge {\underset{\_}{B}}_0$, DE is self-confirming and characterized by

$\begin{array}{cc}\begin{array}{ccc}{\theta }^{DE}& =& \min [\frac{B_0-(\hat{T}-G)q}{(1-\alpha )B_0-\gamma (\hat{T}-G)q},1]\hfill \\ q_b^{DE}& =& \{\begin{array}{cc}q(1-\gamma )\hfill & if{\theta }^{DE}=1\\ \frac{(1-\alpha -\gamma )q{B}_0}{(1-\alpha )B_0-\gamma q(\hat{T}-G)}\hfill & otherwise\end{array}\hfill \\ B^{DE}& =& B_0/q_b^{DE}\hfill \\ T^{DE}& =& \{\begin{array}{cc}G+\alpha B^{DE}\hfill & if\theta =1\\ \hat{T}\hfill & otherwise\end{array}\hfill \end{array}& \left(15\right)\end{array}$

Finally, there are multiple equilibria when financing needs are within the region

${\underset{\_}{B}}_0\le B_0<{\overline{B}}^0$

which is non-empty for Φ > 0.

Proof. See Appendix A.1. ◼

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 $\underset{\_}{\theta }$ given by (8) which is increasing in the fixed cost of default Φ and independent of the government’s period 2 debt burden B, (b) a non-linear optimal haircut schedule θ (B) described by (6) and Table 1, and (c) a market financing schedule B (θ, B₀) given by (12).

Equilibria across initial debt

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (11), (8) and Table 1.

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Equilibria across initial debt

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (11), (8) and Table 1.

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Equilibria across initial debt

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (11), (8) and Table 1.

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The market financing 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 financing need B₀. The optimal haircut schedule is also upward sloping in the interior region $\theta \in (\underset{\_}{\theta },1)$ because the government’s optimal policy plan prescribes a constant primary surplus $(\hat{T}-G)$ at a level pinned down by marginal distortions as explained in Section 2.1.

Figure 1 illustrates the equilibrium solutions at levels of financing need $B_0<B_0^{\prime }<B_0^{\prime \prime }$ — whereby a higher financing need causes a shift to the right in the market financing schedule. Equilibrium occurs at the combinations of (B, θ) where the market financing and optimal haircut schedules intersect. At B₀, these schedules only intersect at $(B,\theta )=(\frac{B_0}{q},0)$ such that there is a unique no default equilibrium denoted as NDE. In this equilibrium, the government has access to funds at the risk-free rate q⁻¹. In contrast, there are multiple equilibria at $B_0^{\prime }$ with NDE′ pointing at an equilibrium with no default and DE′ at an equilibrium with default in state L. The source of multiplicity here is the discontinuity of the optimal haircut in the region $[0,{\underset{\_}{\theta }}_L]$, which stems from the fixed default cost Φ > 0. Finally, at $B_0^{\prime \prime }$, there is a unique equilibrium DE′′ at the upper corner for haircuts θ = 1 such that $B^{D{E}^{\prime \prime }}={\left[q(1-\gamma )\right]}^{-1}{B}_0^{\prime \prime }$.⁶

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.

3 The effect of tranching when senior debt is riskless

In this section, we reconsider the model equilibrium by allowing for a share 0 ≤ ω ≤ 1 of government bonds to have senior status. To begin with, we find 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 reflecting expected returns as per the debt pricing condition (10). The market financing schedule can then be written as

where θ is the haircut on junior bonds. The government’s period 2 budget constraint and default condition (for junior bonds) become, respectively

in reflection of full repayment of senior bonds.

In an interior solution, the discretionary government internalizes only the budgetary cost a of a marginal increase in the haircut on junior bonds. This leaves the trade-of between tax distortions and default costs same as in Section 2.1 with (5) determining the primary surplus $(\hat{T}-G)$ independently of the senior tranche size ω. 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 by

which is increasing in ω.

Note also that tranching reduces the revenues gained from default at a given haircut. With a fixed default cost Φ > 0, this increases the minimum haircut $\underset{\_}{\theta }$ required to make (interior) default optimal. $\underset{\_}{\theta }$ is implicitly defined by the expression

and positively related to ω while remaining independent of B.

For $\hat{\theta }<\underset{\_}{\theta }$, the optimal policy plan is free of default and remains identical to the case without tranching. For $\hat{\theta }>1$ and

the optimal policy plan involves complete default θ = 1 on junior bonds. In this case, tax revenues are given by the expression

and naturally increasing in the share of senior bonds ω, which must still be repaid in full. Finally, Table 2 summarizes the revised optimal policy plan under tranching, where the first two columns describe the government’s optimal haircut schedule for junior bonds

Table 2:

Government’s optimal plan under tranching

Table 2:

Government’s optimal plan under tranching

$\text{If }\hat{\theta }>1,z(G+B)>z(G+(\omega +(1-\omega )\alpha )B)+\mathrm{\Phi }+\alpha (1-\omega )B$:	θ = 1	T = G + [ω + (1 − ω)α]B
$\text{If }\hat{\theta }\in [\underset{\_}{\theta },1]$:	$\theta =\hat{\theta }$	$T=\hat{T}=z^{\prime -1}\left(\frac{\alpha }{1-\alpha }\right)$
Otherwise :	θ = 0	T = G + B

View Table

Table 2:

Government’s optimal plan under tranching

$\text{If }\hat{\theta }>1,z(G+B)>z(G+(\omega +(1-\omega )\alpha )B)+\mathrm{\Phi }+\alpha (1-\omega )B$:	θ = 1	T = G + [ω + (1 − ω)α]B
$\text{If }\hat{\theta }\in [\underset{\_}{\theta },1]$:	$\theta =\hat{\theta }$	$T=\hat{T}=z^{\prime -1}\left(\frac{\alpha }{1-\alpha }\right)$
Otherwise :	θ = 0	T = G + B

Observe that, on the one hand, tranching moves the optimal haircut towards complete default on junior bonds (i.e. the upper corner solution with θ = 1) since both the interior and minimum haircut expressions $(\hat{\theta },\underset{\_}{\theta })$ increase in ω. On the other hand, with sufficiently high ω, the default condition in the upper corner given by (21) is not satisfied such that tranching may eliminate default.⁷ Elaborating on these two effects, 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.

3.1 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 financing needs B₀) nor the government’s debt bill B. This stems from our earlier finding 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 offsets the move to a risk-free price q in the senior tranche.

Proposition 2 Let $\underset{\_}{\omega }$ denote the minimum senior tranche size that leads to a corner solution in the default equilibrium (DE) such that

$\begin{array}{cc}\begin{array}{c}\underset{\_}{\omega }\equiv \frac{(1-\gamma )(\hat{T}-G)-\alpha \frac{B_0}{q}}{(1-\alpha )\frac{B_0}{q}-\gamma (\hat{T}-G)}\in \left[0,1\right)\\ \frac{\partial \underset{\_}{\omega }}{\partial B_0}<0\end{array}& \left(22\right)\end{array}$

For all $\omega <\underset{\_}{\omega }$, we can write

$\begin{array}{cc}& T^{DE}=\hat{T}\\ {\theta }^{DE}& =\frac{1}{1-\omega }\frac{\frac{B_0}{q}-(\hat{T}-G)}{(1-\alpha )\frac{B_0}{q}-\gamma (\hat{T}-G)}<1\\ & \frac{\partial {\theta }^{DE}}{\partial \omega }>\frac{\partial \underset{\_}{\theta }}{\partial \omega }>0\forall B_0>{\underset{\_}{B}}_0\\ & \frac{\partial B^{DE}}{\partial \omega }=\frac{\partial {\overline{B}}_0}{\partial \omega }=\frac{\partial {\underset{\_}{B}}_0}{\partial \omega }=0\end{array}$

where $(B^{DE},\underset{\_}{\theta },{\overline{B}}_0,{\underset{\_}{B}}_0)$ are respectively given by (15), (20), (13) and (14).

Proof. See Appendix A.2. ◼

This finding 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, ωq + (1 − ω) q_b, and the debt bill B are independent of ω such that tranching has no impact on government incentives to default. Constancy of the marginal (budgetary) default cost α and hence the (interior) primary surplus $(\hat{T}-G)$ are crucial for this result; were α to increase (decrease) in θ, tranching would lead to a rise (fall) in the primary surplus and lower (higher) government borrowing costs and incentives to default.

We use Figure 2 to provide further intuition. At senior tranche size ω, the market financing and optimal haircut schedules intersect twice, leading to multiple equilibria. First, there is a no default equilibrium (NDE) with no haircut on public debt and bonds issued at the risk-free price B^NDE = B₀/q. Second, there is a default equilibrium (DE) with a haircut θ^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.

Tranching at an interior equilibrium

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market financing schedule reflects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.

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Tranching at an interior equilibrium

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market financing schedule reflects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.

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Tranching at an interior equilibrium

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market financing schedule reflects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.

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Increasing the size of the senior tranche to ω′ < ω has three distinct effects. First, it pivots the market financing schedule B (θ, B₀, ω) 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 $\underset{\_}{\theta }$ to ${\underset{\_}{\theta }}^{\prime }$. 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 fixed default cost Φ. Third, it causes an upward shift in the optimal haircut schedule θ (B, ω) 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.

NDE naturally remains unchanged in response to the rise in ω as there is no distinction between senior and junior tranches without default risk. In contrast, the default equilibrium moves from DE to DE′ due to the first and third effects. As explained above, the haircut on junior bonds rises from θ^DE to a level θ^DE′ that exactly offsets any movements in the government’s debt bill B^DE. Moreover, Proposition 2 indicates that θ^DE rises by a greater amount than the minimum haircut $\underset{\_}{\theta }$. Therefore, equilibrium boundaries are not affected and tranching is ineffective at eliminating debt crises.

3.2 Non-neutrality

The irrelevance result above is conditional on an interior solution θ^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 ω drives the optimal haircut towards the upper corner. Proposition 2 establishes that there is a threshold $\underset{\_}{\omega }$ such that, for any share of the senior debt in excess of it, DE has a corner solution with θ^DE = 1. We now show that, at the corner solution, the findings for the interior case are reversed: increasing ω reduces the government’s borrowing costs and the total debt bill B, and eventually eliminates default. These findings are formally stated by Proposition 3 below. This proposition also shows that the minimum senior tranche share ${\omega }^{*}\in (\underset{\_}{\omega },1)$ required to eliminate default increases in the government’s financing need B₀.

Proposition 3 For $\omega >\underset{\_}{\omega }$, the default equilibrium has a corner solution characterized by

$\begin{array}{c}{\theta }^{DE}=1\\ T^{DE}=G+(\omega +(1-\omega )\alpha )B^{DE}\\ B^{DE}=\frac{B_0}{q(1-\gamma (1-\omega ))}\end{array}$

such that

$\begin{array}{ccc}\frac{\partial T^{DE}}{\partial \omega }& >& 0\\ \frac{\partial B^{DE}}{\partial \omega }& <& 0\end{array}$

There exists a minimum senior tranche ω^* above which there is no default, implicitly defined by the expression

$\begin{array}{cc}\begin{array}{cc}z(G+\frac{1}{1-\gamma (1-{\omega }^{*})}\frac{B_0}{q})=& z(G+\frac{{\omega }^{*}+(1-{\omega }^{*})\alpha }{1-\gamma (1-{\omega }^{*})}\frac{B_0}{q})\hfill \\ & +\mathrm{\Phi }+\frac{(1-{\omega }^{*})\alpha }{1-\gamma (1-{\omega }^{*})}\frac{B_0}{q}\hfill \end{array}& \left(23\right)\end{array}$

such that

$\begin{array}{c}\underset{\_}{\omega }<{\omega }^{*}<1\\ \frac{\partial {\omega }^{*}}{\partial B_0}>0\end{array}$

Proof. See Appendix A.3. ◼

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 ineffectiveness result from Modigliani and Miller (1958) no longer holds.

There are two distinct channels through which tranching affects 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 ω. As bonds in the senior tranche trade at a higher price q > q^b, tranching reduces the government’s average borrowing costs (ωq + (1 − ω) 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 + (ω + (1 − ω) α) B) under default increase with a rise in ω. Together, these two channel decrease the potential reduction in tax distortions by defaulting on public debt and therefore the government’s temptation to do so. Given the fixed cost of default, a sufficiently large senior tranche then prevents default.

Finally, we use Figure 3 to contrast the effects of tranching in a corner solution with the interior case shown in Figure 2. The baseline scenario with multiple equilibria (NDE, DE) is identical in the two figures but ω″ in Figure 3 is sufficiently large to move the minimum haircut $\underset{\_}{\theta }$ to the upper corner. Consequently, the optimal haircut schedule θ (B₀, ω) loses its interior region and jumps directly from no default to a complete haircut at a boundary determined by (21). An increase in ω moves the default boundary to the right while the market financing schedule B (θ, B₀, ω) pivots leftward as in Figure 2. When the senior tranche is large enough, the market financing and optimal haircut schedules no longer intersect above θ = 0, and the default equilibrium DE is eliminated.

Tranching to eliminate default

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market financing schedule reflects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.

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Tranching to eliminate default

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market financing schedule reflects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.

• Download Figure
• Download figure as PowerPoint slide

Tranching to eliminate default

Citation: IMF Working Papers 2018, 104; 10.5089/9781484353691.001.A001

Note: The market financing, minimum haircut and optimal haircut schedules are respectively given by (16), (20) and Table 2. The market financing schedule reflects the total debt bill while the minimum haircut and the optimal haircut schedule pertain to junior bonds.

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Note that, as before, NDE is not affected 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 of-equilibrium and both junior and senior bonds are priced at risk free level q_b = q.

4 Risky senior debt

In this section, we move away from the extreme assumption that the senior tranche is non-defaultable. 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 financing schedule now are:

where θ_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 taxes $\hat{T}$ pinned down by the same first order condition (5) as before. Similarly, in a corner solution with complete default on both tranches (θ_s = θ = 1), taxes are given by

Proposition 4 describes a “senior default equilibrium” (SDE) where the government optimally defaults on both tranches.

Proposition 4 Let SDE denote an equilibrium where the government defaults on the senior tranche. SDE is characterized by