This paper examines how tranching, or issuing different classes of debt at different levels of seniority, can affect government incentives to default on debt and vulnerability to debt crises. The paper develops a model where a government chooses optimal tax rates and haircuts on debt to maximize welfare. It finds that tranching is only effective at reducing default risk when the senior tranche is large enough to push the government into a "corner solution" where it fully defaults on the junior tranche. With an intermediate senior tranche size, tranching has no effect due to risk neutrality and convex tax distortions. However, making default on the senior tranche sufficiently costly can enable tranching to eliminate self-fulfilling debt crises even with an

1. Debt Seniority and Self-Fulfilling Debt Crises
Anil Ari1 Giancarlo Corsetti2,5 Luca Dedola3,4,5
1International Monetary Fund
2University of Cambridge
3Danmarks Nationalbank
4European Central Bank
5CEPR
April 6, 2018
Disclaimer: The views expressed are those of the authors only and do not represent the views of the IMF, its Executive Board,
IMF management, the ECB, the Eurosystem, Danmarks Nationalbank or any institution to which the authors are affiliated.

2. This Paper
Questions:
- How does tranching affect government incentives to default?
- Can changing the seniority structure of government debt reduce
vulnerability to debt crises?
- Why do countries tranch their debt?

3. This Paper
Questions:
- How does tranching affect government incentives to default?
- Can changing the seniority structure of government debt reduce
vulnerability to debt crises?
- Why do countries tranch their debt?
Motivation: Recent debate on “Eurobonds” in the euro area
- Proposals for senior (blue) and junior (red) bonds
- Risk sharing, demand for safe assets
- This paper: costly default on senior tranch

4. This Paper
Questions:
- How does tranching affect government incentives to default?
- Can changing the seniority structure of government debt reduce
vulnerability to debt crises?
- Why do countries tranch their debt?
Motivation: Recent debate on “Eurobonds” in the euro area
- Proposals for senior (blue) and junior (red) bonds
- Risk sharing, demand for safe assets
- This paper: costly default on senior tranch
Many examples of countries (implicitly) tranching public debt
- Bonds issued under English or US law
- Seniority of official lending

5. Evidence on seniority
Source: Schlegl, Trebesch & Wright (2016)

6. This Paper
Approach:
- Bare-bones model of self-fulfilling debt crises
- Risk neutrality, no risk sharing or safe asset demand
- Government chooses haircut optimally but cannot pre-commit
- Trade-off: Default costs vs. tax distortions

7. This Paper
Approach:
- Bare-bones model of self-fulfilling debt crises
- Risk neutrality, no risk sharing or safe asset demand
- Government chooses haircut optimally but cannot pre-commit
- Trade-off: Default costs vs. tax distortions
Contribution:
- Default on senior tranch is (more) costly
- Tranching as a commitment device
- Can prevent default on both senior and junior tranch

8. This Paper
Approach:
- Bare-bones model of self-fulfilling debt crises
- Risk neutrality, no risk sharing or safe asset demand
- Government chooses haircut optimally but cannot pre-commit
- Trade-off: Default costs vs. tax distortions
Contribution:
- Default on senior tranch is (more) costly
- Tranching as a commitment device
- Can prevent default on both senior and junior tranch
Preview of results:
- Effects of tranching depend non-linearly on size of senior tranch
- Ineffective when senior tranch is too small/large (Modigliani-Miller)
- Intermediate senior tranch size: eliminate default on both tranches

9. Numerical illustration
% senior tranch Haircut on senior Haircut on junior Avg. sov. yield
0% - 77% 9.47%
10% 0% 86% 9.47%
30% 0% 100% 8.61%
50% 0% 0% 1.01%
80% 72% 100% 9.47%

10. Related Literature
Sovereign debt and default:
Calvo (1988), Cole & Kehoe (2000), Bolton & Jeanne (2009),
Lorenzoni & Werning (2013), Nicolini et al. (2015), Corsetti &
Dedola (2016), Hatchondo et al. (2016, 2017),
Eurobonds:
Risk sharing vs. moral hazard: Delpla & von Weizs¨acker (2010),
European Commission (2011), Muellbauer (2011), Philippon &
Hellwig (2011), German Council of Economic Experts (2012)
Demand for safe assets: Beck et al. (2011), Garicano & Reichlin
(2014), Brunnermeier et al. (2016, 2017), B´enassy-Qu´er´e et al.
(2018), Lane & Langfield (2018)
This paper: costly default on senior tranch

11. Outline
1. Motivation
2. Bare-bones model of debt crises
3. Model with tranching
4. Numerical example
5. Conclusion

12. Model
Agents: households and government
Two periods: uncertainty about fundamentals in period 2
- State H: High output, no default risk
- State L: Low output, government decides whether to default

13. Market financing & asset allocation (Period 1)
Government: start with (exogenous) financing need B0
- Market financing by selling discount bonds B
qbB = B0
Households: start with endowment W0, allocate between
- Safe asset K at price q
- Government bonds B at price qb
Budget constraint
qK + qbB = W0

14. Taxation & default (Period 2, State L)
Govenment has choice set
1. Haircut on sovereign bond: 0 ≤ θ ≤ 1
- Fixed default cost: Φ > 0
- Fractional budgetary cost: αθ , 0 ≤ α ≤ 1
2. Taxation: T ≥ 0
- Distortionary with convex deadweight loss Z (T)
Z (.) > 0 , Z (.) > 0
Budget constraint
T − G = (1 − θ) B + αθB
Regularity condition

15. Household’s Portfolio Problem
Risk neutral: government bonds priced at expected return
qb = q (1 − γθ)
- Anticipations of default reduce sovereign bond prices qb
- Government has to sell more bonds B to meet financing need B0
Market financing schedule
B =
B0
qb
=
B0
q (1 − γθ)
Optimization problem Regularity condition

16. Market financing schedule

17. Market financing schedule
Rise in initial financing need B0

18. Government’s Problem
Benevolent: maximize household consumption
Discretionary: cannot commit to period 2 policy (take qb as given)
max
T,θ
Y − Z (T) − T + K + (1 − θ) B − 1{θ>0}Φ
subject to
T − G = (1 − θ) B + αθB (Budget constraint)
0 ≤ θ ≤ 1 (Boundary constraints)

19. Interior solution
Taxes determined by marginal distortions and fractional default costs
Z ˆT =
α
1 − α
⇒ Doesn’t depend on B
Haircut is increasing in B
ˆθ =
1
1 − α
1 −
ˆT − G
B
Fixed default cost: minimum haircut θ to make default optimal
Z ˆT + Φ = Z ˆT +
(1 − α) θ
1 − (1 − α) θ
ˆT − G
−
αθ
1 − (1 − α) θ
ˆT − G
⇒ Independent of B, increasing in Φ

20. Optimal Policy Plan
Condition Haircut Tax
No default (lower corner) ˆθ < θ θ = 0 T = G + B
Interior θ ≤ ˆθ ≤ 1 θ = ˆθ T = ˆT
Full default (upper corner) ˆθ > 1 θi = 1 T = G + αB
Corner solutions
- Haircut constrained at boundary
- Taxes increase with B

21. Optimal haircut schedule

22. Equilibrium
Rational expectations equilibrium
Type and uniqueness of equilibria depend on financing need B0

23. Equilibrium
Rational expectations equilibrium
Type and uniqueness of equilibria depend on financing need B0

24. Equilibrium
Rational expectations equilibrium
Type and uniqueness of equilibria depend on financing need B0

25. Outline
1. Motivation
2. Bare-bones model of debt crises
3. Model with tranching
4. Numerical example
5. Conclusion

26. Tranching
Share ω of government bonds in a senior tranch
- Non-defaultable (to be relaxed later)
- Priced at q same as risk-free asset
Government budget constraint
T − G = ωB + (1 − ω) (1 − θ (1 − α)) B
Market financing schedule
B =
B0
ωq + (1 − ω) qb

27. Market financing schedule
Rise in ω reduces borrowing costs holding qb constant

28. Optimal haircut schedule
Interior solution:
- Primary surplus not affected (pinned down by marginal tax distortions)
Z ˆT =
α
1 − α
- Increase haircut on junior tranch to pay senior tranch
ˆθ =
1
(1 − ω) (1 − α)
1 −
ˆT − G
B
Corner solution:
- Can no longer raise haircut on junior tranch
- Increase tax revenues to pay senior tranch
T = G + [ω + α (1 − ω)] B
Minimum haircut (fixed default cost):
- Less revenues gained from default at a given haircut
- Larger haircut required to make default optimal under fixed cost

29. Optimal haircut schedule
Shift up in interior region and minimum haircut
Hit upper corner with full default at lower debt level

30. Equilibrium: Tranching at interior solution
No default eq: no impact since senior and junior tranch equivalent
Default eq: Move from DE to DE
Rise in haircut. Borrowing costs & default incentives exactly same

31. Equilibrium: Tranching at interior solution
No default eq: no impact since senior and junior tranch equivalent
Default eq: Move from DE to DE
Rise in haircut. Borrowing costs & default incentives exactly same

32. Equilibrium: Tranching at interior solution
No default eq: no impact since senior and junior tranch equivalent
Default eq: Move from DE to DE
Rise in haircut. Borrowing costs & default incentives exactly same

33. Why is tranching ineffective at interior solutions?
1. Convex tax distortions vs. constant marginal cost of increasing haircut
Optimal taxation independent of senior tranch size
Z ˆT =
α
1 − α
Raise haircut on junior bonds instead of increasing taxes
2. Risk neutrality
Fall in junior bond prices exactly higher prices on senior bonds
Government’s borrowing costs stay same
⇒ Modigliani-Miller outcome

34. Equilibrium: Tranching to corner solution
Move to corner solution with complete default on junior tranch
(ω ≥ ω)
Fall in borrowing costs, revenue gained from default
Eliminate default equilibrium (DE doesn’t exist)

35. Equilibrium: Tranching to corner solution
Move to corner solution with complete default on junior tranch
(ω ≥ ω)
Fall in borrowing costs, revenue gained from default
Eliminate default equilibrium (DE doesn’t exist)

36. Equilibrium: Tranching to corner solution
Move to corner solution with complete default on junior tranch
Fall in borrowing costs, revenue gained from default
Eliminate default equilibrium (DE doesn’t exist)

37. Why is tranching effective at corner solutions?
1. Cannot raise haircut on junior tranch any further
Primary surplus increases with size of senior tranch
T − G = (ω + (1 − ω) α) B
↑ ω raises revenues (and tax distortions) during default
2. Fall in the government’s average borrowing costs, debt bill B
Junior bond prices do not rise further while senior bonds trade at
risk-free price
B =
B0
ωq + (1 − ω) qb
Less revenue (and tax distortions) needed to avoid default
⇒ Fall in incentive to default on junior bonds

38. Default on senior tranch
Government may optimally default on senior tranch
- Additional fixed default cost Φs ≥ 0
- Budgetary cost remains the same αs = α
Default on senior tranch: back to Modigliani-Miller
- Tax revenues same as interior solution
- Borrowing costs and default decision same as without tranching
Determine max senior tranch size ω without default on senior tranch
- Increasing in Φs and decreasing in B0
- Φs = 0: ω coincides with ω that leads to corner solution
- Φs > 0: intermediate region where tranching has effect.
- With high enough Φs, can reach ω∗
that eliminates default.

39. Outline
1. Motivation
2. Bare-bones model of debt crises
3. Model with tranching
4. Numerical example
5. Conclusion

40. Without tranching: Haircuts and equilibria
Calibration

41. Without tranching: Debt bill
B increasing in B0 (below Debt-Laffer curve peak)
Bad equilibrium: B jumps up

42. Tranching
Maximum senior tranch size same as corner boundary with Φs = 0
As Φs rises, can eliminate default at higher B

43. Tranching
Maximum senior tranch size same as corner boundary with Φs = 0
As Φs rises, can eliminate default at higher B

44. Tranching
Maximum senior tranch size same as corner boundary with Φs = 0
As Φs rises, can eliminate default at higher B

45. Conclusion
Tranching may reduce vulnerability to sovereign debt crises
- Directly affects government incentives to default
- Default costs on senior tranch important
Effects highly non-linear in size of senior tranch
- Ineffective when senior tranch size is too small/large
- Too small: redistributes revenues from junior to senior tranch
- Too large: government defaults on both tranches
- Intermediate region: may prevent default on both tranches

46. Three states
Modigliani-Miller is not state by state
Enough to hit corner in worst state realization

47. Continuous support of shocks
Three regions across shock σ
Rise in ω shifts region boundaries
Can solve for optimal ω

48. Continuous support of shocks
Three regions across shock σ
Rise in ω shifts region boundaries
Can solve for optimal ω

49. Continuous support of shocks
Three regions across shock σ
Rise in ω shifts region boundaries
Can solve for optimal ω

50. Continuous support of shocks
Three regions across shock σ
Rise in ω shifts region boundaries
Can solve for optimal ω

51. Calibration
Based on Corsetti & Dedola (2016)
Parameterize Z (T) = ϕT2, E [Y ] = 1
Parameter Value Target
q 0.99 1% risk-free rate / discount factor
Φ 0.10 10% of GDP PDV
α 0.10 10% of GDP PDV
γ 0.10 regularity condition
ϕ 0.14 ˆT − G = 0.40
G 0 (without loss of generality)
Back

52. Household’s Portfolio Problem
Households consume only in period 2
Ci = Yi − Zi (Ti ) − Ti + K + (1 − θi ) B − Φi
Household’s problem
max
B,K
(1 − γ) CH + γCL
s.t. qK + qbB = W0
Risk neutral: government bonds priced at expected return
qb = q (1 − γθL)
Back

53. Default on senior tranch
Haircuts
θSDE
= 1
θSDE
s =
1
ω (1 − αs)
ω + α (1 − ω) −
TSDE − G
BSDE
Debt bill
BSDE
=
B0
q [1 − γ + γω (1 − θSDE
s )]
Taxes
Z TSDE
=
αs
1 − αs
Default condition
Z G + (ω + (1 − ω) α) BSDE
= Z TSDE
+ φs
+ ωαsθSDE
s BSDE
Back

54. Aside: Regularity condition
Probability of state H without default above
1 − γ > α
⇒ Always stay on the left side of Debt Laffer curve
dB
dB0
> 0
p2

55. Regularity condition: why do we need it?
Calvo (1988). Debt Laffer curve
May default in all states, no fixed cost of default
p2

56. Regularity condition: why do we need it?
Bad equilibrium has unusual characteristics
↑B0 increases bond prices, reduces debt burden in bad equilibrium
p2

57. Regularity condition: why do we need it?
Regularity condition but no fixed cost of default
Unique equilibrium
p2

58. Regularity condition: why do we need it?
This paper: Regularity condition and fixed cost of default
Multiple equilibria on left side of debt Laffer curve
p2