Government Debt Maturity Structure, Fiscal Policy, and Default - - PowerPoint PPT Presentation

Government Debt Maturity Structure, Fiscal Policy, and Default

Sergii Kiiashko

Kyiv School of Economics and National Bank of Ukraine

August, 2019

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 1 / 21

Government Debt Maturity Structure

Source: Bloomberg. Data collected on 17th of October, 2017.

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 2 / 21

Introduction

Motivation: Empirical facts: decaying maturity profile No consensus on how debt maturity should be managed: more short-term debt or long-term debt? No theory that explains the decaying profile of maturity:

◮ Lucas and Stokey (1983): endogenous risk-free interest rates but no default ⇒ flat

maturity structure

◮ Aguiar et al (2018): endogenous default but exogenous risk-free rates ⇒ issue only

• ne-period debt

Questions:

1

What is the optimal government debt maturity structure in an environment with lack of commitment and opportunity to default, if both risk-free interest rates and risk premiums are endogenous?

2

Are predictions of this model consistent with empirical observations?

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 3 / 21

This Project

A la Lucas and Stokey (1983) model: closed economy: government borrows from households lack of commitment strictly concave utility over consumption ⇒ risk-free interest rates are endogenous With opportunity to default as in Aguiar, Amador, Hopenhayn, and Werning (2018): default risk is continuously increasing in outstanding debt ⇒ risk premiums are endogenous Results:

1

Solution is time-consistent: government follows ex ante optimal fiscal policies conditional

• n no default.

2

In the presence of sufficient default risk, the optimal maturity structure has a decaying profile.

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 4 / 21

Outline

1

Introduction

2

A Three-Period Example

3

Markov Perfect Competitive Equilibrium

4

Optimal Maturity Structure

5

Numerical Exercises

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 5 / 21

A Three-Period Example

Three periods A governemnt is endowed with τ every period and maximizes θ0ω(g0) + ω(g1) + ω(g2) No initial debt but θ0 > 1 so there is incentive to borrow Government borrows from lenders (b1

0, b2 0, b2 1)

Important features:

◮ lack of commitment ◮ risk-averse lenders with u(ct), ct + gt = 1 ◮ default risk in period 2 with π′(b2

1) ≥ 0

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 6 / 21

Modified Commitment Problem

Consider a planner who can commit to policy in period 1. To simplify notation, let st = τ − gt, ct = 1 − τ + st, b2

1 = s2

Optimization problem is max

s0, s1, s2

θ0ω(τ − s0) + ω(τ − s1) + EV2(τ − s2) s.t. s0 + u′(1 − τ + s1) u′(1 − τ + s0) s1 + u′(1 − τ + s2) u′(1 − τ + s0) (1 − π(s2))s2 = 0 The maturity structure is irrelevant. Let (s⋆

0 , s⋆ 1 , s⋆ 2 ) denote the optimal policy. Can the government with lack of commitment

achieve the planner’s allocation?

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 7 / 21

Optimal Maturity Structure

Case of commitment: max

s1, s2

ω(τ − s1) + EV2(τ − s2) s.t. u′(1 − τ + s1)s1 + u′(1 − τ + s2)(1 − π(s2))s2 = −u′(1 − τ + s0)s0 Case of no commitment (same utility function but different budget constraint): s.t. u′(1 − τ + s1)s1 + u′(1 − τ + s2)(1 − π(s2))s2 = u′(1 − τ + s1)b1

0 + u′(1 − τ + s2)(1 − π(s2))b2

Optimal maturity structure of debt should satisfy: ∂u′(1 − τ + s⋆

1 )

∂s1 △s1 · b1

0 + ∂u′(1 − τ + s⋆ 2 )(1 − π(s⋆ 2 ))

∂s2 △s2 · b2

0 = 0

⇒ b1 b2 =

∂u′(1−τ+s⋆

2 )(1−π(s⋆ 2 ))

∂s2 ∂u′(1−τ+s⋆

1 )

∂s1

· MRS⋆

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 8 / 21

Optimal Maturity Structure

Case of commitment: max

s1, s2

ω(τ − s1) + EV2(τ − s2) s.t. u′(1 − τ + s1)s1 + u′(1 − τ + s2)(1 − π(s2))s2 = −u′(1 − τ + s0)s0 Case of no commitment (same utility function but different budget constraint): s.t. u′(1 − τ + s1)s1 + u′(1 − τ + s2)(1 − π(s2))s2 = u′(1 − τ + s1)b1

0 + u′(1 − τ + s2)(1 − π(s2))b2

Optimal maturity structure of debt should satisfy: ∂u′(1 − τ + s⋆

1 )

∂s1 △s1 · b1

0 + ∂u′(1 − τ + s⋆ 2 )(1 − π(s⋆ 2 ))

∂s2 △s2 · b2

0 = 0

⇒ b1 b2 =

∂u′(1−τ+s⋆

2 )(1−π(s⋆ 2 ))

∂s2 ∂u′(1−τ+s⋆

1 )

∂s1

· MRS⋆

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 8 / 21

Optimal Maturity Structure

Case of commitment: max

s1, s2

ω(τ − s1) + EV2(τ − s2) s.t. u′(1 − τ + s1)s1 + u′(1 − τ + s2)(1 − π(s2))s2 = −u′(1 − τ + s0)s0 Case of no commitment (same utility function but different budget constraint): s.t. u′(1 − τ + s1)s1 + u′(1 − τ + s2)(1 − π(s2))s2 = u′(1 − τ + s1)b1

0 + u′(1 − τ + s2)(1 − π(s2))b2

Optimal maturity structure of debt should satisfy: ∂u′(1 − τ + s⋆

1 )

∂s1 △s1 · b1

0 + ∂u′(1 − τ + s⋆ 2 )(1 − π(s⋆ 2 ))

∂s2 △s2 · b2

0 = 0

⇒ b1 b2 =

∂u′(1−τ+s⋆

2 )(1−π(s⋆ 2 ))

∂s2 ∂u′(1−τ+s⋆

1 )

∂s1

· MRS⋆

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 8 / 21

Discussion

Lucas and Stokey, 1983: suppose π = π′ = 0 ⇒ s⋆

1 = s⋆ 2 ⇒ b1 0 = b2

if maturity is not flat, say b1

0 > b2 0, government has incentive to increase s1 and decrease s2.

Aguiar et al., 2018: suppose u′ = const ⇒ b2

0 = 0

if b2 > 0, government has incentive to increase s2 and decrease s1 This model: ST debt is manipulated by changes in risk-free interest rate LT debt is manipulated by changes in rf interest rate AND default premium ⇒ generally, once government changes surplus, LT interest rates respond stronger than ST ⇒ b1

0 > b2 0 > 0

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 9 / 21

Overview of the Model

Representative household: values private consumption ct and government spending gt: E

∞

• t=0

βt (u(ct) + θtω(gt)) endowed with 1 unit of consumption

• ptimally chooses consumption and savings (government bonds)

Government: benevolent (same preference) collects τ in taxes

• ptimally chooses government spending and issues bonds bt+k

t

, qt+k

t

is price lacks commitment (makes decisions every period) and can default if defaults receives V def

t

which is stochastic and continuously distributed Resource constraint: ct + gt ≤ 1 ∀t. Assumptions: θ0 > θ1 = θ2 = ... = θT = 1, ω′(τ) ≥ u′(1 − τ).

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 10 / 21

Overview of the Model

Representative household: values private consumption ct and government spending gt: E

∞

• t=0

βt (u(ct) + θtω(gt)) endowed with 1 unit of consumption

• ptimally chooses consumption and savings (government bonds)

Government: benevolent (same preference) collects τ in taxes

• ptimally chooses government spending and issues bonds bt+k

t

, qt+k

t

is price lacks commitment (makes decisions every period) and can default if defaults receives V def

t

which is stochastic and continuously distributed Resource constraint: ct + gt ≤ 1 ∀t. Assumptions: θ0 > θ1 = θ2 = ... = θT = 1, ω′(τ) ≥ u′(1 − τ).

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 10 / 21

Markov Perfect Competitive Equilibrium

State variables: bt−1 = (bt

t−1, bt+1 t−1, ...) and V def t

Timing: 1. Default decision 2. Fiscal and Debt Decisions Primary budget surplus: st = τ − gt Vt(bt−1) = max

st, bt

• u(1 − τ + st) + θtω(τ − st) + β · E max

Vt+1(bt); V def

t+1

• s.t.

st + qt(st, bt) · bt

• budget surplus and issued debt

≥ (1, qt(st, bt)) · bt−1

• utstanding debt

qt+k

t

(st, bt)

• price of bond maturing at date t + k

= β u′(1 − τ + s⋆

t+1(bt))

u′(1 − τ + st)

• risk-free part

· (1 − πt+1(bt))

• default premium part

· qt+k

t+1

• s⋆

t+1(bt), b⋆ t+1(bt)

• price of bond maturing at date t + k tomorrow

where πt+1(bt) = Prob(V def

t+1 > Vt+1(bt)) is default risk.

MPCE: value Vt(bt−1), policy s⋆

t (bt−1), b⋆ t (bt−1), and pricing functions qt(st, bt).

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 11 / 21

Markov Perfect Competitive Equilibrium

State variables: bt−1 = (bt

t−1, bt+1 t−1, ...) and V def t

Timing: 1. Default decision 2. Fiscal and Debt Decisions Primary budget surplus: st = τ − gt Vt(bt−1) = max

st, bt

• u(1 − τ + st) + θtω(τ − st) + β · E max

Vt+1(bt); V def

t+1

• s.t.

st + qt(st, bt) · bt

• budget surplus and issued debt

≥ (1, qt(st, bt)) · bt−1

• utstanding debt

qt+k

t

(st, bt)

• price of bond maturing at date t + k

= β u′(1 − τ + s⋆

t+1(bt))

u′(1 − τ + st)

• risk-free part

· (1 − πt+1(bt))

• default premium part

· qt+k

t+1

• s⋆

t+1(bt), b⋆ t+1(bt)

• price of bond maturing at date t + k tomorrow

where πt+1(bt) = Prob(V def

t+1 > Vt+1(bt)) is default risk.

MPCE: value Vt(bt−1), policy s⋆

t (bt−1), b⋆ t (bt−1), and pricing functions qt(st, bt).

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 11 / 21

Modified Commitment Problem

Planner commits to fiscal policies, but cannot commit to repay debt. Fiscal plan s0 = (s0, s1, ..., st, ...): a sequence of contingent budget surpluses. Value of fiscal plan at t is defined recursively Wt(st) = u(1 − τ + st) + θtω(τ − st) + β · E max Wt+1(st+1 ∈ st), V def

t+1

• Planner’s problem in period T is to maximize utility subject to the dynamic budget constraint:

max

{sT } WT (sT )

s.t.

∞

• k=T

βk−T u′

k · PrT+k T

· sk

• ≡ST

≥

∞

• k=0

βk−T u′

k · PrT+k T

· bT+k

T−1

• ≡BT−1, T

where PrT+k

T

(sT+1) = T+k

t=T+1 Prob(Wt(st) ≥ V def t

) - probability of repaying bond with maturity k issued in period T.

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 12 / 21

Modified Commitment Problem

Planner commits to fiscal policies, but cannot commit to repay debt. Fiscal plan s0 = (s0, s1, ..., st, ...): a sequence of contingent budget surpluses. Value of fiscal plan at t is defined recursively Wt(st) = u(1 − τ + st) + θtω(τ − st) + β · E max Wt+1(st+1 ∈ st), V def

t+1

• Planner’s problem in period T is to maximize utility subject to the dynamic budget constraint:

max

{sT } WT (sT )

s.t.

∞

• k=T

βk−T u′

k · PrT+k T

· sk

• ≡ST

≥

∞

• k=0

βk−T u′

k · PrT+k T

· bT+k

T−1

• ≡BT−1, T

where PrT+k

T

(sT+1) = T+k

t=T+1 Prob(Wt(st) ≥ V def t

) - probability of repaying bond with maturity k issued in period T.

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 12 / 21

Optimal Maturity Structure

FOC of planner at period 0 : θt+1ω′

t+1 − u′ t+1

θtω′

t − u′ t

=

∂ u′

t+1·(st+1−bt+1 −1 )

∂st+1

+

∂Prt+1 t ∂st+1

Prt+1

t

(St+1 − B−1, t+1)

∂ u′

t ·(st−bt −1)

∂st

FOC of planner at period T ≤ t: θt+1ω′

t+1 − u′ t+1

θtω′

t − u′ t

=

∂ u′

t+1·(st+1−bt+1 T−1)

∂st+1

+

∂Prt+1 t ∂st+1

Prt+1

t

• St+1 − BT−1, t+1
• ∂

u′

t ·(st−bt T−1)

∂st

Optimal maturity structure: θt+1ˆ ω′

t+1 − ˆ

u′

t+1

θt ˆ ω′

t − ˆ

u′

t

=

∂ ˆ u′

t+1·(bt+1 −1 −bt+1 T−1)

∂st+1

+

∂ ˆ Prt+1 t ∂st+1

ˆ Prt+1

t

• B−1, t+1 − BT−1, t+1
• ∂

ˆ u′

t ·(bt −1−bt T−1)

∂st

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 13 / 21

Optimal Maturity Structure

FOC of planner at period 0 : θt+1ω′

t+1 − u′ t+1

θtω′

t − u′ t

=

∂ u′

t+1·(st+1−bt+1 −1 )

∂st+1

+

∂Prt+1 t ∂st+1

Prt+1

t

(St+1 − B−1, t+1)

∂ u′

t ·(st−bt −1)

∂st

FOC of planner at period T ≤ t: θt+1ω′

t+1 − u′ t+1

θtω′

t − u′ t

=

∂ u′

t+1·(st+1−bt+1 T−1)

∂st+1

+

∂Prt+1 t ∂st+1

Prt+1

t

• St+1 − BT−1, t+1
• ∂

u′

t ·(st−bt T−1)

∂st

Optimal maturity structure: θt+1ˆ ω′

t+1 − ˆ

u′

t+1

θt ˆ ω′

t − ˆ

u′

t

=

∂ ˆ u′

t+1·(bt+1 −1 −bt+1 T−1)

∂st+1

+

∂ ˆ Prt+1 t ∂st+1

ˆ Prt+1

t

• B−1, t+1 − BT−1, t+1
• ∂

ˆ u′

t ·(bt −1−bt T−1)

∂st

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 13 / 21

Optimal Maturity Structure

FOC of planner at period 0 : θt+1ω′

t+1 − u′ t+1

θtω′

t − u′ t

=

∂ u′

t+1·(st+1−bt+1 −1 )

∂st+1

+

∂Prt+1 t ∂st+1

Prt+1

t

(St+1 − B−1, t+1)

∂ u′

t ·(st−bt −1)

∂st

FOC of planner at period T ≤ t: θt+1ω′

t+1 − u′ t+1

θtω′

t − u′ t

=

∂ u′

t+1·(st+1−bt+1 T−1)

∂st+1

+

∂Prt+1 t ∂st+1

Prt+1

t

• St+1 − BT−1, t+1
• ∂

u′

t ·(st−bt T−1)

∂st

Optimal maturity structure: θt+1ˆ ω′

t+1 − ˆ

u′

t+1

θt ˆ ω′

t − ˆ

u′

t

=

∂ ˆ u′

t+1·(bt+1 −1 −bt+1 T−1)

∂st+1

+

∂ ˆ Prt+1 t ∂st+1

ˆ Prt+1

t

• B−1, t+1 − BT−1, t+1
• ∂

ˆ u′

t ·(bt −1−bt T−1)

∂st

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 13 / 21

Discussion

Maturity structure serves as a commitment device: no deviations from ex ante optimal fiscal policy can decrease market value of outstanding debt. Perturbation: marginal increase st and decrease st+1, t ≥ 1, keeping Wt(st) Risk-free interest rates: u′(1 − τ + sk) changes for k = t, t + 1 Default premium: default risk at t + 1 changes due to change in Wt+1(st+1), but default risk at t does not change because Wt(st) is constant Assymetry: long-term interest rates are more elastic than short-term Proposition: under certain reasonable assumptions we can analytically prove that bT

T−1 > bT+1 T−1 > ... > bT+k T−1 > ... > 0.

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 14 / 21

Benchmark Case

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 15 / 21

Shock to Taste Parameter

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 16 / 21

Changing Debt-to-GDP

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 17 / 21

Changing Decaying Rate of Maturity Structure

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 18 / 21

Short Initial Debt

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 19 / 21

Long Initial Debt

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 20 / 21

Summary

Main results: Maturity is used to discipline the future governments Asymmetry: longer-term interest rates are more elastic than shorter-term rates Generally, maturity structure has a decaying profile Numerical exercises can be performed to study role of initial maturity structure, debt-to-GDP ratio, unexpected shock to risk premium etc.

Sergii Kiiashko (Kyiv School of Economics and National Bank of Ukraine ) Optimal Maturity Structure August, 2019 21 / 21