A Framework for Debt-Maturity Management Saki Bigio (UCLA) Galo Nuo - - PowerPoint PPT Presentation

A Framework for Debt-Maturity Management

Saki Bigio (UCLA) Galo Nuño (Bank of Spain) Juan Passadore (EIEF) June 2019

How should a country manage its debt?

◮ Large-stake but highly complex problem

◮ Large dimensional state-space: vector of bonds of different maturity ◮ Several risks: income, interest-rate, default

◮ Current literature

◮ use standard bonds = ⇒ curse of dimensionality ◮ forces use of consols ◮ typically a maximum of two consols

What we do...

◮ Small open economy with a continuum of bonds,

◮ Exogenous paths for income and (risk-free) interest rates

◮ Solution using infinite dimensional calculus

◮ Realistic debt structure ◮ Analytic expressions, rich dynamics

◮ Highlight liquidity cost

◮ Price impact of issuances ◮ Microfounded from OTC frictions

A simple formula for issuances

◮ Solution: in each maturity issuance = (price impact)−1 × value gap, where value gap = market price - domestic valuation market price and domestic valuation is the price of a bond using the aggregate stochastic discount factor (domestic discount). ◮ This formula holds even as the price impact → 0 (no indeterminacy)

(Some) related literature

◮ Preferred Habit Models: Duffie, Garleanu, Pedersen (2005), Vayanos Vila

(2010)

◮ Sovereign Debt: Eaton Gersovitz (1981), Bulow Rogoff (1988), Cole and

Kehoe (2000), Chatterjee Eyigungor (2012), Arellano Ramanarayanan (2012), Hatchondo Martinez (2009), Sanchez, Sapriza, Yurdagul (2018), Bocola and Dovis (2018), Aguiar Amador Hopenhayn Werning (2018)

◮ Debt Management w/ Distortionary Taxes: Lucas Stokey (1983),

Angeletos (2002), Buera Nicolini (2004), Bhandari Evans Golosov Sargent (2017), Debortoli Nunes Yared (2017, 2018), Faraglia Marcet Oikonomou Scott (2018)

◮ Corporate finance: Leland Toft (1996), , Chen et al. (2012), He Milbradt

(2016), Manuelli Sánchez (2018)

◮ Continuous Time - Heterogeneous Agents: Lucas Moll (2014), Nuno

Moll (2017), Nuno Thomas (2017)

Maturity management with liquidity costs

Environment: shocks, preference and state

◮ Exogenous paths: income y(t), short rate ¯ r(t) ◮ Preferences: V0 =

∞

e−ρtu(c(t))dt and u(x) ≡ c1−σ − 1 1 − σ ◮ State: debt f (τ, t), expiration τ ∈ [0, T]

Environment: constraint set

◮ Budget Constraint: c (t) = y (t) +

T

q (τ, t, ι) ι (τ, t)dτ

• issuance

− f (0, t)

principal

−δ

T

f (τ, t) dτ

• coupons

. ◮ Bond price: q (τ, t, ι) = ψ(τ, t)

bond price

− 1 2 ¯ λι

• liquidity cost

ψ(τ, t) ◮ PDE constraint: ∂f (τ, t) ∂t = ι (τ, t) + ∂f (τ, t) ∂τ ; f (τ, 0) = f0(τ)

Environment: bond price

◮ Bond Price ψ(τ, t):

◮ Short rate ¯ r(t) path and no-arbitrage: ψ(τ, t) = e− τ

0 ¯

r(t+s)ds + δ

τ e− s

0 ¯

r(t+z)dzds

◮ In PDE form: ¯ r(t)ψ(τ, t) = δ + ∂ψ ∂t − ∂ψ ∂τ ; ψ(0, t) = 1

Environment: liquidity cost

◮ Primary / secondary bond markets ◮ Primary: walrasian auction of ι (τ, t)

◮ Primary dealers only ◮ Cost of capital ¯ r (t) + η ◮ Auction price q(τ, t)

◮ Secondary: OTC market

◮ µ · yss order flow ◮ Client valuation ψ(τ, t)

◮ Auction price (small issuances): q(τ, t, ι) ≃ ψ(τ, t) − 1 2 ¯ λι

• λ(ι)

ψ(τ, t) where ¯ λ ≡ η µyss .

Government problem

Perfect Foresight

V [f (·, 0)] := max

{ι(τ,t)}t∈[0,∞),τ∈[0,T]

∞

t

e−ρ(s−t)u(c(s))ds s.t. c (t) = y (t) − f (0, t) +

T

[q (τ, t, ι) ι (τ, t) − δf (τ, t)] dτ ∂f ∂t = ι (τ, t) + ∂f ∂τ ; f (τ, 0) = f0(τ)

Solution

◮ Optimal issuances ι(τ, t): ι = 1 ¯ λ

• (Price impact)−1

ψ (τ, t) − v (τ, t)

ψ (τ, t)

• Value gap

◮ Domestic valuation v(τ, t) solves the "individual trader” price-PDE: r(t)

• ρ+σ

.

c(t)/c(t)

v (τ, t) = δ + ∂v ∂t − ∂v ∂τ , if τ ∈ (0, T] v (0, t) = 1

Frictionless benchmark

¯

λ = 0

• ◮ If ψ arbitrage free:

r(t) = ¯ r(t), ψ(τ, t) = v(τ, t)

◮ Discount Factor pins consumption:

.

c(t) c(t) = −ρ − ¯ r(t) σ ◮ Indeterminate maturity!

Calibration

◮ Liquidity cost ¯ λ ≡

η µyss = 7.08

◮ Arrival rate η = 0.0011: US primary dealers exhaust 60 percent of their inventory at the end of one week (Fleming and Rosenberg, 2009) ◮ Spread η = 0.015: AA-, A- and BBB-option-adjusted-spreads of bonds from the 5 largest US banks

◮ Spanish data

◮ Discount factor ρ = 0.0416: average Spanish Government Net Debt

• f 46 percent (1985-2016).

◮ Maximum bond maturity T = 20

◮ Other

◮ Rest of the parameters taken from literature (Aguiar and Gopinath, 2006) ◮ Steady state risk-free interest rates and coupons ¯ rss = δ = 0.04 ◮ Risk aversion σ = 2, ◮ Steady state output yss = 1

Asymptotic values - as function of ¯ λ

0.5 1 1.5 0.5 1 0.5 1 1.5 0.04 0.0405 0.041 0.0415 5 10 15 20 0.05 0.1 0.15 0.2 5 10 15 20 0.5 1 1.5 2

Steady State: model versus Spanish data (July 2018)

2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2 4 6 8 10 12 14

Income shocks: consumption smoothing tilts duration upwards

Impulse response to an unanticipated 5 percent decline in income

• 10

10 20 30 40 4 4.2 4.4 4.6 4.8 5

• 10

10 20 30 40 0.94 0.95 0.96 0.97 0.98 0.99 1

• 10

10 20 30 40 40 45 50 55 60

• 10

10 20 30 40 5.8 5.9 6 6.1 6.2 6.3 6.4

• 10

10 20 30 40 0.005 0.01 0.015 0.02 0.025 0.03

• 10

10 20 30 40 0.05 0.1 0.15 0.2 0.25 0.3

Interest rate shocks: a race between bond-price reaction and consumption smoothing

Impulse response to an unanticipated increase from 4 to 5 percent in rates

• 10

10 20 30 40 4 4.2 4.4 4.6 4.8 5

• 10

10 20 30 40 0.94 0.95 0.96 0.97 0.98 0.99 1

• 10

10 20 30 40 40 45 50 55 60

• 10

10 20 30 40 5.8 5.9 6 6.1 6.2 6.3 6.4

• 10

10 20 30 40 0.005 0.01 0.015 0.02 0.025 0.03

• 10

10 20 30 40 0.05 0.1 0.15 0.2 0.25 0.3

Introducing risk

The challenge

◮ What about Risk?

◮ Complication: fixed point in sets of functions (akin to heterogeneous-agent models with aggregate shocks)

Introducing risk

◮ Only one shock ◮ Shock arrival ∼ Exp(φ) ◮ After shock:

◮ Draw: {y (0) ,¯ r(0)} ∼ F ◮ Transition deterministic: {y(t),¯ r(t)} → {yss,¯ r ss},

◮ Risky Steady State (RSS):

◮ Steady state of the model before shock arrival

Pre-shock - Risky Steady State (RSS)

◮ Risk Adjusted Valuation: ρvrss (τ) = δ + ∂vrss(τ) ∂τ + φ

• E
• v (τ, 0) u′ (c(0))

u′ (crss)

• − vrss(τ)
• ◮ Intuition:

◮ Jump in v (τ, 0) and c(0) given by frss (τ) ◮ Marginal-utility ratio: "exchange-rate” between states

Income shock: a race between consumption smoothing and self-insurance

Impulse response to a 5 percent decline in income

5 10 15 20 0.01 0.02 0.03 0.04

• 10

10 20 30 40 0.96 0.97 0.98 0.99 1

• 10

10 20 30 40 20 30 40 50 60

• 10

10 20 30 40 5.8 6 6.2 6.4 6.6 5 10 15 20 0.94 0.95 0.96 0.97 0.98 0.99 1 5 10 15 20 0.94 0.95 0.96 0.97 0.98 0.99 1

Interest rate shock: hedging too costly, bond-price reaction dominates

Impulse response to an increase from 4 to 5 percent in rates

5 10 15 20 0.01 0.02 0.03 0.04

• 10

10 20 30 40 0.96 0.97 0.98 0.99 1

• 10

10 20 30 40 20 30 40 50 60

• 10

10 20 30 40 5.8 6 6.2 6.4 6.6 5 10 15 20 0.94 0.95 0.96 0.97 0.98 0.99 1 5 10 15 20 0.94 0.95 0.96 0.97 0.98 0.99 1

The option to default

Default

◮ Option of default when a shock arrives ◮ At date of default

◮ Draw value of autarky from distribution Θ (·)

◮ Debt policy

◮ Under commitment (no debt dilution)

Default

◮ Domestic valuation: ˆ r (t) ˆ v (τ, t) = δ + ∂ˆ

v ∂t − ∂ˆ v ∂τ

+ φ

• EX

s

• (Θ (V (t)) + Ω(t))U′(c(t))

U′(ˆ c(t))v (τ, t)

• − ˆ

v (τ, t)

• ◮ Bond prices:

¯ r(t) ˆ ψ (τ, t) = δ + ∂ ˆ

ψ ∂t − ∂ ˆ ψ ∂τ

+ φ

  

No-default probability

• Θ (V (t))

+ ψ (τ, t, Xt) − ˆ ψ (τ, t)

  

Revenue-echo effect

◮ Main term: Ω(t) = θ (V (t)) U′ (ˆ c (t)) . . .

T t

max{t+m−T,0}

e− t

z (¯

r(u)−ˆ r(u))du...

ψ (m, t) 2¯ λ

 1 −

• ˆ

v (m + t − z, z) ˆ ψ (m + t − z, z)

2  dzdm

Revenue-echo effect

time maturity T t

{m, t} − bonds vintage {τ0, t0} impact on default at t: θ · U′ · v(τ, t) impact on bond price {m1, t} impact on bond price {m2, t}

Figure: Illustration of the revenue echo effect (axes inverted).

Income shock: (i) bond-price reaction through risk premium, (ii) echo effect, (iii) self-insurance and (iv) insurance through default

Impulse response to a 5 percent decline in income

5 10 15 20 0.01 0.02 0.03 0.04

• 10

10 20 30 40 0.96 0.97 0.98 0.99 1

• 10

10 20 30 40 20 30 40 50 60

• 10

10 20 30 40 5.8 6 6.2 6.4 6.6 5 10 15 20 0.9 0.92 0.94 0.96 0.98 1 5 10 15 20 0.9 0.92 0.94 0.96 0.98 1

Conclusion

◮ Main contribution: new approach to study maturity

◮ Liquidity costs ◮ Continuum of bonds of any class

◮ Forces at play

◮ Consumption smoothing vs. bond-price reaction ◮ Self-insurance and hedging ◮ Revenue-echo effect

◮ Future: lack of commitment

Appendix

Solving it

Step 1. Lagrangian L (ι, f ) =

∞

e−ρtu (c(t)) dt +

∞ T

e−ρtj (τ, t)

• −∂f

∂t + ι (τ, t) + ∂f ∂τ

• dτdt,

c(t) = y (t) − f (0, t) +

T

[q (t, τ, ι) ι (τ, t) − δf (τ, t)] dτ.

Solving it

Step 2. First-Order Condition and Envelope

Marginal Value

• u′(c)
• q (τ, t, ι) + ∂q

∂ι ι (τ, t)

• =

Marginal Cost

−j (τ, t)

ρj (τ, t) = −δu′ (c (t)) + ∂j ∂t − ∂j ∂τ , if τ ∈ (0, T] and j (0, t) = −u′ (c (t)) .

Transitional Dynamics: overview

◮ Input: c(t) ◮ Discount Factor: r(t) = ρ + σ

.

c(t) c(t) ◮ Valuations: r(t)

Value of Debt

v (τ, t) =

Coupon

• δ

+

dv(τ,t) dt

• ∂v

∂t − ∂v ∂τ ◮ Optimal Issuance: ι (τ, t) = 1 ¯ λ

ψ(τ, t) − v(τ, t)

ψ(τ, t)

• ◮ Output: c(t) (follows from KFE+BC)