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Long-Run Sovereign Debt Composition: An Analytic Ergodic Framework with Explicit Maturity Structure

Christopher Cameron

Abstract

This paper describes a discrete-time model of regularly-issued sovereign debt dynamics under a deficit-driven nominal debt growth regime that explicitly accounts for granular maturity. New issuance follows fixed allocations across a finite maturity ladder, and the government budget constraint determines total borrowing endogenously. In the deterministic baseline, we identify a sustainability condition for convergence to a steady-state and derive closed-form steady portfolio shares, as well as key metrics for steady cost and risk (proxied as one-period rollover ratio). Extending the model to a stochastic recurrence equation (SRE) driven by interest rates and (normalized) deficits that are stationary and mean-reverting, and using a future-cashflow state representation of debt, we identify an analogous condition for ergodic convergence to a unique invariant distribution. This implies that metrics calculated by Monte Carlo debt simulations driven by factors with these properties will recover the ergodic means of the underlying system, independently of initial conditions, provided the simulation horizon is sufficiently long. Analytical formulae for expectations of certain key metrics under this invariant distribution are derived, and agreement with simulation is observed. We find that the introduction of stochastic interest-rate/deficit correlation into the framework leads to intuitive correction terms to their deterministic-baseline counterparts.

Long-Run Sovereign Debt Composition: An Analytic Ergodic Framework with Explicit Maturity Structure

Abstract

This paper describes a discrete-time model of regularly-issued sovereign debt dynamics under a deficit-driven nominal debt growth regime that explicitly accounts for granular maturity. New issuance follows fixed allocations across a finite maturity ladder, and the government budget constraint determines total borrowing endogenously. In the deterministic baseline, we identify a sustainability condition for convergence to a steady-state and derive closed-form steady portfolio shares, as well as key metrics for steady cost and risk (proxied as one-period rollover ratio). Extending the model to a stochastic recurrence equation (SRE) driven by interest rates and (normalized) deficits that are stationary and mean-reverting, and using a future-cashflow state representation of debt, we identify an analogous condition for ergodic convergence to a unique invariant distribution. This implies that metrics calculated by Monte Carlo debt simulations driven by factors with these properties will recover the ergodic means of the underlying system, independently of initial conditions, provided the simulation horizon is sufficiently long. Analytical formulae for expectations of certain key metrics under this invariant distribution are derived, and agreement with simulation is observed. We find that the introduction of stochastic interest-rate/deficit correlation into the framework leads to intuitive correction terms to their deterministic-baseline counterparts.
Paper Structure (31 sections, 5 theorems, 77 equations, 11 figures, 4 tables)

This paper contains 31 sections, 5 theorems, 77 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Deterministic Baseline
  3. Model Setup
  4. Exponential Deficits and Growth-Regime
  5. Backward Recursion and Steady-State
  6. Steady-State Portfolio Shares
  7. Steady Interest-Cost Ratio
  8. Steady Rollover Fraction
  9. Relationship to Sustainability Condition
  10. Optimization and Frontier
  11. No-Growth Limit $\gamma \to 1^+$
  12. Stochastic Extension
  13. Stochastic Model Setup
  14. Future Cashflow Representation
  15. Normalized Linear SRE and Ergodic Convergence
  16. ...and 16 more sections

Key Result

Proposition 2.1

The system defined by ass:state--ass:exp_def is deficit-driven with growth rate $\gamma$ if $\Phi(\gamma, {r}, {f}) < 1$, where $\Phi(\cdot)$ is given by eq:feedback. In this case, the normalized issuance converges to $\tilde{N}_\infty = D_0 /(1- \Phi(\gamma, {r}, {f})) > 0$. In other words, for lar

Figures (11)

  • Figure 1: Feedback function $\Phi$ and steady $WAC$ formula calculated for $\gamma=1.045$, $r = (.02,.03,.05)^T$ under various issuance allocations. In this example the long-tilted allocation leads to $\Phi > 1$ (the formula for $WAC$ leads to $WAC>g$), and so debt dynamics are interest-driven rather than deficit-driven. The other two allocations satisfy $\Phi<1$ for this $\gamma$, thus attain steady-state under deficit-driven asymptotics.
  • Figure 2: Illustrative single-path realization of $Q_t$ and $I_t$ (left- and right-axis, respectively) to $t=100$ of the baseline normalized SRE, along with their means $E(Q)$ and $E(I)$ (dotted lines).
  • Figure 3: Single-path realization of the one-period rollover fraction $(Q_1/Q)_t$ along with its invariant mean $\theta_1$ (dotted line).
  • Figure 4: Time-averages $\sum Q_t / T$ and $\sum I_t / T$ vs. $T$ of $Q_t$ and $I_t$ respectively from various randomly-chosen initial conditions, along with the ergodic means $E(Q)$ and $E(I)$ (dotted lines).
  • Figure 5: Fan-chart of 15th to 85th percentile of realized $Q_t$ paths, along with median (solid line) and theoretical $E(Q)$ (dotted line).
  • ...and 6 more figures

Theorems & Definitions (11)

  • Proposition 2.1: Deficit-Driven Regime
  • proof
  • Proposition 2.2: Convergence to Steady-State Shares
  • proof
  • Remark
  • Remark
  • Remark
  • Proposition 3.1: Deficit-Driven Regime and Ergodicity
  • Remark
  • Proposition 3.2: Stochastic Invariant Mean
  • ...and 1 more