[Paper Review] Long-Run Sovereign Debt Composition: An Analytic Ergodic Framework with Explicit Maturity Structure

Christopher Cameron|arXiv (Cornell University)|Feb 23, 2026

Credit Risk and Financial Regulations0 citations

TL;DR

The paper presents a discrete-time, full-maturity ladder model of sovereign debt under deficit-driven growth, derives deterministic steady-state portfolio shares, and extends to a stochastic ergodic framework with mean-reverting rates and deficits, identifying conditions for convergence to a unique invariant distribution.

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.

Motivation & Objective

• Motivate debt management questions about how maturity structure affects cost, risk, and sustainability under regular issuance.
• Develop a disaggregated, full-maturity ladder model that endogenizes issuance under a deficit-growth regime.
• Derive closed-form steady-state portfolio shares and cost/risk metrics in the deterministic baseline.
• Extend the framework to a stochastic recurrence equation with mean-reverting rates and deficits and establish ergodic convergence to an invariant distribution.
• Provide analytical and simulation-backed insights for debt management decisions and potential extensions.

Proposed method

• Define a discrete-time sovereign debt model with M maturities and a fixed issuance allocation across tenors f_j.
• Impose a deficit-driven growth regime with exponential deficit growth D_t and normalize variables to study long-run behavior.
• Derive a backward-recursion leading to a linear system with a Leslie-matrix structure and obtain closed-form steady-state shares θ_j.
• Express steady-state cost as a weighted average coupon (WAC) and a steady one-period rollover fraction θ_1.
• Formulate a deterministic sustainability condition Φ(γ,r,f)<1 and show convergence to a unique steady state when satisfied.
• Extend to a stochastic setting with AR(1) interest rates and deficits, and reframe the model as a linear stochastic recurrence equation using a future-cashflow representation to prove ergodic convergence under a similar sustainability condition.

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 rath

Experimental results

Research questions

• RQ1 Under a deficit-driven growth regime, what are the long-run steady-state debt composition shares across the maturity ladder?
• RQ2 How do fixed issuance allocations influence steady-state costs (WAC) and rollover risk in the deterministic baseline?
• RQ3 What sustainability condition guarantees convergence to a steady state in the deterministic model?
• RQ4 How does introducing stochastic mean-reverting rates and deficits affect convergence, and can an invariant distribution be characterized?
• RQ5 Do simulations of the stochastic model align with theoretical invariant-distribution metrics for long horizons?

Key findings

• In the deterministic baseline, normalized new issuance converges to a positive steady level and steady-state portfolio shares θ_j depend only on issuance weights f and deficit growth γ.
• Coupons and the coupon structure do not affect the long-run portfolio composition under deficit-driven growth; they influence the sustainability condition but not θ_j.
• A closed-form expression for steady-state shares θ_j is derived and a simple WAC formula is obtained, with θ_1 (the rollover) linked to τ_j weights.
• Sustainability requires a deficit-driven growth condition Φ(γ,r,f)<1, equivalent to WAC<g, ensuring convergence to a unique steady state from any initial condition.
• When extending to a stochastic framework with mean-reverting rates and deficits, a linear SRE in the future-cashflow space yields ergodic convergence to a unique invariant distribution under a generalized Φ<1 condition using Foster-Lyapunov drift arguments.
• Numerical examples show agreement between Monte Carlo simulations and invariant-distribution expectations, with intuitive corrections to deterministic baselines when incorporating rate-deficit correlations.

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).