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Econ-447 · economics · Economics

Optimal sovereign-debt maturity

LLM status Partially solved unverified

Statement

Original subject group: Fiscal policy and sovereign debt

Audit classification: Explicit published open question. Rollover-risk measurement and maturity evidence gaps are explicit; the dynamic optimal-control problem is editorial. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Consider a sovereign with stochastic output $y_t$, inherited short- and long-term debt $(b_t^S,b_t^L)$, and a specified cost of default. Short debt matures next period; long debt has declared coupon and decay rate. Investors price both claims using a stated discount factor, recovery rule, and information structure. Government decisions jointly determine primary surpluses, default, and new issuance; equilibrium prices $q^S,q^L$ therefore depend on the policy and debt state. Let $m_t\in[0,1]$ be the fraction of new funding raised at long maturity. For a fixed initial state, determine a policy $m^*(y,b^S,b^L)$ maximizing \[ E\sum_{t\ge0}\beta^t [u(c_t)-\mathcal D_t], \] where $c_t$ is resident consumption after real default expenses, $\beta\in(0,1)$ the resident discount factor, and $\mathcal D_t$ additional nonconsumption disruption loss measured in utility units. Incorporate term premia, refinancing needs, and the opportunity to adjust fiscal policy after future information. Establish comparative statics and quantitatively disciplined welfare gains relative to fixed maturity shares. Separate fundamental insolvency from rollover failure through explicit equilibria or identification restrictions. Validate model-implied maturity and spread responses rather than fitting average debt composition alone. The target is optimal maturity in a declared endogenous-price environment; existing models and practical maturity strategies already solve important special cases, while empirical measurement and generalization remain unfinished.

Write net new one-period issuance as $n_t^S$, long issuance as $n_t^L$, and long debt survival as $1-\delta$, $0<\delta<1$, with coupon $\kappa_L\ge0$. Without default, $b_{t+1}^S=n_t^S$ and $b_{t+1}^L=(1-\delta)b_t^L+n_t^L$. Specify the budget $c_t=y_t-\text{real government purchases and default resource costs}-b_t^S-(\delta+\kappa_L)b_t^L+q_t^Sn_t^S+q_t^Ln_t^L$, adjusted for the declared output and recovery effects of default. Define the funding maturity share as $m_t=q_t^Ln_t^L/(q_t^Sn_t^S+q_t^Ln_t^L)$ only when gross new funding is positive and both issuances are nonnegative; permit buybacks through separate controls. A Markov policy needs every price-relevant state, including a refinancing/sentiment state if present. A maturity rule alone cannot maximize welfare unless fiscal and default policies are optimized jointly or held explicitly fixed.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. No self-fulfilling rollover risk. Impose a unique equilibrium with repayment risk determined solely by fundamentals. Characterize optimal issuance maturity given term premia and endogenous default, then compare with an exogenous fixed-price benchmark.
  2. Sentiment and liquidity buffer. Add a refinancing sunspot and a reserve buffer as an additional control. Quantify how long maturity and liquid reserves substitute or complement one another in avoiding crisis under a stated equilibrium selection.

References and source audit

  • Kris James Mitchener; Christoph Trebesch. Sovereign Debt in the 21st Century. 2021. Locator: Section6, printed p.35, paragraphs on empirical rollover risk and maturity; corrected from Section7. Primary text checked. https://www.ifo.de/DocDL/cesifo1_wp8959.pdf.
Rollover-risk measurement and maturity evidence gaps are explicit; the dynamic optimal-control problem is editorial.
  1. Kris James Mitchener; Christoph Trebesch. 2021. Sovereign Debt in the 21st Century. Section6, printed p.35, paragraphs on empirical rollover risk and maturity; corrected from Section7. https://www.ifo.de/DocDL/cesifo1_wp8959.pdf.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-447-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-447

Liquidity insurance and the endogenous-price obstacle

Status: Partial; the full catalogue problem remains unresolved.

Abstract

A liquidity-loss benchmark yields an exact funding-maturity rule and an explicit reserve comparative static. A rational-pricing example then shows two financing outcomes at the same fundamentals, explaining why the benchmark is insufficient for sovereign debt with endogenous prices. A finite dynamic model admits a Bellman-residual welfare certificate, but verifying government optimization is separate from verifying lender pricing.

The full target and an insurance benchmark

The catalogue asks for jointly optimal maturity, fiscal, and default decisions under endogenous bond prices. Every price-relevant state, including reserves and sentiment when relevant, belongs in the Markov state. Funding maturity shares are defined using proceeds rather than face values.

First isolate one channel. At date zero the government must raise $B>0$ funding units. Long maturity supplies fraction $m\in[0,1]$ and short maturity supplies $1-m$. Prices are summarized by a reduced-form long-maturity premium $\tau mB$, with $\tau\ge0$. A liquidity event at the short refinancing date has probability $p>0$. Reserves $R\ge0$ reduce uncovered short obligations to \[ x(m,R)=\max\{B(1-m)-R,0\}. \] In the event, disruption costs $cx^2/2$, with $c>0$. All costs use the same date-zero monetary numeraire. The objective is \[ J(m;R)=\tau mB+\frac{pc}{2}x(m,R)^2. \tag{1} \] Funding, reserves, premia, and liquidity-event probabilities are fixed inputs for this calculation. There is no separate fundamental default or endogenous pricing channel yet.

The exact maturity rule

Theorem: For $\tau>0$, the unique minimizer of (1) is \[ m^*(R)=\max\left\{0,1-\frac RB-\frac{\tau}{pcB}\right\}. \tag{2} \] For $\tau=0$, the full set of minimizers is $[\max\{0,1-R/B\},1]$.
Proof: The squared positive-part function is convex. On the positive-exposure region, \[ J'(m)=\tau B-pcB\{B(1-m)-R\},\qquad J''(m)=pcB^2>0. \] The stationary point is the interior expression in (2), always below one. If positive, it is below the zero-exposure boundary and is the unique global minimum. If nonpositive, the derivative at zero is nonnegative and the objective increases thereafter. On the zero-exposure region its slope is $\tau B>0$, so no additional minimum appears. When $\tau=0$, precisely the policies eliminating exposure have zero objective, which gives the stated interval. ∎
For $B=100$, $R=20$, $p=0.1$, $c=0.01$, and $\tau=0.02$, formula (2) gives $m^*=0.6$. Short funding is $40$ and uncovered exposure is $20$. The premium is $1.2$, expected disruption loss is $0.2$, and total cost is $1.4$. All-short funding instead costs $3.2$, and all-long funding costs $2$. These inputs are illustrative rather than estimated.

At an interior optimum, $\partial m^*/\partial R=-1/B$ and $\partial m^*/\partial p=\tau/(cp^2B)>0$. Thus reserves replace long maturity unit for unit as liquidity insurance in this particular technology. This conclusion need not hold if reserves change confidence or insolvency. Joint reserve optimization also needs a carrying cost; treating $R$ as free would change the economic problem.

An explicit rational-pricing multiplicity

Consider a two-date government project requiring at least $F>0$ in date-zero proceeds. The government offers fixed face amount $D>0$ to risk-neutral lenders with discount $\delta_L\in(0,1)$. A funded project yields terminal output $Y\ge D$; otherwise resources and recovery are zero. In the productive state the additional utility loss from default exceeds $D$, so the government repays. A subthreshold funding amount cannot operate the indivisible project.

The rational price of the proposed issuance satisfies \[ q=\delta_L{\mathbf 1}\{qD\ge F\}. \tag{3} \]

Proposition: When $0<F\le\delta_LD$, both $q=0$ and $q=\delta_L$ solve (3). If $F>\delta_LD$, only $q=0$ solves it.
Proof: The right side takes only zero and $\delta_L$, so these are the only possible solutions. At zero, funding is below $F$, the project fails to operate, and recovery is zero. At $\delta_L$, funding suffices precisely when $\delta_LD\ge F$, in which case repayment makes the price correct. ∎
The low-price outcome can represent unsuccessful debt offers followed by no issuance, rather than a government voluntarily issuing worthless claims for zero proceeds. The conditional pricing problem for the proposed issuance is unchanged. Fixed issuance is imposed to expose the price complementarity; joint issuance optimization would require an expanded model.

This is not a complete model of all sovereign rollover crises. It does show why a fixed smooth premium cannot automatically summarize an endogenous-price environment. A maturity or reserve change may move the government across a financing threshold or select another equilibrium branch. A large repayment incentive in the productive state alone does not eliminate the funding failure.

The model also distinguishes a resource shortfall from a coordination failure: if $\delta_LD<F$, even the favorable price cannot finance the project; if $\delta_LD\ge F$, the productive equilibrium exists alongside failure. Observing a failure alone does not reveal which of these parameter regions generated it without information about counterfactual funding capacity.

A conditional dynamic certificate

Suppose a finite model has now been fully specified, including a price schedule. Let $S$ be finite and let each state have a finite nonempty menu of joint fiscal, default, reserve, and issuance actions. Let $g(s,a)$ be bounded resident utility and $P(s'\mid s,a)$ the transition kernel. The inherited long-debt decay and short-debt rollover are encoded in this state transition. For resident discount $\beta\in(0,1)$, define \[ (TV)(s)=\max_a\left\{g(s,a)+\beta\sum_{s'}P(s'\mid s,a)V(s')\right\}. \] The inequality $|\max_a u_a-\max_a v_a|\le\max_a|u_a-v_a|$ makes $T$ a sup-norm contraction of modulus $\beta$. Iteration therefore has a unique fixed point $V^*$: differences between successive iterates decay geometrically, completeness gives a limit, and continuity gives the fixed-point equation. Two fixed points would have distance at most $\beta$ times their distance and hence coincide.

If an approximate value satisfies $\|T\widehat V-\widehat V\|_\infty\le\epsilon$, then \[ \|\widehat V-V^*\|_\infty\le\frac{\epsilon}{1-\beta}. \tag{4} \] Indeed, add and subtract $T\widehat V$ and use contraction to obtain $d\le\epsilon+\beta d$.

Let $\widehat a$ be greedy for $\widehat V$. Its policy operator has the same residual at $\widehat V$, so the same argument bounds $\|\widehat V-V^{\widehat a}\|_\infty\le\epsilon/(1-\beta)$. The triangle inequality yields \[ \|V^*-V^{\widehat a}\|_\infty\le\frac{2\epsilon}{1-\beta}. \tag{5} \] This gives a rigorous conditional welfare certificate and optimizes fiscal and default decisions alongside maturity rather than optimizing a share in isolation.

But it does not verify lenders. An endogenous-price solution additionally requires that prices equal discounted repayment and resale values under the computed policy. Small Bellman error at an arbitrary cheap-credit schedule only certifies government optimization in that schedule. Equation (3) shows that even exact rational prices can be multiple, so selection must also be specified.

Precise remaining gap

The attempt solves the quadratic insurance benchmark, demonstrates an endogenous-price obstruction, and derives a conditional finite-model numerical certificate. It does not identify actual rollover probabilities or premia, compute a joint sovereign equilibrium, establish general comparative statics, or quantify real welfare gains from maturity management.

A continuous debt-state application would also need approximation-error bounds beyond the finite-grid Bellman residual. Empirical maturity and spread responses could reject the benchmark but would not by themselves identify a sentiment state. These omissions keep the catalogue problem unresolved. The contraction proof is standard dynamic programming, stated here to clarify the separate conditions that a credible computational claim must satisfy.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves a quadratic maturity and reserve insurance benchmark, exhibits rational-price financing multiplicity, and proves conditional finite-model Bellman welfare certificates. Joint endogenous-price sovereign equilibrium, actual rollover probabilities, continuous-state error control, and empirically disciplined maturity welfare gains remain unestablished.

References

  • [mt] K. J. Mitchener and C. Trebesch (2021). Sovereign Debt in the 21st Century, CESifo Working Paper 8959, section 6. The primary survey supports the rollover-risk measurement and maturity research agenda; the quadratic model is an explicit specialization. https://www.ifo.de/DocDL/cesifo1_wp8959.pdf.

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