🇬🇧 English · 🇪🇸 Español

Sovereign default and restructuring

What is independently checked

For short-term debt (delta=1, kappa=0), deterministic default (sigma_eps=0) and the Arellano output penalty, independent calculations check equations, prices and policies. The archived Fortran solver and MATLAB measurement code are reproduced; the public baseline also passes its registered grid criterion. Exact reproduction of the published moment table is not claimed: see the original-code comparison and grid evidence.

For illustrative comparative statics on this validated model, see the sovereign-risk policy brief.

PureMacro 6.1.0 adds separate small-grid references for the long-term smoothed operator, restructuring and the collateral planner. Published extension moments and their economic grid convergence remain open. dq_db differentiates the fitted price interpolant, not the entire equilibrium. Monotonicity is optionally imposed; the independent reference disables it.

Debt and default conventions

Positive \(b\) denotes debt; negative \(b\) denotes saving. In the short-term model, repayment consumption is \(c=y-b+q(b',y)b'\). The government compares

\[ V^R(b,y)=\max_{b'}\{u(y-b+q(b',y)b')+\beta E[V(b',y')\mid y]\} \]

with default value

\[ V^D(y)=u(h(y))+\beta E[\theta V(0,y')+(1-\theta)V^D(y')\mid y]. \]

Zero taste-shock dispersion gives the deterministic maximum; positive dispersion uses log-sum-exp and logistic choice probabilities. The Arellano cost is \(h(y)=\min(y,0.969\bar y)\) with stationary mean income \(\bar y\). Reentry wipes out debt, including when the grid allows saving.

After default in quarter \(t\), the first reentry draw occurs in \(t+1\). A success permits repayment or default in that quarter, at zero inherited debt. Both the solver's diagnostic simulation and result.simulate() use this timing and start at zero debt. Solver-generated grids include an exact zero node; on a grid spanning saving and borrowing the closest interior node is replaced by zero, preserving its endpoints and node count. Such grids require n_b >= 3.

The 6.1 development API also accepts debt_grid=coordinates: a finite, strictly increasing vector with at least two nodes and an exact zero. It copies the coordinates, preserves every node, and overrides n_b, b_min and b_max. This permits finer spacing in sensitive debt regions while keeping the original support. Maturity restructuring preserves supplied grids; their price tangents use spacing-weighted PCHIP derivatives, clipped at zero. Calls without debt_grid retain the existing grids and numerical results.

Parameter	Meaning	Reference case
`sigma_eps`	Default taste-shock dispersion	`0.0`, deterministic
`delta`, `kappa`	Maturity rate and coupon	`1.0`, `0.0`, one-period debt
`y_penalty_type`	Default-output specification	`"arellano"`
`theta`	Market-reentry probability	`0.282`
`enforce_monotonicity`	Project bond prices to a monotone schedule	`False` for the independent comparison
`policy_search`	`"dense"`, or `"monotone"` for large short-term grids (certified; same values)	`"dense"`

import numpy as np
from puremacro.sovereign import solve_sovereign_default, evaluate_debt_restructuring

baseline = solve_sovereign_default(
    delta=1.0, kappa=0.0, sigma_eps=0.0,
    y_penalty_type="arellano", b_min=-0.4, b_max=0.4,
    n_b=41, n_y=9, howard_steps=0, enforce_monotonicity=False,
    tol=1e-10, max_iter=10000,
)
assert baseline.converged
print(baseline.residual, baseline.spread_y_corr)
print(baseline.metadata["spread_y_corr_status"])

The iteration residual measures the implemented fixed-point update. Independent equation residuals, finer grids and longer simulations are separate checks. The spread–income correlation is a raw simulation statistic; insufficient observations or variation produce NaN with status metadata. An unweighted income/debt-grid correlation is a different statistic.

The built-in diagnostics use 1,500 quarters, seed 42, no burn-in and repayment periods only. result.simulate(T=1500, seed=42) reproduces their observation count, mean spread and spread–income correlation. Income has stationary mean one. Spreads use 4 * max(0, quarterly_yield - r_star); the paper comparison instead compounds quarterly returns and uses predefault windows. The simulation_*, spread_sample, spread_annualization and income_normalization metadata record these choices. Use longer simulations and the explicit grid checks for quantitative work; that guide gives the policy_search="monotone" call for the converged grid. The path's default flag includes entry and continuing exclusion; its mean is not a default-entry frequency. The validation study counts entries separately.

A government-spending version of the short-term model, with spending, primary-balance and debt-to-output policies, is in the fiscal block guide.

Restructuring experiments

A haircut changes the debt state. maturity_extension is a fractional duration increase, with delta_new = delta / (1 + maturity_extension). A value of 1.0 doubles expected maturity; it does not add one year. The calculation preserves the original income discretization, default cost, recovery and solver settings. Recovery paths use the model's actual default output and zero-debt reentry.

identity = evaluate_debt_restructuring(
    baseline, haircut=0.0, maturity_extension=0.0,
    b_distress=0.2, y_distress=1.0, horizon=4,
)
np.testing.assert_allclose(
    identity.metadata["res_restructured"].q_bond,
    baseline.q_bond, atol=1e-8, rtol=0,
)
scenario = evaluate_debt_restructuring(
    baseline, haircut=0.2, maturity_extension=0.0,
    b_distress=0.2, y_distress=1.0, horizon=4,
)
print(scenario.spread_reduction_bps)
print(scenario.pre_default_prob, scenario.post_default_prob)

The signed spread change and default probabilities refer to the selected state. Very low prices can generate extreme yields. Neither spread compression nor welfare improvement is guaranteed for arbitrary counterfactual inputs. A zero-change identity check establishes calibration preservation, not independent validation of all restructuring economics.

Experimental extensions

Long-term debt: an independent small-grid fixed point matches values, prices and debt-choice probabilities. This is the existing smoothed quantal policy operator, not a deterministic optimum or published-moment replication. Simulation now draws from the pricing operator's debt-choice probabilities; policy_b reports their mode. The risk-free price is \([\delta+(1-\delta)\kappa]/(r^*+\delta)\).
Collateral model (solve_sudden_stop): exhaustive independent policy checks pass for binding and loose constraints. Amplification uses average consumption drops at identical simulated states, without prescribed values; undefined cases return NaN. Grid binding and simulated crisis frequencies remain distinct. This illustrative planner is not Mendoza's complete model.
Restructuring: haircuts reuse the equilibrium, maturity changes preserve calibration, unconverged counterfactuals are rejected, and the recovery path starts at the selected income state. Small-grid post-restructuring states match an independent reference.
Term structure: survival-claim pricing uses the full debt-choice distribution. The legacy risk_premium_component is a discrete-compounding adjustment, not an identified economic risk premium.
Stochastic debt-sustainability analysis (simulate_stochastic_dsa): when frequency="quarterly" and exact_compounding=True, interest and growth paths compound per quarter using r / 4 and g / 4 under default rate_convention="apr" or (1 + r)**(1/4) - 1 and (1 + g)**(1/4) - 1 under rate_convention="effective". Annual paths (frequency="annual") are unchanged.

See extension equations and measured diagnostics for tolerances, frozen evidence and remaining economic limits.

Validation workflow

flowchart TD
    A[Income process and debt grid] --> B[Repayment and default values]
    B --> C[Default choice and bond prices]
    C --> B
    C --> D[Check residuals and independent reference]
    D --> E[Simulate raw statistics]
    D --> F[Preserve calibration in counterfactual]
    E --> G[Grid convergence and published moments]
    F --> G

Regeneration tools and pinned reference provenance are under tools/reference_validation/ and tests/fixtures/research_reliability/. Notebook 73 provides an exploratory interface tour. The published short-term reference and public-calibration grid study are complete; the extension checks above are a separate, narrower result. Further quantitative policy claims need extension-specific grid convergence and empirical moments.