Implementation Guide for the Sovereign Debt Crisis Scenario Set

Abbreviations and Acronyms

Executive Summary

This guide sets out how the chief risk officer of a global financial institution can implement the scenario set of the paper on fiscal capacity and sovereign debt crises in the Group of Seven (G7), with China modeled alongside.¹ The scenario set covers eight economies over 2026 through 2036: a baseline, a crisis scenario under each of two regimes of U.S. policy, regime M (U.S. monetary policy active) and regime F (U.S. fiscal policy active), and variants of 14 kinds that remove the Eurosystem’s ceiling, add a Bank of Japan ceiling, apply the central banks’ announced portfolio run-off, change China’s perimeter or price it as a market issuer, add an artificial-intelligence (AI) adoption productivity path with its components, let the primary balance respond to debt, set one short-rate share for all economies, place six economies on their projected primary balances, and take the lower measures of the euro members’ short-rate shares. The baseline is the path on which the market does not coordinate on a crisis equilibrium under the trend growth of the central calibration (the parameter values on which the scenarios are built), so that an economy is on its crisis equilibrium in it only in a year in which that equilibrium is the only one. A crisis year is a year the economy spends on the crisis equilibrium of the paper’s pricing equation, whether the market coordinates on it while a benign equilibrium exists or the benign equilibrium has disappeared, leaving the crisis equilibrium the only one; the distinction matters for use, because the share of paths with at least one year on the crisis equilibrium and the share of paths on which the market coordinates bear on different questions.

Each economy’s fiscal capacity, the largest primary surplus its government can sustain in the model, is read from its long record; its fiscal limit, the present value of that capacity, is evaluated at the neutral rate with the short-rate share of its debt and its central bank’s share as measured; and its primary balance follows a rule on its projected path. At their 2026 debt the United States, France, and Japan stand inside their crisis zones, where a crisis equilibrium exists beside a benign one, and the United Kingdom beyond its own. In the central scenarios the United States, the United Kingdom, France, and Japan are on the crisis equilibrium in every year from 2027 under both regimes, and Italy and Germany enter under regime M. Across the paper’s 10,000 simulated paths, the United States spends at least one year on the crisis equilibrium on 54 percent of paths, the market coordinates on 21 percent, and a U.S. credit event, a default drawn on the path, occurs on 27 percent; conditional on any crisis-equilibrium year, the median peak 10-year Treasury yield is 12.1 percent and the median 2036 debt ratio 141.7 percent of gross domestic product (GDP). Two baselines have a spread of their own: the United Kingdom’s is on the crisis equilibrium in every year and France’s from 2032, so year-by-year differences against them read the stress effect net of a spread the baseline already includes, and Section 4 states how to read the level of the risk instead. The workbook holds the full macroeconomic series of six scenarios (the baseline and the two crisis regimes of the central calibration and their three AI-sensitivity counterparts), the stress deltas with the affected economies flagged, the baseline-property record, the summaries of the central and the AI-adoption simulations and the shares of paths of all 20, and the records of the constructions, each tab named in Section 6.

1 Purpose and Scope

The paper builds and estimates the model; this guide translates its released outputs into use. The release is the paper’s version of 7 November 2026, its records, and its workbook. Every number printed here is one the release prints or one computed from its records as the guide states, and the guide estimates nothing of its own. The workbook that accompanies this guide is part of the deliverable, and the guide references its tabs directly; the Drawn_Parameters tab records the release, the classes of value drawn from it, and the digests that identify the 20 simulations and the nested panel (Section 3.4), so the provenance of every series is checkable from the workbook alone.

The guide covers three uses: scenario analysis on the deterministic paths, with the caveat that the baselines of two economies have a spread (Section 4); path-share and severity inputs for risk appetite and stress-test calibration, from the simulations (Section 5); and the variant scenarios and simulations as sensitivity checks on the central results (Sections 2 and 5). It does not cover currency positions, since the paper has no exchange rate, and it does not cover portfolio mapping, which depends on the institution’s book. Section 2 defines a crisis year and the scenario set, and Section 3 states the constructions that set the scenarios’ inputs, at the level an implementation needs in order to reproduce those inputs and to read what a result owes to each; the paper gives the sources, the estimation, and the 14 limitations in full. Section 4 reads the macroeconomic series, Section 5 the shares of paths and severity, Section 6 names the workbook’s tabs, and Section 7 states the limitations that bear on use.

2 The Crisis Definition and the Scenario Set

A crisis year, in the paper’s definition, is a year the economy spends on the crisis equilibrium of its sovereign-pricing equation. Two routes lead there. Where debt lies above the paper’s operative boundary of the crisis zone, the smallest debt at which its pricing equation has a crisis equilibrium, the equation has three roots: the benign root, an unstable root, and the crisis root. The two stable roots are the equilibria, benign and crisis, and the market can coordinate on the crisis equilibrium; where debt and the rate structure deteriorate further, the benign root ceases to exist, and the crisis equilibrium is the only one, with no coordination involved. Both routes produce crisis-level pricing, and the paper reports them together as years on the crisis equilibrium, with the coordinated subset beside them. For implementation the distinction matters: coordination is the self-fulfilling route, set off by the sunspot, a random event on which investors coordinate, and its share responds to the central-bank backstops; the route of sole survival is fiscal arithmetic. A central bank’s ceiling on the spread, by raising the limit at which the default hazard (the annual probability of default that prices the spread) is assessed, removes the crisis equilibrium on the paths where the lifted limit crosses the debt and caps its spread on the rest.

Table 1 lists the 51 deterministic paths of the release, 37 scenarios and the 14 hazard-free paths they are read against, and names the workbook tab that holds each path’s full macroeconomic series where they are held. The six scenarios whose series the workbook holds are the central baseline, the two crisis regimes, and the three AI-sensitivity counterparts; the remaining paths are the references, the AI path’s decompositions, and the variants. The AI-adoption productivity path adds two components to the central calibration, an increment to trend growth and a capacity channel that moves each economy’s fiscal capacity with the deviation of realized 10-year growth from its projected trend, raising it with the increment and lowering it with a crisis’s growth loss, and its decompositions run each component alone and the channel at the ends of its estimated band. China’s debt is the International Monetary Fund (IMF) staff’s augmented general government debt, which adds local government financing vehicles and government-guided funds to the official perimeter, and a variant puts it on the general government perimeter. The Crisis_Years tab holds the years on the crisis equilibrium of every path, the Variants_Summary tab the shares of paths of every simulation, and the paper’s own workbook holds the full series of every path (its Macro_Long tab).

One rule of coordination applies to every economy. In a crisis scenario the sunspot fires in every year from 2027: an economy coordinates on its crisis equilibrium in the first year in which one exists and stays on it while it exists, the United States and every other economy alike, and an economy whose crisis equilibrium ceases to exist is not coordinated again. In a baseline the sunspot never fires, so a baseline has a crisis year only where the crisis equilibrium is the only one. The two regimes are configurations of U.S. policy. Under regime M the Federal Reserve follows an active Taylor rule (its policy rate responding more than one for one to inflation) bounded at zero and the U.S. primary balance rises with the sovereign spread; under regime F the U.S. fiscal authority does not respond to the spread, the Federal Reserve’s rule is passive (less than one for one), and the price level bears the adjustment over the maturity of the debt. The other seven economies keep an active monetary rule and the fiscal rule of Section 3.3 under both.

Table 2 sets out the central outcomes in the baseline and under both regimes. Under both regimes the United States, the United Kingdom, France, and Japan are on the crisis equilibrium in every year from 2027 through 2036; under regime M Italy enters in 2032 and Germany in 2035, and both are on it through 2036; under regime F neither enters; Canada and China have no such year in any central run. For the United States the regimes differ in the outcome and not in the years: 2036 debt is 195.1 percent of GDP under regime M against 121.2 under regime F, on a baseline of 120.1. Regime F holds the ratio near its baseline through inflation that rises to 9.9 percent in 2036, where under regime M prices fall by 0.8 percent in that year, and the 10-year Treasury yield reaches 13.5 percent in 2036 under regime F against 9.6 under regime M.

The Eurosystem’s ceiling of 1.5 percentage points, the only one the central calibration includes, caps the spread on the crisis equilibrium and, on the deterministic paths, removes the crisis equilibrium only from Italy, in the years before 2032. Under regime M France is on the crisis equilibrium in every year with the ceiling and without it; Italy is on it in 5 years from 2032 with the ceiling against 9 from 2028 without it; Germany is on it in 2 years from 2035 with the ceiling and without it. The ceiling is available in a year while, in the year before, the primary balance net of the automatic stabilizers stood no more than 1 percentage point of GDP below its projected path and the debt did not exceed the fiscal limit. It binds for France in 2027, for Italy in 2031 and 2032, and for Germany in 2034 and 2035, at a spread of 1.5 percentage points, the final year of each span being the economy’s first year on the crisis equilibrium, and it is withdrawn from 2028 for France, 2033 for Italy, and 2036 for Germany; the spreads of the three then stand at 4.2, 4.1, and 3.4 percentage points in 2036 under regime M. The ceiling’s effect on the shares of paths is in Section 5. Japan, whose central bank has had no standing ceiling since yield curve control ended in March 2024, and the United Kingdom, which has no backstop, are on the crisis equilibrium in every year under both regimes.

3 The Constructions Behind the Scenarios

This section states the constructions that set each scenario’s inputs: the fiscal capacity, the fiscal limit and the two shares it reads, the fiscal rule, what the simulations draw, the records that measure what a result owes to each construction, the capacity that would remove the crisis equilibrium, and the order in which the records are reproduced.

3.1 The Fiscal Capacity

The fiscal capacity $p{b}_i^{max}$ is the largest primary surplus government $i$ can sustain in the model, and the fiscal limit is proportional to it. For seven economies it is read from the record: the largest average of 10 consecutive years of the balance before net interest, in percent of GDP, on a record read from 1950 or from the record’s first year where that is later, with every receipt from abroad that the source identifies excluded from the balance in the year it is recorded. The record is the general government’s for six of them and the federal government’s for the United States, as its debt and its balance are. Table 3 gives each capacity with the window that gives it. The windows of the United Kingdom, France, and Germany lie before 1970, and the other four begin in 1980 or later. The capacities of Germany and the United Kingdom are lower bounds on the balance before net interest in their windows, the accounts of those years netting other income with the interest received. China’s capacity is not read from a record of surpluses: it is the median ratio of capacity to revenue across the six general-government economies, 4.2 percent of revenue, applied to China’s general government revenue of 25.8 percent of GDP in 2024. The first year from which the record is read is a choice, and a capacity is a statistic of the realized record: a government that has not needed a larger surplus has not shown that it could not run one.

3.2 The Fiscal Limit and the Measured Shares

The fiscal limit is the present value of the capacity at the real interest-growth differential,

with $d_{i,t}^{*}$ unbounded where ${𝜚}_{i,t}\le 0$, and with the rate at which the limit discounts the capacity

Here ${\overline{g}}_i$ is terminal real trend growth, $r_i^{*}$ the economy’s neutral real rate, $s_{i,t}$ the sovereign spread, $y_{i,t}$ the 10-year yield and ${\pi }_{i,t}^e$ expected inflation, so that $y_{i,t}-{\pi }_{i,t}^e$ is the real 10-year yield, $h_i$ the share of the debt stock the central bank holds, and ${\omega }_i$ the share of the stock it does not hold that pays the short rate. The same two shares set the marginal financing rate of the debt identity, the year-to-year law of motion of the debt, $(1-h_i)[{\omega }_i(i_{i,t}^s+s_{i,t})+(1-{\omega }_i)y_{i,t}]+h_i{i}_{i,t}^s$ with $i_{i,t}^s$ the policy rate, toward which the effective rate on the stock moves by the fraction $1/m_i$ a year, $m_i$ being the average maturity.

Three properties of equations (1) and (2) bear on implementation. The limit’s short-rate leg is the neutral rate, and its other legs are the year’s spread and real 10-year yield, which the present value capitalizes as though permanent; its growth side is the terminal trend; and the neutral rate is fixed for each economy in every scenario and every simulated path. The two shares are measured for each economy at the end of 2025, bills and short-rate floaters over the stock the central bank does not hold and the central bank’s holdings over the stock, each from the debt office’s and the central bank’s own records; the Federal Reserve’s share is the one the projections of the Congressional Budget Office (CBO) use, China’s short-rate share is measured over its book-entry treasury bonds, and its central bank’s share is held at zero. China’s spread is administered rather than priced by the market: the holders of its debt are largely state-directed banks under capital controls, and the spread stands at zero; the paper reports China, which stands outside the G7, in an addendum. Neither share is fitted to a published projection and neither is drawn. And the rate side includes the candidate spread on the market-held stock: a higher spread lowers the limit, a lower limit raises the default hazard, and the hazard prices the spread, which is the route through which the pricing equation has a crisis root (Section 3.5). In that equation the default hazard is the probability that the realized limit lies below the debt, a logistic in the excess of the debt over the limit with the dispersion of the fiscal limit as its scale, capped by the bound on the default hazard; the spread compensates the expected loss, the hazard times the loss given default; and the U.S. convenience yield and the pass-through of U.S. long rates enter through the 10-year yields.

Table 4 gives the shares, the neutral rate, and each economy’s position at its 2026 debt. The short-rate shares run from 3.47 percent for China and 4.56 for the United Kingdom through 26.89 for the United States. The lower the share, the more of the limit’s rate side is the real long yield, which includes the term premium. The variant that gives all eight economies the U.S. share measures what the measured shares contribute: in it France’s baseline has no year on the crisis equilibrium and the United Kingdom’s has 8 from 2029, against 5 and 10 at the measured shares. For Germany, France, and Italy the share the model uses is the measure on which the Eurosystem holds no bill and no floater; the lower measures, over the whole stock, are 5.24, 7.16, and 9.62 percent, and the variant at those measures puts France’s baseline on the crisis equilibrium from 2031, one year sooner. At their 2026 debt three economies stand inside their crisis zones, the United States, France, and Japan: the 2027 pricing equation has two equilibria, so a crisis is available and coordination selects it. The United Kingdom stands beyond its zone, the crisis equilibrium being the only one, and four economies stand below theirs, with no crisis root, China among them with no boundary under its administered spread.

3.3 The Fiscal Rule on the Projected Path

The primary balance (a surplus is positive) follows

where $p{b}_{i,t}^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}$ is the projected primary balance of the year, $d_{i,t-1}^{\mathrm{h}\mathrm{f}}$ debt on the hazard-free path, $y_{i,t}^{\mathrm{g}\mathrm{a}\mathrm{p}}$ the output gap, $a$ the automatic stabilizers at 0.4 percentage points of GDP per percentage point of gap, and $c$ the response to last year’s spread at 0.25 percentage points of GDP per percentage point. The response $b$ to debt is 0 in the central scenarios and in every simulation, a stated assumption: the paper’s panel estimate does not separate the response from zero, and the response is not drawn. A variant sets it at 0.046, the panel estimate on the sample through 2007, and leaves the crisis years of every baseline as they are. Under regime F the responses $b$ and $c$ are zero for the United States.

The projected path is the CBO’s current-law primary balance for the United States, between $-$2.5 and $-$1.7 percent of GDP over 2027 through 2036, and the 2026 balance held in every year for the other seven: $-$2.8 percent of GDP for Germany, $-$2.7 for France, 0.8 for Italy, $-$1.2 for the United Kingdom, $-$2.4 for Canada, $-$1.7 for Japan, and $-$10.0 for China. A variant places the six other G7 economies on the primary balances the IMF’s World Economic Outlook (WEO) projects through 2031, held after; in it France’s baseline has no year on the crisis equilibrium, and the United Kingdom’s is on it in every year as at the center. The Eurosystem’s conditionality (Section 2) reads this projected path.

3.4 The Draws, the Simulations, and the Nested Panel

A stochastic path draws 13 quantities once and, in each year, the output-gap innovations, the sunspot, and the default draws. Table 5 lists the quantities. One principle governs them: a quantity with a measurement is not drawn, a quantity with an estimate is drawn from the estimate, and a quantity with neither is drawn uniformly on a stated range centered on its central value. The capacity is drawn uniformly 1 percentage point of GDP either side of each economy’s historical capacity, with no floor; the supports are in Table 3, the lowest lower end is France’s at 0.06 percent of GDP, and none reaches zero. The width is a convenience and does not measure the uncertainty of a capacity. In a simulation the sunspot is a random event: in each year it fires with the path’s drawn probability, for the eight economies at once, and an economy not on its crisis equilibrium coordinates on it in a year the sunspot fires if one exists. An economy on the crisis equilibrium stays on it while it exists; one whose crisis equilibrium has ceased to exist returns to the benign one and can be coordinated again by a later firing, where the deterministic scenarios do not coordinate it again.

Table 6 lists the 20 simulations of the release. Every simulation uses the same parameter draws and the same shocks, so two simulations differ by the change the second one makes and by nothing else. Of the simulations, 6, the central simulation and the five that change a construction, are read statistic by statistic in Section 5, which also reads the sunspot and loss-given-default simulations; 10 hold one of 5 drawn quantities at an end of its range on every path; 2 hold the sunspot’s probability at an end of its range; 1 selects by the sunspot alone, an economy on a sole crisis root returning to the benign root when that root reappears; and 1 sets the loss given default at 10 percent on every path, the haircut of an illustration the paper cites.

The nested panel crosses 1,000 parameter draws with 10 shock sequences each, 10,000 paths on random streams of its own, and splits the variance of each outcome into the part that lies between parameter draws and the part that lies between shock sequences under one draw. For the indicator that a path has at least one year on the crisis equilibrium the parameter part runs from 65 percent for France to 85 for Italy and China (the final column of Table 10), with jackknife standard errors of 1.1 through 2.4 percentage points; for the United States it is 74 percent, against 46 for the coordination indicator and 90 for 2036 debt. A share of paths in this guide is therefore, for the larger part, a statement about which parameter values produce a crisis year and, for the smaller part, a statement about the shocks.

3.5 What a Result Owes to a Construction

Two records of the release measure the dependence of the results on the constructions above, and a third sets the baseline beside published projections; the workbook holds them in its Limit_Counterfactual, Capacity_Thresholds, and Baseline_Comparison tabs. The first reads the pricing equation of the United States in 2027 under regime M on a grid of spreads. With the limit as the model forms it, the pricing equation has three roots: a spread of zero, an unstable root at 2.28 percentage points, and the crisis equilibrium at 3.14, where the limit stands at 70.3 percent of GDP against the debt the crisis root produces by the end of the year, 105.3. With the limit held at that value the crisis root is the only root; with the limit held at its benign value, which is unbounded, or discounted without the candidate spread, zero is the only root. The slope of the pricing equation’s map at the crisis equilibrium is 0.608 with the limit as the model forms it and 0.0225 with the limit held, so the route from the spread through the financing rate and output to debt accounts for 3.7 percent of it and the limit’s response to the spread the rest. Run whole with the limit discounted without the candidate spread, the model puts no economy on the crisis equilibrium in the baseline or under either regime, the United Kingdom in 2036 excepted. The crisis root of the pricing equation rests on the limit’s being discounted at the spread being solved for.

The second record gives, for each economy, the largest capacity at which a run has a crisis year (Table 7), from which the distance between a historical capacity and the capacity at which a result changes is read. The historical capacity of the United States, 1.91 percent of GDP, lies 0.83 percentage points of GDP below the largest capacity at which its regime-M run has a crisis year and 1.10 percentage points of GDP above the largest at which its baseline does.

The third record sets the baseline beside the published projections it is not fitted to. U.S. baseline debt in 2036 is 120.1 percent of GDP against the CBO’s 120.2, with a root-mean-square difference of 0.41 percentage points over 2027 through 2036 and a largest difference of 0.63; net interest in 2036 is 4.4 percent of GDP against the CBO’s 4.6. The model’s ratios are over calendar-year GDP and the CBO’s over fiscal-year GDP. For the six other G7 economies baseline debt in 2031 stands above the WEO’s projected net debt, by 0.6 percentage points for Italy through 28.5 for the United Kingdom (123.1 against 94.5 percent of GDP). The primary balance held at its 2026 value is the largest term of the gap for the United Kingdom and France, at 15.4 and 8.7 percentage points of gaps of 28.5 and 8.0; the United Kingdom’s gap also includes the spread on the crisis equilibrium in its baseline, through an interest term of 5.0 percentage points and a growth term of 12.2. For Canada and Japan the largest term is the WEO’s own residual between its net lending and the change in its net debt, 7.8 and 9.6 percentage points of gaps of 11.8 and 11.2.

3.6 The Resolving Capacity

For the six economies Table 8 lists, the release solves the capacity that would remove the crisis equilibrium: the smallest capacity that, announced in a given year and held, leaves the economy no year from the announcement on in which a crisis equilibrium exists. Each trial is a run of the whole model, so the limit, the yield, the debt, and the coordination all move with the trial, and each solved value is confirmed by a scan of the capacity over a grid, which establishes that every capacity above it also removes the crisis equilibrium. The requirement does not fall with the announcement year in any row of the table. Table 7 gives the largest capacity that leaves a crisis year when set from 2027 and read as a counterfactual record, and Table 8 the smallest capacity that, announced and priced, leaves no crisis equilibrium. For the United States announced in 2027 it is 2.75 percent of GDP under regime M and 2.67 under regime F, against a historical capacity of 1.91; announced in 2033 it is 5.06 and 3.40.

Table 9 gives the distribution of the requirement announced in 2027 across the resolution panel, 1,000 of the central simulation’s parameter draws run apart from the 20 simulations of Table 6, for each economy that panel covers, over the draws under which a crisis equilibrium exists for the economy from the announcement on; the draws under which none exists require nothing and are counted apart. For the United States the median over the draws that require a capacity is 3.74 percent of GDP under regime M, with a 5th percentile of 1.80 and a 95th of 7.60.

3.7 Reproducing the Records

An implementation that reproduces the release computes the estimation records first (the capacities, the shares, and the estimate of the fiscal multiplier), since the model reads them; the deterministic paths, each simulation, and the nested panel are then independent computations, and the resolving capacities printed in Table 8 are read from the solved values. Every random number of a simulation comes from two fixed seeds, one for the parameter draws and one for the shocks, spawned by block of draws and by path, so a path has the same numbers in whatever order the paths are computed; the nested panel has two seeds of its own.

4 Reading the Macroeconomic Series

Each Macro tab of the workbook holds seven series for each of the eight economies over 2026 through 2036: real GDP growth, inflation, nominal GDP growth, the policy rate, the 10-year sovereign yield, the output gap in percent of potential, and equity valuation. The scenario set and the tab structure are the same for China as for the seven economies of the G7, so a pipeline reads the same series for each, although the paper reports China in an addendum (Section 3.2). The paper has no exchange rate, each economy’s real exchange rate against the dollar being taken to stay on its hazard-free path, so the tabs hold no currency series. Equity is a deviation from the paper’s hazard-free path, not a level, so the baseline tab shows deviations of at most 0.03 percent for the six economies whose baselines spend no year on the crisis equilibrium; the baselines of the United Kingdom and France have a spread and with it an equity loss against that reference; a crisis tab shows the crisis effect against the same reference, and its difference from the baseline tab is the effect of the event in percentage points of the reference level. As a percentage of the baseline’s own level, which the deviation of the equity index in the Macro_Long tab of the paper’s own workbook includes (Section 6), the effect is larger wherever the baseline stands below the reference: in 2036 the United Kingdom’s baseline stands 52.3 percent below the reference and its regime-M path 68.1 percent below it, a difference of 15.8 percentage points and a fall of 33.2 percent from the baseline’s level; France’s stand 44.9 and 65.4 percent below it, a difference of 20.4 percentage points and a fall of 37.1 percent. Rates and yields are levels in percent; the output gap is a level in percent of potential.

The Deltas tabs difference each crisis scenario against its baseline for four series, the 10-year yield, real GDP growth, the output gap, and inflation, and are the input a scenario engine reads, with one qualification the workbook flags in its own note column. The baselines of two economies have a spread of their own. The United Kingdom’s is on the crisis equilibrium in every year, the benign equilibrium existing in none, with a baseline spread that rises to 4.1 percentage points and a baseline 10-year yield running from 8.1 through 9.3 percent. France’s is on it from 2032, at the Eurosystem’s ceiling of 1.5 percentage points in that year and above it from the year after, the ceiling withdrawn, with a baseline 10-year yield running from 3.7 through 8.1 percent. A year-by-year delta against such a baseline reads the stress effect net of the spread the baseline already includes: the U.K. regime-M yield delta ranges from 0.2 through 2.0 percentage points across the decade against a regime-M yield between 8.4 and 11.3 percent, and France’s from 0.8 through 4.2 percentage points against a regime-M yield between 5.3 and 10.6 percent. For those two economies the delta measures what the scenario adds, and the level paths measure what the economy bears—the crisis scenario’s yields, growth, and gap used directly, or differenced against a flat reference of the institution’s choosing that is disclosed as such—and the Baseline_Property tab gives the year-by-year root state, spread, and yield of every baseline so the property is visible rather than inferred. The other six baselines spend no year on the crisis equilibrium and price a spread that prints as zero, and their deltas read as stress effects without qualification; two of them, the United States’ and Japan’s, have in every year a crisis equilibrium the baseline is not on, which the Baseline_Property tab shows as two equilibria.

5 Shares of Paths and Severity

The stochastic simulation is the input for risk appetite and severity calibration. Its shares are shares of simulated paths under the ranges the paper’s assumption tables state, not forecasts and not probabilities estimated from a record: most ranges are ranges of convenience, centered on the central value, and not estimates, the two policy regimes are drawn with equal probability, which no record supports, and a reader holding different ranges obtains different shares. Table 10 reports four statistics per economy from the central simulation of 10,000 paths: the share of paths with at least one year on the crisis equilibrium by 2036 (the crisis share), the share on which the market coordinates while a benign equilibrium exists, and the median years on the crisis equilibrium across all paths and across the paths with at least one such year. The first bears on risk appetite; the second is the self-fulfilling subset and suits questions about backstop design and contagion; the two medians measure persistence. To size a stress window on how long a crisis lasts once it occurs, take the conditional median, which gives a window of 9 years for the United States and of 2 through 10 years for the others: the unconditional median is zero wherever the crisis share is below one half, five of the eight economies in Table 10, so a window sized on it would be no years for those economies. A fifth column, from the nested panel of Section 3.4, gives the part of the first statistic’s variance that lies between parameter draws. The shares of the United Kingdom and France, 80 and 77 percent, the two largest, belong to the two economies whose baselines are on the crisis equilibrium on the central path. The drawn trend growth separates their paths: among the paths on which trend growth is drawn above its projected path the crisis share is 61.3 percent for the United Kingdom and 54.9 for France, against 99.6 and 98.6 percent among the rest. China’s 10 percent is the share of paths on which the benign equilibrium ceases to exist in at least one year, its administered spread standing at zero in those years.

Conditional on at least one U.S. crisis-equilibrium year, the median onset is 2028, the median years on the crisis equilibrium 9, the median peak 10-year Treasury yield 12.1 percent with a 95th percentile of 18.1, and the median 2036 debt ratio 141.7 percent of GDP with a 95th percentile of 300.4. By regime, the median peak yield is 9.8 percent under regime M and 14.5 under regime F, and the median 2036 debt ratio 188.2 and 118.3 percent of GDP; the pooled medians mix the two regimes, which are drawn with equal probability. The 95th percentile of the debt ratio lies among paths with at least seven years on the crisis equilibrium: of the 269 paths at or above it, 268 are regime-M paths and 190 have a drawn credit event, which the simulation records without restructuring the debt, so the ratios on those paths are those of a debt on which the event has occurred and has not been resolved. A U.S. credit event occurs on 27 percent of paths, every one of them a path with at least one crisis-equilibrium year and together 51 percent of the paths with at least one such year, and a credit event for at least one of the seven other economies on 68 percent. Across all paths, crisis or not, the median 2036 U.S. debt ratio is 119.3 percent of GDP and the 95th percentile 255.8. The Stochastic_Central tab holds the full summary and conditional blocks, and the paper’s own workbook (its Stochastic_Summary tab) the blocks by regime; a severity calibration that needs quantiles beyond the ones printed here reads them there.

The variants move these statistics in ways that identify what each mechanism contributes, and the Variants_Summary tab holds the full set. Differences between shares are taken before rounding. Removing the Eurosystem’s ceiling raises the coordination shares of Germany, France, and Italy—Italy’s from 3 to 13 percent—and their crisis shares by 4.5 through 6.5 percentage points across the three, since the ceiling raises the limit at which the hazard is assessed and removes the crisis root on the paths where the lifted limit crosses the debt; adding a Bank of Japan ceiling of 1.0 percentage points, the variant the paper includes because the Bank has no standing ceiling, lowers Japan’s coordination share from 25 to 5 percent and its crisis share by 17.3 percentage points. The announced run-off of the central banks’ portfolios raises Japan’s, Italy’s, France’s, and Germany’s crisis shares by 5.2 through 17.2 percentage points and the coordination shares of Japan, Italy, and Germany—Japan’s from 25 to 33 percent, Germany’s from 0 to 1—while France’s coordination share is lower by 0.2 percentage points, at 19 percent against 20; the U.K. crisis share rises by 5.6 percentage points and the U.S. share by no more than one percentage point. Pricing China as a market issuer raises its coordination share to 33 percent and its crisis share to 39, and leaves the other economies unchanged. The AI-adoption productivity path, a sensitivity of the paper and not its central calibration, lowers the U.S. crisis share from 54 to 52 percent; the coordination share, at 21 percent, and the share with a credit event, at 27, each print alike with and without it; on its deterministic paths U.S. debt ends 2036 at 193.7 percent of GDP under regime M and 114.9 under regime F, against a baseline of 118.2. With the loss given default at 10 percent on every path the U.S. crisis share is 32 percent and its coordination share 1, against 54 and 21 in the central simulation. With the sunspot’s probability held at the low and at the high end of its range the U.S. crisis share is 44 and 60 percent and the coordination share 7 and 30; across the 10 simulations that hold another drawn quantity at an end of its range the U.S. crisis share moves by at most 2.1 percentage points, and Table 6 gives every simulation and economy.

6 The Workbook

The workbook is part of this deliverable, and the guide references it throughout. The Log registers every tab. Drawn_Parameters names the upstream release, the classes of value drawn from it, the digests of the 20 simulations and of the nested panel, and the crisis-year definition. Scenario_Index lists all 51 paths with the tab holding each path’s full series where they are held. Crisis_Years holds the onset, the years on the crisis equilibrium, the coordinated subset, the peak yield, and 2036 debt for every economy on every path. The six Macro tabs hold the full series of the six scenarios of Table 1; the two Deltas tabs difference each regime against its baseline with the rows of the affected economies flagged; Baseline_Property gives, for every baseline, the year-by-year root state, spread, and yield; Stochastic_Central and Stochastic_AI hold the summary and conditional statistics of the central and the AI-adoption simulations with their digests; and Variants_Summary holds the crisis share and the coordination share for all 20 simulations. The constructions of Section 3 have a tab each: Capacity, with the windows and the supports of the draw; Limit_Inputs, with the two shares, the neutral rate, the boundaries, and the projected primary balance by year; Draws; Nested_Panel, with the two parts of the variance of every outcome; Limit_Counterfactual; Capacity_Thresholds; Baseline_Comparison, with the terms of each gap; Resolution, with the solved and the reported resolving capacities and the resolution panel’s quantiles; and Reproduction. The source file for the macroeconomic series is the paper’s own workbook, which holds them at full precision: its Macro_Long tab holds every scenario, economy, series, and year in one long table with the level and the deviation from the scenario’s own baseline side by side for each of its series (the seven of this workbook’s Macro tabs, seven more read from the model under their symbols, and three level indices), its Macro_Quarterly tab expands the baseline and the two central regimes to quarters under a stated rule, and its Macro_Definitions tab states the timing convention, the deviation rules, and a note on severity; this workbook holds the same series rounded to four decimals for reading, and its Log says so.

7 Limitations for Use

The paper’s 14 limitations govern any use of these scenarios, and the following bear on implementation directly. The fiscal capacities that set the crisis boundaries are 10-year maxima from the record, extreme statistics resting in the U.S. case on one episode, so the boundaries move with the capacity read: the U.S. corner boundary, the paper’s closed-form boundary of the crisis zone at the largest crisis yield (the operative boundary of Table 4, which the paths obey, lies above it), moves from 53 percent of GDP at the historical capacity to 83 and 138 at capacities of 3 and 5 percent, and the shares of paths also move with the capacity read; Table 7 gives the capacity at which each central result changes. The neutral rate that enters the limit is fixed at a published estimate for each economy and is not drawn, so the shares of paths reflect none of its uncertainty. The Eurosystem’s ceiling is an assumption modeled from a charter and announcements that were not tested by purchases at scale, so the shares of paths for Germany, France, and Italy depend on it, and the variant without the Eurosystem’s ceiling shows the shares without it; Japan has no ceiling in the central calibration, and the variant with a Bank of Japan ceiling shows the shares with one. The two baselines that have a spread are read as Section 4 states. The U.S. crisis of the central scenario under regime M is conditional on two assumptions, the bound on the default hazard and the loss given default, which together bound the spread and act on the paths through their product alone, so that halving either gives the same paths: at half its central value either leaves the United States with no crisis year. The same two parameters set the severity of a crisis, since the hazard is bounded and the high root lies close to the bound on the spread. The central paths are conditional on no credit event: the cumulative probability of a U.S. credit event along the central path is 62 percent under regime M and 59 under regime F, and that of a U.K. credit event 64 percent under either regime. The AI-adoption scenarios are a sensitivity of the paper, not its central calibration, and results drawn from them have that status.