Debt and default
What debt costs, how it changes every day, and what happens when the markets stop lending. This page covers equations E10, E11 and E16.
In plain words
Every day, the government's debt grows by the interest it costs and shrinks by the primary balance (E11). The interest rate on it follows the market rate, but slowly, as the debt is rolled over (E10). The closer the debt gets to the country's limit, the more lenders charge, and beyond that limit they stop lending, which pushes the country into default (E16).
In the game
Every month, the dashboard publishes the debt, its rate and the risk of default. Default is never the player's choice. It is forced on them and brings down the government half the time, so the game is lost when the government that falls is the player's.
The basics
When the government spends more than it takes in, it borrows by issuing bonds, debt securities that savers, banks or funds buy from it. Public debt is the sum of everything the government owes, and it is compared with GDP to judge its weight. Debt of 110% means that the government owes the equivalent of a little over thirteen months of the country's output.
On this debt, the government pays interest that adds to its spending. Yet the weight of the debt also depends on growth, because if GDP grows faster than interest makes the debt grow, the ratio between the two can fall, even with a small deficit.
Lenders watch the risk. The higher the debt goes, the higher the rate they ask for, which makes the debt heavier still, and near a certain limit they stop lending. This is default. The government then wipes out part of what it owes, but it pays dearly for it, since it spends several years unable to borrow, its economy slows and its government is often brought down.
E10. The rate at which the government borrows, and the effective rate on its debt
In plain words
The government borrows at its country's risk-free rate, to which a premium is added. This premium includes what it was already paying at the start, 0.05 points per point of debt above its starting debt, 0.17 points per point of deficit above its starting deficit, and the price of default risk, though all these premiums together cannot exceed 25 points.
Its debt does not take on this rate all at once, because each loan keeps the rate at which it was taken out. The average rate on the debt, called the effective rate, therefore catches up with the market rate only as old loans mature. In an advanced country, 20% of the debt is rolled over each year.
Take a risk-free rate of 3%. If the debt has risen 20 points above its starting level and the deficit 2 points, the premium is 0.05 × 20 + 0.17 × 2 = 1.34 points, and the market rate 4.34%.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(i^{S}\) | the country's risk-free rate | 3% |
| \(i^{M}\) | the market rate, at which the government borrows today | 4.34% |
| \(\sigma_0\) | the premium the country was paying at the start | 0 |
| \(b_d\), \(b_f\) | the premium per point of debt and per point of deficit | 0.05 and 0.17 |
| \(\rho^{D}\) | the price of default risk (E16) | |
| \(\sigma_{\max}\) | the cap on all premiums | 25 points |
| \(i^{e}\) | the effective rate, the average cost of the debt | |
| \(\lambda\) | the share of the debt rolled over each year | 20% |
Show the full formula and its sourcesThe exact form at the daily step, every parameter with its source, and what the equation leaves out.
with iS the country's risk-free rate; σ₀ = max(ie₀ − (r* + π̄), 0) the premium the country paid at the start, f the primary deficit, B the previous day's borrowing; ρD the price of default risk and σR the spread that bonds pay after a restructuring (E16). When the state is a net lender (D < 0), its savings earn the market rate from the next day (ω = 1). A country shut out of the markets rolls over what falls due at the risk-free rate plus σR.
| Parameter | Value | Status | Source |
|---|---|---|---|
risk_premium_debt b_d | 0.05 points of rate per point of debt | Baldacci and Kumar 2010: 31 advanced and emerging countries, 1980-2008, about 5 basis points; Laubach 2009, United States: 3 to 4 | |
risk_premium_deficit b_f | 0.17 points per point of deficit | Baldacci and Kumar 2010: 17 basis points; Laubach 2009: about 25 | |
risk_premium_cap σ_max | 25 points | of the order of the highest sovereign spread in the euro area (Greece, 10-year rate of 29.24% in February 2012); beyond that, it is default | |
debt_renewal_rate λ | 0.20 / 0.29 / 0.04 per year | advanced: Checherita-Westphal and Domingues Semeano 2020 (ECB), one extra point on the 10-year rate passes as 0.2 points into the effective rate in the same year; emerging: 25 to 33% per year, deduced from the OECD (Global Debt Report 2026: 36% of emerging-market bonds mature in less than 3 years); low-income: debt treated as an ordinary IDA credit, repaid at 4% per year from the 7th to the 31st year (IDA21 terms, 1 January 2026) |
Why. Each loan keeps the rate at which it was issued. The effective rate follows the market at the pace at which the debt matures and the state borrows again. A country that runs up debt or widens its deficit pays more (Laubach 2009; Baldacci and Kumar 2010).
Three decisions, to be validated. 1) The premium never falls below zero: a state does not borrow below the policy rate, even when its debt falls. 2) Debt that was cheaper than the market at the start (concessional loans, arrears, low rates of past years) is rolled over at the market rate as it falls due. Japan, whose net interest costs only 0.05% of its debt, sees this rate rise slowly towards the market. 3) The real market rate is capped like the one in E1. Because it is smoothed, the policy rate can stay above falling inflation for a long time, and the debt would pay for this lag for years.
Rate convention. All rates in the model are annual and compound daily. One day of interest is (1 + i)1/365.25 − 1, so that a year of days gives exactly i, as for growth and prices.
What it simplifies. All debt is in national currency and at a fixed rate. There is no foreign-currency debt and no indexed debt. The premium that foreign-currency debt would add is an unknown, held at zero. There is no credit rating.
E11. The accounts, where every euro paid is received
In plain words
The game does not just keep track of percentages; it also counts in money. Every day, the government spends, collects revenue and pays interest, and whatever it is short of, it borrows from the private sector. Because each payment is recorded twice, once for the payer and once for the receiver, each row of the table adds up to zero, which lets the game check every day that no euro goes missing.
For a GDP of 1,000 billion a year, spending of 55% comes to 1.51 billion a day, and revenue of 51% to 1.40 billion. With debt of 110% of GDP at 2%, interest costs 0.06 billion a day, so on that day the government borrows 0.17 billion. Its debt rises by the same amount, and the private sector holds the same amount of extra bonds.
Each day, in money (YN is nominal GDP; the superscript m denotes an amount in money, not a share of GDP):
| Flow | State | Private sector | Sum |
|---|---|---|---|
| primary spending | −*G*ᵐ | +*G*ᵐ | 0 |
| revenue | +*T*ᵐ | −*T*ᵐ | 0 |
| interest | −*I*ᵐ | +*I*ᵐ | 0 |
| borrowing (change in debt) | +B | −B | 0 |
| sum | 0 | 0 |
Show where this table comes from and what it leaves out
Why. This is the transaction-flow matrix of Godley and Lavoie (2007, chapter 2). Each payment is written down at the payer and at the receiver; each row sums to zero (what is paid is received), and so does each column (what a sector receives and does not spend, it lends). The state's debt and the securities held by the private sector are kept separately, and must stay equal. A default (E16) is not a flow but a revaluation. The haircut removes the same sum from the debt and from the securities; it is counted separately (written_off), and the check verifies that the debt equals the starting debt, plus the flows, minus the haircuts. What it simplifies: only two sectors; the rest of the world will be a third column when trade arrives, and the banks and the central bank will be further columns.
E16. The confidence crisis and default
In plain words
Every country has a debt limit beyond which the markets stop lending. The study that estimates these limits covers only advanced countries, and gives 159.7% of GDP for France and 173.1% for the United States. In the last twenty points before the limit, called the risk zone, the probability that the markets stop lending within the year rises from 0 to 1, and lenders charge for this risk in the premium. Default comes when the debt reaches the limit, or earlier, on an unlucky day. Nobody, on the other hand, estimates the limit of emerging and low-income countries, so it is derived from that of advanced countries. The studio acknowledges that the model therefore produces too few defaults among emerging countries, whose risk starts, at the median, only 67 points above their starting debt, whereas more than half of the defaults of middle-income countries from 1970 to 2001 happened with external debt below 60% of their gross national product.
A default wipes out 37% of the debt. The country pays for it by spending five years unable to borrow, with slower growth and a government that falls half the time.
France starts with debt of 110% of GDP, against a limit of 159.7%. Its risk starts at the highest of three levels, the IMF's high-risk threshold (85% for an advanced country), its starting debt and its limit minus twenty points, which gives 139.7% here. The probability that the markets stop lending within the year reaches 14% at 145% debt, then 53% at 150%. With a risk-free rate of 3%, and given that lenders to an advanced country recover 90% of their stake in a default, they then ask for 11.6 extra points of premium. The crisis can break out on any day, with a risk of about 0.2% a day at 150%. A country that already starts inside its risk zone, or even beyond its limit, such as Singapore, the Maldives or Senegal, receives a limit set 63.4 points above its starting debt, the smallest gap between debt and limit that the study records. The studio set this rule because, without it, such a country defaulted in its very first year.
In the formula:
| Symbol | What it is | France |
|---|---|---|
| \(p\) | the probability, over one year, that the markets stop lending | 53% at 150% |
| \(d\) | the debt, as a % of GDP | |
| \(\bar d\) | the country's debt limit | 159.7% |
| \(\ell\) | the debt level at which the risk starts | 139.7% |
| \(d^{IMF}\) | the IMF's high-risk threshold | 85% |
| \(d^{ref}\) | the starting debt, or the debt after the last default | 110% |
| \(w\) | the width of the risk zone | 20 points |
| \(\rho^{D}\) | the premium lenders ask for this risk | 11.6 points |
| \(\theta^{R}\) | the share of the debt that lenders recover in a default | 90% |
Show the full formula and its sourcesThe exact form, every parameter with its source, and what the equation leaves out.
Each country has a debt limit d̄. Beyond it, the markets no longer lend. Below it, the probability that they stop lending within the year rises over the last points:
(x bounded to [0, 1]); the risk is paid for in the premium (E10):
Default comes when the debt reaches the limit, or earlier, on an unlucky day. Each day it happens with probability 1 − (1 − p)1/365.25, drawn at random. This is the confidence crisis. It wipes out 37% of the debt; the country is shut out of the markets for 5.1 years, spends no more than its structural revenue, and loses 2.6 points of potential growth in the first year and 1.2 points in the following years; its new bonds then pay a spread that declines. Its government falls with a probability of one in two, drawn at random. If it is the player's government, the player has lost.
| Parameter | Value | Status | Source |
|---|---|---|---|
debt_limit d̄ | advanced: the Ghosh et al. limit for 19 countries (France 159.7, United States 173.1, Germany 170.0…), their median 170.7 for the others; emerging 140.6; low-income 110.5 (% of GDP) | Ghosh, Kim, Mendoza, Ostry and Qureshi 2013, table 3, model limit, 23 advanced countries. Emerging and low-income: no estimate; the group's threshold times the limit / threshold ratio of advanced countries (170.7 / 85 = 2.01) | |
debt_threshold dIMF | 85 / 70 / 55% of GDP | IMF 2013 (sustainability analysis for market-access countries), table 7: high-risk thresholds, 85 (advanced) and 70 (emerging); low-income: 55, the medium-capacity threshold of the LIC-DSF (IMF and World Bank 2018), which applies to the present value of debt | |
| dref | the starting debt, or the debt after the last default | on the first day, the markets lend at the rate the data show: risk only grows above that level | |
debt_risk_width w | 20 points of GDP | the zone where the probability rises from 0 to 1: Ghosh et al. find that rates rise abruptly near the limit, over a zone shorter than the support of the primary-balance shock (p. 18 and 32), without publishing its width | |
starting_fiscal_space | 63.4 points | the smallest fiscal space in the table of Ghosh et al. (United States: 173.1 − 109.7): the space given to a country that starts in its risk zone or beyond its limit, since the markets do lend to it (Japan, Italy, Singapore) | |
recovery_rate θR | 0.90 / 0.63 / 0.63 | advanced: 90% (Ghosh et al., after Benjamin and Wright); others: 1 − 0.37, the average haircut of Cruces and Trebesch | |
haircut | 37% of the debt | Cruces and Trebesch 2013, table 1: average haircut (Sturzenegger-Zettelmeyer) of 180 restructurings, 1970-2010 (standard deviation 27) | |
exclusion_years | 5.1 years | same source: average time until a partial return to the market (median 3 years) | |
post_default_spread_first, _later | 0.0675 then 0.0316 points per point of haircut | same source: spread of new bonds in the 1st year, then in years 4 and 5; a straight line in between, nothing after five years | |
default_growth_first_year, _after | 2.6 then 1.2 points of growth | Borensztein and Panizza 2009, table 2: −1.184 − 1.388 in the 1st year, −1.2 in the years of default, nothing afterwards | |
default_fall_probability | 0.5 | same source, p. 21-22: the head of the executive changes in the year of default or the next one in 11 cases out of 22 |
Why. In Ghosh et al., governments raise their primary balance when debt rises, then less and less (“fiscal fatigue”). Beyond a limit, no balance is enough any more, and the markets stop lending; the lender charges for the probability of that day at the price that arbitrage gives (their equation 7). The consequences of default are those the literature measures, on average over decades of restructurings.
Three choices, to be validated. 1) Risk only starts above the starting debt. Without this, France and the United States would default at random around 2035, with their debt barely higher than today. 2) The risk zone is 20 points wide below the limit. Ghosh et al. describe an abrupt rise without publishing a width. 3) A country that already starts in this zone (Singapore, the Maldives, Senegal) receives the smallest fiscal space in the table of Ghosh et al. above its debt. Without this, it would default in the first year.
What it simplifies. Nobody estimates the limit of emerging and low-income countries; it is deduced from that of advanced countries. The threshold of low-income countries applies to the present value of debt, which is lower than the model's nominal debt; those of Reinhart, Rogoff and Savastano apply to external debt, which the model does not distinguish. The player does not choose to default. Default is imposed. A default does not come with the currency crisis that follows 84% of defaults (Reinhart 2002), nor with the inflation that this crisis brings (unknown currency_crisis_inflation). The sovereign credit rating does not exist yet. During exclusion, what the recession adds to the spending share is allowed to slide, as arrears. Closing this deficit as well would deepen the recession that dug it, without end (Macao would fall to −120%).