Debt Sustainability Equations
The Core Equation
“Future debt relative to revenue is a function of 1) spending more or less than one makes in revenue, 2) the ‘compounding’ of one’s existing debts, and 3) revenue growth.” — a-00076
Future Debt/Revenue = f(primary deficit, interest × existing debt, revenue growth)
Three levers:
- Primary deficit (spending ex-interest vs revenue): positive = debt grows
- Debt compounding (interest rate × existing debt): more debt + higher rates = faster growth
- Revenue growth (denominator): faster growth = lower ratio
The Key Metric: Debt-to-Revenue, Not Debt-to-GDP
“I am looking at debt-to-revenue rather than debt-to-GDP. That is because GDP doesn’t matter for the government’s finances unless it is tapped into because what matters are its actual cash flows.” — a-00077
Use debt/revenue for fiscal stress; debt/GDP for structural comparisons.
Government revenue ≠ GDP. At ~17-20% of GDP for the US, 1 of government cash flow.
The r vs g Rule
“The expected level of nominal interest rates relative to nominal growth rates tells us how debt and debt service are likely to grow or shrink.” — a-00078
r > g → debt/income expands automatically, even with zero primary deficit
“If nominal interest rates are at the same level as nominal income growth and a government is running no primary deficit (i.e., revenue = spending excluding interest), the debts will stay the same relative to the incomes. But if interest rates are higher than income growth, then the debt burdens of existing debts will increase.” — a-00080
Stability condition: r = g AND primary balance = 0
Debt spiral condition: r > g with any deficit at all
Quantitative Thresholds
Stability requires satisfying ALL of: a-00079
- r ≤ g (interest rate ≤ growth rate), OR
- Primary surplus large enough to offset r-g spread
US Baseline (2025)
“Nominal income is growing at 3.9% a year, interest rates are 3.5%, and debt levels start at 580% of government income.” — a-00082
| Metric | Value |
|---|---|
| Debt/Revenue | 583% |
| Projected 2035 | 648% |
| Nominal Growth (CBO) | 3.9% |
| Effective Interest Rate | 3.5% |
At these numbers, r (3.5%) < g (3.9%) — barely stable. Any shock to rates (+1%) or growth (-1%) flips the trajectory to spiral. a-00084
FX Impact of Monetization
“Mechanically, pushing down interest rates usually causes the currency to sell off. If you are getting less interest in the meantime because interest rates fell, the new deal is strictly worse for the investor holding that currency.” — a-00083
Rate cuts to manage debt → currency depreciation. This is the transmission channel from debt monetization to FX.
Tension The r-g framework assumes stable demand for government debt. Once net selling begins (see sovereign-debt-stress), rates rise regardless of CB desire, potentially blowing up the r-g equation.