New on FinoLib · Volume III: Finance Encyclopedia
I have just published a new article on FinoLib, my free finance education library: Debt Sustainability Analysis: r vs g Explained. It is part of Volume III: Finance Encyclopedia. Below is a summary of the topic in plain, expert language, so you can follow the idea even before you open the full article.
The idea in plain words
Debt sustainability comes down to a race between two numbers: r, the interest rate on the debt, and g, the growth rate of the economy. If r is below g, a government can run a modest deficit before interest and the debt ratio can still fall, because the economy outgrows the debt. If r is above g, the government must run a surplus before interest just to keep the ratio from rising.
Key points
■At 80% debt, r = 3% and g = 5%, a 0.5% primary deficit takes the ratio to 78.98% in a year; at r = 4% and g = 2% it rises to 82.07%.
■The debt-stabilizing primary balance is pb* = d × (r − g) ÷ (1 + g): 1.57% of GDP in the bad case.
■Over ten years the two cases end 32.0 points of GDP apart with identical fiscal policy.
■Average r lags market rates: the US average marketable rate rose from 1.424% (January 2022) to 3.475% (August 2026).
A closer look
The single most important relationship in assessing whether a government’s debt trajectory is sustainable is the comparison between r (the average interest rate the government pays on its debt) and g (the country’s nominal GDP growth rate). When g exceeds r, a government’s debt-to-GDP ratio can actually fall over time even while running a primary deficit (a deficit before interest payments), because the economy is growing faster than the debt burden itself. When r exceeds g, the reverse dynamic takes hold — debt-to-GDP tends to rise over time unless the government runs an offsetting primary surplus large enough to compensate.
A worked example
Worked Example — The r-g Dynamic
The arithmetic behind r versus g is the debt-dynamics identity. With d the debt-to-GDP ratio, r the average nominal interest rate on the debt, g nominal GDP growth and pd the primary deficit as a share of GDP: d_t = d_t−1 × (1 + r) ÷ (1 + g) + pd_t Interest makes the debt grow by (1 + r); growth makes the economy that carries it grow by (1 + g); the primary deficit adds new borrowing on top. Take a country with debt at 80% of GDP, r = 3%, g = 5% and a primary deficit of 0.5% of GDP, a 2-percentage-point favorable gap.
Decision rule
Compute the debt-stabilizing primary balance, pb* = d × (r − g) ÷ (1 + g), and compare it with the government’s actual primary balance. If the actual balance is weaker than pb* by more than about 1% of GDP, the debt ratio is on a rising path; treat any claim of sustainability as resting on a future policy change, and ask what it is.
The costliest mistake
Projecting today’s favorable r − g forward for a decade. In the table above, a government that assumes Scenario A (r = 3%, g = 5%) when Scenario C (r = 4%, g = 2%) arrives is off by 32.0 points of GDP after ten years with no change in its own policy.
Why It Matters
This single r-versus-g relationship is precisely why central bank interest rate decisions (Volume I’s Part 2) have such enormous, direct fiscal consequences well beyond their immediate effect on inflation and growth — a sustained period of rates rising faster than growth can flip a stable, sustainable debt trajectory into a steadily worsening one, entirely independent of any change in the government’s own spending or tax policy.
Where Experts Disagree: Does r Below g Make Debt Cheap?
In his 2019 presidential address to the American Economic Association, “Public Debt and Low Interest Rates,” Olivier Blanchard showed that since 1950 the US one-year rate had been below nominal growth in every decade except the 1980s, and the ten-year rate in four of seven decades. If that persists, he argued, debt rollovers “may well be feasible” and “public debt may have no fiscal cost”: a government can let old debt be refinanced and still see the ratio fall.
Recap
The debt-dynamics identity d_t = d_t−1 × (1 + r) ÷ (1 + g) + pd_t shows why the gap between the interest rate and nominal growth drives the debt ratio: on the same 0.5% primary deficit, an 80% ratio ends ten years later at 70.60% when r is 2 points below g and at 102.61% when it is 2 points above.
What the full article covers
The full article includes worked examples, a clear decision rule, the one mistake that costs people the most, a recap, a reveal-on-click quiz and the official sources behind every figure.
Volume III: Finance Encyclopedia
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The complete article, Debt Sustainability Analysis: r vs g Explained, with every worked example, diagram, quiz and source, is free to read in Volume III, Finance Encyclopedia on finolib.com.
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