Compounding debt service
A Socratic walk-through of compounding debt service — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why can a government's borrowing costs rise in a year when it borrows nothing new?
Imagine a finance minister who does something almost unheard of: she balances the primary budget. Every programme this year is paid for out of this year's taxes. No new project is financed by borrowing. And at the end of the year the interest bill has gone up — in the United States, net interest reached roughly $881 billion in fiscal 2024, more than the defence budget, after years in that neighbourhood of restraint would still not have held it flat.
That seems to break a simple intuition: costs follow purchases, so if you buy nothing you should owe nothing more. Where does the intuition go wrong?
Reasoning it through
REASONING #Ask first what the interest bill is actually charged on. Not on this year's borrowing — on the accumulated stock of debt from every previous year. So the very first thing to notice is that "borrows nothing new" describes the flow, while the bill is levied on the stock. A year of restraint stops the stock growing from new spending; it does not make the stock disappear.
But a constant stock at a constant rate should give a constant bill. So we need something that changes even while the stock stands still. There are two such things, and they are different in kind.
The first is repricing. Government debt is not one perpetual loan; it is thousands of securities with fixed maturities. When a three-year note issued at 0.5 percent matures, the money to repay it is raised by selling a new security at today's rate. No net new borrowing has occurred — the stock is unchanged — but that slice now costs what the market charges now. Roughly a third of marketable U.S. Treasury debt matures within a year, and the weighted average maturity of the marketable stock has run around six years, so a large fraction reprices annually. The average rate the government pays is therefore a lagging average of past issuance, drifting toward current market rates as the cheap old paper runs off. A minister who borrows nothing in a year when rates have doubled will still watch her bill climb, because she has no choice about refinancing what comes due.
The second thing is the one that makes this a story about exponential growth rather than merely a story about rates. Ask: where does the money to pay the interest itself come from? If the primary budget is exactly balanced — revenue covers programmes and nothing more — then it does not come from taxes. It is borrowed. And now the stock is larger by the amount of this year's interest, so next year's interest is charged on that larger base, and the year after on a base larger still.
That is compounding, in the strict sense: a quantity whose rate of increase is proportional to its own size. Left alone it does not grow by addition, it grows by multiplication. The standard way economists write the condition is that the debt-to-GDP ratio drifts upward whenever the interest rate on the debt exceeds the growth rate of the economy, unless a primary surplus is run to offset the gap. The critical word there is surplus, not balance — balance is not enough, because balance leaves the interest to feed itself.
So the intuition failed on two counts at once: it attached the cost to a flow rather than a stock, and it assumed a stock that is left alone stays the same size.
The analogy
THE ANALOGY #It is the household version of a credit card whose minimum payment is not being made. Cut up the card so no new purchase is possible, and the balance still grows every month, because the interest that was charged has been rolled onto the balance and is now itself earning interest. The purchases stopped; the growth did not, because the growth was never coming from the purchases.
a household must eventually repay in full, while a government with its own currency and an indefinite life never repays the principal — it refinances forever, so what matters is not whether the debt is extinguished but whether it grows faster than the economy that services it, which is a comparison a card balance has no equivalent of.
Clarifying the model
THE MODEL #Two clarifications keep this honest.
The first is what "borrows nothing new" was allowed to mean. If it means a zero primary deficit, the interest is still added to the stock and everything above holds. If it means a zero overall deficit — taxes covering programmes and the whole interest bill in cash — then the stock genuinely does stand still, and only the repricing channel remains. Both are real situations, and the arithmetic differs; conflating them is where most confusion about this comes from.
The second is that the growth is not unconditional. Compounding at rate r is offset by an economy growing at rate g, because the burden that matters is the ratio of debt to the income available to service it. When growth exceeds the effective interest rate, a country can run modest primary deficits indefinitely and still see its debt ratio fall — which is roughly what happened to the very large post-war debts of the United States and United Kingdom. That relationship reverses when rates rise above growth, which is exactly why the same debt stock can feel harmless for a decade and alarming in the next.
A picture of it
THE PICTURE #How to readtwo paths raise the bill without any new spending — the maturing slice being refinanced at today's rate, and the loop at the bottom, where unpaid interest rejoins the stock and enlarges the base the next bill is charged on.
What became clearer
WHAT CLEARED #Debt service is not the price of what a government is buying now; it is the price of what it bought before, recalculated at today's rates, plus interest on interest it never paid in cash. Because the second part is proportional to the balance itself, the cost has a growth engine inside it that runs whether or not anyone is spending — and only a primary surplus, or growth faster than the interest rate, ever switches that engine off.
Where to go next
ONWARD #- Debt maturity management: whether issuing long to lock in low rates is worth the higher yield it usually costs.
- How inflation redistributes between borrower and lender, and why index-linked bonds remove that escape route.
- Fiscal rules that target the primary balance rather than the headline deficit, and what they get right.
Key terms
TERMS #| Term | What it means |
|---|---|
| Primary balance | revenue minus spending, excluding interest payments; the measure that isolates current policy from the legacy of past borrowing. |
| Debt service | the interest owed on the outstanding stock in a period, distinct from repayment of principal. |
| Rollover | issuing new securities to repay maturing ones, which leaves the stock unchanged but reprices that slice at current rates. |
| Weighted average maturity | the average time to maturity across the debt stock, which governs how fast the interest cost tracks market rates. |
Every term the collection defines is gathered in the glossary.