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Monetary systems · Lesson 13

Debt sustainability is a conditional calculation

Separate interest costs, primary balances, growth, and refinancing risk.

By the end: Calculate a debt ratio and state what its arithmetic does not establish.

Reconcile the stock first

Begin with debt outstanding, add interest that must be financed and the primary deficit, and subtract any primary surplus. This simplified debt law assumes no privatization receipts, valuation changes, cash-buffer changes, or other stock-flow adjustments. It is a financing identity for the specified case, not a complete government budget model.

The primary deficit is spending excluding interest minus revenue. The effective interest rate on the debt stock need not equal today's policy rate or the yield on newly issued debt. Old fixed-rate bonds keep their contractual payments until maturity. Refinancing gradually changes the average cost, while short-term or floating-rate debt transmits rate changes faster.

Separate a year's borrowing from the debt that comes due

A debt stock records outstanding obligations at a date. A deficit records a shortfall of revenue against spending during an interval. Start with a paper example in currency units. A treasury owes 240 on January 1. During the year it receives revenue of 90 and spends 96 excluding interest. Subtract 90 from 96 to find a primary deficit of 6. If interest costs 12, total spending is 108 and the overall deficit is 18. Assuming no other adjustments and financing this shortfall by borrowing, debt ends at 258. Adding interest to the overall deficit again would count the same expense twice.

The 12 of interest is a payment to creditors, not a reduction of the amount originally borrowed. A creditor who receives it gains income and, if paid through a bank, a deposit. The treasury's cash account falls while the recipient bank gains reserves and credits that deposit. A preceding bond issue or tax receipt has separate entries that fund the treasury account under the applicable rules. These steps explain payment mechanics. They do not imply that interest recipients spend all their income or that treasury cash can become negative whenever the arithmetic requires funding.

Now suppose 80 of the opening debt matures during this same year. Maturity is the date when a contract requires principal repayment. If the treasury has no cash buffer and replaces that debt with new bonds, it must issue 98 altogether. Of this, 80 replaces principal that comes due and 18 finances the overall deficit. Opening debt 240 plus issuance 98 minus principal repayment 80 gives ending debt 258. Gross issuance is therefore 98, but the net increase is only 18. A large financing operation can coexist with a much smaller annual deficit.

On the creditor's books, purchasing a new bond exchanges a deposit for a claim on the treasury. Repayment of a maturing bond makes the opposite exchange. Neither exchange alone gives the creditor a gain equal to the principal. Interest and any difference between purchase price and repayment are separate. This also distinguishes refinancing from a central-bank purchase of an existing bond. When a central bank buys from a nonbank, the seller exchanges a bond for a deposit and its bank receives reserves. The treasury need not receive cash from that secondary-market sale, and its outstanding principal does not disappear.

Debt can change for reasons outside the deficit. In another paper case, opening debt is 240 and the overall deficit is 18, but the treasury also borrows 7 to build a cash buffer. Ending debt becomes 265 and treasury cash rises by 7. The reconciliation is opening debt plus deficit plus the cash adjustment. Acquiring a financial asset can require similar funding without being ordinary consumption spending. Conversely, using an existing cash buffer can reduce new borrowing. Before comparing reported debt changes with deficits, check whether the statistics use the same institutional coverage, accounting period, and valuation basis.

Then compare debt with nominal income

A debt-to-GDP ratio divides a currency stock by annual nominal output in the same currency. If debt starts at 100, interest is 3, and the primary deficit is 2, ending debt is 105. If nominal GDP also rises from 100 to 105, the ratio stays at 100%. Debt increased, but the ratio did not. Always check the numerator and denominator separately.

Nominal growth includes real growth and price changes. Inflation can increase nominal tax bases and reduce the real burden of existing fixed nominal debt, but it may also raise refinancing rates, indexed spending, and currency risk. It is not a costless way to erase obligations. A stable debt ratio is not by itself evidence of adequate public services, equitable taxation, or financial stability.

Derive a stable ratio using the right denominator

A debt ratio needs a dated numerator and a defined income flow. Annual nominal GDP is the currency value of final domestic production during a year. Debt of 240 divided by annual GDP of 400 gives 0.60, conventionally reported as 60% of GDP. GDP is not government revenue and is not a cash account available to repay creditors. The ratio compares obligations with the scale of domestic economic activity. Revenue of 90 in the preceding paper case would produce a different ratio, 240 divided by 90, about 2.67 times annual revenue. Neither calculation tells us which resources the treasury can legally collect.

Continue with opening debt 240 and annual GDP 400. Let the effective annual interest rate be 5%, nominal GDP growth be 4%, and the primary deficit be 6 currency units. First calculate interest as five hundredths of 240, giving 12. Add it and the primary deficit to opening debt to obtain 258. Separately, 4% of 400 is 16, so ending annual GDP is 416. Finally divide 258 by 416. The result is about 62.02%. Comparing debt 258 with opening GDP 400 instead would give 64.5%, a different question using a different denominator.

To hold the original 60% ratio, find the debt compatible with the new GDP before finding the budget balance. Sixty percent of 416 is 249.6. Opening debt plus interest would be 252, so a primary surplus of 2.4 reduces that amount to the target. The surplus enters the debt calculation with a negative sign. This is a conditional requirement, not a recommendation to cut spending or raise taxes by that amount. If a budget change also changes GDP, the target must be recalculated using that different GDP path. The arithmetic does not hold behavior fixed in the real economy.

There is a shorter way to understand this result without memorizing a formula. Keeping the ratio constant permits debt to grow at the same percentage rate as nominal GDP. Four percent of the opening debt 240 is 9.6, the permitted debt increase. Interest adds 12, exceeding that allowance by 2.4. A primary surplus removes the difference. If interest were instead 2%, its addition would be 4.8 and a primary deficit of 4.8 would still leave the total increase at 9.6. This comparison depends on the opening debt stock, not on comparing the two percentage rates alone.

Budget balances stated as percentages require another denominator check. Two percent of opening GDP 400 is 8 currency units. Two percent of ending GDP 416 is 8.32. Substituting either amount as a primary deficit produces a different debt result from the 6-unit case above. The worked model below uses a primary deficit in currency units, although rates and growth are percentages. Its opening debt and GDP are both 100, which makes some numerical values look interchangeable. That coincidence disappears when opening debt and GDP differ. Write the currency amount next to each percentage before combining a balance with a debt stock.

Work through the accounts and assumptions

1. Accumulate debt
New debt=D×(1+r)+p

D is opening debt, r the effective nominal rate as a decimal, and p the primary deficit in currency units. A primary surplus is negative p. Other stock-flow adjustments are excluded.

2. Divide by new nominal GDP
New ratio=New debtY×(1+g)

Y is opening nominal GDP and g its growth rate as a decimal. Multiply the ratio by 100 to express a percentage.

Change the assumptions, not the accounting

Opening debt 100 currency units; opening annual nominal GDP 100. Each preset replaces the whole one-year experiment.

StepAssumption or calculationResult
Accumulate debt100 + 3 interest + 2 primary deficit105 units
Grow nominal GDP5% growth on 100105 units/year
Debt / annual GDP105 / 105100%
Effective rates apply to the whole opening stock in this model. No endogenous growth response, currency revaluation, cash adjustment, or default probability is calculated. A constant ratio is not proof of sustainability.
One-year reconciliationCurrency units
Opening debt100
Interest at 3%+3
Primary deficit+2
Ending debt105
Ending nominal GDP105
Debt / GDP100%
No stock-flow adjustments. The primary deficit is 2 currency units, not 2% of ending GDP.

Read the presets as separate conditional years

The three buttons above replace the assumptions for one year starting with debt 100 and annual nominal GDP 100. They do not advance a timeline. Select Higher financing cost and then Stable ratio. The latter still starts at 100, not at the debt reached under the former. There are no debt or growth sliders here and no hidden accumulation. This matters when interpreting the comparison. It isolates different assumptions applied to the same opening position, rather than showing what happens when a government changes course after a bad year. A multiple-year path requires additional paper calculations.

In Stable ratio, interest adds 3 and the primary deficit adds 2, while GDP adds 5. Think of the 5 debt increase as the sum of two financing needs. In Higher financing cost, interest adds 8 and the unchanged primary deficit adds 2, while GDP adds only 2. Relative to the first case, ending debt is 5 units higher and GDP is 3 units lower. Both differences raise the ratio. The labels do not establish that higher interest caused lower growth. The model selects them together as assumptions; it contains no equation linking one to the other.

For an additional paper sensitivity check, change only interest in Stable ratio from 3% to 4%. Interest becomes 4, debt becomes 106, and GDP remains 105. Dividing gives about 100.95%, an increase of about 0.95 percentage points from the original 100% ratio. Now restore 3% interest but lower growth from 5% to 4%. Debt remains 105 and GDP becomes 104, producing about 100.96%. Similar movements in the final ratio can therefore have different causes. Inspecting only the ratio would miss whether the change came through financing costs or the size of the economy.

A second paper example shows why today's market yield need not equal the effective rate. Suppose opening debt 100 consists of 80 fixed-rate debt paying 2% and 20 paying 7%, both outstanding throughout the year. Interest is 1.6 on the first part and 1.4 on the second, totaling 3. The effective rate is therefore 3%, not 7%. If the market yield for new borrowing rises, the old fixed payments do not immediately change. Floating-rate reset dates, maturity dates, and the timing of new issuance determine how costs adjust. The calculator simplifies these details into one rate on opening debt.

For a repeated-year paper calculation, use the Stable ratio ending amounts as the next opening amounts. Suppose the same 3% effective interest and 5% nominal growth apply again, with another primary deficit of exactly 2 currency units. Interest is now 3.15 on debt 105, so debt becomes 110.15. GDP 105 grows by 5.25 to 110.25. The ratio is about 99.91%, slightly below 100%. Holding the primary deficit at 2 units differs from holding it at 2% of opening GDP. The latter would be 2.10 and would keep debt equal to GDP in this particular example.

Sustainability includes uncertain future choices

Debt arithmetic is sensitive to growth, interest rates, primary balances, and contingent liabilities such as bank rescues. A country may face a rollover problem before its long-run arithmetic looks impossible if investors refuse to refinance maturing debt on affordable terms. Maturity structure and currency denomination therefore matter alongside the debt ratio.

An issuer of its own floating currency with mostly domestic-currency debt has different nominal payment options from a foreign-currency debtor or a currency-union member. Legal constraints and central-bank independence still matter. A projection should show assumptions and adverse scenarios rather than announce a universal safe debt threshold. Our calculator illustrates one year, not a probability of default or a fiscal-policy prescription.

Distinguish payment risk from a favorable projection

A treasury can have a modest annual interest bill and still face a large payment date. Imagine debt 100 with 60 principal due next month. Expected annual revenue may cover annual spending, but that does not supply 60 of cash next month. Refinancing depends on buyers, acceptable terms, payment arrangements, and legal borrowing authority. Extending maturity can spread payment dates without reducing total principal. This is a liquidity question about timing. Whether future revenue and spending can support the debt over time is a different question. A stable projected debt ratio does not guarantee either condition on every payment date.

The currency of the obligation changes the payment problem. In an additional paper case, a government owes 50 foreign currency units. Quote the exchange rate as domestic units per foreign unit. At 2, the domestic equivalent is 100. If the quote rises to 3, the equivalent becomes 150 even without new borrowing or interest. This is domestic depreciation. With domestic nominal GDP unchanged, the debt ratio rises through currency valuation. That stock-flow adjustment is excluded from the calculator. Issuing domestic claims cannot directly deliver the foreign settlement asset. The borrower needs foreign receipts, foreign funding, reserves, or another agreed means of settlement.

A government owing its own floating currency instead faces no such mechanical conversion of its domestic face debt after depreciation. It still faces law, institutions, refinancing conditions, and the real purchasing power of payments. A national government in a currency union does not individually control issuance of the common settlement asset. These are distinct arrangements, not small variations in a single debt ratio. Nor is central-bank liquidity the same as government revenue or loss-absorbing capital. A loan adds an asset and an obligation to its recipient; whether support is available and who bears losses require separate institutional analysis.

Some obligations enter public debt only after a specified event. A guarantee is a promise to pay if another borrower fails under the guarantee's terms. If a previously excluded guarantee requires the treasury to borrow and pay 15, debt can rise even though the earlier baseline omitted that cost. The creditor receives payment, while the government takes on financing needs and possibly a recovery claim. Whether the transaction is recorded as spending, a financial acquisition, or another adjustment depends on its substance and statistical treatment. A useful stress case states the assumed event and its accounting rather than silently changing the primary deficit.

To investigate a debt claim, reconstruct a short reconciliation on paper. Identify opening principal, interest, the primary balance, and any cash or valuation adjustment. Record maturities separately so gross financing does not become a second deficit. Then state the GDP path and calculate the ratio. Compare a case with changed financing costs against a case with changed growth, keeping other assumptions fixed where possible. Finally list what the calculation leaves unresolved, such as refinancing access or foreign-currency funding. This procedure can expose an inconsistent argument without pretending to estimate default probabilities or decide which fiscal choices a country should make.

Exercises

A worse refinancing environment

Debt starts at 100 and nominal GDP at 100. Effective interest is 8%, the primary deficit is 2 units, and nominal GDP grows 2%. Find ending debt and the debt ratio.

Hint

Apply growth to GDP separately from the debt calculation.

Worked answer
ItemResult
Debt: 100 + 8 + 2110
GDP: 100 × 1.02102
Debt / GDP107.84%
The effective rate is an assumption, not a mechanical consequence of one policy announcement.

Debt becomes 110. GDP becomes 102. The debt ratio is 110 divided by 102, about 107.84%. This scenario assumes the entire effective debt cost is 8%; a policy-rate increase does not automatically reprice all existing debt that way.

Stabilize this ratio

Opening debt and GDP are each 100. Effective interest is 6% and nominal GDP growth is 3%. What primary balance holds the ratio at 100%, under the model?

Hint

Target ending debt equal to ending GDP.

Worked answer
Debt without primary adjustmentTarget debtRequired primary balance
106103Surplus 3
A budget adjustment could itself affect growth, which this conditional calculation holds fixed.

GDP ends at 103, so debt must also end at 103. Interest adds 6 to opening debt 100, so a primary surplus of 3 is needed. Whether that surplus is feasible, desirable, or compatible with the assumed growth rate is a separate economic question.

Sources and further reading

These references support the mechanisms and identify institutional context. They are optional; the explanations and worked answers above stand alone. Numerical examples are teaching cases, not live financial data or personal financial advice.

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