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QuantiaQuantitative economics
Data model QM-01 · Debt

Debt Sustainability Model

Debt dynamics, the snowball term and the balance that holds the ratio flat

Reference QM-01Inputs 7 assumptionsOutputs 4 metrics, 2 exhibitsRuns In your browser
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Project a public debt ratio forward, decompose what is moving it, and read off the primary balance that would hold it flat. The model separates the part driven by policy from the part driven by arithmetic, which is usually where the argument is.

The model executes entirely in the browser. No assumption entered here is transmitted or retained. Use Copy results to export the full assumption set and output for use in a note or spreadsheet.

What the model does

Public debt as a share of GDP moves for three reasons and only three. The government runs a primary deficit or surplus. The existing stock accrues interest faster or slower than the economy grows in nominal terms. And things happen that never pass through the deficit at all: currency revaluation, bank recapitalisation, arrears, privatisation receipts.

The model is the standard debt dynamics identity, iterated forward.

dt = dt-1 × (1 + r) / (1 + g) − pbt + sfat d is debt as a share of GDP, r the effective nominal interest rate on the stock, g nominal GDP growth, pb the primary balance as a share of GDP, and sfa the stock-flow adjustment.

Rearranged, the change in the ratio separates cleanly into a snowball term and a policy term.

Δdt = dt-1 × (r − g) / (1 + g) − pbt + sfat The first term is the snowball. It is favourable whenever nominal growth exceeds the effective interest rate, and it compounds on the existing stock, which is why the level of debt matters and not just the flow.

The debt-stabilising primary balance

Setting the change to zero gives the primary balance that would hold the ratio flat.

pb* = d × (r − g) / (1 + g) + sfa Read this as the fiscal effort required to stand still. The gap between the actual primary balance and pb* is the adjustment needed, in percentage points of GDP.

This single figure does more analytical work than any debt level. A country with 130 percent debt and a favourable differential can be on a declining path while a country with 60 percent and an adverse differential is not. The level tells you about vulnerability to a shock. The differential tells you about direction.

Why the stress path is separate

The effective interest rate on a debt stock is a weighted average of coupons set over many years. It moves slowly, because only maturing debt reprices. That is why the stress scenario applies a shock to the rate rather than replacing it: the model shows what happens if the marginal cost of borrowing rises while the average is still catching up, which is the mechanism through which a rate shock actually transmits to a sovereign.

The distance between the baseline and the stress path at the horizon is a useful summary of refinancing exposure. A short average maturity produces a wide gap. A long one produces a narrow one.

Applications

  • Sovereign and credit analysts. Read the stabilising primary balance and the adjustment gap directly, and use the stress path as a first-pass measure of refinancing exposure before going to the maturity profile.
  • Macroeconomic researchers. The identity is the one used in the fiscal sustainability literature, with every parameter exposed. Useful for teaching the snowball intuition, since the sign flip at r equals g is visible on the chart.
  • Corporate and project finance teams. The same arithmetic governs any leveraged balance sheet where debt is measured against a growing base. Substitute nominal revenue growth for g and the cash interest rate for r.

Limitations

Every model is a simplification and this one is explicit about which simplifications it makes.

  • It assumes constant parameters over the horizon. Real interest rates, growth and primary balances are correlated with each other and with the cycle, and the model does not capture that feedback.
  • There is no exchange rate. For a sovereign with substantial foreign currency debt, valuation effects should be entered through the stock-flow adjustment rather than assumed away.
  • There is no fiscal reaction function. It does not assume governments tighten when debt rises, which is why the paths can run away in a manner no real government would tolerate.
  • The effective rate is a single number. In practice it is an average over a maturity profile, and the speed at which a market rate feeds into it depends on that profile.

Use in client and published work

The model is free to use and the specification above is published so that results can be reproduced independently. Figures generated here should be cited with the model reference and assumption set, as with any other calculation. Extension, recalibration to a specific market, or integration with proprietary data are all within scope of a standard engagement: contact@quantiaconsulting.online.