Why Use a Fiscal Stress Test?
A debt projection depends on assumptions about interest rates, economic growth, fiscal balances, and the valuation of foreign-currency obligations. A stress test makes those assumptions explicit and asks how the projected debt path changes when one or more of them worsen. It is a structured comparison, not a claim that an adverse scenario will occur or a probability estimate of its occurrence.
The useful question is not simply whether a debt ratio rises. It is which assumptions cause it to rise, by how much, and whether the government must also find substantial cash to refinance maturing obligations. The calculations below use a deliberately simple one-period framework, then apply it across scenarios. They are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The Basic Debt-Ratio Projection
First suppose all debt is denominated in domestic currency, the entire opening debt stock accrues interest at rate \(i_t\), and nominal GDP grows at rate \(g_t\). Assume there are no stock-flow adjustments, valuation changes, or other debt-creating transactions. The primary surplus is measured in currency units relative to \(Y_t\). The debt stock and GDP therefore satisfy \(B_t=(1+i_t)B_{t-1}-p_tY_t\) and \(Y_t=(1+g_t)Y_{t-1}\).
The interest factor multiplies opening debt, while GDP growth determines the denominator used to express that debt as a ratio. A higher interest rate raises the projected ratio; higher nominal GDP growth lowers it, all else equal. A primary surplus subtracts from debt relative to current GDP, and a primary deficit adds to it. This is a simplifying projection equation: maturity structure, inflation surprises, exchange-rate revaluations, and other stock-flow adjustments must be added when material.
For a foreign-currency component, depreciation can raise the domestic-currency value of debt even before considering the primary balance. Suppose \(q\) is the share of opening debt denominated in foreign currency, \(i_d\) is the rate on domestic-currency debt, \(i_f\) is the rate on foreign-currency debt, and \(e\) is the increase in the domestic-currency price of foreign currency during the year. With a common primary balance and no other adjustments, a one-year projection is:
The foreign-currency debt is first increased by interest and then translated at the new exchange rate. This expression assumes \(q\) is the opening debt share, and that the rates and exchange-rate change apply over the same year. It is not appropriate to treat \(e\) as an interest rate or to apply it to domestic-currency debt.
Worked Scenarios
Worked Example: Projecting a Baseline and a One-Year Adverse Shock
A hypothetical government begins with debt equal to 60% of GDP. In the baseline, the effective annual interest rate is 5%, nominal GDP growth is 3%, and the primary surplus is 1.5% of current-year GDP in each of the next three years. There are no exchange-rate effects or stock-flow adjustments. Find the three closing debt ratios. Then find the first-year ratio under an adverse scenario with a 7% interest rate, 1% nominal GDP growth, and a primary deficit of 0.5% of GDP.
Method. Apply the basic debt-ratio equation once per year, using each year’s closing ratio as the next year’s opening ratio. The baseline factor is \(1.05/1.03=105/103\approx1.019417476\).
Baseline, year 1. The opening ratio is \(0.60\), so \(b_1=(105/103)(0.60)-0.015=0.611650485-0.015=0.596650485\). The closing debt ratio is about 59.67%.
Baseline, year 2. Use the unrounded year-one ratio: \(b_2=(105/103)(0.596650485)-0.015\approx0.608235932-0.015=0.593235932\). The closing ratio is about 59.32%.
Baseline, year 3. Continue from the unrounded year-two ratio: \(b_3=(105/103)(0.593235932)-0.015\approx0.604755076-0.015=0.589755076\). The closing ratio is about 58.98%. The ratio declines over the baseline projection because the stated primary surplus more than offsets the effect of the interest-growth differential.
Adverse first year. The primary balance is a deficit, so \(p_1=-0.005\). Thus \(b_1=(1.07/1.01)(0.60)-(-0.005)\approx0.635643564+0.005=0.640643564\), or about 64.06%. This is approximately 4.40 percentage points above the baseline first-year ratio of 59.67%.
Check and trap avoided. A primary deficit increases debt; it must not be subtracted as though it were a surplus. The adverse calculation adds 0.005. The result also exceeds the opening 60% ratio, which is consistent with the combined effect of a higher interest rate, slower growth, and a primary deficit.
Worked Example: Isolating a Currency-Valuation Shock
A hypothetical government has an opening debt ratio of 50%. Twenty percent of the opening debt is foreign-currency debt. The effective annual interest rates are 4% on domestic debt and 3% on foreign debt. Nominal GDP grows by 5%, the primary surplus is 1% of current-year GDP, and the domestic currency depreciates by 10% against the foreign currency. Find the closing debt ratio, then compare it with the result if there were no depreciation.
Method. Apply the foreign-currency projection equation, using \(q=0.20\). The domestic debt gross factor is \(1.04\); the foreign debt gross factor including depreciation is \(1.03(1.10)=1.133\).
The weighted debt factor before dividing by GDP growth is \((0.80)(1.04)+(0.20)(1.133)=0.832+0.2266=1.0586\). Therefore \(b_t=(1.0586/1.05)(0.50)-0.01\approx0.504095238-0.01=0.494095238\). The closing ratio is about 49.41%.
With no depreciation, the foreign gross factor is \(1.03\). The weighted factor is \((0.80)(1.04)+(0.20)(1.03)=0.832+0.206=1.038\). The closing ratio is \((1.038/1.05)(0.50)-0.01\approx0.484285714\), or about 48.43%.
The depreciation shock adds approximately \(0.494095238-0.484285714=0.009809524\), or 0.98 percentage points, to the closing ratio. The surplus reduces the ratio in both scenarios, but it does not cancel the upward valuation effect of depreciation.
Check and trap avoided. The 10% depreciation applies to the foreign-currency component, not to the full debt stock. Applying it to all debt would overstate the direct currency effect. The calculation also includes foreign interest: the correct foreign factor is \((1+i_f)(1+e)\), not \(1+i_f+e\).
Worked Example: Estimating Gross Financing Needs
A hypothetical government begins the year with debt equal to 60% of prior-year GDP. Its effective annual interest rate is 4%, nominal GDP grows by 2%, and its primary surplus is 0.5% of current-year GDP. Principal equal to 8% of current-year GDP matures during the year. Assume no other financing items. Calculate gross financing needs as a share of current-year GDP.
Method. Gross financing needs here consist of interest payments, principal amortization, and the primary deficit. The interest bill is based on opening debt, so convert the opening debt ratio from prior-year GDP units to current-year GDP units: \(0.04(0.60)/(1.02)=0.024/1.02\approx0.0235294\).
The primary balance is a surplus, so its contribution to financing needs is \(-0.005\). Add the three components: \(0.0235294+0.08-0.005=0.0985294\). Gross financing needs are approximately 9.85% of current-year GDP.
Check and trap avoided. A surplus lowers the amount that must be financed, whereas maturing principal still needs refinancing unless it is repaid from available funds. The interest term is \(0.04(0.60)/1.02\), not \(0.04(0.60)\), because the requested denominator is current-year GDP.
How to Design and Interpret a Stress Test
A clear stress test names the baseline, states which assumptions change, and reports results in comparable units. The following sequence prevents a scenario from becoming a collection of disconnected numbers.
Record the opening debt ratio, the currency composition if relevant, and the period of projection. State whether balances are measured against current-year or prior-year GDP.
List interest rates, nominal GDP growth, primary balances, exchange rates, and any stock-flow adjustments. Use the same definitions and timing in every scenario.
Change one assumption to isolate its effect, or state all changes in a combined adverse scenario. For a multi-year projection, carry forward unrounded ratios and apply each year’s assumptions to that year.
Report the change in percentage points, identify the assumptions driving it, and check financing needs separately. A debt-ratio path alone does not show the timing of cash that must be raised.
A useful comparison table keeps the scenario inputs and outputs visible. For example, it can show the baseline and adverse interest rate, growth rate, primary balance, exchange-rate change, and closing debt ratio. A one-variable-at-a-time comparison helps identify sensitivity; a combined scenario captures the possibility that several conditions worsen together. These answer different questions and should not be confused.
Stress-test results are conditional: if the stated assumptions hold, the equations imply the reported path. They are not forecasts, and a simple deterministic scenario does not assign likelihood to its inputs. In this framework, interest rates are imposed rather than determined by the debt path, so the calculations do not model an endogenous feedback from rising debt to borrowing costs. The interpretation should remain within those assumptions.
Common Errors and Practical Checks
Several errors can materially distort a result even when the formula is familiar:
- Reversing the primary-balance sign. A positive primary surplus is subtracted; a primary deficit is represented by a negative \(p_t\) and therefore raises projected debt.
- Mixing nominal and real growth. The formula shown uses nominal interest and nominal GDP growth. Do not substitute a real growth rate without first making the quantities consistent.
- Using the opening ratio repeatedly. In a multi-year path, each year begins with the previous year’s closing ratio. Rounding intermediate ratios can also accumulate avoidable error.
- Applying depreciation to the wrong debt share. Currency depreciation changes the domestic value of foreign-currency obligations, not domestic-currency obligations.
- Leaving out refinancing needs. A debt ratio is a stock measure. Gross financing needs also include maturing principal and can be large even if the ratio is stable.
As a final reasonableness check, verify the units, the signs, and the direction of each effect. Holding other assumptions fixed, a larger primary surplus should lower the projected ratio; higher nominal growth should reduce the opening-debt contribution; and depreciation should increase the domestic value of a positive foreign-currency debt share. These checks do not replace arithmetic, but they often reveal a setup error.
Check Your Understanding
For each problem, state the relevant assumptions and retain unrounded values until the final answer.
- A government begins with debt equal to 55% of GDP. The effective annual interest rate is 3%, nominal GDP growth is 4%, and the primary surplus is 1% of current-year GDP. Find the closing debt ratio. (Answer: 53.47%)
- Opening debt is 40% of GDP, of which 25% is foreign-currency debt. The domestic and foreign effective annual interest rates are 3% and 5%, respectively; depreciation is 8%, nominal GDP growth is 4%, and the primary surplus is 0.5% of current-year GDP. Find the closing debt ratio. (Answer: 40.12%)
- Opening debt is 70% of prior-year GDP. The effective annual interest rate is 5%, nominal GDP growth is 3%, the primary deficit is 1% of current-year GDP, and principal equal to 6% of current-year GDP matures. With no other financing items, find gross financing needs as a share of current-year GDP. (Answer: 10.40%)
- Debt begins at 50% of GDP. For two consecutive years, the effective annual interest rate is 4%, nominal GDP growth is 2%, and the primary surplus is 0.5% of current-year GDP. Assuming no other adjustments, find the debt ratio after year 2. (Answer: 50.97%)
- Opening debt is 45% of GDP, with a foreign-currency share of 30%. The domestic and foreign effective annual interest rates are 4% and 6%, depreciation is 5%, nominal GDP growth is 3%, and the primary deficit is 1% of current-year GDP. Find the closing debt ratio. (Answer: 47.39%)