Integrated Public-Debt Sustainability Workshop

From Debt Accounting to a Sustainability Assessment

A debt-sustainability calculation is only as useful as its accounting and assumptions. A government can have a manageable projected debt ratio but face a large near-term refinancing need; it can also meet current payments while its debt ratio rises under the assumed fiscal path. An integrated assessment therefore tracks both the debt stock and the cash flows needed to service or refinance it.

This workshop combines debt accounting, debt-ratio projections, primary balances, adverse interest and growth assumptions, refinancing needs, and national saving. The methods are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. All figures below are hypothetical.

Definitions and conventions: Let $B_t$ be nominal public debt at the end of period $t$, $Y_t$ nominal GDP, and $b_t=B_t/Y_t$ the debt-to-GDP ratio. Let $i_t$ be the effective nominal interest rate on opening debt, $g_t$ nominal GDP growth, and $PB_t$ the primary surplus in dollars. A surplus is positive; a primary deficit is negative. Let $SFA_t$ be a stock-flow adjustment, positive when it adds to debt. A primary balance excludes interest payments.

The basic accounting identity assumes that interest accrues on opening debt during the period and that the primary balance and any stock-flow adjustment affect end-of-period debt. It is written in nominal terms: the interest rate and GDP growth rate must both be nominal, or both be expressed consistently in another unit of account. If there is no stock-flow adjustment, set $SFA_t=0$.

B_t=(1+i_t)B_{t-1}-PB_t+SFA_t

Dividing by current GDP converts the identity into a ratio projection. Because $Y_t=(1+g_t)Y_{t-1}$, the opening debt expressed relative to current GDP is $B_{t-1}/Y_t=b_{t-1}/(1+g_t)$. Define $p_t=PB_t/Y_t$ and $s_t=SFA_t/Y_t$. Then:

b_t=\frac{1+i_t}{1+g_t}b_{t-1}-p_t+s_t

This equation is a compact way to organize a forecast, not a prediction independent of assumptions. Its interest rate should represent the effective cost applied to the debt stock, while market refinancing rates can affect that cost gradually as debt matures. A large exchange-rate valuation change, recognition of a contingent liability, or other stock-flow adjustment can also move debt independently of the primary balance.

Why the Debt-Ratio Formula Works

The key mechanism is that debt carried from the previous period grows with interest, while the economy’s nominal GDP—the denominator of the debt ratio—grows at rate $g_t$. The primary surplus then pays down debt, and a positive stock-flow adjustment adds to it. To see the result directly, substitute $B_{t-1}=b_{t-1}Y_{t-1}$ and $Y_t=(1+g_t)Y_{t-1}$ into the debt identity:

Write end-of-period debt.
$B_t=(1+i_t)B_{t-1}-PB_t+SFA_t$.

Divide every term by current GDP.
$B_t/Y_t=(1+i_t)B_{t-1}/Y_t-PB_t/Y_t+SFA_t/Y_t$.

Replace the opening-debt ratio.
$B_{t-1}/Y_t=b_{t-1}/(1+g_t)$, giving $b_t=[(1+i_t)/(1+g_t)]b_{t-1}-p_t+s_t$.

For a constant debt ratio and no stock-flow adjustment, set $b_t=b_{t-1}=b$ and $s_t=0$. Rearranging gives the exact primary-surplus ratio that stabilizes the debt ratio under this one-period model:

p^*=\frac{i-g}{1+g}b

The commonly used approximation $p^*\approx(i-g)b$ can be useful when rates are small, but it is not the exact expression. If the interest rate exceeds nominal GDP growth, a positive primary surplus is required to stabilize a positive debt ratio. If growth exceeds the interest rate, the stabilizing primary balance can be a deficit. This result assumes the rates and primary balance used in the equation apply as stated; it does not by itself establish that the government can borrow at those rates or meet its payments when due.

Worked Example: Reconcile Debt, Interest, and the Primary Surplus

A hypothetical government begins the year with debt of $600 billion and GDP of $1,000 billion. The effective nominal interest rate on opening debt is 4%, nominal GDP growth is 5%, and the primary surplus is 0.5% of current-year GDP. There is no stock-flow adjustment. Find end-of-year debt and its ratio to GDP. The method is to calculate the dollar stock first, then verify it using the ratio equation.

Step 1: Current-year GDP. $Y_1=1{,}000(1.05)=\$1{,}050$ billion.

Step 2: Interest and primary surplus. Interest is $0.04(600)=\$24$ billion. The primary surplus is $0.005(1{,}050)=\$5.25$ billion.

Step 3: End-of-year debt. $B_1=600+24-5.25=\$618.75$ billion.

Step 4: Debt ratio. $b_1=618.75/1{,}050=0.5892857$, or 58.93% when rounded to two decimal places.

Check using the ratio equation. $b_1=(1.04/1.05)(0.60)-0.005=0.5892857$, which matches the dollar calculation. The common mistake this method avoids is subtracting 0.5% of opening GDP rather than the stated 0.5% of current-year GDP.

Stabilizing Balances and Stressing the Projection

A stabilization calculation is a benchmark, not a full fiscal plan. It asks what primary surplus would keep the ratio unchanged for the specified period, given the assumed interest rate and growth rate. A multi-period projection instead applies the debt-ratio equation period by period, using each period’s opening ratio, fiscal balance, rates, and stock-flow adjustment.

Worked Example: Find the Stabilizing Surplus and Test a Fiscal Plan

A government has a debt ratio of 60%. Assume an effective nominal interest rate of 6%, nominal GDP growth of 3%, and no stock-flow adjustment. Find the exact primary-surplus ratio that stabilizes debt. Then assess a plan with a primary surplus of 1.25% of GDP.

Method: first solve for the stabilizing balance, then substitute the proposed balance into the projection. The exact stabilizing surplus is:

p^*=\frac{0.06-0.03}{1.03}(0.60) =\frac{0.018}{1.03} =0.0174757

Thus the required surplus is 1.7476% of GDP, or about 1.75%. The proposed surplus is $p=0.0125$, so:

b_1=\frac{1.06}{1.03}(0.60)-0.0125 =0.6174757-0.0125 =0.6049757

The projected ratio is 60.50% to two decimal places, an increase of about 0.50 percentage points. The calculation is consistent with the stabilizing benchmark: the proposed surplus is about 0.50 percentage points of GDP smaller than required, and the ratio rises. The common mistake is to use $(i-g)b=1.80\%$ as if it were exact; the exact formula gives 1.7476% under the stated timing convention.

Stress analysis changes one or more assumptions and then recomputes the path. A higher interest rate raises the debt-growth factor; weaker nominal GDP growth also raises it by slowing the denominator. If both occur together, the effects compound. A stock-flow adjustment should be shown separately rather than hidden inside the primary balance, because it has a different source and may signal a distinct fiscal exposure.

Worked Example: Combine a Growth and Interest Shock with Refinancing Needs

Use a hypothetical opening debt of $600 billion and GDP of $1,000 billion. In year 1, the effective interest rate is 4%, nominal GDP growth is 5%, and the primary surplus is 0.5% of current GDP. In year 2, the interest rate rises to 7%, nominal GDP growth falls to 2%, the primary surplus remains 0.5% of current GDP, and a cash-funded stock-flow adjustment adds 1% of current GDP to debt. Assume 30% of opening year-2 debt matures, there is no cash buffer, and maturing principal must be refinanced. Find the year-2 debt ratio and gross financing need.

Step 1: Finish year 1. GDP is $1{,}000(1.05)=\$1{,}050$ billion. Interest is $0.04(600)=\$24$ billion, and the primary surplus is $0.005(1{,}050)=\$5.25$ billion. Thus $B_1=600+24-5.25=\$618.75$ billion and $b_1=618.75/1{,}050=0.5892857$.

Step 2: Compute year-2 GDP and fiscal flows. $Y_2=1{,}050(1.02)=\$1{,}071$ billion. Interest on opening debt is $0.07(618.75)=\$43.3125$ billion. The primary surplus is $0.005(1{,}071)=\$5.355$ billion, and the stock-flow adjustment is $0.01(1{,}071)=\$10.71$ billion.

Step 3: Project year-2 debt. $B_2=618.75+43.3125-5.355+10.71=\$667.4175$ billion. Therefore $b_2=667.4175/1{,}071=0.6231723$, or 62.32% to two decimal places.

Step 4: Calculate gross financing need. Maturing principal is $0.30(618.75)=\$185.625$ billion. Under the stated cash-funding assumption, gross financing need is maturing principal plus interest, less the primary surplus, plus the cash-funded stock-flow adjustment:

185.625+43.3125-5.355+10.71 =\$234.2925\text{ billion}

As a share of year-2 GDP, this is $234.2925/1{,}071=0.21876$, or 21.88% to two decimal places. The common mistake is to treat the 30% maturing share as new debt added to the debt stock. Maturing principal must be financed, but refinancing it does not itself increase the debt stock; it is included here because the question asks for financing need.

Debt, National Saving, and Investment

Public borrowing also relates to national saving, but the accounting link should not be mistaken for a mechanical forecast of investment. In a simplified open-economy accounting identity, national saving equals domestic investment plus the current account balance. A current-account surplus means that national saving exceeds domestic investment; a deficit means domestic investment exceeds national saving.

Accounting identity: $S_N=I+CA$, where $S_N$ is national saving, $I$ is domestic investment, and $CA$ is the current account balance. National saving combines private saving and government saving. For a stated comparison, any conclusion about how investment changes must specify what happens to the current account and other parts of the economy.

Worked Example: Separate the Saving Identity from a Causal Claim

In a hypothetical economy, private saving is $210 billion, government saving is $-\$30$ billion, and the current account surplus is $20 billion. A fiscal change reduces government saving by a further $12 billion, while private saving rises by $4 billion. Calculate national saving and investment before and after the change, assuming the current account remains $20 billion.

Before the change. National saving is $210-30=\$180$ billion. From $S_N=I+CA$, investment is $I=180-20=\$160$ billion.

After the change. Government saving is $-30-12=-\$42$ billion and private saving is $210+4=\$214$ billion. National saving is $214-42=\$172$ billion. With the current account held at $20 billion, investment is $172-20=\$152$ billion.

National saving falls by $8 billion, and investment also falls by $8 billion under the assumed unchanged current account. This is an accounting comparison with a stated closure assumption, not a universal causal result. If the current account changed, investment need not move by the same amount. The common mistake is to infer a one-for-one investment effect without saying what happens to private saving and foreign lending.

How to Read the Combined Results

A coherent assessment keeps distinct quantities distinct. The debt ratio summarizes the debt stock relative to output. The primary balance describes the non-interest fiscal contribution to debt change. The gross financing need measures cash that must be raised over a period, including principal that is due for refinancing. National saving describes a broader economy-wide flow and requires additional assumptions before it can be used to infer investment.

Check units and timing. Use rates and GDP growth over the same period, and be explicit about whether a balance is measured against current or opening GDP.
Show the assumed primary-balance sign. In this tutorial, a surplus is positive and reduces debt; a deficit is negative and increases it.
Report the shock channels separately. Identify changes in interest cost, output growth, the primary balance, and stock-flow adjustments rather than combining them into an unexplained change in debt.
Do not equate a stable ratio with easy refinancing. Maturity structure, available cash, and market access affect near-term financing needs even when the projected debt ratio is stable.
State the economic closure. An identity such as $S_N=I+CA$ does not determine by itself how a fiscal change is divided between investment and the current account.

Key takeaway: Begin with the nominal debt identity, convert it consistently into a debt-ratio path, and test the result under stated assumptions. Then assess refinancing needs and saving effects separately; neither is captured by the debt ratio alone.

Check Your Understanding

Work each problem using the sign convention that a primary surplus is positive. Unless stated otherwise, assume no stock-flow adjustment.

A government has opening debt of $800 billion and opening GDP of $2,000 billion. The effective nominal interest rate is 4%, nominal GDP growth is 3%, and the primary surplus is $20 billion. Find end-of-year debt and the debt ratio. (Answer: Debt is $812 billion; GDP is $2,060 billion; debt ratio is 39.42%.)
The debt ratio is 45%, the effective nominal interest rate is 5%, and nominal GDP growth is 2%. Find the exact primary-surplus ratio that stabilizes the debt ratio for one period. (Answer: 1.3235% of GDP.)
The opening debt ratio is 70%, the effective nominal interest rate is 3%, nominal GDP growth is 4%, and the government runs a primary deficit of 1% of current GDP. Find the next-period debt ratio. (Answer: 70.33%.)
Opening debt is $500 billion, 20% of that debt matures this year, interest due is 6% of opening debt, and the primary deficit is $8 billion. With no cash buffer or stock-flow adjustment, find gross financing need and express it as a percentage of GDP of $900 billion. (Answer: $138 billion, or 15.33% of GDP.)
Private saving is $240 billion, government saving is $-\$40$ billion, and the current account surplus is $10 billion. A policy change improves government saving by $12 billion and reduces private saving by $3 billion. If the current account remains unchanged, find national saving and investment after the change. (Answer: National saving is $209 billion and investment is $199 billion.)

What You'll Learn