National Debt Explorer

The debt is a number everyone has heard and almost nobody can size. Four other numbers decide what it means — the interest rate on it, how fast the economy grows, the primary balance, and what is actually paid against it. Move them and watch.

Worked example — National Debt Explorer

Debt per person	$117,130
Output per person, per year	$90,027
Years of output per person	1.30
Debt to GDP	130.1%
Interest per year	$1.38T
Payment that only covers interest	$1.38T
Primary balance that holds the ratio still	−0.94%
Ratio at the horizon	158.8%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

Below $1.38 trillion a year, it is never repaid 🖖

Interest is the rate times the balance: 3.447% on $40.03 trillion is $1.38 trillion a year, before a cent of principal moves. Any annual payment below that leaves next year’s balance larger than this year’s, and one exactly on it leaves the balance where it was, so the answer to “how long” is not a big number — it is never. Just above the line the arithmetic is nearly as strange: $1.40 trillion takes 125 years, $1.50 trillion takes 74.5, $2 trillion takes 34.6. Drag the payment slider through the red line and the curve goes vertical.

Growth moves the ratio; the repayment slider never enters it 🖖

The ratio steps by d(1+r)/(1+g) − pb, so the sign of r − g decides whether a surplus is needed at all. At a 3.447% rate against 4.2% nominal growth, a primary deficit of 0.94% of GDP holds the ratio exactly still — forever, while repaying nothing. Invert the two and the arithmetic inverts with them: 5.5% against 3% needs a 3.16% surplus to stand still, and a balanced budget carries the ratio to 267.1% in thirty years. Neither case is about paying the debt off.

A share per person is not a burden until you divide by an income 🖖

$117,130 each is the figure that travels. $90,027 is what each person produces in a year, and the ratio of the two is 1.30 — the debt-to-GDP number again, arrived at from the other end. That is the entire content of the per-person framing: the ratio multiplied by output per head. Quoting it without the income is quoting a numerator. Whether 130% is comfortable is a different question, and this page does not answer it.

Problems solved in full

Where the debt ratio ends up, once the thirty-year projection runs out 8 steps
Leave every dial where the today preset puts it: interest at 3.447%, nominal growth at 4.2%, a primary deficit of 2% of GDP, and no policy change ever. The panel says the ratio goes from 130.1% now to 158.8% in thirty years. Keep running the clock. Does it keep climbing?
1. One year of the ratio is one line, and the important thing about that line is its shape: this year’s ratio multiplied by a number, minus the primary balance. That is an affine map, not a linear one, and the constant term is what gives it somewhere to settle. Give the multiplier a name, because everything below is about whether it is bigger or smaller than one.
  
  $d_{n + 1} = d_{n} \frac{1 + r}{1 + g} - p b = φ d_{n} - p b, φ = \frac{1 + r}{1 + g}$
2. Put the two rates in. Growth is running above the interest rate, so φ comes out just under 1. Each year the existing debt shrinks slightly against a larger economy; the deficit pushes the other way, and the two effects are what the map balances.
  
  $φ = \frac{1.03447}{1.042} = 0.992774 < 1$
3. An affine map whose multiplier is anything but one has exactly one place it can sit still. Set next year equal to this year and solve for it.
  
  $d^{*} = φ d^{*} - p b ⟹ d^{*} = \frac{- p b}{1 - φ}$
4. A primary deficit of 2% means pb = −0.02, so −pb is +0.02, and 1 − φ is 0.007226. Divide. The ratio does not run away: it converges, to a little over 276% of GDP, and stops.
  
  $d^{*} = \frac{0.02}{0.0072265} = 2.7676 = 276.8% of GDP$
5. That gives the whole path in closed form, with no simulation. The gap between where you are and where you are going shrinks by a factor φ every year, so the ratio is the fixed point plus a decaying remainder.
  
  $d_{n} = d^{*} + (d_{0} - d^{*}) φ^{n}$
6. Check it against the panel before trusting it. d₀ is 40.033 over 30.77, and n is 30.
  
  $d_{30} = 2.7676 + (1.30104 - 2.7676) (0.992774)^{30} = 1.5878 = 158.8%$
7. So how long is “converges”? The gap halves every ln(½)/ln φ years. Thirty years closes 19.6% of the gap. The projection the tool draws covers a fifth of the distance; half of it is not closed until year 96, and an exponential never closes all of it.
  
  $t_{1/2} = \frac{ln \frac{1}{2}}{ln 0.992774} = 95.6 years$
8. Now change the one thing that decides everything. Click rate above growth: 5.5% interest against 3% growth, and a primary budget in exact balance. φ crosses one. With the primary budget in balance the fixed point sits at zero, and φ above one makes it repel instead of attract, and the ratio grows geometrically from wherever it happens to start.
  
  $φ = \frac{1.055}{1.03} = 1.024272 > 1, t_{2} = \frac{ln 2}{ln φ} = 28.9 years$
Answer

It converges — to 276.8% of GDP — and the 158.8% the panel prints at thirty years is a fifth of the way there. Everything turns on whether φ = (1+r)/(1+g) is above or below one, which is the sign of r − g and nothing else. The panel’s stabilising-balance card is that same comparison written as a required surplus.

The contrast between the two presets is the whole subject. A permanent 2% primary deficit with growth beating the rate settles at 277% and stays there. A primary budget in perfect balance with the rate beating growth settles nowhere: 130% becomes 267% in thirty years and doubles every 28.9 years after that. Of these two countries, the one that never balances its budget is the stable one. Repayment enters neither calculation, because this is the arithmetic of a ratio and the numerator is only half of it.
The borrowing that leaves a debt ratio exactly where it was 7 steps
Load the growth above the rate preset: 3% interest, 5% nominal growth, a primary deficit of 1.5% of GDP. The stabilising card reads −2.48%. Work out what that is in dollars, then decide which of the panel's two clocks a headline about the debt belongs to.
1. The card is a share of GDP, and a share of GDP is a ratio of two things that are both moving. Start from where it comes from: the balance that leaves next year's ratio equal to this year's. At a 3% rate against 5% growth it is negative, so this country can run a permanent primary deficit of 2.48% of GDP and never move its debt ratio at all.
  
  $p b^{*} = d \frac{r - g}{1 + g} = 1.30104 \times \frac{0.03 - 0.05}{1.05} = - 0.02478 = - 2.48%$
2. Now put it in money. The primary balance is measured against next year's GDP, so multiply by that — and watch the GDP cancel. What is left is the debt times the gap between the two rates, and it does not care how large the economy is. A permanent deficit of 801 billion dollars.
  
  $p b^{*} G (1 + g) = \frac{D}{G} \cdot \frac{r - g}{1 + g} \cdot G (1 + g) = D (r - g) = 40.033 \times (- 0.02) = - 0.801$
3. That is the primary balance, which is the budget before interest. Add the interest bill the panel prints, 1.20 trillion, and you have what the Treasury actually borrows in a year to stand still. It comes to 2.00 trillion, and it is exactly 5% of the debt.
  
  $r D - p b^{*} G (1 + g) = 1.201 + 0.801 = 2.002 = g D = 0.05 \times 40.033$
4. Not a coincidence of these numbers. If the ratio is unchanged then the debt and the economy have grown by the same factor, so new borrowing is the growth rate times the debt. The interest rate has vanished and so has the primary balance; whatever those two are, they can only rearrange which pocket the money comes out of.
  
  $\frac{D ^{'}}{G ^{'}} = \frac{D}{G} ⟺ D^{'} = D (1 + g) ⟺ D^{'} - D = g D$
5. Now look at the other half of the page. The payment slider is at 2.00 trillion and the payoff card says 31.0 years. It is the same 2.00 trillion. One panel is calling it money leaving the Treasury every year and the other is calling it money arriving, and both are printed on this page at once.
  
  $n = \frac{ln ( P / ( P - r D ) )}{ln ( 1 + r )} = \frac{ln ( 2/0.799 )}{ln 1.03} = 31.0, g D = 2.002$
6. The preset is not running the stabilising deficit — it runs a smaller one, 1.5% instead of 2.48% — so the ratio falls. Thirty years on the card reads 107.6%, down from 130.1%.
  
  $d_{30} = 1.0759 = 107.6%, G_{30} = 30.77 \times 1.0 5^{30} = 132.99$
7. And the debt itself, in dollars, over those same thirty years. The economy compounds at 5% and is 4.32 times bigger; multiply by the ratio and the debt is 143 trillion. It has more than tripled while the number that measures whether it is a problem went down.
  
  $D_{30} = 1.0759 \times 132.99 = 143.1, \frac{143.1}{40.033} = 3.57$
Answer

2.00 trillion dollars of fresh borrowing a year, which happens to be the figure the payment slider is set to. Borrow that much and the ratio never moves; borrow less, as this preset does, and it falls to 107.6% while the debt climbs to 143 trillion.

Two identities carry across every setting. Borrowing that holds the ratio still is g·D, and the primary surplus that supports it is D(r − g). Both lose the GDP, which is what makes them usable: at the interest rate above growth preset the surplus works out at 1.00 trillion a year against an interest bill of 2.20 trillion, so borrowing is capped at 1.20 trillion, which is 3% of the debt — the growth rate again. A debt is allowed to grow at the speed of the economy and no faster. Which makes the annual dollar increase, the number every headline reports, a quantity that is supposed to rise, and one nobody can read without being told the growth rate beside it.

$borrowing = g D, surplus needed = D (r - g) : 40.033 \times 0.025 = 1.00$

References (4)

The debt outstanding, its split between public and intragovernmental holdings, and the date it was true U.S. Department of the Treasury, Fiscal Data, "Debt to the Penny", record date 2026-08-20.
The average interest rate the debt actually carries, which sets the interest bill and the repayment threshold U.S. Department of the Treasury, Fiscal Data, "Average Interest Rates on U.S. Treasury Securities", Total Interest-bearing Debt, 2026-07-31: 3.447%.
Nominal GDP and population, the two denominators the tool argues about World Bank Open Data, NY.GDP.MKTP.CD and SP.POP.TOTL, United States, 2025. Licensed CC BY 4.0.
The debt-ratio recurrence d(1+r)/(1+g) - pb and the stabilising primary balance Escolano, J. (2010). A Practical Guide to Public Debt Dynamics, Fiscal Sustainability, and Cyclical Adjustment of Budgetary Aggregates. IMF Technical Notes and Manuals 10/02.

By the 42Math team · Updated 2026-09-03

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

today - $40.03T is 130.1% of GDP; interest alone runs $1.38T a year, and $2.00T a year clears the debt in 34.6 years.
a payment that never finishes - $1.35T a year is below the $1.38T interest bill, so the answer is never — the balance grows while you pay.
growth above the rate - With 5% growth against a 3% rate, holding the ratio still would take a primary deficit of 2.48% of GDP. This preset runs a smaller one, and thirty years takes 130.1% down to 107.6%.
rate above growth - A 5.5% rate against 3% growth needs a surplus of 3.16% of GDP just to stand still; a balanced budget reaches 267.1% in thirty years.