RainbowStats · Economic research · 28 October 2026

Real Rates, Sovereign Debt, and the Cost of Repricing

Japan changes the cross-country fit. The interest bill reminds us that the path of rates matters too.

In Why Central Bankers Should Be Derivative Traders, I examined sovereign debt through a derivatives trader’s lens: leverage, uncertainty, policy constraints, and the way risk becomes visible in market prices.

One finding deserved another look. Japan strongly influenced the international comparison. Removing it made the debt/GDP–bond-volatility relationship more clearly upward. The obvious follow-up was to ask whether the same thing happens with real interest rates.

It does, in a limited descriptive sense. In our 2023 sample, removing Japan strengthens the upward relationship between debt/GDP and inflation-adjusted ten-year yields. The relationship is still modest, and it becomes weaker after adjusting nominal yields for observed inflation.

I also suspect that the speed of the rise in rates matters. A country that accumulated debt under low financing costs may face a different adjustment when those costs reset quickly. Our earlier work on the federal interest burden gives that suspicion a concrete place to start.

The same countries, a new measure

The new RainbowStats command uses the same thirteen countries and IMF general-government gross debt snapshot as the Merton comparison. The United States is included in both samples. The second comparison removes only Japan and re-estimates both fitted curves.

REAL_RATES_DEBT_GDP()

The countries are the United States, Italy, France, Greece, Spain, Germany, Japan, Canada, Switzerland, Ireland, the Netherlands, Denmark, and Australia. The common snapshot is 2023; it is not a ranking of today’s rates.

For each country, I calculate:

Adjusted yield (%) = 100 × [(1 + nominal yield/100) / (1 + inflation/100) − 1]

The nominal input is the mean of twelve observed monthly ten-year sovereign yields. The inflation input is the same year’s annual headline CPI inflation. Debt is measured consistently as IMF general-government gross debt/GDP.

This is a trailing-inflation real-yield proxy. It is not a real policy rate, a yield on an inflation-linked bond, or a ten-year yield adjusted for expected inflation over the bond’s remaining life. Those distinctions matter when interpreting negative values.

Japan changes the picture again

Without Japan, the linear slope is approximately 0.01384 percentage points of adjusted yield per percentage point of debt/GDP. A ten-point difference in debt/GDP therefore corresponds to about 0.138 percentage points—13.8 basis points—in this fitted comparison. That is a sample association, not the estimated effect of issuing additional debt.

The quadratic improves ordinary R², as adding a regressor necessarily can. But its adjusted R² falls below the straight line’s in the twelve-country sample. The evidence for useful extra curvature is weak.

Japan’s 2023 debt ratio is 220.3%, its mean ten-year nominal yield about 0.563%, and its inflation-adjusted proxy −2.620%. The corresponding U.S. values are 120.0%, 3.957%, and −0.153%. Japan occupies an influential high-debt, low-yield position.

The earlier Merton post discusses the Bank of Japan’s bond purchases and yield-curve control. Those institutions are reasons to investigate Japan’s position. This scatter does not measure how much of that position they explain. Both charts belong in the analysis.

Inflation adjustment changes the story

In the twelve-country sample, the nominal-yield linear R² is 0.465, compared with 0.270 for the inflation-adjusted proxy. Higher nominal yields can reflect compensation for inflation as well as real financing conditions, term premiums, credit concerns, and liquidity.

The seven-country ECB check supplies another view. Using national headline HICP rates for the euro-area borrowers gives a linear R² of 0.274. The quadratic’s adjusted R² is negative. These borrowers share a currency and central bank but have separate sovereign yields.

The HICP check stays separate because its mean of monthly year-over-year inflation rates differs from the World Bank annual CPI measure. It is a sensitivity check, not a substitute smuggled into the main sample.

My additional suspicion: the speed of the rise matters

A single-year comparison cannot tell us what happens when a large debt stock meets a rapid change in financing costs. That is a question about the path of adjustment.

Three processes can operate on different clocks. Market yields change quickly. Fixed-rate debt resets as it matures or new borrowing is issued. Inflation can fall faster than nominal yields, raising a trailing-inflation real-rate measure even without another nominal-rate increase.

For the United States, we can inspect an actual inflation-indexed market yield separately from the cross-country proxy. The chart below uses DFII10, the ten-year inflation-indexed Treasury yield, alongside the nominal ten-year yield. Its movement is a different observation from nominal yield minus current CPI.

Interest outlays rose from about $352.3 billion in FY2021 to $970.1 billion in FY2025. Their share of total federal outlays increased from 5.17% to 13.84%. A larger interest bill can narrow fiscal room even when a debt/GDP chart changes relatively little. The dollar increase also reflects the debt stock and its composition; it cannot all be assigned to rate changes.

A simple refinancing calculation shows the potential scale. If debt equals 120% of GDP, a one-percentage-point increase in the average nominal financing rate adds approximately 1.2% of GDP to annual interest expense once it applies to the entire stock, holding debt and GDP fixed. If only one-fifth of that stock resets in the first year, the direct first-year effect is about 0.24% of GDP. These are arithmetic illustrations, not estimates of the U.S. maturity schedule.

This is why the speed of rate increases, the maturity profile, new deficits, and the effective coupon should be examined together. The comparison of rate levels across countries does not test that interaction.

We have already examined part of the interest-expense connection

In Federal Debt and the Cost of Money: A Nonlinear Relationship, our quarterly U.S. model linked changes in federal interest payments/GDP with Treasury-yield changes, growth, lagged debt, and persistence. The reported yield-change coefficient was positive, with a HAC statistic of 6.90; the lagged burden-change coefficient was about 0.532, with a HAC statistic of 6.50.

That supports investigating delayed adjustment. The earlier model was an instrument-projected SUR/FGLS system approximation, subject to instrument and specification limitations. It did not isolate a separate causal effect of the speed of rate increases.

The next test should ask whether refinancing exposure makes rapid rate increases especially costly at high debt ratios. That requires rate changes over explicit windows, maturity or rollover measures, new borrowing, growth, and lagged interest expense. Including those variables would let us examine the mechanism suggested here.

What the new result adds

The Merton comparison asked how leverage and uncertainty might become visible in volatility and spreads. This extension asks about the level of an inflation-adjusted yield. The interest-expense work asks how financing conditions enter the government’s cash-flow burden over time.

Those are connected questions, but they measure different outcomes. The new evidence is that Japan materially affects the 2023 cross-country real-yield comparison, and that the upward association outside Japan is modest. It is not evidence of a universal debt threshold or proof that debt caused the recent increase in real rates.

My working hypothesis is that the rate path and refinancing schedule help determine how quickly a large debt stock becomes an expensive one. The growing interest bill makes that a practical question for the next model.

Commands and replication notes

MERTON_DEBT_GDP()
REAL_RATES_DEBT_GDP()

To reproduce the separate U.S. rate and budget charts:

LINE_CHART(SERIES_BETWEEN(GS10,20190101,20260930),SERIES_BETWEEN(DFII10,20190101,20260930))
LINE_CHART(SERIES_BETWEEN(FYOINT%FYONET,20190101,20250930))

The rate command displays the native monthly nominal series and daily inflation-indexed series; Figure 3 aggregates the daily observations into monthly means. Fiscal-year dates represent fiscal year ends. Budget ratios are not substituted for the general-government debt definitions in the cross-country study.

Source observations were inspected on October 28, 2026. They may be revised, and the IMF debt snapshot can contain historical estimates. No missing debt year or incomplete yield year was substituted into the thirteen-country comparison.

References and data

1. Igoe, Bill (2026). Why Central Bankers Should Be Derivative Traders. Earlier Merton and international debt comparison.
2. Igoe, Bill (2026). Federal Debt and the Cost of Money: A Nonlinear Relationship. Quarterly interest-burden system and its limitations.
3. IMF general-government gross debt/GDP snapshot, retrieved October 3, 2026. The thirteen-country study uses its 2023 observations.
4. OECD/FRED IRLTLT01 ten-year yield series; World Bank/FRED FPCPITOTLZG annual headline CPI inflation; ECB national headline HICP series ICP.M.country.N.000000.4.ANR.
5. Federal Reserve via FRED: GS10 and DFII10, nominal and inflation-indexed ten-year Treasury yields.
6. OMB via FRED: FYOINT, Federal Outlays: Interest, and FYONET, Federal Net Outlays. Both are fiscal-year figures in millions of dollars.