Debt Service Ratios in 32 Economies, 1999–2026: Exact Bounds on the Interest-Rate Sensitivity of the BIS Formula and Evidence That High Debt Service Precedes Slower Growth

The debt service ratio (DSR) is the share of income that borrowers spend on interest and principal. The Bank for International Settlements (BIS) computes it for the private non-financial sector of 32 economies with an annuity formula. I prove four properties of the BIS instalment factor: it lies between max(i, 1/s) and i + 1/s, its slope in the interest rate starts at (s+1)/(2s) and never exceeds one, and it is convex. Convexity follows from a variance bound (Bhatia–Davis). A corollary bounds the effect of a one-point rise in the average rate on the DSR between about 0.51 and 1 times the debt-to-income ratio; at 15-year maturities it raises the DSR by 5.4–7.4% of its level. In 2026-Q1 India's private-sector DSR is 12.5% of income, 0.6 points above its 1999–2026 mean of 11.90%, with credit at 105.1% of GDP; India's DSR and credit ratio have a correlation of 0.882. In a panel of 829 country-years, a DSR one point above the country's own mean is followed by GDP growth 0.21 points lower the next year (s.e. 0.04), or 0.11 points with year fixed effects. The estimate survives controls for credit growth, the exclusion of crisis years and the omission of any single economy. Seventeen of the 32 economies are above their own mean in 2026-Q1.