The Mortgage Payment as a Bond: Payment Semi-Elasticity Equals Modified Duration, with Bounds, and an Exact Decomposition of US Mortgage Affordability, 1984–2025

For a fixed-rate amortising loan, the percentage change in the monthly payment for a one-point rise in the annual rate equals the modified duration, in years, of the level payment stream at that rate. I prove this, prove that the log payment is concave in the rate because duration falls as rates rise, and use concavity to bound the exact effect of a finite rate change. For a 30-year loan a one-point rise lifts the payment by 13.24% at 3% and by 10.29% at 7%. I then decompose US mortgage affordability, measured as the annual payment on an 80% loan against the median sales price of houses sold divided by median household income (FRED data on the Freddie Mac 30-year rate, the Census median sales price and Census median household income), exactly into price, income and rate terms. The payment-to-income ratio fell from 40.2% in 1984 to a low of 19.8% in 2020 and then rose to 33.1% in 2023 and 29.1% in 2025. Between 2021 and 2023 the log ratio rose by 0.420, of which the rate term was 0.442, prices 0.108 and income growth −0.130: the deterioration was almost entirely the rate. The same algebra measures lock-in: the payment on a 100,000 USD loan at 3% supports only 61,864 USD at the latest weekly rate of 7.24% (29 October 2026).