Duration (finance)

In finance, duration is a measure of how the price of a fixed-income instrument responds to a change in interest rates. It is used to compare rate risk across bonds and to construct hedges, and is often paired with convexity and the price value of a basis point. Duration-based estimates work best for small, parallel shifts in the yield curve.

Macaulay duration is the present-value-weighted average time to the cash flows and links payment timing to interest-rate risk. Modified duration expresses the first-order percentage price change for a stated compounding convention. When yields vary by maturity, Fisher–Weil duration discounts each payment at its own spot rate; Key rate duration isolates sensitivity at selected maturities; and effective or option-adjusted duration estimates sensitivity for instruments with cash flows that depend on rates.

History and terminology

Early development

The idea of duration was set out by Frederick Macaulay in a National Bureau of Economic Research study in 1938. He defined a time-weighted average of the present values of cash flows and used it to summarize a bond’s timing and rate sensitivity. In actuarial work, Frank Redington linked duration to immunization and added convexity to improve protection against larger moves in yields.

Extensions

With a term structure of rates, discounting each payment at its own spot rate preserves the present-value weighting and gives a first-order hedge for a small parallel shift of the zero curve.