Duration (finance) 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.

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 summarise a bond’s timing and rate sensitivity. In actuarial work, Frank Redington linked duration to immunisation and added convexity to improve protection against larger moves in yields.

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. This is the Fisher–Weil formulation. To handle non-parallel moves, practitioners report localised sensitivities at selected maturities using key rate durations. Option features led to effective or option-adjusted duration, estimated by small curve shifts in a pricing model while the option-adjusted spread is held constant. These uses are standard in index and reporting methodologies.

In modern texts “duration” can mean different but related measures. Macaulay duration is the present-value-weighted average time to payment. Modified duration is the first-order percentage change in price for a small change in the stated yield and compounding. Money or dollar duration is ${\displaystyle P\times D_{\text{mod}}}$. DV01, PV01 and PVBP express the price change per basis point. In the UK gilts market, modified duration is often called “volatility” in index guides and factsheets.

This section uses the following conventions. Times ${\displaystyle t_{i}}$ are in years. The nominal yield to maturity is ${\displaystyle y}$ with ${\displaystyle m}$ compounding periods per year. Cash flows are ${\displaystyle C_{i}}$. The price as a function of yield is $${\displaystyle P(y)\;=\;\sum _{i=1}^{n}{\frac {C_{i}}{{\bigl (}1+y/m{\bigr )}^{mt_{i}}}}\,.}$$

Define the present values ${\displaystyle \mathrm {PV} _{i}=C_{i}\,(1+y/m)^{-mt_{i}}}$ and weights ${\displaystyle w_{i}=\mathrm {PV} _{i}/P(y)}$, which sum to one. Macaulay duration is the present-value-weighted average time to the cash flows: $${\displaystyle D_{\text{Mac}}\;=\;\sum _{i=1}^{n}t_{i}\,w_{i}\;=\;{\frac {\sum _{i}t_{i}\,\mathrm {PV} _{i}}{\sum _{i}\mathrm {PV} _{i}}}\,.}$$ It summarises payment timing. For a zero-coupon bond that pays only at time ${\displaystyle T}$, ${\displaystyle D_{\text{Mac}}=T}$. For a level-coupon bond it lies between zero and final maturity.

To link timing to price sensitivity, differentiate price with respect to yield. Modified duration is the first-order sensitivity of price to a small parallel change in ${\displaystyle y}$: $${\displaystyle D_{\text{mod}}\;=\;-\,{\frac {1}{P(y)}}\,{\frac {\mathrm {d} P}{\mathrm {d} y}}\;=\;{\frac {D_{\text{Mac}}}{1+y/m}}\,.}$$ For a small change ${\displaystyle \Delta y}$ the approximation is $${\displaystyle {\frac {\Delta P}{P}}\;\approx \;-\,D_{\text{mod}}\,\Delta y\,.}$$