Duration (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 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.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. 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.Terminology and market usage
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. 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.Definition and intuition
This section uses the following conventions. A fixed-income instrument has cash flows at times . The last cash flow includes the bond redemption. The nominal yield to maturity is with compounding periods per year. The price as a function of yield isDefine the present values and weights, which sum to one. Macaulay duration is the present-value-weighted average time to the cash flows:
It summarises payment timing. For a zero-coupon bond that pays only at time,. 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 :
For a small change the approximation is
With continuous compounding at rate, pricing is and
These relations keep notation consistent across compounding conventions.
Analogy
Imagine a long plank set along a timeline that begins today. Each future cash flow is a small weight placed on the plank at the cash flow time. Heavier weights correspond to cash flows with larger present values. If you slide a single support under the plank to the point where the system balances, that balance point is the time-centre of all the weights.If most of the weight lies far along the plank the balance point sits further from today and the bond is more sensitive to a change in yields. If weight is concentrated near the start through high coupons or short maturity the balance point moves inward and sensitivity falls. This time-centre corresponds to Macaulay duration.
Now tilt the ground by a very small amount. The plank drops a little and, for such a small tilt, the vertical drop at the balance point is almost exactly proportional to the tilt. That proportional response mirrors modified duration, which gives the first-order change in price for a small change in yield.
With a larger tilt the motion does not remain proportional because the plank follows a curve. The extra curvature in the response explains convexity and shows why the second-order term matters for larger yield moves or for cash-flow patterns that make the curve more pronounced.
If the ground does not tilt uniformly but is raised or lowered under specific years, different parts of the plank move by different amounts. That picture matches shifts in the term structure and motivates measures such as Fisher–Weil duration and key-rate durations, where sensitivity depends on which maturities move.
Worked examples
Assume maturity years and yield with annual compounding. ThenA 25-basis-point change in yield gives
- Level-coupon bond
Price and cash-flow weights:
Macaulay duration:
Modified duration:
A 50-basis-point rise in yield implies
Term-structure intuition
When the term structure is not flat, discounting each payment at its own zero-coupon rate preserves the weighting idea in Macaulay’s statistic and leads to the Fisher–Weil refinement for parallel shifts of the zero-rate curve. Non-parallel movements are analysed with key-rate durations in later sections.Formal derivation
Let a fixed-income instrument pay cash flows at times ,. The last cash flow at time includes the redemption. With a yield to maturity compounded times per year, the price as a function of yield isWrite the present values and define weights so that.
Differentiating with respect to gives
Hence the modified duration is
where the Macaulay duration is the present-value-weighted average time
For a small change, the first-order approximation is
These relations assume fixed cash flows and a small parallel move in the quoted yield.
Continuous compounding
If pricing uses a continuously compounded rate, thenWith weights,
Thus modified and Macaulay duration coincide under continuous compounding.
Term-structure version (Fisher–Weil)
When the term structure is not flat, discount each cash flow at its own zero-coupon rate. For a parallel shift to the zero curve,Define spot-discounted values and weights. Differentiating at gives
the Fisher–Weil duration, which preserves present-value weighting with a full term structure.
Money duration and DV01
These identities are widely used in portfolio reporting and regulation.Properties and portfolio duration
For fixed, positive cash flows:- Duration rises with final maturity and falls as the yield rises.
- Higher coupons shorten duration relative to a zero-coupon with the same maturity.
- Portfolio duration is the present-value–weighted average of component durations:
Macaulay duration
Named for Frederick Macaulay, Macaulay duration is the present-value-weighted average time to a bond’s cash flows. It treats each payment’s time as a “location” and weights it by that payment’s present value. The denominator equals the bond’s price.Definition
Let cash flows be at times ,. Write present values and price asDefine weights, which sum to one. Macaulay duration is
Basic properties
For instruments with fixed, positive cash flows and times,with equality only when there is a single payment. Thus a zero-coupon bond maturing at has, while a level-coupon bond has strictly between the first coupon date and final maturity.
Relation to other duration measures
Under a quoted yield to maturity compounded times per year,which links the time-average concept to the first-order price sensitivity used in hedging. If discounting uses spot rates at each maturity, the same weighted-average form with spot-discounted present values gives the Fisher–Weil duration; when the curve is flat and conventions match, it equals.
Duration and weighted average life (WAL)
averages payment times using principal amounts only and does not discount. Macaulay duration averages using present values and includes both coupons and principal. For an interest-only or bullet structure with small coupons the two figures can be close, yet they differ in general because duration reflects discounting and coupon timing.Modified duration
Modified duration is a price-sensitivity measure. It is the percentage derivative of price with respect to yield, so it captures the first-order change in price for a small parallel change in the quoted yield.Continuous compounding
When the yield is expressed with continuous compounding at rate, the Macaulay duration equals the modified duration:so under continuous compounding.
Periodic compounding
In most markets yields are quoted with compounding periods per year. With the nominal yield to maturity and,This relates the time-average concept to the elasticity used for hedging and reporting.
Units and the small-change formula
Macaulay duration has units of time. Modified duration is unitless and acts as a semi-elasticity. For a small change in the annual yield,For a 100-basis-point change the approximate percentage price change is.