Duration

Duration is the present-value-weighted average time to receive a bond's cash flows. It measures price sensitivity to yield changes: a bond with duration D loses roughly D% in price for each one-percentage-point rise in yield.

ByHoang TruongUpdated July 3, 2026

See it move

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A bond with a modified duration of 6 years approximates its price sensitivity as %ΔP ≈ −duration × Δyield. If yields rise 0.5 percentage points, price falls about 6 × 0.5% = 3%. Longer duration means larger swings; high-coupon bonds have shorter duration than zero-coupon bonds of the same maturity because they return value sooner.

Where it fits

SubjectCorporate FinanceAdvancedTopicBond & Equity ValuationAdvanced

The formula

LaTeX

D_{mac} = \frac{1}{P} \sum_{t=1}^{T} \frac{t \cdot CF_t}{(1+y)^t}

Variables

$D_{m} a c$: Macaulay duration in periods
$t$: time period of each cash flow, numbered 1 to T
$C F_{t}$: cash flow received at time t (coupon, plus face value at maturity)
$y$: yield to maturity per period (decimal)
$P$: current bond price (sum of all present-valued cash flows)

Each cash flow is weighted by its present-value share of total price and by its timing. A zero-coupon bond's Macaulay duration equals its maturity; coupon bonds always have a shorter duration than their maturity.

LaTeX

D_{mod} = \frac{D_{mac}}{1 + \frac{y}{m}}

Variables

$D_{m} o d$: modified duration: direct measure of price sensitivity per unit yield change
$D_{m} a c$: Macaulay duration in periods
$y$: annual yield to maturity (decimal)
$m$: number of coupon payments per year (1 for annual-pay bonds)

For annual-pay bonds (m = 1), D_mod = D_mac ÷ (1 + y). D_mod states directly: if yields rise by 1 percentage point, the bond price falls by approximately D_mod percent.

LaTeX

\frac{\Delta P}{P} \approx -D_{mod} \times \Delta y

Variables

$Δ P / P$: percentage change in bond price
$D_{m} o d$: modified duration
$Δ y$: change in yield to maturity in decimal form (e.g. 0.005 for a 50 basis-point rise)

Linear approximation; accurate for small yield moves. For large shifts, convexity adds a positive second-order correction — the actual price always falls less (or rises more) than the duration estimate alone predicts.

Check yourself

PracticeCORE

Bond X is a 5-year zero-coupon bond. Bond Y is a 5-year bond paying a 10 per cent annual coupon. Both are priced at the same yield to maturity. Which bond has the longer Macaulay duration, and why?

Select an answer to check your understanding.