Skip to content
DP
Debt & Yield Curvesintermediate

Duration

Duration measures how sensitive a bond's price is to a change in its yield. Macaulay duration is the present-value-weighted average time until the bond's cash flows arrive, quoted in years. Modified duration rescales it into a price sensitivity, the approximate percentage price change for a 1% move in yield. Because price and yield move inversely, a higher duration means a larger price swing for the same yield change.

Try it yourself

Duration & Convexity

Duration draws a straight tangent to the price-yield curve. Convexity adds the curvature the tangent misses, so it tracks the true price more closely as the yield moves.

$30$56$81$106$131$1570%4%8%12%16%20%Yield (%)Price ($)$100.00True priceDuration (tangent)Duration + convexity
Price
$100.00
Macaulay dur.
8.11 yr
Modified dur.
7.72 yr
Convexity
75.00
At Δy = +1.0% (new yield 6.0%)
True new price$92.64
Duration estimate$92.28error -0.36
Duration + convexity$92.65error +0.01
Adding the ½·C·Δy² term shrinks the error: convexity captures the curve the tangent misses, and because the price-yield curve bows toward the origin it always sits above the tangent.

Why it matters

Think of duration as the balance point of a bond's cash flows along the time axis. Cash arriving sooner pulls the balance point earlier, so high-coupon and short-dated bonds have low duration, while a zero-coupon bond, whose only cash flow is at maturity, has the highest duration for its maturity. Duration is also the slope of the price-yield curve. A 3-year 5% bond at par has a modified duration near 2.72, so a 100 bp yield rise knocks roughly 2.72% off its price.

Formulas

Macaulay duration
D=1P∑t=1nt⋅CFt(1+y)tD = \frac{1}{P}\sum_{t=1}^{n} t \cdot \frac{CF_t}{(1+y)^t}
The PV-weighted average time to each cash flow CFtCF_t, in years. PP is the bond price and yy the yield. Hull (2022), Ch. 4 (Interest Rates).
Modified duration
Dmod=D1+y/mD_{\text{mod}} = \frac{D}{1 + y/m}
mm is the number of compounding periods per year. Under continuous compounding Dmod=DD_{\text{mod}} = D.
First-order price change
ΔPP≈−Dmod Δy\frac{\Delta P}{P} \approx -D_{\text{mod}} \, \Delta y
A straight-line (tangent) estimate. It is accurate for small yield moves and needs a convexity correction for large ones.

Worked examples

Scenario

3-year bond, face A$100, 5% annual coupon, YTM 5% (so it trades at par, price A$100). Find its Macaulay and modified duration.

Solution

Discounted cash flows are 4.76, 4.54, 90.70 at t=1,2,3t = 1, 2, 3. Macaulay D=(1×4.76+2×4.54+3×90.70)/100=2.86D = (1 \times 4.76 + 2 \times 4.54 + 3 \times 90.70)/100 = 2.86 years. Modified Dmod=2.86/1.05=2.72D_{\text{mod}} = 2.86/1.05 = 2.72. A 100 bp rise is predicted to move the price by −2.72×0.01=−2.72%-2.72 \times 0.01 = -2.72\%, to about A$97.28. The actual reprice is A$97.33; the small gap is convexity.

Scenario

Compare a 5-year zero-coupon bond with a 5-year coupon bond. Which has the larger duration?

Solution

The zero has its only cash flow at maturity, so its Macaulay duration equals its maturity exactly: 5 years (modified duration 4.76). Any 5-year coupon bond has a shorter duration, because its coupons place value before maturity and pull the balance point earlier. This is why duration, not maturity, is the right gauge of interest-rate risk.

Common mistakes

  • ✗Duration is just the time to maturity. Only for a zero-coupon bond. For a coupon bond, duration is the PV-weighted average time to all cash flows, which is always less than maturity.
  • ✗Duration gives the exact price change. It is a first-order (linear) approximation. It is accurate for small yield moves but overstates the loss and understates the gain on large moves, because the true price-yield relationship is curved. The fix is convexity.
  • ✗A higher coupon raises duration. The opposite. A higher coupon delivers more value earlier, pulling the balance point in and lowering duration. Low-coupon and zero-coupon bonds are the most rate-sensitive.

Revision bullets

  • •Macaulay duration is the PV-weighted average time to cash flows, in years
  • •Modified duration Dmod=D/(1+y/m)D_{\text{mod}} = D/(1 + y/m) is the percent price change per 1% yield move
  • •ΔP/P≈−Dmod Δy\Delta P / P \approx -D_{\text{mod}} \, \Delta y is the first-order estimate
  • •A zero-coupon bond's duration equals its maturity
  • •Higher coupon and shorter maturity give lower duration
  • •Duration is linear; large moves need a convexity correction

Quick check

Two bonds share a 10-year maturity. Bond A pays a 2% coupon, Bond B pays an 8% coupon. Which has the higher duration?

A bond has a modified duration of 7. Yields rise by 50 bp. The first-order price change is approximately

Connected topics

Zero RatesBond PriceConvexityCoupon/Yield

Sources

  1. Hull (2022), Ch. 4
    Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
    Defines Macaulay and modified duration and the first-order price-sensitivity relationship.
  2. Fabozzi (2021), Ch. 5
    Fabozzi, Frank J. Bond Markets, Analysis, and Strategies. 10th ed. MIT Press, 2021. ISBN 978-0-262-04627-3.
    Detailed treatment of duration measures, dollar duration, and portfolio duration.
  3. Tuckman and Serrat (2022)
    Tuckman, Bruce and Angel Serrat. Fixed Income Securities: Tools for Today's Markets. 4th ed. Wiley, 2022. ISBN 978-1-119-83555-0.
    Develops duration as a hedging and risk measure alongside DV01 and key-rate duration.
How to cite this page
Dr. Phil's Quant Lab. (2026). Duration. Derivatives Atlas. https://phucnguyenvan.com/concept/duration
← Back to the atlasSee in the network →