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Duration

Duration is the dominant measure of a bond’s sensitivity to interest rates. Macaulay duration is the weighted-average time to receive the bond’s cash flows, with weights equal to each cash flow’s share of present value. Modified duration rescales it into a price-sensitivity figure, so that the percentage price change is approximately minus modified duration times the change in yield. Duration rises with maturity and falls with a higher coupon or yield, which is why long, low-coupon bonds are the most rate-sensitive. It gives investors a single number to estimate price risk and to immunize a portfolio against rate moves.

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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

Duration answers a practical question. If yields move a little, how much does my bond price move? Macaulay duration first finds the bond’s effective time horizon by weighting each payment by how much of the value it carries. Modified duration then turns that horizon into a sensitivity, so a duration of seven means a one percentage point rise in yield knocks roughly seven percent off the price. Long maturities and small coupons push value far into the future, lengthening duration and making the bond swing more when rates move.

Formulas

Macaulay duration
DMac=∑t=1Nt PV(Ct)PD_{\mathrm{Mac}} = \sum_{t=1}^{N} t\,\frac{\mathrm{PV}(C_t)}{P}
Each time tt is weighted by the present value of that period’s cash flow as a fraction of price PP. The result is the present-value-weighted average time to the cash flows.
Modified duration
Dmod=DMac1+yD_{\mathrm{mod}} = \frac{D_{\mathrm{Mac}}}{1+y}
Dividing Macaulay duration by one plus the per-period yield yy converts a time measure into a price-sensitivity measure.
Price change approximation
ΔPP≈− Dmod×Δy\frac{\Delta P}{P} \approx -\,D_{\mathrm{mod}}\times \Delta y
The percentage price change is approximately minus modified duration times the yield change. The minus sign reproduces the inverse price-yield relationship.

Worked examples

Scenario

A bond has a modified duration of Dmod=6.5D_{\mathrm{mod}}=6.5 and trades at A$980. Yields rise by 50 basis points, that is Δy=0.005\Delta y = 0.005. Estimate the new price.

Solution

The approximate percentage change is −6.5×0.005=−0.0325-6.5\times 0.005 = -0.0325, a fall of 3.25 percent. In dollars the price drops by ΔP=0.0325×980=31.85\Delta P=0.0325\times 980 = 31.85, so the estimated new price is Pnew=980−31.85=948.15P_{\mathrm{new}}=980 - 31.85 = 948.15, about A$948. The negative sign confirms that a rise in yields lowers the price.

NoteThis linear estimate is accurate for small yield moves. For large moves it overstates the fall because it ignores curvature, which convexity corrects.

Common mistakes

  • ✗Duration is just the time until the bond matures. Duration is the present-value-weighted average time to all cash flows, so a coupon bond’s duration is shorter than its maturity. Only a zero-coupon bond has duration equal to maturity.
  • ✗Duration gives the exact price change for any yield move. The duration rule is a first-order, straight-line approximation. It is accurate for small changes but errs for large ones because the true price-yield curve bends.
  • ✗A higher coupon raises duration. A higher coupon shortens duration because more value arrives early, making the bond less rate-sensitive, not more.
  • ✗Macaulay and modified duration are interchangeable numbers. They differ by a factor of one plus the yield. Macaulay duration measures time, while modified duration measures price sensitivity.

Revision bullets

  • •Duration is the leading measure of interest-rate sensitivity
  • •Macaulay duration is the present-value-weighted average time to cash flows
  • •Modified duration converts that into a price-sensitivity number
  • •Percentage price change is about minus modified duration times yield change
  • •Duration rises with maturity and falls with higher coupon or yield
  • •A zero-coupon bond has duration equal to its maturity

Quick check

A bond has a modified duration of 8. If its yield rises by one percentage point, its price will change by approximately

Holding maturity fixed, raising a bond’s coupon rate will

Connected topics

Bond BasicsBond PricingBond YieldYield CurveConvexity

Sources

  1. Brailsford, Heaney & Bilson (2015), Ch. on interest-rate risk
    Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
    Defines Macaulay and modified duration and the duration-based price-change rule.
  2. Bodie, Kane & Marcus (2021), Ch. 16
    Bodie, Z., Kane, A., & Marcus, A. J. Investments. 12th ed. McGraw-Hill Education, 2021.
    Develops duration as the key interest-rate sensitivity measure and its determinants.
How to cite this page
Dr. Phil's Quant Lab. (2026). Duration. Derivatives Atlas. https://phucnguyenvan.com/concept/im-duration
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