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Interest Rates & Term Structureintermediate

Interest-Rate Risk & Duration

Even a default-free bond is risky. If interest rates rise, its price falls, so a holder who sells before maturity can lose. Duration measures that sensitivity. Modified duration gives the approximate percent price change for a 1 percentage point (100 bp) change in yield, and longer-maturity, lower-coupon bonds have the most interest-rate risk.

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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 is the present-value-weighted average time until you get your money back. The longer you wait on average, the more a change in the discount rate moves today’s price. A long zero-coupon bond is the most exposed of all.

Formulas

Price change from duration
ΔPP≈−Dmod Δy\frac{\Delta P}{P} \approx -D_{\text{mod}} \, \Delta y
Modified duration DmodD_{\text{mod}} is the percent price change per 1 percentage point change in yield yy. It is a first-order estimate, so large yield moves also need a convexity correction.

Worked examples

Scenario

A 3-year bond with a 5% annual coupon trades at par. Its modified duration is 2.72. Estimate the price effect of a 100 bp rise in yields.

Solution

Change in price is about -2.72 times 0.01, roughly -2.7%. The price falls from $100 to about $97.3. Longer bonds would fall by more.

Common mistakes

  • ✗A government bond has no risk. It has no default risk, but it still carries interest-rate risk if sold before maturity.
  • ✗Maturity alone measures risk. Duration, which also reflects the coupon, is the right gauge, and a high coupon shortens duration.
  • ✗Duration gives the exact price change. It is a first-order approximation that drifts on large yield moves, where a convexity correction is needed.

Revision bullets

  • •Rising rates lower bond prices (interest-rate risk)
  • •Duration is the PV-weighted average time to cash flows
  • •Modified duration: percent price change per 1% yield move
  • •Longer maturity and lower coupon mean more risk

Quick check

Which bond has the most interest-rate risk?

Connected topics

Bond PricingTerm Structure

Sources

  1. Mishkin (2018), Ch. 4
    Mishkin, F. S. The Economics of Money, Banking, and Financial Markets. 12th ed. Pearson, 2018. ISBN 978-1-292-26885-9.
    Covers the distinction between return and yield, interest-rate risk, and duration.
How to cite this page
Dr. Phil's Quant Lab. (2026). Interest-Rate Risk & Duration. Derivatives Atlas. https://phucnguyenvan.com/concept/mb-interest-rate-risk
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