Bond Convexity Explained: The Curvature Beyond Duration

A bond’s market price is the present value of its expected cash flows. When the discount yield changes, those cash flows are repriced. For a plain fixed-rate bond with fixed payments and no embedded option, price generally moves in the opposite direction from yield: a higher yield means a lower price, and a lower yield means a higher price.

The relationship is not a straight line over a wide range. On a price-versus-yield graph, the curve bends. The slope at the current yield tells you how responsive price is right there; the bend tells you how that responsiveness changes as yield moves farther away. CFA Institute describes duration as a linear approximation and convexity as a complementary measure of this curvature in its [yield-based bond convexity overview]({source:cfaBondConvexity2026}).

Modified duration estimates the percentage price change for a small change in yield. Its first-order approximation is ΔP/P ≈ −D × Δy, where Δy is written as a decimal. If modified duration is 6 and yield rises by 0.01, the estimate is −6 × 0.01, or about −6%. For a small move, the tangent line at today’s price can be a useful shortcut.

That straight-line estimate treats equal-sized rises and falls in yield as equal-sized price changes in opposite directions. A curved price-yield relationship does not behave exactly that way. Duration alone is most useful as a local estimate; it becomes less complete as the assumed yield change grows. FINRA also frames duration as a sensitivity measure, not a maturity date, in its [bond duration explainer]({source:finraDurationExplainer}).

Convexity is the second-order part of the price response. In a common modified-duration convention, the approximation is ΔP/P ≈ −D × Δy + ½ × C × (Δy)². The first term is duration’s straight-line estimate. The second term adds the curvature adjustment. Use decimal yields: 100 basis points (bp) is 1 percentage point, or 0.01; 25 bp is 0.0025.

For an option-free fixed-rate bond, yield-based convexity is positive. That positive term raises the estimated price change for either direction of a yield move: it makes an estimated gain larger when yield falls and makes an estimated loss smaller when yield rises, compared with duration alone. This describes sensitivity under the stated model; it does not say that higher convexity guarantees a better investment return.

Suppose a hypothetical bond portfolio has modified duration 6 and convexity 45, using the decimal-yield convention above. If its relevant yield rises by 100 bp, then Δy = +0.01. Duration contributes −6 × 0.01 = −6.00%. Convexity contributes ½ × 45 × 0.01² = +0.225%. Together, the estimate is about −5.775%, before income, spread changes, fees, and other effects.

If yield instead falls by 100 bp, duration contributes +6.00% and convexity again contributes +0.225%, for an estimated +6.225%. Duration alone would give a symmetric +6% and −6%. On a hypothetical $10,000 position, the two estimates are about −$577.50 and +$622.50. They are arithmetic illustrations, not forecasts or promised outcomes.

The positive-convexity result above assumes fixed cash flows with no embedded options. A callable bond gives the issuer a right to repay the bond early. When yields fall, a call can become more attractive to the issuer, limiting how much the bond price rises. Mortgage-backed securities can also see faster prepayments when refinancing becomes more appealing, so their expected cash flows change with rates.

Near an embedded option’s exercise region, a callable bond can show negative convexity: its upside from falling yields may be constrained while its downside from rising yields remains. CFA Institute discusses effective duration and effective convexity for callable and putable bonds and notes that a near-the-money callable bond can have negative convexity in its [embedded-options guide]({source:cfaEmbeddedOptions2026}). Do not apply the plain-bond example mechanically to these securities.

Portfolio duration and convexity summarize how a collection of bond cash flows responds to a yield move. A commonly used approach weights each holding’s measure by its portfolio value. That summary is useful when comparing portfolios under similar assumptions, but it usually treats the yield curve as shifting in parallel: every relevant maturity yield moves by the same amount.

Market yield curves can instead steepen, flatten, or change shape at selected maturities. A portfolio with cash flows concentrated at different points on the curve can react differently even when two portfolios have similar aggregate duration and convexity. Key-rate duration or other curve-specific measures can help describe those exposures. Convexity is not the same as the curvature of the yield curve itself.

Before comparing a bond fund’s convexity, check the measure’s definition, units, as-of date, and whether it is modified, approximate, or effective convexity. For a bond with embedded options, effective measures rely on a model that reprices expected cash flows as yields change. Two providers can report figures that are not directly comparable if their models, curve bumps, holdings dates, or conventions differ.

Also separate interest-rate sensitivity from credit-spread, default, liquidity, and currency risks. A yield change in the formula is the change in the yield input used by that measure, not automatically a central-bank policy-rate move or a change in every market yield. For a basic ETF sensitivity example, see bond ETF duration and rate risk; for futures-versus-forward settlement timing, see the separate convexity adjustment guide.

Duration and convexity help answer a bounded question: under a specified yield move and the measure’s assumptions, how might price sensitivity differ from a straight-line duration estimate? They do not predict a future yield, total return, recovery time, or whether a bond belongs in a portfolio.

State the yield shock, its units, the duration and convexity convention, and which risks are held constant. Treat the output as an approximation and revisit the inputs when market prices, holdings, cash-flow expectations, or the yield curve change. A single convexity number describes curvature around a reference point; it is not a complete risk score.

Keep estimated price sensitivity separate from realized total return. Duration and convexity describe how a price on one valuation date may respond to a specified yield shock. Actual return over a holding period may also reflect coupon income, roll-down, credit-spread changes, transaction costs, and changes in expected cash flows. When comparing the initial estimate with the later return, align the measurement period and risk factors, then identify which components explain the gap. That review can assess the approximation; it cannot turn it into a return promise.