Bond convexity measures how the relationship between a bond’s price and its yield curves. Duration estimates the first move in price; convexity adjusts that estimate for the curve. For an option-free fixed-rate bond, positive convexity generally means a larger price gain when yields fall and a smaller price loss when yields rise than duration alone predicts, all else equal.
How convexity affects bond prices
Bond prices and yields generally move in opposite directions: when market yields rise, the price of an existing fixed-rate bond tends to fall; when yields fall, its price tends to rise. The size of the move depends on the bond’s cash flows and the change in yields.
Think of duration as the straight-line tangent to a bond’s price-yield curve at its current yield. It provides a first-order estimate of price sensitivity. Convexity describes how the curve bends away from that tangent as yields move. This curvature matters more for larger yield changes, and it helps explain why bonds with similar duration can respond differently. CFA Institute’s 2026 curriculum explains yield-based convexity as an adjustment to duration; a conceptual treatment also appears in a 1988 Financial Analysts Journal article on callable bonds.
Estimate a price change with duration and convexity
A common second-order approximation is:
Percentage price change ≈ −(modified duration × change in yield) + ½ × convexity × (change in yield)²
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Use the yield change as a decimal and match the convexity convention to the measure being used. For example, a change of 1 percentage point is 0.01. Modified duration is the first-order term; the convexity term adjusts for curvature. This is a local estimate, not an exact price forecast. The measures depend on the bond’s cash flows and yield assumptions.
For a large rate move, a bond with embedded options, or a shift that changes different parts of the yield curve by different amounts, direct repricing or an appropriate curve-based model is more reliable than treating the formula as exact. CFA Institute’s material on curve-based and empirical fixed-income risk measures discusses the importance of selecting a measure suited to the exposure and scenario.
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What positive convexity means
Option-free fixed-rate bonds have positive convexity. All else equal, the duration-only estimate understates the gain from a yield decline and overstates the loss from a yield increase. That is a more favorable price response in either direction, but it does not make the bond risk-free or guarantee a positive total return.
Convexity is a comparison of price sensitivity, not a complete measure of investment results. Credit spreads, changing credit quality, reinvestment, and the path of rates can also affect what an investor earns.
What affects a bond’s convexity?
For otherwise similar option-free fixed-rate bonds, these general comparisons apply:
- Maturity: Longer time to maturity generally corresponds to greater convexity.
- Coupon: A lower coupon generally corresponds to greater convexity.
- Yield: A lower yield to maturity generally corresponds to greater convexity.
These are all-else-equal tendencies, not rules for comparing unlike bonds. Cash-flow timing, embedded options, credit spreads, and the size and shape of a rate move can change the comparison. Check the bond’s terms and the assumptions behind any quoted risk measure.
Why callable and putable bonds can behave differently
An embedded option can change the shape of a bond’s price-yield curve. A callable bond gives the issuer the right to repay the bond under specified terms. As rates fall and calling becomes more attractive, the call can limit the bond’s price appreciation. A callable bond near the point where the option matters may therefore have negative convexity.
A putable bond gives the investor a right to sell the bond back under specified terms. Near the point where that option matters, a putable bond may have positive convexity. CFA Institute’s 2026 curriculum on bonds with embedded options describes these differences.
For bonds whose cash flows can change as rates change, effective duration and effective convexity are designed to account for that rate sensitivity. Their results depend on the valuation model and the rate shocks used. Do not compare an embedded-option bond’s yield-based convexity mechanically with an option-free bond’s measure as though their cash flows respond in the same way.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Yield-based versus curve-based risk measures
Yield-based duration and convexity estimate how a bond’s price changes when its yield changes, holding the cash-flow assumptions in the measure. Curve-based measures instead evaluate changes in the underlying yield curve; empirical measures use observed relationships. The appropriate choice depends on the bond, its cash flows, and the risk scenario.
A portfolio’s weighted-average duration or convexity may implicitly assume a parallel shift, in which yields at different maturities move by the same amount. Actual yield-curve changes are often nonparallel. A single portfolio measure can therefore miss how exposures at different maturities respond; curve-based analysis is more suitable when the shape of the move matters.
A published example of the price-yield relationship
The SEC’s June 26, 2013 investor bulletin illustrates the inverse relationship with a $1,000 face-value bond paying a 3% coupon and originally having ten years remaining. In the illustration, the bond is priced at $1,000 when market rates are 3%; after one year, with nine years remaining and market rates at 2%, its price is $1,082. This is a specific published illustration, not a market-wide statistic or a prediction for other bonds.
Do these 3 things before closing this tab:
1Fix the driver behind crashes, sound loss and screen glitches2Clear out junk files and repair common Windows errors3Scan for outdated or missing drivers - takes under a minuteThe SEC also describes yield to maturity as a widely used way to compare bonds, with assumptions attached to holding a bond to maturity. See its fixed-income investor bulletin and explanation of corporate bonds.
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