Duration alone underprices a bond when yields fall and overprices it when yields rise. Convexity fixes that error.
The price-yield curve for an option-free bond is convex (curved, bowed toward the origin). Duration draws a tangent straight line at the current yield. For small yield changes the line is close enough. For large changes the line undershoots the true price on both sides because the actual curve lies above the tangent. Interpret convexity as a measure of that curvature: the larger a bond's convexity, the more its true price curve pulls away from duration's straight-line estimate.
KEY: Convexity is always positive for option-free bonds. That positivity means the duration estimate understates the price gain when yields fall and overstates the price loss when yields rise. The convexity adjustment corrects in your favor in both directions.
Reprice the bond up and down by the same small yield shock, then plug into the formula below.
Common mistakes
- Forgetting the ½ in the convexity term. The formula is ½ × Con × (Δy)², not Con × (Δy)². Trap: a problem with Con = 100 and Δy = 0.01 should add 0.50%, not 1.00%.
- Squaring the percentage instead of the decimal. Δy of 50 bps is 0.005, not 0.50. Squaring gives 0.000025, not 0.25. Trap: an answer that overstates the convexity adjustment by 10,000 times.
- Dropping the negative sign on the duration term. ModDur enters as a negative product when yields rise. The convexity adjustment is positive in both directions; duration is signed.
Bottom line
- Convexity captures the curvature of the price-yield relationship that duration's straight tangent line misses, and it is always positive for option-free bonds
- Full price change ≈ −ModDur × Δy + ½ × Convexity × (Δy)²; the duration term is signed while the convexity term is always positive
- Approximate Convexity = (P₋ + P₊ − 2P₀) / (P₀ × (Δy)²), using the same yield shock applied for duration
- Money convexity = annual convexity × full price; multiply by (Δy)² to get the dollar curvature effect
Exam shortcut
When a problem gives ModDur and Convexity with a Δy, write the two-term equation immediately and plug both pieces. Never report just the duration term. Convert basis points to decimals before squaring: 50 bps is 0.005, and 0.005² is 0.000025. For portfolio questions, sum market values first, divide each bond's value by the total to get weights, then weight durations and convexities separately before combining.
The full lesson (about 1,928 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- yield-based convexity portfolio properties
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