Duration alone underprices an option-free bond whether yields fall or rise. Convexity fixes that error.
The price-yield curve for an option-free bond is convex (curved, bowed toward the origin). Duration draws a straight tangent line at the current yield. For small yield changes the line is close enough. For large changes the line falls short of the true price on both sides, because the actual curve lies above the tangent. Convexity measures that curvature. The larger a bond's convexity, the further 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.
You calculate approximate convexity from the same two repriced values you use for approximate modified duration. Reprice the bond with the yield a little lower and a little higher, then compare the average of those two prices with today's price.
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%. In dollars, Example 3's trap loss is −$1,394,000 instead of −$1,469,000.
- 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
- Percentage change in full price ≈ −ModDur × Δy + ½ × Convexity × (Δy)²; multiply by the full price for dollars. The duration term is signed, and for option-free bonds the convexity term is positive
- Approximate Convexity = (P₋ + P₊ − 2P₀) / (P₀ × (Δy)²), using the same yield shock applied for duration and prices carried to four decimals
- Convexity computed in periods is annualized by dividing by m² (4 for semiannual)
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. If the question gives only prices, reprice both ways at four decimals and compute duration and convexity from the same three prices.
The full lesson (about 3,830 words, 26 min read) adds 9 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- yield-based convexity portfolio properties
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