Duration vs Convexity: First- and Second-Order Bond Risk

Key Takeaways

• Duration is the first-order (linear) estimate of a bond's price change for a small yield move: %ΔPrice ≈ −modified duration × Δyield.
• Convexity is the second-order correction for the curvature duration misses; for plain vanilla bonds it is positive and always improves the estimate.
• Because of positive convexity, duration overstates the loss when yields rise and understates the gain when they fall, the error works in the holder's favor.
• Duration suffices for small changes; convexity matters for large yield moves, long-dated bonds, and choosing between bonds of similar duration.

What It Is

Duration (specifically modified duration) measures the percentage change in a bond's price for a 1% change in yield. A modified duration of 7 means a 1% rise in yield produces roughly a 7% price fall. It is a straight-line approximation of the price-yield relationship at the current yield.

Convexity measures how much the price-yield line curves. The true relationship between price and yield is not linear, it bows. Convexity quantifies that bow and is used as a correction term on top of the duration estimate. For standard option-free bonds convexity is positive; for callable bonds and mortgage-backed securities it can turn negative in some yield ranges.

The Intuition

Draw the price-yield curve for a bond: it slopes down (prices fall as yields rise) and it is bowed toward the origin. Duration is the tangent line at today's yield, a good fit right next to the current point, but it drifts away from the true curve as you move further. Convexity says the true curve sits above the tangent line in both directions, which is why the real price is always a little better for the holder than the duration-only estimate: smaller losses on the way up, bigger gains on the way down.

How It Works

The two combine into a single price-change estimate:

%ΔPrice ≈ (−modified duration × Δyield) + (½ × convexity × Δyield²)

The first term is the duration effect (linear); the second is the convexity effect (curvature). Because the convexity term squares Δyield, it is negligible for tiny moves but grows quickly for large ones, which is exactly when the duration-only estimate breaks down. Positive convexity always adds to the price (the term is positive whether yields rise or fall), which is why bond investors prize it.

Worked Example

A bond has a modified duration of 7 and a convexity of 80.

Yields rise 1% (Δy = +0.01):

• Duration effect: −7 × 0.01 = −7.0%
• Convexity correction: ½ × 80 × (0.01)² = +0.4%
• Estimated price change: −6.6%

Yields fall 1% (Δy = −0.01):

• Duration effect: −7 × (−0.01) = +7.0%
• Convexity correction: ½ × 80 × (0.01)² = +0.4%
• Estimated price change: +7.4%

Duration alone would have said ±7.0% symmetrically. Convexity turns that into a −6.6% loss but a +7.4% gain, the same size rate move helps more than it hurts. That asymmetry is the tangible value of positive convexity, and it grows with the size of the rate move.

Common Mistakes

1. Using duration for large yield moves. The linear estimate drifts badly for shocks above ~1%; without the convexity term you overstate losses and understate gains.
2. Assuming convexity is always positive. Callable bonds and MBS exhibit negative convexity when rates fall (the borrower prepays or calls), which hurts the holder, the opposite of the vanilla case.
3. Comparing bonds on duration alone. Two bonds with identical duration can have different convexity; the higher-convexity bond outperforms when rates move sharply either way.
4. Confusing the duration measures. Macaulay duration (in years), modified duration (%/yield), and effective duration (for bonds with options) are related but not interchangeable, use the right one for the job.

Frequently Asked Questions

Q: What is the difference between duration vs convexity in simple terms? Duration is the straight-line estimate of how much a bond's price moves for a small yield change. Convexity is the correction that accounts for the curve, and it matters most for large rate moves.

Q: Why does convexity matter if I already have duration? Duration is only accurate for small yield changes. For large moves it overstates losses and understates gains; the convexity term restores accuracy and captures the favorable asymmetry.

Q: Is convexity always good for a bondholder? Positive convexity is favorable, gains exceed losses for equal-sized rate moves. But callable bonds and mortgage-backed securities can have negative convexity, which is unfavorable when rates fall.

Q: How do duration vs convexity combine in one estimate? %ΔPrice ≈ (−modified duration × Δyield) + (½ × convexity × Δyield²). The first term is linear; the second adds the curvature correction that grows with the square of the yield move.

Q: When can I ignore convexity? For small yield changes and short-dated bonds the convexity term is tiny and duration alone is fine. It becomes important for long-dated bonds, large rate shocks, and bonds with embedded options.

Sources

1. Investopedia. "Duration." https://www.investopedia.com/terms/d/duration.asp
2. Investopedia. "Convexity." https://www.investopedia.com/terms/c/convexity.asp
3. Investopedia. "Interest Rate Risk." https://www.investopedia.com/terms/i/interestraterisk.asp

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.