Convexity - The Second-Order Curvature of Bond Prices
By EC Assets Research Team, Fixed Income Research · Published · Updated
Convexity — Convexity is the second-order measure of a bond's rate sensitivity - the curvature that duration's straight-line estimate misses. Positive convexity helps the holder symmetrically (more gain when yields fall, less loss when they rise); negative convexity, as in callable bonds and mortgages, does the reverse.
What Convexity Measures
Convexity is the second-order measure of a bond's interest-rate risk - the curvature that duration, a straight-line estimate, leaves out. The true relationship between a bond's price and its yield is not a line but a curve, bowed toward the origin. Duration is the slope of that curve at the current yield; convexity is how fast the slope itself changes. Together they give a far more accurate price estimate than duration alone:
%ΔP ≈ −D_mod × Δy + ½ × C × (Δy)²
The first term is the duration effect; the second is the convexity correction. Because the (Δy)² term is always positive for a normal bond, convexity is a friend to the holder: it adds to the gain when yields fall and subtracts from the loss when yields rise.
Why Positive Convexity Is Valuable
The price-yield curve of a standard (option-free) bond bends in the investor's favour. Read off the consequences:
- When yields fall, the price rises by more than duration predicts.
- When yields rise, the price falls by less than duration predicts.
That asymmetry is a genuine, free benefit - which is why, all else equal, investors pay up for convexity (accepting a slightly lower yield) and why long-maturity, low-coupon bonds, which carry the most convexity, behave so explosively when rates move. Convexity is the bond market's analogue of an option's gamma: positive convexity, like long gamma, rewards large moves in either direction.
Worked Example
Take a bond with modified duration 8.0 and convexity 100, and consider a large 100 bps (Δy = 0.01) move.
Yields rise 1%: −8.0 × 0.01 + ½ × 100 × 0.01² = −0.080 + 0.005 = −7.5% Yields fall 1%: −8.0 × (−0.01) + ½ × 100 × (−0.01)² = +0.080 + 0.005 = +8.5%
Duration alone would have predicted a symmetric ±8.0%. Convexity bends both outcomes in the holder's favour: the loss is cushioned to 7.5% and the gain amplified to 8.5%. The bigger the yield move, the more that (Δy)² term matters - which is exactly why duration is dangerous to rely on for large shocks.
Negative Convexity - When the Curve Bends the Wrong Way
Some bonds have convexity that works against the holder. Callable bonds and mortgage-backed securities embed short options: when rates fall, the issuer calls the bond or homeowners refinance, capping the price upside and pulling cash back early. Their price-yield curve flattens or bends backward as rates drop - negative convexity.
[!warning] Negative convexity is the dangerous mirror image: the holder loses more when yields rise than they gain when yields fall - precisely the wrong asymmetry. Mortgage-backed securities are the classic case, and the forced re-hedging of negative-convexity books ("convexity hedging") can amplify moves in the Treasury market itself.
Why It Matters for Institutional Investors
- Accuracy for large moves. In a sharp rate shock, a duration-only estimate can be materially wrong. Risk systems for any serious bond book carry both duration and convexity.
- A priced characteristic. Convexity is not free in the market - positive convexity typically costs a little yield, and harvesting negative convexity (selling it, as MBS investors do) is a way to earn extra carry in exchange for the unfavourable asymmetry. Knowing which side you are on is essential.
- Barbell vs bullet. A barbell portfolio (short- and long-maturity bonds) has more convexity than a bullet (concentrated at one maturity) of the same duration, so it outperforms when yields move a lot and underperforms when they sit still. The choice is an explicit convexity bet.
- Liability-driven investing. Pension and insurance hedges must match convexity as well as duration, or the hedge drifts as rates move - leaving a residual exposure exactly when markets are volatile.
[!key] Duration tells you the slope; convexity tells you how the slope changes. Positive convexity helps you symmetrically - more gain, less pain - and is worth paying for. Negative convexity (callables, MBS) does the opposite and must be compensated by extra yield.
References
- Fabozzi, F. J. (2021). Bond Markets, Analysis, and Strategies (10th ed.). MIT Press.
- Tuckman, B., & Serrat, A. (2011). Fixed Income Securities (3rd ed.). Wiley.
- Ilmanen, A. (2011). Expected Returns. Wiley. (Chapters on bond risk premia and convexity.)
- CFA Institute. Fixed Income: The Term Structure and Interest-Rate Risk. CFA Program Curriculum.
Frequently asked questions
What is the difference between duration and convexity?
Duration is the first-order, straight-line estimate of how a bond's price responds to yield changes - the slope of the price-yield curve. Convexity is the second-order term that captures the curve's curvature - how the slope itself changes. Adding convexity to duration gives a much more accurate price estimate for large yield moves.
Why is positive convexity good for a bondholder?
Because it bends both outcomes in your favour: when yields fall, the price rises by more than duration predicts, and when yields rise, the price falls by less. That favourable asymmetry is a real benefit, which is why investors will accept a slightly lower yield to obtain more convexity.
What is negative convexity?
It is the unfavourable curvature found in bonds with embedded short options - callable bonds and mortgage-backed securities. When rates fall, issuers call the bonds or borrowers refinance, capping the upside and returning cash early. The holder loses more when yields rise than they gain when yields fall - the opposite of what an investor wants.
How is convexity like an option's gamma?
Both measure the curvature of a price response and both reward large moves. A positively convex bond, like a long-gamma option position, gains from volatility in yields regardless of direction. A negatively convex bond behaves like a short-gamma position - it suffers from large moves and must often be re-hedged in a way that amplifies them.
When do I actually need to account for convexity?
For small yield changes, duration alone is accurate enough. Convexity matters for large rate moves, where the duration estimate becomes materially wrong, and for liability-driven hedging, where a hedge must match convexity as well as duration to avoid drifting out of balance when rates move.
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