Bond Convexity: The Curvature in Price-Yield Relationships

Key Takeaways

• Positive convexity means price gains when yields fall exceed price losses when yields rise by the same amount.
• Callable bonds and MBS can show negative convexity, where the call or prepayment option caps price upside.
• Convexity adds a second-order correction to duration: add 0.5 × convexity × (yield change)² to the duration estimate.
• Higher convexity is priced into the market; you earn a slightly lower yield in exchange for the favorable curvature.

What It Is

If you plot a bond's price against its yield, the line is not straight. It bends. Duration is the slope at a single point on that curve. Convexity is how fast the slope itself is changing.

For most standard bonds, the curve bends away from the rate axis in a favorable way. Prices rise faster when yields fall than they drop when yields rise by the same amount. That property is positive convexity. Some bonds, notably callables and mortgage-backed securities, can show negative convexity in certain rate ranges, meaning prices gain less on the way down than they lose on the way up.

The Intuition

Duration is a first derivative. It answers the question "how much does price move per basis point of yield" at the current price. But the answer is only exact for an infinitesimal move. For a real 50 or 100 basis point shock, the price-yield curve has bent enough that duration alone misses part of the move.

Convexity captures the bending. Add a convexity term to the duration estimate and you get a much better approximation over larger shocks. This is why fixed-income desks quote both numbers side by side.

Positive convexity is generally desirable. Two bonds with the same duration are not equivalent: the one with higher convexity gains more when rates fall and loses less when they rise. Traders will pay up for that property, so convexity is priced into the market.

How It Works

The combined duration-and-convexity price estimate is:

dP / P = -D x dy + 0.5 x C x (dy)^2

Where:

dP / P  = percentage change in price
D       = effective (or modified) duration
C       = convexity
dy      = change in yield, in decimal form (e.g. 0.01 for 1 percent)

The first term is the linear duration estimate. The second term is the convexity adjustment. Because it is multiplied by (dy)^2, its contribution grows quickly for large rate moves and is negligible for tiny ones.

Effective convexity, the one used for bonds with embedded options, is computed numerically just like effective duration:

Effective Convexity = (PV- + PV+ - 2 x PV0) / (PV0 x (dCurve)^2)

The numerator asks whether the two shocked prices average out above the starting price (positive convexity) or below it (negative convexity).

Worked Example

Take a bond with effective duration 7.0 and convexity 60. Rates rise 100 basis points (dy = 0.01). Estimate the price change.

dP / P = -7.0 x 0.01 + 0.5 x 60 x (0.01)^2
       = -0.0700 + 0.0030
       = -0.0670  (or -6.70 percent)

Duration alone predicted a 7 percent loss. Convexity adds back 30 basis points of protection, because the price-yield curve bends favorably on the downside.

Now imagine a callable bond of similar duration but with effective convexity of -20 because the call is near the money. If rates fall 100 bps, the estimate is:

dP / P = -7.0 x (-0.01) + 0.5 x (-20) x (0.01)^2
       = 0.0700 - 0.0010
       = 0.0690  (or 6.90 percent)

A non-callable with positive convexity would have gained more than 7 percent. The call option is silently capping the upside.

Common Mistakes

1. Ignoring convexity on large yield moves. For a 10 basis point wiggle, duration alone is fine. For a 100 basis point shock, duration under-predicts the gain when rates fall and over-predicts the loss when they rise. Add the convexity term whenever the move is meaningful.

2. Assuming all bonds have positive convexity. Callables and mortgage-backed securities frequently show negative effective convexity around par. Pricing them with positive-convexity intuition leads to bad hedges.

3. Mixing convexity scales. Textbooks sometimes report convexity as a raw number, sometimes scaled by 100 or by (1+y)^2. Always check how your data vendor defines it before plugging into the price-change formula.

4. Treating convexity as a separate signal. Convexity is not a trade on its own. It is a correction to duration. Two bonds with similar duration can still differ in convexity, and that is the signal. Looking at convexity in isolation is meaningless.

5. Forgetting convexity is priced. Higher convexity sells for a lower yield all else equal. You are not getting free curvature; you are paying for it in the coupon.

Frequently Asked Questions

Why is zero-coupon bond convexity higher than a coupon bond of the same maturity? A zero-coupon bond has a single cash flow at maturity, making its price-yield curve bend more sharply. Coupon bonds return some value early through coupons, which have lower sensitivity to rate changes and dampen the overall curvature. Stripping away coupons concentrates all the interest-rate exposure at the final date, maximizing convexity.

Does positive convexity guarantee a positive return? No. Positive convexity means you lose less on rate rises and gain more on rate drops relative to a duration-only estimate. But if rates rise significantly, the bond still loses value even with positive convexity. Convexity improves outcomes at the margin; it does not prevent losses.

How does negative convexity affect mortgage-backed securities? When rates fall, homeowners refinance their mortgages, returning principal to MBS investors ahead of schedule. This prepayment shortens the bond's duration right when you would prefer it to be long to benefit from falling rates. The asymmetry creates negative effective convexity, meaning MBS gain less in rally scenarios than non-callable bonds of similar stated duration.

Can a portfolio be constructed to maximize convexity? Yes, through a barbell strategy that concentrates holdings at very short and very long maturities rather than intermediate ones. Long-maturity bonds contribute high convexity; short-maturity bonds reduce duration without eliminating convexity. A portfolio with the same duration as a bullet portfolio but a barbell structure will have higher convexity and different performance in large rate moves.

Why do traders say convexity is "gamma" in bond markets? The analogy comes from options theory. Delta is the first-order price sensitivity to the underlying, and gamma is the second-order change in delta. Duration is the bond-market delta; convexity is the bond-market gamma. Traders who are long convexity (positive gamma) benefit from large moves in either direction, while those who are short convexity prefer low-volatility environments.

Sources

1. AnalystPrep. "Calculate and Interpret Bond Convexity." https://analystprep.com/cfa-level-1-exam/fixed-income/calculate-interpret-convexity/
2. AnalystPrep. "Compare Effective Convexities of Callable, Putable and Straight Bonds." https://analystprep.com/study-notes/cfa-level-2/compare-effective-convexities-of-callable-putable-and-straight-bonds/
3. Carpenter, J. (NYU Stern). "Convexity (Debt Instruments and Markets)." https://pages.stern.nyu.edu/~jcarpen0/courses/b403333/06convexity.pdf
4. CFA Institute. "Yield-Based Bond Duration Measures and Properties." https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/yield-based-bond-duration-measures-and-properties

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.