Convexity measures the rate of change of duration with respect to yield. While duration provides a linear approximation of price sensitivity, convexity captures the curvature: the fact that bond prices rise more when yields fall than they decline when yields rise by the same amount.
For a standard non-callable bond, convexity is always positive. This means the duration estimate understates price gains when yields fall and overstates price losses when yields rise by the same amount. Convexity matters most for large yield moves. For small moves of a few basis points, the linear duration approximation is sufficient. For moves of 50 bps or more, ignoring convexity produces meaningful pricing errors, especially for long-dated instruments.
Traders who are long convexity benefit from volatility because they gain more from favorable moves than they lose from unfavorable ones. The market prices this asymmetry through lower yields on high-convexity instruments.
The full second-order Taylor expansion of the price-yield relationship is:
Price change (%) = -D x dy + 0.5 x C x (dy)^2
where D is modified duration, C is convexity, and dy is the yield change expressed in decimal form.
Consider a 10-year Treasury with modified duration D = 8.0 and convexity C = 70. If yields rise 100 bps (dy = 0.01), the calculation runs:
The convexity term adds 0.35% of buffer against the duration-implied loss. For a 200 bp shock, the convexity correction grows to +1.40%, which is the difference between a -16.00% and a -14.60% outcome. The correction is non-trivial for portfolios measured in billions.
For yield decreases, the same math applies in reverse. A 100 bp fall produces a price gain of approximately +8.35%, not +8.00%, again due to the positive convexity term. This asymmetry, more gain on the down-yield side and less loss on the up-yield side, is what portfolio managers mean when they say they are buying convexity.
Convexity scales roughly with the square of modified duration. Approximate convexity values at current market yield levels are:
The 30-year bond carries roughly 4x the convexity of the 10-year, consistent with the squared-maturity relationship. YCP's duration pages display modified duration for each tenor in real time. Combining those readings with the approximate convexity values above lets analysts estimate second-order price sensitivity without a full bond calculator.
Callable bonds and mortgage-backed securities can exhibit negative convexity in certain yield ranges. When yields fall far enough, the issuer is more likely to call the bond or the homeowner is more likely to refinance, capping the price appreciation that would otherwise accrue to the holder.
MBS are the most common source of negative convexity in institutional fixed-income portfolios. As rates decline, prepayment speeds accelerate, shortening the effective duration and preventing the price from reaching the level a comparable bullet bond would achieve. This prepayment optionality is one reason the mortgage spread includes an option-adjusted premium. YCP's mortgage page tracks current mortgage rates and spreads, which reflect this embedded cost.
Callable corporates behave similarly. The price-yield curve for a callable bond flattens or even curves back downward near the call price, meaning the holder effectively sold a call option to the issuer at issuance in exchange for a higher coupon.
Convexity is not free. Investors who hold call-free, long-duration Treasuries are long convexity and pay for it through lower yields relative to callable alternatives of similar stated maturity.
Investors who hold MBS or write options on rates are short convexity. They collect a higher yield or option premium but bear an asymmetric risk profile: they underperform in large yield moves in either direction. This position requires active hedging with rate options or swaptions to manage the gamma exposure.
Duration-extension trades on YCP's duration pages show the modified duration profile across the curve. Pairing those readings with convexity estimates informs both the size of the convexity drag or benefit and the hedge ratio needed to neutralize it.
What is the difference between duration and convexity?
Duration is the first derivative of price with respect to yield, giving the linear approximation of price sensitivity. Convexity is the second derivative, capturing how that sensitivity itself changes as yields move. Duration tells you the slope of the price-yield curve at a given point. Convexity tells you whether that slope is increasing or decreasing.
Does convexity always benefit the bondholder?
For standard non-callable bonds, yes. Positive convexity means the price-yield relationship curves in the holder's favor. For callable bonds and MBS, convexity can be negative in certain yield ranges, working against the holder when rates move sharply in either direction.
How do I use convexity in practice?
Apply the second-order price change formula when evaluating large yield shocks. For daily mark-to-market on small moves, modified duration alone is adequate. For stress testing scenarios of 100 bps or more, add the convexity correction to avoid systematically underestimating the gain and overestimating the loss.