Convexity

Duration provides a linear approximation of how a bond's price responds to yield changes. For small yield moves, this approximation is reasonably accurate. For larger moves, the actual price-yield relationship curves away from the duration-based linear estimate — that curvature is convexity.

Positive convexity, which characterizes most standard government and investment-grade corporate bonds, means that price increases for a given yield decline are larger than price decreases for an equivalent yield rise. This is a desirable property: the bond performs better than its duration predicts in falling-rate environments and worse by less than duration predicts in rising-rate environments. All else equal, investors prefer bonds with higher convexity.

Convexity can be calculated as the second derivative of the price-yield function divided by price. A bond with longer maturity and lower coupon (such as a long-duration zero-coupon Treasury) has higher convexity than a shorter-maturity, higher-coupon bond. This means long zero-coupon bonds outperform their duration-adjusted expectations in volatile rate environments — a key reason why long-duration government bonds are valued in pension fund and insurance liability-matching strategies.

In the US Treasury market, convexity is a constant topic of discussion because the Federal Reserve's asset purchase programs and the issuance profile of Treasuries affect the convexity profile of the outstanding stock of government bonds. Large-scale MBS purchases by the Fed, for example, removed negative-convexity securities from private hands — a factor some analysts cite as having suppressed volatility in interest rate markets during the QE era.

For equity investors, convexity is primarily relevant when evaluating bond holdings in a balanced portfolio. A portfolio of high-convexity bonds provides better downside protection in rising-rate environments and captures more upside in falling-rate environments relative to a lower-convexity portfolio with similar duration.