Convexity
Convexity is a measure of the curvature of the relationship between a bond's price and its yield, serving as a second-order correction to Modified Duration. A bond with higher convexity gains more in price when yields fall and loses less when yields rise compared to a lower-convexity bond with the same duration — making convexity a desirable property in bond portfolios.
The price-yield relationship of a bond is not a straight line — it is a curve, bowing outward toward the origin. Modified Duration captures the slope of this curve at any given point (a first-order approximation), but as yields move significantly, the slope itself changes. Convexity captures this rate of change in the slope — it is, mathematically, the second derivative of the bond's price with respect to yield. A bond with high convexity is more curved, meaning price gains from falling yields are larger than the price losses from equivalent yield increases.
Consider two bonds with identical Modified Durations of 8 years but different convexities. Bond A has convexity of 80, while Bond B has convexity of 40. If yields fall by 2%, Bond A will appreciate more than Bond B. If yields rise by 2%, Bond A will fall in price less than Bond B. In all scenarios of large yield movements, the higher-convexity bond outperforms. This is why convexity is sometimes described as a 'free lunch' — all else being equal, investors rationally prefer higher convexity. In efficient markets, this preference means higher-convexity bonds generally trade at slightly higher prices (lower yields), with investors paying an implicit premium for the convexity advantage.
In the Indian government securities market, longer-maturity bonds typically exhibit higher convexity. A 30-year G-Sec will have significantly greater convexity than a 10-year G-Sec, even after accounting for the higher duration of the 30-year bond. During the extraordinary bond market rally of 2020 — when the RBI cut the repo rate by 115 basis points and purchased G-Secs aggressively — long-duration gilt fund investors benefited from this convexity: the price gains as yields fell were amplified beyond what Modified Duration alone would have predicted.
For sophisticated fixed income portfolio managers in India, convexity is a critical portfolio construction parameter alongside Modified Duration. Mortgage-backed securities, which were not prevalent in India but common in the US, exhibit 'negative convexity' — their price does not rise proportionately when rates fall due to prepayment risk. In the Indian context, callable bonds (bonds the issuer can redeem before maturity) also exhibit negative convexity above the call price, a nuance important for institutional investors evaluating non-standard fixed income structures. Most retail investors encounter convexity indirectly through the outperformance of their long-duration gilt funds during sustained rate-cut cycles.