Curvature of the price-yield relationship, correcting duration's linear approximation and explaining why a duration-matched portfolio stays exposed to large moves.
Modified duration assumes price varies linearly with yield, which is false: the relationship is curved, and the curve is convex. Concretely, a fall in rates gains more than duration predicts, and a rise loses less, so convexity always favors the holder of a fixed cash flow asset. It becomes decisive in two situations. Large rate moves first, where the second-order term is worth several percentage points and ignoring convexity distorts any loss measure. Matching next: two portfolios of equal duration but different convexity behave identically for an infinitesimal shock and differently for a real one, which is precisely the third immunization condition Redington stated. An annuity liability concentrated at thirty years has high convexity, and a bond portfolio of the same duration built from closely spaced maturities has less: the convexity gap is a residual risk no duration measure reveals.
Bond portfolio with modified duration 8.5 and convexity 95. For a 200 basis point rate rise, duration alone predicts a 17.0% loss. The convexity term contributes 0.5 times 95 times 0.02 squared, a 1.9 point gain, hence a real loss near 15.1%. On a 4 billion euro portfolio the gap between the two estimates is worth 76M EUR.
convexity, courbure prix-taux, terme du second ordre, convexité du passif