Convexity
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. It doesn't: the relationship is curved, and the curve is convex. A fall in rates gains more than duration predicts, a rise loses less, and convexity always favors the holder of a fixed cash flow asset. It matters in two cases. Large rate moves first: the second-order term is worth several percentage points, and ignoring it distorts any loss measure. Matching next: two portfolios of equal duration but different convexity behave the same under an infinitesimal shock and differently under a real one. That is the third immunization condition Redington stated. An annuity liability concentrated at thirty years has high convexity; a bond portfolio of the same duration built from closely spaced maturities has less. The gap is a residual risk no duration measure reveals.
Textbook case: 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.
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