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Duration alone is a straight-line approximation of a curved price-yield relationship, so it overstates losses when yields rise and understates gains when they fall. The convexity term is the second-order correction that fixes both.
Second-order price change
ΔP/P ≈ -D_mod·Δy + ½·C·Δy²
Because Δy is squared. Positive convexity helps a bondholder in both directions, which is why it is worth paying for.
For large yield moves and long-dated or low-coupon bonds, where convexity is high.