A bond has a modified duration of 7.2 and a convexity of 65 per 100 of price. Yields rise by 120 basis points. Compute the percentage price change including the convexity term, and say how much of the answer that term contributes.

A bond has a modified duration of 7.2 and a convexity of 65 per 100 of price. Yields rise by 120 basis points. Compute the percentage price change including the convexity term, and say how much of the answer that term contributes.

Approach: Apply the second order Taylor expansion of price in yield, taking the duration term first and then the convexity term with its factor of one half.

-8.17%. The second order expansion is dP/P = -D_mod * dy + 0.5 * C * dy^2. The duration term is -7.2 * 0.012, which is -8.64%, and the convexity term is 0.5 * 65 * 0.012^2, which is +0.47%. Adding them gives a price change of -8.17%, so convexity recovers 47 of the 864 basis points the duration term alone predicted. Convexity helps a long position for a move in either direction, so the same bond gains 9.11% on a 120 basis point rally against the 8.64% the duration estimate would give. That asymmetry is why a positively convex bond outperforms a duration-matched one whenever yields move a long way, and why its buyer accepts a lower yield for it.

Follow-up: A mortgage-backed security has a duration of 7.2 and a convexity of -40. What are its price changes for the same two moves and what causes the sign?

Key concepts: modified duration, convexity, Taylor expansion, price change.