A bond has modified duration 8 and convexity 90. Yields fall 150 basis points. What is the total price change, how much of it is the convexity term, and what does the asymmetry against a 150 basis point rise cost the holder?

A bond has modified duration 8 and convexity 90. Yields fall 150 basis points. What is the total price change, how much of it is the convexity term, and what does the asymmetry against a 150 basis point rise cost the holder?

Approach: Take the second order Taylor expansion of price in yield and evaluate both signs of the move to expose the asymmetry.

13.01%. The second order expansion gives dP/P = -D*dy + 0.5*C*dy^2 = 8*0.015 + 0.5*90*0.000225 = 0.12 + 0.010125 = 13.01%, so convexity contributes 1.01 points, 7.8% of the move. A 150 basis point rise gives -12.00 + 1.01 = -10.99%, so a round trip of equal size in both directions leaves 2.02 points of gain, which is the entire economic content of positive convexity. That asymmetry is paid for through carry: a convex bond yields less than a comparable one of the same duration, and the annual give up is the price of a payoff that behaves like a long option on rates. Convexity is worth most when realised yield volatility is high relative to what the yield give up implies.

Follow-up: What annual yield give up makes the convexity of this bond fairly priced at 100 basis points of yield volatility?

Key concepts: convexity, modified duration, second order expansion, carry.