# effective convexity

Par Value Security Market Value Current Price Price If Yield Change Down 100 bp Up 100 bp \$25,000,000 4.75% due 2010 \$25,857,300 105.96 110.65 101.11 \$40,000,000 5.85% due 2025 \$39,450,000 98.38 102.76 93.53 \$65,000,000 Bond portfolio \$65,307,300 Karstein responds by saying, Estimates of effective duration and effective convexity derived from binomial models are very robust to the size of the rate shock, so I would not expect the estimates to change significantly. Part 5) In regard to the effect of a change in the size of the rate shock on the duration and convexity estimates, Karstein is: A) incorrect in her analysis of the effect on both bonds. B) correct only in her analysis of the effect on the 4.75% 2010 bond. C) correct only in her analysis of the effect on the 5.85% 2025 bond. D) correct in her analysis of the effect on both bonds.

Your answer: B was correct! Duration and convexity estimates for bonds without embedded options will not be significantly affected by changing the size of the rate shock from 100 basis points to 50 basis points. However, for bonds with embedded options, the size of the rate shock can have a significant effect on the estimates. We know from Part 3 that the 2025, 5.85% bond exhibits significant negative convexity, which is consistent with a callable bond. The 2010, 4.75% bond has positive convexity, even when yields are significantly below the coupon rate and the bond is trading at a substantial premium. That suggests the 2010, 4.75% bond has no embedded options. We would expect that changing the size of the rate shock would have a significant effect on the 2025, 5.85% callable bond, but not on the 4.75% 2010 bond. Therefore, Karstein is correct in her analysis of the 4.75% bond, but not the 5.85% bond.

For some reason I can’t get a positive convexity for the 4.75% bond as the solution suggested. Anyone?

That’s because it’s not + convex. 105.96 lies to the right (above) the line connecting the two points.

Joey do you mean the solution is incorrect? “The 2010, 4.75% bond has positive convexity, even when yields are significantly below the coupon rate and the bond is trading at a substantial premium.”