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BD123
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Joined: November 8th, 2014, 11:37 am

Zeros and Convexity

October 26th, 2016, 4:56 pm

Hi all,
I'm reading through the old Salomon Brothers Yield Curve paper and I am trying to reconcile these two statements:
1. "In particular long-term zeros exhibit very high convexity"
2. "There are convexity differences between bonds that have the same duration. A barbell position (with very dispersed cash flows) exhibits more convexity than a duration-matched bullet bond. The reason is that a yield rise reduces the relative weight of the barbell's longer cash flows, shortening the barbell's duration, limiting its losses when yields rise and enhancing its gains when yields decline. Of all bonds with the same duration, a zero has the smallest convexity because its cash flows are not dispersed: thus, its Macaulay duration does not vary with the yield level."
If a zero has the smallest convexity out of a group of bonds that have the same duration, why go out of your way in the first point to emphasize that longer term zeros have very high convexity? Should point 1 say long term zeros have very high convexity, exceeded only by even longer maturity/duration matched coupon paying bonds which have an even higher convexity? What details am I missing that would reconcile these two sentences? Would be grateful for any insights here.
 
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MHill
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Joined: February 26th, 2010, 11:32 pm

Re: Zeros and Convexity

October 27th, 2016, 1:00 pm

I'm guessing the first statement compares bonds with the same maturity, and looks at how modified duration (not Macaulay duration) changes with the yield.
The second statement talks about Macaulay duration.  A zero-coupon bond's Macaulay duration is always it's time to maturity, no matter what.  It also is comparing bonds with the same duration.  So any coupon bonds with the same duration as the zero-coupon must have longer time to maturity.