# The relation between coupon and convexity

Here are three statements:

1. A lower coupon bond exhibits higher duration.

2. The higher the coupon rate, the lower a bond’s convexity. Zero-coupon bonds have the highest convexity.

3. Given particular duration, the convexity of a bond portfolio tends to be greatest when the portfolio provides payments evenly over a long period of time. It is least when the payments are concentrated around one particular point in time.

And we have the relation $$\dfrac{\Delta B}{B} = -D\Delta y + \dfrac{1}{2}C(\Delta y)^2.$$

I understand above three statements as given two coupon paying structures with same maturity and same principle, then at a intersection point $(y_0,B_0),$ we compare their duration, convexity?

And can anyone proof why in formula?

• The question is then how to show that $\Delta B/B = (...)$? – caverac Nov 26 '17 at 10:32
• @caverac just proof those three statements – A.Oreo Nov 26 '17 at 18:21
• I'm voting to close this question as off-topic because they are classical homework excercises. OP also does not want to go into details, he wants "just the proff of those three statements". – vanguard2k Nov 27 '17 at 14:58
• @vanguard2k sorry, I have looked for a lot of paper, but no one showed a reliable results, most of explanations are unconvincing I think. – A.Oreo Nov 27 '17 at 15:20
• @A.Oreo After careful inspection, I tend to agree. I retracted my close vote and upvoted the question. Especially a formal proof of point 3 is probably not easy at all. – vanguard2k Nov 27 '17 at 15:37