# 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 $\Delta B/B = (...)$? Commented Nov 26, 2017 at 10:32
• @caverac just proof those three statements Commented Nov 26, 2017 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". Commented Nov 27, 2017 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. Commented Nov 27, 2017 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. Commented Nov 27, 2017 at 15:37