# 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

## 1 Answer

I can only hint some ways that you can prove it... since the actual prove can be tedious...

1. This one should be obvious from observing any version of the duration formula. For example, http://www.investinganswers.com/financial-dictionary/bonds/duration-1288

You can try to show that if we increase M (maturity payment) and decrease C (coupon payment) in a way that P (price) does not change, the duration will increase. The proof for this should be simple but rather too much typing for me here, so I will skip it for this post...

1. This is a bit more writing than I want... I think we need to take the derivate of

again with respect to y to show this relationship. One hint I have is to run some Taylor Series Approximation first then take the derivative to simplify the math.

1. We can come up with a model of a cash flow stream, say that a bond gives a payout over 3 years as (x,10-2*x,x), we can analyze the marginal effect on the convexity as we increase/decrease y.