Bond Convexity and Maturity

What the reasoning for why bond convexity increases with maturity. Heuristic explanations are somewhat better as I would like a fundamental understanding.

Also what causes a more convex bond to be preferable when interest rates are more volatile? I cannot see nor understand the dynamics driving this problem.

Think of a zero coupon bond - the pv_zero (t years) $= \frac{\rm{pmt}}{(1+r)^t}$

As t increases the compounding effect of that discount increases (the larger the price change)

As for rate vol - convexity brings about a couple of preferable properties - as rates decrease (rates rally), bond a and bond b which have = duration, but bond a high higher convexity, will have a higher sensitivity to the rate change (px up). Conversely, bond a will also have a lower sensitivity to a rate increase (rates sell off)

So vol up, convexity becomes a preferable property to have.

• @AlexC Can you post this as an answer so that I can upvote it... – Helin Apr 28 '18 at 22:38

You can understand convexity by working out a simple example numerically yourself.

Consider two bond portfolios: P1= consists of a 6 year zero coupon bond. P2= half in a 2 year ZCB, half in a 10 year ZCB.

By construction the two portfolios have same duration but different convexity. Now analyze what happens to price in 4 cases: small increase in yield, small decrease in yield, big increase in yield, big decrease in yield. It takes 15 minutes in Excel, including drawing a chart. Finally draw conclusions. What is the effect of convexity, how does knowledge of convexity modify what we knew from duration.