How Bond Convexity Relates to Maturity and Interest Rate Volatility
Summary
The document gives an intuitive explanation of bond convexity using zero-coupon bonds and portfolios with equal duration. For a zero-coupon bond, a longer maturity makes price more responsive to yield changes because the discount rate compounds over a longer period. This helps explain why maturity is associated with greater convexity, although the discussion is qualitative rather than a full derivation.
It also explains the appeal of convexity when rates are volatile: for two positions with equal duration, the more convex position gains more when yields fall and loses less when yields rise. A second answer proposes comparing a single intermediate-maturity zero-coupon bond with a portfolio split between shorter and longer zero-coupon bonds, then examining price changes under small and large yield moves in both directions. That numerical exercise is illustrative; the document supplies no computed results, and actual bond behavior can depend on cash flows and other features.
Key ideas
- Longer maturity increases the compounding effect of yield changes on a zero-coupon bond’s price.
- Duration estimates price sensitivity to small yield changes, while convexity describes how that sensitivity changes.
- At equal duration, higher convexity improves outcomes when yields move substantially in either direction.
- Comparing zero-coupon portfolios with equal duration can make convexity effects easier to see.
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# Bond Convexity and Maturity
# 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.
## Answer by cifc (score 2, accepted)
https://quant.stackexchange.com/a/39483
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.
## Answer by Alex C (score 2)
https://quant.stackexchange.com/a/39486
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.