Bond Convexity, Carry, and the Cost of Positive Gamma
Summary
The document explains why greater bond convexity is not automatically better when comparing portfolios with equal value and duration. Convexity contributes a positive second-order price effect for yield changes in either direction, but this comparison omits the effect of time passing. The response relates convexity to gamma and carry or theta, arguing that a locally yield-hedged portfolio with positive gamma has negative theta under an absence-of-arbitrage relationship.
As a result, higher convexity can improve relative performance when yields move substantially, while its carry cost can weigh on returns when yield changes are small. The same logic applies whether yields rise or fall: the convexity benefit depends on the size of the move, not only its direction. The explanation is qualitative and gives no numerical illustration; its conclusions depend on the stated equal-value, equal-duration comparison and the assumed relationship between carry, yield variance, and convexity.
Key ideas
- Convexity adds a positive second-order price effect for yield changes in either direction.
- Comparing convexity alone ignores the return impact of time passing and carry.
- For portfolios with equal value and duration, positive gamma is associated with negative theta under the described no-arbitrage relationship.
- Higher convexity can outperform for large yield moves but may underperform when yields change little.
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# High convexity vs low convexity bond definition
# High convexity vs low convexity bond definition
Isn't high convexity always better than low convexity bond from the formula that $$\frac {ΔB} B=-D \frac {Δy} {1+y} + \frac 1 2 CΔy^2$$
Since $\frac 1 2 CΔy^2$ is positive no matter what so the price change will be more positive when there is a positive change in interest rate and a less negative price change when there is a negative change in IR? So doesn't this mean high convexity is absolutely better than lower? Obviously this is wrong that is why I am confused and because in my textbook it says "If you increase convexity of a portfolio and duration stays the same. You will make money if there is a large change in yields and lose money otherwise!" and "More convex bonds will have lower expected returns, especially when there is small change in yield." How would you even make money if thee is a large decrease change in yields?
Please help. thanks.
## Answer by Antoine Conze (score 3)
https://quant.stackexchange.com/a/38668
Do not forget the effect of passing time (the theta) on your portfolio.
If two portfolios have the same value and duration, then the portfolio made up of the difference has locally zero sensitivity to yields and is delta hedged. Since the sum of (modified) theta (the derivative to time minus the position funding, in essence the carry) and local yield variance times 1/2 gamma (the second derivative to yields, in essence the convexity) is zero (this follows from the same absence of arbitrage argument that is used to derive for instance the Black & Scholes formula for options), a positive gamma implies a negative theta.
Thus if the first portfolio has a higher convexity than the second portfolio, in a situation where yields changes are small the first portfolio has a lower return than the second one, while in a situation where yields changes are large the first portfolio has a higher return than the second one.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.