# 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.

## 1 Answer

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.