# 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?