Say that I have two bonds and one of them has positive convexity and the other negative. Which one is better (assuming that you only care about convexity)? I understand that high convexity is desirable because the bondholder can benefit more from a drop in the interest rate than an identical increase in the interest rate. Does that mean negative convexity is bad, because investors are more affected by a raise in the interest rate?
I am going to assume that the only thing you are interested in is convexity and the many other aspects as well as the suitability of focusing on a single measure are not addressed. In such a general setting more positive convexity provides, as you have already outlined, for the potential to increase prices at a faster rate as a response to interest rate declines. In addition you want to consider where your bond sits on the convexity spectrum and your interest rate expectations. Quote from comments to the article above with regard to expectations of rising interest rates:
It depends what side of the convexity curve your bond resides. For any given duration, you would want HIGH convexity if you are on the right hand (the flattening) part of the curve.
In summary: high, absolute, positive convexity is most likely desirable while high, absolute, negative convexity is most likely less desirable given stable or falling interest rates. The distinction between level and direction of convexity are important as well the expectations regarding interest level and volatility, among other things such as the position in the covexity spectrum.
negative convexity, most likely, will imply that bond has embedded option. i.e. bond holder sells call option to bond issuer. therefore you'll have negative gamma position = collect option premium and short volatility.