Negative Convexity

What Does Negative Convexity Mean?
When the shape of a bond's yield curve is concave. A bond’s convexity is the rate of change of its duration, and is measured as the second derivative of price with respect to yield.

Most mortgage bonds are negatively convex. 

Investopedia explains Negative Convexity
Callable bonds are negatively convex at lower yields than the yield at which the bond is likely to be called.

One property of a non-callable bond is that as interest rates fall, its price will increase. However, with a callable bond, as interest rates fall, the incentive for the issuer to call the bond at par increases; therefore, its price will not rise as quickly as the price of a non-callable bond. 

The price of a callable bond might actually drop as the likelihood that the bond will be called increases. This is why the shape of a callable bond's curve of price with respect to yield is concave or "negatively convex."