Consider the following statements:
Statement 1: All other things remaining the same, a bond with higher convexity will outperform one with lower convexity in both bull and bear markets.
Statement 2: Callable bonds exhibit negative convexity at low yields, but traditional fixed‐rate bonds and putable bonds exhibit positive convexity at all yield levels.
Which of the following is most likely?

The greater the convexity of a bond, the greater the price appreciation when yields decrease and the lower the price depreciation when yields increase.
Callable bonds exhibit negative convexity at low yield levels. As it becomes increasingly likely that the issuer will exercise the embedded call option, the call price acts as a cap on the bond’s value and the slope of the price‐yield profile flattens (the rate of change of duration becomes negative). Traditional fixed‐rate bonds and putable bonds always have positive convexity.