Consider the following statements:

Statement $1$: At yield levels close to the bond's coupon rate, an investor in a callable bond has more to gain from a decrease in yields than she has to lose from an increase in yields.

Statement $2$: The larger the change in yields the more inaccurate the price estimate based on duration alone, and the lower the convexity adjustment.

Which of the following is most likely?

A$.$ Both Statements are incorrect

B$.$ Only Statement $1$ is correct

C$.$ Only Statement $2$ is correct