Convexity implies that duration predictions:A$)$ I and III onlyConvexity is a measure that quantifies the curvature of the relationship between a bond's price and its yield. It helps to refine the accuracy of duration predictions by accounting for the non$-$linear relationship between bond price and yield changes.I. Duration predictions underestimate the $\%$ increase in bond price when the yield falls: This statement is correct. Duration provides an estimate of the percentage change in bond price for a given change in yield, assuming a linear relationship. However, due to the convex nature of the price$-$yield relationship, duration tends to underestimate the actual percentage increase in bond price when yields fall.III. Duration predictions overestimate the $\%$ increase in bond price when the yield falls: This statement is incorrect. Convexity adjustment helps to refine duration predictions by accounting for the convex shape of the price$-$yield relationship. As a result, convexity adjustments help to correct the underestimation of the percentage increase in bond price when yields fall, rather than overestimating it$.$Therefore, the correct answer is A$)$ I and III only.