With 360$^{\circ}$ video, only a limited fraction of the full view is displayed at each point in time. This has prompted the design of streaming delivery techniques that allow alternative playback qualities to be delivered for each candidate viewing direction. However, while prefetching based on the user's expected viewing direction is best done close to playback deadlines, large buffers are needed to protect against shortfalls in future available bandwidth. This results in conflicting goals and an important prefetch aggressiveness tradeoff problem regarding how far ahead in time from the current playpoint prefetching should be done. This paper presents the first characterization of this tradeoff. The main contributions include an empirical characterization of head movement behavior based on data from viewing sessions of four different categories of 360$^{\circ}$ video, an optimization-based comparison of the prefetch aggressiveness tradeoffs seen for these video categories, and a data-driven discussion of further optimizations, which include a novel system design that allows both tradeoff objectives to be targeted simultaneously. By qualitatively and quantitatively analyzing the above tradeoffs, we provide insights into how to best design tomorrow's delivery systems for 360$^{\circ}$ videos, allowing content providers to reduce bandwidth costs and improve users' playback experiences.