In the Fröhlich effect, the perceived onset of a moving stimulus is displaced in the direction of motion. Previously, we observed that pointing movements produced a Fröhlich effect only when the onset position was highly predictable. Here, we show that relative judgments are not affected by spatial predictability if the relative judgment task is performed in isolation. However, when the two tasks vary randomly from trial to trial, effects of spatial predictability carry over to the perceptual task. Thus, observers' intentions before stimulus onset determine the way position signals are processed. An account in terms of sensory and motor maps is discussed.