Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian
motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$.
This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq
T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$
adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex
function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang"
type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is
negative, and $\tau^*\equiv T$ is the drift is positive.