  This paper studies the utility maximization problem with changing time
horizons in the incomplete Brownian setting. We first show that the primal
value function and the optimal terminal wealth are continuous with respect to
the time horizon $T$. Secondly, we exemplify that the expected utility stemming
from applying the $T$-horizon optimizer on a shorter time horizon $S$, $S < T$,
may not converge as $S\uparrow T$ to the $T$-horizon value. Finally, we provide
necessary and sufficient conditions preventing the existence of this
phenomenon.
