  For a zero-delayed random walk on the real line, let $\tau(x)$, $N(x)$ and
$\rho(x)$ denote the first passage time into the interval $(x,\infty)$, the
number of visits to the interval $(-\infty,x]$ and the last exit time from
$(-\infty,x]$, respectively. In the present paper, we provide ultimate criteria
for the finiteness of exponential moments of these quantities. Moreover,
whenever these moments are finite, we derive their asymptotic behaviour, as $x
\to \infty$.
