For a zero-delayed random walk on the real line, let τ(x), N (x) and ρ(x) denote the first passage time into the interval (x, ∞), the number of visits to the interval (−∞, x] and the last exit time from (−∞, 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 → ∞.