We study a one dimensional Brownian motion moving among Poisson points of constant intensity ν > 0. We introduce the “annealed δ - Lyapounov exponent” β δ ( c). Here “annealed” refers to the fact that averages are both taken with respect to the path and environment measures. The exponent β δ ( c) measures how costly it is for the Brownian motion to reach a remote location while it receives a penalty “proportional” to c ∈ (0, ∞) for spending too much time at Poisson points, and when the particle can pick its own time to perform the displacement. We derive a formula for β δ ( c), which shows that for all c ∈ (0, ∞), β δ ( c) < ν. We conjecture that in general this is also true for the one dimensional “annealed Lyapounov exponent”, introduced by Sznitman, which is an analogue object to β δ ( c).